Magnonic Band Research Articles

Dispersion of collective magnonic modes in stacks of nanoscale magnetic elements

We report a numerical study of the dispersion of collective magnonic modes in magnonic crystals formed by stacks of magnetostatically coupled magnetic nanoelements. The calculations reveal that the sign of the magnonic dispersion is determined by the spatial character and ellipticity of precession for the eigenmodes of the isolated elements that give rise to the magnonic bands. We identify a critical value of the ellipticity at which the dispersion of the collective magnonic modes changes sign. The critical value is independent of the magnetic parameters and shape of the elements but is a characteristic of their arrangement (superstructure).

Effect of Interrow Magnetic Coupling on Band Structures of 2-D Magnonic Crystal Waveguides

We report a micromagnetic investigation of the propagation of spin waves in bi-component magnonic crystal waveguides. The investigated nanostructured waveguides are in the form of regular square lattice arrays of circular Fe dots embedded in a yttrium iron garnet matrix. Magnonic bandgaps of the order of 10 GHz are observed in both calculated frequency power spectra and dispersion curves of spin wave. The effect of the variation of the period along the direction perpendicular to spin wave propagation on magnonic band structures of magnonic crystal waveguides has been investigated. Micromagnetic simulations show that the first bandgap is more sensitive to variations of period than the second bandgap.

Berry phase and thermal transport coefficients in magnon systems

We theoretically calculate transverse thermal transport coefficients of the magnon system in the clean limit by the linear response theory in analogy with the electron. We find that there are additional correction terms to what are calculated by the usual Kubo formula. These correction terms arise because external fields change not only the distribution function but also the current operators in thermal transport. We show that the total thermal transport coefficients are expressed by the Berry curvature in momentum space; therefore the thermal Hall effect of magnon is totally due to the property of the magnon band structure.

Reprogrammable magnonic crystals formed by interacting ferromagnetic nanowires

Spin-wave (SW) modes are addressed which are confined in thin individual Ni80Fe20 nanowires with widths ranging from 220 to 360 nm. In periodic arrays with an edge-to-edge separation of down to 100 nm, confined modes of neighboring nanowires are found to couple coherently and form allowed minibands and forbidden frequency gaps. This gives rise to a one-dimensional magnonic crystal. We present all-electrical SW spectroscopy data and micromagnetic simulations. We find that the nanowire arrays allow us to reprogram the relevant magnonic band structure via the magnetic history. A forbidden frequency gap of up to about 1 GHz is controlled by an in-plane magnetic field being as small as a few mT.

Effect of Interdot Separation on Collective Magnonic Modes in Chains of Rectangular Dots

The behavior of collective spin excitations in chains of rectangular NiFe dots is studied as a function of interdot separation. Dots have thickness of 40 nm and lateral dimensions of 715 × 450 nm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . They are put side by side along the major axis and the interdot separation is varied in the range 55-625 nm. Brillouin light scattering experiments have been performed at normal incidence (exchanged wave vector q = 0) and with the external magnetic field applied along the chain length. A satisfactory interpretation of the experimental data is achieved by magnonic bands calculations based on the dynamical matrix method. Such calculations have been performed at both the center and the border of the first Brillouin zone, in the case of Bloch wave vector q parallel to the applied field. In this way we can predict the amplitude of modes frequency oscillation (magnonic band), which is an important property to identify the behavior of a one-dimensional magnonic meta-material.

Calculation of spin wave spectra in magnetic nanograins and patterned multilayers with perpendicular anisotropy

We study spin-wave excitations in multilayered magnetic nanograins composed of a stack of planes with perpendicular magnetic anisotropy. The inhomogeneity is modeled through the multiple repetition of a unit cell composed of layers of two different magnetic materials. The magnetic inhomogeneity along the central axis is found to split the frequency spectrum of magnetostatic excitations into two bands, while into the number of bands in the spectrum of exchange spin waves. We show that this difference in behavior is a result of the underlying long- and short-range interactions, respectively. We describe a way of increasing the role of the dipolar interactions in the formation of magnonic bands in patterned magnetic multilayers with perpendicular anisotropy, which can allow one to obtain ferromagnetic resonance spectra with two strong absorption peaks in low and high frequencies up to the sub-THz range. Our findings open a new area for modeling the spin-wave spectra of patterned magnetic multilayers with potential applications, and for studying the role of the microscopic magnetic structure in forming spin-wave spectra.

