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arXiv:cond-mat/0508163 · Journal of Statistical Mechanics Theory and Experiment

Simulation results for an interacting pair of resistively shunted Josephson junctions

Philipp Werner, Gil Refael, Matthias Troyer

Published Dec 5, 2005• 8 citations
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Abstract

Using a new cluster Monte Carlo algorithm, we study the phase diagram and critical properties of an interacting pair of resistively shunted Josephson junctions. This system models tunnelling between two electrodes through a small superconducting grain, and is described by a double sine-Gordon model. In accordance with theoretical predictions, we observe three different phases and crossover effects arising from an intermediate coupling fixed point. On the superconductor-to-metal phase boundary, the observed critical behaviour is within error-bars the same as in a single junction, with identical values of the critical resistance and a correlation function exponent which depends only on the strength of the Josephson coupling. We explain these critical properties on the basis of a renormalization group (RG) calculation. In addition, we propose an alternative new mean-field theory for this transition, which correctly predicts the location of the phase boundary at intermediate Josephson coupling strength.

DOI
10.1088/1742-5468/2005/12/p12003

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Using a new cluster Monte Carlo algorithm, we study the phase diagram and critical properties of an interacting pair of resistively shunted Josephson junctions. This system models tunnelling between two electrodes through a small superconducting grain, and is described by a double sine-Gordon model. In accordance with theoretical predictions, we observe three different phases and crossover effects arising from an intermediate coupling fixed point. On the superconductor-to-metal phase boundary, the observed critical behaviour is within error-bars the same as in a single junction, with identical values of the critical resistance and a correlation function exponent which depends only on the strength of the Josephson coupling. We explain these critical properties on the basis of a renormalization group (RG) calculation. In addition, we propose an alternative new mean-field theory for this transition, which correctly predicts the location of the phase boundary at intermediate Josephson coupling strength.
We present a cluster algorithm for resistively shunted Josephson junctions or similar physical systems, which dramatically improves sampling efficiency, and apply it to the superconductor-to-metal transition in a single junction. Measuring the temperature dependence of the zero bias resistance, we confirm that the critical point does not depend on the strength of the Josephson coupling. However, we find that the correlation exponents vary continuously along the phase boundary, indicating that the Schmid-Bulgadaev transition is a line of fixed points.
Based on the two-band free-energy functional, we have investigated phase transitions of a three-dimensional granular superconductor through the Monte Carlo method and finite-size scaling analysis. In the ordered lattices for this superconductor consisting of Josephson-coupled grains, we obtain an intermediate phase above the critical intraband coupling strength while maintaining a constant interband coupling coefficient. This phase is distinct from both the low-temperature superconducting phase and the high-temperature normal phase, characterized by the absence of long-range phase coherence solely for the first band. This numerical prediction can be confirmed through magnetic susceptibility calculations, and the phase diagram is qualitatively consistent with the mean-field theoretical analysis. Additionally, we have considered the effects of interband interaction between the nearest-neighbor grains and random disorders in diluted lattices with three different site occupancies. When the occupation probability exceeds the percolation threshold, the results are qualitatively similar to those of ordered lattices, and the intermediate phase likewise appears by adjusting the intraband Josephson coupling coefficient. In contrast, below the percolation threshold, such a phase does not exist because the long-range phase coherence for both bands disappears.
We study the proximity effect in the Hubbard model coupled to BCS superconductors describing a single-layer strongly correlated electron system in a phase-biased Josephson junction. We find two distinct gapped solutions, a Mott-like insulating (M phase) and a proximitized superconducting phase (S phase), separated by a first-order transition with hysteresis. In the M phase, the large correlation-induced charge gap strongly suppresses the critical current, while the S phase behaves as a 0-junction, with a proximitized gap that closes for ϕ = π to yield a correlated metal. Phase bias and junction transparency can thus serve as tuning knobs to switch between conducting and insulating regimes. Working within the dynamical mean-field theory using the numerical renormalization group as the impurity solver, we associate M- and S-phase solutions with the doublet and singlet fixed points of the underlying superconducting Anderson impurity problem. We obtain detailed insight into the spectral structure on all energy scales. In the M phase, the self-energy has sub-gap resonances symmetrically located around the Fermi level resulting from the splitting of the “mid-gap pole” found in Mott insulators; this structure accounts for phase insensitivity.
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 ppp are considered, where pc{p}_{c}pc​ is the percolation threshold. For p>pcp>{p}_{c}p>pc​ and zero temperature, the $I\ensuremath{-}V$ 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 ${T}^{1+{\ensuremath{\nu}}_{T}},$ with a thermal-correlation length exponent ${\ensuremath{\nu}}_{T}$ consistent with the corresponding value for the diluted XY\mathrm{XY}XY model at pc.{p}_{c}.pc​.
Double-barrier Josephson junctions, with outer electrodes of superconducting Nb and a central thin layer of non-superconducting Al, are being developed as self-shunted junctions for high-frequency superconducting integrated circuits. This paper presents a simplified phenomenological model of the behaviour of such junctions (often called SINIS, for superconductor-insulator-normal-insulator-superconductor) based on the resistively shunted junction model, using Josephson coupling between the layers and the time-dependent Ginzburg-Landau equation to describe the dynamics of the weak induced energy gap in the Al layer. This analysis shows how non-hysteretic I - V characteristics can be obtained but suggests some limitations in their high-frequency performance. Predictions for critical current generally agree with prior experiments. The technological implications of this model are discussed.