chapter 9 – electronic structure of solids – p. 1 - 9. electronic...

Chapter 9 – Electronic Structure of Solids – p. 1 -

9. Electronic Structure of Solids

9.1 Reminder: Free electron model of a solid

We briefly consider the electronic structure of a free electron metal, i.e. a system of non-

interacting electrons in a constant potential (e.g. a alkali metal such as Na ([1s2][2s22p6]3s)

would be an appropriate example):

The eigenfunctions of the momentum operator

( ) ( )

( ) ( ) ( )

=⇒=∂∂

=

(diverges)

:solution

ˆ

xik

xk

x

xx

x

x

NeNexxpx

xi

xpxp

ϕϕϕ

ϕϕ

h

The solutions are eigenfunctions of the Hamiltonian as well:

( ) ( )xExH xϕϕ =ˆ with ( )0 choose weconst,

2ˆ21ˆ

=

+=V

xVpm

H

( ) ( ) ( )xmkxp

mxH x ϕϕϕ

2ˆ

21ˆ

222 h

== ( ) ( ) ( )xmkxp

mxH ϕϕϕ

2ˆ

21ˆ

222 h

==⇐

In 3 dimansions: ( ) rkizkykxki eNex zyxrr

== ++ )(ϕ

In order to avoid boundary effects, cyclic boundary conditions are introduced, i.e. the system

of N atoms is artificially copied to yield an infinite crystal:

Periodic boundary conditions for crystal of size L3:


=

=

=

LL

L00

0

0

00

000

ϕϕϕϕ or LikLikLik zyx eeee ===0

lead to the quantization in k space

zz

yy

xx

nLknLknLk

π

ππ

222

=

==

with ,...2,1,0,, ±±=zyxn , i.e.

i.e the states are equally distributed over k space with a distance 2π/L, i.e. the density of states

per volume in k space decreases with increasing L:

The corresponding energy quantization is:

( )2222

22 2zyx nnn

LmE ++=

πh

The states are filled up to the Fermi energy EF, the surface of states with E=EF represents a

sphere in k space (in the limiting case of infinite L).


9.2 Space groups and subgroups

We would like to interpret the electronic structure of a solid in term of its symmetry

properties. We start by considering the space group of the crystal which contains the set of all

Seitz operators (space group operators) tpr

of the group. As defined before

tr

are translations of the lattice (lattice vectors of fractional lattice vectors for glide planes and

screw axes) and

p is a point group symmetry operation.

Some rules for calculations with Seitz operators are (for proof see e.g. Altmann):

• Operation on a vector: trprtprrrr

+=

• Product: ttppptptprrrr

+= ''''

• Inverse operator: tpptprr 111 −−−

−=

We consider symmorphic space groups first, i.e. those which are derived without the use of

glide planes or screw axes. For such a space group G it is always possible two write the space

group operations as

0pTETprr

= .

Here TEr

are translations by a lattice vector. The operations 0p represent the point group

P of the space group G (see section 2.8). Now we consider any subgroup P’ of the point

group P containing the elements 0'p . It can be shown that the complete set of operators

0'pTEr

again forms a group G’ (closure, inverse, identity, associative law; for proof see e.g.

Wherrett), and therefore G’ is a subgroup of G ( GG ⊂' ). In particular E is a subgroup of any

point group and the group T with the elements


TEr

is a subgroup of any (symmorphic) space group G. T is denoted as the translational subgroup

( GΤ ⊂ ).

It is important to note that for any group it is always possible to choose the basis functions

which span an irrep of the group such that these functions also belong to irreps of a subgroup

of the group (e.g. in case of the atom with the symmetry 23 SR ⊗ the wave functions nlmΨ can

be chosen such that they also belong to irreps of the subgroup vC∞ and allow the assignment

of a quantum number m). Sometimes, of course, irreps of higher dimension split into irreps of

lower dimension by going to the subgroup.

For this reason we will apply the following strategy: We will identify the basis functions

which belong to the irreps of the translational subgroup first and construct from those the

functions which belong to the irreps of the space group.

