Direct observation of geometrical phase transitions in mesoscopic superconductors by scanning tunneling microscopy

Using scanning tunneling microscopy, we mapped the distribution of the local density of states in a single crystal superconductor heterostructure with an array of submicron normal metal islands. We observe the coexistence of strongly interacting multiquanta vortex lattice with interstitial Abrikosov vortices. The newly formed composite magnetic flux structure undergoes a series of phase transitions between different topological configuration states. The vortex configuration states are strongly dependent on the number of flux quanta and the nanoscale confinement architecture of the mesoscopic superconductor. Here, we present images of vortex phase transitions due to confinement effects when the number of magnetic flux quanta in the system changes. The vortex dynamics in these systems could serve as a model for behavior of confined many-body systems when the number of particles changes.     

Helical instability of a straight Abrikosov vortex in an anisotropic superconductor

Because of attraction of the parallel currents forming an Abrikosov vortex, the vortex energy per unit length decreases, under bending of the vortex, by a quantity proportional to the square of the curvature. Solving the London equation in an approximation allowing for this effect makes it possible to calculate the energy of an Abrikosov vortex in the form of a helix whose length and pitch are much larger than the correlation length, whose curvature is small compared to the reciprocal London length, and whose slope in relation to an axis coinciding with the direction in which the vortex energy is the highest is also small. When the anisotropy is large, which is characteristic of high-T _c superconductors, the energy of such an Abrikosov vortex is lower than that of a straight Abrikosov vortex. Certain consequences of the fact that the Abrikosov vortices in a high-T _c superconductor are helical are discussed. Among these is a phase transition that breaks the symmetry between Abrikosov vortices shaped like right-and left-hand helixes in relation to the magnetic field. Zh. éksp. Teor. Fiz. 111, 1869–1878 (May 1997)     

Quasi-local action of curl-less vector potential on vortex dynamics in superconductors

Studies of the Abrikosov vortex motion in superconductors based on time-dependent Ginzburg–Landau equations reveal an opportunity to detect the values of the Aharonov–Bohm type curl-less vector potentials without closed-loop electron trajectories encompassing the magnetic flux.     

Pinning of vortices in a Bose-Einstein condensate by an optical lattice

We consider the ground state of vortices in a Bose-Einstein condensate. We show that turning on a weak optical periodic potential leads to a transition from the triangular Abrikosov vortex lattice to phases where the vortices are pinned by the optical potential. We discuss the phase diagram of the system for a two-dimensional optical periodic potential with one vortex per optical lattice cell. We also discuss the influence of a one-dimensional optical periodic potential on the vortex ground state. The latter situation has no analog in other condensed-matter systems.     

Electrodynamics of Abrikosov vortices: the field theoretical formulation

Electrodynamic phenomena related to vortices in superconductors have been studied since their prediction by Abrikosov, and seem to hold no fundamental mysteries. However, most of the effects are treated separately, with no guiding principles.We demonstrate that the relativistic vortex worldsheet in spacetime is the object that naturally conveys all electric and magnetic information, for which we obtain simple and concise equations. Breaking Lorentz invariance leads to down-to-earth Abrikosov vortices, and special limits of these equations include for instance dynamic Meissner screening and the AC Josephson relation. On a deeper level, we explore the electrodynamics of two-form sources in the absence of electric monopoles, in which the electromagnetic field strength itself acquires the characteristics of a gauge field. This novel framework leaves room for unexpected surprises.     

Scanning tunneling spectroscopy in the superconducting state and vortex cores of the beta-pyrochlore KOs2O6

We performed the first scanning tunneling spectroscopy measurements on the pyrochlore superconductor KOs2O6 (T(c)=9.6 K) in both zero magnetic field and the vortex state at several temperatures above 1.95 K. This material presents atomically flat surfaces, yielding spatially homogeneous spectra which reveal fully gapped superconductivity with a gap anisotropy of 30%. Measurements performed at fields of 2 and 6 T display a hexagonal Abrikosov flux line lattice. From the shape of the vortex cores, we extract a coherence length of 31-40 A, in agreement with the value derived from the upper critical field H(c2). We observe a reduction in size of the vortex cores (and hence the coherence length) with increasing field which is consistent with the unexpectedly high and unsaturated upper critical field reported.     

