Constraints on the observation of half flux quantum magnetoresistance oscillations in mesoscopic rings of Sr2RuO4
Sr$_2$RuO$_4$, a spin-triplet, $p$-wave superconductor, is predicted to feature half flux quantum states in a doubly connected geometry. To measure the modulation of the free energy of the half flux quantum states, we fabricated small crystal based device of mesoscopic superconducting rings of Sr$_2$RuO$_4$ with thin wall thickness by photolithography and a focused ion beam, and carried out the magnetoresistance oscillation measurements. To lower the free energy cost of half flux quantum states, a constant in-plane field is applied during the oscillation measurements. However, based on the results of about ten ring samples, the theoretically predicted transport signatures of the half flux quantum state have not been detected, suggesting the constraints on the energy difference between the half and the full flux quantum states. We discuss means of further reducing the energy cost and stabilizing the states.