9.3 The translational subgroup and its representations

We investigate the translational subgroup T using cyclic boundary conditions with N unit

cells in any direction. With the primitive translations T1, T2, and T3 we obtain:

( )( )( ) ET

ET

ET

N

N

N

=

=

=

3

2

1

3

2

1

Thus the operations in the group T are

,...,,...,,,...,,,...,, 211

32

331

22

221

12

11321 TTTTTTTTTTTE NNN −−−=T

As all translation operators commute, we can express T as the direct product of the three one

dimensional translational groups:

321 TTTT ⊗⊗=


This allows us to build up the character table of T from the table of a cyclic group as derived

in section 5.4. The general character table of a cyclic group of order N is:

E 1T 2T K 1−NT

0Γ 1 1 1 1

1Γ 1 ( )11ε ( )21ε ( ) 11 −Nε

2Γ 1 ( )12ε ( )22ε ( ) 12 −Nε

M

1−ΓN 1 ( )11−Nε ( )21−Nε ( ) 11 −− NNε Ni

eπ

ε2

−=

In general the character of the j-th irrep for the operation Tm is:

( ) Nimj

mj eTπ

χ2

−= .

The characters of the irreps of the three dimensional translation group follow directly from the

direct product group properties ( 321 jjj Γ⊗Γ⊗Γ=Γ ):

( )

++−

⊗⊗ = 3

33

2

22

1

11

321

2N

jmN

jmN

jmi

jjj eTπ

χr

with ,...,,; 210332211 ±±=++= imamamamT rrrr representing an

arbitrary lattice vector.

9.4 The k-space

We define a vector

3

33

2

22

1

11

Nbj

Nbj

Nbjk

rrrr

++=

such that

( ) Tkijjj eTrrr

−⊗⊗ =321χ

This demands that the following condition for kr

is fulfilled:


( )

++=++

++

3

33

2

22

1

11332211

3

33

2

22

1

11 2N

jmN

jmN

jmamamamNbj

Nbj

Nbj

πrrr

rrr

.

Therefore, the basis vectors ibr

for kr

must be chosen such that

ijji ab πδ2=⋅rr

.

In three dimensional space, it is relatively straightforward vectors to calculate the vectors ibr

,

if the basis vectors of the translational lattice iar are known:

( )321

321 2

aaaaab rrr

rrr

×⋅×

= π ; ( )132

132 2

aaaaab rrr

rrr

×⋅×

= π ; ( )213

213 2

aaaaab rrr

rrr

×⋅×

= π

The lattice defined by the basis vectors ibr

with

,...,,; 210332211 ±±=++= inbnbnbnKrrrr

is denoted as the reciprocal lattice.

The space defined by the reciprocal basis vectors ibr

R∈++= ikbkbkbkk ;332211

rrrr

is denoted as the reciprocal space or k-space.

It is important to note the following properties of the k-space:

• The irreps of T are represented by discrete points in k-space:

33

32

2

21

1

1 bNjb

Njb

Njk

rrrr++=


• A unit cell in reciprocal space contains 321 NNN k-points, which belong to irreps of T.

• All irreps if T are contained in one unit cell. Two vectors kr

and Kkrr

+ which differ

only by a reciprocal lattice vector Kr

are indistinguishable from a group theoretical

point of view, i.e. they belong to the same irrep as

( ) ( ) ( )( ) ( )TeeeT kTkimnmnmnTkiTKkiKkrr rrrrrrrrrr

χχ π ==== −+++−+−+ 3322112 .

• There are points on the surface of the reciprocal space unit cell, which differ by a

vector Kr

. These points represent identical irreps and have to be considered only once.

As a consequence only on ½ of the faces, ¼ of the edges and 1/8 of the corner points

are significant.

(9.1: example: Reciprocal lattice of a rectangular 2D lattice)

9.5 Brillouin zones

Instead of using a primitive cell, it is conventional to construct the unit cell in reciprocal space

in form of the so called proximity cell. The strategy is as follows: starting at the origin, (1) we

draw lines to the next neighbouring lattice points in k-space and (2) construct planes normal

to these connecting lines containing the central point of the lines. The smallest enclosed

volume is the proximity cell in k-space.

• The proximity cell and the primitive cell contain exactly equivalent k-points (taking

into account the equivalence of kr

and Kkrr

+ ), i.e. the proximity cell contains all

irreps of T.

• We denote the proximity cell in k-space as the first Brillouin zone. Higher Brillouin

zones can be constructed by not using the nearest neighbours but the next nearest

neighbours in the above construction. Note that the second, third etc. Brillouin zones

are not significant from a group theoretical point of view, as they contain the same

irreps as the 1st.

