Chemistry Reference
In-Depth Information
We consider means of solving eqs (2.9) and (2.10) in the problems at the
end of this chapter. Even without solving exactly, we can deduce sev-
eral important results concerning coupled quantum wells and molecular
bonding from these equations:
(1)
Isolated well limit
:As
b
, and the square quantum wells become
widely separated from each other, we expect the energy levels to approach
those for an isolated well. This is confirmed by noting that both tanh
→∞
(κ
b
)
and coth
(κ
b
)
→
1, as
b
→∞
, so that both eq. (2.9) and eq. (2.10) then take
the form
2
k
2
(κ
−
)
sin
(
ka
)
+
2
κ
k
cos
(
ka
)
=
0
(2.11)
which is just the condition (eq. (1.54)) to determine the energy levels of an
isolated quantum well.
(2)
Bonding and anti-bonding levels
: Conversely, we see from eqs (2.9) and
(2.10) that as the interwell separation
b
decreases the coupled well energy
levels evolve continuously from the isolated single quantum well levels.
This is illustrated in fig. 2.2(a) and (b) where we plot the evolution of
energy levels in a shallow well and a deep well as a function of
b
.As
b
decreases, the doubly degenerate levels start to move apart, one going up
and the other down in energy, with the splitting increasing for decreas-
ing separation
b
. This splitting explains the origins of chemical bonding.
Two electrons (of opposite spin) can occupy each energy level in an atom
(a)
1.0
(b)
6
0.8
4
0.6
0.4
2
0.2
0.0
0
0
5
10
15 2
Well separation (Å)
4
6
8
10
Figure 2.2
Variation of allowed energy levels as a function of the separating barrier
width
b
for (a) two coupled 'shallow' quantum wells (each of width
a
=
6Å
and depth
V
0
=
1.0 eV), and (b) two 'deep' quantum wells (also of width
a
=
6 Å, but of depth
V
0
=
6.0 eV). As the separation
b
→∞
, each of
the shallow wells has one confined energy level, while each of the deeper
wells in (b) has three confined states.
