1,721,093 research outputs found
Theory of spin modes in the vortex state -- Presentazione orale by R. Zivieri - Conferenza internazionale
No full textA theoretical model for the calculation of the quantized spectrum of spin modes frequencies in cylindrical magnetic dots of radius ranging from the nanometric to the micrometric scale in the vortex ground state is presented [1,2]. We take into account the core energy showing its effect on the spin dynamics. The spin mode eigenfrequencies are
calculated in the diagonal approximation. The spin mode eigenfrequencies are
calculated in the diagonal approximation by means of
the exact calculation of the dipolar field [2].
[1] R. Zivieri and F. Nizzoli, “ Theory of spin
modes in vortex-state ferromagnetic
cylindrical dots”, Phys. Rev. B, 71, 014411-
1-5 (2005); R. Zivieri and F. Nizzoli, Phys.
Rev. B, 75, 1(E) (2006)
[2] R. Zivieri, “Dipolar fields in vortex-state
spin dynamics”, in preparation -- Presentazione orale by R. Zivieri - Conferenza internazional
Theory of spin modes in the vortex state -- Presentazione orale by R. Zivieri - Conferenza internazionale
No full textA theoretical model for the calculation of the quantized spectrum of spin
modes frequencies in cylindrical magnetic dots of radius ranging from the
nanometric to the submicrometric scale in the vortex ground state is presented
[1]. We take into account of the core energy showing its effect on the spin
dynamics. At zero applied field the surface modes at lower frequencies present
as radial eigenvectors Bessel functions of high orders (m . 1), while the axially
symmetric modes at higher frequencies correspond to zero order Bessel
functions. The out-of-core and core dipolar field components are calculated
exactly for both axially symmetric and diametrically symmetric vortex spin
modes and the effect on the spin dynamics is studied. A comparison of this
calculation with the spin modes frequencies obtained using the local
approximation is performed both considering a dependence of the dynamic
magnetization from the dot thickness and assuming uniform magnetization.
The dynamics of volume modes in the vortex state is also quantitatively studied
within this model. The frequency splitting of the m = + -1 modes [2] is explained
in terms of the different energy contribution of the two corresponding spin
modes dipolar fields and compares well with available experimental data. An
expression for the gyrotropic mode, classified in this framework as a pseudo-
Goldstone excitation because of the presence of a static core field, is derived
and its frequency is compared to the measured one for various aspect ratios.
The model is generalized [3] to determine the spin dynamics in the vortex state,
but under an external applied field. Both the regime when the vortex centre is
inside the dot and that when it is outside it are considered and their effect on
the quantized spin modes is studied. It is shown that the spin modes profiles are
affected by the broken symmetry induced by the external field. The spin modes
frequencies also depend from additional effective fields absent at vanishing
applied field.
[1] R. Zivieri and F. Nizzoli, Phys. Rev. B 71 (2005) 014411
[2] R. Zivieri in preparation
[3] R. Zivieri in preparation -- Presentazione orale by R. Zivieri - Conferenza internazional
Metamaterial Properties of Magnetic Nanostructures - Invited presentation by R. Zivieri - Conferenza internazionale
No full textIn recent years the study of low-dimensional magnetic systems has become topical for achieving a deep understanding of the underlying physics of magnetic nanostructures and for its potential technological applications. Very recently, great attention has been devoted to the investigation of the effective and metamaterial properties of magnetic nanostructures with special regard to magnonic crystals, a class of periodic magnetic systems. In this talk it is shown according to micromagnetic simulations and analytical calculations that magnonic crystals exhibit effective properties directly linked to the static and dynamic properties of collective modes. Some possible applications based on the effective properties for tailoring new magnetic devices are suggested [1].
