Landau-Lifshitz Equation Research Articles

LOCAL ENERGY ESTIMATES FOR THE MAXWELL–LANDAU–LIFSHITZ SYSTEM AND APPLICATIONS

We study the Maxwell–Landau–Lifshitz system without exchange energy. First, we prove an [Formula: see text] estimate for the linear wave equation and apply this local energy estimate to obtain a bound on the curl of the electromagnetic field, uniformly in time and locally in space. Next, we prove strong convergence results, when the time t tends to ∞ or when the speed of light tends to ∞ (which corresponds to the quasi-stationary approximation). Finally, we establish a stability result with respect to the damping parameter of the Landau–Lifshitz equation.

Frustrated vortex in a two-dimensional antiferromagnet

The interaction of a magnetic vortex with the frustration created by a magnetic defect is investigated in a discrete Heisenberg model of a two-dimensional antiferromagnet with easy-plane anisotropic exchange. Numerical solutions are obtained for the static Landau-Lifshitz equations describing the spin distribution in a system with magnetic frustration and a vortex. It is found that the energy of the magnet is minimum in the case when the center of the vortex coincides with the position of the magnetic impurity. It is shown that as a result of the attraction between the vortex and frustration, a two-dimensional solitonic bound state localized at the magnetic defect—a frustrated vortex—arises in the magnet. The energy of such a vortex is lower than that of the free vortex, and this effect can be manifested in features of the behavior of the EPR linewidth in two-dimensional magnets.

Error estimates for a semi-implicit numerical scheme solving the Landau–Lifshitz equation with an exchange field

We study the Landau–Lifshitz (LL) equation describing the evolution of spin fields in continuum ferromagnets. We consider the 3D case when the effective field arising in the LL equation includes exchange interaction, the most challenging case. This setting corresponds to the pure isotropic case without a demagnetizing field. We derive some regularity results for the exact solution to the LL with Neumann-type boundary conditions. We modify the numerical scheme studied by A. Prohl in two dimensions and we prove error estimates for this scheme in three dimensions.

A numerical method for the Landau–Lifshitz equation with magnetostriction

AbstractWe present a numerical scheme for Landau–Lifshitz–Gilbert equation coupled with the equation of elastodynamics. The considered physical model describes the behaviour of ferromagnetic materials when magnetomechanical coupling is taken into account. The time‐discretization is based on the backward Euler method with projection. In the numerical approximation, the two equations are decoupled by a suitable linearization in order to solve the magnetic and mechanic part separately. The resulting semi‐implicit scheme is linear and allows larger time‐steps than explicit methods. We prove stability and error estimates for the presented time discretization in 2D. Finally, we test the accuracy of the scheme on an academic numerical example with known exact solution. Copyright © 2005 John Wiley & Sons, Ltd.

Magnetic response of nanostructured systems: A ferromagnetic resonance investigation

Ferromagnetic resonance (FMR) measurements probe the response of magnetic systems within the nanosecond-regime due to an excitation within the microwave regime. Due to the high sensitivity of FMR this technique is well suited for the investigation of nanostructures and ultrathin magnetic films or multilayers. As the resonance condition is determined by internal fields like anisotropy fields or interlayer coupling fields within layered structures, FMR experiments give direct and quantitative access to these quantities based on an analysis that uses the Landau–Lifshitz equation of motion. This will be demonstrated for the case of Ni–Cu–Ni films grown epitaxially on Cu(100) substrates and for highly monodisperse Co–CoO core–shell particles of about 10 nm diam. In case of the films the unique possibility to grow and measure the samples within an ultrahigh vacuum environment is presented.

Switching probabilities for single-domain magnetic particles

Solving the stochastic Landau-Lifshitz equation numerically, we compute as a function of time $t$ the probability per unit time, ${P}_{s}(t)$, that a classical, single-domain magnetic particle with an easy uniaxial anisotropy and a collinear applied magnetic field will reverse its magnetization (``switch'') via thermal activation over the energy barrier. The ${P}_{s}(t)$ curves increase with $t$ for small $t$, achieving a maximum at some time ${\ensuremath{\tau}}_{P}$ before decaying exponentially with time constant ${\ensuremath{\tau}}_{D}$ at long time, as per the standard Neel-Brown picture. Both ${\ensuremath{\tau}}_{P}$ and ${\ensuremath{\tau}}_{D}$ increase (the latter exponentially) with increasing barrier height; ${\ensuremath{\tau}}_{P}$ grows logarithmically with ${\ensuremath{\tau}}_{D}$, consistent with a recent phenomenological ``energy-ladder'' model, and experiments on submicron-sized magnetic thin films.

