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arXiv:cond-mat/0011186 · Physical review. B, Condensed matter

Finite-size effects and dynamical scaling in two-dimensional Josephson junction arrays

J. Holzer, Richard S Newrock, Christopher J. Lobb, Tahar Aouaroun, Steven Herbert

Published Apr 18, 2001• 29 citations
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Abstract

In recent years many groups have used Fisher, Fisher and Huse (FFH) dynamical scaling to investigate and demonstrate details of the superconducting phase transition. Some attention has been focused on two dimensions where the phase transition is of the Kosterlitz-Thouless-Berezinskii (KTB) type. Pierson et al. used FFH dynamical scaling almost exclusively to suggest that the dynamics of the two-dimensional superconducting phase transition may be other than KTB-like. In this work we investigate the ability of scaling behavior by itself to yield useful information on the nature of the transition. We simulate current-voltage (IV) curves for two-dimensional Josephson junction arrays with and without finite-size-induced resistive tails. We find that, for the finite-size effect data, the values of the scaling parameters, specifically the transition temperature and the dynamical scaling exponent z, depend critically on the magnitude of the contribution that the resistive tails make to the IV curves. In effect, the values of the scaling parameters depend on the noise floor of the measuring system.

DOI
10.1103/physrevb.63.184508

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In recent years many groups have used Fisher, Fisher and Huse (FFH) dynamical scaling to investigate and demonstrate details of the superconducting phase transition. Some attention has been focused on two dimensions where the phase transition is of the Kosterlitz-Thouless-Berezinskii (KTB) type. Pierson et al. used FFH dynamical scaling almost exclusively to suggest that the dynamics of the two-dimensional superconducting phase transition may be other than KTB-like. In this work we investigate the ability of scaling behavior by itself to yield useful information on the nature of the transition. We simulate current-voltage (IV) curves for two-dimensional Josephson junction arrays with and without finite-size-induced resistive tails. We find that, for the finite-size effect data, the values of the scaling parameters, specifically the transition temperature and the dynamical scaling exponent z, depend critically on the magnitude of the contribution that the resistive tails make to the IV curves. In effect, the values of the scaling parameters depend on the noise floor of the measuring system.
A numerical study of the neutral two-dimensional (2D) lattice Coulomb gas is performed to examine dynamic scaling, finite size effects, and the dynamic critical exponent zzz of the Kosterlitz-Thouless-Berezinskii transition. By studying large system sizes (L=100L=100L=100), we show z=2.0±0.2z=2.0 \pm0.2z=2.0±0.2 using Fisher-Fisher-Huse (FFH) scaling. We also present evidence that the vortex correlation length is finite below the transition temperature, in contrast to conventional wisdom. Finally, we conclude that previous findings for a variety of experimental systems that z≃5.6z\simeq 5.6z≃5.6 using FFH scaling indicates that finite size effects are ubiquitous in 2D superconductors and Josephson Junction Arrays.
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.
There has been ongoing debate over the critical behavior of two-dimensional superconductors; in particular for high Tc{T}_{c}Tc​ superconductors. The conventional view is that a Kosterlitz-Thouless-Berezinskii transition occurs as long as finite size effects do not obscure the transition. However, there have been recent suggestions that a different transition actually occurs which incorporates aspects of both the dynamic scaling theory of Fisher, Fisher, and Huse and the Kosterlitz-Thouless-Berezinskii transition. Of general interest is that this modified transition apparently has a universal dynamic critical exponent. Some have countered that this apparent universal behavior is rooted in a newly proposed finite-size scaling theory; one that also incorporates scaling and conventional two-dimensional theory. To investigate these issues we study dc voltage versus current data of a 12-\AA{}-thick ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ film. We find that the newly proposed scaling theories have intrinsic flexibility that is relevant to the analysis of the experiments. In particular, the data scale according to the modified transition for arbitrarily defined critical temperatures between 0 and 19.5 K, and the temperature range of a successful scaling collapse is related directly to the sensitivity of the measurement. This implies that the apparent universal exponent is due to the intrinsic flexibility rather than some real physical property. To address this intrinsic flexibility, we propose a criterion which would give conclusive evidence for phase transitions in two-dimensional superconductors. We conclude by reviewing results to see if our criterion is satisfied.
The value of the dynamic critical exponent z is studied for two-dimensional superconducting, superfluid, and Josephson junction array systems in zero magnetic field via the Fisher-Fisher-Huse dynamic scaling. We find $z\ensuremath{\simeq}5.6\ifmmode\pm\else\textpm\fi{}0.3$, a relatively large value indicative of nondiffusive dynamics. Universality of the scaling function is tested and confirmed for the thinnest samples. We discuss the validity of the dynamic scaling analysis as well as the previous studies of the Kosterlitz-Thouless-Berezinskii transition in these systems, the results of which seem to be consistent with simple diffusion (z=2)(z=2)(z=2). Further studies are discussed and encouraged.
We revisit the scaling properties of the resistivity and the current-voltage characteristics at and below the Berezinskii-Kosterlitz-Thouless transition, both in zero and nonzero magnetic fields. The scaling properties are derived by integrating the renormalization group flow equations up to a scale where they can be reliably matched to simple analytic expressions. The vortex fugacity turns out to be dangerously irrelevant for these quantities below Tc{T}_{c}Tc​, thereby altering the scaling behavior. We derive the possible crossover effects as the current, magnetic field, or system size is varied, and find a strong multiplicative logarithmic correction near Tc{T}_{c}Tc​, all of which is necessary to account for when interpreting experiments and simulation data. Our analysis clarifies a longstanding discrepancy between the finite size dependence found in many simulations and the current-voltage characteristics of experiments. We further show that the logarithmic correction can be avoided by approaching the transition in a magnetic field, thereby simplifying the scaling analysis. We confirm our results by large-scale numerical simulations, and calculate the dynamic critical exponent zzz, for relaxational Langevin dynamics and for resistively and capacitively shunted Josephson junction dynamics.
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.