Fermi energy

In physics, some particles (for example electrons) follow the Pauli exclusion principle – i.e. that no two particles may occupy the same state at the same time. When a number of electrons are put into a system, electrons will occupy higher energy levels when the lower ones are filled up. Loosely speaking, the Fermi energy is the energy of the highest occupied state at zero temperature. It is given the symbol E_F. Fermi energy is a concept which finds application in semiconductor theory and device design.

The Fermi surface is the surface in the momentum space where the total excitation energy equals the Fermi energy. It can have a non-trivial topology. Roughly speaking, it divides the occupied states from the empty ones.

Fermi energy was named after Enrico Fermi who, with Paul Dirac, derived the Fermi-Dirac statistics. These statistics allow one to predict the behaviour of large numbers of electrons under certain circumstances, especially in solids. The equations of quantum mechanics would otherwise be too hard to solve in such situations.

The Fermi energy of a three-dimensional, non-interacting, non-relativistic Fermi gas (or free electron gas) is related to the chemical potential by the equation

$\mu = \epsilon _F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{\epsilon _F}\right) ^2 + \frac{\pi^4}{80} \left(\frac{kT}{\epsilon _F}\right)^4 + ... \right]$

where ε_F is the Fermi energy, k is the Boltzmann constant and T is temperature. Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic Fermi temperature E_F/k. The characteristic temperature is on the order of 10⁵K for a metal, hence at room temperature (300K), the Fermi energy and chemical potential are essentially equivalent. This is significant since it is the chemical potential, not the Fermi energy, which appears in Fermi-Dirac statistics.

Fermi energy

See also