Semiconductor Cheat Sheet Ver 1

Understand and apply principles of operation and design of modern electronic devices: (i) (i) (ii) (iii (iii))

(iv) (v) (vi) (vii) (viii)

Equ Equation tions s desc descrribing ing dev deviice op operati ation; on; Appropriate de device mo models; Facto actors rs that that dete determ rmin ine e dev devic ice e per perfo form rman ance ce

Formulas Example problems or!ed out  "he steps used used in the problem problem listed in in order #eminders #eminders of things to loo! out for in doing a problem Any rules used to solve problems

Electronic Properties of Silicon Quantum Theory The Waveform of a Small Particle $avefunction % hich describes particle distribution or amplitude as a function of space and time coords& (eiθ = cos θ + i sin θ to split.)

WAVE

FUNCTION

WAVE NUMBER 

ANGULAR  FREQUENCY

k =

2 π 

 λ

←

ω = 2 π 

v ' frequency

←

SCHRODINGER  ←

EQUATION

λ ' avelength

←

ℏ= h ∕   2 π    h is plan!s constant V(x is the potential energy of the particle m is its mass

 "he *intensity+ of the ave and represents the probability that a particle exists at a certain point in space& CONSERVED

PARTICLE AKA: NORMALIZATION CONDITION , AS IN PROBABILITY TEORY!

!

∴

" ,E - . / 0 1 E " E $ A 2 E F U 3 - " 4 . 3 ←

For the in5nite square ell problem&

3ote: 0lane aves cannot be normali6ed in this manner because they have in5nite extent&

"x#ectation an$ Uncertainty 4f in the lowest energy level of well:

AVG POSITION OF



A PARTICLE

HEISENBERG

3ote that the expectation value ill alays be L/2 due to the mirror image symmetry about the center of the potential ell&

Uncertainty beteen position ( x) and momentum (#)& -an never !no the exact position and momentum of a particle at the same time&

UNCERTAINTY PRINCIPLE ENERGY – TIME

%t is the time interval required for an appreciable change to occur in the properties of the system under study

UNCERTAINTY RELATION

&tom' an$ the Perio$ic Tale "ner)y *e)eneracy

4n 7 and 8 dimensions is that several eigenfunctions have the same value of energy&  "he value of the ground state energy level E9 (98& e2) is !non as the ioniation energy  of hydrogen  the energy required to completely remove the electron from the hydrogen atom&

IONIZATION ENERGY YDRO"EN

ao 

WAVEFUNCTION HYDROGEN

BOR  RADI#S •

SPIN

s =± ℏ / 2 •

-orresponds to the most prob radius for the electron

∀ energy

eigenfunction hich represents an electron there also exist to possible spin states& (spin up or spin don)& ∴ in addition to the avefunction e must also specify the spin&

Note+  Qualitatively, one can think of the electron wave possessing a circular polarization and thus a localized “current loop” leading to an intrinsic magnetic eld. Clockwise and counter clockwise rotation can be related to the two possible values of spin.

PAULI EXCLUSION PRINCIPLE

3o more than 9 electron can be in any given state (%(xt) and spin state) at the same time&

"lectron' in Cry'tal' The T,o &##roache' to -o$elin) "lectron' in Cry'tal'

.ne approach considers ho the energy levels of  isolated atoms change hen they are brought together to form a crystal&

 "he other approach examines the quantum mechanical properties of electrons in a crystal by attempting to solve the o and thus the crystal is able to conduct electricity&  "he uppermost band is completely 5lled then there is no easy and continuous ay for charge carriers to gain energy because there is a forbidden energy band gap  0 ") before the next band of states becomes accessible& "he crystal is therefore insulating and cannot conduct electrical current&

1an$ "$)e Preliminary ?and Edge @iagram Energy band edge diagram for aemiconductor& Electronhole pairs are created hen carriers are excited from the valence band to the conduction band& ← ← ←

