Two-dimensional Magnonic Crystals Research Articles

Spin-wave mode coupling in the presence of the demagnetizing field in cobalt-permalloy magnonic crystals

We present the results of studies on the non-uniform frequency shift of spin wave spectrum in a two-dimensional magnonic crystal of cobalt/permalloy under the influence of external magnetic field changes. We investigate the phenomenon of coupling of modes and, as a consequence, their hybridization. By taking advantage of the fact that compressing the crystal structure along the direction of the external magnetic field leads to an enhancement of the demagnetizing field, we analyse its effect on the frequency shift of individual modes depending on their concentration in Co. We show that the consequence of this enhancement is a shift in the coupling of modes towards higher magnetic fields. This provides a potential opportunity to design which pairs of modes and in what range of fields hybridization will occur.

Reconfigurable spin-wave properties in two-dimensional magnonic crystals formed of diamond and triangular shaped nanomagnets

Two-dimensional ferromagnetic nanodot structures exhibit intriguing magnetization dynamics and hold promise for future magnonic devices. In this study, we present a comparative experimental investigation into the reconfigurable magnetization dynamics of non-ellipsoidal diamond and triangular-shaped nanodot structures, employing broadband ferromagnetic resonance spectroscopy. Our findings reveal substantial variations in the spin wave (SW) spectra of these structures under different bias field strengths (H) and angles (φ). Notably, the diamond nanodot structure exhibits a variation from nearly symmetric W-shaped dispersion to a skewed dispersion and subsequent transition to a discontinuous dispersion with subtle variation in bias field angle. On the other hand, in the triangular nanodot array a SW mode anti-crossing appears at φ = 15° which is starkly modified with the increase in φ to 30°. By analyzing the static magnetic configurations, we unveil the nature of the SW spectra in these two shapes. We reinforce our observations with simulated spatial power and phase maps. This study underscores the critical impact of dot shape and inversion symmetry on SW dynamical response, highlighting the significance of selecting appropriate structures and bias field strength and orientation for required functionalities. The remarkable tunability demonstrated by the magnonic crystals underscores their potential suitability for future magnonic devices.

Dirac points and flat bands in two-dimensional magnonic crystals with honeycomb–kagome structure

Based on the model of magnonic crystals (MCs) with honeycomb structure, we propose another model of two-dimensional MCs with honeycomb–kagome structure that is a periodic magnetic composite system composed of Fe, Co, or Py ferromagnetic cylindrical scatterers arranged in the EuO matrix as the honeycomb–kagome structure. The band structures of magnons in these systems are studied numerically by using the plane-wave expansion method. The results show that the Dirac points of magnons will be generated at the Brillouin region points if the scatterers are close-packed, that is to say, the edges of cylindrical scatterers are in contact with each other. The frequency of Dirac points can be indirectly adjusted by changing the radius ratio of close-packed cylinders. In addition, in the case of a large difference in the radius between the close-packed cylindrical scatterers, there will be a magnonic flat band in the band structure, which is a phenomenon of so-called compact localized states different from the impurity state in the crystal, and it is formed by the highly interference superposition of spin waves in the honeycomb–kagome structure. The research on the generation and modulation of magnonic Dirac points and flat bands not only expands the research content of condensed matter topological physics but also provides a promising platform for the application of artificial MCs in the fabrication of spin-wave topological devices.

The influence of the demagnetizing field on the concentration of spin wave energy in two-dimensional magnonic crystals

We use the Plane Wave Method to theoretically study thin-film magnonic crystals (MCs) composed of two very common magnetic materials: cobalt and permalloy. In both cases, we consider Co inclusions in the Py matrix and Py inclusions in the Co matrix. An external magnetic field is applied in the plane of the structure, leading to the formation of a demagnetizing field at the interface between the inclusions and matrix. Previous studies have shown that this field strongly affects the spectrum of spin waves, including the position and width of bandgaps. In this study, we exploit the in-plane squeezing of the MC structure to enhance the demagnetizing field. This results in the transfer of low-frequency spin waves from Py to Co, affecting the energy distribution (i.e., the spin-wave profile). The change in the concentration of spin-wave profiles leads to certain peculiarities in the spin-wave frequency spectrum. These include modes repulsion caused by hybridization, which in turn leads to the reordering of modes in the spectrum.

