Electrons in metals: A short guide to the Fermi surface

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Abstract
The theory of electrons in metals starts with a gas of free electrons, where the Fermi surface is a perfect sphere. But soon we introduce the crystal lattice. The sphere is carved along the zone boundaries, energy gaps appear, the Fermi surface is divided into several sheets and becomes horribly distorted. But we learn the significance of “holes” in energy bands, and unmask the mystery of positive Hall coefficients. Hoping then that we might derive E(k) from first principles, we embarked upon arduous calculations of the band structure. These attempts failed; we never seemed able to calculate the energy gaps with sufficient accuracy. Everything depended on the atomic potential, which was clouded in the fog of correlation and exchange effects (the many-body problem). Our only thread of hope was a tenuous argument suggesting that the effective potential was really rather small, and that the Coulomb interaction between the electrons could be treated as a screening effect. But so far we have made little progress in our quest, for we have not yet predicted, or calculated, or worked out, any exact number for any measurable property of any particular metal.

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