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

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BookID 160928 ChapID 05 Proof# 1 - 29/07/09

160 5 Use of Elementary Group Theory in Calculating Band Structure

Summary 562

In this chapter, we ﬁrst reviewed elementary group theory and studied the 563

electronic band structure in terms of elementary concepts of the group the- 564

ory. We have shown that how group theory ideas can be used in obtaining 565

the band structure of a solid. Group representations and characters of two- 566

dimensional square lattice are discussed in depth and empty lattice bands of 567

the square lattice are illustrated. Concepts of irreducible representations and 568

compatibility relations are used in discussing the symmetry character of bands 569

connecting diﬀerent symmetry points and the removal of band degeneracies. 570

We also discussed empty lattice bands of the cubic system and sketched the 571

band calculation of common semiconductors. 572

The starting point for many band structure calculations is the empty lat- 573

tice band structure. In the empty lattice band representation, each band is 574

labeled by  =(l

) where the reciprocal lattice vectors are given by 575



=2π[l

+ l

]

where (l

)= are integers and b

are primitive translations of the 576

reciprocal lattice. Energy eigenvalues and eigenfunctions are written as 577



(k)=

¯h

(k + K

)

and 578



(k, r)=e

ik·r



·r

The Bloch wave vector k is restricted to the ﬁrst Brillouin zone.

579

The vector space formed by the degenerate bands at E(k) is invariant 580

under the operations of the group of the wave vector k. That is, the space of 581

degenerate states at a point k in the Brillouin zone provides a representation 582

of the group of the wave vector k. We can decompose this representation into 583

its irreducible components and use the decomposition to label the states. 584

When we classify the degenerate statesaccordingtotheIRsofthegroup 585

of the wave vector, we are able to simplify the secular equation by virtue of a 586

fundamental theorem on matrix elements: 587

1. The matrix elements of V between diﬀerent IRs vanish, so many oﬀ- 588

diagonal matrix elements are zero. This reduces the determinantal equation 589

to a block diagonal form. 590

2. The diagonal matrix elements Γ

n |V |Γ

n are, in general, diﬀerent for 591

diﬀerent IRs Γ

. This lifts the degeneracy at the symmetry points. 592

Many common semiconductors which crystallize in the cubic zincblende 593

structure have valence–conduction band structures that are quite similar in 594

gross features. This results from the fact that each atom has four electrons 595

outside a closed shell and there are two atoms per unit cell. 596

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BookID 160928 ChapID 06 Proof# 1 - 29/07/09

6 1

More Band Theory and the Semiclassical 2

Approximation 3

6.1 Orthogonalized Plane Waves 4

So far, we have expanded the periodic part of the Bloch function u



(k, r)in 5

a plane wave basis, i.e., 6



(k, r)=











(k)e





·r

(6.1)

It often occurs that the series for u



(k, r) converges very slowly so that many 7

diﬀerent plane waves must be included in the expansion. The reason for this is 8

that plane wave is not a very good description of the valence and conduction 9

band states in the region of real space in which the core levels are of large 10

amplitude. What are the core levels? They are the tightly bound atomic states 11

associated with closed shell conﬁgurations. States outside the core are valence 12

states that are responsible for the binding energy of the solid. For example, 13

consider Table 6.1. 14

Let us deﬁne the eigenfunction 15

|cj =Ψ

(r − R

) (6.2)

to be the core level c (c =1s, 2s, 2p, 3s,...) of the atom located at position

. The valence and conduction band states that we are interested in must 17

be orthogonal to these core states. When we expand the periodic part of the 18

Bloch function in plane waves, it takes a very large number of plane waves 19

to give band wave functions with all the necessary wiggles needed to make 20

them orthogonal to core states. For this reason, the orthogonalized plane waves 21

(OPW) was introduced by Herring and Hill.

We deﬁne 22

 = |k−





c



|k|c



. (6.3)

C. Herring, Phys. Rev. 57, 1169 (1940) and C. Herring, A.G. Hill, Phys. Rev. 58,

132 (1940).

