quasi-fermi level

**Quasi-Fermi Level** is the **thermodynamic construct that extends the equilibrium Fermi level concept to non-equilibrium conditions** — splitting the single equilibrium Fermi level into separate electron (E_Fn) and hole (E_Fp) quasi-Fermi levels whose local values determine carrier concentrations under bias and whose spatial gradients drive carrier currents throughout the device. **What Is the Quasi-Fermi Level?** - **Definition**: Under non-equilibrium conditions (bias applied), electrons and holes no longer share a common Fermi level. The electron quasi-Fermi level E_Fn is defined by n = ni * exp((E_Fn - E_i)/kT), and the hole quasi-Fermi level E_Fp by p = ni * exp((E_i - E_Fp)/kT), where E_i is the intrinsic Fermi level. - **Equilibrium Limit**: At thermal equilibrium, E_Fn = E_Fp = E_F (the single Fermi level), and the mass-action law n*p = ni^2 is recovered as a special case of the quasi-Fermi level definitions. - **Separation and Voltage**: The separation of quasFermi levels at any point is directly related to the local carrier product pn = ni^2 * exp((E_Fn - E_Fp)/kT). At a forward-biased junction, the applied voltage splits the quasi-Fermi levels by q*V_applied. - **Current as Gradient**: Electron current density can be written as J_n = q*n*mu_n*(1/q)*(dE_Fn/dx), showing that current flows wherever the quasi-Fermi level has a spatial gradient — a flat E_Fn means zero electron current regardless of carrier concentration. **Why Quasi-Fermi Levels Matter** - **Band Diagram Interpretation**: Plotting E_Fn and E_Fp on the device energy band diagram provides an immediate visual representation of where and how current flows — gradients show current, flat regions show equilibrium, and the separation of the two levels indicates the degree of non-equilibrium at each point. - **Recombination Driving Force**: The SRH, Auger, and radiative recombination rates are all functions of the product n*p = ni^2 * exp(q*V/kT), where V = (E_Fn - E_Fp)/q is the local quasi-Fermi level separation. Larger separation drives faster recombination to restore equilibrium. - **Open-Circuit Voltage of Solar Cells**: The maximum open-circuit voltage of a solar cell equals the maximum quasi-Fermi level separation achievable under illumination divided by q — a quantity limited by the bandgap, the illumination intensity, and the recombination rates. This makes quasi-Fermi level separation the direct measure of photovoltaic work output. - **TCAD Visualization**: In TCAD post-processing, quasi-Fermi level plots reveal current bottlenecks (steep gradients), injection levels (large separations), and regions of high recombination (converging quasi-Fermi levels) throughout the device, guiding design optimization far more efficiently than current density plots alone. - **LED Emission Control**: In LED active regions, the quasi-Fermi level separation determines the carrier quasi-equilibrium distribution and thus the gain spectrum — the photon energy range over which stimulated emission or spontaneous emission is possible. **How Quasi-Fermi Levels Are Used in Practice** - **TCAD Output**: All major TCAD solvers output E_Fn and E_Fp as primary solution variables alongside carrier density and potential — standard analysis workflows visualize quasi-Fermi levels to diagnose device behavior. - **Analytical Models**: The diode injection condition (minority carrier concentration at the edge of the depletion region proportional to exp(qV/kT)) follows directly from the quasi-Fermi level definition applied at the depletion boundary. - **Solar Cell Diagnostic**: Measuring implied open-circuit voltage (iVoc) from photoluminescence intensity is equivalent to measuring the quasi-Fermi level separation under illumination, providing a powerful contactless characterization of solar cell precursor material quality. Quasi-Fermi Level is **the thermodynamic language for non-equilibrium semiconductor physics** — by extending the concept of a Fermi level to separately describe electron and hole populations out of equilibrium, it provides the most physically transparent lens through which current flow, carrier injection, recombination, and solar cell efficiency can be understood, visualized, and optimized in any semiconductor device operating under bias or illumination.