Rolf & Norman Fundamentals of Semiconductor Physics and Devices

FUNDAMENTALS OF SEMICONDUCTOR PHYSICS AND DEVICES

Rolf Enderlein ihmboldt-bhiuemity BerEin Universi?yof Sao Pauh

NommJ.M. Horing stewwas Institute of Technology

Hoboken, NJ

World Scientific Singapore New Jersey London Hong Kong

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First published 1997 Reprinted 1999

FUNDAMENTALS OF SEMICONDIJC'IOK PHYSICS AND DEVICES Copyright 0 1997 by World Scicntific Publishing Co Pte. Ltd. All rights reserved. Th.is book, or parts thrreof, may ~ O be I reprudrtced in any jurwi or by ony m e w s , electmnir or rnerhirnicirl, incIudinx photocopying. recording or any information storage and retrieval sys:slew now known or m be ii-zvmted, without written permissionfrom !he Publisher.

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Dedication This book is dedicated to the memory of Adele and Werner Enderlein (par ents of R.E.), and to t h e memory of Joseph and Esther Morgenstern (hfor ganstein)(grandparents of N.J. M. 11.).

Vii

Preface People come to technical books with a vast array of daerent needs and requirements, arising from their differing educational backgrounds, professional orientations and career objectives. This is particularly evident in the field of semiconductors, which stands at the juncture of physics, chemistry, electronic engineering, material science and mathematics. No longer just an academic discipline. this field is at the heart of an ongoing revolution in communications, computation and electronic device applications, that innovate many fields and change modern life in myriad ways, large and small. Its profound impact and further potential command interest and attention from all corners of the earth. and from a wide variety of students and researchers. The clear need to address a broadly diversified and variously motivated readership has weighed heavily in the authors’ considerations. It poses a pedagogical problem faced by many teachers of intermediate level courses on semiconductor physics. Generally speaking, every student has previously studied about half of the course materiaL The difficulty lies in the fact that each student’s exposure is likely to have been to a &fleerent half, depending on which lower level courses and teachers they have had, and where the emphasis lay. To accommodate readers with varied backgrounds we start from first principles and provide fully detaiIed explanations and proofs, assuming only that the reader is familiar with the Schrodinger equation. This intensively tutorial treatment of the electronic properties of semiconductors includes recent fundamental developments and is carried through to the physical principles of device operation, to meet the needs of readers interested in engineering aspects of semiconductors, as well as those interested in basic physics. Clarity of explanation and breadth of exposure relating to the electronic properties of semiconductors, from first principles to modern devices, are our principal objectives in this fraddy pedagogical book. We offer full mathematical derivations to strengthen understanding and discuss the physical significance of results. avoiding reliance on ‘hand waving arguments alone. To support the reader’s introduction to the physics of semiconductors, we provide a thorough grounding in the basic principles of solid state physics, assuming no prior knowledge of the field on the part of the reader. An ele mentary discussion of the crystal structure, chemical nature and macroscopic properties characterizing semiconductors is given in Chapter 1. Moreover, we also include an extensive appendix to guide the reader through group theory and its applications in connection with the symmetry properties of semiconductors, which are of major importance. Beside spatially homogeneous bulk semiconductors, we undertake a full exposition of inhomogeneous semiconductor junctions and heterostructures because of their crucial role

iu

Preface

The book has emerged from lectures which the authors presented for physics students at the Humboldt-University of Berlin. Germany, and the State University of Sao Paulo, Brazil, and for physics and engineering physics students at the Stevens Institute of TeFhnology in Hoboken, New Jersey, C.S.A. Part of the book has similarities with the german book "Grundlagen der Halbleiterphysik" ("Fundamentals of Semiconductor Physics") which was written by one of us (R. E.) together with A. Schenk. We are thankful to Dr. Schenk (now at ETH Zurich) for allowing us to use part of his work in the present volume. In writing this book we have had excellent suppoIt from many of our colleagues at our own and other Universities. We are particularly thankful to Prof. Dr. J. Auth (Humboldt-Lniversity Berlin), Prof. Dr. F. Bechstedt (Friedrich-Schiller University Jena), Prof. Dr. W. A. Harrison (Stanford University), Prof. Dr. M. Scheffler (Fritz-Haber Institut, Berlin), Prof. Dr. J. R. b i t e , Prof. Dr. A. Fazzio, and Prof. Dr. J. L. Alves (State University Sao Paulo), as well as to Prof. Dr. H. L. Cui, Prof. Dr. G. Rothberg, Mr. G. Lichtner (Stevens Institute of Technology), and Prof. Dr. G . Gumbs (Hunter College, CWNY, New York), who read parts of the manuscript and contributed helpful suggestions and critical remarks. The technical assistance of Mrs. Hannelore Enderlein is gratefully acknowledged.

RoIf Enderlein

Norman J.M. Horing

Sao Paulo

Hoboken, N J October 1996

Xi

Contents 1 Characterization of sernicond uct ors Inlrnduclion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Atomic structure of ideal crystals . . . . . . . . . . . . . . . . 1.2.1 Cryst.al latlices . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Point groups of equivalent directions arid crystal classes 1.2.3 Space groups and crystal structures . . . . . . . . . . 1.2.4 Cubic semiconductor structures . . . . . . . . . . . . . 1.2.5 Hexagoiial semiconductor st.ructures . . . . . . . . . . 1.3 Chemical nature of semiconductors. Material classes . . . . . 1.3.1 Group IV elemental semiconductors . . . . . . . . . . 1.3.2 111-V semiconductors . . . . . . . . . . . . . . . . . . . 1.3.3 11-VI semiconductors . . . . . . . . . . . . . . . . . . . 1.3.4 Group \'I elemental semiconductors . . . . . . . . . . 1.3.5 IV-VI semiconductors . . . . . . . . . . . . . . . . . . 1.3.6 Other compound semiconductors . . . . . . . . . . . . 1.4 hlacroscopic properties and their microscopic implications . . 1.4.1 Electrical conductivity . . . . . . . . . . . . . . . . . . 1.4.2 Depenclenre of conductivity on the semiconductor state 1.4.3 Optical absorption spectrum and the band modcl of srmicoiiductors . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Electrical conductivity in the band model . . . . . . . 1.4.5 The Hall effect and the existence of positively charged freely mobile carriers . . . . . . . . . . . . . . . . . . . 1.4.6 Seinicondiictors far from thermodynamic equilibrium . 1.1

2 Electronic structure of ideal crystals 2.1 Abcimic cores and vdcnce electrons . . . . . . . . . . . . . . . 2.2 The ciynaniical problem . . . . . . . . . . . . . . . . . . . . . 2.2.1 Schriidiiiger equation for the interacting core and valence dwtl-on system . . . . . . . . . . . . . . . . . . . 2.2.2 Adiabatic approximation . Lattice dynamics . . . . . . 2.2.3 Oneparticle approximation . Oneparticle Schriidinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 General properties of stationary one-rlectron states in a crystal 2,3.1 Syinrnctry properties of the one-electron Schrtidinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 R b c h theorem . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Reciprocal v e c h space and the reciprocal latt.ice . . . 2.3.4 Relation between energy eigenvalues and quasi-wave vector . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5

6

12 14 16

22 28 29 30 31 31

32

32 33 34 35

38 43

45

49 51 51 54 54

57

66 82

82 85

89 94

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Contents

2.4

2.5

2.6

2.7

2.8

Schrodinger equation solution in the nearly-freeelectron approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.4.1 Kon-degenerate perturbat.ion t.heory . . . . . . . . . . 100 2.4.2 Degenerate perturbation theory . . . . . . . . . . . . . 103 Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.5.1 Brillouin zones . . . . . . . . . . . . . . . . . . . . . . 105 2.5.2 Degeneracy of energy bands . . . . . . . . . . . . . . . 116 2.5.3 Critical points and effective masses . . . . . . . . . . . 119 2.5.4 Density of states . . . . . . . . . . . . . . . . . . . . . 123 2.5.5 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 2.5.6 Calculational methods for band structure determination133 Tight binding approximation . . . . . . . . . . . . . . . . . . 140 2.6.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 140 2.6.2 TB theory- of diamond and zincblende type semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 148 2.6.3 sp3-hybrids, total energy and chemical bonding . . . . 165 k . p-met.hod . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 2.7.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 179 2.7.2 Valence bands of diamond structure semiconductors without spin-orbit interaction . . . . . . . . . . . . . 184 . 2.7.3 h t t i n g eT-Kohn model . . . . . . . . . . . . . . . . . . 189 2.7.4 Kana model . . . . . . . . . . . . . . . . . . . . . . . . 200 Band structure of important semiconductors . . . . . . . . . . 211 2.8.1 Silicou . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 2.8.2 Germanium . . . . . . . . . . . . . . . . . . . . . . . . 218 2.8.3 111-V semiconductors . . . . . . . . . . . . . . . . . . 219 2.8.4 IGVI semiconductors . . . . . . . . . . . . . . . . . . . 221 2.8.5 IV-\'I semiconductors . . . . . . . . . . . . . . . . . . 224 2.8.6 Tellurium and selenium . . . . . . . . . . . . . . . . . 224

3 Electronic s t r u c t u r e of semiconductor crystals with p e r t u r bations 225 3.1 Atomic structure of red semiconductor crystals . . . . . . . . 226 3.1.1 Classification of perturbations . . . . . . . . . . . . . . 226 3.1.2 Point perturbations . . . . . . . . . . . . . . . . . . . . 227 3.1.3 Formation of point perturbations and their movenient 235 3.1.4 h e and planar defects . . . . . . . . . . . . . . . . . 240 3.2 One-electron Schrodinger equation for point perturbations . . 241 3.2.1 Electron-core interaction . . . . . . . . . . . . . . . . . 242 3.2.2 Electron-elw?c.lroninteraction . . . . . . . . . . . . . . 245 3.3 Effective mass equation . . . . . . . . . . . . . . . . . . . . . 252 3.3.1 Effectivemass equation for a single band . . . . . . . 253 3.3.2 Multjband effective mass equation . . . . . . . . . . . 259

Contents

...

Xlll

3.4 Shallow levels. Donor and acceptor states . . . . . . . . . . . 265 3.4.1 Hydrogen model . . . . . . . . . . . . . . . . . . . . . 266 3.4.2 Improvements upon the hydrogen model . . . . . . . . 272 3.5 Deeplevds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 3.5.1 General characterization of deep levels . . . . . . . . . 281 3.5.2 Defect molecule model . . . . . . . . . . . . . . . . . .285 3.5.3 Solution methods for the oneelectron Schriidinger q u a tion of a crystal with a point perturbation . . . . . . . 293 3.5.4 Correlation effects . . . . . . . . . . . . . . . . . . . . 301 3.5.5 Resu1t.s for se1ecDed deep centtas . . . . . . . . . . . . 308 3.6 Clean semiconductor surfaces . . . . . . . . . . . . . . . . . . 334 3.6.1 The concept of clean surfaces . . . . . . . . . . . . . . 334 3.6.2 Atomic structure of clean surlaces . . . . . . . . . . . 336 3.6.3 Electronic structure of crystals with a surface . . . . . 354 3.6.4 htomic and electronic structure of particular surfaces 371 3.7 Semiconductor microstructures . . . . . . . . . . . . . . . . .388 3.7.1 Neterojunctions . . . . . . . . . . . . . . . . . . . . . . 388 3.7.2 Microstructures; Fabrication, classifications, examples 396 3.7.3 h*lethodsfor electronic structure calculations . . . . . 409 3.7.4 Elcctronic structure of particular microstructures . . . 420 3.8 Macroscopic electric fields . . . . . . . . . . . . . . . . . . . . 433 3.8.1 Effective mass equation and stationary electron states 434 3.8.2 Non-stationary states . Bloch oscillations . . . . . . . . 437 3.8.3 Interband tunneling . . . . . . . . . . . . . . . . . . . 440 3.8.4 Photon assisted interband tunneling . . . . . . . . . . 442 3.9 Macroscopic magnetic fields . . . . . . . . . . . . . . . . . . . 443 3.9.1 Effective mass equation in a magnetic field . . . . . . 444 3.9.2 Solution of the effective mass equation . . . . . . . . . 452 4

Electron system in t herrnodynamic equilibrium 457 4.1 Fundamentals of the statistical description . . . . . . . . . . . 457 4.2 Calculation of average particle numbers . . . . . . . . . . . . 460 4.2.1 Configuration-independent oneparticle states . . . . . 460 4.2.2 Configuration-dependent one-particle states . . . . . . 462 4.3 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . 469 4.3.1 Total electron concentration . . . . . . . . . . . . . . . 469 4.3.2 Density of states of ideal semiconductors . . . . . . . . 470 4.3.3 Density of states of real semiconductors . . . . . . . . 474 4.4 Free carrier concentrations . . . . . . . . . . . . . . . . . . . . 477 4.4.1 Conservation of total electron number . . . . . . . . . 477 4.4.2 Free carrier concentration dependence on Fermi energy. Law of mass action . . . . . . . . . . . . . . . . . 478 4.4.3 Intrinsic semiconductors . . . . . . . . . . . . . . . . . 482

Contents

xiv

4.4.4 4.4.5 4.4.6 5

Extrinsic semiconductors . . . . . . . . . . . . . . . . 484 Compensation of donors and acceptors . . . . . . . . . 489 More complex cases . . . . . . . . . . . . . . . . . . . 492

Non-equilibrium processes in semiconductors 499 5.1 Fundamentals of the statistical description of non-equilibrium processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 5.2 Systematics of non-equilibrium processes in semiconductors . 505 5.2.1 Temporal inhomogeneity and spatial homogeneity . . 505 5.2.2 Spatial inhomogeneity and temporal homogeneity . . . 506 5.2.3 Space and time inhomogeneities . . . . . . . . . . . . 508 5.3 Generation and annihilation of free charge carriers . . . . . . 509 5.3.1 Generation processes . . . . . . . . . . . . . . . . . . . 510 5.3.2 Unipolar annihilation of free charge carriers: capture 511 at deep centers . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Bipolar annihilation of carriers at deep centers . . . . 517 5.4 Drift current . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 5.5 Diffusion and annihilation of free carriers . . . . . . . . . . . 527 5.6 Equilibrium of free carriers in inhomogeneously doped semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

6 Semiconductor junctions in thermodynamic equilibrium 535 6.1 pn-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 6.1.1 Establishment of thermodynamic equilibrium . . . . . 539 6.1.2 Diffusion voltage . . . . . . . . . . . . . . . . . . . . . 541 6.1.3 Spatial variation of the electric and chemical potentials: Schottky approximation . . . . . . . . . . . . . . 542 6.2 Heterojunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 549 6.2.1 Equilibrium condition . . . . . . . . . . . . . . . . . . 550 6.2.2 Electrostatic potentid . GaAs/Gal-,Al, As heterojunction as an example . . . . . . . . . . . . . . . . . . . . 552 6.3 Metal-semiconductor junctions . . . . . . . . . . . . . . . . . 557 6.3.1 Energy level diagram before establishing equilibrium . 357 6.3.2 Electrostatic potential . . . . . . . . . . . . . . . . . . 559 6.3.3 Schottky barrier . . . . . . . . . . . . . . . . . . . . . 563 6.4 Insulator-semiconductor junctions . . . . . . . . . . . . . . . . 567 6.4.1 Thermodynamic equilibrium . . . . . . . . . . . . . . 567 6.4.2 Influence of interface states . . . . . . . . . . . . . . . 570 6.4.3 Semiconductor surfaces . . . . . . . . . . . . . . . . . 572

7 Semiconductor junctions under non-equilibrium conditions 573 7.1 pn-junction in an external voltage . . . . . . . . . . . . . . . 574 7.1.1 Electrostatic potential profile . . . . . . . . . . . . . . 576

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Contents

7.1.2 7.1.3 7.1.4 7.1.5

Mechamism of current transport through a pn-junction 577 Chemical potential profiles for electrons and holm . . 580 Dependence of current density on voltage . . . . . . . 583 Bipolar transistor'. . . . . . . . . . . . . . . . . . . . . 585 7.1.6 T u n e 1 diode . . . . . . . . . . . . . . . . . . . . . . . 593 7.2 yn-junction in interaction with light . . . . . . . . . . . . . . 595 7.2.1 Photocffect at a pn-junction . Photodiode and photovoltaic element . . . . . . . . . . . . . . . . . . . . . . 595 7.2.2 Laser diode . . . . . . . . . . . . . . . . . . . . . . . . 599 7.3 Metal-semiconductor junction in an external voltage. Rectificrs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 7.4 hwulator-semiconductor junction in an external voltage . . . 612 7.4.1 Field effect . . . . . . . . . . . . . . . . . . . . . . . . 612 7.4.2 Inversion layers . . . . . . . . . . . . . . . . . . . . . . 614

7.4.3 MOSFET . . . . . . . . . . . . . . . . . . . . . . . . .

620

Appendices A Group theory for applications in semiconductor physics

623 A.1 Definitions and concepts . . . . . . . . . . . . . . . . . . . . . 623 A . l .1 Group definition . . . . . . . . . . . . . . . . . . . . . 623 A.1.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 624 A.2 Rigid displacements . . . . . . . . . . . . . . . . . . . . . . . 627 A.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 621 A.2.2 Translations . . . . . . . . . . . . . . . . . . . . . . . . 628 A.2.3 Orthogonal transformations . . . . . . . . . . . . . . . 629 A.2.4 Geometrical interpretation . . . . . . . . . . . . . . . . 631 -4.2.5 Screw rotations and glide re3ections . . . . . . . . . . 632 A.3 Translation. point and space groups . . . . . . . . . . . . . . 635 A.3.1 Lattice translation groups . . . . . . . . . . . . . . . . 635 -4.3.2 Point groups . . . . . . . . . . . . . . . . . . . . . . . 636 A.3.3 Space groups . . . . . . . . . . . . . . . . . . . . . . . 654 A.4 Representations of groups . . . . . . . . . . . . . . . . . . . . 655 A.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 655 A.4.2 Irreducible representations . . . . . . . . . . . . . . . . 661 4.4.3 Products of representations . . . . . . . . . . . . . . . 667 A . 5 Representations of the full rotation group . . . . . . . . . . . 673 4.5.1 Vector representation of the rotation group and generators of infinitesimal rotations . . . . . . . . . . . . 674 A.5.2 Representations for dimensions other than three . . . 676 A.6 Spinor representations . . . . . . . . . . . . . . . . . . . . . . 682 A.6.1 Space-dependent spinors . . . . . . . . . . . . . . . . . 682

Cootents

mi

A.6.2 Representation V I . . . . . . . . . . . . . . . . . . . . 683 2 A.6.3 Irreducible spinor representations . . . . . . . . . . . . 684 A.6.4 Double group method . . . . . . . . . . . . . . . . . . 685 A.7 Projective representations . . . . . . . . . . . . . . . . . . . . 687 A.7.1 Factor systems . . . . . . . . . . . . . . . . . . . . . . 687 A.7.2 Definitions and theorems . . . . . . . . . . . . . . . . 689 A.7.3 Construction of projective representations . . . . . . . 692 A.8 Time reversal symmetry . . . . . . . . . . . . . . . . . . . . . 692 A.8.1 Time reversal operator . . . . . . . . . . . . . . . . . . 693 A.8.2 Additional degenerac!?~of energy eigenvalues . . . . . 694 A.8.3 Additional selection rules for matrix elements . . . . . 697 A.9 Irreducible representations of space groups . . . . . . . . . . . 698 A.9.1 Representations of translation groups . . . . . . . . . 698 A.9.2 Star of wavevectors . . . . . . . . . . . . . . . . . . . . 700 A.9.3 Small point groups and their projective representations 702 A.9.4 Representations of the fufl space group . . . . . . . . . 704 A.9.5 Spinor representations of space groups . . . . . . . . . 706 A.9.6 Implications of time reversal symmetry . . . . . . . . 707 A.9.7 Compatibility . . . . . . . . . . . . . . . . . . . . . . . 712 A.10 Irreducible representations of small point groups . . . . . . . 712 A.10.1 Character tables . . . . . . . . . . . . . . . . . . . . . 712 A.10.2 Multiplication tables . . . . . . . . . . . . . . . . . . 731 A.10.3 Compatibility relations . . . . . . . . . . . . . . . . . 734

B Corrections to the adiabatic approximation

737

C Occupation number representation

'741

Bibliography

747

Index

757

FUNDAMENTALS OF SEMICONDUCTOR PHYSICS AND DEVICES

1

Chapter 1

Characterization of semiconductors

1.1

Introduction

Semiconductors are identified as a unique material group on the basis of their common macroscopir properties, as is done for metals, dielectrics and magnetic materials. The name ‘semiconductor’ stems from the fact that such materials have moderately good conductivity, higher than that of insulators, and lower than that of metals. However, if this were the only property which these materials had in common, the term ‘semiconductor‘ would have only a very weak foundation. But such is not the case. In fact, many materials having conductivity between that of metals and insulators. display simultaneously a series of further common properties. In particular, their conductivity depends very strongly on material staie, for example, on temperature and chemical purity, much more so than in the case of metals. For sufficiently pure semiconductors, the conductivity decays by orders of magnitude while cooling down from room temperature to liquid helium temperature. ,4t absolute zero temperature, semiconductor conductivity almost vanishes, in contrast to the conductivity of metals, which rises modestly with falling temperature. The conductivity of metals reaches its maximum at low temperature, and for superconductors it effectively becomes infinitely large. In regard to the dependence of semiconductor conductivity on the degree of purity, it has been found that a given semiconductor in a very pure state can resemble an insulator. while in a highly polluted state it acts like a metal, among other peculiarities. Furthermore, irradiation with light can cause a transition from insulator-like behavior to metal-like behavior of one and the same semiconductor. There are yet other optical properties shared by semiconductors: The op-

2

Chapter I. Characterization of semiconductors

tical absorption spectra of semiconductors exhibit a threshold - below the threshold frequency, light can pass through practically without losses, while above it the light is strongly absorbed. Moreover, good luminescence properties in the visible and infrared spectral range are also characceristic of many semiconductors. Thus, the identification of a semiconductor material involves several characteristic properties, not just the one of moderately good electrical conduction. One has a semiconductor only if all such properties apply. This criterion excludes ionic conductors, for example, which exhibit conductivity values of the right order of magnitude, but do not display the characteristic temperature dependence. One may justifiably question the extent to which this definition of a semiconductor really makes sense. For example, it is not yet obvious why just the above properties are selected as defining features while others are not, and what internal connections may exist among them. A full answer can only be given by means of a microscopic theory of semiconductor properties which will be developed systematically in the chapters to follow. For the moment, we invite the reader to join us in the recognition that all macroscopic properties involved in the definition of a semiconductor can be traced back to a common microscopic origin, namely the nature of the spectrum of allowed energy levels and the particulars of their population by electrons. To be specific, the permissible energy levels of a semiconductor form bands which are separated by forbidden regions, and the Characteristic electron population of allowed energy levels is such that, at absolute zero temperature, a semiconductor i s characterized by having only completely occupied and completely empty energy bands (no partially filled bands). It is this common microscopic feature which underlies the totality of macroscopic material properties that uniquely define a semiconductor. It also provides the basis for uncovering yet other common macroscopic features of this class of materials, beyond the ones already discussed. For instance, it may be expected that semiconductors should be predominantly solid crystalline materials, since the formation of energy bands with gaps between them is most likely to occur in the crystalline phase. Nevertheless, amorphous and liquid semiconductors cannot be completely rxcluded since a certain regularity of the relative positions of neighboring atoms also exists in the amorphous and liquid phases. Actually, in addition to solid crystalline semiconductors, which are the main reprcsentatives, a series of amorphous semiconductors has also been found silicon and selenium being important examples. Liquid semiconductors are also possible, with melted tellurium among them. Other semiconducting materials, e.g., silicon and germanium, are metals in the liquid phase. In this book we restrict our considerations to solid crystalline semiconductors. The discussion also partially applies to amorphous and liquid sexniconductors, but in most cases modifications are necessary. Even the basic

3

1.1. Introduction

concept o f the quantum mechanical energy spectrum uf electrons has to be defined in tl different way, and a proper treatment of amorphous and liquid materials cannot be acconimodated within the framework of this introduction to semiconductor physics. Interested readers are referred to the extensive literature on this subject (see, for example? Elliot, 1983; Mott and Davis, 11979; Bunch-Bruevich, Enderlein, Ewer, Keiper, Mironov, and Xvyagin, 1984; Adhr, F r h s c h e and Ovahinsky, 1985). T h e microscopic definition discussed above contains no recognizable constraints with regard to the chemical nature of semiconductors. One may expect, therefore, that semiconductors should be distinguished by having a large chemical diversity. This is in fact true semiconductors may be composed of a large variety of chemical elements and compounds. The most fully expAored candidates and those used for technical applications today arc crystalline semiconductors consisting of relatively small chemical u n i t s , i.e. either dements or binary and ternary compounds. Knowledge of thc mxistence of a distinct material group ‘semiconductors’ developed, historically, only relatively late. MPtals have been used by men since antiquity, but semiconductors attracted attention for the Erst time only a century and a half ago. The first reference to a characteristic semiconductor property dates back to Faraday who in 1833 observed an increase of the electric conductivity of silver sulfide with temperature. The exponential form of this increase was discovcrcxl by Hittorf in 1851. The trerm ‘semiconductor’ was introduced in 1911 by Konigsberger and Weiss after a similar term of about the same context had already been employed by Ebert (1789) and Bromme (1851). The late and, initially, relatively slow development of semiconductor physics, is primarily due to the circnnistance that the characteristic properties of semiconductors depend strongly on their degree of purity, more precisely, on the presence 0’1absence of certain chemical elements. This is also the reason that many semiconducting materials in their natural form as minerals do not display the typical properties of a semiconductor they are too heavily polluted and have too many structural defects. Natural diamondrr, for example, are semiconductors only in rare cases. Accordingly, clean fabrication of the materials in the laboratory and the controlled incorporation of chemical elements played a crucial role from the very beginning. The lack of such control in preparation presented fundamental difficulties which had to be overcame in the early days of semiconductor rasearch. The necessary accuracy of composition control, which amounts to one atom in one hundred thousand or less, excccdcd the accuracy that prevailed in chemistry at the t h e by orders of magnitude. It was necessary to raise the accuracy of chemical composition control to a level wherein one could measure one millionth of a mole haction instead of one thousandth. only in this way could reliable results be achieved with semiconductors. Since such accuracy was achieved ~

~

4

Chapter 1. Characterization of semiconductors

only gradually, semiconductor physics, in its early history, was confronted with apparently mysterious phenomena and contradictory results. For example, silicon and germanium were first thought to be metals, until the recognition that the impurity concentrations of certain elements were too large to achieve semiconducting properties. From the very beginning semiconductor physics research was impeded by the need for expensive fabrication of material samples. The Geld could develop only to the extent that good samples could be made available. Naturally, the first samples weie either materials which could b r fabricated relatively cheaply or materials which occurred in nature in a suitable form, albeit not ideal. Among these first materials were metal sulfides, which like lead sulfide in its mineral form as Galena and copper oxide (CuzO) and selenium in their artificially grown form, displayed good semiconducting properties even with relatively strong pollution and structural imperfections. In 1874, Braun discovered that contacts between certain metal sulfides and metal tips fxhibited different electrical resistance upon reversal of the polarity of the applied voltage. Such point contact structures were used in radio receivers as rectifiers at the beginning of our century. One can mark these point contact rectifiers as the first semiconductor devices. Similar rectifying action was also found for selenium and copper oxide. Moreover, a large change of electrical conductivity could also be achieved in these materials through irradiation by light. For selenium, this property was discovered as early as 1852 by Hittorf. Since the beginning of the 20-th century, this effect has also been used practically in the selenium photocell. The first technical use of copper oxide as a rectifier was accomplished in 1926 by Grondahl, followed by rectifiers using selenium. The first practical application of copper oxide in photocells was accomplished in 1932 by Lang. Because of their technological importance, selenium and copper oxide were the first semiconductors to be subjected to more detailed physical investigations. The semiconducting metal sulfides, selenides and tellurides were already known earlier because of their good luminescence properties. In the further exploration of these materials’ semiconducting behavior, luminescence physics partially merged with semiconductor physics. In the mid-nineteen-thirties, the search for a solid-state-based electronic switching element which could replace the vacuum tube was extended to the elemental semiconductors germanium and silicon. The most important results of this research, which turned out to be decisive for the whole further development of semiconductor physics, were the invention of the germaniumbased bipolar transistor in 1949, and the realization of the field effect transistor with the help of silicon at the end of the nineteen-fifties. With the introduction of silicon, the stage was set for the development of semiconductor microelectronics. Later, a similar role was played by compounds involving elements of the third and fifth groups of the periodic table, like gallium

1.2. Atornic structure of ideal m y s t a h

5

arsenide or phosphide, making possible the development of semiconductor optoelectronirs. The broad technical application of its results distinguishes semiconductor physics now from its early days. It is well established that semicondnctors arc exceptionally well-suited for necessary functions in electronics and electrical engineering. This is by no means accidental, but is due to the microscopic nature of semiconductors, which permits the controlled variation of characteristic matprial properties by external means over a wide range of parameters. The great technical importance of semiconductors has made thorough physical investigation of these materials necessary, but it has also justified the high cost involved in their fabrication and study. Owing to both of these aspects semiconductors are thc best explored and understood materials of condensrd matter today. Moreover, a multitude of physical phenomena which occur in other solid state materials may also be observed in semiconductors, often in the most distinctive way. For this reason studies of semiconductors can also provide knowledge of other solid state mattrials. Semiconductors have in fact become model systems for basic research in condensed matter physics. The presentation of the microscopic principles of semiconductor physics will occupy most of this book. The introductory first chapter lies outside of this framework because it involves discussion of the results on a phenomenological basis. The characterization of semiconductors by means of their unique macroscopic features, which we have touched upon above, will be continued in Chapter 1. In this regard, atomic structure will be discuss4 in section 1.2, chemical nature in section 1.3 and macroscopic physical properties in section 1.4. In dealing with macroscopic properties we will not restrict ourselves to mere description, but we will also use them to motivate the microscopic model of semiconductors introduced above. In this connection, we will make the first step towards a microscopic theory of semiconductors in section 1.4. Naturally, this will have to be done in a heuristic way, and many questions postponed until later. The full presentation of the microscopic principles of semiconductor physics will follow in later chapters.

1.2

Atomic structure of ideal crystals

In all solid state materials, including amorphous ones, the neighbors of an arbitrarily selected atom are ordered in a regular way, just as in a molecule. The term short-range order is uscd for this property. The neighbors of a particular atom form its short-range order complex. Semiconductor materials, as they are treated in this book, are crystals. ‘Sheir atomic structure is approximately that of ideal crystals. The latter are distinguished by yet another order, apart from short-range, which is termed long-range order. This

6

Chapter 1. Characterization of semiconductors

means that,, for a given atom, there are remote atoms possessing the same short-range order complexes as the original atom, rand that the positions of the remote atoms are related to the position of the origina1 atom by sirnple transformations. 'l'hp crystal is considered t o be inhitely large in this context. Atoms having identical short-range order complexes are termed equivalent. Equivalent atoms are necessarily of the same chemical species, but chemically identical atoms need not necessarily be equivalent.

1.2.1

Crystal lattices

The transformations displacing atoms into equivalent ones are translations of the crystal by vectors T which form linear combiaations

of three non-coplanar vectors Al, Az. A3 with arbitrary integer coefficients t l ,t 2 , t3. The parallelepipeds of the crystal spanned by the particular displacement vectors Ak, A2, A3 are called unit cells. By putting unit cells together the wholc crystal may be constructed. ' h e size of a unit cell and the number of atoms in it is not fixed by the above definition, and can in fact be taken arbitrarily large, as long 8s it remains finite. The pertinent question is not how large a unit cell can be, but rather how small. The answer to this question leads us t o the definition of the primitive unit cell and the crystal lattice.

Definition The smallest possible unit cell is called a primitive unit cell In the extreme this cell can contain only 1atom, but as a rule, it has several atoms. If there is only one atom per cell, then the short-range order merges into the long-range order. If the unit cell is taken to be a primitive one, the vectors Al, A2, & are some minimal vectors a l , az,w. The parallelepiped spanned by these vectors is a primitive unit cell, Each translation by a vector R of the form case

with integer coefficients r l , r2, r 3 transforms the crystal into itself. One refers to this property as the trunslation~lsymmetry of the crystal. The point set defined by the vectors R forms a spatial lattice called the crystal lattice. The vectors al,a2, are termed primitive lattice vectors, The volume 00of a primitive unit cell may be written as the triple scalar product of al,ag,a,

Ro = al . [ag x

a].

1.2. Atomic structure of ideal cestals

7

While the lattice of a crystal and the volume Qo of its primitive unit cell are well defined, this is not the case for its primitive lattice vectors al,a2,a3 and also it is not true for the form of its primitive unit cell. Any set of linear combinations of the primitive vectors al,a2,a3 which yields a triple scalar product equal to the volume Qo is again a set of primitive lattice vectors, and the parallelepiped spanned by them forms a primitive unit cell. The corners of the parallelepipeds do not necessarily have to lie on lattice points. Each parallelepiped, shifted arbitrarily in space, is again a primitive unit cell. The 'parallelepiped form is also not imperative, as there are also other forms possible. An especially compact primitive unit cell is the socalled Wzgner3eit.z cell. The center of this cell lies on a lattice point and its surface is formed by the perpendicular bisector planes which divide in half the line segments joining the center to adjacent lattice points. Translations which transform a crystal into itself, by definition, do the same for the lattice of the crystal. Here, the translations are through lattice vectors R, called lattice tramtations The set of all lattice translations forms a group. This term describes a mathematical set of elements among which a 'multiplication' is defined that results in products which are also elements of the set. Further properties of a set forming a group are listed in A p pendix A. In particular, there must be an identity element, and the inverse of an element must also be an element of the set. In the case of translations the 'multiplication' is the consecutive application of two of these transformations. Since two consecutive lattice translations constitute yet another lattice translation, and also the requirements of Appendix A are satisfied, the set of all lattice translations of a crystal forms, in fact, a group, called the translation symmetrg, group or simply the tramlation group. Groups of symmetry elements play a central role in the description of the atomic structure and other microscopic properties of crystals. Appendix A provides a thorough discussion of groups as needed in this book.

Point symmetry of lattices We now ask whether there are other possible spatial transformations, besides translations, which transform lattices into themselves. From the outset it is clear that the only transformations which may be considered are those that do not change the distances between lattice points, i.e. mgzd drsplacernents of the lattice (see Appendix A). One can show that, besides translations, there is a second class of transformations fitting this description, namely rotatoom and reflectzons, as well as all products which are compounded from them, such as rotation-reflections, rotatton-znverstonsand znverszon itself. Taken together, they are termed orthogonal traasfomattom. These differ from translations inasmuch as they leave one or several lattice points unchanged, while the remaining points are shifted by vectors depending on

8

Chapter 1 . Characterkstion of semiconductors

their positions. In a t,ranslation, all points are shifted by the same vector, with no points fixed. Rotations transform right-hand4 Cartesian coordinate systems into right-handed systems, but the application of reflections and inversion to right-handed system results in left-handed ones. It turns out that there are orthogonal transformations which transform lattices into themselves. They are called point symmetry operations of lattices. The set of all point symmetry operations of a lattice forms a group, as does the set of all lattice translations. The multiglicslion of two of these operations is again understood to represent their consecutive application. Groups of point symmetry operations are termed point gvoups. In Appendix A we describe thwn in detail. Not all of the various point groups listed in Appendix A are allowed as symmetry groups of crystal lailices, but only parlicular ones which are called holohedral point groups. We will derive them now by demonstrating that lhey must have three special properties. First, all uf these point groups must contain the inversion transformation with r e q e t t to the lattice point R = 0. This may be seen as fullows: Inversion with respect to 0 transforms a lattice point R into -EL Considered joiutly with R, the point -R is a lattice point having -q, -r2, -7-3 as integer coefficients. Therefore, inversion with respect to 0 transforms an arbitrary lattice into itself. It follows immediately that inversion with respect l o any other lattice point will do h e same. Second, it turns out that rotation symmetry axes Ohrough lattice points can only be 2-, 3-, 4- and 6-fold while ri, 7- and more-fold axes are not compatible with the translation symmetry of the lattice. One may prove this as follows: Let C, be a rotation about such an axis t,hrough an angle 2?r/n. We consider a lattice plane perpendicular to this axis and denote a primitive lattice vector of the corresponding planar lattice by f (see Figure 1.1). Rotating it through 2a/a, it becomes C,f, and a rotation by - 2 ? r / R transforms it into Cqlf. If, as w e suppose, Cmbelongs to the point group of the lattice, them s n d o a C;'. Thus both C,f and C i l f are vectors of the plane lattice. The same holds for the sum Cnf i Cg'f ol t,he two vectors. Moreover, C,,f CGLf represents a vector parallel to f. This means that C,f CF'f must be an integer multiple of f. Since the largest possible length of Cnf Cg'f can only he 2 / f J ,one has Cnf f C'L'f = p,F with pn - -2. -LO, 1 or 2. This is indicative t.hat, the relation

+

+ +

p , = 2COS

(?)

must hold for p,. For p n = -2, equation (1.4) yields n 2. For pn -= -1 one has n = 3, for pn - 0 it follows that n = 4 and for p , = 1 l h e solulion is n = 6 . For ppI- 2 equation (1.4) has only the trivial solution n = I . This completes the proof concerning rotation symmetv opexations. For so-caIled gvasi-crystals. which do not exhibit an exact translation symmetry, rotations 1

1.2. Atomic structure of ideal crystals

9

Figure 1.1: On the multiplicity of the rotation axes of crystal lattices.

about &fold and other axes are also possible. Third, one finds that point groups of lattices having 3-, 4- or 6-fold axes of rotation must also necessarily contain mirror planes parallel to each of these axes. The explicit proof of this assertion will not be presented here. All three required properties described above are satisfied by each of exactly se’i;en point groups, namely c%(i), cZ~($), ~ 2 h9( 2g q, e ) ~>3 & $ ) > D4h(&;$). D ~ h ( $ ; z ) and Oh ($3;). The point group notations used here are those of Schonflies and the international notations are given in brackets. Both systems of notation are explained in Appendix A. In summary. the above results mean that exactly seven different point groups are possible for spatial lattices. They define the seven crystal systems: triclinic (Ct),monoclinic, ( C 2 h ) , orthorhombic ( D z h ) , trigonal ( D 3 d ) . tetragonal (D*h), hexagonal (DGh), and cubic (Oh). Bravais lattices

Within a given crystal system, several different types of lattices may exist. Their common property is that they all have the same point symmetry, but they may differ otherwise. Figure 1.2 visualizes them by means of their unit cells. These differences give rise to diflerenf lattice types. The simplest lattices ef a given point symmetry are represented at the far left of each row in Figure 1.2. They axe called primitive lattices. Even these simple lattices are not unambiguously &ermined by their ‘point symmetry. If one, for instance, multiplies all lattice vectors by the same real number, i.e. stretches or compresses the lattices evenly on all sides, the point symmetry remains unchanged. In less symmetrical crystal system one may even change certain length relationships or angles between primitive lattice vectors without disturbing the point symmetry. In the tetragonal system the height of the

10

Chapter 1. Characterization of semiconductors

rectangular parallelepiped in relation to its basis may be changed arbitrarily. Generally speaking, the primitive lattices are determined only up to continuous point transformations preserving their point symmetries. Moreover, starting from a primitive lattice one can produce other sets of regularly ordered points by adding new points to each primitive unit cell in equivalent positions. As before, equivalent refers to positions which are either identical or that differ by a lattice vector. If one places the new points in symmetrical positions, i.e. those which are transformed into themselves or equivalent points by the point symmetry operations, snch as the centers of the primitive unit cells, one obtains a new point set having the same point symmetry as the original lattice. The new point set, in general, no longer forms a lattice, but may be thought of as the union of several lattices placed within each other. In some special cases, however, the result may still be a primitive lattice. Whether this happens or not is a question which must be explored separately in each case. If the answer is positive, one has to examine whether the lattice is only another realization of the original primitive lattice, i.e. whether or not it can be brought back to the original one by a continuous and symmetry preserving transformation. It turns out that both cases may occur. If the lattices cannot be transformed into each other by such a transformation, then this implies that there are two different types of lattices with the same point symmetry. One calls them different Bmvais types or Bravais latttices. According to this definition two Bravais lattices are of the same type if they may be transformed into each other by a continuous and point symmetry preserving transformation, otherwise they are Bravais lattices of different types. As an example, we consider the different Bravais lattices in the case of the cubic crystal system. If one adds the body centers of the primitive unit-cellcubes as additional points to a primitive cubic lattice, the resulting point set has the cubic point symmetry and it forms again a lattice. The same holds if the added points are the centers of the faces of the primitive unit-cell-cubes instead of the body centers. Neither the spacecentered nor the face-centered cubic lattices can be transformed back to a primitive cubic lattice by means of a continuous and symmetry preserving transformation, nor can the two centered lattices be transformed into each other by such a transformation. Therefore, they represent cubic lattices of two new Bravais types. If one adds both face and body centers to the primitive cubic lattice, then the new point set forms again a primitive cubic lattice, however, with a lattice constant equal to half of that of the original lattice. It may be transformed back to the original primitive cubic lattice by means of a continuous and symmetry preserving transformation, thus it does not represent a cubic lattice of a new Bravais type. Altogether one finds three different cubic Bravais lattices, the primitive (p), the body-centered (bc), and the face-centered (fc) ones. Analogous considerations have to be made for the other 6 primitive Iattices.

1.2. Atomic structure of ideal crystals

11

Chic F

Cuhic 1

Cubic P

Tetragonst I

pj$, .# .‘\

Monocfinic

Monoclinic 1

P

Trigonal

R

@

Triclinic

I

Trigonal and hexagonal P

Figure 1.2: Common unit cells of the 14 Bravais lattices.

12

Chapter I , Charactm‘zation of semiconductors

In this w q ,one finds that, in totul, 14 diEerriit Bravais lattices are possible in the 7 crystal systems. The assignment ol Iht.sr lattices to the crystal systems is indicated in Table 1.1. The Bravais lattices themsehps are shown in Figure 1.2.

1.2.2

Point groups of equivalent directions and crystal classes

The lattice of a crystal serves as a conceptional basis for the illustration of its translation symmetry. Lattice points do not necessarily have to be occupied by atoms, which may be bum1 at general points of the primitive unit cells. Generally, a primitive unit cell contains several atoms which may be either chemically identical or different. We denote them here by an atom index 1 which t,akes the values 1 - 1 , 2 , . . . , L , where L is the total number of atoms in a primitive unit cell. For L 2 2 the set of all L atoms is called the basis uf the crystal. In the case I. = 1 one says t.hat the crystal has no basis. ‘I’he position of the 1-th at,om, relative to the corner R of a primitive unit cell, is drscribed by a vrcbor The position Ri of this atom relative t o the origin is then given by

Ri=Rti. (1.5) Without loss of generality one may always set onr of the vectors il, e.g. i l . equal to zero. If the primitive unit cell canlains only 1 atom, it may be placed in o m o f the roriiers of the cell. A crystal without basis may thiis be describd as a laltice whosp points are dl occupied by ahoms. For L 2 2, the crystal may be generated in such a way that one multiples its crystal lattice L-fold, then shifts the resulting sublattices relative to t,he first by, respcctivcly, the vectors 6 , . . . ,?I,, and finally occupies the points of the shift,ed lattices, respectively, with atoms of t,he species 2,. . . , L. With only one atom per primitive unit cell! any point symmetry operation of the lattice will ncccssatily transform the whole crystal into itsdf. For cryst,alswith basis, however, this is not true in general. For this reason it is meaningful to consider, besides the point symmetry operations of the crystal lattice, also orthogonal transformations which map physically equivalent direchns of Ihe crystal into each other, without necessarily bringing the crystal back onto itself. We explain the meaning of ‘physical equivalence’ by using the example of the relation between the vectors of the electric current density j and the electric field strength E in a crystal. Generally, j is a non-lincar fimrtioii of E and, because of crystal snisotropy, the direct.ions of the two vectors may be different. If E and j are transformed from their original directions reht.ive to the crystal into new ones, without changing their relative orientation, the relation betwccn the new j and E, in general, will differ from the relation before rotat,ion. Analogous statements hold for reflections and rotation-reflections. If tbcre are orthogonal transformations

1.2. Atomic structure of ideal crystals

13

Table 1.1: Symmetry classification of crystals. The following abbreviations are used p - primit.ive, bc - body centered, fc face centered, bfc - basis face centered. ~

Crystal system

tnclinic

Holohedral

Braiais

Crystal

Number of space

group

lattk

class

groups

+

r c 1

C,

monoclinic

! I 1

'2h

rhombic

trigonal

tetragonal

I

hexagonal

cubic

I ) p

bc

fc

1

1

14

Chapter 1. Characterization of semiconductors

which leave the relation between the two vectors unchanged - which ultimately must be verified experimentally - the original and the transformed directions are called ‘physically equivalent’. The group of orthogonal transformations which map physically equivalent directions of a crystal into each other, is called the point group of equivalent directions. It defines the crystal class. Each crystal class corresponds to a particular point group of equivalent directions. In crystals without basis the point group of equivalent directions coincides with the holohedral point group. For crystals with a basis this is no longer true in general, and the point groups of equivalent directions are generally subgroups of the holohedral groups. There are as many different point groups of equivalent directions, or crystal classes, as there are different subgroups of the 7 holohedral groups. Using Appendix A one can easily show that their number amounts to 32 (see Table 1.1). Not all crystal classes can be realized in all crystal systems - the point group which defines the crystal class has to be a subgroup of the holohedral group of the crystal lattice. Each crystal class is, however, found in at least one crystal system, several in more than one. In assigning the crystal classes to the different crystal systems one proceeds as follows: A class which exists in several systems is attributed to the one with the lowest common symmetry. In this way one obtains the assignment between crystal systems and crystal classes shown in Table 1.1.

1.2.3

Space groups and crystal structures

It remains for us to explore the symmetry of the crystal as a whole. This consists of the set of all rigid displacements which transform the atoms of the crystal into identical or equivalent positions. It is obvious that symmetry operations which transform equivalent atoms into each other must necessarily also do the same with physically equivalent directions. The converse, however, is not always true - after carrying out s rotation, reflection, rotationreflection or rotation-inversion which transforms a particular direction into a physically equivalent one, the atoms of the crystal are not necessarily also trausformed into equivalent atoms. It may be necessary to add to a rotation yet another translation parallel to its axis (screw-rotation), or to add to a reflection a displacement parallel to the mirror plgraetglide-re~ection),or to add both in the case of rotation-reflections and rotation-inversions. That one records in this way all conceivable symmetry operations follows from a theorem proven in Appendix A, which states that every rigid displacement which is not a pure translation or an orthogonal transformation, must be a screw-rotation or a glidereflection. The parallel displacement P; which follows a rotation about an n-fold axis in a screw-rotation, must be an integer multiple of one n-th of the smallest lattice vector in the direction of the axis. This follows immediately if one

1.2. Atomic structure of ideal crystals

15

considers the n-th power of the screw-rotation, which is also a symmetry element as well. This describes an n-fold repetition of the rotation by 27r/n, i.e. a rotation by 27r, or no rotation at all, followed by the translation n times Since the crystal must pass into an equivalent position by such a transformation, the vector n times p’ must be a whole lattice vector. As it points in the direction of the screw axis, it is necessarily a multiple of the shortest lattice vector in this direction. In the international notation system the multiplicity is described by a lower index a f b e d to the symbol for the rotation axis. This also indicates that the axis is not an ordinary one but a screw axis (with right-handed thread). The symbol 63, for example, means a 6-fold screw axis with a parallel displacement by half the shortest lattice vector in the direction of the axis. For a glidereflection one can similarly show that the translation in the mirror plane must be a superposition of multiples of halves of the smallest linearly independent lattice vectors in this plane. In summary, we may state that the symmetry operations on a crystal are of the following types: translations by lattice vectors, proper rotations, reflections, rotary reflections, rotation-inversions as well as screw-rotations and glide-reflections. Furthermore, all combinations of these transformations are allowed. The set of all symmetry operations on a crystal forms a group. It is called a space group. If the crystal has no screw-rotations or glidereflections as symmetry operations, then its space group necessarily contains the point group of equivalent directions as a subgroup. Such space groups are called symmorphic. Space groups with screw-rotations or glide reflections are called non-symmorphic. The latter do not contain the point group of equivalent directions of the crystal. Each element of a symmorphic space group is the product of an element of its translation group and an element of the point group of equivalent directions. The set of all possible space groups of crystals may be obtained in the following way: One considers, first of all, crystals of the triclinic system. In this case only the primitive Bravais lattice is possible. The corresponding space groups may easily be determined - there are only two. Similarly one proceeds with all other combinations of crystal classes, Bravais lattices and crystal systems, moving in the direction of increasing symmetry. At each stage of counting only the newly occurring space groups are added. In this way one obtains the numbers indicated in Table 1.1. The total is 230. Each of these 230 possible space groups corresponds to a particular crystal structure. By specifying its space group, the structure of a crystal is uniquely determined, except, of course, for changes of the distances between atoms and of angles between lines connecting atoms which do not affect the symmetry. The majority of semiconductors belong to a small selection of the possible crystal structures or space groups. Five are especially important: Their designations in the international system are Fd3m (diamond structure), F W m

+.

16

Chapter 1. Characterization of semiconductors

(zincblende st ruct ure) , F m3m (rocksalt st ruct ure) , P63m c (wurt zite st ructure), and P3121 or P3221 (selenium structure). Each of these crystal structures is more or less closely tied to a particular material group. We consider this connection in more detail in the next section, which deals with the chemical nature of semiconductors. The five crystal structures above belong to just two different crystal systems - t h e diamond, zincblende and rocksalt structures belong to the cubic system and the wurtzite and selenium structures belong to the hexagonal system. We start with the three cubic structures. Table 1.2 summarizes their properties

1.2.4

Cubic semiconductor structures

Crystals having diamond, zincblende and rocksalt structures not only belong to the same crystal system, but also have the same Bravais lattice, namely the face centered cubic, abbreviated as fcc. The fcc lattice is commonly described in a Cartesian coordinate system whose unit vectors e,, ey,e, are taken in the directions of the cubic crystal axes. The lattice constant a is the distance between the lattice points of a primitive cubic reference lattice, which is obtained from the fcc lattices by omitting the face centers. The primitive lattice vectors al,a2,* of the fcc lattice may be chosen in the form

The lattice constants for a series of semiconductors with fcc lattices are listed in Table 1.3. We stress that the cubic lattice constant is neither the distance between two lattice points (it is u / f i for the fcc lattice), nor the distance between two atoms in a crystal with this structure (which is &/4). For the volume Ro of the primitive unit cell one obtains the value u3/4 from equation (1.3). In the case of silicon this yields Ro = 4.00 x cm3. A silicon crystal of volume 1 cm3, therefore, contains 2.5 x primitive unit cells. The lattice does not determine the positions of the atoms. This is done by the basis of the crystal, in which respect the three cubic semiconductor structures differ.

1.2. Atomic structure of ideal crystals

17

Table 1.2: Characteristicz of the five most comnion structure types of seniiconduc tors. 0 < C < h, 0 < n < $. More details are given in the text.

Diamond

Zincblende

Itocksalt

Seleniuni

Wiirtzite

F#m

P3121

P3221

fy ,3; 4 -

2

43m

Gmm

32

Bravais

cubic

cubic

cubic

hexagonal

hexagonal

lattice

fc

fc

fc

P

P

Primitive lattice vectors

I

Basis

I

I

18

Chapter 1. Characterization of semiconductors

Table 1.3: Lattice constants of semiconductors of various types of structures.

1:

Diamond structure C

Si

Ge

5.43

5.65

Zincblende structure

,

I

a-Sn 6.46

AlP

RlAs

AlSb

GaN

Gal’

Cans

CaSb

Id’

4.37

5.47

5.66

6.14

4.54

5.45

5.65

6.13

5.87

InAs

InSb

ZnS

ZnSe

ZnTc

CdTe

HgSe

IlgTe

6.05

6.47

5.43

5.66

6.08

6.42

6.08

6.37

Rocksalt structure

I

I

PbS

PbSe

PbTe

SnTc

CdO

MgS

MgSe

5.93

6.12

6.45

6.3

4.70

5.20

5.46

Wurtzite structure

1:

I

I

ZnS

CdS

CdSe

MgTe

AlN

GaN

InN

3.82

4.14

4.31

4.53

3.11

3.19

3.54

a

1.64

1.62

1.63

1.62

1.60

1.62

1.61

cla

Selenium structure Se

Te

4.36

4.47

a

1.14

1.33

cl a

Diamond and zincblende structure For diamond and zincblende structures, the basis consists of two atoms which, respectively, have the same or different chemical natures. The two atoms are shifted with respect to each other in the direction of the body diagonal of the primitive reference cube by a quarter of its side. If the basis atom 1 is put at a lattice point, then the position of atom 2 is given by

19

1.2. Atomic structure of ideal crystals

Figure 1.3; Spatial model of a crystal having diamond (a) and zincblende structure (b). In the diamond structure all atoms arc of the same chemical clement, in the zincblendc structure the atoms are of two different chemical elements.

0 1A 02 8 3A 0 4B

(1111

3

P

6 Cmll

a IA r, 2 0 3A 40 * 5A 60 (J

cfm

Figure 1.4: Projection of a crystal having diamond and zincblende structure in a (100) plane (left) and a (111) plane (right). Atoms of the same size and darkness lie in the same plane. The vertical sequence of atomic layers is indicated on the right hand side of the projections. The sequence for the (100) plane is repeated after 4 layers, and that for the (111) plane after 6 layers. Crystals with diamond and zincblende structures may also be described as a composite of two interpenetrating superposed fcc lattices displaced with respect to each other by the vector ?2 and with their lattice points occupied by, respectively, chemically equivalent or different atoms. The geometric relations are depicted in Figures. 1.3 and 1.4 in, respectively, a 3-dimensional representation and a plane projection. From Figure 1.4 one can readily see that for both crystal structures the cubic axes put through an atom form 4-fold mirror-rotation or inversion rotation axes (which are the same in this drawing). The body diagonals through an atom are %fold axes of

Chapter I . Characterization of semiconductors

20

Figuw 1.5: Stereograms of the paint groups Oh (left) and 7'6 (right) of, respectively, the diamond and zinchlmde structiires. rotation, and the planrs through opposite cube dges are mirror planes. Both structures are theiefore tramformd into themselves by the point group T d , the symmetry group of a tetrahcdron. It follows that their point groups of equivalent directions are at least T d . For the zincblende structure there are

no further point symmetry operations which transform equivalent directions into each other. The point group of equivalent directions is therefore T d , and the space group turns out to be symmorphic. It is denoted by F13m.

In the raw of diamond structure, thew i s yet another symmptry operation, namely that which transforms the two chemically equivalent atoms of the basis into cach other. It may be described as reflection in a plane perpendicular to a cubic axis, say e,, which cuts the connecting line bel w r n two atoms at its center, followrd hy a translation in that plane by (e/4)(ex ey). One, therefore, has a glidereflection as additional symmetry dement. The space group of diamond structure becomw non-symmorphir in this way The point group of directions of this structure may be obtained from the point group T d of the zincblende structure by adding the reflections in planes perpmdicular to the thrw cubic axes. This yields the full cubic group Oh, as one can easily determine by means of the stereograms of the Iwo groups rn Figure 1.5 (for a n introduction to the stereograms of point groups, see Appendix A). The group o h can be generated from t h e tetrahedral group in yet anothcr way, namely by adding the inversion operation. As an element of the space group, inversion must be relative to the site of an atom and be followed by a tradation through the vector T2 or -?zr depaiding on whether m c considers a 1- or 2-atom. For future reference we bear in mind that for the diamond and zincblende structures, inversion always involves an exchange of the two sublattims.

+

We now extend our discussion to thc neighborhood-relations in the two crys-

21

Figure 1.6: Spatial model of a crystal with rocksalt structure.

tal structures, the so-called coordmatatm Consider atom 1 of thc Lesis of the unit cell at R = 0. The basis atom 2 of this cell is one of its nearebt neighbors. If all rotation-reflections are executed with respect to the axis through atom I, then the basis atom 2 of the primitive unit cell at R = 0 gives rise t o three more basis atoms of type 2. They are all at the same distance from atom 1 as thP 2-atom first considered, but naturally they lie in three other primitive unit cells. This means that each 1-atom has, altogether, four nearest neighboring 2 atoms. Since it would also have been possible to start from a 2-atom in this consideration, the mme also holds with respect to a %atom, and, of course, independently of whether t h 2-atom is of the same chemical nature as the 1-atom (diamond structure) or is not (zincblende structure). The four nearest neighbor atoms lie at the corners of a tetrahedron, whose center is occupied by the atom itself (see Figure 1.3). Their relative positions are given by the vectors (a/4)(1, 1, l),(e/4)(T, 1,l), (a/4) (1,7,l), (a/4)(1,1,i). Thr distanrc betwrrn nearest neighbor atoms is &a/4. For silicon this is 2+35 A . The second nearest npighbors belong to the same sublattice. Therefore, for both structures, they arc atoms of thP same chemical species. They are located at thc nearest-neighbor lattice points of the fcc lattice. relative to the central atom, thus their positions are ( u / 2 } ( * e 9 f ef), ( a / 2 ) ( f e y f e z ) ,(a/2)(*e, f ey). This means that thew are 12 second nearest neighbors in each slructurr. Rocksalt structure The basis of this crystal structure consists, like that of zincblende structure, of two atoms of diffprent chemical nature which arc displaced relative to each other in the direction of the space diagonal of the reference cube. In contrast to zincblende structure, the displawmcnl i s not, however, a quarter but half of the length o€ the space diagonal of the reference cube, whence

22

Chapter 1. Characterization of semiconductors

(1111 1A

1A 01B 4 2A 0 28

028 0

3A 48 5A

0

6B

-

Figure 1.7: Projection of a crystal having rocksalt structure in the (100) plane (left) and the (111) plane (right). The vertical sequences of atomic layers is shown on the right hand side of the projections. In the (LOO) case the stacking repeata itself after 4 Iayem, and in the (111) case after 6 layers. Accordingly, a crystal having rocksalt structure can be described as a composite of two fcc lattices, displaced with respect to ~ a c hother by the vector ?'z, and with their lattice points occupied by chemically different atoms. This structure is illustrated in Figures 1.6 and 1.7. As one can see horn these drawings, the crystal has the full symmetry of the primitive reference cube. The point group of equivalent directions is therefore the full cubic group o h (see Figure 1.5). The space group is symmorphic and is denoted by FmSm. In Ihe direction of a cubic axis, the atoms are separated by a distance of a / 2 , with 1- and 2-atoms alternating. Since shorter distances do not occur, these atoms are ncarest neighbors. Each atom, therefore, has 6 nearest neighbors. The second nearest neighbors are the nearest neighboring atoms of the same fcc sublatticp. Their number amounts to 12, as in the zincblende structure. The relative positions are also the same as in this structure. 1.2.5

Hexagonal semiconductor structures

The primitive lattice vectors of the hexagonal lattice may be written in the form h

ai y- ae,,

a h

a2 - --ex

2

a + -&e,", 2

h

ce,.

Here ek, I$, e," are unit vectors of a cubic coordinate system whose z-axis points in the direction of the c-axis of the hexagonal lattice and whose z-axis is identifird arbitrarily with one of the three symmetric lattice directions in the plane perpendicular to the c-axis. The two primitive lattice vectors a1

1.2. Atomic structure of ideal crystals

23

and a2 in this plane have equal lengths and form an angle of 2x13 with each other. The volume 5-20 of a primitive unit cell is &a2c/2. The lattice constants a and c of some semiconductors with a hexagonal lattice are listed in Table 1.3. Wurtzite structure The basis of crystals with wurtzite structure consists of 4 atoms. We denote them by M I , X I , Mz and X 2 . Each pair of these atoms, namely M I , M2, on the one hand, and XI, X 2 ) on the other hand, are chemically equivalent, while the two pairs are of different elements M and X. If atom M I is sited at a lattice point, then the position vectors of the three others are

fx1= cc e," ,

(1.10)

Here, ( is a parameter which may take any value between 0 and 1/2. A wurtzite type crystal can be understood as a composite superposition of four interpenetrating hexagonal lattices, with the lattice points of two of them occupied by atoms of chemical species M, and two of them by atoms of species X. The two M-lattices are displaced relative to each other by F M ~ ,likewise as the two X-lattices. The displacement of the XI-lattice with respect to the MI-lattice is Fx1. In Figures 1.8 and 1.9 these relations are illustrated in, respectively, spatial and planar displays. The M-atoms are shown in black and the X-atoms in white. From these figures one can easily determine the symmetry of the wurtzite structure. The c-axis through the center of one of the empty hexagons in Figure 1.9 forms a 6-fold screw axis 63: a 2n/6-rotation about this axis must be connected with a parallel displacement by c/2 in order to transform the crystal into itself. This also mean8 that this axis represents a %fold proper axis of rotation. The axis contains six inequivalent mirror planes, among them three proper reflection planes and three glide-reflection planes joined with translations by c/2 in the direction of the c-axis. The corresponding point group of directions is c 6 v (see Figure 1.10). The space group of the wurtzite structure is nonsymmorphic and is denoted by P6amc. Consider the atom M I in the unit cell at R - 0. Its neighbor in the same unit cell is atom X 1 at the site ?XI. Another neighbor of M I is Since the atom Xz at the site -ce,h I Fxz of the unit cell at R - -c$. the c-axis through atom M i is a %fold symmetry axis (sep Figure t.9), one obtains from Xz two additional X-atoms at the same distance but in other unit cells. Altogether, there are four neighboring X-atoms. In order that all four be at the same distance from M I , and, correspondingly, all four be

24

Chapter 1 . Characterization of semiconductors

Figure 1.8: Spatial model of a crystal with ideal wurtzite structure.

0 1 A 028 3A 0

4B

Figure 1.9; Projectionof a crystal with ideal wurtzite structure in the plane normal to the c-axis. The vertical sequence of atoniic layers is shown on the right herid side.

Figure 1.10: Stereograms of the point group structure.

c6~of

the wurtzite

of M I , the distance 1 7x1 I between M I and X I must be equal to the distance I -ce," t F x 2 I between Mi and X2 in the unit cell at R = -ce!. This results in the condition nearest neighbors

c = , 1t 9 ( ;1)

.

(1.11)

If it is fiilfilled, the atom Mi has four nearest neighboru of chemical species X . The same statement is true for M I , with the interchange of M and X , dito for X I and Xa. In semiconductor crystals of the wurtzite type, the condition (1.11) is always satisfied almosl perfectly. This means that the four nearest neighbors of an atom are sited at the corners of a slightly deformed tetrahedron which surrounds the central atom symmetrically, and

1.2. Atomic structure of ideal crystals

25

which is compressed or stretched in the direction of the c-axis. The degree of deformation depends on the ratio cla. For the deformation to vanish, the distance between the M I - and XI-lattice planes perpendicular to the c-axis must be three times the distance between the X I - and M2-lattice planes perpendicular to this axis, just as in the zincblende structure. This yields = 3(1/2- E, are allowed. In turn, the vaniahing of absorption for E < E , can be explained if the energy values between E, and & are forbidden. Such a distribution of allow4 energy values is shown schematically in Figure 1.19. The representation of these values as a function

+

+

+

I .4. Macroscopic properties and their microscopicimpIications

39

Valence bond

O0

X---

HL 0

1 f [El--)

Figure 1.19: Ordering of the allowed

Figure 1.20:

e n e r a v d i m of an ideal semicnnductor (band model). T h e spatial extension of the corresponding electron states is also illtlstrnted.

E,P.

Fermi distribution function f ( E ) as a function of energy at different temperat#ur&.The energy unit is E,, and EF has been set at

of electron coordinate is intended to illustrate the degree of localization of the electron. A straight line across the whole crystal from 0 to d means that the electron at this energy is spread out over the whole crystal. The regions with closely clustered allowed energy values and electron states extended over the whole crystal are called energy bands. The lower band in Figure 1.19 is the valcrice band, and the upper is the conduction band. Between these two bands lies a region of forbidden energy values, called the forbidden zone or the energy gap. The existence of energy bands done is not in itself siifficient, however, for the explanation of the observed absorption spectrum of semiconductors. For this it is also necessary that the valence band be occupied by electrons and the conduction band be empty. If there were no electrons available with energies in the valence band, then there would &o be no electrons to absorb a photon to make the transition to the conduction band possible. If, on the other hand, all states of the conduction band were to be occupied, then no electron could be excited tu this band, and no photon absorb4 in such a transition since, according t o the P a d exclusion principle, the conduction

band could host no further electrons.

40

Chapter I , Characterization of semiconductors

0

0

L X-

0

L X

4

Figure 1.21: Distribution of electrons of an ideal semiconductor over the valence and conduction bands. On the left for T = 0 K , on the right for T > 0 K. The population of energy levels by electrons at equilibrium is determined by the lkrrnz distribution f i n r t i o n f ( B ) . This function represents the probability of finding an electron in an energy levcl E , and it is given by the expression (1.19)

Here E F is the so-calld Fwmz mevyy. Like temperature, the Fermi energy is an intensive thermodynamic state variable, namely the free enthalpy per particle or the chemical potential. A more detailed treatment of the Fermi distribution function will he given in Chapter 4. The probability of orcupation of a particular energy value E: depends decisively on the relative position of E with respect to the Fermi energy. If E < E F holds, and in addition J E- E F J>> kT,then f ( R ) essentially has the value 1. If, on the contrary, E > E F and again IE - EFI >> k T , then f ( E ) is approximately zero. The shape of f(E)is shown schematically in Figure 1.20. The width of the e n e r a region where the transition horn 1 to 0 takes place is cf the order of magnitude kT. The lower the temperature, the more abrupt this transition becomes. At T = 0 K thc transition is steplike. Qccesionally, one says that f(E)has the form of an ‘iceblock’ at T 0 K , which melts at higher temperatures. 111 Figure 1.20 we have also implied that Pi, lies below E F and E, above E F , in order for the valence bend to be almost completely occupied and the conduction band to be almost empty. This means that

41

1.4. hfacroscopic properties and their microscopic implications

Figure 1.22: Absorption of photons by electron transitions from vaIence to conduction band.

0

L X--L

the Fermi energy must be located within the energy gap between Ev and E,, sufficiently far away from the two band edges (measured in units of kT). In this. we have described an essential microscopic property of pure semiconductors. Their Fermi level lies in the energy gap between the valence and conduction bands. The electrons are distributed over the two bands as shown in Figure 1.21. This knowledge provides a complete explanation of the absorption spectrum in Figure 1.18. The explanation is illustrated in Figure 1.22 and needs no further comment. The differences in the properties of semiconductors and metals may be traced back, in essence, to the different positions of the Fermi levels in these two different types of materials. In metals the Fermi level lies within the conduction band. Insulators do not differ from semiconductors qualitatively with respect to their Fermi level positions, i.e. the Fermi levels are found in the energy gap in both cases, but the gap of insulators is typically larger than that of semiconductors. If Eg > 3.5 e V , as a rule, one has an insulator.

1.4.4 Electrical conductivity in the band model We will now show that the temperature dependence of the electric conductivity of pme semiconductors also follows from the energy band scheme discussed above and the position of the Fermi level in it. Again, we employ the relation u = enp. The mobility p can be assumed to be a relatively weakly varying function of temperature, T . This means that the strong exponential T-dependence of CT must be due to the electron concentration IZ,

42

Chapter 1. Characterization of semiconductors

and comparison with equation (1.18j yields, approximately

(1.20) with n o as a factor of dimension ~ r n -depending ~ weakly on temperature. As we know, the electrons of intrinsic semiconductors are statistically dip tributed over the valence and conduction bands in accordance with the Fermi distribution function f(E). At T 0 K the valence band is fully occupied, and the conduction band is completely empty. The electrons in the fully occupied valence band cannot contribute to carrier transport. The reason for this is again the Pauli principlc, according t o which an electron state may be occupied by only one electron. Since all valence states are occupied, no stste change is possible in the valence band by redistribut,ing the electrons. This means that the T = 0 K state will remain unchanged in 8x1 electric Eeld, and consequently no current will flow. However, a current will arise from the relatively few clectrons which, according to thc Fermi distribution function: are populating the conduction band at ftnite temperature. Their concentration can be easily calculated. For a pure semiconductor Ec - EF >> 6T holds. Thus, in the Fermi distribution function j ( E ) of ecpation (l.l!J),out3 m a y neglect the ‘1’ of the denominator for emrgies E in the conduction band, i.e. energies with E > EC Iu so doing, one obtains, approximately, the Boltzmann distribution function 7

(1.21) The concentration of energy levels in the conduction band having energies between E and E dE is p c ( E ) d E , where pc(E) is the so-called density of states of the conduction band, which describes the number of states per unit energy and unit volume. In Chapter 2 we will calculate this quantity explicitly, here its mere existence suffices. In terms of p , ( E ) one may write the electron concentration n of the conduction band in the form

+

(1.22)

If one substitutes f(E)from (1.21) into the integrand of (1.221, it follows that (1.23) where (1.24)

1.4. Macroscopic properties and their microscopic implications

43

may be understood as the effectiae density of states of the conduction band. To ensure the consistency of the expressions (1.20) and (1.23) for n, we identify no = N , and ( E c - E F )= Eg/2,or

E F = Ev

1 + -E 2

g

'

(1.25)

i.e. the Ferini level lies exactly in the middle between the valence and condiistion bands. Strictly speaking, this is only true for T = 0 K , as we will see later. The corrections at higher temperatures are, however, quite smaI1, and they can bc ignored here. Another inaccuracy of the above consideration that. we have ignored is that, at, finite temperatures, the valence band population also changes. so that no longer are all states of this band ocmpied. Under these circumstances t.he electrons of the valence band also make a contribution to the current. The temperature dependence of this contribution is roughly the same as that of the conduction band. The electric charge transport due to the electrons of a not-completely-occupied valence band will be considered more fully later. It is connetted with a remarkable observation of the Hall effect, which we will discuss below. However, we will first clarify how one may understand the strong change of conductivity with impurity concentration in extrinsic semiconductors. For this purpose we again m e the energy band model of Figure 1.19 for an ides1 semiconductor. If arsenic impurity atoms are present in a silicon crystal, then this model has to be altered. Wc will later prove (see Chapter 3) that each of the arsenic impurity atoms g i v ~ rise to an energy level in the forbidden gap, just below the conduct.ian hand edge, and that the electrons in these levels are localized at the site of the impurity atom. For this reason, we have marked these levels in Figure 1.23 by short line segments. At temperature T = 0 K each of these levels is occupied by one dectron, namely the fifth valence elcctron of an arsenic atom replacing a silicon atom having only four valence electrons (see Figure 1.23). If temperature i s increased, these electrons are excited into the conduction band. In this way, bound electrons which formerly could not participate in carrier transport, become freely mobile electrons (see Figure 1.24) which can contribute to transport. If the concentration N D of arsenic atoms is not too large: and the temperature 7' not too law, then practically all arsenic atoms are ionized and the concentration n of free electrons is equal to the that of the arsenic atoms, i.e. one has R = N D . This corresponds to the approximate proportionality between cr and N o observed in Figures 1.16 and 1.17. If the carrier concentration has Ihe value NLI>which at sufficiently high temperatures T i s actually the caw, then the conductivity becomes independent of T. The only remaining source of a temperature dependence for u is the T-dependence of the mobility p. It is this relatively weak temperature effect which shows up

44

Figure 1.23: Ordering of the allowed energy values of electrons in a silicon crys-

tal doped with arsonic (schematically). Each arsenic atom correspond# to 1 locslized energy level.

t

In

a,

P W

G Eg a W

2 I

a

0

L

0 X--c

in Figure 1.17 at high T . With this, the experimental observations relating to carrier transport in semiconductors are explained microscopically, at least in principle. We now proceed to the Hall effect.

T-0

T.0

0

L X-

X-

Figure 1.24; Distribution of the electrons of a silicon crystal doped with arsenic over the allowed energy values. 011the left for T 0 K , OIL the right for T > 0 K . 7

45

*)U 1Phosphorus doped Si

Current 4

Current

-

Boron doped Si

u Currenta)

bl

Current-

Phosphorus doped Si

H ~ Icurrent I

~ a i current i

HOII current

+

fie'd

Hall current

+

Cl

d)

Figurc 1.25: nlrlstration of the Hall effect in two silicon smrples, one doped with phosphorus, and the other with boron.

1.4.5

The Hall effect and the existence of positively charged freely mobile carriers

We consider two semiconductor samples of the extrinsic type hoth madf of the Same material, however, with different types of doping. To be specific, we assume two silicon samples, one doped with phosphorus as a group V dement. and the other with bnroii as a group 111 element (see Figiiie 1.25). Applying a voltage to both samples, as shown in Figure 1.25a and b. a current I will Bow. For both samples it has the same direction, namely from '+' to '-', Consider, ROW, the ~ I I I Ptwo sarriples in a niagnPtic Geld B ( ~ e e Figure 1 . 2 5 ~and d). As is well-known, the Hall effect will be observed in such circumstances, i.e. a current component I H perpendicular to both the electric field E and the magneliu field B will arise. Gnder the wndilforts

46

Chapter 1. Characterization of semiconductors

of Figure 1.25 no current can flow in this direction, and therefore there will be a voltage UH such that the current caused by it will just compensate the Hall current. Experimentally one finds that the Hall voltage UH has different polarities in the two samples. This means that the corresponding Hall currents flow in opposite directions. If we assume that the magnetic field is directed normal to the plane of the figure and that it points into this plane, then the Hall current flows upwards in the boron-doped sample, and downward in the phosphorus-doped one. The Hall effect can also be measured in metals. In this case the direction of the Hall current is the same as in the silicon sample doped with phospho-

How can the dlfemnt behauiior of thesiicon sample dupd with born be understood? The explanation is quite clear if one assumes that, in contrast to metals and the phosphorus-doped silicon sample, the current in the boron-doped sample is not carried by negative charge carriers but by positive ones. This is illustrated in Figure 1.25d. A charge q , which moves with velocity v in a magnetic field B,experiences the Lorentz force

F = 4- [ v x S ] .

(1.26)

C

Since negative charges move to the left, and positive ones to the right, the Lorentz force F, which depends directly on the sign of the charge, has, for both charge signs the same direction, namely upwards. For the boron-doped sample this means a Hall current directed upwards, but in the phosphorusdoped sample the Hall current is downwards, since in this case the charge carriers are negative. This is just the observed behavior, which means that the assumption of positive freely mobile charge carriers is successful in explaining the unusual sign of the Hall current in boron-doped silicon. In this way the experimental observations of the Hall effect reveal a remarkable general property of semicondiictors: For doping with certain atoms, the current is not carried by negative charges as one would expect considering the negative charge of electrons, but by positive ones. In other words, in addition to the electrons as negative freely mobile charge carriers, one also has positive ones in semiconductors under certain conditions. This surprising observation may be understood as ~OUQWS.One may demonstrate - as we will do later explicitly - that boron atoms whi& substitute silicon atoms in tf silicon crystal, give rise to energy levels in the forbidden zone just above the valence band edge. This is illustrated in Figure 1.26. Electrons in such energy levels are localized at the sites of the boron atoms. At very low temperatures these states are not occupied, and the boron atonis are electrically neutral. At finite temperatures electrons from the valence band are excited into the boron levels (Figure 1.27), leaving behind unoccupied states, or occupation hole3 in the valence band. We

1.4. Macroscopic properties and their microscopic implications

Figure 1.26: Ordering of the allowed energy values of electrons in a silicon crystal doped with boron (schematically). Each boron atom corresponds to 1 locali d energy level.

47

t

immediately recognize that these occupation holes of the valence band behave jubt like freely mobile positive charge carriers in an applied electric field. This is illustrated in Figure 1.28. For simplicity only one hole is assumed to be in an otherwisr rompletely occupied valence band, placed at the upper band edge. The population of the bend edge is shown in Figure 1.28 at different points of time. At the beginning, the hole is at the outermost left position. The adjacent electrons experience a force which tends to move them to tho left. Rut only the electron neighboring the hole on its right hand side can follow this force, since all other electrons are blocked because the states to their left are already occupied. The electron immediately on the right hand side of the hole, moves into the hole and leaves behind another hole to the right of the first one. The hole has thus moved one site further to the right in this way. 111 the next time interval this process is repeated, and the hole again moves one site further to the right etc. Thus, in an electric field, the hole moves like a freely mobile electron, but in the opposite dirwtion, as if carrying a positive charge. In summary, holes in the valence band behave like freely mobile positive charge carriers. This qualitative introduction of the concept of holes will later be elaborated by more quantitative considerations. As we have seen above, the sign of the Hall voltage tells one whether the free carriers of an extrinsic semiconductor are negatively or positively charged, Le. whether they are electrons or holes. In the first case one speaks of a n-typr semzcondurtov (the n stands for ‘negative’), in the second of a p-type semarondector ( p for ‘positive’). The antranszc semiconductors intro-

48

Chapter 1. Characterization d semiconductors

L

0

Figure 1.27: Distribution of elect.rons of

1 X-

X-

a silicon crystal doped with boron over

allowed energy values.

Figure 1.28: Empty states (halefi) in Ihe valence band behaye like freely mobile poriitive charge carriem in an electric field.

duced earlier haw neither electrons nor holes from impurity atoms. Their mobile electrons arr generated by thermal excitation of bound electrons from the valence band to the rondiiction band. Since each excited electron leaves behind a hole in the valence bmd. one also has mobile holes in intrinsic semiconductors. Their number is equal to that of the mobile electrons. Holes are also present in n-type semiconductors, but, in very small numbers compared to elwtrons. Analogously, a few electrons occur in p t y p e materials with mob1 of the carriers being holes. One calls the many mobile carriers of extrinsic semiconductors m a j o n l y cariwrs and the few mobile carriers mzrrortty earners.

The Hall effect can also be used for purposes other than the determination of whethrr an extrinsic semiconductor is n- or p-type. The absolute

valup of the Hall voltage determines the concentration of majority carriers. We will veTify this assertion for an n - t y p ~semicondiictor. In this case the rurrcnt density j can be expressed in the form j =- --env. which permits us l o rewrite equation (1.26) as

1

F=-[j

X B ] .

(1.27)

RC

This is the same force caused by the electric field E ~ I= F/(-e). The voltage corresponding t,o EH is by dehition the Hall voltage Ufi. With b as the width of the sample normal to j and 3,the expression below for U x follows:

(1.28)

where

1

Rff = -

(1.29)

PrtC

is the so-called Hall constant. We can write an analogous expression for holes, only the electron concentration has to be replaced by the hole concentration p . Thin;, hy measuring thc Hall vultagr, onr ran also det.ermine the majority carrier concentration.

1.4.6

Semiconductors far from thermodynamic equilibrium

All properties considered hitherto were for semiconductors in a state of thermodynamic equilibrium or in close proximity. However. semiconductors may easily be driven into states far from equilibrium. Here, 'far' means that characteristic macroscopic propexties of the semiconductor deviate strongly from those in equilibrium. To be more specific about these qualitative statements, we examine the example of photo-conduction. Consider a sample of an intrinsic semironductor, shielded against unwanted influence of light. If, in this 'dark state' the conductivity is measured. one obtains a relatively small value. in accordance with the relatively low carrier concentration of an intrinsic semiconductor in thermodynamic equilibrium. However, if one irradiates the sample with light which is absorbed by the semiconductor (see Figure 1.22). the conductivity will rise more or less strongly, depending on the intensity of the light and the magnitude of the absorption coefficient. This is shown schematically in Figure 1.29. assuming realistic conditions. When the exposure of the sample t o light ceases, its conductivity decreases to the original low dark value. Evidently, electrons in the conduction band and holes in the valence band were created by irradiation with light to such an extent that their equilibrium values were exceeded by orders of magnitude. A simple estimate for Figure 1.29

50

Chapter 1. Characterization of semiconductors

Figiire 1.29: Conductivity of an intrinsic semiconductor irradiated with light as a function of intensity I of the absorbed radiation (schematically).

1M - -~

1~-7 -

10-4

I

I

10-3

lo-7

I

10-1

ma

I h o t t om*

r]'

--+

shows that t.he carrier concentration was increased from about 10'' cm4 (equilibrium value) up to c ~ n . -by ~ light of intensity 10 Wcm-'. The electron-hole pairs created by the radiation decay, however, after a short t,ime. 'lhis follows from the fact that continuous irradiation leads to tt stalionary conductivity value, and lienre a constant carricr concentration, and, on the other hand, from the observat.ionthat the conducbjvity decays down to the dark value after switching off the light source. The 1at.ter observat.ion means that, thermodynamic equilibrium i s reestablished by so-called wxnnbinaiion, p7nuesses.

