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.     

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.     

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.     

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.     

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)     

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.     

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.     

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 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.     

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.     

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.     

Ordered states and structural transitions in a system of Abrikosov vortices with periodic pinning

A system of Abrikosov vortices in a quasi-two-dimensional HTSC plate is considered for various periodic lattices of pinning centers. The magnetization and equilibrium configurations of the vortex density for various values of external magnetic field and temperature are calculated using the Monte Carlo method. It is found that the interaction of the vortex system with the periodic lattice of pinning centers leads to the formation of various ordered vortex states through which the vortex system passes upon an increase or a decrease in the magnetic field. It is shown that ordered vortex states, as well as magnetic field screening processes, are responsible for the emergence of clearly manifested peaks on the magnetization curves. Extended pinning centers and the effect of multiple trapping of vortices on the behavior of magnetization are considered. Melting and crystallization of the vortex system under the periodic pinning conditions are investigated. It is found that the vortex system can crystallize upon heating in the case of periodic pinning.     

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.     

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.     

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.     

Hysteresis in the behavior of a long periodically modulated Josephson junction in a magnetic field for not small values of the pinning parameter

The magnetization curve for a long periodically modulated Josephson junction is calculated using the approach based on analysis of the continuous change in the configuration in the direction of the decrease in the Gibbs potential upon cyclic variation of the external magnetic field for not small values of pinning parameter I. It is shown that unlike in the case of small I, when the hysteresis loop is a part of a certain universal curve, the segments of the loops corresponding to a decrease in h in the first and second quadrants (and symmetric to them) pass below the universal loop, the degree of deviation increasing with pinning parameter I. The properties of the hysteresis loops are considered for various amplitudes of the magnetic field variation on the basis of analysis of vortex configurations.     

Optical saturation in the photothermal spectroscopy of fluorescent materials

The dependence of the fundamental and harmonic photothermal (PT) signal on the intensity I ₀ of the illumination source is analyzed. It is shown that both components of the PT signal do not increase indefinitely with I ₀, but at sufficiently high power densities begin to decrease as 1/I ₀. Along with photoacoustic saturation, this defines an upper limit for the sensitivity of spectrometers based on PT detection.     

Magnetization of type-ii superconductors in the range of fields H c 1 ≤ H ≤ H c 2Variational Method

A variational method is proposed to find the magnetic field dependence of the magnetization of type-II superconductors in the mixed state by a self-consistent technique. This model allows for suppression of the order parameter to zero at the centers of Abrikosov vortices and also for the magnetic field dependence of the order parameter. The results can be applied to the entire range of fields H _c 1 ≤ H ≤ H _c 2 for any values of the Ginzburg-Landau parameter $\kappa > 1/\sqrt 2$ . It is shown that in weak fields where κ ? 1 the behavior of the magnetization can be described exactly in the London approximation provided that the correct value of H _c 1 is used. Near the second critical field this dependence shows good agreement with the well-known Abrikosov result. It is also shown that using the concept of isolated vortices and applying the principle of superposition of the fields and currents generated by these vortices to calculate the magnetization gives inaccurate quantitative results even in fairly weak fields. By going beyond these concepts, it was possible to allow more accurately for the influence of the vortex cores on the magnetization behavior in the intermediate range of fields H _c 1 ?H ? H _c 2 and to identify the range of validity of various approximations used widely in the literature.