188
6 Effects of Quantisation
entering the regime of quantum structures. In other words, it means that when one of
the structure’s dimensions becomes comparable to a (quasi-)particle’s wavelength
1
or size confined in it, classical descriptions of the system’s behaviour fail. Such
low-dimensional structures are well described in many text books and the interested
reader is referred to the literature for a more complete and detailed description of
quantum-physical concepts [1–5].
Nowadays, all kinds of properties, such as electronic, optical and mechanical, are
tailored by size reductions towards the respective quantisation length scale. Typical
examples are quantum wells, wires and dots for electrons and their analogues for
light in the form of microresonators (e.g. [6]) and waveguides (e.g. [7]), widely
used in optoelectronic and quantum-technological devices (cf. [8–10]). Simultaneous
electronic and optical confinement is for instance important for strong light–matter
interactions in optical microcavities, giving rise to polariton physics, and the study of
quantum coupling effects, as well as many-body phenomena in solids (cf. [11–15]).
6.2 From Bulk to Zero-Dimensional Structures
The reduction of size is a powerful tool to achieve spatial confinement (of a physical entity, i.e. a fundamental or composite particle) in any of the desired spatial
dimensions. Such a reduction in different dimensions is sketched in Fig. 6.1. Its consequences are directly noticeable in the reciprocal space, the phase space, in which
the possible particle states are defined in terms of energy and momentum. While in
some cases, the sole purpose of size reduction is given by an enhanced spatial confinement of particles, trapping and waveguiding, other scenarios strongly harness the
effects of quantisation introduced typically on the nanoscale for electronic and on the
microscale for photonic features. In the following, the effects of spatial confinement
shall be highlighted regarding the density of particle states and the energy spectrum,
and some of their uses as examples. However, also the trivial effect of localisation
and its use is summarised.
6.2.1 Spatial Confinement
Spatial confinement can serve different purposes and opens up many possibilities
to design and alter device physics. The most typical purpose remains the confinement of charge carriers in a region of interest in terms of localisation, while different
dimensionalities can still give rise to a certain degree of mobility. In contrast to inplane confinement by wells and solely longitudinal mobility in wires, the extreme
case of an atomistic confinement potential of a 0D box with all length scales on the
1 Characterised by the de-Broglie wavelength λ = h/ p = h/mv = h/
√
2m E, the thermal representation of which reads λ dB (T ) =
2π 2 /(mk B T ), using the relationship E = π k B T = p 2 /2m.
6 Effects of Quantisation
entering the regime of quantum structures. In other words, it means that when one of
the structure’s dimensions becomes comparable to a (quasi-)particle’s wavelength
1
or size confined in it, classical descriptions of the system’s behaviour fail. Such
low-dimensional structures are well described in many text books and the interested
reader is referred to the literature for a more complete and detailed description of
quantum-physical concepts [1–5].
Nowadays, all kinds of properties, such as electronic, optical and mechanical, are
tailored by size reductions towards the respective quantisation length scale. Typical
examples are quantum wells, wires and dots for electrons and their analogues for
light in the form of microresonators (e.g. [6]) and waveguides (e.g. [7]), widely
used in optoelectronic and quantum-technological devices (cf. [8–10]). Simultaneous
electronic and optical confinement is for instance important for strong light–matter
interactions in optical microcavities, giving rise to polariton physics, and the study of
quantum coupling effects, as well as many-body phenomena in solids (cf. [11–15]).
6.2 From Bulk to Zero-Dimensional Structures
The reduction of size is a powerful tool to achieve spatial confinement (of a physical entity, i.e. a fundamental or composite particle) in any of the desired spatial
dimensions. Such a reduction in different dimensions is sketched in Fig. 6.1. Its consequences are directly noticeable in the reciprocal space, the phase space, in which
the possible particle states are defined in terms of energy and momentum. While in
some cases, the sole purpose of size reduction is given by an enhanced spatial confinement of particles, trapping and waveguiding, other scenarios strongly harness the
effects of quantisation introduced typically on the nanoscale for electronic and on the
microscale for photonic features. In the following, the effects of spatial confinement
shall be highlighted regarding the density of particle states and the energy spectrum,
and some of their uses as examples. However, also the trivial effect of localisation
and its use is summarised.
6.2.1 Spatial Confinement
Spatial confinement can serve different purposes and opens up many possibilities
to design and alter device physics. The most typical purpose remains the confinement of charge carriers in a region of interest in terms of localisation, while different
dimensionalities can still give rise to a certain degree of mobility. In contrast to inplane confinement by wells and solely longitudinal mobility in wires, the extreme
case of an atomistic confinement potential of a 0D box with all length scales on the
1 Characterised by the de-Broglie wavelength λ = h/ p = h/mv = h/
√
2m E, the thermal representation of which reads λ dB (T ) =
2π 2 /(mk B T ), using the relationship E = π k B T = p 2 /2m.