Time evolution of energies and populations in germanium perturbed by a near-infrared pulse on the atto-second scale

Implementing time-resolved spectroscopy is one of the challenging topics in solid-state physics: indeed, it can be a crucial step to unveil the early time dynamics of fundamental processes in systems such as semiconductors, metals, superconductors and topological materials. Then, it is essential to build theoretical tools in order to understand the experimental results and what actually happens in solid-state systems on a new and finer timescale: the atto-second one. Accordingly, we have developed a theoretical–numerical tool in order to study the time evolution of the electronic populations on the atto-second scale in materials perturbed by a laser pulse. After finding the ab-initio band structure of the material, we compute the maximally-localized Wannier functions and obtain the hopping terms. Then, we study the effect of the laser pulse using the Peierls substitution method and obtain the equation of motion for the electronic populations. Finally, we obtain a dynamical DOS of the system and the corresponding dynamical electronic populations. In this work, as a relevant case study, we report our results for germanium. We identify oscillations in the eigenenergies and in the populations which roughly follow either the field or its square, with some phase shifts with respect to the field