Biomedical Engineering Reference
In-Depth Information
6.4.2
Sign of e
15
from the Piezoelectric Tensor
in (111)-Oriented Zinc Blende Structures
We have recently derived expressions for the stiffness tensor and the elastic energy
in (111)-zinc blende systems along with equations for the piezoelectric polarization
vectors in these structures [
86
]. These equations allow for an efficient calculation of
strain and piezoelectric fields in (111)-oriented zinc blende nanostructures, such
as site-controlled (111)-oriented InGaAs/GaAs QDs, which are most promising
candidates for sources of entangled photon pairs [
87
,
88
].
Moreover, since (111)-oriented zinc blende and
c
-plane wurtzite systems have
several common features in their crystal structure [
86
,
89
], comparison between
these two systems can prove very beneficial. In addition to providing insight into
the key factors that modify strain and built-in fields in (111)-oriented zinc blende
heterostructures, our approach offers furthermore the opportunity to gain insights
into material parameters for the wurtzite AlN, InN and GaN systems, including,
for example, the sign of the shear strain-related piezoelectric constant
e
15
.From
our analysis of the first-order piezoelectric tensor in (111)-zinc blende systems we
obtain, that the piezoelectric coefficients
e
31
and
e
15
should be equal and negative,
e
15
/
2. When comparing these ratios with
ratios derived from literature piezoelectric coefficients, we find very good agreement
for GaN and InN and good agreement for AlN, as long as the literature value of
e
15
is
negative [
86
]. From this we conclude that
e
15
should be negative in wurtzite nitrides.
More details about the method and in-depth discussions are given in [
86
].
e
31
=
1, and also that the ratio
e
33
/
e
31
=
−
6.4.3
Impact of the Sign of
e
15
on the Built-In Potential
in a Single InGaN/GaN QD
In this subsection we study the impact of the sign of
e
15
on the potential in an
isolated QD. Here, the piezoelectric coefficients
e
31
and
e
33
are taken from [
78
]. In
the following we apply the often used
positive e
15
values of Vurgaftman et al. [
44
]
and the
negative
values of Shimada [
78
] for InN and GaN, respectively. We note
that the ratios of the ab initio
e
ij
values calculated by Shimada [
78
] are in good
agreement with the ratios predicted from the comparison between (111)-zinc blende
and (0001)-wurtzite structures discussed in the previous section.
To investigate the influence of the sign of
e
15
on the total potential
φ
tot
,we
have calculated and discussed the different contributions to
φ
tot
in an isolated lens-
shaped InGaN/GaN QD separately in [
90
]. Here, we summarize the overall result
how
e
15
affects the total built-in potential
φ
tot
. Figure
6.5
shows the total built-in
potential
φ
tot
for a line-scan along the
z
-(
c
-) axis through the center of a lens-shaped
In
0
.
25
Ga
0
.
75
N/GaN QD with diameter
d
=
19
.
2 nm and height
h
=
3
.
1nmfor
e
15
>
0
and
e
15
<
0, respectively. One can infer from Fig.
6.5
that the sign of
e
15
has mainly
two effects on the total built-in potential
φ
tot
. Firstly, with
e
15
<
0, the potential
