Scaling analysis of the static and dynamic critical exponents in under, over, and optimally-doped Pr${}_{2-x}$Ce${}_x$CuO${}_{4-y}$ films

We report on current-voltage measurements of the zero-field normal-superconducting phase transition in thin films of Pr${}_{2-x}$Ce${}_x$CuO${}_{4-y}$ as a function of doping. We find that the small size of the critical regime in these materials ($\approx 25$ mK) gives rise to mean-field behavior at the phase transition with a static exponent of $\nu \approx 0.5$foralldopings(incontrasttohole-doped$$\mathrm{Y}\mathrm{B}{\mathrm{a}}_2\mathrm{C}{\mathrm{u}}_3{\mathrm{O}}_{7-\delta }$$). We also find mean-field behavior in the dynamic exponent $z$. This indicates that Pr${}_{2-x}$Ce${}_x$CuO${}_{4-y}$ behaves similarly to conventional superconductors in contrast to other cuprate superconductors. However, as the transition width in our samples decreases, the dynamic critical exponent approaches $z=1.5$, similar to the critical exponent found in hole-doped $\mathrm{Y}\mathrm{B}{\mathrm{a}}_2\mathrm{C}{\mathrm{u}}_3{\mathrm{O}}_{7-\delta }$.