Large-area Josephson Junctions Research Articles

The role of magnetic fields for curvature effects in Josephson junction

The large area Josephson junction is considered. On the basis of Maxwell equations the influence of the magnetic field on fluxion dynamics is considered. The presented studies show that assumptions presumed in the literature do not restrict experimental settings adopted in the considerations of the fluxion movement in the Josephson junction. It is shown that the particular orientation of the magnetic fields is not needed in order to study physical effects of curvature and therefore they do not restrict the experimental arrangements.

Use of Pd–Fe and Ni–Fe–Nb as Soft Magnetic Layers in Ferromagnetic Josephson Junctions for Nonvolatile Cryogenic Memory

Josephson junctions containing ferromagnetic layers are under consideration as the basic elements for cryogenic random access memory. For memory applications, either the amplitude of the critical current or the phase shift across the junction must be controllable by changing the direction of magnetization of one or more of the ferromagnetic layers in the junction. We have measured the critical currents in large-area Josephson junctions containing three ferromagnetic layers. These junctions carry spin-triplet supercurrent. This work addresses the choice of material and optimum thickness for the one soft magnetic layer in such junctions. We have used either a Pd-Fe or Ni-Fe-Nb alloy for the soft layer, and find hysteresis in the low-field “Fraunhofer patterns” due to magnetic switching of the soft layer. The critical current is one order of magnitude smaller in the junctions containing the Ni-Fe-Nb alloy compared to those containing Pd-Fe alloy, which is probably due to strong spin-memory loss in the former. While the large-area junctions studied here are not suitable for memory applications, these experiments lay the groundwork for future studies of submicron junctions where the magnetic state of the junction can be controlled by shape anisotropy.

Possible curvature effects in the Josephson junction

The large area Josephson junction with one compact dimension is considered. A geometry that increases the velocity of the kink propagating along the junction is considered. The geometry described in this paper can be used in order to increase or decrease the speed of the signal in the transmission elements in electronic devices. Reflection of the kink front from the large curvature areas is also confirmed.

The dynamics of the kink in curved large area Josephson junction

A formalism that allows description of the kink motion in an arbitrarily curved large area Josephson junction is proposed. A general formula for the lagrangian density that describes the curved Josephson junction, in small curvature regime, is obtained. Examples of propagation of the kink along the curved Josephson junction are considered.

Magnetic field of Josephson vortices outside superconductors

We consider the structure of Josephson vortices approaching the junction boundary with vacuum in large-area Josephson junctions with the Josephson length {lambda}{sub J} large relative to the London penetration depth {lambda}{sub L}. Using the stability argument for one-dimensional solitons with respect to two-dimensional perturbations, it is shown that on the scale {lambda}{sub J}, the Josephson vortices do not spread near the boundary in the direction of the junction. The field distribution in vacuum due to the Josephson vortex is evaluated, the information needed for the scanning superconducting quantum interference device microscopy.

An explicit numerical scheme for the Sine-Gordon equation in 2+1 dimensions

The paper presents an explicit finite-difference method for the numerical solution of the Sine-Gordon equation in two space variables, as it arises, for example, in rectangular large-area Josephson junction. The dispersive nonlinear partial differential equation of the system allows for soliton-type solutions, an ubiquitous phenomenon in a large-variety of physical problems. The method, which is based on fourth order rational approximants of the matrix-exponential term in a three-time level recurrence relation, after the application of finite-difference approximations, it leads finally to a second order initial value problem. Because of the existing sinus term this problem becomes nonlinear. To avoid solving the arising nonlinear system a new method based on a predictor-corrector scheme is applied. Both the nonlinear method and the predictor-corrector are analyzed for local truncation error, stability and convergence. Numerical solutions for cases involving the most known from the bibliography ring and line solitons are given. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme

This paper proposes a split cosine scheme for simulating solitary solutions of the sine-Gordon equation in two dimensions, as it arises, for instance, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation allows for soliton-type solutions, a ubiquitous phenomenon in a large variety of physical problems. The semidiscretization approach first leads to a system of second-order nonlinear ordinary differential equations. The system is then approximated by a nonlinear recurrence relation which involves a cosine function. The numerical solution of the system is obtained via a further application of a sequential splitting in a linearly implicit manner that avoids solving the nonlinear scheme at each time step and allows an efficient implementation of the simulation in a locally one-dimensional fashion. The new method has potential applications in further multi-dimensional nonlinear wave simulations. Rigorous analysis is given for the numerical stability. Numerical demonstrations for colliding circular solitons are given.

