Rectified vortex motion in an Nb film with a spacing-graded array of holes

Superconducting Nb thin films with a spacing-graded array of holes were prepared by electron beam lithography. Two films with different hole gradients were fabricated. The ac-driven vortices were investigated in Nb superconductors with a spacing-graded array of holes. The measurements revealed pronounced rectified voltage when the vortex lattice is driven by an ac injected current. The rectified voltage is mainly caused by the strength of the vortex–vortex interaction. The rectified motion of a vortex is affected by the pinning potential of the spacing-graded array and the applied magnetic field. The vortex–vortex interaction strength changes the effective pinning landscape of the vortices and an asymmetric potential is formed. Vortices depin easily from high concentration to low concentration of pinning sites. In both samples, the ac-driven vortices exhibit a variety of dynamical responses and the rectified voltage is tunable with the applied magnetic field.     

Commensurability effects in the critical forces of a superconducting film with Kagomé pinning array at submatching fields

Using molecular dynamics simulations, we find the commensurability force peaks in a two-dimensional superconducting thin-film with a Kagomé pinning array. A transport force is applied in two mutually perpendicular directions, and the magnetic field is increased up to the first matching field. Usually the condition to have pronounced force peaks in systems with periodic pinning is associated to the rate between the applied magnetic field and the first matching field, it must be an integer or a rational fraction. Here, we show that another condition must be satisfied, the vortex ground state must be ordered. Our calculations show that the pinning size and strength may dramatically change the vortex ground state. Small pinning radius and high values of pinning strength may lead to disordered vortex configurations, which fade the critical force peaks. The critical forces show anisotropic behavior, but the same dependence on pinning strength and radius is observed for both driven force directions. Different to cases where the applied magnetic field is higher than the first matching field, here the depinning process begins with vortices weakly trapped on top of a pinning site and not with interstitial vortices. Our results are in good agreement with recent experimental results.     

Driving an individual vortex in the presence of a periodic pinning array

Recently it has been demonstrated experimentally that it is possible to manipulate an individual vortex in a type-II superconductor using a magnetic force microscope tip. Using numerical simulations, we investigate the dynamics of a single driven vortex in the presence of a periodic pinning array and other vortices. Remarkably, we find that the effective drag on the driven vortex is reduced at the matching fields, which is opposite from the behavior of the critical current when all the vortices are driven. We discuss this effect in the context of the type of dynamics that occur at matching and nonmatching fields.     

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.     

Dynamical phase transition of a driven vortex lattice with disordered pinning

We have numerically solved the overdamped equation of vortex motion in a two-dimensional driven vortex lattice with disordered pinning, in which the driving Lorentz force, the pinning force due to point defects, the intervortex interacting force, and the thermal fluctuation force are taken into account. It is found that the vortex density and pinning strength are two important factors of affecting the melting transition of a vortex lattice. At low magnetic fields, there exist hysteresis loops of the average vortex velocity and the average pinning force vs. the driving force, from which the feature of a first-order melting transition of the vortex motion can be clearly seen. As the magnetic field is increased beyond a critical value, the hysteresis loops disappear and the melting transition is replaced by a second-order glass transition. We have also studied the influence of intervortex interactions on the vortex melting transition by comparing several forms of repulsive forces between the vortices.     

Pinning effects on the vortex critical velocity in type-II superconducting thin films

We study the influence of artificial pinning centers on the vortex critical velocity in Al thin films deposited on top of a periodic array of Permalloy (FeNi) square rings. We demonstrate that the field dependence of the flux flow velocity strongly depends on the particular magnetic state of the rings. In particular, we find that, even when the rings are in a flux closure state, i.e. with little stray field, the vortex critical velocity shows a non-monotonic magnetic field dependence. This behaviour is in sharp contrast with the results obtained in a reference plain film, with no rings underneath. A comparison with the intrinsic strong pinning Nb films previously studied, suggests an interpretation in terms of a channel-like motion of vortices, here induced by the artificial pinning structure.     

