Wannier-Slater theorem for solids with nonuniform band structure

The Wannier-Slater theorem for the Hamiltonian of a solid in an external field of force is generalized for solids with a nonuniform band structure, such as graded-gap mixed semiconductors. Our formulation is related to that of Gora and Williams, and their effectivemass equation is obtained if a $\overrightarrow{\mathrm{k}}$-space Taylor expansion is permissible. This paper extends a too-restricted, earlier discussion by us in connection with transport in position-dependent band structures. The concept of a graded effective mass $\overleftrightarrow{M}$ and position-dependent band bottom $E_c$ is derived from this treatment. It is pointed out, however, that the concept of a position-dependent effective mass has no strict quantum-mechanical validity. In the correspondence limit, Hamilton's equations lead to an acceleration which contains the effect of an external field ${\overleftrightarrow{M}}^{-1\mathrm{}}·\overrightarrow{\mathrm{F}}$, plus a dissipative term which stems from the deviation in periodicity of the crystal potential.