Rwidr irradiation with light, extreme nun-equilibrium states in semiconductors can also be created in other ways, for example, by putting an n-type semiconductor in contact with a p t y p e semiconductor or with a metal, or by applying voltage to a. semiconductor which previously had been isolated by a thin insulating layer from one of the electrodes. The ability to c r e at,? extreme non-equilibrium states in semiconductors i s extensively used in electsonic devices. Almost all applicat.ionsof semiconductors in such devices rest on this uniquc pQssibility-. Non-equilibrium processes in semiconductors and the most important scmiconductor devices based o n them, such as electric rect.%ers, bipolar and unipolar transistors, tunnel diodes, photodetectmu: solar cells, as well as luminescence and laser diodes, will be dealt with in the second part of this book, i.e. in Chapters 5 , 6 and 7. In the first part of the book, the basic concept.s, discussed above in a heuristic way, will be developed from first principles. This applies to the stationary electron states of an ideal semiconductor (Chapt,er 2), their niodifications by impurity atoms and other deviations from the ideal crystal, as well as by external fields (Chapter 3) and t h e statistical distribution of charge carriers over aiergy levels in thermodynamic equilibrium (Chapter 4).

51

Chapter 2

Electronic structure of ideal crystals In sections 1.2 and 1.3 of the preceding chapter we discussed the spatial order and chemical nature of the atoms in ideal semiconductor crystals. The present chapter is focused on the quantum mechanical energy eigenvalues and corresponding eigenstates of electrons in such crystals. The substance of this subject is summed up under the designation electronic structure. We will show how the simple model of electronic structure of a crystal, the energy band model, which we heuristically introduced in section 1.4, can be rigorously deduced from the Schrodinger equation. The periodic ordering of the atoms of a crystal in space is crucial for this proof. It will also be seen that the eigenstates of electrons in a crystal devolve upon the electron states of the free unbound atoms. This is immediately understandable if one imagines that the crystal is grown from the gas phase, i.e. by chemical bonding of previously isolated individual atoms. In this process the electron states of the atoms will change, of course, but the resulting states, i.e. the electron states of the crystal will also depend on the initial electron states of the isolated atoms prior to crystal formation.

2.1

Atomic cores and valence electrons

Qualitatively, the kind of changes contemplated may be characterized as follows: There is no doubt that the electrons of the outer shells, i.e. the valence electrons, will react most strongly in assembling the isolated atoms of a crystal, for they are the primary agents which bind the atoms into the crystal state (Table 2.1). Whether, and to what extent the electrons of the inner shells also change their states, is harder to predict. One may suppose that such inner shell changes will be comparatively slight, at least for those inner shells which lie much lower energetically than the valence shells (Ta-

52

Chapter 2. Electronic structure of id&

crystsls

Table 2.1: Characterization of atomic cores and valence electrons of main group elements from which semiconducting mat,erialsmay be formed.

Atom

Core Nucleus

B C N

2922p2 2s22p3

13+

P

323p 132222p6

14+

323p2

15-

Ga

3s23p3

3 1+

32AS

222p 18

7f

A1

Ge

Core electrons

5+ 6+

St

Valenw electrons

424p ls22s22p633s23p63d10 3s24p2

3 3+

4Ap3

ble 2.2). Because the electrons of these deep shells are localized so close to tlirir rrspertive nuclei. they feel potential changes produced by siirrouuding atoms as being almost uniform. Strictly speaking, this means that the wavefunctions of these electrons are essentially unaltered. Their eneqy levels shift, however, specifically by the change of the constant potential value arross their localization region. The term d i d d a t P shtjts is iised for t h e s ~ shifts of the inner electron levels. By measuring these shifts m e can obtain information about the chemical nature and geometric striicturc of the environnient of an atom in B solid. In B way, the inner electrotla threby serve as probes.

Here, we arc intermtd in the electronic strizrture of crystals, dnd in this regard the pertinent feature is that the wavefunctions of the inner shcll plectrons 01 the atoms in the crystal undeigo only weak changes. This statement is even better justifid for the wavefiinctions of the nucleons in the atomic nuclei, which remain practically unaffected. Furthermore, if the gown crystal is expoard to certain extrrrial perturbations - heat. prrssure, elwtromagnetir fields - the states of the electrons of the inner shells and those of IhenucIeoas frequently do not change. For the e x h n a l perturbations which are of p r i m interest in semiconductor physics, this is even generally true. Therefore, in determining the elertronic stricture of sanironductor crystals and the influence of exterruel perturbations on them, the states of the inner electron shells and those of thc nuclei, as a rule, can be assumcd to be the same as those of the free atoms. This allows one to consider the atomic nuclei and

2.1. Atomic cores and valence eJectrons

53

Table 2.2: Energy levels -E, and -4 of the valence electrons and - E , of the shallouwt core electrons for some chemical elements which may occur in semiconducting materials. All energies are given in el/. (Afcer Herm.an. and SkiIlman, 1963.)

0

A1

Si

P

S

Zn

Ga

14.1

4.9

6.5

8.3

10.3

3.1

4.9

23.0 29.1

10.1

13.6

17.1

20.8

8.4

11.4

291

405

537

87.5

116

147.4

182

20.7

31.7

7.9

9.5

3.4

4.7

5.9

7.2

8.6

3.5

5.8

C

N

Atom

3

-tp

6.6

9.00 11.5

-t,

12.5

17.5

-ec

195

-cP

6.4

the inner shcll drrctruns jointly as subsystems of the crystal, whose intrrnal structure i s of no further interest sirire it does nut change. The structure is, so to speak, frozen. In this sense t.hc subsystcms composed of atomic iuiclei and inner electrons are elementary building blocks of the crystal. One rekm t,o them as atom.ic ~ 0 : ~ s . Since the crystal aIso contains valence electrons as independent particles, we arrivr at a yic:i.ure which is fundamental for the further analysis - the picture of a cryst,al as system composed of at,omiccores and valence electrons. Somctirnes one refers to this concept, as the frozen-core a p p ~ ~ ~ i i m In ati~~~ Table 2.1 the division into core8 and valence electrons is indicated for some elements from which semiconductor crystals arc made. The frozen nature of the electron states of the cores of a crystal! and their lack of rcsponse to external influences, generally prevails, but as always, there arc exceptions. In heavy metals such like zinc, t.he inner &shells are, energetically, rather close t,o the oubar valence shells (see Tables 2.1 and 2.2). In this case the d-electrons significantly participate in chemical bonding and can no longer be included in t.he core, which is iinchangeahle by definit,iou, Moreowr, the inner shell eledrons of crystals can be excited by means of electromagnetic radiation in thc far UV and X-ray region. This can also occur by means of an electron beam In solid state nuclear reactions, ~ W R the states of the nuclei of the crystal atoms change.

54

Chapter 2. Electronic structure of ideal crystals

2.2

The dynamical problem

2.2.1

Schrodinger equation for the interacting core and valence electron system

In dealing with the atomic structure of crystals in Chapter 1, we found that their atoms are not located at arbitrary positions but at well d e h d locations, specifically at points which are consistent with the existence of a lattice and a unit cell. In this, the atoms were assumed to be point-like and the space between them was empty. The crystal itself was imagined to extend to infinity. We now consider a more realistic model of a crystal. First, we replace the point-like picture of atoms by introducing spatially extended atomic cores and take their centers of gravity as the atom sites. Second, we recognize that the centers of gravity can also move to other positions than those prescribed for the ideal crystal. In this way, we also account for the fact that the atomic cores in crystals can execute oscillations around their equilibrium positions, and that only these equilibrium positions form an ideal crystal. Thirdly, the space between the massive elements of the crystal, i.e. the cores, is now no longer assumed to be empty, as was done before, but we acknowledge that valence electrons are present there. The assumption of infinite extension of the crystal which, of course, is also not exact, will be addressed at a later stage. This assumption excludes effects due to the existence of bounding surfaces. As far as the electrons are concerned, these effects are treated in section 3.6. The atomic cores will be marked by an integer subindex 3 , and the valence electrons by an integer subindex d. Both should start with 1 and run upwards, reaching arbitrarily large values since we are considering an infinite crystal. By means of a conceptual device which we are about to introduce notwithstanding the infinite extent of the crystal - only finite sets of J cores and N valence electrons need be considered. To understand this, we imagine the infinite crystal to be divided into parallelepipeds of macroscopic size in such a way, that their edges are parallel to the primitive lattice vectors a,,az,W of the crystal. These edges are to be given by the vectors Gal, Gaz, G a with G a large integer. Each of these parallelepipeds should contain an equal number J of cores and N of electrons. One calls these parallelepipeds periodicity regions. Of all possible motions of the particles of the infinite crystal, we now select those particular ones for which the cores and electrons in different periodicity regions have the same positions relative to the origin of their own region, and also have the same speeds. In this way the infinite crystal becomes a periodic continuation of one particular periodicity region, and it suffices to describe the motion of the J cores and N electrons of this particular region. If the periodicity regions are made s a -

2.2. The dynamical problem

55

L

0

/; +

0

\

0

Figure 2.1: 13escription of the positiom of the atomic cores ( 0 ) and valence elm trons {a) {left part) as well as the interactions between these particles (right part). ciently large, they will encompass all types of motions of an infinite crystal with desired accuracy. The concept of the periodicity region makes it possible to pass from the original infinite space problem of motion to a finite one without thereby losing the translational symmetry of the infinite crystal. We use Xj t o denote the center-of-mass coordinates of the j-th atomic core, and xi for the position of the i-th electron, which is further assumed to be point-like {see Figure 2.1)- The j-th core mass will be denoted by Mj. Of course, there are only as many dif€erent values of M j as there are chemically different types of atoms in the crystal, so most of the Mj-values are identical. In the case of electrons we can omit the index i from their masses since they have the common mass m. The momentum of the j-th core is called P3,and that of the i-th electron pi, such that

We are interested in the motion of the interacting atomic cores and valence electrons of the infinite crystal, which can only be adequately treated by means of quantum mechanics. The state of the system is described by a wavefunction @, which depends-on the coordinates xi of all electrons and Xj of all atomic cores, as well as on the time t. Since we assume periodicity of the motion with respect to a periodicity region, it suffices to consider @ as a function of the coordinates xi of the N-electrons and the coordinates Xj of the J cores of only one periodicity region. The state of the particles in the remaining periodicity regions can then be described by means of a periodic continuation of this function, i.e. by means of the relation

56

Chapter 2. Electronic structure of ideal crystals

@(XI I

Ga,,

x2 1 ,%:C

. . . , XN i G%, XI 1 C%, X2 t G&, . . . ,X J t G+, t )

with CY 1 , 2 , 3 . We will now wt up the Hamiltonian 7-l of the system of the N-electrons and J atomic cores of a periodicity region. We use Tc and ‘reto denote the kinetic energies of the atomic cores and of the electrons, respectively, and , . . . ,XN, X i , X2,. . . , XJ) to be the potential energy of we define V ( X ~xz, the system. The Harniltonian is the sum of the kinetic and potential energy operators, ~

+ + V.

‘H =- Tc Te

(2.3)

The kinetic energies Xc and T, ran be expressed in terms of the momenta Pj and pa of the cores and electrons as fOllOW8: (2.4)

The potential energy is due to three interactions (see Figure 2.1): (1) ‘lhe repulsive Coulomb interaction of the electrons with each other. The corresponding potential encrgy is denoted by Vce. It depends only on the coordinates of the electrons, as given by

(2) The interaction of the electrons with the atomic cores due to their mutually attractive Coulomb forces, and also due to (repulsive) forces of quantum mechanical origin, which become effective if the valence electron wavefunctions overlap the inner electron shells of the atomic cores. The electron-core interaction potential energy V,, depends on the locations of both the electrons and cores. With respect to the electrons, it is evidently additive, i.e.

. XJ)=

vet = vec(X1, x2,. . . ,X N , x1,XZ,. .

I

vc (xz,. x1,x2,.. . ,XJ), (2.6)

where Vc(xi, X I ,X2, . . . , XJ) is the potential energy of the i-th electron in the field of all cores. (3) The mutual interaction of cores, which at sufficiently large distances is again of Coulomb type. If the distances become small, repulsive forces of quantum mechanical origin also occur. The core-core interaction potential energy will be denoted by V,. It depends only on the locations of the atomic cores, i.e.

2.2. The dynamical problem

57

VCr(X1, XL

vcc

’ ‘ ’ 9

XJ).

(2.7)

Summing the three potential parts (2.5), (2.6) and (2.7), one gets the total potential

v

-

v, + v,, + v,,

(2.8) which determines the dynamical problem of the crystal uniquely. This problem i s described by the time-dependent Schrodinger equation i3 iFL-e = H6, at whose solution may be determined in terms of the eigeiivalue problem for the Hamilt,onittn 7f, which is given by the time-independenl Schrodinger equation

%IJ! = EQ.

(2.10)

The normalization condition for the wavefunction with refereiicc to a pcriodicity region is

(@I*)

d3X1.

. . d 3 x J p q X 1 ,x2,. . . , XN,X I ,

xZ, . . . ,

xj,t ) l 2 = 1.

(2.11) Attempts to solve this eigenvalue problem exactly are hopeless from the very beginning, because it involves a macroscopic system, i.e. a system with about 10” electrons and a similar number of atomic cores, the motions of which are mutually coupled in a rather complex way. One must therefore resort to approximations. Such approximations must first provide the means to reduce the gigantic number of electrons, and secondly, allow for a proper decoupling of the electron and core motions. The second simplification is achieved by the so-called ‘adiabatic approximation’, and the first by the ‘one-particle approximation’. These two approximations will be elaborated below. We begin with the adiabatic approximation, and in the course of the discussion it will also become clear how the somewhat unexpected designation of the latter arises. 2.2.2

Adiabatic approximat ion. Lattice dynamics

The adiabatic approximation (also known as the Born-Oppenheimer approximation) is based on the fact that the mass of the atomic cores is many tens of thousands of times larger than that of the electrons - in Si, e.g., 52 thousand times, and in mercury 368 thousand times. In addition, it takes advantage of the fact that in a crystal the kinetic energy of an atomic core is, on average,

58

Chapter 2. Electronic structure of ideal cry&&

smaller than that of a valence electron. 'l'his can be seen in the following way: If the cores and valence electrons were fwe particles, i.e. if they did not interact, then the average kinetic e n p r o of a core would be approximately (3/2) h7'. That of a valencp dzctron would be about (3/6) Eli where E F is defined as the Fwmi energy of an electron gas of the same density. The daerence between the average kinetic energies of the two types of particles arises from the fact that the electrons obey Fermi statistics, whereas atomic cores obey Boltzmann statistics. For typical concentrations of valence electrons in a crystal of about 10" C W L - ~ the Frrmi e n e r a EF i~of the order of m~gnit~ude of e V , while kT reaches only about 0.1 eV below the melt mg point of the crystal. The average kinetic energy of a core is therefore generally smdlcr than that of an electron. This remains true when the interactions between the electrons and coresi which were omitted above, are taken into account. Writing (M,/2) < X: > and (m/2) < x: > as the avrragP kinetic enmgies of a core and an electron in a crystal, we thus have Mj

-2< X j >

1 7 1 . '

< -. 2

(2.12)

and it, follows that (2.13) Corrmpondingly, one may say that, on statistical average, the cores move much slower than the electrons. This observation plays an important role in the following considerations. To simplify the notation, we replace the N-component sequence o€ the vectors (XI,x2,.. . , X N )by x, and the J-component sequence of vectors (X\,Xz,. . .,XJ)by X, ie. we write

x = (XI,xz,. , ,

XN),

x = (XI,xz,. I

.

,

XJ).

(2.14)

To take advantage of the slow motion of the cores we write the solution q{x,X) of the Schrodinger equation (2.10) for the total crystal. in the form of a product

q x ,X ) = $(x, X )

'

@(XI.

(2.15)

The necessary normalization (2.11) of the total wavefunction *(x, X) with respect to a periodicity region is assured if each oI the two factors of (2.15) is normalized with respect to this region, i.e. if (2.16)

2.2. The dynarnical problem

59

(2.17) are assumed. An analogous statement holds for the periodicity condition (2.2) of @(x,X). To assure overall periodicity with respect to a periodicity and 4(X)separately. An ‘Ansatz’ of the region we assume it for +(x,X) form (2.15) is always possible since it does not assert separability of the variables x and X,but merely splits off a factor +(X)from the wavefunction @(x,X)which depends only on one variable, X, while retaining the full dependenre on both Coordinates in thr second factor +(x,X). The ‘Ansatz’ (2.15) becomes non-trivial if we proceed as follows: Firstly, we assume that $(xiX)is the solution of a Schrodinger equation for electrons,

1%

+ v,,,c]

$(x,X)= U(X)@(X, X),

(2.18)

where, for brevity, we have set

and U(X)is in the nature of an electron energy eigenvaliie. Secondly, we demand that the split-off factor, 4(X), of the total wavefunction (2.15) satisfy a Schrodinger equation in which the coordinates of electrons do not appear. It turns out that such an equation cannot be derived rigorously, but only in a special approximation - the adiabatic approximation which was mentioned above. Yet without any approximation we have

[Te+ Tc + K , ,

+ VCCI +(x,X)+(X)= E $ ( x , X)4(X).

(2.20)

The set {$4) of the eigenfunctions of the Schriidinger equation (2.20) forms an orthonormalized basis set in the Hilbert space of the crystal. Therefore, relation (2.20) is satisfied if it holds for the Fourier type coefficients relative to all basis functions Q’d’ of this set, i.e., if the identity ($’4’lTe

+ Tc + v&ec + Vccl$,O)= E&!+t++

(2.21)

is valid. The necessary simplification concerns the matrix element ($’4’(Tc I$4) of the kinetic energy of cores in this equation. Using relation (2.4) between Tc and the squares Pj of core momenta, and applying the product rule for differentiation we get, first of all,

60

Chapbrr 2. Electronic strncture of ideal crystals

The first two terms on the right-hand side of this q u a t i o n turn out to be small compared to thr kinetic cuergy term of elwtrons in equation c2.20). One has the order of magnitude relations (2.23) (2.24) Here .n/E is a typird value of the core masses M,. The two equations (2.23) and (2.24) are of fundamental importance in crystal dynamics, because they are ultimately responsible for the drroupling of rloctron dynamics from the dynamics of the cores. Therefore we present the proof of thwc equations in Appendix B. Here we proceed a n the assumption that these relations are proven. The terms of rplativeorders of magnitude (rn/M)1/2 2 lov2 and ( m / h f )w l W 4 , will be nPglatrd henceforth. With this the operator T, for the rambined kinetic energies of all cores satisfies the approximate relation Tc[.l$(x,X)9(X)l E=r ${x, X j W G q .

(2.25)

This means that T, effectively does not act on V(X, X).In view of this relation we reconsider the SrhriidingFr equation (2.20) for the crystal, replaring the terms which still depend on x by means of the electron Schrodinger equation (2.18) in terms involving the electron energy eigenvalue U ( X ) . Finally, forming the scalar product with VI, we obtain the relation (Tc

+ VC,(X)+ U ( X ) \ d ( X )= Ern(X).

(2.26)

This rcpresents the SchrGdinger equation for the atomic cores in which the coordinates x of thc electrons do not appear. The state of the elcctron system, however, enters this equation, namely via its energy U(X)which plays the role of a potential (referred to as adiabatze potential). In summary, we h a w reached the following description of the total crystal, viewed as an interact,ing system of atomic cores and electrqns: The subsystemof electrons is describcd by R separate wavefunction qi(x, X), which obeys a Schrijdinger equahion in which the coordinates of the cores enter only as parameters in the potential, but do not occur as differential operators in t,hr kinetic energy. In this way, the motion 01 the electrons is treat,& as if the cores were at. red. Core motion, which does, of course, occur despite its neglect with respect to electron motion, is described by the wavefunction &(XIand the Schrodinger equation (2.26). T h e potential of this q u a t i o n contains, besides the core-core interaction energy, a second contribution IJ(X). This originsles in the interaction of the electrons with

2.2. The dynamical problem

61

the cores, for without such an interaction the eigenvalue U in the electron Schrodinger equation (2.18) would be a constant independent of X, which could be omitted. The potential contribution U(X) caused by the electroncore interaction does not depend, however, on the electron coordinates. It is an average value over all their positions X. The weight with which the various positions x enter this average over the probability I + ( x ,X)1’ d 3 N ~ of finding the electron system in a volume element d 3 N ~at the position x, since equation (2.18) implies that

u(X) = (+(X)ITe

+ vee,ec(X)l+(X)).

(2.27)

One can alternatively express this as follows: The electrons move so fast that they are no longer seen by the cores as point-like particles, but as smeared out over all space. Equation (2.26) for 4(X) thus contains the same assumption as equation (2.18) for + ( x ,X),namely that the cores move much slower than the electrons. In so far as this feature is seen from the point of view of the electrons, equation (2.18) follows, whereas from the point of view of the cores one obtains equation (2.26). Since the relation between the velocities of the cores and the electrons, according to (2.13), is determined by the inverse ratio of their masses, it is clear why this ratio must be small for the two Schrodinger equations (2.18) and (2.26) to hold jointly in an approximate sense. It remains yet to clarlfy what effects are neglected because of the above approximation and why this approximation is called ‘adiabatic’. In quantum mechanics, one understands adiabatic temporal changes of potentials in the sense that the changes proceed so slowly that no quantum mechanical transitions will occur between the discrete quantum states of the potential, which themselves evolve slowly from the initial onset of time variation. The state of the system thus conforms continuously to the evolving new potential values as a function of time, without any transitions to other states. That exactly this situation is described by equation (2.18) and (2.26), may be seen from (2.24). If one considers the previously neglected term in (2.24) of relative order of magnitude ( m / M ) l I 2 ,then the total Hamiltonian 7-t of the crystal has non-vanishing off-diagonal elements (2.28) and quantum mechanical transitions between the different eigenstates $14 and $’+’ of the crystal are recognized to occur. These transitions are caused by the kinetic energy of the cores exclusively. If terms of the order of magnitude ( m / M ) I 1 2 are omitted, then the quantum transitions due to core motion are also neglected. This is equivalent to the assumption that the core motion be adiabatically slow, in the quantum mechanical understand-

62

Chapter 2. Electronic structure of i d 4 crystals

ing of this term. The term ‘adiabatic’ thus refers to the essential character of the approximation in neglecting (rn/M)’/’. This approximation is useful, of course, only as long as transitions between different eigenstates $$ play no important dynamical role. This is actually the case in regard to many crystal properties and phenomena. There are, however, also effects for which this does not hold, notably electric current transport. The fact that the electric conductivity of an absolutely pure crystal does not become excessively large is due in large part to the scattering of carriers from the oscillations of the atomic cores, i.e. to non-adiabatic quantum transitions between different electron and core states. Also, in the recombination of electron-hole pairs mentioned in Chapter 1, these transitions play a decisive role, with the lattice of atomic cores absorbing the energy which is released during recombination. Formally, one may understand non-adiabatic transitions as the result of an interaction between the electrons and the motion of the atomic cores. Since such core motion, as we will see below, represents a superposition of lattice oscillations, also known as ’phonon’ excitations, this interaction is called the electTon-phonon interactaon We have yet to explore how the two Schrodinger equations (2.18) and (2.26) for the electrons and cores can be actually solved. The problem is that both equations, are, at the outset, not completely determined - the one for the cores contains the adiabatic potential U(X),which can be known only after the equation for the electrons has been solved; and the electron equation can be fully d e h e d , however, only if the positions X of cores in the potential Vee,..-,-(x, X) are known. The direct way to overcome this difficulty would be the following: One assumes a particular spatial ordering X’of the cores and uses it to determine for them the potential Ve,ec(x,X’).The latter is then used to solve the electron Schrodinger equation (2.18) (we will not discuss here how this is accomplished, as it will be the subject of the next subsection, 2.2.3). From the solution of the Schrodinger equation (2.18) one obtains the value of the adiabatic potential U at the position X’ of the cores. The same procedure is then applied to all other possible positions X, whereby the adiabatic potential U(X) and the Schrodinger equation (2.26) for the cores are completely determined. This equation can then be used to calculate the core wavefunction t#(X). It follows that the dynamical problem for the crystal as a whole is solved, since one would know its eigenfunctions @(x, X)= y(x, X)q(X). In reality, however, this procedure is unsuccessful. One cannot solve the electron Schrodinger equation for all possible core positions. Therefore, a simplified procedure is necessary. It contains additional approximations, but has the advantage of being feasible in practical terms. In this approach, one ignores the motion of the atomic cores completely and assumes that they are resting in certain equilibrium positions Xq. In reality, they execute oscillations around these positions with amplitudes that become smaller as the temperature of the crystal decreases. However, due to

63

2.2. The dynamical problem

xo

- X"

I.ke,ec(X,Xn)

?

v,v,(xy

-0 ?

no

J

xn+1

I

Figure 2.2: Iterative calculation of the equilibrium positions of the atomic cores. the quantum mechanical phenomenon of zero-point oscillations, such motion remains finite even at absolute zero temperature. The equilibrium positions Xq are unknown at the outset. One can determine them by demanding that the total energy

+

Vo(X) = U(X) VC(X)

(2.29)

of the crystal in equilibrium have a minimum at Xq. Equivalent to this is the requirement that the forces -VxVo(X) on the cores, the so-called Hellman-Feynman forces, vanish at the equilibrium positions:

--vxvo(x)~x,xeq = 0.

(2.30)

Bearing this in mind, we may employ the iteration process below for the solution of the two coupled adiabatic equations (see Figure 2.2): In this pro-

Chapter 2. Electronic structure of ideal crystah

64

cess, one assumes ccrtain trial equilibrium positions Xo, enters them in the This electron Schrodinger equation, and determines the eigerivalue U(Xo). solution is then used to determine the potential Vo(Xo) and the liellmaiiF e y m m forces. Thanks to the Fe'egrman theorem, taken jointly with appropriate analytical transformations, one can determine these forces without numerically calculating Lhe potentid in the environment of X'. After the first iteration cycle, t.he Hcllman-Feynman forces will, in g~neral.not yet vanish. signifying that thc cmes arc still not at equilibrium posibians. By nieans of the non-vanishing f o r c e one det.ermines new trial positions XI. 'The new positions are then substituted again onto the electron Schrudinger equation (z.18)! to calculate a new eigenvalue U(X1),and the latter determines the corresponding Heban-Feynrnan forces. This procedure is to be repeated until the forces become zero. The corresponding core positions are then the equilibrium positions XeQ.In this way one reaches a very important result, the determination of the atomic structure of the crystal. Such structure calculations are successfully carried out currently fw many solid state systems, including a series of semiconductor cryst.als. With regard to semiconductors, it can he shown, for instance, that under normal conditions, Si haas the diamond structure, and that its Lattice constant a iR 5.49 A.

];at tice oscillations a6 atomic structure is concernedd: ouly the equilibrium positions of the atomic cores are of interest. These can be understood as average values (Q I X 14) of the core positions X with respect to the core wavcfmction 4 , However, the wavehnction 9 itself contains considerably more information. It determines the probability distribution for the positions of the atomic cores. That the probabilities of the cores being removed from their equilibrium positions are non-zero is tbe quantum mechanical indication of the existence of lattice oacdllatioru. These oscillations may be describd easily using the S&rodinger eqriatiou (2.29) for the cores. In this: it suffices to expand the potential I/o in a Taylor series with respect to hhe displacements X - Xq from the equilibrium positions Xq,neglecting ternis beyond the square term. The linear term of this expansion vanishes since the potential energy has a minimum at Xq. Thus one obtains

In so far

VO(X) =

vo(xeQ)i-' ( X 2

- X e ~ ) V ~ v X r ~ ' o ( X-~x)e(4x1.

With this potential the Schriidinger equation (2.26) for @(XIreads

(2.31)

2.2. The dynamical problem

65

where. for simplicity, Vo(Xeq) = 0 has been assumed. Equation (2.32) describes a system of coupled harmonic oscillators. The restoring forces are determined by the second derivatives of the potential Vo(X). Using the eigenvectors and eigenvalues of the matrix of restoring forces (actually, of the so-called dynamical matrix which also includes the kinetic energy term), one can easily transform to a system of uncoupled harmonic oscillators. Their motions are called normal mode osciliataom or lattace oscillations,and their excitation quanta are phonom. Phonons are a good example of the introduction of a concept which is of fundamental importance for the dynamics of many body systems, including the many-electron system of a crystal which will engage us in the next subsection. The concept we have in mind here is identified by the terms elementary excztatzons or quast-particles (both terms are commonly used). This concept is based on the possibility of decomposing the motion of a system of mutually interacting particles - in our case of the atomic cores of a crystal - into non-interacting components of motion - phonons in ow case. The phonons or, more generally, the elementary excitations are. so to speak, the elements of n o t z o n of the system, while the atomic cores or, more generally, the actual particles. form the ~ ~ T U C ~ U Telements -Q~ of the system. ,413elementary excitation involves coordinated motion of all structural elements of the system. Conversely, the motion of an individual structural element is a superposition of all elementary excitations - the motion of the atomic cores, for example, is a superposition of all normal mode oscillations or phonons. Besides the oneelectron and one-hole excitations, the phonons are the most important elementary excitations, or quasi-particles, of a crystal. In this book we will deal mainly with electronzc elementary excitations, and will include phonons only if it is otherwise impossible to properly describe electron dynamics. Relevant phonon information will simply be cited without detailed justification, since a thorough development of the theory of phonons is beyond the scope of this book. O w choice of subject matter here is conditioned by the fact that electrons and holes are much more important for understanding the properties of semiconductors as they are used in electronic devices, than are phonons. Readers who are particularly interested in phonons are referred to other books (see, e.g., Born and Huang, 1968; Bilz and Kress, 1979; Bonch-Bruevich and Kalashnikov. 1982). We return now to the Schrodinger equation (2.18) for electrons. In the sense of Figure 2.2, we approximate the core positions X by their equilibrium values Xeq. As far as the latter are concerned, we take the point of view that they are known from experimental structure investigations, e.g., by means of X-ray diffraction. For common semiconductor crystals this is in fact true in all cases. Taking this approach, the potential b’=,- in the electron Schriidinger equation (2.18) is well-defined from the very beginning. To

66

Chapter 2. Electronic structure of ideal

crystals

simplify the notation, we suppress the core coordinates X in the potential = V, + V, henceforth, writing Vec(x, X ) = v,(x). Similarly, WP write the electron wavefunction @(x,X)as $:,(x). Usingequations (2.19), r1.5) and (2.6), the Hamiltonian H = Te V e , , of the N-electron system in equation (2.18 j takes the explicit form

+

2.2.3

One-particle appraximation. One-particle Schrodinger equation

With the Hamiltonian H of (2.331, the N-electron Schrodinger equation (2.18) can be written as HVj,(Xl,x2,.. ., X N ) = Ull(X1, x2,.. . , XN).

(z -34)

The wavefunction ~+4(XI, xp, . . . ,XN) must be periodic and normalized with respect to a periodicity region. The Schrodinger equation (2.34) is impossible to solve directly since it describes an interacting system of electrons having a tremendously large number of particles of order loz2. The goal of this subsection is to provide an approximate description which allows one to reduce the number of particles down to minimum number 1. This will be done by developing a oneparticle Schrijdinger equation whose solutions are rdated to those of the true many electron Schrodinger equation in a welldefined and sdiciently simple way. ~

Hartree approximation

In keeping with the remarks above, we a s s m e the existence of an infinite set of one-particle wavefunctions q ~ l~, 2 . .: .. 'pm, from which the stationary stat- $(xl,x2). . . ,XN)of the N-electron system may be constructed. The p I I ( x ) ,v = 1,2,. . . , m , are, firstly, taken to be periodic with respect to a periodicity region, as the wavefunction @(XI, x2,.. . , xN) itself, i.e. they satisfy the condition

+

pv(x)= pu(x Cajj

j = 1,Z, 3.

(2.35)

Secondly, they are assumed to form a complete orthonormal set of functions in Hillert space, sylribolically (CPv~lPv) = 6uhr.

(2.36)

Employing pY(x)we form wavefunctions for the N-electron system in the folIowing way. We Brst associate each of the N electrons with a particular

2.2. The dynamical problem

67

oneparticle state pv[x), i.e. particle 1 with state pull particle 2 with state puL,etc., up to partick N which is associated with the state. , ,oy Alternatively, WP may say that we occupy state pw with particle 1, state p , with particle 2, etc. Due to the Pauli exclusion principle, each state can host only 1 particle, ignoring spin (which we do at this stage). Thus a given state p1, p2,. . . ,pm may occur among the papdated ones lpy, 'py, . , ,pVNnot more than once. Most of the states will not even orcur o n r ~ i.a, , not at all (bear in mind that there is an i n h i t e number of them). These states remain unoccupied. The set of quantum numbers, (q, vz, . .. , VN),termed configzsratzon, definea the state of the N-electron system uniquely if we understand that the h s t number in this set refers to the state of particle 1, the second to the state of particle 2 etc.. Henceforth, we abbreviate the configuration ( V l , V a l . . . ,w )by (.I. Thirdly. we assign to each configuration (IJ) of the N-electron system a wavefunction $(y)(xl,x2,. XN) which is given by the bllowing product of oneparticle wavefunctions: I

I

${y}(xl,x21.. .XN)= 'pvl(XL)Lp65(X2).. . IP,(XN)

=~LpvJx3)-

(2.37)

j

Disregarding the miitual interaction of the electrons for the moment, the product (2.37) forms an eigenstate of the N-electron system if the oneparticle wavefunctions pV,(xJ)are energy eigenstates of the individual o n e electron subsystem Hamiltonians. This suggests t h e question whether a similar result might be possible for interacting electrons, i.e. whether it will be possible to choose the py,(xj) in such a way that the product statc +ivi obeys the Schriidinger equation H${V)(Xl*

x2,.' ' I X N ) = ~ { V } ~ { Y } ( X lX72 r . '

. 1

XN)

(2.38)

for the fully interacting N-electron system - if not rigorously. then at least in some reasonable approximation. To address this question one may use the variational principle of quantum mechanics. In this procedureI the oneparticle wavefunctions pV,(xj) are determined such that the expectation value of the N-electron Hamiltonian H becomes a minimum for N-eIecctran states of the product form (2.37). Here we take a slightly different approach, and start from the Schrodinger equation (2.38). This procedure has the advantage that one ran ai. once determine whether there is a suitable approximation in which d ~ { may ~ } be written in the product form (2.37), and also barn the nature of that approximation. Considering an M-electron system, we label a particular electron i, and this index can take all values between 1 and N. The Hilbert space of the

68

Chapter 2. Electronic structure of ideal crysWs

system of t.he N - 1remaining elect,rons 1,2,. . . , i - 1,i by the set, of product functions

+ 1,... ,N is spanned

with P I , . . . , pz-l, p o + l . . . . , pjv ranging over all possible values 1,2,, . . , co independently of each other. In this remaining Hilbert space we form the Fourier-type coefficients of the Schrodinger equation (2.381, i.e., we multiply t,his equation by the complex conjugated product function (2.39) and integrate over all XI,x2... . XN with the exception of xz. In this way we obtain

(2.40) Due to the orthogonality of the qv. t h e right-hand side of this equation differs from zero only if the pvalues coincide with the v-values, i.e.. if = q ,.., pz-l = ~ ~ - pcl,+l 1 , = v,+l. .... p~ = v w holds. However, the left-hand side of equation (2.40) differs from zero if p J f v3 for one or several I # t . Thus equation (2.38) cannot hold rigorously, which means that the eigenfunctions ${,,I of H cannot be written exactly as a product {2.37) of oneparticle wavefunctions. This is only possible under the condition that the non-diagonal elements of the Hamiltonian operator on the left-hand side of (2.40) may be neglected. It is this approximation which makes possible the reduction of the N-particle wavefunctions to products of oneparticle wavefunctions. It is called a one-partzcle approozamatton. Strictly speaking, it is the simplest variant of a oneparticle approximation, the so-called Hartwe appronmatzoa. .4 more accurate oneparticle approximation, called the Hartree-Fock apprommatzoa, will be discussed below. Within the framework of the Hartree approximation the equation system (2.40) involves only the diagonal terms with p 3 = vj for each 1 # 2 , and correspondingly takes the form

The diagonal elements in this equation can easily be evaluated as

2.2. The dynamical problem

69

(2.42)

On the right-hand side of this equation only the first three terms depend on the electron coordinates x1 while the last two are constants in this regard. If one substitutes the expression (2.42) into equation (2.41),then the last two terms can be grouped together with U{'(.}to h m the new eigenvalur

Using the abbreviation (2.44)

(2.45) we rewrit,e (2.41) as

The final relation (2.46) has the form of a Schrodinger equation for the 7-th particle where V'{(")(xp)is the potential energy of this particle and EV,is its energy eigenvalue. Beside the potential energy ITc(&) due t o the atomic It is caused by the cores. V'Iv}(x,) also contains the contribution \#'}(x,). mutual interaction of electrons, and is commonly called the Hartree p o t e n t d In explicit form, ITH%{.I (x,)reads (2.47)

70

Chapter 2. Electronic structure of ideal crystals

Here the integration runs over a periodicity region. The Hartree putential V~‘”’(X~) describes the potential energy of the i-tb particle in the Coulomb potential produced by the charge distribution --e Ck+% i(oV,(x’)l2of the remaining particles. The factor of the electron-electron interaction potential (2.5) does not occur in expressions (2.45) and (2.47) for the Hartree potential since each electron pair contribntes only once. Obviously. the Hartree potential and the corresponding energy cigenvalues depend OB the configuration { v } of the N-electron system, and also on the index i of the particle which was removed. The one-particle Schrdinger equation (2.46) derived above for the i-th elpctron, holds for each other electron as well. only with a somewhat different potential. This difference will now be removed, together with the dependence of the H a r t r e potential on the configuration {v} of the N-electron system. We argue as follows: If the number A; of electrons is macroscopically large, as in the case of the electron system of a crystal, and if we consider only oneparticle states which are spatially spread out more or less evenly aver the entire crystal, there is no signifkant difference if we extend the sum over k in equation (2.47) for the potential V;,[”’(x,)t o include k - i . Then the 2-dependence of the potential no longer exists. The emor thereby incurred is of relative order of magnitude l/N. If one considers, on the othcr hand, only states ( v } of the &--electron system which are similar to each other, one may also neglect the {v)-dependence of the potential and replace V{”}(x) by the value for a representative configuration {vo). The question is, does such a representative codguration exist in the case of a semiconductor, and if so. what is it. The answer to the former question is ‘under normal conditions, yes’. For a ‘representative configuration’ in the abovementioned sense, we have the state of the N-electron system with lowest total energy, the socalled ground state. In this state all one-particle states p” with energies Ev below a special energy value (the Fermi energy) are occupied, and the states with energies above are empty. Under normal conditions the states of the N-electron system which occur in semiconductors, and also in other solids, deviate very little from the ground state, Non-normal conditions are associated with large deviations, e.g., such as semiconductors which are displaced to a highly excited state by intense laser irradiation. Excluding such extreme cases, the Hartree potential Vrj“}(x) for the configuration {v) is almost the same as that for the ground state configuration {v’}, and correspondingly we have

4

(2.48)

For brevity, we set

2.2. The dynamlcal problem

71

V(X,) = V&)

t VH(Xt).

(2.50)

The extent to which the approximation of a Configuration independent Hartree potential is valid again depends on the kind of one-particle states involved. For the extended, planewavelike oneparticle states of an ideal crystal this approximation works better than for the localized oncpartick states of a real semiconductor. In the latter case the configuration dependence of the potential may become essential (see Chapter 3 for further discussion). Using (2.50), the Schrodinger equation (2.46) becomes

(2.51) The Hamiltonian of this equation is the same for all particles and no longer depends on the configuration of the N-particle system. Equation (2.51) is therefore the oneparticle Schrodinger equation par excellence, devoid of any reference to a particular particle or configuration of the N-electron system. We may therefore omit the index i in equation (2.51). TJsing the oneparticle Hamiltonian ff

=

P' 2, I v(x),

(2.52)

this equation becomes

H'Pdx) = Evcpv(x).

(2.53)

The Hamiltonian H of equation (2.52) is Hermitian, and it is natural l o assume that its eigenfunctions form a complete orthonormal set in Hilbert space. 'lhis assumption has in fact been made at the outset, with respect to the oneparticle states cpu(x) forming the product wavefunctionu of the N-electron system. In summary, the discussion above has shown the following: Within the framework of the oneparticle approximation, i.e. neglecting non-diagonal ~, . . ,XN) elements of thc Hamiltonian, the product wavefunctions $ J { ~ } ( X x2,. are eigenstates of the N-electron system provided that the oneparticle wavefunctions of the product functions satisfy the oneparticle Schrodinger equation (2.53). Solving this equation and forming thc product wavefunction (2.37), one gets approximate solutions of the N-electron Schrodinger equation. In this way we have reached the goal which was formulated at the outset to replace the N-electron problem by a oneparticle problem whose

72

Chapter 2. Electronic structure of ideal crystals

solution has a well defined and sufficiently simple connection with the solution of the N-electron problem. The idea that the (py(x)are energy eigenstates and the E , are energies of single electrons, underlying the above consideration, needs to be made more precise. Because of the electron-electron interaction, the motion of a particular electron is always tied to the motion of all others, and the energy of an electron is also, in part, energy of interaction. The latter statement manifests itself clearly in relation (2.43) between the one-particle energies E, and the total energy U{.} of the N-electron system, which we will explore in more detail. First of all, it can be further simplified. Using the one-particle Schrodinger equation, one can re-express the terms on the right-hand side of equation (2.43) by one-particle eigenvalues, leading to

The energy of the N-particle system is therefore not just the sum of all oneparticle energies. It is necessary to subtract the Coulomb interaction energy of the particles. Therein is reflected the fact that the E, contain a certain portion of interaction energy with other electrons. This is doubly counted in E , of one-particle energies, once in summing over the particles the sum themselves, and once in summing over their interaction partners, which is done in E, automatically. To correct this, one must subtract the Coulomb interaction energy. This shows that the (p,(x) may be interpreted as states of single electrons only in a generalized sense. In reality the (py(x)describe stationary states of the motion of the N-electron system in which all electrons are involved. These states of motion are not mutually coupled, as in the case of normal oscillations of a system of interacting atomic cores. Using the terminology introduced in that context one may consider the states q , ( x ) as states of quasi-particles or elementary excitations of the N-electron system. The E, are the corresponding quasi-particle or excitation energies. There is, however, a qualitative difference between these elementary excitations of the electron system and the normal oscillations of a crystal. This may be made clear as follows. If one adds to the N-electron system (which we will assume to be in the ground state) one more electron, i.e. if one passes over to a ( N 1)-electron system, then the one-particle Hamiltonian (2.52) does not change within the framework of the approximations made above. The oneparticle wavefunctions pV of the N-electron system therefore also approximately describe the elementary excitations of the system of ( N 1) electrons. This means that an eigenstate of the ( N 1)-electron system may be realized by keeping the previously available N electrons in their one-

xi

+

+

+

2.2. The dynamical problem

73

+

particle states and adding the ( N 1)-th electron in one of the oneparticle states pu* of the N-electron system which were previously not occupied. Thus, by adding an electron, the energy of the system rises approximately by Ey*.This means that the eigenvalue Ev* of the one-particle Ilamiltonian may be understood as the energy of an electron added to the system. This statement is called Koopman h e o r e m . From it, one can learn more about the kind of elementary excitations of the N-electron system that are described by the p,. These are states in which, as always, all electrons of the system are involved, but not all in the same way. Only one of the electrons is moving in such states, while the others play a passive role; they determine the potential in which this movement occurs. One therefore refers to these states as one-particle excitations of the N-electron system, and to their energies as one-particle excitations energies or, in short, one-particle energies. In addition to the one-particle excitations considered above there may yet be others. This can be confirmed by taking a (N -1)-electron system instead of the ( N 1)-electron system. The missing clectron corresponds to a hole in a previously occupied oneparticle state ( p y ~ . The excitation energy of the hole is -EvO, which did not occur among the oneparticle excitations considered above. It therefore represents an additional one-particle excitation. If an electron is removed from state vy and simultaneously an electron is added in state then this corresponds to the excitation of the N-electron system from state ($, Y:, . , . , v k ) into state (v;, v:, ..., I&). The energy difference with respect to the ground state amounts to EV;- F 0 . It corresponds 5 to the excitation energy of a11 electron-hole-pair with the electron in state pu; and the hole in state p 0 . If one excites a second electron from state v1 'p o into state p,,;, the energy difference with respect to the ground state is u!2 (Eq - Ey:) (By;- E e ) , etc. The excitation energies of the N-electron system can thus be written as a linear superposition of oneparticle energics. This is valid only within the one-particle approximation. In a strict seiisc one also has many-particle excitations, which will be considered in more detail below. As far as the one-particle excitations are concerned, there are no others than the ones considered above, at least as long as one ignores spin and the magnetic interaction between electrons.

+

vT,

+

It.is now appropriate to clarify how the oneparticle wavefunctions cpv(x)

111wbe calculated from the Schrodinger equation

(2.53). The potential in Uhis equation, more specifically the Hartree part VH(X), dcpends on the wavcfunctions pV(x)which are involved in the construction of the ground state of the N-electron system. One must know these functions in order to write down the potential and thus define the oneparticle Schrodinger equation. On the other hand, these functions can only be obtained by solving this equation. The situation is similar to that in the preceding section on the coupled Schrodinger equations for the interacting system of electrons and

Chapter 2. Electronic structure of ideal crystals

74

d?

-

n PU

+

V"(X) = t'c(x) VG(X)

Figure 2.3: Self-consistent solution of the oneparticle Schrijdinger equation. atomic cores. As was done there. we may also solve the present problem iteratively (see Figure 2.3). We employ one-particle wavefunctions pf(x) close to the true stationary oneelectron states. r s i n g cp:(x) we determine a POtential t$(x) according to equation (2.491, form the total potential Vo(x) by means of (2.50), and use this to solve the one-particle Schrodinger equation (2.53). The solutions vt(x)are then substituted into formula (2.49), thereby determining new potentials TG(x) and V l ( x ) . With the latter one recalculates the eigenfunctions p:(x) etc. One continues this iterative procedure until the eigenfunctions, and with them also the potential in the following iteration step. no longer change within a specified limit of accuracy. The eigenfunctions and potential are then said to be determined self-consistently. Spin a n d spin-orbit interaction

At this point in our treatment of the oneelectron approximation, it is a p propriate to recognize that electrons h w e a spin, i.e. an internal angular momentum with the two possible values rC/2 and --7i/2 in a given direction. This is to say that electrons are capable of a motion in spin space, beside

2.2. The dynamical problem

75

their motion in coordinate space, which in this context is called orbatal m o ) tion. As orbital motion involves dependence of the wavefunction p ( ~on the space coordinate x,spin motion involves a dependence of p(x, s) on the spin variable s which may take the two possible values s = and s = - z1

4

-

(below the latter value will be written as =_ -;). 'I'hus, in consideration of spin, the wavefunction of an elcrtron changes from - an ordinary vector p(x) in Hilbert space to an element {~(x, p(x, ;)} in the product space of the ordinary Hilbert spare and tht. two-component spin space, a so-called two-component spinor. fiLnrtion, To determine the spinor state of an electron uniquely, the quantum number X which defines the statr must also specify the spin state. If the latter i s independent of the state in coordinate space, this may be done by specifying another quantum number u for the spin motion along with the quantum number v of the orbital motion, i.e. by setting - v , o where u may take-the two values T (spin up) and 1 (spin down). The spinor {px(x, px(x, can then be represented as a product of only one spatially varying function py(x)and a spinor ,yo($)} which does not change in coordinate space. The two spirior components cpx(x, s) can then be written as

i),

i),

i)}

(~~(4)~

(2.55)

In general, however, the orbital and spin motions are coupled. This is mainly due to the fact that, on the one hand, the spin motion is accompanied by a magnetic moment of the electron, and that, on the other hand, the orbital motion gives rise to a magnetic field which couples that magnetic moment. In quantum theory it i s shown that this interaction, which is called spin-ovhit intemction, can be represented by the following additional term H,, in the oneelectron Hamiltouian: Tz

H,, - -[VV(X) 4m c

(2.56)

x p] . (7.

Here V(x) denotes, as before, the periodic crystal potential of equation (2.50)) and 3 is the vector whose three components are Pauli's spin matrices. In spin space one usually refers all quantities to the basis X I = (1,0),X I = ( 0 , l ) . Then the components of are u.=(;

i),

u y - ( i0

-

0')

1 3

0

(2.57)

Taking account of spin and the spin-orbit interaction, the one-particle Schrodingw equation (2.55) in Hartree approximation t,akes the form

76

Chapter 2. Electronic structure of ideal crystals

(2.58)

Spin-orbit interaction i s in fact an important consideration in determination of t,he energy specha of many serniconduct,ors. Hartree-Fock approximation

An obvious drawback of the Hartree approximation is that the wavefunction of the IV-particle system is not antisymmetric with respect to the exchange of two particles, a requirements of the Pauli exclusion principle. 'l'his defect can be easily remedied by replacing h e product wavefiinction (2.37) of the Ilartree state by a linear combination of product wavefunctions with exchaiigd partirle indices and altered signs. In conjunction with this, the spin of the electrons has to be considered, such that the wavefunction of the t-th particle is given by the spinor p ~ , ( x ,si). , The antisymmetric linear cornhatiom of the product waverunctions may he arit,tcn in t h e form of a so-called Slater determinant

(2.59)

In this determinant, an exchange of t.he variables of two electrons leads to the exchange of the two corresponding rows. The sign of the determinant thereby changes, SO that the Slater determinant actually has the requisite aiitisymmnctry propcrty. If t.wo of the quantum numbers XI, X2,. . . , AN are q u a l , then two colunms 01bhP determinant are identical and vanishes. 'l'his means that no states of the N-electron system are allowed with two electrons in the same oneparticle state. "his is just the Pauli principle, automatically enforcd by the use of the det,erminant,alform of the N-part,icle wawfunr tion.

Employing such SlaPer determinants as N-particle wavefunctions, as opp o s d to simple producls, a one-particle Schriidinger equation far pv(x) may be derivd in the same way as before, but the potential in this equation ia

2.2. The dyynmiicd problem

77

somewhat different than that in the Hartree equation. It contains an additional contribution, the so-called exchange potential Vay(x),and the total potential reads

+ VH(4 + W X ) .

V(X) = K(x)

(2.60)

In the case of negligibly small spin-orbit interaction, the orbital state may be characterized by a separate quantum number v t , and the spin state by a separate cpiantum number m i . T h c spinor components cp~,(x,, 8%) are of thr form (2.55) in such circumstances. The sum VH(X) Vx(x) of the Hartree and cxchange potentials can then be written in a relatively simple form. It can be shown thilt theii action on the coordinate dependent factor of the oncparticle wavefunction cpl,(x)takes the form

+

(2.61)

The Erst term on the right-hand side of this cquation is t,heHartree potential. The factor of 2 results from summing over the t,wo spin states associated with each wavefunction cpvk(x).The second term corrcsponds to the exchange potential. Formally, it differs from the first term by exchanging the states at the two positions x and x’. The factor of $ reflects the fact that, firstly, the exchange potential acts only between electrons of the same spin, and, secondly, that for the ground state with total spin 0, half of the electrons are in spin-down states, and half are in spin-up states. In this way the magnitude of the exchange potential is influenced by the existence of electron spin, although its value is the same for spin-up and spin-down states. Equation (2.61) also shows that, unlike the Hartree potential, the exchange potential is non-local. The effect of the exchange potential on the wavefunction p,(x) is represented by an integral operator. In actual calculations one often uses a local approximation for Vx(x). The exchange potential proofs to be attractive, which is to be expected: the anti-symmetric form (2.59) of the total wavefunction ${A} means that the probability of finding two electrons with the same spin at the same position is zero so that one has an ‘exchange hole’ around each electron. This lowering of electron density in the vicinity of an electron results in an attractive potential in addition to the repulsive Hartree potential since the total Hartree wavefunction (2.37) does not account for the exchange hole. The improved oneparticle Schrodinger equation with the potential of (2.60) and (2.61) is called the Hartree-Fock equation, and the oneparticle approximation, which underlies it, is called Hartvee-Fock approximation Thereby, the Hartree and exchange potentials are understood as those for the

78

Chapter 2. Electronic structrrre of idea? crystah

ground state configuration v i of the N-electron system. The effects of the electron-electron interaction, which are still neglected within the HartreeFock approximation with configuration independent Hartree and exchange potentials. are called correlation effects. Correlation effects Correlation effects are, first of all, manifested in the fact that the true oneparticle excitation energies of an N-electron system differ from those in the Hartree-Fock approximation. In particular, these excitation energies depend on the configuration of the system, one has tt, configuration dependence. Secondly, Slater determinants which in Hartree-Fock approximation are considered to be eigenstates of the total Hamiltonian, in fact do not diagonalize tbis Harniltouian ezactlv; there are non-vanishing offdiagonal elements, an effect which is termed configuration interaction. The exact eigenstates of the total Hamiltonian are linear combinations of diflerent Slater determinants, and the corresponding energy eigenvaluea are no longer s u m of oneparticle excitation energies, as had been the case for individual Slater determinants. In other terms, the exart,eigenalates of the N-elwtrronsystem are not oneparticle excitations! but many-particle excitetians. Examples of many-particle excitations include two-particle ezcitation.5 of an electron and a hole which are bound together by their Coulomb Interaction. The excitation energy of such a hound electron-hole pair, the so-called ezcitan, is smaller than that of the excitation energy of a free electron and hole pair, differing by the binding energy of the pair. The reason for the designnation ‘correlation effect’ for this phenomenon is obvious: binding may be understood as a correlation between the positions of the electron and the hole, since their separation by a distance of about a Bohr radius is more probable than all others. This interpretation presents the correct concept of correlation in other cases also the states of the electrons are no longer independent of each other, but, are correlated contrary to the assumptions implicit in the oneparticle approximation. Collective many-particle excitations are excitations of states in which all electrons of the system participate in comparable measure. Examples include the plasma o s c i l - t i a m of an electron system. They form a direct electronic analogy to the lattice vibratious of the atomic cores of a crystal. Their excitation quanta are called plasmons. The consideration of correlation effects stands along the most difficult problems of solid state thmry which, even today, is not completely solved. A comprehensive analysis of this problem i s far beyond the scope of the present book. Readers who are particularly interested in correlation effects will find discussions in a number of textbooks (see, e . g , hbrikosov, Gorkw, md Dzyaloshinski, 1963; Fetter and Walecka, 1971; Ziman, 1974; Callaway, 1976; Madelung, 1978; Harrison, 1981). Below we summarize some results ~

2.2. The dynamical problem

79

which will be needed in Chapter 3. In doing so, we will concentrate on oneparticle excitations, i.e. individual electrons and holes moving in the force field of all other electrons as well as in the force field of the atomic cores. Correlation effects on one-particle excitations. Density functional theory.

Correlation effects on one-particle excitations may be treated by means of the Green's function theory of many particle-systems. The poles of the o n e particle Green's function in the complex energy plane represent oneparticle excitation energies (more strictly speaking, the real parts of these poles are the energy levels, and the imaginary parts are the lifetime broadening energies of the one-particle excitations). The one-particle Green's function is governed by the Dyson equation, which contains correlation effects through the so-called mass or self-energy operator. Another method which works well for oneparticle states involved in the ground state of the many-particle system is known as density functional theory. This method relies on a theorem, the Hohenberg-Kohn theorem, which ensures that the ground state energy Eo of an interacting electron system in an external potential Vc(x) is a functional E o [ n ( x ) ]of the total electron density n(x) of the ground state alone. This implies, first of all, that the total energy E o [ n ( x ) ]depends on the oneparticle wavefunctions only through the ground state density n,(x)and, moreover, that the density enters at every point x, through an integral over X . The oneparticle wavefunctions determine the ground state density by means of the equation (2.62)

where cpvi(x) denote the one-particle states which, in the ground state of the N-electron system, are populated by electrons z = 1,2, ..., N. According to the variational principle of quantum mechanics, the wavefunctions cpv,(x) adjust so that, while keeping their norms (cpudlqv,) constant, the total envrgy Eo[n(x)]is minimized. This requires the vanishing of the variational derivative of the functional E o [ n ( z ) ] E,(cp,Ip,) with respect to p:*(x), where the factors E , are variational parameters, therefore

-xi

(2.63)

In this functional derivative the value of cp:t(x) at a certain point x is taken as an independent variable, with respect to which the common derivative is taken. The total energy functional Eo[n(x)]of the ground statr may be decomposed into several energy contributions, namely, the kinetic energy E ~ & ( x ) ] , the external potential energy E c [ n ( x ) ] the , Hartree energy E w [ n ( x ) ]and the

80

Chapter 2. Electronic structure of ideal crystals

exchange and correlation energies which are usually summed in the exchangecorrelation energy E x c [ n ( x ) ] .Thus

+

+

Eo[n(x)]= Ekin[n(x)l Ec[n(x)l &dn(x)l

+ Exc[n(x)l.

(2.64)

) ] easily obtained as The functionals E c [ n ( x ) ]and E ~ [ n ( x are

(2.65) E ~ [ n ( x= )]

2

//

R R

d3x'd3x

n(x') . n ( x ) Ix' - XI '

(2.66)

The functional E x c [ n ( x ) ]is less obvious. It is usually taken in a local approximation called the local density approximation (LDA). The LDA starts with the homogeneous electron system without any external potential. In this case the density n ( x ) is a constant n in space, and E x c ( n ) reduces to an ordinary function of n. Dividing E x c ( n ) by the total number nR of electrons yields the exchange-correlation energy E X C ( ~of) the free electron gas, per electron. The total exchangecorrelation energy Exc(n) of a weakly inhomogeneous electron gas of density n ( x ) should then be given approximately by the expression (2.67) Finally, the kinetic energy fuIictiona1 E:k.tn[n(x)]has to be specified. By definition, we have (2.68)

Although this expression does not look like a functional of n ( x ) il is indeed possible l o transform it into such a form because all other terms in the total energy functional Eo[n(x)]of (2.64) are functionals of n ( x ) , and the Hohenberg-Kohn theorem enforces this for E & t ( x ) ] . For the waluation of the variational equation (2.63) we do not, however, need the explicit functional form of EkznIn(x)l;expression (2.68) suffices. Its variational derivative with respect to p;,(x) iu given by (2.69)

For the rcmaining functional derivatives, it follows that (2.70)

2.2. The dynamicd problem

81

(2.71) (2.72)

where

(2.73) denotes the ~xcliangPcnrrelationpotential. The latter can be determined if the exchange-correlatiou energy E x c ( n )of the homogeneous electron gas is known as fniictio~iof n. This dependence can hr obtained by calcrilating E>yc(n)numerically for tfifkrent values of n and then lilting the data to appropriate explicit functions. TJsing this procedure. various Pxchang+ correlation potentials have been proposed, for exaniylr

Lkc(x)- -

(:)

1/3

e2n1j3[x)[I

+ 0.7734 z In

with z - rS/2l, where rS = and CI.Bthe Bohr radius (Hedin, Lundqvist, 1971). Suhstitiiting i n h q u a t i o n (2.63) t h e frlnctionrtl drrivatives obtained above, one arrives at (2.75)

with v(x) = VC(X) I Vrr(X)

+ i 0.

(2.2

It follows that

where B(E - Euo) =

1 f o r E > Euo 0 for

E < Euo

(2.221)

is the Heaviside unit step function. If one considers, instead of an isotropic parabolic band, an anisotropic one with three different effective masses m:l, m:2, m:3 corresponding to the principal axes of the effective mass tensor, then the density of states is again given by expression (2.220), but m: must be replaced as follows

(2.222) by the so-called density-of-states-mass m b . This may be seen immediately if one reviews the calculation for the isotropic case. The anisotropic effective mass involves a change only upon substitution of the variables for the components of the k-vector - the factors for the 3 components are different and lead to the modification indicated in equation (2.222). According to expression (2.220) the density of states of a parabolic band with m*, > 0 exhibits square root-like behavior at the lower band edge, and continues following a square root law up to infinitely large energies. The latter property is a consequence of our untenable assumption of parabolic dispersion up to the upper band edge and the replacement of this edge by the point at infinity. In reality the band edge lies at finite energies, and the density of states again falls off to 0 there. It has, qualitatively, the shape shown by the dashed curve of Figure 2.14. The sudden square root-like increase of the DOS at the band edge reflects the fact that the function p(E) has singularities at these energies - the first derivative with respect to E is 0 if one approaches the edge from the forbidden zone, and it is +CXJ if one approaches it from the interior of the band. This is one of the already mentioned van Hove singularities of the DOS. These singularities occur not only at energies where a particular band E,(k) has a minimum, as in the

2.5. Band structure

127

Figure 2.14: DOS of a parabolic isotropic band in the vicinity of its minimum (solid curve). The dashed curve shows the DOS of a more realistic band.

5-

-c =

<

I

I

I

I

4-

3-

I

vl

2-

0

0

-

1-

0 -1

0

1

2

3

4

Energy lorb. unlts)

case considered here, but at all energies corresponding to critical points of the energy bands, thus also at maxima and saddle points. Counting states: 3D, 2D,and 1D density of s t a t e s

The square root enerw dependence of the density of states in the case of a parabolic dispersion law may also be demonstrated in the following, more vivid way (also see Figure 2.15). For simplicity we set E,o = 0. and omit factors which do not depend on k or E. The n m b r r A Z of b m d states with energy between E and E .f A E is the number of different k-points of the finely meshed net which yield energy values in t,his interval. These kvalues lie in a thin spherical shell in k-space [see Figure 2.15) which, because E o(. k2, has radius k~ 6: &.and thickness Ak which, since Ah' CK k&k, is given by

Ak

1

0;

-AE.

fi

(2.223)

For the volume AV of this shell, it follows that AV m k L A k cc A A E .

(2.224)

Since the density of the finely meshed net of k-points of the first BZ i s the same everywhere, the number of k-points in the spherical shell is proportional to its volume AV. Hence AZ cy v%AE and p(E)

o(.

4%.

(22 2 5)

Such considerations can easily be applied to energy bands in 2-dimensiouel (2D)and 1-dimensional (1D)k-spaces. Such k-Bpaces have physical (as well as mathematical) meaning, in particular for electron systems whose free

128

Chapter 2. Electronic structure of ideal crystals

b

Figure 2.15: Counting states in

C

(a)

3D, (b) 2D, and (c) 1D k-space.

motion is confined to one or two dimensions of 3-dimensional space. We wiU encounter examples of such systems in Chapter 3 - surfaces and quantum wells for the 2D-case, and electrons in a homogeneous magnetic fieId for the 1D-case. For 2 0 k-space, the volume of the spherical shell is replaced by the area A F of a circular ring, such that A F a k E A k LX A E and p ( E ) 3: const.

(2.226)

The density of states of a 2D-electron system is therefore independent of E. In the case of a ID k-space: AV is replaced by the length Ak of the k-interval itself. Thus, p ( E ) DC

2.5.5

1

-AE.

fi

(2.227)

Spin

Thus far, spin has been omitted from our discussion of the general properties of stationary oneelectron states of crystals. It turns out. however, that spin and spin-orbit interaction play an important role in most semiconductors. Therefore, the question arises as to how the results derived above for scalar wavefunctions change when electron states are no longer scalar but spinor functions and the spin-orbit interaction H , is taken into account. The starting point to address this question is the one-electron Schrodinger equation (2.58) with spin, which may be written in the following short form

(2.228)

129

2.5. Band structure

The first point we will consider is that of spatial symmetry in the presence of spin. Spatial symmetry Inspecting the explicit form of the spin-orbit interaction operator ( b e y formula ( 2 . 5 6 ) ) , one readily recognizes that TKso has thc full symrwtry of the crystal. Thus it colrimutes with all elements 9 of the space group. Since the one same holds for the spin free part H of the total Hamiltonian !i+ has

Iff

+ H,.

g] = 0

.

(2.229)

To exploit this symmetry property of thv total Hamiltonian w e mu61 know how thc components p(x, s) of a spinor transforin i d e r thr operations g of the space group. Considering translations first. we have

In t.he absence of spin, the cornmulivity of the HanliltoIlittn with translations rise to the Bloch theorrm. In t.he same way that this theorem was proved without spin above. its validity may be also demonstratd here - in the pwsrnrr of spin. It. stdates that the solutions { p ~ k ( x , y ~ ( xf)} , of t.hr Schrodingeer equation (2.228) for the eigenvalue E A ( ~can ) be chosen in the form of Bloch type spinor functions with a particiilar quasi-wavevecbor k as

gave

4).

zti)

where I ~ X ~ [ X . are the spin-dependent Bloch factors. If k is restricted to the first B Z . then E x ( k ) rrprwents a continuous function of k. Thc band index X here refers to both the state of the orbital motion and also to spin state.

Second, we consider point symmetry operations u , whose action in tramforming t,hc components of a spinor is discussed in Appendix -4. The results of Appendix A are applied here in the form

130

Chapter 2. Electronic structure of ideal crystals

with

In this, $, B and p are the Euler angles of the orthogonal transformation a. If the spinor is spatially constant, the transformation just reduces to multiplication by the matrix D r ( a ) . The set of all orthogonal transforniations cy. 2 forins an (infinite) group. Through relation (2.233) this group is assigned a matrix group D i ( a ) . A peculiarity of thm mapping is that it is not unique 1 - a change of the angles cp or $J in (2.233) by 27r leads to a change of the sign of the associated matrix, even though this signifies the identity transformation. One may say that under a rotation through 27r a spinor does not transform into itself, but into its negative. This gives rise to multiplication rules for the representation matrices which deviate from those of the group itself. Thus the representations of the full orthogonal group in the space of two-component spinors are not representations in the ordinary sense, but in a generalized sense. They form projective representatiom with a special factor system (see Appendix A), or, in short, spinor representatiom. By means of the so-called double group, which includes each element, of the fill1 orthogonal group twice, once in the original form, and once multiplied with a rotation through 27r, one may trace back the spinor representations to ordinary representations. The spinor representations are those representations of the double group which do not occur among the ordinary representations of the single g ~ o u p(for the derivation of this result see Appendix A). Up to this point we have only given attention to the transformations of spinors in spin space. The states of electrons are described by spinor fields having positional dependence. These give rise to spinor representations of the (full) orthogonal group which differ, in general, from D i , The totality of spinor representations may be obtained from the ordinary irreducible representations V, and the particular spinor representation D1 of the full 2

orthogonal group. Indeed, if a spinor field (P,(x, s), with s = fk,transforms according to a certain ordinary representation D, in coordinete space, then its transformation in coordinate spin space is governed by the product representation Vi x Vv.If, here, D, encompasses all vcctor representations, 2 then all spinor representations are obtained. Using the concept of spinor representations, the already discussed connection between the eigenfunctions of the crystal Hamiltonian for a given eigenvalue and the irreducible representations of its symmetry group may be generalized in the following way: ~

+

The spinor eigenfunctions of the crystal Hamiltonian W H , having the same energy eigenvalue span a representatdon space of a n irreducible spinor

2.5. Band structure

13 1

Table 2.6: Irreducible spinar representations (notations and dimensions)

of the small point groups of symmetry points and lines of the firat B Z of the fix lattice for crystala with the diamond structure.

representation of the full symmetry group of H group of the crystal).

+ H,,

(namely, the space

The product representations Dr.x V v , taken as representations of the sman point groups of a particular wavevector rather than for the full orthogonal group, are generally reducible. This means that bands which are degenerate without spin by reason of spatial symmetry, may split if spin is taken into account. The magnitude of the energy splitting depends on the spinorbit interaction, without this interaction, degeneracy persists, but as an accidental rather than a symmetry induced degeneracy. The following example illustrates the removal of degeneracy. We consider the valence band of zincblende type semiconductors which is triply degenerate at I'. The product representation in this case is Di x 1-15. It decomposes into the a ', two irreducible spinor representations ra and r 7 of the point group T d of I where rs is of dimension 4 and r7 of dimensiun 2. This means that, due to the spin-orbit interaction, the triply degenerate r15-valence band without spin splits into the $-fold degenerate rs-band and the 2-fold degenerate r7band, accounting for the effects of spin. In Table 2.6 the irreducible spinor representations are shown for some symmetry paints of the first BZ of diamond type semiconductors. A systematic treatment of these representations is given in Appendix A.

132

Chapter 2. Electronic structure of ideal crystals

Time reversal symmetry As the consequences of spatial symmetry are altered by the phenomenology of spin, so is the action of time reversal symmetry changed by spin. First,we $)} will show that the operator K which-transforms a spinor {cpx(x, $), cp~(x, into the spinor K{cpx(x, cpx(x, at reversed time, is given by the relation

i)}

i),

(2.234)

To prove this assertion, one executes the time reversal operation on the spin-orbit interaction operator H,, of equation (2.56) explicitly. The vector a' then transforms into -a*, and p into -p, so that the net change in Hso is the replacement of d by a'*. Employing the operator K , on the other hand, the effect of timc reversal on H , , may be expressed in terms of the similarity For the two expressions to be identical, transformed operator KH,,K-'. K must have the form given above in equation (2.234), apart from a phase factor which remains undetermined. The wavefunction {cpx(x, cpx(x,$)} must be replaced by K{cpx(x, cpx(x, $)} for the Schrodinger equation (2.180) to remain valid. The question, under what conditions time reversal symmetry includes degeneracy of eigenfunctions not accounted for by spatial symmetry alone, is treated in Appendix A in full generality. Here we deal only with a specjal cp~(x, case. We consider a non-symmetric wavevector k. Let {cpxk(x, be an eigenfunction with energy Ex(k). If the point group of directions contains the inversion, then the eigenfunction {cpx-k(x, cpx-k(x, corresponds to the same eigenvalue Ex(k) -= Ex(-k). The time reverse of the Bloch function {cpx-k(x, cpx-k(x, $)} likewise has energy Ex(k) and wavevector k. It is linearly independent of {(Px~(x, cpxk(x, $)}, since

i),

i),

i),

i),

i),

i)} i)}

i),

as one may easily show by evaluating the scalar product (here, one also has to sum over the spin coordinate s ) . Thus, two linearly independent eigenstates of the same energy have the wavevector k. Since k was chosen arbitrarily, it

follows that, due to time reversal symmetry, all bands of inversion-symmetric crystals are at least twofold degenerate.

2.5. Band structure

2.5.6

133

Calculational methods for band structure determination

There are many methods, quasi-analytic and numerical, for calculating the band structure of crystals. In the following we will give an overview. In band structure calculations one is confronted with two problems, firstly with the formulation of the one-elcctron Schrodingcr equation, i.c. with the determination of the effective periodic oneelectron potential V ( x ) of the crystal, and secondly with finding the eigenvalues and eigenfunctions of that equation. The various methods of band structure calculation differ in the manner in which thcsc two problcms are resolved. Hcre we will deal mainly with the second problem, i.e. with the solution of the Schrodinger equation whose potential is known. The first problem, the determination of thc cffectivc oncclcctron potential V ( x ) , has in principle brcn solved in section 2.2, where we dealt with the one-electron approximation for the many-electron system of a crystal. Here, we only address some further dctails of this problem.

Determination of the effective one-electron potential The simplcst way of dealing with thc oneelectron potential V ( x ) is to treat its matrix elements with respect to a particular basis set as empirical parameters rather than integrals to be evaluated from a knowledge of the particular profile of V ( x ) . Having done that, one has an empzmcal methodof band structure calculation. We will provide examples below. If one wants to forego, however, the aid of empirical data and calculate the band structure from first pnnczples, this evasion is unacceptable. The effective one-electron potential V ( x ) of the crystal must then be known as a function of x. Methods which proceed in this way are referred to as ab znztzo methods. Within the frozen core approximation, V(x) describes three interactions of a valence electron. The f i s t is the Coulomb interaction with the atomic nuclei - this part is not problematic. The second is interaction with the (frozen) core electrons - the energy levels and eigenfunctions of the core states of the free atoms have to be determined in advance for us to be able to write down this potential contribution explicitly. It includes the Hartree and exchange potentials as well as the correlation potential of the valence electron-core electron interaction. These potentials are described by expressions which are completely analogous to those for the valence electronvalence electron interaction derived in section 2.2. The third part of the periodic crystal potential accounts for the effect of the remaining valence electrons on the valence electron specifically considered. It depends on the very same eigenfunctions that are to be calculated. Because of this, the calculation of the band structure must be done self-consistently. In many

134

Chapter 2. Electronic structure of ideal crystaIs

cases, the Hartree and HartreeFock approximations fail to give satisfying results, so that correlation effects must also be inchided. As indicated in section 2.2. one way this can be done is by means of the dematy finctional theory in its local approximation (local density approximation or LDA). The LDA-method yields very good results as far as valence bands are concerned, but it fails if applied to the conduction bands. In particular, as already mentioned in section 2.2, it does not reproduce the correct d u e of the fundamental energy gap. A procedure which avoids this failure is the Gmen’s functzon method mentioned in section 2.2. This approach is now being applied more frequently under the name puma-particle method (see, e.g., Bechstedt, 1992). Within this framework the self-eneqy operator is often taken in the so-called GW -approximation [‘G’ stands for ‘Green’s function’ and ‘W’ for the Coulomb potential). We add two further remarks, related to the potential of the atomic cores. Firstly, only for relatively light atoms such as C or Si, may spin and the spin-orbit interaction of the valence electrons be ignored. For the heavier atoms, such as Ge, this interaction is essential and must be incorporated in the effective one-electron potential. Secondly, the decomposition of the total electron system into valence and core electrons need not to be made in the literal sense of these terms. What counts is which electrons are frozen in their atomic states and, therefore, need not be treated self-consistently, and which electrons must be. The latter are ‘valence electrons’ in a more general sense. In 111-V semiconductors, for example, they can also include d-electrons which, of course, do not belong to the valence shell of one of the two elements involved. In the extreme case, all electrons are treated as ‘valence electrons’. Then one has the so-called call electron problem. The solution of this problem involves an extraordinary large numerical effort, so that such all-electron band structure calculations have been performed in o d y a few cases to date. From the physical point of view, they are the most satisfying. With increasing computing power they will become ever more important.

Solution of the Schrodinger equation

The solution procedures can be divided into three groups, firstly! the matrix methods, secondly, the cell methods, and thirdly, the muffin-tin methods. Matrix methods

In the application of matrix methods one represents Bloch type eigenfunctions of a given quasi-wavevector k in a particular basis set consisting of a finite number of functions of the Bloch type. The Hamiltonian of the crystal is thereby represented by a k-dependent complex Hermitian matrix which

2.5. Rand structure

135

has as many columns and rows as the basis set has functions per k-vector. The calculation of band structure is thus traced back to the determination of the eigenvalues and eigenvectors of a k-dependent Hamiltonian matrix. The basis sets employed differ by the number and kinds of functions they contain. One would like to manage with the fewest possible functions, to minimize the numerical effort. The price for this is a loss of precision, because with fewer basis vectors the eigenfunctions are necessarily approximated more crudely than with a larger number - in the extreme, one needs an i n h i t e number of functions. For a fixed number of basis functions the level of precision achieved is higher for basis functions which are better adjusted to represent the eigenstates. The following basis functions prove to be of practical use: -Plane waves Iki-K) with k being a vector of the first BZ and K a reciprocal lattice vector. This constitutes a generalization of the nearly free electron approximation. Bloch sums of atomic orbitals, also called LCAO's (Lznear Combznatzons of Atomzc Orbatuls), or of other localized functions. This includes the tight binding met hod.

~

Bloch functions with Bloch factors for a special wavevector, referred to as Luttznger-Kohn funetzons. The so-called k . p-method uses these functions. In some circumstances, the tight binding and k pmethods may be used to derive analytic expressions for the k-dispersion of energy bands. Such expressions are extremely useful to achieve a physical understanding of band structure. Therefore, we treat these two methods in greater detail below (sections 2.6 and 2.7).

~

Orthogonalized plane waves, called OY W's. An OPW-function IOPW~+K) is obtained from n plane wave Ik+K) by subtracting a certain linear combination of core band eigenfunctions Irk) of the crystal Hamiltonian. Within the frozen core approximation, the core band eigenfunctions (ck) may be taken as Bloch sums of the core states of single atoms. The linear combinations to be subtracted are chosen such that the OPW's are orthogonal to all core eigenstates, so that

-

(2.236) holds. The OPW's must have the form

+

l O l ' W k + ~ )= Ik. K ) - C(cklk

+ K ) Ick)

(2.237)

C

in order to satisfy the orthogonality condition (2.236). Making the expansion functions orthogonal to the core eigenstates accounts for the fact that

136

Chapter 2. Electronic structure of ideal crystals

eigenstates of the Hamiltonian for different energies are always mutually orthogonal. This means that the sought-after eigenstates of the valence electrons of the crystal for a given quasi-wavevector k must be orthogonal to the core eigenstates having the same quasi-wavevector k. For this reason the O P W ' s are much better adjusted to represent the eigenfunctions of the valence electrons than are pure plane waves; correspondingly one needs fewer O P W ' s than plane waves to accurately represent the valence eigenfunctions. A further development of the OPW-method is the pseudopotential method. Pseudopotential method

The idea underlying the pseudopotential method is to transfer the core state orthogonality term in equation (2.237) from the OPW's to the one-electron Hamiltonian H of the crystal. This transfer is done as follows. Consider the core electrons. which we have hitherto taken jointly with the atomic nuclei to form the cores, to be independent particles, just like the valence electrons. This means that the potential created by the core electrons is no longer included in the core potential Vc, but is added to the eEective o n e electron potential 1 ' ~ 15-cof the electron-electron interaction. Because of this reinterpretation of the oneelectron Hamiltonian H , its eigenstates now also include core states, for which we have

+

(2.238)

H i c k ) = E,lck).

The valence band eigenvalues E and eigenfunctions ~a similarly satisfy the eigenvalue equation

The expansion of

~$m with respect to the OPW's

reads

Removal of the core states from the expansion functions mandates changing from the eigenfunctions $ato other functions v p i that have the same expansion coefficients, but plane waves as expansion functions: (2.241)

~$g~

Surprisingly, the artificial wavefunctions are in fact eigenfunct ions of a particular Hamiltonian H p s , having the same eigenvalue E to which the eigenfunctions $aof H correspond. In fact, by applying of H to $I% one immediately Ends that

2.5. Band structure

137

(2.242) H P S= H

+ E ( E - E,)lck)(ckl

.

(2.243)

C

The Hamiltonian HPS is a well-defined linear operator, although it is nonlocal and depends on the eigenvalue E itself. It is called a pseudo-Hamiltonian The are termed pseudo-wavefunctions. Each eigenvalue of HP" is simultaneously an eigenvalue of H , but the reverse statement does not hold. While core levels E, are eigenvalues of H , they are not also eigenvalues of HPS since the pseudo-wavefunctions of the core states vanish. The pseudo-Hamiltonian HPS may be written in a form which clarifies its meaning. To begin with one revokes the re-interpretation of the Hamiltonian H , i.e. considers the core electrons no longer as independent particles which enter the effective one-electron potential VH Vxc of the electron-electron interaction, but includes them again into the atomic cores making them contributors to the core potential V,. Then HPS takes the form

$gk

+

H P s = -p2+ v

2m

H

+ Vxc + Vc+ C ( E - E,)lck)(ckl

.