1. R. Zivieri, L. Giovannini, Photonics Nanostruct. Fundam. Appl. 11, 191 (2013).
2. R. Zivieri, P. Malagò, L. Giovannini, Photonics Nanostruct. Fundam. Appl. 12, 398 (2014)
Effective description of 2D and 3D magnonic metamaterials -- Invited talk by R. Zivieri -- Conferenza internazionale
No full textThe effective properties of two-dimensional (2D) and three-dimensional (3D) magnonic metamaterials are presented according to micromagnetic and analytical calculations. Micromagnetic calculations were performed by using the Hamiltonian-based Dynamical Matrix Method (HDMM) extended to periodic systems. The 2D systems are composed by periodic square arrays of circular antidots (holes) embedded into a permalloy ferromagnetic film. In the calculations both the diameters d of the holes and the array periodicities are in the nanometric range. Two geometries have been investigated: 1) The external magnetic field H is applied along the y direction and perpendicularly to the Bloch wave vector K placed along the x direction in the sample plane. 2) H (K) forms an angle of 45o degrees with respect to the y (x) axis. Magnonic modes dispersion is calculated as a function of K for both extended and localized modes and opening of band gaps at Brillouin zone boundaries is explained in terms of inhomogeneity of the internal field [1,2]. Band gaps are discussed as examples of metamaterial properties of 2D magnonic crystals. In both geometries it is possible to identify, for a given collective mode, a characteristic wavelength which is commensurable with the periodicity of the system [3]. Since collective modes are mainly affected by the finite size of the holes rather than the periodicity and since the characteristic wavelength is much larger than d, the dynamics is described in terms of effective properties and an effective medium approximation is used to model the metamaterial wave in the propagative regime. These properties can be regarded as metamaterial properties [4,5]. In this way, the characteristic wavelength can be regarded as an effective wavelength λeff related to the scattering from the holes of the given collective mode. Interestingly, the effective wavelength, which can be defined for each mode of the spectrum, is not necessarily equal to the Bloch wavelength. In the cases studied the ratio d /λeff <<1 for the whole range of Bloch wave vectors investigated. Correspondingly, also a small wave vector is introduced and important effective rules that do not contradict the Bloch’s theorem, but complete it, are derived. A description of scattering from antidots in terms of momentum conservation is given and the differences with the well-known Bragg diffraction law are outlined [6,7].
The 3D systems are composed by 2D periodic arrangements of circular nanodots of cobalts partially or totally embedded into a permalloy film, but subdivided into a stack of layers along the z-direction accounting for the nonuniform magnetization of the two materials along z [8,9]. Band structure of the most representative collective modes, namely a mode mainly localized in the cobalt dots and a mode prevantly concentrated in the permalloy film is studied. The dependences of band gap amplitudes on the cobalt volume and on the cobalt position with respect to the permalloy film are also discussed. Effective “surface magnetic charges” are introduced to explain the demagnetizing field behaviour associated to the two materials and effective quantities, like an effective magnetization and an effective exchange stiffness constant, are introduced and the dispersion of the corresponding metamaterial wave in the propagative regime is calculated. It is also shown that the interchange between the two materials in the system leads to different band structure of the two most representative collective modes. Finally, a concentration factor is introduced to quantitatively express the localization of the most relevant collective modes [9] in analogy with the corresponding one defined for electromagnetic waves in 2D photonic crystals.
[1] R. Zivieri et al., Physical Review B 85 012403-1-6 (2012).
[2] R. Zivieri, Solid State Physics 63, 151-216 (2012).
[3] R. Zivieri, Proceedings of Metamaterials ’2012, 6th International Congress on Advanced Electromagnetic Materials in Microwave and Optics, 624-626 (2012).
[4] R. Zivieri and L. Giovannini, Metamaterials 6, e127-138 (2012).
[5] R. Zivieri and L. Giovannini, “Effective quantities and effective rules in 2D ferromagnetic antidot lattices” Photonics and Nanostructures - Fundamentals and Applications 11, 191-202 (2013).
[6] R. Zivieri, “Metamaterial description of magnonic modes along ΓM direction in a 2D antidot lattice” in press in Proceedings of the 7th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics – Metamaterials 2013, Bordeaux, France, 16-21 September 2013.
