Radiative recombination

Radiative recombination is the process by which an excited electron in the conduction band recombines with a hole in the valence band, with the energy released as a photon. It is the inverse of optical absorption and is the dominant light-emission mechanism in direct-bandgap semiconductors. The competing process of non-radiative recombination releases the energy as heat or as phonons, with no photon emission.

Rate equation. The radiative recombination rate per unit volume:

$$R_\text{rad} \;=\; B \cdot n \cdot p,$$

where $n$ is the electron concentration, $p$ is the hole concentration, and $B$ is the bimolecular recombination coefficient (units of cm³/s). $B$ depends on material:

Material	Direct/Indirect	$B$ (cm³/s, room T)
GaAs	Direct	$1 \times 10^{-10}$
InP	Direct	$1 \times 10^{-10}$
InGaAs (lattice-matched to InP)	Direct	$1 \times 10^{-10}$
InGaAsP (1.55 μm)	Direct	$1.0 - 1.5 \times 10^{-10}$
InGaN	Direct	$0.5 - 1 \times 10^{-10}$
GaN	Direct	$\sim 1 \times 10^{-11}$
Si	Indirect	$1 \times 10^{-15}$
Ge	Indirect	$5 \times 10^{-14}$

The factor of $10^5$ difference between Si and GaAs is the principal reason direct-bandgap III-V materials dominate semiconductor light sources.

Carrier-density dependence. For a quasi-neutral region with injected excess carrier density $\Delta n = n - n_0 = p - p_0$:

$$R_\text{rad} \;\approx\; B (n_0 + \Delta n)(p_0 + \Delta n) - B n_0 p_0.$$

In the low-injection limit ($\Delta n \ll n_0$ for n-type material): $R_\text{rad} \approx B n_0 \Delta n$ (linear in injection). In the high-injection limit ($\Delta n \gg n_0, p_0$): $R_\text{rad} \approx B (\Delta n)^2$ (quadratic in injection). Laser active regions operate in the high-injection regime.

Radiative lifetime. The lifetime of an excess carrier limited by radiative recombination alone:

$$\tau_\text{rad} \;=\; \frac{\Delta n}{R_\text{rad}}.$$

For high-injection: $\tau_\text{rad} = 1/(B \Delta n)$.

For GaAs at $\Delta n = 10^{18}$ cm⁻³: $\tau_\text{rad} = 10$ ns. This is a typical operating point for semiconductor lasers.

Internal quantum efficiency. The internal quantum efficiency $\eta_i$ is the fraction of injected carriers that recombine radiatively:

$$\eta_i \;=\; \frac{R_\text{rad}}{R_\text{rad} + R_\text{nonrad}} \;=\; \frac{1/\tau_\text{rad}}{1/\tau_\text{rad} + 1/\tau_\text{nonrad}}.$$

The total carrier recombination rate combines all mechanisms:

$$\frac{1}{\tau} \;=\; A + B (\Delta n) + C (\Delta n)^2,$$

where $A$ is the Shockley-Read-Hall (SRH) defect-mediated rate, $B$ is the radiative coefficient, and $C$ is the Auger coefficient. The three terms dominate in different regimes:

• Low injection: SRH ($A$) dominates if defects are present
• Mid injection: Radiative ($B$) dominates
• High injection: Auger ($C$) dominates

For high-quality MQW lasers above threshold, the ABC model predicts:

$\Delta n$	Dominant mechanism	Typical $\tau$
$10^{15}$ cm⁻³	SRH ($A \sim 10^6$ s⁻¹)	1 μs
$10^{17}$ cm⁻³	Radiative ($B \sim 10^{-10}$)	100 ns
$10^{18}$ cm⁻³ (laser threshold)	Radiative (~ 10 ns)	10 ns
$5 \times 10^{18}$ cm⁻³	Auger competing	2 ns
$10^{19}$ cm⁻³	Auger ($C \sim 10^{-28}$ cm⁶/s)	$< 1$ ns

Spontaneous vs stimulated radiative emission. Radiative recombination has two flavors:

• Spontaneous emission: photon emitted in random direction and time; the basic LED mechanism
• Stimulated emission: photon emitted in same direction and phase as an incident photon; the laser mechanism

Both processes derive from the same fundamental transition matrix element. The Einstein A coefficient (spontaneous) and B coefficient (stimulated) are related by the photon density of states.

Photon energy. Conservation of energy gives photon energy:

$$h\nu \;=\; E_g + (E_c - E_F)_\text{electron contribution} + (E_F - E_v)_\text{hole contribution},$$

approximately $E_g$ for low-temperature operation. At high carrier density, band-filling shifts the photon energy upward by 30 – 100 meV — explaining the blue-shift of LED emission peak with current.

Radiative recombination in indirect-bandgap materials. In silicon and germanium, radiative recombination requires a phonon to conserve momentum, making the process 4 – 5 orders of magnitude slower than in direct-gap materials. Strategies to enhance radiative emission in Si:

• Highly strained Si or SiGe: changes band structure to near-direct
• Nanostructured Si: quantum confinement modifies selection rules
• Erbium-doped Si: introduces direct radiative transitions at 1.54 μm
• Hybrid integration: bond direct-gap III-V to silicon (the standard approach for silicon photonic lasers)

Polarization properties. The radiative emission has polarization properties inherited from the band structure:

• Bulk semiconductors: random polarization (degenerate hh and lh bands)
• Quantum wells: TE-polarized emission dominant (compressively strained); TM possible (tensile strained)
• Strained QWs: can be engineered for desired TE/TM ratio
• Quantum cascade lasers: TM-only emission (intersubband transitions)

Why direct-gap radiative emission is fast in III-Vs. The transition matrix element for direct radiative recombination is large because:

• Initial and final state wavefunctions have similar momentum
• No phonon required to conserve momentum
• The dipole matrix element is on the order of $e \cdot 5$ Å — large for atomic-scale transitions

This produces the famous "10 ns lifetime" that makes GaAs and InP the platform of choice for semiconductor light emitters.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 16 (semiconductor materials); Coldren, Corzine & Mašanović, Diode Lasers and PICs (2nd ed., 2012), Ch. 2 — comprehensive treatment of ABC model; Piprek, Semiconductor Optoelectronic Devices (Academic Press, 2003) for the device-physics derivations.