Free Fermi Level Calculator

The Fermi Level Calculator is a compact online tool that computes the fundamental Fermi parameters—Fermi energy, Fermi temperature, Fermi velocity, and Fermi wave number—directly from the conduction electron density. It also functions as a Fermi-Dirac distribution calculator, returning the probability that an electron occupies a given energy level. This makes it a versatile companion for solid-state physics, semiconductor analysis, and material science.

What Is the Fermi Level?

The Fermi level (or Fermi energy) denotes the highest occupied energy state at absolute zero temperature (0 K). At this temperature, every quantum state below the Fermi level is filled with electrons, while those above remain empty. This ordering originates from the Pauli exclusion principle: no two fermions (particles with half-integer spin, such as electrons) may share the same quantum state. When electrons are added to a system, they sequentially occupy the lowest available energy states, and the uppermost filled state defines the Fermi energy. As temperature increases, thermal excitation allows a fraction of electrons to rise above the Fermi level, creating a distribution described by Fermi-Dirac statistics.

Fermi Parameters in the Free-Electron Model

The free-electron (Fermi gas) model treats the conduction electrons as a non-interacting gas. Within this model, the allowed states in momentum space form a sphere—the Fermi sphere—whose radius is the Fermi wave number $k_{\mathrm{F}}$. Given the electron number density $n$ (electrons per unit volume), the central Fermi parameters are expressed as:

• Fermi wave number:

  $$k_{\mathrm{F}} = \left(3\pi^2 n\right)^{1/3}$$

• Fermi energy:

  $$E_{\mathrm{F}} = \frac{\hbar^2 k_{\mathrm{F}}^2}{2m_{\mathrm{e}}} = \frac{\hbar^2}{2m_{\mathrm{e}}} \left(3\pi^2 n\right)^{2/3}$$

• Fermi velocity:

  $$v_{\mathrm{F}} = \frac{\hbar k_{\mathrm{F}}}{m_{\mathrm{e}}}$$

• Fermi temperature:

  $$T_{\mathrm{F}} = \frac{E_{\mathrm{F}}}{k_{\mathrm{B}}}$$

Here $\hbar$ is the reduced Planck constant ($1.054571817 \times 10^{-34}\,\text{J·s}$), $m_{\mathrm{e}}$ is the electron rest mass ($9.10938356 \times 10^{-31}\,\text{kg}$), and $k_{\mathrm{B}}$ is Boltzmann’s constant ($1.38064852 \times 10^{-23}\,\text{m}^2\,\text{kg}\,\text{s}^{-2}\,\text{K}^{-1}$). By entering the number density into the calculator, you immediately obtain all these quantities.

Fermi–Dirac Distribution

The Fermi–Dirac (FD) distribution gives the occupancy probability of a state with energy $E$ at temperature $T$:

$$f(E) = \frac{1}{\exp\left(\frac{E - E_{\mathrm{F}}}{k_{\mathrm{B}} T}\right) + 1}$$

where $e \approx 2.71828$ is Euler’s number. At 0 K, $f(E) = 1$ for $E < E_{\mathrm{F}}$ and $f(E) = 0$ for $E > E_{\mathrm{F}}$, meaning the Fermi function behaves like a step function. As temperature rises, the step smears out, and some electrons populate states above $E_{\mathrm{F}}$. A key property is that at $E = E_{\mathrm{F}}$, the probability is always exactly 0.5, irrespective of temperature. This calculator evaluates $f(E)$ for any chosen energy and temperature, serving as a dedicated Fermi function calculator.

Free Electron Density Table for Metals

Many common metals are well described by the free-electron approximation. The tabulated values below represent the conduction electron densities (number of free electrons per cubic meter) for selected metals. Selecting a material automatically loads its density into the Fermi level calculator, simplifying the retrieval of Fermi parameters.

Metal	Number density ($\times 10^{28}$ electrons/m³)
Cu (copper)	8.47
Ag (silver)	5.86
Au (gold)	5.90
Be (beryllium)	24.7
Mg (magnesium)	8.61
Ca (calcium)	4.61
Sr (strontium)	3.55
Ba (barium)	3.15
Nb (niobium)	5.56
Fe (iron)	17.0
Zn (zinc)	13.2
Cd (cadmium)	9.27
Al (aluminum)	18.1
Ga (gallium)	15.4
In (indium)	11.5
Sn (tin)	14.8
Pb (lead)	13.2

If your material is not listed, you can manually enter its free electron density. The calculator also functions as a free electron density calculator, accepting any numerical input to compute the corresponding Fermi parameters.

Whether you need the Fermi energy of copper, the Fermi temperature of aluminum, or the Fermi velocity of gold, this Fermi level calculator delivers fast, reliable results. It integrates the central equations of the free-electron model and the Fermi–Dirac distribution, offering a comprehensive tool for exploring electron behavior in solids.