Plasma dispersion effect

The plasma dispersion effect is the modification of refractive index and absorption coefficient of a semiconductor by changes in the free-carrier (electron and hole) density. Adding electrons or holes to a semiconductor produces a Drude-model plasma response that simultaneously reduces the real refractive index ($\Delta n < 0$) and increases the absorption (free-carrier absorption, FCA).

Soref-Bennett relations for silicon (Soref & Bennett, IEEE JQE 1987 — the canonical paper for silicon plasma dispersion at 1550 nm):

$$\Delta n \;=\; -8.8 \times 10^{-22} \Delta N - 8.5 \times 10^{-18} (\Delta P)^{0.8},$$ $$\Delta \alpha \;=\; 8.5 \times 10^{-18} \Delta N + 6.0 \times 10^{-18} \Delta P,$$

where $\Delta N$ and $\Delta P$ are the electron and hole density changes in cm$^{-3}$, $\Delta n$ is dimensionless, and $\Delta \alpha$ is the absorption coefficient change in cm$^{-1}$, all at $\lambda = 1550$ nm.

For typical silicon photonic modulator operating conditions ($\Delta N = \Delta P = 10^{18}$ cm$^{-3}$):

• $\Delta n \approx -1 \times 10^{-3}$
• $\Delta \alpha \approx 15$ cm$^{-1}$ (additional 6.5 dB/mm absorption)

This is the irreducible insertion-loss vs phase-shift tradeoff of all silicon plasma-dispersion modulators: producing a useful $\pi$ phase shift requires waveguide length sufficient to accumulate radians of phase change, but the simultaneous absorption increase limits the achievable transmission.

Why silicon uses plasma dispersion despite the loss. Silicon is centrosymmetric — it has no Pockels effect at all. The Kerr effect produces only $\Delta n \sim 10^{-5}$ at telecom wavelengths even at maximum useful field, far too weak for practical modulation. Plasma dispersion is the strongest available linear electro-optic response in silicon.

Modulator architectures using plasma dispersion.

Architecture	Mechanism	$V_\pi \cdot L$	Insertion loss
Carrier injection (forward-biased p-i-n)	Inject carriers across forward-biased junction	0.3 – 1 V·cm	2 – 5 dB
Carrier depletion (reverse-biased p-n)	Sweep carriers out of depletion region	1 – 3 V·cm	3 – 7 dB
Carrier accumulation (MOS capacitor)	Accumulate carriers at oxide-silicon interface	0.5 – 2 V·cm	2 – 4 dB
Microring resonator + plasma dispersion	Same mechanisms, enhanced by Q factor	0.05 – 0.2 V·cm	0.5 – 2 dB

Carrier injection produces the largest $\Delta n$ but is bandwidth-limited by minority carrier lifetime to $\sim 5$ GHz. Carrier depletion is faster ($> 50$ GHz) but produces smaller $\Delta n$ per volt. MOS-capacitor designs achieve good middle-ground performance and are the architecture of choice in some Intel silicon photonic platforms.

Polysilicon and Ge variants. Plasma dispersion is significantly stronger in heavily-doped polysilicon than in crystalline silicon due to its lower mobility (carrier scattering produces larger Drude response). Germanium has different coefficients; the strain-engineered Ge-on-Si modulators that have been demonstrated combine the plasma dispersion with the Franz-Keldysh electro-absorption effect for stronger modulation.

Comparison to other electro-optic mechanisms.

Mechanism	Strength	Materials	Insertion loss penalty
Pockels effect	Strong	LiNbO3, KDP, GaAs, InP	Minimal
Kerr effect	Weak	Si, glass	Minimal
Plasma dispersion	Moderate	Si, Ge, doped semiconductors	Significant
Quantum-confined Stark	Strong	MQW III-V near band edge	Wavelength-locked
Thermal (thermo-optic)	Strong but slow	All	Minimal

The plasma dispersion / loss tradeoff is the fundamental reason silicon photonic Mach-Zehnder modulators have higher insertion loss than LNOI or InP MQW alternatives. Active research continues into pushing plasma-dispersion performance closer to its theoretical limits and into integrating Pockels-active materials (LNOI, BTO) onto silicon.

References: Soref & Bennett, Electrooptical effects in silicon, IEEE JQE 23, 123 (1987); Reed & Knights, Silicon Photonics: An Introduction (Wiley, 2004), Ch. 5 on modulator design.