Semiconductor Physics Using Arrows: Ohm's Law, Density of States, Mobility and Conductivity from the Arrow Maxwell Equations

Abstract

Ohm's law in its usual form, a current proportional to the electric field, does not describe a semiconductor device; therefore Shockley's equations replace it. They rest on empirical parameters: a mobility measured for each material and fitted to the field, a diffusion coefficient tied to it, and the carriers' lifetimes. This paper proposes a more analytical, self-consistent description with arrow numbers. In the arrows, Maxwell's equations carry one more field, a scalar field, and its slope drives the current: the arrow Ohm's law, j = −c∇E₀. Just as the electric field points down the slope of the electric potential, the current flows down the slope of the scalar field. The conductivity is then the ratio of the two slopes, exact at every point, and the mobility is redefined from them, with no collision time and no fitted constant. In a measured infrared photodiode the current obeys it exactly, even where Shockley needs a diffusion current. Therefore, once each carrier's scalar field is known, no separate drift–diffusion model is needed; the same equation also gives Poisson's equation and each carrier's continuity equation. Electrons in a crystal have a speed limit of their own, set by the gap and the band-edge mass. It plays the part of the speed of light, and one speed links gap and mass across a family of semiconductors. Where momentum and velocity point in different directions, as in silicon, the mass is their quotient as arrows: the division that began this series in 2012. The theory meets measurements: the mobility of InSb from 220 to 700 K within 13% and the shift of its absorption edge with doping within 9%, both with nothing fitted; a 32 nm transistor within 1.4%, with the carriers' maximum speed fixed by the band; and graphene, whose massless carriers behave like light. Files: the preprint as submitted to the Journal of Modern Physics (PDF, v0.30.4), its LaTeX source, and the checks: 39 Python scripts that verify every statement marked (Checked.), most with a control that must fail, with the digitised data they use (README.md maps each script to its section). The device simulator ADEVA is archived separately: 10.5281/zenodo.23271129.