Chemistry Reference
In-Depth Information
k
y
k
x
k
+
d
k
k
Figure 5.4
The grid of allowed k-points
(
2
π
n
/
L
,2
π
p
/
L
)
form a reciprocal lattice asso-
ciated with a 2-D crystal of size
L
2
. The two circles of radius
k
and
k
+
d
k
describe contours of constant energy, with
E
=
2
k
2
/
2
m
∗
on the inner
circle.
This makes sense: if the energy,
E
, is changing rapidly with wavevector
k
d
k
large), then there will be fewer states in a given energy range
than when d
E
(
d
E
/
/
d
k
is small. From eq. (5.5), we have for a parabolic band
2
k
m
∗
,sothat
that d
E
/
d
k
=
/
m
∗
n
(
E
)
=
2
k
n
(
k
)
(5.11)
As the density of
k
-states,
n
, depends on the dimensionality of the
structure, so too will the density of states,
n
(
k
)
.
Figure 5.4 shows the grid of allowed
k
-points
(
E
)
(
2
π
n
/
L
,2
π
p
/
L
)
, near the
origin in two dimensions. The two circles of radius
k
and
k
+
d
k
are contours
2
m
∗
on the inner circle.
In
D
dimensions, each
k
-point occupies a volume
2
k
2
of constant energy, with
E
=
/
D
,sothatthe
(
2
π/
L
)
D
.
With two allowed electron states per
k
-value (one spin up, and one spin
down), the density of allowed states is 2
D
number of
k
-states per unit volume of
k
-space is then 1
/(
2
π/
L
)
=
(
L
/
2
π)
D
.
Turning first to the two-dimensional case of fig. 5.4, the area between
the two rings of radius
k
and
k
(
L
/
2
π)
+
d
k
equals 2
π
k
d
k
, so that the number of
allowed states between
k
and
k
+
d
k
,d
N
(
k
)
, is given by
2
2
d
N
(
k
)
=
2
(
L
/
2
π)
π
k
d
k
=
n
2D
(
k
)
d
k
(5.12)
