Abstract
The superconducting transition is studied in wires ``written'' in oxygen-depleted $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{x}$ $(x\ensuremath{\sim}6.4)$ thin films by photodoping with a near-field scanning optical microscope. The enhancement of the superconducting transition temperature observed for wide wires is found to be suppressed for wires below a sample dependent width $(\ensuremath{\sim}1--2\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m})$. This behavior is understood in terms of percolation of the supercurrent through an inhomogeneous distribution of weak links in these samples grown by pulsed laser deposition. As the wire width is reduced the highest ${T}_{c}$ percolation path is cut off, leading to a lower ${T}_{c}$. By modeling the electronic transport data and ${T}_{c}$ versus wire width data using a classical bond percolation model and the Ambegaokar-Halperin thermally activated phase-slip theory for a weak link, we conclude that the density of weak links in these samples is approximately 1 weak $\text{link}∕\ensuremath{\mu}\mathrm{m}$.
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