Elusive Dzyaloshinskii-Moriya interaction in monolayer ${\mathrm{Fe}}_3{\mathrm{GeTe}}_2$

Using symmetry analysis and density functional theory calculations, we uncover the nature of the Dzyaloshinskii-Moriya interaction in monolayer ${\mathrm{Fe}}_3{\mathrm{GeTe}}_2$. We show that while such an interaction might result in small distortions of the magnetic texture in the short range, in the long range it favors in-plane Néel spin spirals along equivalent directions of the crystal structure. Whereas our results show that the observed Néel skyrmions cannot be explained by the Dzyaloshinskii-Moriya interaction at the monolayer level, they suggest that a canted magnetic texture shall arise at the boundary of ${\mathrm{Fe}}_3{\mathrm{GeTe}}_2$ nanoflakes or nanoribbons and, most interestingly, that homochiral planar magnetic textures could be stabilized.

Figure 1

(a) Top and (b) side view of monolayer ${\mathrm{Fe}}_3{\mathrm{GeTe}}_2$. $(x,y,z)$ are the Cartesian coordinates and $(\mathbf{a},\mathbf{b})$ are the equivalent crystallographic directions.

Figure 2

DM vector (black arrows) for various nearest-neighbor interactions: (a) ${\mathrm{Fe}}_1\text{−}{\mathrm{Fe}}_3$, (b) ${\mathrm{Fe}}_1\text{−}{\mathrm{Fe}}_2$, (c) ${\mathrm{Fe}}_1\text{−}{\mathrm{Fe}}_1$, and (d) ${\mathrm{Fe}}_2\text{−}{\mathrm{Fe}}_2$. The chemical elements are designated by the same color code as in Fig. 1. In these figures, we only represented the atoms that contribute to defining the local symmetry of the environment and removed the other elements for better clarity.

Figure 3

Spin spiral dispersion for Néel in plane along $\mathrm{\Gamma }\mathrm{K}$ without (black) and with (red) SOC. The antisymmetric distortion upon turning on SOC is attributed to DMI. The vertical shift due to magnetic anisotropy has been removed manually for clarity, and the exchange stiffness is $A=47$ meV/Fe.

Figure 4

The mirror symmetry normal to the (110) plane (green line) results in a vanishing DM vector along $\mathrm{\Gamma }\mathrm{M}$, whereas the mirror symmetry breaking normal to the (100) plane (black lines) leads to uncompensated clockwise and anticlockwise paths (rounded arrows) and a finite DM vector along $\mathrm{\Gamma }\mathrm{K}$. $D_{i,\alpha }$ denotes the path connecting $i\mathrm{th}$ nearest-neighbor ${\mathrm{Fe}}_2$ sites passing through element $\alpha$.

Figure 5

Contribution of the different elements to the antisymmetric part of the spin spiral dispersion along $\mathrm{\Gamma }\mathrm{K}$ for (a) Néel out-of-plane, (b) Bloch out-of-plane, and (c) Néel in-plane configurations. (d) Kohn-Sham orbitals at the $\mathrm{\Gamma }$ point showing the strong Te $5p_z\text{−}{\mathrm{Fe}}_2 3d_{z^2}$ hybridization responsible for the large perpendicular orbital moment on ${\mathrm{Fe}}_2$.

Figure 6

Example of a planar homochiral spin spiral propagating along the low-symmetry [100] direction (dashed line), characterized by mirror symmetry breaking.