Realization of the thermal equilibrium in inhomogeneous magnetic systems by the Landau-Lifshitz-Gilbert equation with stochastic noise, and its dynamical aspects
It is crucially important to investigate effects of temperature on magnetic properties such as critical phenomena, nucleation, pinning, domain wall motion, coercivity, etc. The Landau-Lifshitz-Gilbert (LLG) equation has been applied extensively to study dynamics of magnetic properties. Approaches of Langevin noises have been developed to introduce the temperature effect into the LLG equation. To have the thermal equilibrium state (canonical distribution) as the steady state, the system parameters must satisfy some condition known as the fluctuation-dissipation relation. In inhomogeneous magnetic systems in which spin magnitudes are different at sites, the condition requires that the ratio between the amplitude of the random noise and the damping parameter depends on the magnitude of the magnetic moment at each site. Focused on inhomogeneous magnetic systems, we systematically showed agreement between the stationary state of the stochastic LLG equation and the corresponding equilibrium state obtained by Monte Carlo simulations in various magnetic systems including dipole-dipole interactions. We demonstrated how violations of the condition result in deviations from the true equilibrium state. We also studied the characteristic features of the dynamics depending on the choice of the parameter set. All the parameter sets satisfying the condition realize the same stationary state (equilibrium state). In contrast, different choices of parameter set cause seriously different relaxation processes. We show two relaxation types, i.e., magnetization reversals with uniform rotation and with nucleation.
Hysteresis dispersion under a varying external field Hex is investigated through numerical simulations based on the Landau-Lifshitz-Gilbert (LLG) equation, indicating the energy dissipation can be determined by W(η) = A (f, H0). A linear relation between area of hysteresis and magnitude of external field is discovered. Evolution of hysteresis is also investigated under oscillating external field.
We propose a systematic and sequential expansion of the Landau-Lifshitz-Gilbert equation utilizing the dependence of the Gilbert damping tensor on the angle between magnetic moments, which arises from multi-body scattering processes. The tensor consists of a damping-like term and a correction to the gyromagnetic ratio. Based on electronic structure theory, both terms are shown to depend on e.g. the scalar, anisotropic, vector-chiral and scalar-chiral products of magnetic moments: $\vec{e}_i\cdot\vec{e}_j$, $(\vec{n}_{ij}\cdot\vec{e}_i)(\vec{n}_{ij}\cdot\vec{e}_j)$, $\vec{n}_{ij}\cdot(\vec{e}_i…
We calculate the magnetization torque due to the spin polarization of the itinerant electrons by deriving the kinetic spin Bloch equations based on the $s$-$d$ model. We find that the first-order gradient of the magnetization inhomogeneity gives rise to the current-induced torques, which are consistent to the previous works. At the second-order gradient, we find an effective magnetic field perpendicular to the spin stiffness filed. This field is proportional to the nonadiabatic parameter $β$. We show that this vertical spin stiffness term can significantly modify the domain-wall structure in f…
We investigate the temperature dependence of the coefficients in the Landau-Lifshitz-Gilbert equation in ferromagnetic GaMnAs by employing the Zener model. We first calculate the hole spin relaxation time based on the microscopic kinetic equation. We find that the hole spin relaxation time is typically several tens femtoseconds and can present a nonmonotonic temperature dependence due to the variation of the interband spin mixing, influenced by the temperature related Zeeman splitting. With the hole spin relaxation time, we are able to calculate the coefficients in the Landau-Lifshitz-Gilbert …
The Suzuki-Trotter decomposition in general allows one to divide the equation of motion of a dynamical system into smaller parts whose integration are easier than the original equation. In this study, we first rewrite by employing feasible approximations the modified Landau-Lifshitz-Gilbert equation for localized spins in a suitable form for simulations using the Suzuki-Trotter decomposition. Next we decompose the equation into parts and demonstrate that the parts are classified into three groups, each of which can be solved exactly. Since the modified Landau-Lifshitz-Gilbert equation from whi…
