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Fractional calculus has been used to describe physical systems with complexity. Here, we show that a fractional calculus approach can restore or include complexity in any physical systems that can be described by partial differential equations. We argue that the dispersion relation contains the required information relating the energy and momentum space of the system and thus fully describes their dynamics. The approach is demonstrated by two examples: the Landau-Lifshitz equation in a 1D ferromagnetic chain, an example of a periodic crystal system with a bounded dispersion relation; and a modified KdV equation supporting surface gravity waves or Euler dispersion, an example of an unbounded system in momentum space. The presented approach is applicable to fluids, soft matter, and solid-state matter and can be readily generalized to higher dimensions and more complex systems. While numerical calculations are needed to determine the fractional operator, the approach is analytical and can be utilized to determine analytical solutions and investigate nonlinear problems.

Ferromagnetic channels subject to spin injection have been theoretically shown to sustain dissipative exchange flows (DEFs). In the strong injection regime, a soliton is stabilized at the injection site, which has been termed a contact soliton DEF or CS-DEF. Here, we investigate the modulation of CS-DEFs as a mechanism to inject magnons into a DEF. By varying the injection current, the parameters connecting the soliton with the DEF via a boundary layer are varied. These lead to a modification in the DEF which is interpreted as a spin piston. The injected magnons follow the expected dispersion relation of DEFs, akin to the Bogoliubov - de Gennes dispersion relation. This work demonstrates that changes in the injected current can pumping magnons along a ferromagnetic channel via dissipative exchange flows.

Artificial spin ices (ASIs) arranged in square formations have been explored from the perspective of reconfigurable magnonics. A new frontier in ASIs is their three-dimensional (3D) extension. Here, we numerically explore the ferromagnetic resonance of square ASIs as each nanomagnet is rotated out of plane into 3D ASIs, in which the vertex gap can be either kept constant or varying. We study both remanent and vortex configurations using a semi-analytical dynamic approach and micromagnetic simulations. We find that both methods show qualitative agreement of the main spectral features. However, there are important limitations. On one hand, the semi-analytical approach relies on a minimal model of the demag field, preventing exact predictions. On the other hand, micromagnetic simulations suffer from sufficient resolution, making the results grid-dependent and extremely slow. Regardless, both methods display tunability as a function of the tilt angle. These results showcase advantages and limitations of both methods and are promising to further our understanding of 3D ASI dynamics.

Magnetic materials host a wealth of nonlinear dynamics, textures, and topological defects. This is possible due to the competition between strong nonlinearity and dispersion that act at the atomic scale as well as long-range interactions. However, these features are difficult to analytically and numerically study because of the vastly different temporal and spatial scales involved. Here, we present a pseudo-spectral approach for the Landau-Lifshitz equation that invokes energy and momentum conservation embodied in the magnon dispersion relation to accurately describe both atomic and continuum limits. Furthermore, this approach enables analytical study at every scale. We show the applicability of this model in both the continuum and atomic limit by investigating modulational instability and ultrafast evolution of magnetization due to transient grating, respectively, in a 1D ferromagnetic chain with perpendicular magnetic anisotropy. This model provides the possibility of grid-independent multiscale numerical approaches that will enable the description of singularities within a single framework.

Arrays of artificial spin ices exhibit reconfigurable ferromagnetic resonance frequencies that can be leveraged and designed for potential applications.However, analytical and numerical studies of the frequency response of artificial spin ices have remained somewhat limited due to the need of take into account nonlocal dipole fields in theoretical calculations or by long computation times in micromagnetic simulations. Here, we introduce Gaenice, a framework to compute magnon dispersion relations of arbitrary artificial spin ice configurations. Gaenice makes use of a tight-binding approach to compute the magnon bands. It also provides the user complete control of the interaction terms included, e.g., external field, anisotropy, exchange, and dipole, making it useful also to compute ferromagnetic resonances for a variety of structures, such as multilayers and ensembles of weakly or non-interacting nanoparticles. Because it relies on a semi-analytical model, Gaenice is computationally inexpensive and efficient, making it an attractive tool for the exploration of large parameter spaces.

