  The observation of the highly unusual half-quantum vortex (HQV) in a single
crystalline superconductor excludes unequivocally the spin-singlet symmetry of
the superconducting order parameter. HQVs were observed previously in
mesoscopic samples of Sr2RuO4 in cantilever torque magnetometry measurements,
thus providing direct evidence for spin-triplet pairing in the material. In
addition, it raised important questions on HQV, including its stability and
dynamics. These issues have remained largely unexplored, in particular,
experimentally. We report in this paper the detection of HQVs in mesoscopic,
doubly connected cylinders of single-crystalline Sr2RuO4 of a mesoscopic size
and the examination of the effect of the in-plane magnetic field needed for the
observation of the HQV by magnetoresistance (MR) oscillations measurements.
Several distinct features found in our data, especially a dip and secondary
peaks in the MR oscillations seen only in the presence of a sufficiently large
in-plane magnetic field as well as a large measurement current, are linked to
the formation of the HQV fluxoid state in and crossing of an Abrikosov HQV
through the sample. The conclusion is drawn from the analysis of our data using
a model of thermally activated vortex crossing overcoming a free-energy barrier
which is modulated by the applied magnetic flux enclosed in the cylinder as
well as the measurement current. Evidence for the trapping of an HQV fluxoid
state in the sample was also found. Our observation of the HQV in mesoscopic
Sr2RuO4 provided not only additional evidence for spin-triplet
superconductivity in Sr2RuO4 but also insights into the physics of HQV,
including its spontaneous spin polarization, stability, and dynamics. Our study
also revealed a possible effect of the measurement current on the magnitude of
the spontaneous spin polarization associated with the HQV.
