Biomedical Engineering Reference
In-Depth Information
and potential arising from the outer surface of the surrounding matrix. We assume
that in an actual structure external charges will accumulate on this outer surface and
counteract this potential, so that it is negligible compared to the field arising from the
interface between the QD and the barrier material. Consequently, the electrostatic
potential
φ
sp
in Eq. (
6.4
), arising from the spontaneous polarization is determined
by the difference
P
QD
P
sp
between the spontaneous polarization of the QD
and the surrounding barrier material and is therefore equivalent to the potential due
to a charge density
Δ
P
sp
=
−
sp
n
distributed over the QD surface. Having established
the form of the built-in potential due to the spontaneous polarization, we turn now
to consider the piezoelectric potential.
Δ
P
sp
n
3
·
Piezoelectric Polarization
In contrast to the spontaneous polarization, the piezoelectric polarization vector can
have nonzero components in all three spatial directions in a QD or a QDM. In
addition,
P
pz
)
is position dependent. Using the integral expression of the strain field, Eq. (
6.1
), in
combination with Maxwell's equations, one can derive equations to calculate the
position-dependent piezoelectric potential
is not a constant vector within the QD, since the strain field
ε
(
r
ij
φ
(
r
)
by evaluating two 2-D integrals
pz
over the QD surface [
38
]:
J
K
x
i
)
2
QD
(
x
i
−
1
d
S
+
d
S
,
n
3
n
3
φ
(
r
)=
·
·
(6.5)
pz
|
r
−
r
|
3
|
r
−
r
|
QD
where the constants
J
and
K
are functions of the piezoelectric coefficient
e
ij
,the
Poisson ratio
ε
0
. When calculating strain and polarization
potentials we apply a linear interpolation for all parameters except for the sponta-
neous polarization, where a quadratic interpolation is applied [
44
].
We will discuss in Sect.
6.3
the general features of the built-in potential arising
from spontaneous and piezoelectric polarization in an isolated nitride-based QD.
However, before discussing this, we give an overview of the theoretical framework
which we use for the electronic structure calculations, including how the effects of
strain and built-in fields are incorporated in the model.
ν
and the initial misfit
6.2.2
Electronic Structure Calculations
To study the electronic properties of isolated QDs and QDMs different approaches
have been used in the literature, ranging from continuum-based models, e.g. single-
band effective mass [
45
,
46
] and multi-band
k
·
p
[
47
-
49
] theory, to atomistic
models such as pseudo-potential [
29
,
50
-
52
] and tight-binding (TB) calculations
[
28
,
53
-
55
]. Here, to analyze the electronic structure of InGaN/GaN QDMs we
apply an empirical
sp
3
TB model [
55
,
56
]. The TB matrix elements are treated as
parameters and are determined by fitting the TB band structures for InN and GaN
