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Micromagnetic modeling has proved itself as a widely used tool, complimentary in many respects to experimental measurements. The Landau-Lifshitz equation provides a basis for this modeling, especially where the dynamical behaviour is concerned. However, this approach is strictly valid only for zero temperature and for high temperatures must be replaced by a more thermodynamically consistent approach such as the Landau-Lifshitz-Bloch (LLB) equation. Here we review the recently derived LLB equation for two-sublattice systems and extend its derivation for temperatures above the Curie temperature. We present comparison with many-body atomistic simulations and show that this equation can describe the ultra-fast switching in ferrimagnets, observed experimentally. [ABSTRACT FROM AUTHOR]

The Landau-Lifshitz (LL) equation, originally proposed at the macrospin level, is increasingly used in Atomistic Spin Dynamic (ASD) models. These models are based on a spin Hamiltonian featuring atomic spins of fixed length, with the exchange introduced using the Heisenberg formalism. ASD models are proving a powerful approach to the fundamental understanding of ultrafast magnetization dynamics, including the prediction of the thermally induced magnetization switching phenomenon in which the magnetization is reversed using an ultra-fast laser pulse in the absence of an externally applied field. This paper outlines the ASD model approach and considers the role and limitations of the LL equation in this context. [ABSTRACT FROM AUTHOR]

A hierarchical multiscale approach to model the magnetization dynamics of ferromagnetic random 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 parameters). These parameters are subsequently used in the Landau-Lifshitz-Bloch (LLB) model for multisublattice magnets to calculate numerically and analytically the ultrafast demagnetization times. The developed multiscale method is applied here to FeNi (permalloy) as well as to copper-doped FeNi alloys. We find that after an ultrafast heat pulse the Ni sublattice demagnetizes faster than the Fe sublattice for the here-studied FeNi-based alloys. [ABSTRACT FROM AUTHOR]

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 dynamical polarization of one material by another can occur. We discuss the differences between the LLB and the Baryakhtar equations, recently used to explain the ultrafast switching in ferrimagnets. The two-component LLB equation forms the basis for the large-scale micromagnetic modeling of nanostructures at high temperatures and ultrashort time scales. [ABSTRACT FROM AUTHOR]

We compute the temperature-dependent spin-wave spectrum and the magnetization for a spin system using the unified decoupling procedure for the high-order Green's functions for the exchange coupling and anisotropy in both the classical and quantum cases. Our approach allows us to establish a clear crossover between quantum-mechanical and classical methods by developing the classical analog of the quantum Green's function technique. The results are compared with the classical spectral density method and numerical modeling based on the stochastic Landau-Lifshitz equation and Monte Carlo technique. As far as the critical temperature is concerned, there is a full agreement between the classical Green's functions technique and the classical spectral density method. However, the former method turns out to be more straightforward than the latter because it avoids any a priori assumptions about the system's spectral density. The temperature-dependent exchange stiffness as a function of magnetization is investigated within different approaches. [ABSTRACT FROM AUTHOR]

The Landau-Lifshitz-Bloch equation is a formulation of dynamic micromagnetics valid at all temperatures, treating both the transverse and longitudinal relaxation components important for high-temperature applications. In this paper we discuss two stochastic forms of the Landau-Lifshitz-Bloch equation. Both of them arc consistent with the fluctuation-dissipation theorem. We derive the corresponding Fokker-Planck equations and show that only the stochastic form of the Landau-Lifshitz-Bloch equation proposed in the present paper is consistent with the Boltzmann distribution at high temperatures. The previously used form does not satisfy this requirement in the vicinity of the Curie temperature. We discuss the stochastic properties of both equations and present numerical simulations for distribution functions and the average magnetization value as a function of temperature. [ABSTRACT FROM AUTHOR]