The volume pinning force for periodical interaction forces between defects and the flux line lattice

The volume pinning force for some forms of the interaction potential defect-flux line is calculated without restrictions on the vortex lattice distance and the interaction range of defects. It is shown that for larger maximum elementary interaction forces, the direct summation of pinning forces is realistic. However, if the interaction range of the defects is smaller than the vortex lattice distance, one obtains a region (about one order of magnitude) in which Labusch's quadratic dependence of the volume pinning force on the elementary interaction force is valid. In the region where the direct summation of pinning forces occur, the volume pinning force is proportional the vortex lattice distance and one obtains an additional magnetic field dependence of the volume pinning force.     

Peak effect in the critical current of type II superconductors with strong magnetic vortex pinning

We perform 2D Langevin simulations studying the peak effect (PE) of the critical current taking into account the temperature dependence of the competing forces. We observe and report that the PE results from the competition of vortex-vortex interactions and vortex-pin interactions which have different temperature dependencies. The simulations reveal that the PE can take place only for certain pinning strengths, densities of pinning centers, and driving forces, which is in good agreement with experiments. No apparent vortex order-disorder transition is observed across the PE regime. In addition, the PE is a dynamical phenomenon, and thermal fluctuations can speed up the process for the formation of the PE.     

Vortex washboard voltage noise in type-II superconductors

In order to characterize flux flow through disordered type-II superconductors, we investigate the effects of columnar and point defects on the vortex velocity/voltage power spectrum in the driven non-equilibrium steady state. We employ three-dimensional Metropolis Monte Carlo simulations to measure relevant physical observables including the force-velocity/current-voltage (I-V) characteristics, vortex spatial arrangement and structure factor, and mean flux line radius of gyration.Our simulation results compare well to earlier findings and physical intuition.We focus specifically on the voltage noise power spectra in conjunction with the vortex structure factor in the presence of weak columnar and point pinning centers. We investigate the vortex washboard noise peak and associated higher harmonics,and show that the intensity ratios of the washboard harmonics are determined bythe strength of the material defects rather than the type of pins present.Through varying columnar defect lengths and pinning strengths as well as magnetic flux density we further explore the effect of the material defects on vortex transport.It is demonstrated that the radius of gyration displays quantitatively uniquefeatures that depend characteristically on the type of material defects presentin the sample.     

Three-dimensional Ginzburg-Landau simulation of a vortex line displaced by a zigzag of pinning spheres

A vortex line is shaped by a zigzag of pinning centers and we study here how far the stretched vortex line is able to follow this path. The pinning center is described by an insulating sphere of coherence length size such that in its surface the de Gennes boundary condition applies. We calculate the free energy density of this system in the framework of the Ginzburg-Landau theory and study the critical displacement beyond which the vortex line is detached from the pinning center.     

NUMERICAL STUDIES ON THE DYNAMICS OF DISORDERED VORTEX LATTICE

Flow behavior of the driven two-dimensional vortex lattice is numerically studied with different densities of randomly distributed pointlike pinning centers. Different features in the curves of velocity-force dependence are found between the elastic and plastic regimes. Scaling fit between force and velocity above the critical driving force can be obtained in the elastic regime but fails in the plastic regime. Transition from the lastic to plastic regimes is accompanied by maximum peaks in the differential curves of velocity-force dependence in the disordered vortex lattice.     

Optical sorting using an array of optical vortices with fractional topological charge

An array of optical vortices with fractional topological charge is generated using a phase-only Talbot array illuminator and used to sort microparticles. Our theoretical analysis and experimental results reveal that when a particle passes through a fractional vortex array, it will be driven by two forces, intensity-gradient force and phase-gradient force, and the cooperation of these two forces can improve its ability in optical sorting because of the special intensity and phase distribution of the fractional optical vortex array. Larger angle separation could be obtained with moderate laser power.     

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.     

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.     

Flux vortices and transport currents in type II superconductors

This article is concerned with the mechanisms by which type II superconductors can carry currents. The equilibrium properties of the vortex lattice are described and the generalized driving force in gradients of temperature and field is derived using irreversible thermodynamics. This leads to expressions for thermal cross effects which can include pinning forces. The field distributions which occur in a range of situations are derived and a number of useful solutions of the critical state given. In particular, the distribution in a longitudinal field is obtained, and the conditions under which force-free configurations can break down by the cutting of vortices discussed. The effects of lattice rigidity on the summation of pinning forces is considered and it is shown that a summation based on statistical arguments uses the same approximations and leads to the same results as a dissipation argument. Theoretical expressions are derived for the vortex pinning interaction to a number of different metallurgical defects. The theoretical models are compared critically with experimental measurements of pinning forces and other related phenomena, such as flux creep, low amplitude vortex oscillations and vortex lattice defect effects. Finally, the implications for technological materials are assessed.     

