Biomedical Engineering Reference
In-Depth Information
the spatial separation of electron and hole wave functions leads to a decrease in the
oscillator strength and therefore to an increase in the radiative lifetimes. Secondly,
the built-in potential gives rise to a red-shift of the exciton emission energies with
respect to the band gap energy. This field-induced separation and red shift is referred
to as the quantum confined Stark effect (QCSE).
We note that the electrostatic built-in potential also leads to an additional lateral
confinement for both electrons and holes. This ensures that the carriers are well
confined within the QD, close to its central axis. Even though our analysis has
been presented for an InN/GaN QD, the same conclusions also hold for GaN/AlN
systems [
7
,
38
,
41
]andfor
c
-plane nanostructures with ternary or quaternary III-N
alloys in the dot and/or barrier regions.
6.3.2
Comparison Between the Built-In Potential in a QD
and a QW
Having established some of the general features of the electrostatic built-in field in
nitride-based QDs, we turn now to discuss in more detail how the built-in potential
changes when the QD geometry and also its aspect ratio is changed. Before looking
at more realistic QD geometries such as lens-shaped or ellipsoidal structures, we
start with the analysis of a cuboid-shaped dot. This simple system is of interest
because a complete analytic solution can be obtained for the surface integrals in
Eqs. (
6.4
)and(
6.5
) for a cuboidal QD [
38
], thereby permitting a straightforward
understanding of how the built-in potential evolves in going from a
c
-plane QW to
a
c
-plane QD of the same height.
We start by considering the spontaneous polarization potential
φ
sp
between the
top and bottom center of a cuboid-shaped QD of height 2
h
and base length 2
B
, with
the dot aspect ratio then given by
F
0 corresponds to the QW
limit. From the analytic solution for the spontaneous polarization
=
h
/
B
,where
F
=
φ
sp
given in [
38
],
we obtain that the potential difference
Δφ
sp
varies between the center of the top and
bottom surfaces of the cuboid for small
F
as [
16
]:
2
h
1
F
C
sp
h
1
F
2
√
2
π
2
√
2
π
P
s
QD
−
P
s
B
)
(
Δφ
≈
−
=
−
.
(6.9)
sp
ε
ε
r
0
Δφ
sp
initially decreases linearly with
F
in a QD compared
to a QW structure of the same height, as shown in Fig.
6.2
a. This is due to the
reduction of the
This result shows that
(
c
-plane) surface area of the QD and therefore to a reduction
of the surface charge compared to that in the corresponding QW system.
The second contribution to the total built-in potential, the piezoelectric part
[
0001
]
φ
pz
,is
modified by two factors when going from a QW to a QD of the same height. Firstly,
as in the case of the spontaneous polarization potential
φ
sp
the finite size of the QD
affects the piezoelectric contribution. Secondly, there is also a strain redistribution in
