Carrier lifetime

Carrier lifetime $\tau$ is the average time interval between generation of a free carrier (electron or hole) and its recombination with a carrier of the opposite type, directly or through a defect level. It is one of the most important time constants in semiconductor optoelectronics, controlling steady-state carrier density at a given injection rate and setting the bandwidth limit of carrier-mediated devices.

Definition. For a carrier density $N$ in steady-state injection $R_{\text{inj}}$ (rate per volume):

$$\frac{dN}{dt}=R_{\text{inj}}-\frac{N}{\tau }=0\Rightarrow N_{\text{ss}}=R_{\text{inj}}\tau \mathrm{.}$$

Larger lifetime $\to$ higher steady-state carrier density for the same injection rate.

Recombination components. The total recombination rate $1\mathrm{/}\tau$ is the sum of contributions from each recombination mechanism:

$$\frac{1}{\tau }=\frac{1}{{\tau }_{\text{SRH}}}+\frac{1}{{\tau }_{\text{rad}}}+\frac{1}{{\tau }_{\text{Auger}}}+\frac{1}{{\tau }_{\text{surface}}},$$

where:

• Shockley-Read-Hall (SRH): recombination through deep-level defects in the bandgap; $1\mathrm{/}{\tau }_{\text{SRH}}=A{N}_{\text{trap}}$
• Radiative: band-to-band recombination emitting a photon; $1\mathrm{/}{\tau }_{\text{rad}}=BN$ (bimolecular)
• Auger: three-body recombination where two carriers recombine and a third carries away the energy; $1\mathrm{/}{\tau }_{\text{Auger}}=C{N}^2$
• Surface: recombination at unpassivated surfaces; depends on surface area and surface recombination velocity

The total lifetime is a function of carrier density due to the bimolecular and Auger terms. At high $N$, $\tau$ drops as Auger dominates.

Typical values:

Why lifetime matters in lasers.

The carrier rate equation in a semiconductor laser is:

$$\frac{dN}{dt}=\frac{I}{qV}-\frac{N}{\tau }-v_{g}g(N)S,$$

where $I$ is current, $V$ is active region volume, $g(N)$ is gain, $v_g$ is group velocity, and $S$ is photon density.

Below threshold (no stimulated emission), the steady-state carrier density is $N=I\tau \mathrm{/}(qV)$. The transparency current is the current required to bring $N$ to the transparency value $N_{\text{tr}}$, so $I_{\text{tr}}\propto N_{\text{tr}}\mathrm{/}\tau$. Shorter lifetime $\to$ higher transparency current $\to$ higher threshold current. This is why high-quality, defect-free III-V epitaxy is critical for low-threshold lasers: SRH lifetime must be long enough that radiative recombination dominates.

Above threshold, carrier lifetime is clamped at a smaller stimulated-emission-dominated value ${\tau }_{\text{stim}}\sim 1\mathrm{/}(v_{g}gS)$, which is typically 1 – 100 ps depending on photon density.

Why lifetime matters in modulators.

Plasma-dispersion modulators based on carrier injection are bandwidth-limited by the carrier lifetime: the response time to add or remove $\mathrm{\Delta }N$ is set by $\tau$. Injection modulators with $\tau =1$ ns are bandwidth-limited to $\sim$ 160 MHz, far below the GHz speeds needed for modern data communications. The solution is carrier depletion mode (electric field sweeps carriers out at the speed of carrier drift, $\sim$ ps timescale) at the cost of smaller available carrier density excursion.

Why lifetime matters in detectors.

Photodetector dark current scales as $1\mathrm{/}{\tau }_{\text{SRH}}$: short carrier lifetime means rapid thermal generation of carriers, raising shot noise. Long carrier lifetime (clean material) is essential for low-noise photodetection. The carrier transit time across the depletion region typically dominates the bandwidth, not lifetime, so the lifetime impact is primarily on dark current.

Measurement. Carrier lifetime is measured by:

• Time-resolved photoluminescence (TRPL): pulse-excite the sample with a short laser, observe PL decay; the time constant equals $\tau$
• Open-circuit voltage decay: in solar cells, the OC voltage decays after illumination is removed, with time constant $\sim \tau$
• Microwave photoconductance decay: pulse-excite, monitor sample conductivity, fit decay
• Pump-probe: pulsed pump generates carriers, time-delayed probe measures transmission recovery

References: Coldren, Corzine, Mašanović, Diode Lasers and Photonic Integrated Circuits, Ch. 2 (rate equations); Sze, Physics of Semiconductor Devices (3rd ed., 2007), Ch. 1 for the canonical material-physics treatment.