192
6 Effects of Quantisation
with increasing energy (third column of Fig. 6.1). The quantum box of a 0D system
shows the characteristic signatures of atoms with completely discrete delta peaks,
where states only exist at the defined energies of the quantum dot (last column of
Fig. 6.1). Accordingly, the D(E) ∝ δ(E − E i ), where i is the quantum number of
the energy level.
While in three dimensions the DOS has its minimum at the bottom of the band,
the feature at the bottom of the band is stronger in fewer dimensions. This influences
heavily optical properties, such as absorption and the inverse process stimulated
emission. Because of the higher density of states at the bottom of the band, lowdimensional structures are strongly favoured for optoelectronic devices.
6.2.3 Discrete Energies
The reduction of a structure’s dimension down to the strong-confinement regime leads
to quantisation effects that give rise to discrete energies for the respective direction.
In the remaining directions without boundaries, particle propagation is not hindered
and thus the free-particle kinetic energy term with the respective wave-vector for such
propagation is preserved. In the case of a 0D system, which corresponds to a strong
3D confinement, atomistic levels are the consequence, similar to an atom’s electron
situated in the (effective) Coulomb potential of its positively-charged atom core.
In fact, the distribution of energy levels strongly depends on the potential shape.
For simplicity, the principle is discussed in common textbooks using rectangular
potentials, with additional examples often providing triangular or parabolic wells
for comparison. Another important factor regarding the energies in a confinement
potential is the height of the walls, its potential barriers.
For every confinement direction, the wave-function must strictly fulfil certain
requirements imposed by fixed boundary conditions (fbc). The solution of a potentialwell Schrödinger equation consists of a sine behaviour (standing wave) for the
restricted direction, e.g.
ψ ∝ sin(k x x)
(6.1)
for confinement in x direction. Consequently, momenta become discrete, i.e.
k x = n x π/L x , (fbc),
(6.2)
where L x is the width of the quantum well in x and n x = 1, 2, . . . are integer quantum
numbers of the system.
In contrast, for those directions with negligible confinement and surface effects—
which holds true for macroscopic bodies—periodic boundary conditions (pbc) are
applied, mimicking a ring-like system where the tail ends up in the head of the ring,
thereby leading to a closed entity resembling an infinite system. Here,
k y = n y (2π/L y ), (pbc)
(6.3)
6 Effects of Quantisation
with increasing energy (third column of Fig. 6.1). The quantum box of a 0D system
shows the characteristic signatures of atoms with completely discrete delta peaks,
where states only exist at the defined energies of the quantum dot (last column of
Fig. 6.1). Accordingly, the D(E) ∝ δ(E − E i ), where i is the quantum number of
the energy level.
While in three dimensions the DOS has its minimum at the bottom of the band,
the feature at the bottom of the band is stronger in fewer dimensions. This influences
heavily optical properties, such as absorption and the inverse process stimulated
emission. Because of the higher density of states at the bottom of the band, lowdimensional structures are strongly favoured for optoelectronic devices.
6.2.3 Discrete Energies
The reduction of a structure’s dimension down to the strong-confinement regime leads
to quantisation effects that give rise to discrete energies for the respective direction.
In the remaining directions without boundaries, particle propagation is not hindered
and thus the free-particle kinetic energy term with the respective wave-vector for such
propagation is preserved. In the case of a 0D system, which corresponds to a strong
3D confinement, atomistic levels are the consequence, similar to an atom’s electron
situated in the (effective) Coulomb potential of its positively-charged atom core.
In fact, the distribution of energy levels strongly depends on the potential shape.
For simplicity, the principle is discussed in common textbooks using rectangular
potentials, with additional examples often providing triangular or parabolic wells
for comparison. Another important factor regarding the energies in a confinement
potential is the height of the walls, its potential barriers.
For every confinement direction, the wave-function must strictly fulfil certain
requirements imposed by fixed boundary conditions (fbc). The solution of a potentialwell Schrödinger equation consists of a sine behaviour (standing wave) for the
restricted direction, e.g.
ψ ∝ sin(k x x)
(6.1)
for confinement in x direction. Consequently, momenta become discrete, i.e.
k x = n x π/L x , (fbc),
(6.2)
where L x is the width of the quantum well in x and n x = 1, 2, . . . are integer quantum
numbers of the system.
In contrast, for those directions with negligible confinement and surface effects—
which holds true for macroscopic bodies—periodic boundary conditions (pbc) are
applied, mimicking a ring-like system where the tail ends up in the head of the ring,
thereby leading to a closed entity resembling an infinite system. Here,
k y = n y (2π/L y ), (pbc)
(6.3)