Phonon-assisted shift-current from first-principles: application to AlN monolayer

In this work, we present a first-principles approach to investigate phonon-assisted shift current in extended systems. Our method combines the special displacement technique with real-time simulations based on the dynamical Berry phase. Merging these two frameworks enables direct access to non-linear optical responses at finite temperatures, while naturally incorporating phonon scattering and phonon-assisted electronic transitions into the time-dependent dynamics. This allows us to capture temperature-dependent features and lattice fluctuations that are essential for a realistic description of non-linear photocurrent generation. As a representative example, we apply our approach to the shift-current response of a monolayer of hexagonal AlN (h-AlN). This system provides an ideal testing ground because it features an indirect band gap and lacks inversion symmetry. Moreover, h-AlN layers of different thicknesses have been fabricated and characterized in a variety of experimental settings, opening the concrete possibility of testing our predictions. Our results demonstrate the relevance of phonon-mediated processes and the importance of temperature in determining the magnitude and spectral features of the shift current in layered materials.