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:

The four to five orders of magnitude 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 $\mathrm{\Delta }n=n-n_0=p-p_0$:

$$R_{\text{rad}}\approx B(n_0+\mathrm{\Delta }n)(p_0+\mathrm{\Delta }n)-B{n}_0{p}_0\mathrm{.}$$

In the low-injection limit ($\mathrm{\Delta }n\ll n_0$ for n-type material): $R_{\text{rad}}\approx B{n}_0\mathrm{\Delta }n$ (linear in injection). In the high-injection limit ($\mathrm{\Delta }n\gg n_0,p_0$): $R_{\text{rad}}\approx B(\mathrm{\Delta }n{)}^2$ (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:

$${\tau }_{\text{rad}}=\frac{\mathrm{\Delta }n}{R_{\text{rad}}}\mathrm{.}$$

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

For GaAs at $\mathrm{\Delta }n={10}^{18}$ 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:

$${\eta }_i=\frac{R_{\text{rad}}}{R_{\text{rad}}+R_{\text{nonrad}}}=\frac{1\mathrm{/}{\tau }_{\text{rad}}}{1\mathrm{/}{\tau }_{\text{rad}}+1\mathrm{/}{\tau }_{\text{nonrad}}}\mathrm{.}$$

The total carrier recombination rate combines all mechanisms:

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

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

With illustrative coefficients $A={10}^6$ s⁻¹, $B={10}^{-10}$ cm³/s and $C={10}^{-28}$ cm⁶/s, the model gives:

The radiative fraction peaks at intermediate density, where $B\mathrm{\Delta }n$ outgrows $A$ before $C\mathrm{\Delta }{n}^2$ overtakes it; the fall at high density is the droop seen in LEDs. Measured coefficients differ by an order of magnitude or more between materials and reports, particularly $C$, which is largest in narrow-gap materials; the Auger recombination entry discusses them.

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:

$$h\nu =E_g+(E_e-E_c)+(E_v-E_h),$$

where $E_e-E_c$ and $E_v-E_h$ are the electron and hole kinetic energies measured from the band edges; this is 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 about four orders of magnitude slower in silicon, and about three in germanium, 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.