188 9 Optical Properties
Figure 9.6 Comparison of the radius of a
particle with the radius of an exciton. When
the exciton radius is smaller than that of the
particle, there is only a minor dependency of
the optical absorption and emission on the
particle size. In the other case, called
“quantum confinement”, exciton formation is
not possible. Now an inverse quadratic
dependency of the wavelength of the emitted
lines is observed.
h
+
d ... parƟcle diameter
2 exciton
d
r
<
2 exciton
d
r
>
e
−
e
−
h
+
ex ci to n
r
Box 9.1 Energy of an Electron in a Small Particle – Quantum Confinement
Mathematically, it is possible to solve the Schrödinger equation for an electron
in a spherical particle. However, the gain of such a complex procedure is
smaller than the benefit in clarity. Therefore, one uses the model of an electron
in a one dimensional box to describe the solution of this problem. Using this
approach, this problem can be simplified even more, as follows:
The “particle in a box” problem can be described with an approach using the
basic laws of quantum mechanics. In a one-dimensional system, the condition
for a standing wave consisting of n half-waves of the wavelength λ n in a box
with the size l is:
n
l
n
λ
2
= .
(9.6)
Substituted into the DeBroglie relationship this leads to:
p mv
h nh
l
n
=
=
=
λ
2
.
(9.7)
In Eq. (9.7), p is the momentum, m the mass and v the velocity of the electron,
h is Planck’s constant. The energy E n and one obtains
E
mv
p
m
n h
mL
n
L
n =
=
=
∝
2
2
2 2
2
2
2
2
2
8
.
(9.8)
Equation (9.8) gives the main characteristic of quantum-confinement systems:
The energy of the electron is inversely quadratically proportional to the particle