"v 0 2alence band edge "c 0 "he conduction band& ") 2 "he to band edges are separated by the band gap

con$uction an$ of a semiconductor $hen electrons are excited from the valence an$ •  "he electron' promoted into the conduction band can no participate in electronic tran'#ort& • 4n addition the electrons in the valence band no have some empty states available for them to also participate in current >o& "he vacant states that are left in the otherise full valence band can be treated as if they ere particles called hole' & 

3ear the top and bottom of bands they can be approximated by parabolas similar to a free particle&

EFFECTIVE MASS

HOLES

mp ' mn

ELECTRONS

mn

ELECTRON ENERGIES

At the top of the valence band At bottom of the conduction band  4n -onduction ?and 4n 2alence ?and 



4f !B ' !BC then the band gap is direct

Intrin'ic 3 "xtrin'ic (N or P Ty#e intrin'ic material the number of electrons in the conduction band is equal to the number of holes in the valence band& • Fermi level lies very close to the middle of the band gap

*onor Im#uritie'+  eective mass and the dielectric constant of the semiconductor (or relative permittivity r) of the atom &cce#tor Im#uritie'+ #eplace the electron eective mass ith the hole eective mass&

BINDING ENERGY

ENERGY LEVEL VACUUM

WORK  FUNCTION

• • •

•

qG

extrin'ic material one type of carrier has a greater concentration (maDority carriers) than the other (minority carriers)&

EB

 the energy of an electron that has been Dust freed from a material&



 "he dierence beteen the Fermi level and EB (similar to the binding energy  of an atom)&

N TYPE /aDority -arrier: Electrons /inority -arrier: ,oles doping of the crystal ith impurity atoms that $onate electron' to the con$uction an$ 0 As o&

THERMAL VELOCIT Y

←

DRIFT VELOCITY

←

τ   0 mean free time+ time interval beteen scattering events "x 0 con'tant electric /el$  in x direction

@escribes ho easily an electron can move in response to an applied electric 5eld& "he electron and hole mobilities (L n and Lp respectively) each depend on the total $o#ant concentration

MOBILITY

DRIFT CURRENT DENSITY

CONDUCTIVITY

σ =¿

R ESISTIVITY

p=1 / σ 

?y considering a bar of material ith crosssectional: ← resistance  8 ← area & ← length  4

OHMS LAW

Scatterin) •

4attice 'catterin) increases ith temperature as the vibrations of the lattice become greater& "his type of cattering causes the mobility to scale as a poer la ith temperature T00n9 ,ith n ty#ically :;< Im#urity or $efect 'catterin)  on the other hand decreases as temperature increases& "his is due to the increased thermal velocity of carriers hich ma!es them less susceptible to interaction ith impuritiesMdefects

*rift Velocity Saturation •

• •

•

At very high electric 5elds the drift velocity becomes comparable to the thermal velocity& NE O "hermal Energy  ,ot -arriers  "his is caused by increased scattering at high 5elds hich limits further increase in the drift velocity& 4n silicon  "he saturation velocity for both electrons and holes is approximately 9BPQ cmMs 

DIFFUSION CURRENT DENSITY

DIFFUSION CONSTANTS

*i?u'ion Current •

•

$hen there is a carrier concentration gradient& charge carriers >o from a region of high concentration to lo concentration resulting in a net current&

(!b)

-----------T OTAL E LECTRON AN D H OLE C URRENT D E N S I T I E S -------(INCLUDING BOTH DRIFT AND DIFFUSION)

@eneration an$ 8ecomination -

EXCESS E  

 nB and pB denote the thermal equilibrium carrier concentrations 

P

CONCENTRATIONS

RECOMBINATION NET RATE

Rn 'Rp ' R

"xam#le' Particle "ner)ie' (Quantum Well Energy of a Free 0article 2

 E=

2

ℏ k 

=ℏω =hv

2m

 p=ℏk =

h  λ



de roglie relation

Energy of a 1oun$ Particle (4n5nite Suantum $ell) 4nside the ell the potential energy is 6ero hereas at the boundaries it rapidly increases to in5nity and the particle cannot escape or exist outside the ell&