A mathematical walk into the paradox of Bloch oscillations

TL;DR

This paper mathematically models how collisions in a semiconductor can enable current flow through Bloch oscillations, explaining a paradoxical phenomenon with a simplified yet accurate approach.

Contribution

It introduces a novel phase-space based ODE model that captures transient Bloch oscillations and their damping, aligning well with experimental observations.

Findings

Model successfully describes Bloch oscillations and damping.

Explains current flow due to collisions, not despite them.

Matches experimental data on semiconductor charge transport.

Abstract

We describe mathematically the apparently paradoxical phenomenon that an electronic current in a semiconductor can flow because of collisions, and not despite them. A transport model of charge transport in a one-dimensional semiconductor crystal is considered, where each electron follows the periodic hamiltonian trajectories, determined by the semiconductor band structure, and undergoes non-elastic collisions with a phonon bath. Starting from the detailed phase-space model, a closed system of ODEs is obtained for averaged quantities. Such a simplified model is nevertheless capable of describing transient Bloch oscillations, their damping and the consequent onset of a steady current flow, which is in good agreement with the available experimental data.