Zero-field superconducting phase transition obscured by finite-size effects in thick ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ films

We report on the normal-superconducting phase transition in thick ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ films in zero magnetic field. We find significant finite-size effects at low currents even in our thickest films $(d=3200\mathrm{\AA })$. Using data at higher currents, we can unambiguously find $T_c$ and $z$, and show $z=2.1\pm 0.15$, as expected for the three-dimensional $\mathrm{XY}$ model with diffusive dynamics. The crossover to two-dimensional behavior, seen by other researchers in thinner films $(d⩽500\mathrm{\AA })$, obscures the three-dimensional transition in both zero field and the vortex-glass transition in field, leading to incorrect values of $T_c$ (or $T_g$), $\nu$, and $z$. The finite-size effects, usually ignored in thick films, are an explanation for the wide range of critical exponents found in the literature.

Figure 1

$I\text{−}V$ curves for a $2100\mathrm{\AA }$ ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ film, with bridge dimensions $20\times 100\mu {\mathrm{m}}^2$, in zero magnetic field. Isotherms are separated by $60\mathrm{mK}$. The dashed line indicates a slope of 1, or ohmic behavior. The error bars are smaller than the points. The inset is $R\left(T\right)$ at $10\mu \mathrm{A}$.

Figure 2

$d\log E∕d\log J$ vs $I$ for the $I\text{−}V$ curves from Fig. 1. Isotherms are separated by $60\mathrm{mK}$. The conventional choice for $T_c$, $91.26\mathrm{K}$, is clearly not a horizontal line. The opposite concavity criterion can be seen at higher currents $(I>40\mu \mathrm{A})$ about $91.44\mathrm{K}$. The inset shows the $91.26\mathrm{K}$ isotherm for three bridge widths on the same film: $20\mu \mathrm{m}$ (solid line), $50\mu \mathrm{m}$ (dashed line), and a $100\mu \mathrm{m}$ (dotted line), which do not agree as a function of $I$.

Figure 3

$d\log E∕d\log J$ vs $J$ for three bridges of different widths on the same $2100\mathrm{\AA }$ film ($20\times 100\mu {\mathrm{m}}^2$, $50\times 250\mu {\mathrm{m}}^2$, $100\times 500\mu {\mathrm{m}}^2$). The crossover to ohmic behavior clearly depends on $J$.

Figure 4

$1∕\sqrt{J_{min}}$ vs thickness for eight different films. The solid line is a weighted least-squares linear fit to the data, with ${\tilde{\chi }}^2=0.41$. The slope of the line gives $c=0.60\pm 0.17$, of the same order as $\gamma \approx 0.2$, as expected.