Chemistry Reference
In-Depth Information
(a)
10
5
0
-5
-10
0
2
4
6
8
10
Energy (eV)
(b)
10
8
6
4
2
0
-
/(
a
+
b
)
0
/(
a
+
b
)
Figure 3.4
(a) The horizontal dashed lines at
+
1 and at
−
1 show the upper and lower
limits of the left-hand side of eq. (3.13). The smooth curve shows how the
right-hand side of eq. (3.13),
f
R
(
E
)
, varies with energy
E
for the case where
a
=
5Å,
b
=
1.5 Å and
V
0
=
10 eV. It can be seen that there are only
certain bands of energy where there can be allowed solutions of eq. (3.13)
(around 1, 4, and 8 eV in this case), separated from each other by energy
gaps where there are no allowed solutions. (b) shows the band structure
in the K-P model, with energy plotted against wavenumber
q
for the case
considered in (a)
(
a
=
5Å,
b
=
1.5 Å and
V
0
=
10 eV). It can be seen that
the band edges occur at
q
=
0or
π/
(
a
+
b
), that is, where the left-hand
side of eq. (3.13) equals
±
1.
with energy,
E
, of the right-hand side of eq. (3.14),
f
R
, superimposed
on a graph of the range of allowed values of the left-hand side, showing
clearly the existence of allowed and forbidden energy regions.
Figure 3.4(b) shows the calculatedband structure for theK-Pmodel, with
E
plotted against
q
in the first Brillouin zone (for which
(
E
)
|
|≤
π/(
+
))
q
a
b
,