(2.244)

C

The last two terms in this expression jointly constitute the so-called pseudopotential V,p"

(2.245) The second term on the right is significant only in the core regions. There, it is preferentially positive, indicating that it repels valence and conduction band electrons away from the cores. One can show that it largely compensates the variation of the core potential V, in these regions. Because of this, the pseudopotential V,p" is relatively smooth throughout the cores. It can he made even smoother if one exploits a property of V,p" which has not yet heen discussed. We refer to the non-uniqueness of the repulsive part of V,p" in equation (2.245). If there the bra-vectors (E - E,)(ckl in (2.245) are r e placed by completely arbitrary functions (while keeping the ket-vectors Ick) unchanged), the eigenvalues of the pseudo-Schrodinger equation (2.242) remain the same, only the pseudo-wavefunctions change. This freedom may be used to make the pseudopotential still smoother, and also fulfill other requirements, such as, for example, norm-conservation of the pseudo-wavefunctions. The smoothness of the pseudoptential makes it possible to restrict the planewave expansion (2.241) of the pseudo-wavefunction to terms having small reciprocal lattice vectors K. Consequently, the representation matrix of the pseudo-Hamiltonian HPS is small and easy to diagonalize. This is the reason

138

Chapter 2. F k t m n i c structure of ideal crystals

that the pseudopotential method is very helpful in calculating valence and conduction band structures of semiconductors. In order to apply this method, the psendopotential must be known explicitly, of course, In principle it can be determined from the defining equation (2.245),with core levels and wavefunctions taken from atomic calculations, and with bra-vectors (ckl substituted by appropriate functions. In practical applications, the pspudopotential V,p"i s replaced by approximate expressions which range from empirical local pseudopotentials with adjustable parameters up to non-local pseudoyotentiah including core states of s-! p- and d-symmetr y. The pseudopotent,ial method is generally successful if the true valence band eigenfunctions are also sufficiently smooth outside ol the core region. This i s the c a ~ eas long as the valence band states are composed mainly of atomic 3- and p-orbitals. If d-orbitals cont,ribute to these states in an essential manner, i.e. if the d-electrons of the atoms are sigmficantly involved in t,he chemical bonding of the cryshl, then the pseudopotential method becomes problematic, because it,s main advantage in having the pseudoeigenfunctions built up from a relatively small number of plane waves, no longer applies. Thus may occur in trhe rase of TII-V and 11-VI wmpound semiconductors whose cations have flat d-levels as in the case of Zn, for exmaple (see Table 2.2). A completely different approach to solving the oneelectron Schrodinger equation is taken in the so-called cell methods. Cell methods

These methods are 3-dimensional generalizations of the method of matching conditions usually employed in solving thr Schrodinger equation for the square well potential and other 1-dimensional potentials. In cell methods? one first determines linearly independent solutiona of the Schrodinger equation within a primitive unit cell for arbitrary energies. Forming Bloch sums with them, one constructs the solution for the total crystal. The require ments that these functions, and their first derivatives, be continuous at the boundaries of the unit cells determines the energy eigenvalues and eigenfunctiuns. This method suffers from the arbitrariness of t,lie choice of the unit cells and the difficulty of satiseying the boundary conditions over the whole surface, i.s. at an infinite number of points. The most natural choice of unit cell is the Wigner-Seitz cell. A further development, of cell methods lies, in a sense, in the mufin-tin methods. Muf&n-tin methods

In the present method one delimits spheres around the atoms, and leaves some empty space around them. The crystal then looks something like

2.5. Band structure

139

thc method’s narnp. Within the spheres one uses the spherimlly syuunelric potentials of the atoms, trnd in the surrounding regions one takes the potential to be uniform. a muffin-tin, whence

In a special mufin-tin method, t,he augmented plane wave ( A P W)-method, the solutions of the Schriidinger equation within the spheres are expanded with respect to angular momentum eigsnfunctians. The radial parts of the expansion are determined by the radial Schrijdinger equation. This is solved for the various angular momentum quantum numbers numerically. Then, the still unknown expansion coefficients are determined by the requirement that t,he soliit,ions within the spheres mist join continuausly with the solutions outside. The latter are plane waves of some wavevector k K. The functions constructed in this way are called augmented plane wawes {APW’s). They are, of course not eigenfunctions of the oneelectron Schrdinger equation of the crystal, but m a y be taken as a basis for them. In contradistinction to bhe basis functions used in matrix methods, the APW’Ystill depend on the unknown energy eigenvalues. An rigenfunction expansion with respect to APW’s leads, as in matrix methods, l o B homogenwus set of equations €or the expansion coefficients. but the matrix elements are, unlike the Hamiltonian in matrix nielhoads, funcbiom of energy. However, the A PW-matrices are in general smaller, as a consequence of the use of better adjusted basis functions. Often, a linearized energy dependence of the matrix element,s yields useful results in the linearized APW o r LAPW method. If, in constructing the APW’s, Gaussian functions are used instead of the angular momentum eigenfunctions, one speaks of rnuff-tin orbitals (MTO ’s), upon which the MTO and LMTO methods rest.

+

The different APW’s are indexed by reciprocal lattice vectors K, in addition to angular momentum quantum numbers. In another muffin-tin method, called the Komnga-Kohn-Rostoker (KKR)-metho$, which takes advantage of the formal scattering theory of quantum mechanics in the Green‘s function formulation, the expansion functions are also angular momentum eigenfunctions between the spheres.

In band structure calculations for semiconductors, some of the methods listed above are used more frequently than others. Among the ab initio procedures, the pseudopotentid method combined with density functional theory in its local approximation, and lately with the Greens’s function method, is particularly important. In addition, also the APW and LMTO methods are used. mainly in their linearized forms. Of practical importance among the empirical procedures are, above d l , the empirical versions of the tight binding and of the pseudopotential methods.

140

2.6

Chapter 2. Electronic structure of ideal crystals

Tight binding approximation

In the nearly-free-electron approximation, the eigenstates of the one-electron Hamiltonian of a crystal are represented by a superposition of plane waves. The non-diagonal matrix elements of the periodic potential with respect to these functions are treated as a small perturbation. Thus the eigenstates are weakly disturbed plane waves in which lhe electrons are spread out almost uniformly over the whole crystal. Such a distribution might be valid for drctrons of tho conduction band, biit it does not correspond to the reality one should expect for valence electrons if one considers the crystal to be formed from previously isolated atoms. In free atoms, the valence electrons are l o c a l i d at their respeclive atomic cores. Although this localization must be partially breached in the crystal, in order for chemical bonding to take place, an almost complete deloralization such as is assumed in the nearlyfree-electron approximation is not to be expected. This suggests a more appropriate approximation which takcs atomic wavefunctions as the basis set and treats the non-diagonal elements with respect to these functions as small perturbations. This approximation is callcd a tzght b m d m g (TB) appro.mmntzon The errors of this approximation are expected to be small if the valence electrons of the crystal are well localized at the atoms. The approximation of nearly free electrons will work very poorly in this case, while it gives good results if the electrons are weakly localized, i.e. when the tight binding approximation is not applicable. In this sense the two approximations are complementary. Results which are obtained from both these approximations may be considered to be independent of any particular approximation, i.e. to be exact. The term ‘exact’ here means within the framework of the simplifications made earlier. One of these simplifications, the approximation of frozen cores introduced in section 2.1, is particularly important because it allows us to deal with only the valence electrons of the free atoms, while the core electrons are incorporated in the atomic cores. In this section we will develop the basic principles of the TB approximation. These principles will be applied to semiconductors of the diamond and zincblende type, and it will be shown that the TB approximation not only is capable of explaining the valence band structure of these crystals, but it also provides insight into their chemical bonding and atomic structures.

2.6.1

Fundamentals

Atomic Orbitals The basis functions of the TB approximation are the one-particle eigenfunctions of the valence electrons of the free atoms, more strictly, of the atoms composing the crystal under consideration. These eigenfunctions are called

2.6. Tight binding approximation

141

atomic oibatnls. The spinor character of the orbitals may be taken into account, but we will omit it here brevity. The mobt important property of the orhit&, which turns out to be decisive for the TB approximation, is their spatial symmetry. The latter i s determined by the symmetry of the Hartree o i HartrecFock potentials of the atomic cores. These potentials are isotropic if all core shells are fully populated by electrons. In the case of atoms forming diamond and zincblende type semiconductors, this condition is always satisfirul. Thus we may assume isotropic core potentials, and the atomic orbitals of given energy eigcnvalues form basis sets of irrediicible representations of the full orthogonal symmetry group. These representations are characterizd by an angular momentum quantum number 1 which may assume all non-negative integral values. The irreducible representation of a given quantum number 1 is (21 1)-fold degenerate. The 21 1 basis functions are distinguished by the magnetic quantum niimbcr 7n which takes all integei values between - 1 and + l . The energy spectrum of the free atom is degenerate with respect to m. Foi the hydrogen atom there is also a d c generacy with respect to 1. This is not due to the spatial symmetry of the potcntial but to its purr Coulomb form. Here this additional degeneracy may not be assumed. For each value o l lhe quantum number 1 one has an infinite set of different energy eigenvalues. These are distinguished by the mdin quantum number n which may take all intcgcr values horn 1 1 to 00. In this way the energy levels E,l of an atom depend on thc two yuanturn numbers r~ and 1 , and the corresponding eigenfunctions dnlm(x)depend on thrw quantum numbers r ~I ,, m. 'l'ht. eigcnfunctions &,jrn(x) can be written as products of the radial wavefunctions &(I x I) and spherical harmonics Km(xl I x I),

+

+

+

(2.246) Here, it is assumed that the tttorriir core is located iit the coordinate origin x = 0. A wavefunction with the quantum number I = 0 is called an s-orbital, one with 1 = 1 a p-orbital, with 1 - 2 a d-orbital etc. In order to represent the eigenstates of the valence electronb: of a crystal, one needs, rigorously speaking, nll orbitals of the cores of its atoms since only the totality of all orbitals forms a complete basis set in Hilbert space. However, not all of these contribute in an essential manner. 'l'he largest contributions are to be expccted from orbitals forming the valence shells of the free atoms. Within the TB approximation one takes only these orbitals into account, This corresponds to a perturbation-theoretic treatment of the Hamiltonian matrix with respect to the atomic orbital basis; only matrix clcmerits between valence orbitals are considered while those involving other orbitals are neglected. For the elemental semiconductors of the fourth group of the periodic table the valence shell orbitals are formed by the four ns-

142

Chapter 2. Electronic structure of ided crystals

and np-states with n = 2 for C , n = 3 for Si, n = 4 for Ge, and n = 5 for a-Sn. In the case of semiconductors composed of different elements, the valrnce shell orbitals of the various atoms must be considered. For GaAs that means the one 4s-state and the three 4p-states of Ga, and the one 4sand t h e Q-states of As. For GaP one bas, besides the abovementioned 4s- and lip-states of Ga, the one 3s-state and the three Spstates of P. The valence shell orbitals used as basis functions need not, of course, be populated by electrons in the case of the free atoms. For Si, for example, two of the three porbitals are empty. Similarly, the eigenstates of the crystal, which will be calculated later by means of the T B approximation, will also not be completely populated. As this applies to quantum mechanics in general, the eigenstates are candidates for posstble population. Whether they are popdated or not. depends on the macroscopic state of the system, e.g., on the temperature of the crystal In the examples considered above the valence shells of the atoms are formed by s- and p-states. This is the typical case for tetrahedral semiconductors composed of elements of the main groups 11, IV, and VI of the periodic table, but it is by no means the only possibility, especially if one also includes other material classes. For body-centered cubic metals such as Cu and Ni, for example, the valence shells are formed by 3d- and 4sstates. In section 2.1 it was already mention4 that d-states may contribute to the valence shell of 11-VI semiconductors with heavy metal atoms such as Zn. Here, we wiU exclusively consider semiconductor materials for which only s- and p-states need to be taken as basis functions. The corresponding spherical harmonics X,(X/ I x 1) are

Using these harmonics, eigenfunctions &oo, $n.cll,&lo, &i-i may be formed , may use the according to equation (2.246). Instead of drill and d ~ ~ 1 - 1one linear combinations

The latter are also energy eigenfunctions because of the degeneracy of qbn1l aiid 0 ~ 1 - 1 with respect to the magnetic quantum number m. For the sake of uniformity, one sets

2.6. Tight binding approximation

143

Figure 2.16: Polar diagrams of the atomic s-and p- orbitals in Cartesian representation.

The eigenfunctions of equations (2.246) to (2.248) will be referred to as spherical orbitals, and those of (2.249), (2.250) as Carte3ian orbitak. The latter are visualized in Figure 2.16. The quantum numbers nlm of the spherical orbitals, as well as n, 6 , z, y, z of the Cartesian, will be abbreviated by a general index a. The orbitals considered above are orthonormalized, i.e. one has ($a’ I $fZ) ‘af,. (2.251) If the atomic core is not located at the coordinate origin, as has been assumed thus far, but at a particular lattice position R+G, the corresponding orbitals will be denoted by &j~(x). They may be traced back to the orbitals &(x) of atoms located at the origin by shifting their arguments in accordance with

Two different orbitals & f j f R ‘ ( X ) and & ~ R ( x ) with identical values of R’ and R as well 8s of j ’ and j , but different values of a‘ and a, are also orthogonal to each other. For R’ f R or j’ # j , i.e. for orbitals at different centers, no such orthogonality exists. Although the two orbitals are localized in different spatial regions, and the integral over the product of the two, the so-callcd ovcrlap integral, turns out to be relatively small, it may not be n e glected because its influence on the energy eigenvalues is of the same order of magnitude as the matrix elements of the Hamiltonian between orbitals at different centers. The latter elements are essential because they are r e sponsible for the bonding between atoms in a crystal and for the splitting of the atomic energy levels into bands. The non-orthogonality overlap integrals must therefore also be taken into account. This may be done directly, by writing down and solving the eigenvalue problem for the crystal Hamiltonian in the non-orthogonal basis set of the atomic orbitals. This procedure

144

Chapter 2. Ektranic structure of ideal crystds

is, Iiowcvcr, quite inconvenient becaiise the matrix of overlap integrals has to be calculated explicitly and diagonalized together with the Hamiltonian matrix. It is more useful to employ a set of orthoganalizd orbitals by forming suitable h e a r combinations of the q a j ~ ( x )Here, . 'suitable' means that the new orbitals should have t,he same spatial symmetries a s thc original atomic nrhitrals Q U j ~ ( xbecause ), these symmetries are the essential properties that allow the matrix elements of the Ilamiltonian to be reduced to a few const.ants. Consequently, the new orbitals must, likewise form basis sets of irreducible representations of the full orthogonal group, each set heing chara c t w i d by a particular angular morrieriturri quantum miniber 1. That there are indeed linear combinations of atomic orbitals possessing these properties, forms the content of Liimdin,'

&(O)

> E s ( 0 ) - for C and Si,

ii) E3(0) < & ( 0 ) < & ( 0 )

iii) &(O) < &(0)

-

Ge,

< &(o) ru-sn. ~

These relations Inem that the E'c;(O)-levelmoves down with respect to the other two levels as the size of the atoms increases.

162

Chapter 2. Electronic structure of ideal crystals

It turns out that the ordering of the eight energy bands at k = 0 remains the same over the entire first BZ. This is important because the positions of the energy bands relative to each other determine the likelihood of their population by electrons. As already mentioned at the beginning of this section, not all of the bands are expected to be populated, just as the s- and p-levels of the free atom whose orbitals were used as basis functions were not completely filled. Electron population of the ground state of the crystal, i.e. at temperature T = 0 , may be obtained as follows: For a simple band, each k-value corresponds to 2 eigenstates of opposite spin. For a periodicity region of volume Q = G3Ro, the first B Z contains, as does every primitive unit cell of reciprocal space, G3 allowed k -values (see section 2.3). A simple energy band therefore has 2 x G3 states. Four such bands are necessary to host the (2 x 4) G3 = 8G3 valence electrons of a periodicity region. In the ground state of the crystal, therefore, the four lowest bands are populated, and the four highest bands are empty. This means that in the case of C , Si and Ge, E l ( k ) , E2(k), E3(k), E4(k) are the populated valence bands, and Es(k),&(k), E7(k), Es(k) are the empty conduction bands. For a - Sn, El(k) and &(k) form valence bands, together with two of the three bands E2(k), E3(k), E4(k). The remaining band of the three is a conduction band. Since it is degenerate at k = 0 with the highest valence band, the energy gap of a-Sn vanishes. The above band assignment allows us to determine the symmetry of the valence and conduction band states at r.

Symmetry of valence a n d conduction band states at

r

We know that degenerate eigenst ate8 of the crystal Hamiltonian having the same energy value form a set of basis functions for an irreducible representation of the small point group for the wavevector k. For k = 0 this group coincide8 with the full point group of cquivalent crystal directions, whch, here, is oh, The dimensions of the irreducible representations are the same as the degrres of degeneracy of the corresponding energy levels. Therefore, the eigenfunctions for & ( 0 ) and &(O) each belong to 1dimensional representations, and those of E2(0) = &(o) = E4(0) and E s ( 0 ) = E s ( 0 ) = E r ( 0 ) each belong to 3-dimensional representations. According to equation (2.286), the deepest valence bend level Ei(0) ha5 the eigenfunction (l/fi)[Is10)+ I s20)]. In order to determine its transformation properties under the operations of the point group Oh, it is useful to decompose oh into two parts, firstly, the tetrahedron subgroup ‘I>containing only elements which are not involved with an exchange of the two sublattices 1 and 2, and secondly, the remainder of oh which is composed of all elemfxttb of T d multiplied by the inversion. Each of the elements of the second part of 01, exchanges the two sublattices. For brevity, we will term

2.6. Tight binding approximation

163

the latter elements 'exchanging', and the former ones 'non-exchanging'. The eigenfunction (l/&)[ls 1 0 ) + 1 sZO)] for E l ( 0 ) transforms into itself under the action of both types of elements, thus it belongs to the unit representasl0)- I s 2 0 ) ] of the Es(O)-level, the tion rl. For the eigenfunction (l/fi){l transformation into itself occurs only under the action of non-exchanging elements, while a factor -1 is generated in the case of exchanging elements. It follows from the character table of the irreducible representations of Oh given in Appendix A, that this transformation corresponds to the representation

r;.

The upper valence band level E z ( 0 ) = E3(O) = E4(0) possesses the three eigenstates ( ~ / f i210)) [ l 1 220)],( l / f i ) { Ylojl I Y ~ o )(1/]&)[1. ~ 1 0 ) - I d o ) ] . Under the action of the non-exchanging elements of oh. these functions transform like vector components. Inversion, which is part of the exchanging elements. reverses the sign of the vector components, and the exchange of the two sublattices also reverses the sign of the whole eigenfunctions. The three eigenfunctions therefore transform as they w o d d under the action of the corresponding non-exchanging elements. The character table of the irreducible representations of Oh in Appendix A shows that this transformation is characteristic of the representation J&. Similarly, one finds that the eigenstates (1/&)[1x10)+ I d o ) ] ,(1/&)[1 y 1 0 ) + I Y ~ O ) ](, 1 / f i ) [ I zl0j-t I ZZO)] belonging to the eigenvalues E s ( 0 ) = E e ( 0 ) - E7(O), transform according to the irrcduciblc representation rl;. We summarize the results of our TI3 band structure calculations as follows: For crystals which have the diamond structure, i.e. for C, Si, Ge, as a rule, the highest valence band is 3-fold degenerate and belangs to the irreducible representation of the point group oh. The lowest conduction band at r exhibits either a similar %fold degeneracy, in which case it belongs to the repraentation f 1 5 (C, Si), or i t is non-degenerate and belongs t o the representation I 'i (Ge). For LY - Sn, the F2-band lies below the I'g -band, which in this way, is partially a conduction band. These results are illustrated graphically iu Figure 2.21.

r&,

Extension t o semiconductors of zincblcnde t y p e

Band structures of diianioand type semiconductors calculated by means of the empirical TB method reflect the essential features of the real valence bands of these materials quite well. In order to apply the TB approximation to semiconductors having the zincblende structure, the Hamiltonian matrix [Z.ZS3) must be modified as follows. Firstly, it has to be recognized that the between orbitals at the mme center depend on matrix elements E~ and whether the center is an atom of chemical species 1 or 2. This means that two different s- and p-energies have to be inserted into the two 4 x 4 diagonal blocks of the matrix (2.283);ti,t i in the block at the upper left, and e ; , :t

164

Chapter 2. Elect,ronic structure of ideal crystals

r,’

-

6,

-

6, r,’ -

r;, -

r,

-

conduction bands

r;,

-

6 diamond

structure

valence bands

zinblende structure

Figure 2.21: Ordering of energy bands at the center 1’ of the fint B Z for semiconductors of diamond and zincblende type. in that at the lower right. Secondly, the relation V$$ -V:$ G -V3w of equation (2.273) cannot be used in the zincblende case because it rests on the chemical identity of lhe two atoms of A unit cell. For the matrix element l 2 , the s-state belongs to a 1 atom, and the p-state to a 2-atom, while for the s-state belongs to a 2-atom, and the p-state to a 1-atom. Therefore, thr matrix element, V& is given by an independent constant rather than by -V:$, (the parameters iSwin Table 2.8 are associated with by meany of equation (2.284). The other excliangr relations in (2.284) remain valid because they refer to matrix elements whose orbitals at the two different atoms belong to the same state. With these two changes, in equation (2.283) the TB Hamiltonian matrix for zincblende type semiconductors becomes

z;

-espc csF

2.6. Tight binding approximation

165

with ESP= (l/fi)c&,. TTsing this matrix, the band structure and the eigenstates of zincblende type semiconductors may be calculated. YIPspatial symmetry of the eigcnstatcs a t thr R Z rentpr is similar to thtlf, which was found for diamond type crystals above, except that the point group Ott has to he r ~ p l a r dby the tetrahedron group T d . The wprPsPntaliun ri, of Oh thereby becomes the representation r15 of T d , I 'i is replaced by rl, and r1.5 remains ri5. In contrast to diamond type crystals, one has practically only one energetic ordering of the conduction bands here - the rl-band being the deepest. 2.6.3

sp3-hybrids, total energy and chemical bonding

Once the bald stnicture is known, the total energy of the valence electrons ran be calculated. Formula (2.54) of section 2 . 1 indicates how this may be done for the ground state of the crystal. One hw to sum the energy levels of all valence electrons! and subsrquently remove the doubly counted electronelectron Coulomb interaction energy from this sum. The total energy of the vakncr elwtrons of the crystal or, strictly speaking, its deviation from the total energy of the valence electrons of the free atoms which were brought together to form the crystal, defines the energy gain due t o chemical bonding. known as coh.esion energy of the crystal. This definition is reasonable because the valence elect,rons are the only parts of the atoms whose stat= change when the crystal is formed. In order to obtain the total e n e r a in closed analytic form, one needs explicit mathematical expressions for the valence band energies st. all paints k of the f i d BZ. However, the TB approximation in the form developed above produces such expressions only at particular symmetry points. To find them everywhere, one must introduce further simplifications. A starting point for this is a formulation of the 'I'B approximation which employs certain linear combinations of 8- and p orbitals as basis functions, rather than the atomic orhihals of definite angular momentum quantum numbers I ! which were used above. These linear combinations are callcd sp'--hgrhr%d orbitals. In contrast to the atomic orbitals, they are not energy eigenfunctions of the atoms.

sp3-hybrid orbitals

The four hybrid orbitals I h~lR), I h z l R ) , I hslR),I hslR) of a 1-atom in the unit cell at R are defind by the equations

Chapter 2. Electronic structure of ideal crystals

166

23

23

23

23

Figure 2.22: Illustration of the wavefunctions involved in chemical bonding in tetrahedrally coordinated semiconductors: sp3-hybridorbitals (a, b), bonding orbitals (c) and anti-bonding orbitals (d).

1

I h31R) = 5 [dslR(X) - # z l r t ( X ) -tb y d x ) - $zlR(X) 1

1

1

I h d R ) = 5 [ $ s ~ R ( x-) 4 z d X ) - &lR(X) + d'.zlR(X) 1 In Figure 2.22 the probability distributions of the four sp3-hybrid orbitals are shown in the form of polar diagrams. The orbitals resemble clubs pointing to one of the nearest neighbor atoms in the sublattice 2, e.g., I hllR) to atom 21, 1 h z l R ) to atom 22 etc.

2.6. Tight binding approximation

167

Similarly, the four sp3-hybrid orbitals I h12R),I h22R), I h32R), I h42R) at a 2-atom in the unit cell at R, are defined by the relations

These orbitals point to an atom of type 1. The following four orbitals are directed to the 1-atom in the unit cell at R: orbital 1 h12Rl+ R) at atom 21, I h 2 2 R 2 f R) at atom 22, I h32R3+ R)at atom 23,and I h42&+ R)at atom 24. Hybrid orbitals at the same center are orthonormalized with respect to each other, as the s- and porbitals from which they are constructed. If the latter are understood in the sense of Liiwdin, orthogonality also holds for hybrid orbitals at different centers. For each unit cell there are 8 associated hybrid orbitals 1 &jR)with t = 1,2,3,4and j = 1,2, as there are 8 atomic orbitals for the two free atoms of a unit cell. From the hybrid orbitals of a given sublattice one may form Bloch sums I htjk) by means of the relation

(2.291) in complete analogy to the Bloch sums q b ~ involving k atomic orbitals in equation (2.255). Due t o the definitions (2.289) and (2.290) of the hybrid orbitals in terms of atomic orbitals, the Bloch sums I h t j k ) may be thought to arise ~ atomic orbitals by means of a unitary transforfrom the Bloch sums # a 3 of mation. The same holds for the Hamiltonian matrix ( h t j k I H I htrg'k) with respect to the basis I htj k); this matrix may be understood as arising horn the above mentioned unitary transformation of the known Hamiltonian matrix ( a j k I H I a'j'k) with respect to the basis daJk. As such, it has the same eigenvalues and eigenfunctions as the original matrix. The implementation of the TB method by means of hybrid orbitals instead of atomic orbitals having a definite angular momentum quantum number is therefore nothing but the solution of the same eigenvalue problem in another representation. This statement holds, of course, only as long as equivalent approximations are made in the two representations. However, the hybrid orbital representation is well s u i t e d for further approximations, beyond those already made earlier. These approximations facilitate the derivation of simple analytical

168

Chapter 2. Electronic structure of ideal crystals

expressions for the eigenvalues and eigenvectors which may be used to explicitly determine the total energy of the valence electrons of a crystal in closed analytical form. Hamiltonian in hybrid-orbital representation Here, we describe the most important additional approximation available in hybrid orbital representation, which utilizes the fact that those particular hybrid orbitals at nearest neighbor atoms which point toward one another will also overlap one another more strongly than all others. In consequence of this, the Hamiltonian matrix elements ( h t l R I H I ht2 R t R) between these orbitals will be the largest. Approximately, one need only consider these elements, while all others may be neglected. The former elements do not depend on t and R, as can be seen from the explicit expressions for the hybrid orbitals (2.289), (2.290). Their common value, denoted by V2, may be determined from (2.289), (2.290) and equations (2.271) to (2.273) as

+

The corresponding matrix elements (h+lk I H I ht2k) between the Bloch sums of hybrid-orbitals follow from VJ by multiplying this quantity with the factor et eak.dt (2 293) ~

In the casc of Hemiltonian matrix elements between hybrid orbitals at the same center, one has to distinguish between diagonal and non-diagonal ele) thc samc value ments. Thc non-diagonal elements ( ~ Q 1RIf 1 h t , ~ R have for all R and orbital quantum numbers t , t'. For the common value V1 one finds by means of (2.289), (2.290) the expression Vl 5 ( k t j R 1 H

1 ht,jR) -

1

- ( c S - E ~ ,)

4

t

# t'

.

(2.294)

with e3 and ep defined in equation (2.266). Since only nearest neighbor atoms are considered, the rrvult is the same as for the matrix element ( h y k I ZI I h t f jk) between the corresponding Rlocli sums. An analogous result holds for the diagonal elements (&jR 1 H 1 ht3R) at the same center. Their common value ctt is ~h

(htjRI H

1

I h y R ) = -(~g 4

-t-

kp).

(2.295)

Again, this result is the same as for the matrix elements ( h u k 1 H 1 h u k ) between the corresponding Uloch sums. Numerical values for th, along with values of Vi and VL, are listed in Table 2.10.

2.6. Tight binding approximation

169

In writing down the Bamiltonian matrix (htjk I H I hrj’k) in the hybridorbital representation, we arrange the eight basis functions in the sequence I h l l k ) , I hzlk), I h s W , I h 4 W , I h l W , I h Z W , I h32k), I h 4 W . Then the matrix ( h t j k I H I httj’k) takes the form

.

(2.296)

This matrix is also known as the Weare-l‘horpe Uarniltonian. Remarkably, its 8 eigenvalues can be obtained in closed analytical form. Denoting them by E,b, E:, i = 1 , 2 , 3 , 4 ,we havve

Table 2.10: Hybrid matrix elements calculated from the TB parameters in Table 2.7 (in eV).

Ge

-8.27

-1.76

-2.98

-8.37

-2.01

-2.76

170

Chapter 2. Electronic structure of ideal crystals

r

X

wovevector

Figure 2.23: Evolution OF the energy bards of semiconductors with the diamond slructura within the TB approximation The right-hand part shows the band struct,ure of Si

calculated by meam of equations (2.297).

Here gl(k) is the structuredependent factor d e b 4 in equation (2.282). The actual positions and k-dispersions of the bands (2.297) are determined by the parameters th, 1’1 and Vz. Using the values for S i given in Table 2-10, one obtains t h bnrrd ~ structure shown in trhf right hmd part of Figure 2.23. In this regard. the 4 bands indicakl by b lie below the energy gap, arid thP 4 bands indicated by u lie above it. This means that in the ground state of t h e crystal. the &lands are fully populated by electrons, i.c. they form the valence bands, and the a-bands are complekly empty, thus they are the conduction bands. Essential features of these relationships are the negative sign of V2, and the validity of the magnitude relation I V1 1>1 “2 I between the absolute values of and Vz. Figure 2.23 also illustrates how the different bands emerge from the atomic s-and p-1evrlb due to the two interactions V1 and V z . Roughly speaking, Vz determines the distance between the cpnters of gravity of the valence and conduction band complexes, and V1 the width of thpse bands. Now wc return to the main goal of this subsection. the ralcuIatjon of the total energy of the crystal. To accomplish this, the energy values of the four valence bands E,b(k)determined above must be summed over all i and k. It

2.0'. Yi'ghht binding approximation

171

turns out that this task may even be carried out analytically if a suitable additional approximation, the so-called bond orbital approsirnation. is made. Bond orbital approximation To introduce this approximation, we review the already solved problem of diagonalizing the Hamiltonian (2.296). but procwd in a sompwhat different way, The diagonalization will now bt. carried out in two steps. In the first step, the eigenvalues and rigenvectors of the matrix (2.296) are calculated without taking account of the VI-terms, i.e. temporarily setting Vl to zero. As may he scen from formula (2.297). this leads to the reduction of the enerffi bands to the two dispersionless levels givw by

5k) belonging t o q, have the components bik)

O,O),

( l / f i ) [ l ,D,O,O,e;,O,

(2.299)

(htlk 1 b4k) = ( l / J z ) ( O , 0: O , l , O , O , O , e : ) ,

and the 4 eigenfunctions I atk) belonging to

(ht2k I agk) = (l/&)(O,O;

(ht2k I aqk) = ( l / & ) ( O , O , O ,

E,

have the components

1,0,0,0:

-ez,

1.0,0,0,

0),

-.;I.

Each of the eigenfunctions I btk) and 1 atk) is a linear combination of Bloch sums of two hybrid orbitals pointing toward one another, or, equivalently, a Bloch sum of the linear combinations of these orbitals. In the case of &states the hybrid orbitals are added, and the corresponding linear combinations I btR) are given by

172

Chapter 2. Electronic structure of ideal crystah

1

I kR)= - 11 h t l 1/2

R)+ I h2Rt

+ R)], t - 1 , 2 , 3 , 4 .

(2.301)

Thc exponential factors eF of the eigenvectors (2.299) and (2.300) were compensated by the et-factors of the Bloch sums. In the case of a-states the hybrid orbitals are subtracted, and the corresponding linear combinations I a , R ) are given by

The polar diagrams of these functions are shown in Figure 2.22. For reasons which will be clarified later, one refers to the orbitals I h t R ) as bondiny and to the orbitals 1 arR)as anhi-bonding orbitals. With the help of bonding and anti-bonding orbitals the eigenfunctions of the Hamiltonian matrix (2.296) with b1' -- 0 may be written in the form 1

I btk) -

eik'R I b t R ) , 1 - 1 , 2 , 3 , 4 ,

aR

(2.303)

(2.304) In the second step of the diagonalization procedure the Vl-terms are included. The Hamilton matrix (2.296) is transformed iuto the basis set of the previously calculated eigenvectors 1 btk) and 1 atk). This results in the (8 x 8)-matrix

(2.305)

with the (4 x 4) -matrices

2.6. Tight binding approximation

173

{2.307)

as block elements. The expression for If, follows from that €or H & if in the latter q, is replaced by E,. The structure factors g t p and &) in (2.306) and (2.307) are defined as follows:

The (4 x 4)-matrix H M couples the various bonding states, and H,, couples the various anti-bop.ding states. The non-diagonal matrix Hab describes the interaction between the two types of states. It gives rise to corrections to the eigenvalues of relative order of magnitude I 1’1 I /2 I V2 I. Considering the actual values of V1 and V2. these corrections are rather small. This suggests treating them as perturbations. The zero-th approximation, i.e. the complete neglect of the interaction between bonding and anti-bonding states, is referred to as the b o d ovbital approximation. Within this approximation the valence and conduction bands follow from separate eigenvalue equations, the former by diagonalizing the matrix Hbbq the latter by diagonalizing Hw. To calculate the total energy of the crystal one needs the total energy of all valence electrons. T h e latter may be calculated by means of formnla (2.54) which expresses the total energy of an interacting electron system by means of its one-particle energies. One has

Er$&f’= 2

E:(k) k

-Ecd

(2.309)

z

where E d means the Coulomb energy of the interacting valence electrons which is counted twice in summing upon all band states. The factor 2 accounts for the two spin states. Within the bond-orbital approximation, the i-sum in (2.309) may be carried out in closed form, with the result

(2.310) To pro>*ethis relation, we write the eigenvalues @(k) in (2.310) as diagonal elements of H M between eigenstates, and note that within the bond orbital approximation the eigenstates for a given wavevector k are linear

174

Chapter 2. Electronic structure of ideal crystals

combinations of the bonding states I bik) only. In this matter, the linear combinations are generated by the unitary transformation which diagonalizes the Hermitian matrix H a of (2.306). If one sums the eigenvalues @(k) over all i , and takes advantage of unitarity, then the diagonal elements of Hw with respect to the eigenvectors become the diagonal elements of this matrix with respect to the bonding states, i.e. q,, as is stated in relation (2.310). One may also prove this relation in another way, using Vieta's theorem, which states that the sum of all zeros of a polynomial of degree n equals the negative of the coefficient of the (n - 1)- th degree term. In the case of the characteristic polynomial of a matrix, this coefficient repre sents the negative of the sum of all diagonal elements, here, therefore, -4q, confirming the validity of equation (2.310).

Total energy and covalent bonding We proceed on the assumption that the atoms are arranged, as above, in the form of a diamond type crystal. The distance d between the nearest neighbor atoms will now be chosen, however, to have a value different from the do-value of the actual crystal. The total energy of this fictional diamond type crystal represents a function EEfAt"l(d) of d. We will demonstrate that E Z Z t a Z ( d )reaches its absolute minimum at the finite distance d = do. This result constitutes a theoretical proof that the atoms bind themselves into the form of a crystal, and that the nearest neighbor distance will have the experimentally observed value. This does not prove the correctness of the diamond structure of the crystal, for that was assumed a priori. To verify this, one must also show that no other crystal structure can yield a lower total energy minimum. We will not address this question, but rely on experience, which indicates that elements of the fourth group of the periodic table, under normal conditions, crystallize into the diamond structure. Our considerations here have the sole purpose of understanding why, in this structure, the total energy reaches a minimum at a finite distance d = do. In other words, we want to understand why chemical bonding should occur at all between atoms of group IV. The total energy E;f$a'(d) of the crystal is composed of the energy of the valence electron system in the field of the atomic cores, as well as the energy of the atomic cores and their mutual electrostatic interaction energy. For the total energy EE$;f'(d)of the valence electrons of a crystal with G3 unit cells, one obtains from the relations (2.309), (2.310) and (2.298) the value

(2.311)

To get the total energy of the crystal, the energy of the atomic cores must be

2.6. Tight binding approximation

175

added to the energy value of equation (2.311). In doing so, one may again use the fact that the core states of the crystal do not differ from those of the free atoms. This means that only the mutual electrostatic interaction of the cores results in a structuredependent energy contribution, while the internal core energies sum to a constant Eo. The core-core interaction energy has, approximately, the same value as the electron-electron interaction energy E d between the valence electrons on different atoms. This is true because the valence electron charge of an atom equals its core charge for the crystals considered here. The corecore interaction energy approximately cancels, of valence electrons therefore, against the negative Coulomb energy -E,1 in expression (2.311). Finally, the total energy of the crystal is given by

(2.312) The d-dependence of this energy is due to the fact that both Eh and Vz depend on d - the hybrid energies Eh have d-dependence as they are defined by the diagonal matrix elements of the Hamiltonian H between Lowdin orbitals which contain overlap integrals between s- and p - orbitals of adjacent atoms, and V2 because this quantity is the matrix element of H between hybrid orbitals at nearest neighbor atoms. V2 decreases with decreasing distance d, corresponding to an attractive force between the atoms. The hybrid energy q, increases as d decreases, corresponding to a repulsive force. For large d, the attraction dominates over the repulsion, and for small d, the repulsion dominates over the attraction. Overall, the total energy E z l b l ( d ) of the crystal varies with d as shown in Figure 2.24 schematically. At the equilibrium distance do, it takes its absolute minimum value. This means that the initially free atoms will not remain free but form a diamond type crystal with nearest neighbor distance do. They experience what is called covalent chemical bonding.

In order to provide a better physical understanding of the nature of covalent chemical bonding, we compare the total energy E g t p l ( d ) of the crystal with the total energy E f $ T of 2G3 free atoms. For the elements of the fourth group of the periodic table with their two electrons in atomic €,-levels and two in atomic cp-levels, one has

(2.313) where Eo, again, accounts for the energy of the atomic cores. The negative difference of the two energies (2.312) and (2.313) represents the cohesion eriergy of the crystal. It is given by the expression

176

Chapter 2. Elecbronic structure of ideal crystals

Figure 2.24: Dependence of energy difference Eza?''' ( d ) -E::+Ts on f,he inter-atomic

crystal atom Etotol -Eta.ral

distance d (schematically).

Formally, the occurrence of a positive cohesion energy is due to the fact that the matrix element V2 o l H bctween hybrid oxbitals at adjacent atoms pointing toward one another is negative, and that I 4v2 I exceeds the energy differericc (cP - cs). The latter difference may he understood as the energy increase of an atom if one of itb two .+electrons i s lifted into a p-state or, equivalently, if its four valence electrons a x put into four sp'-kybrid orbitals rathe1 than into two s- and two p-orbitals. One calls this population the promoted configuration of the atom. In sp'--hybrid states, the electrons of adjacent atoms are capable of pronounced interference. This can be constructive or destructive, depending on whether bonding or anti-bonding states are considered. In the casc of constructive interference, the probability amplitude becomes relatively large in the region between the two atoms and the two electrons of the interfering sp3-4ybrjds undergo a delocalization (see Figure 2.22). In this process, the potential energy of the Coulomb interaction of the two electrons among themselves and with the atomic cores remains almost unchanged. However, their kinetic energy decreases considerably. This may be imderstood in terms of the Heisenberg TTncertainty Principle which tells us that a weaker localization, i.e. a larger positional uncertainty, corresponds to a smaller momentum uncertainty and, therefore, to a smaller kinetic ene~gy. Altogether, the energy of the two electrons decreases, b e cause of constructive interference, in a bonding state. The energy gain per atom amounts to 4 I Vz 1. If it exceeds the energy necessary for promoting an atom into its sp'-state, i.e. if the condition 4 I V2 I> ( e p - F ~ )holds, it is energetically favorable for covalent chemical bonding to occur. As we have seen, quanturti rtieclianical phenonienology is essential in the interpretation of this behavior. Unlike the bonding of electrically diflerent charged ions, covalent bonding between neutral atoms Lannot bc understood in terms of classical physics.

The c:ondition necessary for thc occiirrcnce of bonding eigenstates able to host all vrtlprrcp electrons is the ordering of the newest, neigIi1,ors of an atom on the comers of a tetrahedron, i.e. the diamond structure of the crystal. In this way thc above consideration also justifies focming on the tetrtlhedral crystal structure of diamond type crystals, which was merely assumed at the outset. The atomic structure follows, so to speak, from the electronic structure.

Ionic bonding The ionic contributions to chemical bonding will now be calculated for mat,+ rials having the zincblende structure. A s is well-known, a scries of 111-V and 11-VI compound semiconductors form crystals of this type. For the Ilamiltonian matrix (2.305), the transit,ion from the diamond to the zincblendc structure means that ch in the upper left (4 x 4)-block has to be replaced by the hybrid energy of the 1-atom? and in the lower (4 x 4 )-block by the hybrid energy e i of the 2-atom. With this replacement, the bonding and anti-bonding energy levels become

EL

!+,

~8.

where =c i Thr energy separation betwren the two levels is larger than that of diamond type crystals. This results in an enlargement of the energy gap betwcm the valence and conduclion hands. The bonding and anti-bonding oibilals arp given by the expressions

+ bp.

where we set a p - Vs/dV; T h e f d o r s {1/2)(1- a p ) and (1/2)(1+ctp) in ('2.316) and (2.317) represent the probabiMes of finding an elrctron in the bonding state at atom 1 or 2, respectively. One calls aP the polarity uj bonding orbitals or simply the polardty of bonding. If f f f is deeper than c i , ! then I!?, and also up,art' positive. The electron prpferpntially stays at atom 2. In this way the polarity of bonding orbitals is such that, in the ground state, whcre the electrons occupy only boxrding orbitals, the previously electrically neutral atoms heconie charged. Atom I becomes the positive cation, and atom 2 is the negative anion. The charge of the cation is given by c Z * with Z' = (Z, - 4 t $ap),where 21 is the number of valence electrons at

178

Chapter 2. Electronic structure of ideal crystals

thc free 1-atom. The anion charge is -e(Zz - 4 - 4 a p ) = - e Z * , i.e. the unit cell is neutral. Owing to this redistribution of electron charge, the electron-electron interaction energy to be subtracted from the sum of oneparticle energies, because of double counting, takes a different, value. It is, therefore, no longer completely compensated by the electrostatic interaction energy between atomic cores. This leads to an additional contribution to the total energy of the crystal which may be interpreted as the electrostatic interaction cnergy between anions and cations. Onc calls it the Madelony energy E M & - The general expression for E M a d iu

where the sum extends over a periodicity region. With energy of the crystal is

EMad,

the total

The Madelung energy is negative, i.e. it strengthens chemical bonding. Since the bonding is then pdrtially due l o attractive forces between ions, OTW refers to it as partially zonzc bondzng. The absolute value of the Madelung energy is, on the one hand, proportional to the number GT of unit cells, and on the other hand, inversely proportional to the distance d between two adjacent ions. One therefore sets

(2.320) with cy as the so-called Madelung constant. The latter depends on crystal structure and can easily be calculated numerically. In Table 2.11, the n-values are listed for crystal structures which are observed in materials composed of group IV elements as well as 111-V, 11-VI and I-VII compounds. The value for the wurtzite structiirp in Table 2.11 corresponds to the ideal tetrahedral case with an equivalent cubic lattice constant &a (see Chapter 1). The contribution of the Madelung energy to the total cnergy of a given compound will be larger for larger effwtive charge number Z* of the compound. This results in a tendency of compounds with larger 8* valiirs to crystallize in structures with Madelung constants larger than that of the zincblende structure. Therefore, in passing from the 111-V through the IIVI to the I-VII compounds, one observes a transition from the zincblende structure through the wurtzite to the rocksalt and cesium chloride structures. The cesium chloride structure follows from the rocksalt structure by replacing the two facecentered cubic sublattices by two primitive cubic sublattices, shifted in the same way with respect to each other as in the rocksalt

2.7. k . p -method

179

Table 2.11: Madelung constants for several crystal structures.

Zincblende U’urtzite

1.6381

Rocksalt

1.7476

1.6410

structure, i.e. by ( a / 2 , a / 2 , a/2). With growing polarity of the bonding, the energy gap becomes larger, as mentioned above. This explains the transition from the semiconducting properties of the group IV crystals to the insulating nature of the I-VII compound crystals. In the case of the I-VII compounds, the absolute values of V3 are so large in comparison with V2 that the bonding polarity op is approximately unity. This implies that almost all valence electrons of the compound stay at the anion. Then the crystal consists of positive ions of the group I atoms, which have lost all their valence electrons, and negative ions of the group VII atoms whose valence shells are completely filled. One refers to such crystals as tonac crystals. In this case, the energy gain due to the transfer of electrons from cations to anions, which represents an essentia1 part of the bonding energy and forms the driving force for the formation of ions, no longer depends on the crystal structure. This structure is determined by the hladelung energy only. Therefore, ionic crystals exhibit structures with particularly large hladelung energies, i.e. rocksalt and cesium chloride structures.

2.7

k .p -method

2.7.1

Fundamentals

Luttinger-Kohn functions The k. p-method rests on a particular property of the BIoch type eigenfunctions pV,+(x)of the crystal Hamiltonian H . As we know these functions (which will be denoted below by (xluk) instead of q y k ( x ) ) are the product of an exponential factor exp(ik.x) and the latticeperiodic BIoch factor uvk(x). If one replaces the wavevector k in uYk(x) by a constant ko, while retaining k in the exponential factor, then the resulting functions

1x0

[2.321) are no longer eigenfunctions of H of course, but they do form a complete orthonarmalizPc1 basis set in Hilbert space, as wcll as the Bloch functions. whence ~

(v’k’ko(vkko)- 6 v l u 6 k ~ k ,

(2 322)

~ ( x ’ l v k k o ) ( v k k o l x--) b(x‘ - x).

(2.323)

uk

Thp vdidily of these relations f d o w b directly from the rorripletenyss and orthonormality of the Bloch functions. The (xlvkko) are referred tu as LuftmgGgP7- K u h fiLa~t7onu.T h y arp determined by the Blorh factors u V k ( x ) for the special wavevector ko in contrast to the Bloch functiom which require full knowledge u l u,k(x) lor all wavevectors k. The k . p-method takw advantage of this properky of the Luttinga-Kohn functions. In this method, one represents the Srhriidinger quation for R crystal electron in terms of the complete orthonormaIized set of these functions. The rpsulting matrix elcments of H can be expressed, as we will s e e later, by the matrur elements of H between the Bloch factors Uyk(X) for k = ko. These elements arc, of course, just as little known as the Bloch factors themselves. However, one may take them as empirical parameters. If one does so and inserts values for the parameters. then the Hamiltonian matrix in the Luttinger-Kohn basis is completely determined. Uiagonalizing this matrix yields the eigenvalues and eigenfunctions of the crystal Hamiltonian H for all valiies of k. Tllis means that the k p-method allows 011rto calrulate, from the Bloch matrix elements at only one point kol the eigenr-alues and cigenfunctions over the entirc first €32,i.e. to extrapolate from thc particular point ko t o t h e entire first BZ. ()€ten one is only interested in solutions in the vicinity of a critical point k,e.g. in the vicinity of the valence band ninxinium or the coridurtion band minimimi. Then it is expedient. although not necessary, to icienliljr ko with k,. If k, hes. for example, at the center of the 6rst BZ,as often occurs. one has ko = 0. This choice will be used later. At the outset, ko should still be considered an arbitrary point of the first HZ. In order to accomplish the ahove program. we expand thP Bloch functions (xjuk}with respect to Luttinger Kohn functions (xjpk’ko). On\y terms with k’ = k occur in this expansion because of the lattice translation symmetry of both functions, whence

2.7. k . p -method

181

With this expansion, the Schriidinger equation (2.178) in the Luttinger-Kohn representation bmomes

k .p

- Hamiltonian

‘l’hematrix elements (pkko I H I p’k‘ko) of the llarniltonian bptween T,uttingpr Kohu functions can he t r a d back to matrix elements (pko I p j p’ko) ol the momentum operator p between Bloch functions. if one uses the easily provcn commutation relation [p2,eik-x] = eik-x (p2 f 2fik. p + h2k2) ,

(2.326)

which yields

where we have set

E:(k) = E,(ko)

Ti2 + -(k 2m

-

ko)2.

(2.328)

The matriv on the right-hand side of (2.327) allows for an important rewriting. If one defines Hk.p(k) =

Ho(k)

+ -nah(k

- ko) . p!

(2.329)

with

(PkkolfflP’kko) = (PkoIHk.,(k)lP’ko).

(2.33 1)

The latter relation means that the actual Hamiltonian matrix W in the kdependent Luttinger-Kohn basis Ipkko) equals the representative matrix of a fictional k-dependent Harniltonian Hk.p(k) in the k-independent partial Bloch basis lpkoko) = jpko) for the wavevector k = ko. The k-dependence of the Luttinger-Kohn basis on the left hand-side of equation (2.327) has been transferred to the new Hamiltonian Hk.p(k) on the right-hand side. The SchrGdinger equation (2.325),with this new Hamilt onian, reads

182

Chapter 2. Eiectronic structure of ideal crystals

The components of the eigenvectors Ivk) in (2.332) refer to the LuttingerKohn basis Ipkko), although the operator Hk.p(k)is represented in the Bloch basis Ipko). Solution of the Schrodinger equation (2.332) involves the diagonalization of the matrix (pkolHkE,(k)lp'ko). For k = ko this matrix is automatically diagonal, by virtue of the fact that Bloch functions Ipko) are eigenfunctions of the Hamiltonian Hk.p(kO) = Ho. For k # ko, the Ipko) states are no longer eigenfunctions of Hk.&), so that the matrix (pkolHk,(k)Ip'ko) has off-diagonal elements with respect to the band indices. Formally. one may interpret these non-vanishing elements as arising horn an interaction between different bands. Since this interaction results from the (k - ko) . p-term in Hk.p(k), one calls it the k . pinteractian. In this, the bands which are mutually coupled, are not bands in the sense of the eigenvalues of the actual crystal Hamiltonian H - the latter are uncoupled by definition - they are fictional bands E:(k) defined by equation (2.328). As the point k in Hk.,,(k) approaches ko, the k . p-interaction tends to zero. For k-vectors sufficiently close to ICO, one can treat this interaction with the help of quantum mechanical perturbation theory. Apart fkom the square term in (k - ko) already present in E:(k), this entails a power series expansion of the energy bands E,(k) with respect to (k - ko) about the point ko. The form of the perturbation theoretical expansion depends on whet her the unperturbed bands, i.e. the eigenvalues E,(ko), are degenerate or not. We will first consider the simpler case of non-degenerate bands.

Application to non-degenerate bands. Effective masses

In first, order perturbation theory the eigenvalue E$(k) arising from EE(k) is given by the relation

(2.333) and the Bloch function Ivk)' arising from 1.k)'

E

Ivkko) by the relation

Since the f m t derivatives VkE,(k) of the exact band energies E , ( k ) at ko depend only on linear expansion terms of E,(k) in k - ko,no approximation is needed to obtain the relation OkE,(k)lk, = VkEb(k)lb. Considering

2.7. k . p method

183

(2.3331, this exact relation yields V&v(k)lb = ( h / r n ) ( v b l p l v h ) . This holds the same content as equation (2.193) used above without proof, because ko may be an arbitrary point of the first B Z . In particular, if ko is a critical point kc, i.e. if V&,(k)\k,, = 0 holds, then the k .p-correction vanishes in first order perturbation theory. One must proceed to the second order to get a non-vanishing contribution from the k p-perturbation. The result reads

where, for brevity, we set (I& sion in the form

I p I pkc) = (v 1 p I p ) . We rewrite this expres-

Generalizing the terminology introduced in section 2.6, we call M L 1 the effective mass tensor at the critical point k,. For the diagonal elements of At;' with respect to the principal axis system, one obtains from (2.337) the relation

mz'

This relation connects the effective masses with the matrix elements of the momentum operator between different bands and with the energy separation of bands at the critical point. The tendency indicated is that the absolute values of the effective masses become larger for smaller momentum matrix elements and larger band separations. One expects small effective masses for large momentum matrix elements and small band separations. As far as the band separations are concerned (only for them can one make an easy estimate), we will later find conhmation of this tendency in all concrete cases. For pairs of bands which are closer to each other than to all other bands and, therefore, whose mutual interaction is stronger than that with all other bands, relation (2.338) allows one to also draw a conclusion about the signs of the effective masses. According to it, the energetically higher of the two bands should have a large positive effective mass, and the energetically lower a mass of the same large absolute value but of negative sign. This

184

Chapter 2. Electronic structure of ideal

CFZ.S~~~S

conclusion also proves to be valid in all cases in which the assumptions of this calculation apply.

Band degeneracy Critical points are often symmetry centers or lie on symmetry lines, and at these symmetry points, degeneracy of the energy bands often occurs. If this happens, one must carry out second order k.pperturbation theory €or degenerate bands. In quantum mechanics, perturbation theory for degenerate energy levels is c o m o n l y of first order - the matrix of the perturbing Hamiltonian operator between the degenerate states has to be diagonalized (we remind the reader of the nearly free electron approximation in section 2.4). This procedure does not apply here because the perturbation matrix at critical points vanishes in first order. One must therefore choose a variant of perturbation theory for degenerate energies which works in second order. To this end, one constructs the matrix of the perturbation operator not between degenerate unperturbed eigenstates, as is commonly done. but between the (also degenerate) eigenstates of first order of perturbation theory. By diagonalizing this matrix one obtains the eigenvalues in second order perturbation theory. These are, in general, no longer degenerate. An important case in which the k . p-perturbation matrix between the degenerate unperturbed states vanishes, is the valence band maximum of semiconductors with diamond structure. This case will now be investigated. In doing so, w e initially neglect the spin-oIbit interaction. This approximation is valid for semiconductor materials composed of light elements only, including, for example, Si. For other materials this procedure serves as a zero-order approximation which can be used to proceed further (as we will do below).

2.7.2

Valence bands of diamond structure semiconductors without spin-orbit interaction

As we know from section 2.3, the valence band maximum of diamond type semiconductors is located at the center r' of the first B Z . Therefore, we set & = 0. The maximum is 3-fold degenerate. We denote the three pertinent Bloch functions by IvmO), where m can assume the values 2,g,2. According to section 2.6, these eigenfunctions belong to the irreducible represent ation l?b5 of the cubic group oh. As indicated in Appendix -4. a basis of this representation is formed by the products yz, zx,z y of the components 2,y, z of position vector x. Therefore, with regard to their transformation properties under the action of elements of o h , we may identify IvzO) with yz, IvyO) with zz, and IvzO) with zy. The vector components 2,y, z of the position vector itself transform in accordance with the irreducible representation I'15 of Oh.

2.7. k . p -method

185

For the subgroup T d of oh the two representations F15 and r h 5 coincide. For semiconductors having the zincblende structure, the three degenerate states ( v I O ) , IuyO), IuzO) of the valence band maximum may therefore be associated with I,y, z insofar as their transformation behavior is concerned. In the case of the diamond structure, 2,y, z are merely a short hand notation. The vanishing of the matrix (vmOlplum'0) of the momentum operator between valence band states at I?, anticipated above, may easily be demonstrated using the pertinent criterion for such vanishing given in Appendix A: The operator p transforms according to the irreducible representation I'15 of oh. The matrix (umOlplvm'0) therefore belongs to the reducible representation x r 1 5 x rl,, = I?;, x (rh ri2 I'15 r 2 5 ) , wherein the identity representation does not occur. According to Appendix A this means that the matrix (umOlplvm'0) must vanish. One can also obtain this result by means of inversion symmetry alone. We have chosen the somewhat more troublesome method of proof because it may also be applied in other, less obvious cases, as we will see immediately below.

+ + +

k . p -perturbation theory t o second order with degeneracy

In order to apply degenerate second order perturbation theory, the solutions of Schrodingers equation (2.332) are needed to first order in the k . pperturbation. For the orthonormalieed Rloch valence band eigenstates (vmk)l one finds

(2.339) where we set E,(O) = for brevity, and the degenerate valence band energy E J O ) is denoted by Ev. The third term in (2.339) guarantees the normalization. (:onsidering the sum on p , tlir value 11 = 71m does not need to be specifically excluded because the matrix elements ( p I p I vm') for p = 117n vanish anyway. Expressions of tlic form (2.339) also hold for the approximate Bloch functions Ipk)' of the remaining bands p with p f urn, but we omit an explicit presentation of them here. The states Ivmk)' and Ipk)' with p # t m will now be used as a basis set to represent the IIamiltonian H . The resulting matrix is 'almost' diagonal, because the basis functions are 'almost' eigenfunctions. In particular, the submatrix of the three velence bands is coupled to the remainder of the matrix only by elements of second order in the k . p-perturbation. These elements give rise to corrections of the valence band energies which are only of third order and can be neglected.

186

Chapter 2. Electronic structure of ideal crystals

In second order perturbation theory, the valence band energies Ev and Forresponding Rloch states I&,) therefore follow from an rigenvalue equation which is decoupled from the remaining bands, namely '(vmklH!vm'k)l l(wn'kl&:y)= E ' , '(vrnklE,).

(2.340)

rn,

The initial occurrence of interaction betwren the valence band and the r e maining bands is incorporated in the matrix (u>mk I H I wn'k) in first order perturbation theory. Harniltonian m a t r i x The (3 x 3)-Hamiltonian matrix '(vmklHlvm'k)' of equation (2.340) can be obtained by means of expression (2.339) for the perturbed states Ivrnk)'. A short calculation yields

where

is a fourth-rank tensor. Since the states I p ) = 1 P O } are eigenfunctions of H with eigenvalue EEl= E,(O), one may write (2.342) in the more compact form

With respect to the indices a r P lthe tensor D z i , is symmetric, and with respect to the indices m , m' it is IIermitiaa From equation ('2.345) one can see that D Z k , transforms under symmetry operations of the cubic group Oh according to the $-fold product representation [rb5 x ri5]a x [1'1~ x l ? 1 ~ ] , where the index s denotes the fiyrnmetrical part of the product. According to Appendix A, then contains as many independent elements as the

L?zk,

number of times the identity representation occurs in the product [I?& x riE;lrn x [I'Is x I115Is. Using Appendix A , one finds [I'15 x rl5Is = [I& x r&lS = r1 r12 ra,, which yields [rk5x r& x [rI5x rl& = 3r1 rz 4r12 3c,-k 5T'k5 The tensor D z L , therefore has three independent components. one can show that these correspond t o the three types of non-vanishing

+ +

+ +

+

2.7. k.p -method

187

matrix elements D z z , Ll;& and D Z . We introduce the abbreviations L D g , hl = D$$, and N = D$ f D g . The elements L, M and can be calculated if the Bloch factors are known. In the absence of this informalion, however, we consider L , M and N to IIPempirical parameters (as indicated at the outset) and use their connection with the Bloch factors only to identify some general properti=, such as the fact that they can be chosen real. Since Q B vanish, the Hamiltmian matrix of the all remaining matrix elements Dmm, valence band has the form

Method of invariants The Hamiltonian matrix (2.344) can also be derived in a somewhat different way, whirh leads to the goal more quickly, but is formally more dernanding. One uses the fact that the Hamiltonian matrix l(vmklHlum'k)l can be represented as a linear combination of the 9 matrices of a basis in the product space 1 vmO)(iirn'O 1 which transforms according to the repreaentation [I& x r:,] of the point group oh. This representation is reducible. By decomposing it into i t s irmliicible parts, m e obtains a basis which consists of subbases, each of which belongs to a particular irreducible representation of Oh. Such a matrix basis can easily be constructed by means of the 3dimrnsionel angular niomenturn matrices Iz,Iv,Iz (considered in Appendix A ) and their products, since it is known how these matrices transform, namely according to the pseudovector representation I':s. In the product spacc k,f-fi of the components of the vwtor k, one proceeds in a similar way. One determimes a basis from subbases which transform according to the irreducible parts of the representation [I'15 x r15Iy.The Hamiltonian matrix reprewrits an element in the product space of Ihe two spaces which is invariant under transformations of the point group Oh. Such invariant elements of the product space can be produced by forming scalar pruducts of subbases of the two spaces whi& transform according to the same irreducible repre sentation. As seen in Appendix A, the corresponding scalar products belong to the- identity representation, that is to say, they are invariant. To find the most general Hamiltonian matrix compatible with the symmetry oh, one has to determine all invariants of the product space. I.€ one then multiplies each by a real scalar factor and sums them all, one obtains

188

Chapter 2. Electronic structure of ideal crystals

the most general invariant of the product space and thus the most general Hamiltonian matrix compatible with Oh symmetry. This process is called the method of invariants. It is applicable to arbitrary symmetry groups and degrees of degeneracy, and it quickly leads to the goal if one considers spin and spin-orbit interaction. It also allows one to determine the matrices for perturbing Hamiltonians other than that of the k . p-interaction, such as the interaction between the angular momentum of Bloch electrons and an external magnetic field (see section 3.9) or the interaction with mechanical strain. In this book, we will only use the method of invariants occasionally. A comprehensive outline of the method with several applications i s given by Bir and Pikus (1974). Valence b a n d s t r u c t u r e The eigenvalues of the matrix (2.344) form three valence bands E,l(k), E,n(k), E,3(k). For the three symmetric k-directions [loo], [lll]and [110] the dispersion curves are determined by simple analytical expressions as follows:

E,lp(k) = M k 2 ,

1 E,lp(k) = - [ L 3

+ 2M - N]k2,

E,3(k) = L k 2 ,

1 E,3(k) = - [ L 3

+ 2M + 2 N ] k 2 ,

1

2

+ M + N]k2,

(2.346)

(2.347)

E,l(k) = M k 2 , E,2(k) = - [ L

(2.345)

1

E,3(k) = - [ L 2

+M - N]k2.

(2.348)

Along the two directions [loo] and [lll], the valence band, being triply degenerate at r, splits into two bands, one 2-fold degenerate and one nondegenerate (see Figure 2.25). In the [llO]-direction and also for all more asymmetric k-vectors, no degeneracy remains. This indicates a %fold splitting of the valence band for such k. All bands are parabolic, but evidently, in general, not isotropic. One speaks of a warping of energy bands. In the case of Si, one has L = -5.64, M = -3.60, N = -8.68 in units of ( h 2 / 2 m ) . Using these values, the two degenerate bands E,l/a(k) of equations (2.345)

2.7.

k . p method

189

Figure 2.25: Valence band dispersion for diamond type semiconductors in the vicinity of the I'i5-maximum for different k-directions. or (2.346) have smaller curvatures than the third band E,s(k) in these equations. Thus the first two bands correspond to the heavy holes and the third band to the light holes of Si. Isotropy exists only if N = 0 and L = M . Then, there also is no longer any distinction between light and heavy holes. Conversely, anisotropy grows stronger as the difference between the masses of the two types of holes becomes larger. The results discussed above were obtained without consideration o l spinorbit interaction. However, for most of the diamond and zincblende type semiconductors, the valence band structure is significantly influenced by this interaction (in the case of Si it is small, hut often not negligible). We now proceed to consider the effects of spin-orbit interaction.

2.7.3

Lut t inger-K ohn model

Including of spin-orbit interaction To include spin-orbit interaction the electrons must be treated as particles with spin. The electron states then depend on both the position coordinate x, and on the spin-coordinate 3, and the quantum numbers of the electron states contain provision for both the motion in coordinate and spin state u describing the spin motion. A set of basis functions for representation of the electron states in coordinatespin space may be formed lrom the Bloch functions (xlpk) or the Luttingw-Kohn functions (xlpkO) without spin, augmented by multiplication with the eigenfunctions ( s l u ) of the z-component of the spin operator. Here u means the spin quantum number which can have the two values u (spin up) and cr =I (spin down). We denote the product functions by (sxlpk) or (sx(pkO),respectively. Then, by definition,

-r

190

Chapter 2. Electronic structure of ideal crystah

we have

The total Hamiltonian of the system is obtained by adding the spin-orbit interaction H, to the Hamiltonian H in its absence, where Hgo is given by equation (2.56) as (2.3501

One has to be aware that the (sxlvuk) are eigenfunctions of H , but not of H H,. Correspondingly, the (xlvuko) signify the spin-dependent Luttinger-Kohn functions of H , but not of fi + f l W .T h e matrix repre senlation of the Schrtdinger equation with respect to the spin-dependent Luttinger-Kahn basis reads

+

x(prkOtH

+ H,, Ip’v‘kO)(p’u’kOIE l , ) - E , (pu kO IE,) .

(2.351)

pW

In calcuhting the matrix of H t H,, of (2.351)) an additional k-dependent term appears in comparison with the spin-less case, as a consequence of the fact that H , contains the momentum operator. This term has the same form as the (A/rn)k. p-term arising from H . except that the p-operator is replaced by the operator p+(1/4rnc2)[5 x OV(x)]. The additional term can be taken into account by replacing the operator Ht.p of equation (2.329) (for ko = 0) by the operator Hk.x = H o ( k )

with ii = p

1 + ---[a 4m c=

Tl + -k . a, m

(2.353)

x VV(X)].

With this the Luttinger-Kohn representation of H (pukOlH

(2.352)

+ H,,

becomes

+ H80/p’a’kO) = ( ~ u O I H+~ Hso{p‘u‘O), .~

(2.354)

and the Schrodinger equation (2.351)) takes the form

C ( ~ ( T O I H+ ~HsoIp‘u’O)(p’u‘kOIEu) .~ = E,(pnkOlE,).

(2.355)

P’U’

Up to this point, we have kept the discussion general. Now we wish t o explore the particular consequences of spin-orbit interaction for the previously

2.7. k p method

191

considered valence band states. To this end, we need the matrix elements (vrnuO~Hso~wm’u’O) of H,, between the spin-dependent Bloch states Ivm 0 . 0 )namely, ~ F,

( u ~ u O I H , , I ~ ~ ’ U ’=O ~) ( v v ~ [VV(X) O I x p] Ium‘O) x (aldld). (2.356) 4m c

To evaluate the matrix element (vmOl[OV(x) x p]]vm’Oj in coordinate space we make use of crystal symmetry, as was done before, in the calculation of the matrix elements of the momentum operator p. The operator [VV(x) x pi is a pseudovector and transforms according to the irreducible representation ri5of oh. The entire third-rank tensor (vrnOI[VV(x) x p]lvrn’O) therefore belongs to the reducible representation I’b5 x Ti, x P2,= rl,, x (r2 r12 T’i5+I7L5), in which the identity representation occurs exactly once. The tensor (vmO ! [VV(x) x p] I vm’O) consequently contains one independent constant. This constant coincides with the matrix elements (vyOl[VV(x) x pIz I v z O j = (vzO I [VV(xj x plXI vyO) = (vzOI [VV(x) x p]ylvzO), as well as

+

+

(~.Ol[V(X)XPl*l v y 0 ) = (WYOI [ w x ~ x P l . / ~ = ~ (oV Z) O l PV(XjXp1ytvrO). where (vxOl[VV(x) x plz I u y O ) = -(uyO I [VV(x) x p],lvzO) holds. Because the Bloch factors are real? these elements are pure imaginary. We denote the value of (vyOl[VV(x) x p],jvzO) by (4m2c2/h)(i/3)A, i.e. we set h

4m c

A

(2.357)

,(uyO~[VV(x)x plrlvzO) = i - .

3

Below, we will see that the constant A is the energy splitting of the valence band at J? due to spin-orbit interaction. Applying equation (2.357) and the explicit form of the spin matrices given in (2.57),the matrix (avmOlH,I a’w m ‘0j becomes 0 - i

(vmaOp,,Ivm.’u’O)

A 3

=-

0

0 0

0 0 - 4

1

i

0

0

0

0

0 - 1 i

0

0

0 - 1

o i

0

0

0 - i - i o

0

1

i

0

0

0 0

(2.358)

Here the rows end columns are associated with the basis functions in the sequence Ivz T O j = Iz t), Ivy 7 0 ) = ly tj, . . ., It)z 1 0 ) 3 1s 1). Spinorbit interaction couples orbital states and spin states to each other. At k = 0 the expressions E:(k) and k ii are both zero. The eigenenergies and eigenfunctions of the total Hamiltonian Hk.a H , are therefore also those

+

192

Chapter 2. Electsonic structure of ideal crystals

of H,, alone, so thsl the Schrfidinger equation (2.355) at

c

k - 0 becomes

(vmaO~H,,lvm'a'O)(vm'a'O~E) = E(wamO(E).

(2.359)

u'm'

Eigenfunctions at

r. Angular momcntum

basis.

The matrix (2.355) has a 4-fold degenerate eigenvalue Ev1/2/3/4 =

1 p

(2.360)

with the four corresponding eigenstates, respectively,

IE )

~

v1 -

IEv3) =

1 T ( 1 ,i, o,o, O,O), IEv2)

Jz

-

1 i z(l, - i , O , 0 , 0 , 2 ) , IEv4) = -(O,

Jz

O,O, 1, - i , O ) ,

(2.361)

and a 2-fold degenerate eigerivslue

--A

2 3

(2.3 62)

i 0, 1, 1, i, o), 1E ) - ----(--I, i, o,o, 0, 1). w6 - d3

(2.363)

Ev5/6 -

with the two corresponding eigenstates

1

IEv5) -

-(o, fi

The components of the eigenvectors given by (2.361) to (2.363) refer l o the IwmaO), using basis functions IvmaO). If we abbreviate these by Ima) I z T), J y I), 12 I), Iz J), ly I ) , Iz i), the cigcnvcctors take the form

The eigenvectors lEvt),i - 1,2, . . . , 6 of (2.361) have a simple meaning. The lEvl),IEvz),IEv3), IEv4) are basis functions of the irreducible representation of the cubic group o h . Acrording to Appendix A, these representations

2.7.

k p -method

193

emerge from the representation 213 of the full rotation group if, D Dis~taken as a representation of the subgroup oh with +1 for inversion. It has also been

shown that the basis functions of this representation are the simultaneous 5 €or the eigenvalue j ( J eigenfunctions of the angular-momentum-squared ' 1) (in units R 2 ), and of the z-component J z of J for the eigenvalues m j = 23 , 12 ., - 1 . =L and -$ One therefore denotes the first four eigenfunctions of (2.364) by

+

4.

33

I--) 22

31 + iy t), I--) 4 22

1

= -1rz

3T

I--) 22

=

1

-[& I.

- iy

T)

i

t) + la: + iY 1 1 1 7

= -"-2Iz

+2/2

fi

33 i = 1.22

111%I--)

Jz

-iY

(2.365)

I),

The lEvs),IE,s) are basis functions of the irreducible representation I'y of o h . These representations do not arise from any representation V3 of the full orthogonal group. in particular not from a representation for J = (this happens with in the case of Oh or r:) in the case of T d . but the expectation values of J2 and J z are the same as those in the Dl basis. Therefore one also uses the angular momentum notation €or the last two eigenfunctiona of (2.3641, i.e. one sets

4

rt

Each of the dgenvectors (2.366 and (2.366) is determined only up to a phase factor, which is chosen heie such that the states with negative total angular $I,): 1 follow, respectively: from the states with positive momentum, ?I):, total angular mornenturn, I$$), ,);I ) ; ; 1 by means of time reversal, i e . by forming the comphx conjugate of the original eigenvector and subsequently multiplying it by rP. We refer to the functions Ijmj) of (2.365) and (2.366) henceforth as the angular momentum basis. According to (2.360) and (2.3621, eigenstates having the same eiggmvalue of the the angular momentum squared, J2,also have the same energy eigenvalue, while the rncrgy eigenvalues differ if states with different eigcnvalues of J' are consitirrrd. The valence band: being B-fold degenerate at the r-point if spin is not taken into account, therefore splits into two bands, one with j : and one with j = if the spin-orbit interaction is considcrccl. That such a .ynin,-o)rhitqditting must occiir? one can recognize just by means of a grnnp theoretical analysis of the problem. The six valence hand states at transform in accordance with the &dimensional representation Dl x , ;'I of Oh. Tlicse representations are reducible, accord-

4,

3,

r

ing to Appendix A , as Di x T

= Y$

+ I';.

The size of the splitting is given

194

Chapter 2. Electronic structure of ideal crystaJs

by the constant A, which determines the strength of the spin-orbit interaction. Therefore A is called spin-orbit splitting energy. One may interpret A as the difference of the spin-orbit interaction energy between the states with 3 and those with j = j = 3 As one should expect, the states with larger angular momentum lie energetically above the states with smalIer angular momentum. States with different m j , i.e. with different projections Jz of total angular momentum on the z-axis, but the same J2,have the same spinorbit interaction energies. Therefore the degeneracy of these states remains.

i.

Valence band structure off r

The above statements refer to valence band states at the center r of the first B Z , where the k p-interaction vanishes. Off this interaction is no longer zero and must be taken into account in addition to the spin-orbit interaction. We have seen how this can be done approximately in the preceding section, without consideration of spin. The method used there indicates the following procedure in the presence of spin and spin-orbit interaction: One determines the functions Ipok)' which diagonalhe the operator H k q of (2.352) in first order perturbation theory. In analogy to equation (2.339), one finds for the valence band states Ivmcrk)' the expression

r,

where we use the same abbreviations as in (2.339). Analogous relations hold for the states of the other bands. The functions Ipgk)' form a complete orthonormal set in t a m s of which the Hamiltonian H may be represented. The submatrix with respect to the valence band states (vmmk)' is decoupled from the remainder of the matrix in second order perturbation theory. Since H,, also only couples the valence band states among themselves, but not to states from other bands, the Schrdinger equation (2.355) in this representation reads

The spin-dependent term of k .7i in the eigenstates Ivmcrk)' of (2.367) d e scribes the change of the k .p-interaction due to spin-orbit interaction. '5'ince t,he two interactious are supposed to be weak, this change i s second order small. It will be omitted below. Then we have, approximately,

2.7. k p -nrt.thd

195

9

(vmakIH,,lv7n’a’k)’

(vmaOlH,qolvm’a’O).

Now we use the fact that H,, is diagonal in the angular momentum basis Ijmj) of (2.365) and (2.366). It is clear that this basis follows from Ivma0) E Imu) by a unitary transformation

(2.370)

~ j m j ) C U r r m j m j Ima). ma

The corresponding unitary transformation matrix UmjmJ obtained from the rclations (2.365) and (2.366). One has

can be readily

(2.371)

If one applies this transformation to the Schrodinger equation (2.355), then the matrix (vmaOlH,,(vm’a’O) takes the diagonal form

(hjlff,olj’m;)

=

o $ o o o o + o o o o g 0

0

0

0

-

0

0

0

0

0

0

9

0

’

(2.372)

196

Chapter 2. Electronic structure of ideal crystds

The sum of the two matrices (2.372) and (2.373) is the new Hamiltonian matrix. It has the same eigenvalues as the original matrix, even though its form deviates from that of the original. The difference in form is, above all, that the new matrix is already diagonal at k = 0. Kon-diagonal elements occur for k # 0. Among them, the elements between basis vectors ] j m 3 )and [ j ' r n $ ) with m3 # m i , , but = 3' play a different role than the ones with .f j'. While the influence of the 2-diagonal elements on the eigenvalues is independent of the size of the spin-orbit splitting A, it does depend on it for the j off-diagonal elements. The magnitude of the latter can be estimated as the larger of the two terms Nlki2 or IL - Mllkl'. If one assumes that l L f a ~ ( ~ VIL, - AIl}lk12 /i$ holds. Describing of l h e valence band of such semiconductors by means o l the Luttingrr-Kohn model would entail treating the effect of the remote spin-orbit-split band exactly, while taking iuto account only the energetically closer conduction band by means of perturbation theory. Such a procedure is not meaningful and one must seek a different, more appropriate description. A model which is tailored exactly to such circumstances is the Kana model, which we will now discuss. In this matler, we asbume that the point group of equivalent directions is the tetrahedral group ‘rh, and no longer the cubic group o h as above, thercby encompessing both typeb of semiconduclois, those of zinrldende type EM well as those of diamond type. In the latter case, inversion symmetry still has to be added. This involves a spwialization of the results, which may be casily done, should thc need arise.

2.7.4

Kane model

The Kane model is based on the following assumptions. P’zrstly, it is assumed that the k. p-interaction of the valence band with the deqest conduction band at 1’ is so strong that it must bc treated exactly. Secondly, at r the spiuless valcnce band should have the symmetry T15,and the spinlesu conduction band should have the symmetry rl. This assumption corresponds to the situation which actually exists in semiconductors of Llncblendc type. Thirdly, the interaction of the valence and conduction bands with all remaining bands (referrcd to as ~ e m o t is~ )assumed to be srrrall, so that it rriay be treated by perturbation theory, similar l o the Lultinger-Kohn model in which the interaction of the valence band with all other bands was treated in this way. Here, we will simplify further and neglwt this interaction COHIplelely. Tn addition, we will also neglect the direct k .p-inleiaction among the three r15-valence bands which, as mentioned above, does not rigorously vanish for zincblenrle type crystals, giving risr to k-linmr terms in the Hamiltonian. It turns out that the latter approximation is valid in most cases.

2.7.

k.p -methad

201

Neglect of interaction with remote bands We analyzr the generally valid Ychrbdinger equation (2.351) using the assumptions and approximations discussed, above. considering spin md spinorbit interaction horn the outset. We again denote the thrw valence band indices by urn, ni - I.u, z. The conduction band index c will be augmented by 3 . idirating the 3- or rl-synlm&-y of the conduction band state at (the coincidence of this notation s with that of the s p h variable 9 is unfortunate, but unavoidable, and the reader should keep the dxerent meaning of s dearly in mirid to avoid confusion). The pertinent spin-dependent Bloch functions at k - 0 are IvmuO) and I c s v O ) , rmpectively. The matrix elements of the term Ho(k] of f f k X are given by

r

(vmaO)Ho(k)jvm'dO)= bmm~6umi-k ri2 2 ,

2m

h2 (csdlHo(k)lcsdO) = 6,,)-k 2m (wmOlHo(k)IcsO)= 0.

2

,

(2.382) (2.383)

(2.384)

Iii the otkw o p ~ a t o rterm of A V ~ .namely ~, {Fa/m)k-.?i,WP may neglect thc spin-dqwndcnt part by virtue of the same arguments as in the Luttingerkohn model. The three needed matrix elements of this operatw may then be determined as

(2.385)

The matrix elements (vmOtplzlm'0) of p between the valence band states are wglectctcd in accordance with the assumptions madr above. The diagonal rlemenl (csOlplcs0) of p in the conduction band state IcsO) vanishes exactly. The matrix elements {cs01plvmO) between valence and conduction band states tranuform according to the produrt representation r1 x I115 x r15 trls i l725. Since the unity representation is contained in it = Fl + exactly once, the matrix does not haw to vanish; it contains exactly one independed rlcnwnt. As such, one may chose ( c s O l p , l v d ) and set it equal t o z ( m / h ) P . The nun-vanishing matrix elements of p are then given by the relation

202

Chapter 2. Electronic strr1c:ture of ideal crystals

m

~ c s o ~ p , ~=7~~r r8o~~~ p v = ~ I( cv syo~/ p ) , ~ ~= ~ ti .F,-o- )~

(2.386)

The factor 2 giiarantws that P i s real, if the Bloch factors are real as we assume. The other factor (m/Ta)was introduced for convenience in the final dispersion rclations. With the moinentiim matrix elements of (2.3861, thp Hamiltonian matrix (pu01Hk ,(k)lp’u’O) takes the form (2.387)

Ec

,iPk,

iYk,

0

0

0

0

--iPk,

0

0

0

0

0

0

-iPkg

0

0

0

0

0

0

-iPk,

0

0

0

0

0

0

0

0

0

E C

iPk,

iPk,

iPk,

0

0

0

-iPk,

0

0

0

0

0

0

-iPk,

0

0

0

0

0

0

-iPk,

0

0

I]

where thr lines and columns arp ordprcd in the sequence I2

/s

I), la I), Ig I),

th I. 1). lz 1), IY 11, 12 1).

Finally ihe matrix demerits (puO1Hsolp‘m‘O) nl t h e spiri-orbit interaction operator ITrn haw to lie detrrminpd. For the Flj-vaknrc band rlernents (wmcrOlH,,lwm’a’O), one can adopt the results which were formerly derived for the valence band. because l’i5 coincides with I’15 for the tetrahedral group. There are new matrix elPrnents (vmuOlH,lrsa’O) and (csuQIH,oj cscr‘0) involving the conduction band states. The coordinatedependent factor of the first matrix element transforms according to the representation r15 x l?25 x I’l - I’z+ I’15 r25. The unity representation does not oc cur here, thus this factor vanishes and with it the whole matrix element (umaOIH,,I csa‘O), whence

+

(um.aOlHs”Icsrr’0)= 0.

(2.388)

The caordinatedeyendrnt factor of (csuOlH,olcsdO) belongs to the product representation rlx r 2 5 x rl : rztj arid must therefore likewise wnish,

2.7. k . p -method

203

Thus, the total Hamiltonian Hk.T representation

+ H,,

is gken by the following matrix

, Ec

iPk,

iPk,

iPk,

0

0

0

0

-iPk,

0

-ig A

0

0

0

?

-iPkg

i$

o

o

0

0

O 0

-iPk,

0

0

0

0

--

i+

o

0

0

0

0

Ec

iPkz

iPk,

iPk,

0

0

0

-A 3

--iPk,

0

o

0

-2z

-iPkY

-+

i+

.A

0

0

.A

O

-iPk,

0

0

0

0 \

0

0 a 3

2 5

$

.A

-27

Here, the order of rows and columns is the same as in (2.387). For k = 0 and vanishing Ec, this matrix reduces to the spin-orbit interaction opcrator WSw If one rearranges the rows and columns of this matrix in such a way that those relating to the conduction band states 1s 1) and 1s J) occur in the left upper corner, side by side, then the matrix decomposes into a (2 x 2)block for the conduction bt~ncl,and B (6 x 6)-block for the valence band. The eigenfunctioiis of the two blocks arc simultaneously also e i g e h c t i o n s of the total matrix. T h {2 x 2)-co,nduction band block is already diagonal, i.c. 1s t) and 1s J] are eigenhnctions of the Hamiltonian matrix (2.390) at k - 0. The (6 x G)-vdencr band block is identical with the matrix of the spin-orbit interaction operator H,, of (2.350) for the Luttinger-Kohn model. The eigenfunctions at k - 0 here are therefore also the vectors 1$ms) and i$rn

1)

T

of ihe ctngpubr momentum basis (2.366).