Optical solitons and quantum solitons

We return to the notion of optical solitons as these were first introduced—namely as exact soliton and multi-soliton solutions of the semi-classical self-induced transparency (SIT) equations. The SIT equations, even in their most general inhomogeneously broadened forms, are infinite dimensional, completely integrable Hamiltonian systems, and thus display much mathematical structure, yet their exact soliton solutions display well the nonlinear (self-induced) transparency features of the physical phenomenon. The mathematical feature of complete integrability is handed down from the SIT equations to the nonlinear Schrodinger equations and from this point of view the nonlinear Schrodinger (NLS) equations equations are studied first of all at classical level and then at quantum level. The 'quantum soliton' of the quantum attractive NLS model is defined and its properties investigated in detail, particularly in a comparison with the particular properties ascribed to the 'quantum soliton' as described in a recently published article (Abraham 1999 Phys. World (February) 21) with this title; as suggested in that article, the quantum NLS soliton is seen as a qubit for quantum information purposes. The covariant form of the quantum attractive NLS model is the quantum sine-Gordon model and this is briefly studied in the context of quantum information also; new results for large area Josephson junctions are reported in this context. The quantum soliton of the one dimensional attractive NLS model has apparently been observed in 2002 in a Bose condensed metal vapour (7Li, (Khaykovich et al 2002 Science 296 1290)) and a detailed calculation suggests that the condensate undergoes a 'collapse' from three to one space dimensions thus forming an exact soliton solution of the one dimensional quantum attractive NLS model.

Phase-locking Josephson junctions arrays

We demonstrate that a large area Josephson junction oscillating in the fluxon oscillator mode can be synchronized to other large junctions and simultaneously pump, by emitted radiation, a small area junction. We study the synchronization of the oscillations of the long junctions as a function of relevant experimental parameters such as bias current and stripline coupling characteristics. Experimental results obtained on coupled series arrays of Josephson junctions designed on the basis of our calculations are presented. We have fabricated coupled arrays containing each up to 1500 junctions in order to estimate the usefulness of our calculations for voltage standard devices.

Control of chaotic patterns in a Josephson junction model

The effect of an applied rf signal on the dynamics of a large-area Josephson junction is examined. The problem of controlling spatiotemporal chaotic patterns induced by the external magnetic field is addressed. Chaos control is conducted by a weak spatially distributed force.

An inverse convergence approach for arguments of Jacobian elliptic functions

Computing the value of the Jacobian elliptic functions, given the argument u and the parameter m, is a problem, whose solution can be found either tabulated in tables of elliptic functions [1] or by use of existing software, such as Mathematica, etc. The inverse problem, finding the argument, given the Jacobian elliptic function and the parameter m, is a problem whose solution is found only in tables of elliptic functions. Standard polynomial inverse interpolation procedures fail, due to ill conditioning of the system of the unknowns. In this paper, we describe a numerical procedure based on the convergence of the unknowns of the solution, by the use of arithmetical method, as an alternative way of solving the problem. The method gives very good results with no significant error, in the computation of the argument of the Jacobian elliptic function given the Jacobian elliptic function and the parameter. This new procedure is important in problems involving cavities or inclusions of ellipsoidal shape encountered in the mechanical design of bearings, filters, and composite materials. They are also important in the modeling of porosity of bones. This porosity may lead to osteoporosis, a disease which affects bone mineral density in humans with bad consequences. Also these procedures are of importance in problems encountered in the physics discipline such as in the analysis of the dependence of the maximum tunneling current on external magnetic field for large area Josephson junctions with overlap boundary conditions.

