Penetration of the magnetic field into a 3D ordered Josephson medium at very small values of the pinning parameter

The Meissner state of a 3D Josephson medium is analyzed for stability against small fluctuations of phase discontinuities at contacts. For any form of fluctuations, there exists value I ₀ of pinning parameter I such that the Meissner configuration remains stable if I < I ₀. Reasons why the configuration remains stable at small I are considered. Instability arises when the quadratic form of the second variation of Gibbs potential G is not a positively definite quantity. At small I, the contribution of the Josephson energy to G is small. The second variation of the magnetic energy, the other component of G, is always a positively definite quadratic form. Therefore, instability may arise only if I has a finite value. This statement holds true not only for the Meissner but also for any equilibrium configuration. At I < I ₀, stability persists up to the boundary of the Meissner state. Then, a sequence of plane vortices parallel to the boundary appears throughout the sample. Thus, vortices appearing at I < I ₀ are plane vortices rather than linear. The configurations of currents and the magnetic field profile inside the sample are calculated for I < I ₀. Calculation is based on analyzing the continuous variation of the current configuration toward a decrease in the Gibbs potential.     

Penetration of the magnetic field into a 3D ordered Josephson medium at small values of the pinning parameter

The current configurations and the profile of the magnetic field penetrating into a 3D ordered Josephson medium are calculated for I < I _C . The calculation algorithm (modified for finite-length samples) is based on analyzing the continuous variation of the configuration toward a decrease in the Gibbs potential. This algorithm makes it possible to find a configuration into which the Meissner state passes when I < I _C and an external field slightly exceeds H _max and trace the evolution of this configuration with a further rise in the field. At H > H _max, the magnetic field penetrates into the sample as a quasi-uniform sequence of plane vortices. When H is roughly equal to H ₀/2, where H ₀ is the outer field at which one fluxoid Φ₀ passes through each cell, the plane vortices disintegrate into linear ones centered in cells neighboring along the diagonal. As the field grows, the vortex pattern condenses: zero-fluxoid cells are gradually “filled” starting from the boundary. When the field approaches H ₀, a sequence of plane vortices centered in adjacent rows arises near the boundary. With a further increase in the field, sequences of linear vortices with a double fluxoid form at the boundary. Then, such a scenario is periodically repeated with a period H ₀ in the external field.     

Magnetic field penetration into a 3D ordered Josephson medium and applicability of the bean model

The results of calculation of penetration of an external magnetic field into a 3D ordered Josephson medium, based on analysis of modification of the configuration in the direction of the decrease in its Gibbs potential, are reported. When the external field slightly exceeds the stability threshold, the Meissner configuration is transformed into a periodic sequence of linear vortices, which are parallel to the boundary of the medium and are located at a certain distance from it. There exists a critical value I _C separating two possible regimes of penetration of the external magnetic field into the medium. For I > I _C , for any value of the external field, a finite-length boundary current configuration appears, which completely compensates the external field in the bulk of the sample. At the sample boundary, the field decreases with increasing depth almost linearly. The values of the slope of the magnetic field dependence are rational fractions, which remain constant in finite intervals of I. When the value of I exceeds the upper boundary of such an interval, the slope increases and assumes the value of another rational fraction. If, however, I < I _C , such a situation takes place only up to a certain value of external field H _max. For higher values, the field penetrates into the medium to an infinite depth. These results lead to the conclusion that the Bean assumptions are violated and that Bean’s model is inapplicable for analyzing the processes considered here.     

