Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications

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BookID 160928 ChapID 04 Proof# 1 - 29/07/09

4 1

Elements of Band Theory 2

4.1 Energy Band Formation 3

So far, we have completely neglected the eﬀects of the ion cores on the motion 4

of the valence electrons. We consider “valence electrons” to be those outside 5

of a closed shell conﬁguration, so that 6

1. Na has a single 3s valence electron outside a [Ne] core. 7

2. Mg has two 3s electrons outside a [Ne] core. 8

3. Ga has ten 3d electrons, two 4s electrons, and one 3p electrons outside an 9

[Ar] core. 10

The s and p electrons are usually considered as the “valence” electrons, since 11

they are responsible for “bonding”. Sometimes the mixing of d-electron atomic 12

states with “valence” electron states is important. 13

To get some idea about the potential due to the ion cores let’s consider 14

thesimplecaseofanisolatedNa

ion. This ion has charge +e and attracts 15

an electron via the Coulomb potential. (See Fig. 4.1.) 16

V (r)=−

if r>ion radius. (4.1)

For a pair of Na atoms separated by a large distance, each “conduction elec-

tron” (3s-electron in Na) has a well-deﬁned atomic energy level. As the two 18

atoms are brought closer together, the atomic potentials V (r) begin to over- 19

lap. Then, each electron can feel the potential of both ions. This gives rise a 20

splitting of the degeneracy of atomic levels. 21

For a large number of atoms, the same eﬀect occur. Think of a crystal 22

structure with a nearest neighbor separation of 1 cm. The energy levels of 23

the system will be atomic in character. However, as we decrease the nearest 24

neighbor separation the atomic energy levels will begin to broaden into bands. 25

(See Fig. 4.2.) The equilibrium separation of the crystal is the position at 26

which the total energy of the system is a minimum. In all crystalline solids the 27

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110 4 Elements of Band Theory

Fig. 4.1. Coulomb potential

Fig. 4.2. Schematic illustration of energy band formation

electronic energies form bands of allowed energy values separated by energy 28

gaps (bands of forbidden energy values). These energy bands determine the 29

electrical properties of the solid. 30

4.2 Translation Operator 31

Because the crystalline potential seen by a single electron in a solid is a peri- 32

odic function of position, with the period of the lattice, it is useful to introduce 33

a translation operator T deﬁned by 34

Tf(x)=f(x + a), (4.2)

where f (x) is an arbitrary function of position and a is the period of the

lattice. It is clear that T commutes with the single particle Hamiltonian H 36

H = −

¯h

∂

∂x

+ V (x), (4.3)

because if we let x



= x + a,wecanseethat∂/∂x



= ∂/∂x and V (x



)=V (x). 37

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BookID 160928 ChapID 04 Proof# 1 - 29/07/09

4.3 Bloch’s Theorem 111

One of the most useful theorems of linear algebra for the study of quantum 38

systems states that if two operatorscommute,onecanﬁnd common eigen- 39

functions for them (i.e, they can both be diagonal in the same representation). 40

Let Ψ be an eigenfunction of H and of T 41

HΨ=EΨandT Ψ=λΨ. (4.4)

Here, E and λ are eigenvalues. Clearly applying T to Ψ N times gives

Ψ(x)=λ

Ψ(x)=Ψ(x + Na). (4.5)

If we apply periodic boundary conditions with period N ,thenΨ(x + Na)=

Ψ(x). This implies that 44

Ψ(x)=Ψ(x), (4.6)

or that λ

=1.Thus,λ itself must be the N th root of unity 45

λ =e

2π

, (4.7)

where n =0, ± 1,....Wecanwriteλ as

λ =e

ika

, (4.8)

where k =

2π

× n. Then, it is apparent that two values of k which diﬀer by 47

2π/a times an integer give identical values of λ. As usual, we choose the N 48

independent values of k to lie in the range −π/a < k ≤ π/a, the ﬁrst Brillouin 49

zone of a one dimensional crystal. 50

For more than one dimension, T

translates through a lattice vector R 51

Ψ(r)=e

ik·R

Ψ(r), (4.9)

where k =

2π

). Here n

are integers and b

, b

are primitive translations of the reciprocal lattice. We have assumed a period 54

