Quantum mechanics of electrons in crystals with graded composition

We construct the effective Hamiltonian describing the motion of electrons in compositionally graded crystals which is valid throughout a given energy band and part way into the gaps. Near the edges of a simple or degenerate band this effective Hamiltonian reduces to an effective mass Hamiltonian with position dependent effective mass. Next, we examine more general states—not restricted to the vicinity of a band edge—in crystals with composition and applied potential variation in one direction. We obtain a WKB-type solution for the envelope functions, as well as the appropriate turning point connection rules.