Biomedical Engineering Reference
In-Depth Information
The analyzed QD system is stacked in a semiconductor material, and thus
subjected to lattice vibration of the surrounding material, that is, phonons. The free
phonon modes are described by the Hamiltonian
H ph = q
hw q b q b q ,
(9.4)
where b q , b q are bosonic operatorscreating and annihilating phonon modes with
wave vector q and hw q are the corresponding energies.
Interaction of carriers confined in the DQD with phonons is modeled by the
Hamiltonian
b q +
q
b
= nm q
(
) |
|
,
H DQD ph
F nm
q
n
m
(9.5)
where F nm (
are the coupling constants specific for a given problem (see, e.g.,
[ 48 - 50 ] for a review). Depending on the system under consideration, either only
deformation potential coupling or both deformation and piezoelectric coupling has
to be included.
We also investigate the interaction of the QD system with its radiative surround-
ing. The Hamiltonian of the photon reservoir is
q
)
H rad = k
ω k c k λ
h
c q λ ,
where operators c k λ
, c k λ
create and annihilate photon modes with wave vectors k
and polarization
ω k denote the corresponding energies. In all the cases
discussed here, the spatial separation D of the two dots is of the order of a few
nanometers, that is two to three orders of magnitude smaller than the wavelength
of the radiation with which the QDs interact (for the wide-gap semiconductors
investigated here, the fundamental transition energy is of the order of 1 eV which
corresponds to the wavelength
λ
, while h
240 nm in vacuum). This allows us to describe
the coupling between excitons and the photon environment in the Dicke limit [ 51 ]
where the spatial dependence of an electromagnetic field within the pair of QDs
is neglected. The relevant coupling Hamiltonian in the dipole and rotating wave
approximation is
1
,
k
λ =
g k λ c
H DQD rad = Σ
k λ +
H
.
c
.,
(9.6)
1
,
2
Σ = σ ( 1 + σ ( 2 )
where
is a collective exciton annihilation operator and
h
ω k
g k λ =
·
e λ (
) .
i
ε 0 ε r v d
k
(9.7)
2
 
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