Superexchange-induced valley splitting in two-dimensional transition metal dichalcogenides: A first-principles study for rational design

Abstract

Monolayer transition metal dichalcogenides (TMDs) with spin-valley coupling are a well-studied class of two-dimensional materials with potential for novel optoelectronics applications. Breaking time-reversal symmetry via an external magnetic field or supporting magnetic substrate can lift the degeneracy of the band gaps at the inequivalent $K$ and ${K}^{\ensuremath{'}}$ high symmetry points, or valleys, in the monolayer TMD Brillouin zone, a phenomenon known as valley splitting. However, reported valley splittings thus far are modest, and a detailed structural and chemical understanding of valley splitting via magnetic substrates is lacking. Here we probe the underlying physical mechanism with a series of density functional theory (DFT) calculations of magnetic atoms with varying coverage on the surface of prototypical monolayer ${\mathrm{WSe}}_{2}$ and ${\mathrm{MoS}}_{2}$ TMDs. Near-valence band edge energies for variable magnetic atom height, lateral registry, and magnetic moment are calculated with DFT, and trends are rationalized with a model Hamiltonian with second-order spin-dependent exchange coupling. From our analysis, we demonstrate how large valley splittings may be achieved and that the valley splitting can be understood with a superexchange mechanism, which strongly depends on overlaps of TMD Bloch states at the valley extrema with the localized $d$ states of the magnetic atom, as well as the out-of-plane component of the magnetic moment of the magnetic atom. Our calculations provide a basis for understanding prior measurements of valley splitting and suggest routes for enhancing valley splitting in future systems of interest.