Solid State Physics
269
called energy bands which in general are separated by some energy gaps. The
spread for the inner-lying levels will be small since they are not greatly influenced
by the presence of other atoms, and it will be larger for the outer levels [sec Fig.
8.6 (a)]. Since the spread is determined by the structure of each atom and the
interatomic distance, the density of levels in a band will increase with the number
of atoms (keeping the interatomic distance constant). It may happen that some
outer-lying bands overlap and this will have a profound influence on the
properties of the crystal.
The tight-binding approximation is reliable mainly for narrow bands of
low-lying energy levels, for which the effect of the interatomic interaction is
small.
Nearly Free Electron Approximation
Alternatively, the electrons may be regarded as moving in a periodic potential
which describes their interaction with the ions. Here, the interaction between
the electrons is neglected. For simplicity, the influence of such an interaction is
considered in the one-dimensional lattice.
A one-dimensional periodic potential with period a can be written as
V(x) =
1
exp (2
/ ),
0, 1, ...
2
π
= ±
∑ m
m
V
m i xa m
(8.20)
with V m = V – m . The effect of only the | m | = 1 term is considered, with
V 1 (x) = V 1 cos (2π x/a)
(8.21)
The wave function will satisfy the equation
2
2
1
2
( )
cos (2 / ) ( )
( )
2
∂
−
ψ +
π
ψ = ψ
∂
x V
x a
x E x
m x
(8.22)
The general form of the solutions of such an equation is given by the Bloch
theorem which states that ψ(x) can be written in the form
ψ(x) = exp (ikx)u k (x)
(8.23)
where u k (x) is periodic with period a. If the number of lattice points is N, a
periodic boundary condition is imposed on ψ that ψ k (Na) = ψ k (0). This is
equivalent to closing a linear chain and implies that the allowed values of k are
k =
2 ,
0, 1,...
n n
Na
π
= ±
(8.24)
These allowed values of k are conveniently divided as follows, into what
are known as the Brillouin zones:
k =
2 ,
0, 1, 2, ... /2,1st Brillouin zone
n n
N
Na
π
= ± ±
,
269
called energy bands which in general are separated by some energy gaps. The
spread for the inner-lying levels will be small since they are not greatly influenced
by the presence of other atoms, and it will be larger for the outer levels [sec Fig.
8.6 (a)]. Since the spread is determined by the structure of each atom and the
interatomic distance, the density of levels in a band will increase with the number
of atoms (keeping the interatomic distance constant). It may happen that some
outer-lying bands overlap and this will have a profound influence on the
properties of the crystal.
The tight-binding approximation is reliable mainly for narrow bands of
low-lying energy levels, for which the effect of the interatomic interaction is
small.
Nearly Free Electron Approximation
Alternatively, the electrons may be regarded as moving in a periodic potential
which describes their interaction with the ions. Here, the interaction between
the electrons is neglected. For simplicity, the influence of such an interaction is
considered in the one-dimensional lattice.
A one-dimensional periodic potential with period a can be written as
V(x) =
1
exp (2
/ ),
0, 1, ...
2
π
= ±
∑ m
m
V
m i xa m
(8.20)
with V m = V – m . The effect of only the | m | = 1 term is considered, with
V 1 (x) = V 1 cos (2π x/a)
(8.21)
The wave function will satisfy the equation
2
2
1
2
( )
cos (2 / ) ( )
( )
2
∂
−
ψ +
π
ψ = ψ
∂
x V
x a
x E x
m x
(8.22)
The general form of the solutions of such an equation is given by the Bloch
theorem which states that ψ(x) can be written in the form
ψ(x) = exp (ikx)u k (x)
(8.23)
where u k (x) is periodic with period a. If the number of lattice points is N, a
periodic boundary condition is imposed on ψ that ψ k (Na) = ψ k (0). This is
equivalent to closing a linear chain and implies that the allowed values of k are
k =
2 ,
0, 1,...
n n
Na
π
= ±
(8.24)
These allowed values of k are conveniently divided as follows, into what
are known as the Brillouin zones:
k =
2 ,
0, 1, 2, ... /2,1st Brillouin zone
n n
N
Na
π
= ± ±
,