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arXiv:cond-mat/9709012 · Physical review. B, Condensed matter

Current-voltage characteristics of diluted Josephson-junction arrays: Scaling behavior at current and percolation threshold

Enzo Granato, Daniel Domı́nguez

Published Dec 1, 1997• 15 citations
arXivHTMLPDF

Abstract

Dynamical simulations and scaling arguments are used to study the current-voltage $(I\ensuremath{-}V)$ characteristics of a two-dimensional model of resistively shunted Josephson-junction arrays in presence of percolative disorder, at zero external field. Two different limits of the Josephson-coupling concentration $p$ are considered, where ${p}_{c}$ is the percolation threshold. For $p>{p}_{c}$ and zero temperature, the $I\ensuremath{-}V$ curves show power-law behavior above a disorder-dependent critical current. The power-law behavior and critical exponents are consistent with a simple scaling analysis. At ${p}_{c}$ and finite temperature $T,$ the results show the scaling behavior of a $T=0$ superconducting transition. The resistance is linear but vanishes for decreasing $T$ with an apparent exponential behavior. Crossover to nonlinearity appears at currents proportional to ${T}^{1+{\ensuremath{\nu}}_{T}},$ with a thermal-correlation length exponent ${\ensuremath{\nu}}_{T}$ consistent with the corresponding value for the diluted $\mathrm{XY}$ model at ${p}_{c}.$

