Chiral orbital/spin textures and Edelstein effects in monolayer Janus TMDs

TL;DR

This work demonstrates that monolayer Janus TMDs host both orbital and spin Edelstein effects, arising from an intrinsic out-of-plane electric field that mixes $d$ and $p$ orbitals and generates orbital textures at $\Gamma$, $K$, and $K'$ valleys. Using DFT with SOC and Wannier-based tight-binding modeling, the authors show that orbital textures exist even without SOC (orbital Rashba effect) and that SOC induces a chirality reversal of the orbital texture while simultaneously generating a spin texture with a measurable spin Rashba constant $\alpha_R$. The orbital Edelstein effect is found to be stronger than the spin Edelstein effect across the doping range, and Te-based Janus compounds exhibit the largest responses due to stronger internal fields and enhanced $p$-dominated SOC contributions; the spin Edelstein effect is particularly sensitive to the chalcogen $p$-orbital SOC $\lambda_p$. These insights suggest that tuning internal fields via Janus composition and gate-induced doping can enable efficient orbital- and spin-orbitronic devices in 2D materials.

Abstract

We investigate the orbital and spin Edelstein effect(OEE and SEE) in two-dimensional Janus transition metal dichalcogenides (TMDs) of the form MXX$^\prime$ $(M = Mo,\ W,\ Nb;\ X/X^\prime = S,\ Se,\ Te)$ with the aid of density functional theory calculations and tight-binding model Hamiltonian studies. The chalcogen layers $X$ and $X^\prime$, break the mirror symmetry to introduce an internal electric field $E_{int}$ normal to the plane, which is responsible for OEE and SEE. Our results show that in a non-Janus framework, the wavefunctions at the valence and conduction bands are dominated with the $|x^2-y^2>$, $|xy>$, and $|z^2>$ orbitals. Due to the $E_{int}$ of the Janus system, these orbitals are now intermixed with the $|xz>$ and $|yz>$ orbitals to produce a robust orbital texture around the valleys $Γ,K$ and $K^\prime$. The spin orbit coupling, in addition to the formation of a spin texture, introduces a chirality reversal to the orbital texture. An applied in plane electric field creates both OEE and SEE with the former being one order higher in magnitude. This makes the Janus materials promising for spin-orbitronics. Our work paves the way for further experimental exploration for orbital and spin orbital torque in Janus TMDs.

Figures (10)

• Figure 1: (a) A schematic of the Edelstien effect driven by orbital and spin textures in the momentum space due to an external electric field applied along $\hat{x}$. The green and red regions indicate the accumulation of positive and negative magnetic moments respectively. (b) The magnetoelectric response in the sample due to orbital Edelstein effect (OEE) and spin Edelstein effect(SEE), where $E_{int}$ is the internal electric field, $\vec{E}$ is the applied electric field, and the spin (arrows) and orbital (curved arrows) moments are accumulated on one side of the material. (c) Structure of the monolayer Janus MoSSe compound. The S and Se atoms are in the bottom and top layers respectively, which breaks the mirror symmetry and in turn creates an internal electric field. The lattice translation vectors are shown by $\vec{a}$ and $\vec{b}$. (d) An illustration with several domains for the Janus TMDs, comparing the strength of ORE and SRE, following the strength of the Rashba coefficients listed in Table II. A larger Rashba coefficient would typically lead to a larger Edelstein effect.
• Figure 2: Band structure of monolayer MoSSe in the presence of SOC showing a good match between DFT and the TB model. Here, we have defined the mid gap energy as the zero of the energy. The dominant orbital characters at high-symmetry points for different bands are indicated. Here, $L_\pm$ represents the orbitals $d_{x^2-y^2}\pm i d_{xy}$ and $l_{\pm}$ denotes the orbitals $d_{xz}\pm i d_{yz}$. Due to hybridization and internal mixing, there is a smooth variation in the orbital characters as a function of k-point for each of the bands. The Rashba-split is observed at the high-symmetry points of the bands dominated with $z^2$ character.(b) Zoomed in valence band edge near $\Gamma$, demonstrating the prominent Rashba-split, which is quantified by Rashba-energy split $E_R$, and momentum split $k_R$ respectively. Similar Rashba-splits are observed for the other Janus materials, and the band structures are provided in the supplemental materialsuppl.
• Figure 3: Orbital texture of the MoSSe near (a) $\Gamma$, (b) K and (c) $K'$ valleys, without the SOC. Here, the length of the vectors are scaled in each of the cases for proper visualization of the orbital texture. Importantly, the $\braket{\vec{L}}$ is much stronger near $\Gamma$ as compared to $K/K^\prime$. (d) The y-component of the orbital moment vector ($m_{L,y}^\eta$) is computed as a function of the small momentum $q_x$ around these valleys $\eta$. The results for the rest of the materials are shown in the supplementary materialssuppl. The distribution of $d_{xz}$ and $d_{yz}$ orbital characters near $\Gamma$, for (e) MoS$_2$, (f) MoSSe, and (g) MoSTe, highlighting the orbital mixing.
• Figure 4: Orbital (blue) and spin textures (red) around $\Gamma,K$, and $K^\prime$ valleys in the presence of SOC for monolayer MoSSe. The magnetic moments are computed for the lower and upper Rashba splitted bands separately, which are represented as "L" and "U". The length of the vectors are scaled for visual clarity, with a scaling factor of 50 for the $K$ and $K^\prime$ valleys as compared to the ones for $\Gamma$, which clearly suggests that both the orbital and spin Rashba effects are more prominent near the $\Gamma$ valley.
• Figure 5: Orbital moment $m_{L,y}^{\Gamma}$ along $q_x$ near the $\Gamma$ point. The black solid line presents the result for the case of no SOC, where the $m_L$ is formed by both the degenerate bands. In the presence of SOC, the contribution from upper and lower bands to the total $m_{L,y}$(black-dotted line) are shown with blue and red dotted lines. It shows that the total orbital moment remains unchanged even with the inclusion of SOC, although the upper and the lower bands show opposite values.