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-two dimensional quantum magnetism is clearly highly correlated with superconducting ground states in cuprate-based High T$_c$ superconductivity. Three dimensional, commensurate long range magnetic order in La$_2$CuO$_4$ 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 La$_{2-x}$Sr$_x$/Ba$_x$CuO$_4$ family of superconductors. It has long been known that the onset of superconducting ground states in these systems is coincident with a remarkable rotation of the incommensurate spin order from "diagonal stripes" below x = 0.05, to "parallel stripes" above. However, little is known about the spin correlations at optimal and high doping levels, where the dome of superconductivity draws to an end. 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 two dimensional, 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. Parallel spin stripes stretch across 0.05 < < 0.26, with rapidly decreasing moment size and onset temperatures for x > 0.125. We also show that the low energy, parallel spin stripe fluctuations for optimally doped x = 0.19 display dynamic spectral weight which grows with decreasing temperature and saturates below its superconducting T$_c$. 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.