Biomedical Engineering Reference
In-Depth Information
Again, the benefit of using the surface integral method is that this approach admits
analytic solutions for certain QD geometries, and thereby provides useful insights
into the parameters that influence the magnitude and the shape of the built-in
potential. More details about this method are given in Refs. [
38
,
41
].
In nitride-based heterostructures, we have two contributions to the total built-in
potential. The first one is known as the spontaneous polarization
P
sp
, arising from
the lack of inversion symmetry along the
c
-axis of the wurtzite crystal structure [
42
].
The second contribution, referred to as the piezoelectric polarization
P
pz
, is related
to the local strain in the system [
42
]. The total polarization vector of a wurtzite
nitride system, treating piezoelectric response to first order, is given by [
38
]:
P
tot
P
sp
P
pz
=
+
⎛
⎞
⎛
⎞
0
0
P
sp
2
e
15
ε
13
2
e
15
ε
23
e
31
(
ε
11
+
ε
22
)+
⎝
⎠
+
⎝
⎠
,
=
(6.2)
e
33
ε
33
where
P
sp
is the spontaneous polarization,
e
ij
are the piezoelectric coefficients, and
ε
ij
denotes the strain tensor components.
It is simplest when introducing the surface integral technique to first consider
the contribution
sp
to the total potential arising from the spontaneous polarization,
before then extending the analysis to the piezoelectric contribution
φ
φ
pz
.
Spontaneous Polarization
When deriving the potential originating from the spontaneous polarization, we make
use of the fact that
P
sp
is a constant vector in the nanostructure region and the
surrounding barrier material, respectively, and points in both regions along the
c
-
axis. Using a standard result from electromagnetism theory [
43
], the spontaneous
polarization potential
φ
sp
can be written as [
38
]:
P
QD
P
sp
·
d
S
d
S
B
1
·
sp
φ
sp
=
r
|
+
(6.3)
4
πε
r
ε
0
|
r
−
|
r
−
r
B
|
QD
B
P
QD
P
sp
−
1
sp
d
S
,
≈
n
3
·
(6.4)
r
|
4
πε
r
ε
0
|
r
−
QD
where
P
QD
P
QD
=
n
3
is the spontaneous polarization of the QD material (e.g., InN)
sp
sp
and
P
sp
=
P
sp
n
3
is that of the barrier material (e.g., GaN). In Eq. (
6.3
), the first
integral is taken over the surface of the QD while the second is taken over all the
surfaces of the surrounding barrier material. Here we assume that the QD is buried
within an infinite matrix. The quantity
P
·
d
S
is equivalent to a fictitious charge
P
n
on a surface element
dS
,where
n
is the unit vector normal to the surface [
38
].
When going from Eq. (
6.3
)toEq.(
6.4
), we have neglected the electrostatic field
·