Pinning of planar vortices and magnetic field penetration in a three-dimensional Josephson medium

The behavior of planar (laminar) vortices in a three-dimensional, ordered Josephson medium as a function of the parameter I, which is proportional to the critical junction current and the cell size, is investigated with allowance for pinning due to the cellular structure of the medium. The minimum possible distances between two isolated vortices are calculated. A system of vortices formed in a sample in a monotonically increasing external magnetic field is analyzed. The minimum distance from the outermost vortex to the nearest neighbor is proportional to I ^−1.1. For I⩽1.3 each vortex contains a single flux quantum Φ₀, and the distance between them does not decrease in closer proximity to the boundary but remains approximately constant, implying that the magnetic field does not depend on the coordinate in the region penetrated by vortices. These facts contradict the generally accepted Bean model. The sample magnetization curve has a form typical of type II superconductors. Allowance for pinning raises the critical field H _c and induces a sudden jump in the curve at H=H _c. Zh. Tekh. Fiz. 67, 38–46 (September 1997)     

Penetration of magnetic field into a long periodically modulated Josephson contact

A new approach to magnetic field profiling inside a Josephson contact is suggested. Its essence consists in analyzing continuous variation of a current configuration leading to a decrease in the Gibbs potential. With this approach, one can find a configuration into which the Meissner state turns when an external field slightly exceeds the upper boundary of the Meissner regime and trace the evolution of this configuration with increasing field. Calculations show that there exists critical value I _c of the pinning parameter in the range 0.95–1.00. This critical value separates two possible conditions of magnetic field penetration into the contact. At I > I _c, a near-boundary current configuration completely compensating for the external field inside the contact arises irrespective of the external field strength. At I < I _c, such a situation is observed only until the external field strength exceeds certain value H _max. Higher fields penetrate into the contact indefinitely deep. In nearboundary configurations, the magnetic field drops with increasing depth almost linearly. Its slope k has rational values, which remain constant within finite intervals of I. As I goes beyond a given interval, k rises stepwise and takes on another rational value. When an external magnetic field is switched on adiabatically, configurations with a maximal growth rate of the magnetic field are observed.     

Formation of a quasi-uniform sequence of vortices in a periodically modulated finite-length Josephson contact

The configurations of currents and the profile of a magnetic field penetrating into a finite-length contact at I < I _C are calculated. The computational method is based on analyzing the continuous variation of the current structure leading to a decrease in the Gibbs potential. Such an approach makes it possible to find a configuration that sets in when an external field slightly exceeds H _max and trace the evolution of this configuration with increasing field. It is shown that at H > H _max boundary structures turn into quasi-uniform sequences of vortices the spacing between which oscillates about a mean value decreasing with increasing H. At some values of H, vortices with a number of fluxoids Φ₀ larger by unity start penetrating into the contact in the form of boundary sequences. As the field grows, they produce quasi-uniform sequences, etc. Vortices with the number of fluxoids Φ₀ differing by more than unity can fall into the contact at no field. The penetration of vortices with (k + 1)Φ₀ into a contact each cell of which contains kΦ₀ is fully identical to the penetration of vortices with one Φ₀ into the Meissner configuration. This statement is supported by the almost strict periodicity of mean induction b in the contact versus external field h dependence with a period of 1 along both axes and also by the form of the dependences of the magnetic field in the cells on the cell-boundary distance.     

Commensurability effects in a Josephson tunnel junction in the field of an array of magnetic particles

Commensurability effects have been theoretically studied in a hybrid system consisting of a Josephson junction located in a nonuniform field induced by an array of magnetic particles. A periodic phase-difference distribution in the junction that is caused by the formation of a regular lattice of Abrikosov vortices generated by the magnetic field of the particles in superconducting electrodes is calculated. The dependence of the critical current through the junction I _c on the applied magnetic field H is shown to differ strongly from the conventional Fraunhofer diffraction pattern because of the periodic modulation of the Josephson phase difference created by the vortices. More specifically, the I _c(H) pattern contains additional resonance peaks, whose positions and heights depend on the parameters and magnetic state of the particles in the array. These specific features of the I _c(H) dependence are observed when the period of the Josephson current modulation by the field of the magnetic particles and the characteristic scale of the change in the phase difference by the applied magnetic field are commensurable. The conditions that determine the positions of the commensurability peaks are obtained, and they are found to agree well with experimental results.     

