Interfacial contributions to spin-orbit torque and magnetoresistance in ferromagnet/heavy-metal bilayers

The thickness dependence of spin-orbit torque and magnetoresistance in ferromagnet/heavy-metal bilayers is studied using the first-principles nonequilibrium Green's function formalism combined with the Anderson disorder model. A systematic expansion in orthogonal vector spherical harmonics is used for the angular dependence of the torque. The dampinglike torque in Co/Pt and Co/Au bilayers can be described as a sum of the spin-Hall contribution, which increases with thickness in agreement with the spin-diffusion model, and a comparable interfacial contribution. The magnetoconductance in the plane perpendicular to the current in Co/Pt bilayers is of the order of a conductance quantum per interfacial atom, exceeding the prediction of the spin-Hall model by more than an order of magnitude. This suggests that the “spin-Hall magnetoresistance,” similarly to the dampinglike torque, has a large interfacial contribution unrelated to the spin-Hall effect.

Figure 1

Dependence of the SOT coefficients on the thickness $d_N$ of the heavy-metal layer in Co/Pt and Co/Au bilayers. (a) Dampinglike (DL) $C_{1,-1}^{\left(1\right)}$ and planar-Hall-like (PHL) $C_{2,1}^{\left(1\right)}$ coefficients. (b) Fieldlike (FL) coefficient $C_{1,-1}^{\left(2\right)}$. Blue (red) circles (squares): $C_{1,-1}^{\left(1\right)}$ or $C_{1,-1}^{\left(2\right)}$ for Co/Pt (Co/Au); green lines and diamonds: $C_{2,1}^{\left(1\right)}$ for Co/Pt. Solid (open) symbols: $V_m=1.09$ eV (0.77 eV). Blue (red) lines in (a): Fits of the data for Co/Pt (Co/Au) at $V_m=1.09$ eV (see text).

Figure 2

Dependence of the magnetoconductances ${\mathrm{\Delta }}_{yx}g$ (red symbols) and ${\mathrm{\Delta }}_{yz}g$ (blue symbols) in Co/Pt bilayers on the thickness of the Pt layer calculated at $w=12$ ML. Gray symbols: Convergence tests at $w=16$ ML. The disorder strength is $V_m=1.09$ eV. The error bars show the standard deviation. Dashed line: The prediction of Eq. (4) with the parameters for Co/Pt at $V_m=1.09$ eV from Table 1, scaled by a factor of 10.