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.     

Phase diagram of a lattice of pancake vortex molecules

On a superconducting bi-layer with thickness much smaller than the penetration depth, λ, a vortex molecule might form. A vortex molecule is composed of two fractional vortices and a soliton wall. The soliton wall can be regarded as a Josephson vortex missing magnetic flux (degenerate Josephson vortex) due to an incomplete shielding. The magnetic energy carried by fractional vortices is less than in the conventional vortex. This energy gain can pay a cost to form a degenerate Josephson vortex. The phase diagram of the vortex molecule is rich because of its rotational freedom.     

Generation of electromagnetic radiation in the motion of vortices in magnetically coupled superconducting films

We examine theoretically the generation of electromagnetic radiation in the relative motion of vortex lattices in magnetically coupled films in the dc transformer geometry. We establish the conditions under which the force of mutual pinning of the vortex lattices varies according to a harmonic law as a function of the relative displacement of the vortices in the films within a given range of magnetic field inductions. In this case the equation describing the viscous flow of vortex lattices in magnetically coupled films is the same as the equation of the resistively shunted Josephson junction model. We show that magnetically coupled superconductors exhibit the properties of a Josephson element without any restrictions on the geometrical size of such a system imposed by the coherence length ξ. The frequency f of the electromagnetic radiation generated by the relative motion of vortex lattices in magnetically coupled superconductors depends on the spatial period of the vortex lattices and the velocity of relative vortex motion, which means that the frequency of the radiation can be tuned by applying a magnetic field or a current. Zh. éksp. Teor. Fiz. 113, 1319–1338 (April 1998)     

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)     

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.     

Vortex deformation and breaking in superconductors: a microscopic description

Vortex breaking has traditionally been studied for non-uniform critical current densities, although it may also appear due to non-uniform pinning force distributions. In this article we study the case of a high-pinning/low-pinning/high-pinning layered structure. We have developed an elastic model for describing the deformation of a vortex in these systems in the presence of a uniform transport current density J for any arbitrary orientation of the transport current and the magnetic field. If J is above a certain critical value, J_c , the vortex breaks and a finite effective resistance appears. Our model can be applied to some experimental configurations where vortex breaking naturally exists. This is the case for YBa₂Cu₃O_7?δ (YBCO) low-angle grain boundaries and films on vicinal substrates, where the breaking is experienced by Abrikosov–Josephson vortices (AJV) and Josephson string vortices (SV), respectively. With our model, we have experimentally extracted some intrinsic parameters of the AJV and SV, such as the line tension ? _l and compared it to existing predictions based on the vortex structure.     

Unambiguous detection of monopoles

Summary The fission of a vortex line along the trajectory of a monopole in a superconducting medium is an unambiguous signature of a monopole. The numbern of the (stable) daughter vortices determines the monopole strengthg withg=2nφ/4π where ϕ₀=2.07·10⁻⁷ G cm².

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Riassunto La fissione di una linea di vortice lungo la traiettoria di un monopolo in un mezzo superconduttore è indice sicuro della segnatura di un monopolo. Il numeron di vortici figli (stabili) determina la forza del monopolog cong=nϕ₀/4π, dove ϕ₀=2.07·10⁻⁷ G cm².

     

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.     

Pinning of vortices by the domain structure in a two-layered type II superconductor-ferromagnet system

The effectiveness of magnetic pinning of vortices in a layered system formed by a uniaxial ferromagnet and type II superconductor is considered. It is shown that, irrespective of the saturation magnetization of the ferromagnet, the energy of pinning at the domain structure does not exceed, in order of magnitude, the energy of artificial pinning at a column-type defect. The limitation of pinning energy is caused by the interaction of external vortices with a spontaneous vortex lattice formed in the superconducting film when the magnetization of the ferromagnetic film exceeds the critical value.     

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)     

Numerical research on RF induced zero-crossing steps of stacked Josephson junction in flux-flow state

RF responses of intrinsic Josephson junction stacks in the flux-flow state have been studied to explore vortex motion. We have reported various RF induced effects on flux-flow of Bi₂Sr₂CaCu₂O_8+y intrinsic Josephson junctions. In a pinning free state, a remarkable zero-crossing step appears at a certain voltage on I–V curve, which is closed to the voltage V = NΦ₀f, where N is the number of junction, Φ₀ is the flux quantum, f is the RF frequency, respectively. It is shown that vortex motion phase-locked to external microwaves play an important role in the responses. In this report, we have carried out numerical simulation by using the coupled sine-Gordon equations in order to compare with the experimental results. The numerical simulations reveal that the alternating magnetic field of microwaves drives vortices into the stack and generates the zero-crossing step.     

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.     

