Biomedical Engineering Reference
In-Depth Information
[
]
is able to relax both in the
x
-
y
-plane (
c
-plane) and also along the
-direction,
so that the magnitude of each of the axial strain components is then reduced in the
QD compared to a QW of the same composition (
0001
QD
ii
QW
ii
|
ε
|≤|
ε
|
). The strain-related
piezoelectric potential drop
pz
across the nanostructure is shown as a function
of
F
in Fig.
6.2
b. The full calculation, based on the results given in [
38
], is shown
by the solid line. The (red) dashed line shows the contribution from the first term
on the RHS of Eq. (
6.10
), while the (blue) dashed-dotted line includes both terms
on the RHS of Eq. (
6.10
). From Fig.
6.2
b we see that the piezoelectric potential
drop
Δφ
=
.
4, with approximately equal
contributions to this reduction from the strain redistribution in the QD system and
the reduction of the QD surface area compared to a QW.
A similar analysis was presented for a cylindrical QD in [
41
], which again
showed a significant reduction in the potential drop across such a QD compared to a
QW of the same composition and height. It should be noted when comparing Fig.
6.2
with Fig. 5 in [
41
] that the potential drop in the cylindrical shaped QD analyzed there
was presented in terms of
f
Δφ
pz
is reduced by approximately 60% for
F
0
=
/
R
,where
R
is the radius and
h
the
total
height.
The same analysis can also be carried out for more realistic QD geometries such
as ellipsoidal or lens-shaped InGaN QDs [
68
-
70
]. We will discuss the experimental
findings on QD geometries, dimensions, and composition range in more detail in
Sect.
6.4
. Here, only the overall geometry is important. Using the results given in
[
38
] for an ellipsoidal dot centered at the origin, with semi-major radius
a
and semi-
minor radius
b
, we calculate the potential drop across the nanostructure as a function
of
F
h
a
.
2
=
/
b
The results for the calculated drop in the spontaneous potential
Δφ
sp
across the QD are shown in Fig.
6.3
a, while those for
pz
are shown in Fig.
6.3
b,
with the approximate solutions for small
F
in both cases including terms up to
F
3
. Comparing the results to those for the cuboid-shaped QD (cf. Fig.
6.2
a), we
observe that the drop in the spontaneous potential difference
Δφ
sp
(cf. Fig.
6.3
a) is
clearly increased in the case of the ellipsoid-shaped dot. The same is true for the
drop in the piezoelectric potential difference
Δφ
pz
, as shown in Figs.
6.2
band
6.3
b,
respectively. This further reduction is directly due to the curved shape of the QD
side walls. In a cuboid QD, each point in the dot is below all points on the full
[0001]-oriented top surface, so that all surface points then give contributions of the
same sign to the total potential. With a curved top surface, points near the top of the
dot experience contributions of opposite sign from points on the upper surface of
the dot which are below and above the given point, leading to an overall reduction
in the total potential change.
A similar effect can be seen in Fig.
6.4
, where we plot the normalized variation
of the drop in the potential difference when going from a QW to a lens-shaped
QD as a function of
F
Δφ
=
/
D
,where
h
is the height of the lens and
D
the base
diameter. By comparing the results of the lens-shaped system, the ellipsoidal QD
and the cylindrical shaped dot [
41
] for the same aspect ratio (e.g.,
F
h
=
.
=
0
5;
f
1),
c
2
3
b
2
is missing in Eq. (B13) for
I
1
c
2
b
2
a
2
−
+
z
(
z
±
b
)
2
Note that the term
∓
2
π
sgn
(
z
±
b
)(
z
±
b
)
+
c
2
c
2
z
2
+
given in the appendix of Ref. [
38
] for an ellipsoid-shaped system.
