Carrier injection

Carrier injection is the introduction of electrons (and holes) into a semiconductor region above their equilibrium concentrations, creating the non-equilibrium population that drives optical emission, modulation, or amplification. In semiconductor lasers and LEDs, carrier injection is achieved by forward-biasing a p-n junction; the injected carriers recombine in the active region to produce light.

The forward-biased p-n junction. At equilibrium, a p-n junction has a built-in voltage $V_{bi}$ across the depletion region, opposing carrier flow. Under forward bias $V_a < V_{bi}$:

• The barrier is reduced by $qV_a$
• Electrons from the n-side flow into the p-side
• Holes from the p-side flow into the n-side
• The minority carrier concentrations on each side rise above equilibrium

Specifically, the minority carrier concentration at the depletion-region boundary on the p-side:

$$n_p \;=\; n_{p0} \exp(qV_a / k_B T),$$

with $n_{p0}$ the equilibrium minority electron density. Forward bias of 0.6 V on Si (where $n_{p0} \sim 10^4$ cm⁻³ for $N_A = 10^{17}$): $n_p \sim 10^{14}$ cm⁻³ — many orders of magnitude above equilibrium.

Double heterostructure. For efficient injection in laser diodes, the active region is sandwiched between wider-bandgap p- and n-cladding layers (a "double heterostructure"). This configuration provides:

1. Carrier confinement: the bandgap discontinuity blocks injected carriers from diffusing out of the active region
2. Optical confinement: the lower refractive index of the cladding provides a waveguide

Standard double heterostructures:

Material system	Cladding	Active	Operating wavelength
AlGaAs/GaAs	Al₀.₃Ga₀.₇As	GaAs	850 nm
InGaAsP/InP	InP	In₁₋ₓGaₓAsᵧP₁₋ᵧ	1300 or 1550 nm
InGaP/AlGaInP	(AlₓGa₁₋ₓ)₀.₅In₀.₅P	GaInP	670 nm
AlGaN/GaN	AlGaN	InGaN/GaN MQW	405 – 450 nm
AlGaInAs/InP	AlGaInAs	AlGaInAs MQW	1310 or 1550 nm

Injection efficiency. Not all current injected into the device contributes to light emission. Standard loss mechanisms:

• Carrier leakage over heterobarriers: especially at high T or low cladding bandgap
• Carrier overflow at high injection: when QW states fill up
• Auger-recombination-driven heating: increased Auger at high injection
• Non-radiative recombination in adjacent layers: SRH, surface recombination in cladding

For high-quality lasers, the internal quantum efficiency (fraction of injected carriers that recombine radiatively in the active region) approaches 90 – 95% at moderate injection; it drops at very high injection due to Auger and carrier leakage.

Injection current density. The current density driving the active region:

$$J \;=\; q \cdot d \cdot \frac{N}{\tau},$$

where $d$ is the active region thickness, $N$ is the injected carrier density, and $\tau$ is the carrier lifetime.

For a 6-nm-thick single QW with $N = 2 \times 10^{18}$ cm⁻³ and $\tau = 1$ ns: $J = 192$ A/cm². Times the active area 300 μm × 2 μm = $6 \times 10^{-6}$ cm²: total current $= 11.5$ mA — typical for a 1310 nm DFB laser.

Injection at high speed. When modulating a laser at high frequency, the carrier injection rate must keep up:

• Modulation bandwidth $\propto 1/\tau$: shorter carrier lifetimes (more carriers, more Auger) enable faster modulation but more loss
• Carrier transport through cladding: time for injected carriers to reach the active QWs; typically 10 – 100 ps
• Capacitance limit: device capacitance + driver impedance set an RC time

Typical 25 Gb/s direct-modulation lasers have:

• Carrier lifetime $\sim 0.5$ ns
• Capacitance $\sim 1$ pF
• RC limit $\sim 100$ ps
• Combined modulation bandwidth $\sim 15 - 20$ GHz

Plasma effect and electroabsorption from injection. Injected carriers cause:

• Plasma dispersion effect: $\Delta n \propto -\Delta N$, where $\Delta N$ is the carrier density. For Si at 1550 nm: $\Delta n \approx -5.4 \times 10^{-22} \cdot \Delta N$ for electrons. Basis of plasma-dispersion modulators.
• Free-carrier absorption: $\Delta\alpha \propto \Delta N$. Inevitable loss accompanying injection-based modulation.
• Band-filling: blue-shifts absorption edge
• Bandgap renormalization: red-shifts band edge through many-body effects

Injection in modulators. Mach-Zehnder modulators use carrier injection (forward-biased) or depletion (reverse-biased) to modulate the refractive index of one arm:

Approach	Speed	$V_\pi L$	Insertion loss
Carrier injection (forward bias)	1 – 10 GHz	0.05 – 0.1 V·cm	High (free-carrier absorption)
Carrier depletion (reverse bias)	25 – 100 GHz	0.5 – 3 V·cm	Lower
Pockels effect (LiNbO₃, BaTiO₃)	100+ GHz	2 – 5 V·cm	Very low
Franz-Keldysh / EAM	25 – 100 GHz	small voltage swing	Wavelength-dependent
Plasma effect in graphene	25+ GHz	very compact	Higher

Depletion-based silicon modulators dominate the modern 100G+ silicon photonic landscape because they offer the best speed and lower insertion loss than injection-based.

Injection in solar cells. A solar cell can be viewed as carrier injection in reverse — light injects carriers, which are extracted as current. The same physics (quasi-Fermi-level splitting, diffusion, recombination) applies. Optimum solar cell design seeks to maximize the quasi-Fermi-level splitting (= $V_{oc}$) and the carrier collection efficiency.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 17 (semiconductor lasers); Coldren, Corzine & Mašanović, Diode Lasers and PICs (2nd ed., 2012), Ch. 5 — comprehensive injection analysis; Sze & Ng, Physics of Semiconductor Devices (3rd ed., 2007), Ch. 2 (p-n junction physics).