Strongly-interacting nanomagnetic arrays are ideal systems for exploring reconfigurable magnonics. They provide huge microstate spaces and integrated solutions for storage and neuromorphic computing alongside GHz functionality. These systems may be broadly assessed by their range of reliably accessible states and the strength of magnon coupling phenomena and nonlinearities. Increasingly, nanomagnetic systems are expanding into three-dimensional architectures. This has enhanced the range of available magnetic microstates and functional behaviours, but engineering control over 3D states and dynamics remains challenging. Here, we introduce a 3D magnonic metamaterial composed from multilayered artificial spin ice nanoarrays. Comprising two magnetic layers separated by a non-magnetic spacer, each nanoisland may assume four macrospin or vortex states per magnetic layer. This creates a system with a rich $16^N$ microstate space and intense static and dynamic dipolar magnetic coupling. The system exhibits a broad range of emergent phenomena driven by the strong inter-layer dipolar interaction, including ultrastrong magnon-magnon coupling with normalised coupling rates of $\frac{\Delta \omega}{\gamma} = 0.57$, GHz mode shifts in zero applied field and chirality-selective magneto-toroidal microstate programming and corresponding magnonic spectral control.

Time-resolved ultrafast EUV magnetic scattering was used to test a recent prediction of >10 km/s domain wall speeds by optically exciting a magnetic sample with a nanoscale labyrinthine domain pattern. Ultrafast distortion of the diffraction pattern was observed at markedly different timescales compared to the magnetization quenching. The diffraction pattern distortion shows a threshold-dependence with laser fluence, not seen for magnetization quenching, consistent with a picture of domain wall motion with pinning sites. Supported by simulations, we show that a speed of $\approx$ 66 km/s for highly curved domain walls can explain the experimental data. While our data agree with the prediction of extreme, non-equilibrium wall speeds locally, it differs from the details of the theory, suggesting that additional mechanisms are required to fully understand these effects., Comment: 5 pages, 4 figures; Supplemental Material: 9 pages, 9 figures

The unsteady, nonlinear magnetization dynamics induced by spin injection in an easy-plane ferromagnetic channel subject to an external magnetic field are studied analytically. Leveraging a dispersive hydrodynamic description, the Landau-Lifshitz equation is recast in terms of hydrodynamic-like variables for the magnetization's perpendicular component (spin density) and azimuthal phase gradient (fluid velocity). Spin injection acts as a moving piston that generates nonlinear, dynamical spin textures in the ferromagnetic channel with downstream quiescent spin density set by the external field. In contrast to the classical problem of a piston accelerating a compressible gas, here, variable spin injection and field lead to a rich variety of nonlinear wave phenomena from oscillatory spin shocks to solitons and rarefaction waves. A full classification of solutions is provided using nonlinear wave modulation theory by identifying two key aspects of the fluid-like dynamics: subsonic/supersonic conditions and convex/nonconvex hydrodynamic flux. Familiar waveforms from the classical piston problem such as rarefaction (expansion) waves and shocks manifest in their spin-based counterparts as smooth and highly oscillatory transitions, respectively. The spin shock is an example of a dispersive shock wave, which arises in many physical systems. New features without a gas dynamics counterpart include composite wave complexes with "contact" spin shocks and rarefactions. Magnetic supersonic conditions lead to two pronounced piston edge behaviors including a stationary soliton and an oscillatory wavetrain. These coherent wave structures have physical implications for the generation of high frequency spin waves from pulsed injection and persistent, stable stationary and/or propagating solitons in the presence of magnetic damping. The analytical results are favorably compared with numerical simulations.

Symmetry is a powerful concept in physics, but its applicability to far-from-equilibrium states is still being understood. Recent attention has focused on how far-from-equilibrium states lead to spontaneous symmetry breaking. Conversely, ultrafast optical pumping can be used to drastically change the energy landscape and quench the magnetic order parameter in magnetic systems. Here, we find a distinct symmetry-dependent ultrafast behaviour by use of ultrafast x-ray scattering from magnetic patterns with varying degrees of isotropic and anisotropic symmetry. After pumping with an optical laser, the scattered intensity reveals a radial shift exclusive to the isotropic component and exhibits a faster recovery time from quenching for the anisotropic component. These features arise even when both symmetry components are concurrently measured, suggesting a correspondence between the excitation and the magnetic order symmetry. Our results underline the importance of symmetry as a critical variable to manipulate the magnetic order in the ultrafast regime.