The latter basis should be particularly suitable for solution of the eigenvalue problem for the Hamiltonian matrix (2.490) at k # 0. The matrix (2.390) is, however, so simple, indeed, that one can also obtain the secular equation directly. We will do t h s . before we h r t h w consider the angular momentum basis. To diminate the free elmtron part ( h 2 / 2 m ) k 2from the eigenvalues Elk), we write them in the form h2 2m

E(k) : E’(k) -I--k2 where E’(k) satisfies the following

(2.391)

204

Chapter 2. Electronic structure of ideal crystals

Secular equation

(2.392) The fact that the two factors in round brackets appear squared, signifies an at least 2-fold degeneracy of all eigenvalues. The reason for this is. again, time reversal symmetry jointly with spatial inversion symmetry (we remind the reader that the term of the Hamiltonian which can break inversion symmetry in the case of Td-symmetry has been neglected). Accordingly, one has in general four 2-fold degenerate bands Ei(k).Ei(k),E$(k),Ei(k). It is also noteworthy that in the secular equation (2.392). k enters only in the form of k2. This means that all four bands are isotropic, in contrast to the LuttingerKohn model where a warping of the valence bands occurs. In the case of k = 0, the energy levels of (2.392) are given by

One may draw conclusions from these expressions in regard to the meaning of the four energy bands E,(k): El(k) is the J?s-conduction band. E z ( k ) 2nd Es(k) are the two upper degenerate rs-valence bands at r. and E4(k) corresponds to the spin-orbit-split rT-valence band. The energy separation EL of the r6-conduction band and the I'g-valence bend at. r is obtained as

(2.394) As long as E: is positive, it represents the energy gap E , at I?. The case of negative E,' is discussed below. For one of the two upper valence bands - the one which arises from the vanishing of the first factor of the secular equation (2.392) and which is denoted by i = 2 - the energy El(k) does not depend on k. For E$(k).a k-dependence follows with finite negative curvature, as we will soon see. Thus E;(k) corresponds to a band of (infinitely) heavy holes, and Ei(k) to a band of light holes. If one adds the (Ti2/2m)k2-term, then the band Ez(k) displays a positive curvature. It is relatively small because of the large free electron mass, but the positive sign contradicts what is to be expected for a valence band. This unexpected prediction for E z ( k ) results from the fact that the interaction of the valence band with all remaining bands, except with the deepest conduction band, was completely neglected. In order to treat the heavy hole band correctly, the interaction with remote bands must also be considered at least by perturbation theory as in the Luttinger-Kohn model. This will be done below.

205

Interaction with remote bands

According to the assumptions made at the outset. the interaction with r e mote bands is weak and may be taken into account by means of second order k. p perturbation theory. The 8 x 8 Hamiltonian matrix of cquatiori (2,390) for the conriuclion-valence band complex contains two 4 x 4 blocks of definite spin with rows and columns referring to the conduction band s state and the three 2--,y--, z-valence band statrs without spin, respectively. To include the inteTactjori with remote bands. RII additional 4 x 4 matrix o€ second order in k has to be added to each of these 4 x 4 blocks. Since the interaction bctween conduct ion and valrncc b a l d states contributes alrpady in first order, second order corrections occurring at $2--,sy-, s z - , and zs--, ys-, zs-positions may be omitted. For the ss-element and the 3 x 3 valence band submatrix, second order corrections beconie import ant. Their genpral forms follow from symmetry arguments as above. The correction of the ss-element may be written in the form Ack2, with A , a constant. The perturbation correction to the 3 x 3 valence band submatrix has the general form of the 3 x 3 matrix in qiiation (2.344) with parameters L , M. N &fin-1 like the matrix elements D E , D g in equation (2.341), however, with the conduction band excluded from the summation over { t bewuse this band is not reinotc. The 4 x 4 matrix block thus determined is added to carti of the two 4 x 4 diagonal blocks already present in the 8 x 8 matrix (2.390). Finally, the whole 8 x 8 matrix is subjected to a unitary transformation into a basis set in which the spin-orbit interaction part of the Hamiltonian becomes diagonal. The latter requirement is evidently satisfied by a basis which, as in the Luttinger-Kohn model. contains the 6 angular momentum eigenfunctions I$:), I$$), I:!),,):;I I%+). I;$), and in addition the two conduction band states 1s 1) and 1s 1). This corresponds to a 8 x 8 unitary transformation matrix composed of a 2 x 2 unity matrix block for the two conduction band states, and the 6 x 6 matrix block from equation (2.371) for the six valence band states. Carrying out the unitary transformation one obtains the general 8 x 8 Kane Hamiltonian which applies to any diamond or zincblende type materials, including those which are already well described by the Luttinger-Kohn model. However, even in these cases the Kane model is more precise than the Luttinger-Kohn model, because the valence-conduction band interaction is treated exactly rather than approximately like in the Luttinger-Kohn model. If one uses the Kane model in cases in which the Luttinger-Kohn model already works well. one has to be aware that generally the parameters 71, 7 2 , 73 have different Values in the two models since those of the Luttinger-Kohn model contain the valenceconduction band interaction while those of the Kane model do not so. The general 8 x 8 Kane Hamiltonian is given by rather lengthy expres-

206

Chapter 2. EIwtronic structure of ideal crystals

sions. To avoid these below, the 4 x 4 block matrix of the remote band interaction will be reduced to a special case before proceeding further. We put L = M -- A,, and N = 0, which means physically that the remote bands affect heavy and light holes in the same way, and do not disturb the isotropy of the bands. -Ordering rows and columns in the sequence 1s t), 1s l),I);:, I;$), . . . , I$$), the transformed 8 x 8 Hamiltonian matrix with simplified remote band interaction becomes

U

0

aP,

0

U

D

-iP-

0

V

0

0

0

0

-fiP-

0

0

0

D

0

-i$&P,

0

v

0

0

O

0

V

0

0

0

0

W

0

0

0

0

U'

-i&

-&

(2.3

+

where the notations l'h = (1/&)P(kz f ik,), Pz = Pk,, U = Ec A&', V = (1/3)A A,k2, and W = -(2/3)A A,k2 are used. The eigenvalues of this matrix follow from the secular equation

+

+

2

E'(k) + -A i 3

- A,k2

I-i

E'(k) +

3

This equation only differs from equation (2.392) in that the factors whose vanishing define the conduction and valence bands contain, respectively, the additional terms A,k2 and A,k2. Solution of the secular equation in limiting cases

The zeros of the h s t factor in (2.396) yield, as seen previously, the I's-band of heavy holes (i = 2). However, the dispersion relation for it now reads

A 3

En(k) = -

ii2 + A , k 2+ -k2 2m

(2.397)

2.7.

k.p -metbod

207

By choosing a negative value for of appropriate magnitude A,, the dispersion for heavy holes can be brought into agreement with experimental findings. The zeros of the second factor in (2.396) determine the dispersion of the Fa-conduction band ( i = l ) ,the rs-band of light holes (i = 3), and the spin-orbit-split I'T-band (i = 4). For the conduction band, the dispersion is changed due to the A,k2-term, and for the two valence bands due to the A,k2-term. But, here, these corrections are added to already existing strong dispersion terms. We will therefore neglect them in the following, as we neglected the weak dispersion duc to the free electron term (li2/2m)k2 earlier. Then the eigenvahe equation for the three bands i = I, 3,4reads

- [E'i(k)

+

$1

P2k2 = 0.

(2.398)

This equation will be solved approximately in three limiting cases with re sped to the order of magnitude relations between the energy gap E i and spin-orbit splittiiig energy A, as w ~ l las with respect to the s i g n of E,'. namely firstly for EF >> A, Eg > 0, secondly for EF -CK A, E; > 0, and thirdly for IEiI < A, E i < 0. The significance of a negative value of E i will be discussed while treating the third case. All three cases actually occur in zincblende type semiconductors, as a look at Table 2.12 immediately shows. The first case corresponds to materials with wide energy gaps whose valence band complex could be described just as well by means of the LuttingerKohn model; the second case refers to semiconductors with narrow energy gaps; and the third to materials whose energy gaps vanish.

Case 1: E: z E , >> A,

EF

>o

We consider energy values Ei(k)in the various bands with energy separations IEi(k) - &(O)( from the respective band exbrema which are small compared with EF. For such energies, the conduction band El(k) approximately obeys the equation

[El(k)- E,]E, - P 2 k 2= 0.

(2.399)

and for the two valence bands E3(k) and E4(k) we have

bi(k) -

$1

bi(k)

21

+ TA

E,

I:

+ [&(k) + -

From (2.399) it follows immediately that

P2k2 = 0, i = 3 , 4 . (2.400)

208

Chapter 2. Electronic structure of ideal crystals

(2.401) and from (2.400) we obtain

1 A E3/4(k) = -2[3

+

P2 -k E,

A

P2

2

8

2]*ij[?f-Kk2]+m'

(2.402)

Under the condition (P2/Eg)k2 A, ( b ) E g 0, ( c ) E , A,E;>O

rdl)

1.(F,/P2)

(3/2). (E,/P2)

rs(3)

-(3/2) . ( E g / P ' )

-(3/2) . ( E S / f " )

rd4) 2.8

-3

'

(E,/P2)

ELedefect, and G U Afor ~ the Ga-antisite defect. ~

Interstitials, vacancies and antisite defects are structural point perlurbations, or point defects. The compositional point perturbations, i.e. the

229

3 . 1 . Atomic structure of real semiconductor crystals

Ideal crystal A -Atom

4

0 B-Atom Impurity atom

Substitutional impurity (S)

SA

SB

Vacancy (V)

Interstitial ( I 1

Interst. impurlty

A,-Antisite ___

B,-Antisite

e)

Figure 3.1: Illustration of the most important point perturbations in semiconductors using the example of a crystal with two atoms per unit cell of the same chemical element (left-hand side) and different chemical elements (middle and right-hand side).

230 Chapter 3. Electronic structure of semiconductor cxystals with perturbations

Table 3.1: Electron corifiguratiori of main group elements. In the rightrnost column the respective closed shells are indicated.

impurity atoms, need to be further specified. This will be done next. Classification of impurity atoms

The division of elements into groups, which is commonly used in chemistry, also proves to be helpful for the classification of impurity atoms in semiconductors. This is not surprising because the incorporation of an impurity atom in a crystal indicates a more or less strong chemical bonding. We summarize this group division of chemical elements below. The periodic table consists of two types of groups of elements, the main groups, and the transition groups. Of the first 98 elements, 50 belong to the main groups and 48 to the transition groups. The elements of the main groups in Table 3.1 have in common the feature that electron shells with angular momentum quantum numbers 1 2 2 either do not occur at all or, if they exist, they are completely filled or completely empty, i.e. no partially filled shells of this kind occur. The energetically highest, and thus in general, only partially filled shells of these elements either have 1 = 0 or 1 = 1. Therefore, they are s- and p-shells. Because of the relatively large spatial extension of s- and p-shells in comparison with d- and f-shells, the former are simultaneously also the outer shells of the atoms which are responsible for chemical bonding. One thus also speaks of sp-bonding elements. The rare gas elements are special cases, in which the s- and pshells are also completely occupied. For the elements of the transition groups presented in Tables 3.2 and 3.3, the shells with 1 2 2 are energetically the highest and, thus, in gen-

3.1. Atomic structure of real semiconductor cry&&

231.

Table 3.2: Electron configuration o€ transition dernent,s. In the rightmost column the respective closed shells are shown.

n

Iron m o w ziSc =Ti 3d4~3~ 3d248'

23V

3d34s2

z4Cr 3d5&

wMn 3d"4s2

mF'e 3d'4s2

z7cO

Ni

3d74s2

3d64aa

3a23p6

era1 they are the not completely occupied oms. Because of the relation R 2 I 1 between the main quantum number ?a and the angular mornenturn quantum numbe1 E. the d-shells (I - 2) are possible only for n >_ 3. the f-shells ( 1 - 3) only for n 2 4, etc. Accordingly, one has the shells 36,&, 4f, 5d, Sf, ;1g, 6 4 Sf, 69, fih ptc. Since among the first 98 elemmts of

+

Table 3.3: Electron configuration of rare earths and actinides. In the right column the respective closed shells are indicated.

Actinides

232 Chaptcr 3. Electronic slructure of semiconductor cr,ystals with perturbations

the periodic tablc, however, the 5g-shell already remains unoccupied, only d- and f-shells are to be considered, namely the d-shells 3 d , 44 5 d , 6 d , and the f-shells 4f and 5 f . The filling of the 3d-, 4d- and 5d-shells takes place in the series of transztzon metals (together with the Elling of the 4s-, 5s- and 6s-shells). Among the transition metals, one distinguishrs the iron groixp in which the 3d- and 3s-shells are being filled, the palladium group in which the samc happens with the 4d- and 4.s-shells, and the platinum group where the 5d- and 5s-shells are being Elled. The 4f-shells are being filled in the rare earth elements, and the 5f-shells in the actinides. In comparison with the s- and p-orbitals, the d - and f-orbitals have a smaller extension in spacc, they lie mostly within the s- and p-shells of the same main quantum number

'l'herefore, mainly s- and p-electrons are involved in chemical bonding. This explains the remarkable chemical similarity of thc rare earth elements with each other, a n d a certain similarity of these elements with the elements of the main groups.

7 ~ .

Complexes of point perturbations; associatos Just as atoms arc bound in molecules when it is energeticaqy advantageous, point perturbations also associate if the result is a state with lower energy. They are called poznt perturbatzoa complezeu or u8souutP8. Bonding can occur between various point perturbations: between chemically identical or different impurity atoms, between impurity atoms and point defectu, and among point defects themsclvcs. The associates may consist of two or of several constituents. The diversity of these associates is comparably large to that of molcculcs in chemistry. We give some examples below. Donor-acceptor pairs

In the case of an ionized donor and an ionized acceptor, the lowering of total energy through the formation of a bound complex of associtlteb is particularly obvious the two point perturbations are differently charged and attract each other through electrostatic forces. This leads to the formation of donor-acceptor paws, in which the donor and acceptor atoms occupy neighboring sites in the crystal. In general, the pairs are stable at several possible distances, whirh gives rise to a variety of different donor-acceptor pair complexes. Di- and multi-vacancies

When there are two vacancies, the mechanism for the formation of bound pairs can also be easily undertitood the (internal) surfacc of thc crystal i s reduced if two previously isolated vacancies move together to occupy neigh-

3.1. Atomic structure of red semiconductor crystah

233

boring cryYtal sites. One caIls this associate a ddvacancy. Analogous atatements holds for the association of more than two vacancies, which are called multrvarunczes.

Frenkel defects

If, in a crystal, an atom moves froin a regular site to an interstitial site, then it h v e x behind a vacancy which attracts the interstitial. Thus a defect pair is formed in this process which consists of a self-interstitial and a vacancy. It is called a Freibel defprt. There arr important puint perturbation complexes which occur only in a specific material or matrrial group. We now consider some examples for si snd G A S . Point perturbation complexes in Si

Wr various reasons, hydrogen is often present in Si. In p-type Si, H atoms undergo chemical bonding with the availablc wreptoi atoms. Thereby the electron of the H atom is captured by the acceptor atom: which becomes singly negatively charged. One may also conclude that the acceptor expends i t h hound hole to the H atom ralher than to the valence band because it has lower energy there. In B-doped Si, for example, it negatively charged B-ion and ti positively charged H-ion are formed in this way. The two ions attract each other by- Coulomb forces, which results in the formation of a neutral ( H . B )pair. Of course, the pair will not be able t o arcept an clcrtron which means that the B atom has lost its ability to act as an acceptor. Chalcogen atoms like S, Se, and Te are incorporated in S i not only as single atoms, but tlBo as two-atom molecules. Oxygen in Si enters into bonding with a vacancy, forming a pair which probably constitutes the so called A-renter known from caparity 111ea~iremmtfi.Oxygpn is also involved in a wries of other defect complexes in Si, among others the so-called thermal d a m r s . which are thusly named hPraiise of their origin in thwmal treatment. Point perturbation complexes in G a A s

A prominent defect associate in GaAs is the so-called RX-center, which acts as a donor. It is found in GaAs and also in (Ga, A1)As mixed crystals under appropriate conditions (e.g., high pressure, for more see section 3.5). Originally, the DX-center was attributed to a donor atom, like S&, bound t o another point perturbation whose nature was unhiowii at t!hHt time and, therefow, was denoted by X. Currently, the DX-center is thought to be due to the donor atom alone, more strictly; to a donor atom which is incorporated interstitially but not substitutionally, as commonly happens. Another

234 Chapter 3. Hectronic structure of semiconductor crystals with perturhations typical defect associate in GaAs and other Ill-V semiconductors is the complex formed by an As-antisite defect (a Ga atom on an &-site) associated with an As-interstitial. Now, this complex is believed to form t h e care of the so-called b;L!Lcentei- in GaAs. This center manifests itself as deep level having strong influence OR the electrical and optical properties of GaAs (more detail on the EL2-rcnter may br found in section 3.5). Point perturbation aggregates

I€ the complexes formed

by point perturbations bmome larger and reach mesoscopic size,one refers to them as aggw.qates. Complexes of macroscopic size such as, for example, oxygen or heavy metals in Si cryshls, are called precipitates.

Latticc relaxation

The forces on atoms in the irnmediatc vicinity of

a point perturbation differ frorri those in lhr ideal crystal. They are non-xr?rol in general, at. the ideal crystal sites. Thus the atoms are forced t.o move t.u new equilibrium sitcs. This is known as lattice relamtion (in Figure 3.1 this effect is omitted). The new rqiiilihriiim sites are initially tinknown. In principle, they can be determined by means uf atomic structure calculatious for t h e perturbed crystal. These have t o be performed sirnultanmusly with calculations of the elect,ronic structure, just as is done in self-consistmi. calcidations of the electronic and atomic st.ructwes of ideal crystals described in Chapter 2, section 2.2. HoweveT, there is an impwtant difference between the two cases. For ideal crystals, the calculation of at,omic structure may be avoided since, for t,he latter, complete and reliable experimental data are available. However, in regard t,o the atomic structure of a crystal in the vicinity of a point perturbdion, in many cases, hardly more t,hari t.he symmebry is known from experiment. Thus the self-consktent calculation of the electronic and at.omir:sbriictures must actually be carried out. if lattice relaxat,ion becomes important. Experimental information concerning the symmetry of lattice relaxation may be derivrd from observation of t,he Jahn-TelEev efleet. This eflect results in splitting of the degenerate energy levels of a point perturbation due to a symmetry-lowcring latt.ice relaxation. In the case of a varaiicy: for example, the lattice rehxation can reduce the original tetrahedral symmetry T d to tetragonal symmetv D2d. Such spontaneous symmetry breaking occurs when it leads to lowering of the total energy of the crystal. This mag; happen when, in the unrelaxed state, i.he point perturbation has a c1rgenerat.e level which is only parhially occupied. First. of all, the degenerate level will split off due to the symmetry lowering relaxation. According to perturbation theory,

3.1. Atomic structure of real semiconductor crystals

235

this splitting proceeds such that the center of gravity of the levels remains unchanged. Thus, along with levels shifted up, there are also levels which are shifted down. If only the latter are predominantly occupied, then the energy of the electrons localized at the point perturbation decreases. This energy reduction can compensate the increase in total energy due to the removal of atoms from their equilibrium sites. If this happens the relaxation is energetically favorable and will take place spontaneously. In the case of the Jahn-Teller effect, the displacements of atoms are of the order of magnitude of one tenth of an Angstrom, i.e. they are relatively small. Larger displacements, of the order of magnitude of one Angstrom, are observed at point perturbations for which different atomic structures are stable, depending on external conditions as, for example, the position of the Fermi level. This phenomenon is observed at the D X centers in GaAs and (Ga, A1)As mentioned above (see section 3.5 for more detail).

3.1.3

Formation of point perturbations and their movement

Formation of structural defects All of the above mentioned defects of ideal crystal structure may in fact exist in real semiconductors. There are various reasons for their occurrence, the most important and general being the second law of thermodynamics. According to this law, the thcrrnodynamic cqiiilibrium state of a crystal at temperature T and pressure p , is characterized by a minimum of the Gibbs free energy G = H - TS. Here H is thc enthalpy and S the entropy of the system. The entropy of a macroscopic state is proportional to the logarithm of its microscopic realization probability (or ‘thermodynamic probability’) and the proportionality factor is given by Boltzmann’s constant k. The system considered here is the totality of the atoms of the crystal. Lets assume that there arc only chemically ‘correct’ atoms, and that these are randomly distributcd in space. Then the idedl crystal is formed wherein the atoms move to the regular crystal sites. This corresponds to a very special state of the system. It is extremely improbable compared to the large number of states in which deviations from the ideal configuration of atoms appear as described above. The entropy S of the ideel crystalline state i s smaller, therefore, than that of the imperfect crystalline states. The minimum of enthalpy II i s reached in the ideal crystalline state. Since the entropy S takes larger values in other states, the minimum of H is not necessarily coincident with a minimum of the Gibhs free energy G . Depending on temperature, a minimum of G = H - T S can also be adjusted for a state of the crystal which is less advantageous with respect to enthalpy, but more advantageous with respect to entropy, i.e. for a b t a k with a nonvanishing concentration of structural defects. As a concrete examplcs we

236 Chapter 3. Efectronic structure of semiconductor crystals with perturbations consider a vacancy in an elemental semiconductor. In an ideal crystal. the periodicity region contains a number J of identical atorris. A vacancy is created w h m one of t h e atoms is removed from the crystal. Altogether, there are J different possibilities for such a removal, as many as thme are atorns. For each realization of the irlcal crystal one has, therefore, J realizations of the crystal with B vacancy. II n independent vacancies exist, the number o f realizations is . J ( J - I ) . . ( J - r L + l)/nL The gain of entropy AS compared to the ideal cryatd t hrrefore mntnints to As=kln[

J! n!(J - It)!

1,

In a more rigorous treatment, the entropy of lattice oscillations has yet to bP considered in AS, more accurately, the change of this entropy because of the alteration of the spectrum of lattice oscillations in the presence oI the vacancies. We will neglect this effect in what follows. What must, however, he taken into account is the enthalpy Rf necessary for the formation of n vacancy (at constant pressure). For R independent vacancies the enthalpy requirement is n H f . Altogether, the gain of Gibbs free energy AG due to the formation of n. vacancies becomes

This expression Ims t o be minimized with respect to TL for a h e d value of T and p. According to of Stirling's formula, h { n ! )N n In(n) - n , the minimum coiidition may be written as

Since n T 0'0°

2! -0,Ol CI,

w I w

-0,02

1985.)

- 0.03 -0,04

-0.05 performed explicitly. _ In this - case ~ one has 6 minima, which lie on the cubic axes k,, k,, k , and k,, k,, k , close to the respective X-points. From symmetry considerations it follows that fl(k,lV'lk,)II;k(0)12 has the general form

O b b O b b c b b c b b

b b

c b b c 0 b b b O b b b O c b b

b b c

b b o

7

(3.101)

b = fi(k~lv'lky)l~z(0)12, r

= ~t(kzIv'lkz)l.)IFz(0)12

(3.102)

have been-used. _ _ Lines and columns in (3.101) are written in the sequence k,, k,, kzrk,, k,, k,. The matrix (3.101) has the three eigenvalues

A&I = 4b + C,

AEz

= -2b

AEg - -c,

+

1 - fold C,

2

fold

3 - fold

(3.103)

3.4. Shallow lwels. Donor and acceptor states

275

with the degrees of degeneracy indicated. Thus, inter-valley coupling partially removes the 6-fold degeneracy among the valleys. The sphtting of the ground state donor lrvel in Si due to inter-valley coupling can also be predicted j u s t oti the basis of group theory. One considers the representation of the symmetry group of the envelope e q u e h n (3.99) for si, i.e. of _ o h ) in the space of the 6 reciprocal vector compo__ nents k,, kg,k,, k,, k,,, k,. This representation is reducible. To demonstrate this one may employ the transformation rules of the vector components x , y, z , a,7j, Z under the action of the clcments of o h (summarized in Table A.7 of Apprndix A), realizing that these component? transform in the same way as thc reciprocal vector components considered here. Then one easily finds that the irreducible parts of this representation arc rl ( A , ) . r12 ( E ) and I’z5(Tz). ‘The notations A , E , Tz are commonly used in the theor,- of l o c a l i d states, and WP will also employ thcm here. With this. it is clear that the symmetry of the AEI-state is A l , that of the two AETstatcs is E , and that of the three AE3 states is 12. The size of the splitting depends on the inter-valley nitrtrix elemen1 of the perturbation yolmtial. If one considers only thc first of thc two abovementioned effects which result in matrix elernentv ol considerable size. i.e. the wavevector dependence of screening, and assumes that screening has become fully ineffective at large wavevectors k, - kj,therefore replaring the screened Coulomb potential V’(x) of equation (3.80) by the bare Coulomb potential -(e2/IxI) in (ktlV’lkj), then it follows that (3.104) With this, one obtains b as -2.36 mel’ and c as -1.2 met‘. This yields AEl = -10.8 meV. A E z = 3.6 met‘ and AE3 = 1.2 meV. A 3-fold splitting of the P-donor ground state level in Si has, in fact, been observed experimentally (Aggrawal and Ramdas, 1965). The simple theoretical estimate presented above agrees remarkably well with the experimental splittings of 11.6 meV between the A l - and T2-levels, and of 1.3 meV between the E - and T2-levels (not resolved in Figure 3.5). The results of a numerical treatment of inter-valley coupling shown in Figure 3.5 are even closer to the experimental values. However, the agreement is not as good for the absolute position of the ground state level. The addition of A E l , i.e. of 10.8 meV to the binding energy of 29 meV without inter-valley interaction yields a corrected binding energy of 39.8 meL’ which is closer to the experimental value of 45 meV, for Si:P but still clearly below. Chemical shifts

In Table 3.5 we list experimental binding energies for a number of singly

276 Chapter 3. Electronic structure of semiconductor crystals with perturbations Table 3.5: Experimental binding energies of several shallow donors and acceptors in Si, Ge and GaAs. (After Landoldt-Bornstein, 1982.) Material Si Sb

P Ge

As Sb

Bi

Ga In

13 14 10 13

B A1 Ga In

153 11 11 11 12

GaAs

ionizable donor and acceptor atoms in Si, Ge and GaAs. Not only are the absolute values of these energies striking, but also the fact that they are not equal for donor or acceptor atoms of different chemicals stands out. Contrary to the prediction of the above theory, there are dependencies of binding energies on the chemical nature of the impurity atoms. One refers to these as chemical shifts. The absence of chemical shifts in the calculated binding energies is a consequence of the approximation of the perturbation potential V’(x) as a purely Coulombic potential in the hydrogen model. As we know from section 3.2, the correct perturbation potential of an impurity atom also contains a short-range part, to which all central cell corrections contribute. Among these, there are also contributions which depend on the chemical nature of the donor. The latter are the main reason for the experimentally measured chemical shifts of the donor binding energies. The effects of central ceE corrections are particularly large in the case of the 1s-ground state, as one can recognize from Figure 3.5 for the special case of the P-donor in Si. A weaker influence, and, accordingly, better agreement between experiment and the predictions of a theory omitting central cell effects occurs for the excited states with n = 2 , 3 , . . .. This is to be expected since the excited state wavefunctions have maxima that do not lie at the perturbation center x = 0

3.4. ShalIow levels. Donor and acceptor states

277

(which does happen in the case of the ground state) but further outside. Electrons in the excited states are therefore less affected by changes of the potential in the central cell than are electrons in the ground state. Generally, the occurrence of pronounced chemical shifts means that the perturbation center is no longer shallow and the effective mass theory is no longer applicable for its theoretical treatment (see section 3.5 for further discussion). Acceptors Corrections to the simple hydrogen model are also necessary for acceptors. Of course, the maximum of the valence band lies in the center r of the first B Z for all diamond and zincblende type semiconductors, as is assumed in the hydrogen model. However. this maximum is degenerate: depending on whether spin-orbit interaction is important or not one has, respectively. the %fold degeneracies of the representations l'b5 (diamond) or (zincblende) in the space of scalar functions. or the 4-fold degeneracies of the representations I?$ (diamond) or Ts (zincblende) in the space of twc-component spinor functions. In the vicinity of r, this degeneracy splits off and one has ihree or two anisotropic valence bands. In section 2.7 these were approximated by two isotropic parabolic bands. one for heavy holes and one for light holes. If one also uses this approximation here, then there are two hydrogen-like series of acceptor le\-els, one for each sort of holes. Calculating acceptor binding energies by means of expression (3.95) in the case of Si yields 50 met7 for heavy holes which are expected to form the ground state. In the case of Ge. the result is 17 meV for heavy holes. One can hardy expect these values to be in agreement with experiment. In fact, they differ appreciably from the measured ground state binding energies shown in Table 3.5. Evidently. non-parabolicity and anisotropy of the band structure play an essential role and must be taken into account in realistic calculations. This may be done by means of the multfband effective mass theory developed in the preceding section. Below. we will explain the application of this theory to simply ionizable acceptors in Si. Owing t o the small spin-orbit splitting energy A of 44 meV and the relatively large acceptor binding energies E B (68 me\' for the isocoric impurity atom Al). spin-orbit interaction initially will be neglected. Later this approximation needs to be corrected since E g is not much larger than A. Without spin and spin-orbit interaction the wavefunction ~ A ( x ) is a linear combination of the three Bloch functions (xlumO),m = x. y, z of the representation Ti,, which are enveloped by the * components F,(x) of the envelope function vector F (x). Thus

(3.105)

278 Chapter 3. Electronic structure of semiconductor crystals with perturbations

The corresponding effective mass equation of the I’b5-valence band is given by relation (3.72). If, therein. U ( x ) is identified as the Coulomb potential. (3.80), one obtains, in the case A 2 = -1,

The eiigenvalues and eigeuvectors of this complicated set of equations cannot be obtained, of course, in closed analytical form. One has to use spproxinia tions and numerical calculations. The variational method again represents a reasonable way to proceed. In this approach one starts with a choice of suitable auxiliary functions for the expansion of the multi-component enve=$ lope fundion F (x). This function should have symmetry properties which fit the symmetry of the acceptor state @ A ( X ) to be calculated. The meaning of this will be explained below. We consider a particular acceptor level E n , which in general will be degenerate. It belongs to an irrdiicible reprc sentation r A of the point moup oh. The corresponding aigenhnctions are . . ., and the pertinent multi-component envelope denoted by $ A ~ ( x ) ,@m(x),

*

*

function by ~ , . q l(x), FA? ( x ) ,. W e have to find the representation D A of =$O h to which the set of vectors F A ~ (x),F A ~. ., . must belong in order that the wavefunctions T , ~ A ~ ( x$~z(x):. ), , . formed from these vectors by meam of equation (3.105), actually transform according to the representation r ~ 3

*

The FA^ (x). FA^ (x),. . . belong to a space which differs from the ordinary Hilbert syaccr in that it is riot spanned hy oue-comyonent basis funrlious, but by three-component basis functions. A representation in this space is given by the product of two represt?nt,atinns, one of which corresponds to the 3D rqresentation according to which the components of each of the three= $ * component, fiinctions FA^, FA^ transform separately, and the other is the representation according to which the threecomponent functions transform among each other. One can easily show that the latter representation must +,.q2(x), . . ., forincd according to relation (3.105),b e be r A , so that $,.ql(x), long to the representation r A , as has been supposed. For the representation * * D A of FA^ (x):FA^ (x),. . . it follows that

(3.107)

.

3.4. Shallow levels. Donor and acceptor states

279

This representation is reducible. The irreducible components determine the symmetries of the three-component basis functions to be used for the r e p resentation of the threecomponent envelope function of the acceptor state under consideration. Equation (3.107) yields a remarkable conclusion concerning the symmetry of acceptor states. An envelope function basis vector which transforms according to the unity-representation occurs in the expansion of the accep. when the symmetry r-4 of these tor eigenfunctions $ A ~ [ X ) , @ ~ ( X .) .~only functions is I'h5. The basis vector belonging to the unity-representation is the only one which does not vanish at the central site x = 0. Since the wave function of the ground state will also be non-zero at x = 0: D A must contain the unity-representation if A is the ground state. According to (3.107) this is only possible if the ground state has the symmetry Fh5. Acceptor levels of other symmetry are necessarily excited states. The symmetries of all t h r e e component basis functions involved in the construction of the ground state envelope function are given by the relation

(3.108) shown in Table A.27 of Appendix -4. Relation (3.108) determines the symmetries of the three-component auxiliary functions under rotations, therefore the angular dependencies of these functions. In order to also get their raand ri5-functions with diaI dependencies one expands the rl-, r12-, respect to angular momentum eigenfunctions with the various quantum numbers 6, rn and multiplies the expansion coefficients of different values of I by corresponding radial wavefunctions r1exp(-T/q). These are formed in analogy to the eigenfunctions for the Coulomb potential, but with &dependent localization radii rl. The latter are treated as variational parameters, just like the Coefficients of the auxiliary functions belonging to different irreducible representations. Applying this procedure to the acceptor ground state of Si, a binding energy of 31 me&' is obtained (Schechter, 1962). More recent calculations, taking spin-orbit interaction into account, have resulted in a value of 44 mel/ (Baldereschi and Lipari. 1977), which is very close to the experimental value of 45 meV for 3 in Table 3.5 but, of course, does not account for the pronounced chemical shifts seen in this table. Calculations of acceptor binding energies have also been performed for materials like Ge whose spin-orbit splitting energies are large compared to the acceptor binding energies. Spin has to be taken into account under such circumstances, i.e. the T'& valence band has to be replaced by the two rg- and r$-bands. Owing to the large spin-orbit-splitting energy, however, the spin-orbit-split I'F-band can be omitted. For the expansion of the 4component envelope function of the remaining rg-band one needs auxiliary functions of 2r15 2r25 symmetry (see Table x I'i = ri fk

+ +

+

+

280 Chapter 3. Electronic structure of semiconductor crystals with perturbations (A.28)). In the calculations for Ge, an acceptor binding energy of 10 m e v is obtained, close to the experimental values of 12 nieV for In and 11 meV for all other elements listed in Table 3.5. Obviously, chemical shifts are very small in the case of Ge, unlikr Si where these shifts were found to be very pronounced. The different behavior of acceptors in Ge and Si is undmstandable if one looks at the absolute magnitudes of their binding energies EB: in Ge, E D is siihstantially srrialler than in Si, reflecting the fact that the effective heavy hole mass in Ge is smaller than in Si. This implies that the acceptor wavefunctions of Ge have smaller amplitudes in the central cell than those of Si. Thus the point charge Coulomb potential which neglects central cell corrections giving rise to chemical shifts is expected to work much better for Ge than for Si, as it dow in fact. For the same reason, excited acceptor states in both Ge and Si are well described by the multi-band effective mass theory using a point charge perturbation potential (Balsderesrhi and Lipari, 1978). Multiply ionizable donors and acceptors Additional changes arc to be expected for substitutional impurity atoms whose core charge numbers do not differ by & A 2 - 1 from those of the host atom, as we assumed above, but differ by 2, 3 or more. Then 2, 3 or more electrons or holes are weakly bound to the impurity atom at T - 0 K , and 1, 2, 3 or more electrons can be thermally excited into the conduction or valence bands, leaving behind a 1-, 2-, 3- or morefold ionized impurity atom. The interaction between the carriers bound at the center is described by the Hubbard energy of equation (3.25). Without this interaction the effective mass equat,ion for a non-degenerate isotropic band yields a hydrogenic s e ries of energy levels with binding energy increased by the factor lA21' in comparison with the corrcsponding energy for a simply ionizable donor or acceptor. The Hubbard correction (3.25) results in a shiit of the energy levels towards higher energies by an amount which depends on the number 'IL of electrons or holes bound at the center. We illustrate these remarks using thc example of the S-donor in Si. The binding e n e r a without H a r t r e potential is four times larger than that of the P-donor in this material. In the ground state, 2 electrons are bound at the S-donor, both of which can be placed in the 1.9-level because of 2-fold spindegeneracy. If the Hubbard correction, which in this case is approximately given by

is taken into account, then the 1s-level is shifted up by just this energy Us.By exciting one electron into the conduction band, the neutral S-donor

3.8. Deep lewls

28 1

S(D+) bccarnes a single positively charged S(1+) donor. For the Is-lad 01 h e S ( l 1 ) ioii thr shift in energy by Us do= not occur. Thus this level i s shifted down by I/, in comparison with the 1s-level of tbc S(0 +-) atom. Exprrirneniully, one finds t,hat the ionk-ation energy of the neutral S(O+) donor is 0.31 eV, and the ionization energy of the S(1t) donor j s 0.69 eV (see Figure 3.6). From thefie values a Hubbard energy I i , of 0.28 eV may be deduced. IIowever, both ionization energies are suhstantially larger than the result. 4 x 0.03 el7 = 0.12 eb’ which follows from the hydrogen model. Evident.ly, the central cell short-range poteniid cont,ribution is ewedial in the case of the S-donor in Si. ‘ l l i s donor is a deep center rather than a shallow me.

3.5

Deep levels

3+5.1 General characterization of deep levels As we know from previous sc3ctions, hhc potential of a point perturbat.ion, generally, consists of a long-range (Coulornbic] part and a short-rangp part.. In regard to their ability to bind states, the two potential contributions differ substantially. In the case of the long-range Conlomb potential, hound $Pates exist, for all possible potential strengths, i.e. for arbitrary magnitudes of the point charge and dielect,rir:cobstant. Short-range potentials, on t,hecontrary, must have a minimum st.rengt.h to he able to bind a stale. For example, for a 3-dimensional potential box of depth Vo and radius a, the condition Vo > h2/(8w>.uli) must he fulfilled for a b a u d state to exist (see, e g . ? Schiff, 1968). In this, Heisenberg’s uncertainty principle manifests itself, in that localization due to the potential leads to a non-vanishing expectation value of momentum and 1,hus also of kinelic energy of the particle. Only if the dept8h of the potential box exceeds the expectation value of this kinetic energy! can t.hc potential prevent the partick from escaping the center. (Not.e, however, that in the case of a 1-dimensional potential box, bound states exist for arbitrarily small m d l depths sincc the 1ocalizat.ionand: with this, the average kinetic energy decreases sufficiently- fast with decreasing well dept.h. This indicates already at t,he outset that 1-dimensional models of deep centers are ralher poor). According to wction 3.4, shallow levek ocrur for statps which are bound by a Coulomb potential, i.e., by the long-range contribution of the total perturbation potential. The concornit,ant short-range potential part does not suffice in this case for binding, it only leads to corrections of the binding energy and the eigenfunction of the ground state, k1loa.n from section 3.4 as c e n l m l tell correcfioru. The spatial extension of the eigenfunction is essentially given by the Bohr radius. If t.he strength of the short-range

282 Chapter 3. Electrunin structiire of Bemiconductor crystals w i t h perturbations part of the potential increases, then a point will be reached at which the short range potential is itself able to bind states. Provided the pertinent binding energy i s Iargcr than that due l o the Coulomb potential, binding to the center will be dominated by the short-range potential contribution, and the chararter of the bound state will change. The spatial cxtension of the eigenfunctjon of the ground state is then no longer dekrrninml by the Bohr radius, but by thr lattice constant. The Coulomb potential only leads to corrections to t h r binding energy and the cigmfunctions, which are mainly determined by the short-range potential alone. States which are primarily bound by the short range potential part, are termed d w p . ‘ h e rorresponding la-& arc called deep levels, and the point perturbations at which they occur, are called as deep centers. Literally, the term ‘deep level’ refers to an energy eigenvalue that lies deep in the energy gap, far away from the two band edges, in contrast to shallow levels which lie close to one of these edges. Actually. the location of deep levels deep in thr energy gap is only a particularly important special case. as there are also yet other case. - derp levels can also be close t o or even wzthm one of the bands. The essential features of deep levels are their binding by a short-range potential and, in combination with this, their localization radius bring restricted to magnitude of the lattice constant. Because of their strong localization, deep states resolve the spatially periodic fluctuations of the rrystal potential while the eigenfunctions of shallow levels average them out. The shallow levels can therefore be treated by means of an effective mass equation. in the case of deep levels the requirements of effective mass theory, namely smoothness of the perturbation potmtial and of the corrpsponding wavefunction, are not fulfilled. Contrary to the assumptions of PEective mass theory, these states cannot be s p t h e s i z d by Bbch functions of k-vectors drawn from a small vicinity of a rritical point, and also not from Bbch functions of only one-band. In a theory of deep states, therefore, neither the effective mass is meaningful, nor can such a theory be based on a oneband equation, not even a degenerate multi-band equation. Attempts t o retain the effective mass concept and to consider solely ihe multi-band character of deep level theory met with little success. In this context one can say that skidow levels are eigenvalues of point perturbations which are capable of description by means of effective mass theory (the concomitant short-range potential contribution can be treated by rncans of perturbation thcorg in this rase), while deep levels are eigcnvaliies whose treatment by effective mass theory is impossible. The discussion above is amply intuitive. it anticipates, in part, results which are i n d d plausible, but which have yet to be proven rigorously. This applies, in particular, to the fundamental feature which distinguishes between shallow and deep levels, namely, that short-range potentials can dominate binding only if they exceed a minimum threahold potential strength

283

2

0

1

-2

-4

-6

-2

-4

-G

.3 Acceptors

2

- - - _ aF:

.4

.-

.5 .6 (eV)

2

1

-8

-1OleV)

Atomic Electronegativity Figure 3.6: Ionization energy of substitutional impurity atoms in varioum tedrahedrally coordinate host semironductors as B function of the strength of the pcrturbation potential, measured in terms of valence shell s-level differences between impurity and host atoms in the case of donors, and valence shell p-level differences in the caae of acceptors. Donor energies are reIative to the conduction band edge, and acceptor energies to the valence band edge. (After Vogl, 1981. Reproduced from Boer, 1990. - in the discussion above, the periodic potential of the crystal was omitted

from consideration. An experimental proof of the threshdd behavior of short-range potentials in crystals may be taken from Figure 3.6. There: the ionization emrgies of a series of substitutional impurity a t o m are plotted as a function of the difference! between the atomic valence shdl energy levels of the host and impurity atoms. This difkrence can serve as a measure of the strength of the short-range potential. The fact that for donor states

284 Chapf,er3. Electronic structure of semiconductor crystals with perturbations the diffrrence of 8-levels is taken, and for acccptor states the differenre of p-levels, is not important foi the present discussion (we will return to this point later). With increasing horizontal scparation of the impurity atom from the host atom in the periodic table, the potential strength increases more or less continuously. The ionization energies initially retain their small valucs characteristic of shallow lcvels, Starting at a threshold distanre, they suddenly become substantially larger, a manifestation of the fact that the short-range potential has takm over binding and the level has become deep, .Just like shallow levels, decp levels can also act as donors or acceptors (sometimes even as both of them). Due to their mostly larger separation from the hand edges they are, however, generally less effective than shallow levels in enhancing the concentrations of free charge carriers. They exhibit their greatest efiectiveness in just thp opposite process, the lowering of free carrier concentrations for which, in turn, shallow levels are not very effective. If, in addition to the neutral center, the single negatively charged center also forms a deep level in the gap with sufficient separation from the conduction band edge, the neutral center will capture electrons from the conduction land, which are available there eithrr as equilibrium charge carriers due to n-doping, or ~ L non-eyujlibiium Y cariiers due to optical or other excitation of the semiconductor. In the first case, one has a c o m p e ~ w u t i o nof the donors by the deep center (for more 5ee Chapter 4), and in the second case, a cupture of non-equilibrium electrons by the center (see Chapter 5). A center which forms deep levels in the gap both in the neutral and simply positively charged state, plays an analogous role in regard to the compensation of acceptors and the capture of non-equilibrium holes. Centers which can capture both electrons as well as holes act as catalysts for the radiationless recombination of electron-hole pairs (see Chapter 5). If one wants to enhance radiative recombination as strongly as possible, such as in semiconductor light emitting diodes or laser diodes, non-radiative recombination processes have to be avoided, which involves the elimination of the corresponding deep centers. On the other hand, if one is interested in having a short lifetime for non-equilibrium charge carriers, as in the case of fast transistors and photodetectors, one should intentionally introduce deep centers in order to enhance non-radiative recombination. In general, deep centers play a role of similar importance to that of shallow ones for semiconductor devices, although in a completely different way. Having discussed the general character and importance of deep centers, we now will treat their electronic structure. In subsection 3.5.2 we will develop a simple model of a deep center, the so-called defect-molecule model. In subsection 3.5.3 we treat some methods of solution of the one-electron Schrodinger equation for deep centers. The consequences of many-electron effects in the electronic structure of deep centers will be discussed in subsection 3.5.4. In section 3.5.5 we display experimental and theoretical results

3.5. Deep levels

285

for selected deep centers, among them the vacancy in Si. the substitutional impurity at,oms of the main groups of the periodic table. the group of 3dtransition metals. and the group of rare earths. Also, we discuss the D X center and the EL2-center in GaAs. 3.5.2

Defect molecule model

The 'Tight 3inding' (TB) method developed in section 2.6 represents one of the various procedures for calculating band structures of ideal crystals. Unlike other methods it uses basis functions to represent the Hamiltonian which are localized on the atomic length scale. Since the perturbation potentials of deep centers are localized on the same scale, the TB method should be particulady well suited for such centers. Of course, one must chose the Hamiltonian matrix elements empirically in order t o arrive at useful practical results, and the results also cannot be expected to be very accurate in a quantitative sense. However, the method should be suited to the derivation of simple models that exhibit the essential physical features of real deep centers. The simplest among these models is the so-called defect-molecule model, which we will introduce below. In doing so, many particle effects and lattice relaxation will be ignored. We will mainly use the model to demonstrate the existence of deep levels and to explore the symmetry of the pertinent eigenfunctions. At the outset we have to clarify which of the various tight binding basis sets should be used for the representation of the Hamiltonian in the case of a deep center. En the ideal crystal case considered in section 2.6,the atomic orbitals or hybrid orbitals IhtRj) were not used directly, but rather. we employed wavevector-dependent Bloch sums lukj) or ihtkj) formed from them. This was advantageous because the translation symmetry of the crystal could be exploited in this way. The latter symmetry no longer exists in a crystal with a deep center. so that localized basis functions, i.e. atomic or hybrid orbitals. can also be used without any loss. We select hybrid orbitals because these produce drastic simplifications of the Hamiltonian matrix which, like the Bloch sums of hybrid orbitals in the case of an ideal crystal, allow the eigenvahes of the Hamiltonian to be calculated in closed analytical form. We introduce the defect molecule model using the vacancy as an example. Later, we will apply this model to substitutional impurities from the main groups of the periodic table. As the host crystal we take, in all cases, an elemental semiconductor of group IV,like Si.

Vacancy Figure 3.7 shows part of a Si crystal containing a vacancy. Symmetry consid-

286 Chapter 3. Electronic structure of semiconductor crystals with perturbations

erations make clear that the origin of the vacancy is not important, whether it originated by removal of a Si atom from sublattice 1 or from sublattice 2. Here, we consider the removal of an atom from sublattice 1, and to be still more specific, from the primitive unit cell at R = 0. The perturbation potential V’(x) is the negative of the potential produced by this atom in the crystal. Because of the removal of the 1-atom, the hybrid orbitals lht2Rt),t = 1 , 2 , 3 , 4 , of the four surrounding 2-atoms in the unit cells Rt, pointing inwards, no longer have a hybrid orbital of a 1-atom to which they can bind. They are called danglzng hybmds. The three other hybrid orbitals at a surrounding 2-atom interact with inwardly directed hybrid orbitals at atoms lying still further away (these atoms are not shown in Figure 3.7). The hybrid orbitals at a pal ticular siirrounding 2-atom also interact among themselves, including the one dangling hybrid at this atom. This means that the dangling hybrids are coupled to the entire crystal through nearest neighbor interactions. If interactions between hybrid orbitals at the same atom are omitted, i.e. if the matrix element V1 introduced in section 2.6 is set to zero, which means - cp, then the dangling hybrids are decouplcd from the remainder of the crystal. They still interact only among themselves. Since the atoms at which these hybrids are located are second-nearest neighbors, this interaction is not within the framework of Erst-nearest neighbor interaction which we have been exclusively considering in the treatment of an ideal crystal in section 2.6, but, rather, it ha6 the sense of second-nearest neighbor interaction. The latter must be taken into account here in order to arrive at non-trivial results. Based on the approximations made above, the crystal with a vacancy decomposes into two partial systems which do not interact with each other, first, the partial system of the four interacting dangling hybrid orbitals, one at each of the four atoms surrounding the vacancy, and second, the yarlial system of all remaining hybrid orbitals of the crystal with the vacancy, i.e. the hybrid orbitals of all atoms which are not directly adjacent to the vacancy, and the three hybrids at each one of the four adjacent atoms which ~

&renot alrpady included in the first ptrrtial system. With some ambiguity

3.5. Deep levels

287

we can refer to the first partial system as a vacancy molecule:

and to the the ‘rest-of-crystal’. This designation also encompasses h e term defect molecule model for the tight binding approximation described above. For each hybrid of the rest-of-crystal another hybrid exists in the restof-crystal which points to it. The Hamiltonian matrix elements between the various pairs have the same value, namely the value given above in equation (2.292) defining matrix element T/2. The energy spertriim of the rest-ofcrystal is therefore identical with t , h d of the infinite idea1 cryslal in bhe simplest TB approximation? consisting of the bonding level t b = th - IVzl, and the anti-bonding level E, = E,+ I&I. The splitting of these highly degenerate levels into bands remains incomplete because of the neglect of Ihe interaction between hybrids at. the same &om? i.e. the neglect. of r/l. In calculating the energy spectrum of the first partial system, i.c., the vacancy-molecule, we need the matrix elements of the pertarbed Hamiltanian H + V ’ of the crystal with vacancy. The diagonal elements (ht2RtlH + V‘lht2Rt) are simply the hybrid energies ~h since the elements (h,t2Rt(V‘ lht2Rt) of the vacancy potential V‘ between hybrid orbitals localized off the %mancysit.e are small. One therefore has second

tts

+

In order to obtain the non-diitlgounel (second-nettrest neighbor) matrix elements ( h t 2 R t l H f V”lht32FQ) between different dangling hybrid orbitals t , t‘ f t , one has to recaIl the symmetry of the perturbed Hamiltonian H +V‘. Since, by creating a vacancy in sublattice 1) the two sublattices are no longer eqiiivalent, the symmetry of the crystal is no longer given by thr full cubic point group Oh, but rather by the tetrahedral group y d . Nevertheless, this means that all non-diagonal elements are equal for symmetry reasons, such that

wit,b W as second-nearest neighbor interaction energy. Because of the predominantly negative values of the operator fI V’ acting on hybrid orbitals and the predominantly positive values of the hybrid orbital products. W is expect.ed to be negative. ‘I’he absolute value of W must be determined empirically. The energy levels of the vacancy-molecule are obtained by diagonaliziug the matrix

+

288 Chapter 3. Electronic structurc of scmiconductor crystals with perturbalions

Its eigenvalues Eyzz,4 read (3.113) (3.114)

IEY)=

- IhaZRz)], Jz [IIZL~RI)

1

(3.116)

) - -[ I ~ I ~ R - IJh2R3)] )

(3.117)

1 138"")= [lh12R1)- lh42R9)]. 45

(3.118)

IE,

VaC

~

v5

I

One can easily demonstrate that IEI") belongs to the irreducible representation A1 of the symmetry group Td of the vacancy, and the three functions IE$uc), IEY'), IEiWc)to the irreducible representation T2 of this group (see Table A.6 of Bppendix 4 ) . The eigenfunction of the A1-level resembles an atomic s-orbital of a Si atom. and the three Tz-eigenfunctions are similar to the three p-orbitals of such an atom. Evidently, the sp3-hybridization of the atomic orbitals in a Si crystal is removed at a vacancy. The states are more atom-like. in consequence of the fact that the crystal potential is no longer fully effective in the vicinity of a vacancy. In Figure 3.8. the e n e r e spectrum of the defect molecule model of a vacancy is shown along with that of the rest-of-crystal. The A1-level lies below the Tz-level because of the negative value of Vt-. Whether it lies below or above the bonding level of the crystal depends on whether we have 31W1 < l\Jzl or 31WI > lF'21. The question of whether it is found in the valence band or in the energy gap, cannot be answered within the defect molecule model because. therein, the valence band has shrunk to the bonding energy level. It is just as difficult to decide whether the Tz-level lia in the conduction bend or in the energy gap. Experiment and more exact calculations, which we will discuss later in more detail. show that the -41level lies in the valence band. and the 7'2-level in the energy gap. With this,

289

3.5. Deep levels

Figure 3.8: Energy spectrum of the defect molecule model of ~ 1 .Si-vacancy along with that. of the rest-of-crystal. The distribution of thc 4 electtona of the defect molecule over the energy levels is also shown. the T2-level is the actual deep level of the vacancy. Of the four electrons of lhe neutral vacancy - each of the four dangling bonds yields one two must he hosted by this lcvcl while the other two occupy the A1-level in the ~

valence band. IJsing the terminology introduced above, we may say that the oxidation state of the neutral vacancy is V2+. The defect moleciile model of the vacancy reflects the actual relationships remarkably well. In any case it, provides a qualitatively correct physical picture of the electronic structure of a vacancy in group-IV elemental semicondiictors. Refinements of this picture will be discussed further b e low. Here we treat a second example to illustrate the defect molecule model which emerges from the v.mancy by occupying its empty lattice site with an impurity atom. Substitutional impurity a t o m s w i t h sp3-bonding We consider a substitutional impurity alom in an elemental semiconductor of ~ o u IV, p which we can again imagine as a Si rrystal (see Figure 3.9). Let the siibstituted Si atom be that of sublattice 1 in the primitive unit cell R = 0. Like Si, the impurity atom should belong to one of the main groups 11,111, IV, V, or VI of the periodic table so that the valence shell is formcd by s- and p-orbitals, which all lie energetically higher than any other occupied orbitals (in contrast to rare earth atoms). The perturbation potentials of these impurity atoms in an elemental semiconductor of group I V possess, as we know, both a short-range part and also a long-range Coulomb part. The

290 Chapter 3. Electronic structure of semiconductor crystals with perturbations Figure 3.9: Defect molecule model of a substitutional main group &bonding impurity atom in a tetrahedrally coordinated semiconductor.

L

latter will be omitted from consideration below. This approximation does not affect the answer to the question of whether a particular impurity atom forms a deep level or not. although it influences the actual position of this level in the energy gap. The latter cannot be determined within the defect molecule model anyway. The four sp3-hybrid orbitals of the impurity atom will be denoted by IhtiO), and the four hybrid orbitals of the host crystal pointing in the direction of this atom will be denoted by IhtzR,),t = 1,2.3,4. Xeglecting interactions between the hybrid orbitals at the host atoms, the crystal with a substitutional impurity decomposes, just as before in the vacancy case, into two partial systems, first. the defect molecule with the 8 orbitals jhtiO), jht2Rt),t = 1.2,3.4, and second. the rest-of-crystal with all remaining orbitals. The energy spectrum of the rest-of-crystal again coincides with that of the ideal crystal within the framework of the approximations used here. The Hamiltonian matrix of the defect molecule is composed of elements of the general form fhtjRtIH

+ 17’lht~j’R~~)~

(3.119)

where j and j ’ take the values i and 2 independently of each other. We consider only the most important elements of this matrix, namely

In this, E ; and c i signify, respectively, the hybrid energies of the impurity and host atoms, V2 corresponds to the matrix element of H of equation (2.292) between hybrid orbitals at nearest-neighbor atoms of the ideal crystal

3.5. Deep levels

291

pointing toward each other, and W describes, as in the vacancy case, the second nearest neighbor interaction between differing host atom hybrid orbitals pointing toward the impurity atom. All elements ( h t j R t l H V’lht‘J’Rt,)not listed above are neglected, among them also the Vl-like elements (htzRt(H V’lht,i&,) between different hybrid orbitals at the impurity atom, i.e. with t’ f t. With these approximations, the Hamiltonian matrix of the defect molecule may now be written down in explicit form. In doing so the rows and columns are ordered in the sequence IhliRl), lh2iR2), lh3iR3), lh4iR4), lh12R1)) lh22R2), lh32R3), I@&), and the Hamiltonian matrix is

+

L i 0

0

0

1720

0

0

0

€e,O

0

0

vzo

0

0

0

6 i 0

0

0

v20

o a o v2a o o

€ ; W W W

0

0

W E k W W

v20 0 I$

W W E k W

0 (0

v2o

0 0

+

a o v2

w w w

(3.124)

e;.

This matrix has four distinct eigenvalues, two simple and two triply degenerate. The eigenfunctions of the two simple levels (we distinguish them by indices b and a ) belong to the 1-dimensional irreducible representation A1 of the tetrahedral group T d , and the two triply degenerate (again distinguished by indices b and a ) belong to the 3-dimensional irreducible representation T2 of T d . The corresponding energy levels are, respectively, denoted as

EiT/b

and E&Y:,b, whence

In Figure 3.10 these levels are plotted as functions of the difference ( c i - E); between the hybrid energies of impurity and host atoms. This difference represents a measure of the strength of the short-range perturbation potential of the impurity atom. In order to better understand the physical meaning of the energy levels derived above, it is helpful to consider two limiting cases. First, we assume the impurity atom to coincide with the host atom, which means t i = c.; If one also neglects the second nearest-neighbor interaction energy W , then the energy spectrum of the perturbed crystal of equations (3.125) and (3.126) must coincide with that of the ideal host crystal in the

292 Chapter 3. Electronic structure of semiconductor crystals with perturbations Figure 3.10: Energy levels Eimp of a substitutional main group

sp3-bonding impurity atom in a tetrahedral semiconductor as function of the hybrid energy difference ( t i - e i ) . Horizontal lines indicate the bonding level t b and the anti-bonding level of the host crystal.

and an simplest TB approximation, i.e. a bonding energy level q = ~ h IVzI, anti-bonding energy level E, = € h + IV21 must emerge. This is in fact the case. of the perturbed crystal Thus it is also clear that the two energy levels correspond to bonding and anti-bonding states of A1-symmetry, and the two energy levels E;Y$. to bonding and anti-bonding states of Tysymmetry. The two undisturbed levels € h f IV21 in Figure 3.10 encompass the energy gap of the host crystal. Second, we consider the limiting case of ( E ; - tk) tending toward $00 or -00 which, for a given host crystal, also means t i -+ +m or E ; -+ -00, respectively. In the limit

EiT,,

E i + $00

equations (3.125) and (3.126) yield

(3.127) (3.128) and in the limit Ei

4

--oo

we have

EiT =

tk + 3W,

(3.129)

293

3.5. Deep levels zmp = Ezmp EAlb Tzb

(3.130)

~

-

+ --O0.

EiT

In the limiting case t i -+ +m, the two anti-bonding levels and EkYZ of (3.127) tend, along with t i , toward 00, i.e. leaving the energy spectrum on its high energy side, while the two bonding levels and limit toward, respectively, the two Al- and Tz-levels of the vacancy of equations (3.113) and (3.114). This is understandable because the limiting case EL ---f +m means that no impurity hybrid energy level exists at finite energy and, therefore, there is also effectively no impurity atom. In other words, there is a vacancy. With E ; -+ -m the two bonding energy levels and E$T leave the energy spectrum at its low energy side, while the anti-bonding levels limit toward the two A l - and T2-levels of the vacancy of equations (3.113) and (3.114). This occurs for the same reason as above in the limit t i -+ +m, namely because t i + -00 means that there is a vacancy at the impurity site. For finite positive values of (EL - E;), the bonding Al- and Tz-levels can occur in the energy gap, forming deep levels there, and for finite negative values of (tL-~k) the same can happen with the two anti-bonding Al- and Tzlevels. A look at Figure 3.10 shows that the most favorable candidate for an - 6;)i > 0 impurity level to be in the energy gap is the bonding Tz-level if ( ~ holds, and the anti-bonding A1-level if ( 6 ; - 6 ; ) < 0 holds. More rigorous calculations confirm this conjecture in that they find exactly these levels to be in the gap. Below, we will describe such calculations and discuss methods for solving the one-electron Schrodinger equation (3.8) for the crystal with a point perturbation.

EiY

EkYl

EiY

3.5.3

Solution methods for the one-electron Schrodinger equa. tion of a crystal with a point perturbation

The solution of the Schrodinger equation for a crystal with a point perturbation begins with the determination of the effective one-electron potential VPrt(x).In section 3.2 this task already was addressed in general terms. Two remarkable differences were found in relation to an ideal crystal: First, depends on the electron poputhe effective one-electron potential VPert(x) lation of (localized) one-particle states, and second the atomic structure of the perturbed crystal, which participates in the determination of the potential Vwrt(x),is in many cases initially unknown and it must be selfconsistently calculated jointly with the electronic structure. Below, we will eliminate such additional difficulties of electronic structure calculations for point perturbations by assuming that the population of the center and the atomic structure of the perturbed crystal are known. Thus the potential Vwrt(x) of the center can also be considered to be known. In regard to the

294 Chapter 3. Electronic structure of semiconductor crystals with perturbations electron-electron interaction contribution to V P d ( x ) , this only means that the functional dependence of this contribution to VpeTt(x)on the eigenfunctions of the one-electron Schrodinger equation is known, but not its actual value. The form of this dependence is governed by the particular oneelectron approximation used in the calculations - the possibilities include Hartree, Hartree-Fock, Local-Density-Functional theory or the quasi-particle method. Because of the dependence of the potential on the eigenfunctions of the Schrodinger equation, any solution procedure must be iterated repeatedly until self-consistency is achieved. The three most important procedures are the cluster method, the supercell method and the Green's function method. As a rule, the replacement of the real potential by a pseudopotential - a technique which has been \-cry successful for ideal crystals (see section 2.5) - is also applied in the supercell and Green's function methods. Cluster method Employing the cluster method, the infinite crystal with a point perturbation is replaced by a finite part which contains the point perturbation at its center and which. on the one hand, is large enough that the band structure of the infinite crystal is almost completely devploprd and, on the other hand, is small enough that the Schrodinger equation for it can be solved easily. This finite part of the crystal is called a rlwter. The cluster with the point pprturbatinn at its center is clearly just a large molecule. Its atomic and electronic struclureb can be calculated ubing the methods of quantum chemistry for the determination of the structure o f molecules. In contrast to infinite Lrystals, clusters have a surface. The latter also acts as a perturbation of the periodic potential and can gi%erise to bound slates in the gap. Such statrs must be remowd in a suitable way in order for the cluster method to be applicable. To this end, different procedures h a w been proposed such as, far example, the saturation of the dangling bonds at the surface by hydrogen atoms. Correspondingly, the surface levels are lowered deep below the valence band edge and raised high above the conduction band edge, where they cause no further difficulty.

Supercell method At the boundary of a properly shaped ciuster, one can add an additional cluster of the same kind. If one does this repeatedly and continues the whole proress ad infiniturn, one finally arrives at an infinite crystal. The primitive unit cell of this crystal is now, however, no longer that of the crystal hosting the point perturbation. but it is the cluster, which in this context is called a supeveell Thp crystal is referred to as a supercrystal. The

3.5. Deep levels

295

periodic repetition of the point perturbation in the supercell method causes the discrete d w p levels to become k-dependent bands of h i t e widths. If the supercell is made large enough, these widths are negligibly small and it suffices to calculate the band structure of the supercrystal for one k-point only, e.g., for k = 0. The appealing feature of the supercell method is that the whole apparatus of band structure calculations for ideal crystals can be exploited for the electronic and atomic structure determinations of deep centers. In the cluster and supercell methods one obtains the deep levels of the perturbed crystal by a process in which one numerically calculates the energy spectrum of a model system, i.e., of the cluster in the first method and of the supercrystal in the second. Results for the band structure of the ideal crystal are neither necessary nor useful in either method. Also, the decomposition of the entire effective oneelectron potential into that of an ideal crystal and a perturbation potential need not to be made. In the third method for calculating the electronic structure of perturbed crystals, the Green's function method, this decomposition is essential, and the band structure of the ideal crystal is required. Green's function method The Green's function method employs techniques and insights of quantum mechanical scattering theory. It provides results not only about bound states with energy levels in the gap of the ideal crystal, but also on scattering states having energies in the allowed energy spectrum of the ideal crystal. First we consider the bound states. Bound states: Kostcr-Slater method To explain this method we write down the Schrdinger (3.21)equation for the perturbed crystal once again in a somewhat modified form, [E - HI+ - L"@.

(3.131)

Since we are interested in bound states. the eigenvalues E which WP seek from this equation lie outside of the bands, to be mare precise. in the iundamenlal energy gap between the highest valence band and the lowest conduction band. As tl preliminary we note that equation (3.131) can be formally solved for @ by multiplying both sides by the inverse [E - H1-l of the operator [ E - HI. The inverse operator [ F - H1-l stands in close relation t o the Green's function of the unperturbed crystal. The retarded Green's finctaon O " ( E ) is defined by the equation

GO@)

=

1 Eti6-H'

(3.132)

296 Chapter 3. Electronic structure of semiconductor crystals with perturbations

where 6 is an infinitesimal positive imaginary part which is set to zero after serving to remove singularities at the energy eigenvalues of the unperturbed crystal. This procedure assures that the wavefunction response of the unperturbed crystal conforms to the causality principle, in that the response occurs only after the system has been perturbed. As our present interest is in deep levels in the forbidden part of the energy spectrum of the ideal crystal, not in changes of wavefunctions with energies in the already existing continuous part, no singularities occur in G " ( E ) of equation (3.132) in the range of E of interest to us, and we may ignore 6. Using G " ( E ) ,equation (3.131) may be formally rewritten as [@(E)V' - I]?)

0.

1

(3.133)

Non-trivial solutions me possible only for energies E for which the determinant of the matrix of the operator [ C 0 ( E ) V '- 11, in an appropriate orthonormalized basis set, vanishps, i.r. if

-

Uet[G"(E)V' 11 = 0

(3.134)

holds. The energy eigenvalues E satisfying this equation are deep levels. These levels ran thus be determined by calculating the Green's function G o ( E )of the unperturbed crystal and solving equation (3.134). This p r o w dure is rcfrrred to as Koster Slater method, and equation (3.131) as KosterSlater equation The Green's operator @ ( E ) can be cdriilattul if the band structure E,(k) and the Bloch functions of the ideal crystal are known. The matrix represenlation of d ' ( E ) in the Bloch basis reads

(3.135) With this we only need to know the matrix elements (uk(V'1u'k') of the perturbation potential V' in Bloch representation in order to determine the deep levels using equation (3.134). However, i t would be more expcdicnt for the matrix representation of the perturbation potential to use localized wavefunctions as) for example, atomic orbitals. If one does so, then the matrix representation of the (heen's opertrtoi is not as simple as in the Bloch basis. A particular compromise in choosing a basis set for the representation ol the eigenvalue equation (3.134) is the use of so-called Wunnzer finctzons. Koster-Slater method in Wannier representation

Wannirr functions are linear combinations of all Bloch functions for a given band index v of the form

3.5. Deep levels

297

(3.136) Here the summation is over the whole first BZ,G“ is the number of primitive unit, rdls in a ppriodicity rcgioa. and ‘ t k are certain phase a~glru.If the latter are chosen properly, the Wunnier h c t i o n s IvRJturn out to be well localized in the unit cell at the lattice point FL If the eigenvalue equation (3.134) i s written in terms of this basis, it reads

The matrix elements of the Green’s operator G o ( E )may be obtained by means of (3.143) as

The matrix elements of the potential (vRIV’jv’R’1 are particularly large if R and R‘ are identical, and both are equal to the lattice vector of the cell which hosts the point perturbation. As before, we assume & = 0. Neglecting all other elements we find

Employing (3.138) and (3.139)) equation (3.137) becomes

(3.140) For the particular lattice point R = 0 it reads

{ 3.141) Equation 13.141) forms a closed set of equations for the central cell components ( ~ O ~ I Jof) $ only. If the latter are known, all other components (vRI$) follow at once from equation (3.140). For a non-trivial solution of the homogeneous system (3.141) to exist, its determinant has to vanish separately, i.e.

298 Chapter 3. Electronic structure of semiconductor crystals with perturbations

must hold. The diagonal elements (vOIGo(E)IvO) = G ! ( E ) of the Green's operator in equation (3.142) may be obtained from (3.138) as

1 (vOlG*(E)IvO) =3

G;(E)

G

1 E+ir-EE,(k)'

Ic

(3.143)

The dominant bands in equation (3.142) are those which form the gap, i.e. the uppermost valence band u and the lowest conduction band c. If we consider only them and neglect all others equation (3.142) becomes

or more explicitly

To solve this equation, the host Green's functions G:(E) and G : ( E ) , as well the perturbation potential matrix elements, have t o be known. Below we calculate the host Green's functions for isotropic parabolic bands of finite widths. In expression (3.143) for G : ( E ) we rqlace the k-sum by an integraL This yields

G 0J E )

=

'

-G 3 8x3 , 1

d%i s

t

~

1 ~E

-I- i t - E,(k) '

v

= c, v.

(3.146)

st beiug the volume of a periodicity region. Introducing the identity operator dE'd(b" - E ) into this integral, G:(E:) may be written as with

Jrm

(3.147) whrre p v ( E ) is the density of states per unit energy and spin state

p,(E)

R

d3kd(E - &(k]) h3 1.232

:

, v

= c,w,

(3.148)

299

3.5. Deep levels

which differs from the DOS in equation (2.213) by a factor (0/2). In evaluating p,(E), we approximate the true valence and conduction bands by isotropic parabolic ones, however, taking their band widths A E , to be the same as those of the original bands. This corresponds to the use of an effective mass for each band averaged over the whole first B Z . The bandwidths are introduced by putting the total number of states of the approximate isotropic parabolic bands equal to G3. Then it follows that

(3.149)

3 pc(E) = ~

G3

~

~

.

~- Eg)[ B(E O E, - AE,)] ( E

(3.150)

where B(E) is the unit step function. We substitute these expressions into the Green’s function G ; ( E ) of equation (3.147). The E‘-integral is readily done. The imaginary part ofG;(E) equals (-TI times pv(E). The real parts ReG:(E) and ReG:(E) are given by

-I.;TI’F]

(3.152)

Although complex numbers appear in these expressions, they Eare in fact real. This would hP obvious if we rcplaced the In-function by an arctan-function. We avoid this because it is more convenient to handle the many-valued character of the In-function rather than that of the arctarz-function. Altogether, the host Green’s functions of OUT model depend on three parameters. the energy gap EB, and the two band widths A& and AE,. The latter are measure3 of the awrage kinetic energies of electrons in the vakncce or conduction bands large bandwidths mean large average kinetic energies (or small eEective masses). Later, the G r m ’ s function method in Wannicr representation will be used to address the question of whether or not main group impurity atoms in tetrahedral semiconductors form deep levels in the gay. ~

300 Chapter 3. E1wtmnic Btructure

of serniconductar crystals with perturbations

Scattering states

The perturbation potential also gives rise to changes within the energy bands of the i d e a l rrystal. Of course, the w e r g i a of t h e w bands are still allowed quantum mechanically in the presence of the perturbation, so that in this rwpeet there is 110 change. However, change within the energy hands of the ideal rrysttll IS indured by the perturbation in the form of a modified density nf allowed energy levels pcr unit mnergy, i.e. in the form of a modifid density ol statvs. To calculat? this change it is expedient to introduce the Green's fiirirtion of the perturbed crystal, 1

G ( E ) - I; 4- i6

- [H+ V ' ].

(3.153)

According to formula (2.209). the imaginary part Im T r [ G ( E ) of ] the trace represents (apart from a factor -1/ir) the density of states p ( E ) of the syst,em, here thai of the crystal with the point perturbation. The Green's function of the p e r t u r b 4 crystal oLrys the equation G ( E )= G o ( E ) t G ' ( E ) V ' C ( E ) .

(3.154)

which follows at once from the definition (3.1533 of G'(E). In quantum field tbwry. this relation is known as the D p o n equatzon Using thk equation and performing some simple calculations, the DOS expression (2.209) may be brought into the form 1

d dE

- I m -In D e t [ G ( E ) ] .

p(E)

A

(3.155)

of the unperturbed crystal may be expressed in terms of the unperturbed Green's function Co(E). We seek the change In an analogous way? the DOS P O ( ! ? )

M E ) = P(E)- P O ( W

(3.156)

ol t h r nos clue l o the point perturbation. This change may be obtained from (3.155) and (3.156) as 1 d Ap(E) = --Irrt-lnDDet[l-

(3 157) G'(E)V'I. dE This relation can be used to ralculatc the total change of the DOS in the entire enerRy range between -oo and t o o . Integrating A p ( E ) over this interval, and considering the fact that G ' ( f i ) vanishes for li: -+ f m , yields

x

lm M

~ E A ~ (=.E0.)

(3.158)

3.5. Deep levels

30 1

This relation signiEes that under the action of the perturbation potential, the total number of states remains unchanged. This result already was used in section 3.4 in lhe context of shallow levels, and there it was referred to as Levznson’s theorem. Here, this theorem means that for each new state of the crystal created by the point perturbation, a state which existed without perturbation must cease to exist. The deep states in the energy gap occur therefore at the expense of band states. For each state occurring in the gap, R stale is lost fiom a band, dEAp(h’:)- -

Lap.

dBAp(E).

(3.159)

In the above derivat,ion of Levinson’s theorem, no assumptions about the spatial variation of the perturbation potential were made. This theorem, therefore, also applies for purely long-range potenlials. Thus, we have proven what was anticipated earlier in section 3.4, namely, that for each shallow level in the gap, a state is lost from a band. In Chapter 4,where we will calculate the electron population distribution over the energy levels in the gap and the bands in thermodynamic equilibrium, this result will play an important role. At certain energy values in the bands, the DOS of the perturbed crystal can display maxima or minima. One speaks of these as resonance and antz-resonance state.9. These states emcrge when the short-range perturbation polential can bind or ‘anti-bind’ states at band energies. Since the corresponding localized level is degenrrate with the band energy continuum, the localization of resonance and anti-resonance states differs from that of deep level states in the gap - with increasing distance from the center, the eigenfunctions do not decay exponentially but oscillate with an amplitude decaying to zero arcording to a power law. 3.5.4

Correlation effects

Correlation effects, as discussed in section 2.1, are important for the N electron system of a crystal with oneparticle states localized at a deep centel. One of thcse e f k t s is based on tlie configuration dependence of the Hartlee and exchange potentials. Another results from the fact that Slater determinants, even thosc calrulatcd by means of configuration dependent Hartrw and exchange potentials, are not exact eigenstates of the N-electron Hamiltonian. The exact eigenstates are linear coinbinations of different Sl&r determinants, an rffwt which is referred to as cor~fignralzon znteraetzon (see section 2.1). The two correlation effects, ‘configuration dependenre’ and ‘configuration interact ion’, will be discussed helow for deep centers. In regard to ‘configuration dependence’, we continue the general discussion of section 3.2 here.

302 Chapter 3. Electronic structure

of semiconductor crystals with perturbations

Configuration dependence For an ideal crystal, the energy eigenvalues E , of the onepartick Schrijdinger equation have direct physical meaning. Apart from their sign, they are the ionization energies I,, of the corresponding eigenststes Y , i.e. E , -I,. In this regard, according to section 2.2, the ionization energy Iv is definned as difference

(3.160) of the total energy R t o ~ ( { v ) ”of) the N-electron system in the ionized stair {v}”, and the total energy E t o d ( { v } ’ ]in theground state {v}’. Theionized state (v)” differs from the ground state (v)” in that a particle, which in the ground state of the system occupies a oneparticle state of energy E,, is transferred to a oneparticle state of energy 0 corresponding to the vacuum level. Like in section 2.1 one says that a31 electron is removed from the system. This expression has to he used with care, however, for if taken literally i t misrepresents the charge neutrality of the system. The total energy of the ground state follows, according to formula (2.541,by summing all occupied oneparticle energies, followed by subtraction of the electrostatic interaction energy of the electrons because the latter is doublecounted in forming the sum of one-particle energies. The equation E , = - I , is the content of Koopman’s theorem, which was explained in section 2.1. The essential requirement for the validity of this theorem is the approximate population independence of the Hartree and exchange potentials or, more generally, of the effective one-particle potential. This requirement is not satisfied for a crystal with a point perturbation. The potentials depend on the number n of electrons occupying oneparticle states localized at the center, and so do the energy eigenvalues E , of the one-particle Schrodinger equation of the crystal with a point perturbation. We will denote these oneparticle energies by E:) henceforth to emphasize this dependence explicitly. Of course, the definition (3.160) of the ionization energy is also valid in this case, but it is no longer true that I,, represents the negative oneparticle energy -E$.( That this cannot hold is immediately clear if one recognizes that the eigenvalues IT?) and E P - l ) of the oneparticle Schrodinger equation with. respectively, n and n - 1 electrons at the center differ from each other - the level EP-” is deeper than the level E p ) because the removal of an electron results in the positive core being less strongly screened, so that the remaining electrons are more strongly attracted. In this situation, we say that the electrons at the center relax on the removal of an electron from the center. If one further considers that the total energies of both systems enter in the definition (3.160) of I,, that of the relaxed system with (n- 1)electrons at the center, and that of the unrelaxed

3.5. Deep levels

303

system with n electrons at the center, there is no way to explain why the ) be equal to the negative energy difference E b t d ( { v } v )- E ~ M ( { V } 'should eigenvalue - E Z( ) of the unrelaxed system and not equal to the negative eigenvalue -EP-') of the relaxed system. In reality, the ionization energy E b t a l ( { v } v ) - E ~ M ( { v } ' )is not equal to either one, but lies somewhere in between the two. It can be shown that it is approximately given by the negative of the one-particle energy eigenvalue of the N-electron system with the fictitious number ( n - 1/2) of electrons at the center. This is plausible because the electron, during its removal, feels, so to speak, the potential with n localized electrons half of the time, and feels the potential with ( n - 1) localized electrons there also half of the time. To carry out an exact calculation of the ionization energy of a center which, in its ground state, has n localized electrons, the one-particle energy eigenvalues E P ) are not sufficient. Applying equation (2.54),these oneparticle energies provide the total energy E b ~ ( { v } ' of ) the ground state, but they cannot be used to calculate the total energy Etotal({v}v) of the ionized state with n - 1 localized electrons at the center. To obtain the latter, one also needs the one-particle energies EP-') of the N-electron system with n - 1 localized electrons. In density functional theory, the two total energies follow more directly: one determines the eigenfunctions of the Kohn-Sham equation for the centers with n and n - 1 electrons, then forms the corresponding ground state densities, and evaluates the total energy functional (2.64) using these densities. Ionization, i.e. exciting an electron to the vacuum level, is only one of the various possible one-particle excitation processes of the N-electron system of a crystal having a perturbation center. Generally, one may examine an ' ~ an electron in a formerly unoccupied one-particle excited state { v } ~with state Y' of energy below the vacuum level, and a hole in a formerly occupied one-particle state v. The corresponding excitation energy Iuiv is given by the total energy difference between the excited state { Y } ~ and ' ~ the ground state {v}',

(3.16 1) As in the case of ionization, the excitation energy Iulv is not equal to the one-particle energy difference E,i - E,. Here, we are interested in excitation processes involving changes of the populations of one-particle states localized at the deep center. There are different types of such processes. Firstly, in the final state, all electrons may still be localized at the center, but with one electron having changed its localized one-particle state. Such excitations are called internal transitions of the center. Secondly, an electron originally localized at the center may undergo a transition to the bottom of the conduction band. This is referred to as a donor transition Thirdly, an electron

304 Chapler ,7. Electronic structure of semiconductor crystals with perturbations

Figure 3.11: Reletion between donor and acceptor ionazation levels. from the t,op of thevalencr band may be transferred into a previously empt,y one-particle state Iocalized at the center. This type of excitation is called an acceptor transition. The excitation energy for an electron from the t,op of the valence band to the bottom of the rnnduct,ion band defines the energy gap. As excitation energies of the N-electron syst.em of the perturbed crystal, all transition energies discussed above may be plotted in the same energy scheme, just as if they- were oneelectron levels. But. they are not: they are one-particle ezcitation levels. More strictly speaking, one has donor and acceptor excitation lcvels =1 ~ 1 ~ . excitation levels E$V - E , The donor excitation level for a center D in the neutral charge state D(O) is denoted by n(O/+),and the mceplor excit,ation level for a center A in its neutral charge state A(0) is denoted by A ( O / - ) . If the donor excitation level D ( + / 2 + ) at t.he single positively chargcd donor center D ( + ) also lies in the gap, one has a dovhly i u n i m b k , OT d o u h k donor. If only the excitation level D ( O / + ) lies in the gap, the donor D ( 0 ) is simply ionizable, or a single donor. In the general case, t,here are m d t i p k donors. An analogous terminology i s used for acceptors. Viewing the hand edges as ‘int.erna1vacuum Icvcls’ one 0ft.m refers to donor and acceptor transitions as donor and acceptor ionizut,iuab, and to & curraponding excitation energies as ionization lez~elx..

