Landau Lifshitz Gilbert Dynamics Research Articles

Ultra-Low-Energy Electric Field-Induced Magnetization Switching in Multiferroic Heterostructures

Electric field-induced magnetization switching in multiferroics is intriguing for both fundamental studies and potential technological applications. Here, we review the recent developments on electric field-induced magnetization switching in multiferroic heterostructures. Particularly, we study the dynamics of magnetization switching between the two stable states in a shape-anisotropic single-domain nanomagnet using stochastic Landau–Lifshitz–Gilbert (LLG) equation in the presence of thermal fluctuations. For magnetostrictive nanomagnets in strain-coupled multiferroic composites, such study of magnetization dynamics, contrary to steady-state scenario, revealed intriguing new phenomena on binary switching mechanism. While the traditional method of binary switching requires to tilt the potential profile to the desired state of switching, we show that no such tilting is necessary to switch successfully since the magnetization’s excursion out of magnet’s plane can generate a built-in asymmetry during switching. We also study the switching dynamics in multiferroic heterostructures having magnetoelectric coupling at the interface and magnetic exchange coupling that can facilitate to maintain the direction of switching with the polarity of the applied electric field. We calculate the performance metrics like switching delay and energy dissipation during switching while simulating LLG dynamics. The performance metrics turn out to be very encouraging for potential technological applications.

Consistent energy barrier distributions in magnetic particle chains

We investigate long-time thermal activation behaviour in magnetic particle chains of variable length. Chains are modelled as Stoner–Wohlfarth particles coupled by dipolar interactions. Thermal activation is described as a hopping process over a multidimensional energy landscape using the discrete orientation model limit of the Landau–Lifshitz–Gilbert dynamics. The underlying master equation is solved by diagonalising the associated transition matrix, which allows the evaluation of distributions of time scales of intrinsic thermal activation modes and their energy representation. It is shown that as a result of the interaction dependence of these distributions, increasing the particle chain length can lead to acceleration or deceleration of the overall relaxation process depending on the initialisation procedure.

Precessing vortices and antivortices in ferromagnetic elements

A micromagnetic numerical study of the precessional motion of the vortex and antivortex states in soft ferromagnetic circular nanodots is presented using Landau-Lifshitz–Gilbert dynamics. For sufficiently small dot thickness and diameter, the vortex state is metastable and spirals toward the center of the dot when its initial displacement is smaller than a critical value. Otherwise, the vortex spirals away from the center and eventually exits the dot which remains in a state of in-plane magnetization (ground state). In contrast, the antivortex is always unstable and performs damped precession resulting in annihilation at the dot circumference. The vortex and antivortex frequencies of precession are compared with the response expected on the basis of Thiele’s theory of collective coordinates. We also calculate the vortex restoring force with an explicit account of the magnetostatic and exchange interaction on the basis of the “rigid” vortex and “two-vortices side charges free” models and show that neither model explains the vortex translation mode eigenfrequency for nanodots of sufficiently small size.

New approach to spin dynamics simulation in magnetic system

New approach to spin dynamics simulation in magnetic system is presented. In the approach, we substitute new algorithm for Landau–Lifshitz–Gilbert dynamics, which enable us to achieve the final stable state of magnetic system much faster than traditional spin dynamics simulation. A square-shaped sample with 32 × 32 × 4 size under different conditions is calculated. Our results show the new approach can largely reduce the computational time.

Wave-type dynamics in ferromagnetic thin films and the motion of Néel walls

We investigate the magnetization dynamics in soft ferromagnetic films with small damping. In this case, the gyrotropic nature of Landau–Lifshitz–Gilbert dynamics and the shape anisotropy effects from stray-field interactions effectively lead to a wave-type dynamics for the in-plane magnetization. We apply this result to study the motion of Néel walls in thin films and prove the existence of a travelling wave solution under a small constant forcing.

