1
J. Humlíček FKL II
2. Translational symmetry, reciprocal lattice, Brillouin zones,
density of states
Invariance of infinite crystals with respect to translations (they form an Abelian group of symmetry
operations)
1 1 2 2 3 3 1 2 3, , , integers.nT n a n a n a n n n  
   
(1.1)
14 possible lattices (Bravais lattices). Combined with point symmetries - 7 systems, with a single or more
types:
triclinic (simple),
monoclinic (simple, base centered),
orthorhombic (simple; base centered, face centered, body centered),
tetragonal (simple, base centered),
hexagonal (simple),
rhombohedral (simple),
cubic (simple; face centered, body centered;
abbreviations: SC, FCC, BCC).
2
For an overview of the 230 space groups, see https://en.wikipedia.org/wiki/Crystal_system:
3
The consequences of translational symmetry for the electron states
One-electron stationary states obey the Schroedinger equation
2 2
2
( ) ( ) ( ) ( ) ( ) ( )
2 2
p
H r V r r V r r E r
m m
   
   
         
   
     
(1.2)
with periodic potential V:
( ) ( ) .nV r T V r 
 
(1.3)
Translations preserve Hamiltonian, probability density at the positions r

and nr T 

is the same,
2 2
( ) ( ) .n nr T r   
 
(1.4)
Consequently, the wavefunction can differ by a complex factor of unite magnitude
( )
,ni T
nC e




(1.5)
4
where  is a scalar function of the vector argument. Performing two consecutive translations, nT

and mT

,
results in the multiplication of the wavefunction by the product m nC C  , the function  in Eq. (1.5) has to be a
linear combination of the components of the translation vector, i.e., the scalar product
( ) .n nT k T   
 
(1.6)
Eigenfunctions of the Hamiltonian are therefore associated with vectors k, guaranteeing the required
properties with respect to the translations by the lattice vectors:
( ) ( ) , where ( ) ( ) .ikr
nk k k k
r e u r u r T u r   

   
   
(1.7)
This form is called Bloch function, factorized as the plane wave ikr
e

and the translationally invariant part u.
The k, vector, indexing the Bloch function, is not defined unambiguously. When adding a vector K, which
produces an integer multiple of 2, in the scalar product of Eq. (1.6), the value of the phase factor nC
remains unchanged. Let us look for such vectors in the form of linear combinations of three vectors,
1 1 2 2 3 3 1 2 3, , , integers.qK q b q b q b q q q  
  
(1.8)
5
The condition
2 , integer,q nK T m m  
 
(1.9)
is fulfilled with
1 2 3 2 3 1 3 1 2
0 0 0
2 2 2
, , .b a a b a a b a a
  
     
  
       
(1.10)
0 1 2 3( )a a a   
  
is the volume of primitive cell, based on the three vectors a. Projections of individual
vectors b on the vectors a of thee same index amount to 2, they vanish for different indices:
2 , , 1,2,3.q n qnb a q n  
 
(1.11)
The set of vectors qK

form the reciprocal lattice (the distances in the "reciprocal space" have the physical
dimension of reciprocal length). The reciprocal lattice belongs to the same crystal system as the initial lattice
in the direct space. The simple and base-centered type is also preserved, while the face- and body-centered
types replace each other.
The k-vectors in the Bloch functions are equivalent, whenever they differ by the vector of reciprocal lattice.
We usually select them from the first Brillouin zone (this is the Wigner-Seitz cell of the reciprocal lattice):
6
we choose any of the lattice site as the origin, and draw planes perpendicular to the neighbors at half of their
distance. The first Brillouine zone (1. BZ) is the volume inside the closest planes. The scalar products with
primitive lattice vectors are within the interval of 2, namely,
, 1,2,3 .jk a j     
 
(1.12)
The restriction to finite volume of a crystal is usually implied via the (cyclic) Born-Kármán conditions,
repeating the wavefunctions at large distances:
( ) ( ) , 1,2,3 ,j jr N a r j   
  
(1.13)
where Nj are large positive integers. We expect the properties of such a crystal to be practically the same as
those of a "sufficiently large" crystal, compared with inter-atomic distances. Pointing to these "bulk"
properties, we evidently neglect the influence of surfaces and/or interfaces with different materials. This
approximation usually fails for small crystals (crystallites, nanostructures).
The cyclic boundary conditions (1.13) imply the „quasicontinuum“ of possible k-vectors within the first
Brillouin zone. These are linear combinations of the primitive vectors of the reciprocal lattice,
1 1 2 2 3 3 ,k k b k b k b  
   
(1.14)
with the coefficients kj from −½ to ½ separated by the small step of 1/Nj.
7
In the association of the stationary states of the one-electron Hamiltonian with the vector k, we usually
substitute the quasi-continuum of possible values by a continuum, and use differentiation with respect to its
components and integration over them. Adding a further discrete index (quantum number) n, the
Schroedinger equation (1.2) for the Bloch form of the wavefunction becomes
2 2 22
2
, ,
( ) ( ) ( ) ( ) ,
2 2
ik r ik r
nk n k n
k i k V r e u r E k e u r
m m m
  
       
 
  
 