Dynamics of Coupled Vortices in a Pair of Ferromagnetic Disks

We here experimentally demonstrate that gyration modes of coupled vortices can be resonantly excited primarily by the ac current in a pair of ferromagnetic disks with variable separation. The sole gyration mode clearly splits into higher and lower frequency modes via dipolar interaction, where the main mode splitting is due to a chirality sensitive phase difference in gyrations of the coupled vortices, whereas the magnitude of the splitting is determined by their polarity configuration. These experimental results show that the coupled pair of vortices behaves similar to a diatomic molecule with bonding and antibonding states, implying a possibility for designing the magnonic band structure in a chain or an array of magnetic vortex oscillators.

Theoretical Prediction of a Rotating Magnon Wave Packet in Ferromagnets

We theoretically show that the magnon wave packet has a rotational motion in two ways: a self-rotation and a motion along the boundary of the sample (edge current). They are similar to the cyclotron motion of electrons, but unlike electrons the magnons have no charge and the rotation is not due to the Lorentz force. These rotational motions are caused by the Berry phase in momentum space from the magnon band structure. Furthermore, the rotational motion of the magnon gives an additional correction term to the magnon Hall effect. We also discuss the Berry curvature effect in the classical limit of long-wavelength magnetostatic spin waves having macroscopic coherence length.

Micromagnetic study of spin wave propagation in bicomponent magnonic crystal waveguides

The propagation of spin waves in bicomponent magnonic crystal waveguides has been investigated by micromagnetic simulations. The nanostructured waveguides studied are regular square lattice arrays of circular Fe dots embedded in an yttrium iron garnet matrix. Our simulations show that the waveguides exhibit wide magnonic band gaps of the order of 10 GHz. Band gap tunability, arising from variations in the filling fraction, lattice constant, and applied magnetic field has been demonstrated. Our findings would be of value to the efficient transmission and processing of microwave signals on the nanoscale by means of spin waves.

First-principle description of magnonic PdnFem multilayers

Ab-initio calculations are used to determine the parameters that determine magnonic band structure of PdnFem multilayers (n = 2, m ≤ 8). We obtain the layer-resolved magnetization, the exchange coupling, and the magnetic anisotropy of the Pd-Fe structures. The Fe moment is 3.0 μB close to the Pd layers and 2.2 μB in the middle of the Fe layers. An intriguing but not usually considered aspect is that the elemental Pd is nonmagnetic, similar to Cu spacer layers in other multilayer systems. This leads to a pre-asymptotic ferromagnetic coupling through the Pd (about 40 mJ/m2). Furthermore, the Pd acquires a small moment due to spin polarization by neighboring Fe atoms, which translates into magnetic anisotropy. The anisotropies are large, in the range typical for L10 structures, which is beneficial for high-frequency applications.

Partial band gaps in magnonic crystals

In this work we investigate magnonic band gaps, in the THz frequency range, in periodic and quasiperiodic (Fibonacci sequence) magnonic crystals formed by layers of cobalt and permalloy. Our theoretical model is based on a magnetic Heisenberg Hamiltonian in the exchange regime, together with a transfer-matrix treatment within the random-phase approximation. For periodic arrangements, the bulk band structure is analogous to those found in photonic crystals, while for quasiperiodic multilayers it presents additional pass bands similar to those found in photonic crystals with defects.

