  Odd-parity, spin-triplet superconductor Sr2RuO4 has been found to feature
exotic vortex physics including half-flux quanta trapped in a doubly connected
sample and the formation of vortex lattices at low fields. The consequences of
these vortex states on the low-temperature magnetoresistive behavior of
mesoscopic samples of Sr2RuO4 were investigated in this work using ring device
fabricated on mechanically exfoliated single crystals of Sr2RuO4 by
photolithography and focused ion beam. With the magnetic field applied
perpendicular to the in-plane direction, thin-wall rings of Sr2RuO4 were found
to exhibit pronounced quantum oscillations with a conventional period of the
full-flux quantum even though the unexpectedly large amplitude and the number
of oscillations suggest the observation of vortex-flow-dominated
magnetoresistance oscillations rather than a conventional Little-Parks effect.
For rings with a thick wall, two distinct periods of quantum oscillations were
found in high and low field regimes, respectively, which we argue to be
associated with the "lock-in" of a vortex lattice in these thick-wall rings. No
evidence for half-flux-quantum resistance oscillations were identified in any
sample measured so far without the presence of an in-plane field.