~2~

Below, we will use this terminology Acceptor ionization lcvcls may be traced back to donor ionization levels. In fact, ionizing a neutral acceptor center A(0) leaves this center in singly negative charge state A ( - ) , and a hole appear8 at the valcncc hand edge. If one subjects the centex A(-) E D ( - ) t o a donor transition D ( - / O ) then the center returns t.o its neutral charge slate D ( 0 ) A(O), and an electron appears at the conduction band edge. As a resdt of this t,wo-step ionization process of the center A(O), an electron-bolc pair is excited while the state of the center has not changed. The sum of the two ionizalion energies A ( O / - ) and D ( - / O ) equals, therefore, the (minimum) excitation energy of an electron-hole pair, namely, the gap energy E , (see Figure 3.11). Generally, for an x:cept.nr A ( Q ) in charge state &, one has (3.162)

If one measures thc acceptor ionization level relative to the conduction b u d

3.5. Deep levels

305

edge instead of the valence band edge (as is done in equation (3.162)), thus setting A’(&/(& - 1))= Eg - A(Q/(& - l)),then it follows from (3.162) that

This unified description of donor and acceptor excitation levels makes it possible to decide in simple way whether a given center X represents a donor, an acceptor, both a donor and an acceptor or none of them. One has

+

(a) a pure donor, when the ionization level X(Q/(Q 1))lies in the gap and, simultaneously, the ionization level X((Q - 1)/Q) does not;

(b) a pure acceptor, if X((Q - l ) / Q ) lie in the gap and, simultaneously, X(Q/(Q 1))does not;

+

+

(c) if both levels X((Q - l ) / Q ) and X(Q/(Q 1)) are found in the gap, the center is both a donor and an acceptor. One then calls it amphoterzc; if neither of the two levels X((Q - 1)/Q) and X(Q/(Q the center is neither a donor nor an acceptor.

-

+ 1))lies in the gap

The difference between the acceptor ionization energy X ( ( Q - 1)/Q) and the donor ionization energy X(Q/(Q 1))of an amphoteric center X(Q) is, by definition, the Hubbard energy 1.J. One therefore has

+

Since, coninionly, the Hubbard energy U is positive, the acceptor ionization level commonly lies highcr in thc gap then the donor ionization level. The number of electrons bound at a center in thermodynamic equilibrium depends on the position of the Fermi level with respect to the ionization levels. From the outset it i s clear that the ionization level X(Q/(Q + 1)) also marks that position of the Fermi level at which the charge state of the cenlei changes: if E p lies just above X(Q/(Q I l)),the charge state Q is realized, if E F lies just below X(Q/(Q l ) ) ,the charge state (Q 1) is realized. h o r n this observation one may conclude that the charge state Q occurs when the Fernii Pnergy lies above the ionization level X ( y / ( y t 1)) and simultaneously below the ionization level X((Q- l ) / Q ) , i.e. between the two levels X(Q/(Q 1)) and X((Q - l ) / Q ) . Of course, for this conclusion to be valid, both levels have to be located in the gap.

+

+

+

Configuration interaction If there is degeneracy among the various oneparticle states of an N-electron system, then there is also degeneracy between the Slater determinants formed

306 Chapter 3. Electronic structure of semiconductor crystals with perturbations from one-particle states of this type. Here, we concentrate on those degenerate one-particle states which are localized at the deep center. The existence of such degenerate states was demonstrated in subsection 3.5.2 - the vacancy in Si exhibits a %fold degenerate state transforming according to the irreducible representation T2 of the symmetry group T d of the vacancy, beside a non-degenerate state belonging to the irreducible representation A l . To analyze the consequences of configuration interaction, we use a simple model: a deep center with 3 degenerate localized one-particle states of symmetry Tz, occupied by 2 of the N valence electrons of the crystal; the remaining N - 2 electrons are in extended valence band states. The extended electrons contribute to configuration interaction only little; we disregard them in our further discussion. Neglecting spin-orbit interaction, the Slater determinants for the two electrons of the deep center are products of coordinatedependent and spin-dependent wavefunctions. Since a Slater determinant itself is antisymmetric with the respect to the exchange of two electrons, there are two possibilities for the behavior of the two factors in regard to particle exchange: either the spin-dependent wavefunction is antisymmetric and the coordinate-dependent wavefunction symmetric, or the spin-dependent wavefunction is symmetric and the coordinate-dependent wavefunction antisymmetric. From quantum mechanics it is known that, for a system with two electrons, the first case applies if the total spin S is 0, and the second case if S = 1 (other values of the total spin are not allowed). Using the three wavefunctions Iz), ly), 12) of the 7'2-state, one can combine 6 symmetric two-particle wavefunctions, namely

and 3 antisymmetric two-particle wavefunctions, namely

These are, altogether, 3' = 9, as one should expect. If spin and exchange interaction are considered, the symmetric and antisymmetric wavefunctions have slightly different energies because the exchange energy depends on total spin S , which differs for the two groups of states. Within each group there is degeneracy, however, at least within the framework of the one-particle approximation. If the configuration interaction is taken into account, this degeneracy is removed. The result of this removil can be derived by means of group theory, strictly speaking,

3.5. Deep levels

307

by means of the decomposition of symmetric and antisymmetric product representations into irreducible parts (see Appendix A). For the symmetric product one obtains (T2 x T Z }=~A1 E2 T2, and for the antisymmetric product (2'2 x T2}a= T I . In these relations, the factors on the left-hand sides are representations in oneparticle Hilbert space, while on the righthand side one has representations in two-particle Hilbert space. This means that every state on the left-hand side can host 1electron, and every state on the right-hand side 2 electrons. To avoid confusion, the representations in oneparticle space are denoted by lower-case letters in this context, i.e. by a l , a2,e, tl, t2 etc., instead by upper-case letters A l , A2,E , T I ,Tz etc., which are common in group theory. Here, upper-case letters are used for two- (or, generally, multi-) particle representations. This is chosen in accordance with the notation for free atoms, where s,p, d , f represent oneparticle states, and S, P, D , F represent many-particle states. Below, we will employ the distinction between lower- and upper-case representations wherever an ambiguity might occur. As in the free atom case, the total spin S of the many-electron state is indicated by the spin-multiplicity 2s 1, appended to the representation letter at its upper left. Using the notations introduced above, we may conclude the analysis of our model deep center by stating that configuration interaction will split its {t;}-configuration into the 4 different two-electron energy levels 'A1, ' E , 'T2,and 3T1.

+ +

+

If more than two electrons are localized at the center, and if oneparticle states of different irreducible representations are involved as, for example, in the ground state configuration {aSt;} of the neutralvacancy with 4 electrons, then the corresponding many-electron levels can be obtained in the same way as above. The only difference is that the symmetric and antisymmetric products to be decomposed into irreducible parts have more than two factors and the factors are not necessarily the same. The splitting of the many-particle levels of deep centers of crystals has its counterpart in the fine structure of the many-particle energy levels of free atoms with more than one electron. For such atoms, many-particle states, formed from one-particle states of given total spin S and orbital angular momentum L , having different total angular momenta J , give rise to slightly different energy levels. (Recall that the irreducible representations of the full rotation group, into which the products of the irreducible one-particle representations decompose, are distinguished by J . ) This fine structure splitting has its largest effect for electron shells which are strongly localized, i.e. for d- and f-shells (as opposed to s- and p-shells). One may expect, therefore, that impurity atoms with unoccupied d- and f-shells, i.e. atoms of transition groups of the periodic table, should result in deep levels exhibiting pronounced fine structure splittings. This is in fact the case, as we will see below.

308 Chapter 3. Electronic structure of semiconductor crystak with perturbations 3.5.5

Results for selected deep centers

Below we discuss the structure of several deep centers which are important either from the scientific or technological point of view. Knowledge about these centers is, in every, case the product of combined experimental and theoretical investigations. Since we have thus far treated only theoretical methods, we first present a short overview of the experimental methods.

Experimental methods One can divide the experimental methods for investigation of deep centers into two groups, on the one hand, methods which measure ground state properties of the centers, and on the other hand, methods which give experimental data on center properties in thermally or optically excited states. Among the methods of the first kind are measurements of magnetic properties, like Electron-Paramagnetic Resonance (EPR);Electron-NuclearDouble Resonance (ENDOR) and magnetic susceptibility. These methods provide data concerning the total spin S of the centers and, if anisotropy effects are measured, also spatial symmetries. The chemical identity of the centers can be determined (in addition to other methods) by means of mass spectroscopy or of Rutherford Backscattering (RBS). Measurements of the Extended-State X-ray Absorption Fine Structure (EXSAFS) provide data about the geometrical ordering of the atoms in the vicinity of a point perturbation. To investigate the excitation properties of deep centers, optical and electrical methods are available. Ionization energies can be determined by means of optical absorption spectroscopy and photoconductivity measurements, mainly in the infrared spectral region. The cross-sections of deep centers for emission of free charge carriers can be determined by means of time resolved current or capacitance measurements at pn-junctions or at other depletion layers. By suddenly applying a reverse bias at such a junction, the deep levels are lifted relative to the Fermi level The new equilibrium state of the junction corresponds to fewer electrons in the deep levels than previously. This state does not occur suddenly, however. it adjusts exponentially through emission of electrons from the deep levels into the conduction band (we assume deep donor levels here, the case of deep acceptor levels may be treated analogously). In the conduction band, the electrons are freely mobile carriers and are immediately sucked up by the positive electrode at the nregion. This results in an exponentially decaying current, which for its part leads to an increasing positive charging and. thus. an increasing capacitance of the junction. The decay time of the current and the rise time of the associated capacitance change are determined by the emission probability from

3.5. I?wp let&

309

the dttep centers. Measuring the capacitance rise time yields experimental values for the emission probability. Of particular importatice is the so-called Deep Level Trawaerat Spectroscopy (DLTS), wherein a reverse biased pnjunction is exposed to periodically repeted voltage pulses of forward {deep level filling) polarity. The recovery time for the capacitance change after a filling pulse has been switched off, is measured as a fimrtion of temperature. This fiinction exhibits maxima which, under certain conditions, can be used to determine the ionization energies of the deep centers. Ionization energy values from LILTS measurements and other thermal quililrrium technique are, as a rule, smaller than optically measured values: the lattice has time to rrlax if ionization proceeds thermally, and this lowers thc enerm of the finel stat?. Optical ionization occurs instantaneously, hence lattice relaxation is is not possible. Perhaps the best understood point perturbation is the vacancy in Si, so we initiate our discussion of particular deep centers with it. Vacaricy in Si

The defwt molerule model

of the vscancy was treated in section 3.2. It predicts the existence of two bound one-particle states, a non-degenerate al-state, and a tripiy degenerate t2-state. More rigorous calculations within one-particle approximation (3araff, Schliiter, 1980; Bernholc, Pantelides. 1980) show that the al-level can be excluded as a deep state in the gap because it lies in the valence band (see Figure 3.12a). As such, it is fully occupied having two electrons. The t 2 -level can lie in the gap depending on how many electrons it hosts. ,4t maximum, this can be 6, and at minimum 0. Thus there exist 7 charge states of the vacancy, namely t7(2+),I,’(+), i’(O), I,’(-), V(2-), Ir(3-) and V(4-). The one-particle energies of these centers differ by the Hubbard energy U. A simple estimate yields 0.3 el’ for C3. Thus, many-body effects, more strictly speaking. configuration dependencies of one-particle energies: should be important in the case of the Si-vacancy. Other many-body effects, including configuration mixing, are small, and a description of the vacancy in terms of one-particle states, albeit configuration dependent ones. is approximately justified.

Owing to the value of about 0.3 e V for U , it may be expected that three or four ionization levels codd fit in the Si gap of about 1.1e V . A4ctualIy,the donor levels 5’(+/2+), t’(O/+), and the acceptor levels V ( - / 0 ) and V ( 2 /-1 are found there (as above the acceptor levels are counted relative to the conduction band edge). In Figure 3.12 (part b). calculated positions of these levels are shown. These are not yet final positions because lattice relaxation has not yet been considered. The latter leads to changes which are discussed below (see Figure 3.13). In charge state V(Z+), no electrons are available

310 Chapter 3. Electronic structure

of

semiconductor crystals with perturbations

for the population of the t 2 level, therefore no Jahn-Teller distortion of the vacancy occurs. In charge state V ( + ) , the tz-level is occupied by 1 electron. Its energy can be degraded through a tetragonal Jahn-Teller distortion. The symmetry after the distortion is DM. For this group, the 3-dimensional irreducible representation t 2 splits into the 1-dimensional representation 62 and the 2-dimensional representation c. The bz-level lies energetically below the e-level and is single occupied. In charge state V ( 0 )the additional electron can also be hosted by the bz-level. In order to gain additional energy, the tetragonal Jahn-Teller distortion is strengthened. In charge state V (-), population of the e-level begins. Through a further distortion of the vacancy, which reduces its symmetry from D M to CzVithe e-level splits into two levels and the additional electron is placed in the deeper of the two. This level can also still host the additional electron of the charge state V(2-), with an increase of the C2,-distortion. If one takes account of the energy shifts diie to Jahn-Teller distortion, the resulting level positions are as shown in Figure 3 . 1 2 ~ .'I'he exchange interaction evidently plays only a minor role, so that practically no spin-splitting of the vacancy levels occurs. Considering that the wavefunction of the deep vacancy levels extend as far as the nearest neighbor atoms, this is understandable. A surprising result concerning the level positions in Figure 3.12~is that the donor level V(O/+) lies below the donor Ievel V ( + / 2 + ) . The usual ordering of the ionization level of the more negative charge state above the ionization level of the less negative charge state is thus reversed. Formally. it seems as if the Hubbard energy U would be negative instead of positive for the transition V ( + / 2 + ) . One therefore also calls the vacancy in Si a negatzwe-U center. Of course, the interaction energy between two electrons at the center does not really change its sign, but the increase of ionization energy due to the Jahn-Teller effect on the V ( + / 2 + ) transition amounts to only about halfof that for the V(O/l+) transition since the Jab-Teller effect is absent at the V ( 2 + ) center. If the vacancy was initially in the neutral state, i.e. with Fermi energy lying above the V ( O / + )donor level and below the V ( - / O ) acceptor level, with subsequent lowering of the Fermi energy below the V ( O / + ) level, then the vacancy will initially capture a hole from the valence band and pass into charge state IT(+), and from there, without further change of the Fermi energy, i.e. spontaneously, it will capture another hole passing into charge state V(2+). The occurrence of an effectively negatke iJ at the vacancy in Si was first predicted theoretically (BarrafF. Schliiter, 1980) and later found experimentally in correlated EPR and DLTS measurements. This phenomenon has now been observed at a number of other deep centers.

3.5. Deep levels

311

Figure 3.12: Deep levels of avacsncy in Si: a) Calculated ionization levels without Hubbard corredions and lattice relaxation. b) Levels of a) with Hubbard corrections. c ) Fsxperimetital ionization levels which, in addition, include Jahn-Teller shifts. The numbers give the level distances from the valence band edge in eV. (After Watkins, l S S 4 . )

Main group impurity atoms in tetrahedral semiconductors The elements of the main groups of the periodic table are distinguished by the fact that their valence shells arise born 3- and p-orbitals. The groupIV elements, as well as thc IV-IV,111-VI.and 11-VI binary compounds, form crystals with, respectively, diamond and zincblende structure in most caws. Relow we refer to them as 'tetrahedral semiconductors'. If main group elements appear as impurity atoms in such crystals, then they are chemically bound in a manner similar to that of the hust crystal atoms which they replace, i.e. through sp'-hybrids. That implies that the incorporation of impurity atoms should be mainly substitntional in such crystals. In the case of a compound semiconductor, the lattice site which is prpferred is the one that belongs to the chemically most similar atom of the host crystal, therefore the anion site, if the impurity atom is a nan-metal atom. and. the cation site, if it is a metal atom. The solubility of the main group elements in tetrahedral senlicntiductors is, accordingly, high. It lies between 10" for chemically very similar systems such as GaAs.Al, and l O I 5 ~ r n -for ~

312 Chapter 3. Electronic structure of semiconductor crystals with perturbations

cl v’

D2d

-

Dzd

d) v-o+d -btc

Figurr 3.13: Defect molecule model of a vacancy in Si. Different charge states of the vacancy are shown, taking into account the torresponding Jtthn-Teller distortions. On the right-hand side, the basis functions of the irreducible representations of the deep k v e l ~are indicated (a, b, c Bnd d bband for the dangling hybrids of the 4 surrounding atom). (After Watktns, l U U 4 . ) relatively different systems as, for inst ance, Si:Hg.

The perturbation polrntial of a main group impurity atom in a tetrahedral semiconductor contains, besides the short-range part, as a rule, also a long-range Coulomb part. For isocoric impurity atoms, i.e. atoms whose cores do not deviate too strongly from that of the host atom, the Coulomb potential is in general the main contribution. As shown in section 3.4, this potential leads to shallow donor or acceptor levels. These are the levels principally involved if main group elements are used as doping atoms for

3.5. Deep ler7ek

313

Table 3.6: Experimental deep l e d positions of neutral main group substitutional impurity atoms in Si (in eV). "Dash" indicates that no deep level occurs in the gap €or this particular substitutional impurity, meaning either that no localized state exists at all (C, Ge, Sn, Pb), or that such states exist but are shallow (B, Al, Ga, P, As, Sb. Bi). An empty space means that the correspondingimpurity atom is either not incorporated substitutionally, or that there is no unambiguous experimental data. The neutral vacancy (Vac.) is shown for comparison. [Data compiled from Landoldt-Barnstein, 1982.)

?r

B

Be

hk

C

-

-

A1

Vac.

-

EC

- 0.43

N Ec - 0.14 P -

Zn

Ga

Ge

As

E1, 0.32

-

-

-

+

+ 0.55

Hg Ec - 0.31

S Ec - 0.31 Se

E,

- 0.3

In

Sn

Sb

Te

E, t 0.15

-

-

Ec - 0.20

T1

Pb

Bi

Po

Cd E,

0

E,

+ 0.25

~

-

1

tetrahedral semiconductors. For non-isocoric elements of the main groups, however, the short-range potential dominates in general, and, in particular cases. gives rise to deep levels in the gap (see Table 3.6). If an impurity atom belongs to the same column of the periodic table as the host atom, one says that the two atoms are isovcalent. In this case. the Coulomb potential vanishes completely and the short-range potential remains as the only potential contribution. Isovalent substitutional impurity atoms therefore lead either to deep levels (this occurs if the isovalent host and impurity atoms are chemically dissimilar, as in the case of GaP:P; or ZnTe:O), or no localized levels OCCLU at all (this takes place for chemically very similar isovalent host and impurity atoms as in the case of Si:Ge. which forms an alloy). An important theoretical problem which has yet to be solved for maingroup impurity atoms in tetrahedral semiconductors is to understand why certain elements cause deep levels in the gap while others do not. The impurity problem mentioned was treated above within the defect molecule modeL Although group-IV elemental semiconductors were assumed as host crystals, the general results obtained there also apply to compound semiconductors.

3 14 Chapter 3. Electronic structure of semiconductor

crystah with perturbations

Accordingly, a substitutional sp3-bonding impurity atom in a sp3-bonding host crystal will introduce two bonding levels EL? and E t T of, respectively, a1 and t 2 symmetry, and two anti-bonding levels E:T and E’:; of these symmetries (see Figure 3.10). Which of these levels are located in the gap, and which are not, cannot be decided by means of the defect molecule model. On the assumption that there is a deep level in the gap. one can guess its symmetry as folIows: The perturbation potential V’ = e t - E: has negative values (or is attracting) if the hybrid energy of the impurity atom is lower than that of the substituted host atom, i.e. if the group number of the impurity atom is higher than that of the host atom. Furthermore, V ’ has positive values (or is repelling) if the hybrid energy of the impurity atom is larger than that of the host atom, i.e. for impurity atoms with lower group numbers than the host atom. If, by varying the impurity atom, the perturbation potential V’= - E: takes increasingly negative values starting from zero, then a deep level in the gap will evolve from the conduction band below a certain negative threshold value. If the perturbation potential V‘ takes increasingly positive values starting from zero, then one expects a deep level to arise from the valence band above a certain positive threshold value. Using this observation, a look at. Figure 3.10 indicates that for negative (attracting) perturbation potentials, the deep level should be the anti-bonding u1-leve1, and for positive (repelling) perturbation potentials, the deep level in the gap should be the bonding ta-level.

ER

The above conclusions are essentially confirmed by the more accurate Green’s function tight binding calculations performed by Hjalmarson, Vogl, Wolford, and Dow (1980). Some results of these authors. concerning allevels, are depicted in Figure 3.14. Since the a1-levels arise mainly from atomic s-orbitals, they are plotted against the s-orbital energy of the impurity atoms in Figure 3.14. Concerning the question of whether or not a main group impurity atom gives rise to a deep level, the indications of Figure 3.14 agree surprisingly well with experiment. For the particular case of Si this can be seen by means of comparison of Figure 3.14 with Table 3.6, which summarizes experimental results for this host crystal. The data from both sources agree that no deep levels exist in the case of the isovalent group-IV atoms C, Ge, Sn, Pb. Among the group-V substitutional impurity atoms N, P, ils, Sb, Bi, only IY gives rise to a deep level while all others result in shallow levels (even those do not exist for the isovalent group-IV atoms). In the case of group-\’I atoms, Figure 3.14 indicates deep levels for substitutional 0, S, and Se. while experimentally one also finds a deep level for substitutional Te. For impurity elements left of column IV of the periodic table, like Ga or Zn. the perturbation potential is positive, and the deep level in the gap is expected to be tz-like rather than al-like. Such levels are not well described in the TB calculations quoted above.

315

3.5. Dcep levels

0

>" QI

Y

-1

W

-2

L

I

I

I

I

I

I 11

0

I I!

t

Eandedge A

>

-1

-

Q)

u

w

-

-2

Cation Site -3

I

-30

I

I

-20

1

I

-10

I

A

0

s - Orbital Energy (ev) Figure 3.14: Deep levels for main group substitutional impurity atoms of various tetrahedral semiconductors. The anti-bonding levels of al-symmetry are shown as a function of the s-orbital energy of the impurity atoms. For further discussion, see the main text. (After Hjalmarson, Vogl, Wolford, and Dow, 1980) The fact that the perturbation potential is negative (attracting) for the antibonding al-level and positive (repelling) for the bonding &level allows an important conclusion on the nature of the deep impurity states: in both cases, the corresponding wavefunct ions have larger amplitudes at the surrounding host atoms than at the impurity atom. In this sense these deep impurity states are more host-like than impurity-like. Quantitatively, the al-level positions in Figure 3.14 differ from the experimental ones. This is to be expected because of the great simplifications made in the calculations. many-body effects, particularly, configuration dependencies of the one-electron levels, have been neglected, and lattice relaxation has not been taken into account (Scherz and Scheffler, 1993). To get an idea how these effects would modify the results, we show in Figure 3.15 the expected occupancy of the deep oneelectron levels in the particular case of Si as host crystal, using the level ordering suggested by the above-quoted calculations

316 Chapter 3. Electronic structure of semiconductor crystals with perturbations

d)

el

fI

Figure 3.15: Energy levels and their populations for sp3-bonding main group impurity atoms in elemental semiconductors of group IV,within the defect molecule model. The partial illustrations a) to h) correspond to impurity a t o m of groups I to VII as follows: a - V, b - \’I, c - VII, d - HI, e - 11; f - I. and the defect molecule model. Of the 9 electrons of a Sirgroup-1’ atom molecule (remember that only N results in a deep level in this case), 8 are hosted by the four bonding states and the remaining electron occupies the anti-bonding al-state in the gap. In a 5’t:groupVI atom molecule with 10 electrons, 2 occupy the al-level, and in the 3t:groty-VI atom molecule with 11 electrons, in addition 1 electron has to be placed into the anti-bonding tl-level. This state of the molecule will certainly not be stable. As in the vacancy case, a Jahn-Teller distortion will occur which removes the degeneracy of the ta-level and allows for a lower total energy by occupying levels shifted downwards. The defect molecules of group-111. -11 and -I atoms have. respectively, 7, 6 and 5 electrons. Of them 2 electrons are hosted by the bonding al-level in the valence band. There are, respectively. 5, 4, and 3 electrons to be placed in the bonding tz-state which forms the deep le\Tel in this case. One may also say that this level hosts, respectively, 1. 2, and 3 holes. Again, a Jab-Teller distortion will occur which lowers the total energy. Qualitative

3.5. Deep levels

317

differences from this model occui for elements of the first main group which have no occupied p-states but relatively shallow closed d-shells (Cu, Ag) or d- and f-shells (Au). Foi these atoms, d-electrons participate in chemical bonding with the host crystal. We will discuss this problem in more detail in the context of the transition metal impuiity atoms. The question of whether deep levels exist or not for a particular substi tutional impurity atom, ran he treated arralytzrally, employing the Wannier representalion of the Green’s function derived above. As lhis calculation provides further physical insight into the formation of deep levels, we will discuss it hclow, agdin taking Si as the host crystal. The Chen’s function in Wannier representation has already been determined above (see equations (3.158) and (3.159)). For the evaluation of the Koster-Slatcr equation (3.152) we need the matrix elemeuts V,,,,V,, V ,of the perturbation potential V ‘ in this representation. To calculate them we use the results of the TR approximation of scction 2.5. In the simplest veision of this upproximation, the ILamiltonian of the ideal crystal is diagonalized by the bonding and anti-bonding Bloch functions Ibtk) and In&), t = 1 , 2 , 3 , 4 , of equations (2.303) and (2.304). Thus, in an approximate sense, the bonding and antibonding orbitals IbtR) and latR) are the Wannier functions of the eigenvalue equation (3.152). We will use thmn to grt explicit expressions for V,,,,,V,,, and V,. Fiist, we write down the matrix representation of the unperturbed Hamiltonian H between bonding orbitals. It is given ’by

(btRIHI4R’)

fhGRR‘, h

(3.165)

where E; is the hybrid energy of a host atom. For the perturbed Hamiltonian EZ fV’, one has t h e same matrix elements for all R and R’ with the exception of R R’= 0. The latter elements are (btOlH iV’lbtO) -

1

S(E;

+ EL),

(3.166)

where EL is the hybrid energy of the impurity atom. Siiicc none of the matrix elements of (3.165) and (3.166) depend on t , one may identify IbtO) with the Wannier function of the uppermost valence band. Then, taking the difference of equations (3.166) and (3.165), an expression for (vO1V’(wo) G V,, follows SS

vlf, - v o , vu

s1( 6 th - Eh). h

(3.167)

The factor in (3.166) arises because the Wannier function is equally spread out over the two atomic sites of a unit cell, while the substitution of a n atom occurs at only one site. Analogoiisly, matrix elementu of fI and IT I V’ between anti-bonding orbitals may be used to obtain an expression for (cO(V‘lc0)I_ v,,,and matrix

318 Chapter 3. Electronic structure of semiconductor crystals with perturbations elements between bonding and anti-bonding orbitals yield the expression for (wO1V'lcO) V,. It turns out that all elements are the same, i.e.

v,

= v, = v,, = LG.

(3.168)

We seek deep level solutions Et of the Koster-Slater equation (3.152)located in the gap, i.e. with 0 < Et < EQ. For such energies, the imaginary parts of G:(E'} and G,!(E) are zero: and equation (3.152) transforms into

Solutions of equation (3.169) within the gap do not exist for all possible values of the 4 parameters entering, i.e. the gap energy Eg, the two band widths AE,, A&, and the perturbation potential constant Vo. This is evident if one considers the particular case AE,,v >> Eg,which can be realized. The arguments of the logarithmic functions in G : ( E ) and G:(E) are close to 1 in thia case, and the logarithms themselves become negligibly small. If one assumes, in addition, that A & = AEw,then

+

Re[G:(E) G:fE)] -

(F)--&[,,/=. fi]

(3.170)

With this exprwsion! the deep level condition (3.169) takes the form (3.171) It has solutions Et only in the energy gap 0 < Et < E,, within which Et has to be restricted anyway because the deep level condition (3.170) is valid only there. For negative Vo, one necessarily has Ef > Eg/2,and for positive Vo, Et < E,/2. For Vi -t +m, El approaches the value E,/2, i.e. the deep level is pinned at the midgap position. This corresponds to the pinning of & to the vacancy level discussed in subsection 3.5.2. In order for a solution Et of equation (3.171) to exist at all, lV01 must exceed a minimum value J V O ) ~ ~ ~ given by

(3.172) This minimum value of (Vo(follows from equation (3.171) by identifying Et with the lowest possible deep level position in the gap namely Et = 0 in the case of positive Vo, and by identifying Et with the highest possible value of Et namely E, in the case of negative VO.Considering equation 13.1721, the solution of equation (3.171) may be written in the closed form

3.5. Deep levels

319

Table 3.7: Perturbation potential matrix elements Vo = (1/2)(~: - $) (in eV) for substitutional impurity atoms of main groups in Si, calculated by means of HermanSkillman s- and p-orbital energies. The latter are partially reproduced in Table 2.2. (After Herman and Skillman, 1963.)

B e B

1.57 Mg 2.18 Zn 1.82 Cd 1.91 Hg 1.87

0.08 A1 1.05 Ga 0.88 In 1.14

TI 1.17

C -1.41 Vac 8.28 Ge -0.04 Sn 0.35 Pb 0.47

N 0 -3.04 -4.8 P s -1.12 -2.31

F -6.71

c l

-3.56 As Se Br -0.99 -1.97 -2.98 Sb Te I -0.42 -1.22 -2.03 Bi Po At -0.24 -0.96 -1.68

According to condition (3.172) for a deep level to be formed, the absolute value of the potential and the energy gap must be relatively large, and the bandwidth relatively small. That lVol needs to be large enough is obvious. The bandwidth must be sufficiently small because it corresponds to the average kinetic energy, which militates against binding. The gap is the energy range where the deep level has t o be placed, so it should not be too small. EnIarging the gap and lowering the bandwidth increases the likelihood of forming a deep leveL The condition lVol > IV0lmifi may even serve as a quantitative guide, as the following numerical example for impurity atoms in Si demonstrates. In this case one has Eg = 1.1 eV, and AE, = A E , = 3.3 eV is a reasonable choice. With these values, IVolmiplis 1.21 e V . The perturbation potentials V-0 for impurity atoms of the main groups II to VI of the periodic table, calculated by means of equation (3.167), are listed in Table 3.7. According to our model, a deep level should exist for lV0l > 1.21 eV, and should not exist for IVoi < 1.21 eV. Comparison with experimental results €or deep levels of substitutional impurity atoms in Si shown in Table 3.6 reveals that

320 Chapter 3. Electronic struetrue

of semiconductor crystals

with perturbations

the rritprion is correct in all caws. pxrept for In and TI, which are close to the border line of our model but still on thp side where no deep level6 exist. while experimentally surh 1wrls art- found.

Transition metal impurity atoms The transition metal ('YM)-impurity atoms of the iron group, i.e. those with closed ad-shells (see Table 3.2), are incorporated in Si crystals prdominantly on interstitial sites of tetrahedral symmetry, while the 5d-TM atoms prefer substitutional cation sites. The position of t h e 4d-TM atoms in Si lies between the two, i.e. both interstitial and substitutional incvrpuretions are observed. In III-V and 11-Vl semiconductors, all three groups of TM atoms prefer substitutional incorporation on cation sites. The solubility of most transition metal elements in Si is relatively small, lying in the range of rm-3. Higher values in the range of 101frm.3arc? reached in tetrahedral compound semiconductors, and for M n in GaAs the solubility reaches as high as 10" ~ r n - ~R;In . also is a transition metal which gives rise to alloys with cerhizl 11-VI compound semicondudors, such as, for example, (Zn, b1n)Te or (Cd.hln)Te. Semiconduct,or alloys exist also with Fe and Co. In tetrehdral semiconductors, most of the TM atmns form deep levels. Among them, the levels of 'I'M-elements with closed 3d-shells (the so-called 3d-TM cleinents) are by far the bestst known. Concerning 4d- and 5d-TM imyurily atoms, it is understood that. deep levels also exist in their rasp (Beeler, Andersm. Sdief€lcr, 1985 and 1990; Alves, Leihe, 1986). This is not surprising since the Thl elements are all chemically very similar. Here, wc restrict our considerations to the 3d-TM atoms. Them are the elements Sc! Ti, V, Cr, Mn, Fe, Go and Xi. Sometimes, the neighbors of Ni to the right in the periodic table, CU and Zn? are also included, although, owing t.o their respective 3d1'4s- and 3d1'4s2-confi~~ations. these are not in fact transition metals. In tetrahedral semiconductor crystals, however, they behave similarly. Special interest in the investigation of 3d-Thl atoms results mainly from t,he fact that their deep levels play an import.ant part in electronic devices. They can serve as recorrilrizlation centers through which the lifetime of non-equilibrium carriers is shortened, or as capture centers for h e carriers which partially compensate the effect of dopant atoms. These effects are imclesirable in most caws, thus one must. try to minimize the contamination of the devicw by 3d-'I'M atoms. Occasionally, 3d-TM impurities can elso be useful. 'I'his happens, for example, in the case of Cr in GaAs which, due to its compensating effect! makes the crystals serui-insulitting. There are numcrouq experimental investigations on 3d-'l'h.I impurit.y &oms in tehahedral semiconductors (interested readers are referr4 to the book by Fkurov and Kikoin, 1994). Beside donor and acceptor transitions alsn internal transitions of the 3d-TM imp1irit.y atoms are observed. Below we

3.5. Deep levels

'

32 1

concentrate on the donor and acceptor transitions. Figure 3.16 contains data related to this, as well as showing the ionization energies of the free 3d-TM ions. The latter are larger than the ionization energies of the 3d-Thl atoms hosted by semiconductor crystals by more than one order of magnitude. The case of free atoms is also striking in regard to the large Iyariations of ionization energies between different ionization states. In semiconductor crystals these variations are almost two orders of magnitude smaller so that, as in the case of the Iacancy, ionization levels of several charge states may be found in the e n e r a gap. Considering the substantially stronger localization of d-electrons and the much weaker screening of their interaction in comparison with the valence electrons this is surprising and needs explanation. We will return to this point later. Theoretically. the donor and acceptor ionization levels of substitutional 3d-TM impurity atoms are now well understood. A simple oneparticle model which is supported by ab initio calculations, is a defect molecule with the 3d-TM atom on a substitutional cation site at its center (see Figure 3.17). In this model, the TM atom is represented by the five 3d-orbitals and the one 4s-orbital of its valence shell, and the representation of the crystal is embodied in the four sp3-hybrid orbitals of the four surrounding cation atoms pointing towards the TM atom. The energies of these orbitals will be denoted by Ed, eS and eh, respectively. Considering the tetrahedral sgmme try of the impurity center,we decompose the four sp3-hybrid orbitals of the four surrounding cation atonis pointing lowdrdb the ThI atom into a state with A1-symmetry and three states Itah) with T2-symmetry. The s-orbits1 of tlir TM atom is deformed by the crystal into an orbital of A1-symmetry, and hhe five d-states decompose into two states ).1 with E symmetry and t h r w states It%) with Tz-symetry. The two 01-orbitals interact with each othrr tand give rise to a bonding state dwp in the valenct. band and an antibonding state deep in the conduction band. The two estates remain without bonding, their energy levels are likely to be found either in the gap or in the valence band. The two triply drgeiicratc 12 states Itah) and 1t2d) do interact mutually. 'rhe corresponding interaction matrix elements are of the type j r and bym {see section 2.6). Neglecting bbP-type matrix elements, one triply degrnertlte bonding level Eb nnd one triply degenerate anti-bonding level Ea arise. For these two levels, the same formulas apply 8s those which wmt' derived in section 2.6 for interacting hybrids at nearest neighbor atoms of an ideal crystal. Denoting the Vpp,-type matrix elemen1 by V h d , we hwe

(3.174) where

322 Chapter 3. Electronic structure of semiconductor crystah with perturbations Figure 3.16: Experimental ionizatjon energjes of free 3dTzvl ions (lower part), as well as donor and acceptor ionization energiea of 3d-TM impurity atoms and ions in various host semiconductors (upper five parts). (After Zernger, 1986.j

d'/do d%' d?d2 dyd' d%l' d?d5 d?da d%' d%lnd'o/d9 I

I

I

I

I

I

I

'

1

11

GoAs to/2.4

The corresponding thecomponent wavefunctions It2b) and Itza) of, respectively, energy Eb and E,, are

where we set 1

a2 = - ( I -

s a)'

a 2 = -1( l + - s) . 2

A

(3.177)

323

3.5. Deep levels

c -band

t2a

\

t2

4s 3d

v - band Figure 3.17: Energy level diagram of the defect molecule model for a substitutional 3d-TM impurity atom in a tetrahedral semiconductor. Explanations are given in the text. Since the two bonding and anti-bonding eigenvectors Itzb) and Itza) of equation (3.175) are linear combinations of basis functions with tz-symmetry, they also have t2-symmetry (as already indicated by the notation). It turns out that the bonding tzb-level lies in the valence band, the non-bonding elevel in the valence band or the gay, and the anti-bonding tk-level is in the gap above the e-level or in the conduction band. In regard to deep levels of the 3d-TM atom i s in the energy gap, therefore, the 2-fold degenerate e-level and the abovelying triply degenerate anti-bonding tp,-level are candidates (see Figure 3.17). The eigenfunctions of the e-level are linear combinations of the &orbitals of the TM atom and are therefore strongly localized at the latter. Considering localization of the t a b and tn,-leveh, the position of the two orbital energies F d and t h relative to each other is decisive. If Ed lies deeper than Ch, then a! > ,8 holds, meaning that the bonding tzb-level is mainly formed from the d-ststes of the TM atom, while the anti-bonding tza-level, which forms the deep level, arises to a considerable extent from the tz-components of the four sp3-hybrid orbitalb of the surrounding host crystal atoms. Exactly this picture emerges from the above-mentioned Green’s function ab initio calculations (Zunger, Lindfeldt, 1983). In Figure 3.18 the calculated oneparticle energy levels are shown for some 3d-TM atoms in Gap. The most striking feature of the above description of the electronic structure of substitutional 3d-TM impurities is the existence of deep levels whose wavefunctions are spread out over the surrounding host atoms. These levels

324 Chapter 3. Electronic structure of semiconductor crystals with perturbations Figure 3.18: Calculated oneelectron energy levels of substitutional 3d-TM impurity atoms in Gap. (After Zunger and Lindtfelt, 1983.)

4a

3

*\1-

-jFq

-3 -2

are host-like to a certain extent rather than purely TM atom-like. This picture stands in remarkable contrast to the so-called ionic model, which was long believed to be correct for substitutional TM atoms also. In this model, the deep level is strongly localized at the TM atom. Today there exists clear experimental evidence that this is not the case. The ionic model is based on the so-called ligand field theory. This entails an approach to the deep level problem for impurity atoms which differs fundamentally from the one used in this book. Within ligand field theory, it is assumed that an impurity atom X in the crystal has essentially the same electronic structure as the free X"+-ion, where V + means the oxidation state of the impurity atom in the crystal introduced above. The energy levels and wavefunctions of this ion are weakly disturbed by the crystal field at the impurity site, an effect referred to as crystal field splitting. While this model applies relatively well in the case of ionic crystals (where ch lies deeper than c d ) , it evidently fails in the case of the covalent or partially covalent tetrahedral semiconductor crystals. The oxidation states of ligand field theory are, however, also r e produced in the approach taken here. To demonstrat,e this we consider the example of a Co atom substituting the metal atom in a 111-V compound. There are 9 electrons in the 3d74s2 valence shell of Co, and 5 in the valence shell of the group V atom to which the Co atom binds. 8 of these 14 electrons are hosted by bonding states of the deep center (2 by the bonding al-state, and 6 by the bonding t 2 state). S i x electrons remain for the population of the deep impurity states, i.e. the non-bonding e-states and the anti-bonding t2-states. Since 9 electrons are expected at a neutral Co atom, the oxidation state of Co in a 111-V compound is Co3+. Substitutional Co in a 11-VI compound has the oxidation state Co2+ since one additional valence electron is provided by the group VI atom of the host crystal.

3.5. Deep levels

325

3.0 I

-552

2.0

LT W

z w 1.o

0.0

Figure 3.19: Calculated inultiplet structure for substitutional 3d impurity atoms in ZnS. [After Faatio, Caldas, and Zungcr, 2984) Ligand field theory accounts for the fine structure of the deep impurity levels, which is not explainrd by the approach taken above. Although this approach provides qualitative understanding of deep 3d-TM centers in tetrahedral semiconductors, the excitation energies it yields are not correct quantitatively, particularly not for the internal transitions of the 3d-TM centers. To be quantitatively correct, this approach must be refined by including many-body effects, in particular, configuration interaction. While the latter effect does not play an important role for main group impurity atoms, as we have scen above, it becomes important for T M impurity atoms with their strongly localized open d-shells (as was specultited in section 3.5.4 using the analogy with frcc I'M atoms). Figure 3.19 demonstrates this in thc case of TM atoms in ZnS.

The differences between the donor or acceptor ioniaation energies of a particular TM atom in different host crystals exhibit interesting behavior. Experimentally one finds that these diffeiences are almost independent of the atom considered. Therefore the ' ionization-encrgy-versus-'I'M atom curves' of different semiconductors are parallel t o each other and ran be translated to overlay by rigid displacements along the energy axis (Caldas, Fazzio, Zunger, 1984; Langer, Heinrich, 1985). In Figure 3.20 this fact is shown for acceptor

326 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.20: (a) Experimental acceptor ionization energies of 3d-TM impurities in 111-V semiconductors. (b) Experimental donor (open symbols) and acceptor (filled symbols) ionization energies of 3d-TM atoms in IFVI semicoIiducton. (After Langer and Heinrich, 1985.)

and donor ionization levels of a 3d-TM impurity atom in a series of I11 V and 11-VT compound semiconductors. The rigid displacement along the energy axis has an important physical meaning as we will discuss further below. Here, we will provide a simple explanation for the similarity of the 'ionization-energy-versus-TM atom' curves in different semiconductors (Tersoff, Harrison, 1986). In this discussion we will use the defect molecule model of 3d-TM impurity atoms presented above, in addition, however, the dependence of the energy levels f d and Eh on their population will be taken into account in a self-consistent way. Let n d , n h and n:, n: be the electron numbers in, respectively, the atomic d- and sp3-hybrid orbitals, the former after charge transfer between the impurity atom and the crystal has taken place, and the latter before. The dependence of the levels Ed and ~h on population may, in the linear approximation of equation (3.25), be written as Ed

= t d0

+ Ud(nd -

r i 0d ) ,

Eh = €1,0

tJh(nh - nh). 0

(3.178)

Here c,: :E denote the energy levels without charge transfer, and [Jd, U h are, respectively, the Hubbsrd energies of the d- and the h-states of the sp3hybrids. The electron numbers in the bonding and anti-bonding ta-states will, respectively, be denoted by rib and n,, and the electron number for the estates by ne. Of the total number of electrons of the defect molecule, n d = nbu! 2

+ nap2 in,

(3.179)

3.5. Deep levels

327

are in d-states, and

(3.180)

n h = n b P 2 +,,a2

are in h-states. The bonding T2-level Eb lies in the valence band, its three states are therefore completely occupied and it holds n b = 6. The difference 6 = (Ed - € h ) / 2 between the d- and h-states adjusts self-consistently: if Eh increases, then 6 becomes small and, with this, also ,B and n h are small. As a consequence of this, n d increases which, according to (3.178), results in an increase of € d and, hence, an increase of 6. Self-consistency is then achieved if the two equations (3.178) for the level positions t d and Eh are satisfied with values n d and n h from equations (3.179) and (3.180), respectively, and values a 2 , P2 from (3.177). For the level difference 6 = ( € d - ~ h ) / this 2 condition yields

" ci)

,(nu

+uh

1 6 + 6 ) - -(nu 2 - 6)-A - n:)]

(3.18 1)

where 260 = denotes the level distance without charge transfer. The crucial point at this juncture is that [ I d , because of strong localization of the &electrons and wcak scrcening of their interaction, has very large values (on the order of magnitude 10 e V ) . Therefore, the factor multiplying U d in equation (3.181) must practically vanish itself in order to satisfy this equation, leading to the approximate rclation

(E!

1

+ 6 ) + s1( n u- 6 ) A-6

i ( ~ a

0

1 ne - n d w 0,

(3.182)

which uniquely determines 6. According to this relation, the d-level adjusts in the crystal in such a way that the charge transfer between the crystal and the d-shell practically vanishes (of course, it cannot vanish completely because adjustrrierii of the levels requiies a bit of rhargc to bc transferred). This adjustment is called the 'neutrality level'. According to equation (3.181), the self-consistent value of h and, thus, also the energy separation tId-€h = 26 between the d- and < h level, is indepmrlent of Ch. Since t h is the only quantity which changes substantially in the series of the 111-V and 11-VI-cornpound semiconductors, the separation between Ed and Eh is the same in different semiconductors. This also holds for the anti-bonding tza-level dnd for thc non-bonding e-level, i.e. for the two levels which are candidatev fol deep levels in the gap - also, their clihtance from c h is independent of Eh. For the ta,-level this follows from equation (3.174), and for the e-level from the fact that its position is determined by the location of the d-level in the c ~ y s tal. We thus arrive at the conclusion that the deep levels of 'I'M atoms in

328 Chapter 3. Electronic structure of semiconductor crystals wilh perturbations the gap are ‘pinned’ at the hybrid level F ? ~of the surrounding host anions. If the coupling hetwccn the anion-hybrids and the rest-of-crystal is taken into account, the anion-hybrid energy Eh has to be replaced by the average cation-anion hybrid energy of the crystal. The rebUlt8 above explain lhe experimental finding that the ‘deeplevelversus-Thl atom-curves’ for two different semiconductors ran be tranbleted to overlay each other by rigid displacements. Moreover. they associate the magnitude of the displa~ementswith the difference between the average hybrid energies of the two semiconductors. Thls quantity has a close relation to the valence band discontinuity at a semiconductor heterostnicture. as will be discussed iu more delail in section 3.8 The large Hubbard energies of the atomic d-states of the ‘I’M impurity atoms and thFir pinning at lhc dangling hybrid lev& of the surrounding atoms are also contributo~yto the striking phenomenology noted above. in that the deep levels of TM atom8 depend only relatively weakly on their population, so that levels of several charge states can fit in the gap. In fact, if an additional electron is put in the deep tzo-lwel, and, with this, also part ofit IS added to the TM atom, the d-level of the latter will be shifted up and the bondmg tB-level will be depolarized ( a becoming smaller and /3 larger in equtliiori (3.177)). This means that electron charge density will flow from the Thl atom to the surrounding host atoms. in almost the same measure as was added t o the T M atom when the additional electron was placed in the deep ant i-bonding &-level. Since the Hubbard energies of the s- and p-shells of the host atoms are substantially smaller than those of the &orbitals of the Thl atoms, the deep TM-levels are shifted up only by several tenths of an e V rather than 10 el, for the case of the free TM-ion if one more electron is p l a ~ e dat the center (Haldan, Anderson, 1976). Cu, Ag, Au in Si These three elements (abbreviated below as ‘Nhl‘, for noble metal) play an important part in silicon device technology. As impurity atoms, they constitute effective capture centers for non-equilibrium charge carriers. Their high solubility in Si is remarkable (in the region of cmd3 to ~ r n - ~ at melting temperature), as is their fast diffusion. Their incorporation in the host crystal is predominantly substitutional. The elements tend to form complexes with other elements. for example, Au with 0. Fe or 3d-TM’s. This makes it somewhat difficult to identity the pure substitutional NM centers. Although the elements Cu, Ag and Au belong to the fast main group of the periodic table, their behavior its impurity atoms in tetrahedral semicunductors resembles that of transition metals. This is understandable if one considers that for these elements only the outer s-shell, but not the outer p-shell, is occupied, and that the closed d-sheIl (Cu, Ag) or d - and f-shells

3.5. Deep levels

32 9

(Auj lie energetically relatively shallow below the s- and p-shells. According to calculations by Fazzio, Caldas and Zunger (1985). the anti-bonding t 2 state lies in the energy gap, just as in the case of transition metals, and as in the latter case, this state arises from the interaction of the impurity-derived d-states (more exactly their tg-component) with the tg-components of the dangling bonds of the surrounding host atoms. Because of the relatively weak localization of this deep impurity state, its Hubbard energy is small so that. again. several ionization levels can fit within the gap. For all three K M atoms, the calculations result in a Nh/l(O/+) donor level, and a NM(-/O) acceptor level. In the case of Ag and Au, the amphoteric character has also been conhmed experimentally. Rare earth atoms The rare earth (RE) atoms are among the following elements: Ce, Pr, Nd, Pm, Sm, Eu. Ac. Th. Pa, U. Np, P u , Am, and Cm. The valence shell configuration of the majority of these atoms may be described as 4fn6s2 with the number n of f-electrons varying from 2 for Ce to 14 for Yb (see Table 3.3). Exceptions are Gd and Tb which have one 5d valence electron in addition. The investigation of RE impurities in Si as well as in the 111-V and 11-VI semiconductors has received new stimulus very recently. The reason for this is the luminescence of RE impurity atoms in these crystals in the technologically interesting visible and infrared spectral regions. In equilibrium the RE atoms are installed both substitutionally as well as interstitially. The equilibrium solubility of the RE atoms is, however; rather small. For practical applications, non-equilibrium incorporation techniques like ion implant ation must be used, although, generally, only parts of the implanted atoms are optically or electrically active. For Erbium (Er), impurity concentrations of about lo1’ ~ r n have - ~ been reported in Si, and of about 10l8 C W I - ~ in Ga4s. The technologically interesting luminescence discussed above is caused by internal transitions within the 4f shell of the RE atoms, more strictly speaking, between the various levels arising from the many-particle levels of the 4f-shell under the influence of the crystal field. The 4f-orbitals remain, in fact. almost unchanged by installation of the RE atoms in the crystal. This may be traced back to the fact that these orbitals are strongly localized and, moreover, strongly shielded by the 6s-electrons shutting out influences of the surrounding crystal. Furthermore, for RE elements heavier than Nd, there is also strong shielding by the completely occupied 5s- and 5p-shells lying outside the 4f-shells for these elements. While internal electron transitions within the 4f-shell have been studied already for a long time, transitions to energy levels outside of the 4f-shell, including donor and acceptor transitions into conduction and valence band states of the crystal, have been subjected to more detailed investigation only recently.

330 Chapter 3. Electronic structure of semiconductor crystals with perturbations Ce Pr Nd Fm ' Sm Eu Gd Tb Dy Ho Er Tm Yb 8

!

'

I

'

I

j

1

1

1

I

"

,

6 4

2

5 0 v

P -2 g-4

k

I

E(3+/4+)

-6 -8

-10 -12

I

1I ,

I

!

I

I

/

I

I

,

!

I

I

Ce Pr Nd Rn Sm Eu Gd Tb Dy Ho Er Tm Yb

Figure 3.21: Donor ionization levels of substitutional rare earth impurity atoms in different semiconductors. (After Delerue and Lannoo, 1994.)

A rough but qualitatively correct picture of the electronic structure of substitutional RE impurity atoms in tetrahedral semiconductors may again be obtained by means of a defect molecule model (Delerue and Lannoo, 1991). In this model, the 4f-orbitals neither interact with the 5d- and 6s-orbitals of the RE atom nor do they couple with the orbitals of the surrounding host atoms. The RE atom is represented by its 5d- and 6s-orbitals only. The 4forbitals, nevertheless, are involved in forming deep states because electrons are transferred to or from them. The host crystal is described by the four sp3-hybrid orbitals at the four neighboring atoms pointing towards the RE atom. The model is therefore largely analogous to that of 3d-TM atoms described above: The five 5d-orbitals decompose into two e-orbitals and three tz-orbitals, and the 6s-orbital becomes a al-orbital under the influence of the tetrahedrally symmetric crystal field. The four sp3-hybrid orbitals at the neighboring atoms pointing toward the RE atom split into one a l - and three t2-orbitals. The interaction between the two al-orbitals results in a bonding and an anti-bonding al-state lying deep in the valence and conduction bands, respectively. The e-states remain without bonding to the host crystal, while the host and RE derived t2-states couple to each other forming bonding and anti-bonding tz-states. The differences between 3d-TM and RE impurities are essentially of a quantitative nature. The 5d-levels of the RE atoms are higher in energy and therefore closer to the hybrid levels at the neighboring atoms than are the 3d-levels in the 3d-TM case. Because of this, the e-level and the anti-bonding tz-level of the RE atom (which may lie in the gap in the case of 3d-TM atoms) are lifted into the conduction band.

33 1

3.5. Deep levels

The position of the If-shell with respect to the above levels still has to be determined. In the model by Delerue and Lannoo this is done assuming that the distance between the 4f-shell and the average 5d-level in the crystal is the same as that between the 4f-shell and the 5d-shell in the free RE atom. The ionization levels of the 4J-shell calculated by means of the above defect molecule model are shown in Figure 3.21 for various RE atoms and host crystals. Two diffcrcnt oxidation states of the RE atom, RE3+ and RE2+, have been assumed. One recognizes that, for InP, the oxidation state RE3+ is stabk in most rasrs, because the RE(3-t /4 t ) ionization levels are still below the Fermi level, while the RE(2 t/3+) ionization levels are above it. For CdTe, the oxidation state RE2+ is found to be stable in most rases. With a few exceptions, these predictions are confirmed by experimental investigations. Oxidation states can also be derived in a more direct way from the above defect molecule model. In a 111-V compound host crystal, the RE defect molecule has 2 n 5 electrons, 2 n of them from the HE atom, and 5 of them from the host crystal anion. These electrons orrupy either states in the 4f-shell, or are in bonds between the RE atom and the crystal. The bonding al- and t2-states host 2 f 6 = 8 electrons provided that they are energetically lower than the f-shell, which in fact turns out to be the case. Thus, 2 n 5 - 8 = JZ - 1 electrons remain for the f-shell. The oxidation state of a neutral RE atom is RE3+ in such circumstances, because 3 electrons are missing at the RE atom, the one f -electron and the two s-electrons in bonds with the host crystal. In 11-VI compounds, the oxidation state of a neutral RE atom should be RE2+ according to this consideration, since the host crystal anion delivers 6 electrons instead of 5. A striking feature of the ‘ionization-energy-versus-RE-atom-curves’ for different semiconductors in Figure 3.21 is that they can be brought to overlie each other by rigid relative displacements. This is reminiscent of the ionization energies of the 3d-TM atoms where the same feature is observed. The reason is similar to that of the 3d-TM case: one the one hand, the 4f -levels are pinned electrostatically to the average 5d-levels by means of the self-consistent charge exchange between the d- and the f-shell with its large Hubbard energy, and on the other hand, the 5d-levels are pinned at the average hybrid energy levels.

+ +

+

+ +

As-antisite defect i n G a A s (GaAs: A s G ~ ) The interest in this defect arises mainly from the fact that it is closely connected with the so-called EL2-center which is one of the most common point perturbations in GaAs. The EL2-center is a double donor and plays an important role in GaAs electronics. Through deliberate use of its properties, p-type GaAs, which normally arises in crystal growth, can be transformed into semi-insulating GaAs which is required for the manufacture of GaAs

332 Chapter 3. Electronic structure of semiconductor crystals with perturbations MESFET’s and other electronic devices based on GaAs. However, in regard to materials for light emitting optoelectronic devices, the concentration of EL2-centers has to be kept as small as possible because EL2 acts to quench luminescence. Whether the As-antisite defect GaAs: ASG, is really identical with the EL2-center, or if it is only part of a complex which represents the EL2-center, is still somewhat controversial today. Nevertheless, important properties of the EL2-center , in particular its structural metastability, can be explained in terms of the simple GaAs:Asc, antisite defect alone (Chadi, 1992, Dabrowski, Scheffler, 1992). The GaAs:AsG, antisite defect may be viewed as a special case of a substitutional main group impurity atom with sp3-bonding in a tetrahedral semiconductor. For the latter point perturbation, the defect molecule model was developed in subsection 3.5.3. Accordingly, one has a bonding and an anti-bonding A1-level, and a bonding and an anti-bonding T2-level. From ab initio calculations it is known that the bonding levels lie within the valence band, the anti-bonding A1-level in the gap, and the anti-bonding T2-level in the conduction band. Of the 10 electrons of the defect molecule, 8 can be placed into the two bonding levels within the valence band. The remaining two electrons just suffice to fill the anti-bonding A1-level in the gap. These are the two electrons which, by exciting the crystal, are transferred to the conduction band giving rise to the doubledonor nature of the center. However, the population of the two anti-bonding states is energetically costly. Therefore it is not surprising that for the As atom other positions than the substitutional one may be more favorable for minimization of the total en ergy. In the above mentioned ab initio calculations, it was shown that a displacement of the As atom by about I A in [Ill]-direction, a displacement away of one of the 4 nearest neighbor A s atoms towards an interstitial position about 0.2 A below the plane spanned by the other three As atoms (see Figure 3.22), wsiilts in a relative totdl energy minirrium which lies only 0.25 eV above the absolute minimum of the substitutional As-site. In addition to the stable substitutional state of the defect, there exists, t h e r e fore, a metastable interstitial state. Thc stable state ia separated from the metastable by an energy barrier of about 0.8 eV, and in the reverse direction the barrier amounts to about 0.34 eV. If onr of the two donor electrons at the substitutional center is optically excited, the barrier decreases siibstantially and a thermal transition into the metastable state is possible. In this state the center is not capable of capturing the electron whir11 was previously optically excited into the conduction baud. This electron remains there resulting in the experimentally observed persistent photoconductivity. Only by heating the sample above toom tempwature can the As atom return to Its stable substitutional site. The main reason for the relatively small energy difference between the substitutional and interstitial locations of the As atom iu that, between an

333

3.5. Deep levels

Direction

Figure 3.22: Geometry of the stable and metastable state of the As-antisitc defect in GaAs. (After Dabrowski and SchrJ’ler, 1992.) interstitial As atom and Ohe three adjacent, A s atoms, st,rong sy2-bonds can be formed similar to the bonds in the graphite structure of carbon and that of g a y As. The non-bonding p-orbital of the A s atom, which lics rclatively high enmgetically, does not increase t,he total energy of thc interstif,ial s h t e because it remains largely unoccupied, the two remaining electrons are hosted by the dangling bond of t,hc fourth remotc lying As atom. DX-centers

A metastability similar to that of the C:aAs:AsG,, ailtisite defect is also observed at ot,hcr point perturbations, among them at t,he so-called L)X-cent,ers in GaAs and (Ga,Al)As mixed crystals (Lang, Logan, 1977). The microscopic nature of these centers was a puzzle for a long time. It was only V and clear that they were related to impurity atoms or the main groups l VI of the periodic table which normally are incorporaled subst,itiitionally on cation and anion sitcs, respectively, and form shallow donors. lTndcr certain conditions hydrostatic pressure or strong Ti-type doping in the case of GaAs, and AlAs mole fractions larger than 22% in the case of (Al, Ga)As deep levels emerge from these shallow donors. Originally this transition was attributed to the formation of a defect complex involving the donor atom (a)and ti11 iinknown point, perturballon (X). Now, it is clear t,hat the D X center j u s t represents another state ol 1he shallow donor. In contrast to the lat,t,er,the donor atom of a DX-ccntcr is not neutral, but singly negatively charged, and its installation does not occur at a substitutional site but at an interstitial site. If the special Conditions mentioned above are not fuliilled, t,hen the shallow donor represents a stable state of the impurity atom, the DX-state is metastbble. However, as for the EL2-center, thc rclalive total energy minimum of the DX-state lies only slightly (about 0.2 e V ) above thc absolute minimum of the shallow donor state. The reason for this energy balance is the sainc as in the E L 2 case, namely the energetically costly populalion of the anti-bonding A1-level at the shallow donor. In the singly negatively charged state which the donor takes under high doping, (and ev-

334 Chapter 3. Electronic structure of semiconductor crystals with perturbations idently also under the other above-cited conditions), 2 electrons must be hosted by this level. The energy costs thereby become so high that the DXstate is energetically more favorable for the impurity atom than the shallow donor state. The donor atom - to be concrete we will identify it below with a Si atom - moves into the interstitial position shown in Figure 3.22, where it can enter into sp2-bonding with the 3 surrounding As atoms. The nonbonding porbital of the Si atom remains largely unoccupied, instead the more deeply lying dangling hybrid orbital of the fourth more remote lying As atom becomes occupied (Scherz, Scheffler, 1993). This model explains in a natural way why the DX-center exhibits persistent photoconductivity. By optically exciting an electron at the DX-center into the conduction band, the center passes into its neutral charge state. This means that the substitutional shallow donor is a stable state of the point perturbation. Therefore, optical excitation transfers the DX-center into a shallow donor in its ground state, and the donor is not able to capture the excited electron from the conduction band. It remains there for a longer time and gives rise to persistent photoconducting. Only after thermal excitation does the center return to its original metastable state. The structural metastability observed at the DX- and EL2-centers is not restricted to these point perturbations, but represents a relatively common phenomenon in tetrahedral semiconductors. It arises, evidently, from the fact that (depending on the number of electrons to be placed at the center), either the sp3- or the sp2-hybridization of the s- and p-orbitals of the impurity atom allow for a lower total energy of the perturbation center, jointly with the fact that the two kinds of hybridizations lead to different atomic structures - the sp3-hybrids to the tetrahedral diamond structure, and the sp2- orbitals to the graphite structure. Whether or not a metastable state actually exists for a particular point perturbation can, however, only be decided by ab initio calculations.

3.6

Clean semiconductor surfaces

3.6.1

The concept of clean surfaces

Every solid is bounded by a surface. Nonetheless, the model of an infinite solid which neglects the presence of a surface works very well in many cases. Why is this possible? The reason is, first, that in many cases one deals with properties, such as transport, optical, magnetic, mechanical or thermal properties, to which all the atoms of the solid contribute more or less to the same extent, and, secondly, that there are many, many more atoms in the bulk of a solid than at its surface, provided the solid is of macroscopic size. In the case of a silicon cube of 1 cm x 1 c m x 1 cm, for example,

3.6. Clean semiconductor surfaces

335

one has 5 x loz2 bulk atoms and only 4 x 1015 surface atoms. Beside the above mentioned bulk properties, there are other properties or processes, like crystal growth, oxidation, etching or catalysis, which are determined by the surface atoms only. In these rases, the model of an infinite solid does not apply, of course. Moreover, in semiconductors, bulk properties like transport are often not controlled by all atoms but only by dopant atoms. Then the number N s of surface atoms per cni’ i s to be compared with the number of dopant atoms. For a semiconductor sample of thickness d , area 1 cm2, and doping concentration N D , the number N o x d of dopant atoms per c7n2 can easily become as small as the number IV, of surface atoms. This means that the transport properties of such a semiconductor sample will depend on its surface. In the history of semiconductor physics, this was recognized at a very early stage. Examples which demonstrate this are the electric rectification at a semiconductor-metal contact (discovered by F. Braun in 1874), and the unsuccessful attempts to build a field effect transistor in the late thirties of this century. The failure was caused by electron states localized at the surface which captured all the electrons induced by the extcrnal voltage. ‘l’he surfaces used in the early field effect experiments were far from perfect. They were made by cutting or cleaving a semiconductor sample in air. If at all, they were clean and smooth in a macroscopic sense, but not so microscopically. They exhibited surface roughness on the 100 nm scale, structural defwts on the 1 nm scale, and impurity atoms at and below the surface. The surface states responsible for the difficulties of the early field effect transistor were due to these imperfections. In the present section we will deal with surfaces which are free of such imperfections. Perfect surfaces of semiconductor crystals in this sense necessarily represent a particular lattice plane occupied only by chemically ‘correct’ atoms at regular sites. No impurity atoms are allowed above or below the surface. The surface is reduced to what it means ideally, the termination of the crystal. Perfwt surfaces in this sense cannot, of course, be realized in practice. One may only try to approximate them so closely that the existing imperfections do not essentially change the properties of the surface as compared to the properties which a perfect surface would have, if it really could be made. One calls such almost perfect surfaces clean. Although this term refers only to chemical composition, it also implies structural perfection or atonzc smoothness. There are essentially t h r w ways to manufactwe clean surfaces. All thrw need ultrahigh vacuum (UHV), i.e. pressures below Torr: (i) Treatment of imperfect surfaces by ion bombardment and thermal annealing (generally in several cycles). (ii) Cleavage under UHV conditions (only surfaces which are cleavagc planes

336 Chapter 3. Electronic structure of semiconductor crystals wilh perturbations of the crystal can be made in this way, of course). (iii) Epitaxial growth of crystal layers by means of molecular beam epitaxy (MUE, see section 3.7). Although most surfaces used in practice are not clean but imperfect, studies of clean surfaces are also of practical importance. This is due to the fact that imperfect surfaces are such complex subjects that it is very difficult to approach them directly. One first has to reduce the level of COEApleXity by considering clean surfaces for investigation. Later, the complexities are introduced step by step and examined for changes. In this way, real imperfect surfaces may also be undeistood. Clean surfaces are, howevei, important themselves. It was already pointed out that these are the surfaces upon which epitaxial crystal growth proceeds in MUE. In this section, we will deal with the atomic and electronic structure of clean semiconductor surfaces. The basic principles will be treated in subsection 3.6.2 for atomic structure and in subsection 3.6.3 for electronic structure. Particular suxfuces are discussed in subsection 3.6.4 taking Si and GaAs as examples.

3.6.2 Atomic structure of clean surfaces A geometrical construction which is of special significance in describing crystal surfaces is that of lattice planes, which we will now describe. Lattice planes of 3D crystals Such planes are usually denoted by Miller indices ( h k l ) where h, k , 1 are the integer reciprocal axis intervals given by the intersections of the lattice planes with the three crystallographic axes. The symbol (loo), for example, dcnotes lattice planes perpendicular to the cubic x-axis, (111)means lattice plunes perpendicular to the space diagonal in the first octant of the cubic unit cell, and (110) denotes lattice planes perpendicular to the face diagonal in the first quadrant of the ry-planc of the cubic unit cell. In the case of trigonal and hexagonal lattices, four crystallographic axes are considered (three instead of two perpendicular to the c-axis). The lattice planes then are characterized by four indices ( h k i l ) instead of three. The first three, however, are not independent of each other, since i h k = 0. The (hkil) are sometimes termed Bravais ~ T L ~ ~ C C R . A particular geometrical plane can also be characterized by its normal direction. To define lattice planes in this manner, it is convenient to write the normal direction as a linear combination of the primitive vectors bl,b2, b3 of the reciprocal lattice introduced in scction 2.4, with integer coefficients

+ +

h l , Ik2, jL3,

337

3.6. Clean semiconductor surfaces

n - h l b l + h2bz

+h ~ b 3 .

(3.182)

Hrre, reciprocal lattice vectors are understood without the factor 27~in definition (2.122). If the normal direction n is given, the roefkients ( h l h ~ h 3 ) are only determined up to a common integer factor. We chose this factor such that h l , h 2 , hs have no common divisor. Then the coeffirients hi,h2, h 3 are the Miller indices of the lattice plane under consideration, however, r e ferring to the primitive lattice vectors al,a 2 , a 3 as coordinate axes rather than to the crystallographic axes. The latter are parallel to the piimitive lattice vectors only in the case of primitive Bravais lattices. For centered laltires, like the face rentrred cubic one, they have different directions, and the coefficienls ( h l h ~ h 3differ ) from the corrirnon Miller indices. Tf necessary, one can rasily switrh from one representation to the other. The ( h l h 2 h 3 ) are, apart from a co~linionfactor, obtained by multiplying the ( h k l ) with the matrix which transforms the three nun-primitive crystallographic axis vectors into thc three primitive lattice vectors. Using the above characterization of lattice planes by their normal direction n, the lattice points

Ro = rloa1-k r20az

+ '7'30ag

(3.183)

of the lattice plane perpendicular to n which contains the zero point, may be defined by the equation

n & E h17.10

+ h2r20 + h3r30 = 0.

(3.184)

The unique character of this equation lies in the fact that only integer solutions r10,r2O1r3o are allomwl. In mathematics it is called a Dzophantin equation.

Whereas the Miller indices definr an infinite family of parallel lattice planes, equation (3.184) yields only a single plane, namely that member of the infinite family which contains the zero point. One can show that all lattice planes Ri = w a l + v z a 2

+w

a ~

(3.185)

of the infinite family of parallel plancs are obtained by replacing the right hand side of quation (3.181) by arbitrary integers 1,

(3.186)

The points of the lattice plane defined by equation (3.184) form a 2D lattice. The primitive vectors of this lattice will be denoted by fi and f2. This i s done in such a way that the three vectors f1, f2, n form a right hand coordinate system. The f1, f2 may be expressed in terms of the primitive lattice

338 Chapter 3. Electronic structure of semiconductor cryst& with perturbations

Table 3.8: Primitive surface lattice \rectors and stacking vectors of low index surfaces of semiconductor crystals.

n

Crpst'alStructures

I

I1

Diamond, Zincblende, Rocksalt.

1 WurtBite, Selenium

vectors a l , a z , q of the crystal under consideration. In order to determine the coefficients of this representation, the Diophantin equation (3.184) has to he s o l v d , which ran be done by Incans of the Euclidean algorithm (set, Bechstrdt and Enderlein, 1988j. The piirriitive vectors fi, fz obtained in this way are shown in Table [3.8) for several Iow index lattice planes of the 5 common scrniconductor crystal structures. IJsing thc lattice vectors fi. f2 and ailitrary integers ~ 1 ~ 5 the 2 , lattice plane given by relations (3.183) and (3.184) can bc represented as

Ro = S l f i

+ sgh.

(3.187)

The lattice planes Rr defined by equations (3.185 and (3.186) can be written in the form

where f.3 is a vcctor complementing F1 and f 2 to form a s ~ of t primitive lattice vrctors Fi, f2, F3 of the 313 bulk lattice of the crystal. The vector €3 can br detcrinined from thr Diophantin equation (3.186) for 1 1 in just the same way, i . ~ .by means of the Euclidean algorithm. BS the vectors Fi,fid were obtained above. The choice of $3 is not uniquc. of course, and any vector fi which differs from f 3 hy a vector aithiii the latticr plan? can also be used. We call f 3 the stacking vector because it determines how the lattice planes arc stacked in the crystal. The considerations above show that the primitive unit cell of a crystal may be chasm as a parallelrpiyed with one of its pair6 of parallel laws parallel to a gwen lattice plane. This implies that the structure of a crystal ~

3.6. Clem semiconductor surfaces

339

Figure 3.23; Construrtion of a crystal from its lattice planes. The lattice points of a given plane are occupicd by identical atoms. Different planes may host different atoms if the crystal has a basis. In the figure, a crystal with o~ilyone atom per primitive unit cell is shown. may be characterized as consisting or parallel lattice planes which are displaced with respect to each other and which are each o c c i i p i d hy atoms of a particular species (see Figure 3.23). 'I'o be more specific, the lattice plane Ro is occupied by atoms of species 1, the next plane, displaced by 6 with respect to the first onc, is occupied by atoms of species 2, etc., and the plane displaced by FJ is occupied by atoms of type J . It may happen that two OT more atoms of the basis are located at the same lattice plane. In that case an ntomic layer consists of two or more basis atoms. The lattice plane Ro 7j completev the constsuction of a crystal slab which, in the vertical direction, encompasses exactly one yrimitivc bulk unit cell. This slab is callcd a p r i m i t i v e crystal slab. A lattice plane occupied by atoms is referred to nb: a t o m i c layer. The second primitive crystal slab begins with a lattice plane occupied by atoms of species 1 and is displaced with respect to the zero-th plane of the first layer by the stacking vector f3, followed by a plane with atoms of species 2 which is displaced by f3 t ?2 etc., the last plane of the second primitive crystal slab being occupied by atoms of species J and displaced by f3 T'J, The second primitive crystal slab is followed by another J planes dlvplaced with respect to the first slab by 2f.j instead of f3. The crystal can therefore be thought of as consisting of successive primitive crystal slabs sitiihtcd one above the other, each of which consists of atomic layers containing difkrmt types of atoms and which are laterally displaced with respect to onc another. The location of an individual atom can bc specified by the number 1 of the primitive crystal slab, the numbers j of the atomic sublattice and the integer coordinates s1, s2 witahinthc lattice plane. The position R(j, I , s1, s2) of an atom j in crystal slab 1 can therefore be written as

+

+

340 Chapter 3. Electronic structure of semiconductor crystals with perturbations

(3.189) The complete set of locations of atoms in an infinite 3D crystal can be obtained by assigning all possible integer values from --oo to +m for 1, s 1 , s 2 , and all integer values from 1 to J for j . The normal direction n, upon which the whole construction of lattice planes is based, is arbitrary in the case of an infinite 3D crystal. Any choice of n yields the same crystal. We can immediately employ the above representation of an infinite crystal in describing a crystal with a surface. The normal direction n is then, however, fixed by the direction perpendicular to the surface. We start with the description of a crystal having an ideal surface. The exact meaning of ‘ideal’ in this context is explained immediately below.

Ideal crystal surfaces Regular crystal sites

We consider a crystal surface given by a lattice plane perpendicular to n as defined by equation (3.184). A crystal having such a surface may be generated from an infinite crystal of infinite extension by removing all atomic layers above the surface and retaining those below. Since the forces acting on atoms situated beneath a lattice plane in an infinite crystal are also partially due to the atoms located above the plane, we can, in general, expect that the forces acting on atoms in a crystal with a surface should differ from those acting in an infinite crystal. The deviation from the infinite case, however, diminishes with increasing distance of atoms from the surface and we can thus assume that the forces acting on, and hence the positions of, atoms deep inside the crystal bulk are, to a good approximation, the same as those in an infinite crystal. This is, however, not true for atoms situated near the surface, and the forces acting on them are appreciably different, resulting in displacements of atomic positions with respect to those of the infinite crystal. We will discuss these displacements below. Here, we assume that they are not present and, correspondingly, the atoms at and immediately below the surface have the same positions as they would in an infinite crystal. The term ideal surface is used to refer to this configuration. The atoms of a crystal ) having an ideal surface are thus located at the positions R(j, 1, ~ 1 , s ~given by equation (3.189), however, only positions below the surface plane are occupied. These obey the relation

n . R(j, 1 , s 1 , 5 2 ) = I + n . Tj 5 0.

(3.190)

The surface or first atomic layer is obtained if the left hand side of this relation is taken to be zero. A solution of (3.190) is 1 = 0 and = 0.

3.6. Clean semiconductor surfaces

34 1

The latter condition signifies j = 1 since 71 = 0 is assumed. Thus, the first atomic layer corresponds to the particular lattice plane perpendicular to n which goes through zero and whose lattice points are occupied by basis atoms of species 1. There may be other ?j beside ?I which, although not being zero themselves, have a zero projection n .3with respect to n. Then the basis atoms of this species j are also located in the first atomic layer. They are shifted with respect to the atoms of species 1 by a vector Tj parallel to the surface. Such multiple-species occupancy of an atomic layer occurs, for instance, in the case of (110) surfaces of diamond and zincblende type crystals. In this case one has two atoms in each primitive unit cell of the 2D lattice of a lattice plane, in the case of Si, for example, Si atoms, and in the case of GaAs, 1 Ga atom and 1 As atom. The distinguished role of atomic species 1 in the above considerations results from the choice of the origin - it has been placed at an atomic site of species 1. Of course, the coordinate system may be shifted in such a way that its origin coincides with the location of any other basis atom. In each case a different surface is obtained. Even if the basis atoms are chemically identical the surfaces may differ from each other in a topological sense. In the case of diamond type crystals, for instance, two topologically different (111) surfaces exist, one with three nearest neighbors above and one below the surface and another with one above and three below. The latter surface is more stable than the former one and the latter is meant if one refers to a (111) surface. For other crystal structures and surfaces the situation is similar. If topologically different surfaces exist for a given set of Miller indices, generally, one of them is more stable than any other, and that is the one which is commonly realized in experiment and studied theoretically. We first consider the translation symmetry of an ideal surface and the corresponding lattice. Translation symmetry and lattices of ideal crystal surfaces

The translation symmetry of a crystal with an ideal surface can be derived from the following observations: (i) Only translations within the surface lattice plane perpendicular to n are admissible. Any translation leading out of this plane would alter the spatial location of the surface and thus would not transform the system ‘crystal with surface’ into itself. The symmetry group of translations and, correspondingly, also the lattice of a crystal possessing a surface are, therefore, only 2-dimensional, although the crystal with surface is 3-dimensional in extent. (ii) Only those translations are admissible which transform each atomic layer into itself. The construction of a crystal, layer by layer, as described above implies that if a particular translation transforms the first atomic layer into

342 Chapter 3. Electronic structure of semiconductor crystals with perturbations itself, this also holds for any other layer. It immediately follows that the group of translations of a crystal with a surface is identical to the translation group of the first atomic layer. The lattice vectors r of a crystal possessing an ideal surface perpendicular t o the normal direction n are therefore those of equation (3.187), so that

r =slf1+

s2f2

(3.191)

holds. The vectors f1, f 2 are the primitive lattice vectors of the 2D lattice of the ideal surface. The primitive unit cell of the crystal with surface has the form of a prism bounded above by the parallelogram spanned by f1, f 2 and extending to minus infinity in the direction of -f3. For the low-index surfaces of the common semiconductor crystal structures, the three vectors f1, f2, f 3 are listed in Table 3.8. As in the 3D case, 2D lattices may be divided into crystal systems and Bravais lattices according to their symmetry with respect to rotations and reflections. We now consider the possible plane crystal systems and plane Bravais lattices. Their point symmetry elements are necessarily rotations about axes which are perpendicular to the 2D lattice plane, and reflections at lines within the 2D lattice plane. Thus, the point groups are either C , ( n ) or C,, (nm, nmm). Since the rotation through 180' is always a symmetry element of a plane lattice, only even n are allowed. The highest value of n may be readily obtained from the derivation of the possible rotation symmetry axes of 3D crystals in Chapter 1. There, n 5 6 was found. Thus, the possible multiplicities of rotation symmetry axis of plane lattices are n = 2,4, and 6. A lattice which only contains a 2-fold symmetry axis is either a completely general oblique lattice or a rectangular lattice. The point groups of these lattices are, respectively, C 2 (2) and C2, (2mm). Lattices with a 4-fold symmetry axis also possess 4 reflection lines which are rotated through 45' with respect to each other. The point group of such a lattice is therefore C4v (4mm). Similarly, lattices with a 6-fold axis have 6 reflection lines which meet at an angle of 30'. In this case the point group is c 6 v (6mm). There are thus 4 different plane crystal systems - the oblique with holohedral point group 2, the rectangular with holohedral point group 2mm, the quadratic with holohedral point group 4mm and the hexagonal with holohedral point group 6mm (see Figure 3.24). The possible plane Bravais lattices are obtained as follows. First, one takes the four primitive lattices with unit cells which are parallelograms having either no particular symmetry, or that of a rectangle, a square or an equilateral hexagon. Then one adds additional points to each of the unit cells of these lattices in such a way that their point symmetries are not lowered. Only one new lattice is obtained in this way, namely the 'body' centered

3.6. Clean semiconductor surfaces

343

oblique

rectangular

quadratic

hexagonal

P

P

P

P

C

Figure 3.24: The 5 plane Bravais lattices of the 4 plane crystal systems. rectangular lattice. This lattice cannot be transformed by continuous and symmetry preserving transformations in the primitive rectangular lattice, nor in any other primitive lattice. So it forms an additional Bravais lattice. In all other cases, the addition of points either leads back to the primitive lattice or results in no lattice at all (i.e. it creates a crystal with a basis). We conclude that five plane Bravais lattices can be realized within the framework of the 4 plane crystal systems: only the primitive lattice in the oblique case, the primitive and the centered in the rectangular case, and again only the primitive in the quadratic and hexagonal cases (see Figure 3.24). With the last statements we have completed the symmetry classification of the plane lattices of crystal surfaces. The lattice types of the ideal low

index surfaces of the common semiconductor structures are summarized in Table 3.9. We now turn our attention to the symmetries of crystals with surface as a whole. Point and space group symmetries of ideal crystal surfaces

We wish to establish, on the one hand, point groups which transfer equivalent directions of a crystal with surface into one another and, on the other hand, space groups which transform the crystal with a surface into itself. In the latter case this implies that not only does the 2D lattice transform into itself, but so do equivalent atoms occupying the primitive unit cells. These atoms are located at positions R(j,I, sir92) given by equation (3.189). To express the 2D nature of the translation symmetry of a crystal with a surface ~) explicitly, we denote the atomic positions by Rjl(s1, s2) R(j,I , ~ 1 , s and write

344 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Table 3.9: Structural properties of ideal low index surfaces of the five common semiconductor crystal structures. Column 6 and 7 give, respectively, the number of atomic layers of an irreducible crystal slab and the number of basis atoms per layer. Bravais Lattice

Point Group

Space Group

Irr. Slab

Basis

hexagonal

3m

p3ml

6

1

square

2mm

p2mm

4

I

p-rectangular

2mm

p2mg

2

2

hexagonal

3m

p3m1

6

1

square

2mm

p2mm

4

I

p-rectangular

m

plml

2

2

hexagonal

3m

p3ml

6

1

square

4mm

P h m

2

2

prectangular

2mm

p2mm

2

2

hexagonal

3m

p3ml

4

1

prectangular

m

plml

4

2

prectangular

m

P W

4

4

hexagonal

1

Pl

3

1

(ioio)

p-rectangular

1

Pl

6

1

(1120)

p-rectangular

2

P21

4

1

(ioio)

where

represents the basis of the crystal with surface. This basis contains an infinite number of vectors corresponding to the infinite number of atoms in a primitive unit cell. The point and space group elements of a crystal with surface must leave the surface invariant, i.e. only 2D symmetry groups need to be taken into consideration. Here, as in the case of translation symmetry, we therefore also have the situation that, although a crystal bounded by a surface extends in three dimensions, its point and space groups are only 2-dimensional. The point groups of directions of crystals with plane surfaces are necessarily subgroups of the holohedral point groups of the plane lattices, i.e. subgroups of the point groups 2,2mm, 4mm and 6mm. There are exactly

345

3.6. Clean semiconductor surfaces

a 2

@ .. 2 mm

0c3 1

.4 m m

(Ii

m

4

@ ..

a 6mm

6

@ 3m

(I> 3

Figure 3.25: The 10 point groups of equivalent directions of crystals with plane surface.

