3.2 Matter Excitations
73
(3.6) by the term in brackets, whereas → (ω, k) and related quantities become
k-dependent in case of spatial dispersion [87].
8 For an introductory purpose and for
clarity, schematic diagrams of these dispersion relations are shown in Fig. 3.6c, d
based on the Lorentz oscillator model for the dielectric function (a, b). Note that
in (3.6), the sum over different resonances (counting index i) with their specific
frequencies and homogeneous linewidths is considered for the dielectric function,
which in the case of a single isolated resonance in a given spectral region can be
approximated without sum.
Similar to the introductory overview given in [159], a brief summary is analogously provided here: When coupling to a photon mode is impossible—either due to
dipole-forbidden transitions or due to the absence of light states at high momentumspace vectors—bare excitons are found in the lattice of the host medium, also known
as dark excitons (indeed, there can be different types of dark excitons, see [160]
and references therein). In contrast, the optical resonance of this quasi-particle state,
which is a consequence of exciton–polariton formation, has its peculiar energy (for
simplicity it is often just referred to as the exciton peak): it is on the one hand
less energetic than the direct band-gap transition, due to the binding energy of the
electronically-neutral bound system, which can be described in the two-particle picture with a free-particle dispersion and excitonic effective mass (cf. Fig. 3.4c). On
the other hand, the optical transitions occurring only at very low momenta within the
range of light states (within the light cone) exhibit a shifted energy with respect to
the bare exciton resonance, since the bright (i.e. optically visible) exciton is usually
discussed in the picture of polaritons (see [85, 87, 161–163] and references therein).
9
The shift of the longitudinal branch compared to the commonly discussed transversal
exciton strongly depends on the oscillator strength. It can range from 0.08 meV for
GaAs [164] over tens of meV to even ≈1 eV for transitions in molecular crystals
[165].
The polaritons represent the quantisation of the polarisation—i.e. they are the
eigen-states (basis vectors) of the light–matter Hamiltonian [113]—and are mixed
states of photon modes and the relevant material resonance. One should note that,
here, the coupling with the ordinary light cone (photon states) in the material is
discussed in contrast to cavity–polaritons that are the product of coupling with a
cavity mode. In a typical PL scenario, the excitation takes place in the upper polariton (UP) branch within the light cone [87, 166, 167]. From there, the excitation
scatters/relaxes to lower lying states in the upper or lower polariton (LP) branch.
8 This is not detailed here, as it would be quickly out of the scope of this work, but briefly addressed
for clarity. Nevertheless, it shall be noted that while f and γ likely are considered k-independent,
the references provided in Chap. 5 of [87] experimentally show a dependency for dipole-allowed
transitions. In the sketched polariton dispersions of Fig. 3.6c, d, the literature model only considered
a parabolic dispersion ω 0 (k) with prefactor A.
9 Note that a typical semiconductor PL spectrum with ‘bright’ modes can be dominated by boundexciton-complexes emission and phonon sidebands as well (see [87]), and the emission at the
exciton resonance does not necessarily represent coherent excitons, but rather incoherent excitons
(microscopic polarisation) with a significant contribution from electron–hole plasma towards higher
densities (cf. [113, 117, 118]).
73
(3.6) by the term in brackets, whereas → (ω, k) and related quantities become
k-dependent in case of spatial dispersion [87].
8 For an introductory purpose and for
clarity, schematic diagrams of these dispersion relations are shown in Fig. 3.6c, d
based on the Lorentz oscillator model for the dielectric function (a, b). Note that
in (3.6), the sum over different resonances (counting index i) with their specific
frequencies and homogeneous linewidths is considered for the dielectric function,
which in the case of a single isolated resonance in a given spectral region can be
approximated without sum.
Similar to the introductory overview given in [159], a brief summary is analogously provided here: When coupling to a photon mode is impossible—either due to
dipole-forbidden transitions or due to the absence of light states at high momentumspace vectors—bare excitons are found in the lattice of the host medium, also known
as dark excitons (indeed, there can be different types of dark excitons, see [160]
and references therein). In contrast, the optical resonance of this quasi-particle state,
which is a consequence of exciton–polariton formation, has its peculiar energy (for
simplicity it is often just referred to as the exciton peak): it is on the one hand
less energetic than the direct band-gap transition, due to the binding energy of the
electronically-neutral bound system, which can be described in the two-particle picture with a free-particle dispersion and excitonic effective mass (cf. Fig. 3.4c). On
the other hand, the optical transitions occurring only at very low momenta within the
range of light states (within the light cone) exhibit a shifted energy with respect to
the bare exciton resonance, since the bright (i.e. optically visible) exciton is usually
discussed in the picture of polaritons (see [85, 87, 161–163] and references therein).
9
The shift of the longitudinal branch compared to the commonly discussed transversal
exciton strongly depends on the oscillator strength. It can range from 0.08 meV for
GaAs [164] over tens of meV to even ≈1 eV for transitions in molecular crystals
[165].
The polaritons represent the quantisation of the polarisation—i.e. they are the
eigen-states (basis vectors) of the light–matter Hamiltonian [113]—and are mixed
states of photon modes and the relevant material resonance. One should note that,
here, the coupling with the ordinary light cone (photon states) in the material is
discussed in contrast to cavity–polaritons that are the product of coupling with a
cavity mode. In a typical PL scenario, the excitation takes place in the upper polariton (UP) branch within the light cone [87, 166, 167]. From there, the excitation
scatters/relaxes to lower lying states in the upper or lower polariton (LP) branch.
8 This is not detailed here, as it would be quickly out of the scope of this work, but briefly addressed
for clarity. Nevertheless, it shall be noted that while f and γ likely are considered k-independent,
the references provided in Chap. 5 of [87] experimentally show a dependency for dipole-allowed
transitions. In the sketched polariton dispersions of Fig. 3.6c, d, the literature model only considered
a parabolic dispersion ω 0 (k) with prefactor A.
9 Note that a typical semiconductor PL spectrum with ‘bright’ modes can be dominated by boundexciton-complexes emission and phonon sidebands as well (see [87]), and the emission at the
exciton resonance does not necessarily represent coherent excitons, but rather incoherent excitons
(microscopic polarisation) with a significant contribution from electron–hole plasma towards higher
densities (cf. [113, 117, 118]).