Numerical calculation of elastic pinning parameters by point pins

One-dimensional numerical simulations of the pinned vortex lattice have been carried out using an inverted model in which widely spaced and weak point pinning centres are distorted by a rigid lattice. This allows us to simulate a much larger array than if we had to find the position of each vortex separately. Using steepest descent and molecular dynamics methods, the Labusch parameter is found to have similar power law dependencies on the density of pins and elementary pinning force to the bulk pinning force. This implies that the maximum reversible displacement is a constant and it is found to be roughly 0.2 of a vortex spacing. The results are in agreement with the Larkin-Ovchinnikov theory of collective pinning. Quasi-static shifting of the pins over the rigid vortex lattice allowed the size of avalanches in the flow state to be measured. The distribution of avalanche sizes displays power law behaviour over several decades.     

Collective multivortex states in periodic arrays of traps

We examine the vortex states in a 2D superconductor interacting with a square array of pinning sites. As a function of increasing pinning size or strength we find a series of novel phases including multivortex and composite superlattice states such as aligned dimer and trimer configurations at individual pinning sites. Interactions of the vortices give rise to an orientational ordering of the internal vortex structures in each pinning site. We also show that these vortex states can give rise to a multistage melting behavior.     

Resistive state of superconducting structures with fractal clusters of a normal phase

The effect of morphologic factors on magnetic flux trapping and critical currents in a superconducting structure, which presents a type II percolation superconductor with pinning centers, is considered. The role of pinning centers is played by fractal clusters of the normal phase. The properties of these clusters are analyzed in detail: their statistics is studied, the distribution of critical currents of depinning is found, and the depen-dences of the main statistical parameters on the fractal dimension are obtained. The effect of fractal clusters of the normal phase on the electric field caused by the motion of the magnetic flux after the vortices have been broken away from pinning centers is considered. The current-voltage characteristics of superconducting structures in a resistive state are obtained for an arbitrary fractal dimension. It is found that the fractality of the boundaries of normal-phase clusters forces magnetic flux trapping, thereby increasing the critical current.     

Critical vortex line length near a zigzag of pinning centers

A vortex line passes through as many pinning centers as possible on its way from one extremety of the superconductor to the other at the expense of increasing its self-energy. In the framework of the Ginzburg-Landau theory we study the relative growth in length, with respect to the straight line, of a vortex near a zigzag of defects. The defects are insulating pinning spheres that form a three-dimensional cubic array embedded in the superconductor. We determine the depinning transition beyond which the vortex line no longer follows the critical zigzag path of defects.Received: 23 July 2004, Published online: 26 November 2004PACS: 74.80.-g Spatially inhomogeneous structures - 74.25.-q General properties; correlations between physical properties in normal and superconducting states - 74.20.De Phenomenological theories (two-fluid, Ginzburg-Landau, etc.)     

Flux vortices and transport currents in type II superconductors

This article is concerned with the mechanisms by which type II super-conductors can carry currents. The equilibrium properties of the vortex lattice are described and the generalized driving force in gradients of temperature and field is derived using irreversible thermodynamics. This leads to expressions for thermal cross effects which can include pinning forces.

The field distributions which occur in a range of situations are derived and a number of useful solutions of the critical state given. In particular, the distribution in a longitudinal field is obtained, and the conditions under which force-free configurations can break down by the cutting of vortices discussed.

The effects of lattice rigidity on the summation of pinning forces is considered and it is shown that a summation based on statistical arguments uses the same approximations and leads to the same results as a dissipation argument. Theoretical expressions are derived for the vortex pinning interaction to a number of different metallurgical defects. The theoretical models are compared critically with experimental measurements of pinning forces and other related phenomena, such as flux creep, low amplitude vortex oscillations and vortex lattice defect effects. Finally, the implications for technological materials are assessed.