The magnetostatic modes in planar one-dimensional magnonic crystals with nanoscale sizes

The plane-wave method is used to determine the spin-wave spectra in nanoscale one-dimensional magnonic crystals formed by a periodic lattice of ferromagnetic stripes. The dynamic demagnetizing field in a new formulation is used in these calculations. The calculated spectra of spin waves are compared with experimental data taken from literature and a good agreement is obtained. The dependence of the spin-wave spectra on magnetic material parameters consisting of magnetic stripes is systematically studied. We proved that the experimentally found magnonic band structure with two magnonic gaps is of magnetostatic nature.

Collective spin modes in chains of dipolarly interacting rectangular magnetic dots

We present a combined experimental and micromagnetic study of spin excitations in chains of dense magnetic dots. The samples consist of long chains of rectangular dots with rounded corners having lateral dimensions of 715 \ifmmode\times\else\texttimes\fi{} 450 nm${}^{2}$ and 1025 \ifmmode\times\else\texttimes\fi{} 450 nm${}^{2}$, respectively. Chains are composed of magnetic elements put side by side along either their major or minor axis with edge-to-edge separation below 100 nm. The frequency dispersion of the spin-wave excitations was measured by Brillouin light-scattering technique as a function of the transferred wave vector directed along the chains of dots and for an external magnetic field applied perpendicularly to the transferred wave vector in the dots plane. Evidence is given of collective excitations in the form of Bloch waves propagating through the chains characterized by magnonic energy bands. Micromagnetic calculations, performed by using the dynamical matrix method, enable us to satisfactorily reproduce the frequency dispersion of collective spin modes as well as to visualize the spatial profile of the dynamic magnetization inside the dots. We also propose a general rule to understand the frequency dispersion of collective modes starting from the relative phase of dynamic magnetization in the facing sides of adjacent dots.

Frequency in Middle of Magnon Band Gap in a Layered Ferromagnetic Superlattice

The frequency in middle of magnon energy hand in a five-layer ferromagnetic superlattice is studied by using the linear spin-wave approach and Green's function technique. It is found that four energy gaps and corresponding four frequencie in middle of energy gaps exist in the magnon band along Kx direction perpendicular to the superlattice plane. The spin quantum numbers and the interlayer exchange couplings all affect the four frequencies in middle of the energy gaps. When all interlayer exchange couplings are same, the effect of spin quantum numbers on the frequency ωg1 in middle of the energy gap Δω12 is complicated, and the frequency ωg1 depends on the match of spin quantum numbers in each layer. Meanwhile, the frequencies ωg2, ωg3, and ωg4 in middle of other energy gaps increase monotonously with increasing spin quantum numbers. When the spin quantum numbers in each layer are same, the frequencies ωg1, ωg2, ωg3, and ωg4, all increase monotonously with increasing interlayer exchange couplings.

Analysis of collective spin-wave modes at different points within the hysteresis loop of a one-dimensional magnonic crystal comprising alternative-width nanostripes

The Brillouin light-scattering technique has been applied to study collective spin waves in a dense array of dipolarly coupled ${\text{Ni}}_{80}{\text{Fe}}_{20}$ stripes of alternating widths, during the magnetization reversal process. Both the saturated ``ferromagnetic'' state, where the magnetizations of wide and narrow stripes are parallel, and the ``antiferromagnetic'' state, characterized by an antiparallel alignment of the static magnetization in adjacent stripes, have been analyzed. The experimental data provide strong evidence of sustained collective excitations in the form of Bloch waves with permitted and forbidden magnonic energy bands. The measured frequencies as a function of the exchanged wave vector have been satisfactorily reproduced by numerical simulations which enabled us to calculate the spatial profiles of the Bloch waves, showing that some of the modes are preferentially localized in either the wide or the narrow stripes. We estimated the expected light-scattering cross section for each mode at different magnetic ground states, achieving a good agreement with the measured intensities. The alternating-width stripes system studied here represents a one-dimensional artificial magnonic crystal with a complex base and can be considered as a model system for reprogrammable dynamical response, where the band structure of collective spin waves can be tailored by changing the applied magnetic field.

