Quasi-Fermi level

Under non-equilibrium conditions (forward bias, optical pumping, current injection), the single Fermi level of equilibrium splits into two effective levels: the quasi-Fermi level for electrons ($E_{Fn}$) and the quasi-Fermi level for holes ($E_{Fp}$). The quasi-Fermi level concept extends equilibrium statistics to mildly non-equilibrium systems, valid when each carrier type is internally thermalized but the two are out of equilibrium with each other.

Definition. Under conditions where carrier-carrier scattering thermalizes electrons among themselves and holes among themselves (on $\sim 100$ fs timescales) but interband recombination is slow (ns to μs), the population of conduction-band states is described by:

$$f_n(E)=\frac{1}{1+\exp [(E-E_{Fn})\mathrm{/}{k}_{B}T]},$$

with a separate $E_{Fn}$. Similarly, for holes in the valence band:

$$f_p(E)=1-\frac{1}{1+\exp [(E-E_{Fp})\mathrm{/}{k}_{B}T]},$$

with a separate $E_{Fp}$. Here $f_p$ is the probability that a valence-band state is empty, that is, holds a hole: it approaches 1 for $E$ well above $E_{Fp}$ and falls toward 0 well below it, so holes collect near the top of the valence band.

Quasi-Fermi-level splitting and injection. Carrier concentrations under non-equilibrium:

$$\begin{array}{rl}{\displaystyle n}& {\displaystyle =N_c\exp [-(E_c-E_{Fn})\mathrm{/}{k}_{B}T],}\\ {\displaystyle p}& {\displaystyle =N_v\exp [-(E_{Fp}-E_v)\mathrm{/}{k}_{B}T]\mathrm{.}}\end{array}$$

The product:

$$np=n_i^2\exp [(E_{Fn}-E_{Fp})\mathrm{/}{k}_{B}T]\mathrm{.}$$

In equilibrium $E_{Fn}=E_{Fp}=E_F$ and $np=n_i^2$. Under forward bias of a p-n junction:

$$E_{Fn}-E_{Fp}=q{V}_a,$$

where $V_a$ is the applied bias. So $np=n_i^2\exp (q{V}_a\mathrm{/}{k}_{B}T)$. The diode equation's exponential factor traces directly to the quasi-Fermi-level splitting.

Bernard-Duraffourg condition for optical gain. Net stimulated emission at a photon energy $h\nu$ requires that the upper electronic state has higher occupation than the lower:

$$E_{Fn}-E_{Fp}>h\nu \mathrm{.}$$

This is the foundational requirement for net gain in any semiconductor laser. The condition derives from comparing stimulated emission ($f_n\cdot f_p$, an electron above and a hole below) with stimulated absorption ($(1-f_n)(1-f_p)$); positive net gain requires the former to exceed the latter, which simplifies to $E_{Fn}-E_{Fp}>h\nu$.

Transparency condition. When $E_{Fn}-E_{Fp}=h{\nu }_{\text{laser}}$, the material has zero net gain at $h{\nu }_{\text{laser}}$ and is "transparent" at this wavelength. The carrier density at this point is the transparency carrier density, and the current density that sustains it is the transparency current density. Above transparency, the material has gain; below, it absorbs.

For typical InGaAsP at 1550 nm:

• Transparency carrier density: $\sim 1.0-1.5\times {10}^{18}$ cm⁻³
• Transparency current density: $\sim 50-80$ A/cm² per QW

Why the Bernard-Duraffourg condition works. At equilibrium, $E_{Fn}=E_{Fp}=E_F$, and absorption exceeds stimulated emission at every photon energy above the bandgap, so the material absorbs. The Bernard-Duraffourg condition shifts the balance toward emission only when carriers are pumped to populate the upper band more than the lower band.

This is the semiconductor analog of "population inversion" in atomic systems. In an atomic 4-level laser, the upper laser level is preferentially populated and the lower level is rapidly emptied; in a semiconductor, states near the conduction-band edge fill with electrons and states near the valence-band edge fill with holes.

Quasi-Fermi levels in a forward-biased p-n junction. Under forward bias, current flows; the quasi-Fermi levels become spatially-dependent quantities:

• In the bulk regions: $E_{Fn}$ and $E_{Fp}$ are nearly equal to the equilibrium $E_F$ (carriers are majority)
• Across the depletion region: both quasi-Fermi levels are roughly flat
• Beyond each depletion edge: the minority-carrier quasi-Fermi level merges back into the majority level as the injected carriers recombine, so $E_{Fn}$ in the p-region falls toward $E_{Fp}$ and $E_{Fp}$ in the n-region rises toward $E_{Fn}$

The current depends on carrier diffusion driven by gradients in the quasi-Fermi levels:

$$J_n=n{\mu }_n\frac{d{E}_{Fn}}{dx},J_p=p{\mu }_p\frac{d{E}_{Fp}}{dx}\mathrm{.}$$

These are the standard semiconductor drift-diffusion equations rewritten in quasi-Fermi-level form.

Quasi-Fermi-level engineering in lasers. Above the laser threshold, the carrier density is "clamped": additional pump current produces additional photons (output power), not additional carriers. The quasi-Fermi-level splitting is also clamped at:

$$E_{Fn}-E_{Fp}=h{\nu }_{\text{laser}}+(\text{small terms})\mathrm{.}$$

This is why a semiconductor laser's junction voltage is "pinned" above threshold; the terminal voltage still rises with current through the series resistance. The Bernard-Duraffourg condition is satisfied at the lasing wavelength, and any additional injection just produces more photons.

Quasi-Fermi levels under optical pumping. In an optically-pumped semiconductor, absorbed pump light generates electron-hole pairs. The same quasi-Fermi-level concept applies; their splitting is set by the pump-generated carrier density.

The fundamental thermodynamic limit on solar cell open-circuit voltage:

$$V_{\text{oc}}=\frac{E_{Fn}-E_{Fp}}{q},$$

with the splitting determined by pump rate and recombination. The Shockley-Queisser limit derives from analyzing the radiative recombination floor in this quasi-Fermi-level framework.

Limitations of the quasi-Fermi level approach. The concept breaks down when:

• Carrier-carrier scattering is too slow: at low carrier densities or low temperatures, the assumption of internal thermalization fails
• Hot-carrier conditions: high optical pump produces a non-thermal distribution; effective $T_c>T_{\text{lattice}}$
• Very fast dynamics: sub-100 fs after pulsed excitation, the distribution isn't yet thermalized
• Heterojunctions with rapid carrier transfer: cross-junction transfer can outpace intraband thermalization

For most semiconductor device operation (CW laser, modulated laser at GHz, photodetector at moderate intensity), the quasi-Fermi-level picture is accurate.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 16 – 17 (semiconductor optics and lasers); Coldren, Corzine & Mašanović, Diode Lasers and PICs (2nd ed., 2012), Ch. 4 (canonical treatment of Bernard-Duraffourg condition); Bernard & Duraffourg, Phys. Status Solidi 1, 699 (1961) for the original derivation; Pierret, Semiconductor Device Fundamentals (1996) for the comprehensive carrier-statistics treatment.