Two-dimensional Josephson Junction Research Articles

Dynamics and chaos of a current driven two-dimensional Josephson junctions array under [formula omitted] magnetic field

We derive the equations of motion for a two-dimensional capacitive Josephson junctions array in the presence of both a DC current and a magnetic field f= 1 3 of the quantum flux Φ 0 . The ground state symmetry of an N× N array is assumed to hold for all currents, then by using the resistively and capacitively shunted junction equations, a model system of four-coupled non-linear second-order differential equations is derived. The system has the form β c x ″+ x ′+∇ U=0, where U is a four-dimensional potential and β c is the Stewart–McCumber parameter. The dynamics can be viewed as the motion of a massive particle sliding under the action of the potential in a four-dimensional configuration space with a friction proportional to its speed. There are three distinct branches: one below the critical current I c where the static zero voltage solution is stable; the second branch which originates from the static solution through a Hopf bifurcation and where a total voltage develops along the direction of the applied current and across the array (instantaneous Hall voltage), (the latter means vortices moving perpendicular to the current and constitutes a flux-flow like regime); and a third branch above the synchronization current I s , where the motion of the junctions synchronizes and the motion of the vortices ceases with zero Hall voltage. For a wide range of β c , the second branch shows chaotic dynamics of extremely rich complexity. A pervasive feature is the presence of antimonotonicity, i.e., reversals of period doubling cascades.

Investigations on phase locking in a two-dimensional Josephson junction array within the Werthamer and the RCSJ model

The synchronization properties of a simple two-dimensional Josephson array consisting of two coupled SQUID cells are studied within the Werthamer as well as the RCSJ model. Special emphasis is placed on the role of inductances arranged perpendicular and parallel to the current bias direction for the phase locking behavior. The general behavior within the Werthamer model is found to be similar to that within the RCSJ model. However, there are quantitative differences, e.g. an enhanced phase shift between the voltage oscillations within one cell and a shift of the parameter range for the in-phase regime between different cells towards lower values of the McCumber parameter in the Werthamer model.

Perturbation simulation for two-dimensional Josephson junction array with resonator structure

A numerical perturbation method to simulate phase-locking behavior for resonator structure-type Josephson junction arrays is introduced. Simulations are completed for one-dimensional parallel, one-dimensional series, and two-dimensional arrays showing new and interesting phenomena. The simulation results provide a clear view of the processes by which the Josephson junctions become mutually phase-locked. Optimum structure parameters for oscillator circuit designs are determined using these simulation results.

Cherenkov Radiation of Vortices in a Two-Dimensional Annular Josephson Junction

A two-dimensional annular Josephson junction is considered in the framework of the s2 1 1dD sine-Gordon equation. By numerical simulations it has been found that once vortex velocity exceeds the critical value a resonance excitation of linear traveling waves appears. The critical velocity of vortices and the radiation field structure are determined. The current-voltage characteristic of such a junction is presented. It has been shown analytically that this effect is brought about by the Cherenkov synchronism between linear waves of the whispering mode type and moving vortices. All analytical results obtained in this study are in good agreement with the numerical experiment data. In conclusion, a possibility of using the discovered effect for generation of microwave radiation is discussed. [S0031-9007(98)05755-X]

How to achieve in-phase locking in small-inductance Josephson junction ladder arrays

We present the results of an analytical study of phase locking in externally loaded two-dimensional Josephson junction ladder arrays with small, but non-vanishing ring inductances. A Lyapunov stability based condition is found controlling realization of the radiating in-phase oscillation regime.

Erratum: Disorder in two-dimensional Josephson junctions [Phys. Rev. B55, 14 499 (1997)

An effective free energy of a two dimensional (i.e. large area) Josephson Junctions is derived, allowing for thermal fluctuations, for random magnetic fields and for external currents. We show by using replica symmetry breaking methods, that the junction has four distinct phases: disordered, Josephson ordered, a glass phase and a coexisting Josephson order with the glass phase. Near the coexistence to glass transition at s=1/2 the critical current is ~ (area)^{-s+1/2} where s is a measure of disorder. Our results may account for junction ordering at temperatures well below the critical temperature of the bulk in high T_c trilayer junctions.

Duality in Josephson junction arrays

Various properties of mesoscopic two-dimensional Josephson junction arrays are reviewed. Particular attention is paid to structure of the topological excitations, charges and vortices, which are shown to be dual to each other. This duality persists in the presence of external magnetic fields and offset charges, which influence vortices and charges in an equivalent way. A double-layer junction array is also considered, where an even further reaching duality is discovered.

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.

Disorder in two-dimensional Josephson junctions

An effective free energy of a two-dimensional (i.e., large area) Josephson junction is derived, allowing for thermal fluctuations, random magnetic fields, and external currents. We show by using replica-symmetry-breaking methods that the junction has four distinct phases: disordered, Josephson ordered, a glass phase, and a coexisting Josephson order with the glass phase. Near the coexistence to glass transition at s= the critical current is \ensuremath{\sim}( area${)}^{\mathrm{\ensuremath{-}}\mathrm{s}+1\mathrm{/}2}$ where s is a measure of disorder. Our results may account for junction ordering at temperatures well below the critical temperature of the bulk in high-${\mathrm{T}}_{\mathrm{c}}$ trilayer junctions.

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.

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.

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.

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.

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.

Comparative dynamics of two-dimensional shorted arrays and continuous stacked Josephson junctions

Multilayer structures of Josephson junctions are discussed both in the continuous and the discrete case. For the continuous case some recent results are shown. For two-dimensional shorted arrays, which account for the discrete limit, a model is presented. Analytical and numerical calculations show typical features of coupled sine-Gordon systems such as the splitting of the limit velocity and modes of different symmetry which correspond to the different velocities. The discretization, however, adds interesting features, the most prominent of them being a strong interaction between solitons and discreteness-induced plasma waves. {copyright} {ital 1997} {ital The American Physical Society}

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.

2D Josephson junction

The paper is devoted to the determination of processes in the two-dimensional Josephson junction via numerical solution of 2+1 dimensional sine-Gordon equation (sGe) with suitably chosen boundary conditions.