Scaling analysis of the static and dynamic critical exponents in Pr$_{2-x}$Ce$_x$CuO$_4$ films as a function of doping

We investigate the static and dynamic critical exponents of the electron-doped superconductor Pr$_{2-x}$Ce$_x$CuO$_4$. Our results are based on current vs.\ voltage measurements in zero-field of the normal-superconducting phase transition in Pr$_{2-x}$Ce$_x$CuO$_4$ films as a function of doping. We find that these materials posses an unusually small critical regime ($\sim$25mK) that gives rise to mean-field behavior at the phase transitions and a static critical exponent of about $\nu\sim$0.5 for all dopings. This is quite unexpected when compared to the critical behavior seen in well-known hole-doped superconductor YBa$_2$Cu$_3$O$_7$, where $\nu\sim$2/3. In addition, mean-field behavior is also exhibited in the dynamic critical exponent ($z$). We find that Pr$_{2-x}$Ce$_x$CuO$_4$ behaves not like other cuprate superconductors, but similarly to conventional superconductors in this regard. Only as transition width decreases to zero does the dynamic critical exponent ($z$) approach the value found in YBa$_2$Cu$_3$O$_7$.