Vortex Ground State Research Articles

Critical velocity for the vortex core reversal in perpendicular bias magnetic field

For a circular magnetic nanodot in a vortex ground state, we study how the critical velocity vc of the vortex core reversal depends on the magnitude H of a bias magnetic field applied perpendicularly to the dot plane. We find that, similarly to the case H=0, the critical velocity does not depend on the size of the dot. The critical velocity is dramatically reduced when the negative (i.e., opposite to the vortex core direction) bias field approaches the value, at which a static core reversal takes place. A simple analytical model shows good agreement with our numerical result.

Precise probing spin wave mode frequencies in the vortex state of circular magnetic dots

We report on detailed broadband ferromagnetic resonance measurements of azimuthal and radial spin wave excitations in circular Permalloy dots in the vortex ground state. Dots with aspect ratio (β=height over radius) varied from 0.03 to 0.1 were explored. The frequency splitting of two lowest azimuthal modes was observed. The experimentally observed dependence of the frequency splitting on β was reasonably well described by dynamic splitting model accounting the spin waves and vortex gyrotropic mode interaction.

Non-Abelian Magnetic Monopole in a Bose-Einstein Condensate

Recently, an effective non-Abelian magnetic field with a topology of a monopole was shown to emerge from the adiabatic motion of multilevel atoms in spatially varying laser fields [J. Ruseckas, Phys. Rev. Lett. 95, 010404 (2005)10.1103/PhysRevLett.95.010404]. We study this monopole in a Bose-Einstein condensate of degenerate dressed states and find that the topological charge of the pseudospin cancels the monopole charge resulting in a vanishing gauge invariant charge. As a function of the laser wavelength, different stationary states are classified in terms of their effect to the monopole part of the magnetic field and a crossover to vortex ground state is observed.

On the effective action of center vortices in continuum Yang-Mills theory

The effective action of sufficiently smooth center vortices in continuum Yang-Mills theory is investigated taking into account some of their basic properties observed in lattice simulations. The obtained action is given by the action of an ideal vortex configuration, which is local on the vortex worldsheet, nonlocal quantum corrections due to virtual gluon exchanges, which can be reliably calculated using perturbation theory, and the contribution of Abelian magnetic monopoles. Smooth center vortices appear to have negative rigidity. This implies that, in agreement with lattice data, in the ground state of the theory center vortices are not regular surfaces.

Frequencies of Radially Symmetric Excitations in Vortex State Disks

Submicron ferromagnetic disks with a vortex ground state exhibit interesting and potentially useful dynamic properties arising from excitations on the ground state. In particular, a magnetic pulse applied perpendicular to the vortex plane will excite radially symmetric modes. Previous calculations of the frequencies of these modes based on the linearized Landau-Lifschitz equation and the magnetostatic Green's function give eigenfrequencies proportional to the square root of the aspect ratio radic( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L/R)</i> , where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> is the disk thickness and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> is the disk radius. However, experimental frequency data show significant deviation from the square root dependence. An improved calculation of the frequency is done through a collective variable approach by exploiting the high symmetry of these modes. In the linear approximation, this approach gives the main contribution to the frequency proportional to radic(( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L/R</i> )ln( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R/L</i> )), which is closer to the observed aspect ratio dependence.

Dynamic Origin of Azimuthal Modes Splitting in Vortex-State Magnetic Dots

A spin-wave theory explaining experimentally observed frequency splitting of dynamical spin excitations with azimuthal symmetry of a magnetic dot in a vortex ground state is developed. It is shown that this splitting is a result of the dipolar hybridization of three spin-wave modes of a dot having azimuthal indices |m|=1: two high-frequency azimuthal dipolar modes of the in-plane part of the vortex with indices m = +/-1 and a low-frequency (Goldstone-like) gyrotropic mode, describing translational motion of the vortex core and having index m = +1. The analytically calculated magnitude of the frequency splitting is proportional to the ratio of the dot thickness to its radius and quantitatively agrees with the results of time-resolved Kerr experiments.

