Like orbitals in atoms and molecules, persistent currents may appear in larger aggregates. Though classically, equilibrium magnetic moment of non-rotating objects is forbidden, it is known that equilibrium quantum circulating currents are allowed. Such currents were predicted to be periodic with the external magnetic field. It is shown that in general, whenever a periodic component is found, there must be a large aperiodic component forcing aperiodic contribution to the equilibrium magnetic moment as a function of the external field.
We have studied non-equilibrium phase transitions in the vortex lattice in superconducting MgB2, where metastable states are observed in connection with an intrinsically continuous rotation transition. Using small-angle neutron scattering and a stop-motion technique, we investigated the manner in which the metastable vortex lattice returns to the equilibrium state under the influence of an ac magnetic field. This shows a qualitative difference between the supercooled case which undergoes a discontinuous transition, and the superheated case where the transition to the equilibrium state is conti…
The vortex lattice in MgB2 is characterized by the presence of long-lived metastable states, which arise from cooling or heating across the equilibrium phase boundaries. A return to the equilibrium configuration can be achieved by inducing vortex motion. Here we report on small-angle neutron scattering studies of MgB2, focusing on the structural properties of the vortex lattice as it is gradually driven from metastable to equilibrium states by an AC magnetic field. Measurements were performed using initial metastable states obtained either by cooling or heating across the equilibrium phase tra…
We study the relaxation dynamics of flux lines in dirty high-temperature superconductors using numerical simulations of a London-Langevin model of the interacting vortex lines. By analysing the equilibrium dynamics in the vortex liquid phase we find a dynamic crossover to a glassy non-equilibrium regime. We then focus on the out-of-equilibrium dynamics of the vortex glass phase using tools that are common in the study of other glassy systems. By monitoring the two-times roughness and dynamic wandering we identify and characterize finite-size effects that are similar, though more complex, than …
In this communication we argue that the Vortex Lattice Melting scenario fails to explain several key experimental results published in the literature. From a careful analysis of these results we conclude that the Flux Line Lattice (FLL) does not melt along a material- and sample-dependent boundary $H_j(T)$ but the opposite, it de-couples from the superconducting matrix becoming more ordered. When the FLL depinning is sharp, the difference between the equilibrium magnetization $M_{eq}(T,H)$ of the pinned and unpinned FLL leads to the observed step-like change $ΔM_{eq}(T,H)$. We demonstrate that…
We report a crucial experimental test of the present models of the peak effect in weakly disordered type-II superconductors. Our results favor the scenario in which the peak effect arises from a crossover between the Larkin pinning length and a rapidly falling elastic length in a vortex phase populated with thermally excited topological defects. A thickness dependence study of the onset of the peak effect at varying driving currents suggests that both screw and edge dislocations are involved in the vortex lattice disordering. The driven dynamics in 3D samples are drastically different from tho…
A numerical scheme to study the mixed states in a mesoscopic type-II superconducting cylinder is described. Steady-state configurations and transient behavior of the magnetic vortices for various values of the applied magnetic field H are presented. Transitions between different multi-vortex states as H is changed is demonstrated by contour plots and jumps in the B vs H plot. Evolving a uniformly-superconducting initial state using the simplest set of relaxation equations shows that the system passes through nearly metastable intermediate configurations, while seeking the final minimum-energy,…
Externally driven glassy systems may undergo non-equilibrium phase transitions (NEPTs). In particular, ac-driven systems display special features, like those observed in the vortex matter of $\mathrm{NbSe_2}$, where oscillatory drives reorganize the system into partially ordered vortex lattices. We provide experimental evidence for this dynamic reorganization and we show an unambiguous signature of its connection with an NEPT driven by ac forces. We perform a scaling analysis and we estimate critical exponents for this transition. Our results share similarities with some glass-to-viscous-liqui…