Interaction and pinning of plane vortices in a three-dimensional Josephson medium and possible distances between two isolated vortices

Within a continuous vortex model, exact expressions are obtained for the Josephson and magnetic energies of plane (laminar) vortices, as well as for the energy and force of pinning by cells in a three-dimensional Josephson medium. If the porosity of the medium is taken into account, the Josephson and magnetic energies of the vortex differ from those for the continuum case. The contributions to the pinning energy from the Josephson and magnetic energies have opposite signs. An algorithm for numerically solving a system of difference equations is proposed in order to find the shape and the energy of the vortex in its stable and unstable states. The continuous vortex model is shown to fail in predicting correct values of the Josephson and magnetic energy of the vortex, as well as of the pinning energy components. Expressions for the least possible distances between two isolated vortices are obtained for a small pinning parameter. Analytical results are in close agreement with computer simulation. An algorithm for numerically solving a system of difference equations is proposed in order to find the least possible distances between two isolated vortices when the pinning parameter I is not small. The minimal value of I at which the center-to-center distance N of the vortices equals three cells is 1.428; for N=2, I _min=1.947. At I>2.907, the vortices can be centered in adjacent cells.     

Pinning of vortices and penetration of the magnetic field into a periodically modulated long Josephson contact

The upper field of the Meissner regime, H _up, and overheat field H′_c1, above which vortices start penetrating into a Josephson contact, are calculated throughout the range of pinning parameter I. The stability of likely configurations is investigated. It is shown that H _up = H′_c1 at any I. The existence of a single vortex centered at the extreme cell in the contact is demonstrated to be a possibility. At I > 3.69, such a vortex may exist even in a zero magnetic field. At 1.48 < I < 3.69, this vortex can exist in an external field in the range from some H _v to H _up. At I < 1.48, the vortex cannot exist under any conditions. From the equality of H _up and H′_c1 at any I, the conclusion is drawn that penetration of vortices into any Josephson medium is conditioned by the need to satisfy flux quantization conditions. Here, not the forces of vortex pinning at defects in the medium but quantization requirements are of major importance, which are satisfied in specific quantum ways rather than by meeting equilibrium conditions for vortices, forces, etc.     

Stability of the Meissner state in a 3D ordered Josephson medium

The stability of the Meissner state of a 3D Josephson medium against combinations of phase jump small fluctuations at contacts is considered. Expressions for the elements of the quadratic form matrix for the second variation of the Gibbs potential are derived. Overheat field values and forms of fluctuations causing instabilities are found. Ratio H _S1/H _S2, where H _S1 is the overheat field and H _S2 is the maximal field at which the Meissner state still exists, grows with increasing pinning parameter I, varying between 0.84 and 1. Almost at all pinning parameters, critical fluctuations represent rapidly decreasing (inward to the sample) periodic alternating-sign structures one cell wide. When the pinning parameter is very small (I < 0.1), such an instability is absent. In this range of I, ratio H _S1/H _S2 is close to unity.     

Pinning of pancake vortices in a three-dimensional Josephson medium and structure of the vortex lattice in the “critical” state

A system of pancake vortices formed near the boundary of a sample in a monotonically increasing external magnetic field is calculated with allowance for pinning due to the cellular structure of the medium for various values of the pinning parameter I, which is proportional to the critical current of the junction and the cell diameter. The shortest distance from the outermost vortex to the nearest neighbor is proportional to I ⁻¹¹. It is shown that the pinning parameter has a critical value I _c separating two regimes with different types of critical states. For I<I _c the external magnetic field has a threshold value H _t(I), above which the field immediately penetrates the interior of the junction to an infinite distance. For I>I _c the magnetic field decays linearly from the boundary into the interior of the junction. The value obtained in the study, I _c=3.369, differs from the value of 0.9716 postulated by other authors. The dependence of the slope of the magnetic field profile near the boundary on I is determined. It is shown that the slope is independent of I in intervals 2πk<I<2πk+π. Fiz. Tverd. Tela (St. Petersburg) 39, 1958–1963 (November 1997)     

Chemomagnetism,magnetoconcentration effect,and “fishtail” anomaly in chemically induced granular superconductors

Within a 2D model of Josephson junction arrays (created by a 2D network of twin boundary dislocations with strain fields acting as an insulating barrier between hole-rich domains in underdoped crystals), a few novel effects expected to occur in intrinsically granular material are predicted, including (i) Josephson chemomagnetism (chemically induced magnetic moment in zero applied magnetic field) and its influence on a low-field magnetization (chemically induced paramagnetic Meissner effect) and (ii) the magnetoconcentration effect (creation of oxygen vacancies in applied magnetic field) and its influence on a high-field magnetization (the chemically induced analogue of the “fishtail” anomaly). The conditions under which these effects can be experimentally measured in nonstoichiometric high-T_c superconductors are discussed.     

