Chemistry Reference
In-Depth Information
Bloch's theorem and
k
·
p
theory
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) then we
can use perturbation theory to calculate the band structure near that
k
value. We use
k
=
0, the
p
theory in Chapters 4 and 5 to explain various aspects of
the electronic structure of semiconductors. A general introduction to first
and second order perturbation theory is given in Appendix C.
The Hamiltonian,
H
0
, in a periodic solid is given by
·
2
2
m
∇
=−
2
H
0
+
V
(
r
)
(E.1)
with
V
(
r
+
R
)
=
V
(
r
)
, as discussed in Chapter 3. We also saw in Section 3.2
ψ
n
k
(
)
how the eigenstates,
, can be written using Bloch's theorem as the
product of a plane wave, e
i
k
·
r
, times a periodic function,
u
n
k
(
r
, with asso-
ciated energy levels,
E
n
k
. For a particular value of
k
, say
k
0
, Schrödinger's
equation may be written as
r
)
2
2
m
∇
−
2
e
i
k
0
·
r
u
n
k
0
(
e
i
k
0
·
r
u
n
k
0
(
H
0
ψ
(
r
)
=
+
V
(
r
)
(
r
))
=
E
n
k
0
(
r
))
n
k
0
(E.2)
We presume that we know the allowed energy levels
E
n
k
0
at
k
0
and now
wish to find the energy levels,
E
n
k
, at a wavevector
k
close to
k
0
, where
−
2
2
m
∇
2
e
i
k
·
r
u
n
k
e
i
k
·
r
u
n
k
+
V
(
r
)
(
(
r
))
=
E
n
k
(
(
r
))
(E.3)
To emphasise that we are interested in values of
k
close to
k
0
,wemay
rewrite eq. (E.3) as
−
e
i
(
k
−
k
0
)
·
r
2
2
m
∇
2
e
i
k
0
·
r
u
n
k
E
n
k
e
i
(
k
−
k
0
)
·
r
e
i
k
0
·
r
u
nk
+
V
(
r
)
(
(
r
))
=
(
(
r
))
(E.4)
