this is quite a simplified situation, the conclusions, drawn by using such onedimensional box, are correct. The assumption of infinite walls is justified in the case
of insulating ceramic materials and, in most cases with good approximation, for
semiconductors, too. This is an idealized description of a free electron in a
nanoparticle with the diameter d. Generally, this problem of quantum confinement
in a particle is solved using the Schr€ odinger equation; however, a “particle in a box”
can be described with a simplified 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 l in a box with the size L (representing
the particle diameter) is:
nl
2
¼ L
ð9:4Þ
When substituted into the DeBroglie relationship, this leads to:
p ¼ mv ¼
h
l
¼
nh
2L
ð9:5Þ
where p is the momentum, m is the mass, v is the velocity of the electron, and h is
Planck’s constant. Now one can calculate the energy E n , obtaining:
E n ¼
mv
2
2
¼
p
2
2m
¼
h
2
2l
2 m
¼
n
2
h
2
8mL
2 ¼ k
n
2
L
2
ð9:6Þ
Equation (9.6) states the main characteristic of quantum confinement systems, as
it shows that the energy of the electron is inversely quadratic to the particle diameter.
This describes the blue shift of the absorption edges with decreasing particle size.
The energy difference DE between two quantum levels n and n þ 1 describes the
energy of an emitted photon:
DE ¼
n þ 1
ð
Þ
2 À n
2
h
i
h
2
8mL
2
¼
2n þ 1
ð
Þ h
2
8mL
2
leading to l ¼
8mcL
2
2n À 1
ð
Þ h
ð9:7Þ
Equation (9.7) clearly shows the blue shift with decreasing size L of the box, which
is equivalent to the particle diameter d, or the wavelength of the emitted photon
increases quadratically with the particle size d. However, it must be borne in mind
that the blue shift is not only observed in the quantum confinement case.
The solution of the Schr€ odinger equation leads to deeper insights. In order to
solve the “particle in a box” problem, potential walls limiting the particle must be
assumed. In the case of an insulating particle these walls are infinitely high and,
therefore, it is impossible for electrons to tunnel to the space outside the particle.
With decreasing height of the potential wall, the probability of finding electrons
outside the particle is increasing, as the electrons are tunneling into the space
outside the box. On the left-hand side of Figure 9.6 the electron density distribution
in an insulator is shown for quantum levels 1, 2, and 3. For a metallic particle, the
same data are plotted at the right-hand side of Figure 9.6. Independent of
the quantum level, there is a nonzero probability of finding electrons outside the
9.3 Optical Properties Related to Quantum Confinement j211
of insulating ceramic materials and, in most cases with good approximation, for
semiconductors, too. This is an idealized description of a free electron in a
nanoparticle with the diameter d. Generally, this problem of quantum confinement
in a particle is solved using the Schr€ odinger equation; however, a “particle in a box”
can be described with a simplified 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 l in a box with the size L (representing
the particle diameter) is:
nl
2
¼ L
ð9:4Þ
When substituted into the DeBroglie relationship, this leads to:
p ¼ mv ¼
h
l
¼
nh
2L
ð9:5Þ
where p is the momentum, m is the mass, v is the velocity of the electron, and h is
Planck’s constant. Now one can calculate the energy E n , obtaining:
E n ¼
mv
2
2
¼
p
2
2m
¼
h
2
2l
2 m
¼
n
2
h
2
8mL
2 ¼ k
n
2
L
2
ð9:6Þ
Equation (9.6) states the main characteristic of quantum confinement systems, as
it shows that the energy of the electron is inversely quadratic to the particle diameter.
This describes the blue shift of the absorption edges with decreasing particle size.
The energy difference DE between two quantum levels n and n þ 1 describes the
energy of an emitted photon:
DE ¼
n þ 1
ð
Þ
2 À n
2
h
i
h
2
8mL
2
¼
2n þ 1
ð
Þ h
2
8mL
2
leading to l ¼
8mcL
2
2n À 1
ð
Þ h
ð9:7Þ
Equation (9.7) clearly shows the blue shift with decreasing size L of the box, which
is equivalent to the particle diameter d, or the wavelength of the emitted photon
increases quadratically with the particle size d. However, it must be borne in mind
that the blue shift is not only observed in the quantum confinement case.
The solution of the Schr€ odinger equation leads to deeper insights. In order to
solve the “particle in a box” problem, potential walls limiting the particle must be
assumed. In the case of an insulating particle these walls are infinitely high and,
therefore, it is impossible for electrons to tunnel to the space outside the particle.
With decreasing height of the potential wall, the probability of finding electrons
outside the particle is increasing, as the electrons are tunneling into the space
outside the box. On the left-hand side of Figure 9.6 the electron density distribution
in an insulator is shown for quantum levels 1, 2, and 3. For a metallic particle, the
same data are plotted at the right-hand side of Figure 9.6. Independent of
the quantum level, there is a nonzero probability of finding electrons outside the
9.3 Optical Properties Related to Quantum Confinement j211