The basis of the semiclassical description of electron transport in solids

Abstract

The theoretical account of electron transport in crystalline solids is often taught to graduate students through a semiclassical equation, which is a classical kinetic equation amended with elements of quantum physics in a non-systematic way. This blending of classical and quantal elements accounts for many experimental features of electron transport, save for a few ones observed over short distances. For students acquainted with solid-state quantum mechanics, it is preferable to derive the semiclassical equation from quantum dynamics so as to pinpoint the shortcomings and approximations underlying the semiclassical picture of transport. This paper explores the pathway leading from quantum dynamics to semiclassical kinetics within a simplified model not exceeding the proficiencies of a graduate student in physics or electronic engineering. In this model an electron moves in a crystal under the joint influences of the periodic crystal potential, an external electrostatic field and a set of lattice imperfections haphazardly distributed in the crystal volume. The first influence is treated exactly by means of standard energy-band theory and the other two are handled perturbatively. Following a procedure already tested in free-space issues, we replace Schrödinger’s wave function in position space with a Wigner function defined in the phase (position–pseudomomentum) space of semiclassical kinetics. The dynamical equation on the Wigner function is worked out and approximated so as to retrieve the semiclassical equation. The errors incurred in making use of the latter are assessed quantitatively. It is explained how a time-irreversible evolution equation—governing a phase-space occupancy—arises from reversible Schrödinger dynamics—governing a complex-valued probability amplitude. Besides clarifying the foundation of the semiclassical picture of electron transport and providing an explicit quantum-coherence correction to that picture, this paper may help students in general physics to better grasp, in a concrete issue, the interplay between classical and quantum concepts.