3.1 Where Strong Interactions with Light Matters
67
Note that in most practical systems and experiments, one is not in the strong but
weak-coupling regime and observes or studies behaviour such as linear absorption,
irreversible spontaneous emission, ASE/lasing, Purcell effect etc., typically the case
for interactions of (e.g. incoherent, broadband, unconfined) light with low-oscillator
strength optical dipole transitions (e.g. electron–hole plasma, microscopic polarisation) and strongly-dissipative systems (e.g. leaky cavities, bulk) or systems with high
dephasing rate (e.g. with high scattering rates, phonon coupling) (cf. [7, 87, 107]).
Semiconductor Crystal Situation
For the semiconductor crystal interacting with a monochromatic light field, the manybody Hamiltonian taking into account excitations of the mechanical system (lattice)
H ph , the electronic system H X , the light field H F , and two interaction terms H F−X and
H X−ph concerning light and excitons as well as excitons and phonons, respectively,
reads:
H =
H ph + H X + H F + H F−X + H X−ph
.
(3.3)
For negligible phonon contributions, this Hamiltonian reduces to a similar one as
in (3.2). Here, the role of the atom is taken by the exciton, the quantum of matter
excitation, with H X = E Q e
†
Q e Q (Q is the centre-of-mass momentum). The obvious
similarity can be easily seen from the time-independent eigen-value equation [87,
108] for the Wannier-like Coulomb-bound electron–hole pair (a composite quasiparticle) in matter:
−
2
∇
2
2μ
−
e
2
4ππ |r |
ϕ α (r ) = E α ϕ α (r ),
(3.4)
whereas μ, = r 0 , e and r denote the reduced two-particle system’s mass, the
permittivity, the electron charge and the electron–hole distance, respectively. E α and
ϕ α (r ) are the exciton’s eigen-energies and (envelope) wave-functions, respectively,
for the quantum numbers α = n, l, m (main quantum number, angular momentum
and magnetic quantum number for atoms, respectively) very similar to the situation
for the hydrogen atom. Quantitatively different by the reduced mass and the dielectric
screening in the solid, this equation (also referred to as Wannier equation) yields
a kinetic free-particle term for the centre-of-mass motion and a series of (bound)
hydrogenic states in the Coulomb potential referred to as excitonic Rydberg series,
in analogy to the atom Rydberg series. Thus, exciton binding energies are in most
semiconductors about 2–3 orders of magnitude smaller than the Rydberg energy
13.6 eV. Of course, the situation is drastically altered in 2D semiconductors due to
of the interested reader is also drawn to an effect with practical relevance for nano-(opto-)mechanical
systems known as the Casimir effect, one explanation of which involves the pressure from virtual
photons in 3D/bulk, or the underpressure in 1D-confined space regarding virtual photons—the other
explanation involving van-der-Waals interactions, the explanation of which from a quantum physics
point involves vacuum fluctuations (here, an interesting connection to 2D materials, also referred
to as vdW materials, can be established).
67
Note that in most practical systems and experiments, one is not in the strong but
weak-coupling regime and observes or studies behaviour such as linear absorption,
irreversible spontaneous emission, ASE/lasing, Purcell effect etc., typically the case
for interactions of (e.g. incoherent, broadband, unconfined) light with low-oscillator
strength optical dipole transitions (e.g. electron–hole plasma, microscopic polarisation) and strongly-dissipative systems (e.g. leaky cavities, bulk) or systems with high
dephasing rate (e.g. with high scattering rates, phonon coupling) (cf. [7, 87, 107]).
Semiconductor Crystal Situation
For the semiconductor crystal interacting with a monochromatic light field, the manybody Hamiltonian taking into account excitations of the mechanical system (lattice)
H ph , the electronic system H X , the light field H F , and two interaction terms H F−X and
H X−ph concerning light and excitons as well as excitons and phonons, respectively,
reads:
H =
H ph + H X + H F + H F−X + H X−ph
.
(3.3)
For negligible phonon contributions, this Hamiltonian reduces to a similar one as
in (3.2). Here, the role of the atom is taken by the exciton, the quantum of matter
excitation, with H X = E Q e
†
Q e Q (Q is the centre-of-mass momentum). The obvious
similarity can be easily seen from the time-independent eigen-value equation [87,
108] for the Wannier-like Coulomb-bound electron–hole pair (a composite quasiparticle) in matter:
−
2
∇
2
2μ
−
e
2
4ππ |r |
ϕ α (r ) = E α ϕ α (r ),
(3.4)
whereas μ, = r 0 , e and r denote the reduced two-particle system’s mass, the
permittivity, the electron charge and the electron–hole distance, respectively. E α and
ϕ α (r ) are the exciton’s eigen-energies and (envelope) wave-functions, respectively,
for the quantum numbers α = n, l, m (main quantum number, angular momentum
and magnetic quantum number for atoms, respectively) very similar to the situation
for the hydrogen atom. Quantitatively different by the reduced mass and the dielectric
screening in the solid, this equation (also referred to as Wannier equation) yields
a kinetic free-particle term for the centre-of-mass motion and a series of (bound)
hydrogenic states in the Coulomb potential referred to as excitonic Rydberg series,
in analogy to the atom Rydberg series. Thus, exciton binding energies are in most
semiconductors about 2–3 orders of magnitude smaller than the Rydberg energy
13.6 eV. Of course, the situation is drastically altered in 2D semiconductors due to
of the interested reader is also drawn to an effect with practical relevance for nano-(opto-)mechanical
systems known as the Casimir effect, one explanation of which involves the pressure from virtual
photons in 3D/bulk, or the underpressure in 1D-confined space regarding virtual photons—the other
explanation involving van-der-Waals interactions, the explanation of which from a quantum physics
point involves vacuum fluctuations (here, an interesting connection to 2D materials, also referred
to as vdW materials, can be established).