Noise spectral power and non-equilibrium dynamics of vortices in the mixed state of Bi2Sr2CaCu2Oy

A model is proposed to explain the observed noise spectral power generated by the moving vortex lattice in Bi₂Sr₂CaCu₂O_y single crystals. The analysis is based on the observations that the noise generated is due to the vortex velocity rather than the thermodynamic phase of the material. The motion of vortices induces a gradual transition of the vortex lattice from the plastic flow to the uniform flow with increasing magnetic field. The influence of the disorder on the dynamics of vortices is such that a uniform distribution in the activation barrier width will exist. This distribution will affect the attempt frequency of the flux line hopping and consequently the spectral shape of the noise generated. The model provides physical basis to the fact that the observed 1/f-noise power corresponds to the smaller velocity of the vortex lattice, while the deviation from the 1/f-spectrum corresponds to the larger velocity region. The results of the simulation are compared to recent experimental measurements where good agreement has been achieved.     

Muon spin rotation in heavy-electron pauli-limit superconductors

The formalism for analyzing the magnetic field distribution in the vortex lattice of Pauli-limit heavy-electron superconductors is applied to the evaluation of the vortex lattice static linewidth relevant to the muon spin rotation (??SR) experiment. Based on the Ginzburg-Landau expansion for the superconductor free energy, we study the evolution with respect to the external field of the static linewidth both in the limit of independent vortices (low magnetic field) with a variational expression for the order parameter and in the near H _c2 ^P (T) regime with an extension of the Abrikosov analysis to Pauli-limit superconductors. We conclude that in the Ginzburg-Landau regime in the Pauli-limit, anomalous variations of the static linewidth with the applied field are predicted as a result of the superconductor spin response around a vortex core that dominates the usual charge-response screening supercurrents. We propose the effect as a benchmark for studying new puzzling vortex lattice properties recently observed in CeCoIn₅.     

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.     

Semiclassical dynamics of vortices in 2D easy-plane ferromagnets

Semiclassical dynamics of magnetic vortices in 2D lattice models of easy-plane ferromagnets is investigated. It is shown that the low-energy part of the spectrum of vortices treated as quantum excitations of the system exhibits a nontrivial structure. The simplest spectrum is observed for standard magnetic vortices, in which magnetization at long distances from the center of a vortex is parallel to the basal plane. In this case, the spectrum has a band structure consisting of several nonintersecting bands, whose number is determined only by the value of atomic spin S and lattice symmetry. For purely 2D magnets with a single spin per unit cell, the number of bands is S or 2S for integral and half-integral values of spin S, respectively. For a lattice with the basis with an even number 2n of spins per unit cell, the number of bands is 2nS for any spins. The situation radically changes for vortices in the cone state as compared to standard vortices, for which the magnetization at a long distance from the center of a vortex rotates in the easy plane of the magnet. Vortices in the cone state are formed under the action of a constant external field perpendicular to the easy plane of the magnet. As a rule, the spectrum for such vortices is not a standard band spectrum and forms a set such that a forbidden energy value can be found in any small neighborhood of an allowed value, and vice versa. The possibility of an oscillatory motion of a vortex under the action of a constant external force is indicated (analog of Bloch oscillations of electrons in crystals). Possible realization of these effects in other ordered media with vortices is considered.     

Attraction between pancake vortices in the crossing lattices of layered superconductors

The intervortex interaction is investigated in very anisotropic layered superconductors in tilted magnetic field. In such a case, the crossing lattice of Abrikosov vortices (AVs) and Josephson vortices (JVs) appears. The interaction between pancake vortices, forming the AVs and JVs, produces the deformation of the AV line. It is demonstrated that, as a result of this deformation, a long range attraction between AVs is induced. This phenomenon is responsible for the dense vortex chain formation. The vortex structure in the weak perpendicular magnetic field is the vortex chain phase, where only a small part of JVs is occupied by AVs.     

Vortex liquid crystals in anisotropic type II superconductors

In an isotropic type II superconductor in a moderate magnetic field, the transition to the normal state occurs by vortex lattice melting. In certain anisotropic cases, the vortices acquire elongated cross sections and interactions. Systems of anisotropic, interacting constituents generally exhibit liquid crystalline phases. We examine the possibility of a two step melting in homogeneous type II superconductors with anisotropic superfluid stiffness from a vortex lattice into first a vortex smectic and then a vortex nematic at high temperature and magnetic field. We find that fluctuations of the ordered phase favor an instability to an intermediate smectic-A in the absence of intrinsic pinning.     

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.     

