Landau-Lifshitz-Gilbert Equation Research Articles

Homogenization results for a Landau–Lifshitz–Gilbert equation in composite materials with transmission defects

We study the homogenization of Landau–Lifshitz–Gilbert equation in a ϵ-periodic composite material formed by two constituents, separated by an imperfect interface , on which we prescribe the continuity of the conormal derivatives and a jump of the solution proportional to the conormal derivative, by means of a coefficient of order . We use the periodic unfolding method together with extension operators for handling the nonlinearities to identify the limit problem when tuning up the parameter γ in .

Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse

We investigate the shape anisotropy effect on the magnetization reversal of a nanoparticle driven by a down-chirp microwave field pulse (DCMP). Based on the Landau–Lifshitz–Gilbert equation, numerical results reveal that the microwave amplitude, initial frequency, and chirp rate of DCMP decrease with the increase of shape anisotropy. For a certain shape of the nanoparticle, the reversal time is significantly reduced. These findings can be attributed as the shape anisotropy opposes the magnetocrystalline anisotropy, and thus the energy barrier decreases. The result of damping dependence of reversal indicates that there exists an optimal damping situation at which magnetization is fastest. Moreover, the required microwave field amplitude can be lowered by applying the spin-polarized current simultaneously. The usage of an optimum combination of DCMP and current is suggested to achieve cost-effective and faster switching. So these findings might be useful to realize the fast and power-efficient data storage device.

A massively parallel time-domain coupled electrodynamics–micromagnetics solver

We present a high-performance coupled electrodynamics–micromagnetics solver for full physical modeling of signals in microelectronic circuitry. The overall strategy couples a finite-difference time-domain approach for Maxwell’s equations to a magnetization model described by the Landau–Lifshitz–Gilbert equation. The algorithm is implemented in the Exascale Computing Project software framework, AMReX, which provides effective scalability on manycore and GPU-based supercomputing architectures. Furthermore, the code leverages ongoing developments of the Exascale Application Code, WarpX, which is primarily being developed for plasma wakefield accelerator modeling. Our temporal coupling scheme provides second-order accuracy in space and time by combining the integration steps for the magnetic field and magnetization into an iterative sub-step that includes a trapezoidal temporal discretization for the magnetization. The performance of the algorithm is demonstrated by the excellent scaling results on NERSC multicore and GPU systems, with a significant (59×) speedup on the GPU using a node-by-node comparison. We demonstrate the utility of our code by performing simulations of an electromagnetic waveguide and a magnetically tunable filter.

Efficient domain wall motion in asymmetric magnetic tunnel junctions with vertical current flow

In this paper, we study the domain wall motion induced by vertical current flow in asymmetric magnetic tunnel junctions. The domain wall motion in the free layer is mainly dictated by the current-induced field-like torque acting on it. We show that as we increase the MTJ asymmetry, by considering dissimilar ferromagnetic contacts, a linear-in-voltage field-like torque behavior is accompanied by an enhancement in the domain wall displacement efficiency and a higher degree of bidirectional propagation. Our analysis is based on a combination of a quantum transport model and magnetization dynamics as described by the Landau–Lifshitz–Gilbert equation, along with comparison to the intrinsic characteristics of a benchmark in-plane current injection domain wall device.

Comparison of Two Approaches for Microstructural Modeling of the Ferromagnetic Alloys Behavior

In this letter, two approaches to describe the behavior of a ferromagnetic alloy in an external magnetic field within the framework of the micromagnetism theory are compared. The first one is based on the minimization of total magnetic energy. Another approach is based on describing the dynamics of the process using the Landau–Lifshitz–Gilbert equation. We have considered the problems of magnetization of a single martensite variant of the Ni <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> MnGa alloy within the framework of the theory of magnetism, in which one of the structural elements is a domain wall of a certain thickness, using bothapproaches. The solutions to these boundary value problems were obtained numerically by the finite-element method using the FeniCS package. The evolution of the magnetic structure was investigated. Magnetization curves were obtained in the external magnetic field directed at different angles to the anisotropy axis.

