Correlations Among STM Observables in Disordered Unconventional Superconductors

New developments in scanning tunneling spectroscopy now allow for the spatially resolved measurement of the Josephson critical current $I_c$ between a tip and a superconducting sample, a nearly direct measurement of the true superconducting order parameter. However, it is unclear how these $I_c$ measurements are correlated with previous estimates of the spectral gap taken from differential conductance measurements. In particular, recent such experiments on an iron-based superconductor found almost no correlation between $I_c$ and the spectral gap obtained from differential conductance $g=dI/dV$ spectra, reporting instead a more significant correlation between $I_c$ and the the coherence-peak height. Here we point out that the correlation--or the lack thereof--between these various quantities can be naturally explained by the effect of disorder on unconventional superconductivity. Using large scale numerical simulations of a BCS $d$-wave pair Hamiltonian with many-impurity potentials, we observe that "substitutional" disorder models with weak pointlike impurities lead to a situation in which the true superconducting order parameter and $I_c$ are both uncorrelated with the spectral gap from $dI/dV$ measurements and highly correlated with the coherence-peak heights. The underlying mechanism appears to be the disorder-induced transfer of spectral weight away from the coherence peaks. On the other hand, smooth impurity potentials with a length scale larger than the lattice constant lead to a large positive correlation between the true superconducting order parameter and the spectral gap, in addition to a large correlation between the order parameter and the coherence-peak height. We discuss the applicability of our results to recent Josephson scanning tunneling spectroscopy experiments on iron-based and cuprate high-temperature superconductors.