Hamiltonian and Lagrangian Dynamical Matrix Approaches Applied to Magnetic Nanostructures

Two micromagnetic tools to study the spin dynamics are reviewed. Both approaches are based upon the so-called dynamical matrix method, a hybrid micromagnetic framework used to investigate the spin-wave normal modes of confined magnetic systems. The approach which was formulated first is the Hamiltonian-based dynamical matrix method. This method, used to investigate dynamic magnetic properties of conservative systems, was originally developed for studying spin excitations in isolated magnetic nanoparticles and it has been recently generalized to study the dynamics of periodic magnetic nanoparticles. The other one, the Lagrangian-based dynamical matrix method, was formulated as an extension of the previous one in order to include also dissipative effects. Such dissipative phenomena are associated not only to intrinsic but also to extrinsic damping caused by injection of a spin current in the form of spin-transfer torque. This method is very accurate in identifying spin modes that become unstable under the action of a spin current. The analytical development of the system of the linearized equations of motion leads to a complex generalized Hermitian eigenvalue problem in the Hamiltonian dynamical matrix method and to a non-Hermitian one in the Lagrangian approach. In both cases, such systems have to be solved numerically

On the Travelling Wave Solution for the Current-Driven Steady Domain Wall Motion in Magnetic Nanostrips under the Influence of Rashba Field

Spin-orbit Rashba effect applies a torque on the magnetization of a ferromagnetic nanostrip in the case of structural inversion asymmetry, also affecting the steady domain wall motion induced by a spin-polarized current. This influence is here analytically studied in the framework of the extended Landau-Lifshitz-Gilbert equation, including the Rashba effect as an additive term of the effective field. Results of previous micromagnetic simulations and experiments have shown that this field yields an increased value of the Walker breakdown current together with an enlargement of the domain wall width. In order to analytically describe these results, the standard travelling wave ansatz for the steady domain wall motion is here adopted. Results of our investigations reveal the impossibility to reproduce, at the same time, the previous features and suggest the need of a more sophisticated model whose development requires, in turn, additional information to be extracted from ad hoc micromagnetic simulations

Curved domain walls dynamics driven by magnetic field and electric current in hard ferromagnets

Abstract The propagation of curved domain walls in hard ferromagnetic materials is studied by applying a reductive perturbation method to the generalized Landau–Lifshitz–Gilbert equation. The extended model herein considered explicitly takes into account the effects of a spin-polarized current as well as those arising from a nonlinear dissipation. Under the assumption of steady regime of propagation, the domain wall velocity is derived as a function of the domain wall curvature, the nonlinear damping coefficient, the magnetic field and the electric current. Threshold and Walker-like breakdown conditions for the external sources are also determined. The analytical results are evaluated numerically for different domain wall surfaces (planes, cylinders and spheres) and their physical implications are discussed

Power and linewidth of propagating and localized modes in nanocontact spin-torque oscillators

Integrated power and linewidth of a propagating and a self-localized spin wave modes excited by spin-polarized current in an obliquely magnetized magnetic nanocontact are studied experimentally as functions of the angle $\theta_e$ between the external bias magnetic field and the nanocontact plane. It is found that the power of the propagating mode monotonically increases with $\theta_e$, while the power of the self-localized mode has a broad maximum near $\theta_e = 40$ deg, and exponentially vanishes near the critical angle $\theta_e = 58$ deg, at which the localized mode disappears. The linewidth of the propagating mode in the interval of angles $58<\theta_e<90$ deg, where only this mode is excited, is adequtely described by the existing theory, while in the angular interval where both modes can exist the observed linewidth of both modes is substantially broadened due to the telegraph switching between the modes. Numetical simulations and an approximate analytical model give good semi-quantitative description of the observed results.Comment: 8 pages, 6 figure

Oscillatory periodic pattern dynamics in hyperbolic reaction-advection-diffusion models

In this work we consider a quite general class of two-species hyperbolic reaction-advection-diffusion system with the main aim of elucidating the role played by inertial effects in the dynamics of oscillatory periodic patterns. To this aim, first, we use linear stability analysis techniques to deduce the conditions under which wave (or oscillatory Turing) instability takes place. Then, we apply multiple-scale weakly nonlinear analysis to determine the equation which rules the spatiotemporal evolution of pattern amplitude close to criticality. This investigation leads to a cubic complex Ginzburg-Landau (CCGL) equation which, owing to the functional dependence of the coefficients here involved on the inertial times, reveals some intriguing consequences. To show in detail the richness of such a scenario, we present, as an illustrative example, the pattern dynamics occurring in the hyperbolic generalization of the extended Klausmeier model. This is a simple two-species model used to describe the migration of vegetation stripes along the hillslope of semiarid environments. By means of a thorough comparison between analytical predictions and numerical simulations, we show that inertia, apart from enlarging the region of the parameter plane where wave instability occurs, may also modulate the key features of the coherent structures, solution of the CCGL equation. In particular, it is proven that inertial effects play a role, not only during transient regime from the spatially-homogeneous steady state toward the patterned state, but also in altering the amplitude, the wavelength, the angular frequency, and even the stability of the phase-winding solutions

