Spin transport and magnetoresistance in ferromagnet/superconductor/ferromagnet spin valves

We consider spin transport and spin relaxation in superconductors using the quasiclassical theory of superconductivity. We include spin relaxation due to spin-orbit interaction as well as magnetic impurities, and show that the energy dependence of the spin-flip rate is different for these two mechanisms. In ferromagnet-superconductor-ferromagnet systems made of Co and Al, interface resistances can be small compared to bulk resistances. This simplifies the description of transport in $\mathrm{Co}∕\mathrm{Al}∕\mathrm{Co}$ spin valves, for which we numerically calculate the temperature and Al length dependence of the magnetoresistance.

Figure 1

Spatial variation of the pairing potential at $T∕T_c=0.40$ scaled by ${\Delta }_0$. Inset, density of states at positions 0.0, 0.2, and $0.6\mu \mathrm{m}$ into the $S$ wire. The curve evaluated at $x=0.0\mu \mathrm{m}$ is identical to the normal state DOS (flat curve).

Figure 2

Temperature dependence of resistance in parallel geometry for spin-orbit (solid lines) and magnetic impurity (dashed lines) induced spin flip. The thick curves are with Al length $1200\mathrm{nm}$, and the thin curves with Al length $900\mathrm{nm}$. $T_c$ is lowered by the presence of the magnetic impurities.

Figure 3

Temperature dependence of the magnetoresistance for spin-orbit (solid lines) and magnetic impurity (dashed lines) induced spin flip. The top set of curves is a system with Al length $900\mathrm{nm}$, and the lower curves with Al length $1200\mathrm{nm}$.

Figure 4

Top panel, spatial dependence of the spin current for the $\mathrm{F}/\mathrm{S}/\mathrm{F}$ structure with Al length $1.2\mu \mathrm{m}$ at $T∕T_c=0.40$ (spin-orbit induced spin-flip). The normal state $(T>T_c)$ spin current is shown with the dotted line. The $F\text{‐}S$ interfaces are at $x=0\mu \mathrm{m}$ and $x=1.2\mu \mathrm{m}$. Bottom panel, spatial dependence of the quasiparticle charge current (solid line) and supercurrent (dashed line) for the same system as in the top panel. Normal state current is shown with dotted line.

Figure 5

The spatial dependence of the spin potential for the $\mathrm{F}/\mathrm{S}/\mathrm{F}$ structure with Al length $1.2\mu \mathrm{m}$ at $T∕T_c=0.40$ (solid line) and $T>T_c$ (dotted line) (spin-orbit induced spin flip). The $\mathrm{F}/\mathrm{S}$ interfaces are at $x=0\mu \mathrm{m}$ and $x=1.2\mu \mathrm{m}$.