Split Mode Method for the Elliptic 2D Sine-Gordon Equation: Application to Josephson Junction in Overlap Geometry

We introduce a new type of splitting method for semilinear partial differential equations. The method is analyzed in detail for the case of the two-dimensional static sine-Gordon equation describing a large area Josephson junction with overlap current feed and external magnetic field. The solution is separated into an explicit term that satisfies the one-dimensional sine-Gordon equation in the y-direction with boundary conditions determined by the bias current and a residual which is expanded using modes in the y-direction, the coefficients of which satisfy ordinary differential equations in x with boundary conditions given by the magnetic field. We show by direct comparison with a two-dimensional solution that this method converges and that it is an efficient way of solving the problem. The convergence of the y expansion for the residual is compared for Fourier cosine modes and the normal modes associated to the static one-dimensional sine-Gordon equation and we find a faster convergence for the latter. Even for such large widths as w=10 two such modes are enough to give accurate results.

Bound states of the elliptic sine-Gordon equation

We give a detailed study of finite-energy solutions to elliptic sine-Gordon (SG) equation in the plane with point-like singularities. These bound-state solutions (in a sense of scalar field theory) with only one singularity at the origin demonstrate a target-like annular soliton pattern at large distance from the origin. An effective radius of this pattern is calculated both analytically and numerically for the case of axial symmetric solutions. The analytic study is based on an isomonodromic deformation method for the third Painlevé equation, which distinguishes bound-state solutions as separatrices in a manifold of general (infinite-energy) solutions. Exact analytic estimates give us a tool to study bounded-state solutions to the non-integrable SG equation with forcing. Namely, for large intensity at the singularity we derive a critical value of forcing, which governs the existence and stability of the bound-state solutions. This plays a crucial role for two concrete physical applications dealing with large area Josephson junctions and nematic liquid crystals in a rotating magnetic field. For both examples we compute critical values of field and driving forces which enables the formation of modes with finite energy. These numerical computed critical values correlate well with computer simulations and experimental data.

Bound state solutions of the elliptic sine-Gordon equation

We present a detailed investigation of finite-energy solutions with point-like singularities of the elliptic sine-Gordon equation in a plane. Such solutions are of the bound-state type in the sense of scalar field theory. If the solution has a unique singularity, then it behaves as a soliton-like annular wave packet at a large distance from the singularity. The effective radius of this wave packet is evaluated both analytically and numerically for axially symmetric solutions. The analytical investigation is based on the method of isomonodromy deformations for the third Painleve equation, which singles out these solutions as separatrices of the manifold of general solutions (with infinite energy). Exact analytical estimates provide a tool for investigating bound-state solutions of the nonintegrable sine-Gordon equation with a nonzero right-hand side. More precisely, for large-intensity fields at the singularity, we derive the critical forcing that allows the existence and stability of a bound state. As an illustration, we consider two applications: a large-area Josephson junction and a nematic liquid crystal in a rotating magnetic field. For each of the examples, we evaluate the critical values of the field that allow finite-energy regimes. These are in good agreement with numerical and experimental data.

Two-dimensional effects in Josephson junctions: Static properties.

We studied the dependence of the maximum tunneling current on external magnetic field for large area Josephson junctions with overlap boundary conditions. We used direct numerical solutions and developed a split Fourier mode method to study the electromagnetic behavior. The steady-state pattern consists of two terms, the first of which satisfies the in-line-like bias current boundary conditions and has zero fluxon content. The second is treated in a mode expansion in the y direction and only two terms are sufficient to reproduce the direct solution of the two-dimensional junction for widths up to 2 p times the Josephson characteristic length. @S1063-651X~96!08007-5#

Stability analysis of a two-dimensional uniaxial vortex glass.

The simplest model of a vortex glass is considered which is applicable for the description of a twodimensional uniaxial vortex crystal formed by the fluxon lines in a large area Josephson junction with inhomogeneous width. The analysis is performed in replica representation in terms of a free-energy functional which depends on the renormalized correlation function. The properties of different solutions of the Dyson equation are considered, the main attention being devoted to investigating the stability of these solutions. In particular the solution with the one-step replica symmetry breaking which corresponds to the absolute maximum of free energy is shown to be always stable ~when it exists at all!. The unimportance of higher-order corrections for the form of the asymptotic behavior of the correlation function in the phase with the one-step replica symmetry breaking is also demonstrated. I. INTRODUCTION The discovery of the high-Tc superconconducting materials have essentially increased the possibilities for the experimental observations of various phenomena related with the presence of vortices in superconductors ~vortex lattice melting, pinning, creep and so on!. This has led to the active development of theoretical investigation of these phenomena and the appearance of many new ideas ~for a recent review see Ref. 1!. In particular a suggestion has been made that at low temperatures a phase should exist in which the motion of the vortices is quenched by disorder and therefore the linear resistance is absent. 2,3 The properties of this phase ~including the multitude of metastable states separated by the diverging barriers! are expected to be more or less analogous to those of widely discussed infinite-range spin-glass models 4 and therefore it is usually called a vortex glass. It has been suggested 2,5 that the simplest model which allows one to analyze the large scale properties of a vortex crystal ~or charge-density wave! interacting with a random potential can be described by the Hamiltonian