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162 6 More Band Theory and the Semiclassical Approximation

t1.1 Tabl e 6.1. Electron conﬁgurations of core states and valence states of Na, Si, and

Cu atoms

t1.2 Atom Core states Valence states

t1.3 Na 1s

, 2s

, 2p

and higher

t1.4 Si 1s

, 2s

, 2p

, 3p

and higher

t1.5 Cu 1s

, 2s

, 2p

, 3s

, 3p

, 4s

and higher

Here, |w

 is an OPW, |k is a simple plane wave, and the sum is over 24

all core levels on all atoms in the crystal. The core levels are solutions of the 25

Schr¨odinger equation 26



−

¯h

∇

+ V

(r) −E



=0 (6.4)

where V

(r) is the atomic potential for the atom located at r

. Because the 27

core levels are tightly bound, this potential is essentially identical to the value 28

of the periodic crystalline potential in the unit cell centered at r

. 29

It is clear from (6.3) that |w

 is orthogonal to the core levels since 30

cj|w

 = cj|k−





c



|kcj|c



, (6.5)

but the core levels themselves satisfy

cj|c



 = δ



(6.6)

This gives cj|w

 = 0. In an OPW calculation, the periodic part of the Bloch 32

function is expanded in OPW’s instead of in plane waves. This improves the 33

convergence. 34

6.2 Pseudopotential Method 35

We can think of the operator P deﬁned by 36



|cjcj| (6.7)

as a projection operator. It gives the projection of any eigenfunction |φ onto 38

the core states. If we expand the wave function Ψ

in OPW’s we can write 39

|Ψ

 =



k+K

 =(1− P)



|(k + K). (6.8)

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BookID 160928 ChapID 06 Proof# 1 - 29/07/09

6.2 Pseudopotential Method 163

Let us deﬁne |φ

 =



|k + K as the pseudo-wavefunction. Clearly, 41

we have 42

|Ψ

 =(1− P) |φ

. (6.9)

We note that |φ

 is the plane wave part of the OPW expansion. Both |cj 43

and |Ψ

 are solutions of the Schr¨odinger equation 44



−

¯h

∇

+ V (r)



Ψ=EΨ, (6.10)

with eigenvalues E

and E(k), respectively. Let us substitute |Ψ =(1−P) |φ 45

into (6.10). This gives 46



−

¯h

∇

+ V (r) −E



(1 − P) |φ =0. (6.11)

Recall that

P |φ =



|cjcj|φ. (6.12)

Therefore, we have

HP |φ =



H |cjcj|φ



|cjcj|φ. (6.13)

We use this in the Schr¨odinger equation to obtain



−

¯h

∇

+ V (r) −E



|φ +



(E − E

) |cjcj|φ =0. (6.14)

We deﬁne an eﬀective potential or pseudopotential by 51

W (r)=V (r)+



(E − E

) |cjcj|. (6.15)

The ﬁrst term in the pseudopotential is just the usual periodic crystalline 53

potential. The second term is a nonlocal repulsive potential. 54

W (r)φ(r)=V (r)φ(r)+



(E − E

)Ψ

(r) cj|φ(r



)





[V (r



)δ(r − r





(E − E

)Ψ

(r)Ψ

∗



)]φ(r



(6.16)

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BookID 160928 ChapID 06 Proof# 1 - 29/07/09

164 6 More Band Theory and the Semiclassical Approximation

It is clear that W is nonlocal since only the ﬁrst term involving the periodic 56

potential contains a δ-function. The second term 57



(E − E

) |cjcj| (6.17)

is repulsive as opposed to an attractive potential like V (r). We can see this

by evaluating φ|V

|φ for any function φ. We ﬁnd that 59

φ|V

|φ =



(E − E

) |φ|cj|

. (6.18)

Because |φ|cj|

is positive and the valence–conduction band energies E are, 60

by deﬁnition, larger than core levels, 61

φ|V

|φ > 0. (6.19)

Therefore, V

cancels a portion of the attractive periodic potential V .The 62

diagram shown in Fig. 6.1 is a sketch of what the periodic potential V (r), 63

the repulsive part of the pseudopotential V

, and the full pseudopotential 64

look like. 65

A number of people have used model pseudopotentials in which the poten- 66

tial is replaced by the one shown in Fig. 6.2. The pseudopotential W (r)is 67

taken to be a local potential which has (1) a constant value V

inside a core 68

or radius d and (2) the actual potential V (r)forr>d.BothV

and d 69

are used as adjustable parameters to ﬁt the energy bands to experimental 70

observation. 71

W(r)

V(r)