[7] R. Zivieri, “Effective scattering of collective modes in two-dimensional magnonic crystals” submitted to New Journal of Physics.
[8] R. Zivieri and P. Malagò, “Effective properties of a three-dimensional permalloy/cobalt binary system”, in press in Proceedings of the 7th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics – Metamaterials 2013, Bordeaux, France, 16-21 September 2013.
[9] R. Zivieri, P. Malagò and L. Giovannini, “Band structure of magnonic modes in three-dimensional permalloy-cobalt binary systems: a micromagnetic study”, submitted to Physical Review B
Metamaterial Description of Magnetic Nanostructures - Invited presentation by R. Zivieri - Conferenza internazionale
No full textIn recent years the study of low-dimensional magnetic systems has become topical for achieving a deep understanding of the underlying physics of magnetic nanostructures and for its potential technological applications. Very recently, great attention has been devoted to the investigation of the effective and metamaterial properties of magnetic nanostructures with special regard to magnonic crystals, a class of periodic magnetic systems. In this talk it is shown, according to micromagnetic simulations and analytical calculations, that magnonic crystals exhibit metamaterial properties directly linked to the static and dynamic properties of collective modes. Some possible applications based on the effective properties for tailoring new magnetic devices are suggested [1,2].
1. R. Zivieri, L. Giovannini, Photonics Nanostruct. Fundam. Appl. 11, 191 (2013).
2. R. Zivieri, P. Malagò, L. Giovannini, Photonics Nanostruct. Fundam. Appl. 12, 398 (2014)
Magnon modes in vortex-state ferromagnetic disks and rings -- Presentazione poster by R. Zivieri - Conferenza nazionale
No full textThe effect of the core formation on the vortex-state frequencies of spin modes in cylindrical magnetic disks at zero applied magnetic field is evaluated by means of an analytical approach [1]. In order to do this we force the static magnetization to lie on the disk surface also in the core region of finite size with the exclusion of an infinitesimal area in the disk centre where the magnetization is perpendicular to the disk plane. Then we calculate the frequencies of spin modes corresponding to this configuration. Due to dipolar effects the frequencies of the radial modes corresponding to this configuration are upshifted with respect to the ones determined in the ground state configuration for disks of nanometric and submicrometric size. Instead the core formation leads to an increase of the energy stored in one mode of the m = ± 1 doublet of azimuthal modes and a decrease of the energy stored in the other mode also for disks of micrometric size. The effect of core removal in the corresponding vortex-state magnetic rings with inner radius of finite size is also investigated and its influence on spin dynamics is compared with the previous effect.
The frequency behavior of the most representative mode of the spectrum, the fundamental mode, is approximately described by means of static demagnetizing contributions. This description reminds qualitatively the one given for the F mode in a cylindrical dot in the saturated state with in-plane magnetization [2]. The frequencies of the most representative vortex modes at zero applied magnetic field calculated by means of the analytical model are compared with the ones of a recent micromagnetic model based on the dynamical matrix method for cylindrical nanodots [1]. The analytically calculated frequency splitting of the m = + 1 and m = -1 doublet compares well with available Time Resolved Kerr Microscopy and Brillouin Light Scattering data for different aspect ratios [3].
