How to obtain complex transition dipole moments satisfying crystal symmetry and periodicity from ab-initio calculations

TL;DR

This paper introduces a method to compute complex transition dipole moments that are continuous, symmetric, and periodic in k-space from ab-initio calculations, improving the accuracy of nonlinear optical simulations in solids.

Contribution

It presents a way to choose a smooth-periodic gauge for TDMs, ensuring their proper symmetry and periodicity, which was previously challenging due to phase issues in ab-initio methods.

Findings

Correctly treated TDMs reproduce symmetry-based harmonic generation features.

Enhanced even harmonics in ZnO due to improved TDM calculations.

Revealed the importance of gauge choice in nonlinear optical simulations.

Abstract

Transition dipole moments (TDM) between energy bands of solids deserve special attention nowadays as intense lasers can easily drive non-adiabatic transitions of excited electron wave packets across the Brillouin zones. The TDM is required to be continuous, satisfying crystal symmetry, and periodic at zone boundaries. While present day ab-initio algorithms are powerful in calculating band structures of solids, they all introduced random phases into the eigenfunctions at each crystal momentum k. In this paper, we show how to choose a ``smooth-periodic'' gauge where TDMs can be smooth versus k, preserving crystal symmetry, as well as maintaining periodic at zone boundaries. Based on band structure and TDMs in the ``smooth-periodic'' gauge calculated from ab-initio algorithms, we revisit high-order harmonic generation from MgO which exhibits inversion symmetry and ZnO which has broken symmetry. The symmetry properties of TDMs with respect to k ensure the absence of even-order harmonics in system with inversion symmetry, while the TDM in the `smooth-periodic'' gauge for ZnO is shown to enhance even harmonics that were underestimated in previous simulations. These results reveal the importance of correctly treating the complex TDMs in nonlinear laser-solid interactions which has been elusive so far.