3.5 Quasi-Fermi levels

Quasi-Fermi Levels

In equilibrium, a single Fermi level describes both electron and hole populations. But real semiconductor devices operate out of equilibrium: light shines on solar cells, voltage biases LEDs, current flows through transistors. Quasi-Fermi levels extend the Fermi level concept to these non-equilibrium situations by assigning a separate Fermi level to electrons and to holes. The separation between these two levels turns out to be directly tied to the voltage across a device, making quasi-Fermi levels central to understanding how semiconductor devices generate, emit, or convert energy.

Fermi Level vs. Quasi-Fermi Levels

At thermal equilibrium, the Fermi level $E_F$ is the energy at which the occupation probability is 50%. It's a single value that governs both electron and hole statistics simultaneously.

When you drive the system out of equilibrium (by applying a voltage or shining light), electrons and holes are no longer described by the same Fermi level. Instead, each carrier type gets its own:

• $E_{Fn}$: the electron quasi-Fermi level
• $E_{Fp}$: the hole quasi-Fermi level

Each quasi-Fermi level plays the same mathematical role as $E_F$ did in equilibrium, but now separately for each carrier population. The key physical idea is that electrons thermalize among themselves (and holes among themselves) much faster than electrons and holes recombine with each other. So each population individually looks like it's in equilibrium, just described by a different Fermi level.

Carrier Concentrations and Quasi-Fermi Levels

The carrier concentration formulas from equilibrium statistics carry over directly, with $E_F$ replaced by the appropriate quasi-Fermi level. Using the Boltzmann approximation (valid when the quasi-Fermi levels are more than a few $kT$ from the band edges):

$n = N_c \exp\!\left(\frac{E_{Fn} - E_c}{kT}\right)$

$p = N_v \exp\!\left(\frac{E_v - E_{Fp}}{kT}\right)$

where:

• $N_c$, $N_v$ are the effective densities of states in the conduction and valence bands
• $E_c$, $E_v$ are the conduction and valence band edges
• $k$ is Boltzmann's constant, $T$ is absolute temperature

Notice what these equations tell you: pushing $E_{Fn}$ closer to $E_c$ (or above it) increases $n$ exponentially. Pushing $E_{Fp}$ closer to $E_v$ (or below it) increases $p$ exponentially. That exponential sensitivity is why even modest quasi-Fermi level shifts produce large changes in carrier concentration.

Quasi-Fermi Levels in Equilibrium

At equilibrium, both quasi-Fermi levels collapse to the single Fermi level:

$E_{Fn} = E_{Fp} = E_F$

This is the condition when no external voltage or illumination is applied. The $np$ product reduces to the familiar equilibrium relation $np = n_i^2$, and there's no net current flow.

Quasi-Fermi Levels Under Bias

Applying a voltage splits the quasi-Fermi levels. The relationship is:

$qV = E_{Fn} - E_{Fp}$

where $q$ is the elementary charge and $V$ is the voltage across the device (or more precisely, across the region of interest).

This is one of the most useful results in device physics. It means you can read the local voltage directly from a band diagram by measuring the vertical gap between $E_{Fn}$ and $E_{Fp}$.

Under forward bias on a p-n junction, $E_{Fn}$ rises above $E_{Fp}$, injecting excess carriers into the junction region. Under reverse bias, the splitting is negative (or equivalently, the equilibrium depletion is enhanced), and carrier concentrations drop below their equilibrium values.

Separation of Quasi-Fermi Levels

The magnitude of $E_{Fn} - E_{Fp}$ is a direct measure of how far the system is from equilibrium.

• Zero separation: equilibrium, $np = n_i^2$
• Small separation: low injection, minority carrier concentration slightly elevated
• Large separation: strong non-equilibrium, both carrier populations significantly enhanced

Using the carrier concentration formulas, you can show that:

$np = n_i^2 \exp\!\left(\frac{E_{Fn} - E_{Fp}}{kT}\right)$

So the $np$ product exceeds $n_i^2$ whenever the quasi-Fermi levels are split. This excess $np$ product is what drives recombination and, in devices like solar cells, is what you're trying to maximize.

Quasi-Fermi Levels and Recombination

Recombination rates depend on carrier concentrations, which are set by the quasi-Fermi levels. For band-to-band recombination, the rate scales with the $np$ product. Since $np \propto \exp\!\left(\frac{E_{Fn} - E_{Fp}}{kT}\right)$, a larger quasi-Fermi level splitting means a higher recombination rate.

In regions where both $E_{Fn}$ is close to $E_c$ and $E_{Fp}$ is close to $E_v$ (meaning both carrier concentrations are high), recombination is strongest. This is exactly what happens in the active region of an LED or at the junction of a forward-biased diode.

Quasi-Fermi Levels in Depletion Regions

Inside the depletion region of a p-n junction, the quasi-Fermi levels are generally not flat. The strong built-in electric field sweeps carriers across this region, and the quasi-Fermi levels vary with position to reflect that transport.

However, under low-injection conditions and moderate bias, a common and useful approximation is that the quasi-Fermi levels are approximately flat through the depletion region. This is because the depletion region is narrow and carries relatively little recombination. The flat quasi-Fermi level approximation is what lets you connect the quasi-Fermi level splitting at the junction to the applied terminal voltage.

