Nanoscale Supersolidity in $^4$He Adsorbed on a C$_{20}$ Molecule

We have studied adsorption of $^4$He on the surface of a single C$_{20}$ fullerene molecule using the path-integral Monte Carlo method. For a full incorporation of the surface corrugations on the molecular surface the $^4$He-C$_{20}$ interaction is treated with a sum of empirical helium-carbon interatomic pair potentials. Radial density distributions show layer-by-layer growth of $^4 $He, and a detailed analysis of energetics and angular density distributions reveals that the strongly-bound first layer, located at a distance of $\sim 4.9$~\AA~from the center of the C$_{20}$ molecule, is in various quantum states as the number of $^4$He atoms changes. This layer, when completed with 32 atoms, is found to be a commensurate solid with an icosahedral lattice structure. We observe that near the completion of the first layer, mobile vacancies can be activated at a low temperature of $T=0.31$ K, which results in a finite superfluid fraction as well as a crystalline order. This is a manifestation of vacancy-based supersolidity on a nanometer scale. Finally we analyze the effects of $^3$He impurities on the superfluidity of the $^4$He adlayer on a C$_ {20}$.