1

Chapter Seven Energy bands

What determines if the crystal will be a metal, an insulator, or a semiconductor ?

Band structures of solids

Free electron model – neglect the interactions of electrons with ions and other electrons.

Failings :⊗ The distinction between metals, semiconductors, and insulator.⊗ The positive value of Hall coefficient.⊗ Magneto-transport .⊗ …..

Band structure – take interactions into account

Independent electron picture – treat individual carrier in equations of motion, transport

2

Free conduction electrons in the box

Electron gas – electrons are completely “free of the nuclei”

Nuclei disappear – empty background

Attractive potential around each nucleus.

Real crystal – potential variation with the periodicity of the crystal

3

Nearly free electron model

-- add the periodic potential of the ion cores to free electrons

Band Electrons

Two consequences :

(1) Bloch’s theorem – solution to Schrödinger equation is of the form

( )rkiexp )r(u)r(ψ kkrrrr

•=

)auauaur(u)Tr(u )r(u 332211kkkrrrrrrr

+++=+=

periodic function due to periodic potential

Bloch wave function

plane wave function

lattice vectors in real space

Mixing free and bound charactersFree : extend through the whole crystalBound : modulated by ion core interaction

4

(2) Energy gaps – open gaps in bands of state

“forbidden energy” due to Bragg reflection of Bloch waves

A free electron

kF

aπ

kF << kBZ and λ >> aElectron wave function samples many atoms

U=constant

“Free Electrons”aπ

−

BZ boundary

aπk

3n V

N3πk

BZ

1/33/12

F

≥

≈

= 1/3v

a3

valence electron #Note :

Typical metals v>1 and hence,kF ~ kBZ.

5

kFkF <~ kBZ and λ > ~ a

aπ

aπ

−

kF

aπ

−aπ

kF > kBZ and λ ~ aDiffraction of Bloch waves

-- Bragg scatterings

“Energy gaps”

6

1D electron wave function : electron in a linear chain of lattice constant a

U

xa

atom

rkik e~ψ

rr• plane wavesk small (λ>>a)

( )( )

aπxsin2rksin2

aπxcos2rk2cos

~

e e~ψ rk-irkik

≈

=•

≈

=•

±

−

+

••

ψ

ψ

ii rr

rr

rrrr standing wavesNormalization

k = ±π/a (BZ)

Bragg reflection

7

ψfor

aπxsin

L2

ψfor aπxcos

L2

~

density y probabilitelectron ψ

2

2

2k

−

+ High density at atoms

Low density at atoms naxatom =

Energies due to potential energy U(x)

aπxsin dx U(x)

L2ψ dx U(x)U

aπxcos dx U(x)

L2ψ dx U(x)U

L

0

2L

0

2

L

0

2L

0

2

∫∫

∫∫

==

==

−−

++

probability densityψ+

ψ- x

8

a x2πdx U(x)cos

L2

aπxsin-

aπxcosdx U(x)

L2UUE

L

0

L

0

22g

∫

∫

=

=−≡ −+

k

ε

BZ zone

EgForbidden

energy zone

0

Free electrons : U=0

k

ε

0

Energy difference

Result :

Standing wave at the zone boundary.

Energy gap – energies at which no wave can travel through crystal

9

∫=a

0G cos(Gx) dx U(x)

a1U

∴ Eg = 2U2π/a = 2UG1Energy gap is equal to the Fourier component of crystal potential.

( )( )∑

∑∑

>

>

=

+==

0GG

0G

-iGxiGxG

G

iGxG

GxcosU2

eeUe UU(x)G : reciprocal lattice vector

w/. integer n

=

a2n π

expanding potential U(x) in Fourier series : U(x)=U(x+a)

Inverse

−=

a x2cosUU(x) o

πFor instance,

U(x)

x( )

oo

L

0og

UL21U

L2

a x2πcos

a x2πcosUdx

L2E

=

=

−= ∫

10

( )

( )

( )

2

U2m

/

4LU

2L

2m/

L2

aπxcos

ax2cosdx U

aπxcosdx

2m/

L2

axcos

ax2cosU

dxd

m2axcosdx

L2HψdxψE

o22

o

22

L

0

2o

L

0

222

L

0

L

0o2

22

−=

−

=

∫

−∫

=

∫ ∫

−−

== +

∗++

a

a

a

π

π

ππ

πππ

h

h

h

h

k

ε

BZ zone0

aπ

aπ

−

ψ+

ψ-

εo

εo+Uo/2

εo-Uo/2( )m2

a/ 22 πε h=o

( )

( )2

U2m

/

axsin

ax2cosU

dxd

m2axsindx

L2HψdxψE

o22

L

0

L

0o2

22

+=

∫ ∫

−−

−== −

∗−−

a

ii

π

πππ

h

h

11

Back to Schrödinger equation of a periodic potential

( )rkiexp )r(u)r(ψ kkrrrr

•=1928, Bloch’s theorem :

Bloch wave function for an electron in the potential

The central idea : A particle in a periodic potential must has this form such that the translation operators commute with Hamiltonian.

