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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.

In binary and ordinal response models outlying covariates as well as incoherent responses may affect the reliability of the maximum likelihood estimators and that of the derived inferential procedures. However the various link functions, which provide the relationship between the linear predictor and the probabilities of the response categories, differ in terms of sensitivity to anomalous data. The current paper derives conditions useful to evaluate the properties of the link functions with respect to robustness, either when the covariates are outlier free or when extreme design points may occur. The main results show that – by an appropriate choice of the link function – robust estimators, with a bounded influence function, can be easily derived from the usual likelihood function, while preserving the predictive ability of the fitted model.

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

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.

First, it is shown that during electromagnetic transients in COMPASS-U the poloidal field coils must drain sizeably the current from the vessel and, therefore, reduce disruption forces and their duration. Next, the role of poloidal eddy current (which is absent in some approaches) in the dynamics of vertical and radial forces is found to be essential. Finally, to verify the CarMa0NL modelling for COMPASS-U, the numerical results are cross-validated with general analytical predictions (Pustovitov 2015 Nucl. Fusion 55 113032): the computed vertical force on the tokamak wall is found to be almost zero during rapid ( jump-like) transients, as it should be because of strong skin-effect. This test proves the credibility of the simulation model and computational realization 2021 IAEA, Vienna.

The NTMpy code package allows for simulating the one-dimensional thermal response of multilayer samples after optical excitation, as in a typical pump-probe experiment. Several Python routines are combined and optimized to solve coupled heat diffusion equations in one dimension, on arbitrary piecewise homogeneous material stacks, in the framework of the so-called three-temperature model. The energy source deposited in the material is modelled as a light pulse of arbitrary cross-section and temporal profile. A transfer matrix method enables the calculation of realistic light absorption in presence of scattering interfaces as in multilayer samples. The open source code is fully object-oriented to enable a user-friendly and intuitive interface for adjusting the physically relevant input parameters. Here, we describe the mathematical background of the code, we lay out the workflow, and we validate the functionality of our package by comparing it to commercial software, as well as to experimental transient reflectivity data recorded in a pump-probe experiment with femtosecond light pulses., 9 pages, 5 figures

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.

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.

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.

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.

In the operation of tokamaks, hard-to-predict fast transient events, called disruptions, are often responsible for severe energy deposition on the structures surrounding the plasma. In this paper, first of all we derive an overall energy balance for a disruptive plasma, then we use an evolutionary equilibrium code (CarMa0NL) to reproduce the macroscopic plasma evolution during a disruption of a sample high-aspect-ratio circular plasma, analysing the energy fluxes which take place during the event. We show that the variations of toroidal magnetic energy are compensated by one portion of the Poynting vector flux, and that dissipated heat during the Thermal Quench is mainly related to the loss of internal energy. Moreover, we develop an analytical model providing physical insight on the energy deposition on the wall during the Current Quench, related to the poloidal magnetic energy, in a quantity depending on the time scale of the event compared to the electromagnetic time constants.