Chapter 3 – Making Light Work in Biology 85
states that relate not only to the material properties of their crystals but also to their
physical dimensions and shapes. These include primarily quantum rods and QDs. The
physics of their fluorescence properties are similar, and we use QDs as an exemplar here
since these, in particular, have found significant applications in biophysics. They have
advantages of being relatively bright, but their diameter is roughly an order of magni
tude smaller than micro-/nanospheres. QDs have a photostability >100 times that
of most organic dyes and so can be considered not to undergo significant irreversible
photobleaching for most experimental applications. They thus have many advantages for
monitoring single biomolecules. They are made from nanocrystal alloy spheres typically
of two to three components (cadmium selenide (CdSe) and cadmium telluride (CdTe)
are the most common) containing ~100 atoms. They are ~3–5 nm in core diameter
(Figure 3.4c) and have semiconductor properties, which can undergo fluorescence due to
an exciton resonance effect within the whole nanocrystal, with the energy of fluorescence
relating to their precise length scale dimensions. An exciton is a correlated particle pairing
composed of an electron and electron hole. It is analogous to the excited electron state of
traditional fluorophores but has a significantly longer lifetime of ~10−6 s.
The fluorescence emission spectrum of a QD is dependent on its size. A QD is an example
of quantum confinement of a particle in a box in all three spatial dimensions, where the particle
in question is an exciton. In other words, QDs have size-dependent optical properties. The 1D
case of a particle in a box can be solved as follows. The Schrödinger equation can be written as
(3.33)
−
+
( )
=
( )
h
m x
V x
x
E
x
2
2
2
2
8π
ψ
ψ
d
d
( )
where
ψ(x) is the wave function of a particle of mass m at distance x
V is the potential energy
E is the total energy
h is Planck’s constant
For a “free” particle, V is zero, and it is trivial to show that a sinusoidal solution exists, such
that if the probability of being at the ends of the “1D box” (i.e., a line) of length a is zero, this
leads to allowed energies of the form
(3.34)
E
n h
ma
n =
2
2
2
8
where n is a positive integer (1, 2, 3, …); hence, the energy levels of the particle are discrete or
quantized. This formulation can be generalized to a 3D Cartesian box of dimensions a, b, and
c parallel to the x, y, and z axes, respectively, which yields solutions of the form
(3.35)
E
h
m
n
a
n
b
n
c
n
x
y
z
=
+
+
2
2
2
2
2
2
2
8
QDs, however, has a spherical geometry, and so Schrödinger’s equation must be solved in
spherical polar coordinates. Also, the different effective masses of the electron me and hole
mH (mH > me in general) need to be considered as do the electrostatic ground state energy and
the bulk energy of the semiconductor, which leads to
(3.36)
E
E
h
m
m
q
a
bulk
e
H
r
=
+
+
−
2
2
0
8
1
1
1 8
4
.
πε ε
where
q is the unitary electron charge
Ebulk is the bulk semiconductor energy