Free-carrier absorption (FCA)

Free-carrier absorption is the absorption of below-bandgap photons by free electrons or holes already present in the conduction or valence band. The carrier absorbs the photon, gaining energy, and conserves momentum via simultaneous interaction with a phonon or ionized impurity. FCA is the natural companion to the plasma dispersion effect — both arise from the same Drude-model response of carriers to the optical electric field.

Functional form. For a semiconductor with free electron density $N$ and hole density $P$, the FCA coefficient is approximately:

$$\alpha_\text{FCA} \;=\; \sigma_n N + \sigma_p P,$$

where $\sigma_n$ and $\sigma_p$ are the absorption cross-sections for electrons and holes. The cross-sections scale with wavelength as $\sigma \propto \lambda^p$, with exponent $p \approx 1.5 - 3.5$ depending on the dominant scattering mechanism (acoustic phonon, optical phonon, or ionized-impurity scattering each give different exponents).

Silicon FCA at 1550 nm (Soref & Bennett, IEEE JQE 1987):

$$\alpha_\text{FCA} \;=\; 8.5 \times 10^{-18} N + 6.0 \times 10^{-18} P, \quad [\text{cm}^{-1}],$$

with $N, P$ in cm$^{-3}$. For typical silicon modulator carrier densities $N = P = 10^{18}$ cm$^{-3}$: $\alpha_\text{FCA} \approx 14.5$ cm$^{-1}$, contributing 6.3 dB/mm of loss.

Material comparison at 1550 nm:

Material	$\sigma_n$ (cm²)	$\sigma_p$ (cm²)	Notes
Silicon	$8.5 \times 10^{-18}$	$6.0 \times 10^{-18}$	Standard reference values
Germanium	$\sim 10^{-17}$	$\sim 10^{-17}$	Stronger than Si
InGaAs (telecom)	$10^{-17} - 10^{-16}$	$10^{-17} - 10^{-16}$	Strong; limits SOA efficiency
InP	$\sim 10^{-17}$	$\sim 10^{-17}$	Significant
GaAs	$\sim 5 \times 10^{-18}$	$\sim 5 \times 10^{-18}$	Moderate

Why FCA matters.

• In silicon plasma-dispersion modulators: FCA contributes the dominant insertion loss when the modulator is driven to high carrier densities. This is the fundamental tradeoff in plasma-dispersion devices — getting larger $\Delta n$ requires injecting more carriers, but more carriers also raise FCA loss.
• In semiconductor optical amplifiers: free electrons and holes that contribute to gain also contribute to FCA at the operating wavelength. The net gain is reduced by intra-band FCA, lowering the maximum achievable on-off ratio.
• In laser cavities: doped contact regions of laser diodes have high carrier density and contribute optical loss. This sets a limit on minimum cladding thickness — too thin a cladding lets the mode penetrate into the contact and incur FCA loss.
• In photodetectors: FCA in undepleted regions reduces the effective absorption length, lowering responsivity.
• In silicon photonic devices at high power: TPA-generated carriers absorb additional light via FCA, producing the dominant power-handling limit in silicon photonics.

Wavelength dependence in silicon. Silicon FCA scales approximately as $\lambda^{1.5}$ to $\lambda^{2}$ — so FCA is larger at longer wavelengths. At 1310 nm, the coefficients are roughly 60 – 70% of the 1550 nm values. At 2 μm (silicon photonic mid-IR), FCA is $\sim$ 2× the 1550 nm values, partially offsetting the benefit of zero TPA at that wavelength.

Mitigation in silicon photonic modulators:

• Carrier depletion mode instead of injection: lower steady-state carrier density at any operating point reduces FCA penalty
• Lateral p-n junctions with high doping near the junction but light doping in the mode core
• Resonator-enhanced modulation: ring resonators effectively recycle the optical field through the active region many times, achieving the needed phase shift with less carrier-density excursion

Measurement. FCA cross-sections are extracted by measuring transmission loss vs known doping level (in doped reference samples) or by pump-probe spectroscopy where the pump generates known carrier density and the probe measures absorption increase. Modern values reported in the literature have $\pm 20$% scatter.

References: Soref & Bennett, Electrooptical effects in silicon, IEEE JQE 23, 123 (1987); Pankove, Optical Processes in Semiconductors (Dover, 1971), Ch. 3 — the canonical treatment of carrier absorption mechanisms.