Fermi Level Calculator - E_F from Density and T

The fermi level calculator solves E_F from electron density in metals and semiconductors, using E_F = (hbar^2 / 2 m_e) (3 pi^2 n)^(2/3) for a free electron gas, E_C - E_F = k T ln(N_C / n) for an n-doped semiconductor, E_F,i = mid-gap + (k T / 2) ln(N_V / N_C) for an intrinsic semiconductor, and f(E) = 1 / (exp((E - E_F) / (k T)) + 1) for the Fermi-Dirac occupancy.

• • Solid-state physics homework and exam problems: Compute the Fermi energy of copper, silver, gold, or aluminum from its conduction-electron density.
• • Semiconductor device design and doping studies: Translate a target donor density n in silicon or GaAs into E_C - E_F for a pn-junction or Schottky-barrier layout.
• • Intrinsic semiconductor Fermi level: Compute E_F,i for an undoped semiconductor from E_g, N_C, and N_V at a chosen temperature.
• • Fermi-Dirac occupancy checks: Read off f(E) at any energy offset to verify the Maxwell-Boltzmann tail or the degenerate-doping limit.

The four modes cover the two regimes where the Fermi level is taught: the free-electron metal case and the band-gap semiconductor case (a Boltzmann-tail formula in N_C, n, T, N_V, E_g).

Fermi level sits in the same quantum-statistical neighborhood as the discrete hydrogen-atom energy levels, so when you need the bound-state picture for a single electron, our Bohr Model Calculator covers the Bohr radius, energy levels, and transition wavelengths.

The fermi level calculator reads the inputs that match the active mode, takes natural logs of the carrier ratios, multiplies by the thermal energy kT, and returns the Fermi level in electron volts. The free-electron branch uses (hbar^2 / 2 m_e) (3 pi^2 n)^(2/3); the semiconductor branch uses the Boltzmann tail with the effective density of states.

• n: Conduction-electron density in m^-3. Used in free-electron and semiconductor modes.
• T: Absolute temperature in Kelvin. All four modes scale the result by kT.
• N_C, N_V: Effective densities of states in the conduction and valence bands in m^-3.
• E_g: Band gap of the semiconductor in eV. Intrinsic mode only.
• E - E_F: Energy offset from the Fermi level in eV. Occupancy mode only.
• hbar, m_e, k: CODATA 2019 values used by the free-electron branch.

The free-electron result depends on density through the cube root of n, so doubling n raises E_F by a factor of 2^(2/3) ≈ 1.59.

According to NIST CODATA 2019, the reduced Planck constant is 1.054571817e-34 J*s and the electron rest mass is 9.1093837015e-31 kg, which are the values used in the free-electron Fermi energy branch.

According to NIST CODATA 2019, the Boltzmann constant is 1.380649e-23 J/K, the exact fixed value in the 2019 SI redefinition that sets the kT scale for every Fermi level calculation.

Four ideas explain every number on the fermi level calculator result panel.

The same natural-log dependence on the carrier-to-density-of-states ratio drives the same physics for holes, with E_F - E_V = kT ln(N_V / p).

Fermi-Dirac statistics treats electrons as quantum particles, and the same quantum description sets the Compton wavelength. When you need the electron-scattering counterpart, our Compton Wavelength Calculator covers the photon-electron wavelength shift.

Five short steps cover the four modes the fermi level calculator supports.

1. 1 Pick the solve mode: Free electron for metals, Semiconductor for n-doped carrier density, Intrinsic for an undoped band gap, or Fermi-Dirac for an occupancy.
2. 2 Enter the conduction-electron density n: Free-electron and semiconductor modes need n in m^-3. Metals are around 8.5e28 m^-3.
3. 3 Enter the temperature T in Kelvin: All four modes need T because the Fermi-Dirac distribution is defined on absolute temperature.
4. 4 Enter the band-specific inputs: Semiconductor mode needs N_C. Intrinsic mode also needs N_V and E_g. Occupancy mode needs E - E_F in eV.
5. 5 Read the result and supporting values: The result panel shows the mode-dependent answer plus kT in eV, and k_F and v_F in free-electron mode.

A solid-state homework problem gives n = 8.5e28 m^-3 for copper at room temperature and asks for E_F. Switch to free-electron mode, type 8.5e28, leave T at 300 K, and read E_F = 7.03 eV. For silicon doping at n = 1e22 m^-3 and N_C = 2.8e25 m^-3, switch to semiconductor mode and read E_C - E_F = 0.2052 eV.

The semiconductor branch uses the Boltzmann-tail form n = N_C exp(-(E_C - E_F) / kT), the same exponential law that the Maxwell-Boltzmann ideal-gas speed distribution follows, so when you need the gas-side counterpart, our Ideal Gas Calculator covers PV = nRT and the Maxwell speed distribution.

A focused fermi level calculator keeps the algebra and the constant values out of the way so the physics is the only thing on the screen.

• • Four solid-state modes in one form: Free-electron Fermi energy, doped-semiconductor band position, intrinsic Fermi level, and Fermi-Dirac occupancy preview.
• • CODATA 2019 constants preloaded: hbar, m_e, and k come from NIST CODATA 2019, so answers agree with textbook references.
• • kT shown alongside every result: The thermal energy kT in eV appears on every result panel.
• • Fermi-surface reference values: k_F and v_F are returned alongside E_F in free-electron mode.
• • Fermi-Dirac tail check built in: The occupancy preview returns f(E) at any energy offset.

The Maxwell-Boltzmann approximation in semiconductor mode is valid when E_C - E_F > 3 kT, the non-degenerate limit.

The free-electron Fermi energy depends on the cube root of n and fixes the Fermi-Dirac distribution width, while the photon counterpart is the Planck distribution that our Blackbody Radiation Calculator uses for spectral radiance.

Three variables drive the Fermi level value, and three limitations tell you when to be careful.

• • The semiconductor and intrinsic branches use the Boltzmann-tail form, valid when E_C - E_F > 3 kT. Heavily doped cases need a full Fermi-Dirac integration.
• • The free-electron branch uses the electron rest mass, not a band effective mass. For transition metals or doped semiconductors the band effective mass should be folded into n.
• • N_C and N_V scale as T^(3/2). A textbook 300 K reference is the safest choice unless you have a T-dependent fit.

The same exponential suppression drives the Boltzmann tail, which is why semiconductor carrier statistics simplify to n ≈ N_C exp(-(E_C - E_F) / kT).

According to HyperPhysics, the conduction-electron density of copper is about 8.5e28 m^-3 and its free-electron Fermi energy is about 7 eV, the textbook reference value for the metal-side branch of this calculator.

The Maxwell-Boltzmann tail exp(-(E_C - E_F) / kT) has the same exponential form as the Arrhenius factor, so when you need the chemistry-side counterpart of the same temperature law, our Activation Energy Calculator solves Ea from a rate constant.