Vortex Lattice Dynamics Research Articles

Integrability and Chaos in Vortex Lattice Dynamics

This paper is concerned with the problem of the interaction of vortex lattices, which is equivalent to the problem of the motion of point vortices on a torus. It is shown that the dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate configurations are found and their stability is investigated. For two vortex lattices it is also shown that, in absolute space, vortices move along closed trajectories except for the case of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero total strength are considered. For three vortices, a reduction to the level set of first integrals is performed. The nonintegrability of this problem is numerically shown. It is demonstrated that the equations of motion of four vortices on a torus admit an invariant manifold which corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices on this invariant manifold and on a fixed level set of first integrals are obtained and their nonintegrability is numerically proved.

Topological defect dynamics of vortex lattices in Bose-Einstein condensates

Vortex lattices in rapidly rotating Bose--Einstein condensates are systems of topological excitations that arrange themselves into periodic patterns. Here we show how phase-imprinting techniques can be used to create a controllable number of defects in these lattices and examine the resulting dynamics. Even though we describe our system using the mean-field Gross--Pitaevskii theory, the full range of many particle effects among the vortices can be studied. In particular we find the existence of localized vacancies that are quasi-stable over long periods of time, and characterize the effects on the background lattice through use of the orientational correlation function, and Delaunay triangulation.

Disorder-induced vortex lattice melting in a Bose-Einstein condensate

We study the vortex lattice dynamics in presence of single impurity as well as random impurities or disorder. We show that in presence of a single impurity the vortex lattice gets distorted and the distortion depends on the position of the single impurity with respect to the positions of the vortices in the impurity free Abrikosov vortex lattice and also the strength of the impurity potential. We then show that a new type of giant hole with hidden vortices inside it can be created in the vortex lattice by a cluster of impurities. In presence of random impurity potential or disorder the vortex lattice melts. We show that the vortex lattice also melts in presence of pseudorandom potential generated by the superposition of two optical lattices. The absence of long-range order in the melted vortex lattice is demonstrated from the structure factor profile and the histogram of the distances between each pair of the vortices.

Proximity effects and vortex dynamics in superconducting heterostructures

Vortex lattice dynamics has been studied in thin Nb superconducting films sputtered on top of a dense triangular array of V dots with and without an intermediate SiO2 insulating layer. While the insulating layer modifies only slightly the Nb film corrugation, it reduces superconducting commensurability effects (CE) substantially. This implies that superconducting commensurability is dominated by proximity effects. Moreover, the HC2 (T) phase diagram of the sample without an insulating layer shows a parabolic temperature dependence near TC and critical temperature oscillations with the periodicity of the matching field. Therefore, strong proximity effects locally suppress superconductivity leading to a superconducting mesh. When the proximity effect is decreased by an insulating layer, HC2(T) follows the expected linear T dependence.

Anomalies in vortex lattice dynamics driven by induced ac currents in superconducting films with magnetic arrays of two-fold symmetry

We study the dynamics of the vortex lattice driven by ac induced currents in the critical state regime, for T > 0.70 TC. The samples are superconducting films grown on top of two-fold symmetry array of magnetic dots. In these heterostructures, the induced ac currents flow parallel to the short and to the long side of the pinning array in different areas of the samples simultaneously. This behavior produces remarkable effects in the vortex lattice dynamics. First of all, periodic features are observed in the ac susceptibility versus applied magnetic field measurements which are related to matching effects between the vortex lattices and the magnetic array. However, the vortex lattice reconfiguration observed in magnetotransport experiments is absent. Some of these features are revealed as maxima instead of being minima, indicating higher mobility at certain matching fields. Competing unstable vortex configurations could lead to increase vortex mobility precluding the reconfiguration transition. At high temperatures, where the matching effects show up, the magnetic permeability of the dots is the mechanism that governs the JC(T) behavior. Moreover, the temperature dependence of the pinning force FP(T) shows a temperature crossover related to an unexpected enhancement in vortex mobility. Vortex–vortex interaction and the interplay between trapped and interstitial vortices are a hint to explain these phenomena.

