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.     

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.     

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.     

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.     

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)     

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.     

Controlled vortex motion in multiple interpenetrating pinning arrays

We study the effect of multiple interpenetrating pinning arrays on the vortex motion in the presence of an ac driving force, f _d (t), by using extensive molecular dynamics (MD) simulations. Firstly, the response to a square ac wave f _d (t) has been explored for the vortices interacting with a periodic square pinning array which has different pinning strengths and sizes. The effect of the type of an ac drive and its amplitude on the oscillatory dynamics of vortices have been investigated in detail. For very low displacements of the vortices, we have found that the single-particle model can produce results analytically similar to the ones obtained by the MD simulations. It is shown that the collective motion of vortices can be controlled easily by varying the number of multiple interpenetrating square pinning lattices (N _SPSL). A regular sequence of peaks has been observed for N _SPSL = 3 in the time evolution of the average velocity of the vortices (i.e., V? _{x - t} curves). The number of peaks (N _peak) strongly depends on the magnitude of f _d (t), and increases with increasing the magnitude of f _d . The close relation between N _peak and f _d is considered as an indication of controlling vortex motion in a multiple periodic pinning structure. Finally, the variation of the power spectrum of noise S(ν) with N _SPSL has been investigated. For N _SPSL = 3, it has been found that the plastic motion of the row of vortices evolves at low frequencies, i.e., 1/ν behavior, whereas, at high frequencies, S(ν) shows a typical behavior of Gaussian white noise.     

Two dimensional vortex structures in a superconductor slab at low temperatures

The dynamics of a two dimensional chain like structure of vortices is studied in the model of nonlinear time dependent Ginzburg–Landau equations (TDGL). The transition between different linear chains of vortices in a superconducting homogeneous slab with both surfaces in contact with a thin layer of metallic material is analyzed. The magnetization curve, vortex number, vortex configurations and modulus of the order parameter are studied as a function of the external magnetic field. We show how these vortex configurations are affected by the extrapolation length b (de Gennes boundary conditions), Ψ due to the proximity effects in a mesoscopic sample of area d_x × d_y, where d_y = 60ξ(0) and d_x varies discretely from 30ξ(0) to 12ξ(0). Possible connection with recent theoretical results in a two dimensional system of charged particles is discussed.     

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.     

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.     

Multi-vortex versus interstitial vortices scenario in superconducting antidot arrays

In superconducting thin films, engineered lattice of antidots (holes) act as an array of columnar pinning sites for the vortices and thus lead to vortex matching phenomena at commensurate fields guided by the lattice spacing. The strength and nature of vortex pinning is determined by the geometrical characteristics of the antidot lattice (such as the lattice spacing a₀, antidot diameter d, lattice symmetry, and orientation) along with the characteristic length scales of the superconducting thin films, viz., the coherence length (ξ) and the penetration depth (λ). There are at least two competing scenarios: (i) multiple vortices sit on each of the antidots at a higher matching period and (ii) there is nucleation of vortices at the interstitial sites at higher matching periods. Furthermore, it is also possible for the nucleated interstitial vortices to reorder under suitable conditions. We present our experimental results on NbN antidot arrays in the light of the above scenarios.     

Comparison on plasticity of current-driven vortex lattices in as-grown and heavy ion irradiated microbridge Bi2Sr2Ca1Cu2O8+δ single crystals

From the I-V characteristics for as-grown and irradiated Bi₂Sr₂CaCu₂O_8+δ single crystals at T=5 K in a magnetic field applied parallel to the c axis, we have seen two types of vortex dynamics near the depinning threshold. For the as-grown sample, at low field, the I-V curves show steps that clearly indicate a “fingerprint phenomenon” since they reflect the current dependence of differential resistance R_d=dV/dI. This can be ascribed to vortices flow through uncorrelated channels for the highly defective lattice. As fields sufficiently increase, these peaks merge, giving broader ones, indicating a crossover from filamentary strings to braid river. In contrast, in the irradiated sample, the pinning is found to be individual at low magnetic fields and collective when the vortex-vortex interactions are involved. Our result suggests a dynamic nature of the peak effect, in agreement with recent numerical simulations and experimental works.     

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)     

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.     

Transition from thermally activated to regular flow of magnetic flux vortices in HTSC

The effect of structural inhomogeneities in a superconductor on a vortex medium flow in weak magnetic fields at temperatures varying from 78 to 83 K for various bias current densities is investigated by using transport measurements of Bi₂Sr₂CaCu₂O_8+x thin-film microbridges. The results obtained are analyzed on the basis of the theories of flux creep and the regular flow of vortices. It is shown that the current dependences of the effective potential for vortex pinning can be satisfactorily described in the framework of two statistical models, one of which was proposed earlier by the authors. Both models cover the regimes of thermally activated and regular flow of vortices as limiting cases. The wide transition region in which the creep and regular vortex flow processes simultaneously occur due to a large dispersion in the pinning energy distribution. It is found that when the magnetic field exceeds a certain value, the average value and dispersion of the pinning potential decrease sharply, so that the conditions of regular flow set in even for small values of the bias current. This fact is attributed to the destruction of vortex lines into two-dimensional segments.     

Superconducting slab in contact with thin superconducting layer at higher critical temperature

Within the framework of nonlinear time dependent Ginzburg–Landau equations (TDGL) we study the properties of a mesoscopic superconducting film with both surfaces in contact with a thin superconducting layer at a higher critical temperature. The properties of the layer are taken into account by the de Gennes boundary conditions via the extrapolation length b. We assume that the magnetic field is parallel to the multilayer interfaces. We obtain magnetization curves and calculate the spatial distribution of the superconducting electron density using a numerical method based on the technique of gauge invariant variables. This work tests both the rectangular cross-section size and b limit for the occurrence of vortices in a mesoscopic sample of area d_xxd_y where d_y = 80ξ(0) and dx varies discretely from 20ξ(0) to 3ξ(0). Our data also show a linear behavior of the magnetization curve and a power-law of order parameter modulus in limit b → 0_-.     

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.     

Excess current in point contacts on two-band superconductor MgB2 in magnetic field

A series of I(V) characteristics and bias-dependent differential resistance dV/dI(V) curves of point contacts made between a single crystal of two-band superconductor MgB₂ and Cu were measured in magnetic fields up to 9 T. The magnetic field dependences of the excess current in the I(V) curves were obtained and analyzed using Koshelev and Golubov's [Phys. Rev. Lett. 90, 177002 (2003)] theoretical results for the mixed state of a dirty two-band superconductor. Introducing a simple model for the excess current in the point contact in the mixed state, our data can be qualitatively described using the theoretical magnetic field dependence of the superconducting order parameter of the σ and π-bands and the averaged electronic density of states in MgB₂.     

Top-quark-mass enhancement in a seesaw-type quark mass matrix

We investigate the implications of a seesawtype mass matrix, i.e.,M _f?m _LM _F ^?1 m _R, for quarks and leptonsf under the assumption that the matricesm _L andm _R are common to all flavors (up-/down- and quark-/lepton-sectors) and the matricesM _F characterizing the heavy fermion sectors have the form [(unit matrix)+b _f (a democratic matrix)] whereb _f is a flavor parameter. We find that by adjusting the complex parameterb _f, the model can provide thatm _t?m _b while at the same time keepingm _u～m _d without assuming any parameter with hierarchically different values betweenM _U andM _D. The model with three adjustable parameters under the “maximal” top-quark-mass enhancement can give reasonable values of five quark-mass ratios and four KM matrix parameters.