Development of magnetization characteristic simulator by LLG equation using particle model Introduction of thermal field in renormalized system

The spin-dynamics simulator for calculating a hysteresis loss is useful for a highly efficient design of the electric motor. We reported a basic calculation method using the Landau-Lifshitz-Gilbert equation and a new treatment of the exchange interaction for particle spins larger than atomic spins. In this report, we introduce a random magnetic field as thermal effect applied for the renormalized exchange interaction.

Anomalous Gilbert damping and Duffing features of the superconductor-ferromagnet-superconductor φ0 Josephson junction

We demonstrate unusual features of phase dynamics, IV-characteristics and magnetization dynamics of the $\varphi_0$ Josephson junction at small values of spin-orbit interaction, ratio of Josephson to magnetic energy and Gilbert damping. In particular, an anomalous shift of the ferromagnetic resonance frequency with an increase of Gilbert damping is found. The ferromagnetic resonance curves show the Duffing oscillator behaviour, reflecting the nonlinear nature of Landau-Lifshitz-Gilbert (LLG) equation. Based on the numerical analysis of each term in LLG equation we obtained an approximated equation demonstrated both damping effect and Duffing oscillator features. The resulting Duffing equation incorporates the Gilbert damping in a special way across the dissipative term and the restoring force. A resonance method for the determination of spin-orbit interaction in noncentrosymmetric materials which play the role of barrier in $\varphi_0$ junctions is proposed.

Fast magnetization reversal of a magnetic nanoparticle induced by cosine chirp microwave field pulse

We investigate the magnetization reversal of single-domain magnetic nanoparticle driven by the circularly polarized cosine chirp microwave pulse (CCMP). The numerical findings, based on the Landau–Lifshitz–Gilbert equation, reveal that the CCMP is by itself capable of driving fast and energy-efficient magnetization reversal. The microwave field amplitude and initial frequency required by a CCMP are much smaller than that of the linear down-chirp microwave pulse. This is achieved as the frequency change of the CCMP closely matches the frequency change of the magnetization precession which leads to an efficient stimulated microwave energy absorption (emission) by (from) the magnetic particle before (after) it crosses over the energy barrier. We further find that the enhancement of easy-plane shape anisotropy significantly reduces the required microwave amplitude and the initial frequency of CCMP. We also find that there is an optimal Gilbert damping for fast magnetization reversal. These findings may provide a pathway to realize the fast and low-cost memory device.

Equilibria and precession in a uniaxial antiferromagnet driven by the spin Hall effect

We systematically study the stationary and precessional states in a uniaxial antiferromagnet under the dampinglike spin–orbit torque (SOT). The stable regions for all equilibria are defined by the linear stability analysis. In the regions without any stable equilibrium, we confirm that a stable conical precession may exist. Invoking symmetry arguments, we rigorously reduce the coupled Landau–Lifshitz–Gilbert equations to a single-vector one, which allows us to derive the analytic expressions of the lower and upper thresholds, the frequency, and the amplitude of precession for a weak uniaxial anisotropy. Its frequency is of the order of THz and increases almost linearly with the current. Further analysis reveals that the precession is mainly propelled by the exchange interaction in the promise that the SOT balances the damping one in average. Moreover, the investigation uncovers that a weak anisotropy can improve the frequency tunability, and a small damping benefits lowering the exciting current. Finally, the analytic expressions are verified by comparing with numerical simulations of the original Landau–Lifshitz–Gilbert equations.

Micromagnetic modeling of magnon coherent states in a nonuniform magnetic field

Quantum magnonics is an emerging research field, with great potential for applications in a magnon based quantum technologies, including quantum computing, processing and encoding information. Magnon correlation and quantum entanglement are the main concepts in many quantum technologies under development. Of particular interest is the magnon Bose condensation. The emerging current question is the applicability of quasi-classical Landau–Lifshitz–Gilbert equations to describe the coherent state of magnons. We performed micromagnetic modeling of the magnetization dynamics at a high angle of deviation in an inhomogeneous magnetic field. We have obtained solutions that are well coincided with the properties of experimentally observed magnon BEC states. This result will be needed for the calculation the properties of magnetic devices on the basis of quantum magnonics.

