Spirals and skyrmions in antiferromagnetic triangular lattices

We study realizations of spirals and skyrmions in two-dimensional antiferromagnets with a triangular lattice on an inversion-symmetry-breaking substrate. As a possible material realization, we investigate the adsorption of transition-metal atoms (Cr, Mn, Fe, or Co) on a monolayer of ${\mathrm{MoS}}_2$, ${\mathrm{WS}}_2$, or ${\mathrm{WSe}}_2$ and obtain the exchange, anisotropy, and Dzyaloshinskii-Moriya interaction parameters using first-principles calculations. Using energy minimization and parallel-tempering Monte Carlo simulations, we determine the magnetic phase diagrams for a wide range of interaction parameters. We find that skyrmion lattices can appear even with weak Dzyaloshinskii-Moriya interactions, but their stability is hindered by magnetic anisotropy. However, a weak easy plane magnetic anisotropy can be beneficial for stabilizing the skyrmion phase. Our results suggest that Cr/${\mathrm{MoS}}_2$, Fe/${\mathrm{MoS}}_2$, and Fe/${\mathrm{WSe}}_2$ interfaces can host spin spirals formed from the ${120}^{\circ }$ antiferromagnetic states. Our results further suggest that for interfaces, such as Fe/${\mathrm{MoS}}_2$, the Dzyaloshinskii-Moriya interaction is strong enough to drive the system into a three-sublattice skyrmion lattice in the presence of experimentally feasible external magnetic field.

Figure 1

(a) Three adsorption sites (H, ${\mathrm{T}}_{\text{S}}$, and ${\mathrm{T}}_{\text{Mo}}$) on top of monolayer ${\text{MoS}}_2$. Mo and S atoms are represented by blue and orange spheres, respectively; (b) the same unit cell with the Dzyaloshinskii-Moriya vector depicted by the black arrows centered on the bond of neighboring, interacting atoms. The convention taken for the direction on bonds is counterclockwise in a triangular plaquette with Mo atom at its center. The vectors a and b are the primitive vectors of the hexagonal Bravais lattice.

Figure 2

Projected band structures and densities of states of (a) $\mathrm{Co}/{\mathrm{MoS}}_2$, (b) $\mathrm{Cr}/{\mathrm{MoS}}_2$, (c) $\mathrm{Mn}/{\mathrm{MoS}}_2$, (d) $\mathrm{Mn}/{\mathrm{WS}}_2$, (e) $\mathrm{Fe}/{\mathrm{MoS}}_2$, and (f) $\mathrm{Fe}/{\mathrm{WSe}}_2$. The red, blue, and green colors in the band-structure plots represent the partial weights of Mo or W states, the majority-spin states of the magnetic atom, and the minority-spin states of the magnetic atom, respectively. Black lines in the DOS plots show the total DOS; blue and green, respectively, majority-spin and minority-spin partial DOS for the magnetic atom.

Figure 3

Phase diagram as a function of DMI and effective anisotropy of AFM triangular lattice at zero temperature in the absence of magnetic field. The horizontal axis indicates the in-plane DMI and the vertical axis indicates the effective uniaxial anisotropy. The plot shows the ground states of (Cr, Fe, Co, ${\text{Mn)/MoS}}_2$, ${\text{Mn/WS}}_2$, and ${\text{Fe/WSe}}_2$. The colors mark the different phases, i.e., the light blue region corresponds to the Néel phase and the pink region is the spiral phase. The plotting range was chosen to distinctly display all six cases in Table 1.

Figure 4

A snapshot of spin spiral in a triangular AFM obtained by simulated annealing for $K_{\text{eff}}=0.4$, $D_{\parallel }=0.2$, $h=0$, and $T=0.02$ on a $75\times 75$ triangular lattice with periodic boundary conditions. The letters on the left side indicate the plotted sublattice. The colors indicate the orthogonal to the plane component of the spins.

Figure 5

Structure factor $S_{\parallel }\left(\mathbf{q}\right)$ of a triangular AFM from Monte Carlo simulations. The vertical and horizontal axes represent the reciprocal space coordinates $k_{\text{x}}$ and $k_{\text{y}}$ in units of $1/L$. The plot is for sublattice A; those for B and C are almost identical. (a) The plot corresponds to spiral phase realizable for $K=0.4$, $D_{\parallel }=0.2$, $h=0$, and $T=0.02$. (b) The plot corresponds to SkX phase realizable for $K=0$, $D_{\parallel }=0.2$, $h=3.1$, and $T=0.02$.

Figure 6

[(a) and (b)] Specific heat and topological susceptibility as a function of temperature for different values of the external magnetic field, $D_{\parallel }=0.2$, $K=0$. [(c) and (d)] Specific heat and topological susceptibility as a function of temperature for different values of the external magnetic field, $D_{\parallel }=0.2$, $K=0.1$.

Figure 7

Magnetic field–temperature ($h\text{−}T$) phase diagram of a triangular lattice AFM obtained by parallel tempering Monte Carlo simulations on a lattice of $75\times 75$ atoms. The red circles correspond to peaks of the specific heat and the blue squares correspond to peaks of the topological susceptibility. (a) Phases SP, SkX, and VL are realized when $D_{\parallel }=0.2$ and $K=-0.05$. (b) Phases SP, SkX, and VL are realized when $D_{\parallel }=0.2$ and $K=0$. (c) Phases SP, UUD, SkX, and VL are realized when $D_{\parallel }=0.2$ and $K=0.05$. (d) Phases SP, Y, UUD, SkX, and VL are realized when $D_{\parallel }=0.2$ and $K=0.1$. The color indicates the total topological charge in a region of $75\times 75$ spins.

Figure 8

A snapshot of SkX in a triangular AFM obtained by simulated annealing for $K=0$, $D_{\parallel }=0.2$, $h=3.1$, and $T=0.02$ on a $75\times 75$ triangular lattice with periodic boundary conditions. The letters on the left side indicate the plotted sublattice. The colors indicate the orthogonal to the plane component of the spins.

Figure 9

Fraction of temperatures $f\left(T\right)$ moving from the lowest to the highest temperature as a function of the temperature index in (a), and as a function of temperature in (b). The calculation corresponds to parameters: $D_{\parallel }=0.2$, $K=0.1$, and $h=0.8$.

Figure 10

Spin configurations for extracting parameters in the model Hamiltonian: [(a) and (b)] in-plane coplanar ${120}^{\circ }$ spin configurations with an opposite chirality. (c) an out-of-plane coplanar ${120}^{\circ }$ spin configuration. [(d) and (e)] Coplanar ${120}^{\circ }$ spin configurations in a $3\times 1$ unit cell.