Energy bands for electrons in crystals

1

Energy bands for electrons in crystals

2

Outline

Recall the solution for the free electron gas in a box of size

LLL

• States classified by k with E(k) = ħ2k2/2m

• Periodic boundary conditions lead to k= integer (2 / L), etc.

• Pauli Exclusion Principle, Fermi Statistics

• Electrons in crystals

• First step - NEARLY free electrons in a crystal

• Simple picture of degenerate perturbation lead to energy gaps

• Band structures in solids

3

Schrödinger Equation

• How can we solve the Schrödinger Eq.

where V(r) has the periodicity of the crystal?

2

2( ) ( ) ( )

2V r r E r

m

− + =

• We will consider simple cases as an introduction

♦ Nearly Free Electrons

4

Consider one dimensional example

• If the electrons can move freely on a line from 0 to L (with no

potential),

Of which we have seen before:

• Schrödinger Eq. in 1-d with V = 0

• Solution with f (x) = 0 at x = 0, L

0 L

2 2

2( ) ( )

2

dx E x

m dx − =

2 2

*

0

exp( ) 2( ) , , 0,1,2, 3, ...

( ) ( ) 1, ( )2

L

ik xx k m m

LL

kx x dx E k

m

= = =

= =

Boundary conditions

5

Electrons on a line

• For electrons on a line, the energy is just the kinetic energy

• Values of k fixed by the line, k = m (2/L), m = 0,1,...

2 2( ) / 2E k k m=

E

kkF0-kF

Filled

states

Empty

states

• The lowest energy state is for electrons to fill the lowest statesup to the Fermi energy and Fermi momentum kF with twoelectrons in each state.• This is a metal– the electrons can conduct electricity asdiscussed before.

6

Why some materials insulators or semiconductors?

• To answer this question we need consider the effect of

crystal on electrons.

• In a crystal, the nuclei are arranged in a periodic crystalline

array.

• This leads to the modification of the energies of the

electrons and the different behaviors in different crystals.

7

Electrons in Crystals

• Simplest extension of the Electron Gas model

Very weak potentials

with crystal periodicity

• Nearly Free electron Gas –

– Very small potential variation with the periodicity of the crystal

• We will first consider electrons in one dimension.

8

Electrons in a periodic potential

2

02

pH

m=

for free electrons

2 2

02

kE

m=

a weak periodic potential

0( )H H V r= + V( )=V( )r r R+

31' exp[ ( ') ] (r)k V k i k k r V d r

V= −

This is zero unless the difference is a reciprocal lattice vector.

9


0 0 0( ) +E k E k V k E V= = +

1st order perturbation theory:

V0 is assumed to be zero for simplicity.

only a constant energy-

shift

2nd order perturbation theory:

2

0

' 0 0

'( ) +

( ) (k')k k G

k V kE k E

E k E= +

=−

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Degenerate perturbation theory

A divergence in perturbation theory

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Degenerate Perturbation Theory

0( )k H k E k=

0( )k G H k G E k G+ + = +

*

Gk H k G V+ =

Gk G H k V+ =

k k G = + +

*

0

0

( )

( )

G

G

E k VE

V E k G

= +

2

0 0( ( ) )( ( ) ) 0

GE k E E k G E V− + − − =

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Degenerate Perturbation Theory

( ) ( )E k E k G= +

22

0( ( ) ) 0

GE k E V− − =

At the zone boundary

0( ) ( )

GE k E k V

=

There is a gap opening up at

the zone boundary.

