Electronic Band Structures

electrons in solids: in a periodic potential due to the periodic arrays of atoms electronic band structure: electron states in the energy versus wavevector space

reciprocal lattice of a crystal structure in the k-space

periodic array of lattices in real space

spatial Fourier transformation

Reciprocal Lattices and the First Brillouin Zone

reciprocal lattice : expression of crystal lattice in Fourier space

t

reciprocal lattice

FT in time

FT in space

Distribution of electrons

Ex) covalent bonding crystal

++ + +

++ + +

( + ) = ( )n r T n r

Any periodic function can be expanded as a Fourier series.

A crystal is invariant under any translation of the form :

, , :a b c crystal

axesPhysical property of the crystal is invariant under :

T

: electron number density ( )n r

1 2 3T m a m b m c

00

( ) cos 2 / sin 2 /p pp

n x n C px a S px a

For a 1-D system, expand n(x) in Fourier series

2p/a : ensures that n(x) has a period a.reciprocal point in the reciprocal lattice or Fourier space of the crystal.

( )n x a

00

cos 2 / 2 sin 2 / 2p pp

n C px a p S px a p

00

cos 2 / sin 2 / ( )p pp

n C px a S px a n x

( ) exp( 2 / )pp

n x n i px ain a compact form

np : complex number

p pn n

The spatial Fourier transform produces the wavevectors.

2( ) expp

p

pxn x n i

a

2 p

a

These wavevectors form discrete points in k-space.

0

: reciprocal lattice vector

G

G

2 / a 4 / a 6 / a

definition of primitive vectors of reciprocal lattice

reciprocal lattice vector

In a 3-D system,

Reciprocal lattice vectors

2 , 2 , 2

b c c a a bA B C

a b c a b c a b c

: primitive vectors of the crystal lattice

, , a b c

uca b c V

2i j ija A

1 2 3G l A l B l C

( ) expG

G

n r n iG r

2

( ) exppp

pxn x n i

a

( )n r

2i j ija A

( ) expG

G

n r n iG r

( + ) = expG

G

n r T n i r T G

exp

GG

n ir G iT G

1 2 3 1 2 3, T m a m b m c G l A l B l C

1 1 2 2 3 3( + ) exp 2G

G

n r T n ir G i m l m l m l

expG

G

n ir G

exp 2 integer 1i

scattering amplitude

the difference in phase factor

Diffraction Conditions

The set of reciprocal lattice vectors determines possible x-ray reflections.

G

( ) ( xpp )x ee i kF dVn r k r dVn r i k r

k

k

k

k : scattering

vector

k k k

exp ( )

GG

F dVn i G k r

( ) exp( )G

G

n r n iG r

( )exp( )F dVn r i k r

If ,G

k G F Vn

diffraction condition : k G

If , 0k G F

1

0

2exp

a

p pp

p p xn a n dx i

a

:integer

2( ) expp

p

pn x n i x

a

1

0

2( )exp

a

p

pxn a dx n x i

a

p pIf

0

2exp exp 2 1 0

2

a p p x adx i i p p

a i p p

p pn np pIf

Inversion of Fourier series

p pSince

is an integer,

exp 2 integer 1i

( ) exp( )G

G

n r n iG r

1uc cell

( )expGn V dV n r iG r

( ) exp exp ( )GG

n r T n iG r iG T n r

reciprocal lattice axis

2 , 2 , 2

b c c a a bA B C

a b c a b c a b c

1 2 3G l A l B l C

1 2 3 1 2 3exp expiG T i l A l B l C m a m b m c

1 1 2 2 3 3exp 2 1i m l m l m l

defined as a Wigner-Seitz unit cell in the reciprocal lattice

The waves whose wavevector starts from origin, and ends at this plane satisfies diffraction condition

Brillouin construction exhibits all the wavevectors which can be Bragg reflected by the crystal

k

Wigner-Seitz unit cell

22k G G

Brillouin zone

1st Brillouin zone

2nd B. Zone

The first brilloun zone is the smallest volume entirely enclosed by planes that are the perpendicular bisection of the reciprocal lattice vectors drawn from the origin

A

B

x

y

zlattice constant : a

primitive basis vectors

Reciprocal lattice to simple cubic lattice

primitive vectors :

ˆ ˆ ˆ, , a ax b ay c az

volume : 3a b c a

c

b

a

reciprocal primitive vectors

2ˆ2

b cA x

aa b c

2ˆ2

c aB y

aa b c

2ˆ2

a bC z

aa b c

1st Brillouin zone : planes that perpendicular at the center of 6 reciprocal vectors

1 1 1ˆ ˆ ˆ, ,

2 2 2A x B y C z

a a a

32

a

volume :

primitive basis vectors

Reciprocal lattice to BCC lattice

a

b

c

primitive vectors :

1ˆ ˆ ˆ

2a a x y z

1ˆ ˆ ˆ

2b a x y z

1ˆ ˆ ˆ

2c a x y z

volume : 31

2a b c a

reciprocal primitive vectors :

2 2 2ˆ ˆ ˆ ˆˆ ˆ, , A y z B x z C x y

a a a

32

2a

volume :

general reciprocal lattice vector :

First Brillouin zone of bcc

1st Brillouin zone : the shortest G

2 2 2ˆ ˆ ˆ ˆˆ ˆ; ;y z x z x y

a a a

1 2 3G l A l B l C

2 3 1 3 1 3

2ˆ ˆ ˆl l x l l y l l z

a

primitive basis vectors of fcc

Reciprocal lattice to FCC lattice

primitive vectors :