10 such groups. Their stereograms are shown in Figure 3.25. This implies the existence of 10 different crystal classes which are associated with corresponding crystal systems as indicated in Figure 3.25.

The possible plane space groups can be found as follows (see Figure 3.26). To start with, it is evident that each of the 10 point groups of equivalent directions combined with the corresponding associated lattice gives rise to a space group. The space groups p l , p 2 1 1 , p l m l , p 2 m m , p 4 , p 4 m m , p 3 , p 3 m l , p 6 and p 6 m m originate in this manner. Since the point groups of the rectangular crystal system are each associated with two Bravais lattices, primitive and centered, we find two further space groups, c l m l and c 2 m m . In the case of the point group 3 m , there are two different possibilities of positioning the reflection lines relative to the hexagonal lattice vectors, either through the vertices of the equilateral hexagon of the Wigner-Seitz cell, as assumed in the case of p 3 m 1 , or such that ,hey bisect its edges. In the latter case one

346 Chapter 3. Electronic structure of semiconductor crystals with perturbations

crystal svstem

oblique 2

rect angular

Bravais lattice

ooint

1

soace

LLZ

p-rect a n g ul o r

6

2mn

I I p'g'

Ip- I c-rectangular

Figure 3.26: The 17 space groups of crystals with plane surface. has the group p31m as a 13-th space group. One should further note that the point group of directions of a crystal remains unchanged if, in its space group, a glide reflection line is substituted for an ordinary reflection line. One must therefore examine the 13 space groups already established to determine whether the substitution of a reflection line m by a glide reflection line g (i.e. a reflection in m in conjunction with a translation 7' by half of the shortest lattice vector parallel to m ) leads to a new space group. One easily finds that this is not the case for the hexagonal crystal system. In the quadratic crystal system it is possible to substitute a system of glide reflection line for one but not both of the non-equivalent reflection line system. This yields the additional space group p4mg. The additional space groups in the case of the

3.6. Clean semiconductor surfaces

347

2mn

c2mn p-square

4

-

p4

4mn p4mn

I I

I

,-hexagonal

3

p3m1

I

Figure 3.26: Continuation: The 17 space groups of crystals with plane surface. primitive rectangular crystal system are p l g l (from p l r n l ) and p2rng,p2gg (from p2mrn). The centered rectangular and the oblique crystal systems do not give rise to additional space groups. In the case of crystals with plane surface, there are therefore a total of 17 different space groups. Four or them involve glide reflections. i.e. they are non-symmorphic. The 2D point and space groups of the various surfaces of a given crystal may be derived from the 3D point and space groups of the infinite crystal under consideration. They are, in fact, the subgroups of the 3D groups which contain only those symmetry elements which transform lattice planes

348 Chapter 3. Electronic structure of semiconductor crystak with perturbations situated parallel to the surface into themselves. More precisely, they contain elements which transform directions in a lattice plane into equivalent directions in that plane as far as point groups are concerned, and transform atomic layers located parallel to the surface into themselves if space groups are under consideration. The 2D point and space groups of the three lowest index surfaces of the five common crystal structures are summarized in Table 3.9, columns 4 and 5. The point and space group symmetry of an ideal crystal surface may also be derived from the projection of the crystal onto its surface. In Figure 3.27 such projections are shown for the three low index surfaces of diamond and zincblende type crystals. Atoms from different layers below the surface are illustrated differently with sizes as indicated. After some number of atomic layers (6 for (lll),4 for (loo), and 2 doubly occupied layers for ( l l o) ) , the projections repeat themselves. A crystal slab which contains the minimum number of atomic layers necessary for completing the projection, is called an irreducible crystal slab. The point and space group symmetries of a crystal with an ideal surface are those of its irreducible crystal slab.

Relaxed and reconstructed surfaces Surface-induced atomic displacements

In the preceding section the atomic structure of crystal surfaces was considered under the assumption that the atoms of the crystal bounded by a surface occupy the same positions Rjl(s1,s2) as they did in the infinite 3D crystal, if the former is generated from the latter by removing one half of it. As already noted, this assumption is actually not valid. The atoms of the surface layer experience different forces than those acting in the bulk of the crystal, and are thus subjected to displacements from their original sites in the crystal bulk. Since the forces acting on the atoms of the second layer are in part determined by the positions of the atoms in the first, these forces are also subjected to changes accompanied by displacements in the second layer and so on for each successive layer. All one can assume is that the displacements decrease from one layer t o the next and vanish altogether at a depth that is relatively far from the surface. Here, we present a more detailed description of the surfaceinduced displacements of atoms and discuss the resulting altered symmetries as compared to those of ideal crystal surfaces. Translation symmetry of relaxed and reconstructed surfaces

We denote the displacements of atoms due to the formation of the surface by 6Rjl(sl, 4, and the new positions of atoms in the crystal with surface

349

3.6. Clean semiconductor surfaces

(1111

coiii

(1101 (111)

(100)

(110)

0 1 A 0 1 A 0 1 A O2B 0 2 6 016 0 3A 0 3A 2A

Figure 3.27: Projections of a diamond or zincblende type crystal onto its low index surfaces. sz). Then by R’j~(s1,

6Rjl(sl, s2) 4 0 f o r I -+ -m.

(3.195)

The displacements 6Rjl may be divided into two classes with regard to their effect on translation symmetry. If the latter is not affected, the displacements are termed r-e-eluzation. In this case equivalent atoms in different primitive unit cells are displaced in the same way, i.e.

350 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.28: Surface relaxation (a) and surface reconstruction (b) extending up to the second atomic layer. A 2 x 2 reconstruction is shown in part (b).

BRjl(s1, s2) = 6Tji‘jl

for all

51, s2.

(3.196)

Only the vectors 61 of the basis are altered, the lattice vectors remain unchanged in this case (see Figure 3.28a). If, on the other hand, the translation symmetry is altered, the displacements are termed reconstruction. In this case equivalent atoms in different unit cells are not all displaced in the same manner, i.e. bRjl(s1,s2) depends on sl,s2. Both the basis and the lattice are changed. We first consider the changed lattice translation symmetry of reconstructed surfaces (see Figure 3.28b). In describing reconstruction it is useful to divide the crystal with a surface into two slabs parallel to the surface, an upper slab containing the atomic layers with displaced atoms, and a lower slab encompassing all other layers, i.e. layers with non-displaced atoms. The upper slab is sometimes called a ‘selvedge’, here we use the terms ‘surface slab’ and ‘bulk slab’ (or simply ‘bulk’) for the upper and lower slabs, respectively. By Ts and Tb we denote, respectively, the plane translation symmetry groups of the surface and bulk slabs. The translations which transform the crystal with surface into itself must belong to both groups of translations, Ts as well as Tb. The translation group T of the whole crystal with surface is thus the intersection T = Ts fl Tb

(3.197)

of Ts and Tb. Alternatively, one can say that T is the largest common

subgroup of both groups Ts and Tb. There are two possibilities, either T only consists of the identity translation, which means that the lattices defined by Ts and Tb are non-commensurate and the crystal with surface does not possess any lattice translation symmetry, or T contains more elements than just the identity, in which case one says that the two lattices derived from Ts and Tb are commensurate. The lattice associated with the T is called the coincidence lattice. If, in particular, Tsis a subgroup of Tb then T is equal

3.6. Clean semiconductor surfaces

35 1

to T,, i.e. the coincidence lattice is identical to the lattice T, of the surface slab. If T, is not a subgroup of Tb then T cannot be equal to T, and is necessarily a proper subgroup of T,, i.e. it is smaller than T,. One can thus distinguish between the following three cases with regard to the translation symmetry of crystals having a reconstructed surface: (i) no translation symmetry, (ii) translation symmetry exists but is smaller than that of the surface slab, (iii) translation symmetry exists and is the same as that of the surface slab.

If one assumes that, among the various conceivable surface reconstructions with a given degree of translation symmetry, that particular reconstruction will take place which allows for maximum translation symmetry of the crystal with surface (the system ‘surface slab plus bulk slab’), then only the third of the above possibilities can be realized. A formal proof of this assumption does not exist, and it is probably not valid without exceptions, however, as a rule, it generally yields correct results. Using the above conclusions we are able to determine the primitive lattice vectors of the reconstructed surface in terms of the primitive lattice vectors f1, f 2 of the ideal surface. The latter are, by definition, also the primitive lattice vectors of the bulk slab of the crystal with surface. Let be fi and fi the primitive lattice vectors of the reconstructed surface slab. They may be linearly composed of f 1 , f 2 according to

+ 912f2 f; = Q2lfl + 922f2. f: = Qllfl

(3.198) (3.199)

Here, the coefficients Q i k , i, k = 1,2, of the transformation matrix Q are, at the outset, arbitrary real numbers. They need to be rational if and only if the two lattices derived from f i , f 2 and f ’ l , f’2 have a coincidence lattice, i.e. in case (ii) above. If the coincidence lattice is identical to the lattice derived from fi, fi, i.e. in the particularly interesting case (iii), it follows that the Qik must have integer values. In this case, the lattice derived from f : , f i is simultaneously the lattice of the reconstructed surface. We thus arrive at the important conclusion that in the only case of practical interest (iii) above, surface reconstruction can be described by a 2 x 2 matrix with integer elements. The two most common forms of this type of reconstruction have a special notation (Wood notation): (1) The non-diagonal elements vanish, i.e. f: and fi are parallel to f 1 , f 2 , respectively, and their lengths are integer multiples of the respective lengths of the latter. We thus have

352 Chapter 3. Electronic structure of semiconductor crystak with perturbations

f: = nfi,

f& = mf2 ,

(3.200)

with n and m being integers. The primitive unit cell of the surface slab contains n x m primitive unit cells of the bulk slab. This is said to constitute an n x m reconstruction An n x m reconstructed crystal surface of a particular material C parallel to a lattice plane with Miller indices (hlcl) (or ( h k i l ) in the case of hexagonal symmetry) is characterized by the symbolic notation C(hlc1) n x m.

(3.201)

(2) The off-diagonal elements of Q are not equal zero, i.e. fi is not parallel to fi and/or fi is not parallel to f2. The angles L(f!, fi) and L(f&fz), which are in general not equal to each other, are assumed to be equal in the case under consideration. This means that the two vectors f1, f2 (with tails joined at the same point) can be transformed into the two corresponding vectors = L(f4, f2) = a about an axis which is perpendicular to the surface, with a subsequent rescaling of f1, f2 by the factors lfiI/fll and ~ f ~ ~respectively. / ~ f ~ ~ A, (hlcl) surface of a particular material C reconstructed in this way is characterized by the symbolic notation

f;, fi by a rotation through the same angle L(f;, fi)

lfil x C(hkl) lfil

141 - a. -

(3.202)

If21

and lfil/lf21 are in general irrational in contrast to the The factors lf~l/lf~l qik, which in the case considered here are integers. Examples of both of the special reconstruction forms discussed, as well as for the general reconstruction form in case (iii), are shown in Figure 3.29. Sometimes, in the notation (3.201), n x m is replaced by p - n x m or c - n x m. The lattice vectors fi = nfi and fi = mfz are then not necessarily primitive as originally assumed in the notation (3.201), and in addition to primitive ( p ) reconstructed surface lattices, also centered ( c ) ones are possible. This can only take place, however, for rectangular surface lattices. Thus, the modified notation applies only to this case, although it is also sometimes used (formally incorrectly) for square lattices. In the rectangular case the notation c - n x m describes a type of reconstruction which is not covered by one of the two notations (3.201), (3.202), and which can otherwise only be characterized by the 2 x 2 matrix Q itself. For square reconstructed lattices the c - n x m notation is just a simpler description of a reconstruction of type (3.202). The point symmetry of a reconstructed surface lattice is generally lower than that of the ideal surface lattice from which it is derived. This implies that the crystal with reconstructed surface belongs to a different plane

3.6. Clean semiconductor surfaces

353

Figure 3.29: Three different types of surface reconstructions: (a) 1 x 2, (b) 3h x 31 -30°, (c) general type, the Wood notation does not apply in this case, the matrix notation does with 411 = 5, 412 = -1, 421 = 2, 422 = 2. crystal system than the crystal with ideal surface. For example, the 2 x 1 reconstruction of a square lattice leads to a rectangular lattice. The same holds for the 2 x 1 reconstruction of a hexagonal lattice which also results in a rectangular lattice.

A further item is worth mentioning, concerning the surface reconstruction itself. It follows from the point symmetry of the crystal with ideal surface itself. If the latter belongs to the square crystal system, i.e. if it has a square lattice and one of the two point symmetry groups 4 m m or 4, the directions of the two primitive lattice vectors are symmetrically equivalent. A particular reconstruction which increases the surface unit cell in the direction of fl by a factor n and in the direction of f 2 by a factor m , is equivalent to another reconstruction which does the same for, respectively, the symmetrically equivalent vectors f2 and fi (Figure 3.30). An analogous statement holds for an ideal surface of the hexagonal crystal system, having a hexagonal lattice and one of the point groups 6mm, 6 , 3 m or 3. In this case, three symmetrically equivalent direction exist (Figure 3.30). If there is no physical reason which makes one of the different symmetrically equivalent reconstructions more likely than another, they will take place simultaneously in different regions of the surface. The result is the formation of domains of otherwise identical, but differently oriented, reconstructed unit cells. Due to the domain structure, the overall translation symmetry of the surface is destroyed. Structural imperfections of a more local nature occur where the boundaries of such domains meet.

354 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.30: Symmetrically equivalent 2 x 1 reconstructions in the case of ideal surfaces belonging to the square (left) and hexagonal crystal systems. Point and space symmetries

The point and space symmetries of relaxed or reconstructed surfaces are generally of lower degree than those of the corresponding ideal surfaces. They do not only depend on the point and space symmetries of the irreducible crystal slab as in the case of an ideal surface, but also on the point and space symmetries of the relaxed or reconstructed surface slab. If the surface slab consists of more layers than the irreducible crystal slab, then its point and space groups can be taken to determine the point and space groups of the whole crystal with relaxed or reconstructed surface. If the surface slab contains fewer layers than the irreducible crystal slab it is expedient to add further atomic layers (which then do not contain displaced atoms) to make up the difference. The point and space group elements of the thus extended surface slab which are simultaneously point and space group elements of the irreducible crystal slab form, respectively, the point and space groups of the whole crystal with relaxed or reconstructed surface. Similarly like this was done for the translation symmetry group above, one may argue that the point and space groups of the relaxed or reconstructed surface slab should be subgroups of the point and space groups of the corresponding ideal surface. If this is the case, the point and space groups of the relaxed or reconstructed surface slab are, respectively, the point and space groups of whole crystal with relaxed or reconstructed surface.

3.6.3

Electronic structure of crystals with a surface

The electrons and cores of a crystal with a surface undergo the same interactions as the electrons and cores of an infinite bulk crystal, the only difference being that the cores and electrons of the removed semi-infinite crystal above the surface are missing, and the positions of the cores just below the surface

3.6. Clean semiconductor surfaces

355

differ from those in the infinite bulk. The two basic approximations of the theory of the interacting electron-core system of an infinite crystal, namely the adiabatic approximation and the one-electron approximation, are not influenced by these modifications, so they may also be used in the presence of a surface. In the electronic structure calculations of infinite bulk semiconductor crystals, the core positions commonly are taken as input data. This is possible because these positions are crystal sites of high symmetry which are well-known from X-ray diffraction experiments. For crystals with a surface, this can no longer be assumed. The positions of atoms in the surface layers of relaxed or reconstructed surfaces are crystal sites of lower symmetry. In many cases, they are not known or only incompletely known from experimental investigations, for these investigations are more difficult and less precise than X-ray diffraction in the case of bulk crystals (a short review of these methods will be given further below). X-ray diffraction cannot be applied to surfaces because it lacks surface sensitivity. In such circumstances, the positions of atomic cores at surfaces are to be treated as output, rather than input, data of the electronic structure calculations. The way that this can be accomplished was discussed in section 2.2 in general terms, and in section 3.5 with respect to point defects. It involves, first, the calculation of the total energy of the electron-core system for a variety of different sets of core positions and, second, the minimization of the total energy with respect to these sets. The minimum set gives the core positions which really apply. The most critical part of the total energy is the energy of the electron system. In order to obtain it, the one-electron energies of the crystal with surface have to be calculated for assumed core positions. Formally, this involves the same task as in the case of bulk crystals, namely, the calculation of stationary one-electron states for given core positions. Below, we will demonstrate how this problem can be solved in the case of crystals with surfaces. The remaining parts of the procedure for determining the atomic and electronic structures of surfaces, the calculation of the entire total energy including the core-core interaction energy, and the minimization of the total energy with respect to the core positions, will not be treated here because it is mainly a numerical task. The assumption of a priori known core positions is valid if ideal surfaces are considered. In this case they are the infinite bulk positions. The electronic structures of ideal surfaces are important as reference data for the electronic structures of relaxed and reconstructed surfaces. Thus we will also deal with them.

One-electron Schrodinger equation The one-particle Schrodinger equation for the wavefunction $E(x) of an

356 Chapter 3. Electronic structure of semiconductor crystals with perturbations electron in a crystal with surface has the same general form

(3.203) as in the case of an infinite bulk crystal (see equation 2.53). The oneelectron potential V(x) may be written as the sum

of the electron-core interaction potential Vc(x), and an effective one-electron potential Ve(x)due to electron-electron interaction. For Ve(x),any of the one-electron approximations introduced in section 2.1 may be used. The core potential Vc(x)follows from the corresponding expression for an infinite crystal if the summation is restricted to cores within and below the surface. Using the surface adapted notation 5 If3 r for the position of the j-th basis atom of the primitive bulk unit cell at the bulk lattice point r If3, j = 1 , 2 , . . . , J , 1 = 0, -1,. . . , -00, this leads to

+ +

-W

+

J

(3.205) where q(x) is the potential of a core of species j located at R = 0. The core potential Vc(x)and, hence, also Ve(x)and V ( x ) have the translation symmetry of the 2D surface lattice, so that

v(x)= V(x + r) .

(3.206)

As in the 3D bulk case, this symmetry can be used to derive the Bloch theorem. Bloch theorem

This theorem states that the energy eigenfunctions $E(x) of the Schrodinger equation (3.203) may be chosen simultaneously as eigenfunctions of the surface lattice translation operators t,.. This allows one to write these functions in the form of Bloch functions @Q(x) with a 2D quasi-wavevector

4 = 91g1 + 9282,

(3.207)

where q1,92 are arbitrary real numbers, and gl and gz are the primitive lattice vectors of the reciprocal surface lattice, defined by the relations

3.6. Clean semiconductor surfaces

35 7

fi . g k = 27T6&, i, k = 1,2,

(3.208)

The latter equations are solved by the vectors g1 = N-lf2 x

[fl x

f2],

g2 = N-lfl x [f2 x

fl]

(3.209)

with N = (1/2n)[(f1 . fl)(fZ.fz) - (fi . f2)’] as normalization constant. The two vectors gl, gz of (3.209) lie in the plane spanned by the two primitive surface lattice vectors f1, fz, i.e., within the surface. The same statement applies to the 2D wavevector q. As in the 3D bulk case it is convenient to introduce a region of macroscopic size with respect to which the eigenfunctions $J,E~(x) can be assumed to be periodic. In the case of a crystal with surface this region forms a parallelogram spanned by the edge vectors Gfl, Gf2, with G being a large integer. The area RII of the periodicity region is G21f1 x f2l. The Bloch functions normalized to it may be written as

1

Q,Eq(x)= -e2qx

fi

UEq(X),

(3.210)

where U Q ( X ) is the Bloch factor, which has the periodicity U E ~ ( X )= U E ~ ( X + r) of the surface lattice and is normalized with respect to a primitive surface unit cell. To guarantee the periodicity of the Bloch functions $ ~ ~ ( xof) (3.210) with respect to the periodicity parallelogram, the wavevectors q must have the form

(3.211) with p1,pz as integers. This means that the q-vectors must belong to a finely-meshed lattice similar to that of Figure 2.4. For the macroscopically large values of G which we assume, q is practically continuous, although the number of different q-values within a given region of q-space is finite.

Surface Brillouin zones and surface energy bands The energy eigenvalues E of a particular Bloch type eigenfunction of equation (3.203) depend on q. If g varies over the whole infinite space, as we assume here, E represents a unique function E ( q ) of q. This description corresponds to the extended zone scheme in the case of an infinite bulk crystal considered in section 2.4. As in the latter case, one may switch from the extended to the reduced zone scheme in which q varies only over a primitive unit cell of the reciprocal surface lattice. Any other point of the i n h i t e qspace may be written as q + g where g is a surface reciprocal lattice vector.

358 Chapter 3. Electronic structure of semiconductor crystals with perturbations In the reduced scheme, the energy function E(q+ g) over the whole q-space is replaced by the manifold of functions E,(q) E ( q g ) defined only over a primitive unit cell. As in the bulk case, there is a distinguished choice of the primitive unit cell, as we will see below. Using first order perturbation theory with respect to the periodic potential V(x), one may demonstrate that, in the extended scheme, E(q) represents a continuous function of q everywhere, except for the q-vectors with lql = lq + gl or

+

q . g + -1 g2 = 0,

2

(3.212)

where g is an arbitrary surface reciprocal lattice vector. Equation (3.212) defines lines in the 2D q-space which are analogous to Bragg reflection planes in the 3D k-space of a bulk crystal. On Bragg reflection lines, the energy function E(q) has discontinuities. These lines may be used to define 2D Brillouin zones in just the same way as was done in the 3D case in section 2.4. One speaks of surface Brillouin zones (BZs). The surface B Z s of the square lattice have, in fact, already been used in section 2.4 as an illustration for the 3D case. Of particular importance is the f i r s t surface B Z . It may also be defined as the Wigner-Seitz cell of the reciprocal surface lattice. Since there are 5 different plane Bravais lattices and, hence, 5 different reciprocal surface lattices, there are also 5 different first surface B Z s . They are shown in Figure 3.31. Their shapes are the same as those of the WignerSeitz cells of the corresponding direct lattices since the Bravais types of the direct and reciprocal surface lattices always coincide. The first surface B Z s are, by definition, free of Bragg reflection lines. Thus the energy function E(q) is continuous within these zones. Furthermore, any higher surface B Z of order p may be reduced to the first surface B Z , and the energy function E(q) in the p-th surface B Z can be folded back to the first surface B Z . There, it forms a continuous function E,(q) which is called a surface energy band. Thus we may state that the energy eigenvalues of a crystal with surface form energy bands over the first surface BZ. There are surface bands of different types, regarding their relations to the energy bands of 3D bulk crystals without surface. Below we characterize these differences in a qualitative way. Types of eigenstates Bulk states

Consider an infinite bulk crystal, and cleave it into two semi-infinite crystals with a surface parallel to a particular lattice plane. The spectra of energy

3.6. Clesn semiconductor surfaces

359

t'

tY

Figure 3.31: Surface B Z s of the 5 plane lattices: (a) oblique, (b) p-rectangular, (c) c-rectangular, (d) square, (e) hexagonal. Symmetry lines and points are also shown, and their notations are introduced. eigenvalues of the two semi-infinite crystals will contain all energy levels which were already eigenvalues of the infinite bulk crystal before cleaving. This implies that each crystal with surface possesses an energy eigenvalue spectrum which partially is made up from energy eigenvalues of the infinite bulk crystals from which it is derived. The eigenfunctions of the crystal with surface corresponding to these eigenvalues, if examined at positions outside of the crystal, will decay exponentially with increasing distance from the surface, while inside they will practically be the same as those of the infinite crystal, i.e. they will exhibit undamped oscillations throughout the whole semi-infinite crystal, as do the eigenfunctions of the infinite bulk crystal (see Figure 3.32). Eigenstates of a crystal with surface exhibiting such properties are called bulk states. The corresponding energy eigenvalues form bulk state surface energy bands Ep(q). These bands can be obtained by projecting the bulk bands E?"(k) of the infinite crystal onto the first surface B Z . Here, 'projecting' means that one assigns to a particular point q of the first surface B Z all bulk band energies EF"(k) corresponding to k-vectors of the first

360 Chapter 3. Electronic structure of semiconductor crystals with perturbations

f

I

x

C

.-0

P

-w

W

0

w

+

C 3

C

W

surface state

9-

0

x-

Figure 3.32: The three different types of electron energy eigenstates of a crystal with surface below the vacuum level (schematically). bulk B Z having the same projection q on the surface B Z plane, but with various components kl perpendicular to it. Formally, this can be expressed by the relations

The bulk state surface band index p in (3.213) replaces the bulk quantum numbers v k l . As kl varies continuously, @ also does. This means that a particular band of the infinite bulk crystal gives rise to a continuum of surface bands. In this way, the infinitely large number of atoms in a primitive unit cell of the crystal with surface manifests itself. We illustrate the projection of bulk bands onto the surface B Z using the (100) surface of a diamond type crystal as an example. The band structure is taken from the empty lattice model introduced in section 2.4. As a first step, the bulk B Z is to be projected onto the plane of the first surface B Z (see Figure 3.32). In doing so, one notes that part of the projected bulk B Z lies in the second surface B Z . This has to be folded back to the first surface B Z , together with its energy values. In this way we obtain the projections of the lowest three empty lattice bands shown in Figure 3.33. In addition to q-vectors for which all energy values are allowed, there are also q-vectors occurring for which certain energy values are forbidden. This peculiarity is found in other, more realistic cases as well, it represents a general feature of the projected bulk band structure. In the forbidden energy regions states may occur which are localized at the surface.

3.6. Clean semiconductor surfaces

Figure 3.33: First (100) surface B Z (shaded area) together with the projected first bulk B Z for a crystal with fcc Bravais lattice.

361

‘I

Surface states

Such states are to be expected for the same reason that localized states are observed in the case of point or 0-dimensional perturbations. In contrast to the latter, surfaces constitute 2-dimensional perturbations. The localization occurs, therefore, only in 3 - 2 = 1 dimension, namely that perpendicular to the surface. If the energy lies in the forbidden region of the projected bulk band structure, the decay of the eigenfunction towards the bulk proceeds exponentially (see Figure 3.32). The states are then called bound surface states and the corresponding energy bands E,(q) are bound surface bands. Besides these, one has surface resonances and antiresonances. The latter occur at energies in the allowed region of the projected bulk band structure and give rise to resonant or antiresonant surface bands Ep(q). They manifest themselves in an increase (resonance) or decrease (antiresonance) of the density of states. The eigenfunctions at these energies are also localized at the surface, but decay less rapidly towards the bulk (according to a power law) than the exponentially decaying bound surface states. Antiresonances are necessary in order to satisfy Levinson’s theorem, which holds for surfaces as well as for point perturbations. If bound surface states exist, the antiresonances must compensate the increase of the total DOS in the previously forbidden energy region.

Implications of symmetry for surface band structure The spatial symmetry of a crystal with surface has implications for the possible degrees of degeneracy of surface energy bands Ep(q), p = p, (T,p, at a

362 Chapter 3. Electronic structure of semiconductor crystals with perturbations Figure 3.34: Projection of the empty lattice band structure of a fcc bulk crystal onto the first (100) surface B Z . Three bulk bands with reciprocal lattice vectors K = 0 (1 = 0), K = ( 2 n / a ) ( l l l ) ( I = 1) and K = ( 2 n / a ) ( m ) ( I = -1) are considered.

-u1l

9

-2

7

5n

%

E? a, C

W

5

3

1

r

K

r

given wavevector q. It also results in symmetry relations between the values of EF(q)for different q values, and it determines the spatial symmetries of the eigenfunctions $i)w(~).The key for such conclusions are, in analogy to the infinite bulk case, the irreducible representations of the space group of the given crystal with surface. This is based on the fact that the eigenfunctions for a particular energy eigenvalue form a basis set of an irreducible representation of this group. Such a representation may be characterized by the star {q} of the wavevector q and the irreducible representations of the small point group of q with the factor system of equation (A. 157). The dimensions of the irreducible representations determine the possible degrees of degeneracy of an energy band E,(q) at the point q. Moreover, at all points of the star {q}, E,(q) has the same value. Since the 10 possible point groups of equivalent directions have at most 12 elements (this happens in the case of C6v(6mm)),and since the small point groups of the symmetric q-points are in general even smaller than the corresponding point groups of equivalent directions, only irreducible representations with dimensions equal to 1 or 2 will appear. 2D representations are likely to occur in cases where the small point groups are, on the one hand, large enough, and on the other hand, the corresponding space groups contain glide reflections. Then nontrivial factor systems arise for points q on the boundary of the first BZ,

3.6. Clean semiconductor surfaces

pig1

r, -A;-?;--Z;r, -A\-x:-z;--M,--z,-

363

~r~r+-~2---~-~;-X;--~;-M2-Z;-

x,-A,-

I

I

I

~--A~;-&-z;-M,-zF-

x2 x1

I

Figure 3.35: Symmetry and degeneracy of the surface energy bands for the 5 space groups of the p-rectangular Bravais lattice. [After Terzibaschian and Enderlein, 1986.)

and 1D representations might not be possible at all. This happens for 2 of the 5 space groups of the primitive rectangular Bravais lattice, which below will be studied as an example. In Figure 3.35, the possible types of band structures are shown on certain symmetry lines of the p-rectangular surface B Z . In the case of the space group p2mg, which applies for (110) surfaces of diamond type crystals, only 2D representations exist at M and X . The two 1D representations on the 2-line connecting these points belong to the same energy eigenvalue because of time reversal symmetry. For the space group p2gg, one has 2D representations only at X and X'. The representations on the connecting line X - 2 - M - 2' - X ' are lD, but the corresponding energy eigenvalues are degenerate because of time reversal symmetry. For the three remaining space groups p2mm, p l m l , and p l g l , the representations at all symmetry points are 1D.

Numerical methods for calculation of the electronic structure of surfaces

A 3D crystal with surface may be characterized as a crystal with a 2D lattice and primitive unit cells extending infinitely in the direction perpendicular

364 Chapter 3. Electronic structure of semiconductor crystals with perturbations to the surface. This point of view gives rise to the so-called slab method for calculating the electronic structure of crystals with surfaces. Slab method

The electronic structure of a crystal with a 2D lattice may be calculated by means of any of the methods known from band structure calculations of 3D crystals, with the exception that the dimensionality of the lattice has to be changed from 3 to 2. In the tight binding method, for example, one has to use Bloch sums of atomic orbitals &(x) over all points r of the 2D surface lattice, rather than over all points R of the 3D bulk lattice. With the surface adapted notation, r+G+Zf3, for a regular crystal site, the orbital a localized at such a site is given by &(x - r - 3 - 1f3) E &jl,.(x), The corresponding Bloch sums &jlp(x)are defined as

(3.214) The number of different Bloch sums is infinitely large even if a finite number of orbitals is used per atom, because the integer 1 counting the primitive crystal slabs, runs from 0 to -co. Consequently the Hamiltonian is given by an infinite matrix in the atomic orbital representation &jl,.(x). Some specific approximation is necessary to transform this to a finite matrix. One possibility is to consider a slab of the crystal, i.e. to cut off the semi-infinite crystal at a particular lattice plane parallel to the surface and discard the remainder. One may say that, in the direction perpendicular to the surface, the true semi-infinite crystal is replaced by a cluster (see Figure 3.36). The latter has two plane surfaces, one of them being real (that of the semi-infinite crystal), and the other one (obtained by cutting the semi-infinite crystal) not. This differs from the cluster method in the case of point defects discussed in section 3.5 where the whole cluster surface is artificial. The slab method may be combined with any band structure calculational method for infinite bulk crystals. Its combination with the tight binding method will be illustrated below by means of an example. We consider the ideal (111) surface of a diamond type crystal which according to subsection 3.6.2, has a hexagonal Bravais lattice. Instead of the one s- and three p-orbitals per atom we use the four hybrid-orbitals, i.e. we replace a in the Bloch sum (3.214) by ht, t = 1 , 2 , 3 , 4 . Only nearest neighbor interactions are taken into account. An illustration of this model is given in Figure 3.37. To get the matrix elements of the Hamiltonian H between Bloch sums, we first need these elements between the localized hybrid orbitals +htjlT Ihtjlr). The diagonal elements are

(3.215)

3.6. Clean semiconductor surfaces

365

Figure 3.36: Illustration of the slab method for surface band structure calculations.

Figure 3 . 3 7 Defect molecule model of the ideal (111) surface of a crystal with diamond structure. and the non-diagonal elements between different hybrids at the same atom j = j ’ are

(htllrlH(htJ1lr)= VI ,

t

# t’ .

(3.216)

The two types of matrix elements in (3.215) and (3.216) are equal for atoms at the surface and in the bulk. This is not true for elements between hybrids located at different atoms. Let j = 1,1= 0, designate the surface atom layer and consider the elements (ht10rJH]ht20rt)between the hybrid ht located at the surface atom 10r and the hybrid lht20x-t) at its nearest neighbor 20rt below the surface, pointing toward (htlOr). Since no nearest neighbor hybrid exists for the hybrid )hllOr) pointing out of the surface, all nearest neighbor elements involving IhllOr) vanish. The other three elements with t = 2 , 3 , 4 are equal to the parameter V2 introduced in equation (2.292),

366 Chapter 3. Electronic structure of semiconductor crystals with perturbations Using equations (3.215), (3.216) and (3.217), the matrix elements between the corresponding Bloch sums (3.214) may be formed. They are

(htlOq(Hlht,lOq)= v1 , (htIOqlHlht20q) = etV2,

t

# t’

.

t = 2,3,4 ,

(3.219) (3.220)

where e t is given by equation (2.240), with dt being a vector which points from the surface atom 10r toward its nearest neighbor atom 20rt in the direction of the hybrid t. We have a

d2 = -(lTT), 2

d3 =

a - -

-(lll), 2

d4 =

a(TT1). 2

(3.221)

The hybrids lht201-i) at the nearest neighbor atoms pointing td the surface atom lor, are coupled to the other hybrids at these atoms, and these hybrids interact with hybrids at more remote atoms. Thus an infinite matrix would occur if we would not restrict consideration to a slab, as we in fact do. Here, we will go one step further and neglect all couplings between the hybrids at the nearest neighbor atoms. Then the 7 Bloch sums IhtlOq), t = 1,2,3,4, and (ht20q),t = 2 , 3 , 4 , are completely decoupled from the rest of the Bloch sums (see Figure 3.67). In this way we arrive at a simplified model of the crystal with surface which effectively reduces it to the two first atomic layers and treats the atoms of the second layer only in an approximate way. This model represents an analog of the defect molecule model in the case of a point perturbation. If the basis functions are arranged in the order IhllOq), IhzlOq), h l o q ) , h l o q ) , lh220q), lh320q), lh420q), then the Hamiltonian matrix is

(3.222)

The eigenvalues of this matrix are plotted in Figure 3.38 for different symmetry lines of the first surface B Z of the hexagonal lattice of Figure 3.31.

3.6. Clean semiconductor surfaces

r

367

M

K

r

W avevec tor Figure 3.38: Band structure of the ideal (111) surface of a diamond crystal within the defect molecule model. The TB parameters are V1 = 2.13 e V , V2 = 6.98 e V . The hybrid energy Eh is used as the energy origin. The three lowest and three highest bands are bulk state surface bands. They arise from the three hybrids of the surface atom, which bind this atom back to its nearest neighbors in the second layer. More strictly, they correspond to the 3 back bonding and antibonding states. The band in the gap is due to the dangling hybrid of the surface atom. It forms a bound surface band. If we were to consider more than 2 atomic layers, the region where the bulk bands in Figure 3.38 occur would be covered by more bulk state surface bands, while the bound surface band in the gap would hardly change. Supercell method

The slab considered in the preceding subsection may be repeated periodically in the direction perpendicular to the surface, simultaneously inserting several layers of vacancies between two neighboring slabs (see Figure 3.39). This structure may be considered to be an infinite repetition of the original crystal with surface, each repetition being approximated by a finite slab of several atomic layers embedded between vacancy layers. This arrangement represents a 3 D supercrystal composed of 1D supercells (bear in mind that in the case of point defects in section 3.5, we similarly considered a

368 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.39: Supercrystal obtained by a periodic repetition of supercells. The latter are composed of crystal slabs embedded between vacancy layers.

3D supercrystal composed of 3 D supercells). The band structure of the supercrystal is approximately the same as that of the original crystal with surface, provided the vacancy slabs are thick enough to suppress coupling between neighboring crystal slabs, and the latter slabs are thick enough to simulate a semi-infinite crystal. The bands Ep(q) of the crystal with surface are obtained from the bulk bands EF(k) of the supercrystal by plotting the latter on the surface B Z . In this, k l may be chosen arbitrarily because any dispersion of EF(k) = E F ( q , k l ) with respect to k l would indicate a coupling between the slabs, which has been excluded. The band structure of the supercrystal may be obtained by means of any 3 D band structure calculation method without modification. This makes the supercell method particularly appealing. Combined with the pseudopotential method, as well as the local density functional or quasi-particle approximations, it represents the most important calculational method for electronic and atomic structure determinations of surfaces. Defect model and Green’s function method

A crystal with surface may also be viewed as a 3 D crystal with a 2D perturbation. The significance of this characterization is the following: Consider first an ideal infinite 3 D crystal, then remove some number of neighboring atomic layers parallel to the surface under consideration or, equivalently,

3.6. Clean semiconductor surfaces

369

Figure 3.40: Illustration of the defect method for calculating surface energy band structure. create the same number of vacancy layers (see Figure 3.40). What remains are two identical semi-infinite crystals which are only displaced with respect to each other. They do not interact provided the number of vacancy layers is large enough, which we will assume. The electronic structures of the two semi-infinite crystals coincide, and are identical with that of the considered crystal with surface. As in the case of a single vacancy, the states bound at the defect may be obtained by means of the Greens function G o ( E )of the unperturbed 3D crystal, more strictly, by the vanishing of the determinant of [Go(E)V’ - 11 (see equation (3.134)). Here, the perturbation potential V’(x) represents the difference of the potential energy of an electron in the perturbed and unperturbed crystals. It is the negative of the sum of the potentials of the removed atoms or the sum of all vacancy potentials. It has the 2D lattice symmetry of the surface so that

~ ’ ( x=) V’(X

+ r).

(3.223)

If only nearest neighbor interactions are taken into account, a single layer of vacancies is enough to decouple the two semi-infinite crystals. For the analysis of the bound state condition (3.74), one has to use a particular basis set, just as in the case of a point perturbation. Wannier functions are again a possible choice, but here they involve localization only in one direction, namely that perpendicular to the surface. We denote these functions by IvqRl) where RI is the component of a 3D lattice vector R perpendicular to the surface, such that

370 Chapter 3. Electronic structure of semiconductor crystals with perturbations

(3.224) The matrix representation of the Green’s function Go(E) with respect to this basis set is

For the perturbation potential V1(x) we take that of a single layer of vacancies located at RI = 0. Then the matrix representation of V1(x) is

where VL,,(q) = (vq01V’Jv’qO)has been used. The relevant matrix elements (vqOIGoV1- 1lvlqO) of GoV1- 1 take the form

with

(3.228) According to equation (3.134), the determinant of the matrix (3.227) must vanish for energy eigenvalues E in the gap of the ideal crystal, i.e. Det [G:(E, q)VL,,(q) - 6,,,]

= 0.

(3.229)

If one compares this equation with the corresponding relation (3.94) for deep centers, it will be noted that the q-dependence which occurs here was absent there. This dependence in the case of surfaces causes the eigenvalues in the gap to form ‘deep’ bands rather than deep levels (which was the case for point perturbat ions). Transfer matrix method

Finally, still another method for surface band structure calculations should be mentioned. It is based on the transfer matrix concept of quantum mechanics. The transfer matrix M ( E ) is formed from solutions of the Schrodinger equation upon one unit cell of the bulk crystal for particular boundary conditions in the direction perpendicular to the surface. The energy E is arbitrary first of all. Transferring the wave function from the surface unit cell to the

3.6. Clean semiconductor surfaces

371

n - th unit cell below the surface can be done by applying the n - t h power W ( E ) of M ( E ) . Bound surface states are obtained for such energy values E for which M n ( E ) decays exponentially with n. The practical use of this

method seems to be limited, however.

3.6.4

A t o m i c and e l e c t r o n i c structure of particular surfaces

In this subsection we will deal with the atomic and electronic structures of some important semiconductor surfaces, including the various reconstruction states of the Si (111) surface, the Si (100) surface, the (110) surface of GaAs and other 111-V compound semiconductors as well as the (111) and (100) surfaces of GaAs. It is advisable to treat the atomic structure together with the electronic structure because of the close relation between the two kinds of structures, as was pointed out earlier. We begin with a short introduction to the experimental methods of surface structure analysis, again referring to both the atomic and electronic aspects. Experimental methods for surface s t r u c t u r e analysis Experimental methods for determining the atomic structure of bulk crystals are all based on the interaction of waves with the atomic cores and valence electrons of the crystal. If the wavelength is of the order of the distance between the atoms, i.e. of the order of magnitude of 1 A, the crystal constitutes a 3D diffraction lattice and diffraction maxima will occur in prescribed directions in space. The crystal structure may be determined from the positions and intensities of these maxima. The various experimental methods differ primarily in the nature of the waves employed. X-rays are by far the most important for determining the structure of bulk crystals. A wavelength in the region of 1 corresponds to a photon energy in the range of 10 k e V . Electron and neutron waves are also diffracted by bulk crystals. Electron energies in the region of 100 eV and neutron energies of 0.1 e V are required for wavelengths in the d region. Since they are neutral particles, X-ray photons and neutrons interact only relatively weakly with the crystal. They can pass through crystals of macroscopic thickness and be backscattered from them from macroscopic depths within them. X-rays and neutrons thus yield information on all atomic layers of a crystal, including those at the surface. Since the number of surface layers is extremely small in comparison to the total number of layers, the diffraction patterns are dominated by the bulk of the crystals. The interaction of electrons with the atomic cores and the valence electrons of a crystal is significantly stronger than that of photons and neutrons. Electrons having energy less than 100 k e V can not pass through a crystal of

372 Chapter 3. Electronic structure of semiconductor crystals with perturbations macroscopic thickness. Experimentally, one therefore has only the backscattering available and then only elastically backscattered electrons can be employed in forming diffraction patterns. These originate at a depth which, on average, is equal to the inelastic mean free path of the electrons. This varies relatively independently of the particular crystal under consideration from 4 A to 10 for energies of 20 eV to 300 eV. Electron diffraction in this energy region is thus hardly suitable for examining bulk crystals but it can be readily employed in studying crystal surfaces. The diffraction of low energy electrons in the region of 100 eV is in fact the most intensively used method for surface structure determination. It is referred to as LEED (Low Energy Electron Diffraction). The principle of LEED may be explained as follows. We consider an incident electron wave with the wavevector ki. The interaction with the crystal generates scattered waves with wavevectors k,. The scattering potential has the translation symmetry of the surface lattice, thus its Fourier components differ from zero only for vectors g of the reciprocal surface lattice. This means that only those scattered wavevectors k, can occur whose components k,ll parallel to the surface differ from the parallel component kill of ki by a reciprocal surface lattice vector g, hence (3.230) There is no relation between the components of k, and ki perpendicular to the surface because there is no translation symmetry of the scattering potential in this direction. In writing down equation (3.230) we have implicitly assumed that only one scattering event takes place. This relation also applies, however, to multiple scattering processes. This is important because electrons scattered back from the surface have, as a rule, experienced many scattering events, in contrast to X-ray photons which typically have been scattered only once. This difference is due to the above mentioned fact that electrons interact with the crystal much more strongly than do X-ray photons. The electrons measured in LEED are elastically scattered. One therefore has Ik,/=

lkzl .

(3,231)

A solution of the two equations (3.230), (3.231) always exists for given vectors ki and g (this is in remarkable contrast to coherent scattering of electrons from 3D bulk crystal, which can only occur if k, lies on a Bragg reflection plane). The solution of equations (3.230) and (3.231) can be readily carried out using the construction shown in Figure 3.41. The points at which the vertical lines passing through the reciprocal lattice points g intersect the

3.6. Clean semiconductor surfaces

Figure 3.41: LEED maxima.

373

Construction of

sphere Jk,]= lkzl, determine the directions in which diffraction maxima occur. There is exactly one maximum for each reciprocal lattice point g. The reciprocal surface lattice can thus be read immediately from the distribution of the diffraction maxima on the registration screen. The direct surface lattice is the reciprocal of the reciprocal surface lattice. Some typical LEED images are shown in Figure 3.42. The bright points correspond to the reciprocal lattice of the ideal surface, and the less bright points to the finer reciprocal lattice of the reconstructed surface. In this way it is relatively easy to determine the surface lattice by means of LEED. To obtain the actual positions of atoms is more diacult. One needs additional experimental and theoretical information about the intensity of the diffraction maxima as a function of the energy of the incident electrons (dynamical LEED). Besides LEED, there are other methods for surface structure analysis which, although they are not a substitute for LEED, can supplement it. These methods include diffraction of energetic electrons in the region of some 10 k e V , known as ‘Re5ection High Energy Electron Diffraction’ ( M E E D ) , diffraction of X-rays incident almost parallel to the surface, and diffraction of slow Helium atoms (of M I00 meV). Scattering of energetic ions ( M 1 M e V ) is used in techniques like ‘Rutherford backscattering’ (RBS) and ‘ion channeling’. Imaging procedures of significance are transmission electron microscopy (TEM), scanning tunneling microscopy (STM) and atomic force microscopy (AFM). The latter two methods have become particularly important. Like for atomic structure determinations, a variety of methods exist to study the electronic structure of surfaces, in particular the bound surface states in the energy gap of the bulk crystal. The most powerful and universal method is photoemission spectroscopy (PES). This method relies on the external photoeffect in which an electron is emitted from the crystal by

374 Chapter 3. Electronic structure of semiconductor crystals with perturbations

x6

Figure 3.42: LEED pictures of six differently prepared GaAs (100) surfaces.(After Drathen, Ranke and Jacobi, 1978.)

absorbing a photon of sufficiently high energy. The emitted photoelectrons are spectrally decomposed with respect to their kinetic energies. The thus obtained energy spectrum of photoelectrons maps the density of states of occupied electron levels of the crystal. To enhance surface states and discriminate bulk states, photoelectrons with kinetic energies around 50 eV are used whose inelastic mean free path is only about 5 A and which can therefore only come from this depth below the surface. These electron energies correspond to photon energies which are not substantially larger, i.e. in the far ultraviolet region. The term UPS (Ultraviolet Photoemission Spectroscopy) is used in this context. The only practically suitable radiation source in this energy region is the electron synchrotron. By measuring angular resolved photoemission spectra (ARUPS), the wavevector dispersion of the bound surface energy bands can be determined. To study moccu-

3.6. Clean semiconductor surfaces

11101

375

hi01

Figure 3.43: Geometry of the ideal Si (111) surface (left) and of the Si (111) 2 x 1 surface according to the buckling model (right). pied surface states one may use inverse PES in which electrons captured by such states emit photons. Beside photoemission, a variety of other techniques exists which can provide data on surface states. In principle, any experimental technique which probes the electronic structure of bulk crystals can be employed for surface electronic structure investigations, provided it can be made surface-sensitive. This applies to optical reflectivity, electrical transport, photoconductivity, and capacity measurements, as well as electron energy loss spectroscopy (EELS). Experimental techniques like field effect measurements fulfill this requirement from the very beginning. Controlling the energy of tunneling electrons in scanning tunneling microscopy, surface states can be resolved spatially and energetically (scanning tunneling spectroscopy). Experimental techniques which primarily measure the electronic structure, can also provide data on the atomic structure. The solid state shifts of core levels (see section 2.1) are an example. These shifts differ for atoms in the bulk and at the surface because of the altered atomic structure at the surface. The difference (typically some tenths of an e v ) can be measured by means of PES and UPS. On the other hand, they can be calculated on the basis of a particular surface structure model. By comparing theory and experiment one can evaluate the feasibility of various models of surface structure. The calculation of the total energy is a purely theoretical test of the validity of a particular surface structure model, and it may be used to determine the parameters which can be varied in such a model. If the model has optimized parameters and results in a lower total energy than other models it may be given preference over them.

376 Chapter 3. Electronic structure of semiconductor crystals with perturbations Figure 3.44: Surface band structure of an ideal Si (111) sur-

I '

O

r

face.

t i

*!:!

-5 - *

.. .. ..

,-I I

!ii ::I

..... .*....

I... ll

iii r

I

K

r 2D wave vector

Si surfaces (111) surface

The geometry of the ideal (111) surface of diamond type crystals is illustrated in Figure 3.43 (left). The surface lattice is hexagonal, and the two primitive lattice vectors are fi, f2 of Table 3.8. There is one surface atom per primitive unit cell, and one dangling bond per surface atom. The (111) surface is the cleavage plane of diamond type crystals. By cleaving a Si crystal in UHV at room temperature, one obtains a 2 x 1 reconstructed (111) surface. After annealing it at 500 C , a 7 x 7 reconstruction state evolves, which remains stable at room temperature. The occurrence of a 2 x 1 reconstruction immediately after cleavage is to be expected if one examines the band structure of the ideal (111) surface (see Figure 3.44) and considers, in particular, the electron occupancy of the bound surface band in the fundamental gap. This band arises from dangling sp3-hybrids of surface atoms, and can host 2 electrons per surface unit cell. Since 3 of the 4 valence electrons of a surface atom are in bonds with second-layer atoms, only 1 electron per surface unit cell is left for the bound surface band. Thus this band remains only half-filled. The ideal Si (111) surface is metallic.

3.6. Clean semiconductor surfaces

377

This state is unlikely to be stable, however, i.e. surface reconstruction is likely to take place. Below we discuss a particularly simple reconstruction model, the so-called buckling model (see Figure 3.43, right-hand side) which in the early days of clean semiconductor surface physics was believed to be correct. Later, it was realized that buckling is energetically advantageous only for 111-V compound semiconductor surfaces, while it is not advantageous for group-IV semiconductor surfaces including the (111) surface of Si. To introduce the buckling model we consider doubling of the primitive unit cell in the direction of primitive lattice vector fi, which according to Table 3.8 points in the direction [ l i O ] . The hexagonal lattice with doubled primitive unit cell forms a rectangular lattice with primitive lattice vectors 2f1 -tf2 and f2 The short side of the rectangular primitive unit cell, shown in Figure 3.43 right, is parallel to [Olq,and its long side parallel to [ 2 m . The corresponding first surface BZ is also a rectangle (see Figure 3.31) with its long side, i.e. its r - X-direction, parallel to [Olq,and its short side, i.e its X - M-direction, parallel to [2ii].The rectangular BZ is half as big as the original hexagonal BZ, and each band of the latter gives rise two band in the former, a direct and a back-folded one. There is no gap between these two bands because they arise from the same band of the larger BZ. The surface is still metallic. A gap arises if the so far formal 2 x 1 reconstruction is made real. This can be done by a buckling of the surface, i.e. by alternately raising and lowering atoms in rows parallel to f1 above the surface and below it (see Figure 3.43, right). In this, the three back-bonding hybrids of a raised atoms becomes more p-like, and the dangling hybrid at this atom more s-like simultaneously lowering its energy, while the three back-bonding hybrids at a lowered atom become more sp2-like and the dangling hybrid at this atom more plike simultaneously raising its energy. The two bound surface bands derived from these s- and p-like dangling hybrids are just the bands below and above the gap discussed before. The lower s-like band can host all electrons of the dangling hybrids, while no electrons remain for the population of the upper p-like band. If the total energy of this state were in fact lower than that of the ideal surface, buckling would take place spontaneously, i.e the translation symmetry of the surface would spontaneously be lowered, A similar spontaneous symmetry breaking, the Jahn-Teller effect, was discussed in the context of point perturbations in section 3.5. There, the point symmetry was broken, while no translation symmetry was involved. If the translation symmetry is broken, as in the case of surface reconstruction, one speaks of a Peierls instability or a Peierls t r a n s i t i o n However, as has been indicated at the outset, buckling turns out to be energetically not favorable in the case of Si (111) surfaces. Populating the lower s-like band with two electrons per primitive surface unit cell means transferring charge from the atoms lowered below the surface to the atoms

378 Chapter 3. Electronic structure of semiconductor crystals with perturbations

r?

'P r-

U

[I101

[1?01

Side view

a)

b)

Figure 3.45: a-bonded chain model of the Si (111) 2 x 1 surface (After Pandey, 1982). Part (a) shows the unreconstructed surface in top and side views. The top view in the second row has been rotated with respect to the top view in the first row in order t o allow for the side view below. Part (b) shows the same views of the surface as in part (a), but after reconstruction has taken place.

3.6. Clean semiconductor surfaces

379

raised above. This implies the creation of an electric dipole which is too costly in energy to actually take place. Using the terminology of section 2.2 we may say that correlation effects of electron electron interaction, more strictly speaking, the configuration dependence of oneelectron states, prevents the buckled Si (111) surface to be lower in energy than the ideal one. The reconstruction model which actually applies to the Si (111) surface is the so-called s-bonded chain model, illustrated in Figure 3.45. In this model, second layer atoms in rows parallel to fi f2, i.e. along the [lo3 direction in Figure 3.45a (including atom number 2) are raised into the first layer as shown in Figure 3.45b, breaking their bonds with atoms in the third layer (for example, the 2-5 bond). The dangling bond of the new surface atom (say atom 2) is used to establish bonds with atoms of the first layer (the 2-1 bond in this case). These can only be s-bonds (indicated by double lines in Figure 3.45b) because the dangling bonds are perpendicular to the surface. In this way s-bonded chains occur along the [ l O q direction (Pandey, 1982). The dangling bonds left at the third layer atoms (for example, atom 5) are saturated by hybrids of the first layer atoms which have been lowered down to the second layer (for example, atom 3). The surface is in fact 2 x 1 reconstructed. This may be seen by taking the primitive lattice vectors of the ideal surface to be f 1 + f 2 and -f2. Doubling -f2 yields the rectangular lattice indicated in Figure 3.45b by dashed lines. Its primitive lattice vectors are f1 + f 2 and - 2 f 2 + (fi + f 2 ) = f 1 - f 2 so that the short side of the rectangle is parallel to the chain direction [lOT], and the long side perpendicular to it (parallel to [121]).A peculiarity of the n-bonded chain model is that it has a different bonding topology in comparison with the ideal (111) surface and also with respect to the buckling model. While the latter exhibit rings with 6 mutually bonded atoms (see Figure 3.45a), the former shows alternating rings with 5 and 7 bonded atoms (Figure 3.45b). This is due to the fact that bonds existing at the ideal and buckled surfaces are broken and new bonds are established in the s-bonded chain model. The total energy of this model is clearly below that of the ideal surface (about 0.5 eV per surface atom). Thus it represents a good candidate for the reconstruction of the (111) Si surface. Further evidence is provided by ARUPS and optical measurements. Figure 3.46 shows the wavevector dispersion of the two bound surface bands as obtained from ARUPS measurements together with the calculated dispersion of these bands. The agreement is quite satisfying. The strong dispersion of the bound surface band on the r-X-line and the weak dispersion on the X-M-line of the rectangular surface B Z is easily understandable: the long r-X-side of the rectangular unit cell in q-space corresponds to the short side of the rectangular unit cell in coordinate space, which is also the direction of the s-bonded chains. One expects strong dispersion along the chains and weak for the perpendicular direction, exactly what is seen in Figure 3.46.

+

380 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.46: Dispersion of the bonding (B) and antibonding (A) bound surface state bands along the r - X - M line for the Si (111) 2 x 1 surface. Curves are calculated within the 7r-bonding chain model, points are obtained by means from ARUPS measurements. (After Martensson, Cri-

centi, and Hansson, 1985.)

t2

-2> W

3 1 0

d

Q,

a

x

Yl

g o

W

-1

r

X

nf

20 wave vector

Figure 3.47: Differential reflectivity spectrum of the Si (111) 2 x 1 surface (After Chiarotti, Nannarone, Pastore and Chiaradia, 1971.)

0

93

0.A

0,s Energy (ev)

$5

[

7

3.6. Clean semiconductor surfaces

38 1

Figure 3.48: Polarization dependence of the differential reflectivity spectrum of the Si (111) 2 x 1 surface of Figure 3.45 (taken at its maximum). The solid curve is calculated using the x-bonded chain model, the dashed curve using the buckling model, and the points are experimental data (After Del Sole and Selloni, 1984.) Support for the r-bonded chain model comes also from optical measurements. The differential reflectivity spectrum of the 2 x 1 reconstructed (111) surface of Si is shown in Figure 3.47. There is no doubt that the observed peak at 0.5 eV is due to optical transitions between occupied and unoccupied bound surface bands. Such bands exist both in the buckling model as well as in the 7r-bonding chain model (in the latter one has 7r-bonding and x-antibonding bound surface bands). However, the two models differ in regard to their predictions on polarization dependence of optical reflectivity. According to the 7r-bonded chain model, transitions with light polarized parallel to the chains, i.e. parallel to [lOq, should be allowed and transitions for light polarized parallel to the perpendicular direction [121]should be forbidden, while this should be reversed for the buckling model. The experimentally observed polarization dependence shown in Figure 3.48 is that predicted by .I-bonding chain model. It rules out the buckling model. Besides the 2 x 1 reconstruction, there are other reconstruction states of the Si (111) surface. The 7 x 7 reconstructed surface is the most stable one. The complicated structure of this surface has finally been resolved by combining the results of various experimental methods including STM (see Figure 3.49). The model which accounts for all experimental data utilizes three structural disturbances of the ideal surface, these being dimers (D), adatoms (A) and stacking faults (S). It is referred to as the DAS model

382 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.49: Scanning tunneling microscopy (STM) image of the Si (111) 7 x 7 surface. (After Quate, 1986.) (Takayanagi, 1984). The DAS model of the 7 x 7 reconstructed (111) Si surface is shown in Figure 3.50. (100)surface

The (100) surface is the surface of choice for electronic device applications of Si. The geometry of the ideal (100) surface is shown in Figure 3.51. The

surface lattice is a square one, with primitive lattice vectors f 1 , f2 given in Table 3.8. The primitive unit cell has one surface atom, and each atom has two dangling bonds which point out of the surface like ‘rabbit ears’ (see Figure 3.51a). Each dangling bond is only half-filled as in the case of the (111)surface considered above. Thus, the ideal surface is metallic, and this state is unlikely to be stable. A state of lower total energy can be established by a 2 x 1 reconstruction as follows: The atoms of two neighboring rows parallel to [ O l i ] (or [Oll]) move slightly towards each other in order to allow bonding between two of their four dangling hybrids. This dimerization of the surface gives rise to bonding and antibonding bound surface states, the lower bonding state being completely filled and the upper antibonding state being completely empty. The primitive lattice vector in the direction of f 1 doubles, thus a 2 x 1 reconstruction takes place, and the surface lattice becomes rectangular. The remaining two dangling bonds of a dimer are still half-filled, however, so that the surface is not yet stable. It is stabilized by buckling,

3.6. Clean semiconductor surfaces

383

a1

Figure 3.50: Dimer-Adatom-Stacking-Fault (DAS) model of the Si (111) 7 x 7 surface. The side view (a) is shown to identify the atoms: large shaded circles are adatoms, open circles are surface atoms of the first (large circles) and second (smaller circles) monolayer, solid circles are bulk atoms which do not undergo reconstruction. The top view (b) shows a 7 x 7 surface unit cell and its surroundings. The small circles within shaded circles represent second layer atoms vertically below the adatoms. (After Takayanagi, 1984.) which turns out to be energetically favorable in this case. One of the two atoms of a dimer moves above the surface and one below. The dangling hybrid at the lowered atom, being p-like, takes a higher energy and is correspondingly empty, while the dangling hybrid at the raised atom, being s-like, takes a lower energy and is filled. Rigorous structure calculations essentially confirm this simple tight binding picture. They only add displacements of the dimer atoms parallel to the surface in addition to the perpendicular ones. Due t o the two kinds of displacements, the dimers become asymmetric. The described 2 x 1 reconstruction of the Si (100) surface is therefore called the

384 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Top view

I--

(100)

0

0'- -0

n

It-

-?

0 U

10113

Side view

U

a)

b)

Figure 3.51: Geometry of the ideal Si (100) surface (a), and of the asymmetric dimer model of this surface. asymmetric dimer model. Although the 2 x 1 reconstruction is the most common superstructure of the Si (100) surface, other reconstructions are also observed, for example, 2 x 2, c - 2 x 2, c - 4 x 2. Most of these structures may be traced back to asymmetric dimers as building blocks.

GaAs and other 111-V compounds (110) surfaces

The (110) surface represents the cleavage plane of zincblende type crystals. The best understood (110) surface of all zincblende type 111-V semiconductors is the (110) surface of GaAs. Its surface geometry is shown in Figure

3.6. Clean semiconductor surfaces

Top view

385

Ga o

(110)

As

0

I ?

I I I

I

I

I

1

I

I

5 0 Y

I I

1

I I

I

I

I I

I I

[?I0 1

Side view

0 c7 U

[?I01

a)

b)

Figure 3.52: Geometry of the ideal GaAs (110) surface (a), and of the same surface after relaxation (b). 3.52. The surface lattice is p-rectangular. The primitive lattice vectors are given in Table 3.8. There are two surface atoms in a primitive unit cell, one Ga and one As atom. Two bonds of each surface atom lie within the surface, one is directed back and one is dangling. The two dangling hybrids per unit cell have different energies since they belong to either a Ga- or an As atom. Thus one expects two bound surface bands, one Ga-like and one As-like. These are in fact seen in the band structure of the ideal (110) GaAs surface depicted in Figure 3.53. The lower As-like bound surface band is completely occupied, and the higher Ga-like band is completely empty. Thus the ideal surface is semiconducting. Nevertheless, it does not yet represent the stable state, as the gap between the two bound surface states is too small. It can be enlarged by moving the As atom above the surface, rendering its dangling hybrid more s-like and lowering its energy, and moving the Ga atom below

386 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Ga As (111)

(1001

(1101

-10

r

M

K

r r x '

M

X

r r

i

~

2 D wave vector Figure 3.53: Band structure of the ideal GaAs (110) surface. (After Talwar and Ting, 1992.)

the surface, rendering its dangling hybrid plike and raising its energy (see Figure 3.52). These displacements do not change the translation symmetry of the surface because the two atoms belong to the same unit cell. One therefore has a relaxation instead of a reconstruction of the (110) surface. This is consistent with LEED measurements, which do not show spots other than those due to the ideal prectangular lattice. The experimental value for the rotation angle w of a Ga-As bond with respect to the [ l i O ] direction is close to 30°, in good agreement with structure calculations. Experiment and theory also agree in regard to an essential feature of the band structure of the relaxed (110) surface. As indicated in Figure 3.54, relaxation moves the bound surface bands completely out of the fundamental gap, the As-like band merges into the valence band, and the Ga-like band merges into the conduction band. This implies that perturbations of the surface such as, for example, coverage by an insulator, can easily create surface states in the gap. Such states are in fact present at GaAs/insulator interfaces. They pin the Fermi level and preclude the possibility of making GaAs-based field effect transistors in the same way as the Si-based MISFET (see Chapter 7 for further discussion). Relaxations similar to that of the (110) surface of GaAs are also observed at the (110) surfaces of other 111-Vcompounds. For all materials except Gap, relaxation moves the bound surface states out of the gap (see Figure 3.54).

l

r

3.6. Clean semiconductor surfaces

387

(110)cleavage face GaP GaAs GaSb

-

A

-

-3-

2

h

-0 5

In P

In As

In Sb

-4-

3

u

9 -5-

U

n

g

3 -6x e E -7-

w

Figure 3.54: Energy level diagrams of bound surface states and bulk states of various 111-Vcompound semiconductors below the vacuum level. Solid lines represent experimental results, and dashed regions represent dangling bond surface bands. F'ramed undashed regions indicate bulk bands. (After Bertoni, Bisi, Calandra, and Manghi, 1978.)

Other surfaces

Besides the (110) cleavage surface, investigations have mainly been focused on the low index (111)and (100) surfaces of GaAs. (100) is the preferred surface orientation of GaAs wafers used in device fabrication (partially because the (011) plane perpendicular to this surface represents the cleavage plane of GaAs). The geometrical structures of the ideal surfaces are the same as those of the corresponding Si surfaces shown, respectively, in Figures 3.43 and 3.51. The (111)and (100) surfaces of GaAs differ from the (110) surface of GaAs mainly in regard to the fact that two different surface terminations are possible in their case, one by Ga atoms and another by As atoms. One says that these surfaces are polar, in contrast to the (110) surface, which is said to be non-polar. In forming a polar surface, an electric dipole is created between a Ga-layer and an As-layer which is costly in energy. This explains why for GaAs and other 111-V compound semiconductors, the cleavage plane is neither ( l l l ) , like in the case of group-IV materials, nor (loo), but the non-polar (110) plane. In the latter case the 1:l ratio of Ga- and As atoms is strictly k e d by chemical stoichiometry. For the polar surfaces, there are, however, no stoichiometrical reasons which cause a surface termination by

388 Chapter 3. Electronic structure of semiconductor crystals with perturbations only Ga- or only As atoms. Both kinds of terminations can occur simultaneously at a given surface, and in order to define the surface uniquely one has to specify the percentage of Ga and As it contains. A rough distinction is that between Ga-rich and As-rich surfaces. The structure of a particular polar GaAs surface depends decisively on its termination. For the Ga-rich (111) surface one finds a 2 x 2 reconstruction, x - 23.4'' and for the As-rich case there are f i x 8- 30° and reconstructions in addition. For the Ga-terminated 2 x 2 (111) surface, a model has been proposed with a quarter of the surface atoms missing. The remaining first-layer Ga atoms and the second layer As atoms undergo a buckling, which raises the As atoms close to the surface. The structure of the GaAs (100) surface exhibits an even greater variety, depending on surface termination, surface treatment and temperature. Some of the possible reconstruction states can be seen in the LEED pictures of Figure 3.42. The c - 2 x 8 structure is found for an As-stabilized surface, which is important for MBE-growth because this commonly begins and ends with As-rich conditions. For the Ga-stabilized surface, the c - 8 x 2 reconstruction is found to be stable.

a

3.7

Semiconductor microstructures

Semiconductors with a clean planar surface, considered in the section above, may be thought of as units of two infinite half spaces, one filled with the semiconductor material, and the other being empty. If the vacuum is replaced by a semiconductor material different from the first, then one obtains a semiconductor heterojunction or single semiconductor heterostructure. Figure 3.55 shows an example. Below the plane at z = 0 one has semiconductor material 1, say GaAs, and above it is the material 2, say AlAs. The plane at z = 0 is called the interface. The microstructures to be considered in this subsection are composed of semiconductor heterojunctions. Thus, before addressing microstructures we must deal with het eroj unct ions.

3.7.1

Heterojunctions

Below we describe the electronic structure of semiconductor heterojunctions. The two semiconductor materials are taken to be undoped. Free carrier effects on the electronic structure can be neglected in these circumstances. In the case of heterostructures formed from doped semiconductors, such effects might be important. They are treated in Chapter 6 in a systematic way. Below, we discuss the various stationary one-electron states of undoped heterostructures, omitting free carrier effects.

3.7. Semiconductor microstructures

389

Stationary one-electron states

As in the case of a crystal with a clean surface, heterojunctions of the kind described above possess a 2-dimensional rather than a 3-dimensional lattice translation symmetry. Generally, their one-electron states cp(x) are Bloch states (Pk,,(xI(,z ) in regard to their dependence on the position vector component XIIparallel to the interface, with 2-dimensional quasi-wavevectors kll. The corresponding energy eigenvalues form bands in the 2-dimensional first B Z of the heterostructure. Just as in the surface case, almost all energy eigenvalues of the two infinite bulk materials, i.e. the Bloch bands E,l(k) and E,a(k) of these materials, are also energy eigenvalues of the heterojunction. What may change are the eigenfunctions of these energy bands. Let us fix a particular quasi-wavevector ko. If an energy level E,n(ko) of material 2 does not coincide with any of the allowed energy levels E,l(ko) at ko of material 1, then the eigenfunction belonging to this level and quasi-wavevector will be localized in material 2. There, its wavefunction is spread out uniformly over the whole semi-infinite crystal from z = 0 to z = +co;it forms a bulk state of material 2. This is illustrated in Figure 3.56a by representing energy levels of this kind by lines extending from z = 0 to z = +co. Vice versa, if, at ko, an energy level E,l(ko) of material 1 does not coincide with any of the allowed energy levels E,z(ko) of material 2, it forms a bulk state of material 1, and may be represented by a line extending from z = -co to 0. If there are energy levels at ko which are identical in both materials, i.e. with E,l(ko) = EA(ko), the corresponding wavefunctions will extend over both materials (Figure 3.56b). In the case of identical energy levels E,l(ko) and E,a(kb) at dzflerent wavevectors ko and kb, it is not a priori clear what will happen. If a matching of the corresponding two eigenfunctions and their derivatives at the interface turns out to be possible, then states extending over the whole heterojunction from -cc to f m will exist (Figure 3.56~).If

390 Chapter 3. Electronic structure

Material 1

of semiconductor crystals with perturbations

I

Material

2

P a, L.

Figure 3.56: Various bulk states of a semiconductor heterojunction. no matching is possible, then the two eigenfunctions will remain localized in their respective material regions. One has a situation similar to that in the case of electromagnetic waves propagating between two semi-infinite dielectric media: for certain wavevectors they may propagate from one medium into the other, and for others they are internally reflected, as indicated in Figure 3 . 5 6 ~ .

Besides bulk states, there may be stationary states with energy eigenvalues which are allowed in none of the two materials. The wavefunctions of such states are localized at the interface between the two materials, just like the bound surface states in the case of clean surfaces. They are called bound interjuce states. In analogy to the clean surface case, interface resonances

3.7. Semiconductor microstructures

391

may also occur, their energy levels lie in one of the bands and their wave functions are weakly localized at the interface. If the two materials forming a heterojunction are composed of chemically similar atoms as in the case of GaAs and AlAs and in many other cases of practical importance, then no bound interface states will be possible, because the perturbation at the interface is too weak.

Valence band discontinuity

In many semiconductors, the maximum energy of the valence band occurs at the center r of the first BZ. Below we will restrict ourselves to materials of this kind. Of course, the maximum energy of the valence band itself depends on the material under consideration. This statement may seem to contradict the results of Chapter 2, where this energy was set to zero for all materials. However, this was done in the context of dealing with the band structure of only o n e infinite bulk semiconductor at a time. For such a single semiconductor, the energy origin could be chosen arbitrarily, and we took it to be at the valence band maximum. In the heterojunction of Figure 3.55, two semiconductor materials are involved, but the energy origin of the heterojunction can be fmed only once. If the valence band maximum is taken as zero for one material, it will, in general, differ from zero for the other material. Here, instead of setting it to zero for any material, we select the vacuum level as the common energy origin. This level may be defined as the minimum energy which an electron in an infinite semiconductor sample must have in order to escape. We will denote the valence band edge of a particular material i , i = 1,2, referred to the vacuum level as origin, by E$. Then the minimum energy which must be expended to remove a valence electron from material i , is given by --E$ If the electron should escape by absorbing a photon, then -Eti is the minimum photon energy required (photo-t hreshold energy). For the actual positions of the valence band edges Evl and Ev2 at a heterojunction, the infinite bulk values Etl and Et2 have only an indirect meaning. In fact, for each of the two semiconductors of a heterojunction the adjacent material is foreign and represents an external perturbation. Even if no free charge carriers are available, as we assume, each of the semiconductors reacts to this perturbation by redistributing its electrons, in this case its bound valence band electrons. How this occurs is quite clear physically, and it can also be analyzed in a more rigorous treatment: valence electron charge will flow from the material with the higher valence band edge into the material with the lower valence band edge, thus lowering the total energy of the heterojunction. In this way, an electric dipole layer develops at the interface, having a thickness of several atomic monolayers. Macroscopically, the spatial extension of the dipole layer is zero, but the associated dipole

392 Chapter 3. Electronic structure of semiconductor crystals with perturbations moment has a non-zero macroscopic magnitude. As is well-known from electrostatics, the potential cp exhibits a discontinuity in passing through such a dipole layer. We denote the left boundary potential value by cp1, and the right one by 9 2 . Then the actual positions of the valence band edges E,1 and E,2 at the heterojunction are given by the expressions

(3.232) The difference AE, zz [E,1 - E,2] between the two valence band edges is called valence band discontinuity or valence band offset of the ‘material l/material 2’ heterojunction. By means of equation (3.232) the valence band discontinuity AE, is expressed as

(3.233) Beside the pure bulk contribution [E$ - E&], the band discontinuity also contains the dipole contribution -e[cpi-pg]. The latter generally depends on the properties of the interface. If one were to ignore this dipole contribution, then AE, would obey the so-called ‘transitivity rule’ which states that the AE,-values for a succession of two heterojunctions ‘1/2‘ and ‘213‘ should add up to the AE,-value of the combination ‘1/3’. In practice this ‘rule’ is rarely fulfilled, which points up the importance of the dipole contribution to A E,. The dipole contribution -e[cpl - cpz] may be estimated by means of the tight binding method developed in section 2.6. The bonding energy level € b considered there forms a rough measure for the average valence band energy. In formula (2.315)’ € b is given for a zincblende type semiconductor. The corresponding bonding orbitals IbtR) of equation (2.316) describes the charge transfer from the cation c (there denoted by an upper index ‘1’)to the anion a (there denoted by an upper index ‘2’) in terms of the hybrid energy difference EL - €2. If the hybrid energies were equal, no charge transfer would occur. A semiconductor material i composed of cations ci and anions ai, forms a big molecule of the average hybrid energy 2 ; = (€2 &/2. Thus the charge transfer between material 1 and material 2 is governed by the average hybrid energy difference S i - 2;. If this difference is non-zero, valence electron charge will be transferred into the material with the lower value of 2;. As pointed out above, the charge transfer will result in an electrostatic potential difference at the interface. The potentials on either side of the interface will add to the average hybrid energies and diminish their difference. Charge will flow until the average hybrid energy difference has been equilibrated by the potential jump. This is described by the relation

+

-1

ch - ecpi = F;

-

ecp2,

(3.234)

3.7. Semiconductor microstructures

393

i.e. the average hybrid energy levels on the two sides of the heterojunction

align, and the potential contribution -e[cpl- 1,721 to the valence band offset matches IS; - T i ] . At this point it is advisable to recall a result concerning deep levels of transition metal (TM) atoms obtained in section 3.5. There, it was shown that the difference between the position E+M of a deep level of a particular TM atom in a semiconductor material 2, and the position E&M of the same deep level in a semiconductor material 1, equals the average hybrid energy difference between the two materials,

- E&M = 7; - Sk.

ETM 2

(3.235)

Taking the deep level positions to be measured with respect to the valence band edges E& of the two infinite bulk materials, we denote these positions by E ) ~ , i = 1 , 2 , so that E

)

(3.236)

= ~ E k M - EVi. 0

Considering relations (3.234) to (3.236), the difference [ c $ ~- t+M] between the positions of the deep level in the two materials is just the valence band discontinuity A E , of the two materials, hence AEIJ=

2

[ETM

- 61 ~

~

1

.

(3.237)

This relation reduces the experimental determination of valence band discontinuities to measurements of deep level positions of TM atoms with respect to the valence band edge. Other experimental methods rely on measurements of photoemission, optical or transport properties of heterojunctions. Experimental values for valence band discontinuities of several heterojunctions are shown in Table 3.10. Often, different values are obtained by different methods or even by different authors using the same method. This points up the experimental difficulties in determining valence band discontinuities, and also to the dependence of the discontinuities on the preparation of the heterojunctions. In order to obtain theoretical values for valence band discontinuities, electronic structure calculations are required with an accuracy of less than 0.1 eV. Such an accuracy is difficult to achieve so that, in many cases, calculated valence band discontinuities have also considerable uncertainty. To this day, a great deal of effort is devoted to the task of obtaining more reIiable experimental and theoretical data for AE,, even in the case of the most thoroughly explored heterojunction, that being between GaAs and (Ga,Al)As alloys. O t h e r band discontinuities Knowing the lineup of the valence band edge at I? for a particular heterojunction, the lineups of all other energy levels at I' and off I' can be determined from the known bulk band structures of the two materials. Below we

394 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Junction

Ge/GaAs

InAs/GaSb

GaAs/ZnSe

HgTe/CdTe

InP/CdS

AE,(eV)

0.54

-0.46

0.96

0.02

1.63

AE,(eV)

0.23

0.81

0.29

1.41

-0.39

AEg(eV)

0.77

0.35

1.25

1.43

1.24

demonstrate this taking the conduction band edges of a GaAs/Gal-,Al,As heterojunction as an example. In Figure 3.57 the valence and conduction band edges of the Gal-,Al,As alloy are plotted as a function of x. While GaAs and Gal-,Al,As with x < 0.42 are direct semiconductors with both the valence band maximum and the conduction band minimum at I‘, alloys with z > 0.42 and pure AIAs form indirect materials with the valence band maximum still at r, but the conduction band minimum at the B Z boundary point X . These two cases are also to be distinguished in aligning the conduction band edges of the heterojunction. In the f i s t case (x < 0.42, the conduction band edges Ec1,2 of the two materials are obtained from the relation Ec1 = E v 1

+ E ~ I , Ec2 = E v 2 + Eg2,

(3.238)

where E,l,2 are the gap energies of the two materials. For the conduction band discontinuity AE, = E,2 - E,1 of the ‘material l/material 2’ heterojunction, it follows that AE,

Ec2 - Ecl =

-AEv

+ AEg,

(3.239)

where AE, = E,2 - E,1 is the discontinuity of the energy gap. If AE, and AE, are known, AE, can be calculated by means of relation (3.239). The gap discontinuity AE, follows from the gap energies Egl,Eg2 of the two infinite bulk materials. The lineup thus obtained for the conduction band edge of a GaAs/Gal_,Al,As heterojunction with x < 0.42 is schematically plotted in Figure 3.58a, together with the lineup of the valence band edge. Often, the two band offsets AE, and AEc are expressed in percent of the gap discontinuity AE, by means of the ratios Qv = AE,/AE, and Q c = AE,/AE,. Because of AE, A E , = AE,, one has Q , Qc = 1. In the second case, i.e. for GaAs/AlAs or GaAs/Gal-,Al,As heterojunctions with alloys having an indirect gap (z > 0.42), the conduction band

+

+

3.7. Semiconductor microstructures

Figure 3.57: Valence band edge at

r together with the lowest conduction band levels at I? and X for Gal-xA1xAs alloys of varying composition x. The Fe(1 /2+) acceptor level is used as energy origin following the discussion in the main text. ( A f t e r Langer and Heinrich, 1985.)

395

f

2.0

c

3

c

1.6

+

-.+ ---.-

Fe(1+/2+)

0.4

Composition x

Ec2

X r

I

tc2

r

r

I

Ev2

Ev2

Figure 3.58: Conduction and valence band lineups for a GaAs/Gal-,Al,As heterojunction. In part (a) ( x < 0.42) the alloy gap is direct, and in part (b) (x > 0.4!2), the alloy gap is indirect. In the latter case, the r and X levels forming the conduction band edges in, respectively, GaAs and Gal-,AI,As are also plotted in the respective other material where they do not form the conduction band edge.