Superconducting heterostructures: from antipinning to pinning potentials

We study vortex lattice dynamics in a heterostructure that combines two type-II superconductors: a niobium film and a dense triangular array of submicrometric vanadium (V) pillars. Magnetic ac susceptibility measurements reveal a sudden increase in ac penetration, related to an increase in vortex mobility above a magnetic field, , that decreases linearly with temperature. Additionally, temperature independent matching effects that occur when the number of vortices in the sample is an integer of the number of V pillars, strongly reduce vortex mobility, and were observed for the first and second matching fields, and . The angular dependence of , and shows that matching is determined by the normal applied field component, while is independent of the applied field orientation. This important result identifies with the critical field boundary for the normal to superconducting transition of V pillars. Below , superconducting V pillars repel vortices, and the array becomes an ‘antipinning’ landscape that is more effective in reducing vortex mobility than the ‘pinning’ landscape of the normal V sites above . Matching effects are observed both below and above , implying the presence of ordered vortex configurations for ‘antipinning’ or ‘pinning’ arrays.

An efficient spectral method for computing dynamics of rotating two-component Bose–Einstein condensates via coordinate transformation

In this paper, we propose an efficient and accurate numerical method for computing the dynamics of rotating two-component Bose–Einstein condensates (BECs) which is described by the coupled Gross–Pitaevskii equations (CGPEs) with an angular momentum rotation term and an external driving field. By introducing rotating Lagrangian coordinates, we eliminate the angular momentum rotation term from the CGPEs, which allows us to develop an efficient numerical method. Our method has spectral accuracy in all spatial dimensions and moreover it can be easily implemented in practice. To examine its performance, we compare our method with those reported in the literature. Numerical results show that to achieve the same accuracy, our method takes much shorter computing time. We also apply our method to study issues such as dynamics of vortex lattices and giant vortices in rotating two-component BECs. Furthermore, we generalize our method to solve the vector Gross–Pitaevskii equations (VGPEs) which is used to study rotating multi-component BECs.

Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

In this paper, we begin with the nonlinear Schrödinger/Gross–Pitaevskii equation (NLSE/GPE) for modeling Bose–Einstein condensation (BEC) and nonlinear optics as well as other applications, and discuss their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, and dispersion relation to soliton solutions. Then, we review and compare different numerical methods for solving the NLSE/GPE including finite difference time domain methods and time-splitting spectral method, and discuss different absorbing boundary conditions. In addition, these numerical methods are extended to the NLSE/GPE with damping terms and/or an angular momentum rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate a quantized vortex lattice dynamics in a rotating BEC are reported.

Low Temperature Vortex Dynamics in Superconducting Nb Films Containing Square and Rectangular Arrays of Ni Nanodots

We examine the vortex lattice (VL) dynamics in superconducting Nb films containing square and rectangular arrays of Ni nanodots, using ac susceptibility techniques. A remarkable robust feature is the increase in pinning that occurs for magnetic fields that create integer n or half integer flux lines per pinning cell. This phenomenology has been reported in transport experiments performed very close to the sample critical temperature TC. In our contactless experiments we determined vortex mobility and pinning properties in an extended temperature range 0.65 <; T/TC <; 0.99. This was possible after growing larger samples, in the square millimeters range, with top down techniques. For the square pinning array, matching up to n = 8 was observed and pinning for n = 3, 4 and 5 show different characteristics, in agreement with published simulations. For the rectangular array, some of the matching orders are missing and we do not find clear evidence of a reconfiguration transition, where vortex commensuration changes from a rectangular cell to a square cell, as reported near TC in earlier transport experiments.

Vortex lattice motion in the flux creep regime on asymmetric pinning potentials

Vortex lattice dynamics is investigated in the adiabatic regime in superconducting/magnetic nanostructures for strong and weak asymmetric pinning potentials, provided by a rectangular array of Ni triangles and a kagomé-like array of Ni dots, respectively. The AC voltage response of the AC-driven superconducting vortex lattice is measured in addition to the DC voltage response. The results allow us to link voltage rectification effects with the vortex lattice dynamics. In particular, for the weaker pinning kagomé-like array, a high-temperature zero critical current regime is identified, in which both AC and DC voltage responses can be described in terms of a simple model of thermal activation of single vortices over finite asymmetric barriers.

