Magnetic susceptibility and superfluid density of inhomogeneous superconducting films: the Tc-mosaic model

Whole-sample and mutual-inductance part-sample measurements of the linear magnetic susceptibility, $χ$, of superconducting films are routinely used to obtain the London penetration depth, $λ(T)$, assuming uniform superconductivity. However, spatially resolved probes show that many films of current interest are inhomogeneous on micrometer scales. Here we ask whether a distribution of local transition temperatures in a fully gapped film can produce an apparent line-nodal, $T$-linear $λ(T)$. The film is modeled as a random Voronoi mosaic of domains with $T_c$ distributed between $T_{c,\min}$ and $T_{c,\max}$ with s-wave (also d-wave, for comparison) London response inside each domain. We show that detectable screening appears only below the percolation temperature, $T_p$, and the transition broadens with the spread of $T_c$. For s-wave domains, $λ(T)$ remains activated for any $T_{c,\min}>0$. Only for $T_{c,\min}=0$ does it become a power law, $Δλ\propto T^n$ with $n\approx1.7$, but then the onset of screening drops to $T_p=T_{c,\max}/2$. Weighting the $T_{c}$ distribution towards $T_c\to0$ (more low-$T_{c}$ domains) can bring $n$ closer to 1 ($n\approx1.2$), but simultaneously pushes the apparent transition, $T_p$, more than an order of magnitude below $T_{c,\max}$. For d-wave domains, the linear term survives, and its relative slope is enhanced almost fourfold for $T_{c,\min}=0$. Hence, a $T$-linear $λ(T)$ measured in a film with screening onset within a factor of a few of $T_{c,\max}$ by whole-sample or mutual-inductance part-sample measurements cannot originate from $T_c$ mosaicity in a fully gapped superconductor.