Non-uniform spin wave softening in 2D magnonic crystals as a tool for opening omnidirectional magnonic band gaps

By means of the plane wave method we study spin wave dynamics in two-dimensional bi-component magnonic crystals based on a squeezed hexagonal lattice and consist of a permalloy thin film with cobalt inclusions. We explore the dependence of a spin wave frequency on the external magnetic field, especially in weak fields where the mode softening takes place. For considered structures, the mode softening proves to be highly non-uniform on both the mode number and the wave vector. We found this effect to be responsible for the omnidirectional band gap opening. Moreover, we show that the enhancement of the demagnetizing field caused by the squeezing of the structure is of crucial importance for the non-uniform mode softening. This allows us to employ this mechanism to design magnonic gaps with different sensitivity for the tiny change of the external field. The effects we have found should be useful in designing and optimization of spin wave filters highly tunable by a small external magnetic field.Comment: Final versio

Vortices in two-dimensional nanorings studied by means of the dynamical matrix method

This paper concerns an investigation of the spin wave excitations in magnetic nanoparticles. We provide a detailed derivation of the theoretical method for the determination of the normal modes of confined magnetic systems based on a discrete lattice of magnetic moments. The method is based on the damping-free Landau–Lifshitz equation and general enough to be utilized for the magnetic system of any dimensionality, magnetic structure, shape, and size. As an example we explore the influence of the competition between exchange and dipolar interactions on the spectrum of normal modes as well as on the stability of the vortex state in two-dimensional nanorings. We show the lowest-frequency mode to be indicative of the dipolar-to-exchange iterations ratio. We also study behavior of the fundamental mode and present the influence of both, the discreteness of the lattice and the dipolar-to-exchange iterations ratio, on its hybridization with azimuthal modes. We complete the paper with a selective review of the spin wave excitations in circular dots to compare with the results obtained for the rings

Influence of nonmagnetic dielectric spacers on the spin wave response of one-dimensional planar magnonic crystals

The one-dimensional planar magnonic crystals are usually fabricated as a sequence of stripes intentionally or accidentally separated by non-magnetic spacers. The influence of spacers on shaping the spin wave spectra is complex and still not completely clarified. We performed the detailed numerical studies of the one-dimensional single- and bi-component magnonic crystals comprised of a periodic array of thin ferromagnetic stripes separated by non-magnetic spacers. We showed that the dynamic dipolar interactions between the stripes mediated by non-magnetic spacer, even ultra-narrow, significantly shift up the frequency of the ferromagnetic resonance and simultaneously reduce the spin wave group velocity, which is manifested by the flattening of the magnonic band. We attributed these changes in the spectra to the modifications of dipolar pinning and shape anisotropy both dependent on the width of the spacers and the thickness of the stripes, as well as to the dynamical magnetic volume charges formed due to inhomogeneous spin wave amplitude

On the Formulation of the Exchange Field in the Landau-Lifshitz Equation for Spin-Wave Calculation in Magnonic Crystals

The calculation of the magnonic spectra using the plane-wave method has limitations, the origin of which lies in the formulation of the effective magnetic field term in the equation of motion (the Landau-Lifshitz equation) for composite media. According to ideas of the plane-wave method the system dynamics is described in terms of plane waves (a superposition of a number of plane waves), which are continuous functions and propagate throughout the medium. Since in magnonic crystals the sought-for superposition of plane waves represents the dynamic magnetization, the magnetic boundary conditions on the interfaces between constituent materials should be inherent in the Landau-Lifshitz equations. In this paper we present the derivation of the two expressions for the exchange field known from the literature. We start from the Heisenberg model and use a linear approximation and take into account the spacial dependence of saturation magnetization and exchange constant present in magnetic composites. We discuss the magnetic boundary conditions included in the presented formulations of the exchange field and elucidate their effect on spin-wave modes and their spectra in one- and two-dimensional planar magnonic crystals from plane-wave calculations

Magnonic crystals — prospective structures for shaping spin waves in nanoscale

We have investigated theoretically band structure of spin waves in magnonic crystals with periodicity in one(1D), two- (2D) and three-dimensions (3D). We have solved Landau–Lifshitz equation with the use of plane wave method, finite element method in frequency domain and micromagnetic simulations in time domain to find the dynamics of spin waves and spectrum of their eigenmodes. The spin wave spectra were calculated in linear approximation. In this paper we show usefulness of these methods in calculations of various types of spin waves. We demonstrate the surface character of the Damon–Eshbach spin wave in 1D magnonic crystals and change of its surface localization with the band number and wavenumber in the first Brillouin zone. The surface property of the spin wave excitation is further exploited by covering plate of the magnonic crystal with conductor. The band structure in 2D magnonic crystals is complex due to additional spatial inhomogeneity introduced by the demagnetizing field. This modifies spin wave dispersion, makes the band structure of magnonic crystals strongly dependent on shape of the inclusions and type of the lattice. The inhomogeneity of the internal magnetic field becomes unimportant for magnonic crystals with small lattice constant, where exchange interactions dominate. For 3D magnonic crystals, characterized by small lattice constant, wide magnonic band gap is found. We show that the spatial distribution of different materials in magnonic crystals can be explored for tailored effective damping of spin wave

Magnonic Metamaterials

A large proportion of the recent growth of the volume of electromagnetics research has been associated with the emergence of so called electromagnetic metamaterials1 and the discovered ability to design their unusual properties by tweaking the geometry and structure of the constituent “meta-atoms”. For example, negative permittivity and negative permeability can be achieved, leading to negative refractive index metamaterials. The negative permeability could be obtained via geometrical control of high frequency currents, e.g. in arrays of split ring resonators, or alternatively one could rely on spin resonances in natural magnetic materials, as was suggested by Veselago. The age of nanotechnology therefore sets an intriguing quest for additional benefits to be gained by structuring natural magnetic materials into so called magnonic metamaterials, in which the frequency and strength of resonances based on spin waves (magnons) are determined by the geometry and magnetization configuration of meta-atoms. Spin waves can have frequencies of up to hundreds of GHz (in the exchange dominated regime) and have already been shown to play an important role in the high frequency magnetic response of composites. Moreover, in view of the rapid advances in the field of magnonics, which in particular promises devices employing propagating spin waves, the appropriate design of magnonic metamaterials with properties defined with respect to propagating spin waves rather than electromagnetic waves acquires an independent and significant importance

Vortices in two-dimensional nanorings studied by means of the dynamical matrix method

This paper concerns an investigation of the spin wave excitations in magnetic nanoparticles. We provide a detailed derivation of the theoretical method for the determination of the normal modes of confined magnetic systems based on a discrete lattice of magnetic moments. The method is based on the damping-free Landau–Lifshitz equation and general enough to be utilized for the magnetic system of any dimensionality, magnetic structure, shape, and size. As an example we explore the influence of the competition between exchange and dipolar interactions on the spectrum of normal modes as well as on the stability of the vortex state in two-dimensional nanorings. We show the lowest-frequency mode to be indicative of the dipolar-to-exchange iterations ratio. We also study behavior of the fundamental mode and present the influence of both, the discreteness of the lattice and the dipolar-to-exchange iterations ratio, on its hybridization with azimuthal modes. We complete the paper with a selective review of the spin wave excitations in circular dots to compare with the results obtained for the rings