Two-dimensional Josephson Junction Arrays Research Articles

Field distribution in thin superconductors: effect of sample shape

Flux distribution in a finite-sized thin superconductor in transverse magnetic field is strongly influenced by the sample shape. This is investigated by direct numerical simulation of the field distribution in a two-dimensional Josephson junction array (JJA). The effect of non-local interaction of fields and currents in thin continuum superconductor is considered by including the mutual inductance between all pairs of plaquettes in the array. In small applied fields, for a rectangular array, the flux penetration occurs predominantly through the middle of the edges. The line originating from the corners (discontinuity lines) at which the flux distribution shows sharp ridges, form strong barriers to the flux-motion, gives rise to independent domains. In large external field (field greater than that requred for full-penetration of the array), the low-field distribution crosses over to a new kind in which vortices accumulate close to the discontinuity lines. This field distribution closely resembles the ‘mountain-pass’ type, observed by Brawner et al. [Nature 358, (1992) 567]. A qualitative explanation to this is presented by considering the influence of the discontinuity lines on the penetrating vortices. Effect of inner boundaries on the flux distribution in a multiply-connected geometry is studied by simulating the square array with a central hole in transverse magnetic field. The field-distribution is influenced by the interaction of the discontinuity lines originating from the external and internal boundary which depends upon the dimension and orientation of the central hole region.

Theory of phase locking in simple two-dimensional Josephson junction arrays with small inductances

Several elementary two-dimensional Josephson junction arrays have been investigated mainly analytically with the special emphasis on phase locking. For this purpose, a novel perturbation scheme appropriate for the handling of small loop inductances has been developed and applied to (i) a single SQUID cell, (ii) two cells coupled by an inductive line, and (iii) an externally loaded two-cell configuration with more general internal shunt. While internal in-phase locking within the cells can easily be achieved as long as the normalized ring inductances are small compared to unity, the inter-cell coupling tends to favor an antiphase regime. A mechanism is proposed for circumventing this problem and the boundaries separating both regimes are determined. The influence of an external load is considered as well.

Phase-locked one- and two-dimensional Josephson-junction arrays as millimeter and submillimeter wave generators

The results of numerical simulations of phase-locked 1D and 2D Josephson-junction arrays are reported. It is shown that the multi-junction structures with non-local junction interaction provide phase-locked oscillation state within wide critical current margins /spl Delta/I/sub c/ up to 40...50%. The effective non-local interaction in 2D structure based on 4-junction interferometer cell can be provided by the regular single flux quantum array flow across it. The wide tolerable spread in /spl Delta/I/sub c/ allows one to realize both (i) phase-locked 2D high-T/sub c/ Josephson-junction array with the oscillation frequency F close to characteristic value F/sub c/ and (ii) based on Nb tunnel Josephson-junction technology 1D array with the voltage-controlled phase-locked oscillation frequency within wide frequency band from 0.4 F/sub c/ to 1.2 F/sub c/.

High-frequency properties of two-dimensional Josephson junction arrays

Two-dimensional arrays of Josephson junctions have been shown to operate in a state in which the microwave oscillations of the junctions are phase-coherent. However, the performance of an array is strongly dependent on both the details of the load-coupling circuit and the characteristic parameters of the junctions. We have measured two-dimensional arrays of resistively shunted Josephson tunnel junctions which are capacitively coupled to a nearby detector junction. Changing the critical current of the junctions in our arrays, while keeping the junction parasitics and array coupling roughly the same, results in an unexpected change in device operation. We have also investigated variations of the coupling circuit. Numerical simulations of the detection circuit enable us to estimate the power output from the array and to optimize the parameters of the system.

Intrinsic mechanism of phase locking in two-dimensional Josephson junction networks in presence of an external magnetic field

We present numerical simulations of the dynamics of two-dimensional Josephson junction arrays to study the mechanism of mutual phase locking. We show that in the presence of an external magnetic field two mechanisms are playing a role in phase locking: feedback through the external load and internal coupling between rows due to microwave currents induced by the field. We have found the parameter values (junction capacitance, cell loop inductance, impedance of the external load) for which the interplay of both these mechanisms leads to the in-phase solution. The case of unloaded arrays is discussed as well.

