Fermi Energy

Fermi energy, usually denoted $E_F$, is the energy marking the boundary between occupied and unoccupied single-particle states in the zero-temperature ground state of a system of noninteracting fermions. For a gas with a continuous spectrum, it equals the zero-temperature chemical potential, measured using the same energy reference. It is a central concept in statistical mechanics and condensed-matter physics, particularly in describing electrons in metals. The closely related term Fermi level often denotes the chemical potential at any temperature, so the two terms are not always interchangeable. (damtp.cam.ac.uk)

Physical origin

Fermi energy arises from the Pauli exclusion principle: identical fermions cannot occupy the same complete single-particle quantum state. Consequently, cooling a collection of fermions to absolute zero does not place every particle in the lowest-energy state. Instead, particles occupy successively higher states until all have been accommodated. The upper boundary of this filling defines the Fermi energy. (damtp.cam.ac.uk)

An electron has two possible spin projections. Thus, in an unpolarized system, a spatial state can accommodate two electrons with different spin states without violating exclusion. Even without interactions, a many-electron system therefore possesses a nonzero ground-state kinetic energy. This is not thermal energy: it remains when thermal excitations disappear. (ocw.mit.edu)

Fermi energy and Fermi level

In thermal equilibrium, the mean occupation of an independent fermion state of energy $E$ follows Fermi–Dirac statistics:

where $T$ is temperature, $\mu$ is the chemical potential, and $k_B$ is the Boltzmann constant. As $T$ approaches zero, this distribution becomes a step: states below $\mu$ are occupied and states above it are empty. For a continuous spectrum at fixed particle density,

At positive temperature, some states above the chemical potential are occupied, so there is no sharp “highest occupied energy.” (schwartz.scholars.harvard.edu)

Terminology depends on context. In ideal-gas calculations, Fermi energy usually means a fixed zero-temperature scale, whereas Fermi level means $\mu(T)$. Semiconductor literature frequently uses $E_F$, and sometimes “Fermi energy,” for the finite-temperature Fermi level. An energy value must also specify its reference: the bottom of an electronic band and the vacuum level are different choices. (schwartz.scholars.harvard.edu)

For a semiconductor, the Fermi level can lie inside a band gap, where no electronic states exist. It is therefore not necessarily the energy of an occupied electron. In a finite system or a gapped spectrum, identifying the chemical potential with the highest occupied state requires care; the simple equality is most directly applicable to a macroscopic system with available states at the occupation boundary. (schwartz.scholars.harvard.edu)

Three-dimensional ideal Fermi gas

Consider $N$ noninteracting, nonrelativistic fermions of mass $m$ in volume $V$, with number density $n=N/V$. Their dispersion relation is

where $\hbar=h/(2\pi)$ is the reduced Planck constant. If $g$ is the number of equally populated internal states, counting occupied states gives

For unpolarized spin-$\tfrac12$ particles, $g=2$, yielding

The energy reference here is the minimum of the free-particle spectrum. Increasing density raises $E_F$, while increasing mass lowers it at fixed density. (damtp.cam.ac.uk)

The associated Fermi momentum and velocity are

These describe the boundary states, not the momentum and speed of every particle. For this particular three-dimensional model, the ground-state internal energy and pressure are

Thus, the mean energy per particle is $3E_F/5$, not $E_F$. The residual pressure is called degeneracy pressure. These coefficients depend on the dimensionality and dispersion relation and are not universal. (ocw.mit.edu)

Fermi surface and density of states

At zero temperature, occupied free-particle states fill a sphere in momentum space, often called the Fermi sea. Its boundary is the Fermi surface. In a crystal, electronic energies instead form bands $E_a(\mathbf{k})$; within an independent-electron description, the Fermi surface consists of points satisfying

It can have several disconnected sheets and need not be spherical. Consequently, Fermi energy alone does not specify a material’s electronic structure. (damtp.cam.ac.uk)

The density of states, $D(E)$, measures how many states are available per unit energy. It connects the occupation distribution to the particle number:

At zero temperature, this becomes an integral over occupied states up to $E_F$. Determining the Fermi energy in a solid therefore requires both its band spectrum and its electron population; the free-gas formula is not a general formula for arbitrary materials. (damtp.cam.ac.uk)

Temperature scale and low-temperature behavior

The Fermi temperature is defined by

Here $E_F$ is understood as an energy scale relative to the spectrum minimum. When $T\ll T_F$, the gas is strongly quantum-degenerate; when $T\gg T_F$, ideal-gas behavior approaches the classical regime. Metallic electron Fermi energies are commonly of order electronvolts, making room temperature low compared with their Fermi temperature. (damtp.cam.ac.uk)

At low temperature, changes in occupation occur mainly within an energy interval of order $k_BT$ around the Fermi level. The Sommerfeld expansion gives, for a three-dimensional ideal gas at fixed density,

and

The electronic heat capacity is therefore proportional to temperature rather than the temperature-independent classical value. These expressions assume $T\ll T_F$ and a smooth density of states near the occupation boundary. (damtp.cam.ac.uk)

Applications and limitations

In metals, Fermi energy provides the characteristic scale for electronic excitations. The quantum free-electron model explains the small, approximately linear electronic heat capacity and supplies a starting point for electrical and thermal transport. Quantitative transport also requires information about bands, velocities, and scattering; it cannot be inferred from $E_F$ alone. (ocw.mit.edu)

In semiconductors, the position of the Fermi level relative to band edges determines equilibrium electron and hole concentrations. Doping generally moves it upward for n-type material and downward for p-type material relative to the intrinsic level. Its position must be calculated together with charge neutrality and impurity ionization. (ocw.mit.edu)

In white dwarfs, electron degeneracy pressure helps resist gravitational compression. At sufficiently high densities, electron motion becomes relativistic, so the nonrelativistic energy–momentum relation and pressure formulas must be replaced. (damtp.cam.ac.uk)

The elementary picture also omits the periodic crystal potential and electron interactions. Band theory replaces the free-particle spectrum with material-specific bands, while interacting systems require a many-body description rather than literal filling of independent electron levels. The meaning of a quoted Fermi energy must therefore be read together with its model and energy reference. (live.ocw.mit.edu)

Historical development

The concept takes its name from Enrico Fermi, who formulated statistics for particles obeying the exclusion principle in 1926. The application of fermionic statistics to metallic electrons became central to Sommerfeld’s quantum electron theory of metals, discussed in contemporary literature by 1928. This replaced classical occupation statistics with quantum filling and established the occupation boundary as a fundamental scale in electron theory. (nobelprize.org)

References

1. Lecture 17 — Electronic, Optical and Magnetic Properties of Materialsocw.mit.edu
2. Physics 181 Lecture Notes — Matthew D. Schwartzschwartz.scholars.harvard.edu
3. Lecture 1: Free Electron Modelocw.mit.edu
4. Chapter 4: Identical Particlesocw.mit.edu
5. Band Theory of Solidslive.ocw.mit.edu
6. Enrico Fermi — Biographicalnobelprize.org
7. Sommerfeld’s Electron-Theory of Metalspmc.ncbi.nlm.nih.gov