Chemistry Reference
In-Depth Information
10
(a)
(b)
8
6
4
2
0
-
/
L
0
/
L
-
/
L
0
/
L
Wavevector,
q
Figure 3.7
Comparison of the exact band structure for the K-P potential (solid lines)
and the band structure calculated using the TB method (dashed lines).
(a) Widely separated wells (
a
=
5Å,
V
0
=
5 eV,
b
=
3 Å), where agree-
ment is excellent. Note that the TB method is used here to fit the two
lowest bands only, as the next band cannot be formed as a linear combina-
tion of states confined in an isolated quantum well. (b) Moderate barrier
widths (
b
=
1.5 Å), where the agreement is still very good, although it can
be seen that the two sets of results are beginning to diverge for the second
band.
the state with wavevector
q
in the
m
th band as
+
+
(
)
E
m
E
m
2
V
m
cos
qL
E
mq
=
(3.27a)
1
+
2
S
m
cos
(
qL
)
(3.27b)
where we assume that the overlap term
S
m
and the self-energy shift
=
E
m
+
2
V
m
cos
(
qL
)
E
m
are sufficiently small that they can be neglected.
Figure 3.7 compares the exact K-P and the TB band structure calcu-
lated using eq. (3.27a) for two particular cases, first, where the wells are
widely separated (large
b
) and second, when the wells have been brought
closer together (moderate
b
). It is clear that the TB band structure (dashed
lines) is in excellent agreement with the exact solution (solid lines) for the
large
b
case. The agreement still remains very good in the second case, for
moderate
b
, particularly for the lowest band.
The agreement in fig. 3.7(b) could be further improved by 'parameteris-
ing' the model, that is, choosing TB interaction parameters
V
m
and orbital
self-energy shifts,
E
m
such that the TB band structure is then fitted to be in
good agreementwith the exact bands. This iswhat generally occurs inprac-
tical applications in solid state physics, where neither the basis functions
nor interaction parameters are calculated exactly but are instead found