(9.2: example: Brillouin zone of a rectangular 2D lattice, 2nd and 3rd Brillouin zones)


(9.3: example: Brillouin zones a sc, bcc and fcc lattice)

Note: The centred unit cell is advantageous as it better reflects the symmetry of the energy

bands as we will see later (time reversal symmetry). This is immediately apparent considering

the free-electron model (section 9.1).

In solid state physics it is common to use specific notations for special points, lines etc. of a

Brillouin zone. We consider two cases:

(1) 2D square lattice (a general point is denoted as G) (from G. Burns):

(left: constant energy contours, see section 9.7)

(2) 3D fcc, bcc, simple orthorhombic lattice (from G. Burns):


9.6 Bloch functions

Now, we try to find the symmetry adapted basis function for the translational group T. As

functions belonging to different k-vectors also belong to different irreps and therefore can

have different energy eigenvalues, the energy should also be labelled by k:

( ) ( )rErHkkk

vvrrr ψψ =ˆ .

The symmetry adapted wave function belonging to irrep k must obey the transformation

property:

( ) ( ) ( ) ( ) Tkikkkk

errTrTErr

rrrrvvrvr

−=Γ= ψψψ

This is the case if the wavefunctions are of the form of so called Bloch functions:

( ) ( ) ( ) ( ) )periodiciy latticewith (function with Truruerurkk

rkikk

rvvvvrr

rr

rr +==ψ

Proof:

( ) ( )( )( ) ( )

( ) ( )

( ) Tkik

Trkik

Trkik

k

k

kk

er

eru

eTru

Tr

rTE

rTErTE

rr

r

rrr

r

rrr

r

r

r

rr

v

v

rv

rv

vr

vrvr

−

−

−

−

=

=

−=

−=

−=

=

ψ

ψ

ψ

ψψ

1

(9.4: example: consider a 1D chain of atoms and draw the Block function at different points of in the 1st Brillouin zone)

Every set of eigenfunctions ( )rk

vrψ forms a so called energy band in the sense that for a

macroscopic crystal the spacing between the k-values is extremely narrow. Of course there

can be more than one band which allows us to classify the eigenfunctions as:

( ) ( )rErHknknkn

vvrrr ψψ =ˆ .

9.7 Energy band symmetries and the star of k


We now consider the effect of a space group operation G∈WRr

on a Bloch function krψ

(restricting ourselves to symmorphic groups). We would like to investigate the effect of a

translation TEr

on k

WR r

rψ . For this purpose, we express the space group operation product

as (see rules in section 9.2):

TREWRTWRWRTErrrrrr

1−=+=

Now we try to find the irrep of T, which k

WR r

rψ belongs to:

( ) ( )

k

TkiR

TkiR

kTkiR

kTkiR

kTRRkiR

kTRki

kk

WRe

e

eWR

eWR

eWR

eWR

TREWRWRTE

r

rr

rr

r

rr

r

rr

r

rr

r

rr

rr

r

rr

r

r

r

rrrr

ψ

ψ

ψ

ψ

ψ

ψψ

−

−

−

−

−

−

−

=

=

=

=

=

=

−

−

)prefactor

on thenot and r scoordinateon act operations(symmetry

operation)symmetry any toinvariant isit Therefore ctors.between ve anglesor

vectorsoflength changenot must operation symmetry (a 1

1

1

It follows that:

kRk

WR rr

rψψ =

i.e. the operation of the space group operator TRr

generates from the Bloch function ( )rk

vrψ

another Bloch function ( )rkR

vrψ . This means that all functions generated constitute a

representation of G, and if this representation is irreducible, all functions generated are

degenerate. Sometimes, however, these sets of functions contain redundancies. Therefore we

define the star of k:


• We define the star of k as the set of non equivalent k-vectors generated by the

operation of kRr

0 . The number of points in star of k is equal to the order of the

point group P for a general point and lower for higher symmetry points.

• For a general point in k-space, we can generate a basis for a irrep of the space group

by using all Bloch functions belonging to the star of k. The functions are transformed

into each other under the operations of the point group ( )∑ Γ=m

Gmnkk

TRTRmn

rrrr ψψ .

(for special points in k-space, one more detail to be considered, as we will see in the

next section).

• The points belonging to the star of k represent points of equal energy of a band.

Therefore, energy bands k

E r have the symmetry of the point group P of the space

group G.

