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Here we report the possibility to excite ultra-short spin waves in ferromagnetic thin-films by using time-harmonic electromagnetic fields with terahertz frequency. Such ultra-fast excitation requires to include inertial effects in the description of magnetization dynamics. In this respect, we consider the inertial Landau-Lifshitz-Gilbert (iLLG) equation and develop analytical theory for exchange-dominated inertial spin waves. The theory predicts a finite limit for inertial spin wave propagation velocity, as well as spin wave spatial decay and lifetime as function of material parameters. Then, guided by the theory, we perform numerical micromagnetic simulations that demonstrate the excitation of ultra-short inertial spin waves (20 nm long) propagating at finite speed in a confined magnetic nanodot. The results are in agreement with the theory and provide the order of magnitude of quantities observable in realistic ultra-fast dynamics experiments., Comment: The following article has been accepted by Physical Review B. After it is published, it will be found at https://journals.aps.org/prb/. Revised version, 9 pages, 6 figures. Changes made in v2: added some references, minor edits and corrections

The impact of granular microstructure in permanent magnets on eddy current losses are investigated. A numerical homogenization procedure for electrical conductivity is defined. Then, an approximated simple analytical model for the homogenized conductivity able to capture the main features of the geometrical and material dependences is derived. Finally eddy current losses analytical calculations are given, and the two asymptotic expressions for losses in the stationary conduction limit and advanced skin effect limit are derived and discussed., 5 pages, 7 figures

The emergence of exceptional points (EPs) in the parameter space of a (2D) eigenvalue problem is studied in a general sense in mathematical physics. In coupled systems, it gives rise to unique physical phenomena upon which novel approaches for the development of seminal types of highly sensitive sensors shall leverage their fascinating properties. Here, we demonstrate at room temperature the emergence of EPs in coupled spintronic nanoscale oscillators, coming along with the observation of amplitude death of self-oscillations and other complex dynamics. The main experimental features are properly described by the linearized theory of coupled system dynamics we develop. Interestingly, these spintronic nanoscale oscillators are deployment-ready in different applicational technologies, such as field, current or rotation sensors, radiofrequeny and wireless devices and, more recently, novel neuromorphic hardware solutions. Their unique and versatile properties, notably their large nonlinear behavior open up unprecedented perspectives in experiments as well as in theory on the physics of exceptional points. Furthermore, the exploitation of EPs in spintronics devises a new paradigm for ultrasensitive nanoscale sensors and the implementation of complex dynamics in the framework of non-conventional computing.

We have numerically solved the Landau-Lifshitz-Gilbert (LLG) equation in its standard and inertial forms to study the magnetization switching dynamics in a $3d$ thin film ferromagnet. The dynamics is triggered by ultrashort magnetic field pulses of varying width and amplitude in the picosecond and Tesla range. We have compared the solutions of the two equations in terms of switching characteristic, speed and energy analysis. Both equations return qualitatively similar switching dynamics, characterized by regions of slower precessional behavior and faster ballistic motion. In case of inertial dynamics, ballistic switching is found in a 25 % wider region in the parameter space given by the magnetic field amplitude and width. The energy analysis of the dynamics is qualitatively different for the standard and inertial LLG equations. In the latter case, an extra energy channel, interpreted as the kinetic energy of the system, is available. Such extra channel is responsible for a resonant energy absorption at THz frequencies, consistent with the occurence of spin nutation., Comment: 9 pages, 6 figures

We apply a non-abelian gauge field approach to generalize the micromagnetic energy description of a ferromagnet. This approach, without further assumption takes into account three different energy terms: the well known exchange term, a chiral ones and an intrinsic anisotropy term. In the non-abelian gauge field approach the covariant gauge derivative plays a key role. The results is the emergence of a Dzyaloshinskii–Moriya like energy term under two conditions: the first one is the pure gauge field background and the second one is the presence of a static electric field. Moreover this approach allows to reach a more deep understanding of the micromagnetics theory if rewritten in a gauge invariant formulation. In this paper clearly emerges the interpretation of the voltage controlled magnetic anisotropy (VCMA) mechanism.

