Role of Interstitial Pinning in the Dynamical Phases of Vortices in Superconducting Films

We analyze the vortex dynamics in superconducting thin films with a periodic array of pinning centers. In particular, we study the effect of anisotropy for a Kagomé pinning network when longitudinal and transverse transport currents are applied. By solving the equations of motion for the vortex array numerically at zero temperature, we find different phases for the vortex dynamics, depending on the pinning and driving force. An unusual sequence of peaks for driving force along and perpendicular to the main lattice axes is observed for the differential resistance, reflecting the anisotropy of the transport properties and the complex behavior of the vortex system. This behavior may be understood in terms of interstitial pinning vacancies, which create channels of vortices with different pinning strengths.     

Ginzburg–Landau Study of Superconductor with Regular Pinning Array

The vortex distributions and dynamics in superconductors with triangular and honeycomb pinning arrays are investigated by numerical simulation of the two- dimensional (2-D) time-dependent Ginzburg–Landau equations. Periodic boundary conditions are implemented through specific gauge transformations under lattice translations. We model the pinning sites as holes. The simulation results at different magnetic fields are presented. For film with regular triangular pinning array, the vortices are all captured within the holes for a wide range of magnetic fields. For film with regular honeycomb pinning array, the interstitial vortices appear at relatively low magnetic fields. With an increase of magnetic field, the new vortices may enter the holes again and keep the number of vortices at the interstitial positions unchanged. These results confirm our explanations of the experimental results we obtained earlier.     

Critical Currents and Melting Temperature of a Two-Dimensional Vortex Lattice with Periodic Pinning

The critical current and melting temperature of a vortex system are analyzed. Calculations are made for a two-dimensional film at finite temperature with two kinds of periodic pinning: hexagonal and Kagomé. A transport current parallel and perpendicular to the main axis of the pinning arrays is applied and molecular dynamics simulations are used to calculate the vortex velocities to obtain the critical currents. The structure factor and displacements of vortices at zero transport current are used to obtain the melting temperature for both pinning arrays. The critical currents are higher for the hexagonal pinning lattice and anisotropic for both pinning arrays. This anisotropy is stronger with temperature for the hexagonal array. For the Kagomé pinning lattice, our analysis shows a multi stage phase melting; that is, as we increase the temperature, each different dynamic phase melts before reaching the melting temperature. Both the melting temperature and critical currents are larger for the hexagonal lattice, indicating the role for the interstitial vortices in decreasing the pinning strength.     

Numerical simulation of collective phenomena in Josephson junction networks

Collective pinning phenomena of vortices in Josephson junction networks (JJN) in magnetic fields are studied using numerical simulations. We consider two kinds of structure of JJN: a ladder (Josephson junction ladder, JJL) and a two-dimensional square array (Josephson junction array, JJA). For the JJL and JJA with distribution of the strengths of junction critical currents, we investigate the critical current of vortex depinning in the presence of bias currents. On the basis of a theory of collective vortex pinning, it is found that the critical currents for both JJL and JJA show a universal scaling behavior.     

Flux vortex dynamics and electric fields in matched pinning systems

The pinning of flux vortices in type II superconductors has been the subject of extensive research. Certain experiments have attempted to investigate this problem by the use of specially prepared pinning structures consisting of regular arrays of pinning centers. In this paper a theory relating to such experiments is described. This theory is based on the existence and properties of defects in an otherwise perfect vortex lattice which is commensurate with a pinning array consisting of a triangular lattice of holes in a superconducting thin film. A quantitative treatment predicts the existence and position of substructure on the critical current versus magnetic field curves in addition to the main peaks previously predicted to occur when the vortex and hole lattices are exactly matched. The theory also qualitatively describes the overall shape of these curves. An analysis of the temperature dependence of this substructure shows broad agreement with existing experimental results. The application of this theory to future experiments should allow a detailed investigation of vortex lattice elasticity and flux flow.     

Vortex equilibrium in film flow near a defect

A simple model based on numerical simulations in bulk is used to examine the pinning characteristics of vortices in helium films adsorbed in the vicinity of substrate defects. The film profile in the absence of a vortex is first calculated including both the Van der Waals potential of a bump defect and surface tension. The displacement and ultimate de-pinning of a vortex line constrained within the profile is then followed as a function of an imposed superfluid flow. The behavior of the vortices lends insight into recent experimental results sensitive to the presence of pinned vortices and indicates that the relevant pinning sites are atomic in dimension.     

Statics and Dynamics of Vortex Matter with Competing Repulsive and Attractive Interactions

We numerically examine vortex matter with repulsive interactions at short range and attractive interactions at long range in the presence of periodic pinning arrays. Such competing vortex interactions are predicted to occur in multiband superconductors or type-I and type-II hybrid materials. For weak pinning, the vortices form cluster states, while for strong pinning, the vortices form uniform states in which flux is evenly distributed among the pinning sites. As a function of external drive, the weak pinning system exhibits clump depinning with no structure changes, while the strong pinning system depins into a disordered fluctuating state followed by a transition to a partially aligned stripe state. For weak pinning, there is a single peak in the differential conductivity, while for strong pinning there are two peaks, indicating that transport curves could be used to distinguish between vortex systems with purely repulsive interactions and those with additional long range attractive interactions.     