)auauaurU()Tr U()rU( 332211rrrrrrr

+++=+=Periodic potential

( )( )( ) ( )( )Tki)expr(ψ

Tkiexprki)expr(u

Trkiexp )Tr(u)Tr(ψ

k

k

kk

rrr

rrrrr

rrrrrrr

•=

••=

+•+=+

Bloch states are not momentum eigenstates i.e. P ≠ h kThe allowed states can be labelled by a wavevectors k.

Band structure calculations give E(k) which determines the dynamical behaviour.

12

Kronig-Penny ModelSquare well periodic potential by Kronig and Penney

0 a a+b-b

U(x)Uo

U(x) :

0 where 0<x<a

Uo where –b<x<0-a-b

ba

x

εψ(x)U(x)ψ(x)dxψ(x)d

2m 2

22

=+−h

2mKε w/.BeAeψ(x) a,x0

22iKxiKx h

=+=<< −

2mqUε w/.DeCeψ(x) 0,xb-

22

oqxqx h

−=+=<< −

square well U(x)

A+B = C+D

iK(A-B) = q(C-D)(1) x=0

Boundary conditions :

13

How about other boundaries ?x = a? What is ψ ( a<x<a+b ) ?

x = -b? What is ψ ( -a-b<x<-b ) ?

( ) ( ) b)ik(ae0xbbaxa +<<−Ψ=+<<ΨBy Bloch theorem( ) ( ) b)ik(aex0bxba +<<−Ψ=−<<−−Ψ a

( ) b)ik(aqbqbiKxiKx eDeCeBeAe +−− +=+

( ) ( ) b)ik(aqbqbiKxiKx eDeCeqBeAeiK +−− −=−(2) x=a

Solving (1) and (2)

kb)(kacoss(Ka)cosh(qb)con(Ka)sinh(qb)si2qK

Kq 22

+=+

−

14

The result can be simplified by a periodic delta potentialb → 0 and Uo→ ∞

Hence, bUo is finite and 2mbq

2mqεbbU

2222

ohh

→

+= remain finite !

(ka)coscos(Ka)sin(Ka)2Ka

baq2

=+

2Kq

2qKq

2qKKq 222

=→−

and sinh(qb) → qb

P : a measure of strength of the barrier

(ka)coscos(Ka)Ka

sin(Ka)P =+

( )

≈

−==

∞→→ 2o02

o2 mabUmabεU2abqP

hl

h oUbim

P→0 implying that K→k free electron

15

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-2

0

2

4

6

8

10

12

14

P=4π, 2π, 1

Ka (π)

-1 ≤ (P/Ka)sin(Ka)+cos(Ka) ≤ 1

cos(Ka)=1cos(Ka)=-1

Ka=2nπKa=(2n+1)π

no solution

cos(Ka)Ka

sin(Ka)P +

1cos(ka) =

1cos(ka) −=

16

(ka)coscos(Ka)Ka

sin(Ka)P =+

Discontinuity occurs at Ka=nπ (corresponding to ka=nπ ) where n∈Z

2

22222

2maπn

2mKε hh

==

1

4

9

16

Energy band

Energy gap

Energy gap

gap

Discontinuous !

17

In general, in a periodic potential

εψ(x)U(x)ψ(x)dxψ(x)d

2m 2

22

=+−h

∑=G

iGxG e UU(x)

∑=k

ikxkeCψ(x)Let where

=

L2nk π

∑=−k

ikxk

22

2

22

eCk2mdx

ψ(x)d2m

hh

∑∑∑ =

G k

ikxk

iGxG

G

iGxG eCeUψ(x)eU

∑∑ ∑∑ =+k

ikxk

k G

ikxk

iGxG

ikxk

2

k

2

eC ε eC eU eCk2mh

ikx

k GGkG

k G

G)xi(kkG

k

ikxk

22

e CUe CU e Cεk2m ∑ ∑∑∑∑

−=−=

− −

+h

18

0CUCε2m

kG

GkGk

22

=+

− ∑ −

hTherefore,

A set of simultaneous linear equation that connects coefficients Ck-Gfor all reciprocal lattice vectors G.

