3.2 Matter Excitations
69
the surpassing of a laser threshold condition (see [121] for a definition of a laser
threshold).
5 The lower the threshold, the more efficient the laser.
3.2.1 Excitons as Composite Bosons
In contrast to higher-density conditions, the picture looks different at the quantum
limit of single-particle excitation. Here, the fundamental excitation of matter is characterised by the formation of a hydrogen-like quasi-particle [122] consisting of an
excited negatively-charged electron in the conduction band (CB) Coulomb-bound to
a positively charged “hole”—a defect electron in the valence band (VB) (Fig. 3.4).
These Rydberg-like excitons
6 deliver spectral features very similar to their atomic
counterparts, as investigations on cuprous-oxide excitons in bulk crystals with their
large binding energies revealed [123, 124]. In a diluted ‘gas’, the description of these
quasi-particles as ideal bosons and individual systems holds true, since the screening effects from other uncorrelated carriers and excitation-induced dephasing can be
neglected. In quantum wells, excitons, which are composite bosons, are attractive for
investigations of collective properties and quantum phenomena [125, 126], such as
Bose–Einstein condensation [127–131]. However, in many semiconductor systems,
exciton condensation was not possible due to the typically unfavourable conditions.
On the one hand, condensate droplets formation can take place in potential landscapes of real
7 quantum wells, leading to a system without superfluidity, referred to
as Bose glass. On the other hand, excitons are prone to disorder and density-dependent
dephasing effects which make the formation of a long-range order unlikely in many
practical systems even at ultralow temperatures.
5 To achieve lasing conditions, either an optical pump density or an electrical current density exceeds
the threshold density, which according to laser rate equations fundamentally depends on the optical
losses τ C and the light–matter interaction strength W i f ∝ |M| 2 ∝ d. Here, W i f is the transition rate,
which can be expressed through the transition matrix element M = f | d · E |i. i and f are initial
and final state, respectively. The transition rate is proportional to the emitter’s dipole moment d,
thus, optical gain as well.
6 Coulomb-bound electron–hole pairs in solids are quasi-particles with hydrogen-like Rydberg series
of energy levels, resembling rather a positronium atom than hydrogen due to the similarity of the
masses of electron and hole, and the fact that positronium is also a particle–anti-particle compound,
whereas in hydrogen, an electron is bound to the much heavier proton (itself a composite particle built
from quarks). The exciton binding energies can vary between weak for Wannier–Mott-like species
to strong for Frenkel-like species, with their wave-function delocalised over many lattice sites or
strongly localised, respectively. The oscillator strength of the dipole determines the interaction
strength with electromagnetic waves. In low-dimensional crystal structures, quantum confinement
can raise the binding energies with respect to the bulk crystal.
7 Real refers to the situation in practical systems achieved by semiconductor growth technology in
contrast to ideal systems.
69
the surpassing of a laser threshold condition (see [121] for a definition of a laser
threshold).
5 The lower the threshold, the more efficient the laser.
3.2.1 Excitons as Composite Bosons
In contrast to higher-density conditions, the picture looks different at the quantum
limit of single-particle excitation. Here, the fundamental excitation of matter is characterised by the formation of a hydrogen-like quasi-particle [122] consisting of an
excited negatively-charged electron in the conduction band (CB) Coulomb-bound to
a positively charged “hole”—a defect electron in the valence band (VB) (Fig. 3.4).
These Rydberg-like excitons
6 deliver spectral features very similar to their atomic
counterparts, as investigations on cuprous-oxide excitons in bulk crystals with their
large binding energies revealed [123, 124]. In a diluted ‘gas’, the description of these
quasi-particles as ideal bosons and individual systems holds true, since the screening effects from other uncorrelated carriers and excitation-induced dephasing can be
neglected. In quantum wells, excitons, which are composite bosons, are attractive for
investigations of collective properties and quantum phenomena [125, 126], such as
Bose–Einstein condensation [127–131]. However, in many semiconductor systems,
exciton condensation was not possible due to the typically unfavourable conditions.
On the one hand, condensate droplets formation can take place in potential landscapes of real
7 quantum wells, leading to a system without superfluidity, referred to
as Bose glass. On the other hand, excitons are prone to disorder and density-dependent
dephasing effects which make the formation of a long-range order unlikely in many
practical systems even at ultralow temperatures.
5 To achieve lasing conditions, either an optical pump density or an electrical current density exceeds
the threshold density, which according to laser rate equations fundamentally depends on the optical
losses τ C and the light–matter interaction strength W i f ∝ |M| 2 ∝ d. Here, W i f is the transition rate,
which can be expressed through the transition matrix element M = f | d · E |i. i and f are initial
and final state, respectively. The transition rate is proportional to the emitter’s dipole moment d,
thus, optical gain as well.
6 Coulomb-bound electron–hole pairs in solids are quasi-particles with hydrogen-like Rydberg series
of energy levels, resembling rather a positronium atom than hydrogen due to the similarity of the
masses of electron and hole, and the fact that positronium is also a particle–anti-particle compound,
whereas in hydrogen, an electron is bound to the much heavier proton (itself a composite particle built
from quarks). The exciton binding energies can vary between weak for Wannier–Mott-like species
to strong for Frenkel-like species, with their wave-function delocalised over many lattice sites or
strongly localised, respectively. The oscillator strength of the dipole determines the interaction
strength with electromagnetic waves. In low-dimensional crystal structures, quantum confinement
can raise the binding energies with respect to the bulk crystal.
7 Real refers to the situation in practical systems achieved by semiconductor growth technology in
contrast to ideal systems.