The emergence of exceptional points (EPs) in the parameter space of a non-hermitian (2D) eigenvalue problem is studied in a general sense in mathematical physics, and has in the last decade successively reached the scope of experiments. In coupled systems, it gives rise to unique physical phenomena, which enable novel approaches for the development of seminal types of highly sensitive sensors. Here, we demonstrate at room temperature the emergence of EPs in coupled spintronic nanoscale oscillators and hence exploit the system’s non-hermiticity. We describe the observation of amplitude death of self-oscillations and other complex dynamics, and develop a linearized non-hermitian model of the coupled spintronic system, which properly describes the main experimental features. Interestingly, these spintronic nanoscale oscillators are deployment-ready in different applicational technologies, such as field, current or rotation sensors, radiofrequeny and wireless devices and, more recently, novel neuromorphic hardware solutions. Their unique and versatile properties, notably their large nonlinear behavior, open up unprecedented perspectives in experiments as well as in theory on the physics of exceptional points. Furthermore, the exploitation of EPs in spintronics devises a new paradigm for ultrasensitive nanoscale sensors and the implementation of complex dynamics in the framework of non-conventional computing.

The Landau-Lifshitz equation governing magnetization dynamics is written in terms of the amplitudes of normal modes associated with the micromagnetic system's appropriate ground state. This results in a system of nonlinear ordinary differential equations, the right-hand side of which can be expressed as the sum of a linear term and nonlinear terms with increasing order of nonlinearity (quadratic, cubic, etc.). The application of the method to nanostructured magnetic systems demonstrates that the accurate description of magnetization dynamics requires a limited number of normal modes, which results in a considerable improvement in computational speed. The proposed method can be used to obtain a reduced-order dynamical description of magnetic nanostructures which allows to adjust the accuracy between low-dimensional models, such as macrospin, and fully spatially discretized micromagnetic models. This new paradigm for micromagnetic simulations is tested for three problems relevant to the areas of spintronics and magnonics: directional spin-wave coupling in magnonic waveguides, high power ferromagnetic resonance in a magnetic nanodot, and injection-locking in spin-torque nano-oscillators. The case studies considered the validity of the proposed approach as a new systematic method to obtain an intermediate order dynamical model based on normal modes for the analysis of magnetic nanosystems. The time-consuming calculation of the normal modes has to be done only one time for the system. These modes can be used to optimize and predict the system response for all possible time-varying external magnetic fields. This is of utmost importance for various applications where a fast and accurate system response is required, such as in electronic circuit simulation software where an accurate response of the magnetic device has to be predicted.

We introduce a normal form for the equation which describes nonlinear oscillations of spin-torque nano-oscillators (STNO). The general form of the equation might be reduced to the equation for STNO with uniformly magnetized free layer or to the equation for STNO with free layer in a magnetic vortex state. The equation is used to study the injection-locking problem. The bifurcation diagram for this problem is derived by using as bifurcation parameters the frequency and the amplitude of the ac excitation. Bifurcations lines corresponding to local bifurcations, i.e. the Saddle-node and the Hopf bifurcations, are derived by using linearization, while the lines corresponding to the Homoclinic bifurcation, which is global in nature, are derived numerically. The diagram exhibits universal features in terms of the shape and position of bifurcation lines. When the frequency of the ac excitation is relatively close to the free-running frequency of the oscillator the system is in synchronized state. For small ac amplitudes, the synchronization is lost for values of the ac frequency which imply the crossing of saddle-node bifurcations. For sufficiently larger values of the ac amplitude and for ac frequency smaller that the free-running frequency, the loss of the synchronized state occurs through a more complex bifurcation pattern which involves a sequence of two homoclinic bifurcations and an Hopf bifurcation. The value of frequency corresponding to loss of stability of the synchronized state can be determined analytically through linearization. The above sequence of bifurcations is connected to the presence of a special point in the bifurcation diagram, referred to as Takens-Bogdanov bifurcation.

Demagnetization in a thin film due to a terahertz pulse of magnetic field is investigated. Linearized LLG equation in the Fourier space to describe the magnetization dynamics is derived, and spin waves time evolution is studied. Finally, the demagnetization due to spin waves dynamics and recent experimental observations on similar magnetic system are compared. As a result of it, the marginal role of spin waves dynamics in loss of magnetization is established., 5 pages, 6 figures

Magnetic tunnel junctions with free layer thermally stabilized by the combined action of perpendicular magneto-crystalline and shape anisotropy are considered. The spin-transfer torque driven switching dynamics is studied. Analytical formulas for switching time statistical distributions and write-error rate are derived under the assumption of ballistic dynamics. The analytical results are compared with full micromagnetic and macrospin simulations based on stochastic Landau-Lifshitz-Gilbert-Slonczewski equation. This comparison reveals quantitative agreement with the proposed analytical theory.

