Spin-wave Band Structure Research Articles

Hopfion based magnonic crystal

We conduct a numerical analysis to explore the magnonic band structure of spin waves (SWs) within a unique magnonic crystal. The latter is composed of a waveguide hosting a linear array of hopfions, evenly spaced along its center, generated through micromagnetic simulations. The computed dispersion relationship of spin waves (SWs) in the envisioned hopfions magnonic crystal, illustrating the frequency versus wave vector, demonstrates a periodic attribute, a hallmark of conventional magnonic crystal band structures. Likewise, the computed magnonic spectra reveal both permitted frequency bands and restricted frequency bandgaps. These results may inspire additional exploration into the diverse functionalities offered by magnonic crystals built on periodic Hopfion lattices.

Impact of the interfacial Dzyaloshinskii-Moriya interaction on the band structure of one-dimensional artificial magnonic crystals: A micromagnetic study

We present the results of a systematic micromagnetic study of the effect of the Dzyaloshinskii-Moriya interaction (DMI) on the spin wave band structure of two one-dimensional magnonic crystals (MCs), both with the same periodicity p = 300 nm, but different implementation of the DMI modulation. In the first system the artificial periodicity was achieved by modulating the interfacial DMI constant D, while in the second system also the sample morphology was modulated. Due to the folding property of the band structure in the dispersion relations of the magnonic crystals it is possible to extend the sensitivity of Brillouin light scattering towards weak DMI strength (D in the range from 0 to 0.5 mJ/m2), by measuring the frequency splitting of folded modes in high-order artificial Brillouin zones, since the splitting increases almost linearly with the band index. For relatively large values of the DMI (D in the range from 1.0 to 2.0 mJ/m2) the spin waves dispersion relations present flat modes for positive wavevectors, separated by forbidden frequency gaps whose amplitude depend on the value of D. These frequency gaps are more pronounced for the sample with morphology modulation. The non-reciprocal, localised, spatial profiles of these modes in both MCs are discussed with reference to spin waves in plain films and in isolated stripes of the same thickness.

Measuring the dispersion relations of spin wave bands using time-of-flight spectroscopy

We develop a generic all-inductive procedure to measure the band structure of spin waves in a magnetic thin stripe. In contrast to existing techniques, our method works even if several spin wave branches coexist in the investigated frequency interval, provided that the branches possess sufficiently different group velocities. We first measure the microwave scattering matrix of a network composed of distant antennas inductively coupled to the spin wave bath of the magnetic film. After a mathematical transformation to the time domain to get the transmission impulse response, the different spin wave branches are viewed as wave packets that reach successively the receiving antenna after different travel times. In analogy with time-of-flight spectroscopy, the wave packets are then separated by time gating. The time-gated responses are used to recalculate the contribution of each spin wave branch to the frequency domain scattering matrix. The dispersion relation of each branch stems from the absolute phase of the time-gated transmission parameter. The spin-wave wave vector can be determined unambiguously if the results for several propagation distances are combined, so as to get the dispersion relations.

Phase resolved observation of spin wave modes in antidot lattices

Antidot lattices have proven to be a powerful tool for spin wave band structure manipulation. Utilizing time-resolved scanning transmission x-ray microscopy, we are able to experimentally image edge-localized spin wave modes in an antidot lattice with a lateral confinement down to <80 nm×130 nm. At higher frequencies, spin wave dragonfly patterns formed by the demagnetizing structures of the antidot lattice are excited. Evaluating their relative phase with respect to the propagating mode within the antidot channel reveals that the dragonfly modes are not directly excited by the antenna but need the propagating mode as an energy mediator. Furthermore, micromagnetic simulations reveal that additional dispersion branches exist for a tilted external field geometry. These branches correspond to asymmetric spin wave modes that cannot be excited in a non-tilted field geometry due to the symmetry restriction. In addition to the band having a negative slope, these asymmetric modes also cause an unexpected transformation of the band structure, slightly reaching into the otherwise empty bandgap between the low frequency edge modes and the fundamental mode. The presented phase resolved investigation of spin waves is a crucial step for spin wave manipulation in magnonic crystals.

