Evidence for vertical line nodes in Sr$_2$RuO$_4$ from nonlocal electrodynamics

By determining the superconducting lower and upper critical fields H$_{c1}$ (T ) and H$_{c2}$(T ), respectively, in a highpurity spherical Sr$_2$RuO$_4$ sample via ac-susceptibility measurements, we obtain the temperature dependence of the coherence length ξ and the penetration depth λ down to 0.04Tc. Given the high sample quality, the observed T$^2$ dependence of λ at low temperatures cannot be explained in terms of impurity effects. Instead, we argue that the weak type-II superconductor Sr$_2$RuO$_4$ has to be treated in the nonlocal limit. By comparing our data with existing theory in that limit, the penetration depth in Sr$_2$RuO$_4$ agrees with a gap structure having vertical line nodes, while horizontal line nodes cannot account for the observation. The work highlights the potential benefits of purifying other unconventional superconductors in order to access the fascinating nonlocal regime in more materials and to determine their Cooper pair wave functions.