In the last decade, nanoscale resistive devices with memory have been the subject of intense study because of their possible use in brain-inspired computing. However, operational endurance is one of the limiting factors in the adoption of such technology. For this reason, we discuss the emergence of current-induced memristance in magnetic materials, known for their durability. We show analytically and numerically that a single ferromagnetic layer can possess GHz memristance, due to a combination of two factors: a current-induced transfer of angular momentum (Zhang-Li torque) and the anisotropic magnetoresistance (AMR). We term the resulting effect the anisotropic magneto-memristance (AMM). We connect the AMM to the topology of the magnetization state, within a simple model of a 1-dimensional annulus-shaped magnetic layer, confirming the analytical results with micromagnetic simulations for permalloy. Our results open a new path towards the realization of single-layer magnetic memristive devices operating at GHz frequencies., Comment: 20 pages + supplementary material; 31 pages in total; 11 figures; Introduction: https://www.youtube.com/watch?v=WbUTtJq8FfM&feature=youtu.be

Magnetic topological defects are energetically stable spin configurations characterized by symmetry breaking. Vortices and skyrmions are two well-known examples of 2D spin textures that have been actively studied for both fundamental interest and practical applications. However, experimental evidence of the 3D spin textures has been largely indirect or qualitative to date, due to the difficulty of quantitively characterizing them within nanoscale volumes. Here, we develop soft x-ray vector ptychography to quantitatively image the 3D magnetization vector field in a frustrated superlattice with 10 nm spatial resolution. By applying homotopy theory to the experimental data, we quantify the topological charge of hedgehogs and anti-hedgehogs as emergent magnetic monopoles and probe their interactions inside the frustrated superlattice. We also directly observe virtual hedgehogs and anti-hedgehogs created by magnetically inert voids. We expect that this new quantitative imaging method will open the door to study 3D topological spin textures in a broad class of magnetic materials. Our work also demonstrates that magnetically frustrated superlattices could be used as a new platform to investigate hedgehog interactions and dynamics and to exploit optimized geometries for information storage and transport applications.

Unsteady nonlinear magnetization dynamics are studied in an easy-plane ferromagnetic channel subject to spin injection at one edge. The model Landau-Lifshitz equation is known to support steady-state solutions, termed dissipative exchange flows (DEFs) or spin superfluids. In this work, by means of numerical simulations and theoretical analysis, we provide a full description of the injection-induced, large-amplitude, nonlinear magnetization dynamics up to the steady state. The dynamics prior to reaching steady state are driven by spin injection, the perpendicular applied magnetic field, the exchange interaction, and local demagnetizing fields. We show that the dynamics take well-defined profiles in the form of rarefaction waves (RW), dispersive shock waves (DSW), and solitons. The combination of these building blocks depends on the interplay between the spin injection strength and the applied magnetic field. A solitonic feature at the injection boundary, signaling the onset of the magnetic "supersonic" condition at the injection edge, rapidly develops and persists in the steady-state configuration of a contact soliton DEF. We also demonstrate the existence of sustained soliton-train dynamics in long time that can only arise in a nonzero applied magnetic field scenario. The dynamical evolution of spin-injection-induced magnetization dynamics presented here may help guide observations in long-distance spin transport experiments., Comment: 5 pages, 4 figures, INTERMAG 2021 conference-related paper

Ferromagnetic channels subject to spin injection at one extremum sustain long-range coherent textures that carry spin currents known as dissipative exchange flows (DEFs). In the weak injection regime, spin currents carried by DEFs decay algebraically and extend through the length of the channel, a regime known as spin superfluidity. Similar to fluids, these structures are prone to phase-slips that manifest as vortex-antivortex pairs. Here, we numerically study vortex shedding from DEFs excited in a magnetic nanowire with a physical obstacle. Using micromagnetic simulations, we find regimes of laminar flow and vortex shedding as a function of obstacle position tunable by the and spin injection sign and magnitude. Vortex-antivortex pairs translate forward (VF regime) or backward (VB regime) with respect to the detector's extremum, resulting in well-defined spectral features. Qualitatively similar results are obtained when temperature, anisotropy, and weak non-local dipole fields are included in the simulations. These results provide clear features associated with DEFs that may be detected experimentally in devices with nominally identical boundary conditions. Furthermore, our results suggest that obstacles can be considered as DEF control gates, opening an avenue to manipulate DEFs via physical defects.