Artificial vortex pinning arrays in superconducting films deposited on highly ordered anodic alumina templates

A simple procedure is described for creating periodic vortex pinning centers in thin superconducting NbN films. We report on three different strategies which involve the use of highly ordered alumina templates. In this approach, NbN thin films are deposited either on the porous face of the template made of a triangular array of nanoholes or on the triangular array of bumps formed by the barrier layer or even on the top of perpendicularly oriented ferromagnetic nanowire arrays obtained by electrochemical deposition, thus forming superconductor-ferromagnet hybrids. In all cases, the ordered template allows NbN films to form a periodic pinning array during its growth. The interpore (or inter-bump) distance ranged between 50 and 100?nm and adjustable pore (or wire) diameter was varied between 30 and 60?nm. Numerous matching effects have been observed up to 2.5?T and are maintained at low temperature. These fields are considerably higher than those typical for periodic pinning arrays made by lithographic techniques, which reflects the benefits of nanostructuring superconductors by using self-organized growth to enhance vortex pinning in a large field and temperature range.     

Vortex Dynamics in a Superconducting Film with a Kagome and?a?Honeycomb Pinning Landscape

The vortex dynamics in a superconducting thin Al film with a periodic Honeycomb or Kagome array of antidots has been investigated by electrical transport measurements. The large values of the superconducting coherence length and penetration depth of the Al films guarantee a maximum of one flux quantum trapped per pinning site. This allows us to directly compare the experimental results with previous theoretical investigations based on molecular dynamics simulations. For the Kagome lattice, two submatching features not anticipated theoretically at H/H ₁=1/3 and 2/3, where H ₁ is the field at which the number of vortices coincides with the number of pinning sites, are observed. Possible corresponding stable vortex patterns are suggested. For the Honeycomb pinning landscape, the commensurability effects are in agreement with the theoretical expectations. A preliminary analysis of the vortex mobility in this lattice shows the presence of a weak vortex guidance.     

Phase Diagram of Vortices in Type-II Superconductors with Periodic Square Columnar Pins

The phase diagram of a two-dimensional vortex system with periodic square columnar pins is studied. For the case of vortex number matching pinning number, we find two scenarios for the freezing transitions. For weak pinning where vortex–vortex interactions dominate, the vortex liquid is frozen into triangular lattice via a first-order phase transition. For strong pinning, the vortex liquid is frozen into a square lattice with all vortices trapped by pins via a second-order phase transition.     

Depinning of Two-Dimensional Vortex System with Square Pinning Array at the Second Matching Field

Depinning of a two-dimensional vortex system at the second matching field B=2B _? in the presence of square pinning array was studied by molecular dynamics simulations. The annealed ground structures and depinning properties are found to be dependent on pinning strength. For strong pinning, we find a two-step depinning transition with a lower depinning force F _c1 and a higher one F _c2. At F _c1 only the interstitial vortices start to move, while at F _c2 the vortices at pinning sites are depinned and all vortices move.     

Vortex Dynamics in Bi2Sr2CaCu2O8 Single Crystal with Low Density Columnar Defects Studied by Magnetic Force Microscope

We have studied vortex dynamics in Bi₂Sr₂CaCu₂O₈ single crystal with low density columnar defects by using a magnetic force microscope. Single crystal Bi₂Sr₂CaCu₂O₈ sample was irradiated by 1.3 GeV uranium ion to form artificial pinning centers along the crystalline c-axis. The irradiation dose corresponded to a matching field of 20 gauss. The radius of an individual vortex is approximately 140 nm, which is close to the penetration depth of this material. Magnetic force microscope (MFM) images show that intrinsic crystalline defects such as stacking fault dislocations are very effective pinning centers for vortices in addition to the pinning centers due to ion bombardment. By counting the number of vortex, we found that the flux trapped at each pinning center is a single flux quantum. At higher magnetic field, the vortex structure showed an Abrikosov lattice disturbed only by immobile vortices located at pinning centers. When increasing or decreasing the external magnetic field, the spatial distribution of vortices showed a Bean model like behavior.     

Pinning Effects in Nb Thin Films with Artificial Pinning Arrays

We summarize some results on the behavior of vortex dynamics and pinning effects in superconducting films with artificial pinning centers. Superconducting thin films with regular arrays of holes were fabricated using electron-beam lithography and reactive dry etching techniques. Vortex dynamics in the mixed state in type II superconductors is strongly influenced by the presence of defects, which act as pinning centers. Periodic critical current matching peaks were observed in magnetotransport measurements. The matching effect is caused by the interplay between the pinning centers and vortex lattice. Therefore, vortex lattice behaviors are changed for different temperatures and the geometry of the pinning centers. Molecular dynamic simulations are made to study this phenomenon. The ground state distribution of vortices obtained from simulations can give a reasonable explanation of the prominent matching peaks we found in the experiments.     