Magnetization of two-dimensional square arrays of nanomagnets

The interplay between dipole-dipole interaction, magnetocrystalline anisotropy, and disorder on the magnetic hysteresis in two-dimensional square arrays of nanomagnets is studied by numerically solving the LandauLifshitz equation. The interaction-induced frustration gives rise to even-odd oscillations in the magnetic properties when the system size is varied. The oscillations persist to remarkably large arrays with fluctuating amplitudes, evidencing the significance of the boundary effects and the competition among a number of quasistable orders of the magnetic moments. The hysteresis is strongly affected by the magnetocrystalline anisotropy and disorder. While both of them broaden the hysteresis loop, the remanence in the disordered system bears a universal value: almost half of the saturation magnetization.

Magnetic soliton and soliton collisions of spinor Bose-Einstein condensates in an optical lattice

We study the magnetic soliton dynamics of spinor Bose-Einstein condensates in an optical lattice which results in an effective Hamiltonian of anisotropic pseudospin chain. A modified Landau-Lifshitz equation is derived and exact magnetic soliton solutions are obtained analytically. Our results show that the time-oscillation of the soliton size can be controlled in practical experiment by adjusting of the light-induced dipole-dipole interaction. Moreover, the elastic collision of two solitons is investigated.

Influence of surface anisotropy on the magnetization precessional switching in nanoparticles

A study of surface anisotropy effects on the precessional switching of nanoparticles is presented. Spherical nanoparticles with uniaxial anisotropy in the bulk and radial anisotropy for spins on the boundary are considered. The multispin dynamics is found by using the Landau–Lifshitz equation with the effective field derived from a Heisenberg-type Hamiltonian. The expressions for critical magnetic fields that guarantee the precessional switching are derived analytically for the case of very strong exchange and weak surface anisotropy. These analytical results are also used to test the numerical approach that is applied to the general case of the problem. The distinct features of the precessional switching in nanoparticles are examined and their dependence on various parameters of the problem is discussed.

Microscopic study of magnetostatic spin waves

A relatively new method is developed to numerically calculate the spin-wave-related properties of a magnetic body of arbitrary shape. Starting with a discrete dipole approximation and the linearized Landau–Lifshitz equation, the resonant frequencies and the associated amplitudes of the individual moments are obtained for all modes; from this information we are able to calculate the energy absorbed by the various modes excited by a position- and time-dependent external magnetic field. The method has been demonstrated for a number of cases including thin disks and rings and for equilibrium configurations ranging from the saturated high-field limit to the vortex states at low fields.

1 ∕ f -type noise in a biased current perpendicular to the plane spin valve: A numerical study

We add the spin momentum-transfer torque to the stochastic Landau–Lifshitz equation and use it to study the noise spectrum as a function of the current and easy axis field for configurations close to equilibrium. The current perpendicular to the plane structure is biased by a constant field perpendicular to the polarization axis of the pinned layer. We show that this structure can exhibit large 1∕f-type noise for frequencies in the microwave regime. This 1∕f noise is not due to the spin torque. The spin torque can only change the amplitude of the noise.

Generalized reversible susceptibility tensor

A theory of reversible susceptibility tensor based on magnetization vector dynamics, as described by the Landau–Lifshitz equation of motion, is given. It is shown that the reversible transverse susceptibility (RTS) is in fact the zero frequency limit of the ferromagnetic resonance (FMR). Thus, the methods which have been developed previously for the theoretical description of FMR may be applied to predict the RTS behavior. The importance of these results resides in the generality of the approach which allows one to find the reversible susceptibility tensor for virtually any magnetic system if an expression for the magnetic free energy density is known.

The vector generalization of the Landau–Lifshitz equation: Bäcklund transformation and solutions

The Bäcklund transformations (BTs) for vector generalization of the Landau–Lifshitz equation are presented. Also periodic and soliton-like solutions from BTs are computed.

Ultra-soft magnetic films: micromagnetism and high frequency properties

The wavelength and amplitude of the longitudinal oscillations of the transversal and longitudinal components of magnetization was obtained from the contrast ripple observed by the Lorentz TEM. The micromagnetic ripple is connected with the internal stray field in the film. The theory, based on Landau–Lifshitz equation, is developed, where the internal stray field is taken into account. The theory predicts that the micromagnetic ripple can broaden the ferromagnetic resonance (FMR) to lower range of the frequencies. For a certain range of the internal stray field a second bump at lower frequencies arrives, which shows up in some previously reported spectra.

