the Fermi surface

The Fermi surface is the coastline of the electron sea. Picture, in wavevector space, all the occupied electron states as a body of water filled to the brim: the surface separating the filled states below from the empty states above, at absolute zero, is the Fermi surface. Almost everything a metal does electrically, thermally, or magnetically is decided by the electrons living right at this shoreline.

Concretely, at zero temperature electrons fill every state with energy E(k) less than the Fermi energy E_F, so the Fermi surface is the constant-energy surface defined by E(k) = E_F in k-space. For a free-electron gas the bands are simple parabolas and the surface is a sphere, the Fermi sphere, of radius k_F = (3 pi^2 n)^(1/3). In a real crystal the periodic potential warps this sphere into intricate shapes, which can bulge, neck, and even break where the surface meets a Brillouin zone boundary. Only electrons within about k_B T of this surface can be thermally excited or scattered, so they alone carry current, conduct heat, and set the specific heat.

This is why the Fermi surface is one of the most measured objects in solid-state physics, mapped experimentally through quantum oscillations such as the de Haas-van Alphen effect. A sharp point of honesty: a Fermi surface exists only for a metal, a system with a partially filled band of fermions. An insulator or intrinsic semiconductor, whose bands are either completely full or completely empty, has no Fermi surface at all.