The understanding of how spins move and can be manipulated at pico- and femtosecond timescales has implications for ultrafast and energy-efficient data-processing and storage applications. However, the possibility of realizing commercial technologies based on ultrafast spin dynamics has been hampered by our limited knowledge of the physics behind processes on this timescale. Recently, it has been suggested that inertial effects should be considered in the full description of the spin dynamics at these ultrafast timescales, but a clear observation of such effects in ferromagnets is still lacking. Here, we report direct experimental evidence of intrinsic inertial spin dynamics in ferromagnetic thin films in the form of a nutation of the magnetization at a frequency of ~0.5 THz. This allows us to reveal that the angular momentum relaxation time in ferromagnets is on the order of 10 ps. Inertial dynamics are observed in a ferromagnet. Specifically, a nutation is seen on top of the usual spin precession that has a lifetime on the order of 10 picoseconds.

We study magnetization oscillations in a uniformly magnetized nanomagnet driven by ac excitations (external fields and injected currents) when the frequency of excitation is close to ferromagnetic resonance frequency. By using separation of time scale and the averaging technique, we derive dynamical equations, which can be studied by the methods of dynamical systems theory. This leads to an analytical description of nonlinear frequency response and to the analysis of relevant bifurcation phenomena. The theoretical results are compared with numerical simulations.

The synchronization of vortex spin-transfer nano-oscillators (both nanopillar and nanocontact structures) with external microwave excitations is considered. A collective variables description is used to derive an analytical reduced-order model for the vortex core dynamics, which is able to predict all the possible oscillation regimes and hysteretic transitions among them. The influence of thermal fluctuations on hysteretic synchronization is studied by full micromagnetic simulations. The numerical results are in good agreement with the theory.

Recent studies on heterostructures of ultrathin ferromagnets sandwiched between a heavy metal layer and an oxide have highlighted the importance of spin-orbit coupling (SOC) and broken inversion symmetry in domain wall (DW) motion. Specifically, chiral DWs are stabilized in these systems due to the Dzyaloshinskii-Moriya interaction (DMI). SOC can also lead to enhanced current induced DW motion, with the Spin Hall effect (SHE) suggested as the dominant mechanism for this observation. The efficiency of SHE driven DW motion depends on the internal magnetic structure of the DW, which could be controlled using externally applied longitudinal in-plane fields. In this work, micromagnetic simulations and collective coordinate models are used to study current-driven DW motion under longitudinal in-plane fields in perpendicularly magnetized samples with strong DMI. Several extended collective coordinate models are developed to reproduce the micromagnetic results. While these extended models show improvements over traditional models of this kind, there are still discrepancies between them and micromagnetic simulations which require further work.

Magnetization dynamics in single-domain particles driven by elliptically polarized ac external fields is studied. It is shown that such dynamics exhibits chaotic and fractal nature at a moderate power level. This is due to the phenomenon of heteroclinic tangle produced by the combined action of ac excitations and saddle-type dynamics. By using the special perturbation technique based on the Melnikov function, analytical formulas for the threshold ac excitation amplitudes necessary to create the heteroclinic tangle are derived as a function of frequency and general elliptic polarization of the applied ac field. Numerical simulations confirm that the heteroclinic tangle implies the erosion of the safe basin around stable magnetization states.

Energy-efficient switching of magnetization is a central problem in nonvolatile magnetic storage and magnetic neuromorphic computing. In the past two decades, several efficient methods of magnetic switching were demonstrated including spin torque, magneto-electric, and microwave-assisted switching mechanisms. Here we experimentally show that low-dimensional magnetic chaos induced by alternating spin torque can strongly increase the rate of thermally-activated magnetic switching in a nanoscale ferromagnet. This mechanism exhibits a well-pronounced threshold character in spin torque amplitude and its efficiency increases with decreasing spin torque frequency. We present analytical and numerical calculations that quantitatively explain these experimental findings and reveal the key role played by low-dimensional magnetic chaos near saddle equilibria in enhancement of the switching rate. Our work unveils an important interplay between chaos and stochasticity in the energy assisted switching of magnetic nanosystems and paves the way towards improved energy efficiency of spin torque memory and logic., Energy-efficient manipulation of spins at the nanoscale can advance magnetic storage and computing technologies. Here the authors show that low-dimensional chaos generated by alternating spin torque can induce magnetic switching in a nanoscale ferromagnet.

