Modified Landau-Lifshitz Equation Research Articles

Electron Paramagnetic Resonance and the Modified Landau–Lifshitz Equation in Strongly Correlated Electronic Systems with Quantum Fluctuations of the Magnetic Moment

Electron Paramagnetic Resonance and the Modified Landau–Lifshitz Equation in Strongly Correlated Electronic Systems with Quantum Fluctuations of the Magnetic Moment

Riemann–Hilbert problem for the modified Landau–Lifshitz equation with nonzero boundary conditions

We study systematically a matrix Riemann-Hilbert problem for the modified Landau-Lifshitz (mLL) equation with nonzero boundary conditions at infinity. Unlike the zero boundary conditions case, there occur double-valued functions during the process of the direct scattering. In order to establish the Riemann-Hilbert (RH) problem, it is necessary to make appropriate modification, that is, to introduce an affine transformation that can convert the Riemann surface into a complex plane. In the direct scattering problem, the analyticity, symmetries, asymptotic behaviors of Jost functions and scattering matrix are presented in detail. Furthermore, the discrete spectrum, residual conditions, trace foumulae and theta conditions are established with simple and double poles. The inverse problems are solved via a matrix RH problem formulated by Jost function and scattering coefficients. Finally, the dynamic behavior of some typical soliton solutions of the mLL equation with reflection-less potentials are given to further study the structure of the soliton waves. In addition, some remarkable characteristics of these soliton solutions are analyzed graphically. According to analytic solutions, the influences of each parameters on dynamics of the soliton waves and breather waves are discussed, and the method of how to control such nonlinear phenomena are suggested.

On Guided Remagnetization in Layered Anoheterostructures

Field-guided magnetic dynamics in magnetic multilayer nanostructures involves interconnection of the control field with localized spin states, which can occur directly or indirectly depending on the nature of the field and spin polarization. At the control electromagnetic field, this interconnection can be directly induced by the photon-induced spin-flip processes and indirectly by a bias field during antiferromagnetic exchange relaxation. The control impact of electric field and electric current on the magnetic states occurs indirectly via the spin polarization and spin current in combination with the exchange interaction of these polarized spins with localized magnetic states. The corresponding description of the magnetic dynamics is based on the modified Landau-Lifshitz equation and spin diffusion equations, taking into account the spin Hall and the inverse spin Hall effects for systems with normal metal sublayers. In the case of the magnetic nanostructures with the Rashba spin-orbit interaction in interfaces, the electric field-controlled magnetization is realized via the Rashba field-induced spin polarization, and its exchange interaction with localized magnetic states. Corresponding description is based on a tight-binding model of spin-orbit-coupled electrons exchange coupled to the localized magnetic states.

Gauge equivalent structure and solitary wave solution for a modified Landau–Lifshitz equation

In this paper, the gauge equivalence between the modified Landau–Lifshitz equation and the perturbed defocusing nonlinear Schrödinger equation is proved from the perspective of geometry of given curvature condition. By using the gauge equivalence and the soliton perturbation theory of defocusing nonlinear Schrödinger equation, the first-order approximate 1-soliton solution to the modified Landau–Lifshitz equation is given.

Current-Induced Magnetization Dynamics in 1-D Bicomponent Magnonic Crystal

Nonlinear localized magnetic excitations in 1-D bicomponent magnonic crystal are investigated under periodic magnetic field of spatially varying strength with spin current. The governing modified Landau–Lifshitz equation with spin current is transformed into the variable coefficient nonlinear Schrodinger (VCNLS) equation using a stereographic projection. In general, the VCNLS equation is nonintegrable and by using similarity transformation technique, the VCNLS equation is reduced into the standard NLS equation with certain integrability conditions. A systematic connection between the solution of VCNLS and NLS equations is established. The solution of VCNLS equation is obtained using the solution of the NLS equation. The excitation of magnetization is in the form of solitons on the oscillatory background with structure similar to the form of spin Bloch waves, and in addition, the spin current controls the soliton velocity. The velocity of soliton is directly proportional to the strength of current density and inversely proportional to the average saturation magnetization value, which is the value of saturation magnetization of the composite material at the interface of the two ferromagnetic materials, which are used to form the bicomponent magnonic crystal.

