One-dimensional Magnonic Crystals Research Articles

Спиновые волны в одномерных магнонных кристаллах с двумя пространственными периодами

The propagation of surface and backward volume magnetostatic waves in a one-dimensional magnonic crystal in the form of a yttrium iron garnet film with two periodic lattices of grooves with different periods etched on its surface was experimentally studied. It is shown that the form of the frequency dependence of the transmission coefficient of magnetostatic waves in such crystal is not a simple superposition of similar dependences for two magnonic crystals with the same lattice periods.

Experimental Observation of Flat Bands in One-Dimensional Chiral Magnonic Crystals.

Spin waves represent the collective excitations of the magnetization field within a magnetic material, providing dispersion curves that can be manipulated by material design and external stimuli. Bulk and surface spin waves can be excited in a thin film with positive or negative group velocities and, by incorporating a symmetry-breaking mechanism, magnetochiral features arise. Here we study the band diagram of a chiral magnonic crystal consisting of a ferromagnetic film incorporating a periodic Dzyaloshinskii-Moriya coupling via interfacial contact with an array of heavy-metal nanowires. We provide experimental evidence for a strong asymmetry of the spin wave amplitude induced by the modulated interfacial Dzyaloshinskii-Moriya interaction, which generates a nonreciprocal propagation. Moreover, we observe the formation of flat spin-wave bands at low frequencies in the band diagram. Calculations reveal that depending on the perpendicular anisotropy, the spin-wave localization associated with the flat modes occurs in the zones with or without Dzyaloshinskii-Moriya interaction.

Управление хаотическими темными диссипативными солитонами огибающей в активном кольцевом резонаторе на основе магнонного кристалла с динамическим дефектом

Dynamic control of chaotic dark dissipative envelope solitons in a microwave active ring resona-tor containing two nonlinear elements: a one-dimensional magnonic crystal (MC) with a dynamic linear defect and a transistor amplifier is performed. Dissipative envelope solitons are formed on a long-wavelength surface magnetostatic wave propagating in the MC and parametrically de-caying into short-wavelength spin waves. A constant electric current flowing through a copper conductor creates a dynamic linear defect along the longitudinal axis of the MC and controls the duty factor of the generated parametric pulses only at the MC forbidden frequency.

Control of Chaotic Dark Dissipative Envelope Solitons in an Active Ring Resonator Based on a Magnonic Crystal with a Dynamic Defect

Dynamic control of chaotic dark dissipative envelope solitons in a microwave active ring resona tor containing two nonlinear elements: a one-dimensional magnonic crystal (MC) with a dynamic linear defect and a transistor amplifier is performed. Dissipative envelope solitons are formed on a long-wavelength surface magnetostatic wave propagating in the MC and parametrically decaying into short-wavelength spin waves. A constant electric current flowing through a copper conductor creates a dynamic linear defect along the longitudinal axis of the MC and controls the duty factor of the generated parametric pulses only at the MC forbidden frequency. Keywords: dissipative solitons, magnonic crystals, spin waves.

Наведенный нелинейный сдвиг частоты активного кольцевого резонатора на магнонном кристалле

The nonlinear shift of the resonant frequencies of a microwave active ring resonator (ARR) induced by a pump wave has been studied for the first time. The resonator was designed according to the scheme in the form of a closed ring containing a spin-wave delay line made on the basis of a one-dimensional magnonic crystal, as well as a microwave amplifier in a feedback circuit. It is shown, that with an increase in the power of the pumping wave, the shift of the resonant frequencies of the ARR increases in different ways depending on the position of the pumping frequency. The maximum induced nonlinear frequency shift is observed if the pumping frequency is at the minimum of the ARR transmission coefficient. A model describing the studied effect is developed.

