  The difference in the density of states for up- and down-spin electrons in a
ferromagnet (F) results in spin-dependent scattering of electrons at a
ferromagnet / nonmagnetic (F/N) interface. In a F/N/F spin-valve, this causes a
current-independent difference in resistance ($\Delta R$) between antiparallel
(AP) and parallel (P) magnetization states. Giant magnetoresistance (GMR),
$\Delta R = R(AP) - R(P)$, is positive due to increased scattering of majority
and minority spin-electrons in the AP-state. If N is substituted for a
superconductor (S), there exists a competition between GMR and the
superconductor proximity effect: in the AP-state the net magnetic exchange
field acting on S is lowered and the superconductivity is reinforced meaning
$R(AP)$ decreases. For current-perpendicular-to-plane (CPP) spin-valves,
existing experimental studies show that GMR dominates ($\Delta R>0$) over the
superconductor proximity effect ($\Delta R<0$) [J. Y. Gu $\textit{et al.}$,
Phys. Rev. B 66, 140507 (2002)]. Here, however, we report a crossover from GMR
($\Delta R > 0$) to the superconductor proximity effect ($\Delta R < 0$) in CPP
F/S/F spin-valves as the superconductor thickness decreases below a critical
value. The results are key to the development of quasiparticle spintronics.
