Frequency Nonreciprocity Research Articles

Spin-wave nonreciprocity at the spin-flop transition region in synthetic antiferromagnets

We investigate the frequency nonreciprocity in CoFeB/Ru/CoFeB synthetic antiferromagnets near the spin-flop transition region, where the magnetic moments in the two ferromagnetic layers are noncollinear. Using conventional Brillouin light scattering, we perform systematic measurements of the frequency nonreciprocity as a function of an external magnetic field. At near zero magnetic field, where the antiparallel alignment of the magnetic moments in the two layers is established, we observe a significant frequency nonreciprocity of up to a few GHz, which vanishes when the relative magnetization orientation switches into the parallel configuration at a large magnetic field. For the intermediate values of the magnetic field, where the system transitions from the antiparallel to the parallel orientation, a nonmonotonous dependence of the frequency nonreciprocity is found, with a maximum frequency shift around the spin-flop critical point. This nontrivial dependence of the nonreciprocity versus field is attributed to the nonmonotonous dependence of the dynamic dipolar interaction, which is the main factor that causes asymmetry in the dispersion relation. Furthermore, we found that the sign of the frequency shift changes even without switching the polarity of the bias field. These results show that one can precisely control the nonreciprocal propagation of spin waves via field-driven magnetization reorientation.

Nonreciprocal collective magnetostatic wave modes in geometrically asymmetric bilayer structure with nonmagnetic spacer.

Nonreciprocity, i.e. inequivalence in amplitudes and frequencies of spin waves propagating in opposite directions, is a key property underlying functionality in prospective magnonic devices. Here we demonstrate experimentally and theoretically a simple approach to induce frequency nonreciprocity in a magnetostatically coupled ferromagnetic bilayer structure with a nonmagnetic spacer by its geometrical asymmetry. Using Brillouin light scattering, we show the formation of two collective spin wave modes in Fe81Ga19/Cu/Fe81Ga19 structure with different thicknesses of ferromagnetic layers. Experimental reconstruction and theoretical modeling of the dispersions of acoustic and optical collective spin wave modes reveal that both possess nonreciprocity reaching several percent at the wavenumber of 22 × 104 rad cm-1. The analysis demonstrates that the shift of the amplitudes of counter-propagating coupled modes towards either of the layers is responsible for the nonreciprocity because of the pronounced dependence of spin wave frequency on the layers' thickness. The proposed approach enables the design of multilayered ferromagnetic structures with a given spin wave dispersion for magnonic logic gates.

Excitation and reception of magnetostatic surface spin waves in thin conducting ferromagnetic films by coplanar microwave antennas. Part I: Theory

A fully self-consistent model for the excitation and reception of magnetostatic surface waves in thin ferromagnetic films by a set of coplanar antennas was developed and implemented numerically. The model assumes that the ferromagnetic film is highly conducting and is interfaced with non-magnetic metallic films, but is also suitable for modeling magneto-insulating films. Perpendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction can be included at both interfaces of the ferromagnetic layer. The model calculates the coupling impedances between the different strips constituting the coplanar antennas. In some situations, this leads to a frequency non-reciprocity between counter-propagating waves even in the case of no asymmetry in the spin-wave dispersion relation. Several intermediate results of the model were checked numerically and the final output of the model, given as the scattering parameters, S11, S12, and S21 of the antenna system, were in good agreement with previous experimental studies.

Spin-wave nonreciprocity and formation of lateral standing spin waves in CoFeB/Ta/NiFe meander-shaped films

Studying the spin-wave (SW) propagation in 3D periodic structures opens new possibilities for joining functional units placed on the different layers of the magnonic circuitry. In the path toward 3D magnonics, the main challenge is the fabrication of large-scale 3D magnetic structures with nanometric precision control of geometry and material composition. In this work, we study the dependence on the Ta spacer thickness of the magnonic band structure, measured by Brillouin light scattering spectroscopy, of CoFeB/Ta/NiFe meander-shaped bilayers fabricated on pre-patterned Si substrate with thickness steps of 50 nm. Both propagating and stationary SW modes are observed. While the frequency of the dispersive mode slightly depends on the Ta spacer thickness, the frequency position of the three stationary modes in the lowest frequency range of the spectra significantly increases by increasing the Ta thickness. Micromagnetic calculations indicate that each of the three stationary modes is composed of a doublet of modes whose frequency separation, within each doublet, increases by increasing the mode frequency. The origin of this frequency separation is ascribed to the dynamic dipolar coupling between the magnetic layers that generate a significant frequency nonreciprocity of counterpropagating SWs. For these reasons, the investigated structures offer potential application as the nonreciprocal versatile interconnections performing the frequency selective regimes of signal propagation in magnonic circuits.

