Chemistry Reference
In-Depth Information
is given by
2
k
2
2
k
2
m
2
2
|
i
·
p
nn
|
+
2
m
+
E
n
k
=
E
n
0
(4.4a)
E
n
0
−
E
n
0
n
=
n
where
i
is a unit vector pointing along the direction
i
, and
p
nn
is the
momentum matrix element linking the
n
th and
n
th basis state of
H
0
(see
Appendix E). From the definition of
m
∗
above, we can then estimate the
effective mass along the direction
i
in the
n
th band,
m
i
,as
m
2
n
=
2
1
m
i
=
1
m
+
2
|
i
·
p
nn
|
(4.4b)
E
n
0
−
E
n
0
n
For real semiconductors, we should take account of the spin-orbit interac-
tion and also of band degeneracies, neither of which were included in the
derivation of eq. (4.4). The practical application of the
k
·
p
model, therefore,
looks rather complicated. However, two main reasons help to ensure its
usefulness (Kane 1966; Bastard 1988):
1 Inmost cases, we only have to deal with a very small number of bands,
whichare close to eachother in energy, and can ignore higher and lower
bands, where the denominator in the summation terms of eq. (4.4) is
large.
2 When we restrict the number of bands in eq. (4.4), and then start
to fit experimental data, such as measured effective masses,
m
∗
,
and measured energy gaps
E
n
0
−
E
n
0
it is found that the momen-
tum matrix elements
p
nn
are remarkably constant between different
semiconductors, with
p
nn
identically zero for many pairs of bands
n
and
n
.
We can consider a very simple (but not totally unrealistic) model of
a direct-gap semiconductor, with zero spin-orbit splitting. The conduc-
tion band consists of a single s-like anti-bonding state at
k
=
0, while the
top of the valence band has pure p-like symmetry, so is threefold degener-
ate. We choose the three p states to point along the three crystal axes, and
label them p
x
,p
y
and p
z
, respectively.
It can be shown from symmetry considerations that the matrix elements
p
ij
linking any pair of the p states are identically zero, while themomentum
matrix element linking s with each p level points along the direction of that
p state, so that
p
sx
=
P
i
say, with
p
sy
=
P
j
and
p
sz
=
P
k
, where
i
,
j
, and
k
are unit vectors along the three crystal axes.
If we assume that the wavevector
k
points along the
x
-direction, then the
matrix elements
k
·
p
nn
corresponding to
k
·
p
sy
and
k
·
p
sz
are identically
