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 current density, induced by the nondeterministic switching. Compared to the Landau-Lifshitz-Gilbert model, our developed model based on the Landau-Lifshitz-Bloch equation enables the systematic study of spin-torque effect and the evaluation of ferrimagnet-based devices.
We present a micromagnetic formulation for modeling the magnetization dynamics and thermal equilibrium in ferrimagnetic materials at low and elevated temperatures. The formulation is based on a mean field approximation (MFA). In this formulation, the ferrimagnet is described micromagnetically by two coupled sublattices with corresponding interactions, including inter- and intra-sublattice micromagnetic exchange as well as four-spin interactions described as an inter-sublattice molecular field with a cubic dependence of the magnetization. The MFA is used to derive a Landau Lifshitz Bloch type e…
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…
The detailed derivation of the quantum Landau-Lifshitz-Bloch (qLLB) equation for simple spin-flip scattering mechanisms based on spin-phonon and spin-electron interactions is presented and the approximations are discussed. The qLLB equation is written in the form, suitable for comparison with its classical counterpart. The temperature dependence of the macroscopic relaxation rates is discussed for both mechanisms. It is demonstrated that the magnetization dynamics is slower in the quantum case than in the classical one.
We derive the Landau-Lifshitz-Bloch (LLB) equation for a two-component magnetic system valid up to the Curie temperature. As an example, we consider disordered GdFeCo ferrimagnet where the ultrafast optically induced magnetization switching under the action of heat alone has been recently reported. The two-component LLB equation contains the longitudinal relaxation terms responding to the exchange fields from the proper and the neighboring sublattices. We show that the sign of the longitudinal relaxation rate at high temperatures can change depending on the dynamical magnetization value and a …
Ultrafast laser-induced magnetic switching in rare earth, transition metal ferrimagnetic alloys has recently been reported to occur by ultrafast heating alone. Using atomistic simulations and a ferrimagnetic Landau-Lifshitz-Bloch formalism, we demonstrate that for switching to occur it is necessary that angular momentum is transferred from the longitudinal to transverse magnetization components. This dynamical path leads to magnetization switching and subsequent ultrafast precession caused by the inter-sublattice exchange field on the nanoscale.
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 $λ$,…
This work demonstrates that the magnetization and angular momentum compensation temperature (TMC and TAMC) in ferrimagnets (FiM) can be unambiguously determined by performing two sets of temperature dependent current switching, with the symmetry reverses at TMC and TAMC, respectively. A theoretical model based on the modified Landau-Lifshitz-Bloch equation is developed to systematically study the spin torque effect under different temperatures, and numerical simulations are performed to corroborate our proposal. Furthermore, we demonstrate that the recently reported linear relation between TAM…
Using the Landau-Lifshitz-Bloch (LLB) equation for ferromagnetic materials, we derive analytic expressions for temperature dependent absorption spectra as probed by ferromagnetic resonance (FMR). By analysing the resulting expressions, we can predict the variation of the resonance frequency and damping with temperature and coupling to the thermal bath. We base our calculations on the technologically relevant L1$_0$ FePt, parameterised from atomistic spin dynamics simulations, with the Hamiltonian mapped from ab-initio parameters. By constructing a multi-macrospin model based on the LLB equatio…
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…
We report on an element- and time-resolved investigation of femtosecond laser induced ultrafast dynamics of 3d and 4f spins in a ferrimagnetic Co80Dy20 alloy as a function of temperature. We observe an increase of the Co3d characteristic demagnetization time and a decrease of the Dy4f demagnetization time when the temperature is approaching the Curie temperature. It suggests that the critical slowing down regime, which affects the laser induced ultrafast dynamics in pure 3d transition metals and 4f rare-earth ferromagnetic layers, vanishes for the Dy sublattice in the CoDy alloy, in line with …
Laser-induced magnetization precessional dynamics was investigated in epitaxial films of Mn$_3$Ge, which is a tetragonal Heusler-like nearly compensated ferrimagnet. The ferromagnetic resonance (FMR) mode was observed, the precession frequency for which exceeded 0.5 THz and originated from the large magnetic anisotropy field of approximately 200 kOe for this ferrimagnet. The effective damping constant was approximately 0.03. The corresponding effective Landau-Lifshitz constant of approximately 60 Mrad/s and is comparable to those of the similar Mn-Ga materials. The physical mechanisms for the …
In this work, we derive the Landau-Lifshitz-Bloch equation accounting for the multi-domain antiferromagnetic (AFM) lattice at finite temperature, in order to investigate the domain wall (DW) motion, the core issue for AFM spintronics. The continuity equation of the staggered magnetization is obtained using the continuum approximation, allowing an analytical calculation on the domain wall dynamics. The influence of temperature on the static domain wall profile is investigated, and the analytical calculations reproduce well earlier numerical results on temperature gradient driven saturation velo…
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 spin transfer torque (STT) in the magnetic multilayer structures with micromagnetic simulations. We implement the STT contribution for the magnetic multilayer structures in addition to the Landau-Lifshitz-Gilbert (LLG) micromagnetic simulators. Not only the Sloncewski STT term, the zero, first, and second order field- like terms are also considered, and the effects of the Oersted field by the current are addressed. We determine the switching current densities of the free layer with the exchange biased synthetic ferrimagnetic reference layers for various cases.