Ultrafast optical pumping of spatially nonuniform magnetic textures is known to induce far-from-equilibrium spin transport effects. Here, we use ultrafast x-ray diffraction with unprecedented dynamic range to study the laser-induced dynamics of labyrinth domain networks in ferromagnetic CoFe/Ni multilayers. We detected azimuthally isotropic, odd order, magnetic diffraction rings up to 5th order. The amplitudes of all three diffraction rings quench to different degrees within 1.6 ps. In addition, all three of the detected diffraction rings both broaden by 15% and radially contract by 6% during the quench process. We are able to rigorously quantify a 31% ultrafast broadening of the domain walls via Fourier analysis of the order-dependent quenching of the three detected diffraction rings. The broadening of the diffraction rings is interpreted as a reduction in the domain coherence length, but the shift in the ring radius, while unambiguous in its occurrence, remains unexplained. In particular, we demonstrate that a radial shift explained by domain wall broadening can be ruled out. With the unprecedented dynamic range of our data, our results provide convincing evidence that labyrinth domain structures are spatially perturbed at ultrafast speeds under far-from-equilibrium conditions, albeit the mechanism inducing the perturbations remains yet to be clarified.

Over the past few years, the study of magnetization dynamics in artificial spin ices has become a vibrant field of study. Artificial spin ices are ensembles of geometrically arranged, interacting magnetic nanoislands, which display frustration by design. These were initially created to mimic the behavior in rare earth pyrochlore materials and to study emergent behavior and frustration using two-dimensional magnetic measurement techniques. Recently, it has become clear that it is possible to create artificial spin ices, which can potentially be used as functional materials. In this Perspective, we review the resonant behavior of spin ices (which is in the GHz frequency range), focusing on their potential application as magnonic crystals. In magnonic crystals, spin waves are functionalized for logic applications by means of band structure engineering. While it has been established that artificial spin ices can possess rich mode spectra, the applicability of spin ices to create magnonic crystals hinges upon their reconfigurability. Consequently, we describe recent work aiming to develop techniques and create geometries allowing full reconfigurability of the spin ice magnetic state. We also discuss experimental, theoretical, and numerical methods for determining the spectral response of artificial spin ices, and give an outlook on new directions for reconfigurable spin ices., Comment: 15 pages, 7 figures

Artificial spin ices are ensembles of geometrically-arranged, interacting nanomagnets which have shown promising potential for the realization of reconfigurable magnonic crystals. Such systems allow for the manipulation of spin waves on the nanoscale and their potential use as information carriers. However, there are presently two general obstacles to the realization of artificial spin ice-based magnonic crystals: the magnetic state of artificial spin ices is difficult to reconfigure and the magnetostatic interactions between the nanoislands are often weak, preventing mode coupling. We demonstrate, using micromagnetic modeling, that coupling a reconfigurable artificial spin ice geometry made of weakly interacting nanomagnets to a soft magnetic underlayer creates a complex system exhibiting dynamically coupled modes. These give rise to spin wave channels in the underlayer at well-defined frequencies, based on the artificial spin ice magnetic state, which can be reconfigured. These findings open the door to the realization of reconfigurable magnonic crystals with potential applications for data transport and processing in magnonic-based logic architectures., Comment: 8 figures

The field of spin hydrodynamics aims to describe magnetization dynamics from a fluid perspective. For ferromagnetic materials, there is an exact mapping between the Landau-Lifshitz equation and a set of dispersive hydrodynamic equations. This analogy provides ample opportunities to explore novel magnetization dynamics and magnetization states that can lead to applications relying entirely upon magnetic materials, for example, long-distance transport of information. This article provides an overview of the theoretical foundations of spin hydrodynamics and their physical interpretation in the context of spin transport. We discuss other proposed applications for spin hydrodynamics as well as our view on challenges and future research directions., Comment: 19 pages, 4 figures, preprint