Systems of nearly parallel, slender vortex filaments in which angular momentum is conserved are an important simplification of the Navier-Stokes equations where turbulence can be studied in statistical equilibrium. We study the canonical Gibbs distribution based on the Klein-Majda-Damodaran (KMD) model and find a divergence in the mean square vortex position from that of the point vortex model of Onsager at high temperature. We subsequently develop a free energy equation based on the non-interacting case, with a spherical constraint, which we approximate using the method of Kac-Berlin, adding …
The thermodynamic $H-T$ phase diagram of Bi$_2$Sr$_2$CaCu$_2$O$_8$ was mapped by measuring local \emph{equilibrium} magnetization $M(H,T)$ in presence of vortex `shaking'. Two equally sharp first-order magnetization steps are revealed in a single temperature sweep, manifesting a liquid-solid-liquid sequence. In addition, a second-order glass transition line is revealed by a sharp break in the equilibrium $M(T)$ slope. The first- and second-order lines intersect at intermediate temperatures, suggesting the existence of four phases: Bragg glass and vortex crystal at low fields, glass and liquid …
Vortex-line equilibrium configurations and critical currents for type-II superconducting films at zero-temperature are studied theoretically. The films are assumed to be of thickness less than or equal to the penetration depth, free of pinning by imperfections, and to be subjected to a magnetic field parallel to the film surfaces and to a transport current perpendicular to the field. By numerical minimization of the exact London theory energy expression, using simulated annealing techniques, the equilibrium configurations are determined with great accuracy over a wide range of fields and curre…
We discuss the use of Langevin molecular dynamics in the investigation of the non-equilibrium properties of disordered vortex matter. Our special focus is set on values of system parameters that are realistic for disordered high-$T_c$ superconductors such as YBCO. Using a discretized elastic line model, we study different aspects of vortices far from thermal equilibrium. On the one hand we investigate steady-state properties of driven magnetic flux lines in a disordered environment, namely the current-voltage characteristics, the gyration radius, and the pinning time statistics. On the other h…
The continuum theory of domain structures in ferromagnetic/superconducting bilayers fails when the equilibrium domain size becomes comparable with effective penetration depth $Λ$. Instead, a lattice of discrete vortices must be considered. Here, we report our results on the discrete vortex lattices in stripe domain structures of ferromagnetic/superconducting bilayers. The vortices are assumed to be situated periodically on chains in stripe domains. We study the configurations containing up to three chains per domain, and calculate their equilibrium energies, equilibrium domain size and vortex …
We study the relaxational dynamics of flux lines in high-temperature superconductors with random pinning using Langevin dynamics. At high temperatures the dynamics is stationary and the fluctuation dissipation theorem (FDT) holds. At low temperatures the system does not equilibrate with its thermal bath: a simple multiplicative aging is found, the FDT is violated and we found that an effective temperature characterizes the slow modes of the system. The generic features of the evolution -- scaling laws -- are dictated by the ones of the single elastic line in a random environment.
The Kopnin mass and the Baym-Chandler mass of the vortex have the same origin. Both represent the mass of the normal component trapped by the vortex. The Kopnin mass of the vortex is formed by quasiparticles localized in the vicinity of the vortex. In the superclean limit it is calculated as linear response exactly in the same way as the density of the normal component is calculated in homogeneous superfluid. The Baym-Chandler mass is the hydrodynamical (associated) mass trapped by vortex. It is analogous to the normal component formed by inhomogeneities, such as pores and impurities. Both con…
The dynamic phase diagram of vortex lattices driven in disorder is calculated in two and three dimensions. A modified Lindemann criterion for the fluctuations of the distance of neighboring vortices is used, which unifies previous analytic approaches to the equilibrium and non-equilibrium phase transitions. The temperature shifts of the dynamic melting and decoupling transitions are found to scale inversely proportional to large driving currents. A comparison with two-dimensional simulations shows that this phenomenological approach can provide quantitative estimate for the location of these t…
We consider the problem of the vortex contribution to thermal properties of dirty d-wave superconductors. In the clean limit, the main contribution to the density of states in a d-wave superconductor arises from extended quasiparticle states which may be treated semiclassically, giving rise to a specific heat contribution δC(H)\sim H^{1/2}. We show that the extended states continue to dominate the dirty limit, but lead to a H \log H behavior at the lowest fields, H_{c1}\ltsim H\ll H_{c2}. This crossover may explain recent discrepancies in specific heat measurements at low temperatures and fiel…
