Dynamics of electrons in 2D materials

During this thesis, we have proposed theoretical and numerical methods to study the behavior of electrons in periodic materials, with a particular interest for 2D materials, like graphene, and topological insulators. The thesis consists in three parts, organized as follows. The first part provides a mathematically proven and simple algorithm to build localized Wannier functions, with the only requirement that the system has vanishing Chern numbers. Based on an explicit and constructive proof of homotopies for the unitary group, the algorithm is able to build localized Wannier for topological insulators such as the Kane-Mele model. The method is validated by numerical tests for several systems. In the second part, we propose an approximation method for Wannier functions that is adapted to the computation of tight-binding Hamiltonians in non-periodic van der Waals heterostructures, that is, layers of 2D materials stacked on top of each other, bound together by the comparatively weak van der Waals forces. This setting is challenging for the usual computational tools of solid-state physics, which rely on the periodicity of crystals. In this context, a first-order approximation is to consider the Wannier functions computed on each layer independently. We therefore propose an approximation scheme for Wannier functions that allows for an efficient computation of tight-binding matrix coefficients, even in the non-periodic case. The third part is theoretical and devoted to the study of independent electrons in a periodic crystal in their ground state, set in motion by a uniform electric field at some prescribed time. We rigorously define the current per unit volume and study its properties using both linear response and adiabatic theory. Our results provide a unified framework for various phenomena such as the quantification of Hall conductivity of insulators with broken time-reversibility, the ballistic regime of electrons in metals, Bloch oscillations in the long-time response of metals, and the static conductivity of graphene.