Ck-G

( )∑=G

xG-kiG-kk eC(x)ψ

ikxk

ikxiGx

GGkk (x)eueeC(x)ψ =

= −

−∑rearrange

iGx

GGkk eC(x)u −

−∑=

( ) (x)u(x)ueeCT)(xu

T)(xu(x)u

kkiGTTxiG

GGkk

kk

===+

+=−+−

−∑1periodicand satisfying

19

( )rkiexp )r(u)r(ψ kkrrrr

•=Significance of ?kr

( )Tkiexp )r(ψ)Tr(ψ kk

rrrrr•=+● under a crystal lattice translation T,

● when periodic potential vanishes, Ck-G meaningless uk(r) is a constant

and (back to a free electron)

● is the crystal momentum of an electron

( )rkiCexp)r(ψkrrr

•= 2mkε

22h=

kr

h

20

Reduced Brillouin zone schemeThe only independent values of k are those in the first Brillouin zone.

Results of tight binding calculation

Results of nearly free electron calculation

Discard for |k| > π/a

Displace into 1st BZ

Reduced Brillouin zone scheme

21

Extended, reduced, and periodic Brillouin zone schemes

Periodic zone Reduced zone Extended zone

All allowed states correspond to k-vectors in the first Brillouin Zone.

Can draw ε(k) in 3 different ways

22kx0 π/a-π/a

ε [0.5h2π2/ma2]

1

5

9

Empty simple cubic lattice, ε(k) in the reduced zonefree electron ( )2

z2y

2x

2

zyx kkk2m

)k,k,ε(k ++=h

( ) ( ) ( ) ( )( )2zz

2yy

2xx

22

2

zyx GkGkGk2m

Gk2m

)k,k,ε(k +++++=+=hh

SC, za

2πya

2πxa

2π)(G llr

++= khhk

2mk2

x2h

( )2m

/a2k 2x

2 π+h

( )2m

/a2k 2x

2 π−h)001(

)100(

)000(

)010( ( )( )2m

/a2k 22x

2 π+h )100(),010(),001(

)110( ( ) ( )( )2m

/a2/a2k 22x

2 ππ ++h )110(),011(),101(

In the 1st BZ

Along [100] direction,

23

For other directions, change kx, ky, kzFor other lattices, must use proper Gs

To get band structure of real crystals, turn on weak periodic potentialBand gap opens up at the BZ boundary

Near the zone boundary ( k=±0.5G1= ± π/a ) G1=2π/a

0CUCε2m

kG

GkGk

22

=+

− ∑ −

h

Assuming that the Fourier components UG1 is small , =U

And at 0.5G1 , k1-G= 0.5G1 and k2 -G= -0.5G1 satisfy that

h2(k-G)2/2m= h2(G1/2)2/2m

U(x)= 2U cos(G1x)Central equation

0ελU

Uελ=

−−( )

( ) 0UCCελ

0UCCελ

2G

2G

2G

2G

11

11

=+−

=+−

−

−CG1/2≠0 and C-G1/2≠0

k=G1/2,

k= -G1/2,U

2G

2mUλ ε

21

2

±

=±=

h

24

Two solutions of energy at zone boundary ±π/a :

One is lower than KE. of the free electron by UThe other is higher than KE. of the free electron by U

creating an energy gap 2U at zone boundary ±π/a

1Uελ

C

C

2G

2G

1

1

±=−

−=−

−±

= x

2Giexpx

2Giexpψ(x) 11and

Standing waves, identical to previous discussion

Near the zone boundary

( )( ) 0UCCελ

0UCCελ

kGkG-k

Gkkk

=+−=+−

−

− 0ελU

Uελ

Gk

k =−

−

−

Ck≠0 and Ck-G≠0

( ) ( ) 0Uλλ ελλ ε 2GkkGkk

2 =−++− −−where λk≡ h2k2/2m

( ) ( ) 22

GkkGkk U

4λλλλ

21ε +

−±+= −

−Two solutions :