Chaotic patterns in a Josephson junction model.

The effect of an applied rf signal on the dynamics of a large area Josephson junction is examined by means of a model based on the sine-Gordon equation. In particular the problem of characterizing spatiotemporal chaotic patterns induced by the external harmonic varying magnetic field is addressed. In general it is found that instantaneous spatiotemporal patterns, i.e., patterns confined to one period of oscillation of the external field correlates poorly. Also the correlation between instantaneous patterns and average patterns is poor. However, the correlation between patterns obtained as averages over, e.g., ten or more consecutive instantaneous spatiotemporal patterns is high. Furthermore, averaged spatiotemporal patterns obtained for different values of the magnetic field amplitude correlate well. The average potential and kinetic energies of spatiotemporal patterns are calculated. In general the energies in the chaotic regions are lower than in regions with periodic response.

DYNAMICS OF ROTATIONALLY SYMMETRIC SINE-GORDON SOL1TONS WITH APPLLCATIONS FOR A LARGE AREA JOSEPHSON JUNCTION

The dynamics of rotationally symmetric sine Gordon solitons of large radius under poten-tial and dissipative perturbations is considered, with the help of a collective-coordinate desc-ription of the soliton and the Lagrangian variation method. The particle-like character of the soliton is emphasized The equation of motlon and in particular, the general momentum of the soliton are obtained in a canonical manner. It is possible to discuss the dynamics of the soliton even without applying the explicit form of the perturbating potential All dyna-mical regimes in the phase space are explored. Such phenomena as the soliron return effect, soliton escaping and saddle point are addressed In the presence of dissipation, the corrected equation of soliton motion is obtained from the generalized variational equation. Finally, the application of the theoretical treatment is considered for the fluxon dynamics in a circularly symmetric Josephson junction. The analytical results are examined by direct numerical simu-lations, fairly good agreements being achieved. it turns out that the presented ueaMnem provides a reliable description of the dynamics of the sol tons considered.

Dynamics of rotationally symmetric solitons in the near-SG field model with application to fluxon motion in a large area Josephson junction: a collective-coordinate description

The dynamics of the ( n+1)-dimensional rotationally symmetric sine-Gordon solitons of large radius under potential perturbations and dissipative pertubations as well, is considered by means of a perturbation treatment based on a collective-coordinate description of the solution and the Lagrangian variational method. The particle-like character of a soliton is emphasized. The equation of soliton motion and, in particular, the generalized momentum of the soliton are obtained in a canonical manner. All kinds of dynamical regimes in the phase space that correspond to various initial conditions are explored. In the presence of dissipation, the corrected equation of soliton motion is obtained from the generalized variation method. Finally the application of the theoretical treatment is considered for the solitons (or fluxons) in a circularly symmetric Josephson junction, and the analytical results are examined by direct numerical simulations with fairly good agreement.

Finite element approximation to two-dimensional sine-Gordon solitons

The paper presents a finite element algorithm for the numerical solution of the sine-Gordon equation in two spatial dimensions, as it arises, for example, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation of the system allows for soliton-type solutions, an ubiquitous phenomenon in a large variety of physical problems. A semidiscrete Galerkin approach based on simple four-noded bilinear finite elements in combination with a generalized Newmark integration scheme is used throughout the paper and is tested in a variety of cases. Comparisons with finite difference solutions show the superior performance of the proposed algorithm leading to very accurate, numerically stable and physically consistent solitary wave solutions. The results support the confidence in the present numerical model which should be capable to treat also more complex situations involving soliton-type interactions.