The elastic depinning transition of vortex lattices in two dimensions

Large-scale numerical simulations are used to study the elastic dynamics of two-dimensional vortex lattices driven on a disordered medium in the case of weak disorder. We investigate the so-called elastic depinning transition by decreasing the driving force from the elastic dynamical regime to the state pinned by the quenched disorder. Similarly to the plastic depinning transition, we find results compatible with a second-order phase transition, although both depinning transitions are very different from many viewpoints. We evaluate three critical exponents of the elastic depinning transition. β = 0.29 ± 0.03 is found for the velocity exponent at zero temperature, and from the velocity–temperature curves we extract the critical exponent δ−1 = 0.28 ± 0.05. Furthermore, in contrast with charge-density waves, a finite-size scaling analysis suggests the existence of a unique diverging length at the depinning threshold with an exponent ν = 1.04 ± 0.04, which controls the critical force distribution, the finite-size crossover force distribution and the intrinsic correlation length. Finally, a scaling relation is found between velocity and temperature with the β and δ critical exponents both independent with regard to pinning strength and disorder realizations.

Superconducting vortex dynamics on arrays with bicrystal-like structures: matching and rectifier effects

Hybrid nanostructures of magnetic Ni dots embedded in superconducting Nb films have been fabricated. The dot arrays show bicrystal-like structures with an interface which separates two different dot arrangements in the same array. Vortex lattice dynamics is studied on these particular bi-arrays using magnetotransport measurements. Commensurability and rectifier (ratchet) effects have been observed. Matching between the vortex lattice and the pinning bi-array is only governed by the interplay between the densities of vortices and pinning centers. In bi-arrays with dissimilar pinning densities, for vortex flow parallel to the boundary, the highest pinning center density governs the matching effects. Hybrid samples with only half of the array with asymmetric potentials show a ratchet effect; that is input ac currents yield a net flow of vortices.

Amorphous and Crystalline Magnetic/Superconducting Hybrids: Interplay Between Periodic Defects and Random Defects

Arrays of nanometric Ni dots embedded in amorphous Mo3Si (a-Mo3Si) or crystalline Nb superconducting films modify the vortex lattice dynamics and the dissipation mechanisms, close to critical temperatures. The vortex lattice force vs. velocity characteristics have been measured in both hybrids. The velocities of the ordered and synchronized vortices are the same in both systems (hybrids based on Nb films or a-Mo3Si films), but remarkably the needed driving forces are one order of magnitude smaller in a-Mo3Si hybrids than in Nb based hybrids. This experimental behavior points out magnetic/superconducting hybrids, based on amorphous superconductors films, could be a promising material for controlling vortex motion with very low driving forces.

Time-resolved stroboscopic neutron scattering of vortex lattice dynamics in superconducting niobium

Superconducting vortex lattices, glasses and liquids attract great interest as model systems of crystallization and as a source of microscopic information of the nature of superconductivity. We report for the first time direct microscopic measurements of the vortex lattice tilt modulus c44 in ultra-pure niobium using time-resolved small angle neutron scattering. Besides a general trend to faster vortex lattice dynamics for increasing temperatures we observe a dramatic changeover of the relaxation process associated with the non-trivial vortex lattice morphology in the intermediate mixed state. This changeover is attributed to a Landau-branching of the Shubnikov domains at the surface of the sample. Our study represents a showcase for how to access directly vortex lattice melting and the formation of vortex matter states for other systems.

Vortex lattice dynamics in Nb films with competing intrinsic random and artificial periodicpinning

We study vortex lattice (VL) dynamics in patterned Nb films containing denseperiodic arrays of sub 50 nm magnetic nanodots or holes, by means of acsusceptibility measurements. For both types of samples, we observe matchingeffects within a wide temperature range, determined by the periodicity of thestrong artificial pinning potential. Below a crossover reduced temperaturet* = 0.75, the ac vortex mobility and matching fields are different for increasing ordecreasing fields. Low temperature field cooling experiments indicate that thehysteretic behavior is not related to different intensities of magnetic inductionB in the ac penetrated region of the samples. The fact that hysteresis appears in both patternedsamples when the intrinsic random pinning becomes stronger, together with the absence ofhysteresis in a reference unpatterned Nb film, suggests that the interplay between randomand periodic pinning, independent of its origin, induces VL metastability belowt*, which leads to the observed irreversibility.