=

( )1

2k k G

= +

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In one dimension

0

2( ) cos( )

xV x V

a

=

( )1

2k k G

= +

exp( )ikxk

L=

exp( )ikxk G

L

−+ =

( ) ( )

( ) ( )

/ 2cos /

/ 2 sin /

i x a i x a

i x a i x a

e e a x a a

e e a i x a a

−

+

−

−

= + =

= − =

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In one dimension

15

Nearby the zone boundary

2

2 2

0

2

2 2

0

2( + ) [( ) ]

2

2( + ) [( ) ]

2

n n nE

a m a a

n n nE

a m a a

+

−

= + +

− = − +

*

0

0

G

G

E VE

V E

+

−

=

2

0 0( )( ) 0

GE E E E V

+ −− − − =

2 222 2 2 22

{ [( ) ] )} ( )2 2

G

n nE V

m a m a

+ − = +

Not on the zone boundary

2 222 2 22

[( ) ] ( )2 2

G

n nE V

m a m a

= + +

δ is small

2 2 2 2

2 2 1( ) [1 ( ) ]

2 2G

G

n nE V

m a m m a V

= +

>1 for weak potentialThe dispersion is quadratic in δ.

2 2 2

2

*( )

2 2G

nE V

m a m

= +*

22 1

1 ( )G

mm

n

m a V

=

17

Nearly free electrons on a line

Energy gap: energies

at which there are no

waves that can travel

through crystal

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Nearly free electrons on a line

plotted in repeated zone scheme

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Why materials insulating (or semiconducting)?

• If there are just the right number of electrons to fill the lower

band and leave the upper band empty, the Fermi level lies in

the gap.

• Electrons are not free to move in the gap!

• Only if one adds an energy as large as the gap energy can an

electron be raised to the upper band where it can move.

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Summary: Electrons in Crystals

• Real Crystal –

Potential variation in accord

with the periodicity of the crystal

Attractive (negative) potential

around each nucleus

• Periodic potential leads to:

– Electron bands – E(k) different from free

electron bands

– Band Gaps

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Outline

• Electrons in crystals


Bloch theorem

• Quantitative calculations for nearly free electrons

Energy bands and standing waves at the Brillouin

zone boundary

Energy gaps

• Tight Binding Model

• Metals vs. insulators – simple arguments

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Bloch theorem (1928)

( ) (R) ( )R

T r C r =

( ) ( ) ( )R

T H r H r R r R = + +

( ) ( )V r V r R= + ( ) ( )H r H r R= +

[ , ] 0R

T H =There are simultaneous eigen-functions of H

and all translation operator.

( ) ( )R

T r r R = +

Translation operator

= ( ) ( )= ( )R

H r r R HT r +

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( ) ( ) ( )R

T r r R r = + =

a ik=

ln (R)=a RC

( ) ( ) exp[ ] ( )R

r R T r ik R r + = =

1 2 1 2R R R RT T T

+=

1 2 1 2C(R ) (R ) (R R )C C= +

1 2 1 2ln (R )+ln (R ) ln (R R )C C C= +

lnC is linear in R.

exp[ ] ( )R

PT i R r=


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• Bloch theorem

( ) ( ) exp( )k k

r u r ik r =

• Key points:

-- Each state is labeled by a wave vector k

-- k can be restricted to the first Brillouin zone:

where is just another

periodic function

( ) ( ) exp[ ( ) ]

'( ) exp( )

k G k G

k

r u r i k G r

u r i k r

+ + = +

=

'( ) ( ) exp( )k k G

u r u r i G r+

=

uk(r) is a periodic function


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• Thus a wave function in a crystal can always be written as

where uk(r) is a periodic function and k is restricted to the first

Brillouin zone

• k becomes continuous in the limit of a large system.

( ) ( ) exp( )k k

r u r ik r =

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Bloch wave packet

Real part of wave function Condensed Matter Physics by Marder (2000)

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Potential energy as a periodic function

• We have seen in Chapter 2 that any periodic function can be

expressed as Fourier series

where G is a reciprocal lattice vector.