An efficient method for the calculation of ferromagnetic resonant modes of magnetic structures is presented. Finite-element discretization allows flexible geometries and location dependent material parameters. The resonant modes can be used for a semi-analytical calculation of the power spectral density of the thermal white-noise, which is relevant for many sensor applications. The proposed method is validated by comparing the noise spectrum of a nano-disk with time-domain simulations.

Magnetization switching in nanoparticles and thin-films is the fundamental issue to deal with in order to obtain high speed and energy-efficient recording devices [1]. The optimization of switching mechanisms is constrained in the framework of the so-called magnetic recording trilemma. On one hand, one would like to have the magnetized bit occupying a smaller area on the recording medium and, at the same time, magnetization remaining stable over long enough time for reliable data retention. These two constraints are competing since thermal stability decreases with decreasing active volume of the magnetic bit. On the other hand, circumventing this issue would require higher coercivity of the magnetic material and, consequently, larger current feeding the write head. However, the maximum current amplitude is constrained by technological limits in the realizations of the pole tips and, thus, one cannot meet the aforementioned requirements. For these reasons, several strategies have been investigated in the last decades to realize fast magnetization switching with greater efficiency, such as microwave-assisted switching [2] and precessional switching [3]. In particular, the latter occurs through the application of a field transverse to the initial magnetization and yields much smaller switching times than conventional switching [4], [5]. However, to achieve successful switching, an extremely precise design of the field pulse is needed to switch off the field at the right moment [6]. Then, the equilibrium magnetization is reached after a relaxation from a high-to low-energy state [7]. This mechanism is probabilistic even when thermal fluctuations are neglected, due to multistability and small dissipation in magnetization dynamics [8]. When also thermal fluctuations are considered, the stochasticity of the switching process is even more pronounced [3]. On the other hand, magnetic recording devices must undergo strict reliability requirements in terms of extremely low write-error rates, which can be realized at expense of the speed of the write process. In this paper, we theoretically analyze the magnetization switching for a single magnetic grain of the recording medium subject to the write head field pulses and room temperature thermal fluctuations, as it is the case of perpendicular magnetic recording. This situation, in the absence of thermal fluctuations and for special symmetry of the magnetic particle, has been studied in a pioneering paper [9] and is referred to as damping switching. In this paper, by using analytical techniques, we derive expressions for the switching times distribution functions in terms of material, geometrical and external field properties. These analytical results provide a tool to quantify the write-error rates as function of design parameters, which may help the optimization of switching processes. To this end, we consider the Landau-Lifshitz-Gilbert (LLG) equation augmented with a thermal field of stochastic nature [10], whose intensity is given by the fluctuation-dissipation theorem. We assume that the magnetization is spatially-uniform during the dynamics, so that the magnetic particle can be described within a macrospin approximation. In the absence of excitation, the energy barrier separating the equilibria is much higher than the thermal energy. This hypothesis is suitable when the uniaxial anisotropy is large enough as it is the case for magnetic recording grains. The external field pulse amplitude is assumed to be above the critical switching field of the device. In this situation, the switching time can be evaluated considering the deterministic magnetization motion acting on a random initial magnetization distribution due to thermal fluctuations. In the absence of external field, the magnetization is distributed according to the stationary solution of the Fokker-Planck equation, which can be simply expressed in terms of the small tilting angle $\theta ( \sin \theta \approx \theta $ with respect to the particle's easy axis. Then, considering a rotationally-symmetric particle (z is the symmetry axis) and neglecting the thermal fluctuations during magnetization evolution, the LLG equation can be integrated by separation of variables [9] in order to determine the switching time $\mathrm {t}_{s}$ defined as the time interval between the application of the field pulse (the initial z-component of the magnetization is $\mathrm {m}_{z0})$ and the time instant where the z-component of the magnetization is equal to a given value $\mathrm {m}_{zf}$. By using appropriate derivation (details will be given in the full paper), it can be shown that the relationship $\mathrm {t}_{s}( \mathrm {m}_{z0} , \mathrm {m}_{zf})$ allows one to derive the probability and cumulative distribution functions for magnetization switching times as function of geometrical, material and excitation parameters. Such functions can be used to compute the write error-rate of the switching process for given switching duration $\mathrm {t}_{s}$. The analytical predictions are compared with macrospin and micromagnetic simulations of magnetization switching (an example computation for a circular nanodot with 30nm diameter, 1nm thickness and perpendicular anisotropy is reported in Fig. 1) in order to show the effectiveness of the approach.