Spin wave dispersion relation engineering by magnonic crystals with arbitrary symmetry

The use of metasurfaces to engineer the response of magnetic materials is of utmost importance in the field of magnon-spintronics. Here, we demonstrate a method to fabricate one- and two-dimensional magnonic crystals with arbitrary symmetry and use it to engineer the amplitude-frequency characteristic of magnetostatic surface spin waves excited in a magnetic material. The technique is based on the gentle microablation of the sample surface by focused femtosecond laser pulses. Tightly focused illumination allows using modest pulse energy while achieving micrometer precision. By raster scanning the incident laser spot on the sample surface, we control the shape and size of the building blocks constituting the unit cell of the crystal along with its symmetry and lattice parameter. Remarkable and controlled changes in the measured transmission characteristics reveal the strong and complex symmetry-dependent interaction of the spin waves with Bravais and non-Bravais lattices. The described single-step microfabrication method facilitates and speeds up the realization of integrated spintronics components and provides an efficient tool to explore complex magnetic dynamics in scattering lattices.

Omnidirectional flat bands in chiral magnonic crystals

The magnonic band structure of two-dimensional chiral magnonic crystals is theoretically investigated. The proposed metamaterial involves a three-dimensional architecture, where a thin ferromagnetic layer is in contact with a two-dimensional periodic array of heavy-metal square islands. When these two materials are in contact, an anti-symmetric exchange coupling known as the Dzyaloshinskii–Moriya interaction (DMI) arises, which generates nonreciprocal spin waves and chiral magnetic order. The Landau–Lifshitz equation and the plane-wave method are employed to study the dynamic magnetic behavior. A systematic variation of geometric parameters, the DMI constant, and the filling fraction allows the examination of spin-wave propagation features, such as the spatial profiles of the dynamic magnetization, the isofrequency contours, and group velocities. In this study, it is found that omnidirectional flat magnonic bands are induced by a sufficiently strong Dzyaloshinskii–Moriya interaction underneath the heavy-metal islands, where the spin excitations are active. The theoretical results were substantiated by micromagnetic simulations. These findings are relevant for envisioning applications associated with spin-wave-based logic devices, where the nonreciprocity and channeling of the spin waves are of fundamental and practical scientific interest.

Reconfigurable spin-wave dynamics in two-dimensional quasiperiodic magnonic crystals

Active control of spin-wave properties is at the heart of on-chip reconfigurable magnonics, which promise development of energy efficient and miniaturized on-chip processing and communication devices in the GHz and sub-THz frequency range. Here, we demonstrated active control of spin-wave spectra in two-dimensional nanoscale antidot lattices arranged in shifted honeycomb and octagonal symmetries by varying the antidot diameter using broadband ferromagnetic resonance. Both these lattices exhibited stark variation in the spin-wave spectra with antidot diameter, besides the variation of the strength and orientation of bias magnetic field. With increasing diameter number of spin-wave modes gradually increased, and the position of mode splitting shifted to higher magnetic field due to ensuing increase in the demagnetizing field governing the spin-wave extension or localization in the lattices. The bias-field angle variation revealed higher-order rotational symmetry characteristic of quasi-periodic structure. Micromagnetic simulations qualitatively reproduced the experimental observations, and simulated spin-wave mode profiles revealed extended and standing spin-wave modes and their mutual conversion with bias-field angle in both lattices. The contrasting variation in the amplitude of rotational symmetry of spin-wave frequency with antidot diameter was interpreted by the variation of spin-wave mode profiles as well as the simulated internal magnetic field variation. Such efficient tunability of spin-wave properties with internal structure and external control parameters and higher-order rotational anisotropy in these atypical magnonic crystals will promote them for applications in on-chip reconfigurable magnonic devices.

