Competing Pair Density Wave and Uniform $d$-wave Superconductivity in Phase Separated 214 Cuprates at the 1/8 Anomaly

Compelling evidence exists for electronic phase separation in cuprate high-$T_c$ superconductors, emerging near 1/8 hole doping. At these dopings and low temperatures, intertwined charge and spin stripes coexist with more uniformly doped regions in the two-dimensional (2$D$) copper-oxide planes. Each region is capable of developing superconducting pairing, either as a pair density wave (PDW) within the stripes or as a uniform $d$-wave condensate ($d$-SC) in the more homogeneous regions. Using neutron scattering on single crystals of La$_{1.875-y}$Nd$_{y}$Sr$_{0.125}$CuO$_4$, we demonstrate that the onset temperatures for spin stripe order ($T_N$) and superconductivity ($T_c$) merge as the average ordered moment vanishes in LSCO ($y = 0$), whereas Nd doping stabilizes static stripe order and suppresses $T_c$. Because the spin stripes possess the same in-plane periodicity (8$a$) as the PDW and establish the framework in which the PDW resides, the stabilization of spin stripe order enhances PDW correlations. Thus, the competition between $d$-wave pairing in the uniform regions and PDW pairing in the stripe-ordered regions can be controlled by the Nd concentration in La$_{1.875-y}$Nd$_{y}$Sr$_{0.125}$CuO$_4$, allowing the superconducting $T_c$ to vary by nearly an order of magnitude at a fixed 1/8 hole doping.