Numerical techniques for the time integration of Landau-Lifshitz magnetization dynamics are considered. In the continuous model, such dynamics implies the conservation of magnetization amplitude and, when dissipation is neglected, even the conservation of free energy, a property which is generally corrupted by the time-discretization method. In this work, two classes of explicit schemes, based on Runge-Kutta and midpoint methods respectively, are introduced. The schemes are termed pseudo-symplectic in that they are accurate to order p , but preserve magnetization amplitude and free energy to order q > p . Numerical tests are performed on the simulation of fast precessional switching dynamics for which an analytical solution is available.

Numerical modeling of random magnetization dynamics driven by a jump-noise process is discussed. In the important case of weak thermal effects, random magnetization dynamics is reduced to stochastic processes on graphs and numerical analysis of these processes are outlined.

The main analytical model for describing the motion of magnetic domain walls is the 1-D model formulated based on the profile of a Bloch wall. This model qualitatively describes the motion of magnetic domain wall in nanowires, while it may fail to match experimental and numerical results quantitatively. In recent years, the 1-D model has been further generalized by the introduction of terms such as spin transfer torques and spin orbit torques. It has also been used to describe the motion of different domain walls, including vortex walls. It seems that in many such attempts, formalisms are not followed accurately and the main assumptions of the model (such as the Bloch wall profile used in developing the model) are underestimated. In this paper, we first derive an analytical model to describe the motion of a tilting Bloch wall in perpendicularly magnetized materials using four collective coordinates. We then compare the energy landscape predicted by this model to that of micromagnetic simulations, highlighting the possibility of using such comparisons to develop corrections for the 1-D model.

An averaging technique is applied to random magnetization dynamics driven by a jump-noise process. This results in the reduction of this dynamics to stochastic magnetic energy dynamics on graphs. The latter eventually leads to a continuous master equation whose validity is not based on or limited by the Kramers-Brown approximation. An eigenvalue technique for solving this master equation is presented and illustrated by numerical examples.

The magnetization dynamics of a ferromagnet can be influenced by the second-order anisotropy contribution when the magnetocrystalline perpendicular anisotropy is of the same order of the out-of-plane demagnetizing field (compensation point). This condition can be fulfilled in magnetic tunnel junctions (MTJs) where the induced interfacial perpendicular anisotropy can be controlled by the thickness of the free layer. This paper describes, within a both micromagnetic and analytical/numerical framework, how switching and self-oscillation of the magnetization are affected by the second order magnetocrystalline anisotropy near the compensation point. We have derived analytical expressions for the switching critical current and the oscillation frequencies that are well reproduced by numerical simulations. Our results can support the design of MTJs for application either in storage and nanoscale signal generators.

The synchronization of nanopillar vortex spin-transfer nano-oscillators with external ac injected current is considered. A collective variables description is used to derive an analytical reduced-order model for the vortex core dynamics. This model is able to predict all the possible vortex oscillation regimes as well as the transition mechanisms among them. In particular, it is shown that hysteretic synchronization occurs at moderate ac currents. The analytical results are in good agreement with micromagnetic simulations.

Random magnetization dynamics driven by a jump-noise process is reduced to stochastic magnetic energy dynamics on specific graphs using an averaging technique. An approach to analyzing stochastic energy dynamics on graphs is presented and applied to the calculation of power spectral density of random magnetization dynamics. An eigenvalue technique for computing the power spectral density under specific cases is also presented and illustrated by numerical results.

A numerical technique for Monte Carlo simulations of random Landau-Lifshitz dynamics driven by a jump-noise process is briefly described. This technique is intrinsically parallel in nature and its implementation on GPUs is discussed and illustrated.