The information about the current-carrying ability of a type-II superconductor can be obtained by studying the flux relaxation and vortex penetration phenomena in the superconductor. In early studies, the infinite series models of the flux relaxation and vortex penetration phenomena were constructed at a vanishing current density and vanishing internal field, respectively. However, this is not the only possibility. Here it is shown that one can reconstruct the theoretical models at the critical points. The new polynomial model of the flux relaxation (vortex penetration) phenomenon was construc…
Using small-angle neutron scattering, we investigated the behavior of a metastable vortex lattice state in MgB2 as it is driven towards equilibrium by an AC magnetic field. This shows an activated behavior, where the AC field amplitude and cycle count are equivalent to, respectively, an effective "temperature" and "time". The activation barrier increases as the metastable state is suppressed, corresponding to an aging of the vortex lattice. Furthermore, we find a cross-over from a partial to a complete suppression of metastable domains depending on the AC field amplitude, which may empirically…
We study numerically correlation and response functions in non-equilibrium driven vortex lattices with random pinning. From a generalized fluctuation-dissipation relation we calculate an effective transverse temperature in the fluid moving phase. We find that the effective temperature decreases with increasing driving force and becomes equal to the equilibrium melting temperature when the dynamic transverse freezing occurs. We also discuss how the effective temperature can be measured experimentally from a generalized Kubo formula.
The small size vortex-antivortex pairs proliferation in type II superconducting film is considered for the wide interval of temperatures below Tc. The corresponding contribution to free energy is calculated. It is shown that these fluctuations give the main contribution to the heat capacity of the film both at low temperatures and in the vicinity of transition.
It is argued that the stress caused by vortex cores in the mixed state of superconductors may result in a field dependent contribution to the free energy and magnetization of measurable levels. For sufficiently strong stress dependence of the critical temperature, $\partial T_c/\partial p$, this contribution may result in the so-called "second peak" in the field dependence of the reversible magnetization, the effect often masked by vortex pinning and creep.
We calculate Ohmic and Hall flux flow conductivities in the mixed state of d-wave superconductors taking into account the spectral flow along the anomalous branch of the states bound to vortex cores. Three regimes of vortex dynamics are found. In the moderately clean limit the contribution of the spectral flow to the conductivities coincides with that for s-superconductors. In the superclean regime in low magnetic fields both Ohmic and Hall conductivities acquire relaxation-time-independent universal values. The contribution of the spectral flow in the superclean regime is suppressed in higher…
A perturbed system relaxes towards an equilibrium given by a minimum in the potential energy landscape. This often occurs by thermally activated jumps over metastable states. The corresponding dynamics is named creep and follows Arrhenius' law. Here we consider the situation where the equilibrium position depends on temperature. We show that this effect occurs in the vortex lattice of the anisotropic superconductor $2\mathrm{H}$-$\mathrm{Nb}\mathrm{Se}_{2}$ when the magnetic field is tilted away from the principal axes, and that it leads to the peculiar appearance of creep when cooling the sam…
We present a detailed study of the equilibrium properties and stochastic dynamic evolution of the U(1)-invariant relativistic complex field theory in three dimensions. This model has been used to describe, in various limits, properties of relativistic bosons at finite chemical potential, type II su- perconductors, magnetic materials and aspects of cosmology. We characterise the thermodynamic second-order phase transition in different ways. We study the equilibrium vortex configurations and their statistical and geometrical properties in equilibrium at all temperatures. We show that at very hig…
Within the non-linear Ginzburg-Landau (GL) theory, we investigate the vortex structure in a superconducting thin film with a ferromagnetic disk on top of it. Antivortices are stabilized in shells around a central core of vortices (or a giant-vortex) with size-magnetization controlled ``magic numbers''. An equilibrium vortex phase diagram is constructed. The transition between the different vortex phases occurs through the creation of a vortex-antivortex pair under the magnetic disk edge.
