| Summary: | Some two-dimensional superconductors like, e.g., LaAlO<inline-formula> <math display="inline"> <semantics> <msub> <mrow></mrow> <mn>3</mn> </msub> </semantics> </math> </inline-formula>/SrTiO<inline-formula> <math display="inline"> <semantics> <msub> <mrow></mrow> <mn>3</mn> </msub> </semantics> </math> </inline-formula> heterostructures or thin films of transition metal dichalcogenides, display peculiar properties that can be understood in terms of electron inhomogeneity at the nanoscale. In this framework, unusual features of the metal-superconductor transition have been interpreted as due to percolative effects within a network of superconducting regions embedded in a metallic matrix. In this work we use a mean-field-like effective medium approach to investigate the superconducting phase below the critical temperature <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> at which the resistivity vanishes. Specifically, we consider the finite frequency impedance of the system to extract the dissipative part of the conductance and the superfluid stiffness in the superconducting state. Intriguing effects arise from the metallic character of the embedding matrix: upon decreasing the temperature below <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> proximity effects may rapidly increase the superfluid stiffness. Then, a rather fragile superconducting state, living on a filamentary network just below <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula>, can be substantially consolidated by additional superconducting regions induced by proximity effect in the interstitial metallic regions. This mean-field prediction should call for further theoretical analyses and trigger experimental investigations of the superconducting properties of the above systems.
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