Landau-Lifshitz Equation Research Articles

The initial-boundary value problem for the Landau–Lifshitz equation with Gilbert damping term

The initial-boundary value problem for the Landau–Lifshitz equation with Gilbert damping term

Partial regular solution to the coupled spin polarized system in three dimensions

We consider the partial regular solution to the coupled spin polarized transport equations in 3D. This coupled system includes the Landau-Lifshitz equation and a quasi-linear parabolic equation describing the diffusion of spin accumulation. Our proof is strictly based on Ginzburg-Landau approximation theory and Moser iterative computation.

Discussion on generalized formulation of spin semiclassical Langevin equation

The stochastic dynamics of spin semiclassical system at finite temperature is usually described by stochastic Landau-Lifshitz equation. In this work, the stochastic differential equation for spin semiclassical system is studied. The generalized formulation of effective Langevin equation and the corresponding Fokker-Planck equation are derived. The obtained effective Langevin equation offers an accurate description of the distribution in the canonical ensemble for spin semiclassical system. When the damping term and the stochastic term vanish, the effective Langevin equation reduces to the semiclassical equation of motion for spin system. Hence, the effective Langevin equation can be seen as a generalization of the stochastic Landau-Lifshitz equation. The explicit expressions for the effective Langevin equation and the corresponding Fokker-Planck equation are shown in both Cartesian coordinates and spherical coordinates. It is demonstrated that, the longitudinal effect can be easily illustrated from the expressions in spherical coordinates. The effective Langevin equation is applied to the simple system of a single spin in a constant magnetic field. Choosing an appropriate form, the Langevin equation can be easily solved and the stationary Boltzmann distribution can be obtained. The correctness of the Langevin approach for the spin semiclassical system is thus confirmed.

Численное моделирование магнетиков «атом-в-атом», закон Блоха 3/2 и третье начало термодинамики

The traditional model of a magnet <<atom-to-atom>> considers the magnetic moments of individual atoms connected by an exchange interaction. The evolution of magnetic moments is described on the basis of the Landau-Lifshitz equation, in which a random Langevin source is introduced, determined by the final temperature of the system. In this case, one of the main problems is the violation of the Bloch 3/2 law and the third law of thermodynamics. The recalculation of the intensity of a random source, carried out by Wuu et al., taking into account magnons provides the correct 5/2 asymptotics for the energy, but does not give the correct asymptotics for the magnetization. To solve this problem, we add a random delta-correlated in time but correlated in space source to the Langevin source. This approach, in conjunction with the Wuu approach, ensures the fulfillment of the third law of thermodynamics and Bloch's 3/2 law.

Nonreciprocal Spin-Wave Propagation in a Ferromagnet With Stepwise Interfacial Dzyaloshinskii–Moriya Interaction

In this article, boundary conditions for Landau–Lifshitz equations at the interface between two ferromagnets with different interfacial Dzyaloshinskii–Moriya interactions (iDMIs) are derived. We calculated and verified the analytical expression for the energy flux density continuity for the spin-wave (SW) propagation through the ferromagnet with stepwise iDMI considering the boundary conditions mentioned. Analytical expressions for reflection, transmission coefficients, and the nonreciprocity factor are calculated in the case of SW propagation through a ferromagnet with the stepwise iDMI. Two different types of nonreciprocity effects for SW propagation through a ferromagnet with the stepwise interface Dzyaloshinskii–Moriya interaction are revealed for opposite orientations of Dzyaloshinskii–Moriya vector. The material parameters of a ferromagnet and the stepwise iDMIs are found for which the extremely high nonreciprocity factor (>10) is expected according to the results of calculations. The results of this article deepen the knowledge about the SW propagation control in magnonic devices.

Correlation between interlayer exchange coupling and domain wall velocity in multilayered magnetic wires

The effect of interlayer exchange coupling (IEC) in {Tb/Co}4/Rh/{Co/Tb}4/Pt multilayered wires on the domain wall (DW) structure and DW velocity was investigated. It was found that the stability of the Néel-type DWs was enhanced and the velocity of the current-induced DW movement increased with an increase in the antiferromagnetic IEC strength. The micromagnetic simulation based on the Landau–Lifshitz equation clarifies that the direction of the magnetic moment in the DWs depends on the antiferromagnetic IEC strength, and the stability of the Néel-type DW increases with an increase in the antiferromagnetic IEC strength. As a consequence of this stability, the spin-orbital torque affects effectively on the magnetic moment of the DWs and the DW velocity increases. These results imply that the DW velocity in magnetic domain-based memories can be controlled by the antiferromagnetic IEC strength.