Vortex chains induced by anisotropic spin-orbit coupling and magnetic field in spin-2 Bose-Einstein condensates

We investigate the anisotropic spin-orbit coupled spin-2 Bose-Einstein condensates with Ioffe-Pritchard magnetic field. With nonzero magnetic field, anisotropic spin-orbit coupling will introduce several vortices and further generate a vortex chain. Inside the vortex chain, the vortices connect to each other, forming a line along the axis. The physical nature of the vortex chain can be explained by the particle current and the momentum distribution. The vortex number inside the vortex chain can be influenced via varying the magnetic field. Through adjusting the anisotropy of the spin-orbit coupling, the direction of the vortex chain is changed, and the vortex lattice can be triggered. Moreover, accompanied by the variation of the atomic interactions, the density and the momentum distribution of the vortex chain are affected. The realization and the detection of the vortex chain are compatible with current experimental techniques.     

Spectroscopy of magnetic excitations in magnetic superconductors using vortex motion

In magnetic superconductors a moving vortex lattice is accompanied by an ac magnetic field which leads to the generation of spin waves. At resonance conditions the dynamics of vortices in magnetic superconductors changes drastically, resulting in strong peaks in the dc I-V characteristics at voltages at which the washboard frequency of the vortex lattice matches the spin wave frequency omegaS(g), where g are the reciprocal vortex lattice vectors. We show that if the washboard frequency lies above the magnetic gap, measurement of the I-V characteristics provides a new method to obtain information on the spectrum of magnetic excitations in borocarbides and cuprate layered magnetic superconductors.     

Vortex sublattice melting in a two-component superconductor

We consider the vortices in a superconductor with two individually conserved condensates in a finite magnetic field. The ground state is a lattice of cocentered vortices in both order parameters. We find two phase transitions: (i) a "vortex sublattice melting" transition where vortices in the field with lowest phase stiffness ("light vortices") lose cocentricity with the vortices with large phase stiffness ("heavy vortices"), entering a liquid state (the structure factor of the light vortices vanishes continuously; this transition is in the 3Dxy universality class); (ii) a first-order melting transition of the lattice of heavy vortices, in a liquid of light vortices.     

Magnetic-field effects on the size of vortices below the surface of NbSe2 detected using low energy beta-NMR

A low energy radioactive beam of polarized 8Li has been used to observe the vortex lattice near the surface of superconducting NbSe2. The inhomogeneous magnetic-field distribution associated with the vortex lattice was measured using depth-resolved beta-detected NMR. Below Tc, one observes the characteristic line shape for a triangular vortex lattice which depends on the magnetic penetration depth and vortex core radius. The size of the vortex core varies strongly with the magnetic field. In particular, in a low field of 10.8 mT, the core radius is much larger than the coherence length. The possible origin of these giant vortices is discussed.     

Vortices in Bose-Einstein condensates: A review of the experimental results

Rotating dilute Bose-Einstein condensates (BEC) of alkali atoms offer a testing ground for theories of vortices in weakly interacting superfluids. In a rotating super-fluid, quantised vortices, with a vorticity h/m, form above a critical velocity. Such vortices have been generated in BEC of alkali atoms by different techniques such as (a) wave function engineering of a two-component BEC, (b) decay of solitons, (c) rotation of a thermal cloud before cooling it below the condensation temperature, (d) stirring with an ‘optical’ spoon, (e) rotating a deformation in the anisotropic trap in which the condensate is trapped and (f) by creating Berry phase by adiabatically reversing the axial magnetic field. Since the core of a vortex is a fraction of a micrometer in diameter, it cannot be directly imaged optically. The condensate with vortices is allowed to ballistically expand till the size increases by one order before the vortices are imaged. Surface wave spectroscopy and the change in aspect ratio of a rotating cloud are the other techniques used. Studies have been made on the creation and dynamics of single vortex and on systems with more than a hundred vortices. Results have been obtained on vortex nucleation, stability of vortex structures, nature of the vortex lattice and defects in such a lattice. Important results are: (a) evidence exists that vortex nucleation takes place by a surface mode instability; but this is not the only mechanism; (b) the vortex lattice is perfectly triangular right up to the edge; (c) in the initial stages of rotation of the cloud a tangled web of vortices is seen; it takes a few hundred milliseconds before the vortices arrange themselves in a lattice; this time appears to be independent of temperature; (d) the decay of vortices appears to arise from the transfer of energy to the rotating thermal component and is dependent on temperature; (e) defects in the lattices such as dislocations and grain boundaries are seen; (f) transverse oscillations (Tkachenko modes) of the vortex lattice have been observed; and (g) giant vortices have been produced. These will be discussed.     