Effects of Cobalt Nanoisland Geometry on Terahertz Negative Refraction: a Numerical Analysis

The architectural framework plays a major role in determining the optical properties of left-handed materials. Most metamaterial so far designed have been based on complicated resonant elements such as split rings and hyperbolic metamaterials. Unfortunately, magnetic response of most of the materials tails off at higher frequencies. Higher frequency excitation in metamaterials requires more complicated designs or toroidal moment. In the proposed magnonic metamaterials, the toroidal excitations are naturally maintained and make it suitable for THz resonances. Magnetism and left-handed behavior are the two extreme phenomena that do not seem to be compatible and the coexistence of both cooperative effects was not foreseen. Terahertz spintronics is an emerging field that bridges the boundary between magnetism and photonics, which make the magnetic material suitable for photonic applications. Here, we present the first numerical realization of a magnonic metamaterial artificial spin ice (ASI), designed to operate at terahertz frequency. The two-dimensional square arrays exhibit a rich spin wave band structure that is tunable both by external electromagnetic fields and the magnetization configuration of individual elements. The ultrafast spin wave resonances have provided a direct gateway to manipulate magnetic field of free-space terahertz pulses. The proposed system consists of two-dimensional square arrays of cobalt nanoislands in four different geometries and the micromagnetic simulations of these systems are carried out using mumax3. From the numerical results, the theoretical prediction of negative permeability is done using Schloemann model.

Magnonic crystal-semiconductor heterostructure: Double electric and magnetic fields control of spin waves properties

The main features of the spin wave band structure formation and manipulation in the magnonic crystal – semiconductor layered heterostructure have been studied. We demonstrate possibility of double electric and magnetic control of band gap characteristics within the spectra of propagating spin waves. It is shown that in the magnonic crystal- semiconductor heterostructure, when a positive polarity voltage is applied to the semiconductor layer, the band gap in the spin-wave spectrum shifts to the low-frequency region, while for the negative polarity the band position is practically not changed. We show also that as the magnitude of the applied magnetic field is increased the shift of the band gap is decreased. Our results show the possibility of integration of magnonics and semiconductor electronics on the base of proposed structure.

Unidirectional spin-wave channeling along magnetic domain walls of Bloch type

From the pioneering work of Winter [Phys. Rev. 124, 452 (1961)], a magnetic domain wall of Bloch type is known to host a special wall-bound spin-wave mode, which corresponds to spin-waves being channeled along the magnetic texture. Using micromagnetic simulations, we investigate spin-waves travelling inside Bloch walls formed in thin magnetic media with perpendicular-to-plane magnetic anisotropy and we show that their propagation is actually strongly nonreciprocal, as a result of dynamic dipolar interactions. We investigate spin-wave non-reciprocity effects in single Bloch walls, which allows us to clearly pinpoint their origin, as well as in arrays of parallel walls in stripe domain configurations. For such arrays, a complex domain-wall-bound spin-wave band structure develops, some aspects of which can be understood qualitatively from the single-wall picture by considering that a wall array consists of a sequence of up/down and down/up walls with opposite non-reciprocities. Circumstances are identified in which the non-reciprocity is so extreme that spin-wave propagation inside individual walls becomes unidirectional.

Interplay between intra- and inter-nanowires dynamic dipolar interactions in the spin wave band structure of Py/Cu/Py nanowires

We have studied both experimentally and theoretically the reprogrammable spin wave band structure in Permalloy(10 nm)/Cu(5 nm)/Permalloy(30 nm) nanowire arrays of width w = 280 nm and inter-wire separation in the range from 80 to 280 nm. We found that, depending on the inter-wire separation, the anti-parallel configuration, where the magnetizations of the two Permalloy layers point in opposite directions, is stabilized over specific magnetic field ranges thus enabling us to directly compare the band structure with that of the parallel alignment. We show that collective spin waves of the Bloch type propagate through the arrays with different magnonic bandwidths as a consequence of the interplay between the intra- and inter-nanowire dynamic dipolar interactions. A detailed understanding, e.g. whether they have a stationary or propagating character, is achieved by considering the phase relation (in-phase or out-of-phase) between the dynamic magnetizations in the two ferromagnetic layers and their average value. This work opens the path to magnetic field-controlled reconfigurable layered magnonic crystals that can be used for future nanoscale magnon spintronic devices.

