Universal critical behavior in single crystals and films of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$

We have studied the normal-to-superconducting phase transition in optimally doped ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$ in zero external magnetic field using a variety of different samples and techniques. Using dc transport measurements, we find that the dynamical critical exponent $z=1.54\pm 0.14$, and the static critical exponent $\nu =0.66\pm 0.10$ for both films (when finite-thickness effects are included in the data analysis) and single crystals (where finite-thickness effects are unimportant). We also measured thin films at different microwave frequencies and at different powers, which allowed us to systematically probe different length scales to avoid finite-thickness effects. dc transport measurements were also performed on the films used in the microwave experiments to provide a further consistency check. These microwave and dc measurements yielded a value of $z$ consistent with the other results, $z=1.55\pm 0.15$. The neglect of finite-thickness, finite-current, and finite-frequency effects may account for the wide ranges of values for $\nu$ and $z$ previously reported in the literature.

Figure 1

Schematic plots of electric field $E$ versus current density $J$. (a) $E\text{−}J$ plot in log-log scale and (b) $d{\text{log}}_{10}\left(E\right)/d{\text{log}}_{10}\left(J\right)$ vs $J$ in semilog scale.

Figure 2

(a) The electric field $E$ versus current density $J$ for an untwinned YBCO single crystal. The dashed line separates the normal and superconducting phases. The isotherms above the dashed line exhibits ohmic behavior at low-current density. The isotherms below the dashed line display the vanishing linear resistivity. Isotherms are separated by 3 mK. (b) Derivative plot of the data in Fig. 2a. The dashed line separates the normal and superconducting phases. The crossing of the dashed line with isotherm of 93.839 K is due to the noise. Both plots indicate $z\approx 1.5$.

Figure 3

Power law fit to Eq. (8) of the ohmic response of the isotherms just above the transition temperature $T_c$. Because $R_L\propto {\left(T/T_c-1\right)}^{\nu \left(z-1\right)}$, we can use this line to find $\nu$. By setting $T_c=93.837\pm 0.003\text{K}$ and $z=1.5\pm 0.20$, we get $\nu =0.71\pm 0.30$.

Figure 4

(Color online) (a) $T_c-T_m$ vs ${\mu }_{0}H$ of an untwinned YBCO single crystal. Here, as $T_c-T_m\left(H\right)\sim H^{1/2\nu }$, we can find $\nu$ from this line without assuming a value for $z$, and find $\nu =0.68\pm 0.10$. The inset is the melting line for the crystal up to 7 T. (b) A similar plot of $T_c-T_g$ vs ${\mu }_{0}H$ for a YBCO thin film $\left(d\approx 150\text{nm}\right)$. From this curve we find $\nu =0.63\pm 0.10$. The inset is the glass transition line for the film up to 6.5 T.

Figure 5

(a) $E\text{−}J$ isotherms for a 150 nm YBCO film in zero magnetic field. The spacing between isotherms is 50 mK. (b) The derivative plot of some selected isotherms from (a). At low-current-density regime, the phase transition is obscured by finite-size effects. The spacing between isotherms is 100 mK.

Figure 6

Schematic plots of (a) Magnitude $\left|{\sigma }_{fl}\right|$ vs $\omega$ in log-log scale and (b) phase ${\varphi }_{\sigma }$ vs $\omega$ in semilog scale at various temperatures around $T_c$, based on Fisher-Fisher-Huse ac scaling (Ref. 1).

Figure 7

(Color online) (a) Magnitude $\left|{\sigma }_{fl}\right|$ and (b) phase ${\varphi }_{\sigma }$ vs frequency at various temperatures for a typical YBCO film (xuh139). The black (upper) lines were measured with $-22\text{dBm}$ power while the red (lower) lines were measured with $-2\text{dBm}$ at the same temperature. (For clarity, only every other isotherm is shown.)

Figure 8

(Color online) $\left|{\sigma }_{fl}\right|$ vs incident microwave power at different frequencies. ( $T=89.140\text{K}$, sample xuh139 below $T_c$)

Figure 9

(Color online) Summary of length scales and finite-size effects in Corbino ac measurements of fluctuation conductivity of YBCO films near $T_c$. The dotted line in the figure gives the boundary $L_J=L_{\omega }$. At low frequency and small current density, the probed length scale $L_{\text{ac}}$ approaches the thickness of the sample.

Figure 10

(Color online) Scaling of phase and magnitude of fluctuation conductivity for sample xuh139 to determine ${\omega }_0\left(T\right)$ and ${\sigma }_0\left(T\right)$. (a) ${\varphi }_{\sigma }$ vs $\omega /{\omega }_0\left(T\right)$; (b) $\left|{\sigma }_{fl}\right|/{\sigma }_0\left(T\right)$ vs $\omega /{\omega }_0\left(T\right)$. Solid lines are the theoretical calculation of $\left|\phantom{[}{S}_{+}\left(y\right)]\right|$ and $2{\varphi }_{S_{+}\left(y\right)}/\pi$ from Eq. (25) for different values of $z$, assuming $D=3$. These lines can be used to determine ${\omega }_0\left(T\right)$ and ${\sigma }_0\left(T\right)$ far above $T_c$. In this figure, only the ${\omega }_0\left(T\right)$ and ${\sigma }_0\left(T\right)$ of the isotherm $T=89.763\text{K}$ are shown to make the measured ${\varphi }_{\sigma }\left[\omega /{\omega }_0\left(T\right)\right]$ and $\left|{\sigma }_{fl}\right|/{\sigma }_0\left(T\right)$ vs. $\omega /{\omega }_0\left(T\right)$ overlap with the theoretical prediction. For all the other isotherms, ${\omega }_0\left(T\right)$ and ${\sigma }_0\left(T\right)$ for each temperature are chosen to connect smoothly to the existing curve of ${\varphi }_{\sigma }\left[\omega /{\omega }_0\left(T\right)\right]$ and $\left|{\sigma }_{fl}\right|/{\sigma }_0\left(T\right)$, respectively, to make all the temperature curves collapse into one smooth and continuous curve. The lowest temperature isotherm (blue) appears on the right side of each plot, while the highest temperature isotherm (black) appears on the left.

Figure 11

(Color online) ${\omega }_0\left(T\right)$ vs $\left|\frac{T-T_c}{T_c}\right|$ and ${\sigma }_0\left(T\right)$ vs $\left|\frac{T-T_c}{T_c}\right|$ for different assumed $T_c$ for sample xuh139 and temperature from 89.297 to 89.763K. The errors of ${\omega }_0\left(T\right)$ and ${\sigma }_0\left(T\right)$ are about the size of the points.

Figure 12

(Color online) dc current-voltage characteristics measurement, performed after the ac experiment on xuh139 in zero magnetic field. (a) $E\text{−}J$ isotherms (50 mK apart), (b) $d{\text{log}}_{10}E/d{\text{log}}_{10}J$ vs $J$ derivative plot (40 mK apart).