The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi gas, the lowest occupied state is taken to have zero kinetic energy, whereas in a metal, the lowest occupied state is typically taken to mean the bottom of the conduction band.
The term "Fermi energy" is often used to refer to a different yet closely related concept, the Fermi level (also called electrochemical potential).
In quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons) obey the Pauli exclusion principle.
Since an idealized non-interacting Fermi gas can be analyzed in terms of single-particle stationary states, we can thus say that two fermions cannot occupy the same stationary state.
These stationary states will typically be distinct in energy.
To find the ground state of the whole system, we start with an empty system, and add particles one at a time, consecutively filling up the unoccupied stationary states with the lowest energy.
As a consequence, even if we have extracted all possible energy from a Fermi gas by cooling it to near absolute zero temperature, the fermions are still moving around at a high speed.
The Fermi energy is an important concept in the solid state physics of metals and superconductors.
It is also a very important quantity in the physics of quantum liquids like low temperature helium (both normal and superfluid 3He), and it is quite important to nuclear physics and to understanding the stability of white dwarf stars against gravitational collapse.
The Fermi energy for a three-dimensional, non-relativistic, non-interacting ensemble of identical spin-1⁄2 fermions is given by[1]
where N is the number of particles, m0 the rest mass of each fermion, V the volume of the system, and
of conduction electrons in metals ranges between approximately 1028 and 1029 electrons/m3, which is also the typical density of atoms in ordinary solid matter.
This number density produces a Fermi energy of the order of 2 to 10 electronvolts.
[2] Stars known as white dwarfs have mass comparable to the Sun, but have about a hundredth of its radius.
The radius of the nucleus admits deviations, so a typical value for the Fermi energy is usually given as 38 MeV.
Using this definition of above for the Fermi energy, various related quantities can be useful.
Other quantities defined in this context are Fermi momentum
These quantities are respectively the momentum and group velocity of a fermion at the Fermi surface.
These quantities may not be well-defined in cases where the Fermi surface is non-spherical.