Curvilinear spin-wave dynamics beyond the thin-shell approximation: Magnetic nanotubes as a case study

Surface curvature of magnetic systems can lead to many static and dynamic effects which are not present in flat systems of the same material. These emergent magnetochiral effects can lead to frequency nonreciprocity of spin waves, which has been shown to be a bulk effect of dipolar origin and is related to a curvature-induced symmetry breaking in the magnetic volume charges. So far, such effects have been investigated theoretically mostly for thin shells, where the spatial profiles of the spin waves can be assumed to be homogeneous along the thickness. Here, using a finite-element dynamic-matrix approach, we investigate the transition of the spin-wave spectrum from thin to thick curvilinear shells, at the example of magnetic nanotubes in the vortex state. With increasing thickness, we observe the appearance of higher-order radial modes which are strongly hybridized and resemble the perpendicular-standing spin waves (PSSWs) in flat films. Along with an increasing dispersion asymmetry, we uncover the curvature-induced nonreciprocity of the mode profiles. This is explained in a very simple, general picture for thick curvilinear shells, considering the inhomogeneity of the emergent geometric volume charges along the thickness of the shell. Such curvature-induced mode-profile asymmetry also leads to nonreciprocal hybridization, which can facilitate unidirectional spin-wave propagation. With that, we also show how curvature allows for nonlinear three-wave splitting of a higher-order radial mode into secondary modes which can also propagate unidirectionally. We believe that our study provides a significant contribution to the understanding of the spin-wave dynamics in curvilinear magnetic systems, and also advertises these for novel magnonic applications.

Temperature-sensitive spin-wave nonreciprocity induced by interlayer dipolar coupling in ferromagnet/paramagnet and ferromagnet/superconductor hybrid systems

Spin-wave (SW) spectra have theoretically been studied in a thin film of a ferromagnet (FM) on a substrate from a paramagnet (PM) (an FM above the critical temperature) or from superconductor (SC). A spin wave propagating in the FM induces the dynamic magnetization and superconducting current in the underlying PM and SC, respectively, which affect the SW propagation by their magnetic fields. As a result of this interaction, the SW spectrum becomes nonreciprocal to depend on the sign of the SW wave vector q. We show that the nonreciprocal contribution to the SW spectra in FM/PM and FM/SC systems is given by the frequency shift of {\Delta}{\omega}(q)={\omega}(q)-{\omega}(-q)=a(T)({\tau}{\cdot}q) with {\tau}=(n{\times}M) being the toroidal magnetic moment, M the FM magnetization, n the unit vector normal to the FM/PM(SC) interface, and a(T) the temperature-dependent constant of a dipole nature, whose sign depends on the substrate type. As the {\Delta}{\omega}(T) dependence is strong at temperatures T close to the critical temperature Tc for the FM-PM or normal metal-SC transition, one gets a possibility to control the frequency SW nonreciprocity with temperature variation near Tc. The dipolar mechanism we propose for SW frequency nonreciprocity is promising for introducing this property of SW propagation into functional devices.

Tunable spin-wave nonreciprocity in synthetic antiferromagnetic domain walls

Nonreciprocal propagation of spin waves provides the possibility to design novel functional magnonic devices. In the present paper, based on micromagnetic simulation, we investigate the properties of the spin waves propagating within magnetic domain walls in a synthetic antiferromagnetic bilayer structure and demonstrate a giant frequency nonreciprocity, of which not only the magnitude but also the sign can be efficiently controlled by an external magnetic field. This results from the field-induced modification of the hybridization between the domain-wall spin wave channels from different magnetic layers. The unidirectional excitation of the gigahertz hybridized domain-wall spin waves by a nanostripline antenna is also demonstrated. Our finding suggests the nanoscale antiferromagnetic domain walls should be a promising platform for programmable nonreciprocal spin-wave devices.