We present a detailed study of the dynamic interactions between Josephson vortices and stacks of pancake vortices in a Bi2Sr2CaCu2O8 single crystal, obtained by measuring the effect of applied inplane magnetic field pulses on the c-axis magnetisation. The predominant interaction is to relax the system towards equilibrium. However, using a highly sensitive AC technique we are able to measure also the forces acting to drive the system away from equilibrium, consistent with the existence of dragging interactions between the Josephson and pancake systems. Such forces were discussed recently as the…
In superconductors with strong coupling between superconductivity and elasticity manifested in a strong dependence of transition temperature on pressure, there is an additional contribution to inter-vortex interactions due to the strain field generated by vortices. When vortex lines are along the $c$ axis of a tetragonal crystal, a square vortex lattice (VL) is favored at low vortex densities, because the vortex-induced strains contribution to the inter-vortex interactions is long range. At intermediate magnetic fields, the triangular lattice is stabilized. The triangular lattice evolves to th…
We study the non-equilibrium dynamical regimes of a moving vortex lattice in the periodic pinning of a Josephson junction array (JJA) for {\it finite temperatures} in the case of a fractional or submatching field. We obtain a phase diagram for the current driven JJA as a function of the driving current I and temperature T. We find that when the vortex lattice is driven by a current, the depinning transition at $T_p(I)$ and the melting transition at $T_M(I)$ become separated even for a field for which they coincide in equilibrium. We also distinguish between the depinning of the vortex lattice …
Nowadays superconductors serve in numerous applications, from high-field magnets to ultra-sensitive detectors of radiation. Mesoscopic superconducting devices, i.e. those with nanoscale dimensions, are in a special position as they are easily driven out of equilibrium under typical operating conditions. The out-of-equilibrium superconductors are characterized by non-equilibrium quasiparticles. These extra excitations can compromise the performance of mesoscopic devices by introducing, e.g., leakage currents or decreased coherence times in quantum devices. By applying an external magnetic field…
A method, using a Corbino disk sample geometry, is described and applied to study of resistive tails of sintered $YBCO$. When the transport current passes radially from the rim of the disk sample to its center, the two component potential drop signal is detected below $T_c$, which is due to (i) quasiparticles $W_q$ and (ii) vortex-core-motion related contribution $W_{ϕ}$. When the contact pairs for $W$ are placed radially, $W_q(r)$ and $W_{ϕ}(r)$ are found to follow the markedly distinctive functional dependences, providing a unique possibility to deconvolute the relative strengths of both con…
The problem of vortex structure in a single Josephson junction in an external magnetic field, in the absence of transport currents, is reconsidered from a new mathematical point of view. In particular, we derive a complete set of exact analytical solutions representing all the stationary points (minima and saddle-points) of the relevant Gibbs free-energy functional. The type of these solutions is determined by explicit evaluation of the second variation of the Gibbs free-energy functional. The stable (physical) solutions minimizing the Gibbs free-energy functional form an infinite set and are …
In type-II superconductors that contain a lattice of magnetic moments, vortices polarize the magnetic system inducing additional contributions to the vortex mass, vortex viscosity, and vortex-vortex interaction. Extra magnetic viscosity is caused by radiation of spin waves by a moving vortex. Like in the case of Cherenkov radiation, this effect has a characteristic threshold behavior and the resulting vortex viscosity may be comparable to the well-known Bardeen-Stephen contribution. The threshold behavior leads to an anomaly in the current-voltage characteristics, and a drop in dissipation for…
We present numerical studies of the dynamics of vortices in the Ginzburg Landau model using equations derived from the gradient flow of the free energy. These equations have previously been proposed to describe the dynamics of n-vortices away from equilibrium. We are able to model the dynamics of multiple n-vortex configurations starting far from equilibrium. We find generically that there are two time scales for equilibration: a short time scale related to the formation time for a single n-vortex, and a longer time scale that characterizes vortex-vortex interactions.
