Biomedical Engineering Reference
In-Depth Information
S
xy
=
2
D
5
ε
12
,
√
2
D
6
=
,
S
xz
ε
13
√
2
D
6
=
,
S
yz
ε
23
where the
D
i
's denote the valence-band deformation potentials, while
a
cp
and
a
ct
are the conduction-band deformation potentials.
1
With this approach, the relevant
deformation potentials for the highest valence and lowest conduction band states
are included directly without any fitting procedure. In the work described below,
the deformation potentials for InN and GaN are taken from hybrid-functional
density functional theory calculations [
63
], and a linear interpolation is used to
obtain the parameters for In
x
Ga
1
−
x
N. Our approach is similar to that used for
the strain dependence in an 8-band
k
·
p
model [
47
], but has the benefit that the
TB Hamiltonian still takes the correct
C
3
v
symmetry of the system into account,
including explicitly the detail of the dot-barrier interface region.
The built-in potential
tot
, arising from both spontaneous and piezoelectric polar-
ization is likewise included as a site-diagonal contribution in the TB Hamiltonian.
This also is a widely used approach [
54
,
64
-
66
].
To perform a realistic calculation of the electronic structure of InGaN/GaN QDs
and QDMs, knowledge about the dot size, shape and composition is required,
as well as a consistent set of input material parameters, such as band offsets,
elastic constants, piezoelectric coefficients, etc. Even though a lot of progress has
been made in recent years on the growth of high-quality nitride samples, there is
still a large degree of uncertainty in some of the key material parameters, such
as the piezoelectric coefficients
e
ij
[
13
]. We briefly review these uncertainties in
the piezoelectric coefficients in Sect.
6.5
and show how these uncertainties affect
the calculated electrostatic built-in fields in isolated QDs and QDMs. However,
before turning to such details, we focus our attention in the next section on more
general aspects of the built-in potential in an isolated dot. This will establish the
foundations for understanding the more complex behavior of the built-in field and
of the electronic structure in stacked InGaN/GaN QDs.
φ
6.3
Built-In Fields in Nitride-Based QDs: General Aspects
and Simple Analytic Solutions
Having outlined the general aspects of the theoretical framework used here to
study isolated and stacked InGaN/GaN QDs with realistic dimensions and indium
contents, it is useful to next consider the general shape and form of the electrostatic
built-in potential of a single InGaN/GaN dot. In the following section we briefly
1
Note that the quantities
a
2
and
a
1
given by Vurgaftman and Meyer in their 2003 review article [
44
]
are not
the conduction-band deformation potentials
a
cp
and
a
ct
, respectively. The quantities
denoted by
a
1
and
a
2
in [
44
]arethe
band gap
deformation potentials, e.g.
a
1
=
a
cp
−
D
1
and