Existence of Partially Regular Solutions for Landau–Lifshitz Equations in ℝ3

We establish the existence of partially regular weak solutions for the Landau–Lifshitz equation in three space dimensions for smooth initial data of finite Dirichlet energy. The construction is based on Ginzburg–Landau approximation. The new key ingredient is a nonlocal representation formula for the penalty term that permits us to take advantage of the special trilinear structure of the limiting nonlinearity.

Current distribution and giant magnetoimpedance in composite wires with helical magnetic anisotropy

The giant magnetoimpedance effect in composite wires consisting of a non-magnetic inner core and soft magnetic shell is studied theoretically. It is assumed that the magnetic shell has a helical anisotropy. The current and field distributions in the composite wire are found by means of a simultaneous solution of Maxwell equations and the Landau–Lifshitz equation. The expressions for the diagonal and off-diagonal impedance are obtained for low and high frequencies. The dependences of the impedance on the anisotropy axis angle and the shell thickness are analyzed. Maximum field sensitivity is shown to correspond to the case of the circular anisotropy in the magnetic shell. It is demonstrated that the optimum shell thickness to obtain maximum impedance ratio is equal to the effective skin depth in the magnetic material.

Self-consistent theory of current-induced switching of magnetization

A selfconsistent theory of the current-induced switching of magnetization using nonequilibrium Keldysh formalism is developed for a junction of two ferromagnets separated by a nonmagnetic spacer. It is shown that the spin-transfer torques responsible for current-induced switching of magnetization can be calculated from first principles in a steady state when the magnetization of the switching magnet is stationary. The spin-transfer torque is expressed in terms of one-electron surface Green functions for the junction cut into two independent parts by a cleavage plane immediately to the left and right of the switching magnet. The surface Green functions are calculated using a tight-binding Hamiltonian with parameters determined from a fit to an {\it ab initio} band structure.This treatment yields the spin transfer torques taking into account rigorously contributions from all the parts of the junction. To calculate the hysteresis loops of resistance versus current, and hence to determine the critical current for switching, the microscopically calculated spin-transfer torques are used as an input into the phenomenological Landau-Lifshitz equation with Gilbert damping. The present calculations for Co/Cu/Co(111) show that the critical current for switching is $\approx 10^7A/cm^2$, which is in good agreement with experiment.

Transverse susceptibility as the low-frequency limit of ferromagnetic resonance

A new theory of transverse susceptibility (TS) based on magnetization vector dynamics, as described by the Landau–Lifshitz equation of motion, is given. It is shown that the traditional TS experiment is, in fact, the zero-frequency limit of the ferromagnetic resonance (FMR). The importance of these results resides in the generality of the approach which allows one to find the TS for virtually any magnetic system if an expression for the magnetic free-energy density is known. Moreover, the effect of the frequency of excitatory AC field on the TS experiments and the effect of energy dissipation through the imaginary part of TS emerge coherently from the new TS model.

Class of exact three dimensional solutions of Landau–Lifshitz equations in simply connected specimens of ferromagnets and antiferromagnets of arbitrary shape with uniaxial magnetic anisotropy

In this work,class of exact three dimensional solutions of Landau–Lifshitz equations in simply connected specimens of ferromagnets and antiferromagnets of arbitrary shape with uniaxial magnetic anisotropy, was found.

Localization properties of pure magnetostatic modes in a cubic nanograin

We investigate the dynamical properties of a system of interacting magnetic dipoles disposed in sites of an sc lattice and forming a cubic-shaped sample of size determined by the cube edge length (N-1)a (a being the lattice constant, N representing the number of dipolar planes). The dipolar field resulting from the dipole-dipole interactions is calculated numerically in points of the axis connecting opposite cube face centers (central axis) by collecting individual contributions to this field coming from each of the N atomic planes perpendicular to the central axis. The applied magnetic field is assumed to be oriented along the central axis, magnetizing uniformly the whole sample, all the dipoles being aligned parallelly in the direction of the applied field. The frequency spectrum of magnetostatic waves propagating in the direction of the applied field is found numerically by solving the Landau-Lifshitz equation of motion including the local (nonhomogeneous) dipolar field component; the mode amplitude spatial distributions (mode profiles) are depicted as well. It is found that only the two energetically highest modes have bulk-extended character. All the remaining modes are of localized nature; more precisely, the modes forming the lower part of the spectrum are localized in the subsurface region, while the upper-spectrum modes are localized around the sample center. We show that the mode localization regions narrow down as the cube size, N, increases (we investigated the range of N=21 to N=101), and in sufficiently large cubes one obtains practically only center-localized and surface-localized magnetostatic modes.