Valley Zeeman effect and Landau levels in two-dimensional transition metal dichalcogenides

Abstract

This paper presents a theoretical description of both the valley Zeeman effect (g-factors) and Landau levels in two-dimensional H-phase transition metal dichalcogenides (TMDs) using the Luttinger-Kohn approximation with spin-orbit coupling. At the valley extrema in TMDs, energy bands split into Landau levels with a Zeeman shift in the presence of a uniform out-of-plane external magnetic field. The Landau level indices are symmetric in the K and K ′ valleys. We develop a numerical approach to compute the single-band g-factors from first principles without the need for a sum over unoccupied bands. Many-body effects are included perturbatively within the GW approximation. Nonlocal exchange and correlation self-energy effects in the GW calculations increase the magnitude of single-band g-factors compared to those obtained from density functional theory. Our first-principles results give spin- and valley-split Landau levels, in agreement with recent optical experiments. The exciton g-factors deduced in this work are also in good agreement with experiment for the bright and dark excitons in monolayer WSe2, as well as the lowest-energy bright excitons in MoSe2 − WSe2 heterobilayers with different twist angles.