Noncollinear exchange interaction in transition metal dichalcogenide edges

Abstract

We study the Ruderman-Kittel-Kasuya-Yosida effective exchange interaction between magnetic impurities embedded on the edges of transition metal dichalcogenide flakes, using a three-orbital tight-binding model. Electronic states lying midgap of the bulk structure have a strong one-dimensional (1D) character, localized on the edges of the crystallite. This results in exchange interactions with 1/r (or slower) decay with distance r, similar to other 1D systems. Most interestingly, however, the strong spin-orbit interaction in these materials results in sizable noncollinear Dzyaloshinskii-Moriya interactions between impurities, comparable in size to the usual Ising and in-plane components. Varying the relevant Fermi energy by doping or gating may allow one to modulate the effective interactions, controlling the possible helical ground state configurations of multiple impurities.

Long range of indirect exchange interaction on the edges of MoS2flakes

“…between magnetic impurities hybridized to these modulated wave functions on the edges, is largely dependent on the energy doping [48], since different states close to a given Fermi level carry different information on spin/spatial structure. This gives rise to complex oscillation patterns on the interaction between impurities.…”

“…Notice that the envelope is further affected by the presence of antinodes in the edge states close to the Fermi level, as seen in figure 4. Although we had seen evidence of unusually long-range interaction on edges of triangular MoS 2 flakes before [48], this was studied only for two midgap states, and we thought the effect fragile. However, as we now see, the long range is rather robust to shifts in the Fermi level and for rather large flakes.…”

“…Although we had found 2D behavior for the interaction between impurities in the interior of flakes before [47], and some evidence of long range interaction for hybridization near the edges [48], we present here the full systematics.…”

“…We use 2λ = 150 meV, in good agreement with DFT calculations [5,45] and experimental values (152 meV [53], 138 meV [54] and 145 meV [55]). With this Hamiltonian and the right choice of boundaries, one can construct a flake like the triangle we consider [48,47,52]. Others TMDs can be similarly modeled, taking the onsite, hopping and SOC parameters from Tables II and III in [45].…”

“…The total effective interaction is a competition between in-plane J XX (= J Y Y ), Ising J ZZ , and DzyaloshinskiiMoriya J DM terms. In TMDs the Ising and DM terms are generated by the SOC; their competition has been recently discussed for bulk samples [31,33] and flakes [48,47]. To calculate the effective J's in our finite sample, we take the difference of triplet and singlet configurations of impurities in the electronic ground state [58,59] as…”