Single Vortex State Research Articles

Flux Dynamics and Vortex Phase Diagram in Tl2Ba2CaCu2O8 System

Experimental data on the field and time dependencies of the isothermal magnetization hysteresis for Tl 2 Ba 2 CaCu 2 O 8 single crystals with fields applied parallel to the c-axis are reported. From the M–H measurements, we determined the first peak field (Hp), the onset field of the second peak (Hon), the second peak field (Hsp), and the irreversible field (Hirr). Using these physical parameters, we have drawn a vortex phase diagram for this system. As observed in BSCCO(2212) system, below the irreversibility line, the H–T plane is divided into three distinct phases: the vortex lattice, the entangle phase, and the quasi 2D pancake vortex states in the temperature interval 30 K ≤T≤60 K .1 From the relaxation of remanent magnetization measurements, we have obtained the flux creep activation barrier U(j) as a function of j using the method developed by Malay et al. Within the collective pinning theory, it is argued that T≤30 K , the pancake vortices are pinned individually, while for T≥42 K , the vortex state is characterized by individually pinned 3D vortex strings. At T~30 K , a crossover from 2D pancake to 3D collectively pinned state was observed and shows a smooth crossover to 3D single vortex state at higher temperatures.

High‐temperature magnetic stability of small magnetite particles

The stability of magnetic domain structures of small grains of magnetite were examined between room temperature and the Curie temperature using a high‐resolution three‐dimensional micromagnetic algorithm. At all times the minimum resolution used was determined by calculating the exchange length. Using an unconstrained model, the single domain (SD) to multidomain (MD) threshold grain size d0 was found to be nearly independent of temperature up to ∼450°C. Above this temperature, d0 was observed to rise sharply. Energy barriers between metastable domain states trapped in local energy minimums (LEM) were determined using a constrained algorithm. Three types of domain structure were considered: SD, vortex, and double vortex (effectively three domain), in a range of grain sizes with side length between 30 and 300 nm. In addition, the effect of varying shape was also considered by examining asymmetric grains with aspect ratios up to 1.4. From the numerical solutions energy barriers between LEM states were determined. It was found that MD grains 300 nm in size display higher stability than smaller SD grains (∼50 nm). Double vortex states were found to be less stable than single vortex states at nearly all temperatures. Blocking temperatures as function of grain size for both symmetric and asymmetric grains were determined and agree well with experimental results. Transdomain thermoremanence analysis indicated that there are a limited number of grain sizes and shapes which will nucleate domain wall‐type structures during cooling. Such nucleation events would cause the total measured remanence to decrease with cooling in conflict with Néel's analytical theory for remanence cooling behavior but in agreement with experimental observations.

Magnetic relaxation inTl2Ba2CaCu2O8single crystals

The time relaxation of the remanent magnetization of a ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}$ single crystal has been carried out. The flux creep activation barrier $U(j),$ obtained using the method developed by Malay et al. [Phys. Rev. B 42, 2639 (1990)], follows a logarithmic behavior in j except in the temperature region $30\mathrm{K}<~T<~42\mathrm{K}.$ In this temperature region, $U(j)$ was found to follow a power law dependence. Within the weak collective pinning theory for strongly layered superconductors, it is argued that for low temperatures, the pancake vortices are pinned individually while for high temperatures, the vortex state is characterized by individually pinned three-dimensional (3D) vortex strings. At $T\ensuremath{\sim}30\mathrm{K},$ due to strong thermal fluctuation effects, a crossover from 2D pancake vortices to a 3D collectively pinned state was observed and shows a smooth crossover to a 3D single vortex state at higher temperatures.

Vortex charge in the superconducting state

A microscopic calculation for the charge distribution in the single vortex state in type II superconductors is presented, in which the charge screening effect is microscopically incorporated within linear response of the scalar potential. The Poisson equation is numerically solved with the Bogoliubov–de Gennes equation in the single vortex state, and the charge screening inside the core is clarified. The Coulomb interaction between core electrons induces the oscillations in the charge distribution inside the vortex core.