Spin-wave focusing induced by dipole-dipole interaction in synthetic antiferromagnets

Under certain conditions, spin waves can be channeled into a broad angular spectrum of wave vectors, where the direction of the group velocity becomes independent of those wave vectors. Such highly focused waves are called caustic waves, whose properties can be manipulated by anisotropies or chiral interactions, like the Dzyaloshinskii-Moriya interaction. In this paper, we theoretically study the focusing features of the spin waves induced by the dipole-dipole interaction in synthetic antiferromagnets. For stacked systems, the dipolar interaction causes a noticeable frequency nonreciprocity when the magnetizations in both films are antiparallelly aligned, and then the focusing properties of the spin waves are enhanced. The role of thicknesses and magnetic graduation along the film's normal are systematically analyzed. We found that the degree of focalization of the spin waves can be manipulated by increasing the layers' thickness. Also, we show that the low- and high-frequency modes exhibit different focalization properties; the low-frequency mode manifests a similar behavior to the heavy-metal/ferromagnet systems with interfacial Dzyaloshinskii-Moriya interaction, while the high-frequency one tends the generate almost reciprocal interference patterns along one axis. In the case of magnetization-graded synthetic antiferromagnets, we demonstrate that the graduation slightly influences the low-frequency mode, while the focusing and nonreciprocal dynamic properties of the high-frequency ones are notoriously altered. The theoretical calculations are compared with micromagnetic simulations, where a good agreement is found between both methods. Our results demonstrate that a synthetic antiferromagnetic system allows for controlling the propagation of spin waves, assisting in the transfer of angular momentum and energy.

Long-range spin-wave propagation in transversely magnetized nano-scaled conduits

Magnonics attracts increasing attention in the view of low-energy computation technologies based on spin waves. Recently, spin-wave propagation in longitudinally magnetized nano-scaled spin-wave conduits was demonstrated, proving the fundamental feasibility of magnonics at the sub-100 nm scale. Transversely magnetized nano-conduits, which are of great interest in this regard as they offer a large group velocity and a potentially chirality-based protected transport of energy, have not yet been investigated due to their complex internal magnetic field distribution. Here, we present a study of propagating spin waves in a transversely magnetized nanoscopic yttrium iron garnet conduit of 50 nm width. Space and time-resolved microfocused Brillouin-light-scattering spectroscopy is employed to measure the spin-wave group velocity and decay length. A long-range spin-wave propagation is observed with a decay length of up to (8.0 ± 1.5) μm and a large spin-wave lifetime of up to (44.7 ± 9.1) ns. The results are supported with micromagnetic simulations, revealing a broad single-mode frequency range and the absence of a mode localized to the edges. Furthermore, a frequency nonreciprocity for counter-propagating spin waves is observed in the simulations and the experiment, caused by the trapezoidal cross section of the structure. The revealed long-distance spin-wave propagation on the nano-scale is particularly interesting for an application in spin-wave devices, allowing for long-distance transport of information in magnonic circuits and low-energy device architectures.

Enhancement of the spin-wave nonreciprocity in antiferromagnetically coupled multilayers with dipolar and interfacial Dzyaloshinskii-Moriya interactions

Spin-wave-based circuits and logic devices have been considered as an alternative to current electronic devices as they approach the physical limit of miniaturization. Asymmetrical propagation of spin waves, also known as nonreciprocity, provides an additional degree of freedom to these spin-wave-based devices, increasing their flexibility. In thin films, nonreciprocity can be induced by the Dzyaloshinskii-Moriya interaction (DMI) at heavy-metal/ferromagnet bilayers, and by the dipolar coupling in multilayers. Here, we show that in an antiferromagnetically coupled multilayer with interfacial DMI, the frequency nonreciprocity induced by the DMI is enhanced when both heavy metals are the same as long as the multilayer remains in an antiparallel state. Furthermore, we show that the interplay between the dipolar and Dzyaloshinskii-Moriya interaction enhances the nonreciprocity of one oscillation mode and reduces the nonreciprocity of the other. Which mode is enhanced depends on the sign of the induced Dzyaloshinskii-Moriya interaction at the interfaces and the magnetic moments of the layers. Finally, we show that it is possible to change the frequency nonreciprocity of Pt/Co/Cu/Co/Pt and Pt/Co/Cu/Py/Pt multilayers by $\ensuremath{\sim}7$ GHz when applying an in-plane magnetic field of 130 mT. This includes a change in the sign of the nonreciprocity, which could be used to control the direction of the flux of information in spin-wave devices.

Possibility of reducing nonlinear distortions of a frequency response of a gas ring laser with a periodic alternating frequency bias

We report a theoretical study of the dependence of the beat frequency of counterpropagating waves on the rotation velocity in a gas ring laser with a periodic alternating frequency bias. Rectangular biases of two types are considered: with constant amplitude and with periodic amplitude modulation. We have found dynamic zones for locking-in the counterpropagating wave frequencies and determined deviations of the frequency response from the ideal one that arise when measuring frequency nonreciprocities, close in magnitude to the amplitude of an alternating frequency bias. It is shown that in the case of a rectangular bias with periodic amplitude modulation, nonlinear distortions of the frequency response can be reduced by an order of magnitude.

