  We theoretically investigate the gap function, superfluid density and the
transition temperature of the superconductivity (SC) on semi-periodic Penrose
lattice, where an attractive Hubbard model is adopted as an example. Firstly,
we clarify that the gap function, density of states and superfluid density are
all positively correlate to the extended degree of single particle states
around the Fermi energy. Secondly, we identify that the paramagnetic component
of the superfluid density does not decay to zero in the thermodynamic limit,
which is completely different from the periodic system. The difference between
the diamagnetic and paramagnetic currents keeps stable with whatever scaling,
which is consistent with recent experimental results that although the
superfluid density is lower than that of the periodic system, the system has
bulk SC. Thirdly, we find that both the superfluid density and SC transition
temperature can be boosted with the increase of disorder strength, which should
be general to quasicrystal but unusual to periodic systems, reflecting the
interplay between the underlying geometry and disorder.