The symmetry background underlying the ring structures of quantum dots and aclassification scheme

Due to the constraints arising from symmetry, each symmetric geometric configurationis accessible only to a specific group of states having specific quantum numbers. On thebasis of the accessibility of the important symmetric configurations of a two-dimensionalN-electrondot and/or of its subsystems, a scheme is proposed for classifying the states. Inthis scheme, each type of state has its own preference for ring structures.Numerical calculations for 9-electron and 19-electron dots have been performed tocheck the scheme. It was found that, on the basis of the scheme, the electronicstructures of low-lying states can be predicted nearly perfectly. The symmetrybackground underlying the Laughlin wavefunction and the single-vortex states hasalso been discussed. Incidentally, an integer can be ultimately resolved as aproduct of prime numbers; this resolution has been used in the classification.

Vortex states in a mesoscopic superconducting triangle

The set of the nonlinear Ginzburg–Landau equations is solved for an Al mesoscopic superconducting triangle of finite thickness. We calculate the distributions of the superconducting phase in the triangle and of the magnetic field in and near the triangle. The distribution of the superconducting phase in the triangle is studied as a function of the applied magnetic field. Possible scenarios of penetration of the magnetic field into the triangle are analyzed. We consider two different states: a single vortex state and a state in the form of a symmetric combination of three vortices and an antivortex with vorticity L a =−2 (“3−2” combination). The free energy calculations show that a single vortex penetrates the triangle through a midpoint of one side. The “3−2” combination turns out to be thermodynamically preferable when the vortices are close to the center of the triangle. Equilibrium is achieved when a single vortex (or each component of the “3−2” combination) is in the center of the triangle.

Micromagnetics simulation of deep-submicron supermalloy disks

The results of recent micromagnetic simulations of deep submicron supermalloy disks are presented. A recent experimental measurement of the hysteresis and magnetic domain structure in supermalloy disks with diameters ranging from 55 to 500 nm and thickness ranging from 6 to 15 nm has been reported. Our micromagnetic simulations show remarkable agreement with the experimental hysteresis loops. The simulation results show that for thin or small diameter disks a single magnetic domain exists with all spins aligned. The hysteresis loop represents free rotation of these spins. For larger diameter disks or as the thickness increases the hysteresis loops change shape due to the appearance of a single vortex state appearing at low applied fields.

Vortex state in superfluid trapped Fermi gases at zero temperature

We discuss various aspects of the single vortex state of a dilute superfluid atomic Fermi gas at $T=0.$ The energy of the vortex in a trapped gas is calculated and we provide an expression for the thermodynamic critical rotation frequency of the trap for its formation. Furthermore, we propose a method to detect the presence of a vortex by calculating the effect of its associated velocity field on the collective mode spectrum of the gas.

Poisson's Equation in the Vortex State of Type II Superconductors

We derive Poisson's equation in the single-vortex state of type II superconductors in a conserved approximation in both dynamical and static cases. The charge and current vertex functions in the ch...

Rotating Bose gas with hard-core repulsion in a quasi-two-dimensional harmonic trap: Vortices in Bose-Einstein condensates

We consider a gas of $N(<~15)$ Bose particles with hard-core repulsion, contained in a quasi-two-dimensional harmonic trap and subjected to an overall angular velocity $\ensuremath{\Omega}$ about the z axis. Exact diagonalization of the $n\ifmmode\times\else\texttimes\fi{}n$ many-body Hamiltonian matrix in given subspaces of the total (quantized) angular momentum ${L}_{z},$ with $n\ensuremath{\sim}{10}^{5}$ (e.g., for ${L}_{z}=N=15, n=240782)$ was carried out using Davidson's algorithm. The many-body variational ground-state wave function, as also the corresponding energy and the reduced one-particle density-matrix $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})={\ensuremath{\sum}}_{\ensuremath{\mu}}{\ensuremath{\lambda}}_{\ensuremath{\mu}}{\ensuremath{\chi}}_{\ensuremath{\mu}}^{*}(\mathbf{r}){\ensuremath{\chi}}_{\ensuremath{\mu}}({\mathbf{r}}^{\ensuremath{'}})$ were determined. With the usual identification of $\ensuremath{\Omega}$ as the Lagrange multiplier associated with ${L}_{z}$ for a rotating system, the ${L}_{z}\ensuremath{-}\ensuremath{\Omega}$ phase diagram (or the stability line) was determined that gave a number of critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{},$ at which the ground-state angular momentum and the associated condensate fraction, given by the largest eigenvalue of the reduced one-particle density matrix, undergo abrupt jumps. For a given N, a number of (total) angular momentum states were found to be stable at successively higher critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{}.$ All the states in the regime $N>{L}_{z}>0$ are metastable. For ${L}_{z}>N,$ the ${L}_{z}$ values for the stable ground states generally increased with increasing critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}},$ and the condensate was strongly depleted. The critical ${\ensuremath{\Omega}}_{\mathrm{ci}}$ values, however, decreased with increasing interaction strength as well as the particle number, and were systematically greater than the nonvariational yrast-state values for the ${L}_{z}=N$ single vortex state. We have also observed that the condensate fraction for the single vortex state (as also for the higher vortex states) did not change significantly even as the two-body interaction strength was varied over several $(\ensuremath{\sim}4)$ orders of magnitude in the moderately to the weakly interacting regime.

