Symmetries and hybridization in the indirect interaction between magnetic moments in ${\mathrm{MoS}}_2$ nanoflakes

We study the Ruderman-Kittel-Kasuya-Yosida interaction between magnetic impurities embedded in $p$-doped transition metal dichalcogenide triangular flakes. The role of underlying symmetries is exposed by analyzing the interaction as a function of impurity separation along zigzag and armchair trajectories, in specific parts of the sample. The large spin-orbit coupling in these materials produces strongly anisotropic interactions, including a Dzyaloshinskii-Moriya component that can be sizable and tunable. We consider impurities hybridized to different orbitals of the host transition metal and identify specific characteristics for onsite and hollow site adsorption. In the onsite case, the different components of the interaction have similar magnitude, while for the hollow site, the Ising component dominates. We also study the dependence of the interaction with the level of hole doping, which supplies a further degree of tunability. Our results could provide ways of controlling helical long range spin order in magnetic impurity arrays embedded in these materials.

Figure 1

(a) Top view of the effective lattice used to simulate the triangular zigzag-terminated TMD nanoflake. Each site represents a Mo atom, including the numbering used to construct the flake. The inset shows a top view of the real representation of the flake, with the Mo (S) atoms shown in dark green (dark yellow). The three hopping directions are given by ${\mathit{a}}_1=a(1,0)$, ${\mathit{a}}_2=a(1/2,\sqrt{3}/2)$, and ${\mathit{a}}_3={\mathit{a}}_2-{\mathit{a}}_1$, where $a$ is the lattice constant. Yellow arrows represent one pair of magnetic moments in the zigzag direction and another pair in the armchair direction. One impurity is held fixed and the other is moved along the corresponding direction, as indicated by red dashed lines. Two independent zigzag (b) and armchair (c) trajectories for both onsite and plaquette triangle-down configurations.

Figure 2

(a) High symmetry directions in the first Brillouin zone of the infinite ${\mathrm{MoS}}_2$ monolayer. The valence band maximum (VBM) at $K$ is shifted to zero energy and the energy levels of the $\mathrm{\Gamma }$ point, and $K$ for spin up and down are shown in dashed lines. The light blue area indicates the direct gap ($\sim 1.6$ eV). (b) Discrete energy levels for a 50-row flake. Edge states generated by the finite size appear in the gap. Inset shows states near the VBM. ${\varepsilon }_{F1}$ and ${\varepsilon }_{F2}$ represent the two different levels of doping (or gating) considered in this work.

Figure 3

(a) Four different impurity trajectories for onsite hybridization, fixing the first impurity on top of a given atom and moving the second one away from the first. In trajectory 1 (red solid line), the first impurity is located at the 10th row on the edge and the second one is moved along the zigzag direction ${\mathit{a}}_2$, starting on the 11th row. Trajectory 2 (dashed blue line) represents a different zigzag direction in the bulk of the flake. For trajectory 3 (orange dotted line), the first impurity is at the bottom corner of the flake, while the second moves up along the armchair direction ${\mathit{a}}_2+{\mathit{a}}_3$. Trajectory 4 (pink dot-dashed line) is shifted laterally with respect to the previous one. (b)–(f) Orbital-resolved magnitude squared of the wave function, for two doping levels and the three different orbitals, as indicated in each panel.

Figure 4

The three components of the effective impurity interaction, scaled by ${(r/a)}^2$, vs relative distance along zigzag directions. All curves correspond to ${\varepsilon }_{F1}$, and onsite hybridization to the orbitals indicated in the panels. The different trajectories are explained in Fig. 3.

Figure 5

The three components of the effective impurity interaction, scaled by ${(r/b)}^2$, vs relative distance along armchair directions, with $b=a\sqrt{3}=|{\mathit{a}}_2+{\mathit{a}}_3|$. All curves correspond to ${\varepsilon }_{F1}$ and onsite hybridization to the orbitals indicated in the panels. Trajectories are explained in Fig. 3. Notice that in panels (a) and (b) there is no DM interaction, since the impurities lie on the line bisecting the flake, where reflection symmetry forbids its appearance.

Figure 6

Maximum interaction strength vs separation of parallel armchair trajectories with respect to the central bisecting line. Curves correspond to ${\varepsilon }_{F1}$ and onsite hybridization. The zero value in $x$ represents trajectory 3 and, as the vertical trajectory moves to the right, $x>0$, it approaches the edge of the flake.

Figure 7

Effective interaction vs relative distance along armchair direction. These results correspond to ${\varepsilon }_{F2}$ and onsite hybridization. (a) When both impurities hybridize to $d_{z^2}$ orbitals, a Heisenberg-like interaction is found. (b) For this pair of orbitals, $XX$ gets out of phase with coinciding Ising and DM terms.

Figure 8

Ising component of the indirect exchange, scaled by ${(r/a)}^2$, for various levels of $p$ doping, with trajectory and orbital hybridization as indicated in the figure. Positive (negative) values correspond to AFM (FM) impurity alignment.

Figure 9

(a) Effective impurity interaction, scaled by ${(r/a)}^2$, vs the relative distance in zigzag direction between the impurities $r$. All curves for ${\varepsilon }_{F1}$ and plaquette triangle-down configuration. The first magnetic impurity is located at bottom corner of the flake (surrounded by sites 1, 2, and 3), and the second one moves along the zigzag edge on the right. (b) The same, but for armchair trajectory 3 (notice that the scales for $XX=YY$ and $XY$ have been amplified 10 and 100 times respectively, for better visualization). Orbitals are indicated in each panel.

Figure 10

Contribution of the two lowest energy particle-hole excitations, obtained from perturbation theory, to the effective impurity exchange components $\mathit{ZZ}$ and $\mathit{XX}$, in the triangle-down plaquette configuration. Trajectories and orbitals are the same as in Fig. 9. The legend for each thin curve, $1\gamma -2{\gamma }^{\prime }$, with $\gamma ,{\gamma }^{\prime }\in \{\mathrm{B},\mathrm{L},\mathrm{R}\}$, indicate the first (1) and second impurity (2) connection to the (bottom, left, right) Mo atom in the respective surrounding triangle. The thick curve indicates the average of the nine onsite terms.