Landau-lifshitz-gilbert Equation
Marguerite Litscher
The various forms of the LLG equation are commonly used in micromagnetics to model the effects of a magnetic field and other magnetic interactions on ferromagnetic materials. It provides a practical way to model the time-domain behavior of magnetic elements. Recent developments generalizes the LLG equation to include the influence of spin-polarized currents in the form of spin-transfer torque.[2]
where α is a dimensionless constant called the damping factor. The effective field Heff is a combination of the external magnetic field, the demagnetizing field, and various internal magnetic interactions involving quantum mechanical effects, which is typically defined as the functional derivative of the magnetic free energy with respect to the local magnetization M. To solve this equation, additional conditions for the demagnetizing field must be included to accommodate the geometry of the material.
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We propose an extension of the Landau-Lifshitz-Gilbert (LLG) equation by explicitly including the role of conduction electrons in magnetization dynamics of conducting ferromagnets. The temporal and spatial dependent magnetization order parameter m(r,t) generates both electrical and spin currents that provide dissipation of the energy and angular momentum of the processing magnet. The resulting LLG equation contains highly spatial dependence of damping term and thus micromagnetic simulations based on the standard LLG equation should be reexamined for magnetization dynamics involving narrow domain walls and spin waves with short wavelengths.
It is well known that the Gilbert relaxation time of a magnetic moment scales inversely with the magnitude of the externally applied field, H, and the Gilbert damping, α. Therefore, in ultrashort optical pulses, where H can temporarily reach high amplitudes, the Gilbert relaxation time can momentarily be extremely short, reaching even picosecond timescales. Here we show that for strong enough ultrashort pulses, the magnetization can respond within the optical cycle such that the optical control of the magnetization emerges by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. Surprisingly, when circularly polarized optical pulses are introduced, an optically induced helicity-dependent torque results. We find that the strength of the interaction is determined by η=αγH/fopt, where fopt and γ are the optical frequency and gyromagnetic ratio, respectively. Our results illustrate the generality of the LLG equation to the optical limit and the pivotal role of the Gilbert damping in the general interaction between optical magnetic fields and spins in solids.
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The behavior of a uniformly magnetized domain of ellipsoidal shape subject to a static external field and oscillatory external driving field is analyzed near bifurcation events. The analysis includes the effects of both linear and circularly polarized driving fields and is performed using numerical simulations of the Landau-Lifshitz-Gilbert (LLG) equation. Under a linearly polarized driving field, the LLG equation is a nonautonomous differential equation which can lead to complex magnetization motions, such as bistability, multiperiodic orbits, quasiperiodicity, and chaos. Under a circularly polarized driving field, the LLG equation can be written in autonomous form by transforming to the frame rotating with the driving field. The autonomous nature allows one to perform a fixed-point analysis of the system for select demagnetization factors. Similarities and differences between the driven systems are highlighted through bifurcation diagrams, phase portraits, basins of attraction, and Lyapunov exponents. Magnetization switching, prolonged transients, quasiperiodicity, and chaos are observed with both linearly and circularly polarized driving fields in the magnetic systems investigated.
Zoomed-in view of a small section of Fig. 10 that reveals five different riddled basins of attraction. The white basin corresponds to period-five attractors of various phases, and the remaining basins correspond to the attractors shown in Fig. 10.
Eq. (2.39) does not explain the absorptionline inresonance experiments and the fact that the magnetic momenteventually aligns with field direction. The motion is modified bythe interaction with crystal lattice vibrations, conductionelectrons and other external sources. L.D Landau and E.M. Lifshitz[Landau 35] proposed to add aterm proportional to, which conservesthe magnitude, in order to obtain a phenomenological equation of thespin dynamics. Posteriorly, T. Gilbert suggested to add a viscous force to the equation ofmotion (2.39) [Gilbert 55]. The Gilbert andthe Landau-Lifshitz equations are equivalent with the renormalizationofthe precession and dissipation terms. The Gilbert version ispreferred because it predicts slower motion with increasing damping.The Gilbert equation, converted to Landau-Lifshitz form, is known asthe Landau-Lifshitz-Gilbert(LLG) equation and has the expression:
The second term of right side of Eq. (2.41), thedampingterm, makes the magnetization rotate towards the direction of theeffective field and eventually to be parallel to its direction,reaching the equilibrium. That represents a minimum of the energy.Accordingly, we can use the integration of LLG equation to minimizethe energy [Berkov 93] and thisis the method we mostly use inthis thesis. For this purpose, it is better to use a large value ofthe damping constant and to remove the precession term.
Nanoscale magnetic materials and devices are at the heart of memory and recording technologies ranging from magnetic hard drives to spintronic devices, such as magnetic random access memory (MRAM) and spin transfer torque oscillators. Advanced development of these technologies requires comprehensive computational tools. This dissertation presents a theoretical and micromagnetic study of challenges faced when considering interactions between applied fields and spin-polarized currents with nanomagnetic materials. The study is about solving the generally non-linear Landau-Lifshitz-Gilbert (LLG) equation using its linearized version. The approaches include using a linearized eigenvalue framework, solving a source-excited linearized LLG equation, and using a harmonic balance approach for the study of the higher-harmonic generation in weakly-nonlinear magnetization dynamics problems. The dissertation starts with an introduction to micromagnetics and modeling of spin-torque-driven devices. The following chapters present the eigenvalue based micromagnetic framework for spin-torque-driven devices. It presents an analysis related to the MRAM switching properties, including the critical current, switching time, and magnetization time evolution. It also introduces an optimization approach based on the eigenvalue analysis to reduce the critical current in MRAM. It then extends the eigenvalue analysis to the Fokker-Planck equation framework to study of non-switching probability, namely write error rate, under finite temperature. Next, the dissertation presents a solver for the linearized LLG equation with under time-harmonic applied fields, describing its formulation, numerical implementation, results, and analysis. The linearized LLG equation solver is finally extended to create a harmonic balance solver, which represents the solution as a set of multiple frequency components with an iterative process, which allows computing the excitation coefficients of these components. All the codes are developed in the finite element method framework, which is flexible in handling complex materials and devices, and it is integrated with the high-performance micromagnetic FastMag framework.
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