Detailed investigation of two-dimensional Josephson junction array circuits

We have investigated two-dimensional arrays of overdamped Josephson junctions that have been fabricated in Nb/Al/sub 2/O/sub 3//Nb-technology, together with microwave coupling structures. Different arrays have shown to emit coherent microwave radiation by using microwave-coupled on-chip detectors. Low-temperature scanning electron microscopy is used to study the coupling of the microwave radiation to an on-chip load. From the appearance of Shapiro steps in the current vs. voltage curves of detector junctions for different degrees of disorder in the array-junctions, margins of the spread in the critical currents of the Josephson junctions for coherent oscillation can be drawn.

Capacitance-induced in-phase switching in a basic two-dimensional Josephson junction array

Phase locking in a basic two-dimensional Josephson junction array consisting of two coupled SQUIDs is studied exploiting a new method combining the ideas of harmonic balance with the phase-slip procedure. This permits not only including the effect of non-vanishing junction capacitance, but extending our earlier investigations of the same circuit to arbitrary ring inductances, including inductances l ≈ 1 as well. As a result, our earlier finding that only antiphase oscillations can be stable in such a circuit are modified by observing a transition to the in-phase regime for larger values of the junction capacitance. The transition between both regimes is found to be determined by the resonance condition between ring inductance and junction capacitance.

Helicity modulus, quantum fluctuations, and superconducting transition in a Josephson junction, array

The effect of quantum fluctuations on a phase transition in a two-dimensional Josephson junction array is studied in terms of the two-dimensional XY model. A self-consistent harmonic approximation is used to calculate the linear response of the system to a perturbation by a uniform phase gradient (helicity modulus γ) as a function of dimensionless temperature Θ and of a quantum parameter q appearing due to the finite capacitance of each junction. Calculation of this quantity has permitted us to find the dependence of the Kosterlitz-Thouless transition temperature on q, which within a broad range of q variation is in agreement with the results of a quantum Monte Carlo simulation.

Fast Algorithms for Triangular Josephson Junction Arrays

We develop fast algorithms for the numerical study of two-dimensional triangular Josephson junction arrays. The Dirac bra-ket formalism is introduced in the context of such arrays. We note that triangular arrays can have both hexagonal and rectangular periodicity and develop algorithms for each. Boundaries are next introduced and fast algorithms for finite arrays are developed.

Complex impedance of a two-dimensional Josephson junction array

Simulations on the two-dimensional XY model are performed. The results are compared to experiments on a Josephson junction array for the case of a triangular lattice well below the Kosterlitz-Thouless temperature in a weak perpendicular magnetic field. Comparisons between experiments, existing theories and the simulations suggest that the characteristic features of the complex impedance seen in the experiments are due to vortex correlations induced by the presence of vortices with opposite vorticity.

Two-dimensional Josephson junction arrays with dc drives: The fixed-point regime

A fixed-point analysis of two-dimensional Josephson junction arrays subject to dc drives is carried out analytically, at a zero temperature and in a zero magnetic field. Conditions for the existence and stability of such behavior are established, by studying in detail, the isolated triangular and square plaquette. The results are subsequently generalized to Josephson junction ladders and rectangular arrays which could contain linear defects. We deduce a closed form inductive expression for the critical surface in the space of phase configurations. All the analytic results are checked against numerical simulations.

Magnus force in superfluids and superconductors

The forces on the vortex, transverse to its velocity, are considered. In addition to the superfluid Magnus force from the condensate (superfluid component), there are transverse forces from thermal quasiparticles and external fields violating the translational invariance. The forces between quasiparticles and the vortex originate from interference of quasiparticles with trajectories on the left and on the right from the vortex like similar forces for electrons interacting with the thin magnetic-flux tube (the Aharonov-Bohm effect). These forces are derived in the Born approximation for phonons from the equations of superfluid hydrodynamics, and for BCS quasiparticles from the Bogolyubov-de Gennes equations. The effect of external fields violating translational invariance is analyzed for vortices in the two-dimensional Josephson junction array. The symmetry analysis of the classical equations for the array shows that the total transverse force on the vortex vanishes. Therefore the Hall effect which is linear in the transverse force is absent also. This means that the Magnus force from the superfluid component exactly cancels with the transverse force from the external fields. The results of other approaches are also brought together for discussion.

Theory of phase-locking in generalized hybrid Josephson-junction arrays

A recently proposed scheme for the analytical treatment of the dynamics of two-dimensional hybrid Josephson junction arrays is extended to a class of generalized hybrid arrays with ''horizontal'' shunts involving a capacitive as well as an inductive component. This class of arrays is of special interest, because the internal cell coupling has been shown numerically to favor in-phase synchronization for certain parameter values. As a result, we derive limits on the circuit design parameters for realizing this state. In addition, we obtain formulas for the flux-dependent frequency including flux-induced switching processes between the in-phase and anti-phase oscillation regime. The treatment covers unloaded arrays as well as arrays shunted via an external load.

