Fermi surface

Theory

Consider a spin-less ideal Fermi gas of ${\displaystyle N}$ particles. According to Fermi–Dirac statistics, the mean occupation number of a state with energy ${\displaystyle {ϵ}_i}$ is given by

${\displaystyle ⟨n_i⟩=\frac{1}{e^{({ϵ}_i-\mu )/k_{\mathrm{B}}T}+1},}$

where

• ${\displaystyle ⟨n_i⟩}$ is the mean occupation number of the ${\displaystyle i}$th state
• ${\displaystyle {ϵ}_i}$ is the kinetic energy of the ${\displaystyle i}$th state
• ${\displaystyle \mu }$ is the chemical potential (at zero temperature, this is the maximum kinetic energy the particle can have, i.e. Fermi energy ${\displaystyle E_{\mathrm{F}}}$)
• ${\displaystyle T}$ is the absolute temperature
• ${\displaystyle k_{\mathrm{B}}}$ is the Boltzmann constant

Suppose we consider the limit ${\displaystyle T\to 0}$. Then we have,

${\displaystyle ⟨n_i⟩\to \{\begin{array}{ll}1& ({ϵ}_i<\mu )\\ 0& ({ϵ}_i>\mu )\end{array}.}$

By the Pauli exclusion principle, no two fermions can be in the same state. Additionally, at zero temperature the enthalpy of the electrons must be minimal, meaning that they cannot change state. If, for a particle in some state, there existed an unoccupied lower state that it could occupy, then the energy difference between those states would give the electron an additional enthalpy. Hence, the enthalpy of the electron would not be minimal. Therefore, at zero temperature all the lowest energy states must be saturated. For a large ensemble the Fermi level will be approximately equal to the chemical potential of the system, and hence every state below this energy must be occupied. Thus, particles fill up all energy levels below the Fermi level at absolute zero, which is equivalent to saying that is the energy level below which there are exactly ${\displaystyle N}$ states.

In momentum space, these particles fill up a ball of radius ${\displaystyle k_{\mathrm{F}}}$, 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. In reciprocal space, the Fermi surface of an ideal Fermi gas is a sphere of radius

${\displaystyle k_{\mathrm{F}}=\frac{p_{\mathrm{F}}}{\hslash }=\frac{\sqrt{2m{E}_{\mathrm{F}}}}{\hslash }}$,

determined by the valence electron concentration where ${\displaystyle \hslash }$ is the reduced Planck 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. Figure 2 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 ${\displaystyle {\mathbf{k}}_z}$ direction. Often in a metal, the Fermi surface radius ${\displaystyle k_{\mathrm{F}}}$ 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 ${\displaystyle \mathbf{k}}$ is allowed to have arbitrarily large values or a reduced-zone scheme where wavevectors are shown modulo $\frac{2\pi }{a}$ (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 ${\displaystyle \mathbf{k}}$ there is an appropriate number of reciprocal lattice vectors ${\displaystyle \mathbf{K}}$ subtracted that the new ${\displaystyle \mathbf{k}}$ now is closer to the origin in ${\displaystyle \mathbf{k}}$-space than to any ${\displaystyle \mathbf{K}}$. 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.