( ) exp( )G G

f r f iG r=

( ) exp( )G G

U r U iG r= *

G GU U

−=

G GU U

−=

• Thus

since U(r) is real,

• If the crystal is symmetric, i.e., U(-r) = U(r), then

( ) ( )f r T f r+ =

• To check: A periodic function satisfies

where 1 2 3 1 1 2 2 3 3

( , , ) : integeri

T n n n n a n a n a n= + +

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Schrödinger equation

• In a perfect crystal

2

2[ exp( )] ( ) ( )

2 G GU iG r r r

m− + =

2, 0,1,2, ...k m m

L

= =

• Note that we do NOT assume is periodic! It is a wave!

• What is k?

Similar to the problem of electrons in a box, we assume (r) is

periodic in a large box of volume L3. This leads to

( ) ( ) exp( )k

r C k ik r = • Now expand

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Schrödinger equation - continued

• The Schrödinger equation becomes

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( )[ exp( ) exp[ ( ) ]

where ( /( )exp( ), 2 )

kk G

k k

GC k i k r U i k G r

k i mC k r k

+ +

= =

( ) ( ) ( ) 0k GG

C k U C k G − + − =

• To satisfy the equal sign for any given r, the coefficient in the

above equation must be equal to zero, we then obtain the central

equation (eq. 7-27):

[( ) ( ) ( )]exp[ ] 0kk G G

C k U C k G ik r − + − =

• This can be written as, by re-labeling the sums,

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Central equation

• How do we interpret the equation?

( ) ( ) ( ) 0k GG

C k U C k G − + − =

( ) ( ) exp[ ( ) ]k G

r C k G i k G r = − −

Yet to be determined

• If UG ≠ 0 then each k is mixed with (k – G) where G is a

reciprocal lattice vector. The wave function is

2 2( ) ( ) exp( ), ( / 2 )

k kr C k ik r m k = = =

• If UG = 0 (no potential – free electrons) then each k is

independent and each wave function is

• Consider the case where the potential UG is very weak. Then we

can find a good approximate solution. This is the “nearly free

electron” approximation.

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Solving central equation

( ) ( ) ( ) 0k GG

C k U C k G − + − =The central equation

• For k near the Brillouin zone boundary, i.e., k → /a, the wave

exp(ik·r) is mixed strongly with exp[i(k-G)·r], where G is the one

(and only one) vector that leads to

│k│~ │( k – G )│

• Let U = UG =U-G for that G. Retain only equations in the central

equation that contain C(k) and C(k-G)

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Solving central equation

• Leads to two coupled equations:

( ) ( ) ( ) 0

( ) ( ) ( ) 0

( )0

( )or

k

k G

k

k G

C k U C k G

C k G U C k

U

U

−

−

− + − =

− − + =

−=

−

The central equation ( ) ( ) ( ) 0k GG

C k U C k G − + − =

2 2 1/21 1( ) [ ( ) ]

2 4

and ( ) [( ) / ] ( )

k k G k k G

k

U

C k G U C k

− −= + − +

− = −

• Solution:

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Example in 1D

• At the zone boundary: k = /a, k-G = -/a and

2 2( / 2 )( / )

k k Gm a

−= =

kU

= • Solution for k = /a:

Each root describes an energy band

2 2 1/2(1 / 2)( ) [ (1 / 4)( ) ]

k k G k k GU

− −= + − +

the energies are given by

( ) [( ) / ] ( ) ( )k

C k G U C k C k − = − = • Eigenvectors:

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Example in 1D

( ) ( )exp( )

( )[exp( ) exp( )]

( )[exp( / ) exp( / )]

kx C k ik x

C k ikx ikx iGx

C k i x a i x a

=

= −

= −

• The wave function:

• The same solution as the a weak periodic potential

• Even though the potential is extremely strong, the electrons

still behave almost as if they do not feel the atoms at all.

• They still almost form plane-wave eigenstates with the

only modification being the periodic Bloch function.

