Chemistry Reference
In-Depth Information
and
1
m
∗
W
d
ψ
1
m
B
d
ψ
W
d
z
B
d
z
=
(5.2b)
This second relation is a generalisation of the earlier condition inChapter 1,
that d
d
z
, to the casewhere themass changes on crossing the
boundary. Equation (5.2b) ensures what is referred to as the 'conservation
of probability current density' between different layers (Bastard 1988). The
confinement energies are then found by solving
k
tan
kL
2
ψ
/
d
z
=
d
ψ
/
W
B
m
∗
W
m
B
κ
=
(5.3a)
for states of even parity, while for states of odd parity
k
cot
kL
2
m
∗
W
−
=
m
B
κ
(5.3b)
where
L
is the well width, with
k
2
2
m
∗
W
E
2
and
2
2
m
B
(
2
.
=
/
κ
=
E
c
−
E
)/
E
c
is the conduction band offset, and the zero of energy is at the bottom
of the well.
We can extend the analysis to quantumwires and dots. If we assume an
infinite confining potential (
V
in barrier) and rectangular wires and
dots, then the allowed energy states of the confined electrons are given by
=∞
h
2
i
2
8
m
∗
L
z
+
2
2
m
∗
(
k
x
+
k
y
)
E
i
(
k
x
,
k
y
)
=
Quantum well
i
2
L
z
+
h
2
8
m
∗
j
2
L
y
2
k
x
2
m
∗
+
E
i
,
j
(
k
x
)
=
Quantum wire
(5.4)
i
2
L
z
+
h
2
8
m
∗
j
2
L
y
+
k
2
L
x
E
i
,
j
,
k
=
Quantum box
where
L
z
,
L
y
, and
L
x
are the confining dimensions,
i
,
j
,
k
are
the quantum confinement numbers,
k
x
,
k
y
are the wavevector components
along the unconfined directions,
m
∗
is the carrier effective mass, and we
assume the zero of energy at the confined layer conduction band edge.
=
1, 2,
...
5.3 Density of states in quantum wells, wires, and dots
Limiting the electron motion to fewer dimensions dramatically modifies
the electron energy spectrum, leading to an enhancement of the density of
states near the band edge. To see this, we need to calculate the electronic
