R. Zivieri

A strategy for the design of skyrmion racetrack memories.

Magnetic storage based on racetrack memory is very promising for the design of ultra-dense, low-cost and low-power storage technology. Information can be coded in a magnetic region between two domain walls or, as predicted recently, in topological magnetic objects known as skyrmions. Here, we show the technological advantages and limitations of using Bloch and Néel skyrmions manipulated by spin current generated within the ferromagnet or via the spin-Hall effect arising from a non-magnetic heavy metal underlayer. We found that the Néel skyrmion moved by the spin-Hall effect is a very promising strategy for technological implementation of the next generation of skyrmion racetrack memories (zero field, high thermal stability, and ultra-dense storage). We employed micromagnetics reinforced with an analytical formulation of skyrmion dynamics that we developed from the Thiele equation. We identified that the excitation, at high currents, of a breathing mode of the skyrmion limits the maximal velocity of the memory.

Combined Frequency-Amplitude Nonlinear Modulation: Theory and Applications

In this paper, we formulate a generalized theoretical model to describe the nonlinear dynamics observed in combined frequency-amplitude modulators whose characteristic parameters exhibit a nonlinear dependence on the input modulating signal. The derived analytical solution may give a satisfactory explanation of recent laboratory observations on magnetic spin-transfer oscillators and fully agrees with results of micromagnetic calculations. Since the theory has been developed independently of the mechanism causing the nonlinearities, it may encompass the description of modulation processes of any physical nature, a promising feature for potential applications in the field of communication systems.

Metamaterial Properties of One-Dimensional and Two-Dimensional Magnonic Crystals

Abstract In this chapter, some recent results on the metamaterial properties of one- and two-dimensional magnonic crystals are presented. These results were obtained by means of a recently developed micromagnetic approach, the so-called Hamiltonian-based dynamical matrix method, formulated for studying the dynamical properties of isolated magnetic particles and generalized to periodic magnetic systems. The method is essentially based on an eigensystem problem according to which the equations of motion are written in terms of the second derivates of the energy density evaluated at the equilibrium. The formalism of the differential scattering cross-section associated with each collective mode and generalized to periodic systems is also outlined. Applications of the micromagnetic formalism to one- and two-dimensional magnonic crystals are presented and the metamaterial properties found according to the method are illustrated and discussed. First, metamaterial properties of one-dimensional magnonic crystals represented by chains of rectangular dots are discussed by showing the calculated frequency dispersion for two scattering geometries, namely with the external magnetic field either perpendicular or parallel to the Bloch wave vector in the plane of the system. As a further example of metamaterial properties typical of chains of rectangular dots, calculated and measured bandwidths and band gaps between adjacent collective modes are shown and discussed. Second, application of the Hamiltonian-based dynamical matrix method is reviewed for presenting the most important metamaterial properties in two prototypes of two-dimensional magnonic crystals: (a) arrays of close-packed disks of Permalloy coupled via dipolar magnetic interaction and (b) arrays of circular antidots embedded into a Permalloy ferromagnetic film. For the array of disks, the calculation of the band structure and the band behavior is discussed in terms of an effective wave vector. For the arrays of AD, frequency dispersion of extended and localized modes are presented and the opening of band gaps is explained by studying the inhomogeneity of the internal field according to a recently developed analytical model. A few examples of micromagnetic calculations performed on ferromagnetic arrays of antidots with different periodicities are presented.

Magnetic properties of submicron circular permalloy dots

Both the static and the dynamical magnetic properties of a square array of circular permalloy dots, characterized by a magnetic vortex configuration of the magnetization, have been investigated by means of magneto-optical Kerr effect and Brillouin light scattering (BLS) from thermally excited spin waves. The measured hysteresis loop can be satisfactorily reproduced by micromagnetic simulations, showing that the vortex configuration is stable over a wide range of applied field. The high frequency response of the dots was analyzed by BLS measurements performed under an external magnetic field intensity large enough to uniformly magnetize the dots. Evidence is given of a marked discretization of the spin wave spectrum with respect to the case of the continuous permalloy film, where only one peak, corresponding to the Damon-Eshbach mode, was detected. The experimental frequencies have been compared to those calculated using a recently developed analytical model for a flat uniformly magnetized cylindrical dot.

Spin wave modes in submicron cylindrical dots

The dynamic properties of a squared array of cylindrical Ni81Fe19 dots with thickness L=50 nm, radius R=100 nm, and separation 2R have been investigated by Brillouin light scattering. The sample was prepared by means of electron-beam lithography and evaporation in ultrahigh vacuum. The lateral confinement of spin waves within each dot causes a marked discretization of the spin wave spectrum. Several discrete peaks were measured in the saturated state as a function of both the incidence angle of light and the applied magnetic field. The detected modes are classified as surface dipolar and bulk magnetostatic modes at frequencies higher and lower than the Kittel uniform mode, respectively.

