6.2 Electrons in a Periodic Potential
137
(a)
(b)
(c)
4
3
2
1
0
X
X W
L
K X
Fig. 6.2 Dispersion of free electrons (empty lattice calculation, U = 0, shown in the first Brillouin zone) in a a onedimensional lattice (G = n 2π/a), b a simple cubic lattice (G = (h, k, l) 2π/a) and c in a fcc lattice. The energy is
measured in units of the energy at the X-point, E X = ( 2 /2m)(π/a) 2 . The shaded circle in (c) represents the region
where the band gap develops for finite periodic potential U = 0
1
2m
(p + k)
2
+ U (r)
u nk (r) = E nk u nk (r) ,
(6.6)
which is easy to see from p = −i
First, we discuss the simplest case of a periodic potential, U ≡ 0. This calculation is also called the
empty lattice calculation. The solution of (6.6) is then just constant, i.e. u k = c and k (r) = c exp(ikr).
The dispersion of the free electron is then given by
E(k) =
2
2m
k
2
,
(6.7)
where k is an arbitrary vector in the reciprocal space. k
is a vector from the Brillouin zone such that
k = k
+ G with a suitable reciprocal lattice vector G. Because of (6.5) the dispersion relation can be
written also as
E(k) =
2
2m
(k
+ G)
2
,
(6.8)
where k
denotes a vector from the Brillouin zone. Thus, many branches of the dispersion relation arise
from using various reciprocal lattice vectors in (6.8).
The resulting dispersion relation for the free electron is shown in Fig. 6.2a for a one-dimensional
system (k
and G are parallel) and in Fig. 6.2b for the simple cubic lattice (in the so-called reduced
zone scheme). In Fig. 6.2c, the (same) dispersion of the free electron is shown for the fcc lattice.
6.2.3 Non-Vanishing Potential
Now the effect of a non-vanishing periodic potential on electron motion will be discussed. A simple,
analytically solvable model that visualizes the effect of a periodic potential on the dispersion relation of
the electrons and the formation of a (one-dimensional) band structure with gaps is the Kronig-Penney
model [71] which is discussed in the Appendix F.
137
(a)
(b)
(c)
4
3
2
1
0
X
X W
L
K X
Fig. 6.2 Dispersion of free electrons (empty lattice calculation, U = 0, shown in the first Brillouin zone) in a a onedimensional lattice (G = n 2π/a), b a simple cubic lattice (G = (h, k, l) 2π/a) and c in a fcc lattice. The energy is
measured in units of the energy at the X-point, E X = ( 2 /2m)(π/a) 2 . The shaded circle in (c) represents the region
where the band gap develops for finite periodic potential U = 0
1
2m
(p + k)
2
+ U (r)
u nk (r) = E nk u nk (r) ,
(6.6)
which is easy to see from p = −i
First, we discuss the simplest case of a periodic potential, U ≡ 0. This calculation is also called the
empty lattice calculation. The solution of (6.6) is then just constant, i.e. u k = c and k (r) = c exp(ikr).
The dispersion of the free electron is then given by
E(k) =
2
2m
k
2
,
(6.7)
where k is an arbitrary vector in the reciprocal space. k
is a vector from the Brillouin zone such that
k = k
+ G with a suitable reciprocal lattice vector G. Because of (6.5) the dispersion relation can be
written also as
E(k) =
2
2m
(k
+ G)
2
,
(6.8)
where k
denotes a vector from the Brillouin zone. Thus, many branches of the dispersion relation arise
from using various reciprocal lattice vectors in (6.8).
The resulting dispersion relation for the free electron is shown in Fig. 6.2a for a one-dimensional
system (k
and G are parallel) and in Fig. 6.2b for the simple cubic lattice (in the so-called reduced
zone scheme). In Fig. 6.2c, the (same) dispersion of the free electron is shown for the fcc lattice.
6.2.3 Non-Vanishing Potential
Now the effect of a non-vanishing periodic potential on electron motion will be discussed. A simple,
analytically solvable model that visualizes the effect of a periodic potential on the dispersion relation of
the electrons and the formation of a (one-dimensional) band structure with gaps is the Kronig-Penney
model [71] which is discussed in the Appendix F.