Low-Dimensional Magnetic Systems

The interest in the nanoscale properties of low-dimensional magnetic systems has grown exponentially during the last decades and has attracted the attention of both experimentalists and theorists. The state of the art of these investigations has indeed allowed to give valuable insights into the underlying physics of complex magnetization dynamics driven by magnetic fields, electric currents and thermal effects. At the same time, such studies have found, in relatively short times, several applications at industrial level in the field of spintronics and magnonics as magnetic memories, microwave oscillators, modulators, sensors, logic gates, diodes and transistors. The goal of this special issue is to offer a variety of recent developments on this topic by gathering contributions arising from several specialists in the field of nanomagnetism. The strength of this issue lies indeed on its "variety": the properties of these systems are, in fact, investigated from the viewpoint of physicists, engineers and mathematicians. Also, the issue encloses studies carried out at both mesoscopic and atomic scales, as well as results of both theoretical approaches (analytical, numerical and, in some cases, even "hybrid") and experimental observations. The covered topics range from the micromagnetic modeling of domain wall motion, dynamics of vortex structures, phase-locking phenomena in spintronic oscillators, experimental techniques for realizing heterostructures based on magnon-induced spin transfer torque, band structure and exchange field in the Landau-Lifshitz equation for magnonic crystals, gap and gapless structures in fractional quantum Hall effect, semiclassical description of anisotropic magnets and classical critical behaviour of Heisenberg ferromagnets. More specifically, within the subject dealing with domain walls, for example, the structure of complex cross-tie/vortex wall structures in soft films has been studied in detail by using micromagnetic simulations whereas the influence of the Rashba spin-orbit coupling on the current-induced dynamics has been investigated analytically. Regarding the exchange interaction governing the dynamics in magnonic crystals, a full analytical calculation of the exchange field acting on spin-wave dynamics from themicroscopic Heisenbergmodel has been performed. Attention has been also devoted to the study of thermodynamics in the case of classical planar ferromagnets close to the zero-temperature critical point. Two reviews are also included in this special issue. The first one deals with two hybrid micromagnetic tools, based on Hamiltonian and Lagrangian approaches, to model the spin-dynamics in laterally confined magnetic systems. The second one is mostly devoted to the micromagnetic analysis of static and dynamic properties of magnetic domain walls in materials exhibiting perpendicular anisotropy

Experimental evidence of self-localized and propagating spin wave modes in obliquely magnetized current-driven nanocontacts

Through detailed experimental studies of the angular dependence of spin wave excitations in nanocontact-based spin-torque oscillators, we demonstrate that two distinct spin wave modes can be excited, with different frequency, threshold currents and frequency tuneability. Using analytical theory and micromagnetic simulations we identify one mode as an exchange-dominated propagating spin wave, and the other as a self-localized nonlinear spin wave bullet. Wavelet-based analysis of the simulations indicates that the apparent simultaneous excitation of both modes results from rapid mode hopping induced by the Oersted field.Comment: 5 pages, 3 figure

Oscillatory transient regime in the forced dynamics of a spin torque nano-oscillator

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 figure

Excitation of spin waves by a current-driven magnetic nanocontact in a perpendicularly magnetized waveguide

It is demonstrated both analytically and numerically that the properties of spin wave modes excited by a current-driven nanocontact of length $L$ in a quasi-one-dimensional magnetic waveguide magnetized by a perpendicular bias magnetic field ${H}_{e}$ are qualitatively different from the properties of spin waves excited by a similar nanocontact in a two-dimensional unrestricted magnetic film (``free layer''). In particular, there is an optimum nanocontact length ${L}_{\mathrm{opt}}$ corresponding to the minimum critical current of the spin wave excitation. This optimum length is determined by the magnitude of ${H}_{e}$, the exchange length, and the Gilbert dissipation constant of the waveguide material. Also, for $Ll{L}_{\mathrm{opt}}$ the wavelength \ensuremath{\lambda} (and the wave number $k$) of the excited spin wave can be controlled by the variation of ${H}_{e}$ (\ensuremath{\lambda} decreases with the increase of ${H}_{e}$), while for $Lg{L}_{\mathrm{opt}}$ the wave number $k$ is fully determined by the contact length $L$ (k\ensuremath{\sim}1/L), similar to the case of an unrestricted two-dimensional free layer

Excitation of self-localized spin-wave "bullets" by spin-polarized current in in-plane magnetized magnetic nano-contacts: a micromagnetic study

It was shown by micromagnetic simulation that a current-driven in-plane magnetized magnetic nano-contact, besides a quasi-linear propagating ("Slonczewski") spin wave mode, can also support a nonlinear self-localized spin wave "bullet" mode that exists in a much wider range of bias currents. The frequency of the "bullet" mode lies below the spectrum of linear propagating spin waves, which makes this mode evanescent and determines its spatial localization. The threshold current for the excitation of the self-localized "bullet" is substantially lower than for the linear propagating mode, but finite-amplitude initial perturbations of magnetization are necessary to generate a "bullet" in our numerical simulations, where thermal fluctuations are neglected. Consequently, in these simulations the hysteretic switching between the propagating and localized spin wave modes is found when the bias current is varied.Comment: 27 pages, 5 figures, paper submitted to Physical Review