Position-dependent effective mass for inhomogeneous semiconductors

A systematic approach is adopted to extract an effective low-energy Hamiltonian for crystals with a slowly varying inhomogeneity, resolving several controversies. It is shown that the effective mass m(R) is, in general, position dependent, and enters the kinetic energy operator as -∇[m${\mathrm{}\left(\mathrm{R}\right)\mathrm{}}^{\mathrm{-}1}$]∇/2. The advantage of using a basis set that exactly diagonalizes the Hamiltonian in the homogeneous limit is emphasized.