       
(1.15)
where we can remove the plane wave, ikr
e

, obtaining the equation for the periodic part u. Its solutions can be
limited to the primitive cell, with the boundary conditions provided by the periodicity of u.
(Quasi)continuous change of the eigenenergy E with changing the k-vector within the 1. BZ and a given
value of n is confined to an interval, denoted as n-th band; the dependence on the magnitude of k along a
given direction in the 1. BZ is called dispersion of energy in this direction. The set of bands of a given
crystal is called bandstructure. The bandstructure can contain an interval with no allowed energies, called
forbidden band, or gap. A rudimentary representation of bandstructure is the following:
8
Energy bands in semiconductors or
insulators with a larger (left) and
smaller (right) gap. The lower bands
are occupied, the upper ones are
empty.
The occurrence of gap in semiconductors and insulators is substantial, making this way of depicting the
bandstructure appropriate. For bulk material, using "position" in the crystal is irrelevant. However, in dealing
with heterostructures (different materials connected with "atomic precision"), specifying the position is
important. For example, we can idealize the heterostructures as being composed of different materials with
distinct bandstructures. Their behavior is determined by the band alignment at interfaces; the upper figure
EgEg
ENERGIE
POLOHA
9
can represent one of the possible alignment of the occupied and free bands. The two materials can form a
heterostructure described by a horizontal shift of the bandstructures until brought in contact.
Density of states
More complete information about the bandstructure is contained in the density of states, D(E). The density is
related to the energies, i.e., it represents the number of states,, P(E, E+dE), in the energy interval between E
and E+dE, divided by the width of the interval, dE:
( , d )
( ) .
d
P E E E
D E
E

 (1.16)
Dernoting by Nn(E) the number of states from the lowest energy of the n-th band (a safe choice is -∞) up to
the argument E, we can utilize the (quasi)continuous intervals of the k -vectors and energies E and produce
the derivative of this function; the density of states results as the sum over all bands n:
d ( )
( ) ( ) .
d
n
n
n n
N E
D E D E
E
   (1.17)
10
Prototypical densities of states of Si, Ge and -Sn (gray tin) are plotted in the following figure. We can
reduce them to the simplest form by forming the occupied band from about -13 eV to zero for all of the three
substances. Following this, we get the gap of about 1.2 eV (Si), 0.7 eV (Ge) and zero for -Sn. Larger
energies belong to the set of empty bands.
11
Density of states: calculated using nonlocal
pseudopotentials, Chelikowsky and Cohen,
PRB (1976). Experimental data from R.Pollak
et al., PRL (1973) for Si, Grobman et al., PRL
(1972) for Ge.
12
Equation (1.16) provides a convenient way of calculating the density of states. Since the allowed values of
the k-vector are equidistant, treating them as a quasicontinuum leads to the number Nn(E) in the form of the
volume limited by the area of En(k) = E, divided by the volume belonging to a single state in the reciprocal
space. Consequently, the contribution of the n-th band to the density of states results as the derivative of this
ratio with respect to the energy E:
3
( ) 31 2 3 0
3 3
( )
1 2 3 0
d
d d
( ) = d .
(2 )d (2 ) d
n
n
E k E
n
E k E
k
N N N
D E k
E E
N N N
 











(1.18)
The volumes in reciprocal space can by calculated fairly easily for some of the dispersions of energy. a
simple example is provided by the quadratic dispersion above a minimum band energy,
22 22
* * *
0 * * *
( , , ) , , , >0 .
2
yx z
n x y z x y z
x y z
kk k
E k k k E m m m
m m m
 
     
 

(1.19)
This would be the dependence of energy of free electron on its momentum, if the effective masses * * *
, ,x y zm m m
were independent of the direction, and equal to the mass of the electron in free space. The crystalline
potential leads to the possibility of the inertia of electrons differ in different directions, and differ from the
13
free electron case. The equi-energy areas for the dispersion of Eq. (1.19) are ellipsoids, with volumes
calculated easily using the transformed coordinates in the reciprocal space,
' 3 * * * 3 '
*
, , , , d d .
j
j x y z
j
k
k j x y z k m m m k
m
  
 
(1.20)
The volume integral in Eq, (1.18) is the volume of 3-dimensional sphere of the squared radius of 2(E-E0)/ħ2
,
amounting to (4/3)[2(E-E0)/ħ2
]3/2
. The differentiation with respect to E leads to the contribution of the band
(1.19) to the density of states in the form
* * *1 2 3 0
0 03
4 2
( ) pro .
(2 )
n x y z
N N N
D E m m m E E E E



  
 (1.18)
For energies below E0 , the density of states vanishes (there are no states in such a band).
The parabolic dispersion occurs for free electrons, where the three effective masses are the same and equal to
the fundamental physical constant m0. The system of free (noninteracting) electrons can be placed in a
infinitely weak periodic potential, which does not change the dispersion relations; however, the symmetry of
the potential matters, and we arrive at the concept of "empty lattice“.
Empty FCC lattice, respecting the symmetry of its potential, leads to the band structure depicted in the
following figure:
14
Bandstructure of the empty FCC lattice,
from
Dresselhaus, Dresselhaus & Jorio,
Group Theory, Application to the
Physics of Condensed Matter.
15
Standard symbols are used for the points and directions of high symmetry (this is the usual notation of
irreducible representation of the corresponding symmetry group):
Symbols for the 1. BZ of the FCC structure.
What is the density of states for the empty FCC lattice?