Current-induced vortex oscillations in an extended thin-film with point-contact geometry are considered. The synchronization of these oscillations with a microwave external magnetic field is investigated by a reduced order model that takes into account the dynamical effects associated with the significant deformation of the vortex structure produced by the current, which cannot be taken care of by using the standard rigid vortex theory. The complete phase diagram of the vortex oscillation dynamics is derived and it is shown that strong hysteretic behavior occurs in the synchronization with the external field. The complex nonlinear nature of the synchronization manifests itself also through the appearance of asymmetry in the locking frequency bands for moderate microwave field amplitudes. Predictions from the reduced order model are confirmed by full micromagnetic simulations.

Magnetization dynamics in uniformly magnetized particles subject to time-harmonic (AC) external fields is considered. The study is focused on the behavior of the AC-driven dynamics close to saddle equilibria. It happens that such dynamics has chaotic nature at moderately low power level, due to the heteroclinic tangle phenomenon which is produced by the combined effect of AC-excitations and saddle type dynamics. By using analytical theory for the threshold AC excitation amplitudes necessary to create the heteroclinic tangle together with numerical simulations, we quantify and show how the tangle produces the erosion of the safe basin around the stable equilibria.

The precessional switching process of a magnetic nanoparticle subject to external field pulses applied along the hard-axis is considered. The critical field pulse amplitude necessary to realize the switching is determined. Then, the analytical solution of magnetization switching dynamics is derived in the lossless limit by using elliptic functions. Moreover, expressions for the field pulse duration tolerances which guarantee successful switching are also obtained. The theoretical predictions are verified by macrospin numerical simulations of ultra-fast magnetization switching.

It is demonstrated that the Slonczewski spin-torque term can be naturally derived from the equation for magnetization dynamics driven by a jump-noise process. The central point of the derivation is the modification of transition probability rate of the jump-noise process caused by spin-polarized current injection. This modification results in two distinct terms in the expected value of the jump-noise process: the traditional one corresponding to the Landau–Lifshitz damping and another one representing the Slonczewski spin-torque term.

It is rigorously demonstrated that spatially uniform magnetization oscillations in sufficiently small ellipsoidal nanoparticles can be unconditionally stable with respect to spatially nonuniform perturbations. The proof reveals the dominant stabilizing effect of exchange field which is mathematically manifested through the Poincare inequality.

An analytical approach to the study of ferromagnetic resonance in nanomagnets is discussed in the case when radio-frequency magnetic fields and spin-polarized electrical currents are simultaneously present. Current-controlled foldover effects in ferromagnetic resonance are predicted and analyzed.

An overview is given of the various approaches that have been proposed for the interpretation of spin-transfer-driven magnetization dynamics. Models of critical currents and critical fields for switching as well as for the onset of magnetization oscillations are discussed, together with methods for the construction of field-current stability diagrams. Finally, the role of thermal fluctuations is analyzed. Particular emphasis is given to the study of uniformly magnetized nanomagnets, which represents an essential step before moving to the numerical computation of more complex micromagnetic configurations.

Magnetization dynamics in uniformly magnetized nanomagnets excited by time-harmonic (AC) external fields or spin-polarized injected currents is considered. The analysis is focused on the behaviour of the AC-excited dynamics near saddle equilibria. It turns out that this dynamics has a chaotic character at moderately low power level. This chaotic and fractal nature is due to the phenomenon of heteroclinic tangle which is produced by the combined effect of AC-excitations and saddle type dynamics. By using the perturbation technique based on Melnikov function, analytical formulas for the threshold AC excitation amplitudes necessary to create the heteroclinic tangle are derived. Both the cases of AC applied fields and AC spin-polarized injected currents are treated. Then, by means of numerical simulations, we show how heteroclinic tangle is accompanied by the erosion of the safe basin around the stable regimes.

Stochastic magnetization dynamics in uniformly magnetized nanomagnets is considered. The system is assumed to have rotational symmetry as the anisotropy axis, the applied field, and the spin polarization are all aligned along an axis of symmetry. By appropriate integration of the Fokker-Planck equation associated to the problem, the stochastic differential equation governing the evolution of the angle between the magnetization orientation and the symmetry axis is derived. The drift terms present in this equation contain a noise-induced drift term, which, in combination with drift terms of deterministic origin, can be written as the derivative of an effective potential. Superparamagnetic-like transitions are studied in connections with the bifurcations of the effective potential as temperature and excitation conditions are varied.

