Biomedical Engineering Reference
In-Depth Information
Fig. 10.2
FRET process in a QDM (
a
) QDM schematics; two dissimilar
disk-shaped
vertically
stacked QDs. (
b
) Ground state excitons in each QD are represented by an electronic state (
blue
)
and a hole state (
red
) in the conduction band (CB) and valence band (VB), respectively. A FRET
interaction,
V
F
, de-excites an exciton on QD1 and transfers its energy to create an exciton on QD2.
(
c
) The FRET process represented as a single transition between two excitons
1
10
X
and
0
01
X
.Here
an exciton is denoted by
e
1
e
2
h
1
h
2
X
, with
h
i
,
e
i
being the occupation numbers on the
i
th QD
Within the envelope wave function approximation, the single carrier wave functions
are given by the products
(
)=
φ
(
)
(
)
(
)
, contains
the slow variation of the wave function amplitude over the QD, and most of the
important properties of single carrier states.
u
i
ψ
r
r
u
i
r
. The envelope part,
φ
r
i
i
i
(
)
is the Bloch wave function which
possesses the lattice periodicity, and it is important for deriving the optical transition
dipole moments, and many-body interactions. Therefore,
r
d
3
r
1
ψ
e
,
n
(
r
1
=
r
1
)
r
1
ψ
h
,
n
(
r
1
)
(10.2)
d
3
r
2
ψ
h
,
n
(
r
2
=
r
2
)
r
2
ψ
(
r
2
)
,
(10.3)
e
,
n
where
n
is the label for single particle states.
Using the results above, it is possible to separate the contributions arising from
the envelope and Bloch parts of the wave function. This results in a simplified
expression for the Forster coupling,
e
2
3
R
2
(
μ
1
·
V
F
=
−
πε
0
ε
r
R
3
O
1
O
2
[
μ
1
·
μ
2
−
R
)(
μ
2
·
R
)]
(10.4)
4
where
O
1
(
2
)
are the overlap integrals of the envelope wave functions for QD1 and
QD2, respectively,
d
3
r
O
i
=
φ
e
(
r
i
)
φ
h
(
r
i
)
.
(10.5)