Magnetic Vortex State Stability, Reversal and Dynamics in Restricted Geometries

Magnetic vortices are typically the ground states in geometrically confined ferromagnets with small magnetocrystalline anisotropy. In this article I review static and dynamic properties of the magnetic vortex state in small particles with nanoscale thickness and sub-micron and micron lateral sizes (magnetic dots). Magnetic dots made of soft magnetic material shaped as flat circular and elliptic cylinders are considered. Such mesoscopic dots undergo magnetization reversal through successive nucleation, displacement and annihilation of magnetic vortices. The reversal process depends on the stability of different possible zero-field magnetization configurations with respect to the dot geometrical parameters and application of an external magnetic field. The interdot magnetostatic interaction plays an important role in magnetization reversal for dot arrays with a small dot-to-dot distance, leading to decreases in the vortex nucleation and annihilation fields. Magnetic vortices reveal rich, non-trivial dynamical properties due to existance of the vortex core bearing topological charges. The vortex ground state magnetization distribution leads to a considerable modification of the nature of spin excitations in comparison to those in the uniformly magnetized state. A magnetic vortex confined in a magnetically soft ferromagnet with micron-sized lateral dimensions possesses a characteristic dynamic excitation known as a translational mode that corresponds to spiral-like precession of the vortex core around its equilibrium position. The translation motions of coupled vortices are considered. There are, above the vortex translation mode eigenfrequencies, several dynamic magnetization eigenmodes localized outside the vortex core whose frequencies are determined principally by dynamic demagnetizing fields appearing due to restricted dot geometry. The vortex excitation modes are classified as translation modes and radially or azimuthally symmetric spin waves over the vortex ground state. Studying the spin eigenmodes in such systems provides valuable information to relate the particle dynamical response to geometrical parameters. Unresolved problems are identified to attract attention of researchers working in the area of nanomagnetism.

Magnetic Vortex State Stability, Reversal and Dynamics in Restricted Geometries

Magnetic vortices are typically the ground states in geometrically confined ferromagnets with small magnetocrystalline anisotropy. In this article I review static and dynamic properties of the magnetic vortex state in small particles with nanoscale thickness and sub-micron and micron lateral sizes (magnetic dots). Magnetic dots made of soft magnetic material shaped as flat circular and elliptic cylinders are considered. Such mesoscopic dots undergo magnetization reversal through successive nucleation, displacement and annihilation of magnetic vortices. The reversal process depends on the stability of different possible zero-field magnetization configurations with respect to the dot geometrical parameters and application of an external magnetic field. The interdot magnetostatic interaction plays an important role in magnetization reversal for dot arrays with a small dot-to-dot distance, leading to decreases in the vortex nucleation and annihilation fields. Magnetic vortices reveal rich, non-trivial dynamical properties due to existance of the vortex core bearing topological charges. The vortex ground state magnetization distribution leads to a considerable modification of the nature of spin excitations in comparison to those in the uniformly magnetized state. A magnetic vortex confined in a magnetically soft ferromagnet with micron-sized lateral dimensions possesses a characteristic dynamic excitation known as a translational mode that corresponds to spiral-like precession of the vortex core around its equilibrium position. The translation motions of coupled vortices are considered. There are, above the vortex translation mode eigenfrequencies, several dynamic magnetization eigenmodes localized outside the vortex core whose frequencies are determined principally by dynamic demagnetizing fields appearing due to restricted dot geometry. The vortex excitation modes are classified as translation modes and radially or azimuthally symmetric spin waves over the vortex ground state. Studying the spin eigenmodes in such systems provides valuable information to relate the particle dynamical response to geometrical parameters. Unresolved problems are identified to attract attention of researchers working in the area of nanomagnetism.

Phases of Atom-Molecule Vortex Matter

We study ground state vortex configurations in a rotating atom-molecule Bose-Einstein condensate. It is found that the coherent coupling between the atomic and molecular condensates can render a pairing of atomic and molecular vortices into a composite structure that resembles a carbon dioxide molecule. Structural phase transitions of vortex lattices are also explored through different physical parameters including the rotational frequency of the system.