N for periodic boundary condition with L

= Na

, L

= N a

, L

= N a

, 55

and values of n

, n

are chosen to restrict k to the ﬁrst Brillouin zone. 56

4.3 Bloch’s Theorem 57

We have just demonstrated that for a one-dimensional crystal with N-atoms 58

and periodic boundary conditions 59

T Ψ(x) ≡ Ψ(x + a)=e

ika

Ψ(x), (4.10)

where the N independent values of k are restricted to the ﬁrst Brillouin zone.

We can deﬁne a function 61

(x)=e

−ikx

Ψ(x). (4.11)

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112 4 Elements of Band Theory

It is apparent that 62

(x)=T {e

−ikx

Ψ(x)} =e

−ik(x+a)

Ψ(x + a)=e

−ikx

Ψ(x)=u

(x).

Therefore, we can write

Ψ(x)=e

ikx

(x), (4.12)

where u

(x) is a periodic function i.e. u

(x + a)=u

(x). This is known to 65

physicists as Bloch’s theorem, although it had been proven sometime earlier 66

than Bloch

and is known to mathematicians as Floquet’s theorem. 67

4.4 Calculation of Energy Bands 68

There are two very diﬀerent starting points from which one can approach 69

energy bands in solids. The ﬁrst approach is to start with atomic orbitals and 70

to form linear combinations which satisfy Bloch’s theorem. The second is to 71

start with a Sommerfeld free electron gas picture (for the electrons outside 72

a closed shell core) and to see how the periodic potential of ions changes 73

the ε

= k

¯h

/2m free electron dispersion. The ﬁrst approach works well for 74

systems of rather tightly bound electrons, while the second works well for 75

weakly bound electrons. We will spend a good deal of time on the “nearly free 76

electron” model and how group theory helps to make the calculations easier. 77

Before doing that, we begin with the ﬁrst approach called the “tight-binding 78

method” or the LCAO (linear combination of atomic orbitals). 79

4.4.1 Tight-Binding Method 80

Suppose that a free atom has a potential V

(r), so that a “conduction 81

electron,” like the 3s electron in sodium, satisﬁes the Schr¨odinger equation 82



−

¯h

∇

+ V

(r) −E



φ(r)=0. (4.13)

Here, E

is the atomic energy level of this conduction electron. When atoms 83

form a crystal, the potentials of the individual atoms overlap, as indicated 84

schematically in Fig. 4.3. 85

solid

Fig. 4.3. Tight-Binding Potential

Felix Bloch, Z. Physik 52, 555 (1928).

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4.4 Calculation of Energy Bands 113

In the tight-binding approximation, one assumes that the electron in the 86

unit cell about R

is only slightly inﬂuenced by atoms other than the one 87

located at R

. Its wave function in that cell will be close to φ(r − R

), the 88

atomic wave function, and its energy close to E

. One can make a linear 89

combination of atomic orbitals φ(r −R

) as a trial function for the electronic 90

wave function in the solid. 91

To satisfy Bloch’s theorem, we can write 92

(r)=

√



ik·R

φ(r −R

). (4.14)

Clearly, the translation operator operating on Ψ

(r)gives 94

(r)=Ψ

(r + R

)

ik·R

√



ik·(R

−R

)

φ(r −R

+ R

)=e

ik·R

(r).