25

Introducing a new parameter , difference bet. k and zone boundarykK −≡2G~

( ) ( ) 22

GkkGkkk U

4λλλλ

21ε +

−±+= −

−

2

2

2GK~

2GK~

2GK~

2GK~K~ U

4

λλλλ

21ε +

−

±

+=

−+

−+

2GkK~ −≡

+=

−+

+=

+

−+ 4GK~

22GK~

2GK~

4λλ

21 2

22222

2GK~

2GK~ mm

hh

( )222

4222

2

42

2GK~

2GK~

GK~42

GK~2GK~

16λλ

21

mmhh

=

−−

+=

−

−+

λ2mK~4

2G

2m2mK~4

222222 hhh=

=

1st

26

2

2222

2222

22

K~

Uλ

2mK~41Uλ

2mK~

Uλ2mK~4

4GK~

2mε

hh

hh

+±

+=

+±

+=

( )

±+±=

+±

+≈

U2λ1

2mK~Uλ

Uλ

2mK~4

211Uλ

2mK~ε

22

2

2222

K~

h

hh

( ) ( )( )xGkiexpCikxexpCψ(x) G-kk −+=

kGk

kk

k

Gk

ελU

Uελ

CC

−−=

−−=

−

−

2nd band

0 k

-1

+1

2nd band

1st band

free e

lectro

n

ε

k0 π/a=G/2

1st band

U<0

-

+

λ-U

λ+U

k

Gk

CC −

Gk

k

CC

−

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Nearly free electron (Free electron in a weak periodic potential)

Dispersion relation : ε(k)

• Each segment of ε versus k is an energy band

• Energy bands are separated by an energy gaps : 2UG

• Number of states per band (k’s are discrete due to periodic boundary)

A linear crystal contains N primitive cells of lattice constant a

Na=L and k = 0, ±2π/L , ±4π/L , ±6π/L

D(k)= L/2π

∆k(BZ)=2π/acellband N

aL

a2π

2πL∆k(BZ) D(k)N ====

one state (two electrons) per unit cell

two spins

28

Summary

Effects of periodic potential in 1D crystals

(1) Modify ψ(x) : ψ(x)=uk(x) eikx Bloch wave(2) Modify ε(k) : dispersion / Band gaps

k

ε

0

+ U(x)

π/a-π/a

ε

k0

# of k-states per band = L/a = Ncell# of electron states per band = 2L/a =2Ncell

even

29

In terms of band theory of solids,

the absence of metallic conductivity implies that no partially filled bands.

In insulator, every band is either completely filled or completely empty.

Monovalent 1/2 band filled (metal)Divalent 1 band filled (insulator ?)Trivalent 3/2 band filled (metal)

Crystal with an odd number of electrons per cell

must be metallic.

30

31

Crystal with an even number of electrons per cell may be either metallic or insulating.

MetalsOverlapping bands

Be,Mg,Ca,Sr,Ba…

Both 3S and 3P bands are partially filled.

3S

3P

Valence bandconduction

bandfilled

empty

εFε

ε

D(ε)

D(ε)

Intrinsic gap

InsulatorsNo overlap of bands

Si, Ge, …

Valence band filled

Conduction band empty

32-1.0 -0.5 0.0 0.5 1.00

2

4

6

8

10

12

14

16

(11) (11)

(11)

(01)(01)(00)

(11)

(10)

ε (8π2 h

2 /9m

a2 )

kya (4π/3)

(10)

[01]-1.0 -0.5 0.0 0.5 1.00

2

4

6

8

10

12

14

(11) (11)

(11)

(01)(01)

(00)

(11)

(10)

ε (2π2 h

2 /3m

a2 )

kxa (2π/31/2)

(10)

[10]

Two dimensional hexagonal latticeIn the reciprocal space

y2x3

2 ;y2x3

221 aa

baa

b ππππ+−=+=

rr

21 bnbmGrrr

+=A

B

C

0) ,3

2C( ),34 B(0, ),0 ,

34(A

aaaπππ

x

y

y2n)(mx3

2n)(maaππ

++−=

33

Two dimensional square latticeIn the reciprocal space

y2 ;x221 a

ba

b ππ==

rr

21 bnbmGrrr

+=

),C( ), B(0, ),0 ,(Aaaaaππππ

AB C

x

y

-1.0 -0.5 0.0 0.5 1.00123456789

101112

(11)(11)

(01) (01)

(10)

(11)

(10)

(00)

ε (π

2 h2 /2

ma2 )

kx (π/a)

(11)

Square Lattice along [10]

[10]-1.0 -0.5 0.0 0.5 1.00123456789

101112

(11)

(11)

(01)(01)(10)

(11)

(10)

(00)

ε (π

2 h2 /2

ma2 )

kxky (π/a,π/a)

(11)

Square Lattice along [11]

[11]

y2nx2maaππ

+=