Fig. 6.1. A sketch of various potentials

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6.3 k · p Method and Eﬀective Mass Theory 165

Fig. 6.2. A model pseudopotential

6.3 k · p Method and Eﬀective Mass Theory 72

Often in discussing properties of semiconductors it is more important to have 73

a single analytic description of the band structure very close to a conduction 74

band minimum or valence band maximum than to have detailed numerical 75

calculations of E

(k)andΨ

throughout the Brillouin zone. One approach 76

that has proven to be useful is called the k · p method. We know that Ψ

= 77

ik·r

(r) is a solution of the Schr¨odinger equation 78



+ V (r) − E



(r)=0. (6.20)

By substituting the Bloch wave form for Ψ

, it is easy to see that u

(r)satisﬁes 79

the Schr¨odinger equation 80



(p +¯hk)

+ V (r) −E



(r)=0. (6.21)

For k = 0 (i.e., at the Γ-point) this equation can be written as



+ V (r) −E



(r)=0. (6.22)

There are an inﬁnite number of solutions u

(1)

, u

(2)

, u

(3)

, ..., u

(n)

, ... with 82

energies at the Γ-point E

(1)

,..., E

(n)

,.... Here, the superscript (n) is a band 83

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BookID 160928 ChapID 06 Proof# 1 - 29/07/09

166 6 More Band Theory and the Semiclassical Approximation

index and the subscript 0 stands for k = 0. For any ﬁxed value of k the set 84

of functions u

(n)

(r) form a complete orthonormal set in which any function 85

with the periodicity of the lattice can be expanded. Therefore, we can use the 86

set of function u

(n)

(r) as a basis set for a perturbation expansion of u

(m)

for 87

k = 0 and for any band m. By this, we mean that we can write 88

(m)



(m)

(k)u

(n)

. (6.23)

The Schr¨odinger equation for u

(m)

can be written as 89



¯h

k ·p +

¯h

+ V (r) −E





(m)

(k)u

(n)

(r)=0. (6.24)

We omit the band superscript (m) for simplicity. We know that



+ V (r)



(n)

= E

(n)

; therefore, we can write 91



(n)

¯h

− E





(n)

(r)+

¯h

k ·p



(n)

(r)=0. (6.25)

Take the scalar product with u

(m)

remembering that

(m)

(n)

= δ

.This 92

gives 93



(m)

+ ε

− E





(m)



¯h

k · p



(n)

=0. (6.26)

This is just a matrix equation of the form 95

⎛

⎜

⎝

(1)

+ ε

− E

(1)

(2)

···

(2)

(1)

(2)

+ ε

− E

···

(3)

(1)

(3)

(2)

···

⎞

⎟

⎠

⎛

⎜

⎝

⎞

⎟

⎠

= 0 (6.27)

Here, H

¯h

k · p,wherep = −i¯h∇, ε

¯h

, and we have put 96

(n)

|p|u

(n)

= 0. This last result holds forcrystals with a center of sym-

metry because parity is a good quantum number and p is an operator that 98

changes parity. If this matrix element does not vanish it must be added 99

to ε

. 100

If we consider k to be small (compared to

), then if

(m)



¯h

k · p



(n)

101

does not vanish, it is usuallyquite small compared to



(n)

− E

(m)



.When

102

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6.3 k · p Method and Eﬀective Mass Theory 167

the oﬀ-diagonal elements are treated by perturbation theory, the resulting 103

expression for E

(n)

is written as 104

(n)

= E

(n)

¯h





k ·

(n)

|p|u

(l)



(n)

− E

(l)

. (6.28)

This can be rewritten as

105

(n)

= E

(n)

¯h

k · m

∗

−1

· k, (6.29)

106

where the inverse eﬀective mass tensor (for the band n)isgivenby 107

∗

−1

= m

−1



(n)

(l)

(n)

− E

(l)

. (6.30)

108

Because u

(n)

is a periodic function with period a, the lattice spacing, the 109

matrix element

(n)

(l)

is of the order of

¯h

if it does not vanish by 110

symmetry considerations. Thus for two coupled bands separated by an energy 111

gap ΔE 112

∗

 1+2

¯h

/ma

ΔE

(6.31)

Since a  3 × 10

−8

cm,

¯h

 10 eV, but typical gaps in semiconductors can 113

be as small as 10

−1

eV. Thus, in small-gap semiconductors it is very possible 114

to have 115

∗

¯h

/ma

ΔE

 10

−2

Eﬀective masses of 0.1 −−0.01 m are not at all unusual in semiconductors.

