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In the unconventional superconductor Sr$_2$RuO$_4$, uniaxial stress along the$[100]$ direction tunes the Fermi level through a Van Hove singularity (VHS) inthe density of states, causing a strong enhancement of the superconductingcritical temperature $T_\textrm{c}$. Here, we report measurements of the Londonpenetration depth $\lambda$ as this tuning is performed. We find that thezero-temperature superfluid density, here defined as $\lambda(0)^{-2}$,increases by $\sim$15%, with a peak that coincides with the peak in$T_\textrm{c}$. We also find that the low temperature form of $\lambda(T)$ isquadratic over the entire strain range. Using scanning tunneling microscopy, wefind that the gap increases from $\Delta_0 \approx 350~\mu$eV in unstressedSr$_2$RuO$_4$ to $\Delta_0 \approx 600~\mu$eV in a sample strained to near thepeak in $T_c$. With a nodal order parameter, an increase in the superconductinggap could bring about an increase in the superfluid density through reducedsensitivity to defects and through reduced non-local effects in the Meissnerscreening. Our data indicate that tuning to the VHS increases the gapthroughout the Brillouin zone, and that non-local effects are likely moreimportant than reduced scattering.

Superconducting Penetration Depth Through a Van Hove Singularity: Sr$_2$RuO$_4$ Under Uniaxial Stress