The energy of a state Ψ

(r)isgivenby 95

Ψ

|H|Ψ



Ψ

|Ψ



, (4.15)

where H is the Hamiltonian for an electron in the crystal, and

Ψ

|Ψ

 =



j,m

ik·(R

−R

)



rφ

∗

(r − R

)φ(r − R

). (4.16)

If we neglect overlap between φ(r − R

)andφ(r − R

), the d

r integration 97

gives δ

j,m

and the sum over j simply gives a factor N, the number of atoms 98

in the crystal. 99

The Hamiltonian for an electron in the solid contains the potential V (r). 100

Let’s write 101

V (r)=

V (r − R

)+V

(r − R

). (4.17)

In other words,

V (r −R

) is the full potential of the solid minus the potential 102

of an atom located at R

. It’s clear from Fig. 4.3 that V

is larger than V (r) 103

in the cell containing R

so that

V (r − R

) is negative. Since 104



−

¯h

∇

+ V

(r − R

) − E



φ(r −R

)=0, (4.18)

105



ik·(R

−R

)



rφ

∗

(r −R

)



V (r −R

)



φ(r −R

(4.19)

In the ﬁrst term, E

is the constant value of the atomic energy level and it can 106

be taken out of the integration. All that remains in the integral is Ψ

|Ψ

 107

which is 1, so the ﬁrst term is just E

. We can deﬁne 108

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114 4 Elements of Band Theory

α = −



rφ

∗

(r −R

)

V (r −R

)φ(r − R

), (4.20)

and

109

γ = −



rφ

∗

(r −R

)

V (r −R

)φ(r − R

). (4.21)

In the deﬁnition of γ we assume that the only terms that are not negligible

110

are terms in which R

is a nearest neighbor of R

. Then, we have 111

= E

− α −γ





ik·(R

−R

)

(4.22)

112

where the sum is over all nearest neighbors of R

. We chose minus signs in the 113

deﬁnition of α and γ to make α and γ positive (since

V (r − R

) is negative). 114

Now look at what happens for a simple cubic lattice. There are six nearest 115

neighbors of R

located at R

±aˆx, R

±aˆy,andR

±aˆz. Substituting into 116

(4.22) gives 117

= E

− α −2γ (cos k

a +cosk

a) . (4.23)

Because γ is positive

118

Min

= E

− α −6γ, (4.24)

and

119

Max

= E

− α +6γ. (4.25)

The result is sketched in Fig. 4.4.

120

For k  π/a 121

 E

− α −6γ + γa

; k

= k

+ k

= E

Min

¯h

∗

The eﬀective mass m

∗

¯h

2γa

.Asγ decreases, the band width ΔE gets smaller 122

and the eﬀective mass near E = E

Min

increases. 123

[1,1,1]

Fig. 4.4. Tight-binding dispersion for simple cubic lattice

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4.4 Calculation of Energy Bands 115

4.4.2 Tight Binding in Second Quantization Representation 124

Suppose a system of free electrons is described by the Hamiltonian 125



†

, (4.26)

where ε

¯h

is the kinetic energy. In the presence of a periodic potential 126

V (r)=



iK·r

, we can write the potential energy of the electrons as 127





k,K

†

k+K

. (4.27)

Now, introduce the operators c

and c

†

which annihilate or create electrons 128

at site R

. 129

√



ik·R

. (4.28)

The inverse transformation is

130

√



−ik·R

. (4.29)

Substitute the latter equation into H

to obtain 131



†

ik·(R

−R

)

. (4.30)

Deﬁne

132



ik·(R

−R

)

. (4.31)

Then

133



†

. (4.32)

Now, look at H



134





k,K



†

i(k+K)·R

−ik·R



Knm





ik·(R

−R

)



iK·R

†

Since



ik·(R

−R

)

= δ

,wehave 135





K,n

iK·R

†

. (4.33)

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116 4 Elements of Band Theory

But we note that



iK·R

= V (R

), and hence H



becomes 136





V (R

†

. (4.34)

Adding H

and H



gives 137

H =





+ V (R

)



†



n=m

†



†



n=m

†

, (4.35)

138

where ε

= T

+ V (R

) represents an energy on site n and T

denotes the 139

amplitude of hopping from site m to site n. Starting with atomic levels ε

and 140

allowing hopping to neighboring sites results in energy bands, and the band 141

width depends on the hopping amplitude T

. Later we will see that starting 142

with free electrons and adding a periodic potential V (r)=



iK·r

also 143

results in energy bands. The band gaps between bands depend on the Fourier 144

components V

of the periodic potential. 145

4.5 Periodic Potential 146

Because the potential experienced by an electron is periodic with the period 147

of the lattice, it can be expanded in a Fourier series 148

V (r)=



iK·r

, (4.36)

where the sum is over all vectors K of the reciprocal lattice, and

149



rV(r)e

−iK·r

For any reciprocal lattice vector K

150

K · R =2π ×integer,

if R is any translation vector of the lattice. Thus

151

V (r + R)=



iK·(r+R)



iK·r

= V (r).

