Electron effective mass in solids-A generalization of Bardeen's formula

A formula is derived for the effective mass of an electron in a crystal which replaces the sum over excited states in the usual sum rule by an integral over the surface of the unit cell. The integrand of the surface integral involves the wave function(s) at the symmetry point or band extremum k₀ and a second solution of Schroedinger's equation at the same energy but satisfying inhomogeneous boundary conditions on the cell surface. The procedure is applicable regardless whether there is a degeneracy at k₀, and spin-orbit coupling may be taken into account. The result thus represents a generalization of Bardeen's formula for the effective mass of an s-band at k = 0 in the Wigner-Seitz spherical approximation to an arbitrary band at any point k₀, using the polyhedral cell. A related variational principle for the components of the effective mass matrix is also derived.