Lecture contents - University at Albanysoktyabr/NNSE508/NNSE508_EM-L9-bands-Bloch.pdfNNSE 508 EM Lecture #9 Nearly free electrons: 2D bands 10 Irrelevant to dimensionality, the following

NNSE 508 EM Lecture #9

1

Lecture contents

• Bloch theorem

• k-vector

• Brillouin zone

• Almost free-electron model

• Bands

• Effective mass

• Holes


2

Translational symmetry: Bloch theorem

)()( RrVrV 332211 amamamR

)()(2

2

rErrVm

p

If V(r) is a periodic function:

One-electron Schrödinger equation (each state can accommodate up to 2 electrons):

)()( ruer k

ikr

k

The solution is :

)()( Rruru kk

where uk (r) is a periodic function:

From:

• Linearity of the Schrödinger

equation

• Fourier theorem

22)()( Rrr kk

Quasi-wavevector k is analogous to a

wavevector for free electrons (V=const)

uk might be not a single valence electron

function but is close to linear combination of

valence electron wavefunctions

Important :


3

• Introduced k-vector quantum number for

periodic potential (to enumerate states)

• Momentum is not conserved (not a quantum

number), however quasi-momentum is

conserved

• k-vector can be considered to lie in the first

Brillouin zone

• Solution with periodic boundary conditions

gives eigen-functions un,k for a given k which

forms orthogonal basis (compare with Fourier

expansion)

• n –values enumerate bands

• Electron occupying level with wavevector k in

the band n has velocity (compare to group

velocity)

Bloch theorem: consequences

)()( Rruru kk

)()()(1

2

22

ruErurVkim

kkk

)()( ruer k

ikr

k

)()(2

2

rErrVm

p

knkn Eru ,, ),(

)(1

)( kEkv nkn

22 2

2E k n

m a


4 Reciprocal space (1D)

)()( ruer k

ikr

k Wavefunction of an electron in crystal :

1D reciprocal lattice vector :

m

kE

2

22

ma

b0

2

1D free electrons “band structure” is:

k’ k’-b

'

' ' ' 2,( ) ( ) ( ) ( )ik r ikr ibr ikr

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

periodic function

First Brillouin zone:

00 ak

a

2nd band


5 Diamond or zinc-blende structures

• 4(Ga) + 4(As)=8 atoms in a cubic

unit cell

• 1+1 =2 atoms in a primitive unit cell

Brillouin zone (FCC):

Primitive unit cell (FCC):

0,1,12

1,0,12

1,1,02

03

02

01

aa

aa

aa

332211 bmbmbmb

...,,2 32

321

321

bb

aaa

aab

Primitive unit cell in a reciprical space

(1st Brillouin zone)

4

a


6

Two wells: Illustration of Bloch Theorem

E

E211121 |||| VVE

212

1

0 L

x

)()(2

100 axx

a

)()(2

100 axx

)()( ruer k

ikr

k

12

1

0

0 )(2

1

m

ikmak maxe

How?

1

0

0)(

1

0

0)( )(

2

1)(

2

1

m

maxikikx

m

xxmaik maxeemaxe

equivalentarea

kandk

periodicxuk

2

!)(

ak

,0

0 0

0 0

1( ) ( ) , k=0

2

1( ) ( ) , k=

22

x x a

x x a


7

Free electrons

)()()(2

)(ˆ2

rErrVm

rH

• Time-independent Schrödinger equation:

• Solution - plane wave

• Though free electron wave functions do not

depend on the structure of solid, they can be

written in the form of Bloch functions

• For any propagation vector k’ we can find

in the first Brillouin zone

• Then wave function (Bloch function) and

energy:

• For these wave functions we can plot the band

diagram, which become periodic with 2/a

with energy

0( )V r V const

'( ) ik rr e 2 2

0

'

2

kE V

m

'k k G

( ) ikr iGrk r e e

22

02

E V k Gm

periodic function


8

Nearly free electrons: bandgap

2

( ) ( ) ( )2

V r r E rm

• Introduce weak periodic potential

• Let’s simplify the problem: 1D potential with

just one Fourier component:

