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The Fermi surface is the surface in k-space separating filled from empty states at T=0, defined by |k| = k_F = (3π^2 n)1/3. The Fermi energy E_F = ℏ^2 k_F2/(2m) depends on number density. Most low-temperature properties of metals (specific heat, magnetic susceptibility, transport) are dominated by states near the Fermi surface.
From your study of the ideal Fermi gas at T=0, you know that electrons fill available states from the bottom up, obeying the Pauli exclusion principle. In the free-electron model, the quantum states are labeled by wave vector k, with energy E(k) = ℏ²|k|²/(2m). At absolute zero, electrons occupy all states with |k| less than some maximum value k_F, and all states with |k| > k_F are empty. The Fermi surface is the boundary between these two regions in k-space — a sphere of radius k_F for free electrons. Everything interesting in a metal happens at or near this surface.
The Fermi wave vector k_F is set entirely by the electron number density n: counting the states inside the Fermi sphere (and including the factor of 2 for spin) gives n = k_F³/(3π²), so k_F = (3π²n)1/3. The Fermi energy E_F = ℏ²k_F²/(2m) is then fixed by n as well. For a typical metal like copper with n ≈ 8.5 × 10²⁸ m⁻³, this gives E_F ≈ 7 eV, corresponding to a Fermi temperature T_F = E_F/k_B ≈ 80,000 K. This enormous energy scale is purely a quantum mechanical effect: electrons are forced into high-energy states not by thermal agitation but by the Pauli exclusion principle. At room temperature (T/T_F ≈ 0.004), the thermal energy is tiny compared to E_F, which is why the T=0 picture remains a good starting point for most metal physics.
The dominance of the Fermi surface in low-temperature properties follows from this same logic. At small but nonzero temperature, only electrons within about k_BT of E_F can be thermally excited — those much deeper in the Fermi sea lack available empty states nearby to move into. This means only a fraction T/T_F of electrons participate in thermal processes. The electronic specific heat is therefore linear in T (c_e ∝ T, not the classical equipartition value 3k_B/2 per electron), and magnetic susceptibility is temperature-independent (Pauli paramagnetism), both in sharp contrast to classical predictions. These are direct signatures of the sharp Fermi surface.
In real metals, the Fermi surface is rarely a perfect sphere: the crystal lattice imposes a periodic potential that distorts E(k) away from the free-electron parabola, reshaping the surface into complex geometries — necks, pockets, sheets. The topology and shape of the Fermi surface control electrical conductivity (which electrons carry current), optical properties (which photon frequencies are absorbed), and magnetic behavior. Measuring the Fermi surface via de Haas-van Alphen oscillations or angle-resolved photoemission spectroscopy (ARPES) is one of the central experimental tools in condensed-matter physics, because the Fermi surface is the fingerprint of a metal's electronic structure.