Chemistry Reference
In-Depth Information
into the quantum well, leaving the ionised impurity centres in the barrier,
typically over 100Å from the well.
It is also possible by modulation doping to achieve 2-D conduction at
a heterojunction between two layers of differing band gap. Consider, for
instance, doping a thin layer of AlGaAs with an areal doping density,
N
s
,
with the doping layer separated by an undoped spacer layer of width
w
from a neighbouring GaAs region (fig. 5.6b).
It is energetically favourable to transfer the doping electrons into the
narrower band gap GaAs layer. If
n
s
electrons are transferred across per
unit area, this will leave a fixed positive charge associated with the ionised
impurity atoms, and induce a built-in electric field,
E
, in the spacer layer,
of magnitude
en
s
ε
E
=
(5.18)
ε
0
r
with a consequent linear variation in potential across the spacer layer
(fig. 5.6c).
The heterojunction potential and confined state energies should be
determined self-consistently, as the confined electron states, and their
wavefunctions,
, while the poten-
tial variation is in turn determined by the electron spatial distribution
(proportional to
|
ψ(
ψ(
z
)
, will depend on the potential
V
(
z
)
2
at the heterojunction.
It is beyond our scope to calculate these confined state energies self-
consistently, but we can get a qualitative understanding of the behaviour
of carriers at a heterojunction by approximating to the potential and using
the variational method introduced in Chapter 1 and Appendix A.
We presume the electrons are confined wholly within the narrow band-
gap layer, and so set
V
z
)
|
)
. Near the interface, the
electric field due to the fixed charge, eq. (5.18), is largely unscreened, and
so we let the conduction band edge energy,
V
=∞
at the interface
(
z
=
0
)
(
z
)
, vary as
e
2
n
s
z
ε
(
)
=
>
V
z
z
0
(5.19)
ε
0
r
The trial wavefunction,
f
(
z
)
, must satisfy the conditions
f
(
0
)
=
0, and
f
(
z
)
→
0as
z
→∞
. We choose
f
(
z
)
=
0
z
≤
0
b
3
2
z
e
−
bz
/
2
=
z
>
0
(5.20)
b
3
1
/
2
is chosen so that
where the normalisation constant,
(
/
2
)
∞
d
zf
∗
(
z
)
f
(
z
)
=
1
(5.21)
−∞