Pinning of linear vortices and possible distances between them in a 3D ordered Josephson medium with a nonzero structural factor

On the basis of the fluxoid quantization conditions, we derive a system of equations describing the current configuration of two interacting linear vortices in a 3D ordered Josephson medium in the entire range of possible values of structural factor b. The axes of these vortices are located in the middle row of an infinite strip with a width comprising 13 meshes. We propose a method for solving this system, which makes it possible to calculate the current configurations exactly. The critical values of pinning parameter I _d are calculated, for which two linear vortices can still be kept at a distance of d meshes between their centers in the entire range of possible values of parameter b. The formula describing the I _d(b) dependences for various values of d is derived. The dependences of the maximal pinning force F on parameter I for various values of b are analyzed. It is shown that for the same value of I, larger values of b correspond to larger maximal pinning forces.     

On the field-induced diaelastic effect in a small Josephson contact

An analog of the diaelastic effect is predicted to occur in a small Josephson contact with Josephson vortices manifesting itself as magnetic field induced softening of the contact shear modulus C(T, H). In addition to Fraunhofer type field oscillations, C(T, H) is found to exhibit pronounced flux driven temperature oscillations near T _C .     

Line vortex pinning and spacing in a three-dimensional ordered Josephson medium

A method of calculating the configuration of two line vortices interacting in a three-dimensional ordered Josephson medium and a minimal distance between them at a given pinning parameter is proposed. The axes of the vortices lie in the middle row of an infinite slab 9 or 13 cells thick with different conditions at the boundaries of the slab. Away from the centers of the vortices, the system of finite-difference equations becomes linear. Fluxoid quantization conditions in cells near the centers of the vortices serve as boundary conditions. An exact solution is approached by iterations in those phase discontinuities which cannot be considered small. This technique provides a much higher calculation accuracy and offers a wider domain of applicability than the earlier methods. Critical values I _d of the pinning parameter at which two initial vortices keep given spacing d between them are calculated. For various vortex configurations, maximal pinning forces are calculated as functions of the pinning parameter and the distance to the nearest vortices. It is shown that the pinning force decreases near parallel vortices and increases near antiparallel ones.     

Equilibrium states of planar vortices in a three-dimensional josephson medium and meaning of the pinning-energy concept

Various ways of specifying the pinning-energy concept for planar vortices in a three-dimensional cellular Josephson medium are analyzed. It is shown that, for values of the pinning parameter I that are not small, a universal characteristic of vortex interaction with the lattice cannot be found, since the displacement of a vortex distorts its shape. At small values of I, the maximum pinning force can be chosen for such a characteristic. Two equilibrium states of a vortex are analyzed for stability. It is revealed that the state of higher energy is not inevitably unstable. A correct analysis of stability must be based on exploring a quadratic form that describes the energy of a current configuration. Such an investigation is performed for the equilibrium state of a vortex. At small values of the pinning parameter, the vortex state of higher energy is quasistable.     

