Biomedical Engineering Reference
In-Depth Information
Data presented in Table
4.2
show that for CdSe/ZnS quantum dots with two
ZnS monolayers
k
q
values follow a monotonous function drastically decaying with
the QD core diameter. For comparison, CdSe QDs without a ZnS shell show a
much stronger quenching than the respective core-shell analogue. With respect to
porphyrins, it was found [
62
,
90
,
101
,
127
] that the observed QD PL quenching
in “QD-H
2
P” nanoassemblies does not show a remarkable dependence on the
electronic structure and properties of attached dyes, and a non-specified quenching
mechanism has to be postulated, which does not depend on the electronic structure
of the attached dye. Therefore we concentrate only on those aspects that are related
directly to the outer interface of a QD as a function of size and shell thickness.
At the outer interface of the “QD-H
2
P” nanoassembly two contributions are
important: (1) the presence of “binding” orbitals such as the lone pair of a pyridyl
ring of the porphyrin or pyridine itself and (2) the evanescent wave function of the
confined exciton “leaking” out of the core (and the ZnS shell) of the QD. From
a physicochemical point of view it means that upon interaction of (m-Pyr)
4
-H
2
P
with the QD surface, the electron wave function may be locally modified (e.g., via
inductive and/or mesomeric effects [
90
]) forming a surface local state capable to
trap the electron of the photogenerated electron-hole pair. So, as in any related
quantum mechanical problem the wave function of the delocalized electron may
extend beyond an imposed barrier or, in other words, the corresponding particle
may tunnel through the barrier. With respect to QDs, quantum confinement of the
excitonic wave function in its most simple version confines the wave function of the
exciton (electron-hole pair) to a spherical box [
148
].
The general scheme of our model is shown in Fig.
4.21
a. In addition to the
coordination of a pyridyl ring to the surface the electron wave function at the outer
interface of the QD is presented schematically. Correspondingly, all observed size
dependence of the PL quenching (as expressed by the quenching rate
k
q
) should be
directly related to the amplitude of the electron wave function at the interface. The
calculation of changes of charge densities is therefore reduced to an evaluation of
the radial probability function
r
2
2
(
r
) for the electron as a function of the CdSe
core diameter and ZnS shell thickness. Now the QD PL quenching rate becomes
ψ
2
k
q
(
r
)=
C
ψ
(
r
)
.
(4.12)
ψ
a
in a spherical core/shell QD are
determined by the Schr odinger equation (the particle-in-a-box model [
149
]) by
h
2
2
The single-carrier envelope wave functions
r
2
1
2
∂
)
+
V
a
(
r
)
ψ
=
E
a
ψ
,
(4.13)
a
a
m
a
∗
(
∂
r
where the index
a
h
represents electron and hole, respectively. Solutions were
obtained by applying continuity relations for the single-carrier wave functions at
the interfaces between the CdSe core (
i
=
e
,
=
1), ZnS shell (
i
,
j
=
2), and matrix (
j
=
2),
respectively, by assuming that
