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:
Chapter 9 – Electronic Structure of Solids – p. 2 -
=
=
=
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).
Chapter 9 – Electronic Structure of Solids – p. 3 -
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
Chapter 9 – Electronic Structure of Solids – p. 4 -
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 ⊗⊗=
Chapter 9 – Electronic Structure of Solids – p. 5 -
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:
Chapter 9 – Electronic Structure of Solids – p. 6 -
( )
++=++
++
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++=
Chapter 9 – Electronic Structure of Solids – p. 7 -
• 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)
Chapter 9 – Electronic Structure of Solids – p. 8 -
(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):
Chapter 9 – Electronic Structure of Solids – p. 9 -
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
Chapter 9 – Electronic Structure of Solids – p. 10 -
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:
Chapter 9 – Electronic Structure of Solids – p. 11 -
• 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
Chapter 9 – Electronic Structure of Solids – p. 12 -
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
Chapter 9 – Electronic Structure of Solids – p. 13 -
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
Band symmetry notation Point group of k symmetry
X1 A1
Chapter 9 – Electronic Structure of Solids – p. 14 -
X2 A2
X3 B1
X4 B2
• Special line: ∆, Σ, Ζ Group of k: Cs
Band symmetry notation Point group of k symmetry
∆1 Σ1 Ζ1 A’
∆2 Σ2 Ζ2 A’’
• General point: G Group of k: C1
Band symmetry notation Point group of k symmetry
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
Chapter 9 – Electronic Structure of Solids – p. 15 -
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
Chapter 9 – Electronic Structure of Solids – p. 16 -
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 .
Chapter 9 – Electronic Structure of Solids – p. 17 -
9.11 Examplex of band structures and Fermi surfaces
(Cu band structure, from S. L. Altman)
(Cu Fermi surface, from S. L. Altman)
Chapter 9 – Electronic Structure of Solids – p. 18 -
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)):
Chapter 9 – Electronic Structure of Solids – p. 19 -
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).