Calculation of the vortex pinning energy in a long periodically modulated Josephson contact

An analytical method for evaluating the Josephson and magnetic energies of a vortex, as well as the vortex pinning energy and its components, in a long periodically modulated Josephson contact is suggested. The method allows one to take into account the variation of the vortex shape with the position of the vortex. The results obtained with this method are much closer to those of exact computer analysis compared with results of the conventional techniques. The discrepancy between the exact shape of the vortex and a function obtained by solving a differential equation approximating the exact difference equation is studied.     

高温超导体中约瑟夫森涡旋流阻的振荡效应

通过数值计算耦合sine-Gordon方程组研究高温超导体中约瑟夫森涡旋的运动,得到约瑟夫森涡旋电压和流阻随平面磁场和驱动电流的变化规律.固定驱动电流,约瑟夫森涡旋电压和流阻随着磁场的增大出现周期性的振荡行为,振荡周期与每层约瑟夫森结中进入一个磁通量子相对应.分析和阐明了产生这种周期性振荡的原因. 关键词： 约瑟夫森涡旋 涡旋格子 高温超导     

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.     

Current-voltage characteristics for a Peierls-conducting point contact

Current-voltage (J-V) and differential-conductivity-voltage ( dJ/dV-V) characteristics are analytically calculated at zero temperature for a point contact consisting of: two Peierls conductors P ( = 1, 2) separated by an insulator (I). Here P is a conductor with charge density wave (CDW). The J-V and dJ/dV-V characteristics depend on the CDW phases ( = 1, 2) in the mean field approximation. To calculate them analytically we assumed, = ≡Δ where ( = 1, 2) are the energy gaps of P ( = 1, 2). The current J has a discontinuous jump at eV = 2Δ for ϕ ₁ = ϕ ₂≠ 0. The differential conductivity dJ/dV has a singularity at eV = 2Δ for ϕ ₁ = ϕ ₂≠ 0. The relation J(V,ϕ ₁,ϕ ₂) = - J(- V,ϕ ₁ + π,ϕ ₂ + π) is obtained. Received 4 July 2001 and Received in final form 13 September 2001     

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.     

Development of the Meissner state instability in an ordered 3D Josephson medium

A new approach based on analysis of continuous configurational modification in the direction of a decrease in the Gibbs potential is proposed for computing the penetration of an external magnetic field in an ordered 3D Josephson medium. The configuration to which the Meissner state passes when the external field slightly exceeds the Meissner stability threshold is determined. This configuration contains a periodic sequence of linear vortices with centers lying in an alternating cell, parallel to the boundary, and located at a certain distance from it. A further increase in the field reveals that the 3D medium behaves like a long periodically modulated Josephson junction. However, the critical value I _C of the pinning parameter for a 3D medium, which lies in the interval 0.7–0.8, is lower than the analogous value I _C = 0.9716 for a long junction. The values of H _max for I < I _C , as well as the steepness of the decrease in the magnetic field at the boundary for I > I _C , are higher in the 3D medium than in a long junction. For very large values of I, the field penetrates the boundary region not as a 2D lattice of linear vortices, but as a 1D lattice of plane vortices, which are mathematically equivalent to the vortices in a long junction.     

Structure of the mixed state in type II superconductors and its connection with Josephson effects in long superconducting bridges

The interaction energy between vortices and the structure of the vortex lattice in superconducting layers is calculated. It is shown that the one-dimensional vortex lattice (with vortices in the centre of the layer) is stable in a relatively large interval of magnetic fields above the flux penetration field. This result is true not only for very thin layers (with dimensions smaller than the penetration depth), but also for layers withd>. The results can be helpful at explaining some unusual experimental results on Josephson effects in long superconducting bridges.     

Static and dynamic behaviours of multivortex states in a superconducting sample with mesoscopic pinning sites

This preliminary work has focused on the static transitions between the multivortex states interacting with square arrays of the mesoscopic pinning sites in superconducting samples. Our results were obtained from an extensive series of numerical simulations as functions of the magnetic field, pinning radius, and sample size. We have presented a wide range of multivortex configurations from commensurate dimer states to more concentric vortex shells at the matching fields. The stability of these states was also studied by means of the current-voltage V(I) curves which illustrate dynamic phase transitions as a function of applied driving force. These transitions manifested themselves as either a sudden jump in velocity or a nonlinear increase with velocity fluctuations in V(I) curves. We have investigated whether that the phase transitions between the pinned regime and the elastic flow regime are indicative of the stability of the initial vortex states. The variety of intermediate flow phases is attributed to large pinning size (reentrant behavior), strong commensurability and caging effects. In particular, three-shell vortex structures were obtained in the presence of larger pinning sites at adequate matching magnetic fields.