Biomedical Engineering Reference
In-Depth Information
smaller than the strain domain due to the strong spatial confinement of the electron
and hole wave functions inside the QD region. The electronic domain has fixed
boundary conditions in all spacial directions. The atoms at the surface are passivated
according to our published approach [
38
].
The inter-band optical transition strengths between the electron and hole energy
states are computed using Fermi's golden rule by calculating the squared absolute
value of the momentum matrix elements summed over spin degenerate states
[
33
,
35
]:
E
i
|
[
n
2
T
E
i
−
H
i
=
| <
,
H
]
|
H
i
> |
(5.3)
where
H
is the single particle tight binding Hamiltonian in the sp
3
d
5
s
∗
basis,
E
i
is
an electron energy state,
H
i
is a hole energy state, and
n
is a selected polarization
direction. The polarization-dependent optical modes are calculated by rotating the
polarization vector
n
(
x
+
y
)cos
+
z
cos
=
φ
sin
θ
θ
along the appropriate
90
◦
and
45
◦
,forthe
θ
=
φ
=
direction in the polar coordinates: for the TE
[
110
]
:
90
◦
and
135
◦
, and for the TM
[
001
]
:
0
◦
. Here the angles
θ
=
φ
=
θ
=
TE
:
φ
and
[
110
]
θ
are measured from the [100] and [001] axis in the polar coordinate system.
5.3
Electronic Properties
5.3.1
Hydrostatic and Biaxial Strains
Figure
5.2
plots the hydrostatic
∈
H
=
∈
xx
+
∈
yy
+
∈
zz
(dotted lines) and biaxial
∈
B
=
∈
xx
+
∈
yy
−
∈
zz
(solid lines) strain profiles along the [001] direction through
the center of the quantum dot systems shown in Fig.
5.1
. The hydrostatic strain
exhibits a very slight change from the SQD to the QDM-4. The biaxial strain,
however, significantly changes as the size of the QDM increases. For the single
QD layer, the biaxial strain is highly negative inside the QD region. This is typical
for any flat-shaped InAs QD compressively grown on GaAs substrate.
As the vertical size of the QDM is increased by adding QD layers, the biaxial
strain at the center of the molecule reduces. For the QDM-4, as shown in the
Fig.
5.2
e, the biaxial strain at the center of the molecule approaches zero. The reason
for such behavior of the biaxial strain is that in general the InAs unit cells inside the
QD region tend to fit over the GaAs matrix by an in-plane compression and an
elongation along the [001] direction. This results in highly negative biaxial strain as
is observed for the SQD system in Fig.
5.2
a. However, when the size of the QDM
increases, the unit cells of InAs around the center of the molecule feel lesser and
lesser compressive force from the surrounding GaAs. As a result, the vertical lattice
constant of the InAs starts matching with the GaAs and hence the biaxial strain tends
to change its sign around the middle of the QD stack. Similar strain profiles were
reported in an earlier study related to the columnar QDs by Saito et al. [
8
].
2