Recent theoretical studies have demonstrated the possibility to excite and sustain noncollinear magnetization states in ferromagnetic nanowires. The resulting state is referred to as a spin-density wave (SDW). SDWs can be interpreted as hydrodynamic states with a constant fluid density and fluid velocity in systems with easy-plane anisotropy. Here, we consider the effect of the nonlocal dipole field arising from the finite thickness of magnetic thin films on the spatial profile of the SDW and on the associated magnon dispersion. Utilizing a hydrodynamic formulation of the Larmor torque equation, it is found that the nonlocal dipole field modulates the fluid velocity. Such a modulation induces a magnonic band structure unlike the typical dispersion relation for magnons on uniform magnetization. The analytical results are validated by micromagnetic simulations. Band gaps on the order of GHz are numerically observed to depend on the SDW fluid velocity and film thickness for realistic material parameters. The presented results suggest that SDWs can find applications as reconfigurable magnonic crystals., Comment: 5 pages, 5 figures

Nonlocal compensation of magnetic damping by spin injection has been theoretically shown to establish dynamic, noncollinear magnetization states that carry spin currents over micrometer distances. Such states can be generically referred to as dissipative exchange flows (DEFs) because spatially diffusing spin currents are established by the mutual exchange torque exerted by neighboring spins. Analytical studies to date have been limited to the weak spin injection assumption whereby the equation of motion for the magnetization is mapped to hydrodynamic equations describing spin flow and then linearized. Here, we analytically and numerically study easy-plane ferromagnetic channels subject to spin injection of arbitrary strength at one extremum under a unified hydrodynamic framework. We find that DEFs generally exhibit a nonlinear profile along the channel accompanied by a nonlinear frequency tuneability. At large injection strengths, we fully characterize a novel magnetization state we call a contact-soliton DEF (CS-DEF) composed of a stationary soliton at the injection site, which smoothly transitions into a DEF and exhibits a negative frequency tuneability. The transition between a DEF and a CS-DEF occurs at the maximum precessional frequency and coincides with the Landau criterion: a subsonic to supersonic flow transition. Leveraging the hydraulic-electrical analogy, the current-voltage characteristics of a nonlinear DEF circuit are presented. Micromagnetic simulations of nanowires that include magnetocrystalline anisotropy and non-local dipole fields are in qualitative agreement with the analytical results. The magnetization states found here along with their characteristic profile and spectral features provide quantitative guidelines to pursue an experimental demonstration of DEFs in ferromagnetic materials and establishes a unified description for long-distance spin transport.

Stripe domains are narrow, elongated, reversed regions that exist in magnetic materials with perpendicular magnetic anisotropy. Stripe domains appear as a pair of domain walls that can exhibit topology with a nonzero chirality. Recent experimental and numerical investigations identify an instability of stripe domains in the long direction as a means of nucleating isolated magnetic skyrmions. Here, the onset and nonlinear evolution of transverse instabilities for a dynamic stripe domain known as the bion stripe are investigated. Both non-topological and topological variants of the bion stripe are shown to exhibit a long-wavelength transverse instability with different characteristic features. In the former, small transverse variations in the stripe's width lead to a neck instability that eventually pinches the non-topological stripe into a chain of two-dimensional breathers composed of droplet soliton pairs. In the latter case, small variations in the stripe's center results in a snake instability whose topological structure leads to the nucleation of dynamic magnetic skyrmions and antiskyrmions as well as perimeter-modulated droplets. Quantitative, analytical predictions for both the early, linear evolution and the long-time, nonlinear evolution are achieved using an averaged Lagrangian approach that incorporates both exchange (dispersion) and anisotropy (nonlinearity). The method of analysis is general and can be applied to other filamentary structures., Comment: 8 figures, 13 pages