Topological, non-topological and instanton droplets driven by spin-transfer torque in materials with perpendicular magnetic anisotropy and Dzyaloshinskii-Moriya Interaction.

The interfacial Dzyaloshinskii–Moriya Interaction can modify the topology of droplets excited by a localized spin-polarized current. Here, we show that, in addition to the stationary droplet excitations with skyrmion number either one (topological) or zero (non-topological), there exists, for a fixed current, an excited mode with a non-stationary time behavior. We call this mode “instanton droplet”, which is characterized by time domain transitions of the skyrmion number. These transitions are coupled to an emission of incoherent spin-waves that can be observed in the frequency domain as a source of noise. Our results are interesting from a fundamental point of view to study spin-wave emissions due to a topological transition in current-driven systems, and could open the route for experiments based on magnetoresistance effect for the design of a further generation of nanoscale microwave oscillators.

Lagrangian formulation of the linear autonomous magnetization dynamics in spin-torque auto-oscillators

Abstract A Lagrangian formalism is used to find steady-state solution of the Landau–Lifshitz–Gilbert–Slonczewski equation corresponding to the linear autonomous dynamics of a magnetic auto-oscillatory system subject to the action of a spin-polarized electric current. In such a system, two concurrent dissipative mechanisms, arising from the positive intrinsic dissipation and the negative current-induced one, take place simultaneously and make the excitation of a steady precessional motion of the magnetization vector conceivable. The proposed formulation leads to the definition of a complex generalized non-Hermitian Eigenvalue problem, both in the case of a macrospin model and in the more general case of an ensemble of magnetic particles interacting each other through magnetostatic and exchange interactions. This method allows to identify the spin-wave normal modes which become unstable in the presence of the two competing dissipative contributions and provides an accurate estimation of the value of the excitation threshold current.

Spin-Hall nano-oscillator with oblique magnetization and Dzyaloshinskii-Moriya interaction as generator of skyrmions and nonreciprocal spin-waves

Spin-Hall oscillators (SHO) are promising sources of spin-wave signals for magnonics applications, and can serve as building blocks for magnonic logic in ultralow power computation devices. Thin magnetic layers used as “free” layers in SHO are in contact with heavy metals having large spin-orbital interaction, and, therefore, could be subject to the spin-Hall effect (SHE) and the interfacial Dzyaloshinskii-Moriya interaction (i-DMI), which may lead to the nonreciprocity of the excited spin waves and other unusual effects. Here, we analytically and micromagnetically study magnetization dynamics excited in an SHO with oblique magnetization when the SHE and i-DMI act simultaneously. Our key results are: (i) excitation of nonreciprocal spin-waves propagating perpendicularly to the in-plane projection of the static magnetization; (ii) skyrmions generation by pure spin-current; (iii) excitation of a new spin-wave mode with a spiral spatial profile originating from a gyrotropic rotation of a dynamical skyrmion. These results demonstrate that SHOs can be used as generators of magnetic skyrmions and different types of propagating spin-waves for magnetic data storage and signal processing applications.

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

The behavior of collective spin excitations in chains of rectangular NiFe dots is studied as a function of interdot separation. Dots have thickness of 40 nm and lateral dimensions of 715 × 450 nm2. They are put side by side along the major axis and the interdot separation is varied in the range 55-625 nm. Brillouin light scattering experiments have been performed at normal incidence (exchanged wave vector q = 0) and with the external magnetic field applied along the chain length. A satisfactory interpretation of the experimental data is achieved by magnonic bands calculations based on the dynamical matrix method. Such calculations have been performed at both the center and the border of the first Brillouin zone, in the case of Bloch wave vector q parallel to the applied field. In this way we can predict the amplitude of modes frequency oscillation (magnonic band), which is an important property to identify the behavior of a one-dimensional magnonic meta-material.

Localized spin modes in ferromagnetic cylindrical dots with in-plane magnetization

A study of spin localized excitations in ferromagnetic tangentially magnetized dots of cylindrical shape and of nanometric size is presented. A recently formulated variational theory permits us to study the most representative localized spin modes of the spectrum. One of these, the fundamental mode, is mainly localized in the central part of the dot endfaces and is analogous to the Kittel uniform mode in ellipsoids. We also investigate the dynamical properties of spin modes localized in the lateral part of the dot endfaces along the direction of the applied magnetic field, studying the dependence of their localization on the variational parameter and the applied magnetic field. Finally, a comparison of the calculated frequencies of some of these localized modes with available experimental data is performed.