Vortex states of a superconducting film from a magnetic dot array

Using Ginzburg-Landau theory, we find novel configurations of vortices in superconducting thin films subject to the magnetic field of a magnetic dot array, with dipole moments oriented perpendicular to the film. Sufficiently strong magnets cause the formation of vortex-antivortex pairs. In most cases, the vortices are confined to dot regions, while the antivortices can form a rich variety of lattice states. We propose an experiment in which the perpendicular component of the dot dipole moments can be tuned using an in-plane magnetic field. We show that in such an experiment the vortex-antivortex pair density shows broad plateaus as a function of the dipole strength. Many of the plateaus correspond to vortex configurations that break dot lattice symmetries. In some of these states, the vortex cores are strongly distorted. Possible experimental consequences are mentioned.     

Spatial line nodes and fractional vortex pairs in the Fulde-Ferrell-Larkin-Ovchinnikov vortex state of spin-singlet superconductors

A Zeeman magnetic field can induce a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in spin-singlet superconductors. Here we argue that there is a nontrivial solution for the FFLO vortex phase that exists near the upper critical field in which the wave function has only spatial line nodes that form intricate and unusual three-dimensional structures. These structures include a crisscrossing lattice of two sets of nonparallel line nodes. We show that these solutions arise from the decay of conventional Abrikosov vortices into pairs of fractional vortices. We propose that neutron scattering studies can observe these fractional vortex pairs through the observation of a lattice of 1/2 flux quanta vortices. We also consider related phases in noncentrosymmetric superconductors.     

Flux flow in current driven mesoscopic superconductors: size effects

Flux-flow phenomena in a superconducting mesoscopic stripe submitted to an appliedcurrent and external magnetic field is studied. The time-dependent Ginzburg-Landauequations are solved numerically to obtain the electric and magnetic response of thesystem. It is shown that the I-V curves, for the wider strips, present a universalbehaviour. The dependence of the flux-flow resistivity on the magnetic field and widthallow us to propose a criterion characterizing, both, the macroscopic and mesoscopicregimes. The power spectrum of the average voltage permits identifying the effect ofsurface currents in vortices movement. Based on the maximum value of the power spectrumfirst harmonic we propose a geometric condition for matching between the sample dimensionsand the vortex lattice parameter.     

Deconfinement in the two-dimensional XY model

The unbinding of vortex-antivortex pairs for the classical two-dimensional XY model in a magnetic field is studied. A single such pair is connected by a string of overturned spins, leading to linear confinement. We show that this system supports two phase transitions, one in which closed strings proliferate, and a second in which vortices unbind. The transitions are shown to be dual to one another, and are remarkably continuous. Possible consequences for a variety of systems are discussed.     

Vortex lattice generation using interferometric techniques based on lateral shearing

Interferometric techniques that use lateral shearing for generation of a regular network of optical vortices are reported. These techniques are simple, robust, and less sensitive to vibrations compared to the existing interferometric methods of vortex lattice generation. Vortex lattice is created using three-wave interference of plane as well as spherical waves. Theory of vortex lattice formation using phasor approach is provided. The location of vortices in the interference field is described in terms of spatial frequencies of the interfering waves. The presence of the vortices in the interference field is confirmed experimentally by producing fork fringes, as vortex signatures. Results of simulation studies are presented to support experimentally recorded interferograms.     

Spin waves,vortices, fermions,and duality in the Ising and Baxter models

Field-theoretic methods are applied to a number of two-dimensional lattice models with Abelian symmetry groups. It is shown, using a vortex + spin-wave decomposition, that the Z_p^? Villain models are related to a class of continuum field theories with analogous duality properties. Fermion operators for these field theories are discussed. In the case of the Ising model, the vortices and spin-waves conspire to produce a free, massive Majorana field theory in the continuum limit. The continuum limit of the Baxter model is also studied, and the recent results of Kadanoff and Brown are rederived and extended.     

Multi-vortex versus interstitial vortices scenario in superconducting antidot arrays

In superconducting thin films, engineered lattice of antidots (holes) act as an array of columnar pinning sites for the vortices and thus lead to vortex matching phenomena at commensurate fields guided by the lattice spacing. The strength and nature of vortex pinning is determined by the geometrical characteristics of the antidot lattice (such as the lattice spacing a₀, antidot diameter d, lattice symmetry, and orientation) along with the characteristic length scales of the superconducting thin films, viz., the coherence length (ξ) and the penetration depth (λ). There are at least two competing scenarios: (i) multiple vortices sit on each of the antidots at a higher matching period and (ii) there is nucleation of vortices at the interstitial sites at higher matching periods. Furthermore, it is also possible for the nucleated interstitial vortices to reorder under suitable conditions. We present our experimental results on NbN antidot arrays in the light of the above scenarios.