Irreducible first Brillouin zone for two-dimensional magnonic crystals with asymmetric complex lattices

Two-dimensional (2D) magnonic crystal (MC) with an asymmetric complex basis is proposed in this paper, and its band structures are calculated in the whole area of the first Brillouin zone (BZ). This kind of MCs is composed of two different atoms in the unit cell, and the symmetry of the unit cell is broken due to changes in the position of the second atom, so the irreducible part of the BZ is no longer the small area [Formula: see text] for square lattice, and it must be expanded to the whole first BZ. Only by investigating the whole first BZ, can we get the true full band-gap for this kind of MCs.

Observation of spectral narrowing and mode conversion in two-dimensional binary magnonic crystal

We study binary magnonic crystal, where Ni80Fe20 nanodots of two different sizes are diagonally connected forming a unit and those units are arranged in a square lattice. The magnetization dynamics of the sample is measured by using time-resolved magneto-optical Kerr effect microscope with varying magnitude and orientation (ϕ) of the bias magnetic field, which is applied at about 10° angle from the sample plane. Interestingly, at ϕ = 0°, the spin-wave mode profiles show frequency selective spatial localization of spin-wave power within the array. With the variation of ϕ in the range 0° < ϕ ≤ 45°, we observe spectral narrowing due to localized to extended spin-wave mode conversion. Upon further increase of ϕ, the spin-wave modes slowly lose the extended nature and become fully localized again at 90°. We have extensively demonstrated the role of magnetostatic stray field distribution on the rotational symmetries obtained for the spin-wave modes. From micromagnetic simulations, we find that the dipole-exchange coupling between the nanodots leads to remarkable modifications of the spin-wave mode profiles when compared with arrays of individual small and large dots. Numerically, we also show that the physical connection between the nanodots provides more control points over the spin-wave propagation in the lattice at different orientations of bias magnetic field. In particular, the directional propagation of spin-wave is crucial for the application in coupler. Overall, we envision this binary magnonic crystal to have potential applications in magnonic devices such as spin-wave waveguide, filter, coupler, and other on-chip microwave communication devices.

Reversible tuning of omnidirectional band gaps in two-dimensional magnonic crystals by magnetic field and in-plane squeezing

By means of the plane wave method, we study nonuniform, i.e., mode- and k-dependent, effects in the spin-wave spectrum of a two-dimensional bicomponent magnonic crystal. We use the crystal based on a hexagonal lattice squeezed in the direction of the external magnetic field wherein the squeezing applies to the lattice and the shape of inclusions. The squeezing changes both the demagnetizing field and the spatial confinement of the excitation, which may lead to the occurrence of an omnidirectional magnonic band gaps. In particular, we study the role played by propagational effects, which allows us to explain the k-dependent softening of modes. The effects we found enabled us not only to design the width and position of magnonic band gaps, but also to plan their response to a change in the external magnetic field magnitude. This allows the reversible tuning of magnonic band gaps, and it shows that the studied structures are promising candidates for designing magnonic devices that are tunable during operation.

Anisotropic spin-wave dispersion in two-dimensional Ni80Fe20 diatomic nanodot array

Abstract Artificially patterned magnonic crystals serve as a promising candidate for the emerging research fields of magnonics and spintronics. Here, we investigate the spin-wave dispersion in a diatomic nanodot lattice made of Ni80Fe20 nanodots of two different diameters placed in close proximity to form a binary magnonic crystal with a complex double-dot unit cell. The frequency dispersion of SW eigenmodes experimentally measured by the Brillouin light-scattering technique is in good agreement with the numerically calculated band structure derived from the linearized Landau-Lifshitz equation. Due to the change of the nature of interaction among the dots in two mutually perpendicular orientation of the applied bias magnetic field, magnonic band structure, including the slope of the dispersion curves, varies significantly. The anisotropic SW propagation is also studied numerically by local excitation of spin-wave dynamics in this system. This phenomenon is further explained by the calculated iso-frequency contours. Efficient manipulation of spin waves in this new type of two-dimensional magnonic crystal is promising for the development of nanoscale magnonic and spintronic devices.

