Chemistry Reference
In-Depth Information
density of states for a
D
-dimensional crystal, whose sides are of length
L
,
and in which the dispersion near a band edge is given by the parabolic
dispersion,
2
k
2
2
m
∗
=
E
(5.5)
Not all values of
k
are allowed in eq. (5.5). We require that the allowed
solutions of Schrödinger's equation satisfy the boundary conditions appro-
priate to the given potential. For a given crystal, we therefore require that
the wavefunctions decay to zero at the crystal surfaces. The existence of
the surface implies that, strictly speaking, Bloch's theorem should not be
applied within a finite crystal. But we know that Bloch's theorem works
and is very useful for describing many crystal properties. We would like
to maintain Bloch's theorem, while still recognising that we have a finite
crystal of size
L
. We can do so by introducing periodic boundary condi-
tions on the crystal, requiring that the amplitude and derivative of each
wavefunction are equal at
x
=
0 and
x
=
L
,sothat
e
i
k
x
L
=
1
(5.6)
with the allowed values of
k
x
then given by
k
x
L
=
2
π
n
(5.7a)
or
k
x
=
2
π
n
/
L
(5.7b)
where
n
is an integer. Likewise, in a cube of side
L
, we have
k
y
=
2
π
p
/
L
and
k
z
L
, where
p
and
q
are both integers.
We can then define the density of states function,
n
=
2
π
q
/
, such that the
number of allowed energy states, d
N
, between energy
E
and
E
(
E
)
+
d
E
is
given by
=
(
)
d
N
n
E
d
E
(5.8)
The number of allowed states in this energy range will depend directly on
the number of states whose allowed wavevectors fall in this range, that is,
n
(
E
)
d
E
=
n
(
k
)
d
k
(5.9)
where
n
is the density of allowed
k
points.
Re-arranging eq. (5.9), the density of electronic states,
n
(
k
)
(
E
)
, is then
given by
d
k
d
E
=
n
(
k
)
n
(
E
)
=
n
(
k
)
(5.10)
d
E
/
d
k