Critical properties of two-dimensional Josephson-junction arrays with zero-point quantum fluctuations.

We present results from an extensive analytic and numerical study of a two-dimensional model of a square array of ultrasmall Josephson junctions. We include the ultrasmall self and mutual capacitances of the junctions, for the same parameter ranges as those produced in the experiments. The model Hamiltonian studied includes the Josephson, $E_J$, as well as the charging, $E_C$, energies between superconducting islands. The corresponding quantum partition function is expressed in different calculationally convenient ways within its path-integral representation. The phase diagram is analytically studied using a WKB renormalization group (WKB-RG) plus a self-consistent harmonic approximation (SCHA) analysis, together with non-perturbative quantum Monte Carlo simulations. Most of the results presented here pertain to the superconductor to normal (S-N) region, although some results for the insulating to normal (I-N) region are also included. We find very good agreement between the WKB-RG and QMC results when compared to the experimental data. To fit the data, we only used the experimentally determined capacitances as fitting parameters. The WKB-RG analysis in the S-N region predicts a low temperature instability i.e. a Quantum Induced Transition (QUIT). We carefully simulations and carry out a finite size analysis of $T_{QUIT}$ as a function of the magnitude of imaginary time axis $L_\tau$. We find that for some relatively large values of $\alpha=E_C/E_J$ ($1\leq \alpha \leq 2.25)$, the $L_\tau\to\infty$ limit does appear to give a {\it non-zero} $T_{QUIT}$, while for $\alpha \ge 2.5$, $T_{QUIT}=0$. We use the SCHA to analytically understand the $L_\tau$ dependence of the QMC results with good agreement between them. Finally, we also carried out a WKB-RG analysis in the I-N region and found no evidence of a low temperature QUIT, up to lowest order in ${\alpha}^{-1}$

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.

Antiphase locking in a two-dimensional Josephson junction array

We consider theoretically phase locking in a simple two-dimensional Josephson junction array consisting of two loops coupled via a joint line transverse to the bias current. Ring inductances are supposed to be small, and special emphasis is taken on the influence of external flux. It is shown that in the stable oscillation regime both cells oscillate with a phase shift equal to π (i.e., antiphase). This result may explain the low radiation output obtained so far in two-dimensional Josephson junction arrays experimentally.

Charge representation of a small two-dimensional Josephson-junction array in the quantum regime.

Using the charge representation, we calculate the ground state energy and the critical current of a small two-dimensional Josephson junction array subject to both charge and magnetic frustration. In the quantum regime the ground state of the array is a superposition of charge states, allowing a supercurrent to flow through the circuit. Both the ground state energy and the critical current can be tuned by the two frustrations. We show that the notion of a vortex is compatible with a charge representation of the array. \textcopyright{} 1996 The American Physical Society.

Fast algorithms for Josephson-junction arrays: Busbars and defects.

We critically review the fast algorithms for the numerical study of two-dimensional Josephson junction arrays and develop the analogy of such systems with electrostatics. We extend these procedures to arrays with busbars and defects in the form of missing bonds. The role of boundaries and of the gauge choice in determining the Green's function of the system is clarified. The extension of the Green's-function approach to other situations is also discussed. \textcopyright{} 1996 The American Physical Society.

Chaos and dynamics of vortices in Josephson junction arrays

The approach to spatiotemporal chaos in two-dimensional (2D) Josephson junction arrays (JJAs) subjected to DC current drives is reviewed. The role of vortices in describing the various transitions and types of dynamical behaviour that occur is explored. Some analogies with hydrodynamics are found.

Response of Josephson-junction arrays and granular superconductors near the superconductor-insulator transition

We study the response of two-dimensional Josephson-junction arrays and granular films of superconducting material near the superconductor-insulator transition. Close to the transition the system is described by a Ginzburg-Landau-Wilson free energy functional for the global superconducting order-parameter. We consider different situations: First we study the effect of local ohmic shunts. They yield nonohmic dynamics for the order parameter. The conductivity at the transition is nonuniversal within this model. Then we discuss a boson-fermion model which yields ohmic dynamics for the order parameter. A possible realization for this scenario is the Andreev scattering process at the surface of the superconducting grains. This model leads to a universal conductivity at the transition. Finally, in the absence of damping we evaluate the universal conductivity in an ϵ-expression. The result is in good agreement with existing Monte Carlo data.