Controlled evolution of spin waves in unconventional defective honeycomb antidot lattices

Abstract Efficacious control of spin wave dynamics is demonstrated in two-dimensional magnonic crystals in the form of defective honeycomb antidot lattices. The spin wave dynamics of this system is investigated using broadband ferromagnetic resonance spectrometer. A stark modification in magnonic spectra is achieved with the modulation in the lattice spacing and with the bias magnetic field strength and its orientation. A rich band of spin wave modes are obtained for the lattice with smallest periodicity, which finally converges to two modes in case of the largest one. The spin wave modes in all lattices possess a strong anisotropic behaviour with a six- and two-fold rotational symmetries as a function of the bias field orientation originating from the strong modulation of internal field distribution at different regions of such unorthodox system. The experimentally observed spin wave modes have been qualitatively well reproduced by micromagnetic simulations and the simulated mode profiles unravel the presence of diverse genres of extended and localized standing modes in these unusual magnonic crystals. These observations are significant for tunability and anisotropic propagation of spin waves leading towards the development of efficient magnonic devices competent with the future microwave communication systems.

Nonuniform Spin-Wave Softening in Two-Dimensional 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.

Tunable geometrical frustration in magnonic vortex crystals

A novel approach to investigate geometrical frustration is introduced using two-dimensional magnonic vortex crystals. The frustration of the crystal can be manipulated and turned on and off dynamically on the timescale of milliseconds. The vortices are studied using scanning transmission x-ray microscopy and ferromagnetic resonance spectroscopy. They are arranged analogous to the nanomagnets in artificial spin-ice systems. The polarization state of the vortices is tuned in a way that geometrical frustration arises. We demonstrate that frustrated polarization states and non-frustrated states can be tuned to the crystal by changing the frequency of the state formation process.

Efficient Modulation of Spin Waves in Two-Dimensional Octagonal Magnonic Crystal.

Efficient tunability of magnonic spectra is demonstrated in two-dimensional ferromagnetic antidot lattices with different lattice constants arranged in the octagonal lattice which can be considered as quasi-periodic magnonic crystals due to the presence of broken translational symmetry. The precessional dynamics of these samples are investigated in the frequency domain with the help of broadband ferromagnetic resonance spectrometer. A rich variation in the spin wave spectra is observed with the variation of lattice constant as well as the strength and orientation of the bias magnetic field. A broad band of spin wave modes are observed for the denser array, which finally converges to two spin wave modes for the sparsest one. In addition to this, the most intense spin wave frequency shows an 8-fold anisotropy with a superposition of weak 4- and 2-fold anisotropy, which arises due to the angular variation of the magnetostatic field distribution at different regions of the octagonal lattice. Micromagnetic simulations qualitatively reproduce the experimentally observed modes, and the simulated mode profiles reveal the presence of different types of extended and quantized standing spin wave modes in these samples. The observations are important for the tunable and anisotropic propagation of spin waves in magnonic crystal based devices.

Pseudo-One-Dimensional Magnonic Crystals for High-Frequency Nanoscale Devices

Periodically patterned composite ferromagnetic nanostructures known as magnonic crystals (MCs) are promising for spintronic and other spin-based technologies, as they manipulate spin waves in analogy to photonic crystals' manipulation of light. The authors create an array of asymmetric sawtooth waveguides that presents two different one-dimensional MCs in perpendicular directions, yet not a two-dimensional MC. Its magnonic band structure and gap are tuned by rotating the applied magnetic field, and its edge modes may also be controlled, to yield, say, a GHz-frequency nanoscale filter.

Extremely flat transmission band of forward volume spin wave using gold and yttrium iron garnet

An extremely flat transmission band for a forward volume spin wave (SW) propagating in an SW waveguide composed of yttrium iron garnet film and SW absorbers was obtained, using the finite integration technique. Three-dimensional analysis of the calculated results showed that the transmission ripples of the SWs propagating in the waveguide were caused by SW interference, especially that due to two standing waves originating from the waveguide edge–edge and the waveguide–antenna edge. To suppress these waves, SW absorbers composed of gold film were introduced and the resultant positions and shapes were investigated precisely. Hence, an extremely flat transmission band was obtained. The results of this study have potential application in the development of one- and two-dimensional magnonic crystals for integrated SW devices.

Shape- and Interface-Induced Control of Spin Dynamics of Two-Dimensional Bicomponent Magnonic Crystals.

