School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

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6.4 ELECTRONIC BAND STRUC-TURES

Dongwoo, Shin

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

2

Contents

6.4.1. Reciprocal Lattices and the First Brillouin Zone

6.4.2. Bloch’s Theorem

6.4.3. Band Structures of Metals and Semiconduc-tors

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6.4.1. Reciprocal Lattices and the First Brillouin Zone

• Reciprocal Lattice – Crystal is a periodic array of lattices

Performing a spatial Fourier transform

– Reciprocal Lattice • Expression of crystal lattice in fourier space

t

Reciprocal lat-tice

FT for time

FT for space

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Reciprocal Lattice

• Primitive Vector for a simple orthorhombic lattice

• Reciprocal primitive vectors

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Reciprocal Lattice

*a

b-c plane

Next lattice plane

d*aa

* b ca

a b c

* 1| |a

d

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First Brillouin Zone

• The smallest of a Wigner-Seitz cell in the reciprocal lattice

The reciprocal lat-tices (dots) and cor-responding first Bril-louin zones of (a) square lattice and (b) hexagonal lattice.

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The first Brillouin zone of a FCC structure

Face-centered cubic

K Middle of an edge joining two hexagonal faces

L Center of a hexagonal face

UMiddle of an edge joining a hexagonal and a square face

W Corner point

X Center of a square face

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The first Brillouin zone of a FCC structure

-Primitive basis vectors of the face-centered cubic lat-tice

- Primitive vectors :

b

c

a

ˆ ˆ( )2

ˆ ˆ( )2

ˆ ˆ( )2

aa y z

ab x z

ac x y

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The first Brillouin zone of a FCC structure

- Reciprocal primitive vec-tors :

General reciprocal lattice vector:

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The first Brillouin zone of a FCC structure

2ˆ ˆ ˆ( )( )x y z

a

2 2 2ˆ ˆ ˆ( )( 2 ) ; ( )( 2 ) ; ( )( 2 )x y z

a a a

• 1st Brillouin zone : the short-est

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6.4.2 Bloch’s Theorem

• Hamiltonian Operator (for the one-electron model)From (3.68),

“The one-electron Schrödinger equa-tion”

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Bloch’s Theorem

- The periodicity of the lattice structure :

( ) ( )

U r U r R

- can be expanded as a fourier series :( )

U r

𝑈 ( �⃗� )=∑⃗𝐺

𝑈�⃗�𝑒𝑖 �⃗�∙ �⃗� h𝑊 𝑒𝑟𝑒𝐺𝑖𝑠𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑟𝑒𝑐𝑖𝑝𝑟𝑜𝑐𝑎𝑙𝑙𝑎𝑡𝑡𝑖𝑐𝑒𝑣𝑒𝑐𝑡𝑜𝑟

- The solution of the Schrödinger equation for a peri-odic potential must be a special form :

Where is a periodic function with the periodicity of the lattices

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Central Equation

• The wavefunction can be expressed as a Fourier se-ries:

From the one-electron Schrödinger equation, the coeffi-cients of each Fourier component must be equal on both sides of the equation.

2 2

02

k G k G

Ge

kE C U C

m

: Central equa-

tion

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Central Equation for 1-D

• From the first Brillouin zone 22

k G G 2 2 2 2 21 1 1( ) ; ( ) ( ) ( )2 2 2

k G k G G G G

/ 2 /k G a

0

0

( ) 0

( ) 0

E E C UC

E E C UC

2 2

01,

2 2

e

kG E

m where

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Central Equation for 1-D

0

0

( )0

( )

E E U

U E E

20 ( / )

2 e

aE E U U

m

- Near the zone boundary

0

0

( ) 0

( ) 0

k k k G

k G k G k

E E C UC

E E C UC

- Nontrivial solutions for the two coefficient

1/ 2

0 0 0 0 2 21 1( ) ( ) ( )

2 4k k G k k GE k E E E E U

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Standing Waves

- Wave function at the zone edge

( / ) ( / )1( )

2i x a i x ae e

L

( ) 2 / cos( / )

( ) 2 / sin( / )

x L x a

x i L x a

0E E U

Forming 2 standing waves

h𝑤 𝑒𝑟𝑒𝐿𝑖𝑠 h𝑙𝑒𝑛𝑔𝑡 𝑜𝑓 h𝑡 𝑒𝑐𝑟𝑦𝑠𝑡𝑎𝑙

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Electron band structure

(a) The extended-zone scheme (b)The reduced –zone scheme

- Representation of the electronic band struc-ture

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6.4.3 Band Structures of Metals and Semiconductors

• Band Structures of Metals– Copper outermost configuration :– Electron in the s band can be easily excited from

below the to above the “CONDUCTOR”– Interband transition The absorption of photons will cause the electrons in thes band to reach a higher level within the same band.

1 104 3s d

- Calculated energy band structure of copper

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Band structure of semiconductor

Calculated energy band structure of Sili-con

Calculated energy band structure of GaAs

- Interband transitions : The excitation or relaxation of electrons between subbands- Indirect gap : The bottom of the conduction band and the top of the va-lence band do not occur at the same k

- Direct gap : The bottom of the conduction band and the top of the va-lence band occur at the same k

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Band structure of semiconductor

2 2 2 2

* *( ) ( )2 2e C h V

e h

k kE k E and E k E

m m

- Energy versus wavevector relations for the car-riers

,g

d Ev

dk

- Effective mass1

g

dEv

dk