Inhomogeneous Superconductivity in YBa$_2$Cu$_3$O$_y$ and La$_{2-x}$Sr$_x$CuO$_4$ Above $T_c$

An exciting development in the immense research effort focused on resolving the origin of high-$T_c$ superconductivity, is the growing experimental evidence for signatures of superconductivity in cuprate materials at temperatures far above $T_c$. Recent STM experiments on Bi$_2 $Sr$_2$CaCu$_2$O$_{8+\delta}$ have provided new insight into the precise nature of these pairing correlations, by revealing the occurrence of nanometre-sized pairing regions above $T_c$. Whether nanoscale inhomogeneous superconductivity is universal to the cuprates, and whether $T_c$ is driven by Kosterlitz- Thouless physics or Josephson coupling between nanometre-sized superconducting regions are matters of current debate. Very recently we have used $\mu$SR to probe the local response in the bulk of YBa$_2$Cu$_3$O$_y$ and La$_{2-x}$Sr$_x$CuO$_4$ single crystals to a large applied magnetic field (H = 7 T). At temperatures above $T_c$, we detect a spatially inhomogeneous magnetic field that tracks the hole-doping dependences of both $T_c$ and the superfluid density at $T = 0$ K. Our experiments are inconsistent with the field inhomogeneity above $T_c$ being caused by electronic magnetic moments or a vortex liquid. Instead they are explained by the existence of nanometre-size superconducting regions with a local $T_c$ that exceeds the bulk $T_c$. In YBa$_2$Cu$_3$O$_y$, we detect a spatially inhomogeneous response to field that persists beyond $T = 200$ K, indicating that the basic ingredients for superconductivity near room temperature already exist in spatially localized regions of this material. A lingering question is the origin of the weak magnetism detected earlier in YBa$_2$Cu$_3$O$_y$ by zero-field $\mu$SR, at temperatures below the pseudogap temperature $T^*$.