The stress caused by vortices in tetragonal superconductors contributes to the intervortex interaction which depends on vortex orientation within the crystal, on elastic moduli, and is attractive within certain angular regions even in fields along the $c$ crystal axis. For sufficiently strong stress dependence of the critical temperature, this contribution may result in distortions of the hexagonal vortex lattice for $\bm H||\bm c$. In small fields it leads to formation of a square vortex lattice with a fixed $H$ independent spacing. This should be seen in the magnetization $M(H)$ as a discont…
At low enough temperatures and high densities, the equilibrium configuration of an ensemble of ultrasoft particles is a self-assembled, ordered, cluster-crystal. In the present work we explore the out-of-equilibrium dynamics for a two-dimensional realization, which is relevant to superconducting materials with multi-scale intervortex forces. We find that for small temperatures following a quench, the suppression of the thermally-activated particle hopping hinders the ordering. This results in a glass transition for a monodispersed ensemble, for which we derive a microscopic explanation in term…
We consider the problem of the vortex contribution to thermal properties of dirty d-wave superconductors. In the clean limit, Volovik has argued that the main contribution to the density of states in a d-wave superconductor arises from extended quasiparticle states which may be treated semiclassically, giving rise to a specific heat contribution δC(H)\ltsim H^{1/2} . We show that the extended states continue to dominate the dirty limit, but lead to a H \log H behavior at the lowest fields, H_{c1}\ll H\ll H_{c2}. This crossover may explain recent discrepancies in specicific heat measurements at…
The vortex lattice in a Type II superconductor provides a versatile model system to investigate the order-disorder transition in a periodic medium in the presence of random pinning. Here, using scanning tunnelling spectroscopy in a weakly pinned Co0.0075NbSe2 single crystal, we show that at low temperatures, the vortex lattice in a 3-dimensional superconductor disorders in two steps across the peak effect. At the onset of the peak effect, the equilibrium Bragg glass transforms into an orientational glass through the proliferation of dislocations. At a higher field, the dislocations dissociate …
The contribution to the vortex lattice energy which is due to the vortex-induced strains is calculated covering all the magnetic field range which defines the vortex state. This contribution is compared with previously reported ones what shows that, in the most part of the vortex state, it has been notably underestimated until now. The reason of such underestimation is the assumption that only the vortex cores induce strains. In contrast to what is generally assumed, both core and non-core regions are important sources of strains in high-$κ$ superconductors.
We examine energy relaxation of non-equilibrium quasiparticles in different vortex configurations in ``dirty'' $s$-wave superconductors. The heat flow from the electronic subsystem to phonons in a mesoscopic superconducting disk with a radius of the order of several coherence lengths is calculated both in the Meissner and giant vortex states using the Usadel approach. The recombination process is shown to be strongly affected by interplay of the subgap states, located in the vortex core and in the region at the sample edge where the spectral gap $E_{\rm g}$ is reduced by the Meissner currents.…
The melting transition of the vortex lattice in highly anisotropic, layered superconductors with commensurate, periodic columnar pins is studied in a geometry where magnetic field and columnar pins are normal to the layers. Thermodynamic properties and equilibrium density distributions are obtained from numerical minimizations of an appropriate free-energy functional. We find a line of first-order transitions that ends at a critical point as the pin concentration is increased. A simple Landau theory providing a semi-quantitative explanation of the numerical results is proposed.
The vortex state at the lower critical field, H_{c1}, in clean anisotropic superconductors placed in an external field tilted with respect to the axis of anisotropy (c-axis) is considered assuming two possible arrangements: dilute vortex-lines or dilute vortex-line chains. By minimizing the Gibbs free energies in the London limit for each possibility we obtain the corresponding lower critical fields as a function of the tilt angle. The equilibrium configuration at H_{c1} for a given tilt angle is identified with that for which H_{c1} is the smallest. We report results for parameter values typi…
We show that some experimentally observed features of vortex matter in high temperature superconductors may be interpreted in simpler ways than it is usually done. In particular, we consider magnetic flux creep at low temperatures as well as the irreversibility line in the H-T phase diagram. We also discuss a new approach to the analysis of the equilibrium magnetization in the mixed state of type-II superconductors and we suggest an alternative configuration for the mixed state in magnetic fields close to the upper critical field.
