Current-voltage scaling of a Josephson-junction array at irrational frustration.

Numerical simulations of the current-voltage characteristics of an ordered two-dimensional Josephson junction array at an irrational flux quantum per plaquette are presented. The results are consistent with an scaling analysis which assumes a zero temperature vortex glass transition. The thermal correlation length exponent characterizing this transition is found to be significantly different from the corresponding value for vortex-glass models in disordered two-dimensional superconductors. This leads to a current scale where nonlinearities appear in the current-voltage characteristics decreasing with temperature T roughly as T 2 in contrast with the T 3 behavior expected for disordered models. 74.50.+r, 74.60.Ge, 64.60.Cn Typeset using REVTEX 1 There has been an increasing interest in two-dimensional Josephson-junction arrays as a model system to study disorder and frustration effects as found in high-Tc superconductors and spin-glass systems. In Josephson-junction arrays, frustration without disorder can in principle be introduced by applying an external magnetic field on a perfect periodic array. The frustration parameter f , the number of flux quantum per plaquette, is given by f = φ/φo, the ratio of the magnetic flux through a plaquette φ to the superconducting flux quantum φo = hc/2e, and can be tuned by varying the strength of the external field. Frustration can be viewed as resulting from a competition between the underlying periodic pinning potential of the array and the periodicity of the vortex lattice. At a rational value of f , the ground state is a commensurate pinned vortex lattice leading to discrete symmetries in addition to the continuous U(1) symmetry of the superconducting order parameter. In particular, for f = 1/2 which has been intensively studied both experimentally and theoretically, a superconducting phase transition takes place at finite temperatures with an interplay of U(1) and discrete Z2 symmetry . At irrational values of f , the behavior is much less understood since the vortex lattice is now incommensurate with the array. The ground state consists of a disordered vortex pattern lacking long range order which can also be regarded as a vortex-glass state without disorder. One can not completely rule out a possible glass transition at finite temperatures as has been suggested by Halsey but other arguments suggest a zero temperature transition. In any case, on the basis of a close analogy between this system and gauge glass models of disordered superconductors, one expects metastable states with long relaxation times and a nonzero critical current at zero temperature. In fact, measurements of I-V characteristics in two-dimensional superconducting wire networks at an irrational frustration have been interpreted within the scaling analysis of a vortex glass transition as in three-dimensional disordered superconductors where there is evidence of a finite-temperature glass transition. In the case of superconducting wires, the behavior is likely to be dominated by a mean field transition which should correspond to the behavior at higher dimensions. This could provide a possible explanation for the observed scaling behavior of a finite-temperature transition. On the other hand, no evidence was found in 2 experiments on two-dimensional proximity-coupled Josephson-junction arrays where phase fluctuations are expected to be more important. In disordered superconductors, studies of different models of the vortex glass tend to agree that in two dimensions a vortex glass transition takes place only at zero temperature which is supported both by numerical simulations and experiments. Although being completely different in the nature of their ground states, the similarity of the behavior of these two systems, specially regarding slow relaxation vortex dynamics and a possible zero temperature transition in low dimensions, strongly suggests that a two-dimensional array at irrational f should behave as a zero-temperature vortex glass. In this case, the appropriate scaling of the I-V characteristics should be the one corresponding to a zero temperature transition while the different nature of the ground states should be reflected in the value of the critical exponents as a different universality class. It seems therefore worthwhile to investigate to which extent an array at irrational f can be described as a zero temperature vortex glass. In this work, we present simulations of the current-voltage characteristics of a Josephson junction array at an irrational flux quantum per plaquette f = (3 − √ 5)/2, a golden irrational, and an scaling analysis which assumes a zero temperature vortex glass transition. The results are consistent with the scaling assumption and allow for an estimation of the thermal correlation length critical exponent ν characterizing the zero temperature transition. This critical exponent is found to be significantly different from the corresponding value for vortex-glass models in disordered two-dimensional superconductors. As a result, the current density scale, Jnl ∼ T 1+ν , where nonlinearities appear in the current-voltage characteristics decreases with temperature roughly as T 2 in contrast with the T 3 behavior expected for disordered models. This could provide a signature of the behavior at irrational frustration in experimental studies of ordered arrays of Josephson junctions. We considered an ordered array of Josephson junctions defined on a square lattice. The I-V characteristics of the array was computed using an overdamped Langevin moleculardynamics as described by Falo et al. One allows for a capacitance to the ground Co in addition to a shunt resistance R0 between superconducting grains. In the overdamped limit, 3 this particular dynamics reduces to the standard resistance shunt junction (RSJ) model commonly used in dynamical simulations. The Langevin equations can be written as Co dθi dt + 1 Ro ∑ j d(θi − θj) dt = −Ic ∑ j sin(θi − θj −Aij) + I i + ∑