396 Chapter 3. Electronic structure of semiconductor crystals with perturbations edges of the two materials occur at different k-points, that of GaAs at r, and that of the Gal-,Al,As alloy at X . Recalling the discussion about the stationary electron states of heterojunctions at the outset of this section, it becomes evident that it is no longer meaningful to plot the conduction band edge throughout the whole heterojunction as before. What may be plotted is the lineup of the r and X levels forming the band edges in one of the two materials. This is done in Figure 3.58b. The illustration indicates that for the conduction band states of GaAs with wavevector r, the Gal-,Al,As alloy region forms a barrier, and for the conduction band states of the alloy at X , the GaAs region does so. Types of heterojunctions

Consider two direct gap semiconductors with the valence and conduction band edge at I?. There are several qualitatively different possibilities for the lineup of the two band edges E,i and Eci, i = 1 , 2 (see Figure 3.59). The conduction band edge of one material, say of materiai 1, may lie below the conduction band edge of material 2, while simultaneously the valence band edge of material 1 may lie above that of material 2. This case is referred to as heterojunction of type I. In this case the gap of material 1 is located entirely within the gap of material 2. If both the conduction and valence band edges of a particular material, say, again 1, are, respectively, below the two edges of material 2, one has a heterojunction of type II. The staggered type II heterojunction occurs when we have Ec2 < E,1, and also E,1 < Ec2, E,1 < Eva holds, and the misaligned case applies if Ec2 > E,1. In the staggered case the heterojunction still has a gap, while in the misaligned case the conduction band of material 1 overlaps with the valence band of material 2 so that the gap of the heterojunction disappears. Sometimes, type ZZI heterojunctions are defined. These are heterojunctions of type I with zero energy gap in material 1. A look at Table 3.10 shows that the heterojunction Ge/GaAs is of type I, CdS/InP of type 11- staggered, InAs/GaSb of type 11-misaligned, and HgTe/CdTe is of type 111.

3.7.2

Microstructures: Fabrication, classifications, examples

Heterojunctions cannot be fabricated by simply putting together two separately made semiconductor samples with plane surfaces. If one would proceed in such a way the result would be a completely rough, polluted interface, incapable of hosting electron and hole states which extend upon both materials, a property which has been assumed above and which turns out to be crucial for the electronic behavior of a heterojunction. In actual practice, the second material must be placed on top of a crystal made from the first by continuing the crystal growth, a process referred to as epitaxial growth. In

3.7. Semiconductor microstructures

397

Type staaaered

+ >r P W

EC2

C

W

Ev2

tvi

misaligned A

x

F

d

15

0 material 1

material 2

0

2 material 1

2

material 2

Figure 3.59: Heterojunctions of type I, I1 and 111. this kind of growth the underlying crystal, the so-called substrate, imposes its structure onto the growing layer, in contrast to ordinary deposition of an evaporated material which commonly results in a non-regularly structured layer. For epitaxial growth to be possible, the two materials must have similar crystallographic structures, and their lattice constants must be close to each other.

Epitaxial growth

The fabrication of heterojunctions by means of epitaxy suggests to proceed from simple structures consisting only of the substrate and the epitaxial layer, to more complex structures by growing a second epitaxial layer of another material on top of the first, a third layer on top of the second etc. If one does so, one obtains double and multiple heterostructares. The term superlattice is used if alternating layers of two materials are grown with equal thicknesses for layers of the same material (see Figure 3.60). Two epitaxial growth techniques are particularly important in this context, namely Molecular Beam Epitaxy (MBE) and Metal Organic Vapor Deposition (MOCVD).

398 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.60: Heterostructures as grown by epitaxy: single heterostructure (a), double heterostructure (b), multiple heterostructure (c), superlattice (d). The principle of MBE is shown in Figure 3.61. The whole growth process takes place in a vacuum chamber under UHV conditions (typically about 2 x 10-l' T o w ) . In this chamber, the substrate is placed in front of a number of effusion cells which contain the various chemical elements to be deposited on the substrate. If one, for example, wants to grow a AlAs layer on top of a GaAs substrate, two cells are required, one for A1 and one for As, and their shutters must be open at the same time. Doping of the layer may be achieved by opening a third effusion cell containing the dopant atoms. If one wants to proceed with a GaAs layer on top of the AlAs layer, one needs a fourth cell with Ga. In order to grow a GaAs/AlAs superlattice, the shutters of the Ga and A1 cells have to be opened and closed alternatively while the As shutter has to be kept open all the time. There are many parameters which influence the growth process, in particular, the temperature of the substrate, the flux from the effusion cells, and the partial pressures of the various elements involved. The atoms or molecules from the effusion cells may have different sticking coefficients on the substrate. Arsenic atoms, for example, stick much less well on GaAs then Ga atoms. Thus the As partial pressure must be much larger than the Ga partial pressure in the epitaxy of GaAs on a GaAs substrate. The fact that MBE proceeds entirely in UHV may be exploited in controlling the growth process. One may employ characterization techniques which require UHV, such as RHEED for studying surface perfection, or mass spectroscopy for analyzing the composition of the residual gas in the growth chamber (for more on MBE see, e.g., Herman and Sitter, 1989). Unlike MBE, MOCVD takes place in a chemical reactor at atmospheric

3.7. Semiconductor microstructures

LIQUID NITROGEN COOLED SHROUDS

FLUORESCENT SCREEN

399 RHEED GUN

/

MAIN SHUTTER

TO VARIABLE SPEED MOTOR ANDSUBSTRATE HEATER SUPPLY

Figure 3.61: Principle of MBE growth.[After Cho and Cheng, 1981.) pressure or slightly below. The atoms to be deposited, say A1 and As, are provided by metal-organic gases, Al(CH3)3 (trimethyl aluminum) and AsH3 (Arsin) in this case. The A1 and As atoms are liberated from their compounds by means of a pyrolytic process which takes place on top of the heated substrate. After this the atoms are chemically bound to the substrate. In order to proceed with a layer of different chemical composition, say GaAs, Al(CH3)3 has to be replaced by Ga(CH3)3 (trimethyl gallium) in the growth reactor. MOCVD, unlike MBE, operates close to equilibrium conditions. The growth velocity in MOCVD is typically somewhat larger than in MBE because more atoms are provided to the growing layer in MOGVD than in MBE. This makes MOCVD particularly suitable for producing devices based on heterostructures. MBE is more universal and more responsive to control than MOCVD. The advantages of both methods are combined in MOMBE (Metal organic MBE): the growth process takes place in an UHV chamber, as in MBE, but the atoms to be deposited are provided by the pyrolysis of

400 Chapter 3. Electronic structure of semiconductor crystals with perturbations metal-organic compounds, as in MOCVD. By means of these and some other growth techniques it became possible to grow high-quality heterostructures of many elemental and compound semiconductors, as well as from semiconducting alloys. The interfaces of these structures can be made almost abrupt, meaning that the transition from one material to the other occurs essentially within one atomic layer. Unwanted impurities and structural defects can be excluded to a large extent, both at the interface as well as in the bulk. One can grow layers up to several millimeter thicknesses and beyond but, what is more important from the physical point of view, one can also grow very thin layers, down to the ultimate limit of one atomic monolayer. It turns out that the electronic structures of multiple heterostructures differ from those of the constituent materials if the layer thicknesses reach the nanometer range. In subsection 3.7.4 we will prove this rigorously and describe the modified electronic structures in greater detail. Here, we will start with a qualitative discussion. We consider a double heterostructure formed by a GaAs layer embedded between two Gal_,Al,As layers with x < 0.42 (see Figure 3.62a). The electron states of this heterostructure with energies close to the conduction band bottom will be spatially confined to the GaAs layer. The latter is called a quantum well (QW) in this context, and the alloy layers are referred to as barriers (for a systematic introduction of these concepts see subsection 3.7.4). The confinement of electron states in the quantum well raises their energy and creates discrete levels, just like for a particle in a potential box. The same statement applies to holes in the case of the double heterostructure of Figure 3.62a. If, instead of type I GaAs/Gal-,Al,As double heterostructures, we consider those of type I1 (see Figure 3.62b,c), electrons and holes are no longer confined to layers of the same material, but to layers of different materials. The central layer in Figure 3.62b which forms a well for electrons is a barrier for holes, and the two outermost layers which are barriers for electrons are (semi-infinite) wells for holes. In the misaligned case of Figure 3.62c, an energy region exists where the stationary states of the double heterostructure are mixtures of the electron states of the two outermost layers and the hole states of the central layer. Another example of multiple heterostructures with modified electronic properties is provided by superlattices composed of alternating (and sufficiently thin) layers of GaAs and Gal-,Al,As with z < 0.42 (see Figure 3.62d). In this case electrons from a GaAs well layer may tunnel through the neighboring Gal_,Al,As barrier layer reaching the next GaAs well, from which they may tunnel through the following Gal_,Al,As barrier layer, and so on. This leads to the formation of Bloch states and an additional energy band structure superposed upon the bulk band structure. It is called a miniband structure. Qualitatively, the same behavior is expected for su-

Semiconductor microstructures

40 1

a)

d) Type II-staggered

Type

II - misaligned

P Q,

5 c

2

b

2

Figure 3.62: Double heterostructure of type I acting as a quantum well both for electrons and holes (a), and of type I1 acting as a quantum well for electrons and a barrier for holes (b, c). In the misaligned case of type I1 (c) an energy region occurs where electrons and holes may coexist. Parts (d), (e) and (f) of the figure show the correponding superlattices giving rise to minibands on top of the bulk band structure. Further discussion is given in the main text. perlattices of the staggered type I1 shown in Figure 3.62e; solely the gap becomes indirect in coordinate space in this case. For misaligned type I1 superlattices shown in part f of Figure 3.62, Bloch states are formed from the electron and hole states of the respective wells. It is not surprising that the layer thicknesses must lie in the nanometer range for the confinement and tunneling effects discussed above to occur: 1 nm = 10 A is close to the distance between nearest neighbor atoms in a natural crystal (in GaAs the nearest neighbor distance is about 2.5 A). Heterostructures with such thin layers possess, so to speak, an artificial atomic superstructure which is likely to result in a modified electronic structure.

402 Chapter 3. Electronic structure of semiconductor crystals with perturbations The term artificial semiconductor microstructures or just semiconductor m i crostructures is therefore used in this context. The term nanostructures is also common. Besides epitaxially grown planar heterostructures and superlattices having a 1-dimensional artificial microstructure (in the growth direction), systems with a 2- or 3-dimensional microstructure are also investigated. The number of non-microstructured dimensions determines the number of spatial degrees of freedom of electrons and holes - these systems are termed quasi %, 1-, and 0-dimensionaL Quasi 1-dimensional systems are also referred to as quantum wires, and quasi 0-dimensional as quant u m dots. Such systems may be fabricated by means of an additional lateral structuring of epitaxially grown thin layers. The more appealing nutural growth of quantum wires and dots forms an area of active research at present. One refers to this as self-organized growth (Leonard, Krishnamurthy, Reaves, Denbaars, and Petroff (1993); Christen and Bimberg (1990); Notzel, Ledentsov, Daweritz, Hohenstein, and Ploog (1991); Zrenner, Butov, H a p , Abstreiter, Bom, and Weiman (1994); Stutzmann (1995)). In this book we concentrate on planar microstructures. The modified electronic structure of semiconductor microstructures r e sults in transport and optical properties which differ from those of the constituent bulk materials. Moreover, these properties may be tuned to a certain extent by varying the layer thicknesses and chemical compositions of the materials. The possibility of tailoring their properties makes semiconductor microstructures extremely interesting subjects for micro- and optoelectronics. Devices may be created with performance data superior to those of conventional electronic components, or with functions not accessible at all to elements made of bulk materials. High Electron Mobility Transistors ( H E M T s ) and Quantum Well (QW) laser diodes are tangible examples of this that already exist. The concept of semiconductor microstructures was first introduced by Esaki and Tsu (1970). Today, investigations of artificial microstructures are the most active area of semiconductor physics. Before we examine the electronic structure of these systems in more detail, we will present a short overview of the various kinds of planar semiconductor microstructures. Besides heterostructure systems discussed exclusively hitherto, doping microstructures will also be covered. In these, the spatial variation of material composition is replaced by the spatial variation of doping. We start with microstructures composed of different materials, which are by far the most important ones. Compositional microstructures As a guide to finding material combinations from which compositional microstructures with modified electronic structure may be formed, one can use a diagram which plots the energy gaps of the various semiconductors against

3.7. Semiconductor microstructures

3.5

' ZnS '

I

\

403

I

I

I

I

I

I

I

_ .-

MnSe

3.0 h

2 g

2.0

>r

1.5

2

1.0

v

2.5

0

P w

0.5 0.0

5.4

5.6

5.8

6.0

6.2

Lattice constant (

6.4

6.6

LI

Figure 3.63: Energy gap versus lattice constant for diamond and zincblende type semiconductors. Full lines represent alloys with direct gaps, and dashed lines represent alloys with indirect gaps. their lattice constants. Such a diagram is shown in Figure 3.63 for elemental and compound semiconductors of diamond and zincblende structure. Lines connecting two different materials indicate the gap energies of alloys made of these materials. The composition of an alloy is related to its lattice constant by means of Vegard's rule which linearly interpolates between the lattice constants of the two alloy components. For two direct gap materials 1 and 2, the gap difference AE, provides some hints about their band edge discontinuities. If the gap discontinuity AE, vanishes, then band edge discontinuities may, but need not, occur. If a gap discontinuity does exist, then at least one of the two band edges must also exhibit a discontinuity. In general, both the valence and conduction band discontinuities AE, and AE, differ from zero and their magnitudes reflect the gap discontinuity to a certain extent. Lattice mismatch

As the materials shown in Figure 3.63 are all of zincblende type, their main

404 Chapter 3. Electronic structure of semiconductor crystals with perturbations structural differences lie in their different cubic lattice constants a. Ideally, in epitaxial growth, the lattice constant a1 of the substrate (material 1)should be equal to the lattice constant a2 of the layer (material 2). If they differ, the layer material will not grow with exactly the same lattice parameters as would a free standing bulk crystal of this material, but with a lattice constant parallel to the layers equal to that of the substrate. One says that the layer grows in a pseudomorphic phase. The adjustment of the parallel lattice constant creates a mechanical strain in the layer. The corresponding strain energy increases the total energy of the layer, and this increase grows larger as the layer becomes thicker. Past a critical thickness d,, the accommodation of the lattice mismatch by strain becomes energetically more costly than the formation of lattice defects, in particular of dislocations. The strain will then be released by the formation of dislocation lines (see section 3.2). As long as one stays below d,, only a few dislocations occur and the layer will have good structural perfection despite the strain. One speaks of strained layers. The critical thickness depends on the magnitude of the lattice mismatch. Suppose that a layer of a cubic material 2 is growing on a (100) surface of a layer of cubic material 1, and that the cubic lattice constants a l , a:! of the two materials are different. The relative deviation f = (a:! - a l ) / a l of these constants is termed lattice misfit and measured in percent. For a lattice misfit f of a few percent the critical thickness may reach values not much smaller than 100 A. If f is less than a few tenths of a percent, the lattice mismatch has only little effect and, in many cases, it can be completely neglected. One speaks of lattice matched heterostructures. Otherwise one has lattice mismatched heterostructures. In order to grow lattice mismatched heterostructures it is important to know the distribution of strain between layers of different thicknesses. Such layers may be the substrate and the epitaxial layer, but also a buffer layer on top of the substrate which is used to accommodate part of the lattice mismatch strain. We suppose again a layer of cubic material 2 which is growing on a (100) surface of a layer of cubic material 1. The non-vanishing components of the stress tensors ~ ( i of) the two layers i = 1 , 2 are a,(i) and ayy(i)with aee(i) = ayy(i),and the non-vanishing strain components are e x s ( i ) , eyY(i) and e z z ( i ) with e r X ( i ) = eyY(i). The independent stressstrain relations for a particular layer read

“ 2 4 4

= cll(i)€zz(i)

+ c12(i)[err(i) + €yy(i)],

(3.240)

with cll(i) and clz(i) being the elastic stiffness constants of the cubic layer of material i. Because m t z ( i ) = 0, it follows from (3.240) that

3.7. Semiconductor microstructures

405

(3.241) For the parallel strain components e z Z ( i ) and e y 3 / ( i ) one has

4 2 ) - &(1) = tyy(2)- E v y ( l ) = t,

t

=

-a2 . - ai ai

(3.242)

The total strain E defined in equation (3.242) equals the lattice misfit f divided by 100%. Using the stress and strain components, the total elastic energy Eda of the two layers may be calculated, with the result

(3.243) where ci =

[Cll(i)

+ (1 - Ki)Cl2(i)l.

(3.244)

Minimizing Eela with respect to txZ(2)(or ~ ~ ~ yields ( 1 )the ) conditions

(3.245) According to these relations, the thin layer is more heavily strained than the thick one, provided the elastic constants of the two layer materials are comparable. This means, in particular, that the substrate will be almost unstrained and the epitaxial layer will accommodate almost the whole misfit strain. This was anticipated in the discussion above. Another general point to be mentioned in the context of lattice mismatched heterostructures concerns the effect of strain on the band structure. From theoretical considerations and experimental studies it is well known that such effects can be quite large (Bir, Pikus, 1974). To give an example, we consider a Si layer on top of a Ge substrate. The strain components in the Si layer are e X x = cyy = 0.04, corresponding to a lattice misfit of 4% (see below), and e Z z = -0.03, using the elastic stiffness constants c11 = 16.1 x 1011 dyn/crn2 and c12 = 6.4 x 1011 dynlcm’ of Si. The degenerate heavy-light hole valence band maximum of Si splits by 0.31 el/ under this strain. To produce the same splitting by means of uniaxial strain applied from outside, a pressure of 73 Kbar would be necessary. This example shows that considerable changes of the band structure are to be expected because of the lattice mismatch strain in heterostructures. In this context, energy levels in different materials or at different points of the first B Z can shift in dserent ways. One may take advantage of this to adjust the band discontinuities of a heterostructure to specified conditions. The strain becomes, so to speak, an additional degree of freedom for tailoring the properties of a heterostructure.

406 Chapter 3. Electronic structure of semiconductor crystals with perturbations Particular compositional microstructures GaAs/(Ga,Al)As

GaAs/Gal-,Al,As multiple heterostructures are the prototypes for lattice matched type I microstructures. The valence and conduction band edges of the Gal-,Al,As alloys have already been discussed in the context of Figure 3.57. Here, we add some quantitative estimates. The valence band heterojunction scales almost discontinuity A E v of the GaAs/Gal-,Al,As linearly with x according to the relation AE, = 0.45 x x eV. For the gap discontinuity A E , we have A E , = 1.25 x x e V , as long as we consider the direct region x < 0.42. This gives A E , = 0.85 x x eV for the conduction band discontinuity. With this, the valence band discontinuity takes the constant ratio Q v = (45/125)% = 36% of the gap discontinuity, and the conduction band discontinuity has the constant ratio Q, = 64%. Beside providing model systems for basic research, GaAs/Gal-,Al,As microstructures are used in devices like HEMTs, heterojunction bipolar transistors (HBPT), and QW laser diodes. (In,Ga)As/Ga( Sb,As)

The Inl-,Ga,As/GaSbl-yAsy material system forms heterostructures of type 11. The two band edges are lower in Inl-,Ga,As than in GaSbl-,Asy for all compositions x and y. For small values of x and y, especially for InAs/GaSb with x = y = 0, one has a misaligned type I1 heterostructure. The conduction band edge of InAs is 0.14eV below the valence band edge of GaSb. Superlattices based on InAsfGaSb heterostructures have been the subject of intensive basic research. Under certain conditions, these SLs exhibit metallic behavior. For Inl-,Ga,As/GaSbl-yAsY heterostructures with larger x and y, including the trivial GaAs/GaAs homostructure, one has staggered type I1 heterostructures, for which the conduction band edges are above the valence band edges in both materials. Perfect lattice matching is achieved if y = 0 . 9 1 8 ~ 0.082.

+

Si and (Si,Ge)

The lattice constants of Si and Ge are, respectively, 5.43 A and 5.65 A. This gives the lattice mismatch of 4 %, which has already been used above. The critical thickness d, amounts to about 30 A which is rather small. Therefore, one also considers heterostructures between Si as substrate and Sil-,Ge, alloys with low Ge content (x < 0.5) as epitaxial layers. In this case d, can be 100 A or larger depending on the actual value of x. (Si, Ge) alloys may also be used as substrates. Consider, for example, a Sil-,Gey substrate on which a Sil-,Ge, alloy layer is grown first, followed by a pure Si layer. Both layers are strained in this case. For 0 < y < x, the strain in the alloy layer

3.7. Semiconductor microstructures

407

is compressive, and that in the Si layer tensile. If the same Si/Sil-,Ge, double layer is grown on Si substrate, the whole mismatch strain occurs in the alloy layer while the Si layer remains unstrained. Because of the different strain distributions, the band alignment of a Si/Sio.~Ge0.5heterostructure on top of a Sio.75Geo.25 substrate differs from the band alignment of the same Si/Si0.5Ge0.5heterostructure on top of a Si substrate. The latter is of type I, with both the highest valence band edge and the lowest conduction band edge in the alloy. The former still has its highest valence band edge in the alloy, but the conduction band edge of Si is shifted down below the conduction band edge of the alloy by the tensile strain in the Si layer. Thus the Si/Si0.5Ge0.5 heterostructure on Si0.75Ge0.25 substrate is of type 11. The type I1 Si/Sil-,Ge, heterostructures are better suited for applications in electronic devices (in particular HEMTs), because here the free electrons are hosted by the pure Si layer, where the mobility is much higher than in the alloy layer. (Ga,In)(As,P)/InP

Quaternary alloys of composition Inl--rGa,AsyP1-y may be thought to be formed of the four binary compounds I d s , InP, GaAs and Gap, with composition ratios given, respectively, by (1-z)y, (1-z)(l-y),zy, and z(1-9). Generalizing Vegards rule, the lattice constant of the alloy may be estimated as 6.058(1- z)y 5.869(1- z)(1- y) 5 . 6 5 3 ~ ~5.451~(1- y). It matches with that of InP if the relation z = 0.189y/(0.418 - 0.013~)holds. Varying y between 0 and 1, the lattice matched alloy transforms from pure InP to an Inl-,Ga,As alloy with I = 0.47. Lattice matched Ino,53Gao,47As/InP microstructures may be used to fabricate HEMTs. The fundamental energy gap of the lattice matched Inl-,Ga,As,P1-, alloy varies almost linearly between 1.35 eV for InP and 0.85 eV for Ino,53Gao,47As, therefore, covering light emission wavelengths down to the technologically important near infrared region. Using Ino.53Gao.47As/InP based microstructures, laser diodes and photodetectors may be fabricated for optical fiber communication at 1.55 p m wavelength, which is the wavelength with minimum losses in quartz-based fibers.

+

+

+

(Zn, Cd) (Se,S) /GaAs

ZnSe and GaAs form an almost unstrained type I heterostructure. If a Znl-,Cd, S,Sel-, alloy of a certain composition z , y is combined with a Znl-,tCd+! SY!Sel-,~ alloy of another composition x’,y’, one obtains a strained heterostructure of type I. Blue-green laser diodes have been fabricated from such structures, more strictly, from Zn(S,Se)/(Zn,Cd)Se/Zn (S, Se) double heterostructures, embedded between other 11-VI-compound layers, deposited on top of a GaAs substrate.

408 Chapter 3. Electronic structure of semiconductor crystab with perturbations (Hg,Cd)Te/CdTe

HgTe and CdTe are lattice matched because the lattice misfit is smaller than 0.4 %. Being a zero gap material, HgTe forms a type I11 heterostructure with CdTe. The valence band offset turns out to be rather small ( M 0.02 e v ) . In Hgl-,Cd,Te alloys the gap opens at 2 = 0.15. For 0.15 < 2 < 1 the alloys are direct gap materials with the valence and conduction band edges at.'I In this range, Hgl-,Cd,Te/CdTe forms type I heterostructures which, for sufficiently small 2 , can be applied in infrared detectors. (Pb,Sn)Te/PbTe

The lattice misfit between PbTe and SnTe is about 2 %, so that strain effects in Pbl-,Sn,Te/PbTe heterostructures are not negligible. Pbl-,Sn,Te alloys are direct gap materials with the conduction and valence band edges on the first B Z boundary at L. For PbTe, the L$ state forms the valence band edge, and the Lg-state forms the conduction band edge (see Figure 2.35). For SnTe, this level ordering is inverted. Thus, the gap of Pbl-,Sn,Te alloys must go through zero for some composition TO. At 77 K , one has zo M 0.4. Thus, Pbl-, Sn,Te/PbTe heterostructures are of type I for z < 20,of type I11 for z = zo, and again of type I for 2 > 2 0 , however, with inverted band edges of the well material in this case. Depending on strain, type I1 situations also seem to be possible. Microstructures based on Pbl-, Sn,Te/PbTe have a potential for applications in infrared laser diodes. Doping microstructures nipi-structures

Semiconductor samples with alternating n- and p-type doped layers may exhibit properties similar to those of superlattices composed of two different materials (Esaki, Tsu, 1970; Dohler, 1972). Such doping superlattices may be thought of as periodic arrays of pn- and np-junctions with alternating negatively and positively charged intrinsic (i) regions between the neutral n- and p-layers. In this context they are sometimes referred to as nipi - structures. The oscillating space charge distribution gives rise to an oscillating electrostatic potential which modulates the valence and conduction band edges as shown in Figure 3.64a. If the modulation period approaches the nanometer range, minibands arise just as in the compositional superlattices considered above. Doping superlattices are principally of type I1 because the two band edges, at a given position, are shifted by the same amount of energy, namely the electrostatic energy of an electron. Their periods, generally, cannot be made smaller than 10 nm because of the unavoidable diffusion of dopant atoms. The most common nipi-structures are those based on GaAs.

409

3.7. Semiconductor microstructures

m +

a)

-

+

-

-1

+

-

b)

+

+

+

I

z

-

-

I

I

+ + + +

+ + + +

+ + + +

v

Z

Figure 3.64: Valence and conduction band lineup for doping microstructures in a direct gap material for a) a nipi-structure and b) a n-type &doping structure. &doping structures

By means of MBE, spikes of dopant atoms of only a few nanometer width may be constructed in an otherwise undoped sample. One speaks of planar doping or 6-doping (see, e.g., Schubert, 1994). Like nipi-structures, 6-doping structures are always of type 11. The sheet of ionized dopant atoms forms an electrostatic potential well which binds the emitted free (majority) carriers, i.e. electrons in the case of n-type &doping (see Figure 3.64b), and holes in the case of p-type &doping. The energies of these carriers become quantized just as in the case of a quantum well. The potential well experienced by the majority carriers represents a barrier for the minority carriers. In GaAs, n-type &doping has been achieved, for example, by means of Si, and p-type 6-doping by means of Be.

3.7.3

Methods for electronic structure calculations

The theoretical methods which were developed to calculate the electronic structure of bulk crystals in Chapter 2 and of clean surfaces in Chapter 3 are also suitable for artificial semiconductor microstructures. Below, we will discuss this in the case of compositional superlattices (SLs).

Bulk methods We consider SLs composed of two zincblende type materials as, e.g., GaAs and AlAs. The interface is taken to be parallel to a (001) lattice plane. A GaAs layer of the SL contains a number 2n of (001) lattice planes alternatively occupied by Ga and As atoms, and the AlAs layer contains a number 2m of (001) lattice planes alternatively occupied by Al- and As atoms. The notation (GaAs),/(AlAs), is used for such a SL. The primitive unit cell of

410 Chapter 3. Electronic structure of semiconductor crystah with perturbations

[IIO]

0 Ga

OAs

8At

Figure 3.65: Primitive unit cell of a (GaAs)3/(AlAs)3SL. the SL is spanned by the primitive lattice vectors of a (001) lattice plane, namely, A1 = (a /2 )(e z ey),A2 = (u/2)(eZ - ey),and another vector A3. Provided that (n+m) is an integer, A3 may be taken as A3 = (a/2)(n+m)ez. The pertinent primitive vectors of the reciprocal lattice are then given by

+

27r B1

= -(ez a

27r - ey), B2 = -(ez a

+ ey),

4n B~ =

+

(n

+m)a

ez.

(3.246)

The volume of the first B Z of the SL is a ( n m)-th fraction of that of the first bulk B Z . In Figures 3.65 and 3.66we show, respectively, the primitive unit cell and the first B Z of a (GaAs)g/(AlAs)~SL as an example. The stationary states of the SL are Bloch states with quasi-wavevectors k of the first SL B Z . The energy eigenvalues form bands in this B Z , referred to as minibands in regard to k-dispersion parallel t o the SL axis, and as subbunds if k-dispersion parallel to the SL layers is considered. The minibands arise from bulk bands folded back upon the first SL B Z . As the bulk B Z encompasses ( n m ) first SL B Z s , one bulk band gives rise to ( n m) SL minibands. The minibands are separated by m i n i g a p s which occur at the Bragg reflection planes of the reciprocal SL lattice perpendicular to the z-axis. The minigaps become wider for stronger perturbations of the bulk crystals due to the superlattice structure, i.e. the larger the difference of the periodic potentials of the two bulk crystals, and the shorter the SL period is (as long as it does not become too short), the wider the minigaps open. All methods for determining band structure of bulk crystals can also be used to calculate the band structure of SLs, the only difference being the larger size of the primitive unit cell. Figure 3.67 shows the results of such a calculation performed by means of the TB method. The valence band dispersion of a (GaAs)a/(Ga0,~A10,3As)3SL is plotted along the r - Z symmetry

+

+

3.7. Semiconductor microstructures

41 1

1".

Figure 3.66: First B Z of the (GaAs)3/(AlAs)3 SL of Figure 3.65. line of the first SL B Z and, to demonstrate the folding character of the SL band structure, also along the r - X line of the first bulk B Z . Important properties of the corresponding SL eigenfunctions are illustrated in Figure 3.68 for a (GaAs)7/(AlAs)T SL. The lowest valence and conduction band states are represented with respect to their spatial variations and symmetry characters. While the valence band states are well localized in the GaAs layers and are dominantly composed of GaAs-r-states, both AlAs-X-states and GaAs-r-states contribute to the conduction band states of the SL. This illustrates the general discussion of subsection 3.7.1 about the existence of eigenstates of heterojunctions which arise from bulk states with different k-vectors in the two materials . Bulk methods may also be used to calculate the electronic structure of non-periodic microstructures such as GaAs/AlAs single heterostructures or (Ga, Al)As/GaAs/(Ga, A1)As double heterostructures. In these cases one may use a procedure similar to the slab method for surfaces: At the outset, one defines a slab with a sufficiently thick but finite GaAs layer and forms a single heterostructure with a sufficiently thick but finite AlAs layer, and then repeats this slab periodically. In the case of the (Ga, Al)As/GaAs/(Ga, A1)As double heterostructure, one embeds the GaAs layer between thick but finite (Ga, A1)As layers, and repeats the slab thusly formed periodically. In this way an SL is simulated whose electronic structure can be calculated by means of any of the bulk band structure calculation methods. However, in many cases the application of bulk methods to artificial microstructures is not appropriate. These methods yield the totality of energy bands of a microstructure in all parts of the first B Z , while in most cases only a small number of minibands are of interest because only these undergo changes in comparison with the back-folded bulk bands. Moreover, even the

412 Chapter 3. Electronic structure of semiconductor crystals with perturbations

to

-

3 -1 A

ol

!i

W

-2

-3

-4

-F;

-€

Figure 3.67 Valence band structure of a (GaAs)3/(Gao.7Alo.3As)3SL calculated by means of the TB method, and plotted upon the r - 2 symmetry line of the first SL B Z (left) and the I?-X line of the first bulk B Z (right). (After Riicker, Hanke, Bechstedt, and Enderlein, 1986.) changes of these few minibands are small. The bulk methods are often not accurate enough to reproduce them sufficiently well. The situation is similar to the case of the shallow levels of impurity atoms - if one would try to obtain these levels by means of a full bulk band structure calculation method the same difficulties would occur. Fortunately, another method, namely the effective mass theory, exists in the shallow level case. It is particularly well tailored to calculate the small changes of electronic structure occurring at the conduction band minimum and the valence band maximum, where the kinetic energy is small enough to allow the perturbation potential to produce measurable effects. The situation in the case of artificial microstructures is comparable: the perturbation potential is relatively weak, and changes of the band structure are expected only for selected bands and in the vicinity

3.7. Semiconductor microstructures

GaAs

413

A1As

r

X

Figure 3.68: Localization of the lowest valence and conduction band states of the SL at the center r of the first SL B Z (left), and the symmetry character of these states (right), for a (GaAs)7/(A1As)7SL. While the valence band states are well localized in the GaAs layers and are dominantly composed of GaAs-r-states, conduction band states extend over both layers, and are composed of both AlAsX-states and GaAs-I'-states. h l = -0.11 eV,h2 = -0.19 e V , el = 1.75 eV,e2 = 1.77 eV, e3 = 1.88 eV.(After Rucker, 1985.) of critical points. There are, however, also differences between the shallow level problem and the microstructure problem. The perturbation potential, i.e. the difference of the periodic oneelectron potentials of the two m a t e rials of a heterostructure, is far from being smooth on the atomic length scale. Even if one averages out the microscopic potential fluctuations over a unit cell, the average potential difference has still an abrupt change at an interface. For this reason one has to determine at the outset whether or not the effective mass theory developed in section 3.3 can in fact be applied to artificial microstructures. We will address this question below (for a detailed discussion see, e.g., Burt, 1992).

414 Chapter 3. Electronic structure of semiconductor crystab with perturbations Effective mass theory for microstructures Consider a single heterostructure. There is no doubt that the effective mass theory may be applied to each of the two infinite half spaces of this structure filled with material 1 for z < 0 and with material 2 for z > 0, provided the conditions of validity of effective mass theory are satisfied in each of the two regions separately, which we will assume. Accordingly, we restrict ourselves to eigenstates of the heterojunction having energies in the vicinity of the band edges E,1 and E,2 of the two bulk materials. Both edges should occur at the centers I? of the respective bulk B Z s . The two band edges at are assumed to be non-degenerate and isotropic (the degenerate case will be treated separately). Furthermore, we suppose that, at critical points off r, all bulk band energy levels are far removed from the energies of the eigenstates under consideration. Then these eigenstates are composed of only bulk states with k-vectors in the vicinity of r, just as is assumed in effective mass theory (in fact this theory assumes composition of bulk states from the vicinity of a particular critical point which need not necessarily be f). If the eigenstates were composed of bulk states from different parts of the B Z , say from f and X as happens in the case of the conduction band states of the (GaAs)7/(AlAs)i. SL shown in Figure 3.68, then the effective mass theory could not be applied. If, however, an eigenstate of the heterostructure may be formed only from bulk states at an off-center critical point like X , then the effective mass theory is applicable as well. Non-degenerate band edges

In the case under consideration, the eigenfunctions & l ( x ) and $ 4of~ the) two material regions 1 and 2 having energies, respectively, close to the band edges Evl and E,2 at f,may be written as

The two envelope functions F,l(x) and Fv2(x) obey the effective mass equations

x in m a t e r i a l 1,

(3.248)

F V ~ ( X=) E , F , ~ ( x ) , x in m a t e r i a l 2.

(3.249)

F,l(x) = E,F,i(x),

Formally, these equations may be written as one equation for the envelope function

3.7. Semiconductor microstructures

415

(3.250) of the whole heterostructure by introducing a z-dependent effective mass m * ( z ) and a z-dependent band edge E,(z) defined, respectively, by

We write this one equation as

(3.252) The true wavefunction $,(x) of the heterostructure is given by the expression

(3.253)

(3.254) is the Bloch factor of the heterostructure. Clearly, equation (3.252) holds only within the two material half spaces, but not at the interface. Thus it determines the possible envelope function solutions for z < 0 and z > 0. At z = 0, the solutions for the two half spaces must be connected in an appropriate way to form an envelope eigenfunction for the entire heterostructure. To do so, matching conditions are required, which we will now discuss. Firstly, we derive a condition for the change of the envelope function across the interface. To this end we consider the interface behavior of the Bloch factor ud(x) in equation (3.253) for the total wavefunction &,(x). If this factor was continuous at the interface, the envelope function F,(x) also would have to be continuous there since the total wavefunction @,(x) must be continuous everywhere. Thus the condition Fv(z,Y, 0 - 6 ) = F,(z, Y, 0

+6)

(3.255)

should hold in the limit 6 --+ 0. Unfortunately, no general proof exists for the continuity of the Bloch factor at a heterostructure interface. The only case in which such a continuity is assured is that of a 'heterostructure' composed of two identical materials. Having this obvious result in mind, it is commonly

416 Chapter 3. Electronic structure of semiconductor crystals with perturbations argued that the Bloch factor should be continuous at an interface, at least in an approximate sense, if the two materials of the heterostructure are not too different from each other with respect to their energy band structures. This happens, for example, in the case of GaAs/Gal-,Al,As heterostructures with sufficiently low z-values. It turns out, however, that effective mass calculations based on envelope function continuity, also yield correct results for heterostructures made of materials with considerably different band structures. Thus, similarity of band structures, while being sufficient, does not seems to be necessary for the applicability of effective mass theory to heterostructures. In fact, there have been various attempts in the literature to justify the continuity condition (3.255) without using the band structure similarity argument. In our opinion, it is essential to realize that the Bloch factor ul/o(x) in equation (3.252) occurs in an eigenfunction of the entire heterostructure rather than in an eigenfunction of the two infinite bulk crystals, as is tacitly assumed in the above discussion. For an infinite bulk crystal, the Bloch factor is that particular solution of equation (2.136) which obeys the lattice periodicity condition. Without demanding satisfaction of this condition, there exists a variety of solutions of equation (2.136). In a heterostructure, the two Bloch factors at the interface need not be periodic with respect to z because the lattice periodicity in the z-direction is perturbed. This freedom may be used to satisfy the continuity condition for the Bloch factor at the interface. Secondly, we derive a condition for matching the interface values of the first derivative of the envelope function with respect to t. To this end we use the physically obvious fact that the probability current density joined with the total wavefunction $ J ~ ( X )must be the same on the left and right of the interface. This must also hold after averaging with respect to a primitive unit cell, i.e. with respect to a region where the envelope function is almost constant. Applying the well-known quantum mechanical expression for the current density one obtains two contributions. One is due to the gradient of the Bloch factor and it vanishes after averaging. The other contribution arises from the gradient of the envelope function. It is multiplied by the squared modulus of the Bloch factor, which yields 1 after averaging because of normalization. This means that the average current density in state $v(x) is obtained by applying the quantum mechanical expression to the envelope function Fv(x)alone rather than to the total wavefunction &,(x). Since the average current density must be continuous at the interface, and since the envelope function itself has this property because of relation (3.255), we arrive at the conclusion that the first derivative of the wavefunction must obey the relation

3.7. Semiconductor microstructures

417

which is known as the BenDaniel-Duke boundary condition. This condit,ion follows automatically if the Schrodinger equation (3.252) is also applied at z = 0 after the kinetic energy term has been replaced in accordance with

_-

h2

2m*(z)

v2

-.+

h2 --v.-

2

v.

m*(z)

(3.25 7)

With this replacement the effective mass equation (3.252) becomes

Integrating this equation with respect to z from -6 to +6 one obtains the boundary condition (3.256). The kinetic energy operator of the Schrodinger equation (3.257) is Hermitian, as any reasonable kinetic energy operator must be in order to avoid complex energy eigenvalues. The original form -[l/2m*(z)]V2 of the kinetic energy operator is not Hermitian, and must be rejected. Its replacement by the right hand side of relation (3.256) cannot be justified, however, by the hermiticity demand alone since there are also other ways to introduce hermiticity. In fact, one can easily demonstrate that any operator of the form -(F,2/4)[m*a(z)Vm*~(z).Vm*~(z)+m*~(z)Vm*~(z).Vm*a(z)] is Hermitian if a,p, y are real numbers obeying the relation (Y ,B y = -1. This implies that the boundary condition (3.256) involves more than just the hermiticity of the kinetic energy operator. In the effective mass equation (3.258) the band edge plays the role of an external potential. Since it depends only on z , and since the effective mass does so also, the dependence of the envelope function on the component XI[ of x parallel to the layers may be taken in the form of a plane wave of parallel wavevector kll, therefore as

+ +

(3.259) For F u ( z ) the Schrodinger equation follows as

where EL is given by EL = E , -

h2

-k i 2m*,

with l/E; as the average inverse effective mass (m;:'

(3.261)

+ m;T1)/2.

418 Chapter 3. Electronic structure of semiconductor crystals with perturbations Degenerate band edges @

In the case of degenerate band edges E v ( z ) ,the Hamiltonian’s H i of the two materials i = 1,2, are of the general form (3.262) derived in equation (3.73), and the components Fym(X) of the envelope func* tions F~ ( x )obey the effective mass equations

[-~g$r(z)aaaat + ~ v ( z ) 6 m m iFumt(x) ] = ~ v ~ v r n ( x z) ,# 0, m’ a d

(3.263) with DE$(z) = DYZAl for z < 0, and Dg$l(z) = D;ELl for z > 0. The solution of equation (3.263) may again be taken as a plane wave (3.264)

=+

parallel to the layers. The boundary conditions for F v ( x ) at the interface follow in the same way as in the non-degenerate case. The z-dependent * * factor F~ (2) of F,, ( x )in equation (3.264) must be continuous at z = 0, i.e.

*

*

F U (z)lz=-6 = F u

(3.265)

(z)I~=+6

must hold for 6 + 0. To determine the condition for the derivative with respect to z , one defines the Hermitian Hamiltonian matrix

(3.266) ($

which exists for any value of z including z = 0 and equals H 1 for z < 0 and e H z for z > 0. Integrating the effective mass equation for this generalized Hamiltonian with respect to z over a small interval across the interface leads to the conclusion that the expressions

must be continuous at the interface z = 0. Below, these general results are specified for the fourfold degenerate rs valence band edge of diamond and zincblende type materials. Using the 4 x 4 Luttinger-Kohn Hamiltonian of equation (3.74), the effective mass equation (3.263) becomes

3.7. Semiconductor microstructures

419

(3.268)

The quantities Q ( z ) ,f i ( z ) ,g ( z ) , ? ( z ) here are the differential operators

k(z)=

--a [y2(zj(k: - k i ) - 2i73(z)k,kg] .

(3.272)

The constants M , L , N in equations (3.75) to (3.78) have been replaced by the Luttinger parameters y1,72,y3 using equations (2.381). From equation (3.267) one finds that the following combinations of the components F m ( z ) and their derivatives FLm(z) with respect to z must be continuous at the interface z = 0:

420 Chapter 3. Electronic structure of semiconductor crystals with perturbations

L

XJ

Figure 3.69: Geometry of the SL whose stationary electron and holes states are calculated in the text.

3.7.4 E l e c t r o n i c structure of particular microstructures Compositional superIat tices and q u a n t u m wells We consider a SL composed of two zincblende type semiconductor materials 1 and 2 with the conduction band edges Ecl and Ec2 located at r. To be specific, we may associate ‘1’with GaAs, and ‘2’ with a Gal-,Al,As alloy for z < 0.42. The thicknesses of the two material layers are denoted by d l and d 2 , and the lattice constant in z-direction by d , with d = d l d 2 ( s e e Figure 3.69). We want to calculate the stationary electron states of this SL with energies in the vicinity of the conduction band edges Ecl and E,2 using the effective mass theory derived above. These states are the ones which would host the free electrons of the SL, if there were any. We suppose that there are none, as we did before.

+

Electron states Effective mass equation and its solutions

For the zincblende type materials we are considering, the conduction bands are non-degenerate and isotropic in the vicinity of I’. The two effective electron masses m:l and mE2 generally differ, but we will initially assume that they are equal, denoting their common value by m;. In the case of SLs composed of GaAs and Gal_,Al,As this is a reasonable approximation. Later, we discuss the modifications which occur if m;l and m:2 are different. The effective mass equation for the electron envelope function Fc(x)of the

3.7. Semiconductor microstructures

421

thus specified SL follows from equation (3.252) if we set v = c and

Ec(z) =

Egl

+AE,

f o r Id

< z < ld+dl

f o r Id

+ d l < z < (1 + l ) d .

Here 1 is the integer index of the various SL unit cells, -co

(3.274)

<

1

<

00.

The envelope function F,(x) may be taken in the form Fckl, (x)of equation (3.260) describing a plane wave of wavevector kll parallel to the layers. The z-dependent factor F,(z) of Fckl,(x)obeys the equation

where E; is given by EA = E , -

(3.276)

In solving this equation, the boundary conditions

Fc(ld

+ d l - 0 ) = F,(ld + d l + 0 ) ,

d

-Fc(ld dz d

-F,(ld dz

+d l

d - 0) = -F,(ld

dz

+ 0),

-

(3.277)

have to be satisfied as well as the normalization condition with respect to 2i = Ld ( L denotes the number of SL the periodicity interval of length C unit cells in a periodicity region of volume 0). Applying the Bloch theorem to the SL under consideration it follows that F,(z) may be written as a lattice-periodically modulated plane wave F,(z)

= Fck(z) = Lfe ii k r U c k ( z ) ,

(3.278)

where k is the quasi-wavevector component parallel to z , and U c k ( z )is the superlattice Bloch factor. The component k varies within the (1-dimensional) first B Z of the superlattice between - r / d and r l d . It must have the form ( 2 a / L d )x (0, f l , f 2 , . . .) in order to guaranty the periodicity of F&) with respect to the periodicity interval. The Bloch factor U c k ( z ) of the superlattice has to be distinguished from the Bloch factor uco(x) of the two bulk

422 Chapter 3. Electronic structure of semiconductor crystals with perturbations materials defined in equation (3.254). The occurrence of two different Bloch factors reflects the fact that two periodic potentials are present in a superlattice, one due to the natural crystal structure, and one due to the artificial superstructure. The total wavefunction &(x) &k(x) contains the product of both. It reads

(3.279) The effective mass equation (3.275), the matching conditions (3.277), and the Bloch condition (3.278) define an energy eigenvalue problem which differs from the well-known Kronig-Penney problem only by notation. The latter problem constitutes an exercise of elementary quantum mechanics and is commonly used to demonstrate the existence of energy bands. In the case of SLs this exercise takes on real physical meaning. In the following we sketch its solution. We restrict ourselves to energy values 0 < E' < A E , or E,1 < E < E,1 AE, (7i2/2mr)ki, meaning energies below the lowest allowed energy in material 2 for a given value of kll. According to classical mechanics, an electron with such an energy cannot penetrate within the superlattice layers made of material 2, it will be confined to layers consisting of material 1. If it hits the interface to material 2, then it will be reflected back to the interior of its material 1 layer. In quantum mechanics, the reflection is not complete, there is a certain probability for the electron to tunnel through a material 2 layer and reach the next neighboring material 1layer. Although the probability for an electron to stay in material 1 is not unity, as in classical mechanics, but smaller, it is still much larger than that for material 2. As has already been mentioned, one uses the terms quantum wells for the layers of material 1, and bamiers for the layers of material 2. In the wells and barriers the wavefunction F,]F(z)is given by different expressions. For the wells 0 < z < d l , Schrodinger's equation (3.276) yields

+

+

F,]F(z)= ale iKZ

+ ble-iKz,

0 < z < dl,

(3.280)

with K as a real number which determines the energy E' by means of the relation E ,I = - h2 K2

(3.281)

2 m ~ '

and a l , bl as coefficients which are still to be determined. For the barrier d l < z < d one has Fck(z)= a2enz

where we have set

+ b2e-nz,

dl

< z < d,

(3.282)

3.7. Semiconductor microstructures

2

KE.=

2mE -(AE, Ti2

423

(3.283)

- EL).

The four coefficients a l , b l , a2, b2 are governed by a set of linear equations which follows from the 4 matching conditions (3.277). These equations read

For this set to have a non-trivial solution, the determinant of the matrix of exponentials must vanish. This yields the secular equation

cos(Kd1) cosh(tid2)

tc2 - K~ +sin(Kd)l sinh(~d2)= cos(kd). 2Kn:

(3.285)

Allowed K - and &-valuesmust satisfy equation (3.285). The values of K and K are not independent of each other, both are determined by the energy E L through equations (3.281) and (3.283), respectively. Thus equation (3.285) represents a condition for EL. It coincides with that of the Kronig-Penney problem. As we know from the solution of the latter, there are discrete energy eigenvalues EL, n = 1,2,. . ., for a given quasi wavenumber k. They split into energy bands E&(k), separated by energy gaps, if k varies within the first B Z . To obtain the total energy eigenvalues E, = Ecn(k) of the superlattice, (Ti2/2m*)kll to according to equation (3.276), one still has to add E,1 E L ( k ) , with the result

+

Em(k) = E&(k)

h2 + Egi + -kll 2mr

(3.286)

Unlike the Kronig-Penney problem, these energy bands exhibit an additional dispersion with respect to the wavevector component kll along the SL layers. Commonly, the energy bands and gaps for a fixed value of kll, but with varying values of k , are termed minibands and minigaps. Bands which arise for fixed IF and varying kll are termed subbands. The kll or subband dispersion is due to the natural crystal structure, while the k or miniband dispersion follows from the artificial microstructure. In Figure 3.70, central part, the sub- and minibands of a SL are schematically plotted. The effective masses of the minibands become negative at the boundaries of the superlattice B Z in the case of odd band numbers n, and at the center of the B Z for even n.

424 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.70: Mini- and subbands of a superlattice. Also shown are the two limiting cases of extremely wide and/or high barriers and of narrow and/or flat barriers. In the lower part of the figure the wavefunctions are illustrated. Multiple quantum well versus superlattice behavior

Figure 3.70 also shows two enlightening limiting cases. If the thickness d2 of the barriers between the wells becomes large, and/or the barriers AE, are high, then the hyperbolic terms in equation (3.285) dominate over the cos(kd)-term on the right hand side which is responsible for the k-dispersion. Neglecting the latter, the minibands degenerate into discrete energy levels, termed sublevels. These levels are just the discrete energy eigenvalues of a single quantum well. Obviously, the superlattice decomposes into isolated quantum wells in the limiting case under consideration. The electron states are almost completely confined within a well, and practically zero within the barriers. One uses the term multiple quantum well for a superlattice in this limit. For infinitely high potential walls the secular equation (3.285) becomes sinKd1 = 0. The solutions of this equation are K-values of the general form K = (7r/dl)nwhere n is an integer. They give rise to the energy eigenvalues 2

Elcn =

2mE

, n = 1 , 2 , .. . ,

(3.287)

3.7. Semiconductor microstructures

425

and Bloch factors

(3.288) The confinement to the well region is complete in this limit. Assuming d l = 100 A and mE = 0.067 m , one gets a value of 57 meV for the first energy level ELl. This means that the ground state of the quantum well is shifted up by this energy amount as compared to the conduction band bottom of the infinite bulk crystal. This shift is a consequence of the localization of an electron within the quantum well which leads to a momentum uncertainty because of Heisenberg's uncertainty principle. We consider next the opposite limiting case, that of extremely narrow and/or flat barriers (see Figure 3.70). In zero-th approximation the secular equation (3.285) takes the form k = K . In this limit the minigaps vanish completely, and the superstructure which distinguishes the superlattice from a natural crystal has no effect at all. In the first non-vanishing approximation with respect to the SL potential, gaps are opened at the Bragg reflection points k = ( n n / d ) of the 1-dimensional reciprocal lattice of the SL. For the energy splitting between the two bands E&+l(n/d) and E & ( n / d ) , the perturbation theory developed in section 2.4 yields E&+1

(:)

-E L

(:)

1

= -AEc 7rn

sin ( y d g ) .

(3.289)

According to this expression, the minigaps are proportional to the conduction band discontinuity AE,. They decrease with band number n. Thus, for large n, there are no substantial energy gaps even if sufficiently wide and high barriers exist. One may say that for minigaps to occur the product of wavevector k = ( x n / d ) and superlattice period d should not be large compared to 1. In other words, the superlattice period d should not be large in comparison with the electron wavelength X = 2 r / k corresponding to k. This proves and specifies the expectation stated at the outset of this section, that substantial changes of electronic structure are to be expected if the superstructure occurs on a sufficiently small length scale. The length scale turns out to be that of the de Broglie wavelength X of an electron. For a conduction band electron of silicon, having an energy equal to the average thermal energy $kT at room temperature, one gets A =" 20 nm. Similar values are obtained for other semiconductors. Thus the layer thicknesses of superlattices have to be on the order of, or smaller than, 10 nm in order that superlattice effects on free carriers may become important at room temperature. The limit of narrow and flat barriers is the true superlattice case, as distinguished by the coupling of quantum wells with one another. The en-

426 Chapter 3. Electronic structure of semiconductor crystals with perturbations ergy eigenfunctions also have non-vanishing values within the barriers, in contrast to multiple quantum wells where these values are almost zero. In a superlattice, an electron having a particular allowed energy, may tunnel from its quantum well through the barrier to the neighboring well. Since there it encounters the same allowed energy value from which it may tunnel back, the tunneling will be amplified, as in a Fabry-Perot resonator in the case of light. One has so called r e s o n a n t tunneling. It causes the discrete energy levels of isolated quantum wells to broaden into bands. Electron density of states of SLs and QWs

The general expression (2.212) for the density of states (DOS) p ( E ) of a 3D crystal may be immediately applied to a SL. To this end one has to replace the energy band structure E,(k) in formula (2.212) by the expression given in equation (3.286). Then the DOS p s ~ ( Eof) the SL is obtained as

To evaluate this expression further, the miniband dispersion EA(k) has to be known. In the particular case of isolated QWs the dispersion vanishes completely, and the DOS of equation (3.291) takes the form of the DOS P M Q W ( E )of a MQW structure,

(3.291) Each subband n gives rise to a step-like partial DOS, being zero below the band bottom E L E g l , and non-zero but constant above. Such behavior of the DOS of an electron which is free to move only in two dimensions was already found in section 2.5. The summation in expression (3.291) for p ~ over ~ all w subbands results in a staircase-like DOS as schematically shown in Figure 3.71. The heights of the steps have the constant value ( m * / n 2 h 2 ) ( n / d )while , the step widths depend on n. If the barriers of the QW are taken to be infinite, then the energy levels EAn are given by ( h . 2 / 2 m E ) ( n ~ / d 1 and ) 2 , the step widths of the staircase scale with ( n / d ~ ) ~ . For very wide QWs, i.e. in the limit d l -+ 00, the staircase-like DOS transforms into the smooth square-root-like DOS p 3 ~ ( Eof ) an electron free to move in three dimensions. Equation (3.291) yields

+

(3.292)

3.7. Semiconductor microstructures

427

4

7

3

3

0 1

3

0 2

Q

1

Figure 3.71: Electron density of states PQW of a QW as function of energy. with f = limd,,,(dl/d) being the geometrical fraction of material 1 regions in the infinite MQW. The staircases of the DOS in Figure 3.71 are always below or at the limiting fractional 3D DOS f p j ~ ( E ) .The case when they are at the DOS occurs for energies E at the subband bottoms. Refinements

In the above treatment of the electron states of a SL, the effective masses mzl and mE2 were assumed to be the same in the two materials. If this is not the case, the following modifications will occur. First, in the boundary conditions (3.277) for the derivative ( d F , / d z ) , different mass factors will appear on the two sides of the interface and they will not cancel. For the secular equation (3.285) this means that all K ' s except those which are factors in arguments of trigonometric functions, have to be replaced by (m&'mal)K. Second, in solving the secular equation, the relations (3.281) and (3.283) between K and E i and, respectively, n and E' must be written with m; and mE2 instead with mE. Finally, the kll-dispersion of the total energy eigenvalue in relation (3.286) becomes different in the two material layers. ~ the This results in an additional contribution (h2/2)[mZ-'(z) - E ~ - ' ] k to SL potential, as may be seen from equation (3.260). The additional term introduces a kll-dependence of the envelope function F& which otherwise depends only on the z-component k . Another simplification made in the above treatment of the electron states of a SL, was the assumption that no electrons are available to occupy the SL

428 Chapter 3. Electronic structure of semiconductor crystals with perturbations

t

0.5 x10’2cm-2

6 xlO’*cm-*

1.5 x10’2cm-2

Figure 3.72: Self-consistently calculated potential profiles and energy levels for a Alo.aGao..rAs QW of 100 A width containing different numbers of electrons per cm-2.

bands. If there are such electrons present, the potential seen by an electron will change due to its interaction with other electrons in the SL bands. As in the case of bulk semiconductors, this potential change may be decomposed into a Hartree and an exchange-correlation part. The expressions derived for these potentials in Chapter 2 also apply here if the stationary states of the bulk crystal are replaced by the stationary states of the SL. As the latter states are only known after the effective mass equation has been solved, the inclusion of the Hartree and exchange-correlation potentials must be performed in a self-consistent way. The net effect of the two potential parts is repulsive, thus the SL potential well will flatten. For a given number of electrons in SL states, this flattening will be enhanced by greater localization of the electrons in the well regions. It will be particularly pronounced for isolated QWs. In Figure 3.72 we show self-consistently calculated potential profiles and energy levels for a Alo.3Gao.7As QW of 100 A width, containing . well bottom bends up with different numbers ng of electrons per ~ r n - ~The rising n s , and the energy levels are shifted to higher energies.

Hole states We consider a QW structure composed of zincblende type well and barrier materials which are both described within the Luttinger-Kohn model. The effective mass equation for the hole states of such a QW with energies close to the well bottom follows from equation (3.268) by installing there the valence band edge profle E,(z):

0

forOa

=

80

60

0

40

0

>

20

0

0

10 20 30 40

SO 60 70

I

D

E [kV/cml well depth in the p-type case as compared to the n-type case results mainly from the fact that heavy holes are more strongly localized at the ionized dopant sheet and thus are more effective in screening out the sheet potential then the lighter electrons.

Macroscopic manifestations of the electron and hole states of semiconductor microstructures The modifications of electron and hole states of semiconductor microstructures manifest themselves in the macroscopic electronic properties of these structures. The more detailed treatment of these properties is beyond the scope of this volume. Here we will give only two characteristic examples, one concerning electric transport and another concerning optical properties. Figure 3.75 shows the calculated drift velocity V d versus electric field characteristics of GaAs/ AlAs SLs with the electric field E directed along the SL axes. Above a critical field value at which the V d peaks to its maximum value, the drift velocity decreases with increasing field strength, i.e, the differential conductivity becomes negative. Such a behavior, which originally was predicted by Esaki and Tsu (1970), has been confirmed in more rigorous calculations by Lei, Horing and Cui (1991). Experimentally, a sublinear increase of V d with E has been found above a critical field value, as shown in Figure 3.76. If corrected for effects of the Ohmic contacts and macroscopic electric field inhomogeneities, the experimental V d versus E characteristics clearly reveals the negative differential drift velocity predicted by Esaki and Tsu. The underlying physics can be understood easily. Consider a single electron in the lowest SL miniband EAl(b). In the presence of an electric field parallel to the SL axis, the electron is accelerated

432 Chapter 3. Electronic structure of semiconductor crystals with perturbations Figure 3.76: Experimental current-voltage characteristics of various GaAs/AlAs SLs with the electric field parallel to the SL axes. (After

300K

>-

5

-

Sibille, Palmier, Wang, and Mollot, 1990.)

2

4

6

0

3

BIAS (VI

until it reaches the inflection point of Eil(k). Thereafter, its effective mass becomes negative, causing deceleration instead of acceleration. Since the electron is inevitably scattered by phonons and impurities, its velocity takes a stationary value in a dc field. The latter increases with increasing field strength E if the corresponding k-value is below the inflection point, and it decreases with increasing E if the k-value of the electron is above this point. However, a superlattice contains a gas of many electrons, populating the miniband E:(k). For low field strengths, most of the electrons have low k-momenta and positive effective masses. Thus, the total current density of the gas increases if E is further increased. Simultaneously, the electron gas is heated up by the electric field, which implies that more electrons populate k-points with a negative effective mass and less with a positive one. This reduces the current increase if E raises further because a larger subgroup of electrons is decelerated instead of being accelerated by the field increase. If the subgroup of electrons having negative effective mass is sufficiently large, further increase of the field strength will actually decrease the current. This explains the negative differential drift velocity seen in Figure 3.75. An example of an optical experiment is shown in Figure 3.77. There, the absorption spectra of GaAs/Al,Gal-,As QW structures of different wen widths are shown. The staircase-like shape of these spectra reflects the DOSs of the electron and hole subbands of a QW which above have been shown to be staircase-like as well. Also other features of the QW band structure observed above, as the widening of the fundamental energy gap and the shift of the steps towards higher energies with decreasing well width, can be clearly seen in the spectra of Figure 3.77. The enhancement of absorption at the step edges, which correspond to optical transitions from the tops of hole subbands to the bottoms of electron subbands, is due to the Coulomb

433

3.8. Macroscopic electric fields

Photon energy ( e v ) ---w Figure 3.77: Absorption spectra of GaAs/Al,Gal-,As

QWs of various well widths.

(After Dingle, Wiegmann, and Henry, 1974.)

interaction between electrons and holes (excitonic effects). Beside absorption spectra, photoluminescene spectra are extensively used for the experimental characterization of semiconductor microstructures. In photoluminescene, one observes mainly radiative transitions between the electron and hole ground state levels of an (undoped) QW because only these are considerably populated. Due to the enlarged energy separation of these levels, a QW emits light at shorter wavelength than the corresponding bulk material does. In the case of GaAs this effect is particularly striking - the bulk crystal emits radiation in the non-visible infrared region, while a GaAs-quantum well of 50 A width emits visible red light.

3.8

Macroscopic electric fields

Macroscopic electric fields in semiconductors are the central focus of many device applications and give rise to important physical phenomena in these materials. For instance, if one applies an external electric field to a semiconductor sample, the response yields information about the band structure, impurity states and other microscopic properties of the material. The electric transport properties and electro-optic effects are particularly well suited

434 Chapter 3. Electronic structure of semiconductor crystals with perturbations to extract this information. Often, electric fields in semiconductors need not, however, be applied externally as they are already present internally, either because of spatially inhomogeneous doping such as in the case of a pn-junction, or because of other spatial inhomogeneities as surfaces or interfaces. In the following theoretical description, the source of the electric field will play a role only inasmuch as we assume that the field is spatially uniform. In practical terms, this means that the field should not change appreciably within a periodicity region, i.e. within a characteristic length of G x a. Since the limit of the infinitely extended semiconductor is effectively reached in good approximation with G M 100 the characteristic length need not to be larger than about 0.01 ,urn. Fields which change only little over this very small distance can be considered as homogeneous. In addition to excluding spatial inhomogeneities of the electric field, we will also exclude temporal changes in our discussion below. This only means that the frequencies of these changes should be small compared with the characteristic frequencies of the electrons of the semiconductor.

3.8.1

Effective mass equation and stationary electron states

A semiconductor in a homogeneous external electric field E represents a perturbed crystal in the sense of sections 3.1 and 3.2. The presence of such a field can be described by adding a perturbation potential V’(x) to the one-electron Hamiltonian H of the ideal crystal. This potential is defined as the difference of the energy of a crystal electron in the presence of the electric field and without it, and thus is given by

V’(x)= e E . x.

(3.294)

As in the case of point perturbations considered in sections 3.4 and 3.5, the perturbation potential V’(x)of equation (3.294) does not possess lattice translation symmetry. Moreover, V’(x) diverges at infinity. The latter fact implies that, unlike to the case of point perturbations, an infinite crystal in a homogeneous electric field cannot be replaced by a perturbed supercell whose periodic repetition forms an unperturbed supercrystal. The crystal in an electric field has to be treated as what it in fact is, namely an infinite system with 2-dimensional lattice symmetry perpendicular to the field, and no lattice symmetry along the field direction. If the field strength E is not too large then the perturbation potential (3.295) fulfills the smoothness condition of effective mass theory of section 3.3, which here reads (3.295) e I E I a -Eg,and oscillates with decreasing z for e E z < -Eg. On the decaying side, the total energy E g of an electron is smaller than its potential energy e E z . Thus, this region is classically forbidden, it represents a potential barrier for electrons. The envelope function is nonzero only because electrons tunnel into this barrier. The envelope function for holes behaves similarly, it decays within the hole barrier with e E z < 0, and oscillates in the classically allowed hole region with e E z > 0. The Schrodinger equations (3.300), (3.301) and their solutions (3.303) and (3.304) describe, respectively, stationary electron and hole states in an electric field. Electrons or holes in such states do not carry an electric current, more strictly speaking, the expectation value of the current density operator in the field direction vanishes. Although this does not mean that a current flow cannot be described at all by means of stationary states (if the electron system is characterized by a statistical ensemble with respect to stationary states, then non-zero non-diagonal elements will occur in the expectation value of the current density operator), one may suspect, however, that in addition to stationary states there will be yet other states which are better suited to the condition of a non-vanishing current. Such states do in fact exist, they are not eigenstates of the Hamiltonian but rather are non-stationary states cp,(x, t ) which devebp in time not only by means of a timedependent phase factor. They correspond to the classical trajectories of an electron in an electric field, with momentum growing linearly in time.

437

3.8. Macroscopic electric fields

0.4 C

0 .u g3

0.3 (D

rn

0.2

-e N

Lc

g

0.1

g

0.0

-0

P c 2 rt

W

v)

Y

-0.1

-0.2

-Eg/eE

0

Z

Figure 3.78: Envelope functions of stationary electron and hole states in an electric field, taken at energy cC = E, = 0. Oc = OU = 8.

3.8.2

Non-st at ionary states. Bloch oscillations

The envelope functions F,(x,t)of these states are time-dependent, so one has

(3.306)

$v(x, t ) = ~ V O ( X ) F V ( Xt ,) .

To determine F,(x, t ) ,a time-dependent effective mass equation is required. This may be obtained from the corresponding time-independent equation (3.296), wherein the energy E, on the right hand side is replaced by the operator i b ( a / a t ) . The resulting equation reads

+

a

{&,(--iV) eE . z} Fv(x,t ) = iTi-F,(x, t ) .

at

(3.307)

In solving this equation we assume that the electron was in a Bloch state (p,k(x) at t = 0. Then the initial condition for F,(x,t ) may be stated as FJX, t = 0) s F,/C(X, t = 0) =

-.ik.x

G

'

(3.308)

Here, the envelope function at t = 0 has been chosen such that its normalization integral with respect to the infinite interval over which the crystal in field direction extends, is given by a &function with respect to k. The solution F,k(x, t ) of equations (3.307) and (3.308) is given by

438 Chapter 3. Electronic structure of semiconductor crystals with perturbations

(3.309) with kt = k - ZEt. fi

(3.3 10)

This is readily verified by a straightforward calculation. Owing to the envelope function (3.309) and the previously made approximation U& Uuk, the total non-stationary wavefunction & , k ( x , t ) evolving from a Bloch state (pvk(x)at t = 0 remains a Bloch state for t > 0 also, however, with a time-dependent quasi-wavevector kt,

=

(3.311) This important result may also be established without approximating the Bloch factor U v k by u,o as is done in effective mass theory. The only approximation needed is the neglect of interband transitions induced by the electric field. Equation (3.311) states that the quasi-wavevector k of a Bloch state in the presence of an electric field changes with time in just the same way as does the ordinary wavevector of a free electron. Differing from the latter, the quasi-wavevector is restricted to the first B Z . However, relation (3.310) results in kt-values which can also lie outside the first B Z . This means that in the above derivation we inadvertently changed from the reduced to the extended zone scheme. According to the general considerations of Chapter 2, one may recover the description in terms of the first B Z by subtracting a suitable reciprocal lattice vector K(t). For electric fields in symmetry directions of the first B Z , there will be a particular point in time t = T(E) at which, with increasing t , the reduced wavevector kT - K(T) equals the wavevector kt at t = 0 for the first time. This means that the time-dependent Bloch state (3.311) returns, at t = T , to the Bloch state at t = 0 (not counting a phase factor). The same holds for time points 2T,3T etc. The time development of Bloch states in an electric field is therefore periodic, hence the term Bloch oscillations. The period T of these oscillations may be obtained from (3.310) as

(3.312) The Bloch oscillations affect not only the wavefunctions but also the pertinent energies E,(kt), meaning that Bloch electrons execute periodic motion within their energy bands. Oscillations also occur in the velocity of Bloch electrons, defined by the quantum mechanical expectation value < ( d x / d t ) >= (qJvhI(dX/dt)IVvk,)of

3.8. Macroscopic electric fields

439

the velocity operator ( d x l d t ) with respect to the timedependent Bloch function Pvk,. using the identity (dxldt) = ( l / m ) p , the velocity < ( d x / d t ) > follows by means of the previously derived relation (2.193) for the expectation value ((p,hJpJ(p,h)of the momentum operator p. The timeaverage of the velocity < ( d x / d t ) > obtained in this way, taken over a Bloch period, turns out to be zero. The same result holds for the timeaveraged current of the electrons of a crystal in an external electric field, it also vanishes. That this is actually not the case is due to perturbations of the Bloch oscillations by collisions of electrons with phonons, impurities and other point perturbations. To prove this statement we consider a field strength of lo4 V / c m and a primitive reciprocal lattice vector K typical of semiconductors. The Bloch period T is of the order of magnitude lo-'' s in such circumstances. The time rv between two collisions is substantially smaller than T , typically of the order of magnitude s. Thus, an electron starting a Bloch oscillation will soon be scattered and its momentum will be distributed randomly over all k-space. In other terms, no periodic motion can develop in the presence of collisions. We conclude that collisions are not only responsible for the fact that the current of a free electrons, which would otherwise be infinitely large, remains finite, but also for the fact that an otherwise vanishing average current of a Bloch electron does not actually go to zero. There is still another reason that Bloch oscillations cannot be observed in ordinary semiconductor crystals. It is due to the fact that the time dependent Bloch states (3.311) decay in time by themselves since the exponential factor oscillates with increasing frequency. This may easily be seen for quasiwavevectors k close to a critical point. For zero wavevector component k, in the field direction, the phase of the exponential contains a term (1/3)e:t3 (0, is the electro-optic frequency of equation 3.305). If (1/3)0:t3 is considerably larger than 1, the exponential oscillates so fast that the time dependent Bloch states (3.311) average out to zero almost completely. One may say that these states have a finite lifetime 0;' due to the electric field. For nottoo-large field strengths, e;' is small compared with the period T of Bloch oscillations, just as the time T, between two collisions above was found to be small with respect to T . This implies that, in actual crystals and under normal conditions, Bloch oscillations are not observable even without collisions, just due to the finite field induced lifetime. For the artificial superlattices considered in the previous section 3.7, the primitive lattice vectors and thus the periods T are much shorter than those in natural crystals. In such circumstances T may become larger than T~ and OF1,and Bloch oscillations may in fact be observed. The negative differential electric conductivity of a superlattice parallel to its axis, reported in section 3.7, also may be understood as a manifestation of Bloch oscillations. If one considers only k-vectors close to critical points of the band structure, then the energy bands E,(k) can be taken in parabolic approximation.

440 Chapter 3. Electronic structure of semiconductor crystals with perturbations This yields, approximately,

(3.313) With the neglect of collisions, an equation similar to Newton’s law of motion m t -d 2 < x > = -eE, dt2

(3.314)

follows from (3.313). It differs from the ordinary Newton’s equation in that the free electron mass is replaced by the effective mass of the respective band v. For electrons from the valence band the effective mass is negative. The pertinent particles with positive mass, the holes, behave like particles with positive charge e. 3.8.3

Interband tunneling

In the effective mass equation (3.296), the possibility of electric field induced coupling between the valence and conduction bands is neglected. In reality such coupling exists due to the non-vanishing interband matrix elements of the potential V’(x) = e E . x. Taking account of these off-diagonal elements yields a description of electrons undergoing quantum mechanical transitions between the valence and conduction bands. For reasons of energy and momentum conservation, the initial and final state energies E,, Ec and wavevector components k,l, k,l perpendicular to the field have to be equal in such a transition. According to equation (3.299) this means E, = cC. The quantum mechanical probability W , for field induced interband transitions is proportional to the overlap integral between the two envelope functions F,,vkvl(x) and FECkci(x) with identical values for E, tC and k w l ,kc-. If, in particular, transitions of electrons between the two band edges, i.e. with E, = ec = 0 and k ,l = k,l = 0, are considered, the overlap integral has to be taken with respect to the product Ai([E,+ e E ~ ] / A e , ) A i ( [ - e E z ] / A e , ) . This integral is non-zero since valence and conduction band states having the same energies E, = eC = 0 differ from zero simultaneously for almost all values of z , as may be seen from Figure 3.106. The reason for the non-zero overlap is the tunneling of the valence and conduction band electrons into their respective barriers. This explains why field induced interband transitions are referred to as i n t e r b a n d t u n n e l i n g (the term Z e n e r t u n n e l i n g is also used). Performing the overlap integral of A i ( [ E , e E z ] / ~ e , ) A i ( [ - e E z ] / A e , ) with respect to z one arrives at an expression proportional to the square of the Airy-function Ai3(E,/he,) where 8, is a frequency analogous to the electro-optic frequencies e , , of equation (3.305), formed here, however with the reduced effective mass mT, = mZ,m:/(mz m:) of electrons and holes

+

+

441

3.8. Macroscopic electric fields

instead of their simple effective masses m,*,,. One thus obtains the interband transition probability W , as W,, oc A i 2

(z) .

For realistic field strengths E, is measurably larger than TLB,,.

(3.315) Therefore,

the asymptotic approximation

(3.316) for the Airy-function applies. To understand what this result means physically we may argue as follows. Expression (3.315) for the transition probability W , was derived using the stationary electron states in an electric field of subsection 3.8.2. The same expression follows if one uses the nonstationary states of subsection 3.8.3. In this, the electro-optic energy Ti@,, may be understood as the energy uncertainty of an electron-hole pair due to its finite lifetime in the presence of an electric field. Without such an uncertainty interband tunneling would not be allowed by reason of energy conservation: the final state in the conduction band would differ in energy from the initial state in the valence band by the gap energy E,. The energy uncertainty relieves the need to satisfy energy conservation and permits the interband transition probability W , to be nonzero. The non-stationary approach also provides a rough estimate of the proportionality factor in relation (3.315). The probability W,, for an electron to tunnel per second from the valence into the conduction band at k,l = kl = 0, may be obtained from Ai2(E,/7iB,,) by multiplying this expression with the frequency 1/T of Bloch oscillations. This is plausible since this frequency indicates how often a valence band electron, per second, reaches the upper band edge, from which it can tunnel into the conduction band. It is just the probability for this tunneling step which the Airy-function expression A i 2 ( E g / W , ) represents. The complete expression for W , thus reads W , = -Ai2 1

T

(2).

(3.317)

Although the interband matrix element (p,kJeE.xlp,k) of the perturbation potential V'(x) does not enter this expression explicitly, as one would expect, it occurs implicitly through the lattice constant a in T , which has the same order of magnitude as (p,k[xIp,k). According to expression (3.315), the tunneling probability W,, grows larger as the energy uncertainty hOCw approaches the gap energy E,. It is small if the condition fit',

0, we have

(4.45) The density of states of the valence band is thus seen to have square root-like dependence at the upper band edge and formally also continues to increase according to a square-root law for energies deeper in the band (see bottom part of Figure 4.2). The dashed curve again shows the more realistic shape. The entire density of states pideaz(E)of an ideal semiconductor having one valence and one conduction band is given by the expression

In the energy gap the density of states pideal(E)vanishes, as a direct con-

4.3. Density of states

473

sequence of the fact that no stationary electron states exist there (see Figure 4.2). Finally, we write down the total electron concentration n&%: of the ideal semiconductor having one conduction band and one valence band. From equation (4.42) it follows that

Bands with more complex structures

In the case of the elemental semiconductors Si and Ge, the conduction band edges consist of several valleys, and the iso-energy surfaces of the valleys are anisotropic. This has consequences for p,(E). For Si, for example, the conduction band density of states p p ( E ) takes the form

The valence band edges of diamond and zincblende type materials are degenerate. The degree of degeneracy depends on whether spin-orbit splitting is important or not. In the case of Si, the spin-orbit splitting energy A is small in absolute units (44 m e V ) , but still large enough to compete with k T (25 meV for T = 300 K ) . Thus, for temperatures that are not extremely high, most of the holes are hosted by the I?$ valence band while the spinsplit I?$ band remains practically devoid of holes. This means that there are two hole states per k-vector (apart from the 2-fold spin degeneracy), one for heavy holes, and one for light holes. Thus , the corresponding hole density of states p p of Si, calculated in the isotropic parabolic approximation of equation (4.45), is given by the expression

The same expression applies for Ge and GaAs. In cases where non-parabolicity effects are essential, the energy dependence of the density of states no longer follows a square-root law. This is surely true for energies far removed from the critical point. In narrow gap semiconductors this occurs, however, even close to the critical point. For example, as shown in section 2.7, (see equation (2.404), the electrons and light holes in semiconductors having small energy gaps are, within the Kane model, described by the dispersion relation

4 74

Chapter 4. Electron system in thermodynamic equilibrium

(4.50) The conduction band density of states p P ” ( E ) can be derived from equation (4.50) as follows: First of all, the general relation p p ( E )=

4

k 2 S ( E - E,(k))

27r

(4.51)

is still applicable. Here, we introduce E, as new integration variable, with the function k ( E c ) taken from the dispersion law (4.50). In this way, we obtain p p ( E ) for E-values above the conduction band edge at Ec = E , as follows:

+4

Kane(E) Pc

1

27r2

(-)3 2P2

3/2

(.

-

.-/

5 - 3) 3

2

(4.52) If E lies close to E,, then p p ( E ) given by (4.52) takes the form of expression (4.44) for a parabolic band. To see this, the effective mass rn; has to be replaced by the expression from Table 2.13. That is to be expected, because the band structure is also almost parabolic for energies close to the edge within the Kane model.

4.3.3

Density of states of real semiconductors

We now consider doping the ideal semiconductor discussed above with N D donor atoms and N A acceptor atoms per cm3 to obtain a keal semiconductor’. Each donor atom is assumed to give rise to a 1-fold ionizable donor level E D , and each acceptor atom gives rise to a 1-fold ionizable acceptor level EA. Excited donor or acceptor states are omitted from this consideration. The assignment of the donor and acceptor levels to impurity atoms is not essential in what follows. In principle, these levels can be generated by any simply ionizable point perturbation, including structural defects, with the same results. This also implies that the levels need not be shallow, they may be deep, although the shallow case is more typical than the deep case for the considerations below. The electron system of a semiconductor with simply ionizable donors and acceptors may be thought to arise from that of the corresponding ideal semiconductor with one electron added per donor atom, and one electron is removed per acceptor atom. The electron concentration nL7; of the real semiconductor is thus given by the relation

4.3. Density of states

475

+

The concentration (1/0)&=D,A < k 1 kT-+ > of electrons occupying 72 the donor and acceptor levels may be calculated by means of equations (4.23), (4.24), and (4.25), with the result

As in the context of the total electron concentration (4.42) of an ideal semiconductor, we introduce here the density of states pTal(E) of the real semiconductor. The contribution arising in the energy gap is denoted by p E $ ( E ) and we set

For this relation to correspond to equation (4.54), pi$ the expression pgap Teal ( E ) = NDG(E - E D )

must be defined by

+ N A ~ ( E- E A ) .

(4.56)

The densities of states of the valence and conduction bands of the real semiconductor will be denoted by pLd(E) and p:"'(E), respectively. The band states of real semiconductor are, of course, no longer pure Bloch states, but Bloch states perturbed by scattering from the impurity atoms. Similar to pure Bloch states, they are, however, still spread out over the entire crystal. Correlation effects therefore play no important role in their occupancy. This means that the total electron concentration in the bands of the real semiconductor may be expressed by p;T'(E) in the same way as the total electron concentration of the ideal semiconductor was expressed by p Z d ( E ) in equation (4.47). Therefore, ,Teal

total =

1, 0

dE P:az(E)

f(E) + /Or d E PF%)

f(-w

Eg 00

+I,dE

PEW

f*(E) .

(4.57)

The band densities of states pLY'(E) of the real semiconductor differ from p ; Y ' ( E ) in accordance with the considerations of sections 3.4 and 3.5, where it was shown that localized states occur at the expense of the band states

476

Chapter 4. Electron system in thermodynamic equilibrium

(Levinson's theorem). The number of conduction band states therefore decreases by the number of donor states, and the number of valence band states decreases by the number of acceptor states, related to the volume unit in each case. Thus we have 0

dE pLal(E) =

dE p i k d ( E )

- NA,

(4.58)

--M

(4.59) The entire density of states

of the real semiconductor satisfies the conservation law of the density of states, such that

l, m

dEp'"l(E) =

Jrn dEpiM(E). -m

(4.61)

Concerning the band state densities p,':'(E), we initially know, apart from the integral relations (4.58) and (4.59), nothing further than that it must reproduce p Z z ( E ) if N D and N A are zero. Below, we will see that it is possible, under certain conditions, to replace the real state densities p T f ( E ) approximately by the ideal state densities p : p z ( E ) even when N D # 0 and N A f 0. In the following, we calculate the average electron concentrations in the energy bands and the donor and acceptor levels separately. Such separate calculations of the concentrations are necessary because the electrons in these different parts of the energy spectrum behave quite differently. The electrons localized at an impurity atom, for example, do not contribute to electric charge transport. The electrons of the conduction band do, just as do the electrons of the valence band, the latter, however, provide such a contribution in a different sense, not as electrons, but as holes. Therefore, facilitate to calculation of the conductivity of a semiconductor, one needs the average electron concentrations in the conduction band, in the donor and acceptor levels, and in the valence band, separately. The same remarks apply to the calculation of other measurable quantities.