Spin-wave propagation spectrum in magnetization-modulated cylindrical nanowires

Spin-wave propagation in periodic magnetization-modulated cylindrical nanowires is studied by micromagnetic simulation. Spin wave scattering at the interface of two magnetization segments causes a spin-wave band structure, which can be effectively tuned by changing either the magnetization modulation level or the period of the cylindrical nanowire magnonic crystal. The bandgap width is oscillating with either the period or magnetization modulation due to the oscillating variation of the spin wave transmission coefficient through the interface of the two magnetization segments. Analytical calculation based on band theory is used to account for the micromagnetic simulation results.

Reconfigurable wave band structure of an artificial square ice

Artificial square spin ices are structures composed of magnetic elements arranged on a geometrically frustrated lattice and located on the sites of a two-dimensional square lattice, such that there are four interacting magnetic elements at each vertex. Using a semi-analytical approach, we show that square spin ices exhibit a rich spin wave band structure that is tunable both by external magnetic fields and the configuration of individual elements. Internal degrees of freedom can give rise to equilibrium states with bent magnetization at the edges leading to characteristic excitations; in the presence of magnetostatic interactions these form separate bands analogous to impurity bands in semiconductors. Full-scale micromagnetic simulations corroborate our semi-analytical approach. Our results show that artificial square spin ices can be viewed as reconfigurable and tunable magnonic crystals that can be used as metamaterials for spin-wave-based applications at the nanoscale.

Magnonic Band Structure in a Skyrmion Magnonic Crystal

We present a numerical investigation of the magnonic band structure of spin waves (SWs) in a novel magnonic crystal consisting of a waveguide with a linear array of periodically spaced skyrmions created along its center by micromagnetic simulations. The interfacial Dzyaloshinskii–Moriya interaction (DMI) induced the presence of skyrmions causes a periodical magnetization modulation of the waveguide, which can be dynamically controlled by changing either the strength of the external magnetic field or the period of the skyrmion lattice. The diameters of the skyrmions are highly dependent on the strength of the applied magnetic field, and they can exist for a wide magnetic field range depending on the density of the DMI. The calculated dispersion relation of SWs in the proposed skyrmion magnonic crystal, frequency versus wave vector, exhibits a periodical property, which is the characteristic feature of the band structure of conventional magnonic crystals. Similarly, the calculated magnonic spectra exhibits allowed frequency band and forbidden frequency bandgaps. Our findings could stimulate further exploration on multiple functionalities provided by magnonic crystals based on periodic skyrmion lattices.

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 waves.

Spin-wave dynamics in permalloy/cobalt magnonic crystals in the presence of a nonmagnetic spacer

In this paper, we theoretically study the influence of a non-magnetic spacer between ferromagnetic dots and ferromagnetic matrix on the frequency dispersion of the spin wave excitations in two-dimensional bi-component magnonic crystals. By means of the dynamical matrix method we investigate structures inhomogeneous across the thickness represented by square arrays of Cobalt or Permalloy dots in a Permalloy matrix. We show that the introduction of a non-magnetic spacer significantly modifies the total internal magnetic field especially at the edges of the grooves and dots. This permits the manipulation of the magnonic band structure of spin waves localized either at the edges of the dots or in matrix material at the edges of grooves. According to the micromagnetic simulations two types of end modes were found. The corresponding frequencies are significantly influenced by the end modes localization region. We also show that, with the use of a single ferromagnetic material, it is possible to design a magnonic crystal preserving properties of bi-component magnonic crystals and magnonic antidot lattices. Finally, the influence of the non-magnetic spacers on the technologically relevant parameters like group velocity and magnonic band width are discussed.

Universal dependence of the spin wave band structure on the geometrical characteristics of two-dimensional magnonic crystals.

In the emerging field of magnon-spintronics, spin waves are exploited to encode, carry and process information in materials with periodic modulation of their magnetic properties, named magnonic crystals. These enable the redesign of the spin wave dispersion, thanks to its dependence on the geometric and magnetic parameters, resulting in the appearance of allowed and forbidden band gaps for specific propagation directions. In this work, we analyze the spin waves band structure of two-dimensional magnonic crystals consisting of permalloy square antidot lattices with different geometrical parameters. We show that the frequency of the most intense spin-wave modes, measured by Brillouin light scattering, exhibits a universal dependence on the aspect ratio (thickness over width) of the effective nanowire enclosed between adjacent rows of holes. A similar dependence also applies to both the frequency position and the width of the main band gap of the fundamental (dispersive) mode at the edge of the first Brillouin zone. These experimental findings are successfully explained by calculations based on the plane-wave method. Therefore, a unified vision of the spin-waves characteristics in two-dimensional antidot lattices is provided, paving the way to the design of tailored nanoscale devices, such as tunable magnonic filters and phase-shifters, with predicted functionalities.