Vortex formation in a stirred bose-einstein condensate

Using a focused laser beam we stir a Bose-Einstein condensate of 87Rb confined in a magnetic trap and observe the formation of a vortex for a stirring frequency exceeding a critical value. At larger rotation frequencies we produce states of the condensate for which up to four vortices are simultaneously present. We have also measured the lifetime of the single vortex state after turning off the stirring laser beam.

An investigation of magnetic reversal in submicron-scale Co dots using first order reversal curve diagrams

First order reversal curve (FORC) diagrams are a powerful method of investigating the physical mechanisms giving rise to hysteresis in magnetic systems. We have acquired FORC diagrams for an array of submicron-scale Co dots fabricated by interference lithography. These dots reverse magnetization through the nucleation and annihilation of a single-vortex state. Using FORC diagrams, we are able to obtain precise values for the nucleation and annihilation fields involved in magnetic reversal. Our results indicate, however, that there are actually two distinct paths for vortex annihilation: When a complete magnetic reversal takes place, a vortex enters on one side of a dot and exits out the opposite side. But if the magnetization is returned to its original orientation before a complete reversal has occurred, then the vortex will exit on the same side from which it has entered. We are unable to obtain a precise field value for this later path of annihilation; however, it is shown that, statistically, the vortex annihilates with greater ease when it exits out the same side from which it has entered.

Magnetic domain structure and magnetization reversal in submicron-scale Co dots

We present a magnetic force microscopy (MFM) analysis of arrays of submicron-scale Co dots fabricated by interference lithography. The dots are thin (180–300 Å) and elliptical in shape. MFM reveals that these structures relax into highly ordered remanent states whose symmetry and configuration are governed by their shape anisotropy. In particular, when the dots are saturated along their long-axis, a uniformly magnetized state persists at remanence. However, when the dots are saturated along their short-axis, they relax into a single-vortex state in which the circulation can have either sign. Both states are characterized by smoothly varying magnetization patterns and a high degree of uniformity across the array. We attribute the ordered behavior of these structures to the film microstructure, which allows the shape anisotropy to dominate over magnetocrystalline anisotropy. By imaging a series of minor-loop remanent states, we show that magnetization reversal in these structures occurs via the nucleation and annihilation of a single vortex. Magnetic hysteresis loop measurements are consistent with these observations and provide additional details. Furthermore, we present the results of micromagnetic simulations, which are in excellent agreement with both the MFM images and the hysteresis loop measurements.

Superconducting states and depinning transitions of Josephson ladders

We present analytical and numerical studies of pinned superconducting states of open-ended Josephson ladder arrays, neglecting inductances but taking edge effects into account. Treating the edge effects perturbatively, we find analytical approximations for three of these superconducting states -- the no-vortex, fully-frustrated and single-vortex states -- as functions of the dc bias current $I$ and the frustration $f$. Bifurcation theory is used to derive formulas for the depinning currents and critical frustrations at which the superconducting states disappear or lose dynamical stability as $I$ and $f$ are varied. These results are combined to yield a zero-temperature stability diagram of the system with respect to $I$ and $f$. To highlight the effects of the edges, we compare this dynamical stability diagram to the thermodynamic phase diagram for the infinite system where edges have been neglected. We briefly indicate how to extend our methods to include self-inductances.

