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

130 5 Use of Elementary Group Theory in Calculating Band Structure

Order of a group: If a group G contains g elements, it is said to be of order g. 24

Abelian group: A group in which all elements commute. 25

Cyclic group: Agroupofg elements, in which the elements can be written 26

A, A

,...,A

g−1

= E.

Class When an element R of a group is multiplied by A and A

−1

to form 27



= ARA

−1

,whereA and A

−1

are elements of the group, the set of 28

elements R



obtained by using every A belonging to G is said to form a 29

class. Elements belong to the same class if they do essentially the same 30

thing when viewed from diﬀerent coordinate system. For example, for 31

4 mm there are ﬁve classes: 32

(1) E,(2)R

◦

and R

−90

◦

,(3)R

180

◦

,(4)m

and m

,(5)m

and m

−

Rearrangement theorem If G = {E,A,B,...} is the set of elements of a group, 34

AG = {AE,AA,AB,...} is simply a rearrangement of this set. Therefore 35



R∈G

f(R)=



R∈G

f(AR). 36

Generators If all the elements of a group can be expressed in form A

, 37

where m and n are integers, then A and D are called generators of the 38

group. For example, the four operators of 2 mm can all be expressed in 39

terms of R and m

such as E = R

= m

, R = R, m

= m

, m

= R

. 40

5.2.1 Some Examples of Simple Groups 41

Cyclic Group of Order 42

n Consider an n-sided regular polygon. Rotation by R

2π

× j with j = 43

0, 1, 2,...,n− 1 form a group of symmetry operations. The generator of this 44

group is R

= rotation by

2π

. 45

G = {R

,...,R

= E} (5.4)

Symmetry Operations of an Equilateral Triangle 47

G = {E,R

120

−120

} (5.5)

Here, R

120

and J

are generators of G. In this case, we have three classes of 48

{E}, {R

120

, R

−120

},and{J

, J

}. (See Fig. 5.1.) 49

Fig. 5.1. Equilateral triangle

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5.2 Review of Elementary Group Theory 131

Fig. 5.2. Rectangle

Fig. 5.3. Square

Symmetry Group of a Rectangle 50

G = {E,R,m

} (5.6)

Here, R and m

are generators of G, and each element belongs to a diﬀerent 51

class since x and y directions diﬀer. (See Fig. 5.2.) 52

Symmetry Group of a Square 53

G = {E; R

(≡ R

) ,R

−90

(≡ R

);R

180

(≡ R

);m

; m

−

} (5.7)

Here, R

and m

are generators, and classes are discussed earlier. (See 54

Fig. 5.3.) 55

Other Examples 56

Groups of matrices, e.g., (1) n × n matrices with determinant equal to unity 57

and (2) n × n orthogonal matrices. 58

5.2.2 Group Representation 59

A group of matrices that satisfy the same multiplication as the elements of the 60

group is said to form a representation of the group. To be concrete, suppose a 61

group G = {E,A,B,...} of symmetry operations operates on some function 62

Ψ(x, y, z). These operations give us the set of functions. 63

EΨ,AΨ,BΨ,.... (5.8)

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132 5 Use of Elementary Group Theory in Calculating Band Structure

which form a vector space that is invariant under the operations of the group. 64

By this, we mean that the space of all functions of the form 65

Φ=c

EΨ+c

AΨ+c

BΨ+··· (5.9)

where c

are complex numbers is invariant under the operations of the group. 66

We can choose a basis set Ψ

with j =1, 2,...,l≤ g,whereg is the order of 67

the group. Then for any φ belonging to this vector space we can write 68

φ =



j=1

. (5.10)

For any element R ∈G,

RΨ



k=1

(R)Ψ

, (5.11)

where D(R) is a matrix. The set of matrices D(R)(foreachR ∈G)forma

representation of G. 71

5.2.3 Examples of Representations of the Group 4 mm 72

Under the operations of the eight elements of the group 4 mm, x always 73

transforms into ±x or into ±y as shown in Table 5.1. 74

Representation Γ

Consider the function Ψ

= x

+ y

.Itisobviousthat 75

under every operation belonging to G,Ψ

is unchanged. For example, 76

180

= R

180

+ y

)=(−x)

+(−y)

= x

+ y

=Ψ

. (5.12)

