Fermi surface

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

meaning

Area of ​​constant energy of a primitive cubic lattice in the 1st  Brillouin zone

The Fermi surface is a surface of constant energy, and not in ordinary spatial space , but in reciprocal space . This is the momentum space that is obtained purely mathematically by a Fourier transformation from the spatial space. The use of this abstract concept of space has many advantages in describing crystalline systems, e.g. B. the reflections in the X-ray structure analysis can be assigned directly 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 (at absolute zero ) are only occupied up to a certain energy, the Fermi energy . The set of points pointed to by the momentum vectors of electrons with the Fermi energy form a closed surface or a few closed surfaces called the Fermi surface (s). With their help, many electronic and magnetic properties of the metal can be described. For example, only the electrons with Fermi energy and thus on the Fermi surface contribute to the electric current .

The Fermi surfaces of the alkali metals as well as 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 spheres. In the case of Cu, Ag and Au, however, the Fermi surfaces in the 111 directions each have a “neck” to the edge of the Brillouin zone; Small "necks" also occur with  Cs . The surfaces and necks can be z. B. measured experimentally using the De Haas van Alphen effect . ${\displaystyle \langle }$ ${\displaystyle \rangle }$

In the simplest case, ferromagnetic metals have two Fermi surfaces because of the two possible orientations of the electron spin .

Insulators and undoped semiconductors do not have a Fermi surface because their Fermi energy falls into the band gap and there are therefore no electronic states whose energy is equal to the Fermi energy. By introducing additional charge carriers into a semiconductor ( donor or acceptor doping ), however, the Fermi level can be shifted and thus the formation of a Fermi surface can be forced.

This also gives rise to what is probably the most precise definition of the term “metal” in the sense of a differentiation from other (solid) substances: A metal is a solid with a Fermi surface. According to this definition, liquid mercury (and melts of other “metals”) would not be a metal.

Fermi sphere

In a free electron gas , the states in the reciprocal space are successively filled up energetically, i.e. H. Starting with a wave vector up to a limit wave vector , the states are each occupied with two spin settings, which is referred to as the Fermi wave vector. The states in reciprocal space are therefore all within a sphere, the Fermi sphere . The electrons on the surface of the Fermi sphere have the energy ${\displaystyle k=0}$ ${\displaystyle k=k_{F}}$ ${\displaystyle k_{F}}$

${\displaystyle E_{F}={\frac {\hbar ^{2}k_{F}^{2}}{2m_{e}}}={\frac {1}{2}}{\frac {p_{F}^{2}}{m_{e}}}}$,

where is also referred to as Fermi impulse . ${\displaystyle p_{F}=\hbar k_{F}}$

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

${\displaystyle V_{F}={{4\pi } \over {3}}k_{F}^{3}={{4\pi } \over {3}}\left({{2m_{e}E_{F}} \over {\hbar ^{2}}}\right)^{3/2}}$.

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

Web links

• Graphic representation of the Fermi surfaces
• Summary of Fermi surfaces ( Memento from September 29, 2007 in the Internet Archive ) (PDF; 295 kB)

literature

• Charles Kittel: Introduction to Solid State Physics. 1st edition 1953 to 14th edition 2005, ISBN 0-471-41526-X (German introduction to solid state physics. Oldenbourg, ISBN 3-486-57723-9 )