Low-energy antiferromagnetic spin fluctuations limit the coherent superconducting gap in cuprates

Motivated by recent attention to a potential antiferromagnetic quantum critical point at $x_c\sim 0.19$, we have used inelastic neutron scattering to investigate the low-energy spin excitations in crystals of La$_{2-x}$Sr$_x$CuO$_4$ bracketing $x_c$. We observe a peak in the normal-state spin-fluctuation weight at $\sim20$~meV for both $x=0.21$ and 0.17, inconsistent with quantum critical behavior. The presence of the peak raises the question of whether low-energy spin fluctuations limit the onset of superconducting order. Empirically evaluating the spin gap $\Delta_{\rm spin}$ in the superconducting state, we find that $\Delta_{\rm spin}$ is equal to the coherent superconducting gap $\Delta_c$ determined by electronic spectroscopies. To test whether this is a general result for other cuprate families, we have checked through the literature and find that $\Delta_c\le\Delta_{\rm spin}$ for cuprates with uniform $d$-wave superconductivity. We discuss the implications of this result.