Unconventional Approach to Orbital-Free Density Functional Theory Derived from a Model of Extended Electrons

An equation proposed by Levy, Perdew and Sahni (Phys. Rev. A 30:2745, 1984) is an orbital-free formulation of density functional theory. However, this equation describes a bosonic system. Here, we analyze on a very fundamental level, how this equation could be extended to yield a formulation for a general fermionic distribution of charge and spin. This analysis starts at the level of single electrons and with the question, how spin actually comes into a charge distribution in a non-relativistic model. To this end we present a space-time model of extended electrons, which is formulated in terms of geometric algebra. Wave properties of the electron are referred to mass density oscillations. We provide a comprehensive and non-statistical interpretation of wavefunctions, referring them to mass density components and internal field components. It is shown that these wavefunctions comply with the Schrödinger equation, for the free electron as well as for the electron in electrostatic and vector potentials. Spin-properties of the electron are referred to intrinsic field components and it is established that a measurement of spin in an external field yields exactly two possible results. However, it is also established that the spin of free electrons is isotropic, and that spin-dynamics of single electrons can be described by a modified Landau-Lifshitz equation. The model agrees with the results of standard theory concerning the hydrogen atom. Finally, we analyze many-electron systems and derive a set of coupled equations suitable to characterize the system without any reference to single electron states. The model is expected to have the greatest impact in condensed matter theory, where it allows to describe an N-electron system by a many-electron wavefunction Ψ of four, instead of 3N variables. The many-body aspect of a system is in this case encoded in a bivector potential.

Modeling of Thermally Assisted Magnetodynamics

Thermomagnetic recording uses local heating of ferromagnetic media to locally decrease coercivity and change saturation magnetization of the material and alleviate magnetization reversal. A modified Landau-Lifshitz equation is proposed as a phenomenological model to allow for changes in magnetization magnitude and is studied both analytically and numerically.

Modelling of toroidal moment switching and dynamics in ferromagnetic nanorings

Switching between right and left vortex magnetization states in annular ferromagnetic nanostructures, was studied. The study was performed by numerically solving the modified Landau-Lifshitz equation with consideration of the effect of spin transfer and full-scale consideration of the magnetostatic field, exchange energy, and anisotropy energy. The dynamics of magnetization reversal of the ferromagnetic nanoring, caused by the electric current flowing perpendicularly to the object plane and penetrating the structure, was studied taking into account two mechanisms of the current effect on magnetization: by the Oersted field and spin transfer. It was found that the presence of the spin polarization both perpendicular and parallel to the nanoring plane decreases the critical current at which the object is switched by an order of magnitude. It was shown that the toroidal moment \( \vec T \) is a convenient characteristic for describing magnetization reversal processes in annular magnetic nanostructures.

Spin-transfer-induced magnetization switching in magnetic tunnel junctions

Micromagnetic simulations for spin-transfer-induced magnetization dynamics in nanoscale low-resistance magnetic tunnel junctions (MTJs) are performed by using the modified Landau-Lifshitz equation in which a spin-transfer model for MTJs is included. Calculations show that critical switching current density (Jc) is about (4–8)×106A∕cm2, consistent with the experimental data, lower than the values found in metallic spin-valve nanopillars (typically in the order of ∼107A∕cm2). A slight asymmetry of the critical current is observed in the magnetization switching loop, due to the asymmetry of the spin-torque factor. The switching time is related to the applied current, and our calculation results show that a longer switching time, 10–15ns, is needed when the applied current density J is close to Jc, and it decreases down to 1ns for J>2×107A∕cm2.

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.

Soliton solution of continuum magnetization equation in a conducting ferromagnet with a spin-polarized current.

Exact soliton solutions of a modified Landau-Lifshitz equation for the magnetization of conducting ferromagnet in the presence of a spin-polarized current are obtained by means of inverse scattering transformation. From the analytical solution the effects of spin current on the frequency, wave number, and dispersion law of spin wave are investigated. The one-soliton solution indicates obviously current-driven precession and periodic shape variation as well. The inelastic collision of solitons, by which we mean the shape change before and after collision, appears due to the spin current. We, moreover, show that complete inelastic collisions can be achieved by adjusting spectrum and current parameters. This may lead to an potential technique for shape control of spin wave.

Stress dependence of magnetization processes: Reversals and relaxation inFexCo85−xB15amorphous ribbons

The effect of stress on the domain-wall dynamics for magnetization reversal and relaxation in ${\mathrm{Fe}}_{x}{\mathrm{Co}}_{85\ensuremath{-}x}{\mathrm{B}}_{15}$ amorphous ribbons was investigated by inductance voltage measurements in the absence of applied field after pulse saturating the sample. Optical Kerr-effect and three-dimensional neutron depolarization experiments have been exploited. Residual domains at pinning sites are the main sources of the domains of reversed magnetization. Such a reversal is driven by an internal field through the motion of more than two walls, nucleated at residual domains and/or ribbon edges, and propagating towards the ribbon center. The number of active walls depends on the regularity of the domain structure. An external tensile stress decreases the voltage induced by the change of the magnetization of the sample and delays the ``depinning time ${t}_{c}''$ of the reverse domain walls. A modified Landau-Lifshitz equation was used to interpret the stress dependence of the magnetization reversal and relaxation. We found that ${t}_{c}$ is proportional to the coercive field, confirming the validity of the proposed model.

