Fermi surface

In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied electron states from unoccupied electron states at zero temperature.

[1] The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of electronic energy bands.

The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state.

Additionally, at zero temperature the enthalpy of the electrons must be minimal, meaning that they cannot change state.

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.

[8] 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.

A material whose Fermi level falls in a gap between bands is an insulator or semiconductor depending on the size of the bandgap.

Materials with complex crystal structures can have quite intricate Fermi surfaces.

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 three-dimensional case the reduced zone scheme means that from any wavevector

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.

Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields

and occur because of the quantization of energy levels in the plane perpendicular to a magnetic field, a phenomenon first predicted by Lev Landau.

In a famous result, Lars Onsager proved that the period of oscillation

is related to the cross-section of the Fermi surface (typically given in Å−2) perpendicular to the magnetic field direction

Thus the determination of the periods of oscillation for various applied field directions allows mapping of the Fermi surface.

Observation of the dHvA and SdH oscillations requires magnetic fields large enough that the circumference of the cyclotron orbit is smaller than a mean free path.

The most direct experimental technique to resolve the electronic structure of crystals in the momentum-energy space (see reciprocal lattice), and, consequently, the Fermi surface, is the angle-resolved photoemission spectroscopy (ARPES).

An example of the Fermi surface of superconducting cuprates measured by ARPES is shown in Figure 3.

In this way it is possible to probe the electron momentum density of a solid and determine the Fermi surface.

ACAR has many advantages and disadvantages compared to other experimental techniques: It does not rely on UHV conditions, cryogenic temperatures, high magnetic fields or fully ordered alloys.

However, ACAR needs samples with a low vacancy concentration as they act as effective traps for positrons.

Fig. 1: Fermi surface and electron mo­mentum density of copper in the reduced zone schema measured with 2D ACAR . [ 6 ]
Fig. 2: A view of the graphite Fermi surface at the corner H points of the Brillouin zone showing the trigonal symmetry of the electron and hole pockets.
Fig. 3: Fermi surface of BSCCO measured by ARPES . The experimental data shown as an intensity plot in yellow-red-black scale. Green dashed rectangle represents the Brillouin zone of the CuO 2 plane of BSCCO .