We present a general finite element linearized Landau-Lifshitz-Gilbert equation (LLGE) solver for magnetic systems under weak time-harmonic excitation field. The linearized LLGE is obtained by assuming a small deviation around the equilibrium state of the magnetic system. Inserting such expansion into LLGE and keeping only first order terms gives the linearized LLGE, which gives a frequency domain solution for the complex magnetization amplitudes under an external time-harmonic applied field of a given frequency. We solve the linear system with an iterative solver using generalized minimal res…
Starting from the Dirac-Kohn-Sham equation we derive the relativistic equation of motion of spin angular momentum in a magnetic solid under an external electromagnetic field. This equation of motion can be written in the form of the well-known Landau-Lifshitz-Gilbert equation for a harmonic external magnetic field, and leads to a more general magnetization dynamics equation for a general time-dependent magnetic field. In both cases with an electronic spin-relaxation term which stems from the spin-orbit interaction. We thus rigorously derive, from fundamental principles, a general expression fo…
Keffer-like form of the symmetric Heisenberg exchange integral: Contribution to the Landau--Lifshitz--Gilbert equation and spin wave dispersion dependence
Dynamic simulations of spin-transfer and spin-orbit torques are increasingly important for a wide range of spintronic devices including magnetic random access memory, spin-torque nano-oscillators and electrical switching of antiferromagnets. Here we present a computationally efficient method for the implementation of spin-transfer and spin-orbit torques within the Landau-Lifshitz-Gilbert equation used in micromagnetic and atomistic simulations. We consolidate and simplify the varying terminology of different kinds of torques into a physical action and physical origin that clearly shows the com…
We investigate in details the inertial dynamics of a uniform magnetization in the ferromagnetic resonance (FMR) context. Analytical predictions and numerical simulations of the complete equations within the Inertial Landau-Lifshitz-Gilbert (ILLG) model are presented. In addition to the usual precession resonance, the inertial model gives a second resonance peak associated to the nutation dynamics provided that the damping is not too large. The analytical resolution of the equations of motion yields both the precession and nutation angular frequencies. They are function of the inertial dynamics…
The magnetostatic mode (MSM) spectrum of a 300$μ$m diameter single crystalline sphere of yttrium iron garnet is investigated using broadband ferromagnetic resonance (FMR). The individual MSMs are identified via their characteristic dispersion relations and the corresponding mode number tuples $(nmr)$ are assigned. Taking FMR data over a broad frequency and magnetic field range allows to analyze both the Gilbert damping parameter~$α$ and the inhomogeneous line broadening contribution to the total linewidth of the MSMs separately. The linewidth analysis shows that all MSMs share the same Gilbert…
Electric susceptibility of antiferromagnetic multiferroics with cycloidal spin order at magnetoelectric effect associated with collinear component of spins
Spin torques play a crucial role in operative properties of modern spintronic devices. To study current-driven magnetization dynamics, spin-torque terms providing the action of spin-polarized currents have previously often been added in a phenomenological way to the Landau-Lifshitz-Gilbert equation describing the local spin dynamics, yet without derivation from fundamental principles. Here, starting from the Dirac-Kohn-Sham theory and incorporating nonlocal spin transport we rigorously derive the various spin-torque terms that appear in current-driven magnetization dynamics. In particular we o…
An essential property of magnetic devices is the relaxation rate in magnetic switching which depends strongly on the damping in the magnetisation dynamics. It was recently measured that damping depends on the magnetic texture and, consequently, is a non-local quantity. The damping enters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter $α$, that does not, in a straight forward formulation, account for non-locality. Efforts were spent recently to obtain Gilbert damping from first principles for magnons of wave vector $\mathbf{q}$. However, to the best of ou…
A theoretical model based on the Landau-Lifshitz-Bloch equation is developed to study the spin-torque effect in ferrimagnets. Experimental findings, such as the temperature dependence, the peak in spin torque, and the angular-momentum compensation, can be well captured. In contrast to the ferromagnet system, the switching trajectory in ferrimagnets is found to be precession free. The two sublattices are not always collinear, which produces large exchange field affecting the magnetization dynamics. The study of material composition shows the existence of an oscillation region at intermediate cu…