Reconfigurable magnonic crystal consisting of periodically distributed domain walls in a nanostrip

We study spin wave propagation in a new type of magnonic crystal consisting of a series of periodically distributed magnetic domain walls in a nanostrip by micromagnetic simulation. Spin wave bands and bandgaps are observed in frequency spectra and dispersion curves. Some bandgaps are caused by the Bragg reflection of the spin wave modes at the Brillouin zone boundaries, while others originate from the coupling between different incident and reflected spin wave modes. The control of the spin wave band structure by changing the magnetocrystalline anisotropy or applying an external magnetic field is studied. Increasing the magnetocrystalline anisotropy leads to an increase of the bandgaps. The external field applied perpendicular to the nanostrip gives rise to a doubling of the domain-wall magnonic crystal period. As a result, more bandgaps appear on the frequency spectra of propagating spin waves. The results presented here may find their use in the design of reconfigurable magnonic devices.

Effect of hole shape on spin-wave band structure in one-dimensional magnonic antidot waveguide

We present the possibility of tuning the spin-wave band structure, particularly the bandgaps in a nanoscale magnonic antidot waveguide by varying the shape of the antidots. The effects of changing the shape of the antidots on the spin-wave dispersion relation in a waveguide have been carefully monitored. We interpret the observed variations by analysing the equilibrium magnetic configuration and the magnonic power and phase distribution profiles during spin-wave dynamics. The inhomogeneity in the exchange fields at the antidot boundaries within the waveguide is found to play a crucial role in controlling the band structure at the discussed length scales. The observations recorded here will be important for future developments of magnetic antidot based magnonic crystals and waveguides.

Forbidden Band Gaps in the Spin-Wave Spectrum of a Two-Dimensional Bicomponent Magnonic Crystal

The spin-wave band structure of a two-dimensional bicomponent magnonic crystal, consisting of Co nanodisks partially embedded in a Permalloy thin film, is experimentally investigated along a high-symmetry direction by Brillouin light scattering. The eigenfrequencies and scattering cross sections are interpreted using plane wave method calculations and micromagnetic simulations. At the boundary of both the first and the second Brillouin zones, we measure a forbidden frequency gap whose width depends on the magnetic contrast between the constituent materials. The modes above and below the gap exhibit resonant spin-precession amplitudes in the complementary regions of periodically varying magnetic parameters. Our findings are key to advance both the physics and the technology of band gap engineering in magnonics.

Bragg diffraction of spin waves from a two-dimensional antidot lattice

The spin-wave band structure of a two-dimensional square array of NiFe circular antidots (hole diameter 120 nm, periodicity 800 nm) is investigated. Brillouin light scattering experiments and band structure calculations, carried out by means of the dynamical matrix method, provide evidence for either extended or localized magnonic modes. Both families exhibit band gaps at Brillouin zone boundaries, attributed to Bragg reflection. Their calculated magnitude agrees with the one obtained by using an analytical model that takes into account the periodic variation of the internal field. This is in contrast to antidots in photonics and electronics, where the back-reflection is directly caused by the presence of holes. The results are important for advancing research on nanostructured two-dimensional magnonic crystals.

Spin-Wave Band Structure in 2D Magnonic Crystals with Elliptically Shaped Scattering Centres

Spin waves in 2D periodic magnetic nanocomposites are studied by means of the plane wave method. The effect of the ellipticity and in-plane rotation of the scattering centers on the band structure is investigated, to indicate new possibilities of fine tuning of spin-wave filter passbands.

Magnetic superlattice with two-dimensional periodicity as a waveguide for spin waves

We describe a simple method of including dissipation in the spin-wave band structure of a periodic ferromagnetic composite, by solving the Landau-Lifshitz equation for the magnetization with the Gilbert damping term. We use this approach to calculate the band structure of square and triangular arrays of Ni nanocylinders embedded in an Fe host. The results show that there are certain bands and special directions in the Brillouin zone where the spin-wave lifetime is increased by more than an order of magnitude above its average value. Thus, it may be possible to generate spin waves in such composites which decay especially slowly, and propagate especially large distances, for certain frequencies and directions in $\mathbf{k}$ space.