116

The simple perturbation theory breaks down when there are a number 117

of almost degenerate bands at the point in k-space about which the k · p 118

expansion is being made. In that case, it is necessary to keep all the nearly 119

degenerate states in the matrix Schr¨odinger equation and refrain from using 120

second-order (nondegenerate) perturbation theory. One example of this is the 121

Kane Model

used frequently in zincblende semiconductors (like InSb, InAs, 122

GaSb, GaAs, etc.). In these materials, there are four bands that are rather 123

close together (see Fig. 6.3). If spin–orbit coupling is included (it can be impor- 124

tant in heavy atoms) one must add to the periodic potential V (r)theatomic 125

spin–orbit coupling 126

¯h

(σ ×∇

V ) · p. (6.32)

E.O.Kane,J.Phys.Chem.Solids1, 82 (1956); ibid. 1, 249 (1957); Semiconductors

and Semimetals, Vol. 1, pp.75-100 (Academic, New York, 1966).

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BookID 160928 ChapID 06 Proof# 1 - 29/07/09

168 6 More Band Theory and the Semiclassical Approximation

Fig. 6.3. Schematics of the band structure of zincblende semiconductors near the

Γ point. Nonparabolicity of the valence band is neglected here

Then the four bands become eight (including the spin splitting) and ¯hk ·p is 127

replaced by Π · p,whereΠ =¯hk +

¯h

σ ×∇

V .The8× 8matrixmust 128

be diagonalized to obtain a good description of the conduction–valence band 129

structure near the Γ point. 130

6.4 Semiclassical Approximation for Bloch Electrons 131

When we discussed the Sommerfeld model of free electrons, we discussed the 132

motion of electrons in response to electric ﬁelds and temperature gradients 133

which introduced r-dependence into the equilibrium distribution function 134

(ε)=

exp

[ε−ζ(r)]/k

T (r)

. (6.33)

The eigenfunctions of the Sommerfeld model were plane waves, so the prob-

135

ability that an electron was at a given position r was independent of r. 136

Therefore, the r-dependence in f

(ε) only made sense if we introduced the 137

idea of localized wave packets deﬁned by 138

(r,t)=





g(k − k



i(k



·r−ω



, (6.34)

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BookID 160928 ChapID 06 Proof# 1 - 29/07/09

6.4 Semiclassical Approximation for Bloch Electrons 169

where g(k−k



)  0if|k − k



| is larger than some value Δk.BytheHeisenberg 139

principle 140

ΔkΔx

∼

> 1, (6.35)

so that the electron can be localized in a region Δx of the order of (Δk)

−1

. 141

We must have 142

1. Δx  a, the atomic spacing. 143

2. Δx  L the distance over which the potential φ(x)=eEx or the 144

temperature T (x) changes appreciably. 145

Thus, the semiclassical wave packet picture can be applied only to slowly 146

varying (in space) perturbations on the free electrons. 147

In the presence of a periodic potential, we have Bloch states (or Bloch 148

electrons) described by 149

(r)=e



·r

(r) (6.36)

and

150

E = ε

(k) (6.37)

Here, k is restricted to the ﬁrst Brillouin zone, and there is a gap between

151

diﬀerent energy bands at the same value of k, i.e., ε

(k) − ε



(k)=E

GAP

152

(k) =0. 153

The semiclassical wave packet picture can be used to describe the motion 154

of Bloch electrons in a given band in response to slowly varying perturbations 155

by taking 156

(r)=





(k − k



)Ψ



(r) (6.38)

with g

(k − k



)  0if|k − k



| > Δk. Then, the standard expression for the 157

group velocity of a wave packet gives 158

(k)=

¯h

∇

(k) (6.39)

as the velocity of a Bloch electron of wave vector k in the nth band. In the

159

presence of a force F, the work done in moving an electron wavepacket a 160

distance δx is written by 161

δW = F · δx = F · v

δt (6.40)

But this must equal the change in energy

162

δW = E

(k + δk) − E

(k)

= ∇

(k) · δk =¯hv

kδt (6.41)

Equating (6.40) and (6.41) gives

163

k =¯h

−1

F. (6.42)

164

165