Biomedical Engineering Reference
In-Depth Information
The inter-band dipole matrix elements, connecting the electron and hole Bloch wave
functions,
u
e
(
h
)
(
r
)
,aregivenby
e
u
e
(
d
3
r
i
.
μ
i
=
r
i
)
r
i
u
h
(
r
i
)
(10.6)
Using the axial symmetry of the problem, and assuming QDs with transition
dipole moments along the in-plane direction, the expression for
V
F
reduces to
μ
μ
1
2
V
F
=
−
πε
0
ε
r
R
3
κ
.
(10.7)
4
In the model,
2
are not calculated explicitly, since more accurate values are
readily available from experiments [
12
-
14
]. The interaction strength of
V
F
depends
on the degree of exciton confinement within each QD (via the electron-hole wave
function overlap). It also depends on the relative orientation of their dipole moments,
κ
=
μ
1
·
μ
2
μ
1
μ
2
μ
1
and
μ
; assuming in-plane and parallel permanent dipoles,
κ
∼
1. Furthermore,
2
h
V
F
Θ
in this approximation the FRET exciton hopping rate,
K
DA
=
, depends on
the spectral overlap
between the donor emission and acceptor absorption [
15
,
16
].
However, the excitonic levels in self-assembled QDs are spectrally narrow, which
typically makes FRET signatures difficult to observe in experiments. On the other
hand, the energy mismatch,
Θ
X
12
, between the bottom (
1
10
X
)andtop(
0
01
X
) exciton
levels, depends on growth conditions, so QDs might be grown to have near resonant
exciton transitions [
17
,
18
]. For QDMs such that
Δ
V
F
, the resonant energy
transfer is nearly coherent “exciton hopping”; on the other hand, when
Δ
≤
X
12
V
F
,
the transfer is incoherent, requiring phonon-assisted transitions via auxiliary excited
levels [
19
].
Δ
X
12
10.4
Effective Hamiltonian
In what follows, we consider a QDM formed by vertically stacking two self-
assembled InGaAs/GaAs QDs, a top QD (T) and a bottom QD (B) separated
by a distance
d
. We assume QDs grown by the Stranski-Krastanov method
on a GaAs host matrix, so the barrier material separating the dots is mostly
GaAs with InGaAs/GaAs alloying on the QD boundaries [
20
]. Exciton transition
energies in single QDs are critically affected by strain-induced modifications to the
single particle conduction and valence band offsets. In addition, the QDM has an
intrinsic broken inversion symmetry, which leads to a difference between the hole
confinement potential experienced at the base of the top dot from that experienced at
the base of the bottom dot [
21
]. This difference suppresses interdot hole tunneling,
which typically is measured to be an order of magnitude less than electron tunneling.
Atomistic pseudopotential calculations of exciton wave functions on strained QDMs
have shown that holes are mostly heavy-hole like, experiencing a higher interdot