Imaging Spin‐Wave Damping Underneath Metals Using Electron Spins in Diamond

AbstractSpin waves in magnetic insulators are low‐damping signal carriers that can enable a new generation of spintronic devices. The excitation, control, and detection of spin waves by metal electrodes is crucial for interfacing these devices to electrical circuits. As such, it is important to understand metal‐induced damping of spin‐wave transport, but characterizing this process requires access to the underlying magnetic films. Here it is shown that electronic sensor spins in diamond enable imaging of spin waves that propagate underneath metals in magnetic insulators. This capability is then used to reveal a 100‐fold metal‐induced increase in spin‐wave damping. The damping enhancement is attributed to spin‐wave‐induced electrical currents as well as, above a certain frequency, three‐magnon scattering processes. This interpretation is supported by deriving expressions for the current‐induced damping and the three‐magnon threshold from the Landau–Lifshitz–Gilbert equation that agree well with the observations. The detection of buried scattering centers further highlights the technique's power for assessing spintronic device quality. These results open new avenues for studying metal – spin‐wave interactions and provide access to interfacial processes such as spin‐wave injection via the spin‐Hall effect.

Study of type-III intermittency in the Landau–Lifshitz-Gilbert equation

We have studied a route of chaos in the dissipative Landau–Lifshitz-Gilbert equation representing the magnetization dynamics of an anisotropic nanoparticle subjected to a time-variant magnetic field. This equation presents interesting chaotic dynamics. In the parameter space, for some forcing frequency and magnetic strength of the applied field, one observes a transition from a regular periodic behavior to chaotic dynamics. The chaotic dynamics, close to the bifurcation, are characterized by type-III intermittency. Long epochs of quasi-regular dynamics followed by turbulent bursts. The characterization of the intermittencies has been done through four different techniques. The first method is associated with the computation of the Lyapunov exponents that characterize the chaotic regime. The second and third methods are associated with the statistics of the duration of the laminar epochs prior to a turbulent burst. The fourth method is associated with the subharmonic instability present in those laminar epochs and quantified through a Poincaré section method. At the end of the manuscript, we compare the result obtained by the different techniques and discuss the methods’ limitations.

Dispersion relation of spin wave in chiral helimagnet under stationary optical vortex radiation

We investigate a dispersion relation of a spin wave of an effectively one-dimensional chiral helimagnet CrNb3S6, which is under stationary radiation of a resonant optical vortex beam coaxial to that spiral magnetic order axis. We derive a classical magnetic structure by solving Landau–Lifshitz–Gilbert equation and calculate the collective excitation on it. We show that the optical vortex radiation makes the energy gap at k ∼ 0, and the exceptional points appear in the dispersion relation. Because such a dispersion relation implies the dynamical properties with instability, our result implies a possibility to control spin wave transport.

Wong\u2013Zakai Approximation for Landau\u2013Lifshitz\u2013Gilbert Equation Driven by Geometric Rough Paths

We adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.

Colossal stability of antiferromagnetically exchange coupled nanomagnets

Bistable nanomagnets store a binary bit of information. Exchange coupled nanomagnets can increase the thermal stability at low dimensions. Here we show that the antiferromagnetically (AFM) coupled nanomagnets can be highly stable at low dimensions than that of the ferromagnetically coupled nanomagnets. By solving stochastic Landau–Lifshitz–Gilbert equation of magnetization dynamics at room temperature, we analyze the stability of the exchange coupled nanomagnets in the presence of correlated, uncorrelated, and anti-correlated noise. The results show that the correlated noise can make the stability of the AFM coupled nanomagnets very high. Such finding will lead to very high-density non-volatile storage and logic devices in our future information processing systems.

Second harmonic generation in thin permalloy film

Second harmonic generation versus strength and direction of the applied static magnetic field was measured for a thin permalloy (Ni80Fe20) film in a microstrip line at a driving frequency of 1 GHz and maximum input power of ∼110 mW. The measurements revealed two peaks in the double frequency signal—in the low static field (∼10 Oe) and the high one (∼45 Oe). To explain these findings, a macrospin model of a thin magnetic film with in-plane uniaxial magnetic anisotropy was considered. A perturbation expansion of the Landau–Lifshitz–Gilbert equation provided an explanation of the experimental data. The analysis of the model revealed that the low-field peak was caused by the longitudinal second-order magnetization component and the high-field peak by the transversal one. It was also shown that the uniaxial magnetic anisotropy of the film and the dependence of the magnetic damping parameter on the applied field play an important role in the process of the second harmonic generation. The results obtained give insights into some peculiarities of the nonlinear magnetization dynamics that are important in the development of magnetic film-based devices in the field of microwave signal processing and manipulation.