Existence and uniqueness of weak solutions to the Landau–Lifshitz equation for large ferromagnetic bodies

In this paper, we follow the exchange model considered by Han, Tan, and Yang in [SIAM J. Math. Anal., 53(6), 7024–7061 (2021)] to consider the nonexchange case. More precisely, we study the weak solutions of the two‐ or three‐dimensional Landau–Lifshitz equation describing the nonexchange energy model in large ferromagnetic bodies. It is different from the classical Landau–Lifshitz equation because there is no regularization effect of the exchange energy. We prove the existence and uniqueness of global weak solutions. As a by‐product, we obtain the local time ‐stability of the weak solution.

Effect of radiation-reaction on charged particle dynamics in a focused electromagnetic wave

The effect of radiation-reaction force on the dynamics of a charged particle in an intense focused light wave is investigated using the physically appealing Hartemann-Luhmann equation of motion. It is found that, irrespective of the choice of initial conditions, radiation reaction force causes the charged particle to cross the focal region, provided the particle is driven into regions where the radiation reaction force dominates over the Lorentz force, thus enhancing the forward energy gained by the particle from the intense light wave. This result is in sharp contrast to the well known result, derived in the absence of radiation reaction forces, where for certain initial conditions the particle reflects from the high intensity region of the focused light wave, thereby losing forward energy. From the perspective of energy gain, our studies clearly show that the parameter space for forward energy gain which is reduced by ponderomotive effects is compensated by radiation reaction effects. These results, which are of relevance to the present day direct laser acceleration schemes of charged particle, also agrees with that obtained using the well known Landau-Lifshitz equation of motion.

Heterogeneous Multiscale Methods for the Landau–Lifshitz Equation

In this paper, we present a finite difference Heterogeneous Multiscale Method for the Landau–Lifshitz equation with a highly oscillatory diffusion coefficient. The approach combines a higher order discretization and artificial damping in the so-called micro problem to obtain an efficient implementation. The influence of different parameters on the resulting approximation error is discussed. Further important factors that are taken into account are the choice of time integrator and the initial data for the micro problem which has to be set appropriately to get a consistent scheme. Numerical examples in one and two space dimensions and for both periodic as well as more general coefficients are given to demonstrate the functionality of the approach.

Existence and Stability of Cylindrical Symmetric Static Solutions to the Landau–Lifshitz Equation for Multidirectional Ferromagnets

Existence and Stability of Cylindrical Symmetric Static Solutions to the Landau–Lifshitz Equation for Multidirectional Ferromagnets

Magnetic neutron scattering from spherical nanoparticles with Néel surface anisotropy: analytical treatment.

The magnetization profile and the related magnetic small-angle neutron scattering cross section of a single spherical nanoparticle with Néel surface anisotropy are analytically investigated. A Hamiltonian is employed that comprises the isotropic exchange interaction, an external magnetic field, a uniaxial magnetocrystalline anisotropy in the core of the particle and the Néel anisotropy at the surface. Using a perturbation approach, the determination of the magnetization profile can be reduced to a Helmholtz equation with Neumann boundary condition, whose solution is represented by an infinite series in terms of spherical harmonics and spherical Bessel functions. From the resulting infinite series expansion, the Fourier transform, which is algebraically related to the magnetic small-angle neutron scattering cross section, is analytically calculated. The approximate analytical solution for the spin structure is compared with the numerical solution using the Landau-Lifshitz equation, which accounts for the full nonlinearity of the problem. The signature of the Néel surface anisotropy can be identified in the magnetic neutron scattering observables, but its effect is relatively small, even for large values of the surface anisotropy constant.

Magnetic neutron scattering from spherical nanoparticles with Néel surface anisotropy: atomistic simulations.

