Fermi Surface - Theory

Theory

Consider a spinless ideal Fermi gas of particles. According to Fermi–Dirac statistics, the mean occupation number of a state with energy is given by

where,

• is the mean occupation number

• is the energy of the state

• is the chemical potential (which is the maximum energy the particle can have at zero temperature, i.e. Fermi energy )

Suppose we consider the limit . Then we have,

By the Pauli exclusion principle, no two particles can be in the same state. Therefore, in the state of lowest energy, the particles fill up all energy levels till, which is equivalent to saying that is the energy level below which there are exactly states.

In momentum space, these particles fill up a sphere of radius, the surface of which is called the Fermi surface

The linear response of a metal to an electric, magnetic or thermal gradient is determined by the shape of the Fermi surface, because currents are due to changes in the occupancy of states near the Fermi energy. Free-electron Fermi surfaces are spheres of radius

determined by the valence electron concentration where is the reduced Planck's constant. A material whose Fermi level falls in a gap between bands is an insulator or semiconductor depending on the size of the bandgap. When a material's Fermi level falls in a bandgap, there is no Fermi surface.

Materials with complex crystal structures can have quite intricate Fermi surfaces. The figure illustrates the anisotropic Fermi surface of graphite, which has both electron and hole pockets in its Fermi surface due to multiple bands crossing the Fermi energy along the direction. Often in a metal the Fermi surface radius is larger than the size of the first Brillouin zone which results in a portion of the Fermi surface lying in the second (or higher) zones. As with the band structure itself, the Fermi surface can be displayed in an extended-zone scheme where is allowed to have arbitrarily large values or a reduced-zone scheme where wavevectors are shown modulo (in the 1-dimensional case) where a is the lattice constant. In the three dimensional case the reduced zone scheme means that from any wavevector there is an appropriate number of reciprocal lattice vectors subtracted that the new now is closer to the origin in -space than to any . Solids with a large density of states at the Fermi level become unstable at low temperatures and tend to form ground states where the condensation energy comes from opening a gap at the Fermi surface. Examples of such ground states are superconductors, ferromagnets, Jahn–Teller distortions and spin density waves.

The state occupancy of fermions like electrons is governed by Fermi–Dirac statistics so at finite temperatures the Fermi surface is accordingly broadened. In principle all fermion energy level populations are bound by a Fermi surface although the term is not generally used outside of condensed-matter physics.