SrTiO3基片上Tl-2212双晶约瑟夫森结的动态特性及噪声影响研究

在SrTiO₃（STO）基片上制作了Tl-2212双晶约瑟夫森结,并对其进行了微波辐照下的I-V特性测试,观察到了夏皮罗台阶,符合约瑟夫森电压-频率关系.利用数值仿真对约瑟夫森结建立了RCSJ模型,仿真结果与实验数据符合较好,利用此模型深入研究了噪声对结动态特性的影响,解释了噪声影响下结的微波感应台阶幅度减小和极小值展宽现象,提出了有效噪声温度为工作温度和外部噪声的等效温度之和. 关键词： RCSJ模型 噪声 Tl-2212双晶约瑟夫森结     

Chaotic dynamics of resistively coupled DC-driven distinct Josephson junctions and the effects of circuit parameters

Two distinct Josephson junctions (JJs) connected with a constant coupling resistance R_cp are theoretically considered to investigate the overall dynamics below and above the critical current I_c. The circuit model of the device is driven by two DC current sources, I₁ and I₂. Each junction is characterized by a nonlinear resistive–capacitive junction (NRCSJ). Having constructed the circuit model, time-dependent simulations are carried out for a variety of control parameter sets. Common techniques such as bifurcation diagrams, two-dimensional attractors and Lyapunov exponents are applied for the determination of chaotic as well as periodic dynamics of the superconducting junction devices. According to the findings, two states (namely superconducting and ordinary conducting) are determined as functions of the source currents. The chaotic current which flows through R_cp exhibits a very rich behavior depending on the source currents I₁ and I₂ and junction capacitances C₁ and C₂. The device characteristics are summarized by a number of three-dimensional phase diagrams in the parameter space. In addition, for certain parameters, hyper-chaotic cases with two positive Lyapunov exponents are encountered. In contrast to earlier studies claiming the need for a sinusoidal feeding current for generating a chaotic signal, our circuitry can easily generate one via a DC source.     

Static characteristics of Josephson interferometers

This investigation deals with the range in operating currents for which a Josephson interferometer, sometimes also referred to as Superconducting QUantum Interference Device (SQUID), may remain in the zero-voltage Josephson condition. An interferometer consists of one or more inductive loops each of which contains two Josephson junctions or other weak links. Two types of current are considered. Gate currentI _gpasses the junctions in parallel. Control currentI _cgenerates magnetic flux via inductive coupling in the loops. Zero-voltage operation is possible within certain areas of theI _g,I _cplane. These areas are manifestations of flux-quantum states and their boundary lines are referred to as static characteristics. In view of the nonlinearity of the constituting equations, not all their formal solutions are physically realizable. A stability analysis yields criteria which permit the identification of realizable operating conditions. The static characteristics comprise operating conditions where the limit of stability is reached. To obtain the static characteristics, linearized equations may be utilized if theLI _o product, a measure for the size of an interferometer, is large compared to the flux quantumΦ ₀, whereL is the inductance per loop, andI _o the maximum Josephson current per junction. As a general method of solving system of transcendental equations, continuation is discussed. The utilization of continuation for obtaining interferometer characteristics is explained. It is shown that some changes in the gate-current feed arrangement are equivalent to shearing the characteristics in theI _g,I _cplane. Analytical results are given on extrema, inflexion points, and singularities in the shape of cusps which conceptually relate to the existence and connectivity of flux-quantum states. Experimental static characteristics are presented on two-and four-junction interferometers. They are in agreement with characteristics computed on the basis of simple lumped circuit models. Relevant circuit parameters are obtained from the experimental characteristics.     

Characteristics of the Surface-Intrinsic Josephson Junction

During the fabrication of intrinsic Josephson junctions (IJJs) with Bi₂Sr₂CaCu₂O_8+δ (BSCCO) single crystals, the superconductivity of the surface Cu-O layer is degraded because of a deposited metal film on top of the stack. Thus, the characteristics of the surface junction consisting of the surface Cu-O double layers remarkably differ from those of the junctions deep in the stack, which will be referred to as ordinary IJJs. The electrical transport characteristics of the surface junction, such as I-V, I _c′-T, and R-T, show that the critical temperature T _c′ of the surface junction is always lower than that of ordinary IJJs, and that the change of its critical current I _c′ with temperature is different from that of ordinary IIJs. Furthermore, by shunting the surface junction resistively, we are able to observe the AC Josephson effect at 3-mm waveband. Translated from Chinese Journal of Low Temperature Physics, 2005, 27(2) (in Chinese)