Chemistry Reference
In-Depth Information
2
In the
repeated zone
scheme, we include on the plot several (in principle
all) wavenumbers
k
associated with a given energy state. This gets
over the difficulty of choosing the 'correct'
k
for each state, but it can
be seen from fig. 3.2(b) that the repeated zone scheme contains a lot of
redundant information.
3
Finally, in the
reduced zone
scheme, we choose the wavenumber
k
for
each state such that the magnitude of the wavenumber
k
is minimised.
This scheme has the advantage of providing a simple rule for assign-
ing a preferred
k
value to each state, and gives a simple prescription
for plotting the band structure, as in fig. 3.2(c). We will always use
the reduced zone scheme for plotting band structure in this topic. The
reduced zone is also widely referred to as the first
Brillouin zone
for the
given crystal structure.
3.3 The Kronig-Penney model
3.3.1 Full band structure
We can illustrate many of the basic properties of electrons in a periodic
solid by using the K-P model, where we calculate the band structure of
a periodic array of square wells, each of width
a
, separated by barriers of
height
V
0
and width
b
from each other (fig. 3.3).
From Bloch's theorem, we know the wavefunctions must be of the form
e
i
qx
u
nq
ψ
(
x
)
=
(
x
)
(3.9a)
nq
where
u
nq
(
x
)
=
u
nq
(
x
+
a
+
b
)
(3.9b)
andwherewe have chosen the letter
q
to symbolise the Blochwavenumber.
We first solve Schrödinger's equationwithin the first well to the right of the
V
(
x
)
V
0
-b
0
a
Position,
x
Figure 3.3
The K-P potential: a periodic array of square wells. We choose the wells
here to be of width
a
, each separated by a barrier of height
V
0
and width
b
from its immediate neighbours.