Thus, every operation of G can be represented by the unit matrix

D(E)=D(R

180

)=D(R

−90

)=···= D(m

−

)=1. (5.13)

This set of matrices forms a representation of G that is called the “iden-

tity” representation and denoted by the symbol Γ

(i.e. the representation 79

). Any function f(x, y) that transforms under the group operation R in 80

exactly the same manner as multiplication by the matrix D(R)represent- 81

ing R in the representation Γ

is said to belong to the representation Γ

. 82

t1.1 Tabl e 5. 1. Operations of the group 4 mm on functions of x and y

t1.2 Operation ER

180

−90

−

t1.3 xx−xy−yx−xy −y

t1.4 yy−y −xx−yyx−x

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5.2 Review of Elementary Group Theory 133

Representation Γ

Consider the function Ψ

= xy.ItisobviousthatE,R

180

, 83

, and m

−

operating on Ψ

leave it unchanged but that R

, R

−90

, m

, 84

and m

operating on Ψ

change it to −Ψ

. Therefore the matrices 85

D(E)=D(R

180

)=D(m

−

)=1

D(R

)=D(R

−90

)=D(m

)=−1

form a representation of G. This representation is called the Γ

represen- 86

tation. 87

By constructing Ψ

= x

− y

,Ψ

= xy(x

− y

), and Ψ





,it 88

is easy to construct Table 5.2, which illustrates the sets of matrices D

(R) 89

for each R ∈Gof the group 4 mm. The sets of matrices {D

(R)} for each 90

R ∈Gform representations of the group 4 mm. Functions belonging to the 91

representation Γ

transform under the operations of 4 mm in exactly the same 92

way as multiplying them by the appropriate D

(R). 93

t2.1 Tabl e 5.2. Group representation table of the group 4 mm

t2.2 R

= x

+ y

(R)

= xy(x

− y

)

(R)

= x

− y

(R)

= xy

(R)





(R)

t2.3 E 1111





t2.4 R

180

1111



−10

0 −1



t2.5 R

11−1 −1



−10



t2.6 R

−90

11−1 −1



0 −1



t2.7 m

1 −11−1



0 −1



t2.8 m

1 −11−1



−10



t2.9 m

1 −1 −11





t2.10 m

−

1 −1 −11



0 −1

−10



t2.11 ⇑⇑⇑⇑⇑

t2.12 Γ

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134 5 Use of Elementary Group Theory in Calculating Band Structure

5.2.4 Faithful Representation 94

You will notice that the set of matrices forming the representation Γ

of 4 mm 95

to which the function x

belonged were all identical, i.e. D(E)=D(R

)= 96

D(R

)=···= D(m

) = 1. Such a representation is a homomorphism between 97

the group of symmetry operators and the group of matrices, and it is said to be 98

an unfaithful representation. A representation in which each operation R ∈G 99

is represented by a diﬀerent matrix D(R) is called a faithful representation. 100

In this case, the group of symmetry operations and the group of matrices are 101

isomorphic. 102

5.2.5 Regular Representation 103

If we construct a multiplication table for a ﬁnite group G as shown in Table 5.3 104

for 2 mm and we form 4 × 4 matrices D(E), D(R), D(m

), D(m

) by sub- 105

stituting 1 in the 4 × 4 array wherever the particular operation appears and 106

0 everywhere else, the set of matrices form a representation known as the 107

regular representation.Thus,wehave 108

109

D(E)=

⎛

⎜

⎝

1000

0100

0010

0001

⎞

⎟

⎠

,D(R)=

⎛

⎜

⎝

0100

1000

0001

0010

⎞

⎟

⎠

, etc. (5.14)

110

5.2.6 Reducible and Irreducible Representations 111

Suppose D

(1)

(R)andD

(2)

(R) are two representations of the same group, then 112

D(R) deﬁned by 113

D(R)=



(1)

(R)0

0 D

(2)

(R)



(5.15)

also forms a representation of G. D(R) is called the direct sum of the ﬁrst two

114

representations D

(1)

(R)andD

(2)

