Spin-orbit torques in Co/Pt(111) and Mn/W(001) magnetic bilayers from first principles

An applied electric current through a space-inversion asymmetric magnet induces spin-orbit torques (SOTs) on the magnetic moments, which holds much promise for future memory devices. We discuss general Green's function expressions suitable to compute the linear-response SOT in disordered ferromagnets. The SOT can be decomposed into an even and an odd component with respect to magnetization reversal, where in the limit of vanishing disorder the even SOT is given by the constant Berry curvature of the occupied states, while the odd part exhibits a divergence with respect to disorder strength. Within this formalism, we perform first-principles density-functional theory calculations of the SOT in Co/Pt(111) and Mn/W(001) magnetic bilayers. We find the even and odd torque components to be of comparable magnitude. Moreover, the odd torque depends strongly on an additional capping layer, while the even torque is less sensitive. We show that the even torque is nearly entirely mediated by spin currents in contrast to the odd torque, which can contain an important contribution not due to spin transfer. Our results are in agreement with experiments, showing that our linear-response theory is well-suited for the description of SOTs in complex ferromagnets.

Figure 1

Illustration of the unit cell used in the thin film calculations of the Co/Pt(111) systems. Magnetization $\mathbf{M}$ is in $z$ direction.

Figure 2

(a) Even torkance $t_{yx}^{\mathrm{even}}$ and (b) odd torkance $t_{xx}^{\mathrm{odd}}$ in Co(3)/Pt(7) (), Co(3)/Pt(10) (), Co(3)/Pt(13) (), Co(3)/Pt(15) (), Co(3)/Pt(20) (), Al(1)/Co(3)/Pt(10) (), O(1)/Co(3)/Pt(10) (), Mn(1)/W(9) (), and Mn(1)/W(15) () for $\mathbf{M}$ in the $z$ direction. The product of elementary positive charge $e$ and Bohr radius $a_0$ used as unit of torkance amounts to $e{a}_0=8.478\times {10}^{-30}$ Cm.

Figure 3

Atom-resolved torkance $t_{yx\alpha }^{\mathrm{even}}$ (triangles) and atom-resolved spin-flux coefficient $q_{yx\alpha }^{\mathrm{even}}$ (circles) in (a) Co(3)/Pt(10), (b) O(1)/Co(3)/Pt(10), and (c) Al(1)/Co(3)/Pt(10). Atom-resolved torkance $t_{xx\alpha }^{\mathrm{odd}}$ (triangles) and atom-resolved spin-flux coefficient $q_{xx\alpha }^{\mathrm{odd}}$ (circles) in (d) Co(3)/Pt(10), (e) O(1)/Co(3)/Pt(10), and (f) Al(1)/Co(3)/Pt(10). The broadening was set to $\Gamma =25$ meV. Lines serve as guides for the eye.

Figure 4

Atom-resolved torkances (triangles) and spin-flux coefficients (circles) in Mn(1)/W(15) at $\Gamma =100$ meV. (a) $t_{yx\alpha }^{\mathrm{even}}$ and $q_{yx\alpha }^{\mathrm{even}}$; (b) $t_{xx\alpha }^{\mathrm{odd}}$ and $q_{xx\alpha }^{\mathrm{odd}}$.

Figure 5

Atom-resolved torkance $t_{yx\alpha }^{\mathrm{even}}$ (triangles) and atom-resolved spin-flux coefficient $q_{yx\alpha }^{\mathrm{even}}$ (circles) in Mn(1)/W(9)/Mn(1) at $\Gamma =100$ meV.

Figure 6

Atom-resolved torques (triangles) and spin-fluxes (circles) in Mn(1)/W(9) without applied electric field when the magnetization is rotated away from the easy axis by ${30}^{\circ }$.