Landau-Lifshitz-Bloch equation for ferrimagnetic materials

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.

Figure 1

(Left) Sketch of atomistic regular ferrimagnetic lattice. Each point represents a magnetic moment associated with an atomic site. Magnetic moments of blue points are pointing downwards and red ones upwards. (Right) A macroscopic view of partial average magnetization $m_A=\langle s_A\rangle$ and $m_B=\langle s_B\rangle$ by two macrospins in each sublattice as described by the Landau-Lifshitz-Bloch equation.

Figure 2

Damping parameters ${\alpha }_{\parallel \left(\perp \right)}^{\nu }\left(\vartheta \right)$ (normalized to the corresponding intrinsic values) for a pure ferromagnet (FM), rare-earth (RE) component in a GdFeCo ferrrimagnet and a transition metal (TM) in a ferrimagnet as a function of reduced temperature $\vartheta =T/T_C$ for three different rare earth (RE) concentrations $x$. The blue solid line represents the $x=0$ limit that corresponds to a pure ferromagnet (FM). (Up) The corresponding curves for a $25\%$ concentration of RE. (Middle) The corresponding damping parameters for a $50\%$ alloy. (Bottom) Damping values for $75\%$ RE amount. It can be also seen as a RE doped with a $25\%$ of transition metal (TM).

Figure 3

Different longitudinal relaxation regions for $T/T_C=0.95$ for parameters of the GdFeCo alloy with $x=0.25$.

Figure 4

Comparison between atomistic LLG-Langevin and macrospin LLB calculations of the longitudinal relaxation of the GdFeCo alloy ($x=0.25$) corresponding to the three different relaxation cases in Fig. 3. In the left column, we show atomistic LLG-Langevin multispin simulations and in the right one, the LLB macrospin calculations. The graphs (a) and (b) correspond to the region with ${\tilde{\Gamma }}_{\text{TM}}>0$ and ${\tilde{\Gamma }}_{\text{RE}}<0$. The graphs (c) and (d) correspond to the region with ${\tilde{\Gamma }}_{\text{TM}}<0$ and ${\tilde{\Gamma }}_{\text{RE}}<0$. The graphs (e) and (f) correspond to the region with ${\tilde{\Gamma }}_{\text{TM}}<0$ and ${\tilde{\Gamma }}_{\text{RE}}>0$.

Figure 5

Temperature dependence of the ratio between longitudinal susceptibilities for parameters of the GdFeCo alloy.