(R). A representation which can be written 115

as the direct sum of two smaller representations is said to be reducible. Some- 116

times a representation is reducible, but it is not at all apparent. The reason 117

for this is that if the matrices D(R) form a representation of G,then 118

t3.1 Tabl e 5. 3. Multiplication table of the group 2 mm

t3.2 E

−1

= ER

−1

= Rm

−1

= m

−1

= m

t3.3 EE R m

t3.4 RR E m

t3.5 m

t3.6 m

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

5.2 Review of Elementary Group Theory 135

D(R)=S

−1

D(R)S (5.16)

also form a representation (corresponding to a change in the basis vectors of

119

the vector space on which the matrices act). This similarity transformation 120

can scramble the block diagonal form so that the resulting

D(R) do not look 121

reducible. A representation is reducible if and only if it is possible to perform 122

the same similarity transformation on all the matrices in the representation 123

and reduce them to block diagonal form. Otherwise, the representation is 124

irreducible. Clearly all one-dimensional representations (1 × 1 matrices) are 125

irreducible. 126

5.2.7 Important Theorems of Representation Theory 127

(without proof ) 128

1. Theorem one The number of irreducible representations (IRs) is equal to 129

the number of conjugate classes. 130

2. Theorem two If l

is the dimension of the ith IR and g is the number of 131

operations in the group G 132



= g. (5.17)

133

Example 134

1. 2 mm. There are four operations E, R, m

, m

and there are four classes 135

(remember that because the x and y directions are distinct m

and m

136

belong to diﬀerent classes). From Theorem one, there are four IRs; from 137

Theorem two, they are all one dimensional so that 138



=4=g.

2. 4 mm. There are eight operations falling into ﬁve conjugate classes: E; R

; 139

and R

; m

and m

; m

and m

−

. Therefore, there are ﬁve IRs and four 140

IR’s must be one dimensional and one must be two dimensional so that 141



=8=g

5.2.8 Character of a Representation

142

The character χ of a representation D(R) is deﬁned as 143

χ(R)=



(R) for each R ∈G. (5.18)

144

Because the application of a similarity transformation does not change the 145

trace of a matrix 146

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136 5 Use of Elementary Group Theory in Calculating Band Structure

1. χ(R) is independent of the basis used for the vector space. 147

2. χ(R) is the same for all elements R belonging to the same conjugate class. 148

Thus, we can deﬁne χ(C) to be the common value of χ(R) for all R belonging 149

to conjugate class C.

150

5.2.9 Orthogonality Theorem 151

In trying to determine the IRs and their characters certain orthogonality 152

theorems are very useful. We state them without proof. 153

154



R∈G

(R)χ

∗

(R)=gδij, (5.19)

155

where χ

and χ

are the characters of two representations. This can also 156

be written 157



(C)χ

∗

(C)=gδ

, (5.20)

where n

is the number of elements in the class C. 158

159



(C)χ

∗



C,C



. (5.21)

160

3. If D

(i)

μν

(R)istheμν matrix element of the ith IR for the operation R,then 161

162



R∈G

(i)

μν

(R)



(j)

(R)



−1



μμ



νν



. (5.22)

163

For a unitary representation



(j)

(R)



−1



= D

(j)

∗



(R) so that for unitary 164

representation 165



R∈G

(i)

μν

(R)D

(j)

∗

(R)



μμ



νν



. (5.23)

Some Examples

166

1. 2 mm. We know there are four IRs all of which are one dimensional. We 167

label them Γ

,Γ

.The1× 1 matrices representing each element 168

are given in Table 5.4. 169

2. 4 mm. There are ﬁve IRs; one is two dimensional and the rest are one 170

dimensional as are shown in Table 5.5. 171

The reader should use these simple examples to demonstrate that the orthog- 172

onality theorems hold. 173

In the regular representation, each IR Γ

appears l

times, where l

is the

dimension of the IR Γ

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5.3 Empty Lattice Bands, Degeneracies and IRs at High-Symmetry Points 137

t4.1 Tabl e 5. 4. Irreducible representations of the group 2 mm

t4.2 Γ

t4.3 E 1 1 1 1

t4.4 R 1 1 −1 −1

t4.5 m

1 −11−1

t4.6 m

1 −1 −11

t5.1 Tabl e 5. 5. Irreducible representations of the group 4 mm

t5.2 Γ

t5.3 E 11 1 1





⇒ 2

t5.4 R

11 1 1



−10

0 −1



⇒−2

t5.5 R

, R

11− 1 −1



−10





0 −1



⇒ 0

t5.6 m

, m

1 −11−1



0 −1





−10



⇒ 0

t5.7 m

, m

−

1 −1 −11







0 −1

−10



⇒ 0

5.3 Empty Lattice Bands, Degeneracies 174

and IRs at High-Symmetry Points 175

In Sect. 5.2 we determined the free electron energies and wave functions in the 176