Pillars as antipinning centers in superconducting films

Using the nonlinear Ginzburg-Landau theory we study vortex configurations in a superconducting thin film with a square array of pillars in the presence of a uniform applied magnetic field. The presence of the pillars changes the vortex structures in the superconducting film considerably: a transition between triangular and square vortex lattices takes place with increasing size and/or height of the pillars and vortex lines and vortex clusters can be obtained for particular applied magnetic fields. All of these findings are summarized into an equilibrium vortex phase diagram, which shows the transition between different ground-state vortex configurations as a function of the radius and periodicity of pillars. For larger radius of the pillars vortices start to penetrate the pillars and order in vortex shell structures both inside and around the pillars. The theoretical results are complemented by an experimental study of the vortex configurations in single-crystal based Nb pillars with radius $R\ensuremath{\sim}0.5--2\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$. Using Bitter decoration, concentric shells of vortices are revealed inside the pillars and shell-like structures are found at the interstitial sites, in accordance with the theoretical results. In addition, the observed distortion of the vortex shell structures by weak pinning centers present in our Nb samples is investigated by molecular dynamics simulations. We also show that the transition between predicted vortex states can be obtained from the temperature dependence of local magnetization measurements.

Novel pinning phenomena in Nb thin films with square pinning arrays

Electron-beam lithography and reaction dry etching techniques have been used to fabricate superconductor thin films with regular square hole arrays to explore the vortex pinning behavior. Magnetotransport measurements were carried out by a four-probe technique. Periodic critical current matching peaks are observed. We found that the height of the third matching peak is close to that of the fourth matching peak for film with holes of diameter of 200nm at certain temperatures, and it appears that the third matching peak is missing. Similar peculiar phenomena are observed for films with holes of different diameters, but the missing peaks are at the fourth or fifth matching field for larger holes. We made molecular dynamic simulations to study the ground state vortex distributions and the dynamics of the vortex motion. The simulation results offer good explanations to the features we found in the experiments.

Excitation of Spin Dynamics by Spin-Polarized Current in Vortex State Magnetic Disks

A spin-polarized current with the polarization direction perpendicular to a disk in the vortex ground state will result in renormalization of the effective damping of excitations on this state. As the current is increased to a threshold current Ic the effective damping will be zero and the lowest threshold current corresponds to the vortex gyrotropic mode. For larger values of the current the excitation is a nonlinear gyrotropic mode having nonsmall amplitudes and larger frequency than the linear mode. This effect occurs for any mode of the vortex-state disk, and the value of Ic is proportional to the mode frequency.

Field evolution of the magnetic normal modes in elongated permalloy nanometricrings

The eigenmode spectrum of elongated permalloy rings with relatively wide arms isinvestigated by combined Brillouin light scattering and ferromagnetic resonancemeasurements as a function of the applied field intensity, encompassing both vortex andonion ground states. To reproduce the frequencies and the spatial profiles of themeasured modes we performed micromagnetic simulations which solve the discretizedLandau–Lifshitz–Gilbert equation in the time domain and calculate locally theFourier transform. This allowed us to correlate the field dependence of differentmodes to their localization inside different portions of the rings. With the ringsin the vortex ground state, in addition to radial, fundamental, and azimuthalmodes, a localized mode, existing in the element portions where the internalfield assumes its minima, has been measured and identified. This latter mode,whose frequency decreases for increasing field intensity, becomes soft near thetransition from vortex to onion state and determines the change in symmetryof the magnetic ground state. After the transition, it is replaced by two edgemodes, localized on the internal and external boundary of the rings, respectively.

Superconducting Transitions in Wire Network under Spatially Modulated Magnetic Field

Superconducting wire network subjected to a uniform and a checkerboard-patterned magnetic field is investigated. The checkerboard field is created by an array of ferromagnetic dots placed on top of the superconducting network. Measurements of the current–voltage ( I – V ) characteristics have revealed the canonical Kosterlitz–Thouless–Berezinskii (KTB) transition at zero magnetic field. The superconducting transition at α=1/2 (α being the frustration parameter) is studied in detail, and compared with the theoretical predictions on the fully frustrated XY model which has a double degeneracy in the ground state vortex configuration. The I – V characteristics for α=1/2 is consistent with the scenario of kink–antikink unbinding at antiphase domain boundaries first proposed by Korshunov [Phys. Rev. Lett. 88 (2002) 167007]. Introduction of the checkerboard field with amplitude β to the α=1/2 case lifts the double degeneracy. The effect of non-zero β on the nature of superconducting transition at α=1/2 is consis...