Diffusion of a magnetic skyrmion in two-dimensional space

Two-dimensional magnetic skyrmions are particle-like magnetic domains in magnetic thin films. The kinetic property of magnetic skyrmions at finite temperature is well described by the Thiele equation, including a stochastic field and a finite mass. This study examines the validity of the constant-mass approximation by comparing the Fourier spectrum of the Brownian motions described by the Thiele equation and the Landau–Lifshitz–Gilbert equation. Then, the expression of the diffusion flow is derived from the four-dimensional Fokker–Planck equation by considering Chandrasekhar's solution for the Thiele dynamics. Finally, a method to observe the gyro-diffusion is discussed.

An observable effect of spin inertia in slow magneto-dynamics: increase of the switching error rates in nanoscale ferromagnets * *Invited paper.

The Landau–Lifshitz–Gilbert (LLG) equation, used to model magneto-dynamics in ferromagnets, tacitly assumes that the angular momentum associated with spin precession can relax instantaneously when the real or effective magnetic field causing the precession is turned off. This neglect of ‘spin inertia’ is unphysical and would violate energy conservation. Recently, the LLG equation was modified to account for inertia effects. The consensus, however, seems to be that such effects would be unimportant in slow magneto-dynamics that take place over time scales much longer that the relaxation time of the angular momentum, which is typically few fs to perhaps ∼100 ps in ferromagnets. Here, we show that there is at least one very serious and observable effect of spin inertia even in slow magneto-dynamics. It involves the switching error probability associated with flipping the magnetization of a nanoscale ferromagnet with an external agent, such as a magnetic field. The switching may take ∼ns to complete when the field strength is close to the threshold value for switching, which is much longer than the angular momentum relaxation time, and yet the effect of spin inertia is felt in the switching error probability. This is because the ultimate fate of a switching trajectory, i.e. whether it results in success or failure, is influenced by what happens in the first few ps of the switching action when nutational dynamics due to spin inertia hold sway. Spin inertia increases the error probability, which makes the switching more error-prone. This has vital technological significance because it relates to the reliability of magnetic logic and memory.

Extended micromagnetic model for the detection of superparamagnetic labels using a GMR vortex sensor

A self-consistent micromagnetic model is proposed for simulating interactions between a superparamagnetic label particle and a low-noise GMR vortex sensor, focusing on most common operations such as label detection and saturation. For this purpose, we evaluate the combined action of vortex stray field and the applied external field.By solving the Landau–Lifshitz-Gilbert equation we find that a superparamagnetic label with a diameter of 60 nm can be successfully detected by a magnetic vortex structure assisted with an external magnetic field of 100 mT. Subsequently, the self-consistent spin diffusion model is applied to calculate electric response of a standard low-noise GMR vortex sensor to the presence of magnetically-saturated label. It is shown that the stray field of the activated label produces a detectable potential difference in the GMR sensor of 27.1 to 27.8 mV, the magnitude of which depends on location of the label with respect to the sensor surface.

Magnetization dynamics of antiferromagnetic metals of PtMn and IrMn driven by a pulsed spin-transfer torque

Antiferromagnets are promising materials for memory devices owing to their ultrafast spin dynamics. For efficient control of antiferromagnets, a direct interaction between the spin current and local magnetization of the sublattice is required. In this study, we demonstrated that a short-pulsed spin current can induce fast dynamics of metallic antiferromagnets of PtMn and IrMn via spin-transfer torque on the magnetization of sublattices. We employed two methods to generate a short-pulsed spin current, namely ultrafast demagnetization of a ferromagnet and optical spin polarization of a heavy metal. The magnetization dynamics were measured using the time-resolved magneto-optical Kerr effect and were analyzed using the Landau–Lifshitz–Gilbert equation. Our results provide important evidence of the direct interaction between the magnetization of antiferromagnets and spin current.