Chemistry Reference
In-Depth Information
kL
2
That is,
n
2D
. Substituting this into eq. (5.11), we find that the
density of allowed electron states near a band edge in a 2-D structure of
area
L
2
is given by
(
k
)
=
/π
L
2
m
∗
π
(
)
=
n
2D
E
(5.13)
with the density of states per unit area,
g
2D
(
E
)
, then given by
m
∗
h
2
4
π
g
2D
(
E
)
=
(5.14)
We can go through a similar analysis in both the 1-D and 3-D cases. The
density of states per unit volume,
g
3D
(
E
)
, in a bulk crystal is given by
2
m
∗
h
2
3
/
2
E
1
/
2
g
3D
(
E
)
=
4
π
(5.15)
while in 1-D, the density of states per unit length,
g
1D
(
E
)
is given by
2
m
∗
)
1
/
2
2
(
E
−
1
/
2
g
1D
(
E
)
=
(5.16)
h
Figure 5.5 illustrates schematically the change in the density of states,
g
,
as the electron motion is limited to fewer dimensions in a semiconduc-
tor structure. The most striking feature observed as the dimensionality is
(
E
)
Bulk
Quantum well
Quantum wire
Quantum box
E
E
E
E
g
(
E
)
g
(
E
)
g
(
E
)
g
(
E
)
Figure 5.5
The change in the density of states,
g
(
E
)
, as the number of confining dimensions
is increased.
