Carrier density

Carrier density is the number of free electrons ($N$) or holes ($P$) per unit volume of a semiconductor, quoted in cm⁻³. Undoped silicon at room temperature holds about 10¹⁰ cm⁻³ of each; doped contact layers hold 10¹⁷–10¹⁹ cm⁻³ of one type; and the injected electron and hole densities in the active region of a laser diode are of order 10¹⁸ cm⁻³, roughly one carrier in every cube 10 nm on a side. In III-V lasers the injected electron and hole densities are nearly equal, because the active region is undoped or lightly doped and charge neutrality holds, so a single symbol $N$ usually stands for both.

The quantity matters because gain, refractive index and recombination all depend on it. Optical gain appears once the electron and hole quasi-Fermi levels are separated by more than the photon energy, which happens at the transparency density $N_\text{tr}$. For InGaAsP and InGaAs quantum wells at 1.3–1.55 µm, published values of $N_\text{tr}$ cluster around 1–2 × 10¹⁸ cm⁻³, with the exact figure depending on well width, strain and temperature.

Steady state from the injected current

Below threshold, every injected carrier eventually recombines, and the rate balance in an active layer of thickness $d$ is

$$\frac{J}{q\,d} = \frac{N}{\tau} \quad\Rightarrow\quad N = \frac{J\,\tau}{q\,d},$$

where $J$ is the current density through the layer and $\tau$ the carrier lifetime. For five 8 nm wells ($d = 40$ nm), $\tau = 2$ ns and $J = 1$ kA/cm²,

$$N = \frac{(1000)(2\times10^{-9})}{(1.602\times10^{-19})(4\times10^{-6})}$$

which gives $N \approx 3.1 \times 10^{18}$ cm⁻³. Read the other way, the same structure reaches $1.5 \times 10^{18}$ cm⁻³ at about 480 A/cm², in line with the transparency current densities of five-well stacks. The formula with a constant $\tau$ is an approximation: the lifetime itself falls as $N$ rises, through the bimolecular radiative term $BN$ and the Auger term $CN^2$, so the current needed to raise $N$ grows faster than linearly. Current that leaks over the barriers or spreads outside the stripe also contributes to $J$ without contributing to $N$.

Threshold and clamping

The laser reaches threshold when the modal gain balances the internal and mirror losses, $\Gamma g(N_\text{th}) = \alpha_i + \alpha_m$, where $\alpha_i$ is the internal loss. With the logarithmic gain model used in the threshold current density entry, $N_\text{th}/N_\text{tr}$ is typically 1.5–3 for telecom multi-quantum-well lasers, so threshold densities of 2–5 × 10¹⁸ cm⁻³ are common.

Above threshold the carrier density clamps near $N_\text{th}$. Any extra carriers raise the gain above the loss, the photon density grows, and stimulated emission removes the excess within tens of picoseconds; the additional current therefore goes into photons. Clamping is approximate: spatial hole burning, gain compression and heating let the average density creep up slowly with current, and the clamp is what makes the junction voltage and the spontaneous emission from the sides of the chip saturate at threshold.

Index changes and modulators

Free carriers also lower the refractive index and add absorption. In a laser this coupling of gain and index is described by the linewidth enhancement factor and produces chirp under direct modulation. In silicon it is the basis of the plasma dispersion effect: by the Soref–Bennett relations at 1550 nm, adding 10¹⁸ cm⁻³ each of electrons and holes changes the index by about −3.0 × 10⁻³ and adds 14.5 cm⁻¹ (63 dB/cm) of absorption. Injection modulators push carriers into an intrinsic region by carrier injection; depletion modulators sweep carriers out of a doped pn junction under reverse bias, changing the density by a smaller amount but in picoseconds rather than nanoseconds. The plasma dispersion calculator evaluates these relations for any density.

Measurement and pitfalls

Carrier density is rarely measured directly in a working laser. It is inferred from the current through $N = J\tau/(qd)$ with a lifetime taken from small-signal impedance or optical modulation response below threshold, from the blue shift of the gain peak (band filling), or from the index change seen as a wavelength shift of Fabry–Pérot modes with current below threshold. Doping densities in passive layers come from Hall or capacitance–voltage measurements. Three errors recur: using the total stack thickness when the gain model is written per well, treating $\tau$ as constant over a wide current range, and assigning all of the measured current to the active region when some of it leaks around it.

Common questions

What is a typical carrier density in a laser diode?

About 2–5 × 10¹⁸ cm⁻³ in the active region at threshold for InP- and GaAs-based quantum-well lasers, and 1–2 × 10¹⁸ cm⁻³ at transparency, and the density stays close to its threshold value at all currents above threshold.

Why does carrier density stop rising above threshold?

Because the gain is tied to the carrier density and the gain cannot exceed the loss in steady state. Extra carriers are converted into stimulated photons as fast as they arrive, so the extra current raises the output power instead of the density.

References: L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012); G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993); R. A. Soref and B. R. Bennett, "Electrooptical effects in silicon," IEEE J. Quantum Electron. 23, 123 (1987); S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007).