Fermi Level Calculator — Semiconductor Physics
Find the Fermi energy level position above the valence band for any semiconductor material. Enter the bandgap, temperature, intrinsic carrier concentration, and (for doped material) the donor or acceptor concentration to get the Fermi level in eV and the resulting carrier concentrations.
Semiconductor type
K
eV
cm⁻³
Intrinsic semiconductor — EF shift from mid-gap: 0 eV
- 1
Thermal energy kT
8.617×10⁻⁵ eV/K × 300 K = 0.02585 eV - 2
Fermi level (intrinsic mid-gap)
1.12 ÷ 2 = 0.5600For an undoped semiconductor, E_F sits at the midpoint of the bandgap.
How does this calculator work?
The Fermi level in a doped semiconductor shifts from mid-gap by ΔEF = kT·ln(N_d/n_i) for n-type or −kT·ln(N_a/n_i) for p-type, where k = 8.617×10⁻⁵ eV/K. At 300 K, kT ≈ 0.026 eV. For Si with N_d = 10¹⁶ cm⁻³ and n_i ≈ 1.5×10¹⁰, E_F sits about 0.35 eV above mid-gap.
Formula
How this is calculated
In a semiconductor, the Fermi level (E_F) is the electrochemical potential of electrons — the energy at which the Fermi-Dirac probability function equals 0.5. For an intrinsic (undoped) semiconductor, E_F sits approximately at the midgap: E_i ≈ E_v + E_g/2, where E_g is the bandgap energy. Adding donors (n-type doping) shifts E_F upward toward the conduction band by ΔE_F = kT·ln(N_d/n_i); adding acceptors (p-type) shifts it downward by kT·ln(N_a/n_i). Here k is the Boltzmann constant (8.617×10⁻⁵ eV/K) and n_i is the intrinsic carrier concentration.
This calculator assumes complete ionisation of dopants (valid at room temperature for Si, Ge, GaAs) and that the doping concentration exceeds n_i by a comfortable margin (the charge-neutrality approximation n ≈ N_d for n-type). The minority carrier concentration follows from the mass-action law: n·p = n_i². The ±kT shaded region on the distribution indicates the thermal smearing around E_F where the Fermi-Dirac occupation function transitions between near-1 and near-0.
Values for n_i and effective density-of-states masses are material and temperature-dependent and differ from the 300 K defaults for other temperatures or materials — update the intrinsic concentration field accordingly. The simplified mid-gap approximation for E_i is exact only when the effective masses of electrons and holes are equal; in practice the offset is small (a few meV for Si).
Frequently asked questions
The Fermi level is the energy level at which a quantum state has a 50% probability of being occupied by an electron. In a doped semiconductor it marks the balance between available electrons (donors) and holes (acceptors), and its position relative to the band edges controls electrical conductivity.
Donor atoms supply extra electrons, raising the electron concentration above n_i. Because more states near the conduction band are filled, the Fermi level rises by kT·ln(N_d/n_i) above the intrinsic level. The heavier the doping, the closer E_F gets to the conduction band edge.
kT (≈ 0.026 eV at 300 K) is the thermal energy per electron. It sets the energy scale over which the Fermi-Dirac distribution transitions from occupied to empty states. A Fermi level within a few kT of a band edge means that band is partially populated by thermally excited carriers.
TG we-Calculate Editorial Team. (2026). Fermi Level Calculator — Semiconductor Physics [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/fermi-level-calculator
TG we-Calculate Editorial Team. "Fermi Level Calculator — Semiconductor Physics." TG we-Calculate. 2026. https://we-calculate.com/calculator/fermi-level-calculator.
TG we-Calculate Editorial Team, "Fermi Level Calculator — Semiconductor Physics," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/fermi-level-calculator
@misc{wecalculate_fermi_level_calculator, title = {Fermi Level Calculator — Semiconductor Physics}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/fermi-level-calculator}}, year = {2026}, note = {TG we-Calculate} }
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