Magnetism induced by nonlocal spin-entangled electrons in a superconducting spin-valve

In the traditional view, the magnetic moment appearing in the superconducting region is induced by equal-spin triplet superconducting correlations in superconductor ($S$) ferromagnet ($F$) heterostructure with noncollinear magnetization. In this paper, we represent that in $NSF_1F_2$ ($N$--normal-metal) spin-valve structure the induced magnetic moment emerging in both the $S$ and $N$ regions can also be generated by Cooper pair splitting: one electron coherently tunnels from the $S$ layer into the $F_1$ layer, and the other one stays in the $S$ layer or tunnels into the $N$ layer. Two electrons are spatially separated from each other but their total spin ground state is entangled in this process. In contrast, the magnetic moment induced by the equal-spin triplet correlations hardly penetrates from the $S$ layer into the $N$ layer. In particular, by tuning the size of the exchange field and the thickness of the $F_1$ layer, one may control the direction of the induced magnetic moment in the $N$ layer. This interesting phenomenon can be attributed to the phase-shift obtained by the spin-entangled electrons. Our theoretical proposal will offer an effective way to control the entanglement of the nonlocal electrons, and also may provide possible explanations for previous and recent experimental observations [Stamopoulos et al 2005 Phys. Rev. B 72 212514; Ovsyannikov et al 2016 J. Exp. Theor. Phys. 122 738; Flokstra et al 2016 Nat. Phys. 12 57].