Biomedical Engineering Reference
In-Depth Information
techniques are available to calculate the strain and the electrostatic built-in fields.
These approaches range from density functional theory-based calculations for small
systems (few thousands atoms) [
25
,
26
], through to atomistic descriptions based on
semi-empirical force field models [
27
-
30
], which can treat up to several millions
of atoms, and on to continuum-based descriptions [
27
,
31
,
32
], which in turn can
deal with even larger systems. Using a continuum-based description, analytic results
can be derived by assuming isotropic and homogenous elastic properties. Despite
the simplifications involved, these techniques provide considerable insight into the
strain field of a variety of different nanostructures [
33
-
36
].
On the same footing, assuming isotropic and homogenous material parameters,
Davies [
37
] and Williams et al. [
38
] derived real space surface integral methods to
describe the electrostatic built-in fields in cubic and wurtzite-based semiconductor
nanostructures. We summarize here how these real space surface integral methods
can be used to calculate and understand the behavior of the strain and built-in
potentials in nitride-based heterostructures.
6.2.1.1
Calculation of the Strain Field
To calculate the strain field in a QD system, we apply the surface integral method
discussed in detail in Refs. [
35
,
37
]. This approach is based on Eshelby's theory
of inclusion [
39
] to express the stress and strain for a QD embedded in an infinite
medium as an integral over the nanostructure under consideration. The strain tensor
components
for isotropic and homogenous elastic constants can be written as
integrals over the surface of the QD [
35
,
37
]:
ε
ij
(
r
)
x
i
)
)=
δ
ij
ε
0
χ
QD
+
ε
0
A
4
QD
(
x
i
−
d
S
,
ε
ij
(
r
n
j
·
(6.1)
π
|
r
−
r
|
3
where the primed quantities denote points on the surface of the dot,
(
x
1
,
x
2
,
x
3
)
≡
(
χ
QD
is defined as the dot characteristic function,
which is equal to 1 inside the QD and zero outside. The constant
A
x
,
y
,
z
)
,
ε
0
is the initial misfit and
1
+
ν
1
=
,where
−
ν
ν
is the Poisson ratio [
38
]. The unit vector along the
j
-direction is denoted by
n
j
.
The strain field of a QDM can be calculated by a superposition of the strain fields
obtained from Eq. (
6.1
) for two independent dots centered at different positions. We
note also for a cuboid QD that the strain components
ε
ii
at a given point are directly
related to the solid angles subtended by the two cuboid surfaces perpendicular to
n
i
.
This simple relation can be very useful for the design and analysis of different QD
structures [
36
,
40
].
6.2.1.2
Calculation of the Built-In Potential
To treat strain and built-in potential on the same theoretical footing we use the
surface integral method developed by Williams et al. [
38
] to achieve a realistic
description of the electrostatic built-in potentials in nitride-based QDs and QDMs.
