Chemistry Reference
In-Depth Information
The third valence band is generally called the spin-split-off band. The
lowest conduction band in a direct gap semiconductor always has a low
effective mass;
m
c
is always smaller than
m
LH
, as can be seen in Table 4.1.
It can also be observed that
m
c
and
m
LH
increase with increasing energy
gap,
E
g
.
Clearly, these trends in semiconductor band structure must have a phys-
ical basis, which could be elucidated using either the tight-binding or
pseudopotential method introduced in Chapter 3. In practice, however,
the observed trends are best explained using a semi-empirical technique,
called
k
p
theory, which will also prove very useful when considering top-
ics such as doping and low-dimensional semiconductor structures later in
this and in the next chapter.
·
4.2
k
·
p
theory for semiconductors
k
p
theory is a perturbation method, whereby if we know the exact energy
levels at one point in the Brillouin zone (say
k
·
point) thenwe can
use perturbation theory to calculate the band structure near that
k
value.
The zone centre energy levels have been determined experimentally for
many semiconductors, as listed in Tables 4.1 and 4.2. A general introduc-
tion to first and second order perturbation theory is given in Appendix C,
while the
k
=
0, the
·
p
model is summarised here and described in more detail in
Appendix E.
From Bloch's theorem, Schrödinger's equation in a periodic solid can be
written as
H
0
ψ
(
r
)
=
E
n
k
ψ
(
r
)
(4.2)
n
k
n
k
where
H
0
is the
k
-independentHamiltonian actingon
ψ
n
k
, the
k
-dependent
wavefunction e
i
k
·
r
u
n
k
(
associated with the state with energy
E
n
k
.We
show in Appendix E how it is possible to transform eq. (4.2) to give a
k
-dependent Hamiltonian,
H
k
r
)
e
−
i
k
·
r
H
0
e
i
k
·
r
, acting on the wavefunction
=
u
n
k
(
r
)
.
H
k
is given by
H
k
H
k
=
H
0
+
2
k
2
2
m
+
m
k
+
=
H
0
·
p
(4.3)
where
p
is the momentum operator,
p
. We expect for small values
of
k
that the energy levels
E
n
k
of
H
k
will be very close to the known energy
levels
E
n
0
of
H
0
. We can then use standard perturbation theory to estimate
that the dispersion along the
i
-direction near the band edge at energy
E
n
0
=−
i
∇