(9.5: example: star of k for some 2D lattices, irrep for the space group; basis for an irrep)

9.8 The group of k

So far we have generated the bases for the representations of a space group G by formation of

symmetry adapted functions for the translational subgroup T and generation of the star of k.

This means that kr

is the only symmetry label:

( ) ( ) ( ) ( ) Tkiknkknkn

erTrrTErr

rrrrvrvvr

−=Γ= ψψψ

For special points in k-space, however, we can define an additional symmetry label in the

following fashion: Instead of the translational subgroup we can a larger subgroup (containing

more symmetry information) which is the so called the space group of k Gk. Gk contains

those operations TRk

rr which leave a particular k-vector unchanged and all translations. The

operations 0k

Rr constitute the so-called point group of Pk. We now try to find the


irreducible representations of Gk which should transform sets of Bloch functions into each

other:

( )∑ Γ=m k

Gmnkmnkk

TRTR krr

rrrr

ψψ

(In analogy to the construction of a basis for G in the previous section).

The characters can be easily calculated:

( )

( )∑ Γ=

∑ Γ=

=

−

m k

Gml

Tkikm

m k

Gmnkm

knkknk

Re

RTE

RTETR

k

k

0

0

0

r

rr

r

rr

rrrr

r

rr

rr

ψ

ψ

ψψ

This means that

( ) ( )0k

GTkik

G ReTR kk r

rr

rrr

rΓ=Γ −

i.e. we can classify the irreps of the space group of k by two symmetry labels (see e.g.

Wherrett for details): (1) the label kr

related to the translational subgroup T and (2) by the

irrep of the point group of k. The hierarchy of subgroups is indicated below:

The final strategy for the preparation of the symmetry adapted basis function for a

(symmorphic) space group is therefore:

1. preparation of Bloch functions

2. symmetry adaptation of Block functions according to the group of k

3. formation of the basis functions by construction of the star of k


Remark: the previous arguments hold for symmorphic groups only. For non-symmorphic

groups, the situation is complicated by the fact that the entire point group of the crystal is not

a subgroup of the space group (see textbooks).

(9.6: example: square 2D lattice, different points in BZ, group of k, symmetry adaptation, star of k, guessing band trends)

9.9 Band symmetries, splitting and degeneracies

In summary, we can label a crystal electronic state by a k-vector and an irrep of the point

group of k. In solid state physics it is common to use a specific notation for the symmetry of

bands including (1) the point / line label introduced in section 9.5 and (2) a number subscript

(plus superscripts) which indicates the irrep of the point group of k. We will not discuss the

different notations here (see e.g. W. Ludwig, C. Falter for overview). Instead, we will briefly

considering a simple example.

Example: 2D square lattice

• Special Point: Γ, M Group of k: C4V

Band symmetry notation Point group of k symmetry

Γ1 Μ1 A1

Γ2 Μ2 A2

Γ3 Μ3 B1

Γ4 Μ4 B2

Γ5 Μ5 E

• Special Point: X Group of k: C2V


X1 A1


X2 A2

X3 B1

X4 B2

• Special line: ∆, Σ, Ζ Group of k: Cs


∆1 Σ1 Ζ1 A’

∆2 Σ2 Ζ2 A’’

• General point: G Group of k: C1


G1 A1

If we move in k-space from a point of high symmetry to a point of lower symmetry, the

problem is analogous to the subgroup problem treated before (see section 6.8). The irreps of

the higher symmetry group can be reduced in term of the irreps of a subgroup. The relations

between the irreps can be calculated manually (section 3.13) or looked up in a so-called

correlation table. In solid state physics, these correlations are called compatibility relations.

Example: 2D square lattice

Compatibility relations:

Γ1 ...⇒ ∆1 ...⇒ Σ1 ...⇒ G1

Γ2 ...⇒ ∆2 ...⇒ Σ2 ...⇒ G1

Γ3 ...⇒ ∆1 ...⇒ Σ2 ...⇒ G1

Γ4 ...⇒ ∆2 ...⇒ Σ1 ...⇒ G1


Γ5 ...⇒ ∆1+∆2 ...⇒ Σ1+Σ2 ...⇒ 2G1

Μ1 ...⇒ Σ1 ...⇒ Ζ1 ...⇒ G1

Μ2 ...⇒ Σ2 ...⇒ Ζ2 ...⇒ G1

Μ3 ...⇒ Σ2 ...⇒ Ζ1 ...⇒ G1

Μ4 ...⇒ Σ1 ...⇒ Ζ2 ...⇒ G1

Μ5 ...⇒ Σ1+Σ2 ...⇒ Ζ1+Ζ2 ...⇒ 2G1

X1 ...⇒ ∆1 ...⇒ Ζ1 ...⇒ G1

X2 ...⇒ ∆2 ...⇒ Ζ2 ...⇒ G1

X3 ...⇒ ∆2 ...⇒ Ζ1 ...⇒ G1

X4 ...⇒ ∆1 ...⇒ Ζ2 ...⇒ G1

Thus, we can translate the band structure of the pxpy bands in the 2D square lattice (exercise

