Calculate semiconductor Fermi level position, carrier concentrations, and intrinsic properties from band gap, doping, and temperature with band diagram visualization.
The Fermi level calculator determines the position of the Fermi energy in a semiconductor based on band gap, effective density of states, doping concentrations, and temperature. The Fermi level is the most important parameter in semiconductor physics — it determines carrier concentrations, junction potentials, and device behavior.
In an intrinsic (undoped) semiconductor, the Fermi level sits near mid-gap, shifted slightly by the density-of-states asymmetry between conduction and valence bands. Adding donor impurities (n-type doping) pushes the Fermi level toward the conduction band, increasing electron concentration exponentially. Acceptor doping (p-type) does the reverse. At high temperatures, thermal generation of carriers overwhelms doping, and the semiconductor reverts to intrinsic behavior.
This calculator computes Fermi level position, electron and hole concentrations, intrinsic carrier density, and the temperature at which extrinsic behavior is lost. It includes a schematic band diagram, temperature-dependent analysis, and presets for common semiconductors including silicon, GaAs, and germanium.
Fermi level position is one of the quickest ways to see how doping and temperature are shaping a semiconductor. It connects the band diagram to carrier concentration, which makes it useful for device design, process review, and classroom semiconductor problems.
Intrinsic level: Ei = Eg/2 + (kT/2)ln(Nv/Nc). Intrinsic carrier density: ni = √(NcNv) exp(−Eg/2kT). n-type: Ef = Ei + kT ln(n/ni) where n ≈ Nd−Na. p-type: Ef = Ei − kT ln(p/ni) where p ≈ Na−Nd. Mass-action law: n × p = ni².
Result: Ef = 0.917 eV from Ev, n = 1×10¹⁶ cm⁻³
Silicon doped with 10¹⁶ donors/cm³ at 300 K has its Fermi level 0.917 eV above the valence band (0.203 eV below Ec), with electron concentration equal to the donor density in the extrinsic regime.
As donor concentration increases in n-type material, the Fermi level moves upward toward the conduction band. As acceptor concentration increases in p-type material, it moves downward toward the valence band. The shift is often the most direct way to reason about how strongly a semiconductor is doped.
Raising temperature increases intrinsic carrier concentration and eventually weakens the influence of doping. That is why the same material can behave extrinsically at room temperature and much more intrinsically at higher temperature.
Diodes, transistors, and solar cells all rely on the position of the Fermi level. Looking at carrier density and band placement together helps explain why a particular device behaves the way it does under equilibrium.
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The Fermi level (Ef) is the energy at which the probability of electron occupation is exactly 50%. In a semiconductor, it determines the equilibrium carrier concentrations through the Fermi-Dirac distribution.
The intrinsic level shifts toward the band with the larger effective density of states. In silicon, Nv < Nc, so Ei is slightly below mid-gap (about 0.013 eV at 300 K).
The product n × p = ni² holds at thermal equilibrium regardless of doping. Adding donors increases n but decreases p proportionally, keeping n×p constant at a given temperature.
As temperature increases, intrinsic carriers (ni) grow exponentially. When ni exceeds the doping concentration, the Fermi level returns to near mid-gap — the semiconductor behaves intrinsically.
When doping is so heavy that the Fermi level enters the conduction band (n-type) or valence band (p-type), Fermi-Dirac statistics must be used instead of the Boltzmann approximation. This occurs around 10¹⁹–10²⁰ cm⁻³ doping in silicon.
Nc and Nv are effective density of states: Nc = 2(2πm*ekT/h²)^(3/2) and similarly for Nv with hole effective mass. For Si at 300K: Nc ≈ 2.8×10¹⁹, Nv ≈ 1.04×10¹⁹ cm⁻³.