The pseudo-PT symmetric Dirac equation is proposed and analyzed by using a non-unitary Foldy-Wouthuysen transformations. A new spin operator PT symmetric expectation value (called the mean spin operator) for an electron interacting with a time-dependent electromagnetic field is obtained. We show that spin magnetization - which is the quantity usually measured experimentally - is not described by the standard spin operator but by this new mean spin operator to properly describe magnetization dynamics in ferromagnetic materials and the corresponding equation of motion is compatible with the phen…
Using broadband ferromagnetic resonance, we measure the damping parameter of [Co(5 Å)/Pt(3 Å)]${\times 6}$ multilayers whose growth was optimized to maximize the perpendicular anisotropy. Structural characterizations indicate abrupt interfaces essentially free of intermixing despite the miscible character of Co and Pt. Gilbert damping parameters as low as 0.021 can be obtained despite a magneto-crystalline anisotropy as large as $10^6~\textrm{J/m}^3$. The inhomogeneous broadening accounts for part of the ferromagnetic resonance linewidth, indicating some structural disorder leading to a equiva…
The ultra-fast magnetisation relaxation rates during the laser-induced magnetisation process are analyzed in terms of the Landau-Lifshitz-Bloch (LLB) equation for different values of spin $S$. The LLB equation is equivalent in the limit $S \rightarrow \infty$ to the atomistic Landau-Lifshitz-Gilbert (LLG) Langevin dynamics and for $S=1/2$ to the M3TM model [B. Koopmans, {\em et al.} Nature Mat. \textbf{9} (2010) 259]. Within the LLB model the ultra-fast demagnetisation time ($τ_{M}$) and the transverse damping ($α_{\perp}$) are parameterized by the intrinsic coupling-to-the-bath parameter $λ$,…
The modification of the magnetization dissipation or Gilbert damping caused by an inhomogeneous magnetic structure and expressed in terms of a wave vector dependent tensor $\underlineα(\vec{q})$ is investigated by means of linear response theory. A corresponding expression for $\underlineα(\vec{q})$ in terms of the electronic Green function has been developed giving in particular the leading contributions to the Gilbert damping linear and quadratic in $q$. Numerical results for realistic systems are presented that have been obtained by implementing the scheme within the framework of the fully …
Heat assisted recording is a promising technique to further increase the storage density in hard disks. Multilayer recording grains with graded Curie temperature is discussed to further assist the write process. Describing the correct magnetization dynamics of these grains, from room temperature to far above the Curie point, during a write process is required for the calculation of bit error rates. We present a coarse grained approach based on the Landau-Lifshitz-Bloch (LLB) equation to model exchange coupled grains with low computational effort. The required temperature dependent material pro…
Precise modeling of the magnetization dynamics of nanoparticles with finite size effects at fast varying temperatures is a computationally challenging task. Based on the Landau-Lifshitz-Bloch (LLB) equation we derive a coarse grained model for disordered ferrimagnets, which is both fast and accurate. First, we incorporate stochastic fluctuations to the existing ferrimagnetic LLB equation. Further, we derive a thermodynamic expression for the temperature dependent susceptibilities, which is essential to model finite size effects. Together with the zero field equilibrium magnetization the suscep…
Micromagnetic modelling provides the ability to simulate large magnetic systems accurately without the computational cost limitation imposed by atomistic modelling. Through micromagnetic modelling it is possible to simulate systems consisting of thousands of grains over a time range of nanoseconds to years, depending upon the solver used. Here we present the creation and release of an open-source multi-timescale micromagnetic code combining three key solvers: Landau-Lifshitz-Gilbert; Landau-Lifshitz-Bloch; Kinetic Monte Carlo. This code, called MARS (Models of Advanced Recording Systems), is c…
The Barnett effect refers to the magnetization induced by rotation of a demagnetized ferromagnet. We describe the location and stability of stationary states in rotating nanostructures using the Landau-Lifshitz-Gilbert equation. The conditions for an experimental observation of the Barnett effect in different materials and sample geometries are discussed.
We show that recently reported precessing solution of Landau-Lifshitz-Gilbert equations in ferromagnetic nanowires is stable under small perturbations of initial data, applied field and anisotropy constant. Linear stability is established analytically, while nonlinear stability is verified numerically.