[1] R. Zivieri and F. Nizzoli, “Magnon modes in vortex-state ferromagnetic cylindrical dots: from standard disk to ring” in preparation
[2] R. Zivieri and R.L. Stamps, Phys. Rev. B 73 (2006) 144422
[3] R. Zivieri and F. Nizzoli, Phys. Rev. B 78 (2008) 064418; R. Zivieri and F. Nizzoli in “Electromagnetic, Magnetostatic, and Exchange-Interaction Vortices in Confined Magnetic Structures” edited by E.O. Kamenetskii (Transworld Research Network, Kerala, India ) (2008)
Effective Description of Dynamical Properties of Magnetic Nanostructures - Invited talk by R. Zivieri
No full textFerromagnetic materials composing periodic magnetic systems can be described in terms of effective properties and can be regarded as magnetic metamaterials. It is possible to define effective quantities characterizing the ferromagnetic media and the collective mode dynamics like, for instance, an effective magnetization and an effective permeability both in a lossless and in a lossy ferromagnetic medium, an effective wavelength and an effective wave vector and, under some special conditions, an effective diamagnetic behavior of ferromagnetic periodic systems. Moreover, the band structure of different kinds of magnonic crystals can be determined. The aim of this talk is to give an overview of the recent results obtained on the study of effective properties of two-dimensional ferromagnetic nanostructures ranging from ferromagnetic films to two-dimensional periodic systems. Micromagnetic simulations and simple analytical calculations allow to extract the above described metamaterial properties. Some applications based on the effective properties for tailoring new magnetic devices are suggested
Metamaterial Description of 2D Ferromagnetic Nanostructures - Invited talk by R. Zivieri
No full textIn the last years the study of low-dimensional magnetic systems has become topical for its several technological applications but also for a complete understanding of the underlying physics of magnetic nanostructures. Very recently, for their challenging features, great attention has been given to the investigation of the static and dynamical properties of magnetic nanostructures. A special class of magnetic nanostructures is represented by magnonic crystals, periodic magnetic systems characterized by modulated properties. As shown by several theoretical approaches, the ferromagnetic materials composing periodic magnetic systems can be described as metamaterials whose properties arise from their structure rather than from their composition. The aim of this talk is to give an overview of the recent theoretical results obtained, via micromagnetic simulations and analytical calculations, on thin ferromagnetic films and on two-dimensional periodic systems in terms of metamaterial properties
Interplay Between Topology and Dynamics in Magnetic Skyrmions - Invited talk by R. Zivieri - Conferenza internazionale
No full textThe interplay between topology and dynamics is discussed by introducing the notion of topological degeneracy according to which two topological droplet textures (hedgehog-like and vortex-like, respectively) having different ground-state energies are characterized by the same topological charge.
This work was partially supported by MIUR-PRIN 2010-11 Project2010ECA8P3 "DyNanoMag".
[1] T. Moriya, Phys. Rev. Lett. 4, 228 (1960).
[2] R. Zivieri et al., “Topological skyrmion dynamics driven by spin-transfer torque” submitte
Discrete Symmetries and Electrodynamic Description of Topological Magnetic Defects - Invited talk by R. Zivieri
No full textThe study of topological magnetic defects has received in the last years a great impulse thanks to their technological applications. First, a great deal of work has been devoted to the study of vortex- and anti-vortex configurations in classical ferromagnets. These investigations were stimulated by experimental evidence for the flux-closure configuration developed in circular nanomagnets in the absence of an external magnetic field. Secondly, a few recent works on magnetic skyrmions nucleation in ferromagnetic stripes in the presence of Dzyaloshinskii–Moriya interaction and spin currents (either spin-transfer torque or spin Hall) and an external bias field have inspired a strong interest towards the study of this class of topological magnetic defects and their applications. As for other classes of topological defects investigated in condensed matter physics (e.g. superfluid or superconductor vortices), magnetic defects have a quantized vorticity or topological charge (otherwise called skyrmion number in 3D magnetization textures) and this quantization is typical of these defects treated semi-classically. Here, an electrodynamic description of vortices, anti-vortices and skyrmions as the most representative topological defects in magnetic systems is given. This is accomplished by calculating the circulation, the Amperian current density and the vector potential for every defect studied and for different magnetization textures. It is also shown, by means of simple mathematical and graphical arguments, that the spatial reflection symmetry of a magnetic vortex and of a magnetic skyrmion in the vortex-like configuration, the so-called Bloch skyrmion, is broken by the vortex core magnetization. The analogies and differences with the classical hydrodynamic vortices are highlighted. A topological charge conjugation operator is defined and the physical implications of the corresponding symmetry operation are discussed
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