Jovian Dynamics. Part 1: Vortex Stability, Structure, and Genesis

The vertical of Jupiter's atmosphere is probed and isolated by evaluating the stability characteristics of planetary vortices over a wide parameter range. The resulting structures lead to simulating the genesis of single and multiple vortex states in Part I of this paper and the genesis of an equatorial superrotation and midlatitudinal multiple jets in Part II. The stability and genesis of baroclinic Rossby vortices, the vortices associated with long solitary Rossby waves in a stratified fluid, are studied numerically using a primitive equation model with Jovian and oceanic parameters and hypo-thermal structures. Vortex stability, that is, coherence and persistence, depends primarily upon latitude location and vertical structure and is used to deduce possible stratifications for Jupiter's atmosphere. The solutions suggest that Jupiter's large-scale motions are confined to a layer of depth h and are bounded by an abyss with an impermeable interface at a depth H, such that h/H≤1/20. Consequently, they also extend earlier results derived with the reduced-gravity, shallow-water model, particularly the explanation for the origin, uniqueness, and longevity of the Great Red Spot (GRS). Beginning at the equator, stable anticyclones are seen to exist only when they have the Hermitian latitudinal form, the Korteweg-deVries longitudinal form, the confined exponential vertical structure exp(Nz/H), and the amplitude range as prescribed by the analytical theory of Marshall and Boyd for N=8. Soliton interactions occur between equatorial vortices of similar horizontal and vertical form. In middle and low latitudes, shallow anticyclones with an exponential structure of N=20 exist quasi-stably for a variety of sizes. Such vortices remain coherent but tend to migrate equatorward (where they disperse) at rates that depend upon their size, location, and vertical structure: large and medium anticyclones propagate primarily westward while migrating slowly, whereas small storms just migrate rapidly and then collapse. The migration of these large, shallow vortices can be reduced, but not stopped, in low latitudes by an easterly jet with the same vertical structure. Anticyclones are stabler when they are thinner relative to the abyss. Thus, when N=60, their migration is sufficiently slow that it can be stopped by a weak easterly jet. Furthermore, absolute stability sets in when N=90 and migration ceases completely for the large, thin anticyclones that now just propagate westward. Such flows may also be usefully represented by a vertical structure that is linear in z for the velocity and static stability in the thin upper layer and vanishes in the abyss. Large, thin (N≥90) anticyclones can exist indefinitely either freely or when embedded within an anticyclonic zone of alternating jet streams of similar vertical structure. This holds true for the confined linear-z representation also. The permanence of GRS-like, low-latitude vortices in Jovian flow configurations occurs in a variety of lengthy calculations with thin structures. Ocean vortices are less persistent because the thermocline is relatively thick. The baroclinic instability of easterly jets is nonquasigeostrophic and takes on the form of solitary rather than periodic waves when the jets have a thin exponential (N≥90) or confined linear-z structure. Such nonlinear waves develop into vortices that exhibit a variety of configurations and evolutionary paths. In most cases multiple mergers tend toward an end state with a single large vortex. Two types of merging occur in which a stronger vortex either catches a weaker one ahead of it or reels in a weaker one from behind. This duality occurs because propagation rates depend as much on local as on global conditions. In a further complication, vortices generated by an unstable easterly tend to have an exponential structure for exponential jets but a first baroclinic eigenmodal structure for confined linear-z jets. Single vortex states resembling the ORS, with sizes ranging from 15° to 50° in longitude and with temperature gradients, velocities, and propagation rates near the observed range, can be generated either directly through the growth of a local front in a marginally unstable easterly jet or indirectly through a series of mergers of the multiple vortices generated by a more unstable easterly jet. Sets of vortices can be produced simultaneously in the anticyclonic zones centered about latitudes −21°, −33°, and −41°, and have the same relative scales as Jupiter's GRS, Large Ovals, and Small Ovals. Thin anticyclones can also be generated at the equator by the action of vortices lying in low latitudes. Equally realistic long-lived vortices can also be generated by jets with structures matching the recent Galileo spacecraft observations by using other hyperbolic forms and greater depth scales.

Andreev bound states in unconventional vortices

We present isolated vortex solutions of the quasiclassical equations in weak coupling for an unconventional superconductor that belongs to the E1g representation of the hexagonal symmetry group. We discuss the structure of the single flux quantum vortex states and the dispersion of Andreev bound states.

Stability of a Vortex near the Surface of Ferromagnetic Superconductors

Near the surface of ferromagnetic superconductors, the free energy difference between the single vortex state and the no vortex state is calculated as a function of the distance between the vortex and the surface for various values of temperature T and external magnetic field H 0 . It is shown that the vortex is bound near the surface at T < T 0 under H 0 higher than a critical value, where T 0 is the characteristic temperature below which the external magnetic field penetrates into the crystal in an oscillatory fashion. The interaction between the vortex and the penetrating magnetic field plays an important role in the stability of the vortex near the surface. A possibility of the self-induced vortex state near the surface is predicted.