Abrikosov-to-Josephson vortex lattice crossover in heavy fermion CeCoIn5

We present torque magnetization measurements on the quasi-2D heavy fermion superconductor CeCoIn₅ at temperatures down to 20 mK and magnetic fields up to 18?T. At orientations with the magnetic field perpendicular to the conducting planes, a prominent vortex lattice peak effect is present at around 0.5H _c2. The peak effect gradually disappears upon rotating the field into the plane parallel orientation. We interpret the absence of the peak effect for the plane parallel case as a transformation of the Abrikosov lattice into a Josephson vortex state, favored by the Pauli paramagnetic limit in CeCoIn₅ together with the unusually large condensation energy. Additionally, we do not observe flux avalanches as found in organic superconductors and suggest that the complete absence of vortex activity in the plane parallel field orientation is crucial for the formation of Fulde–Ferrell–Larkin–Ovchinnikov superconductivity in CeCoIn₅.     

Features of the effect of the Josephson medium parameters on vortex formation and critical current

The response of an intergranular Josephson junction to displacements of an Abrikosov vortex in a superconducting polycrystal is studied theoretically. The vortex filament in the vicinity of the junction excites a tunnel current in the junction and also generates a Josephson vortex with which it merges upon emergence at the surface of the junction. It is shown that the process of the Josephson vortex formation passes through a stage of overcoming a potential barrier, whose height depends on the distance between the Abrikosov vortex and the junction, as well as on the effective thickness of the junction, which is determined by the characteristic grain size, grain anisotropy, and the intensity of the intergranular coupling. The magnetic field dependence of the critical current of the intergranular Josephson junction is determined for various grain and intergranular parameters, as well as for the triangular and square configurations of the Abrikosov vortex lattice. The results indicate that a high degree of texturing in the grain size, anisotropy, and intensity of intergranular coupling is very important for obtaining high critical currents in pure polycrystalline materials.     

Attraction-repulsion transition between abrikosov vortices and Josephson vortices in the strongly layered superconductor Bi2Sr2CaCu2O8

A change in the effect of a frozen magnetic field parallel to the c-axis on rf power absorption, which is associated with the motion of Josephson vortices, is observed in the layered superconductor Bi₂Sr₂CaCu₂O₈ at a low temperature (～15 K). The effect is interpreted as a change in the interaction between an Abrikosov vortex and a Josephson vortex from attraction (at high temperatures) to repulsion (at low temperatures). It is found that the dynamics of the magnetic flux parallel to the ab plane of the single crystal becomes irreversible upon a transition of the superconductor to the layered state. Possible reasons behind the observed effect are considered, one of them being a manifestation of the second superconducting transition in the elementary-excitation spectrum of a d-type superconductor near the core of Abrikosov vortices.     

Magnetic flux inversion in charged BPS vortices in a Lorentz-violating Maxwell–Higgs framework

We demonstrate for the first time the existence of electrically charged BPS vortices in a Maxwell–Higgs model supplemented with a parity-odd Lorentz-violating (LV) structure belonging to the CPT-even gauge sector of the standard model extension and a fourth order potential (in the absence of the Chern–Simons term). The modified first order BPS equations provide charged vortex configurations endowed with some interesting features: localized and controllable spatial thickness, integer flux quantization, electric field inversion and localized magnetic flux reversion. This model could possibly be applied on condensed matter systems which support charged vortices carrying integer quantized magnetic flux, endowed with localized flipping of the magnetic flux.     

Inverted forerunning and wake of the solitary oscillating abrikosov vortex in a magnetic superconductor

The magnetic structure of a solitary oscillating Abrikosov vortex in a magnetic superconductor is investigated. It is shown that the process of the vortex motion in the presence of the magnetic subsystem substantially modifies the shape of the vortex, with the result that there appears an “inverted forerunning” in front of the vortex and an “inverted wake” behind it that is far from its center.     

A single Abrikosov vortex trapped in a mesoscopic superconducting cylindrical surface

We investigate the behaviour of a single Abrikosov vortex trapped in a mesoscopic superconducting cylindrical surface with a magnetic field applied transverse to its axis. In the framework of the time-dependent Ginzburg-Landau formalism we show that, provided the transport current and the magnetic field are not large, the vortex behaves as an overdamped quasi-particle in a tilted washboard potential. The cylindrical thin strip with the trapped vortex exhibits E(J) curves and time-dependent electric fields very similar to the ones exhibited by a resistively shunted Josephson weak link.     

Self-dual Vortices in a Maxwell–Chern–Simons Model with Non-minimal Coupling

We study self-dual vortex solutions in a Maxwell – Chern – Simons model with anomalous magnetic moment. We establish the existence of multivortex solutions and obtain the quantized energy and the magnetic flux. We also prove the uniqueness of solutions when there is only one vortex point.      