Extraction of Dzyaloshinskii-Moriya interaction from propagating spin waves

The interfacial Dzyaloshinksii-Moriya interaction (iDMI) is of great interest in thin-film magnetism because of its ability to stabilize chiral spin textures. It can be quantified by investigating the frequency non-reciprocity of oppositely propagating spin waves. However, as the iDMI is an interface interaction the relative effect reduces when the films become thicker making quantification more difficult. Here, we utilize all-electrical Propagating Spin Wave Spectroscopy (PSWS) to disentangle multiple contributions to spin wave frequency non-reciprocity to determine the iDMI. This is done by investigating non-reciprocities across a wide range of magnetic layer thicknesses (from 4 to 26 nm) in Pt/Co/Ir, Pt/Co/Pt, and Ir/Co/Pt stacks. We find the expected sign change in the iDMI when inverting the stack order, and a negligible iDMI for the symmetric Pt/Co/Pt. We additionally extract a difference in surface anisotropies and find a large contribution due to the formation of different crystalline phases of the Co, which is corroborated using nuclear magnetic resonance and high-resolution transmission-electron-microscopy measurements. These insights will open up new avenues to investigate, quantify and disentangle the fundamental mechanisms governing the iDMI, and pave a way towards engineered large spin-wave non-reciprocities for magnonic applications.

Determining Key Spin-Orbitronic Parameters via Propagating Spin Waves

We characterize spin wave propagation and its modification by an electrical current in Permalloy(Py)/Pt bilayers with Py thickness between 4 and 20 nm. First, we analyze the frequency non-reciprocity of surface spin waves and extract from it the interfacial Dzyaloshinskii-Moriya interaction constant $D_s$ accounting for an additional contribution due to asymmetric surface anisotropies. Second, we measure the spin-wave relaxation rate and deduce from it the Py/Pt spin mixing conductance $g^{\uparrow\downarrow}_{eff}$. Last, applying a \textit{dc} electrical current, we extract the spin Hall conductivity $\sigma_{SH}$ from the change of spin wave relaxation rate due to the spin-Hall spin transfer torque. We obtain a consistent picture of the spin wave propagation data for different film thicknesses using a single set of parameters $D_s=0.25$ pJ/m, $g^{\uparrow\downarrow}_{eff} = 3.2\times 10^{19}$ m$^{-2}$ and $\sigma_{SH}=4\times10^{5}$ S/m.

Spacetime Metamaterials—Part I: General Concepts

This article deals with the general concepts underpinning spacetime metamaterials and related systems. It first introduces spacetime metamaterials as a generalization of (bianisotropic) metamaterials, presented in the holistic perspective of direct and inverse spacetime scattering, where spacetime variance and dispersion offer unprecedented medium diversity despite some limitations related to the uncertainty principle. Then, it describes the fundamental physical phenomena occurring in spacetime systems, such as frequency transitions, nonreciprocity, Fizeau dragging, bianisotropy transformation, and superluminality, allowed when the medium moves perpendicularly to the direction of the wave. Next, it extends some principles and tools of relativity physics, particularly a medium-extended version of the spacetime (or Minkowski) diagrams, elaborates a general strategy to compute the fields scattered by spacetime media, and presents a gallery of possible spacetime media, including the spacetime step discontinuity, which constitutes the building brick of any spacetime metamaterial. Finally, the conclusion section provides a list of 16 items that concisely summarizes the key results and teachings of the overall document. The second part establishes the theory and overviews some current and potential applications of spacetime metamaterials.

Spontaneous phase symmetry breaking in a gas coupled-cavity ring laser

A theoretical model is proposed that describes the lasing dynamics in a gas ring laser (GRL) with coupled cavities. The conditions are found under which spontaneous phase symmetry of counterpropagating waves is broken in the GRL with an antiphase optical coupling of the cavities. It is shown that in the case of spontaneous phase symmetry breaking, two branches appear on the GRL frequency characteristic. In some region of frequency nonreciprocity of a ring cavity, both branches can exist under the same conditions. In this case, radiation bistability appears in the GRL, and hysteresis phenomena can be observed.

