The interaction of light with semiconductors links microscopic quantum electrodynamics with macroscopic optoelectronic devices. Whether a semiconductor absorbs or emits light with high quantum efficiency depends decisively on whether its fundamental bandgap is direct or indirect, and on the formation of Coulomb-bound electron-hole quasiparticles called excitons.
1. Direct Optical Absorption and Fermi’s Golden Rule
When an incident electromagnetic field with vector potential (mathbf{A}(t) = A_0 hat{mathbf{e}} e^{i(mathbf{q} cdot mathbf{r} – omega t)}) impinges on a semiconductor, it perturbs the crystal Hamiltonian via the minimal-coupling momentum operator (hat{H}’ = -rac{e}{m_0} mathbf{A} cdot hat{mathbf{p}}). The transition rate per unit volume from a full valence band state (|v, mathbf{k}
angle) to an empty conduction band state (|c, mathbf{k}’
angle) is evaluated via Fermi’s Golden Rule:
[
W_{vc} = rac{2pi}{hbar} sum_{mathbf{k}, mathbf{k}’} |langle c, mathbf{k}’ | hat{H}’ | v, mathbf{k}
angle|^2 delta(E_c(mathbf{k}’) – E_v(mathbf{k}) – hbar omega)
]
Because the momentum of an optical photon ((q = 2pi/lambda approx 10^5 ext{ cm}^{-1})) is negligible compared to the Brillouin zone dimensions ((pi/a approx 10^8 ext{ cm}^{-1})), photon absorption preserves crystal momentum strictly: (mathbf{k}’ = mathbf{k}). This corresponds to a strictly vertical transition on the band diagram.
Evaluating the optical absorption coefficient (alpha(hbar omega)) for parabolic 3D bands yields the fundamental direct-bandgap absorption profile:
[
alpha_{ ext{direct}}(hbar omega) = rac{C_0}{hbar omega} sqrt{hbar omega – E_g} quad ( ext{for } hbar omega ge E_g)
]
where (C_0) depends on the interband dipole momentum matrix element (P_{cv} = |langle c | hat{mathbf{e}} cdot hat{mathbf{p}} | v
angle|). In direct bandgap semiconductors like GaAs and InP, (alpha) rises steeply to (sim 10^4 – 10^5 ext{ cm}^{-1}) immediately above the band edge, enabling micron-thin solar cells and efficient laser diodes.
2. Indirect Bandgaps: Second-Order Phonon-Assisted Absorption
In indirect bandgap semiconductors such as Silicon (conduction band minima along (Delta), valence band maximum at (Gamma)) and Germanium (minima at (L)), the conduction band minimum and valence band maximum occur at completely different wavevectors (mathbf{k}_c
eq mathbf{k}_v). A direct vertical optical transition would violate momentum conservation.
Consequently, absorption must proceed as a second-order quantum perturbation mediated by the simultaneous absorption or emission of a momentum-conserving phonon with wavevector (mathbf{q} approx mathbf{k}_c – mathbf{k}_v) and energy (hbar Omega_{mathbf{q}}):
[
alpha_{ ext{indirect}}(hbar omega) propto rac{(hbar omega – E_g + hbar Omega_{mathbf{q}})^2}{e^{hbar Omega_{mathbf{q}} / k_B T} – 1} + rac{(hbar omega – E_g – hbar Omega_{mathbf{q}})^2}{1 – e^{-hbar Omega_{mathbf{q}} / k_B T}}
]
Because second-order processes have low probability, indirect absorption coefficients are two to three orders of magnitude smaller than direct ones ((alpha sim 10^2 – 10^3 ext{ cm}^{-1})). This requires silicon solar cells to be hundreds of microns thick and makes silicon an intrinsically poor light emitter.
3. Wannier-Mott Excitons and the Elliott Formula
An optically excited electron in the conduction band and the hole it leaves behind in the valence band attract each other via attractive Coulomb interaction. In semiconductors with high dielectric permittivity, this attractive force binds them into an electrically neutral hydrogenic quasiparticle—the Wannier-Mott exciton.
The exciton Hamiltonian is given by:
[
left[ -rac{hbar^2}{2mu}
abla_{mathbf{r}}^2 – rac{e^2}{4pi psilon_0 psilon_r r}
ight] psi_{ ext{exc}}(mathbf{r}) = E_{ ext{exc}} psi_{ ext{exc}}(mathbf{r})
]
where (mu = rac{m_e^* m_h^*}{m_e^* + m_h^*}) is the reduced effective mass, and (mathbf{r} = mathbf{r}_e – mathbf{r}_h). The resulting bound energy eigenvalues lie just below the fundamental conduction band edge:
[
E_n = E_g – rac{R_y^*}{n^2}, quad n = 1, 2, 3, dots
]
where (R_y^* = rac{mu}{m_0} rac{1}{psilon_r^2} (13.6 ext{ eV})) is the exciton Rydberg energy (typically (4 – 15 ext{ meV}) in 3D semiconductors, but exceeding (300 ext{ meV}) in 2D monolayer TMDs). The Elliott formula proves that excitonic Coulomb enhancement alters the entire absorption spectrum: introducing sharp sub-gap absorption peaks and enhancing the continuum absorption right at the band edge by the Sommerfeld factor.