DOI
10.1103/physrevb.56.14671

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Dynamical simulations and scaling arguments are used to study the current-voltage (I\ensuremath−V)(I\ensuremath{-}V)(I\ensuremath−V) characteristics of a two-dimensional model of resistively shunted Josephson-junction arrays in presence of percolative disorder, at zero external field. Two different limits of the Josephson-coupling concentration ppp are considered, where pc{p}_{c}pc​ is the percolation threshold. For p>pcp>{p}_{c}p>pc​ and zero temperature, the I\ensuremath−VI\ensuremath{-}VI\ensuremath−V curves show power-law behavior above a disorder-dependent critical current. The power-law behavior and critical exponents are consistent with a simple scaling analysis. At pc{p}_{c}pc​ and finite temperature T,T,T, the results show the scaling behavior of a T=0T=0T=0 superconducting transition. The resistance is linear but vanishes for decreasing TTT with an apparent exponential behavior. Crossover to nonlinearity appears at currents proportional to T1+\ensuremathνT,{T}^{1+{\ensuremath{\nu}}_{T}},T1+\ensuremathνT​, with a thermal-correlation length exponent \ensuremathνT{\ensuremath{\nu}}_{T}\ensuremathνT​ consistent with the corresponding value for the diluted XY\mathrm{XY}XY model at pc.{p}_{c}.pc​.
We numerically investigate dynamic critical behaviors of two-dimensional (2D) Josephson-junction arrays with positional disorder in the scheme of the resistively shunted junction dynamics. Large-scale computation of the current-voltage characteristics reveals evidence supporting that a phase transition occurs at a nonzero critical temperature in the strong disorder regime, as well as in the weak disorder regime. The phase transition at weak disorder appears to belong to the Berezinskii-Kosterlitz-Thouless (BKT) type. In contrast, evidence for a non-BKT transition is found in the strong disorder regime. These results are consistent with the recent experiment on positionally disordered Josephson-junction arrays; in particular, the critical temperature of the non-BKT transition (ranging from 0.265 down to the minimum 0.22 in units of EJ∕kB{E}_{J}∕{k}_{B}EJ​∕kB​ with the Josephson coupling strength EJ{E}_{J}EJ​), the correlation length critical exponent $\ensuremath{\nu}=1.2$, and the dynamic critical exponent z=2.0z=2.0z=2.0 in the strong disorder regime agree with the existing studies of the 2D gauge-glass model.
Using resistivity-shunted-junction dynamics, we model bond dilutions in Josephson junction arrays. The current-voltage characteristics are obtained by large-scale numerical simulations. By dynamic scaling analysis to the finite-temperature phase transition, we provide clear evidence for the existence of non-Kosterlitz–Thouless-type phase transition, as described by Kosterlitz et al. [J. Phys. C 6, 1181 (1973); J. Phys. C 7, 1046 (1974)] in a wide range of disorder strengths. Critical temperature and exponents of the phase transitions are estimated, which are further confirmed by the finite-size scaling analysis at the transition temperature.
It has been suggested [Pierson et al., Phys. Rev. B 60, 1309 (1999); Ammirata et al., Physica C 313, 225 (1999)] that for two-dimensional (2D) superconductors there exists a phase transition with the dynamic critical exponent $z\ensuremath{\approx}5.6.$ We perform simulations for the 2D resistively-shunted-junction model and compare the results with the experimental data in Repaci et al. obtained for an ultrathin YBCO sample [Phys. Rev. B 54, R9674 (1996)]. We then use a different method of analyzing dynamic scaling than in Pierson et al., and conclude that both the simulations and the experiments are consistent with a conventional Kosterlitz Thouless (KT) transition in the thermodynamic limit for which z=2.z=2.z=2. For finite systems, however, we find both in simulations and experiments that the change in the current-voltage $(I\ensuremath{-}V)$ characteristics caused by the finite size shows a scaling property with an exponent $\ensuremath{\alpha}\ensuremath{\approx}1/6,$ seemingly suggesting a vanishing resistance at a temperature for which $z={\ensuremath{\alpha}}^{\ensuremath{-}1}.$ It is pointed out that the dynamic critical exponent found in Pierson et al. corresponds to the exponent ${\ensuremath{\alpha}}^{\ensuremath{-}1}.$ It is emphasized that this scaling property does not represent any true phase transition since in reality the resistance vanishes only at zero temperature. Nevertheless, the observed scaling behavior associated with $\ensuremath{\alpha}\ensuremath{\approx}1/6$ appears to be a common and intriguing feature for the finite size caused change in the $I\ensuremath{-}V$ characteristics around the KT transition.
The scaling behavior of the current-voltage characteristics of chiral and gauge-glass models of disordered superconductors, are studied numerically, in two dimensions. For both models, the linear resistance is nonzero at finite temperatures and the scaling analysis of the nonlinear resistivity is consistent with a phase transition at T=0T=0T=0 temperature characterized by a diverging correlation length $\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}{\ensuremath{\nu}}_{T}}$ and thermal critical exponent ${\ensuremath{\nu}}_{T}.$ The values of ${\ensuremath{\nu}}_{T},$ however, are found to be different for the chiral and gauge-glass models, suggesting different universality classes, in contrast to the result obtained recently in three dimensions.
We numerically investigate the intriguing effects produced by random percolative disorder in two-dimensional Josephson junction arrays. By dynamic scaling analysis, we evaluate critical temperatures and critical exponents with high accuracy. It is observed that with the introduction of site-diluted disorder, the Kosterlitz-Thouless phase transition is eliminated and evolves into a continuous transition with power-law divergent correlation length. Moreover, genuine depinning transition and creep motion are studied; evidence for distinct creep motion types is provided. Our results not only are in good agreement with the recent experimental findings but also shed some light on the relevant phase transitions.
Measurements of the IV characteristics of site-diluted Josephson-junction arrays have revealed intriguing effects of percolative disorder on the phase transition and the vortex dynamics in a two-dimensional XY system. Different from other types of phase transitions, the Kosterlitz-Thouless transition was eliminated with the introduction of percolative disorder far below the percolation threshold. Even after the Kosterlitz-Thouless order had been removed, the system remained superconducting at low temperatures by establishing a different type of order. Near the percolation threshold, evidence was found that, as a consequence of the underlying fractal structure, the critical dynamics of the phase degrees of freedom persisted down to zero temperature.