Abrikosov vortices in SF bilayers

We study the spatial distribution of supercurrent circulated around an Abrikosov vortex in an SF bilayer in perpendicular magnetic field. Within the dirty limit regime and circular cell approximation for the vortex lattice, we derive the conditions when the Usadel equations the F-layer can be solved analytically. Using the obtained solutions, we demonstrate the possibility of reversal of direction of proximity induced supercurrents around the vortex in the F-layer compared to that in the S-layer. The direction of currents can be controlled either by varying transparency of the SF interface or by changing an exchange field in a ferromagnet. We argue that the origin of this effect is due the phase shift between singlet and triplet order parameter components induced in the F-layer. Possible ways of experimental detection of the predicted effect are discussed.     

Instability of a triangular Abrikosov lattice at values of the Ginzburg–Landau parameter κ close to unity

The “soft” transverse mode of gapless excitations related to the deformation of a triangular Abrikosov lattice with a single flux quantum per unit cell at an arbitrary value of the Ginzburg–Landau parameter κ is investigated. An Abrikosov lattice with the angle φ = π/3 between the unit cell vectors is shown to be unstable in a narrow range of values, 1 < κ < 1.000634. The excitation spectrum of the mode under consideration at low values of the momentum k (in the k² approximation) is isotropic at k lying in a plane perpendicular to the magnetic field.     

Noncollinear orientation of external magnetic field and flux lines penetrating an isotropic hard superconductor

Penetration by Abrikosov flux lines of an isotropic hard superconductor in the critical state induced by changes in the orientation of external magnetic field has been theoretically investigated. The analysis has been based on the microscopic nonlocal model taking into account forces of bulk and surface pinning, alongside magnetic forces of interaction of the row of penetrating vortices with existing flux lines, Meissner currents, and vortex images. New vortices penetrate a superconductor only when the angle through which the field is rotated is larger than a certain critical value. It has been determined that the alignment of entering vortices is essentially different from that of the applied magnetic field. The feasibility of detecting noncollinearity effects is discussed. Zh. éksp. Teor. Fiz. 114, 1804–1816 (November 1998)     

Electrodynamics of hard superconductors in crossed magnetic fields

A model systematically accounting for the cutting of Abrikosov flux lines has been developed for the critical state of a hard superconductor in crossed dc and ac magnetic fields. The electrodynamic equations have been derived by minimizing the Gibbs free energy calculated using the proposed two-velocity hydrodynamic model. One velocity describes the motion of the vortex lattice as a whole, and the other describes the relative motion of the two intersecting sublattices. The resulting equations yield as special cases the previously known electrodynamic equations for hard superconductors. The model provides a natural explanation for the suppression of dc magnetization by a transverse ac magnetic field observed in our experiments. Zh. éksp. Teor. Fiz. 111, 1071–1084 (March 1997)     

Numerical calculations of the driving force on an Abrikosov vortex

The driving force on an Abrikosov vortex is calculated numerically from the London equation and involved energies for a vortex perpendicular to the screening current near the surface of a superconductor. Compared with previous analytical derivation of the total force, the partial magnetic, kinematic, and external forces are also obtained so that the nature of the driving force may be deeply discussed. It is shown that the force is neither a Lorentz force nor a Magnus force as often believed and that in order to get a correct result, the image effects and the work done by the applied field must be taken into account. A name of London force is suggested for the driving force. A deep understanding of the nature of the driving force on Abrikosov vortices may also be important in the study of vortex pinning and dynamics in type-II superconductors.     

Dynamic vortex Mott transition in triangular superconducting arrays

The proximity-coupled superconducting island arrays on a metallic film provide an ideal platform to study the phase transition of vortex states under mutual interactions between the vortex and potential landscape. We have developed a top-down microfabrication process for Nb island arrays on Au film by employing an Al hard mask. A current-induced dynamic vortex Mott transition has been observed under the perpendicular magnetic fields of $f$ magnetic flux quantum per unit cell, which is characterized by a dip-to-peak reversal in differential resistance d$V$/d$I$ vs. $f$ curve with the increasing current. The d$V$/d$I$ vs. $I$ characteristics show a scaling behavior near the magnetic fields of $f= {1}/{2}$ and $f=1$, with the critical exponents $\varepsilon$ of 0.45 and 0.3, respectively, suggesting different universality classes at these two fields.     

Statistical mechanics of flux lines in high-T c superconductors

A theory of the entangled flux liquids which arise in the new high-T _c superconductors is reviewed. New physics appears because of the weak interplanar couplings and high critical temperatures in these materials. Flux line wandering melts the conventional Abrikosov flux lattice, and leads to an entangled vortex state whose statistical mechanics is closely related to the physics of interacting bosons in two dimensions. The phase diagram as a function of magnetic field and temperature is discussed, and it is argued that an entangled vortex liquid appears just aboveH _c1 at all nonzero temperatures. The decay of vortex line correlations in the entangled liquid state is controlled by the superfluid excitation spectrum of the bosons. Line wandering produces drastic changes in theB(H) constitutive relation nearH _c1.