Meissner state in a 3D ordered Josephson medium

A system of difference equations describing current configurations in a 3D ordered Josephson medium is obtained. The system is based on the fluxoid quantization conditions in the cells. A method for the exact solution to the system is proposed for the Meissner configuration for any values of the parameters. The current configurations for the Meissner regime are computed, as well as the values of the critical fields above which the Meissner configuration does not exist. Approximate analytic solutions are obtained for the system of difference equations, which correctly describe the exact solution for the major part of the range of the parameters.     

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.     

Possible spacing between linear vortices in a 3D ordered Josephson medium

A method is proposed for solving the nonlinear system of equations of fluxoid quantization for two interacting linear vortices. It is shown that the centers of the vortices may lie in adjacent cells only if the pinning parameter I > 0.91, in alternate cells if I > 0.44, and in each third cell if I > 0.25. These critical values are substantially lower than analogous values for planar vortices. It is shown that, as the value of I tends to zero, the minimal spacing between linear vortices does not increase indefinitely, but attains a certain finite value and then remains unchanged. This means that pinning of linear vortices cannot be ignored even for values of I quite close to zero. It is shown that two linear vortices with centers in the neighboring cells along a diagonal may coexist for indefinitely small values of I.     

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.     

Hysteresis in the behavior of a 3D ordered Josephson medium in magnetic field at a small pinning parameter

Using the approach based on analyzing variations in the configuration in the direction of the decreasing Gibbs potential, the magnetization curve for a three-dimensional ordered Josephson medium upon magnetic field cycling has been calculated for the case of a small pinning parameter. It has been demonstrated that, at any turning point, the hysteresis loop is part of some universal curve that is strictly periodic along both axes. The existence of a universal curve and its periodicity have been explained by analyzing the vortex configurations.     

Line vortices in a three-dimensional ordered Josephson medium

Two equilibrium configurations of a line vortex in a three-dimensional ordered Josephson medium are considered: (i) the vortex core is at the center of a cell and (ii) the vortex core is on a contact. Infinite systems of equations describing these configurations are derived. In going to a finite system, the currents far away from the center are neglected. A new technique for solving the finite system of equations is suggested. It does not require smallness of phase discontinuities at all vortex cells and, therefore, can be applied for any values of pinning parameter I down to zero. The structures and energies of both equilibrium states for isolated line vortices are calculated for any I from the range considered. For I >0.3, a vortex can be thought of as fitting a square of 5×5 cells. For lower I, the vortex energy can be expressed as a sum of the energies of the small discrete core and the quasi-continuous outside. The core energy is comparable to the energy of the outside and is a major contributor to the vortex energy when I is not too small. For any I, the energy of the vortex centered on the contact is higher than the energy of the configuration centered at the center of the cell.     

Critical state of Josephson medium

A model of the critical state of a Josephson medium is developed on the basis of the Sonin theory of averaged Josephson medium. The model is used to explain the experimental data on the differential magnetic susceptibility _χd (H) and magnetoresistance R(H) of polycrystalline YBa₂Cu₃O_7?x samples in fields H<100 Oe.     

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.     

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.     

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.     

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.     

Structure and energy of a line vortex in a three-dimensional ordered Josephson medium

Two possible equilibrium configurations of line vortices in a three-dimensional ordered Josephson medium for any value of structural factor b are considered: the center of the vortex coincides with the center of one of the cells and the center of the vortex is on one of the contacts. Infinite sets of equations describing these configurations are derived. The infinite set can be made finite if currents away from the center are neglected. The assumption b = 0 is shown to be valid if pinning parameter I is less than 0.25. For I > 0.25, the structures and energies of both configurations of line isolated vortices are calculated throughout the range of structural factor b. As structural factor b increases, phase jumps at the contacts, currents in the central part of the vortex, and the total energies of the vortices decrease in both configurations. This leads to a decrease in critical field H _c1. For all values of I and b, the energy of the vortex centered on the contact is higher than that of the vortex centered in the middle of the cell.     

Line vortices in a three-dimensional ordered Josephson medium at a small pinning parameter

The structure and energy of a line vortex whose axis is aligned with the symmetry axis of a finite-thickness slab indefinitely long in two directions is calculated by solving a set of linear finite-difference equations. Fluxoid quantization conditions in cells near the center of the vortex serve as boundary conditions. An exact solution is approached by iterations in phase stepwise discontinuities that cannot be considered small. A close similarity between the configuration under study and a periodic sequence (chain) of vortices makes it possible to allow for the effect of the domain boundary on the structure and energy of the vortex. It is shown that, at any width of the slab, one can find a pinning parameter value so small that the vortex cannot be viewed as solitary and contributions from other vortices should be taken into account in calculation. Proceeding in this way, one can find the structure and energy of the vortex however small the pinning parameter is. The total energy of the vortex is its intrinsic energy plus the sum of its energies of interaction with other members of the chain. In turn, the intrinsic energy is the sum of the energies of the small discrete core and quasi-continuous outer shell. It is demonstrated that the energy of the core is a linear function of the pinning parameter and is comparable to the energy of the shell.     

Order of phase transition between the Meissner state and the mixed state

The critical value of the Ginzburg-Landau parameter which separates type-II/1 from type-II/2 superconductors is computed by means of the boson formulation of superconductivity. By taking into account the tail effect of the function c(k), which describes the electrodynamics of the superconducting state, it is shown that the attractive part of the interaction energy between two flux lines indeed vanishes for a critical value of κ. The dependence on Λ = VN(O) is also discussed.     

Different methods of calculating the pinning energy of plane vortices in a 3D Josephson medium: A comparative study

The pinning energy of plane (laminar) vortices in a 3D Josephson medium is calculated within a continuous vortex model considering functions of two types: V=1−cosϕ and V= 2/π⁴ϕ²(2π−ϕ)². The shape and energy of the stable and unstable vortices are found with an algorithm for the exact numerical solution of a set of difference equations. The vortex magnetic and Josephson energies diverge. The magnetic and Josephson components of the pinning energy are close in magnitude but differ in sign; as a result, the total pinning energy is smaller than its components by one order of magnitude. This result is confirmed analytically. An analytical computing method within the continuous vortex model is suggested. This method preserves the difference terms in the energy expression. The magnetic energy found by this method differs from the Josephson energy in magnitude, and the magnetic component of the pinning energy is opposite in sign to the Josephson component. Comparative analysis of the approximate approaches to energy calculation within the continuous vortex model when the difference terms are retained and when they are replaced by derivatives is performed. It is shown that the continuous vortex model gives incorrect values of the Josephson and magnetic components of the pinning energy. The actual values are several tens or several hundreds of times higher than those obtained with the continuous vortex model. Yet, since the Josephson and magnetic components of the pinning energy have different signs, the exact value of the total pinning energy and the approximate value obtained within the continuous vortex model differ insignificantly.     

Flux avalanches in a Josephson medium

Avalanche flux penetration dynamics has been experimentally observed in a Josephson medium, a granular high-T _c superconductor, with a slowly increasing external magnetic field. The observed voltage spikes are associated with the stepwise penetration of the field into the superconductor and obey a power-law size distribution. The results directly confirm the hypothesis of self-organized criticality in such a system.     

Fabrication of 3D proximity coupled Josephson junction arrays

Experimentally realizable 3D arrays of Josephson junctions have been a goal of researchers since 2D Josephson junctions (JJ) arrays were first introduced. In the past, it has proven to be technically impossible to manufacture 3D proximity-coupled arrays. Recent advancements in etching technology have now made fabrication more feasible. In this paper, we present details of our fabrication process.