International audience; Microwave assisted magnetization reversal are studied in the bulk bilayer exchange coupled system. We investigate the nonlinear magnetization reversal dynamics in a perpendicular exchange spring media using Landau-Lifshitz equation. In the limit of the infinite thickness of the system, the propagation field leads the reversal of the system. The reduction of the switching field and the magnetization profile in the extended system are studied numerically. The possibility to study the dynamics analytically is discussed and an approximation where two P-modes are coupled by an interaction field is presented. The ansatz used for the interaction field is validated by comparison with the numerical results. This approach is shown to be equivalent to two exchange coupled macrospins.

Landau-Lifshitz-Gilbert (LLG) equation is the fundamental equation to describe magnetization vector field dynamics in microscale and nanoscale magnetic systems. This equation is highly nonlinear in nature and, for this reason, it is generally solved by using numerical techniques. In this paper, the mid-point rule time-stepping technique is applied to the numerical time integration of LLG equation and the relevant properties of the numerical scheme are discussed. The mid-point rule is an unconditionally stable and second order accurate scheme which preserves the fundamental geometrical properties of LLG dynamics. First, it exactly preserves the LLG property of conserving the magnetization magnitude at each spatial location. Second, for constant in time applied fields, it preserves the LLG Lyapunov structure, namely the fact that the free energy is a decreasing function of time. In addition, in the case of zero damping, the mid-point rule preserves the conservation of the system free energy. The above preservation properties are unconditionally valid, i.e. they are fulfilled for any value of the time-step. Finally, the LLG hamiltonian structure in the case of zero damping is preserved up to the third order terms with respect to the time-step. The main difficulty related to this scheme is the necessity of solving a large system of globally coupled nonlinear equations. This problem has been circumvented by using special and reasonably fast quasi-Newton iterative technique. The proposed numerical scheme is then tested on the standard micromagnetic problem no. 4. In the numerical computations, the spatial discretization is obtained by finite difference technique and the magnetostatic field is computed through the Fast Fourier Transform method.

A uniform mode theory is used to find the parameter range for realizing fast switching below the Stoner–Wohlfarth limit in weakly coupled granular tilted media. The theory is used to estimate the field amplitudes and directions for simultaneous switching of a family of uniformly magnetized and noninteracting grains with dispersion in the easy axis and initial magnetization directions. The effect of interactions is then studied numerically by using a 3D finite element micromagnetic simulations. The numerical results shows that the theory provides rather accurate estimate. In addition, it is verified that in an appropriate range of applied field amplitudes the switching time decreases by decreasing the external field.

Nonlinear magnetization dynamics under spin-polarized currents is discussed by the methods of the theory of nonlinear dynamical systems. The fixed points of the dynamics are calculated. It is shown that there may exist 2, 4, or 6 fixed points depending on the values of the external field and of the spin-polarized current. The stability of the fixed points is analyzed and the conditions for the occurrence of saddle-node and Hopf bifurcations are determined.

Magnetization dynamics in uniformly magnetized ferromagnetic bodies is studied by using Landau–Lifshitz–Gilbert (LLG) equation. This equation is written in a generalized form to take into account the effect of spin-polarized currents. The general properties of the corresponding nonlinear dynamical system are studied in detail. It is underlined that, in many experimental situations relevant to magnetic storage applications, LLG dynamics is a small perturbation of conservative precessional dynamics. In this respect, the conservative case is treated in detail and it is shown that analytical solutions of the conservative dynamics can be derived by using the existence of two integrals of motion. The (perturbed) dissipative LLG dynamics is then studied by developing appropriate perturbation techniques. The general methods of analysis discussed in the first part of the paper are applied to two particular problems: precessional switching and current-driven switching. The accuracy of theoretical predictions are tested by comparing analytical results with numerical solutions.

Magnetization relaxation starting from a generic non-equilibrium state is analytically described. An equation for the energy decay is obtained. On this basis, an approximate expression for the magnetization motion during the ringing process is obtained in terms of Jacobi elliptic functions with time-dependent parameters.