Imaging the Spatial Form of a Superconducting Order Parameter via Josephson Scanning Tunneling Spectroscopy

Motivated by recent experiments, we investigate Josephson scanning tunneling spectroscopy in an s-wave superconductor. We demonstrate that the spatial oscillations in the superconducting order parameter induced by defects can be spatially imaged through local measurements of the critical Josephson current, providing unprecedented insight into the nature of superconductivity. The spatial form of the Josephson current reflects the nature of the defects, and can be used to probe defect-induced phase transitions from an $S=0$ to an $S=1/2$ ground state.