Nonlocal fieldlike spin-orbit torques in Rashba systems: Ab initio study of a ${\mathrm{Ag}}_2\mathrm{Bi}$-terminated Ag(111) film grown on a ferromagnetic Fe(110) substrate

We investigate from first principles the fieldlike spin-orbit torques (SOTs) in a ${\mathrm{Ag}}_2\mathrm{Bi}$-terminated Ag(111) film grown on ferromagnetic Fe(110). We find that a large part of the SOT arises from the spin-orbit interaction (SOI) in the ${\mathrm{Ag}}_2\mathrm{Bi}$ layer far away from the Fe layers. These results clearly hint at a long-range spin transfer in the direction perpendicular to the film that does not originate in the spin Hall effect. In order to bring evidence of the nonlocal character of the computed SOT, we show that the torque acting on the Fe layers can be engineered by the introduction of Bi vacancies in the ${\mathrm{Ag}}_2\mathrm{Bi}$ layer. Overall, we find a drastic dependence of the SOT on the disorder type, which we explain by a complex interplay of different contributions to the SOT in the Brillouin zone.

Figure 1

Illustration of a unit cell of a ${\mathrm{Ag}}_2\mathrm{Bi}$-terminated Ag(111) film grown on ferromagnetic Fe(110). Bi, Ag, and Fe atoms are represented in yellow, grey, and red, respectively.

Figure 2

(a) and (b) Spectral density of states for a ${\mathrm{Ag}}_2\mathrm{Bi}$-terminated Ag film grown on ferromagnetic Fe(110) along the $k$ paths $P-\mathrm{\Gamma }-P^{\prime }$ and $K-\mathrm{\Gamma }-K^{\prime }$ shown in Fig. 3. (c) and (d) Local density of states on the ${\mathrm{Ag}}_2\mathrm{Bi}$ layer. An example of asymmetric band-gap opening is marked by the letter A. The dashed lines indicate the energy $E_{\mathrm{S}}=E_{\mathrm{F}}-0.93$ eV corresponding to the states shown in Fig. 3.

Figure 3

(a) States at the Fermi energy $E_{\mathrm{F}}$ and (b) at the energy $E_{\mathrm{S}}=E_{\mathrm{F}}-0.93$ eV below the Fermi energy marked by the portion of their wave function on the ${\mathrm{Ag}}_2\mathrm{Bi}$ layer.

Figure 4

Distribution of ${\mathcal{T}}_z\left(\mathit{k}\right){\lambda }_x\left(\mathit{k}\right)/\left|\mathit{v}\left(\mathit{k}\right)\right|$ and ${\mathcal{T}}_z\left(\mathit{k}\right){\lambda }_y\left(\mathit{k}\right)/\left|\mathit{v}\left(\mathit{k}\right)\right|$ in the Brillouin zone for the Bi/Ag/Fe film. The Fermi energy was set to its true value $E_{\mathrm{F}}$ and the disorder strength to $\mathrm{\Gamma }=25$ meV.

Figure 5

Distribution of ${\mathcal{T}}_z\left(\mathit{k}\right){\lambda }_x\left(\mathit{k}\right)/\left|\mathit{v}\left(\mathit{k}\right)\right|$ and ${\mathcal{T}}_z\left(\mathit{k}\right){\lambda }_y\left(\mathit{k}\right)/\left|\mathit{v}\left(\mathit{k}\right)\right|$ in the Brillouin zone for the Bi/Ag/Fe film. The Fermi energy was set to $E_{\mathrm{S}}=E_{\mathrm{F}}-0.93$ eV and the disorder strength to $\mathrm{\Gamma }=25$ meV.

Figure 6

(a) Torkance and (b) effective magnetic fields per unit of current density as a function of the impurity concentration ${\overline{c}}_{\mathrm{imp}}$ for $\mathrm{\Gamma }=25$ meV.

Figure 7

(a) Torkance and (b) effective magnetic fields per unit of current density as a function of the disorder strength $\mathrm{\Gamma }$ for ${\overline{c}}_{\mathrm{imp}}=0$ (dashed lines) and ${\overline{c}}_{\mathrm{imp}}=0.1$ (full lines).

Figure 8

Distribution of ${\mathcal{T}}_z\left(\mathit{k}\right){\lambda }_x\left(\mathit{k}\right)/\left|\mathit{v}\left(\mathit{k}\right)\right|$ and ${\mathcal{T}}_z\left(\mathit{k}\right){\lambda }_y\left(\mathit{k}\right)/\left|\mathit{v}\left(\mathit{k}\right)\right|$ in the Brillouin zone for the Bi/Ag/Fe film. The Fermi energy is set to $E_{\mathrm{S}}=E_{\mathrm{F}}-0.93$ eV. The impurity concentration and the disorder strength are set to ${\overline{c}}_{\mathrm{imp}}=0.1$ and $\mathrm{\Gamma }=1$ meV, respectively. The color scale was adapted by a factor of 11.1 to account for the larger longitudinal conductivity as compared to the case where ${\overline{c}}_{\mathrm{imp}}=0$ and $\mathrm{\Gamma }=25$ meV (Fig. 5).