The periodic part of the Bloch function can also be expanded in Fourier series.

152

We can write 153

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4.5 Periodic Potential 117

(k, r)=



(n, k)e

iK·r

, (4.37)

For the moment, let us omit the band index n and wave number k and simply

154

write 155

(r)=e

ik·r

u(r)=



i(k+K)·r

. (4.38)

Use the Fourier expansion of V (r)andu(r) in the Schr¨odinger equation; this

156

gives 157







¯h

(k + K



)





·r





i(k+K



)·r

= E





i(k+K



)·r

. (4.39)

We multiply by e

−i(k+K)·r

and integrate recalling that

iK·r

=Ωδ(K) 158

where Ω is the volume. This gives 159



E − V

−

¯h

(k + K)





H=0

K−H

(4.40)

160

Here, we have set K



= H and have separated the H = 0 term from the other 161

terms in the potential. This is an inﬁnite set of linear equations for the coeﬃ- 162

cients C

. The non-trivial solutions are obtained by setting the determinant 163

of the matrix multiplying the column vector 164

⎛

⎜

⎝

⎞

⎟

⎠

equal to zero. The roots give the energy levels (an inﬁnite number – one for

165

each value of K) in the periodic potential of a crystal. We can express the 166

inﬁnite set of linear equations in the following matrix notation: 167

⎛

⎜

⎝

+ K

|V |K



−E

K

|V |K

··· K

|V |K

···

K

|V |K



+ K

|V |K



−E ···

K

|V |K

···

K

|V |K

K

|V |K

···

+ K

|V |K



−E ···

⎞

⎟

⎠

⎛

⎜

⎝

⎞

⎟

⎠

(4.41)

Here, ε

¯h

(k + K)

and K |V |K



 = V

K−K



,where|K =

√

iK·r

.The 168

object of energy band theory is to obtain a good approximation to V (r)and 169

to solve this inﬁnite set of equations in an approximate way. 170

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118 4 Elements of Band Theory

4.6 Free Electron Model 171

If V

=0forallK = 0, then in the notation used earlier 172

(E − V

− ε

k+K

) C

=0. (4.42)

This is exactly the Sommerfeld model of free electrons in a constant potential

173

.Wecanwrite 174

(0)

= V

+ ε

k+K

(0)

1 For the band K

0 For all other bands

(4.43)

Let us discuss this in detail by considering a simple one-dimensional case, for

175

example, as shown in Fig. 4.5. The allowed values of the Bloch wave vector k 176

are restricted to the ﬁrst Brillouin zone. Values of k outside the ﬁrst Brillouin 177

zone are obtained by adding a reciprocal lattice vector K to k. The labels C

178

refer to the non-zero coeﬃcients for that particular band; e.g. for C

we have 179

(0)

= V

¯h



k +2



2π



(0)

(r)=e

ik·r

(0)

(r).

(4.44)

180

But u

(r)=



(k)e

iK·r

;withC

(0)

(k)=1for| K |= K



2π



· n,

181

we have Ψ

(0)

(r)=e

ik·r

·e

·r

. All of this is simply a restatement of the free 182

electron model in the “reduced” zone scheme (i.e. all Bloch k−vectors are in 183

the ﬁrst Brillouin zone, but energies of higher bands are obtained by adding 184

reciprocal lattice vectors K to k; the periodic part of the Bloch function is 185

iK·r

). 186

Fig. 4.5. One dimensional free electron band