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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. Non-local 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 WSe$_2$, as well as the lowest-energy bright excitons in MoSe$_2$-WSe$_2$ heterobilayers with different twist angles.
Highlights
When a weak external magnetic field is applied to a periodic system, there are two physical effects on the band extrema
The energy expressions for the band extrema contain both the Zeeman term and LLs. One drawback of this approach is that the g-factor used in the Zeeman term is model-dependent, and in particular, does not have the atomic contribution that was suggested in the phenomelogical models. To describe both the valley Zeeman effect and LLs in two-dimensional (2D) transition metal dichalcogenides (TMDs), we propose using a general Hamiltonian including spin-orbit coupling (SOC) for an electron in a periodic potential perturbed by a uniform external magnetic field
We have used the Luttinger-Kohn (LK) approximation to provide a unified description of the Zeeman effect and LLs in 2D TMDs, with both effects being treated on an equal footing within the same general Hamiltonian
Summary
When a weak external magnetic field is applied to a periodic system, there are two physical effects on the band extrema. Kormányos et al have proposed to reduce a multiband k · p Hamiltonian to a single-band model using Löwdinpartitioning [25] In this case, the energy expressions for the band extrema contain both the Zeeman term and LLs. one drawback of this approach is that the g-factor used in the Zeeman term is model-dependent, and in particular, does not have the atomic contribution that was suggested in the phenomelogical models. To describe both the valley Zeeman effect and LLs in two-dimensional (2D) TMDs, we propose using a general Hamiltonian including spin-orbit coupling (SOC) for an electron in a periodic potential perturbed by a uniform external magnetic field. Derivations of the atomic, valley, and cross terms of the orbital magnetic moment are included in Appendix D
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