The Fermi-Dirac staircase occupation of Floquet bands and current rectification inside the optical gap of metals: a rigorous perspective

TL;DR

This paper rigorously demonstrates that a driven Bloch band coupled to a fermionic bath exhibits a staircase occupation distribution, leading to a finite rectified current inside the optical gap of metals, even with negligible relaxation.

Contribution

It provides a rigorous analysis of Floquet band occupations and rectification phenomena, clarifying previous assumptions and connecting to perturbative current expressions.

Findings

Floquet band occupation forms a staircase distribution, not a simple Fermi-Dirac.

Finite rectified current persists within the optical gap even with negligible relaxation.

Rectified current approaches the Jerk current in the small frequency limit.

Abstract

We consider a model of a Bloch band subjected to an oscillating electric field and coupled to a featureless fermionic heat bath, which can be solved exactly. We demonstrate rigorously that in the limit of vanishing coupling to this bath (so that it acts as an ideal thermodynamic bath) the occupation of the Floquet band is not a simple Fermi-Dirac distribution function of the Floquet energy, but instead it becomes a ``staircase'' version of this distribution. We show that this distribution generically leads to a finite rectified electric current within the optical gap of a metal even in the limit of vanishing carrier relaxation rates, providing a rigorous demonstration that such rectification is generically possible and clarifying previous statements in the optoelectronics literature. We show that this current remains non-zero even up to the leading perturbative second order in the amplitude of electric fields, and that it approaches the standard perturbative expression of the Jerk current obtained from a simpler Boltzmann description within a relaxation time approximation when the frequencies are small compared to the bandwidth.