Lock-in thermographic study of spin-wave propagation in magnonic crystals

We have investigated the spin-wave dynamics in a one-dimensional magnonic crystal (MC) with respect to the heat radiation due to the damping of the spin waves. The spin waves were excited by applying microwaves via a wire antenna. The heat induced by the excitation and propagation of spin waves was measured using the lock-in thermography (LIT) technique by periodically modulating the microwave power. The LIT measurements resolved the heat source distributions inside the MC, which is made of an yttrium iron garnet film and periodic grooves with a sub-mm interval, in the backward volume wave geometry. The temperature distribution induced by the spin-wave excitation notably depends on the frequency or wave number of the spin waves, as a result of the formation of rejection bands in the MC. The observed temperature modulation profiles are complicated, but their behavior is consistent with a calculation based on the microwave transmission line approximation of the MCs, demonstrating the applicability of the LIT measurements to the investigation of the spin-wave dynamics in sub-mm scale MCs.

Spectrum of exchange spin waves in a one-dimensional magnonic crystal with antiferromagnetic ordering

Purpose of the study is to show that the conditions for the propagation of exchanged spin waves (ESWs) in an asymmetric superlattice with antiferromagnetically ordered cells depend significantly on the chirality of the precession of the ESW magnetization (polarization, “magnon pseudospin”). Method. When constructing the EWS spectra, the Croning– Penny model (transfer-matrix method) and the Landau–Lifshitz equation are used to determine the nature of the waves in the cells. In the case of a uniaxial medium, there is only one type of ESW, therefore, when fields are joined at the boundary, the conservation of chirality is an essential factor due to which the ESW in one cell is always traveling, and in the other — evanescent. Thus, a superlattice for ESW is an effective periodic “potential” in which asymmetry can be realized either by applying an external field, or by a difference in the thickness and/or physical properties of the cell materials. Results. Based on the analysis of the spectrum, maps of the transmission zones for ESW of different chirality were constructed in three representations — “Bloch wave number – frequency”, “frequency – relative cell thickness”, as well as in the plane of cell wave numbers. It is shown that the presence of asymmetry leads to a difference in the width of the transmission zones for waves of different chirality. For a finite structure, the frequency dependences of the transmission and reflection coefficients of the ESW are plotted. An increase in the attenuation of the ESW near the boundaries of the transmission zones was also found. Conclusion. The results of the study can be used in the design of magnon valves and other devices based on ESW, in which their chirality can be controlled.

Interface modes in planar one-dimensional magnonic crystals

We present the concept of Zak phase for spin waves in planar magnonic crystals and discuss the existence condition of interface modes localized on the boundary between two magnonic crystals with centrosymmetric unit cells. Using the symmetry criterion and analyzing the logarithmic derivative of the Bloch function, we study the interface modes and demonstrate the bulk-to-edge correspondence. Our theoretical results are verified numerically and extended to the case in which one of the magnonic crystals has a non-centrosymmetric unit cells. We show that by shifting the unit cell, the interface modes can traverse between the band gap edges. Our work also investigate the role of the dipolar interaction, by comparison the systems both with exchange interaction only and combined dipolar-exchange interactions.

From the Spin Eigenmodes of Isolated Néel Skyrmions to the Magnonic Bands of Skyrmionic Crystals: A Micromagnetic Study as a Function of the Interfacial Dzyaloshinskii-Moriya Interaction and the Exchange Constants

The presence of interfacial Dzyaloshinskii-Moriya interaction (DMI) may lead to the appearance of Néel skyrmions in ferromagnetic films. These topologically protected structures, whose diameter is as small as a few nanometers, can be nowadays stabilized at room temperature and have been proposed for the realization of artificial magnonic crystals and new spintronic devices, such as racetrack memories. In this perspective, it is of utmost importance to analyze their dynamical properties in the gigahertz range, i.e., in the operation range of current communication devices. In this letter, we exploited the software MuMax3 to calculate the dynamics of Néel skyrmions in the range between 1 and 30 GHz, considering first the eigenmodes of an isolated skyrmion, then the case of two interacting skyrmions, and finally a linear chain, representing a one-dimensional magnonic crystal, whose magnonic band structure has been calculated as a function of the strength of both the DMI- and the exchange-constants, namely <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$A$</tex-math></inline-formula> . The magnonic bands can be interpreted as derived from the eigenmodes of isolated skyrmions, even if hybridization and anticrossing phenomena occur for specific ranges of values of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$A$</tex-math></inline-formula> . Therefore, varying the latter parameters, for instance, by a proper choice of the materials and thicknesses, may enable one to fine-tune the permitted and forbidden frequency interval of the corresponding magnonic crystal.

