Fermi Level Calculator — Fermi Energy, Temperature, Velocity & Wave Vector

What Is the Fermi Level? — Fermi Level Definition

Our Fermi level calculator is a handy tool that helps you determine the Fermi parameters from the number density of electrons. Moreover, you can use this calculator to compute the probability of finding a particle with a specific energy — the so-called Fermi function (the Fermi–Dirac statistics).

The Fermi energy is a physical quantity used to determine the maximum energy of an electron at absolute zero temperature (0 K). All energy levels up to the Fermi level are occupied by electrons, and all levels above it are unoccupied.

The existence of the Fermi level is a consequence of the Pauli exclusion principle, which states that two fermions (e.g., electrons) cannot occupy the same stationary state. Fermions are particles that obey Fermi–Dirac statistics and have half-integer spin (s = 1/2). In other words, the Fermi level is the surface (the Fermi surface) at absolute zero temperature below which no electron will have enough energy to rise above.

To find the Fermi energy, imagine an empty system that we fill with electrons one at a time. Each electron consecutively occupies the lowest-energy available stationary state. When all electrons have been placed, the highest occupied electron energy level is the Fermi energy. At higher temperatures, a certain fraction of electrons gains enough energy to exist above the Fermi level; this number can be estimated using the Fermi–Dirac distribution.

Calculate Fermi Energy, Fermi Temperature, Fermi Velocity and Fermi Wave Vector

We use the Fermi gas model to quantify the Fermi parameters. A Fermi gas is a phase of matter consisting of a large number of non-interacting fermions. In this model, the Fermi surface forms the Fermi sphere, above which no electron can rise at absolute zero temperature. The Fermi level equations are:

Fermi wave vector (wave number): k_F = ∛(3π²n)

Fermi energy: E_F = ℏ²k_F² / (2m_e)

Fermi velocity: v_F = ℏk_F / m_e

Fermi temperature: T_F = E_F / k_B

Where:

• n — Electron number density (number of electrons per unit volume)
• m_e — Electron mass = 9.1093837 × 10⁻³¹ kg
• k_B — Boltzmann constant = 1.380649 × 10⁻²³ m²·kg/(s²·K)
• ℏ — Reduced Planck constant = h/2π = 1.054571817 × 10⁻³⁴ J·s

The Fermi–Dirac Distribution

The Fermi–Dirac distribution (or Fermi–Dirac statistics) gives the probability that an electron occupies a specific energy state. It applies to identical, indistinguishable particles with half-integer spin (fermions), in particular electrons that do not interact with each other (Fermi gas model). Our Fermi level calculator uses the following Fermi function definition:

f(E) = 1 / (e^{(E − E_F)/k_BT} + 1)

Where:

• f(E) — Probability that a particle has energy E
• e — Euler's number ≈ 2.71828
• E — Energy of a particle
• T — Temperature

At absolute zero temperature, every energy level below the Fermi surface is occupied and every level above it is vacant. Increasing the temperature lets some electrons rise above the Fermi energy. Notably, if the energy of the particle equals the Fermi energy (E = E_F), then f(E) is exactly 0.5 regardless of temperature.

Free Electron Number Densities Table

The table below lists the number density of free (conduction) electrons in some common metals. Use the quick-select buttons in the calculator to load these values instantly.

How to Use the Fermi Level Calculator

1. Choose the unit system — Metric (electrons per cubic meter, m⁻³) or American (electrons per cubic inch, in⁻³).
2. Enter the electron number density n, or pick a metal preset to load a typical value.
3. Enter the temperature T (default 300 K) — used for the Fermi–Dirac distribution.
4. (Optional) Enter a particle energy E in eV to compute the Fermi function f(E) — the occupation probability at that energy.
5. Click Calculate to instantly get the Fermi wave vector, Fermi energy, Fermi velocity, and Fermi temperature in both metric and American units.

Frequently Asked Questions

What is the difference between Fermi level and Fermi energy?

At absolute zero (0 K) the Fermi level and the Fermi energy coincide. At finite temperatures the Fermi level (chemical potential) shifts slightly, but for a free electron gas the two terms are often used interchangeably.

Why is f(E) always 0.5 at E = E_F?

When E = E_F, the exponent (E − E_F)/k_BT is zero, so e⁰ = 1 and f(E) = 1/(1+1) = 0.5 for any temperature above 0 K.

What is a typical Fermi energy for metals?

Fermi energies of common metals are on the order of a few electron volts — for example, about 7 eV for copper and 5.5 eV for gold.

What is the Fermi temperature?

The Fermi temperature T_F = E_F/k_B is the temperature equivalent of the Fermi energy. For metals it is typically tens of thousands of kelvin, which is why electrons in metals behave as a degenerate Fermi gas at room temperature.