Biomedical Engineering Reference
In-Depth Information
Fig. 9.2
(
a
) The basis states of a pair of QDs. (
b
) The energy levels in a pair of QDs
Coulomb electron-hole attraction the energy of such states is much lower than the
energy of “dissociated” states (this can change if external electric fields are applied
[
11
,
12
]). In modeling excitonic systems, we also assume that the polarization of the
exciting laser pulse is adjusted in such a way that it corresponds to the polarization
eigenstate of the excitons. This allows us to include only one out of two bright states
in each dot, since in both dots only one fundamental transition is allowed. With these
restrictions, we may describe each QD as a two-level system with the basis states
|
|
denoting an empty dot and a dot occupied with an exciton, respectively.
Consequently, excitons in a pair of QDs may be modeled as a four-level system
with the ground state
0
and
1
|
|
00
referring to two empty dots; single-exciton states are
10
|
and
with electron-hole pairs residing in the lower or upper QD, respectively;
and the “molecular biexciton” state
01
|
11
representing both QDs occupied with single
excitons (Fig.
9.2
a).
Typically, the QDs have different fundamental excitonic transition energies
E
1
and
E
2
for the two QDs. It is convenient to use the average transition energy
E
=
(
+
)
/
Δ
=(
−
)
/
E
1
E
2
2 and transition energy mismatch
E
1
E
2
2 to express the exciton
binding energies of the two QDs,
E
1
=
E
+
Δ
and
E
2
=
E
−
Δ
.
(9.2)
Due to the interaction of static dipole moments of excitons confined in the two QDs,
the energy of the biexcitonic state is shifted from the sum of single-exciton transition
energies
E
1
+
E
2
by
V
B
(biexcitonic shit). Assuming that the energy of the ground
state is zero, the Hamiltonian of the excitons confined in a pair of QDs in this simple
model takes the form
H
DQD
=(
E
+
Δ
)
|
10
10
|
+(
E
−
Δ
)
|
01
01
|
+(
2
E
+
V
B
)
|
11
11
|
V
|
|
,
+
10
01
|
+
|
01
10
(9.3)
where
V
is an amplitude of the coupling between single-exciton states of the system
which may originate either from Coulomb (Forster) interactions or from the tunnel
coupling. The schematic diagram of excitonic states in a DQD is presented in
Fig.
9.2
b.
