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.     

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.     

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.     

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.     

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 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)     

Sliding Abrikosov lattice in a superconductor with a regular array of artificial pinning centers: AC conductivity and criticality at small frequencies

Dynamics of the flux lattice in the mixed state of strongly type-II superconductor near the upper critical field subjected to AC field and interacting with a periodic array of short range pinning centers is considered. The superconductor in a magnetic field in the absence of thermal fluctuations on is described by the time-dependent Ginzburg–Landau equations. For a special case of the δ-function shaped pinning centers and for pinning array commensurate with the Abrikosov lattice (so that vortices outnumber pinning centers) an analytic expression or the AC conductivity is obtained. It is found that below a certain critical pinning strength and for sufficiently low frequencies there exists a sliding Abrikosov lattice, which vibrates nearly uniformly despite interactions with the pinning centers. At very small frequencies the conductivity diverges at the critical pinning strength.     

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.     

NMR study of the displacement of vortices in the superconductor Tl<Subscript>2</Subscript>Ba<Subscript>2</Subscript>CuO<Subscript>6+δ</Subscript> under the action of a magnetic field pulse

A transition from irreversible to reversible displacement of vortices in the high-temperature super-conductor Tl₂Ba₂CuO_6+δ (T _c =78 K) is investigated by the NMR method combined with pulse modulation of the magnetic field in the temperature range from 20 to 35 K. The results also make it possible to estimate the distance between pinning centers in the sample, which lies in the interval from 4 to 10 nm at a temperature of 20 K.     

Attenuation of interfacial pinning enhancement in YBCO using a PrBCO buffer layer

Numerous experimental results have suggested that the J_c of YBa₂Cu₃O₇ (YBCO) films is significantly higher near the film–substrate interface than in the remainder of the film. We previously proposed that this effect is due to interfacial pinning enhancement caused by stress and the resulting misfit dislocations at the heteroepitaxial interface. To test this hypothesis we have used a non-superconducting PrBa₂Cu₃O_7?δ (PrBCO) buffer layer to minimize the lattice mismatch with YBCO. We find that the PrBCO layers lower J_c of the 0.4 μm YBCO films in a predictable way, and that, if sufficiently thick (～0.5 μm), they eliminate interfacial enhancement altogether. Our interpretation of this result is that the defects responsible for interfacial enhancement of flux pinning originate at the bottom of the non-superconducting PrBCO layer, which screens the pinning centers from vortices in YBCO. This result demonstrates that the pinning enhancement arises from stress at the film–substrate interface.     

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.     

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.     

Effect of low-temperature annealing on the critical parameters of highly textured YBa2Cu3O y

The effect of treatment at a temperature of 200°C and the natural aging on the critical parameters of a highly textured yttrium barium cuprate YBa₂Cu₃O_6.9 has been investigated. It has been shown that non-superconducting (at T = 77 K) particles precipitated during phase decomposition of this compound are effective pinning centers. At 200°C, the YBa₂Cu₃O _y compound interacts with atmospheric moisture. This inter-action results in the formation of stacking faults, which also provide pinning of magnetic vortices. The structural changes occurring during low-temperature annealing and natural aging of the compound lead to an increase in the critical current density and the first critical field. The presence of pinning centers of different nature in the structure causes a synergistic effect, which significantly increases the current-carrying capacity of materials, including those in strong magnetic fields.     

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.     

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.     

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)     

Attraction-repulsion transition between abrikosov vortices and Josephson vortices in the strongly layered superconductor Bi2Sr2CaCu2O8

A change in the effect of a frozen magnetic field parallel to the c-axis on rf power absorption, which is associated with the motion of Josephson vortices, is observed in the layered superconductor Bi₂Sr₂CaCu₂O₈ at a low temperature (～15 K). The effect is interpreted as a change in the interaction between an Abrikosov vortex and a Josephson vortex from attraction (at high temperatures) to repulsion (at low temperatures). It is found that the dynamics of the magnetic flux parallel to the ab plane of the single crystal becomes irreversible upon a transition of the superconductor to the layered state. Possible reasons behind the observed effect are considered, one of them being a manifestation of the second superconducting transition in the elementary-excitation spectrum of a d-type superconductor near the core of Abrikosov vortices.     

Features of the melting dynamics of a vortex lattice in a high-T c superconductor in the presence of pinning centers

The phase transition “triangular lattice-vortex liquid” in layered high-T _c superconductors in the presence of pinning centers is studied. A two-dimensional system of vortices simulating the superconducting layers in a high-T _c Shubnikov phase is calculated by the Monte Carlo method. It was found that in the presence of defects the melting of the vortex lattice proceeds in two stages: First, the ideal triangular lattice transforms at low temperature (≃3 K)into islands which are pinned to the pinning centers and rotate around them and then, at a higher temperature (≃8 K for T _c 584 K), the boundaries of the “islands” become smeared and the system transforms into a vortex liquid. As the pinning force increases, the temperatures of both phase transitions shift: The temperature of the point “triangular lattice-rotating lattice” decreases slightly (to ≃2 K)and the temperature of the phase transition “rotating lattice-vortex liquid” increases substantially (≃70 K). Pis’ma Zh. éksp. Teor. Fiz. 66, No. 4, 269–274 (25 August 1997)     

Pinning,thermally activated depinning and their importance for tuning the nanoprecipitate size and density in high Jc YBa2Cu3O7−x films

YBa₂Cu₃O_7?x (Y123) films with quantitatively controlled artificial nanoprecipitate pinning centers were grown by pulsed laser deposition (PLD) and characterized by transport over wide temperature (T) and magnetic field (H) ranges and by transmission electron microscopy (TEM). The critical current density J_c was found to be determined by the interplay of strong vortex pinning and thermally activated depinning (TAD), which together produced a non-monotonic dependence of J_c on c-axis pin spacing d_c. At low T and H, J_c increased with decreasing d_c, reaching the very high J_c ～ 48 MA/cm² ～20% of the depairing current density J_d at 10 K, self-field and d_c ～ 10 nm, but at higher T and H when TAD effects become significant, J_c was optimized at larger d_c because longer vortex segments confined between nanoprecipitates are less prone to thermal fluctuations. We conclude that precipitates should extend at least several coherence lengths along vortices in order to produce irreversibility fields H_irr(77 K) greater than 7 T and maximum bulk pinning forces F_p,max(77 K) greater than 7–8 GN/m³ (values appropriate for H parallel to the c-axis). Our results show that there is no universal pin array that optimizes J_c at all T and H.     

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.