Planar ferromagnetic channels have been shown to theoretically support a long-range ordered and coherently precessing state where the balance between local spin injection at one edge and damping along the channel establishes a dissipative exchange flow, sometimes referred to as a spin superfluid. However, realistic materials exhibit in-plane anisotropy, which breaks the axial symmetry assumed in current theoretical models. Here, we study dissipative exchange flows in a ferromagnet with in-plane anisotropy from a dispersive hydrodynamic perspective. Through the analysis of a boundary value problem for a damped sine-Gordon equation, dissipative exchange flows in a ferromagnetic channel can be excited above a spin current threshold that depends on material parameters and the length of the channel. Symmetry-broken dissipative exchange flows display harmonic overtones that redshift the fundamental precessional frequency and lead to a reduced spin pumping efficiency when compared to their symmetric counterpart. Micromagnetic simulations are used to verify that the analytical results are qualitatively accurate, even in the presence of nonlocal dipole fields. Simulations also confirm that dissipative exchange flows can be driven by spin transfer torque in a finite-sized region. These results delineate the important material parameters that must be optimized for the excitation of dissipative exchange flows in realistic systems., Comment: 20 pages, 5 figures

Recent experiments on spin torque oscillators have revealed interactions between multiple magnetodynamic modes, including mode-coexistence, mode-hopping, and temperature-driven cross-over between modes. Initial multimode theory has indicated that a linear coupling between several dominant modes, arising from the interaction of the subdynamic system with a magnon bath, plays an essential role in the generation of various multimode behaviors, such as mode hopping and mode coexistence. In this work, we derive a set of rate equations to describe the dynamics of coupled magnetodynamic modes in a nano-contact spin torque oscillator. Expressions for both linear and nonlinear coupling terms are obtained, which allow us to analyze the dependence of the coupled dynamic behaviors of modes on external experimental conditions as well as intrinsic magnetic properties. For a minimal two-mode system, we further map the energy and phase difference of the two modes onto a two-dimensional phase space, and demonstrate in the phase portraits, how the manifolds of periodic orbits and fixed points vary with external magnetic field as well as with temperature., Comment: 13 pages, 8 figures; 2 figures (Figs.5 & 6) corrected and redrawn; 2 new figures (Figs.7 & 8) added; Accepted by Physical Review Applied

Systems that exhibit topologically protected edge states are interesting both from a fundamental point of view as well as for potential applications, the latter because of the absence of back-scattering and robustness to perturbations. It is desirable to be able to control and manipulate such edge states. Here, we show that artificial square ices can incorporate both features: an interfacial Dzyaloshinksii-Moriya gives rise to topologically non-trivial magnon bands, and the equilibrium state of the spin ice is reconfigurable with different configurations having different magnon dispersions and topology. The topology is found to develop as odd-symmetry bulk and edge magnon bands approach each other, so that constructive band inversion occurs in reciprocal space. Our results show that topologically protected bands are supported in square spin ices., Comment: 27 pages, 6 figures

Magnetization dynamics in thin film ferromagnets can be studied using a dispersive hydrodynamic formulation. The equations describing the magnetodynamics map to a compressible fluid with broken Galilean invariance parametrized by the longitudinal spin density and a magnetic analog of the fluid velocity that define spin-density waves. A direct consequence of these equations is the determination of a magnetic Mach number. Micromagnetic simulations reveal nucleation of nonlinear structures from an impenetrable object realized by an applied magnetic field spot or a defect. In this work, micromagnetic simulations demonstrate vortex-antivortex pair nucleation from an obstacle. Their interaction establishes either ordered or irregular vortex-antivortex complexes. Furthermore, when the magnetic Mach number exceeds unity (supersonic flow), a Mach cone and periodic wavefronts are observed, which can be well-described by solutions of the steady, linearized equations. These results are reminiscent of theoretical and experimental observations in Bose-Einstein condensates, and further supports the analogy between the magnetodynamics of a thin film ferromagnet and compressible fluids. The nucleation of nonlinear structures and vortex-antivortex complexes using this approach enables the study of their interactions and effects on the stability of spin-density waves., Comment: 23 pages, 7 figures

