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.     

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.     

Relation Between Vortex Pinning Energy and Anderson's Theorem

We discuss the elementary vortex pinning in type-II superconductors in connection with the Anderson's theorem for nonmagnetic impurities. We address the following two issues. One is an enhancement of the vortex pinning energy in the unconventional superconductors. This enhancement comes from the pair-breaking effect of a nonmagnetic defect as the pinning center far away from the vortex core (i.e., the pair-breaking effect due to the non-applicability of the Anderson's theorem in the unconventional superconductors). The other is an effect of the chirality on the vortex pinning energy in a chiral p-wave superconductor. The vortex pinning energy depends on the chirality. This is related to the cancellation of the angular momentum between the vorticity and chirality in a chiral p-wave vortex core, resulting in local applicability of the Anderson's theorem (or local recovery of the Anderson's theorem) inside the vortex core.     

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.     

Vortex lattice melting,pinning and kinetics

The phenomenology of the high T_c superconductors is discussed both at the level of the thermodynamics of melting of the Abrikosov flux lattice and in terms of the melting and kinetics of the flux lattice for a pinned system. We review results on 3D melting obtained by a Monte Carlo simulation approach in which the 2D pancake vortices are treated as statistical variables. We discuss pinning in the context of the strong pinning regime in which the vortex density given in terms of the applied field B is small compared to that represented by an effective field B_pin measuring the pinning center density. We introduce a new criterion for the unfreezing of a vortex glass on increase of magnetic field or temperature, in the strong pinning, small field limit. We model this limit in terms of a single flux line interacting with a columnar pin. This model is studied both analytically and by computer simulation. By applying a tilt potential, we study the kinetics of the vortex motion in an external current and show that the resulting current-voltage characteristic follows a basic vortex glass-like scaling relation in the vicinity of the depinning transition.     

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.     

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.     

Vortex Pinning to a Solid Sphere in Helium II

We report here the results of a computer simulation of quantized vortex pinning in He II at 0 K in the simple situation where a single sphere and a rectilinear vortex are considered. Our simulation shows that a vortex nearby a sphere is captured due to a velocity field produced by the sphere, exciting Kelvin waves. The dependence of the “pinning” on temperature is investigated as well. Finally, the possibility of pinning in PIV experiments, which visualize superfluid turbulence, is discussed.     

On the threshold pinning force in type II superconducting films

The maximum pinning force of a two-dimensional vortex lattice in a random potential is calculated. A connection is established between this threshold pinning force and the potential energy discontinuities due to elastic and plastic instabilities of the vortex lattice. Inspired by recent computer simulations, we assume that the fluctuations in the commensurability between the random potential and the vortex potential breaks the vortex system up into a set of flowing channels in between trapped regions. Two instability mechanisms and their contribution to the threshold force are discussed within this channel-flow picture. We find that three different regimes exist depending on, w, the width of the channels;w=,a ₀w=a ₀ , wherea ₀ is the vortex lattice spacing. Weak pinning superconductors can pass through all three regimes as the reduced magnetic field is varied from 0 to 1, whereas strong pinning compounds can remain in the saturated region (w=a ₀ ) for all values of the field. We compare the expression for the threshold force with experimental results for both strong and weak pinning samples. A satisfactory qualitative agreement is obtained between theory and experiment.     

On the theory of vortex motion in an inhomogeneous superconducting film

A phenomenological theory is presented giving an account of the pinning effect which occurs when vortex motion in a superconducting film is hindered by inhomogeneities of the material. In particular, the theory incorporates the interaction (pinning) potential between a vortex and the material defects, as well as long wavelength distortion of the vortex lattice. A recent observation of quantum interference effects of a moving vortex lattice can be explained successfully. It is shown how the pinning potential and the elastic constants can be obtained from experiment.     

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.     

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.     

The Vortex-Magnetic Dipole Interaction in the London Approximation

The interaction between a vortex in a superconducting film and a magnetic dipole with in- or out-of-plane magnetization is studied within the London approximation. We investigate the magnetic pinning properties of such magnetic dipoles. The dependence of the interaction energy on the parameters of the system is investigated and analytical results are obtained in limiting cases.     