Keffer-like form of the symmetric Heisenberg exchange integral: Contribution to the Landau--Lifshitz--Gilbert equation and spin wave dispersion dependence
The 212 species of structural phase transitions which break macroscopic symmetry are analyzed with respect to the occurrence of time-reversal invariant vector and bidirector order parameters. The possibility of discerning the orientational domain states of the low-symmetry phase by these `vectorlike' physical properties has been derived using a computer algorithm exploiting the concept of polar, axial, chiral and neutral dipoles. It is argued that for species 32 > 3, 422 > 4 and 622 > 6, Bogdanov-Yablonskii phenomenological theory for a ferromagnetic Bloch Skyrmions applies also to the ferroel…
We present a microscopic calculation of magnetization damping for a magnetic "toy model." The magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless band of localized spins, and the magnetization damping is due to coupling of the itinerant carriers to a phonon bath in the presence of spin-orbit coupling. Using a mean-field approximation for the kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we derive Boltzmann scattering integrals for the distributions and spin coherences in the case of an antiferromagnetic exchan…
A hierarchical multiscale approach to model the magnetization dynamics of ferromagnetic ran- dom alloys is presented. First-principles calculations of the Heisenberg exchange integrals are linked to atomistic spin models based upon the stochastic Landau-Lifshitz-Gilbert (LLG) equation to calculate temperature-dependent parameters (e.g., effective exchange interactions, damping param- eters). These parameters are subsequently used in the Landau-Lifshitz-Bloch (LLB) model for multi-sublattice magnets to calculate numerically and analytically the ultrafast demagnetization times. The developed mul…
The effect of inertial spin dynamics is compared between ferromagnetic, antiferromagnetic and ferrimagnetic systems. The linear response to an oscillating external magnetic field is calculated within the framework of the inertial Landau--Lifshitz--Gilbert equation using analytical theory and computer simulations. Precession and nutation resonance peaks are identified, and it is demonstrated that the precession frequencies are reduced by the spin inertia, while the lifetime of the excitations is enhanced. The interplay between precession and nutation is found to be the most prominent in antifer…
Nonlinear localized magnetic excitations in one dimensional magnonic crystal is investigated under periodic magntic field. The governing Landau-Lifshitz equation is transformed into variable coefficient nonlinear Schrodinger equation(VCNLS) using sterographic projection. The VCNLS equation is in general nonintegrable, by using painleve analysis necessary conditions for the VCNLS equation to pass Weiss-Tabor-Carnevale (WTC) Painleve test are obtained. A sufficient integrability condition is obtained by further exploring a transformation, which can map the VCNLS equation into the well-known stan…
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…
We report about element specific measurements of ultrafast demagnetization and magnetization precession damping in Permalloy (Py) thin films. Magnetization dynamics induced by optical pump at $1.5$eV is probed simultaneously at the $M_{2,3}$ edges of Ni and Fe with High order Harmonics for moderate demagnetization rates (less than $50$%). The role of the intersublattice exchange interaction on both longitudinal and transverse dynamics is analyzed with a Landau Lifshitz Bloch description of ferromagnetically coupled Fe and Ni sublattices. It is shown that the intersublattice exchange interactio…
We compare femtosecond pump-probe experiments in Ni and micromagnetic modelling based on the Landau-Lifshitz-Bloch equation coupled to a two-temperature model, revealing a predominant thermal ultrafast demagnetization mechanism. We show that both spin (femtosecond demagnetization) and electron-phonon (magnetization recovery) rates in Ni increase as a function of the laser pump fluence. The slowing down for high fluences arises from the increased longitudinal relaxation time.