Vortex ratchet reversal at fractional matching fields in kagomélike array with symmetric pinning centers

Arrays of Ni nanodots embedded in Nb superconducting films have been fabricated by sputtering and electron beam lithography techniques. The arrays are periodic triangular lattices of circular Ni dots arranged in a kagom\'e-like pattern with broken reflection symmetry. Relevant behaviors are found in the vortex lattice dynamics : i) At values lower than the first integer matching field, several fractional matching fields are present when the vortex lattice moves parallel or perpendicular to the reflection symmetry axis of the array showing a clear anisotropic character in the magnetoresistance curves, ii) injecting an ac current perpendicular to the reflection symmetry axis of the array yields an unidirectional motion of the vortex lattice (ratchet effect) as a result of the interaction between the whole vortex lattice and the asymmetric lattice of dots, iii) increasing the input current amplitudes the ratchet effect changes polarity independently of matching field values. These experimental results can be explained taking into account the vortex lattice density.

Critical behavior of plastic depinning of vortex lattices in two dimensions: Molecular dynamics simulations

Using molecular dynamics simulations, we report a study of the dynamics of two-dimensional vortex lattices driven over a disordered medium. In strong disorder, when topological order is lost, we show that the depinning transition is analogous to a second order critical transition: the velocity-force response at the onset of motion is continuous and characterized by critical exponents. Combining studies at zero and nonzero temperature and using a scaling analysis, two critical expo- nents are evaluated. We find v\sim (F-F_c)^\beta with \beta=1.3\pm0.1 at T=0 and F>F_c, and v\sim T^{1/\delta} with \delta^{-1}=0.75\pm0.1 at F=F_c, where F_c is the critical driving force at which the lattice goes from a pinned state to a sliding one. Both critical exponents and the scaling function are found to exhibit universality with regard to the pinning strength and different disorder realizations. Furthermore, the dynamics is shown to be chaotic in the whole critical region.

Smooth double critical state theory for type-II superconductors

Several aspects of the general theory for the critical states of a vortex lattice and themagnetic flux dynamics in type-II superconductors are examined by a direct variationaloptimization method and widespread physical principles. Our method allows us to unify anumber of conventional models describing the complex vortex configurations in the criticalstate regime. Special attention is given to the discussion of the relation between the fluxline cutting mechanism and the depinning threshold limitation. This is done byusing a smooth double critical state concept which incorporates the so-calledisotropic, elliptical, T and CT models as well-defined limits of our general treatment.Starting from different initial configurations for a superconducting slab in a 3Dmagnetic field, we show that the predictions of the theory range from the collapse tozero of transverse magnetic moments in the isotropic model, to nearly force-freeconfigurations in which paramagnetic values can arbitrarily increase with the applied fieldfor magnetically anisotropic current–voltage laws. Noteworthily, the differencesbetween the several model predictions are minimal for the low applied field regime.

Vortex dynamics in anisotropic traps

We investigate the dynamics of linear vortex lattices in anisotropic traps in two dimensions and show that the interplay between the rotation and the anisotropy leads to a rich but highly regular dynamics.

Intrinsic bulk vortex dynamics revealed by time resolved small angle neutron scattering

Measurements of intrinsic vortex lattice (VL) dynamics in superconductors as for instance VL melting or Bragg glass transitions are typically performed by e.g. macroscopic transport or surface sensitive measurement techniques. Therefore, usually thin superconducting films are used for microscopic measurements of VL dynamics. A direct consequence of using thin films is the strong influence of surface effects and defects, sample quality and geometry. We succeeded to combine time resolved stroboscopic small angle neutron scattering (SANS) with an advanced, time varying magnetic field setup allowing to extend the time window for slow dynamical processes to the range of 10 ms up to several minutes. The new results demonstrate that it is possible to observe directly the intrinsic dynamics of the VL in a bulk niobium single crystal on a microscopic scale without limitations due to surface effects. Field and temperature dependent relaxation times of the VL from 100 to 500 ms could be observed for the first time, allowing to directly measure the VL of the tilt modulus. This new experimental technique provides the possibility to study also the dynamical magnetic properties of various strongly correlated electron systems.