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• The energy is always (K) = (ħ2/2m)K2 for free electron bands

• We define the reduced k by

k = K – G

• Introduce the effects of potential, i.e., band gaps for periodic

potential

Nearly-free electron approximation in general

crystals

36

Brief summary

• We have solved the “central equation” in the “nearly-free

electron approximation”

• The results applied to all types of crystals but we have

assumed the potential is “weak” which is not always true

• Which conclusions will always apply in all crystals?

-- The Bloch theorem

-- Standing waves, energy bands and energy gaps at the BZ

boundary

37

Metals vs. Insulators

• How can we use the results so far to determine which crystals will

be metals? Which are insulators?

38


• How can we know when the Fermi energy will be in the band? in the gap?

• We often need to sum or integrate over k to find total quantities,

for example the total number of filled states in the bands.

39

Sums and integrals over k points

• We can use the idea of periodic boundary conditions on box of

size L L L

Exactly the same as for phonons, electrons in a box,...

• Volume per k point = (2π/L)3

• Total number of k points in BZ:

• Rules:

3 3 3

k-point cell3

2

22

BZV L L

N Na a

L

= = = =

k-point cell

32

k

N N

dkL

=

→

40


• Electrons obey the exclusion principle.

2 electrons per primitive cell of the crystal fill a band. Any

additional electrons must go into the next band, and so forth.

• An odd number of electrons per primitive cell always leads to a

partially filled band –a metal.

• An even number may lead to an insulator.

41


• An even number of electrons per cell may lead to an insulator if

the Fermi energy is in a gap everywhere in the BZ.

42

Fermi sea in 2D square lattice

Weak periodic potential Strong periodic potential

The area of the Fermi sea remains fixed as the strength of the

periodic potential is changed.

43

Equi-energy contours in 2D square lattice

From tight-binding

model

44

Fermi surface

Weak periodic

potential

Strong periodic

potential

45


No periodic potential Strong periodic potential

Metal Insulator

46

Intermediately strong periodic potential

47

Intermediately strong periodic potential

48


• A band holds two electrons per cell of the crystal

• A crystal with an even number of electrons per cell may be an

insulator!

Electrons “frozen”, i.e., electrons can’t move around the

energy gap for any excitations of electrons.

• Therefore a crystal with an odd number of electrons per cell

must be a metal!

49


Examples

• Na: 1 valence electron/atom => 1 valence electron/cell

• NaCl: 1 + 7 = 8 valence electrons/cell

• Xe: 8 valence electrons/cell

• Mg: 2 electrons/cell, bands overlap, not very good metal

• Si: 4 valence electrons/atom => 8 electrons/cell, pure

crystal an insulator at absolute zero.

• CoO: 7 + 8 = 15 valence electrons/cell, but an insulator…

50

Mott-Hubbard insulator

strong correlation, insulating,

atomic limitU t

non-interacting, metallic,

band-limitt U

U W (band width)

metal-to-insulator transition

, ,

j i i i

i j i

H t c c U n n

Hubbard model

51


Indirect band gap direct band gap

52

Direct/Indirect transitions

53

Optical Properties of Insulators

GaAs with 1.44 eV band

gap looks black since it

absorbs all frequencies of

visible light.

CdS with 2.6 eV band gap

looks reddish since it

absorbs violet and blue

light.

54

Optical Properties of Metals

Since metals are very conductive, photons excite the electrons

which then re-emit light.

Noble metals looks shiny because there is no insulating oxidized

layer.

Silver looks brighter than copper and gold because the band

width of silver is greater than that of gold or copper.

Impurity could have a strong influence on their optical properties.

55

Summary

• “central equation”: general for ALL

• We solved the problem in the “nearly-free electron

approximation” where we assume the potential is “weak”

• Some results applied to ALL types of crystals:

-- The Bloch theorem

-- Standing waves and gaps at the BZ boundary

-- Energy bands

• We can predict that most materials must be metals, and

other materials can be insulators – simply by counting

electrons!

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Energy bands for electrons in crystals