Bäcklund Transformation and Dressing Chains for the Landau–Lifshitz Equation

We use the dressing method for the Landau–Lifshitz equation and the breeding procedure to obtain a modified Landau–Lifshitz equation. We find the Darboux transformation (the “dressing”) for the integrable Landau–Lifshitz equation of the anisotropic magnet. We find the Backlund transformations and dressing chains.

On topological soliton dynamics in multidimensional ferromagnetic continuum

A multidimensional model for a ferromagnetic continuum with hydrodynamical properties, which can be regarded as a modified Landau-Lifshitz equation, is presented. The treatment of some physical examples suggests that the fluid vorticity has to be proportional to the magnetic topological current. The model can be written in the Hirota bilinear form. In two spatial dimensions, the existence of a positive definite ‘energy’ functional is shown. The Bogomol'nyi inequality leads to the self-dual equations of the model, which can be expressed by the Liouville equation. By using time-dependent gauge transformations, a wide class of solutions can be generated. These are, in general, associated with the linear problem of the modified Kadomtsev-Petviashvili equation. In some particular cases, the isolated vortices can move along arbitrary trajectories on the plane. The quantization problem of the time-dependent vortex configurations is briefly discussed, in relation to the possible evaluation of their energy spectrum.

A classical theory of superparamagnetic relaxation

A Fokker-Planck equation for the relaxation of a classical ferromagnetic particle coupled to a classical heat bath is derived from the Nakajima-Zwanzig equation. The equation of motion for the mean magnetization of an ensemble of particles is found to be closed only under special circumstances. In the strong motional narrowing limit the equation of motion reduces to the Bloch equations in the limit MH ⪡ k B T, i.e. for small particles, and to the Landau-Lifshitz equation in the opposite limit. For the motional narrowing region in toto the particular case of uniaxial anisotropy is analysed, giving an equation of motion which for large particles reduces to a modified Landau-Lifshitz equation with g-shift and a reduced damping constant. This equation cannot be meaningfully identified with the Gilbert equation. Approximate expressions for superparamagnetic relaxation rates by Kramers' method are obtained for the case of (i) triaxial (i.e. orthorhombic) and (ii) cubic (K +ve and −ve) anisotropy, assuming large energy barriers. The results supplement Brown's expression for uniaxial anisotropy and show a more complicated dependence on the Landau-Lifshitz parameter λ than the linear dependence found by Brown. For small λ the rates tend to constant values compatible with the transition.

Approximate Solution of the Equations of Magnetization Reversal by Coherent Rotation

A complete solution of the modified Landau-Lifshitz equation of magnetization reversal by coherent rotation is presented for α2≫Ha/4πMs, where α is the phenomenological damping constant, Ha is the applied field, and Ms is the saturation magnetization. This solution extends previous work in that it is valid for t≥0, whereas, other solutions have not been valid over this entire range. In addition, from this solution the time dependence of the demagnetizing field is found.

Operation of Magnetic Film Parametrons in the 100- to 500-Mc Regions

An analysis of the behavior of thin magnetic films of Permalloy when used as time variable inductors has been made in terms of a modified Landau-Lifshitz equation. These results show that parametrons using time variable magnetic film inductors can be made to operate at reasonable power levels with large grains per cycle at oscillating frequencies in the 100- to 500-Mc region. A way of fabricating magnetic film parametrons with strip line techniques is shown. Calculations indicate that units with dissipations in the 10 mw range are feasible with existing techniques for operation in the 100- to 500-Mc region. Ways of advantageously using capacitive coupling between parametron units is indicated in which the two new subharmonic states of a magnetic film parametron created by a bias reversel are used.

On the Minimum of Magnetization Reversal Time

A modified Landau-Lifshitz equation is solved for a single-domain sphere and an infinitely-wide thin single-domain sheet of ferromagnetic material neglecting anisotropy. The external magnetic field is switched from one direction to its opposite instantaneously at the initial time and the behavior of the magnetization vector is investigated thereafter. It is shown that there is a critical value of the damping constant corresponding to the minimum value of the (repetitive) magnetization reversal time. If the damping constant is larger than the critical value, the magnetization vector moves slower; if it is smaller, the magnetization vector moves faster but oscillates so that it takes longer time until it comes to a rest at the final position. The critical values of the Landau-Lifshitz damping constant λ are γM for the sphere and 0.013γM for the thin sheet, where γ and M are the gyromagnetic ratio and the magnetization, respectively. The computed minimum switching time for the thin sheet of 4–79 molybdenum Permalloy is of the order of 10−9 sec.