A dilute ensemble of randomly oriented non-interacting spherical nanomagnets is considered, and its magnetization structure and ensuing neutron scattering response are investigated by numerically solving the Landau-Lifshitz equation. Taking into account the isotropic exchange interaction, an external magnetic field, a uniaxial magnetic anisotropy for the particle core, and in particular the Néel surface anisotropy, the magnetic small-angle neutron scattering cross section and pair-distance distribution function are calculated from the obtained equilibrium spin structures. The numerical results are compared with the well known analytical expressions for uniformly magnetized particles and provide guidance to the experimentalist. In addition, the effect of a particle-size distribution function is modelled.

Solitons in low-dimensional magnets: Elementary excitations with a nontrivial dispersion law

Solitons are known to play the role of elementary excitations for one-dimensional ordered systems, like atomic chains with charge or spin ordering. The main characteristic of solitons is their dispersion relation, dependence of soliton energy on the linear momentum. Topological kink-type solitons are the simplest and most important for the description of many physical properties of one-dimensional magnets. Here we provide a detailed analysis of solitons in some general class of magnets, ferrimagnets with the spin compensation point. The nonlinear spin dynamics of ferrimagnets are examined using a nonlinear sigma-model for the antiferromagnetic vector, which is a generalization of the Landau-Lifshitz equation for ferromagnets and sigma-model for the antiferromagnets. The characteristic features of this equation are governed by the value of the compensation parameter, describing the rate of compensation of spins of sublattices. The dispersion relation for kink-type solitons appears to be quite nontrivial, including periodic dispersion law for continuum model of magnet or the presence of ending point for kink spectrum.

Omnidirectional flat bands in chiral magnonic crystals

The magnonic band structure of two-dimensional chiral magnonic crystals is theoretically investigated. The proposed metamaterial involves a three-dimensional architecture, where a thin ferromagnetic layer is in contact with a two-dimensional periodic array of heavy-metal square islands. When these two materials are in contact, an anti-symmetric exchange coupling known as the Dzyaloshinskii–Moriya interaction (DMI) arises, which generates nonreciprocal spin waves and chiral magnetic order. The Landau–Lifshitz equation and the plane-wave method are employed to study the dynamic magnetic behavior. A systematic variation of geometric parameters, the DMI constant, and the filling fraction allows the examination of spin-wave propagation features, such as the spatial profiles of the dynamic magnetization, the isofrequency contours, and group velocities. In this study, it is found that omnidirectional flat magnonic bands are induced by a sufficiently strong Dzyaloshinskii–Moriya interaction underneath the heavy-metal islands, where the spin excitations are active. The theoretical results were substantiated by micromagnetic simulations. These findings are relevant for envisioning applications associated with spin-wave-based logic devices, where the nonreciprocity and channeling of the spin waves are of fundamental and practical scientific interest.

Frequency dynamics of a chain of magnetized rotors: dumbbell model vs Landau–Lifshitz equation

During the past decades magnetic materials and structures that span several length scales have been of interest mainly due to their application in data storage and processing, flexible electronics, medicine, between others. From a microscopic point of view, these systems are typically studied using the Landau–Lifshitz equation (LLE), while approaches such as the dumbbell model are used to study macroscopic magnetic structures. In this work we use both the LLE and the dumbbell model to study spin chains of various lengths under the effect of a time dependent-magnetic field, allowing us to compare qualitatively the results obtained by both approaches. This has allowed us to identify and describe in detail several frequency modes that appear, with additional modes arising as the chain length increases. Moreover, we find that high frequency modes tend to be absorbed by lower frequency ones as the amplitude of the field increases. The results obtained in this work are of interest not only to better understand the behavior of the macroscopic spins chains, but also expands the available tools for qualitative studies of both macroscopic and microscopic versions of the studied system, or more complex structures such as junctions or lattices. This would allow to study the qualitative behavior of microscopic systems (e.g. nanoparticles) using macroscopic arrays of magnets, and vice versa.

