How the Fermi Level Is Calculated
In the free-electron (Sommerfeld) model of a metal, conduction electrons behave as a gas of fermions confined to the material. Because electrons obey the Pauli exclusion principle, no two can occupy the same quantum state, so at absolute zero they stack up from the lowest available momentum state to a maximum energy called the Fermi energy, E_F — the Fermi level measured from the bottom of the conduction band. This calculator computes E_F, the associated Fermi temperature, Fermi velocity, and a degeneracy ratio from the conduction-electron number density n.
Deriving the Fermi energy formula
Filling momentum states two at a time (spin up and spin down) out to a Fermi sphere of radius k_F in momentum space, and matching the enclosed states to the electron density n, gives the Fermi wavevector k_F = (3π²n)^(1/3). The Fermi energy follows from the free-particle dispersion relation E = ħ²k²/2mₑ evaluated at k_F: E_F = (ħ²/2mₑ)(3π²n)^(2/3), where ħ is the reduced Planck constant and mₑ is the electron mass. Copper, with about one free electron per atom and a density of 8.96 g/cm³, has n ≈ 8.5×10²⁸ m⁻³ and E_F ≈ 7.0 eV — a typical value for simple metals.
Fermi temperature, velocity, and degeneracy
Dividing E_F by Boltzmann's constant gives the Fermi temperature, T_F = E_F / k_B, an energy scale expressed in kelvin. For most metals T_F is tens of thousands of kelvin — far above room temperature — so the electron gas is always "degenerate": only the small fraction of electrons within roughly k_B T of E_F can be thermally excited, and Fermi-Dirac statistics (not the classical Maxwell-Boltzmann distribution) govern their behavior. The Fermi velocity, v_F = √(2E_F/mₑ) = ħk_F/mₑ, is the speed of the electrons that sit at the Fermi surface; it sets the scale for electrical conductivity, thermal conductivity, and mean free path in the Drude and Sommerfeld transport models.
Where this model applies
The free-electron formula here describes simple metals with a partially filled conduction band (alkali metals, copper, silver, gold, aluminum). It is less accurate for transition metals, where d-band structure distorts the density of states, and it does not describe semiconductors or insulators: in those materials the Fermi level lies inside a band gap and its position depends on doping concentration and temperature through a different calculation. Always confirm that the electron density you enter reflects the material's actual conduction-electron count (valence electrons per atom × atomic number density), not the total electron count.