4.4. fiee carrier concentrations

4.4

477

Free carrier concentrations

We start with general considerations which refer to both ideal as well as real semiconductors. The object is to derive a condition which fixes the total number of electrons of these systems. 4.4.1

Conservation of total electron number

Consider (4.53) for the total electron concentration nL%t of a real semiconductor. In this relation, the total concentration n&%! of the ideal semiconductor may be taken from equation (4.47), which we write down for T = 0 K . Since the Wrmi level of an ideal semiconductor lies in the energy gap according to the considerations of Chapters 1, one has f(E)= O(EF - E ) for T = 0 K . With this, relation (4.47) takes the form 0

nzdeal total

-

(4.62)

S _ _ d E P"Vd"YE).

This equation says that the total concentration nitg: of electrons equals the total concentration of valence band states of the ideal semiconductor. This statement is, of course, independent of temperature. The total electron concentration of the real semiconductor may therefore be written for any temperature as 0

n Ttotal fd -

1,

d E pid"'(E)

+ N o - NA.

(4.63)

Employing equation (4.58) this yields

(4.64) On the other hand, we have relation (4.57) for p z z ( E ) using (4.56) to obtain

nTzL,

where we replace

(4.65) From this equation we subtract equation (4.64), h d i n g

(4.66)

478

Chapter 4. Electron system in thermodynamic equilibrium

Consider the abbreviations

(4.67)

These quantities have direct physical meaning: n is the concentration of electrons in the conduction band, p the concentration of holes in the valence band, Ngf is the concentration of non-occupied (ionized) and therefore simply positively charged donor atoms, and NX the concentration of occupied and thus simply negatively charged acceptors. The electrons in the conduction band and the holes in the valence band are freely mobile charge carriers, as pointed out in Chapter 1. In these terms, equation (4.66) becomes n- p -

ND+ + N;

= 0.

(4.71)

This expresses the conservation of total electron number for a real semiconductor. The electrons may be redistributed between the four energy regions, i.e. valence band, acceptor levels, donor levels and conduction band, but their total number cannot change thereby. One may also understand relation (4.71) as the neutrality condition for a real semiconductor - the negative charge of the electrons of the conduction band and of the ionized acceptors must compensate for the positive charge of the holes of the valence band and the ionized donors. 4.4.2

Free carrier concentration dependence on Fermi energy. Law of mass action.

We now derive an explicit relation between the concentrations n , p of electrons and holes in the bands, as they are given by equations (4.67) and (4.68), and the Fermi energy EF. The Fermi energy, in turn, is determined by the neutrality condition (4.71). In the non-degenerate case of an exponential Boltzmann distribution for f(E), the position of the Fermi level has no influence on the relative distribution of the electron occupancy over the various available energy levels, only the total electron concentration depends on EF. However for a degenerate electron gas, i.e. in the regime of quantum statistics, the Fermi distribution function f ( E ) mandates that the relative distribution of electrons over the energy levels is also determined by

4.4. nee carrier concentrations

479

the Fermi energy. In particular, this means that the relative distribution depends on the total particle concentration. By changing the total concentration, the ratios of the average particle numbers in the various energy levels can be altered. This is a clear manifestation of the statistical correlation between the electrons of an ideal Fermi gas. It also shows that the chemical potential, or Fermi energy, is much more important in quantum statistics than in classical statistics. In evaluating the integrals of (4.67) and (4.68), we note that the occupancy factors f (E) and [l- f ( E ) ]decrease monotonically with, respectively, increasing or decreasing energy and effectively cut off the integration range. The cutoff energies are separated by an interval of about kT from the respective band edges. We assume that the densities of states of a real semiconductor, which are required in equations (4.67) and (4.68) for the effectively contributing energy intervals, may be approximately replaced by the state densities of the ideal semiconductor. This approximation is justified as long as the concentrations of donors and acceptors is small compared to the concentration of significantly contributing band states. Furthermore, we assume that in the contributing energy intervals, the densities of states (4.44) and (4.45) of the ideal semiconductor, which are predicated on parabolic band structure, are approximately valid. Thus we obtain

(4.72)

(4.73) Defining the Fermi integral as

(4.74) the relations (4.72) and (4.73) may be written as

(4.75) (4.76)

with (4.77)

480

Chapter 4. Electron system in thermodynamic equilibrium.

In the frequently occurring case that the Fermi level lies not only in the energy gap, but is separated by at least several k T from the two band edges, the arguments of the Fermi integrals in (4.75), (4.76) are negative and have absolute values large compared to 1. For such arguments the Fermi integral is approximately given by (4.78) so that

n = Nce(EF-Eg)/kT, ( E , - E F ) >> k T , p = NVevEFfkT , E F

>> k T .

(4.79) (4.80)

The same result for n would be obtained if the Fermi distribution f(E)for the electrons of the conduction band were approximated by the Boltzmann distribution (4.13) from the very beginning, as is possible for ( E - E F ) >> k T . An analogous approximation for the electrons of the valence band is not available, since there E < EF holds. One may apply this approximation, however, to the holes of the valence band, i.e., to 1 - f(E)= e(E~-E)/rCT+ 1 '

(4.81)

For ( E F - E ) / k T >> 1, we have, approximately, (4.82) As in the case of conduction band electrons, the holes of the valence band are also described by a Boltzmann distribution if the Fermi energy is sufficiently remote from the band edges. One may say in this situation that the electron system of the conduction band and the hole system of the valence band are non-degenerate carrier gases. We emphasize that in the case of the valence band this statement refers to the holes and not to the electrons, which are highly degenerate. In further considerations we will always assume that the electrons of the conduction band and the holes of the valence band form non-degenerate carrier systems. The expressions (4.75) and (4.76) for n and p , respectively, provide a simple criterion for the validity of this assumption: The electrons are non-degenerate if n rn; or N , > N,, EF( moves away from the valence band edge, and for rn; > rn; or Nc > N,, away from the conduction band edge. This may be understood as follows: If the effective density of states in the valence band is larger than that in the conduction band, one would have more holes in the valence band than electrons in the conduction band if the Fermi level were to stay in the middle of the gap. It must move away from the valence band edge to assure the equality of n and p . An analogous interpretation can be given in the case of Nc > N,. In short, one can say that the Fermi level is repelled by regions of high density of states, and is attracted by regions of low density of states. We next determine n and p for the case of real semiconductors with donor and acceptor atoms present. In Chapter 1, the designation eztrznsic semiconductors was already introduced to describe them.

484

Chapter 4 . Electron system in thermodynamic equilibrium

Figure 4.3: Intrinsic concentrations of several semiconductors as functions of temperature (calculated values).

&T/OC

IO~/T/K-’

-.

Figure 4.4 Variation of Fermi level with temperature for an intrinsic semiconductor.

4.4.4

Extrinsic semiconductors

n-type-semiconduct ors Initially, we assume that there are only donors, and no acceptors are present, N D # 0 and N A = 0. The neutrality condition (4.71) is then transformed into an equation for the electron concentration n. In this regard, we need ~ ED)] of ionized donors. The donor the concentration Ngf = N D ( population probability f * ( E D ) is given by equation (4.24) as

4.4. fiee carrier concentrations

485

(4.92) where EB denotes the donor binding energy. In general, the donor levels, unlike band energies, do not lie well above the Fermi level. They may be close to, or even below it. This means that the degeneracy of the Fermi distribution, which could be neglected for carriers in the bands, must in general be taken into account in the effective donor distribution function !*(ED). The expression (4.92) for f*(Eo)must therefore be used in its original form. The Fermi energy can be eliminated from it by introducing the electron concentration n given in equation (4.79), with the result

(4.93) (4.94) Apart from the factor 1/2, the quantity nl represents the electron concentration in the conduction band of a non-degenerate ideal semiconductor with a (fictitious) Fermi level at the position of the donor level. If we replace p in terms of nT/n using equation (4.85), then the neutrality condition takes the form

(4.95) From this equation one may easily derive the magnitude relations

(4.96) In a semiconductor containing donors but no acceptors, the electron concentration is always larger than the intrinsic concentration ni, and the latter is larger than the hole concentration p. It is therefore called an n-type semiconductor. The electrons are majority carriers in this case, and the holes are minority carriers. To solve the neutrality condition (4.95) for n, we transform it into the third-order polynomial equation n3

2 + nln2 - (ni2 + n l N D ) n - ninl = 0.

(4.97)

The solution n of equation (4.97) can be obtained in closed analytical form. The minority carrier concentration p follows from n by means of the relation n p = nz. We examine here an approximate solution of equation (4.97) in two typical limiting cases: 1) the case of extremely small donor concentrations N o , in which N D > ni.

486

Chapter 4. Electron system in thermodynamic equilibrium

1) N D nil one then also has N D n >> n:. Under this condition, the last term in (4.97) may be neglected in comparison with the term n l N D n . With this, one has the approximate equation n2

+ n1n - n l N D = 0.

(4.100)

Its positive solution is given by

(4.101) If N D > nil sufficiently small donor concentrations. The latter restriction is necessary in order to

4.4. n e e carrier concentrations

487

b

tion of temperature for N D = loi5 ~ r n - (After ~ . Smith, 1979.)

I

-5 E

*

t

I

I

"I

-- I

7

I

Id6

INTRINSIC RANGE SLOPE = Eg

1

!I

:I - I

$ E ; c

lo"= f

SATURATION RANGE

I I \ \nl

1 1 10'3-

1 1

I

I

1

I

I

1

I

I

3

insure that all donor electrons can be hosted by the conduction band. One terms this case the exhaustion of doping centers or the saturation of carrier concentration. If, conversely, N o >> n1 holds, which means very high donor concentrations and low temperatures, equation (4.101) approximately yields n=

JnTNo.

(4.103)

Because of the assumption nl > ni, and is also still valid for levels E D = ( 1 / 2 ) E g at the middle of the gap for which nl = ( 1 / 2 ) n i follows. The property of impurity atoms to be effective electron donors for the conduction band is not restricted, therefore, to the case where the impurity levels lie just beneath the conduction band edge. Even from the middle of the forbidden zone they can increase n far above the intrinsic concentration. This can change only for yet deeper levels with n1 > ni holds. Employing the relation (4.104) we obtain, for N D

> nl, i.e. for low temperatures and high doping, we find that (4.106)

The dependence of the Fermi level on temperature is shown in Figure 4.6 for n-type germanium with typical values of N D and EB. At T = 0 K the Fermi level lies exactly in the middle between the donor level and the conduction band edge, the donors being only partially ionized. With increasing temperature the degree of ionization increases. In order for this to be possible, the Fermi level must shift to lower energies. This may again be understood as a repulsion of the Fermi level by the group of states with higher density - the effective conduction band density of states is larger than the donor density of states, which equals the donor concentration. p-type semiconductors

All the above considerations concerning carrier concentrations for n-type semiconductors can be transferred without difficulty to semiconductors which

4.4. free carrier concentrations

489

contain acceptors instead of donors, i.e. for which N A # 0 and N D = 0. They are called p-type semiconductors. The holes are the majority carriers in this case, and the electrons are the minority carriers. We omit details of derivation here and only display the results. In place of equations (4.92), (4.93), and (4.94), we have the relations (4.107) (4.108) where E B denotes the acceptor binding energy, and p i is given by p l = 2 N , e - E /kT.

(4.109)

With p instead of n as independent variable, the neutrality condition (4.71) takes the form (4.110) From this, we obtain the equation for p as P3

+ p i p 2 - (ni2 + p i N A ) p - n i2p 1 = 0.

(4.111)

For N A > ni we obtain P=NA

(4.113)

P = & K

(4.114)

if NA > p l . In both cases the minority carrier concentration n is much less than p . For the Fermi level position, we have results analogous to those obtained for an n-type semiconductor. 4.4.5

Compensation of donors and acceptors

Above, we considered real semiconductors which contained only donor or only acceptor atoms, but not both kinds of atoms simultaneously. Now, we consider the presence of both types of impurities in comparably large concentrations. Such a state would hardly be created intentionally, but it may occur unintentionally in the fabrication of a semiconductor material,

490

Chapter 4 . Electron system in thermodynamic equilibrium

where unwanted chemical pollution cannot be avoided. Such pollution could involve, for example, some boron atoms in a silicon crystal which one would like to n-dope by means of phosphorus. Also, technological aspects of the fabrication of semiconductor devices can play a role, for instance, a p-doping of partial areas of a previously n-doped silicon wafer causes the simultaneous occurrence of donors and acceptors. The question arises as to how carrier concentrations adjust under these circumstances in the semiconductor. Do the respective electron and hole concentrations simply add without interfering with each other, or do they strengthen or compensate among themselves completely or partially? A general answer to this question can be readily obtained from the mass action law (4.85). According to this law, the addition of acceptors to a material which previously contained only donors, cannot lead to an increase of the hole concentration without simultaneously lowering the concentration of electrons. The acceptors therefore compensate the effect of the donors to a certain degree. The physical reason for this is illustrated in Figure 4.7. The introduction of acceptor levels slightly above the valence band edge in a material which already contains donor levels slightly below the conduction band edge, changes the ground state of the crystal. Instead of transferring into the conduction band accompanied by a small energy increase, the donor electrons will transfer into the empty acceptor levels with a substantial lowering of the total energy of the crystal, and the acceptors on their part preferentially trap these electrons rather than those from the valence band, since the latter must first be excited. Analogously, holes will move from acceptor levels to the donor levels instead of being excited into the valence band. The capture of the free carriers can be complete, whereupon one speaks of complete compensation If the concentration of acceptors is too small to host all donor electrons, one has a partially compensated n-type semiconductor (p < n < N D ) , and if the concentration of donors is too small to capture all acceptor holes, one has a partially compensated p-type semiconductor (n < p < N A ) . In regard to occupancy statistics, the source of electrons (valence band or donor levels) which occupy the acceptor levels is completely irrelevant. What does matter is only that levels which can be occupied by electrons exist below the donor levels. Instead of shallow acceptor levels these can also be deep levels which do not at all act as acceptors because of their overly large energy separation from the valence band edge. If they can bind electrons, they can still be important for the compensation of donors - acting as trapping centers for electrons or simply as electron traps. Analogous statements hold for the compensation of acceptors by deep levels which can bind holes, one has hole traps. As we have seen in section 3.5, deep levels are caused both by structural defects as well as by chemical impurities. Since semiconductor materials always contain such perturbations - in larger numbers for lower

4.4. fiee carrier concentrations

49 1

Figure 4 . 7 Semiconductors with a) a donor level, b) an acceptor level, and c) with both kinds of levels. In parts a) and b) the upper drawings correspond to the ground state and the lower to the excited state. In part c) the lower drawing corresponds to the ground state, the acceptor level has captured the donor electron. cost fabrication - the complete or partial compensation represents, so to say, the usual state of a semiconductor material. To overcome this requires strong efforts in material cleaning and in crystal growth. But, even then, certain semiconductor materials still resist intentional doping. Known examples occur among the wide-gap 11-VI compound semiconductors, such as ZnS and ZnSe, which can be n-doped with relative ease, but p-doped only with difficulty. This problem was solved only recently, when the application of modern epitaxial growth techniques made it possible to also fabricate pdoped 11-VI-semiconductorswith wide energy gaps. Quantitatively, the free carrier concentrations n , p in the case of both nand p-type doping may be obtained from the same relations which were used in the case of pure n- or p t y p e doping. The Fermi level again follows from the neutrality condition (4.71), which here takes the form

n:

n---n

n1

n+n1 N D

+

nf +pin

N A = 0,

(4.115)

where n and p are to be taken from equations (4.79) and (4.80). The relation (4.115) can be transformed into a fourth-order equation for n or p. Since this is not solvable in compact form, we consider certain limiting cases which allow for simple solutions and provide some physical insight. Under the assumption n ~ n?, the result is

p = (NA- No),

n=

n? (NA- N D )’

if

N A > ND.

(4.119)

Thus a partially compensated n-type semiconductor occurs for N D > N A , and a partially compensated p t y p e semiconductor occurs for N A > ND. In both cases the majority carrier concentration is J N D- NAJ. For not completely ionized donors or acceptors, this simple relationship is no longer valid. In Figure 4.8, partial compensation as a function of temperature is illustrated using the example of n-doped germanium.

4.4.6

More complex cases

Several different single donors and acceptors

In the analysis above, we considered at most two kinds of simply ionizable doping centers simultaneously. But, in fact, more can occur, of course. It is not difficult to expand the theory to this more general case. Each additional type of a simply ionizable donor (described by an index j) introduces an

4.4. f i e e carrier concentrations

493

+

additional term N D j / ( n n j ) into the neutrality condition, and each additional type of such an acceptor (described by an index k ) gives rise to an additional term NAkn/(n: n p k ) . The neutrality condition for this more general case therefore takes the form

+

n2

n - 2 - z ~ D j - nj + ) : N A ~ zpkn = 0. n j n+nj ni + P k n

(4.120)

As before, it determines the position of the Fermi level uniquely. For the doping centers themselves, excited bound states were excluded up to now, and we admitted only two charge states, the ground state and the simply ionized state. If we give up these assumptions, then the neutrality condition (4.71) changes. This means that the position of the Fermi level then depends on N D , N A , N,, N u , and T differently than in the case of centers having excited states. The particulars of this dependence will be analyzed below. In this, we may use the fact that the relations (4.79) and (4.80) between the carrier concentrations n , p and the Fermi level remain the same as for the case of simple impurities.

Excited bound st ates We consider a donor which, as before, is only simply ionizable, but has excited levels ED,, n = 2 , 3 , . . . in addition to the ground state level E D ED^. In section 4.2, the average number < CdmsN D > of~ electrons ~ at ~ such an impurity was calculated. Using equation (4.29) we find that

(4.121) with y* given by equation (4.28). Formally, the occurrence of excited bound states modifies the neutrality condition (4.97) in that nl is replaced by 61 =

y*2n1.

(4.122)

Since y* < f, and therefore 6 1 < n l , the concentration n of free electrons is decreased in general. This is readily understandable since the excited elect,ron states of the donors result in more electrons being bound. For acceptors with excited bound states one gets analogous results. The concentration p l is replaced by $1 = y*-12-lp1.

(4.23)

in the neutrality condition (4.111). Now, j1 > p l holds, which means that the number of free holes is increased. This is immediately clear because

494

Chapter 4. Electron system in thermodynamic equilibrium

the acceptors bind fewer holes if there are excited electron states at the acceptors.

Multiply ionizable centers n-type semiconductors

We consider double donors, such as substitutional sulfur atoms in silicon. The average electron number < fi 1 ND-; > at such centers has been 0, calculated in section 4.2. The result is shown in equation (4.21). To evaluate the neutrality condition

+

n- p -N

D ~= 0,

(4.124)

one needs the hole concentration Ngf localized at the donors. It is given by the relation

+

Substituting < fi I fiD-l > as given by equation (4.21), replacing the DI chemical potential p by E p l Z E bby Eg - E h , and E g by Eg - E i l where E L and E i l are, respectively, the ionization energies of one electron at the 1- or 2-fold occupied center, it follows that

As before, in the case of simply ionizable donors and non-degenerate carrier gases, one can eliminate the Fermi energy from this expression by introducing the electron concentration n given by (4.79). Instead of the one concentration constant n1, two now occur, namely

Because E h > E i l , we always have n; < n;. In terms of these constants, equation (4.126) may be written in the form

(4.128) With this result (and using n!/n for p ) , the neutrality condition (4.124) takes the form

495

4.4. Free carrier concentrations

(4.129) It may be transformed in a fourth-order equation for the determination of n. Because the difficulty of solving this in closed form, we again consider only limiting cases. Generally, the doping concentrations ND >> ni should

be sufficiently large that the intrinsic term in (4.129) can be neglected. If one additionally assumes that n ni and with this also nz >> nT must hold. This means that n* n* also insures that ND >> na. This means that the reserve case also occurs with respect to the 2-fold occupied center. In comparison with a one-particle center having an ionization energy E B equal to the ionization energy EB of the simply occupied two-particle center, the carrier concentration due to the two-particle center is enlarged by the factor 2(n$/n;). The ‘2’ is a consequence of the doubling of the electron number at the non-ionized donors. The factor ( n ; / n ; ) , which exceeds 1, reflects the fact that the ionization of the 2-fold occupied center has a larger probability than that of the simply occupied center. p-type semiconductors

We consider double acceptors such as substitutional zinc atoms in silicon. The neutrality condition reads, in this case, p - - n5 -2N~ P2

P

+ 2PIP + P2

(4.134)

with (4.135) Here EL and Eil refer to electrons (rather than holes) and denote, respectively, the ionization energies of the acceptor with, respectively, 2 holes (no electron) or 1 hole (1 electron) at the center. Because Eil > EL,one always has pT > pz. The neutrality condition will again be solved in limiting cases. In these cases the acceptor concentration NA will always be assumed to be large compared to the intrinsic concentration p i = ni of holes, so that the intrinsic term in (4.134) can be neglected. We first assume that p

(x,t ) = C f ( v k , X , t)(vk)alvk).

(5.5)

uk

In addition to the physical quantities which are of interest in both equilibrium and non-equilibrium, such as the electron concentration and the energy density, non-equilibrium also involves other quantities to be determined, in particular, currents related to equilibrium quantities such as, for example, particle or energy currents. The latter differ from zero‘only in non-equilibrium, and thus are of no interest in equilibrium. Moreover, those quantities whose average values do not vanish in equilibrium must also be reconsidered, because under non-equilibrium conditions they exhibit temporal and spatial variations which are not present in equilibrium. As an example, one may take the total number of free electrons in a spatially homogeneous semiconductor sample. Under non-equilibrium conditions this number may have larger or smaller values than in equilibrium. If one withdraws external influences, equilibrium and its parameter values will be restored by nonequilibrium processes. The two classes of phenomena, the appearance of currents, on the one hand, and, of time and space changes of quantities which are non-zero but constant in equilibrium, on the other hand, constitute the totality of nonequilibrium processes. For semiconductors, both types of processes are important. Their theoretical description can be based on the Boltzmann equation - one solves this equation and then employs the resulting distribution function f(vk, x,t ) to calculate the average values of the various quantities of interest. Alternatively it can be based on empirical equations for the average currents and the temporal or spatial change rates. We refer to these as phenomenological equations. Ohm’s law for electrical conduction and the first and second Fick’s laws for diffusion are examples.

Phenomenological equations and their derivation from the Boltzmann equation Insofar as information about the state of the system is concerned, the Boltzmann equation provides much more detail than the phenomenological equa-

5.1. Fundamentals of the statistical description of non-equilibrium processes 503

tions. The latter can be derived from the Boltzmann equation with additional assumptions, but the reverse, i.e. the derivation of the Boltzmann equation from the phenomenological equations, is not possible. The transition from the Boltzmann equation to phenomenological equations represents, essentially, a coarse graining of the length and time scales on which the nonequilibrium processes proceed. If processes must be analyzed on the space scale of the average mean free path length and the time scale of the mean free flight time, then Boltzmann’s equation is required. If a macroscopic average description suffices, one can use the phenomenological equations. In semiconductor physics, the latter case often applies. Accordingly, we will r e strict our considerations to the phenomenological equations in our treatment of non-equilibrium processes. These equations are valid only within prescribed limits. One can consider these limits, like the equations themselves, to be empirically given. However, deeper insight can be gained by using Boltzmann’s equation. The question to be answered in this context is: Under what assumptions can the phenomenological equations be derived from the Boltzmann equation? The answer depends on the phenomenon under consideration. Equations as, for example, that for the temporal change rate of a given physical quantity, may be obtained by multiplying the Boltzmann equation by the value of this quantity in a Bloch state vk and subsequently summing over all states. The simplest equation of this type is that for the temporal change rate of the free carrier concentration of a single band, which later will explicitly be written down. To obtain this equation from Boltzmann’s equation in the manner indicated above, simplifying assumptions are necessary concerning the quantum mechanical transition probabilities between bands and localized states occurring in the collision term: These probabilities must be approximated by their values in thermodynamic equilibrium. In the derivation of phenomenological equations for particle currents, to consider another example, one assumes that the non-equilibrium state deviates only slightly from a spatially and temporally local equilibrium state. In the latter state, the distribution function depends on the Bloch states only through their energies, and the form of the dependence is that of the Fermi distribution function f ( E ) , just like in global equilibrium. However, the are understood temperature T and chemical potential p occurring in f(E), as functions T(x,t ) and p(x, t) of space and time coordinates. Thus we have,

This particular non-equilibrium state may only be assumed if the temperature and chemical potential are still well defined quantities, albeit in a local sense, not globally. For this assumption to be valid, adjacent parts of the

504

Chapter 5. Non-equilibrium processes in semiconductors

system with measurably different values of temperature and chemical potential are effectively decoupled from each other spatially and temporally. From the microscopic theory of non-equilibrium processes it is known that spatial decoupling (in the statistical sense of this term) occurs over a distance of the order of magnitude of the mean free path length l f , and temporal decoupling occurs over a time interval of the order of magnitude of the mean free flight time t f . The characteristic lengths of the spatial inhomogeneities of T(x, t ) and p(x, t ) may be expressed by the ratio TIIVTI and p/IVpL(, and the characteristic time of the temporal inhomogeneities of these quantities is given by T/laT/atl and p/lOp/OtL(. In order for local equilibrium to be approximately realized, the conditions

If > Ep. On the other hand, the electron captured from the conduction band must make the transition to the valence band more quickly, or, in other words, the deep center must capture a hole from the valence band more quickly, than the time it takes the electron captured by the deep center can be re-emitted back to the conduction band. Thus C p >> En must hold. If different order of magnitude relations exist between the four rates, the deep center can no longer function as a recombination center, but has other effects (see Figure 5.4). We summarize these in the following survey: The deep center works as a)

recombination center

for

Cn >> E p , C p >> En

b)

electron capture center

for

C, >> Ep,

C p >

En,

C, has a non-vanishing

524

Chapter 5. Non-equilibrium processes in semiconductors

component in the direction of the field. Its rate of change

[$ < m i i n > field

1

per second, according to equation (3.110), equals the force impressed by the field on a Bloch electron. whence

[& <

mkkn

1

>

=

-eE.

(5.72)

field

In addition to the electric field there are frictional forces acting on the Bloch electron which arise from the semiconductor itself. The interactions of an electron with lattice vibrations and impurity atoms or defects result in scatterings which act to brake its acceleration along E, producing a frictional effect similar to that in a viscous medium. Often, this frictional resistive force grows stronger in proportion to the average electron speed, so that one may put the corresponding rate [$ < mkxn of momentum change per second due to collisions with phonons and crystal imperfections, proportional to the average momentum, whence

>Iez

(5.73) with l / r i as proportionality factor. If there is a momentum excess at initial time t = 0, < m;xn(0) >, the solution of this equation reads

< mkxn(t) > = < rn:jcn(o) > e-t/T:.

(5.74)

Thus, the momentum excess decays exponentially, relaxing to zero, as one may say. In this context, one terms r;E as the momentum relaxation time. This time should not be confused with the previously introduced electron lifetime 7, for capture or recombination. The two characteristic times describe completely different physical phenomena. In general the momentum relaxation time is shorter than the capture and recombination lifetimes. The total-momentum balance is governed by Newton’s equation of motion,

whence d - < mkxn dt

> = -eE-

1

- < mn * kn

>.

(5.76)

r ;

For steady state dc current flow, the left hand side of this equation vanishes and the concomitant constant value of the average velocity is

5.4. Drift current

525

(5.77) which is referred to as the drifi velocity. For a field strength of 1 Vcm-’ the drift velocity has absolute value (5.78) This quantity is a measure of the mobility of an electron per unit applied force, therefore pn is simply referred to as electron mobility. Using equation (5.78), the drift velocity of (5.77) may be written in the form

< X, > = pnE.

(5.79)

The average electron current density j,, defined as j, = - e n < kn >,

(5.80)

may be re-expressed in terms of p, as

j , = enp,E. For the electron conductivity

(5.81)

as it occurs in Ohm’s law, one t,hus obtains

D,,

on = enp,.

(5.82)

This conductivity expression will be used frequently below. It can be transferred to holes without difficulty, reading ‘Tp =

eppp,

(5.83)

where the hole mobility p p is given by er; Pp=

- - y I

(5.84)

mP

with r; signifying the momentum relaxation time of holes. Since electrons and holes in a semiconductor are always present simultaneously, the total current density is the sum of the electron and hole current densities,

j and the total conductivity

D

=

.in

+j p ,

(5.85)

is the sum of the two partial conductivities, C T = un

+ up.

(5.86)

526

Chapter 5. Non-equilibrium processes in semiconductors

Table 5.3: Electron and hole mobilities for pure semiconductor materials at T = 300 K . (After Landoldt-Bornstein, 1982.)

1000 Si

200

40

100

N=10"

20 Temperature ("C)

I

-50

O

0

50

100

150

Temperature ("C

U 200

1

Figure 5.5: Temperature dependence of the electron and hole mobilities of Si for various doping concentrations.

For extrinsic semiconductors of n- or p-type, one of the two charge carrier concentrations exceeds the other by many orders of magnitude. To a good approximation, the total current density for an n-type semiconductor can be identified with j,, and for a p-type semiconductor with j,. We will tabulate some values of the transport parameters above, and discuss aspects of their characteristic dependencies. Since the charge carrier concentration of a given semiconductor can vary over a wide range, it is more meaningful to discuss p n and p p rather than of u, and up. The mobility unit one commonly uses is cm2V-ls-'. If the mobility is unity in this unit, then a field of 1 Vcm-' causes a drift velocity of 1 cm2v-1s-l x 1 v c m - l = 1 cms-'.

(5.87)

In Table 5.3 the mobilities of electrons and holes are given for some important semiconductors at room temperature. That the mobilities of electrons exceed those of holes, is to be attributed to the smaller effective masses of the electrons in these materials. The mobility depends on the interaction of the carriers with phonons through

5.5. Diffusion and annihilation of free carriers

Figure 5.6: Dependence of the electron and hole mobilities on

5

doping concentration for Ge, Si and GaAs at T = 300 K . (After

N

Sze, 1981.)

52 7

l o T- 3004 K 7

=- 103

-5

,102

Doping concentration ( ~ r n - ~ )

the momentum relaxation time, and the same may be said for temperature dependence. In Figure 5.5 this dependence is shown for Si. The decrease of mobility with rising temperature is to be attributed to the growth of the amplitudes of the lattice oscillations or, in other terms, of the number of phonons. In extrinsic semiconductors the mobility also depends on the doping concentration, because doping atoms constitute collision centers. Generally, the momentum relaxation times, and through them also the mobilities pn and p p , decrease with rising doping concentrations. Examples of such dependencies are shown in Figure 5.6.

5.5

Diffusion and annihilation of free carriers

In this section, we consider the concentrations n and p of electrons and holes to be spatially inhomogeneous. A direct consequence of this assumption is that the total charge density, which consists of the charge density of the free carriers and that of the ionized doping atoms, is no longer zero locally, although the total charge vanishes upon integration over the entire sample. The non-vanishing local charge distribution is accompanied by spatially inhomogeneous electric fields. Later, we will take account of these fields, but for the moment we omit them. To be consistent, we correspondingly ignore the charges carried by electrons and holes. Spatial inhomogeneities of the electron and hole concentrations n ( x ) and

528

Chapter 5. Non-equilibrium processes in semiconductors

p(x) imply the existence of non-vanishing gradients O n ( x ) and Vp(x). The

latter cause spatially inhomogeneous diffusion currents in and ip of, respectively, electrons and holes. Considering first the case of electrons, the diffusion current is given by

in = -DnVn.

(5.88)

Because of its inhomogeneity, this current leads to temporal changes of the local electron concentration, whereby the latter may take values which deviate from local equilibrium. As a result, generation and annihilation processes will take place which tend to restore local equilibrium values. The overall balance of the electron concentration change is described by the continuity equation (5.20). The right-hand side of this equation is specified such that generation processes are omitted while capture and emission processes at deep centers are taken into account. Then the annihilation term - ( d ~ ~ / d t ) ~ ~ equals ~ i h i l (dn/dt),,pture. For small deviations from equilibrium, (an/dt)capture may be identified with expression (5.39), whence dn

-+V.i at

n - no

n-

(5.89)

7,

For the sake of simplicity, we suppose that the electron concentration n ( x ) changes only in x-direction. Then only the x-component of in is non-zero and we denote it by in. In the continuity equation, we eliminate V.in = (&,/ax) by means of the diffusion equation (5.88), with the result

-d -nD

at

d2n nax2

n-no -

rn

.

(5.90)

For steady state this yields

(5.91) The determination of a unique solution of this equation requires specification of boundary conditions, which we choose as follows: at x = 0 a particular non-equilibrium value n(x = 0) = no

+ An0

(5.92)

of the electron concentration is maintained, and n ( x ) will decay to the equilibrium value no at x = 00. In the interval 0 < x < 00, the solution is written in the form

(5.93)

5.5. Diffusion and annihilation of free carriers

Figure 5.7: Profile of the nonequilibrium electron concentration as a function of position x in the presence of diffusion and charge carrier annihilation (schematically).

529

t 0 X-

Then it is readily seen that n(2)= no

+ Ange-"ILn,

(5.94)

where L,=

&.

(5.95)

The solution (5.94) admits the following simple physical interpretation (see also Figure 5.7). Diffusion tends to disassemble existing concentration gradients, transferring part of the distribution near z = 0 further to the right. However, since the lifetime of non-equilibrium charge carriers is limited, this process cannot be fully accomplished, as electrons are captured during the transfer process, leading to the exponentially decaying profile of the electron concentration given in equation (5.94). The characteristic decay length L , can be interpreted as the distance to which an electron diffuses, on the average, before it is captured. One refers to L , as the dzfluszon length. L , is the larger for larger lifetime 7,. One can illustrate the interpretation above by means of a fluid current moving over porous ground. At a particular point x = 0 the fluid current level is maintained at a fked height. In the direction of the current flow, the fluid level drops because fluid is absorbed into the porous ground. Sufficiently far downstream, practically no fluid will arrive because most of the h i d was absorbed into the ground further upstream. Nevertheless, one has a continuous fluid current at x = 0, which provides exactly as much fluid as is absorbed along the entire pat,h between x = 0 and z = 00. The analogy to the diffusion of electrons with finite lifetime is obvious. Furthermore, the current flow at z = 0, in this case i,(z = 0), may be determined from (5.88) and (5.94) as (5.96)

These considerations for electrons can be immediately transferred to holes. In analogy to equation (5.91) one obtains

530

Chapter 5. Non-equilibrium processes in semiconductors

Table 5.4: Diffusion coefficients and diffusion lengths for several pure semiconductors at T = 300 K . Diffusion coefficients are calculated by means of mobility data of Table 5.3 using Einstein's relation.

Si Ge

100.8

GaAs

227.5

10.3

10-8

3.2

(5.97) For boundary conditions analogous to those in equation (5.92), we have

with

(5.99) as the diffusion length of holes. In Table 5.4, diffusion lengths of electrons and holes are listed for several semiconductor materials. The lifetimes r, and rp depend strongly on the degrees of purity and crystallographic perfection of the semiconductors. In s was assumed for T,,~. However, in very pure Table 5.4 a value of and perfect materials, considerably larger values are possible. The diffusion lengths are then much larger than those given in Table 5.4, and they can reach hundreds of microns.

5.6

Equilibrium of free carriers in inhomogeneously doped semiconductors

In this section we consider semiconductors with spatially inhomogeneous doping, and explore the thermodynamic equilibrium state of such a spatially inhomogeneous system. The more general case, which includes semiconductor materials having an inhomogeneous chemical composition, will he discussed in Chapter 6. As before, we can consider the semiconductor to be composed of a subsystem of atomic cores and another of the free charge carriers. Both subsystems are taken to he spatially inhomogeneous here.

5.6. Equilibrium of free carriers in inhomogeneously doped semiconductors 531

Complete equilibrium of the total system can only exist then if the inhomogeneities have equalized out in both partial systems. This means also that the doping of the semiconductor material must have become spatially homogeneous. The process which brings this about, the diffusion of dopant atoms, is so very slow, however, as compared to the carrier processes, that one can completely neglect it. This means that, the subsystem of atomic cores effectively remains in the non-equilibrium state in which it started; this state is frozen, so to say. The subsystem of the free charge carriers, however, will relatively quickly pass into a new stationary state. On the one hand, this state will differ from the equilibrium state of free charge carriers in a homogeneous semiconductor - the concentrations n and p of the electrons and holes will depend on position coordinate x,as a consequence of the inhomogeneity of the dopant concentrations. On the other hand, all current densities of the free charge carriers will also vanish in the inhomogeneous case, just as in the homogeneous case. A stationary final state conforming to this characterization may be understood as a thermodynamic equilibrium state of the free charge carriers of the spatially inhomogeneous semiconductor. The nature of thermodynamic equilibrium in this sense will be discussed below. We consider only spatial inhomogeneities in the x-direction. The com, and j , of the electric current densities of electrons and holes in ponents j this direction are given by the phenomenological equations

(5.100) j , = a,E(x)

-

dP dx

eDp-.

(5.101)

Here E(x) represents the electric field that occurs in consequence of the spatially inhomogeneous charge carrier distributions, initially omitted in the previous section. In equilibrium the current densities of electrons and holes must vanish. That this must hold for the electron and hole currents separately, and not just for the sum, follows from Boltzmann’s equation which states that in equilibrium the partial currents of the subsystems of electrons having a fixed energy must vanish separately (not just after summing over all energy values). The holes are, of course, also a subsystem of the total electron system with particular energy values, namely the energies of the unoccupied states of the valence band. Hence, in equilibrium we have

+

dn 0 = ~ , E ( x ) eDn--, dx

(5.102)

0 = apE(x) - eDp-.dP

(5.103)

dx

532

Chapter 5. Non-equilibrium processes in semiconductors

The two electron current contributions in (5.102), the drift current v n E and the diffusion current eD,(dn/dx), need not vanish separately. The same holds for the hole current contributions a p E and - e D p ( d p / d x ) in equation (5.103). Only the sum of these contributions must be zero. Thus we arrive at the conclusion that in thermodynamic equilibrium the drift and diffusion currents of each type of carrier must exactly compensate each other. Our object now is to transform this observation into a general condition for thermodynamic equilibrium in systems with spatially inhomogeneous doping. To this end we express the field strength E of equations (5.102) and (5.103) in terms of the electrostatic potential cp, and replace c n and up, respectively, by the right hand sides of equations (5.78) and (5.83). Thus, we obtain

o=-

dYJ dx

+ Dn-ddx [Inn ( x ) ],

(5.104)

dcp

d - Dp- [ h p ( ~ ) ] dx

(5.105)

~ n -

O = -pp-

dx

In the case of local equilibrium, the charge carrier concentrations n(x) and p(x) depend on the chemical potential p(x) in just the same way that n and p in expression (4.59) depend on E F , (5.106) Substituting these expressions in (5.104) and (5.105) we have (5.107) (5.108)

In order to satisfy these two relations simultaneously, it is necessary that (5.109) Equations (5.107) and (5.108) will now be applied to a particular spatially inhomogeneous system for which the space variation of the chemical potential is known. This consideration will result in the Einstein relation.

Einstein relation Because the chemical potential for holes is the same as that for electrons, we can restrict ourselves to the latter. We assume that at a given position z = xo there is an infinitely high potential wall for these particles (see Figure 5.8). Electrons can reside on the right side of the wall, but not on the left, as

5.6. Equilibrium of free carriers in inhomogeneously doped semiconductors

533

. .

Figure 5.8: Illustration of the derivation of the Einstein relation.

X

4

aJ I

01

w

I I

xo X

they are reflected by the wall if they move toward the left. The situation is comparable with that of molecules of air above the surface of the earth. As in the latter case, in which the equilibrium vertical distribution of particles is given by the ‘law of atmospheres’, we have here

Comparing equations (5.110) and (5.106) it follows p(z) = ecp(z). Since equation (5.107) applies also in this special case, the factor multiplying ( d p l d z ) must necessarily be ( l / e ) , whence we conclude that (5.111) This equation is the Einstein relation It holds for holes as well as electrons, kT

D, = -

p P .

(5.112)

A generalization of Einstein’s relation for degenerate charge carrier gases is possible, but we will not discuss this here. Condition for thermodynamic equilibrium of free carriers Combining equations (5.111) and (5.107), we obtain the desired general condition for thermodynamic equilibrium of free charge carriers,

9-e d- v = 0 dx

dx

(5.113)

or p ( z ) - ecp(z) = const

(5.114)

The quantity on the left hand side of this equation is referred to as the electrochemical potential. In the case of vanishing electric potential, the electrochemical potential is just the same as the simple chemical potential. This also remains true if the electric potential has a non-zero but spatially

534

Chapter 5. Non-equilibrium processes in semiconductors

constant value which, as we know, is irrelevant. The terms Fermi energy or Fermi level which previously, in the absence of spatially varying electric potential, were used for the simple chemical potential, will now be used for the electrochemical potential. With this terminology, relation (5.114) becomes

(5.115) In thermodynamic equilibrium, the electrochemical potential of the free charge carriers is spatially constant, and the constant value is equal to the Fermi energy. The chemical potential and the electric potential themselves can change, however. It is appropriate to bear in mind here that the initially assumed spatial inhomogeneity of the semiconductor is based on the inhomogeneity of its doping. The formal expression of this inhomogeneity is the spatial variation of the chemical potential. To properly adjust the thermodynamic equilibrium state, the spatial variation of the electric potential must just compensate the spatial variation of the chemical potential. The equilibrium state of equation (5.114) represents an equilibrium in both the local and in the global senses. That in the local sense was presupposed (approximately) at the outset with the use of phenomenological equations for the current densities of the charge carriers. Moreover, the equilibrium state described by (5.114) represents a relative equilibrium, because it was derived under the assumption that the initial spatial inhomogeneity of doping, and thus the inhomogeneity of the chemical potential, remains unchanged. Over very long periods of time, this is assumption ceases to be valid. However, this does not affect the equilibrium condition because it is only essential that the inhomogeneity of doping does not change during the relaxation of the charge carriers towards equilibrium, a condition which is always fulfilled. As one may expect, the equilibrium condition (5.114) or (5.115) will play a central role in the treatment of semiconductor junctions in the next chapter.

535

Chapter 6

Semiconductor junctions in thermodynamic equilibrium The distinguished role of semiconductors in the modern world stems from devices which can be fabricated from these materials and, like transistors and semiconductor lasers, have opened the door to the present computer and communication technologies. The operation of semiconductor devices depends largely on effects which occur at semiconductor junctions, in addition to the properties of homogeneous semiconductors which have been treated almost exclusively in this book thus far. A semiconductor junction is understood as a structure composed of two interconnected layers, of which at least one is a semiconductor material. Various semiconductor junctions are shown in Figure 6.1 schematically. In case (a) the two layers are made of the same semiconductor material, e.g. silicon, but one of the layers is p-doped and the other n-doped. This is called a pn-junction In case (b) two layers of different semiconductor materials are connected. The doping type may, but need not be different. This structure is called a semiconductor heterojunction. We encountered such semiconductor heterostructures above in section 3.7. If a semiconductor layer is combined with a metal layer, then one has a metal-semiconductor junction, and if it is connected with an insulator layer it is an insulator-semiconductor junction Formally, one can also include semiconductor surfaces here, which correspond to vacuumsemiconductor junctions. Each of the semiconductor junctions identified above is associated with an important physical discovery. Several of these discoveries were even honored with Nobel prizes. Moreover, each of these junctions plays a role in electronic devices, and in most cases a semiconductor junction is crucial for operation of the device. Electric rectification at a metal-semiconductor junction, the so-called Schottky contact, was historically the first semiconductor junction to be used technologically. In this particular case, funda-

536

Chapter 6. Semiconductor junctions in thermodynamic equilibrium

a) n -type

b)

cn Semic.

Metal

Figure 6.1: The various semiconductorjunctions. mental general laws of semiconductor junctions were first recognized, particularly by Schottky and Mott. It is thus astonishing that, in this case of the metal-semiconductor junction, a complete microscopic understanding is elusive even now. The pn-junction, more precisely the combination of a pnand a np-junction, underlies the bipolar transistor, for which the inventors Shockley, Bardeen and Brattain received the Nobel prize for physics in 1956. Also, the tunnel diode, for which the Japanese physicist Esaki earned the Nobel prize in 1973, rests on the pn-junction. This device employs electron tunneling between the valence and conduction bands of such a junction. The unipolar field effect transistor (MOSFET) contains, beside a pn-junction, also an insulator-semiconductor junction as an active structural element. In experimental studies of the 2-dimensional electron gas in specially tailored silicon MOSFET's, the quantized Hall effect was discovered by von Klitzing. This accomplishment was honored with the Nobel prize for physics in 1985. The 2-dimensional electron gas and the quantized Hall effect may also be observed in semiconductor heterojunctions. Combining heterojunctions with pn-junctions, efficient light emitting diodes (LEDs) and semiconductor injection lasers may be fabricated. Heterojunctions also form the key structural elements of which the superlattices and quantum wells of section 3.7 are composed. The advent of these systems has opened a completely new and fruitful area in semiconductor physics and electronics, that of semiconductor microstructures.

In general terms, the operation of electronic devices based on semiconductors may be described as follows: The thermodynamic equilibrium of a semiconductor junction is disturbed by an external perturbation, for exam-

537

6.1. pn-junction

ple, by applying a voltage or by exposing the sample to light. The result of the perturbation are large changes of some properties of the junction, for example, its electric resistivity. These changes are the effects employed to advantage in semiconductor devices. To understand the perturbation of an equilibrium state of a semiconductor junction, one must first deal with the equilibrium state itself. This will be done in the present chapter. Nonequilibrium processes will be treated in Chapter 7. To initiate discussion of the equilibrium state, we start with the pn-junction.

pn-junction

6.1

pn-junctions are fabricated such that either the surface region of a homogeneously doped n (or p)-type sample, is p (or n)-doped by means of diffusion or ion implantation, or that a p (or n)-type layer is deposited on a n (or p)-type substrate by means of epitaxy. In the particular case of Si, neutron bombardment may also be used. Due to the nuclear reaction ;gSi n :aSi y -+ p, a previously p-doped sample becomes n-doped in the surface region exposed to the neutrons. In the following theoretical description of the pn-junction we assume homogeneously doped p - and n-regions left and right of the plane normal to the z-axis of a Cartesian coordinate system (see Figure 6.2a). The p region extends to -00 and the n-region to +co. The transition from the p to the n-region takes place abruptly at z = 0. We refer to this position as the ‘nominal transition’. In the transverse plane normal to the z-axis, the two semi-infinite semiconductor samples are taken to extend to infinity in all directions. Thus, there are no dependencies on y and z , provided the semiconductor materials are homogeneous in the transverse plane, which we take to be the case. With this, the problem is effectively one dimensional. The carrier concentration distribution along the x-axis before equilibrium is established is shown schematically in Figure 6.2a. Conceptually, this corresponds to the instant at which the junction is formed, so that the carrier distribution has not yet had enough time to adjust to form a new equilibrium state. Considering that the p-region also has electrons which are minority carriers in it, and that the n-region also has holes as minority carriers there, one must distinguish the region in which these concentrations are meant. We indicate this by affixing a subscript to n and p . The concentrations in the n-region will be denoted by n, and p,, and those in the p-region by np and p p . Thus, n,, and p p are majority carrier concentrations, while np and p, are minority carrier concentrations. The different values of the charge carrier concentrations in the n- and p-regions are related to different values E F , and E F of ~ the bulk Fermi energies in these regions, such that

+

+

+

---f

538

Chapter 6. Semiconductor junctions in thermodynamic equilibrium

1--I

D i f f h o n tc Drift

-ecp (x)

C)

Recombination

I

I

I

I I

I

I I

1

I

I

I

X

I

I

I

Figure 6.2: Non-equilibrium state of a pn-junction (a), from which the establishment of thermodynamic equilibrium develops (b,c).

6.1. pn-junction

6.1.1

539

Establishment of thermodynamic equilibrium

The situation shown in Figure 6.2a certainly does not represent an equilibrium state. The charge carrier concentrations are spatially varying functions n(x) and p ( z ) , and because np o

Cl 0

0

0

0

0

0

0

0

0

0

0

0

0

0

.

0

0

0

0

0-0

+ l o o o 0 /. . . . ( ~ 0 0 0 9

0

.

0

0

4 0

0

0

0

.to

.

.

0

0

0

0 0

0

0

. c

+ ~ o o o o ( . . . .

0

0

0

0

0

0

0

0

0

0

-0

0

0

0

0-0

0

0

.to

0

0

0

0

o c

Figure 7.3: Mechanism of current flow through a pn-junction, a) pn-junction without voltage, b) Injection of minority carriers by applying a (positive) voltage, c) Recombination of the injected minority carriers with available majority carriers, d) Refilling of the emptied band states by majority carriers from the bulk region. There a majority charge carrier current flows. differ from that in an infinite semiconductor sample, i.e. the current in the p-region will be carried mainly by holes, and that in the n-region mainly by electrons. In this matter, one has the peculiar situation that a hole current in the left part of the junction passes over into an electron current in the right part of the junction. The mechanisms which can realize such a transition are the recombination and generation of electron-hole pairs, whence we may conclude that the latter processes should play an important role in current flow through the pn-junction.

In our further considerations we assume CJ > 0. In order for recombination to occur, non-equilibrium carriers must be available. These appear in the p-region in consequence of the fact that electrons move over from the nregion to the p-region because their potential energy in the n-region is lifted under the applied voltage by the amount eU. Therefore they overcome the potential barrier between the n- and p-regions more easily. This transfer of carriers across a potential barrier is called injection In the p-region the electrons are minority charge carriers. Therefore we have an injection of minority charge carriers into thep-region. An analogous process takes place on the n-side of the junction. There, holes from the p-region move over into the n-region, so that one has an injection of minority holes. These relationships are illustrated in Figure 7.3. The injected minority carriers (Figure 7.3b) recombine with the already present majority carriers (Figure 7 . 3 ~ ) . The

7.1. pn-junction in an external voltage

5 79

states in the bands which become unoccupied in this way will be filled by majority carriers from the bulk regions of the junction (Figure 7.3d), where majority carrier currents therefore flow. The magnitudes of these currents are determined by the speeds of the injection and recombination processes. In the stationary state both speeds must be equal - all carriers which are injected must recombine, and all carriers which recombine must have been previously injected. If one reverses the sign of the voltage, corresponding to U < 0, then one has extraction of minority charge carriers instead of injection, and generation instead of recombination. Since extraction and generation processes consume energy, in contrast to injection and recombination wherein energy is released, one must expect the current through the pn-junction to be much smaller for U < 0 than for U > 0. This is in fact the case, as we will formally prove below. The above mechanism for current transport through a pn-junction will now be formulated quantitatively. The total current density j ( x ) consists, according to formula (5.85)) of the electron current density jn(x) and the hole current density jp(x). For the two current constituents the continuity equations (5.20) and (5.21) hold. In the present case, the generation term is zero, and the annihilation term is determined by recombination. In the stationary state one obtains

djp

- = -eR(x). dx

where R ( z ) represents the recombination rate given by equation (5.61). Adding (7.7) and (7.8), it follows that the total current density j ( x ) is free of sources and must be spatially constant, so that j(x) = j n ( 2 )

+ jp(x) zz j

= const.

(7.9)

We consider a particular position X I in the p-region and another particular position x2 in the n-region. The fact that the total current density j is const ant yields (7.10) Integration of equation (7.7) provides the result (7.11) Combining (7.10) and (7.11)) we have

580

Chapter 7. Semiconductor junctions under non-equilibrium conditions

(7.12) Setting (a)

21 =

-oo,z2 =

+m, we find (7.13)

On the other hand, considering (b) z1 = x p , z 2 = zn, it follows that j =j n ( 4

+jp(zn)

+e

/

X,

d 4 z ) .

(7.14)

XP

We will later see that, under certain conditions, the two minority charge carrier contributions jn(-oo) and jp(+oo) in (7.13) can be approximately neglected. The same holds for the recombination contribution in (7.14). Consequently, equation (7.13) means that the total current, in its essence, represents a recombination current. Alternatively, in formula (7.14) the total current is seen to be the sum of the minority carrier currents at the two space charge boundaries. There, they are determined by the injection of minority carriers. The total current represents, therefore, an injection current. The two interpretations are equivalent, but they emphasize different aspects of the total current. With these considerations concerning the mechanism of the current transport, we are sufficiently prepared to calculate the spatial profiles of the chemical potentials of the two types of carriers.

7.1.3

Chemical potential profiles for electrons and holes

We first calculate the charge carrier concentrations n ( z ) and p ( z ) . Once they are known, the chemical potentials follow immediately from relations (7.1) and (7.2). Since the current is due to recombination of injected nonequilibrium minority carriers in the bulk regions, we will restrict our considerations to these carrier concentrations in particular. The space charge region will therefore be omitted initially. Moreover, we will use the fact that the relative change of the majority carrier concentrations caused by injection is substantially smaller than the corresponding relative change of the minority carrier concentrations. For the majority carrier concentrations in the bulk regions, the values without injection, i.e. without an applied voltage, can be used. This means that

7.1. pn-junction in an external voltage

581

n(x) = n,

for

2

> x,,

(7.15)

p(z) = p p

for

x

< xp.

(7.16)

The minority carrier concentrations in the bulk regions are calculated from the combined diffusion-recombination equations (5.81) and (5.87). To solve these equations uniquely, boundary conditions are required, in the case of electrons at -m and xp,and in the case of holes at x n and +oo. For n(-m) and p(00) the equilibrium values

apply. The concentrations at the boundaries of the space charge region may be taken from relations (6.15) for n ( z ) and p ( z ) . Of course, the latter expressions were written down for equilibrium conditions, but they provide approximately correct values for n ( z p ) and p(x,) even when a voltage is applied. The only change to be made is the replacement of [cp(zn)- ( p ( z p ) ] by U D - U instead of by UD. It follows that n(q,) = n,e

-e(UD-u)/kT

=

npe

eU/kT

,

(7.18) (7.19)

For U = 0, the values n ( x p ) and p(;cn) are the minority carrier concentrations np and p , in equilibrium. If U > 0, the factor multiplying n p in (7.18), and the factor multiplying p , in (7.19) is larger than 1, i.e., the minority carrier concentrations exceed the values they would have in the absence of an external voltage. This is the formal expression of the injection of minority charge carriers. How effective injection is can be recognized through the following estimate. With U = 0.25 V and T = 300 K we obtain exp(eU/kT) M elo M 2 x lo4. This is to say that the small voltage of 0.25 V suffices to increase the minority carrier concentrations by more than 10000. The steady state solutions of the diffusion-annihilation equations (5.90) for n(x) and (5.97) for p ( z ) , under the respective boundary conditions (7.15), (7.16) and (7.17), are given by

P(X)

=P,

+ [~(z,)

- P,I~-(x-x*)lLp

,

> ",

(7.21)

where L , and L, are the diffusion lengths of, respectively, electrons and holes. For positive external voltages one has n ( x p ) > n, and p(x,) > p,.

582

Chapter 7. Semiconductor junctions under non-equilibrium conditions

Figure 7.4: Lineup of the quasi Fermi levels of a pn-junction under an applied voltage, (a) flow direction (U > 0 ) , (b) blocking direction (U < 0)). Dashed curves correspond to the unbiased pn-junction. In the space charge region interpolated values are used. The decay of the quasi Fermi levels in the two bulk regions is drawn greatly exaggerated. This means that in the two bulk regions, both minority carrier concentrations are larger than their respective equilibrium values np and p,. The chemical potential p n ( z ) of the electrons in the pregion, is therefore shifted to higher energies with respect to its equilibrium value, and that of the holes in the n-region is shifted to lower values. The same holds for the pertinent nonequilibrium electrochemical potentials

E F ( ~ =) p n ( x ) - ecp(x),

E;(X)

= p P ( z )- ecpG),

(7.22)

which are also referred to as quasi Fermi levels. In Figure 7.4a the spatial variation of the two quasi Fermi levels is shown schematically, together with the valence and conduction band profiles. For a negative voltage U < 0 one has n ( z p ) < np and p ( z n ) < p,. The chemical potential of electrons in the p-region therefore lies below the equilibrium value, and that of the holes in the n-region lies above it. The same is true again for the quasi Fermi levels (see Figure 7.4b). The elevation or depression of these levels in the bulk regions is effective up to a distance from the depletion region which roughly equals the diffusion length of the pertinent minority carrier. In the depletion region between the two bulk regions, we cannot make such statements, or any others, because the above consideration excludes this region. Fortunately, the diffusion lengths are, as a rule, an order of magnitude larger than the width of the space charge

7.1. pn-junction in an external voltage

583

region (see section 5.4), so that this lack of knowledge is not important. The space charge regions function solely as potential barriers over which nonequilibrium carriers are injected. The spatial expansion of the barriers can be neglected in a first approximation.

7.1.4 Dependence of current density on voltage We can now carry out the calculation of the total current density through a pn-junction without difficulty, starting from the relations derived for the recombination current density (expression 7.13, method a) and for the injection current density (expression 7.14, method b). Both methods must lead to the same result. Formally, it would therefore suffice to consider only one of them. We will study both to demonstrate explicitly that the recombination current (7.13) and the injection current (7.14) are in fact identical. Method a From the qualitative discussion of current flow through the pn-junction in subsection 7.1.2, we know that the currents at -co and +co do not differ from the currents in an infinite p or n-type semiconductor. This means, in particular, that the minority carrier currents at -00 and +co are negligibly small. In regard to the remaining integral in (7.13), recombination processes in the bulk regions to the left and right of the depletion region contribute significantly only up to depths which roughly equal the diffusion lengths of the minority carriers - only there do the concentrations of these carriers differ substantially from their equilibrium values, so it is only there that recombination occurs. The diffusion lengths are, as has already been mentioned, generally much larger than the width of the depletion region. Taking advantage of this magnitude relation, we neglect the contribution of the depletion region to the integral in equation (7.13), whence we obtain, approximately,

(7.23) The recombination rate R is given by relations derived above in section 5.2. According to formula (5.64), in the presence of a small excess concentration A n ( x ) of electrons (here, of electrons in the p-region), we have

(7.24) and according to formula (5.66), in the presence of a small excess concentration A p ( z ) of holes (here, such in the n-region), the corresponding relation is

584

Chapter 7. Semiconductor junctions under non-equilibrium conditions

(7.25) with rn and rp given by (5.65) and (5.67) as minority carrier lifetimes of electrons and holes, respectively. For An(.) and A p ( x ) we employ relations (7.20) and (7.21) in the form

~ p ( x=) b ( x n ) - pnle-(z-xJ'Lp,

xn

< 2.

(7.27)

Substituting these formulas in equations (7.24), (7.25) and (7.23) and using relations (7.18) for n ( x p ) and (7.19) for p ( x n ) , we obtain j = js(e

eUlkT

- 1)

(7.28)

with (7.29) The same result is obtained if one proceeds in accordance with method b, which will be verified below. Method b

In equation (7.14) for j we again neglect the recombination integral over the depletion region. The minority carrier current densities j n ( z p )and j p ( x n )at the boundaries of the depletion region follow from the general phenomenological equations (5.100) and (5.101), wherein the electric field strength has to be set zero since the electric potential is constant in the bulk regions. The charge carrier concentrations n ( x ) and p ( x ) are taken from expressions (7.20) and (7.21), yielding

(7.31) Adding these equations and applying relations (7.18) and (7.19), the expression (7.28) follows for j with (7.32) The two expressions (7.32) and (7.29) for j , are identical, since Ln = (see equation 5.95) and L, = (see equation 5.99).

fi

7.1. pn-junction in an external voltage

585

Figure 7.5: Current-voltage characteristic of a pn-diode made of Ge. Note the different voltage scales for forward and reverse biasing. For extremely large negative voltages, the diode undergoes electrical breakdown. (After Seeger, 1973.)

With this observation, the task of calculating the current through a pnjunction under an applied voltage is completely solved, within the framework of the conditions and approximations set forth above. We will now discuss the results. The current-voltage characteristic (7.28) is extremely non-linear. It exhibits the expected asymmetry with respect to a change of the sign of voltage U . For a positive U of a few tenths of a Volt, j is several orders of magnitude larger than j,, while for a negative U of same absolute value, j approaches - j , (see Figure 7.5). The current density j,, which cannot be surpassed at even larger negative voltages, is called the saturation current density. To estimate the size of this current density in the case of Si, we assume typical values for L,,p of 10 pm and for 7n,pof lo-' s. For the minority carrier concentrations, we obtain, from np = n?/NA and p, = n!/ND with ni = lo1' cm-3 and N A = N D = 1OI6 cmV3, the values np = p, = lo4 c ~ L - Using ~. e = 1.6 x 10-lgA s, it follows that j, 10-l' A/cmz. The saturation current density is therefore extremely small. Thus, for U < 0 practically no current flows, the pn-junction blocks the current flow. One says that it is reverse biased or biased in blocking direction, The biasing U > 0 refers to the forward bias or Bow direction because the current density in this direction is orders of magnitude larger than j, as we have seen above. The pn-junction operates as an electrical rectifier. It is called a rectifyingpndiode in this context. In Figure 7.5 a measured current-voltage characteristic of a pn-diode made of Ge is shown.

7.1.5

Bipolar transistor

The pn-junction has important application in the bipolar transistor. This device consists either of two n-regions, which are separated by a p-region, or of two p-regions separated by an n-region (see Figure 7.6). In the first case one speaks of an npn-transistor, and in the second case of a pnp-transistor.

586

Chapter 7. Semiconductor junctions under non-equilibrium conditions

Figure 7.6: Bipolar npn- and pnp-transistor.

m Emitter

Base Collector

Figure 7.7: npn-transistor in the common-emitter configuration (left-hand side). Illustration of the current flow (right-hand side). In the following considerations we confine our attention to the npn-case. In Figure 7.7 the npn-transistor is shown in one of the possible switching modes, called common-emitter configuration (for reasons explained below). The left n-region is connected both with the p-region in the middle as well as with the n-region on the right. The voltage source of the np-circuit puts the left n-region at a potential P E , and the p-region at a potential p ~ The . voltage source in the npn-circuit puts the potential of the right n-region at (pc.We consider the case in which PE

PB

(PCI

(7.33)

and, accordingly,

(7.34) holds. In this case the left pn-junction is biased in the flow direction, and the right in blocking direction (see Figure 7.8). From the left n-region, electrons are injected into the p-region in the middle, while the right n-region extracts electrons from the p-region. One therefore calls the left n-region the emitter, and the right the collector. The p-region in the middle is called the base. Accordingly, the np-circuit will be referred to as the emitter-base circuit or, in short, the base circuit and the npn-circuit as the emitter-collector circuit,

7.1. pn-junction in an external voltage

587

or in short, the collector circuit. Let the current in the base circuit be ig, and that in the collector circuit ic. Then the current i E through the emitter follows from Kirchhoff’s current branching theorem as the sum of the two currents, i~ = iB

+ ic,

(7.35)

One can also say that the emitter current splits in two partial currents, one flowing through the base, and one flowing through the collector (see Figure 7.7 on the right). Our goal is the calculation of the three currents i ~ iB, , ic. In this matter, we can employ the results obtained above for the current flow through an individual pn-junction, with the valence and conduction band edges of the npn-transistor lined up as shown in Figure 7.8. The pn-junction on the emitter side is subject to the flow voltage U g , and the pn-junction on the collector side is subject to the blocking voltage U c - UB. The emitter current i~ is the current flowing through the emitter-side pn-junction. As such it is given by expression (7.12), multiplied by the emitter area A . Accordingly, it consists of the injection current j n ( z p ) Aof the minority electrons into the

588

Chapter 7. Semiconductorjunctions under non-equilibrium conditions

base, the injection current jp(x,)A of the minority holes into the emitter, and the recombination current in the depletion region. Neglecting the latter contribution as before, and using relations (7.30) and (7.31), we have

Here the signs are opposite to those of relations (7.30) and (7.31) above, due to the fact that the emitter-base junction has thep-region on the right-hand side and the n-region on the left, whereas it was the opposite above. If both regions were expanded infinitely, as we always assumed above in the treatment of the pn-junction, then the minority charge carrier concentrations n(x) and p(x) in (7.37) could be replaced by the previously derived expressions (7.20) and (7.21). This procedure would result in expressions for j,(xp) and j p ( z n )which are just the negatives of those in relations (7.30) and (7.31). In the transistor, however, only the emitter can still be considered to be infinitely extended, while the width b of the base must be treated as finite because it is not large in comparison with the diffusion length L, of the minority charge carriers. Thus, only the injection current density jp(x,) of holes may be taken from the previously derived expression (7.31). Adjusting this expression to the relationships at the emitter-base junction, we find along with (7.27), (7.38) The injection current density jn(zp)of electrons, however, must be calculated anew. This can be done on the basis of the following considerations. To start, it is clear that, because of the finite width of the base, only part of the injected minority electrons recombine in the p-region, while the remainder diffuse through this region and reach the depletion region at the collector side of the pn-junction. The negative electric field there pulls the electrons into the bulk region of the collector, from which they are sucked up by the applied positive voltage. The equilibrium value np of the electron concentration is therefore reached not only at x = 00, but it is already realized at x = b. This means that the boundary condition An(x = b) = 0 must now be imposed. The solution of the diffusion-recombination equation (5.91), which accounts for this new boundary condition, results in

(7.39)

7.1. pn-junction in an external voltage

589

Substituting this into (7.37), we find Dn (eeuBjkT - 1) coth[(b - z p ) / L n ] . Ln

j n ( z p ) = enp-

(7.40)

The current ig in the base circuit would be identical with the emitter current iE, and both currents would equal the current through the emitter-base pn-

junction, if the base were to be infinitely expanded. For finite base width, however, the emitter current differs from the base current because the latter only takes contributions from the portion of the electrons injected into the base which also recombine in the base. We denote the pertinent current density as jnT. The first term of formula (7.23) expresses it as

(7.41) The upper boundary of the recombination region, which in (7.23) is located at infinity, is replaced here by the finite base width b. Furthermore, xn and x p are interchanged. The recombination rate R(x) in (7.41) can, as before, be calculated from the minority charge carrier concentration n ( z )= np+An(z) in the p-region by means of relation (7.24). In this, expression (7.39) has to be used for An(x), yielding

The base current ig, like the emitter current in (7.36), also involves the injection current j p ( z n ) Aof holes from the base into the emitter, besides the electron current j,A. Altogether, we therefore have

The collector current ic need not be calculated separately, because it is already determined by Kirchhoff's law (7.35). The main source of this current are electrons which leave the emitter and do not recombine in the base but diffuse further to the collector where they are sucked up. This current contribution is entirely due to the coupling between the collector-base pn-junction and the emitter-base np-junction; it would not appear at all at a single, separate pn-junction. Another current which flows through the collectorbase pn-junction is the true pn-current, which represents the current which would exist if this junction were not coupled to a second one. However, the collector-basepn-junction is biased in blocking direction, so that the true pncurrent is an extraction or generation current, which because of its smallness

590

Chapter 7. Semiconductor junctions under non-equilibrium conditions

Figure 7.9: Characteristics of a bipolar transistor made of Si in the commonemitter configuration. (After Volz, 1986.) may be completely neglected. This approximation was used above tacitly, as we identified the current through the emitter-base pn-junction with the true pn-current of this junction, without adding the hole extraction current flowing in from the collector-base pn-junction on the right-hand side . With this, the three currents i E , i g , ic we sought are fully determined. In Figure 7.9 some characteristic dependencies for i g and ic are shown in the case of a npn-transistor made of Si. We can now proceed to the question of what conditions are needed for the bipolar transistor to operate as an amplifier. As we will see below, significant amplification by a bipolar transistor results when the base current is only a small portion of the emitter current. In order for that to be the case, on the one hand, j,,A must be small. Considering (7.42)) this means that the base width b must be small in comparison with the diffusion length L , of the minority charge carriers. On the other hand, the hole injection current j p ( z , ) A is not allowed to be too large. This can be achieved by low doping of the base in comparison with the emitter, because jp(z,) is proportional to p, (as may be seen from (7.38)) and j n ( z p )is proportional to np (see relation 7.39). If the doping n, of the emitter is substantially higher than the doping p, of the base, i.e. if nn >> p, holds, then it follows from the mass action law that p, > k T , the Fermi distribution (7.77) can be approximated by the Boltzmann distribution, whence (7.79) For Ec(k) we use an isotropic parabolic dispersion law with effective mass m:. Then relation (7.78) takes the form

7.3. Metal-semiconductor junction in an external voltage. Rectifiers

609

The conduction band edge, referred to the vacuum level, lies at E, = - X E, = CI, - X is equal to the Schottky barrier @B. After a short calculation we find so that @

+

i ~ ~ (=0 )

-OB/kT,

4

(7.81)

with (7.82) as the average speed component in positive z-direction and n as the electron concentration in the semiconductor. Somewhat surprisingly, through mE and n quantities characteristic of the semiconductor occur in expression (7.81) for i ~ s ( Oalthough ), the current originates in the metal and one might have expected the appearance of characteristic metal quantities only. The reason is that for U = 0 (which is the case we are considering now) thermodynamic equilibrium exists, and the current from the metal into the semiconductor must equal the current from the semiconductor into the metal. That the latter depends on semiconductor quantities is obvious. 0 ) the semiconductor The interpretation of i ~ ~ (as0 a)current i ~ ~ (from into the metal in thermodynamic equilibrium may also be used to calculate i s ~ ( Uin) the presence of a voltage. For U = 0, first of all, this interpretation yields the expression

If a voltage U is applied to the junction, the conduction band edge of the semiconductor at the interface with the metal shifts from its equilibrium position E, to the new position E, - e U . This change in (7.83) leads to the relation iSM(U)

,eU/kT Z S M ( 0 ) . I

(7.84)

Correspondingly, the total particle current density i ( U ) is (7.85)

For the electric charge current density j = -ei it follows that (7.86) with I -@p,/kT j , = Tenvoe Lf

(7.87)

6 10

Chapter 7. Semiconductor junctions under non-equihbrium conditions

For positive voltages U , the current density j grows exponentially with increasing U , while for negative U of increasing magnitudes, j approaches the saturation current density -js. The latter occurs when 1UI >> kT holds. The Schottky junction therefore operates as a rectifier, with positive voltages corresponding to the flow direction, and negative to the blocking direction. The following estimate shows that j , can assume quite large values. For T = 300 K , N D = 10l6 cm-3 and @B = 0.25 e V , we have j , 0.2 A / c m 2 . From this we may conclude that Schottky junctions are suitable for rectification of relatively strong currents. This distinguishes them from pnjunctions where the saturation current density is commonly much lower (see section 7.1). The reason for this difference is that the saturation current of a pn-junction is due to minority carriers, while the saturation current of a Schottky junction is caused by majority carriers. The conditions of validity for the current-voltage characteristic of a metalsemiconductor junction derived above will now be examined. The application of formula (7.83) to the calculation of the current density iSM(0) reflects the assumption that thermionic emission of electrons from the semiconductor into the metal proceeds unhindered. To appreciate the significance of this assumption one must first recognize that the emitted electrons originate in the bulk region of the semiconductor, for in the depletion region none are available. Unhindered emission can only occur, therefore, if the electrons, during their flight through the depletion region suffer no collisions. That is assured if the mean free path length Zf is larger than the space charge width "B,

lf > X B .

(7.88)

For practical Schottky junctions, operating in blocking direction, this condition is often fulfilled, provided the blocking voltage is not too large. For flow voltages of sufficient magnitude it is always correct. Condition (7.88) excludes the possibility that the depletion region can even approximately be in a local equilibrium state. In particular, chemical and electrochemical potentials cannot be meaningfully defined, not even in a local or 'quasi' sense. This implies that an essential requirement for the applicability of the phenomenological equations (5.14) and (5.15) for the current densities is no longer valid. If the inequality (7.88) is satisfied, drift and diffusion lose their meaning as transport mechanisms in the depletion region of a metalsemiconductor junction. The transport proceeds by electrons flying through the depletion zone unimpeded, which is termed ballistic transport. If, instead of l f > ZB, the condition lf > L , ns is practically independent of U . In order that the accumulated charge density ns can be tuned by means of U as effectively as possible, we must have d Uo, we have

620

Chapter 7. Semiconductor junctions under non-equilibrium conditions

(7.119) With U M 5 V , this yields (dn,/dU) M 1013 crn-'V-l. A voltage increase of 1 V results in about lOI3 electrons being induced into the inversion layer per cm2. Electrons in an inversion layer have been the subject of many physical investigations. Like carriers at a semiconductor heterojunction (see section 6.2), these electrons are confined in a potential well. Just above the bottom ( r i 0.1 e V ) , the width of this well is still smaller than L. Thus, carriers of such energy are freely mobile only parallel to the well. One has a quasi 2-dimensional electron gas. In it, confinement effects, like the formation of subbands, can be observed. By means of an applied voltage, the density of the electron gas can be varied. In this context, the accumulation layer at the interface between SiOz and Si has been of particular interest. Historically, it was the 2D gas of this layer in which carrier confinement effects were studied first, and in which the quantized Hall effect was discovered. 7.4.3

MOSFET

Whether or not the charge density of a semiconductor inversion layer suffices to cause a resistance change of sufficient magnitude to function as a transistor, depends (among other things) mainly on the specific resistivity of the semiconductor material in the absence of an applied voltage. This must be as high as possible. The highest possible resistivity, or the smallest conductivity, of a semiconductor material is observed when it is in its intrinsic state. This fact is used to advantage in the most important type of field effect transistor, the so-called MISFETs (Metal Insulator Semiconductor Field Effect Transistor). Here, one exploits the fact that in a pn-junction a depletion region is formed wherein the charge carrier concentrations have intrinsic values. The p-region is embedded between two n-regions, as shown in Figure 7.22. If one applies a positive voltage between the left n-region (source) and the right n-region (drain) (Figure 7.22), then the left pn-junction is biased in the flow direction, and the right in the blocking direction. Between the two n-regions, one therefore has a p-region which is almost completely depleted of holes. In the theory developed above, this may be taken into account formally by adding a positive prevoltage Uv to the voltage U applied to the insulator-semiconductor junction from outside, i.e. by the replacement

u-+utuv.

(7.120)

For U = 0, the semiconductor region between source and drain, the so-called channel, has high intrinsic resistivity. If a positive voltage U = UG is applied between the bulk of the p-semiconductor (substrate) and the metal layer on

7.4. Insulator-semiconductorjunction in an external voltage

62 1

a1

I

P Substrat

Figure 7.22: Structure of a MISFET (a). In part (b) the switching scheme of the MISFET is shown.

E .. . n v)

Y

-0

4

8

12

16

20

24

Figure 7.23: Influence of the gate voltage UG on the I s 0 - versus - U s 0 currentvoltage characteristics for a n-channel MOSFET of enhancement-type. (After Vzilz, 1986.)

top of the insulator (gate), large enough to create an inversion layer, then the channel becomes a good conductor. The current in the source-drain circuit can be tuned by the voltage UG of the gate-substrate circuit since the electron density of the inversion layer depends on UG. The voltage UG is called gate voltage, and the minimum gate voltage UG necessary to a achieve inversion is the threshold voltage. If one also includes a working resistor in the source-drain circuit and compares the power changes in this resistor with those in the gate-substrate circuit, one finds that the former exceeds the latter appreciably. In this sense, the MISFET operates as an amplifier, just like the bipolar transistor. Because of the insulating layer below the gate electrode, the MISFET has a much larger input resistance than the bipolar

622

Chapter 7. Semiconductor junctions under non-equilibrium conditions

transistor. The output resistance of the MISFET, i.e. that of the source drain circuit, is small because of the accumulation layer between source and drain. If the MISFET is realized using Si as the semiconductor material and SiO2 as the insulator, one has the Metal Oxide Semiconductor Field Effect Transistor, abbreviated MOSFET, which is by far the most important MISFET. The basic requirement that the electrons of the inversion layer shall not be captured by interface states, are met by the MOSFET extremely well. Just as the base of a bipolar transistor can be made either of p or n-material, one also has two possibilities in the case of the unipolar MOSFET - the charge carriers in the conducting channel can either be electrons, as has been assumed thus far, or holes. In the first case one speaks of a n-channel MOSFET, and in the second of p-channel MOSFET. The prevoltage U V ,which was introduced above only formally to simulate the intrinsic state, can really exist for various reasons, e.g., because of positively charged centers within the oxide arising during its formation. The prevoltage can take such large values that inversion exists even without an applied gate voltage. Then the transistor is already in its conducting state at zero gate voltage. To switch it into its blocking state, one must apply a negative gate voltage in the case of the n-channel MOSFET, and a positive gate voltage in the case of the p-channel MOSFET. One says that the transistor operates in the depletion mode. If the transistor is blocked without an applied voltage and changes into the conducting state by applying a positive (for n-channel) or negative (for p-channel) gate voltage, one has the enhancement mode. The current-voltage characteristics of the source-drain circuit of a n-channel-enhancement-mode MOSFET are shown in Figure 7.23 for different gate voltages. One recognizes how the sourcedrain current at a h e d value of U s 0 increases with increasing gate voltage UG. Somewhat unexpectedly, for a k e d gate voltage UG, the current saturates at higher source-drain voltages USD. This is a consequence of the fact that the effective gate voltage in the vicinity of the drain electrode becomes smaller and smaller as the drain potential grows larger. This creates a ‘pinch off’ of the inversion layer, making the current stay constant. The MOSFET represents the most important electronic component of digital circuits in microelectronics. Here it is used as an electrically controllable switch, meaning that its control function is reduced to two states only, one with maximum output power, corresponding to a binary ‘l’,and another having minimum output power, corresponding to a binary ‘0’. Today, MOSFETs can be made as small that millions can be integrated in one single Si chip, more than of any other electronic component. It is this ultra-large scale integration (ULSI) technique which has made modern computer and communication technologies possible.

623

Appendix A

Group theory for applications in semiconductor physics

A.l

Definitions and concepts

A.l.l

Group definition

A group G is defined as a set of elements g which possesses the following properties: 1) There exists an assignment instruction which associates an ordered pair of elements g1 and 92 of the set uniquely with another element which also is a member of the set. One terms such an assignment instruction a 'mul-

tiplication', and writes g 1 . 92 for the associated element. The totality of the assignment instructions forms the multiplication table of the group. This table determines the individual nature of a group.

2) The multiplication obeys the associative law, i.e. for three arbitrary elements 91, 92, 93 one has the identity

3) The set of elements contains the identity element. This is an element e defined by the property that its right or left product with any other element g of the group yields again g. Thus,

624

Appendix A . Group theory for applications in semiconductor physics

4) For each element

4, the set also contains its im-erse element 9-l: such

that g.g-'

=;

g -1 - g - e .

(A.3)

Onc may show that it suffices to dcmand the existence of only the left-sided or only the right-sided inverse. This one-side inverse is thcn uniquely determined and its existcnce automatically leads t o the existence and uniqueness of the other-siticd iiiverse. T h e other-sidd inverse thusly determined is identical with the original oneside in?-erse. Analogous dstements hold €or the identity dement discussed above. The inverse (91 . g2)-' of tl product gl . g:! is given by gz1 . gl1: as one may easily verify by explicit. mult.iplicat.ionof t h e two products. The existence of the inverse ensures that t.he products g1 . g and g2 . g of two elements g1 and g~ with an arbitrary group element g are different, if the elements g1 and 92 differ from each other. The same is true for the re-ordered products g .g1 and 9 92. The number N of elements of a group is callrd its order. Depending on whether N is finite or infinite, one has finiteor iu,~%ittgrotqs. ' h e dements of infinite groups may be either discrehe or continuous. In the latter rase one has a coatinuow group. The group multiplication which i s associative by definition, need not also be commutative. In general: g1 . $2 differs from g2 . gl. If the multiplication is also commutative, the group is call& A h e l i a ~ ~ We now introduce some goup-thcoretical concepts which are n d e d in this book.

A.1.2

Concepts

A suihject of a group which forms a group by itself with respect to the multiplication tahle defined by that of the full group, is termed a subgroup. Subgroups

For finite groups G, the theorem of Lagrange holds, wherein the order 1L" G I is a divisor ol thp order N of the group. Thc proof of this theorem will be briefly sketched because it also provides some insight in other group properties. Let be ,g! = E,g$,. . , g h , the elements of a subgroup G" of G We denote an arbitrary element of G not contained in G' by gz. 'I'he products 92 gi, g 2 . gh,. . . , g2 . gh, of g:! with dl elements of the subgroup G' form a 5et M,