Approximated analytical solution of the Landau–Lifshitz equation in tightly focused laser beams in the ultrarelativistic limit

The Landau–Lifshitz equation is widely considered as the equation, which correctly includes the effects of radiation reaction in the classical motion of an electric charge. Here, we provide an approximated analytical solution of the Landau–Lifshitz equation in the presence of a virtually arbitrary electromagnetic field by making the assumptions that the electron’s initial energy is the largest dynamical energy in the problem. We show that in a regime where radiation–reaction effects are a small correction of the Lorentz dynamics, the four-momentum of the electron can be determined perturbatively in the ultrarelativistic limit. We explicitly compute the electron four-momentum up to the first order and in the experimentally relevant case of an ultrashort, tightly focused laser beam as an external field.Graphical abstract

Spectrum of exchange spin waves in a one-dimensional magnonic crystal with antiferromagnetic ordering

Purpose of the study is to show that the conditions for the propagation of exchanged spin waves (ESWs) in an asymmetric superlattice with antiferromagnetically ordered cells depend significantly on the chirality of the precession of the ESW magnetization (polarization, “magnon pseudospin”). Method. When constructing the EWS spectra, the Croning– Penny model (transfer-matrix method) and the Landau–Lifshitz equation are used to determine the nature of the waves in the cells. In the case of a uniaxial medium, there is only one type of ESW, therefore, when fields are joined at the boundary, the conservation of chirality is an essential factor due to which the ESW in one cell is always traveling, and in the other — evanescent. Thus, a superlattice for ESW is an effective periodic “potential” in which asymmetry can be realized either by applying an external field, or by a difference in the thickness and/or physical properties of the cell materials. Results. Based on the analysis of the spectrum, maps of the transmission zones for ESW of different chirality were constructed in three representations — “Bloch wave number – frequency”, “frequency – relative cell thickness”, as well as in the plane of cell wave numbers. It is shown that the presence of asymmetry leads to a difference in the width of the transmission zones for waves of different chirality. For a finite structure, the frequency dependences of the transmission and reflection coefficients of the ESW are plotted. An increase in the attenuation of the ESW near the boundaries of the transmission zones was also found. Conclusion. The results of the study can be used in the design of magnon valves and other devices based on ESW, in which their chirality can be controlled.

Weak Solutions to a Landau–Lifshitz Equation with a p -Laplacian Operator as an Exchange Field

We are dealing with the classical Landau–Lifshitz equation with an unusual exchange field expressed in terms of a p -Laplacian operator for p greater than 2. We obtain weak solutions to the model by using penalization, compactness arguments, and monotonicity method.

Influences of Anisotropic Equivalent Field and Magnetic Damping Coefficient on Giant Magnetoimpedance Effect of Cylindrical Alloy Fibers: Theoretical Magnetoimpedance Calculations

In this paper, the giant magneto-impedance (GMI) model of a cylindrical alloy fiber was established by the Maxwell equation and Landau–Lifshitz equation to simulate the influence of physical parameters of cylindrical alloy fiber on GMI under different control parameters. MATLAB was employed to calculate the magneto-impedance of cylindrical fibers and draw its curves. We found that when the anisotropic equivalent field of the fiber changes from 10Oe to 50Oe, the peak position of the GMI ratio also moves from about 10Oe to 50Oe, and the peak value gradually increases from 100% to 300%. The GMI ratio increased rapidly with the decrease in the magnetization damping coefficient. Our findings could further guide the design of supersensitive micro GMI sensors by optimally regulating the magnetic damping coefficient, the angle between the external magnetic field and easy axis and the anisotropic equivalent field of cylindrical alloy fibers.

Local well-posedness of the Landau–Lifshitz equation with helicity term

We consider the initial value problem for the Landau–Lifshitz equation with a helicity term (chiral interaction term), which arises from the Dzyaloshinskii–Moriya interaction energy. We prove that it is well-posed locally in-time in the space k̄+Hs for s ≥ 3 with s∈Z and k̄=(0,0,1)t. We also show that if we further assume that the solution is homotopic to constant maps, then local well-posedness holds in the space k̄+Hs for s > 2 with s∈R. Our proof is based on two different approaches: One is the geometric energy method by McGahagan and the other is via the modified Schrödinger map equation. In the present analysis, we exploit a special structure of the helicity term, which enables us to overcome the difficulty in the quadratic derivative nonlinearity of the helicity term.