Chemistry Reference
In-Depth Information
2
P
2
where we have set
E
p
m
. When we include the spin-orbit
interaction, the expressions for the valence and conduction effectivemasses
are modified to account for the revised interactions between the conduc-
tion band and the spin-split-off band, with the effective masses for the
conduction and three valence bands then given by
=
/
2
E
g
+
E
p
3
m
m
c
=
1
1
+
(4.9a)
E
g
+
E
so
2
E
p
3
E
g
m
m
LH
=
1
−
(4.9b)
m
m
so
=
E
p
1
−
(4.9c)
3
(
E
g
+
E
so
)
m
∗
HH
=
and
m
1, as before.
We can use eq. (4.9a) along with the quoted values of
E
g
,
E
so
, and
m
c
in
Tables 4.1 and 4.2 to estimate the value of
E
p
for a range of materials. This is
left as an exercise to the end of the chapter, where it can be observed that the
magnitude of
E
p
is reasonably constant throughout the III-V semiconduc-
tors. It is further possible, using eq. (4.9b) and the calculated values of
E
p
to
estimate the magnitude of the LH masses,
m
LH
, which are then observed
to be in generally good agreement with the experimentally determined
values listed in Table 4.1.
The above calculations serve to show the value of
k
/
p
theory when
applied to tetrahedrally bonded semiconductors. As a further example,
we consider the highest valence and lowest conduction band along the
·
and
p
inter-
action between the conduction and valence band along these directions.
Therefore, the highest valence and lowest conduction band are approxi-
mately parallel for much of
directions, linking X and L respectively to
.Thereisno
k
·
-L, as illustrated in fig. 4.1. The
effective mass along this (longitudinal) direction,
m
l
, is then large at the
X and L points (of the order of the free electron mass), and consequently
there is a much larger density of states associated with the lowest X and L
conduction band minima compared to the lowest
-X and
minimum.
p
interaction in the transverse direction
at X and L (moving perpendicular to the
By contrast, there is a large
k
·
directions), so that
the transverse effective mass,
m
t
is then comparatively small at X and L.
Surfaces of constant energy near X and L are then described by ellipsoids,
as illustrated in fig. 4.4.
and
4.3 Electron and hole effective masses
We introduced the electron effective mass phenomenologically in
Section 4.1, claiming, for example, that for many purposes we can treat
