Electron dynamics in crystalline semiconductors

TL;DR

This paper clarifies the distinction between instantaneous and average electron velocities in crystalline semiconductors, introduces a velocity-based effective mass concept, and explores a two-band model with semi-relativistic analogies.

Contribution

It introduces the velocity mass as a more useful physical quantity than the acceleration mass and applies a two-band model to describe narrow-gap semiconductors and graphene.

Findings

Velocity mass is a scalar for spherical, nonparabolic bands.

The two-band model resembles special relativity.

Electron velocity in transport differs from instantaneous velocity.

Abstract

Electron dynamics in crystalline semiconductors is described by distinguishing between an instantaneous velocity related to electron's momentum and an average velocity related to its quasi-momentum in a periodic potential. It is shown that the electron velocity used in the theory of electron transport and free-carrier optics is the average electron velocity, not the instantaneous velocity. An effective mass of charge carriers in solids is considered and it is demonstrated that, in contrast to the "acceleration" mass introduced in textbooks, it is a "velocity" mass relating carrier velocity to its quasi-momentum that is a much more useful physical quantity. Among other advantages, the velocity mass is a scalar for spherical but nonparabolic energy bands $\epsilon(k)$, whereas the acceleration mass is not a scalar. Important applications of the velocity mass are indicated. A two-band ${\bm k}\cdot {\bm \hp}$ model is introduced as the simplest example of a band structure that still keeps track of the periodic lattice potential. It is remarked that the two-band model, adequately describing narrow-gap semiconductors (including zero-gap graphene), strongly resembles the special theory of relativity. Instructive examples of the "semi-relativistic" analogy are given. The presentation has both scientific and pedagogical aspects.