Spin-wave non-reciprocity in magnetization-graded ferromagnetic films

A theoretical approach has been developed to study the spin-wave dynamics of magnetization-graded ferromagnetic films, where the magnetic properties change along the film thickness. The theory is based on a multilayer approach, where the influence of both long-range dipolar interactions and interlayer exchange coupling between sublayers is included. This allows for instance to describe films with a continuous variation of the saturation magnetization along the thickness. A systematic study is carried out in order to analyze different profiles of the saturation magnetization, which is checked through a test of convergence. It is found that the spin-wave dispersion is significantly modified when the strength of the magnetization changes in the bulk film, where a notable frequency non-reciprocity of two counter propagating spin waves is predicted. This is associated with heterosymmetric mode profiles and a modification of the conventional quantization condition associated to perpendicular standing spin-wave modes. Micromagnetic simulations have been carried out to validate the model, where a perfect agreement is reached between both methods. These results show that magnetization-graded ferromagnetic films can be used to channel and control spin waves, thus promoting different kinds of functionalities for magnon-based devices.

Electrical field enhanced interfacial Dzyaloshinskii-Moriya interaction in MgO/Fe/Pt system

We report electric field control of the interfacial Dzyaloshinskii-Moriya interaction (DMI) in MgO/Fe/Pt multilayers. The interfacial DMI is obtained and quantified by Brillouin light scattering measurements based on the frequency nonreciprocity of spin waves in a ferromagnet. The magnitude of the induced DMI is found to linearly increase as a function of electric field intensity. The efficiency of electric field control of the interfacial DMI characterized as a ratio of the DMI energy change to the electric field, which is found to be equal to 67 fJ/(Vm) with a positive electric field. We further demonstrate that the origin of the enhanced DMI results from the MgO/Fe interface. The Rashba spin-orbit coupling and electric field induced anisotropy at the MgO/Fe interface might be responsible for the enhanced interfacial DMI. Our findings open up a way for exploring the spintronic devices with a tunable DMI.

Interfacial Dzyaloshinskii-Moriya Interaction in Pt/CoFeB Films: Effect of the Heavy-Metal Thickness.

We report the observation of a Pt layer thickness dependence on the induced interfacial Dzyaloshinskii-Moriya interaction in ultrathin Pt(d_{Pt})/CoFeB films. Taking advantage of the large spin-orbit coupling of the heavy metal, the interfacial Dzyaloshinskii-Moriya interaction is quantified by Brillouin light scattering measurements of the frequency nonreciprocity of spin waves in the ferromagnet. The magnitude of the induced Dzyaloshinskii-Moriya coupling is found to saturate to a value of 0.45 mJ/m^{2} for Pt thicknesses larger than ∼2 nm. The experimental results are explained by analytical calculations based on the three-site indirect exchange mechanism that predicts a Dzyaloshinskii-Moriya interaction at the interface between a ferromagnetic thin layer and a heavy metal. Our findings open up a way to control and optimize chiral effects in ferromagnetic thin films through the thickness of the heavy-metal layer.

Frequency response of a gas ring laser with a variable-sign frequency bias in the case of frequency nonreciprocity comparable with the bias amplitude

The dependence of the beat frequency of counterpropagating waves on the rotation rate is studied theoretically and experimentally in a gas ring laser (GRL) with a Zeeman effect-based, variable-sign frequency bias. Use is made of magneto-optical biases of two types, namely, a rectangular (meander) bias and a combined bias consisting of fast and slow meanders. At high rotation rates, when the frequency nonreciprocity determined by the rotation rate is close in magnitude to the amplitude of the variable-sign frequency bias, the dynamic locking bands take maximum values and there arise the most extensive deviations of the frequency response from the ideal response. Comparison of experimental and theoretical results for the widths of dynamic bands that appear in this region of the measured rotation rates shows good agreement between theory and experiment. The results suggest that the frequency response of a GRL in this region can be described by one differential equation for the phase difference of counterpropagating waves.

Probing the Dzyaloshinskii-Moriya interaction in CoFeB ultrathin films using domain wall creep and Brillouin light spectroscopy

We have characterized the strength of the interfacial Dyzaloshinskii-Moriya interaction (DMI) in ultrathin perpendicularly magnetized CoFeB/MgO films, grown on different underlayers of W, TaN, and Hf, using two experimental methods. First, we determined the effective DMI field from measurements of field-driven domain wall motion in the creep regime, where applied in-plane magnetic fields induce an anisotropy in the wall propagation that is correlated with the DMI strength. Second, Brillouin light spectroscopy was employed to quantify the frequency non-reciprocity of spin waves in the CoFeB layers, which yielded an independent measurement of the DMI. By combining these results, we show that DMI estimates from the different techniques only yield qualitative agreement, which suggests that open questions remain on the underlying models used to interpret these results.