Quantum Mechanics of Electrons in Crystals with Graded Composition.

摘要

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. The effective Hamiltonian, constructed from the band structures of uniform crystals, also includes the effects of a slowly varying applied scalar potential U(r). Near the edges of a nondegenerate band, this effective Hamiltonian reduces to an effective mass Hamiltonian with position dependent mass (one of several forms previously appearing in the literature): H sub eff = 1/2 pi(1/m*(r)) sub ij pj + Epsilon(r) + U(r), where Epsilon(r) is the energy of the band edge as a function of position. The analogous effective mass Hamiltonian for degenerate bands is also derived. 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.... Effective Hamiltonian, Band structures, applied scalar potential U(r), Analogous effective mass Hamiltonian for degenerate bands, WKB-Type solution, Envelope functions