Computer simulation of vortex pinning in type II superconductors. I. Two-dimensional simulation

The summation of pinning forces to a volume force exerted on the vortex lattice in type II superconductors allowing it to carry a loss-free current is not fully understood. In order to clarify this question we have started computer simulations of flux pinning. It is shown that for many experimental situations bending of vortices may be neglected since the vortices are too short or pinning is too weak, and thus pinning is two-dimensional. As a first step, two-dimensional pinning simulations will thus be instructive with regard to, say, ribbons of amorphous metals. A general expression for the energy of a vortex-pin system in two and three dimensions is given. The simulation method is presented and illustrated for the isolated pin (with a detailed discussion of the “threshold effect” and of elastic instabilities) and for pin “walls” (grain boundaries) and “nozzles.” Random point pins acting on a perfect or defective vortex lattice are treated in an accompanying paper.     

Asymmetry of the Pinning Force in Thin Nb Films in Parallel Magnetic Field

The pinning force, F _p, is studied in Nb films of different thickness in parallel magnetic field H. The asymmetry in the magnetic field dependence of F _p has been observed for two opposite directions of the transport current. The effect is less pronounced for thin and thick films where, respectively, single vortex pinning and pinning of the internal vortices, is relevant. At intermediate thickness, where the pinning mechanism is mostly caused by surface effects, an asymmetry in the F _p(H) dependence is clearly visible. The different surface barriers that vortices should overcome to enter the sample from opposite sides of the film explain the effect, as confirmed by numerical calculations. These have been obtained by solving the Ginzburg?CLandau equations with asymmetric boundary conditions which take into account the different superconducting properties of the film?Csubstrate and film?Cvacuum interface. Such difference can also explain the reduction of the critical current usually observed in thin films as a function of their thickness.     

The Effects of Pinning on Critical Currents of Superconducting Films

Using molecular dynamics simulations, we analyze the effects of artificial periodic arrays of pinning sites on the critical current of superconducting thin films as a function of vortex density. We analyze two types of periodic pinning array: hexagonal and Kagomé. For the Kagome pinning network we make calculations using two directions of transport current: along and perpendicular to the main axis of the lattice. Our results show that the hexagonal pinning array presents higher critical currents than the Kagomé and random pinning configuration for all vortex densities. In addition, the Kagomé networks show anisotropy in their transport properties.     

Surface Effects on the Dynamic Behavior of Vortices in Type II Superconducting Strips with Periodic and Conformal Pinning Arrays

Using molecular dynamics techniques, we simulate the vortex behavior in a type II superconducting strip in the presence of triangular and two types of conformal pinning arrays, one with a pinning gradient perpendicular to the driving force (C1) and the other parallel (C2), at zero temperature. A transport force is applied in the infinite direction of the strip, and the magnetic field is increased until the rate between the density of vortices (n _v ) and pinning (n _p ) reaches n _v /n _p =?1.5. Our results show a monotonic increase, by steps, of the vortex density with the applied magnetic field. Besides, each pinning lattice presents a different vortex penetration delay. For the triangular pinning array, different than the case of infinite films, here the system exhibits only one pronounced depinning force peak at n _v /n _p =?1. However, the depinning force peak is present for only one value of field, in the range of fields where n _v /n _p =?1 is stable. For the case of conformal pinning arrays, in analogy to what is observed in infinite films, no commensurability depinning force peaks were found. In the present case, the C1 array is more efficient at low fields, all arrays are equivalent in the intermediate fields, and the C2 array is more efficient for high fields. We also show that for the C1 array at high fields, vortices depin following the conformal arches, from the edge to the center. For the C2 array, the depinning force is higher when compared to that of C1, because this particular conformal structure prevents the formation of easy vortex flow channels.     

Stable Vortex Configurations in a Cylinder

We calculate stable arrangements for a single superfluid vortex pinned to the wall of a stationary cylindrical container. We find that, independent of the details of the pinning site, stable vortices must subtend most of the cell horizontally and cannot be vertical or nearly vertical. More generally, the geometry of a container severely limits the possible vortex configurations, making macroscopic trapped vortices less common than previously believed.     

Dynamical Properties of Vortices in a Bose-Einstein Condensate in a Rotating Lattice

We present simulation results of the vortex dynamics in a trapped Bose-Einstein condensate in the presence of a rotating optical lattice. Changing the potential amplitude and the relative rotation frequency between the condensate and the optical lattice, we find a rich variety of dynamical phases of vortices. In particular, when the optical lattice rotates faster than the condensate, the competition between the pinning force and the interactions by nucleated interstitial vortices leads to the melting of vortex lattice, yielding a vortex liquid phase.