Some fullerenes are superconducting but their Tc is pretty low, of the order of a few hundred mK if I remember correctly. However, I don't think that is relevant here.
You can indeed find Phi0/2 flux in some situations. There are a number of ways you can realize this experimentally. The most straightforward way is to use a d-wave superconductor (i.e. all known high-Tc supererconductor). These have an intrinsic phase shift of pi in their order parameter as you move from one lobe to the next you (there are four lobes 90 degrees apart with alternating sign +-+- and nodes inbetween where the gap disappers).
Now, this means that it is possible to use these superconductors (e.g. YBCO) to create "frustrated" situations where an interface has a + lobe on one side and a a - lobe on the other; when you cross the interface you therefore get an additional phase shift of pi. If you now make a ring with one such interface the ground state energy will be such that a phi0/2 flux will be spontaniously generated (to screen the currents in the ring).
This was done in a well know experiement by Kirtley and Tsuei a few years ago, it was an important experiment since it confirmed that the order parameter in high-Tc is ineed d-wave whereas it is s-wave in most low-Tc superonductors (aluminium, niobiúm etc, there are some p-wave superconductors as well).
Nowadays people are actually trying to use phi0/2 quanta in various systems, one idea is to use them to create memories for RSFQ circuits. Hilgenkamp and co-workers at the University of Twente have done a lot of work on this.
You can also, at least in principle, realize systems like this using SFS-junctions etc.
You can find a lot of information of you google the names above,
Look e.g. at
http://cmd.tnw.utwente.nl/research/