Microwave magnetodynamics in ferromagnets are often studied in the small-amplitude or weakly nonlinear regime corresponding to modulations of a well-defined magnetic state. However, strongly nonlinear regimes, where the aforementioned approximations are not applicable, have become experimentally accessible. By re-interpreting the governing Landau-Lifshitz equation of motion, we derive an exact set of equations of dispersive hydrodynamic form that are amenable to analytical study even when full nonlinearity and exchange dispersion are included. The resulting equations are shown to, in general, break Galilean invariance. A magnetic Mach number is obtained as a function of static and moving reference frames. The simplest class of solutions are termed uniform hydrodynamic states (UHSs), which exhibit fluid-like behavior including laminar flow at subsonic speeds and the formation of a Mach cone and wave-fronts at supersonic speeds. A regime of modulational instability is also possible, where the UHS is violently unstable. The hydrodynamic interpr, Comment: 20 pages, 5 figures

This work provides a detailed investigation of the measured in-plane field-swept homodyne-detected ferromagnetic resonance (FMR) spectra of an extended Co/Cu/NiFe pseudo spin valve stack using a nanocontact (NC) geometry. The magnetodynamics are generated by a pulse-modulated microwave current and the resulting rectified dc mixing voltage, which appears across the NC at resonance, is detected using a lock-in amplifier. Most notably, we find that the measured spectra of the NiFe layer are composite in nature and highly asymmetric, consistent with the broadband excitation of multiple modes. Additionally, the data must be fit with two Lorentzian functions in order to extract a reasonable value for the Gilbert damping of the NiFe. Aided by micromagnetic simulations, we conclude that (i) for in-plane fields the rf Oersted field in the vicinity of the NC plays the dominant role in generating the observed spectra, (ii) in addition to the FMR mode, exchange dominated spin waves are also generated, and (iii) the NC diameter sets the mean wavevector of the exchange dominated spin wave, in good agreement with the dispersion relation., Comment: 7 pages, 7 figures

Artificial square spin ices are structures composed of magnetic elements arranged on a geometrically frustrated lattice and located on the sites of a two-dimensional square lattice, such that there are four interacting magnetic elements at each vertex. Using a semi-analytical approach, we show that square spin ices exhibit a rich spin wave band structure that is tunable both by external magnetic fields and the configuration of individual elements. Internal degrees of freedom can give rise to equilibrium states with bent magnetization at the edges leading to characteristic excitations; in the presence of magnetostatic interactions these form separate bands analogous to impurity bands in semiconductors. Full-scale micromagnetic simulations corroborate our semi-analytical approach. Our results show that artificial square spin ices can be viewed as reconfigurable and tunable magnonic crystals that can be used as metamaterials for spin-wave-based applications at the nanoscale., Comment: 20 pages, 6 figures

The rapid development of the magnetic tunnel junction (MTJ) spin torque oscillator (STO) technology demands an analytical model to enable building MTJ STO-based circuits and systems so as to evaluate and utilize MTJ STOs in various applications. In Part I of this paper, an analytical model based on the macrospin approximation, has been introduced and verified by comparing it with the measurements of three different MTJ STOs. In Part II, the full Verilog-A implementation of the proposed model is presented. To achieve a reliable model, an approach to reproduce the phase noise generated by the MTJ STO has been proposed and successfully employed. The implemented model yields a time domain signal, which retains the characteristics of operating frequency, linewidth, oscillation amplitude and DC operating point, with respect to the magnetic field and applied DC current. The Verilog-A implementation is verified against the analytical model, providing equivalent device characteristics for the full range of biasing conditions. Furthermore, a system that includes an MTJ STO and CMOS RF circuits is simulated to validate the proposed model for system- and circuit-level designs. The simulation results demonstrate that the proposed model opens the possibility to explore STO technology in a wide range of applications., Comment: 7 pages, 7 figures

