Semiconductor Bloch equations in Wannier gauge with well-behaved dephasing

TL;DR

This paper analyzes the impact of dephasing operators in semiconductor Bloch equations within the Wannier gauge, identifies stability issues at band crossings, and proposes a modified approach that improves accuracy and efficiency.

Contribution

It introduces a modified dephasing operator in the Wannier gauge that reduces artifacts and computational cost in high-harmonic spectrum simulations.

Findings

Standard dephasing approach causes artifacts at band crossings.

Modified dephasing operator reduces artifacts and speeds up calculations.

The approach marginally affects the high-harmonic spectrum.

Abstract

The semiconductor Bloch equations (SBEs) with a dephasing operator for the microscopic polarizations are a well established approach to simulate high-harmonic spectra in solids. We discuss the impact of the dephasing operator on the stability of the numerical integration of the SBEs in the Wannier gauge. It is shown that the standard approach to apply dephasing is ill-defined in the presence of band crossings and leads to artifacts in the carrier distribution. They are caused by rapid changes of the dephasing operator matrix elements in the Wannier gauge, which render the convergence of the simulation in the stationary basis infeasible. In the comoving basis, also called Houston basis, these rapid changes can be resolved, but only at the cost of a largely increased computation time. As a remedy, we propose a modification of the dephasing operator with reduced magnitude in energetically close subspaces. This approach removes the artifacts in the carrier distribution and significantly speeds up the calculations, while affecting the high-harmonic spectrum only marginally. To foster further development, we provide our parallelized source code.