Materials optimization of the magnonic gap in three-dimensional magnonic crystals with spheres in hexagonal structure

We present the results of plane wave method-based calculations adopted to magnonic band structures for exchange spin waves propagating in three-dimensional magnonic crystals (MCs) composed of two ferromagnetic metals. The crystals under consideration consist of a system of ferromagnetic spheres arranged in sites of a hexagonal lattice and embedded in a ferromagnetic material. Having analyzed all the possible combinations of magnonic crystal component materials from: Co, Ni, Fe, and Py (for spheres and matrix), we find material configurations for which either absolute or partial magnonic gaps occur in the spin-wave spectrum of the MC. We also demonstrate that the opening of a magnonic gap necessitates a sufficiently large contrast of magnetic parameters, and find the exchange length contrast to be the best measure of the capacity of the MC to produce a magnonic gap. Wider magnonic gaps are obtained in MCs in which the exchange length in the sphere material is larger than in the matrix. Among the MCs considered in this study, an absolute magnonic gap is obtained in a crystal with Ni spheres embedded in Fe.

Band gaps in the terahertz frequency range in quasiperiodic one-dimensional magnonic crystals

In this work we investigate magnonic band gaps, in the terahertz (THz) frequency range, in periodic and quasiperiodic (Fibonacci sequence) magnonic crystals formed by layers of Cobalt (Co) and Permalloy (Py). Our theoretical model is based on a magnetic Heisenberg Hamiltonian in the exchange regime, together with a transfer-matrix treatment within the random-phase approximation (RPA). For periodic arrangements the bulk band structure is analogous to those found in photonic crystals, while for quasiperiodic multilayers it presents additional pass bands similar to those found in doped electronic materials.

Band structures of two-dimensional magnonic crystals with different shapes and arrangements of scatterers

The spin-wave band structures of two-dimensional magnonic crystals with different shapes and arrangements of scatterers are investigated numerically. It is shown that the magnonic band gap of spin waves in magnonic crystals can be optimized by changing the shape or the arrangement of scatterers in background materials. The largest absolute gap can be achieved when the scatterer has the same symmetry as that of the coordination polygon of lattice point, such as the case of square rods in square lattices. For a given shape of scatterer, the gap width is the largest when the lattice has the largest coordination number if the scatterer does not reduce the symmetry.

Making a Reconfigurable Artificial Crystal by Ordering Bistable Magnetic Nanowires

Spin-wave excitations (magnons) are investigated in a one-dimensional (1D) magnonic crystal fabricated out of Ni80Fe20 nanowires. We find two different magnon band structures depending on the magnetic ordering of neighboring wires, i.e., parallel and antiparallel alignment. At a zero in-plane magnetic field H the modes of the antiparallel case are close to those obtained by zone folding of the spin-wave dispersions of the parallel case. This is no longer true for nonzero H which opens a forbidden frequency gap at the Brillouin zone boundary. The 1D stop band gap scales with the external field, which generates a periodic potential for Bragg reflection of the magnons.

Control of magnonic spectra in cobalt nanohole arrays: the effects of density, symmetry and defects

Magnetic nanohole arrays are important systems for propagation of magnetic excitations and are among the potential candidates for magnonic crystals. A thorough investigation of magnonic band structures and the effect of the geometry of the array on them are important. Here, we present a systematic micromagnetic simulation study of magnonic modes in cobalt nanohole (antidot) arrays. In particular, we investigate the effects of the areal density and symmetry of the array and defects introduced in the array. The magnonic modes are strongly dependent on the density and the symmetry of the array but are weakly dependent on the defects. We have further investigated the modes in a tailored array consisting of equally wide hexagonal arrays with varying density. The magnonic spectrum of the tailored array contains additional modes above the modes of the constituent arrays due to the appearance of irregular domain structures at the regions joining arrays of two different types. This opens up the possibility of tuning the magnonic bands in magnetic nanohole arrays by careful design of the structure of the array.