Generalized Landau–Lifshitz–Gilbert equation for uniformly magnetized bodies

We consider generalized Landau–Lifshitz–Gilbert (LLG) deterministic dynamics in uniformly magnetized bodies. The dynamics take place on the unit sphere Σ , and are characterized by a vector field v tangential to Σ . By using Helmholtz decomposition on Σ , it is proven that v is uniquely defined by two potentials χ and ψ . Potential χ can be identified with the free energy of the system, while ψ describes non-conservative interactions of the system with the environment. The presence of ψ modifies the usual energy balance of LLG dynamics. Instead of purely relaxation dynamics we may have steady injection of energy through non-conservative interactions. The implications of the new form of the energy balance are discussed in detail.

Ferromagnetic resonance in an ensemble of nanoparticles with randomly distributed anisotropy axes

Spectra of absorbed power as probed by ferromagnetic resonance (FMR) are numerically calculated within a macro-spin model for single domain nanoparticles using Landau–Lifshitz–Gilbert dynamics. Randomly distributed anisotropy axis and a distribution of anisotropy energies result in a significant broadening of the FMR signal as compared to an ensemble of particles all having the same anisotropy. Additionally, a temperature dependence of the shift of the resonance frequency is obtained which is in a qualitative agreement with experimental data on single domain nanoparticles.

Geometrical integration of Landau–Lifshitz–Gilbert equation based on the mid-point rule

Landau–Lifshitz–Gilbert (LLG) equation is the fundamental equation to describe magnetization vector field dynamics in microscale and nanoscale magnetic systems. This equation is highly nonlinear in nature and, for this reason, it is generally solved by using numerical techniques. In this paper, the mid-point rule time-stepping technique is applied to the numerical time integration of LLG equation and the relevant properties of the numerical scheme are discussed. The mid-point rule is an unconditionally stable and second order accurate scheme which preserves the fundamental geometrical properties of LLG dynamics. First, it exactly preserves the LLG property of conserving the magnetization magnitude at each spatial location. Second, for constant in time applied fields, it preserves the LLG Lyapunov structure, namely the fact that the free energy is a decreasing function of time. In addition, in the case of zero damping, the mid-point rule preserves the conservation of the system free energy. The above preservation properties are unconditionally valid, i.e. they are fulfilled for any value of the time-step. Finally, the LLG hamiltonian structure in the case of zero damping is preserved up to the third order terms with respect to the time-step. The main difficulty related to this scheme is the necessity of solving a large system of globally coupled nonlinear equations. This problem has been circumvented by using special and reasonably fast quasi-Newton iterative technique. The proposed numerical scheme is then tested on the standard micromagnetic problem no. 4. In the numerical computations, the spatial discretization is obtained by finite difference technique and the magnetostatic field is computed through the Fast Fourier Transform method.

Numerical integration of Landau–Lifshitz–Gilbert equation based on the midpoint rule

The midpoint rule time discretization technique is applied to Landau–Lifshitz–Gilbert (LLG) equation. The technique is unconditionally stable and second-order accurate. It has the important property of preserving the conservation of magnetization amplitude of LLG dynamics. In addition, for typical forms of the micromagnetic free energy, the midpoint rule preserves the main energy balance properties of LLG dynamics. In fact, it preserves LLG Lyapunov structure and, in the case of zero damping, the system free energy. All the above preservation properties are fulfilled unconditionally, namely, regardless of the choice of the time step. The proposed technique is then tested on the standard micromagnetic problem No. 4. In the numerical computations, the magnetostatic field is computed by the fast Fourier transform method, and the nonlinear system of equations connected to the implicit time-stepping algorithm is solved by special and reasonably fast quasi-Newton technique.

Effective dynamics for ferromagnetic thin films: a rigorous justification

In a thin–film ferromagnet, the leading–order behaviour of the magnetostatic energy is a strong shape anisotropy, penalizing the out–of–plane component of the magnetization distribution. We study the thin–film limit of Landau–Lifshitz–Gilbert dynamics, when the magnetostatic term is replaced by this local approximation. The limiting two–dimensional effective equation is overdamped, i.e. it has no precession term. Moreover, if the damping coefficient of three–dimensional micromagnetics isa, then the damping coefficient of the two–dimensional effective equation isa+ 1/a; thus reducing the damping in three dimensions can actually increase the damping of the effective equation. This result was previously shown by García–Cervera and E using asymptotic analysis; our contribution is a mathematically rigorous justification.

