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

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 $K^{\prime }$ 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.

Figure 1

(a) Side and top view of a monolayer TMD crystal structure with the transition metal in turquoise and the chalcogen in yellow. (b) Diagram of hexagonal 2D Brillouin zone. (c) Spin-valley coupled valence and conduction bands at $K$ and $K^{\prime }$ without (left) and with (right) time-reversal symmetry breaking because of a magnetic field (yellow arrow) or magnetic substrate.

Figure 2

(a) The band structure of pristine ${\mathrm{WSe}}_2$ with the spin character of W in each state indicated in red and blue for positive and negative, respectively. (b) The band structure of ${\mathrm{WSe}}_2$ with $100\%$ coverage of Fe atoms that are 4.4 Å above the W atom, with valley splitting clearly visible. (c) The band structure of ${\mathrm{WSe}}_2$ with $33\%$ coverage of Fe atoms that are 4.4 Å above the W atom, with $K$ and $K^{\prime }$ folded into $\mathrm{\Gamma }$. (d) Schematic of Fe atoms above TMD monolayer indicating $x$ and $z$ displacements. (e) Schematic of Fe atoms above TMD monolayer at $100\%$ coverage (left) and $33\%$ coverage (right).

Figure 3

(a) Schematic illustrating definition of the valence band (VB) splitting. (b) Valence band splitting of ${\mathrm{WSe}}_2$ and ${\mathrm{MoS}}_2$ with Fe atom at $100\%$ and $33\%$ coverage at variable Fe-W separations $z$. Circles and triangles are data points, solid lines are exponential fits. (c) Valence band splitting of ${\mathrm{WSe}}_2$ and ${\mathrm{MoS}}_2$ with Fe atom at $100\%$ coverage with variable lateral position $x$ and with $z$ fixed at 4.4 Å. Lateral position is represented here as a fractional position along the $a$ axis ($x/a$) following the direction of the $a$ axis displayed in Fig. 1. Circles are data points, solid lines are fits to cosines. (d) Schematic of canted Fe atom spin and definition of $z$ component of magnetic moment. (e) Valence band splitting of ${\mathrm{WSe}}_2$ and ${\mathrm{MoS}}_2$ at $100\%$ coverage with the magnetic moment of the Fe atom at variable angles with respect to the $c$ axis, causing the $z$ component of the magnetic moment to take different values. Circles are data points, solid lines are linear fits. (f) Valence band splitting of ${\mathrm{WSe}}_2$ in the presence of different magnetic atoms, Fe, Cr, and Zn. Circles are data points, solid lines are exponential fits.

Figure 4

(a) Schematic of spin-orbit coupling (SOC) splitting at $K$ and $K^{\prime }$ of TMD valence bands in the presence of an Fe atom. (b) Change in SOC splitting at $K$ and $K^{\prime }$ relative to no Fe system for ${\mathrm{WSe}}_2+\mathrm{Fe}$ atom system at $100\%$ coverage and Fe directly above W with interpolation between data points. The absolute value of the SOC splitting in the absence of Fe is 459 meV. (c) Change in SOC splitting at $K$ and $K^{\prime }$ relative to no Fe system for ${\mathrm{MoS}}_2+\mathrm{Fe}$ atom system at $100\%$ coverage and Fe directly above Mo. The absolute value of the SOC splitting in the absence of Fe is 146 meV. See the Supplemental Material for plots of absolute SOC splitting (Fig. S5) [28].

Figure 5

(a) Valence band splitting for ${\mathrm{WSe}}_2$ and ${\mathrm{MoS}}_2$ for $100\%$ coverage and $33\%$ coverage (with the Fe atoms positioned directly on top of the W and Mo atoms), same as Fig. 3, and fits to the valence band splitting model. (b) Fitted values of effective exchange $J$ arising from the superexchange interaction for ${\mathrm{WSe}}_2$ and ${\mathrm{MoS}}_2$ for $100\%$ and $33\%$ coverage.