Fermi surface

The Fermi surface ( named after the Italian physicist Enrico Fermi ) is a construction that is used in solid-state physics to describe the energy states of electrons in a metal.

Importance

The Fermi surface is not in the usual spatial domain, but in the so-called reciprocal space, and there is a constant-energy surface. The reciprocal space is the momentum space, which is obtained purely mathematically by a Fourier transform of the spatial domain. The use of this rather abstract space has many advantages in the description of crystalline systems, such as can be the reflections in the X-ray structure analysis directly assigned to the reciprocal lattice.

In particular, the energy can be represented directly as a function of the momentum of the electrons in reciprocal space. In metals, the energy levels of the conduction band in the lowest energy state are only up to a certain energy, the Fermi energy, busy ( at absolute zero ). The set of points, point to the momentum vectors of the electrons with the Fermi energy, which form a closed surface or a few closed surfaces, the Fermi level (s ) is or are called. With their help, many electronic and magnetic properties of the metal describe. For example, only the electrons contribute to the Fermi energy and thus at the Fermi surface for electricity.

The Fermi surfaces of the alkali metals and the metals Cu, Ag and Au are relatively simple because all conduction electrons lie within the first Brillouin zone. The Fermi surfaces are therefore almost balls. For Cu, Ag and Au, the Fermi surfaces have, however, in the 111- directions, respectively a "neck" to the edge of the Brillouin zone; also occur in Cs small " necks " on. The faces and necks can be measured experimentally, for example by making use of the de Haas -van Alphen - effect.

Ferromagnetic metals have two Fermi surfaces for the two possible orientations of the electron spin in the simplest case.

Undoped semiconductors and insulators do not have Fermi - surface, because the Fermi energy in them falls within the band gap, and thus there is no electron states, whose energy is equal to the Fermi energy. By introducing additional charge carriers ( donor or acceptor ) in a semiconductor, however, the Fermi level can be moved and thus the formation of a Fermi surface are accelerated.

It follows also the probably most accurate definition of the term " metal " in the sense of separation from other ( solid ) materials: A metal is a solid state with a Fermi surface - according to this definition would be liquid mercury ( and melts of other "metals") no metal.

Fermi sphere

In a free electron gas states in reciprocal space are filled successively energy, that is, starting with a wave vector to a boundary wave vector which states are occupied by two spin settings being referred to as Fermi wave vector. Therefore, the states lie in reciprocal space all within a sphere, the Fermi sphere. Electrons on the surface of the ball have the Fermi energy

The volume of the Fermi sphere in reciprocal ( three-dimensional ) space is then

The model of the free electron gas exceeds approximately for metals, particularly alkali metals such as sodium or potassium, as these have only one electron in the unit cell as the free charge carrier. Due to the spherical shape, some physical calculations can be simplified in order to reach a qualitative understanding.