Fermi Level Calculator

About the Fermi Level Calculator

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.

When This Page Helps

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.

How to Use the Inputs

1. Enter the semiconductor band gap energy in eV (1.12 eV for Si at 300 K)
2. Enter the effective density of states Nc (conduction) and Nv (valence) in cm⁻³
3. For doped semiconductors, enter donor (Nd) and/or acceptor (Na) concentrations
4. Set the temperature in Kelvin — typical room temperature is 300 K
5. Use presets for common semiconductor configurations
6. Review Fermi level position, carrier concentrations, and the band diagram

Example Calculation

Tips & Best Practices

• At 300 K, kT ≈ 25.9 meV — each kT of Fermi level shift changes carrier concentration by a factor of e ≈ 2.72
• Silicon becomes intrinsic above ~500–600 K for typical doping levels — limiting high-temperature operation
• GaAs has larger bandgap (1.42 eV) than Si (1.12 eV), so it works at higher temperatures
• For compensated semiconductors (both Nd and Na present), net doping Nd−Na determines carrier type
• Heavy doping (>10¹⁸ cm⁻³) causes band gap narrowing, reducing the effective band gap by 50–100 meV

Doping and the Fermi Level

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.

Temperature Effects

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.

Device Context

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.

Frequently Asked Questions

• What is the Fermi level?

  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.

• Why is the intrinsic Fermi level not exactly at mid-gap?

  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).

• What is the mass-action law?

  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.

• What happens to Fermi level at high 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.

• What is a degenerate semiconductor?

  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.

• How do I find Nc and Nv?

  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.