Biomedical Engineering Reference
In-Depth Information
9.2
The Model and the Methodology
9.2.1
Model
The system under study is composed of two vertically stacked semiconductor QDs
spatially separated by a distance
D
of the order of few nanometers (Fig.
9.1
).
In the simplest approach, the wave functions of carriers confined in the QDs are
modeled by identical anisotropic Gaussians with identical extensions
l
in the
xy
plane and
l
z
along the growth direction for electrons and holes,
4
l
z
l
exp
x
2
y
2
z
2
l
z
1
1
2
+
1
2
Ψ
(
)
∼
−
−
.
r
(9.1)
3
/
l
2
π
This is an approximation, since the QDs formed in the two layers differ slightly in
size and shape, but most of the effects discussed within this chapter do not depend
considerably on the carrier wave function shapes.
A more accurate model can be based on Bir-Pikus 8 band
k
p
Hamiltonian
accounting also for the strain distribution in the system. The strain tensor is found
by minimization of the elastic energy of the system [
43
] and using for calculating
the position-dependent band edges and the effective masses. The electron and hole
wave functions are then found by a combination of Lowdin perturbation theory
[
44
], a generalization of “adiabatic” separation of variables [
45
] and Ritz variational
method [
46
]. While most of the results presented here, in particular those related
to optical properties of the system, rely on the simplified model described above,
the
k
·
p
theory is indispensable to reveal the actual physics of phonon-assisted
tunneling, to be described in Sect.
9.3.1
.
In all the theoretical results presented here, the discussion is restricted to the
ground states of electrons and heavy holes in both dots. Taking spin into account,
this yields 16 electron-hole (exciton) configurations. This superfluous richness
is reduced by restricting the model to spin-bright configurations (see [
47
]fora
discussion of optical selection rules) and to “spatially direct” states, i.e., states for
which electron-hole pairs forming the excitons reside in the same QD. Due to strong
·
Fig. 9.1
Schematic
representation of a pair of
vertically stacked QDs