• Electrons are waves : Bragg reflection occurs at

• In quantum mechanics degenerate states

can split when perturbation is applied:

• Wave functions corresponding to split states will

be linear combinations of :

or in a Fourier series

( ) ( )V r g V r

( ) iGrG

G

V r V e

22

2E k

m

1

2( ) cos

xV r V

a

, 1, 2...k n pa

ka

ka


9

Nearly free electrons: Bandgap

• By first-order perturbation theory:

• Calculating the integral, find bandgap:

• Free electrons (plane waves) don’t interact with the

lattice much until wavevector becomes comparable with

1/a, then they are Bragg reflected and we have

interference between a plane wave and its oppositely

directed counterpart.

• These superpositions are standing waves with the

same kinetic energy, but total energy is different

1

2| cos |

xE V

a

2 211

0

2 2cos cos sin

L

g

V x x xE E E dx V

L a a a


10 Nearly free electrons: 2D bands

Irrelevant to dimensionality, the following properties are valid:

• Within the first zone lie all points of allowed reduced wave vector

• “One-zone” and “many zone” descriptions are alternatives

• All the zones has the same “volume”

• The zone boundaries are the points of energy discontinuity

E-k curves for three different

directions for parabolic band

From Cusack 1963

The first three Brillouin zones of a

simple square lattice



First Brillouin zones for various 3D structures

From Cusack 1963



Electron bands in fcc Al compared to free electron bands (dashed lines)

First Brillouin zone for fcc structure Free electron bands of fcc structure

From Hummel, 2000


13 Band structure for several fcc semiconductors

With diamond structure

From Burns, 1985

With zinc-blende structure


14 Band-structures of Si and Ge


Most essential bands in diamond/ZB

semiconductors

15

From www.ioffe.ru

GaAs


16 Free electrons and crystal electrons

Free electrons

m

kdr

m

iv

*

Kinetic energy: m

kE

2

22

Electrons in solid

*2

)( 20

2

m

kkE

Dispersion near band

extremum

(isotropic and parabolic):

Velocity or group velocity:

)()( ruer k

ikr

k Wave function: Wave function: ikr

k eV

r1

)(

*

)( 0

m

kkv

Group velocity: )(1

kEv k

Velocity at band extremum:

Dynamics (F – force): Dynamics in a band:

Fmdt

dv 1

2

2

2

2

1

*

1;

*

1

111

k

E

mF

mdt

dv

FEFvdt

dE

dt

dvkkkk

(if m* isotropic and parabolic) Force equation:

dt

dk

dt

dpF

Force equation:

dt

dkF Fv

dt

dkE

dt

kdEk

)(


17 Holes

• It is convenient to treat top of the

uppermost valence band as hole states

• Wavevector of a hole = total wavevector of

the valence band (=zero) minus

wavevector of removed electron:

• Energy of a hole. Energy of the system

increases as missing electron wavevector

increases:

• Mass of a hole. Positive! (Electron

effective mass is negative!)

• Group velocity of a hole is the same as of

the missing electron

• Charge of a hole. Positive!

eh kk 0

Hole energy:

Missing electron energy: )()( eehh kEkE

*

22

2)(

e

evee

m

kEkE

*

22

2)(

h

hvhh

m

kEkE

**eh mm

eee eh

eeekhhkh vkEkEv )(1

)(1

hh

e

edt

dk

edt

dk


18

Example: electron-hole pairs in semiconductors

Electron (e-)

Si atom

Hole (h+)

Eg

E

c

Ev

EHP generation : Minimum energy required to break

covalent bonding is Eg.


19

Charge carriers in a crystal

V

E Si atom

qEmaF

hole

qEmaF

electron

Charge carriers in a crystal

are not completely free.

Need to use effective mass

NOT REST MASS !!!

+ -

Lecture contents - University at Albanysoktyabr/NNSE508/NNSE508_EM-L9-bands-Bloch.pdfNNSE 508 EM Lecture #9 Nearly free electrons: 2D bands 10 Irrelevant to dimensionality, the following

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