Long-range singlet proximity effect in ferromagnetic nanowires

Recently a long-ranged superconductor/ferromagnet (S/F) proximity effect has been reported in Co crystalline nanowires [J. Wang, M. Singh, M. Tian, N. Kumar, B. Liu, C. Shi, J. K. Jain, N. Samarth, T. E. Mallouk, and M. H. W. Chan, Nat. Phys. 6, 389 (2010)]. Since the authors take care to avoid the existence of magnetic domains, the triplet character of the long-ranged proximity effect is improbable. Here we demonstrate that in the one-dimensional ballistic regime the standard singlet S/F proximity effect becomes long ranged. We provide an exact solution for the decay of the superconducting correlations near critical temperature $\left(T_c\right)$ and for arbitrary impurities concentration. In particular, we find a specific regime, between the diffusive and ballistic ones, where the decay length is simply the electronic mean-free path. Finally possible experiments which could permit to elucidate the nature of the observed long-ranged proximity effect in Co nanowires are discussed.