Engineering spin-orbit effects and Berry curvature by deposition of a monolayer of Eu on ${\mathrm{WSe}}_2$

Motivated by recent progress in two-dimensional (2D) spintronics, we present a monolayer of Eu deposited on $1\mathrm{H}\text{−}\mathrm{W}{\mathrm{Se}}_2$ as a promising platform for engineering spin-orbit effects and Berry curvature. By first-principles calculations based on density functional theory, we show that $\mathrm{Eu}/\mathrm{W}{\mathrm{Se}}_2$ exhibits intriguing properties such as high magnetic anisotropy, valley-dependent polarization of spin and orbital angular momenta, and Rashba textures. These originate from magnetic and spin-orbit proximity effects at the interface and the interplay between localized $4f$ magnetic moments of Eu and mobile charge carriers of Eu and $\mathrm{W}{\mathrm{Se}}_2$. The analysis of the magnetic properties reveal a ferromagnetic configuration with an out-of-plane easy axis of the magnetization, which favor a pronounced anomalous Hall effect in the proposed system. Thus, we promote $4f$ rare-earth metals deposited on top of a transition-metal dichalcogenides as a promising platform for 2D spintronics.

Figure 1

Structure of the $1\times 1$ unit cell of Eu (purple spheres) monolayer deposited on top of the W atom (gray spheres) of a single layer of $\mathrm{W}{\mathrm{Se}}_2$. Se atoms are indicated by yellow spheres.

Figure 2

(a) Contribution to the local DOS of the $f$ and $d$ electrons of Eu. The total DOS (TDOS) is shown as gray shaded area. (b) Contribution to the local DOS of the $s,p,d$ electrons of Eu, $d$ electrons of W, and $p$ electrons of Se. Both DOS in panels (a) and (b) have been calculated without SOC. (c) Band structure of $\mathrm{Eu}/\mathrm{W}{\mathrm{Se}}_2$ calculated with $\mathrm{DFT}+U$ (blue dashed line represents the majority channel and red dashed line represents the minority channel) and with $\mathrm{DFT}+U+\mathrm{SOC}$ (black solid line). (d) First Brillouin zone with high-symmetry points.

Figure 3

(a) Magnetic anisotropy energy curve: the total energy of the system plotted vs the polar angle $\theta$ of the magnetization measured from the $z$ axis. (b) The energy of the spin-spiral states of a flat spiral, i.e., with cone angle $\beta =\pi /2$, computed for the values of the $\mathbf{q}$ vector along the $\mathrm{\Gamma }-\mathrm{K}-\mathrm{M}$ path, presented with respect to the ferromagnetic ground state at the $\mathrm{\Gamma }$ point.

Figure 4

Spin and orbital texture in $\mathbf{k}$ space at the Fermi surface. (a) Expectation value for the out-of-plane component of the orbital angular momentum at the Fermi surface ${\langle L_z\rangle }_{\mathrm{FS}}$. (b) Magnitude of the expectation value of the in-plane component of the orbital angular momentum ${\langle L_{xy}\rangle }_{\mathrm{FS}}=\sqrt{{\langle L_x\rangle }_{\mathrm{FS}}^2+{\langle L_y\rangle }_{\mathrm{FS}}^2}$. Analogously, the $z$ component and the magnitude of the in-plane component for the spin expectation value at the Fermi surface are shown in panels (c) and (d), respectively. ${\langle L_z\rangle }_{\mathrm{FS}}>(<)0$ and ${\langle S_z\rangle }_{\mathrm{FS}}>(<)0$ [color blue (red)] correspond to the angular-momentum direction (anti)parallel to the spin of Eu-$4f$ electrons.

Figure 5

(a) Band structure around the Fermi energy with color scale indicating the value of the Berry curvature ${\mathrm{\Omega }}_{n\mathbf{k}}$. (b) Berry curvature summed over all occupied states along the $\mathbf{k}$ path $\mathrm{\Gamma }\text{−}\mathrm{K}\text{−}\mathrm{M}\text{−}{\mathrm{K}}^{\prime }\text{−}\mathrm{\Gamma }$.

Figure 6

Anomalous Hall conductivity as a function of the Fermi level.

Figure 7

Comparison of the electronic structure of Eu monolayer on $\mathrm{W}{\mathrm{Se}}_2$ monolayer for two simulation cells: (a) $1\times 1$ unit cell (high coverage of Eu) and (b) $\sqrt{3}\times \sqrt{3}$ unit cell (low coverage of Eu). The corresponding band structures determined neglecting SOC are shown in panels (c) and (d), where blue and red lines indicate majority and minority states, respectively.