Biomedical Engineering Reference
In-Depth Information
phonon dynamics within the Fermi golden rule approach. We study phonon-assisted
transitions between two lowest carrier states. In particular, if these states describe a
carrier localized in different dots, the phonon-assisted relaxation leads to a charge
transfer in the system.
Phonon-assisted relaxation in the QDM in the case of a single electron has been
widely studied in the literature. In [
78
], the system containing two lateral dots cou-
pled by an inter-dot tunneling barrier was investigated. Two lowest electron states
were calculated in a relatively simple model using a perturbation method combined
with unitary transformation. The calculation of the phonon-assisted relaxation rate
in this paper accounts for coupling by a deformation potential as well as coupling by
a piezoelectric field. Furthermore, in [
79
] a similar system was investigated. Here,
the phonon-induced decoherence caused by the piezoelectric field coupling was
calculated within Born-Markov approximation. A laterally coupled DQD system
has been studied in [
80
]. In that paper, the confining potential in the
z
direction
was modeled as an infinite quantum well, while the in-plane confinement was
described as two parabolic wells separated by an inter-dot distance. Calculations
of the phonon-assisted relaxation rate were performed within the Fermi golden rule.
In [
81
] phonon-related processes in the system of stacked QDs were investigated. In
a plane, the potential was modeled as an isotropic parabolic confinement, while in
the growth direction the system was approximated by two rectangular wells with a
finite barrier. The interaction of the electron with LO phonons was described using
self-consistent Green's function formalism combined with degenerate perturbation
approach. Finally, the model of the phonon-assisted relaxation in the case of two
electrons in a lateral DQD was described in [
82
]. The wave functions were assumed
to be a linear combination of Gaussians localized on each dot. In [
83
] a similar
system was investigated. In that work, authors, using non-perturbative quantum
kinetics theory, described carrier-phonon kinetics in the presence of both diagonal
and off-diagonal couplings. The result shows the importance of off-diagonal terms
and indicates the need of using non-Markovian theory. Experimentally, the phonon-
assisted relaxation in the system containing a single hole in a vertically stacked
QDM was measured [
84
], but also in this case only a simple model was used to
interpret the results.
Here, we present the result of our modeling based on Bir-Pikus 8 band
k
p
theory
[
76
,
77
]. The shape of both dots is modeled as a spherical segment with the base
radius of 10 nm and the height 3
·
.
7 nm. The dots are grown on the wetting layer
with thickness of 0
6 nm. Due to the lattice mismatch between dots and matrix
material, the strain appears in the system and affects the band structure. The strain
tensor is calculated by minimization of the elastic energy of the system [
43
]. Using
the strain tensor components, the position-dependent band edges and the effective
masses are found. In the case of a single electron as well as for a single hole, L owdin
perturbation method is used to separate the electron and hole part from the full
Hamiltonian. By generalizing the method of “adiabatic” separation of variables [
46
],
we calculate
z
-dependent components of the wave functions. The radial parts are
then obtained using Ritz variational method [
76
]. The phonon-assisted transition
rates are calculated from the Fermi golden rule, using the carrier wave functions
obtained in this way.
.
