Parallel Spin Stripes and Their Coexistance with Superconducting Ground States at Optimal and High Doping in La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$*

Quasi 2d quantum magnetism is clearly highly correlated with superconducting ground states. 3d, commensurate long range magnetic order quickly evolves to quasi-two dimensional, incommensurate correlations on doping with mobile holes, and superconducting ground states follow for x as small as 0.05 in the LSCO/LBCO family of superconductors. It has been known that superconducting ground states in these systems is coincident with a rotation of the incommensurate spin order from ``diagonal stripes" to ``parallel stripes". However, little is known about the spin correlations at optimal and high doping levels close to the end of superconductivity. Here we present new elastic and inelastic neutron scattering measurements on single crystals of La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ with x = 0.125, 0.19, 0.24 and 0.26, and show that 2d, quasi-static, parallel spin stripes are observed to onset at temperatures such that the parallel spin stripe phase envelopes all superconducting ground states in this system. The elastic order parameter for x = 0.19 also shows plateau behavior coincident with the onset of superconductivity. This set of observations assert the foundational role played by two dimensional parallel spin stripe order and fluctuations in High T$_c$ cuprate superconductivity.