Chemistry Reference
In-Depth Information
for the case with well width
a
=
5Å, barrier width,
b
=
1.5Å, and barrier
height
V
0
10 eV. The energies vary continuously with
q
within each
band, and the band edges always occur at high symmetry points: either
at the centre of the Brillouin zone
=
(
q
=
0
)
, or else at the Brillouin zone
(
=
π/(
+
))
edge
. Although derived using the K-P model, this result
is general for 1-D periodic structures, where the band extrema and band
gaps are always associated with these two high symmetry points. We will
see below that in two and three dimensions band extrema may in addition
occur also at lower symmetry points, due to bandmixing and anti-crossing
effects (see fig. 3.14).
Wewould hopewith eq. (3.13) that as
q
a
b
, the allowed energy levels
should approach those for an isolated finite quantumwell. Equation (3.13)
can be rewritten as
κ
b
→∞
k
−
sin
1
2
k
κ
cos
(
q
(
a
+
b
))
(
)
+
(
)
(κ
)
=
cos
ka
ka
tanh
b
(3.14)
cosh
(κ
b
)
As
κ
b
→∞
, tanh
(κ
b
)
→
1, while cosh
(κ
b
)
→∞
, so that eq. (3.14) then
reduces to
k
−
sin
1
2
k
κ
cos
(
ka
)
+
(
ka
)
=
0
(3.15)
But we saw in Chapter 1 (eq. (1.54)) that this is just the condition
to determine the energy levels for an isolated well. Hence, as for the
double quantum well in Chapter 2, we expect that we should be able
to use the TB (linear combination of atomic orbitals) method to deter-
mine the energy levels in crystalline solids. We also expect that only
the higher energy (valence) levels will contribute to the bonding, while
the deeper (core) levels will be largely unperturbed by the neighbouring
atoms.
3.3.2 High symmetry energy states
We saw in Chapter 1 howwe can use symmetry arguments to simplify the
calculation of the confined state energies in an isolated square well, with
the energy levels for even states given by eq. (1.50):
k
tan
(
ka
/
2
)
=
κ
(3.16a)
and for odd states by eq. (1.51)
k
cot
(
ka
/
2
)
=−
κ
(3.16b)
We can also use symmetry in the K-P model to simplify the calcula-
tion of the confined state energies for the Bloch wavevectors
q
=
0 and
