Quantum Mechanics of Electrons in Crystals with Graded Composition
Abstract
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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1993
- Accession Number
- ADA260294
Entities
People
- M. R. Geller
- W. Kohn
Organizations
- University of California, Santa Barbara