194
6 Effects of Quantisation
Fig. 6.2 Sketch of an energy potential well with relative ground state E 0 and potential barrier
imposed by the energetics of the environment (barrier) E barrier . In the top row, the situation is
schematically drawn for a large well with negligible confinement effect on a massive electronic
particle with quantum-mechanical wavelength λ dB . The quantum particle’s wave-function within
the well does not self-interfere (relevant length scale is the coherence length) and behaves like an
isolated wave-package with a group velocity and kinetic energy of a classical particle. The available
energy states become quasi-continuous with increasing well width L (here in direction x), i.e. they
are so densely packed that their discrete character becomes negligible. The corresponding density of
states for (quasi-)3D is sketched as an example to the right of the figure. In the bottom row, the effect
of size reduction is sketched for the confinement structure. With decreasing well width, the possible
quantum states spread further apart energetically and the number of confined modes reduces. This
is also represented by the schematic density of states for the three well width examples to the right
of the figure. Strong confinement is achieved for potential well widths of the order of λ dB . Too
small features do not exhibit any bound states and the particle remains delocalised. This simple 1D
representation can be easily used to imagine the quantisation effects for 2D and 3D confinement
as well (also see Fig. 6.1). Discrete energy states have been at the core of various quantum devices
and have delivered a strong impact on device functionalities in optoelectronics
Discrete energies or bands are not restricted to electronic systems, as photonic
resonator modes in microresonators and energy stop bands in photonic crystals show.
A 1D representative of an optical crystal is a distributed Bragg reflector, from which
very high reflectivities can be obtained by a sequence of quarter-wavelength layers
(a and b) with alternating refractive index
7 n a/b . By sandwiching an optical half7 Conveniently, this magnitude (i.e. the refractive index of a material) can be coarsely regarded as
inverse to the electronic band gap: the higher the gap, the lower the refractive index. For a more
accurate picture, the interested reader is referred to the Kramers–Kronig relations, see e.g. [21].
Amendment: The higher the refractive index contrast between the alternating layers (materials)
of a Bragg mirror (photonic crystal), the better the overall reflectivity of the photonic stop band
becomes. Note that the fundamental concept applies: Where there is no energy state for a particle,
there can be no penetration of that particle into that specific region. More precisely expressed: If a
physical environment lacks particle-specific energy states (modes) in a certain spectral region, e.g.
by means of periodic arrangement of potential landscapes, such particle with energies within that
range cannot be transmitted through that environment (unless through barrier tunnelling) and will
be fully reflected. Nowadays, even electronic or photonic structures are investigated that can have
6 Effects of Quantisation
Fig. 6.2 Sketch of an energy potential well with relative ground state E 0 and potential barrier
imposed by the energetics of the environment (barrier) E barrier . In the top row, the situation is
schematically drawn for a large well with negligible confinement effect on a massive electronic
particle with quantum-mechanical wavelength λ dB . The quantum particle’s wave-function within
the well does not self-interfere (relevant length scale is the coherence length) and behaves like an
isolated wave-package with a group velocity and kinetic energy of a classical particle. The available
energy states become quasi-continuous with increasing well width L (here in direction x), i.e. they
are so densely packed that their discrete character becomes negligible. The corresponding density of
states for (quasi-)3D is sketched as an example to the right of the figure. In the bottom row, the effect
of size reduction is sketched for the confinement structure. With decreasing well width, the possible
quantum states spread further apart energetically and the number of confined modes reduces. This
is also represented by the schematic density of states for the three well width examples to the right
of the figure. Strong confinement is achieved for potential well widths of the order of λ dB . Too
small features do not exhibit any bound states and the particle remains delocalised. This simple 1D
representation can be easily used to imagine the quantisation effects for 2D and 3D confinement
as well (also see Fig. 6.1). Discrete energy states have been at the core of various quantum devices
and have delivered a strong impact on device functionalities in optoelectronics
Discrete energies or bands are not restricted to electronic systems, as photonic
resonator modes in microresonators and energy stop bands in photonic crystals show.
A 1D representative of an optical crystal is a distributed Bragg reflector, from which
very high reflectivities can be obtained by a sequence of quarter-wavelength layers
(a and b) with alternating refractive index
7 n a/b . By sandwiching an optical half7 Conveniently, this magnitude (i.e. the refractive index of a material) can be coarsely regarded as
inverse to the electronic band gap: the higher the gap, the lower the refractive index. For a more
accurate picture, the interested reader is referred to the Kramers–Kronig relations, see e.g. [21].
Amendment: The higher the refractive index contrast between the alternating layers (materials)
of a Bragg mirror (photonic crystal), the better the overall reflectivity of the photonic stop band
becomes. Note that the fundamental concept applies: Where there is no energy state for a particle,
there can be no penetration of that particle into that specific region. More precisely expressed: If a
physical environment lacks particle-specific energy states (modes) in a certain spectral region, e.g.
by means of periodic arrangement of potential landscapes, such particle with energies within that
range cannot be transmitted through that environment (unless through barrier tunnelling) and will
be fully reflected. Nowadays, even electronic or photonic structures are investigated that can have