Chemistry Reference
In-Depth Information
If we then require the wavefunction and its derivative to be continuous at
the origin, we find
ψ(
0
)
:
A
cos
(
ka
/
2
)
=
C
cosh
(κ
b
/
2
)
(3.18a)
ψ
(
0
)
:
kA
sin
(
ka
/
2
)
=
κ
C
sinh
(κ
b
/
2
)
(3.18b)
Dividing (3.18b) by (3.18a) we obtain for the states in fig. 3.5(a) that
k
tan
(
ka
/
2
)
=
κ
tanh
(κ
b
/
2
)
(3.19a)
A similar analysis gives, respectively, for the states in fig. 3.5(b)-(d)
k
cot
(
ka
/
2
)
=−
κ
coth
(κ
b
/
2
)
(3.19b)
k
tan
(
ka
/
2
)
=
κ
coth
(κ
b
/
2
)
(3.19c)
k
cot
(
ka
/
2
)
=−
κ
tanh
(κ
b
/
2
)
(3.19d)
The results of eq. (3.19) are particularly elegant. They illustrate clearly
how the band edge energies in the solid evolve both from the isolated well
values
(
b
=∞
)
and at the opposite extreme from the 'empty lattice' results
(
b
=
0
)
.
3.4 The tight-binding method
We outline in this section how the TB method can be used to success-
fully calculate the band structure of the K-P model from 'first principles'.
The calculation provides an excellent description of the energy spectrum
for bound states up to relatively small interwell separations,
b
, and also
illustrates several general features of the TB method. We will see how the
magnitude of the Hamiltonian matrix elements linking the atomic levels
in neighbouring quantum wells decreases both with increasing well sepa-
ration, and also as a state becomes more tightly bound within a given well.
We shall also see that the TBmethodworks best for the lowest lying energy
levels, becoming less acccurate for higher-lying excited states. This again
is a general feature of the method.
We consider the periodic array of square wells, illustrated in fig. (3.3),
with the
n
th well defined in the region
nL
b
.
We first solve Schrödinger's equation to find the energy levels
E
m
and
normalised wavefunctions,
<
x
<
nL
+
a
, where
L
=
a
+
φ
(
x
)
for an isolated quantum well defined
m
between 0
a
. We presume that the states in the
m
th energy band of
the K-Pmodel can be formed using a linear combination of the
m
th energy
states in each of the wells. In order to satisfy Bloch's theorem (eq. 3.2), the
wavefunction,
<
x
<
ψ
(
x
)
, for the state in the
m
th band with wavenumber
q
is
mq