Controlled fabrication of periodically arranged embedded nanostructures with strong interelement interaction through the interface between the two different materials has great potential applications in spintronics, spin logic, and other spin-based communication devices. Here, we report the fabrication of two-dimensional bicomponent magnonic crystals in form of embedded Ni80Fe20 nanostructures in Co50Fe50 thin films by nanolithography. The spin wave (SW) spectra studied by a broadband ferromagnetic resonance spectroscopy showed a significant variation as the shape of the embedded nanostructure changes from circular to square. Significantly, in both shapes, a minimum in frequency is obtained at a negative value of bias field during the field hysteresis confirming the presence of a strong exchange coupling at the interface between the two materials, which can potentially increase the spin wave propagation velocity in such structures leading to faster gigahertz frequency magnetic communication and logic devices. The spin wave frequencies and bandgaps show bias field tunability, which is important for above device applications. Numerical simulations qualitatively reproduced the experimental results, and simulated mode profiles revealed the spatial distribution of the SW modes and internal magnetic fields responsible for this observation. Development of such controlled arrays of embedded nanostructures with improved interface can be easily applied to other forms of artificial crystals.

Band-gap structures of two-dimensional magnonic crystals with complex lattices

Magnonic crystals with spin waves as information carriers are the magnetic counterparts of photonic and phononic crystals. The studies of spin waves or magnons in magnonic crystals have attracted increasing attention, especially for the characteristics of band gaps. However, most of the previous work has paid attention to the magnonic crystals with simple lattices. In this paper, the model of magnonic crystals with complex lattices which is composed of two different scatterers of ferromagnetic materials periodically embedded in another kind of ferromagnetic matrix material is proposed for the first time. Then, the plane-wave expansion method is developed by using the idea of super cells, in which the Fourior coefficient of exchange constant in the space of reciprocal lattice vector is analytically derived, and this method can be used to numerically investigate the eigen-properties of spin waves in magnonic crystals with complex lattices. Of course, it can be applied to the fields of other artificial crystals with complex lattices after the corresponding process, such as photonic crystals and phononic crystals. Band structures of two-dimensional magnonic crystal with complex lattices consisting of two different sizes of Fe cylinders alternately arranged in Euo matrix, are numerically calculated by using the above plane-wave expansion method. The behaviors of band gaps of spin waves changing with the total filling fraction of volume f and also with the mismatch of the filling fraction of volume of two Fe cylinders in EuO matrix are numerically studied. The results of magnonic crystals with complex lattices are compared with those of magnonic crystal with simple latticeic. Some conclusions are summarized as follows. In the same filling fraction of volume f, the width of band gap B4, 5 in the magnonic crystal with complex lattice is always larger than that with the simple lattice, but the width of band gap B8, 9 in the complex lattice is less than that in the simple lattice. When f = (fA + fB)/2 = 0.5, the width of band gap B4, 5 increases as the mismatch between fA and fB increases, but the behavior of the gap B8, 9 is opposite. Moreover, some new spin-wave gaps can be generated by changing the mismatch between fA and fB. This is because the gaps in our studied systems result from the mechanism of Bragg scattering of spin wave in periodic ferromagnetic materials. When the mismatch between fA and fB increases, the multiple scattering effects become stronger. All of these results show that the width or the frequency of band gap can be optimized or tuned by using the complex lattice. Such an approach through fabricating complex lattices may open a new scope for engineering and designing the band gaps of magnonic crystals.

Dual Band Magnonic Crystals: Model System and Basic Spin Wave Dynamics

We investigate a special design of two-dimensional magnonic crystal, consisting of two superimposed lattices with different lattice constants, such that spin waves (SWs) can propagate either in one or the other sublattice, depending on which of the two frequency bands they belong to. The SW bands are separated by a very large bandgap (in our model system, 6 GHz), easily tunable by changing the direction of an applied magnetic field, and the overlap of their spatial distribution, for any frequency of their bands, is always negligible. These properties make the designed system an ideal test system for a magnonic dual band waveguide, where the simultaneous excitation and subsequent propagation of two independent SW signals are allowed, with no mutual interference.