Vortex structures in mesoscopic cylinder placed in external magnetic field are studied under the general de Gennes boundary condition for the order parameter corresponding to the suppression of surface superconductivity. The Ginzburg-Landau equations are solved based on trial functions for the order parameter for vortex-free, single-vortex, multivortex, and giant vortex phases. The equilibrium vortex diagrams in the plane of external field and cylinder radius and magnetization curves are calculated at different values of de Gennes "extrapolation length" characterizing the boundary condition fo…
We derive generalized quasiclassical transport equations which include the terms responsible for the Hall Effect in the vortex state of a clean type-II superconductor, and calculate the conductivity tensor for an s-wave superconductor in the high-field regime. We find that below the superconducting transition the contribution to the transverse conductivity due to dynamical fluctuations of the order parameter is compensated by the modification of the quasiparticle contribution. In this regime the nonlinear behaviour of the Hall angle is governed by the change in the effective quasiparticle scat…
The magnetic field distribution, the magnetic flux, and the free energy of an Abrikosov vortex loop near a flat surface of type--II superconductors are calculated in the London approximation. The shape of such a vortex line is a semicircle of arbitrary radius. The interaction of the vortex half--ring and an external homogeneous magnetic field applied along the surface is studied. The magnitude of the energy barrier against the vortex expansion into superconductor is found. The possibilities of formation of an equilibrium vortex line determined by the structure of the applied magnetic field by …
A many-particle system must posses long-range interactions in order to be hyperuniform at thermal equilibrium. Hydrodynamic arguments and numerical simulations show, nevertheless, that a three-dimensional elastic-line array with short-ranged repulsive interactions, such as vortex matter in a type-II superconductor, forms at equilibrium a class-II hyperuniform two-dimensional point pattern for any constant-$z$ cross section. In this case, density fluctuations vanish isotropically as $\sim q^α$ at small wave-vectors $q$, with $α=1$. This prediction includes the solid and liquid vortex phases in …
We present a study of the behaviour of metastable vortex states in mesoscopic superconductors. Our analysis relies on the London limit within which it is possible to derive closed analytical expressions for the magnetic field and the Gibbs free energy. We consider in particular the situation where the vortices are symmetrically distributed along a closed ring. There, we obtain expressions for the confining Bean-Livingston barrier and for the magnetization which turns out to be paramagnetic away from thermodynamic equilibrium. At low temperature, the barrier is high enough for this regime to be…
Topological excitations are believed to play an important role in different areas of physics. For example, one case of topical interest is the use of dual models of quantum cromodynamics to understand properties of its vacuum and confinement through the condensation of magnetic monopoles and vortices. Other applications are related to the role of these topological excitations, nonhomogeneous solutions of the field equations, in phase transitions associated to spontaneous symmetry breaking in gauge theories, whose study is of importance in phase transitions in the early universe, for instance. …
We employ an elastic line model to investigate the steady-state properties and non-equilibrium relaxation kinetics of magnetic vortex lines in disordered type-II superconductors using Langevin molecular dynamics (LMD). We extract the dependence of the mean vortex line velocity and gyration radius as well as the mean-square displacement in the steady state on the driving current, and measure the vortex density and height autocorrelations in the aging regime. We study samples with either randomly distributed point-like or columnar attractive pinning centers, which allows us to distinguish the co…
Resistively-shunted-junction dynamics is applied to the three dimensional uniformly frustrated XY model with randomly perturbed couplings, as a model for driven steady states in a type-II superconductor with quenched point pinning. For a disorder strength p strong enough to produce a vortex glass in equilibrium, we map the phase diagram as a function of temperature T and uniform driving current I. We find that, within a finite current range I_{c1}(T)<I<I_{c2}(T), the system orders into a set of periodically spaced smectic planes of vortex lines. Smectic planes are short range correlated with n…
In the vortex state of a d-wave superconductor, massless Dirac quasiparticles are scattered from magnetic vortices via a combination of two basic mechanisms: effective potential scattering due to the superflow swirling about the vortices and Aharonov-Bohm scattering due to the Berry phase acquired by a quasiparticle upon circling a vortex. In this paper, we study the superflow contribution by calculating the differential cross section for a quasiparticle scattering from the effective non-central potential of a single vortex. We solve the massless Dirac equation in polar coordinates and obtain …