Reliably predicting bit-error rates in realistic heat-assisted magnetic recording simulations is a challenging task. Integrating the Landau-Lifshitz-Bloch (LLB) equation can reduce the computational effort to determine the magnetization dynamics in the vicinity of the Curie temperature. If one aims that these dynamics coincide with trajectories calculated from the atomistic Landau-Lifshitz-Gilbert equation, one has to carefully model required temperature dependent material functions such as the zero-field equilibrium magnetization as well as the parallel and normal susceptibilities. We present…
Antiferromagnets (AFMs) are strong candidates for the future spintronic and memory applications largely because of their inherently fast dynamics and lack of stray fields, with Mn2Au being one of the most promising. For the numerical modelling of magnetic material properties, it is common to use ab-initio methods, atomistic models and micromagnetics. However, each method alone describes the physics within certain limits. Multiscale methods bridging the gap between these three approaches have been already proposed for ferromagnetic materials. Here, we present a complete multiscale model of the …
Magnetization manipulation is becoming an indispensable tool for both basic and applied research. Theory predicts two types of ultrafast demagnetization dynamics classified as type I and type II. In type II materials, a second slower process takes place after the initial fast drop of magnetization. In this letter we investigate this behavior for FePt recording materials with perpendicular anisotropy. The magnetization dynamics have been simulated using a thermal micromagnetic model based on the Landau-Lifshitz-Bloch equation. We identify a transition to type II behavior and relate it to the el…
In conventional micromagnetism magnetic domain configurations are calculated based on a continuum theory for the magnetization which is assumed to be of constant length in time and space. Dynamics is usually described with the Landau-Lifshitz-Gilbert (LLG) equation the stochastic variant of which includes finite temperatures. Using simulation techniques with atomistic resolution we show that this conventional micromagnetic approach fails for higher temperatures since we find two effects which cannot be described in terms of the LLG equation: i) an enhanced damping when approaching the Curie te…
There is little doubt that the magnetization dynamics of ferromagnetic systems is governed by the Landau-Lifshitz-Gilbert equation or its generalization with various spin torques. In contrast, there are several sets of dynamic equations for two-sublattice antiferromagnets (AFMs) in literature that have different forms of dissipative torques and no proper dynamic equations for multi-sublattice AFMs and ferrimagnets in general. Here we introduce the general Rayleigh dissipation functional into the Lagrange equation and derive the proper form of the dissipative torques in the phenomenological equ…
The interaction of polarized light with a spin in the presence of dissipation is shown to be equivalent to a spin transfer process that can cause switching. In plasmas, the spin transfer is dominated by a spin-spin exchange term while at lower energy densities it is dominated by an optical Barnett-like effect. This latter effect is used in conjunction with optical phonons to predict femtosecond magnetization reversal believed to be recently measured in GdCoFe thin films. Conventional approaches based on the Bloch and the Landau-Lifshitz equations do not reproduce this ultrafast switching.
The motion of simple domain walls and of more complex magnetic textures in the presence of a transport current is described by the Landau-Lifshitz-Slonczewski (LLS) equations. Predictions of the LLS equations depend sensitively on the ratio between the dimensionless material parameter $β$ which characterizes non-adiabatic spin-transfer torques and the Gilbert damping parameter $α$. This ratio has been variously estimated to be close to 0, close to 1, and large compared to 1. By identifying $β$ as the influence of a transport current on $α$, we derive a concise, explicit and relatively simple e…
Magnetic domain wall (DW) motion induced by a localized Gaussian temperature profile is studied in a Permalloy nanostrip within the framework of the stochastic Landau-Lifshitz-Bloch equation. The different contributions to thermally induced DW motion, entropic torque and magnonic spin transfer torque, are isolated and compared. The analysis of magnonic spin transfer torque includes a description of thermally excited magnons in the sample. A third driving force due to a thermally induced dipolar field is found and described. Finally, thermally induced DW motion is studied under realistic condit…