Analytical approach to current-driven self-oscillations in Landau–Lifshitz–Gilbert dynamics

Nonlinear magnetization dynamics in uniformly magnetized bodies subject to spin-polarized current, is described by Landau–Lifshitz–Gilbert (LLG) equation with an additional spin-transfer torque term. The resulting magnetization dynamics may exhibit self-oscillatory regimes, i.e. limit cycles. By using the fact that spin-transfer torque and Gilbert damping are small perturbations of the conservative LLG dynamics, the analysis of limit cycles is carried out by an appropriate perturbation method known as Melnikov-function technique. The technique is then applied to the analysis of a typical current-driven switching process in magnetic thin film. Analytical formulas for frequency, amplitude of limit cycles in function of the injected current are derived along with critical values of current which characterized the switching process. Finally, the accuracy of the perturbative technique is tested by comparing analytical results with numerical solutions.

Nonlinear magnetization dynamics and magnetization switching in uniformly magnetized bodies

Magnetization dynamics in uniformly magnetized ferromagnetic bodies is studied by using Landau–Lifshitz–Gilbert (LLG) equation. This equation is written in a generalized form to take into account the effect of spin-polarized currents. The general properties of the corresponding nonlinear dynamical system are studied in detail. It is underlined that, in many experimental situations relevant to magnetic storage applications, LLG dynamics is a small perturbation of conservative precessional dynamics. In this respect, the conservative case is treated in detail and it is shown that analytical solutions of the conservative dynamics can be derived by using the existence of two integrals of motion. The (perturbed) dissipative LLG dynamics is then studied by developing appropriate perturbation techniques. The general methods of analysis discussed in the first part of the paper are applied to two particular problems: precessional switching and current-driven switching. The accuracy of theoretical predictions are tested by comparing analytical results with numerical solutions.

Analysis of instabilities in nonlinear Landau–Lifshitz–Gilbert dynamics under circularly polarized fields

The stability of large spatially uniform magnetization motions driven by circularly polarized fields is studied within the framework of the Landau–Lifshitz–Gilbert equation. It is demonstrated that the occurrence of spin-wave instability of large scale magnetization dynamics may depend on history of its formation.

Spin-wave instabilities in Landau–Lifshitz–Gilbert dynamics

The stability of large rotationally invariant magnetization motions is analytically studied. By using the analytical form of rotationally invariant solutions of the Landau–Lifshitz–Gilbert (LLG) equation and by performing the perturbation analysis around these solutions, spin-wave instability conditions valid for arbitrary frequencies and amplitudes of the driving RF field are derived. It is demonstrated that sufficiently large magnetization motions are always stable with respect to spin-wave perturbations. It is also demonstrated that the occurrence of spin-wave instability of spatially uniform large scale magnetization dynamics may depend on history of its formation.

Bifurcation analysis of Landau–Lifshitz–Gilbert dynamics under circularly polarized field

Uniform solutions of Landau–Lifshitz–Gilbert equation coupled with magnetostatic Maxwell equations are discussed in the case where the problem is rotationally invariant around a certain axis and the external field is circularly polarized in the perpendicular plane. It is shown that a remarkably rich variety of phase portraits is present in the dynamics, with two or four time-harmonic modes rigidly rotating with the field (P modes) and zero, one, or two quasiperiodic modes (Q modes). Different portraits are separated by bifurcation lines of saddle node, Andronov–Hopf, homoclinic-saddle connection, and semistable-limit-cycle type. The complete phase portrait and bifurcation diagram of thin films with negligible crystal anisotropy is presented and discussed.

Nonlinear aspects of Landau–Lifshitz–Gilbert dynamics under circularly polarized field

Landau–Lifshitz–Gilbert dynamics of a particle under circularly polarized field is reduced to exactly solvable autonomous form. Two or four rotating magnetization modes are possible. Stability, bifurcation diagrams, and existence of quasi-periodic modes are discussed. Connection with Stoner–Wohlfarth theory is shown.