Fermi surface

In electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which prevents fermions from all crowding into the same state.

Theory

Formally speaking, the Fermi surface is a surface of constant energy in $\vec{k}$ -space where $\vec{k}$ is the wavevector 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 $\vec{k}_z$ direction. Often in a metal the Fermi surface radius $k F$ is larger than the size of the first spin density waves.

The state occupancy of 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.

Experimental determination

de Haas-van Alphen effect. Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields $H$ , for example the de Haas-van Alphen effect (dHvA) and the Shubnikov-De Haas effect (SdH). The former is an oscillation in Lars Onsager proved that the period of oscillation $Δ H$ is related to the cross-section of the Fermi surface (typically given in $\AA^{-2}$ ) perpendicular to the magnetic field direction $A_{\perp}$ by the equation $A_{\perp} = \frac{2 \pi e \Delta H}{\hbar c}$ . 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. Therefore dHvA and SdH experiments are usually performed at high-field facilities like the High Field Magnet Laboratory in Netherlands, Grenoble High Magnetic Field Laboratory in France, the Tsukuba Magnet Laboratory in Japan or the National High-Field Magnet Lab in the United States.

Angle resolved photoemission. The most direct experimental technique to resolve the electronic structure of crystals in the momentum-energy space (see ARPES is shown in figure.

Two photon positron annihilation. With positron annihilation the two photons carry the momentum of the electron away; as the momentum of a thermalized positron is negligible, in this way also information about the momentum distribution can be obtained. Because the positron can be polarized, also the momentum distribution for the two spin states in magnetized materials can be obtained. Another advantage with De Haas-Van Alphen-effect is that the technique can be applied to non-dilute alloys. In this way the first determination of a smeared Fermi surface in a 30% alloy has been obtained in 1978.

References

N. Ashcroft and N.D. Mermin, Solid-State Physics, ISBN 0-03-083993-9.
W.A. Harrison, Electronic Structure and the Properties of Solids, ISBN 0-486-66021-4.
VRML Fermi Surface Database

Fermi surface

Theory

Experimental determination

References

See Also