Fermi Level Calculator
Calculate the Fermi level position, intrinsic carrier concentration, electron and hole concentrations, and thermal voltage for intrinsic or doped semiconductors. Select a material preset or enter your own bandgap and effective masses. Switch between intrinsic, n-type, and p-type modes. Results update instantly as you type.
What is the Fermi level in a semiconductor?
The Fermi level (EF) is the electrochemical potential of electrons in a solid at thermal equilibrium. At absolute zero, all states below EF are fully occupied and all states above it are completely empty. At finite temperatures, the Fermi-Dirac distribution smears that boundary by roughly kT on each side. In a semiconductor, the position of EF relative to the conduction band minimum (Ec) and valence band maximum (Ev) determines how many electrons and holes exist. When EF lies near midgap the material is intrinsic or lightly doped. When EF moves toward Ec the semiconductor is n-type with electrons as the majority carrier. When EF moves toward Ev the semiconductor is p-type with holes as the majority carrier.
Intrinsic carrier concentration and how temperature affects it
The intrinsic carrier concentration ni is the number of thermally generated electron-hole pairs per cubic centimetre in an undoped semiconductor. It is governed by ni = sqrt(Nc x Nv) x exp(-Eg / (2kT)), where Nc and Nv are the effective densities of states at the band edges and Eg is the bandgap. Because the exponential factor is dominant, ni rises steeply with temperature: for silicon it is about 1.5 x 10^10 cm^-3 at room temperature but exceeds 10^15 cm^-3 above 500 K. This exponential sensitivity is why power devices must operate well below the temperature at which ni becomes comparable to doping, a condition called intrinsic excitation or thermal runaway.
Doping, majority and minority carriers
Adding donor atoms (phosphorus, arsenic) to silicon introduces extra electrons without adding holes, shifting EF toward the conduction band. The electron concentration n approximately equals the donor concentration Nd at room temperature under complete ionisation. The hole concentration p follows from the mass action law: np = ni^2, so p = ni^2 / Nd. For n-type silicon doped at 10^16 cm^-3, n is about 10^16 cm^-3 but p is only about 2.25 x 10^4 cm^-3. Adding acceptors (boron) works symmetrically, shifting EF toward the valence band and making holes the majority carrier. These large carrier ratios are exploited in diodes, bipolar transistors, and MOSFETs, where controlled injection of minority carriers is the key active mechanism.
How to read your results
The primary output is the intrinsic carrier concentration ni, expressed in scientific notation. The intrinsic Fermi level position (EFi - Ev) tells you where the intrinsic reference sits above the valence band: for a symmetric bandgap with Nc = Nv it would be exactly Eg/2, but asymmetric effective masses shift it slightly. The Fermi level shift (EF - EFi) is zero for intrinsic, positive for n-type, and negative for p-type; its magnitude is kT x ln(majority carrier / ni). The thermal voltage kT/q equals about 25.85 mV (0.02585 eV) at 300 K and sets the energy scale for exponential carrier statistics in diodes. If EF - Ev exceeds Eg, the semiconductor has become degenerate (heavily doped or very low temperature) and Boltzmann statistics overestimate carrier concentrations; full Fermi-Dirac integrals are needed in that regime.
Common semiconductor material parameters at 300 K
| Material | Bandgap Eg (eV) | mc/m0 | mv/m0 | ni (cm^-3) |
|---|---|---|---|---|
| Silicon (Si) | 1.12 | 1.08 | 0.81 | 1.5 x 10^10 |
| Germanium (Ge) | 0.66 | 0.55 | 0.37 | 2.4 x 10^13 |
| GaAs | 1.42 | 0.067 | 0.50 | 1.8 x 10^6 |
| InP | 1.35 | 0.077 | 0.60 | 1.3 x 10^7 |
| 4H-SiC | 3.26 | 0.39 | 1.20 | ~10^-9 |
| GaN | 3.39 | 0.20 | 0.80 | ~10^-10 |
Standard values used by textbooks and device simulators. Effective mass ratios are density-of-states values.
Frequently asked questions
Why does the intrinsic Fermi level not sit exactly at midgap?
For the Fermi level to sit exactly at midgap, the effective density of states Nc and Nv would have to be equal, which requires equal effective masses for electrons and holes. In most semiconductors they differ, so EFi is shifted from Eg/2 by (kT/2) x ln(Nv/Nc). For silicon at 300 K this shift is about -0.013 eV, pushing EFi slightly toward the valence band because Nv is smaller than Nc.
What does thermal voltage (kT/q) mean and why does it matter?
The thermal voltage kT/q (where k is Boltzmann's constant, T is absolute temperature, and q is the elementary charge) equals about 25.85 mV at 300 K. It appears in the Shockley diode equation, the minority-carrier diffusion equations, and all carrier statistics. A forward bias comparable to a few times kT/q changes the diode current by orders of magnitude. At higher temperatures kT/q grows, which is why junction voltages and turn-on thresholds decrease as a device heats up.
What is the mass action law and when does it fail?
The mass action law states that in a semiconductor at thermal equilibrium, the product of electron and hole concentrations equals ni^2 regardless of doping: np = ni^2. This powerful relationship lets you find minority carrier concentration from majority carrier concentration. It holds as long as Boltzmann statistics apply (non-degenerate doping), the material is in thermal equilibrium (no illumination, no applied forward bias), and the temperature is uniform. It breaks down under heavy injection (minority carriers become comparable to majority carriers), in degenerate semiconductors (very heavy doping), and out of equilibrium.
What is degenerate doping and when does it matter?
When doping is heavy enough that the Fermi level enters (or closely approaches) the conduction or valence band, the semiconductor is called degenerate. At this point Boltzmann statistics, which assume EF is at least a few kT away from the band edge, are no longer accurate and the full Fermi-Dirac integral must be used. Degenerate regions are intentional in tunnel diodes, ohmic contacts, and the emitter of a bipolar transistor, where extreme carrier density is needed.
How do I calculate the built-in junction potential of a p-n diode?
The built-in voltage Vbi across a p-n junction equals (kT/q) x ln(Na x Nd / ni^2), where Na is the acceptor concentration on the p-side and Nd is the donor concentration on the n-side. At room temperature in silicon with Na = Nd = 10^16 cm^-3, Vbi is about (0.02585) x ln((10^16 x 10^16) / (1.5 x 10^10)^2) which is approximately 0.70 V. This built-in field opposes minority carrier injection and sets the contact potential of the junction.
Why does ni differ so much between silicon, germanium, and GaAs?
ni depends exponentially on -Eg / (2kT). Silicon (Eg = 1.12 eV) has ni about 1.5 x 10^10 cm^-3, germanium (Eg = 0.66 eV) has ni about 2.4 x 10^13 cm^-3, and GaAs (Eg = 1.42 eV) has ni about 1.8 x 10^6 cm^-3 at 300 K. The smaller the bandgap, the more electrons can be thermally promoted across it. This is why germanium-based devices cannot operate at high temperatures and why wide-bandgap materials like GaN (Eg = 3.4 eV) are used in high-temperature power electronics.