190
6 Effects of Quantisation
order of the particle’s (de-Broglie) wavelength entirely suppresses spatial propagation
2 and fully localises a particle.
3 Thereby, this increased certainty regarding the
spatial information leads to a smearing out of the momentum information related to
Heisenberg’s uncertainty principle. The finite momenta in every of those confinement
directions add up energy to the “boxed” particle and lead to a non-zero ground-state
energy. Simultaneously, the density of states and the energy spectrum are altered.
Confined particles are commonly viewed as standing waves regarding their quantummechanically-defined wave-functions between the walls of the confining potential
barriers.
This concept of confinement became particularly successful for electronic particles, such as quasi-free
4 electrons (excited into the conduction band) and holes
(defect electrons in the valence band) in the host crystal, and similarly applies to
compounds of particles (e.g. excitons, polaritons). When the wave-functions of an
excited electron and hole overlap, optical transitions become likely.
5 The higher
the spatial overlap between the particles, the higher the transition probabilities. A
measure of the overlap is conveniently given by the magnitude of a particle’s wavefunction |ψ(r, t)|. Such wave-function ψ(r, t) gives the probability amplitude at
space–time coordinate (r, t) (or in the steady-state picture ϕ(r)). Thus, to be more
precise, the spatial overlap is governed by the probabilities to find the respective
particles at a spot given by the square of their individual wave-function’s absolute
|ψ|
2 . Quantum-mechanically, the transition rate corresponds to the matrix element
for a given excitation/de-excitation in conjunction with the annihilation/creation part
of the electric field taking into account the initial and final states of the system (for
details see for instance [17]). Such confinement scenarios can particularly ramp up the
oscillator strength of a system’s optical resonance based on electron–hole recombination, and the Coulomb-binding energies between oppositely-charged constituents
of composite electronic quasi-particles, such as excitons.
Theoretically, in the case of an infinitely high potential well, the exciton Bohr
radius of confined excitons in 2D quantum wells is halved compared to the bulk
2 Note that the momentum in the quantum box is not zero, which leads to a finite confinement energy.
In other words, while the self-interfering particle is not propagating, its quantum-mechanical phase
is finite and becomes discrete. The wave-package forms standing wave-functions and exhibits an
enhanced probability to be found within the box. Amendment: The discrete nature of wave-vectors
becomes obvious under confinement conditions, and else, available particle states are evidenced as
quasi-continuous in macroscopic structures.
3 The motion of the wave-package is restricted by the potential barriers of the confinement structure.
Amendment: It means that the particle wave-function solutions must fulfil the (resonance) conditions
imposed by the boundaries of the potential box. Only those wave-vectors which result in constructive
interference forming standing waves prevail inside the box.
4 The expression “quasi” takes into account the effect of a periodic potential landscape in the lattice
of a crystal structure on propagating charge carriers.
5 This is characterised by the dipole transition-matrix element M ∝ d, the emitter’s dipole moment.
See Fermi’s golden rule, e.g. in [16, 17]. For an exciton, the dipole length is approximated by the
exciton Bohr radius of the bound electron–hole pair at given main quantum number n. For an atom,
the dipole moment is caused by the position change of electrons in the atom for a given transition
|b → |a (ground to excited state) corresponding to the optical frequency ω ab .
6 Effects of Quantisation
order of the particle’s (de-Broglie) wavelength entirely suppresses spatial propagation
2 and fully localises a particle.
3 Thereby, this increased certainty regarding the
spatial information leads to a smearing out of the momentum information related to
Heisenberg’s uncertainty principle. The finite momenta in every of those confinement
directions add up energy to the “boxed” particle and lead to a non-zero ground-state
energy. Simultaneously, the density of states and the energy spectrum are altered.
Confined particles are commonly viewed as standing waves regarding their quantummechanically-defined wave-functions between the walls of the confining potential
barriers.
This concept of confinement became particularly successful for electronic particles, such as quasi-free
4 electrons (excited into the conduction band) and holes
(defect electrons in the valence band) in the host crystal, and similarly applies to
compounds of particles (e.g. excitons, polaritons). When the wave-functions of an
excited electron and hole overlap, optical transitions become likely.
5 The higher
the spatial overlap between the particles, the higher the transition probabilities. A
measure of the overlap is conveniently given by the magnitude of a particle’s wavefunction |ψ(r, t)|. Such wave-function ψ(r, t) gives the probability amplitude at
space–time coordinate (r, t) (or in the steady-state picture ϕ(r)). Thus, to be more
precise, the spatial overlap is governed by the probabilities to find the respective
particles at a spot given by the square of their individual wave-function’s absolute
|ψ|
2 . Quantum-mechanically, the transition rate corresponds to the matrix element
for a given excitation/de-excitation in conjunction with the annihilation/creation part
of the electric field taking into account the initial and final states of the system (for
details see for instance [17]). Such confinement scenarios can particularly ramp up the
oscillator strength of a system’s optical resonance based on electron–hole recombination, and the Coulomb-binding energies between oppositely-charged constituents
of composite electronic quasi-particles, such as excitons.
Theoretically, in the case of an infinitely high potential well, the exciton Bohr
radius of confined excitons in 2D quantum wells is halved compared to the bulk
2 Note that the momentum in the quantum box is not zero, which leads to a finite confinement energy.
In other words, while the self-interfering particle is not propagating, its quantum-mechanical phase
is finite and becomes discrete. The wave-package forms standing wave-functions and exhibits an
enhanced probability to be found within the box. Amendment: The discrete nature of wave-vectors
becomes obvious under confinement conditions, and else, available particle states are evidenced as
quasi-continuous in macroscopic structures.
3 The motion of the wave-package is restricted by the potential barriers of the confinement structure.
Amendment: It means that the particle wave-function solutions must fulfil the (resonance) conditions
imposed by the boundaries of the potential box. Only those wave-vectors which result in constructive
interference forming standing waves prevail inside the box.
4 The expression “quasi” takes into account the effect of a periodic potential landscape in the lattice
of a crystal structure on propagating charge carriers.
5 This is characterised by the dipole transition-matrix element M ∝ d, the emitter’s dipole moment.
See Fermi’s golden rule, e.g. in [16, 17]. For an exciton, the dipole length is approximated by the
exciton Bohr radius of the bound electron–hole pair at given main quantum number n. For an atom,
the dipole moment is caused by the position change of electrons in the atom for a given transition
|b → |a (ground to excited state) corresponding to the optical frequency ω ab .