reduced zone scheme for a two-dimensional rectangular lattice. The starting 177

point for many band structure calculations is this empty lattice band structure 178

obtained by writing, as (5.3), 179

(k)=

¯h

(k + K

)

where k is restricted to lie in the ﬁrst Brillouin zone and K

is a reciprocal 180

lattice vector of (5.1) 181

=2π[l

+ l

Here, b

are the primitive translation vectors of the reciprocal lattice and 182

=0, ±1, ±2,.... We evaluate the energy at particular symmetry points, e.g. 183

at k = 0, the Γ point, along k

= k

= 0, the line Δ, etc. 184

The group of symmetry operations which leave the lattice invariant also 185

leaves the reciprocal lattice invariant. Suppose we know some wave function 186

. A rotation or reﬂection operation of the point group acting on Ψ

will 187

give the same result as the rotation or reﬂection of k,thatis 188

RΨ

(x)=Ψ



−1



. (5.24)

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138 5 Use of Elementary Group Theory in Calculating Band Structure

Fig. 5.4. STAR of square lattice

Here, we used the fact that applying the same orthogonal transformation to 189

both vectors in a scalar product does not change the value of the product, for 190

example k ·R

−1

r =Rk ·RR

−1

r =Rk ·r. By applying every R ∈Gtoawave 191

vector k, we generate the STAR of k. For a two-dimensional square lattice, all 192

operations leave Γ invariant so Γ is its own STAR. (See Fig. 5.4) For a general 193

point k, there will be g (=8 for 4 mm) points in the STAR of k. The symmetry 194

point X has four points in its STAR; two of these lie along the x-axis and 195

are equivalent because they are separated by a reciprocal lattice vector. The 196

other two points in the STAR of X are not equivalent to the X-point along the 197

x-axis because they are not separated from it by a reciprocal lattice vector. 198

All four points in the STAR of M are equivalent since they are all separated 199

by vectors of the reciprocal lattice. 200

5.3.1 Group of the Wave Vector k 201

The group (or subgroup of the original point group) of rotations and reﬂections 202

that transform k into itself or into a new k vector separated from the original 203

k point by a reciprocal lattice vector belong to the group of the wave vector k. 204

Example 205

For two-dimensional square lattice (Fig. 5.5) we remember that Δ, Z,andΣ 206

denote, respectively, any point on the line from Γ → X, X → M,andΓ→ M . 207

We have the groups of the wave vectors as follows: 208

= G

= {E,R

−

}

= {E,R

}

= {E,m

} (5.25)

= {E,m

}

= {E,m

}

209

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5.3 Empty Lattice Bands, Degeneracies and IRs at High-Symmetry Points 139

Fig. 5.5. First Brillouin zone of a square lattice

When the empty lattice bands are calculated, there are often a number of 210

degenerate bands at points of high symmetry like the Γ-point, the M -point, 211

and the X-point. Because the operations of the group of the wave vector Γ 212

(or M or X) leave this point invariant, one can construct linear combinations 213

of these degenerate states that belong to representations of the group of the 214

wave vector Γ (or M or X,etc.). 215

Example 216

Empty lattice bands for a two-dimensional square lattice are written by 217

(k)=

¯h

(k + K

)

and Ψ

(k, r)=e

i(k+K

)·r

. (5.26)

Here

218

2π

ˆx + l

ˆy) (5.27)

with l

=0, ±1, ±2,.... The empty lattice bands are labelled by the pair 219

of integers (l

). (See Fig. 5.6.) By deﬁning ξ and η by 220

k =

2π

[ξˆx + ηˆy] (5.28)

then we have

221

2ma



(ξ + l

)

+(η + l

)



and Ψ

2πi

[(ξ+l

)x+(η+l

)y]

(5.29)

The parameters ξ and η are restricted to the range [−

]. 222

Exercise 223

Evaluate the energies E

(Γ), E

(X) for energies up to E =

2ma

× 10. 224

Make a sketch of the empty square lattice bands going from Γ → Δ → X. 225

(Use straight lines to connect E

(Γ) to E

(X) ). List the degeneracies at 226

Γ, Δ, and X. For example, see Table 5.6. 227