Mode degeneracy due to vortex core removal in magnetic disks

The mode spectrum of micrometer-sized ferromagnetic Permalloy disks, exhibiting a vortex ground state, is investigated by means of time-resolved scanning Kerr microscopy. The temporal evolution of the magnetization is probed after application of a fast in-plane field pulse. The lowest order azimuthal mode, a mode with only one diametric node, splits into a doublet as the disk diameter decreases. Theoretical models show that this splitting is a consequence of the interaction of the mode with the gyrotropic motion of the vortex core. Our experiments and micromagnetic simulations confirm that by removing the vortex core from the disk, the mode splitting vanishes.

New commensurate vortex structures in type-I and type-II superconducting films with antidot-arrays

For a superconducting film with regular array of antidots, we constructed the equilibrium vortex phase diagram within the Ginzburg–Landau (GL) formalism. Novel ground-state vortex configurations were found depending on the size and periodicity of the antidots and the GL parameter κ, comprising vortex–antivortex structures, giant- and multi-vortex combinations, or lattices of giant-vortices, all commensurate with the square pinning arrays.

Magnetization process of the vortex state in soft magnetic thin square platelets

An analysis of the magnetization process in magnetically ultra-soft thin film elements is presented for the case of square platelets, which show a vortex ground state and have a very low thickness. The analysis is based on micromagnetic calculations for systems with 1 μm edge length, 8–20 nm thickness and for material parameters similar to permalloy, but with negligible magnetic anisotropy. For the case of a field applied along the diagonal of the square platelet the evolution of magnetization, critical fields for the expulsion of the vortex and the leading energy terms have been determined from the numerical simulations. We show that a phase theory approximation, by using the position of the vortex as the only variable for the evolution of the vortex pattern, semi-quantitatively describes the main features of the reversible magnetization process and the stability limit. An effective demagnetization factor for the vortex pattern of the squares is determined from the numerical results. This effective parameter enables a quantitative description of the main properties of the vortex state and its magnetization process. Deviations of this phase-theory approximation from the numerical calculations are traced back to the evolution of inhomogeneous internal magnetization states, in particular wide walls that change their profiles under the applied field.

Multicanonical sampling of vortex states in magnetic nanoelements

An auto-adaptive multicanonical Monte Carlo (MMC) simulation method is suggested and tested on a single-vortex model of magnetic nanoelement. Simulation process consisting of nonequilibrium and equilibrium stages that circumvents ergodicity sampling problems which stem from a potential barrier standing between the vortex and counter-vortex states is proposed. The method is formulated by the means of an effective Hamiltonian with additional term proportional to the overlap of given configuration and bistable ground-state vortex configuration. The self-organized neural network is used to construct the synopsis of the vortex reversal process.

Dynamics of vortex core switching in ferromagnetic nanodisks

Dynamics of magnetic vortex core switching in nanometer-scale Permalloy disk, having a single vortex ground state, was investigated by micromagnetic modeling. When an in-plane magnetic field pulse with an appropriate strength and duration is applied to the vortex structure, additional two vortices, i.e., a circular and an antivortex, are created near the original vortex core. Sequentially, the vortex-antivortex pair annihilates. A spin wave is created at the annihilation point and propagated through the entire element; the relaxed state for the system is the single vortex state with a switched vortex core.

Magnon excitations in vortex-state nanorings

The magnon mode excitation spectrum is calculated using the linearized Landau–Lifshitz equations applied to the ring in the vortex ground state. A combination of analytical techniques with numerical evaluation of integrals is used to obtain the mode frequency, with the effective magnetostatic volume charge as a small perturbation. In general, the modes can be classified according to radial number n and azimuthal number m. It is shown that the frequency increases as n increases, and the frequency decreases as m increases.