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 $K^{\prime }$ 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 ${\mathrm{WSe}}_2$, as well as the lowest-energy bright excitons in ${\mathrm{MoSe}}_2\text{−}{\mathrm{WSe}}_2$ heterobilayers with different twist angles.

• Received 14 February 2020
• Revised 3 July 2020
• Accepted 7 July 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033256

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society