ˆ ˆ ˆ ˆˆ ˆ, , 2 2 2

a a aa y z b x z c x y

volume : 3

4

aa b c

reciprocal primitive vectors

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ, , 2 2 2

A a x y z B a x y z C a x y z

32

4a

volume :

ab

c

Brillouin zone of fcc

1st Brillouin zone : the shortest

2ˆ ˆ ˆx y z

a

2 2 2ˆ ˆ ˆ2 ; 2 ; 2x y z

a a a

volume :

34

a

G

One electron model or nearly free electron model: Each electrons moves in the average field created by the other electrons and ions

Hamiltonian operator

Bloch’s Theorem

e

ip

x

ep i x

i

t

i

t

Recall 2 / 2( , ) i x tx t Ae e

, h E

p h

e/( , ) i Et p xx t Ae

2

e 2px i x

2

e e 2

ip p

i x

22 2e

K2e e2 2

pE

m x m

K pE E E

2 2

K p 2e

( )2

E E E U xm x

periodic potential function( ) ( , , ):U r U x y z

corresponding eigenvalue problem

H E

2 2 2e

K 2e e2 2

pE

m m x

22 2e

K2e e2 2

pE

m x m

In 1-D2 2

2e

ˆ ( )2

H U xm x

In 3-D2

2

e

ˆ ( )2

H U rm

one-electron Schrödinger equation :

periodicity of the lattice structure :

22

e

( )2

U r r E rm

( ) ( )U r U r R

can be expanded as a Fourier series :

( )U r

( ) iG r

GG

U r U e

1 2 3G l A l B l C

solution to Schrödinger equation (eigenfunctions) for a periodic potential must be a special form : Bloch theorem

( )ik r

kr e u r

: a periodic function with the periodicity of lattices

( )ku r

( ) ( )k ku r u r R

( ) ( )ik r R ik R ik r ik R

k kr R e u r R e e u r e r

Central Equation

( )ik r

kr e u r

ik r

kk

r C e

22 2ik r ik r

k kk k

r C ik e k C e

( )( ) i k G r

G kk G

U r r U C e

ik r

G k Gk G

U C e

k k k

one-electron Schrödinger equation :

22 ( )

2 e

U r r E rm

Wavefunction can be expressed as a Fourier series summed over all values of permitted wavevectors:

r

( ) iG r

GG

U r U e

The coefficient of each Fourier component must be equal :

: central equation

2 2

2ik r ik r ik r

k G k G kk k G ke

kC e U C e EC e

m

2 2

e

02 k G k G

G

kE C U C

m

When

( ) 0,U r

2 2 2 20K

e e

02 2

k kE E E

m m

free electron as used in the Sommerfeld theory

ep k

At the 1st Brillouin zone boundaries : due to symmetry

(from diffraction condition )

1st Brillouin zone

22k G G

2 2 2

22 1 1 1;

2 2 2k G k G G G G

2

Gk

a

0 0E E C UC

0 0E E C UC

where2 2

0

e

1,

2 2

kG E

m

Consider the 1-D case when the Fourier components small compare with the kinetic energy of electrons at the zone boundary : Weak-potential assumption

2 2

e

02 k G k G

G

kE C U C

m

Nontrivial solutions for the two coefficient

energy band

0

00

E E U

U E E

2 20 0

e2E E

m

20

e

/

2

aE E U U

m

near the zone boundary : due to symmetry

nontrivial solutions for the two coefficient

energy band

0 0k k k GE E C UC

0 0k G k G kE E C UC

1/ 20 0 0 0 2 21 1

( ) ( ) ( )2 4k k G k k GE k E E E E U

0

00

k

k G

E E U

U E E

standing wave

Time-independent states is represented by standing wave

( / ) cos( / ) sin( / )i x ae x a i x a

standing wavefunctions

in a 1-D weak potential at the edge of the first Brillouin zone; the lower part of the figure illustrates the actual potential of electrons

()Ux

-The upper part of the figure plots the probability density

2

L : length of crystal

0E E U

Standing Wave

( / ) ( / )1

2i x a i x ae e

L

( ) 2 / cos( / )x L x a

( ) 2 / sin( / )x i L x a

Electron band structure

reduced-zone schemeextended-zone scheme

completely free electron without band gap 2

0 2K

e

( )2

E k E km

- Calculated energy band structure of copper

1 104 3s dCopper outermost configuration :

4s 3d

Interband transitions :The absorption of photons will cause the electrons in the s band to reach a higher level within the same band

Band Structures of Metals and Semiconductors

Band structure of Metals

Electron in the s band can be easily excited from below the EF to above the EF: concuctor

Calculated energy band structure of Silicon

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 valence band do not occur at the same k- Direct gap : The bottom of the conduction band and the top of the valence band occur at the same k

Band Structure of Semiconductor

- Energy versus wavevector relations for the carriers

e : electron, h : holeEV : the energy at the top of the valencw bandEe : the energy at the bottom of the conduction band

2 2

e *e

( )2C

kE k E

m

2 2

h *h

( )2V

kE k E

m

Effective Mass

,g

dv

dk

1g

d d dv

dk dk dk

An electron in a periodic potential is accelerated relative to the lattice in an applied electric or magnetic field as if the mass of the electron were equal to an effective mass.

2

2

1 1 1 1gdv d d d d d d dk d dk

dt dt dk dk dt dk dk dt dk dt

work done on the electron by the electric field E in the time interval t

geEv t

1g

dv

dk

dk

dk

g

dv

dk

g

dk v k

dk

geEv teE

k t

dkeE

dt F

2

2

1gdv d dk

dt dk dt

2 2

2 2 2

1 1d F dF

dk dk

12

2 2

1 gdvdF

dk dt

* ,gdvm

dt

12*

2 2

1 dm

dk

and