Twists in ferromagnetic monolayers with trigonal prismatic symmetry

Two-dimensional materials such as graphene or hexagonal boron nitride are indispensable in industry. The recently discovered 2D ferromagnetic materials also promise to be vital for applications. In this work, we develop a phenomenological description of noncentrosymmetric 2D ferromagnets with trigonal prismatic crystal structure. We chose to study this special symmetry group since these materials do break inversion symmetry and therefore, in principle, allow for chiral spin structures such as magnetic helices and skyrmions. However, unlike all noncentrosymmetric magnets known so far, we show that the symmetry of magnetic trigonal prismatic monolayers neither allow for an internal relativistic Dzyaloshinskii-Moriya interaction (DMI) nor a reactive spin-orbit torque. We demonstrate that the DMI only becomes important at the boundaries, where it modifies the boundary conditions of the magnetization and leads to a helical equilibrium state with a helical wave vector that is inherently linked to the internal spin orientation. Furthermore, we find that the helical wave vector can be electrically manipulated via dissipative spin-torque mechanisms. Our results reveal that 2D magnets offer a large potential for unexplored magnetic effects.

Figure 1

(a) Ferromagnets with 2H structure lack inversion symmetry. Nevertheless, the systems do not have an internal DMI field. Only a boundary-DMI field exists, which in equilibrium stabilizes a helical spin phase (arrows). (b) The crystal structure of the 2H monolayer. (c) Shows the atomic configuration at a lattice point. In the TMDs, the red atom represents the transition metal atom and the blue atoms are the chalcogen atoms.

Figure 2

[(a)–(e)] Shows the equilibrium state of the magnetization (arrows) on the 2D disk for ${\varphi }_0=0, \pi /4, \pi /2, 3\pi /4$, and $\pi$, respectively. The density plots illustrate the first-order correction $\tilde{D}{\varphi }_1$ to the magnetic equilibrium state produced by the DMI-induced BCs. For clarity, we have used $\tilde{D}=0.5$.

Figure 3

Vector-field plot of the differential equation in Eq. (17) with the current density applied along the $x$ axis. The red circles indicate unstable fix points for the solution ${\varphi }_0\left(t\right)$, whereas the red dots represent stable fix points. Here, we assume $\mathcal{A}\equiv -\frac{\left|\mathcal{J}\right|\tilde{D}}{{\alpha }_{\perp }}\left(\frac{P\beta }{R}-\frac{\gamma {\mathrm{\Lambda }}_{\mathrm{so}}}{\tilde{D}}\right)>0$.