Recently, the superconductivity in a metal-organic framework (MOF) has been discovered for the first time in copper(II) benzenehexathiolate ([Cu$_3$(C$_6$S$_6$)]$_n$, Cu-BHT). The Cu atoms form a two-dimensional perfect Kagome lattice, which has the potential to host a metallic quantum spin liquid state. Here we present high-precision measurements of in-plane magnetic penetration depth $lambda$ in Cu-BHT films down to 40,mK. The temperature dependence of $lambda$ shows a non-exponential, quasi-linear behavior at low temperatures, suggesting that unconventional superconductivity with low-energy quasiparticle excitations is realized in this system. With the reported non-Fermi liquid behavior, this finding implies that MOFs can provide a flexible platform to investigate the superconducting pairing mechanisms in the presence of spin frustration and strong quantum fluctuations.
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