Vortex Matter in Highly Strained Nb$$_{}$$Zr$$_{25}$$25: Analogy with Viscous Flow of Disordered Solids

We present the results of magnetization and magneto-transport measurements in the superconducting state of an as-cast Nb\(_{75}\)Zr\(_{25}\) alloy. We also report the microstructure of our sample at various length scales by using optical, scanning electron and transmission electron microscopies. The information of microstructure is used to understand the flux pinning properties in the superconducting state within the framework of collective pinning. The magneto-transport measurements show a non-Arrhenius behaviour of the temperature- and field-dependent resistivity across the resistive transition and is understood in terms of a model for viscous flow of disordered solids which is popularly known as the ‘shoving model’. The activation energy for flux flow is assumed to be mainly the elastic energy stored in the flux-line lattice. The scaling of pinning force density indicates the presence of two pinning mechanisms of different origins. The elastic constants of the flux-line lattice are used to estimate the length scale of vortex lattice movement, or the volume displaced by the flux-line lattice. It appears that the vortex lattice displacement estimated from elastic energy considerations is of the same order of magnitude as that of the flux bundle hopping length during flux flow. Our results could provide possible directions for establishing a framework where vortex matter and glass-forming liquids or amorphous solids can be treated in a similar manner for understanding the phenomenon of viscous flow in disordered solids or more generally the pinning and depinning properties of elastic manifolds in random media. It is likely that the vortex molasses scenario is more suited to explain the vortex dynamics in conventional low-T\(_C\) superconductors.     

Temperature and Current Dependent Matching Fields in?Superconducting NbN Film

Patterned superconducting thin films having a periodic array of submicrometric pinning centers have been of great interest due to their excellence for the studies of the vortex pinning mechanisms in the type-II superconductors. Square hole array has been fabricated over a micro-bridge 60 mm??60 mm of NbN thin film by electron beam lithography. Previous works have been carried out in Nb, Pb and Al thin films where the vortex pinning effect is assumed to be small. In this work, we study the matching pinning effect by the artificial hole array in superconducting NbN thin films. We observed the interplay between the vortex quantization and the artificial hole array. Magneto-resistance minima at integer matching fields up to five times of H ₁ (the first matching field corresponding to one vortex inside each hole) and fractional matching fields at 1/2H ₁, 3/2H ₁ and 5/2H ₁ have been observed.     

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.     

Superconducting and Structural Properties of Nb/PdNi/Nb Trilayers

The superconducting and structural properties of S/F/S (Superconductor/Ferromagnet/Superconductor) heterostructures have been studied by means of microwave measurements (1–20 GHz) and x-ray absorption fine structure (XAFS) spectroscopy. Nb/PdNi/Nb trilayers have been studied as a function of F layer thickness. With respect to pure Nb, XAFS analysis shows that the heterostructures exhibit larger structural disorder in the S layers. Microwave measurements show evidence for a progressively weaker vortex pinning with increasing F thickness. However, no clear correlation is found with the local disorder in Nb: the weakest pinning is not in the most disordered trilayer. Therefore, the structural disorder in the superconducting material cannot explain on its own the changes in vortex pinning. We argue that the F layer acts on the superconducting state itself. We propose possible explanations for the observed behavior.     

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.     

Field Dependent Microwave Resistivity in YBa2Cu3O7−δ

We investigate the effects of directional pinning due to the layered structure and to columnar defects on the microwave response in YBa₂Cu₃O_7– films. We present measurements of the field-induced microwave resistivity at 48 GHz and 21 GHz taken in various relative orientations between the microwave currents, dc magnetic field and (a, b) planes. From measurements taken in the Lorentz-force-free configuration we experimentally show the relevance of the magnetic field induced increase of quasiparticle (qp) density. We identify the vortex motion contribution, and extract the vortex parameters. We estimate a pinning frequency of order 30 GHz when the field is aligned to the (a, b) plane. Secondly, we show that the introduction of columnar defects gives rise to a strong pinning along the columns, detectable even at 48 GHz. The pinning frequency appears to be of the same magnitude than for pinning by the layered structure.     

Dynamic Transitions within the Vortex Solid of YBa2Cu3Oy Single Crystals

Transport measurements on YBa ₂ Cu ₃ O _y have revealed strong history dependences in pinning properties of the vortex solid. By using an annealing current to control exposure of the vortex system to the disorder potential, we were able to study the development of such hysteretic responses, across the region of the vortex freezing transition. We have explained the observed behaviour in terms of coexistent vortex liquid and solid phases.