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.

Bistable nanomagnet as programmable phase inverter for spin waves

To realize spin wave logic gates, programmable phase inverters are essential. We image using phase-resolved Brillouin light scattering microscopy propagating spin waves in a one-dimensional magnonic crystal consisting of dipolarly coupled magnetic nanostripes. We demonstrate phase shifts upon a single nanostripe of opposed magnetization. Using micromagnetic simulations, we model our experimental finding in a wide parameter space of biasfields and wave vectors. We find that low-loss phase inversion is achieved, when the internal field of the oppositely magnetized nanostripe is tuned such that the latter supports a resonant standing spin wave mode with an odd quantization number at the given frequency. Our results are key for the realization of phase inverters with optimized signal transmission.

Magnon valve effect and resonant transmission in a one-dimensional magnonic crystal

We theoretically investigate the transmission of exchange-dominated spin waves in a one-dimensional magnonic crystal (MC) with a periodic exchange bias field. By recasting the Landau-Lifshitz-Gilbert equation into an effective Schr\"odinger equation and establishing spin-wave functions, it is found that MCs with upward (up) and downward (down) magnetization, respectively, correspond to the rectangular $N$-fold barriers and wells for magnons. The broadband transmission spectra in up and down states are systematically investigated. We show the phenomena of the magnon resonant transmission in both states and calculate the resonant transmission wave functions, which are related to the magnon density. Our results also show a transmission spectra shift effect (TSSE) between up and down states, which is found to be general in this system. Furthermore, the TSSE is useful to design a type of magnon valve, the magnonic-crystal-based magnon valve (MCMV), which has a large on/off ratio and bandwidth. By high-throughput screening, 125 000 groups of parameters of the MC are calculated, and 1948 parameter groups of high-performance MCMVs are screened out. Our work clarifies the physical details of the exchange-dominated spin-wave transmission in rectangular $N$-fold barriers and wells and also provides a promising route for designing novel magnonic devices.

Spin-Wave Resonance in One-Dimensional Magnonic Crystals by an Example of Multilayer Co–P Films

Layered magnetic films of Co–P alloy, which are sets of laminates with certain atomic and chemical structures, have been investigated by spin-wave resonance. Coatings of two types were synthesized: a magnonic crystal (amorphous Co(P)/fcc Co(P))N and gradient Co(P) films. They exhibit two different modifications of the spin-wave resonance (SWR) spectrum. In the first case, the SWR spectrum is described as Hn(n) ~ n2 with a modification caused by the formation of the first stop band of the magnonic crystal. In the second case, the dependence of resonance fields Hn of the spin-wave modes on mode number n has the form of Hn(n) ~ n2/3. Thermal annealing leads to crystallization (amorphous alloy → fcc) and causes transformation of the layered films to the films of single-phase Co–P alloys, which is accompanied by a change in the shapes of the SWR spectra and occurrence of a “exchange kink” spectral modification.

Wannier-Stark ladder spectrum of Bloch oscillations of magneto-dipole spin waves in graded 1D magnonic crystals

We have used the method of Wannier functions to calculate the frequencies and profiles of spin waves localized in one-dimensional magnonic crystals due to a gradient in the bias magnetic field. This localization of spin waves is analogous to the phenomenon of Bloch oscillations of quantum-mechanical electrons in crystals in a uniform electric field. As a convenient yet realistic model, we consider backward volume magnetostatic spin waves in a film of yttrium-iron garnet in a bias magnetic field comprising spatially uniform, cosine and gradient contributions. The spin-wave spectrum is shown to have the characteristic form of a Wannier–Stark ladder. The analytical results are verified using those obtained using numerical micromagnetic simulations. The physics of spin-wave Bloch oscillations combines the topics of magnonic crystals and graded magnonic index — the two cornerstones of modern magnonics.