At high injection or with significant generation/recombination in the depletion region, this approximation breaks down and the spatial variation of $E_{Fn}(x)$ and $E_{Fp}(x)$ must be accounted for.

Quasi-Fermi Levels and Current Density

One of the most powerful results in semiconductor transport theory is that total current density for each carrier type can be written compactly in terms of the quasi-Fermi level gradient. This single expression captures both drift and diffusion:

$J_n = \mu_n \, n \, \frac{dE_{Fn}}{dx}$

$J_p = \mu_p \, p \, \frac{dE_{Fp}}{dx}$

where $\mu_n$ and $\mu_p$ are the electron and hole mobilities.

These are not two separate equations for drift and diffusion added together. The quasi-Fermi level gradient already encodes both mechanisms. That's the elegance of this formulation:

• If $E_{Fn}$ is flat ($dE_{Fn}/dx = 0$), there is zero electron current, even if there are electric fields and concentration gradients present. The drift and diffusion components exactly cancel.
• A spatial gradient in $E_{Fn}$ means the electron population is not in local equilibrium, and current flows.

You can verify this by expanding the derivative. Starting from $n = N_c \exp\!\left(\frac{E_{Fn} - E_c}{kT}\right)$ and differentiating, you recover the standard drift-diffusion equation $J_n = qn\mu_n \mathcal{E} + qD_n \frac{dn}{dx}$, where $\mathcal{E}$ is the electric field, using the Einstein relation $D_n = \mu_n kT/q$.

Quasi-Fermi Levels in Solar Cells

In a solar cell, absorbed photons generate electron-hole pairs, splitting the quasi-Fermi levels even without an applied voltage. The physics proceeds in a few steps:

1. Photons with energy above the bandgap are absorbed, creating excess electrons and holes.

2. These excess carriers raise $E_{Fn}$ toward $E_c$ and lower $E_{Fp}$ toward $E_v$, increasing the splitting.

3. The splitting $E_{Fn} - E_{Fp}$ at the junction corresponds to the photovoltage: $qV_{oc} = E_{Fn} - E_{Fp}$ at open circuit.

4. This voltage drives current through an external load.

The open-circuit voltage $V_{oc}$ of a solar cell is fundamentally limited by how large you can make the quasi-Fermi level splitting before recombination balances generation. Maximizing $V_{oc}$ means minimizing recombination losses so the splitting stays as large as possible.

Quasi-Fermi Levels in LEDs

In an LED under forward bias, injected carriers recombine radiatively, emitting photons. The quasi-Fermi level splitting determines the emission characteristics:

• The minimum photon energy for stimulated or spontaneous emission is approximately $E_g$ (the bandgap).
• The quasi-Fermi level splitting sets the chemical potential of the emitted photon gas. For net stimulated emission (lasing), you need population inversion, which requires $E_{Fn} - E_{Fp} > E_g$ (the Bernard-Duraffourg condition).
• For spontaneous emission in a standard LED, the peak emission energy is close to $E_g + \frac{1}{2}kT$, and the applied voltage is approximately $V \approx E_g / q$.

Note that the emitted photon energy is primarily determined by the bandgap and the thermal distribution of carriers, not simply equal to $E_{Fn} - E_{Fp}$. The quasi-Fermi level splitting determines the rate and efficiency of emission through the carrier concentrations it controls.

Measuring Quasi-Fermi Levels

Quasi-Fermi levels aren't directly measurable as a single quantity, but several experimental techniques give access to them:

• Photoluminescence (PL) spectroscopy: The high-energy tail of the PL emission spectrum is related to the quasi-Fermi level splitting through a generalized Planck law. Fitting the spectrum lets you extract $E_{Fn} - E_{Fp}$.
• Open-circuit voltage measurements: For a p-n junction or solar cell, $qV_{oc}$ gives the quasi-Fermi level splitting at the junction under illumination.
• Capacitance-voltage (C-V) profiling: Provides information about carrier concentration profiles, from which quasi-Fermi level positions can be inferred relative to the band edges.

Limitations of the Quasi-Fermi Level Concept

The quasi-Fermi level description rests on a key assumption: each carrier population (electrons and holes separately) is internally thermalized and can be described by a Fermi-Dirac distribution, just with different Fermi levels. This works well when:

• Carrier-carrier scattering is fast compared to recombination (typically true)
• The system is in steady state
• Injection levels aren't so extreme that the distribution functions become non-thermal

The approximation can break down in several situations:

• High injection or transient conditions: Carrier distributions may not be Fermi-Dirac shaped (e.g., hot carriers that haven't thermalized yet).
• Carrier trapping: Deep traps can hold carriers out of thermal equilibrium with the band, complicating the picture.
• Surface and interface states: These introduce additional populations that may not be well-described by the bulk quasi-Fermi levels.

Despite these caveats, quasi-Fermi levels remain one of the most practical and widely used tools in semiconductor device analysis. Nearly all device simulation software solves for $E_{Fn}(x)$ and $E_{Fp}(x)$ as the primary unknowns.