Bravais Lattice

6.730 Physics for Solid State Applications Lecture 6: Periodic Structures Tuesday February 17, 2004

Outline • Point Lattices •Crystal Structure= Lattice + Basis •Fourier Transform Review •1D Periodic Crystal Structures: Mathematics

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Point Lattices: Bravais Lattices Bravais lattices are point lattices that are classified topologically according to the symmetry properties under rotation and reflection, without regard to the absolute length of the unit vectors.

A more intuitive definition: At every point in a Bravais lattice the “world” looks the same.

1D: Only one Bravais Lattice

-2a 6.730

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-a

0

a

2a

3a

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2D Bravais Lattices hexagonal rectangular

square

oblique

centered rectangular 6.730

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3D: 14 Bravais Lattices

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Lattice and Primitive Lattice Vectors A Lattice is a regular array of points {Rl } in space which must satisfy (in three dimensions)

The vectors ai are know as the primitive lattice vectors.

A two dimensional lattice with different possible choices of primitive lattice vectors. 6.730

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Cubic

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Spring Term 2004 http://cst-www.nrl.navy.mil/lattice/struk.picts/a_h.s.png

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Body Centered Cubic

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Spring Term 2004 http://cst-www.nrl.navy.mil/lattice/struk.picts/a2.s.png

FACE CENTERED CUBIC

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Spring Term 2004 http://cst-www.nrl.navy.mil/lattice/struk.picts/a1.s.png

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Wigner-Sietz Cell 1. Choose one point as the origin and draw lines from the origin to each of the other lattice points. 2. Bisect each of the drawn lines with planes normal to the line. 3. Mark the space about the origin bounded by the first set of planes that are encountered. The bounded space is the Wigner-Sietz unit cell.

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Wigner-Sietz Cell

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Wigner-Sietz Cell

FCC 6.730

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BCC Spring Term 2004

Crystal Structure = Lattice + Basis

A two dimensional lattice with a basis of three atoms 6.730

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Cubic Structure with atomic basis

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Diamond Crystal Structure

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http://cst-www.nrl.navy.mil/lattice Spring Term 2004

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Zincblende Crystal Structure

http://cst-www.nrl.navy.mil/lattice 6.730

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Diamond Crystal Structure: Silicon

Bond angle = 109.5º

• 2 atom basis to a FCC Lattice • Tetrahedral bond arrangement • Each atom has 4 nearest neighbors and 6.730

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12 next nearest neighbors Spring Term 2004

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Perovskite and Related Structures

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YBa2Cu3O7-x High-Tc Structure

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Fourier Transform Review EE-convention for Fourier Transforms

S(f)

S(t) (1)

(1)

(1)

(1)

(1)

f

t -2T

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-T

0

T

2T

2 T

1 T

0

1 T

2 T

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Physics Convention for Fourier Transforms

In general,

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1D Periodic Crystal Structures

Sa(q)

Sa(x) (1)

(1)

(1)

(1)

(1)

x -2a

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-a

0

a

2a

q 4π a

2π a

0

2π a

4π a

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1D Periodic Crystal Structures M(x) (1) (1)

-2a

-a

0

a

2a

x

=

-2a

0

0

a

(1)

2a

x

-a

0

4π a

2π a

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2π a

a

2a

x

Ma(q)

Sa(q)

= 2π a

(1) (1)

-a

M(q)

4π a

Ma(x)

Sa(x)

0 2π a

4π a

q 4π a

2π a

0 2π a

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Reciprocal Lattice Vectors 1. The Fourier transform in q-space is also a lattice 2. This lattice is called the reciprocal lattice 3. The lattice constant is 2 π / a 4. The Reciprocal Lattice Vectors are

q K-2 6.730

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K-1

K0

K1

K2

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Periodic Function as a Fourier Series Recall that a periodic function and its transform are

Define

then the above is a Fourier Series:

and the equivalent Fourier transform is

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1D Periodic Crystal Structures with a basis M(x) (1) (1)

-2a

-a

0

a

2a

x

=

-2a

0

0

a

(1)

2a

x

d1

-a

d2 0

2π a

4π a

2π a

0 2π a

a

2a

x

Ma(q)

Sa(q)

= 2π a

(1) (1)

-a

M(q)

4π a

Ma(x)

Sa(x)

4π a

q 4π a

2π a

0 2π a

Delta functions still only occur where the reciprocal lattice is, independent of the basis. But the basis determines the weights.

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Atomic Form Factors & Geometrical Structure Factors Ma(x) d1

-a

Different Atoms (1) and (2)

d2 0

a

2a

x

Ma(x) d1

-a

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Same atoms at d1 and d2

d2

0

a

2a

x

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