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.

Band structure of a one-dimensional bilayer magnonic crystal

The band structure of two coupled one-dimensional (1D) bicomponent magnonic crystals is constructed from theory. The model takes into account both the exchange interaction and the long-range dipolar interaction between the magnonic crystals, for which different coupling configurations are analyzed systematically. In comparison with a monolayer 1D bicomponent magnonic crystal, the band structure of the bilayer magnonic crystal is significantly modified and depends on the geometric positioning between the layers. The widths of the band gaps may be altered considerably to induce a flattening of the high-frequency modes or nonreciprocity in frequency of two counterpropagating collective spin waves. In the latter case, the nonreciprocity is mainly observed in modes with frequencies above the first band gap, whereas the low-frequency modes remain almost reciprocal. These theoretical results show that two coupled magnonic crystals are excellent candidates to assist in controlling the collective spin waves, thereby permitting great versatility in designing reconfigurable spin-based devices.

Optical heterodyne imaging of magnetostatic modes in one-dimensional magnonic crystals

We demonstrate a real-space imaging of a heterodyne signal of light that is produced as a result of the Brillouin light scattering by coherently driven magnons in magnetostatic modes. With this imaging technique, we characterize surface magnetostatic modes (Damon-Eshbach modes) in a one-dimensional magnonic crystal, which is formed by patterned aluminum strips deposited on the ferromagnetic film. The modified band structures of the magnonic crystal are deduced from the Fourier transforms of the real-space images. The heterodyne imaging provides a simple and powerful method to probe magnons in structured ferromagnetic films, paving a way to investigate more complex phenomena, such as Anderson localization and topological transport with magnons.

Spin-wave modes in transition from a thin film to a full magnonic crystal

Spin-wave modes are studied under the gradual transition from a flat thin film to a 'full' (one-dimensional) magnonic crystal. For this purpose, the surface of a pre-patterned 36.8 nm thin permalloy film was sequentially ion milled resulting in magnonic hybrid structures, referred to as surface-modulated magnonic crystals, with increasing modulation depth. After each etching step, ferromagnetic resonance measurements were performed yielding the spin-wave resonance modes in backward-volume and Damon-Eshbach geometry. The spin-wave spectra of these hybrid systems reveal an even larger variety of spin-wave states compared to the 'full' magnonic crystal. The measurements are corroborated by quasi-analytical theory and micromagnetic simulations in order to study the changing spin-wave mode character employing spin-wave mode profiles. In backward-volume geometry, a gradual transition from the uniform mode in the film limit to a fundamental mode in the thin part of the magnonic crystal was observed. Equivalently, the first and the second film modes are transform into a center and an edge mode of the thick part of the magnonic crystal. Simple transition rules from the $n^{\mathrm{th}}$ film mode to the $m^{\mathrm{th}}$ mode in the 'full' magnonic crystal are formulated unraveling the complex mode structure particularly in the backward-volume geometry. An analogous analysis was performed in the Damon-Eshbach geometry.

One-Dimensional Magnonic Crystal With Cu Stripes for Forward Volume Spin Waves

One-Dimensional Magnonic Crystal With <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mi>Cu</mml:mi></mml:math> Stripes for Forward Volume Spin Waves

Влияние эффектов самовоздействия на распространение импульсов поверхностных магнитостатических волн в структуре магнонный кристалл--диэлектрик--металл

The influence of self-action effects on the magnetostatic surface waves (MSSW) pulses propagating in the structure of a one-dimensional magnonic crystal - dielectric - metal, where the choice of dielectric thickness forms an anomalous dispersion region that provides the fulfillment of Lighthill criterion on MSSW solitons formation, is studied. It is shown that in those cases when the portion of anomalous dispersion caused by metallization coincides with the frequency range of Bragg resonance, MSSW solitons are not formed.