Momentum dependence of spin–orbit interaction effects in single-layer and multi-layer transition metal dichalcogenides

Abstract

One of the main characteristics of the new family of two-dimensional crystals of semiconducting transition metal dichalcogenides (TMD) is the strong spin-orbit interaction, which makes them very promising for future applications in spintronics and valleytronics devices. Here we present a detailed study of the e ect of spinorbit coupling (SOC) on the band structure of single-layer and bulk TMDs, including explicitly the role of the chalcogen orbitals and their hybridization with the transition metal atoms. To this aim, we combine density functional theory (DFT) calculations with a Slater-Koster tight-binding model. Whereas most of the previous tight-binding models have been restricted to the K and K' points of the Brillouin zone (BZ), here we consider the e ect of SOC in the whole BZ, and the results are compared to the band structure obtained by DFT methods. The tight-binding model is used to analyze the e ect of SOC in the band structure, considering separately the contributions from the transition metal and the chalcogen atoms. Finally, we present a scenario where, in the case of strong SOC, the spin/orbital/valley entanglement at the minimum of the conduction band at Q can be probed and be of experimental interest in the most common cases of electron-doping reported for this family of compounds.

Valley Zeeman effect and spin-valley polarized conductance in monolayerMoS2in a perpendicular magnetic field

“…These parameters will be presented elsewhere [ 26]. We might express that this Hamiltonian provides a very good energy band structure in according to the comparison with those results obtained within the density functional theory simulations [24].…”

“…where λ M = 0.075eV and λ X = 0.052eV stand for the spin-orbit coupling originating from the Mo (metal) and S (chalcogen) atoms, respectively [24]. Notice that s = ± indicates the z−component of the spin degree of freedom.…”

Atomic defect states in monolayers of MoS2 and WS2

“…The hopping parameters, t iα,jβ = ψ iα |H SK |ψ jβ , between atomic orbitals ψ iα and ψ jβ are real Slater-Koster integrals that depend for each orbital pair on the directional cosines of the vector connecting nearest neighbors and on the Slater-Koster TB parameters V ssσ , The intra-atomic spin-orbit interaction acting on both the transition metal and the chalcogen atoms is incorporated in the Hamiltonian via the second term in Eq. (1) which is written as 31 ,…”

“…After the two-band k · p model describing the conduction and valence bands around the two valleys (K and K ′ points) in the hexagonal Brillouin zone of LTMD 27 , several TB models in various approximations have been proposed to reproduce the first-principles band structure of pristine LTMD 23,[28][29][30][31] . Among them, the TB model of Zahid et al 29 , including nonorthogonal sp 3 d 5 orbitals of M and X atoms and spin-orbit coupling, is able to accurately reproduce the first-principles bands for a wide range of energies in the Brillouin zone.…”

Green's function approach to edge states in transition metal dichalcogenides

“…One can straightforwardly include onsite atomic λL · S spin-orbit coupling (SOC) in the tight-binding matrices [34,75] where ± (∓) is for the spin-up, respectively the spin-down component, and λ gives the strength of the SOC [34]. In the three-band tight-binding basis S z remains a good quantum number, and the calculations for the spin-up and the spin-down bands can be performed independently.…”