Magnetic tunnel junction (MTJ) spin torque oscillators (STO) have shown the potential to be used in a wide range of microwave and sensing applications. To evaluate potential uses of MTJ STO technology in various applications, an analytical model that can capture MTJ STO's characteristics, while enabling system- and circuit-level designs, is of great importance. An analytical model based on macrospin approximation is necessary for these designs since it allows implementation in hardware description languages. This paper presents a new macrospin-based, comprehensive and compact MTJ STO model, which can be used for various MTJ STOs to estimate the performance of MTJ STOs together with their application-specific integrated circuits. To adequately present the complete model, this paper is divided into two parts. In Part I, the analytical model is introduced and verified by comparing it against measured data of three different MTJ STOs, varying the angle and magnitude of the magnetic field, as well as the DC biasing current. The proposed analytical model is suitable for being implemented in Verilog-A and used for efficient simulations at device-, circuit- and system-levels. In Part II, the full Verilog-A implementation of the analytical model with accurate phase noise generation is presented and verified by simulations., Comment: 8 pages, 4 figures

Experiments on spin torque oscillators commonly observe multi-mode signals. Recent theoretical works have ascribed the multi-mode signal generation to coupling between energy-separated spin wave modes. Here, we analyze in detail the dynamics generated by such mode coupling. We show analytically that the mode-hopping dynamics broaden the generation linewidth and makes it generally well described by a Voigt lineshape. Furthermore, we show that the mode-hopping contribution to the linewidth can dominate in which case it provides a direct measure of the mode-hopping rate. Due to the thermal drive of mode-hopping events, the mode-hopping rate also provides information on the energy barrier separating modes and temperature-dependent linewidth broadening. Our results are in good agreement with experiments, revealing the physical mechanism behind the linewidth broadening in multi-mode spin torque oscillators.

Magnetic dissipative droplets are localized, strongly nonlinear dynamical modes excited in nano-contact spin valves with perpendicular magnetic anisotropy. These modes find potential application in nanoscale structures for magnetic storage and computation, but dissipative droplet studies have so far been limited to extended thin films. Here, numerical and asymptotic analyses are used to demonstrate the existence and properties of novel solitons in confined structures. As a nanowire's width is decreased with a nano-contact of fixed size at its center, the observed modes undergo transitions from a fully localized two-dimensional droplet into a two-dimensional droplet edge mode and then a pulsating one-dimensional droplet. These solitons are interpreted as dissipative versions of classical, conservative solitons, allowing for an analytical description of the modes and the mechanisms of bifurcation. The presented results open up new possibilities for the study of low-dimensional solitons and droplet applications in nanostructures.

It has been argued that if multiple spin wave modes are competing for the same centrally located energy source, as in a nanocontact spin torque oscillator, that only one mode should survive in the steady state. Here, the experimental conditions necessary for mode coexistence are explored. Mode coexistence is facilitated by the local field asymmetries induced by the spatially inhomogeneous Oersted field, which leads to a physical separation of the modes, and is further promoted by spin wave localization at reduced applied field angles. Finally, both simulation and experiment reveal a low frequency signal consistent with the intermodulation of two coexistent modes., Comment: 20 pages (including supplementary material), 6 figures

Modulation of Spin Torque Oscillators (STOs) is investigated by analytically solving the time-dependent coupled equations of an auto-oscillator. A Fourier series solution is proposed, leading to the coefficients being determined with a linear set of equations, from which a Nonlinear Amplitude and Frequency Modulation (NFAM) scheme is obtained. In this framework, the NFAM features are related to the intrinsic STO parameters, revealing a frequency-dependence of the harmonic-dependent modulation index that allows a modulation bandwidth to be defined for these devices. The presented results expose a rich parameter space, where the modulation and the STO's operation conditions define the observed modulation features. The Fourier-series representation of the time signal is suitable for studying periodic perturbations on the auto-oscillator equation.

We demonstrate that the transient non-autonomous dynamics of a spin torque nano-oscillator (STNO) under a radio-frequency (rf) driving signal is qualitatively different from the dynamics described by the Adler model. If the external rf current $I_{rf}$ is larger than a certain critical value $I_{cr}$ (determined by the STNO bias current and damping) strong oscillations of the STNO power and phase develop in the transient regime. The frequency of these oscillations increases with $I_{rf}$ as $\propto\sqrt{I_{rf} - I_{cr}}$ and can reach several GHz, whereas the damping rate of the oscillations is almost independent of $I_{rf}$. This oscillatory transient dynamics is caused by the strong STNO nonlinearity and should be taken into account in most STNO rf applications., Comment: 4 page, 3 figures