Strong spin-orbit torque effect on magnetic defects due to topological surface state electrons in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$

We investigate the spin-orbit torque exerted on the magnetic moments of the transition-metal impurities Cr, Mn, Fe, and Co, embedded in the surface of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, in response to an electric field and a consequent electrical current flow in the surface. The multiple scattering problem of electrons off impurity atoms is solved by first-principles calculations within the full-potential relativistic Korringa-Kohn-Rostoker (KKR) Green function method, while the spin-orbit torque calculations are carried out by combining the KKR method with the semiclassical Boltzmann transport equation. We analyze the correlation of the spin-orbit torque to the spin accumulation and spin flux in the impurities and unveil the effect of resonant scattering. In addition, we relate the torque to the resistivity and Joule heat production. We predict that the $\mathrm{Mn}/{\mathrm{Bi}}_2{\mathrm{Te}}_3$ is optimal among the studied systems.

Figure 1

The spin polarization of the Fermi surface states of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ film (side view). The red arrow in the middle represents the magnetization $\mathit{M}$ of the magnetic impurity atom.

Figure 2

(a) Illustration of the first quintuple layer of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ in side and (b) top views. The pink (medium-sized) and the gray (large) spheres represent the Te and Bi atoms, respectively. The magnetic impurity atom is depicted in red (small sized sphere). The impurity shows an inward relaxation with respect to the surface Te layer with vertical distance of 0.9 Å, as has been found for Fe impurities by Eelbo et al. [42]. (c) Schematic representation of the random positions of the defects on the surface in one of the random configurations. The filled red (gray-colored in gray-scale) circles represent the magnetic transition metal defects. The filled black and empty circles depict unoccupied surface impurity sites (threefold hollow positions with fcc stacking with respect to the surface layer), respectively inside and outside the disk in which the 51 impurities are embedded.

Figure 3

[(a)–(d)] The $y$ component of the torkance $t_{yy}$ as a function of the $x$ component of the torkance $t_{xy}$, [(e)–(h)] the torkance $t_{yy}$ as a function of the response coefficient of the spin accumulation ${\chi }_{xy}$, [(i)–(l)] the torkance $t_{yy}$ as a function of the response coefficient of the spin flux $q_{yy}$, on the central Cr, Mn, Fe, and Co impurity atom in the presence of 1 defect (squares) and 2% defects concentration for 20 different distributions (circles), embedded in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ surface. The results are scaled to a 2% concentration of defects. The electric field is taken in $y$ direction. The torkance is given in units of $e{\mathrm{a}}_{\mathrm{B}}=9\times {10}^{-5}{\mu }_{\mathrm{B}}$T/(V/cm). The spin accumulation is given in units of $e{\mathrm{a}}_{\mathrm{B}}{\mu }_{\mathrm{B}}{\mathrm{Ry}}^{-1}=3\times {10}^{-10}{\mu }_{\mathrm{B}}$/(V/cm).

Figure 4

The local spin-resolved density of states (DOS) of the Cr, Mn, Fe and Co impurity atoms. The positive $y$ axis corresponds to majority spin and the negative $y$ axis to minority spin.

Figure 5

(a) The average torkance $t$ (circles), spin flux $q$ (triangles), and spin accumulation $\chi$ (squares) on the central atom in the presence of $2\%$ and (b) $5\%$ concentration of Cr, Mn, Fe, and Co impurities embedded in the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ surface. The values are averaged over the 20 different configurations and the error bars indicate the standard deviation of the values.

Figure 6

The resistivity ${\rho }_{yy}$ (circles), the torkance on the central impurity atom multiplied by the resistivity $\tilde{t}=t{\rho }_{yy}$ [Eq. (19)] (triangles), and the angular rotation velocity $\omega$ per unit electric field [Eq. (18)] (squares), averaged over the 20 different configurations, in the presence of 2% defects concentration in the (Cr, Mn, Fe, Co)/${\mathrm{Bi}}_2{\mathrm{Te}}_3$ systems. The electric field is taken in $y$ direction.