Figure 1

Dependence of the spin magnetic moments of W adatoms ${\mu }_S$ (in ${\mu }_B$) for the out-of-plane and in-plane magnetization (a) and magnetic anisotropy energy MAE (b) as a function of the external electric field $\mathcal{E}$. The electric field applied is perpendicular to graphene surface, with negative values of $\mathcal{E}$ corresponding to the electric field in $+z$ direction, i.e., the direction from graphene towards the adatoms. Positive (negative) MAE corresponds to an out-of-plane (in-plane) easy axis. The inset displays the difference between spin densities $\Delta m(r)$ (in units of $\mathrm{\text{electrons}}/{\mathrm{a}.\mathrm{u}.}^3$) for the system without and with an electric field of $-0.13\mathrm{V}/\AA$.Reuse & Permissions

Figure 2

Electronic structure of $4\times 4$ W on graphene. (a) Density of ${\Delta }_1(s,d_{z^2})$, ${\Delta }_3(d_{xz},d_{yz})$, and ${\Delta }_4(d_{xy},d_{x^2-y^2})$ states without SOC. Up and down arrows mark spin-up (red) and spin-down (blue) channels, respectively. ${\Delta }_3$ $[{\Delta }_4]$ states are located predominantly above [below] the Fermi energy, and are shaded in line-shaded dark red (light gray) or blue (dark gray) [dot-shaded red (light gray) or blue (dark gray)] correspondingly. (b) Band structure without SOC along high symmetry lines in the 2D Brillouin zone. Red and blue color of the bands stands for the spin-up and spin-down character, respectively. Circles highlight the points at which a gap will open when SOC is considered. (c) Band structures with SOC for $\mathbf{M}\parallel z$: without electric field (solid line), with positive (dashed) and negative (dot-dashed) electric field of magnitude $0.13\mathrm{V}/\AA$. The inset shows the anomalous Hall conductivity with respect to the position of the Fermi energy $E_F$ for $\mathcal{E}=0$ and $\mathbf{M}\parallel z$. (d) Berry curvature (in units of $\mathrm{a}.\mathrm{u}{.}^2$) distribution of occupied bands in the momentum space (in units of $\frac{2\pi }{a}$, where $a=9.87\AA$ is the in-plane lattice constant of the $4\times 4$ supercell). The Brillouin zone boundaries are marked with solid lines.Reuse & Permissions