Spin-transfer torque and magnetoresistance in superconducting spin valves

We study the spin-transfer torque and magnetoresistance of a $\text{ferromagnet}\mid \text{superconductor}\mid \text{ferromagnet}$ spin valve, allowing for an arbitrary magnetization misorientation and treating both $s$-wave and $d$-wave symmetries of the superconductor. We take fully into account Andreev reflection and also the spin-triplet correlations that are generated when the magnetizations are noncollinear. It is found that the torque and magnetoresistance are both strongly enhanced when topological zero-energy states are present at the interfaces, which is the case for $d$-wave superconductors with a crystallographic orientation of [110] relative to the interface ($d_{xy}$-wave symmetry). Moreover, we find that the magnetoresistance displays a strong oscillatory and nonmonotonous behavior as a function of $d_S/\xi$ where $d_S$ and $\xi$ are the interlayer width of the superconducting region and the superconducting coherence length, respectively. This feature is also attributed to the crossover from layers of size $d_S\sim 2\xi$ to layers of size $d_S⪢2\xi$, where the contribution to transport from zero-energy states gradually vanishes.

Figure 1

(Color online) A spin-valve setup with misaligned magnetizations in the ferromagnetic regions. We consider both $s$-wave and $d$-wave symmetries in the superconducting region.

Figure 2

(Color online) Plot of the spin-transfer torque and its dependence on the misorientation angle $\phi$. We give results for three different symmetries of the superconducting order parameter with interfaces characterized by the barrier parameter $Z$. Here, we have fixed $d_S/\xi =3.0$.

Figure 3

(Color online) Plot of the spin-transfer torque and its dependence on the interface transparency $Z$. Here, we have set $d_S/\xi =3.0$ and $\phi =\pi /2$.

Figure 4

(Color online) Plot of the decay of the transverse ($\hat{\mathit{x}}$ component) part of the spin current in the right ferromagnetic region for several values of the interface transparency $Z$. Here, we have set $d_S/\xi =3.0$ and $\phi =\pi /2$.

Figure 5

(Color online) Plot of the MR for several choices of symmetry for the superconducting order parameter and several barrier transparencies $Z$. Here, we have set $d_S/\xi =3.0$.

Figure 6

(Color online) Plot of the MR for the $d_{xy}$-wave symmetry with $eV/{\Delta }_0=0.01$.

Figure 7

(Color online) Plot of the probability measures for EC and CAR processes as a function of the barrier parameter $Z$ with fixed $eV/{\Delta }_0=0.01$ and $d_S/\xi =3.0$.