9.6) into solid state notation:

9.10 Time-reversal symmetry

So far, we have found that the symmetry of E(k) reflects the symmetry of the crystal.

Therefore, E(k)=E(-k) for crystals with inversion symmetry only. In fact, E(k)=E(-k) for all


crystals. This is a result of the fact that we are dealing with real Hamiltonians (only if the spin

in considered explicitly, H is complex).

In order to show this property, we define a new operator, the conjugator j :

( ) ( ) ( )***ˆ*ˆ ωκωκωκ Ω=Ω=Ω jj .

If the H is real, it is immediately apparent that

ϕϕ jHHj ˆˆˆˆ =

i.e. j commutes with H . Thus j behaves like a symmetry operator, although it is not

associated with a geometric transformation. If ϕϕ EH =ˆ , it is obvious that because

**ˆˆˆˆˆˆˆ ϕϕϕϕϕϕ EHjEjHEjHj =⇒=⇒=

*ϕ is also a solution of the Schroedinger equation to the same energy. Another important

property of j is that it commuted with all symmetry operations ( jRRj ˆˆˆˆ = ; for proof see e.g. S.

L. Altman).

We assume that the solution has the form of a Bloch function and investigate the translation

behaviour of its complex conjugate:

ktki

ktki

kk

je

ej

tEjjtE

r

rr

r

rr

rr

rr

ψ

ψ

ψψ

ˆ

ˆ

ˆˆ

=

=

=−

Thus k

j rψˆ is a degenerate solution belonging to kr

− instead of kr

. It follows that

( ) ( )kEkErr

−= .

This type of symmetry is referred to as time reversal symmetry as ( )t*ψ behaves as the time-

reversed function ( )t−ψ :

( ) ( )( )

( )t

tittitH

∂−∂

−=−∂−∂

=−ψψ

ψ hhˆ and ( ) ( ) ( )t

tittitH

∂∂

−=

∂∂

=**

*ˆ ψψψ hh .


9.11 Examplex of band structures and Fermi surfaces

(Cu band structure, from S. L. Altman)

(Cu Fermi surface, from S. L. Altman)


9.12 Some experimental remarks

The most important experimental tool for the investigation of electronic band structures is

Angular Resolved Ultraviolet Photoelectron Spectroscopy, i.e. UV light is used to emit an

electron from the crystal or the surface / adsorbate. Applying first order time-resolved

pertubation theory and the dipole approximation, the photoemission process is described as:

2

0AipfIr

ˆ~

with 0Ar

the electromagnetic vector potential and p is the momentum operator. We can take

advantage of the fact that in addition to energy conservation there is also conservation of the

parallel momentum of the photoemitted electron upon transition into the vacuum:

Gkk inout

rrr+= ||,||,

( Gr

is a reciprocal lattice vector. Note all electron waves which differ by Gr

belong to the

same irrep and can mix in the crystal potential).

There is, however, no conservation of ⊥kr

as the symmetry is broken by the surface, once the

electron leaves the crystal. Therefore the determination of 2D band structures is much simpler

that the determination of 3D band structures (however, there are various methods of

determining ⊥kr

in practice).

The schematic representation below illustrates the relation of the photoelectron spectrum to

the band structure for the case of Cu in the Γ-X direction (from E. W. Plummer, W.

Eberhardt, Adv. Chem. Phys. 49, 533 (1982)):


For an example for an extremely well investigated adsorbate band structure including a

discussion of symmetry assignments and selection rules, it is referred to the CO(2×1)p2mg on

Ni(110) (H.-J. Freund, H. Kuhlenbeck in Appl. of Synchrotron Radiation, ed. W. Eberhardt,

Springer Ser. Surf. Sci. Vol. 35).

chapter 9 – electronic structure of solids – p. 1 - 9. electronic...

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