Spin and charge currents induced by the spin Hall and anomalous Hall effects upon crossing ferromagnetic/nonmagnetic interfaces

We start closing a gap in the comparison of experimental and theoretical data associated with the spin Hall effect. Based on a first-principles characterization of electronic structure and a semiclassical description of electron transport, we compute the skew-scattering contribution to the transverse spin and charge currents generated by spin and anomalous Hall effect in a Co/Cu multilayer system doped with Bi impurities. The fact that the created currents cross the interface between the two materials strongly influences the efficiency of charge-to-spin current conversion, as demonstrated by our results.

Figure 1

Schematic representation of a unit cell of the investigated multilayer crystal in the Cartesian coordinate system. The applied electric field $\mathit{E}=E\widehat{\mathit{x}}$ leads to a longitudinal charge current $j_x$. With chosen magnetization direction and spin quantization axis along $\widehat{\mathit{z}}$, the SHE and the AHE create the spin current $j_y^z$ and the charge current $j_y$, respectively. Flowing in $y$ direction, these currents cross the interface between cobalt and copper.

Figure 2

Dependence of (a) longitudinal charge conductivity, (b) transverse spin and charge conductivity, and (c) spin and anomalous Hall angle on the Bi impurity position in the ${\mathrm{Co}}_9{\mathrm{Cu}}_7$ supercell. For comparison, the corresponding values for bulk crystals with substitutional Bi impurities [39, 67] are shown by horizontal lines. Graph (a) omits such a guide for the eyes because the associated values in the used units, ${\sigma }_{xx}^{\text{Cu(Bi)}}=0.22$ and ${\sigma }_{xx}^{\text{Co(Bi)}}=0.20$, are very small. The conductivities are shown for an impurity concentration of 1 at. %.