Stochastic ferrimagnetic Landau-Lifshitz-Bloch equation for finite magnetic structures

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 susceptibilities are used in the stochastic ferrimagnetic LLB to simulate a ferrimagnetic ${\mathrm{Gd}}_{30}{\left(\mathrm{FeCo}\right)}_{70}$ particle with a diameter of 5 nm and a height of 10 nm under various external applied fields and heat pulses. The obtained trajectories agree well with those of an atomistic model, which solves the stochastic Landau-Lifshitz-Gilbert equation for each atom. Finally, we apply a 50-fs heat pulse with a maximum temperature of 1193 K to a ${\mathrm{Gd}}_{24}{\left(\mathrm{FeCo}\right)}_{76}$ particle of the same size to test the proposed model for all-optical switching (AOS). We observe switching with a ferromagneticlike state, which was identified to be the key for AOS in GdFeCo. Although the coarse-grained model in general shows remarkably good agreement with atomistic simulations, the computed switching probability is lower than expected. The magnetization trajectories seem to be too far away from equilibrium for ultrashort and very high heat pulses, which are typical for AOS. This leads to the conclusion that either the model needs higher-order corrections to accurately describe AOS, or that the two-spin model is not capable of describing deterministic AOS at all.

Figure 1

Variance of the $y$ component of the normalized sublattice magnetizations of ${\mathrm{Gd}}_{30}{\left(\mathrm{FeCo}\right)}_{70}$ (parameters are given in Table 1) for various values of the exchange $J_{\mathrm{FeCo}-\mathrm{Gd}}$ between FeCo atoms and Gd atoms. The horizontal dashed lines represent the expected variances based on the anisotropy fields of the materials.

Figure 2

Zero-field equilibrium magnetization $m_{\mathrm{e}}$ vs temperature, computed with an atomistic model of ${\mathrm{Gd}}_{30}{\left(\mathrm{FeCo}\right)}_{70}$ (parameters are given in Table 1). The black solid lines show fits, representing an infinite system.

Figure 3

Perpendicular susceptibility ${\tilde{\chi }}^{\perp }$, computed with an atomistic model of ${\mathrm{Gd}}_{30}{\left(\mathrm{FeCo}\right)}_{70}$ (parameters are given in Table 1) from magnetization fluctuations. The black solid lines show fits, representing an infinite system.

Figure 4

Temperature evolution of $\mathrm{\Lambda }$ for ${\mathrm{Gd}}_{30}{\left(\mathrm{FeCo}\right)}_{70}$ (parameters are given in Table 1), based on Eq. (31) with the mean-field expression of Eq. (15) and with the longitudinal susceptibilities obtained by atomistic simulations from fluctuations of the $z$ component of the magnetization.

Figure 5

Temporal evolution of the $z$ component of the normalized magnetization of both sublattices of ${\mathrm{Gd}}_{30}{\left(\mathrm{FeCo}\right)}_{70}$ computed with the proposed coarse-grained ferrimagnetic LLB model and the atomistic code vampire. (a) A constant magnetic field with ${\mu }_0{H}_{\mathrm{ext}}=-0.8\mathrm{T}$ and an angle of $6^{\circ }$ with the $z$ direction is applied. (b) A Gaussian shaped heat pulse is applied (blue solid line, right $y$ axis).

Figure 6

Hysteresis loops of ${\mathrm{Gd}}_{30}{\left(\mathrm{FeCo}\right)}_{70}$ with a field rate of 1 T/ns calculated with the proposed coarse-grained ferrimagnetic LLB model and the atomistic code vampire. (a) Easy axis loop at a constant temperature of (a) 100 K and (b) 500 K. Hysteresis loop with the applied field tilted ${45}^{\circ }$ against the $z$ direction at a constant temperature of (c) 100 K and (d) 500 K.

Figure 7

Switching probabilities of a ${\mathrm{Gd}}_{30}{\left(\mathrm{FeCo}\right)}_{70}$ particle computed from 128 switching trajectories at each $T_{\max }$. In each simulation a constant field and a Gaussian shaped heat pulse according to Eq. (33) with $T_{\min }=300\mathrm{K}$ are applied. Two different field magnitudes ${\mu }_0{H}_{\mathrm{ext}}=-0.5\mathrm{T}$ and ${\mu }_0{H}_{\mathrm{ext}}=-0.8\mathrm{T}$ as well as two different pulse durations $\tau =100\mathrm{ps}$ and $\tau =10\mathrm{ps}$ are compared. Atomistic simulations with vampire are compared with the proposed coarse-grained LLB model. Additionally in (a) the LLB model with a constant anisotropy field [Eq. (6)] is illustrated.

Figure 8

Temperature-dependent material functions for ${\mathrm{Gd}}_{25}{\left(\mathrm{FeCo}\right)}_{75}$ (parameters are given in Table 2). (a) Zero-field equilibrium magnetization $m_{\mathrm{e}}$, (b) perpendicular susceptibility ${\tilde{\chi }}^{\perp }$, (c) $\mathrm{\Lambda }$ computed with the mean field expression of Eq. (15) vs temperature. The black solid lines show fits of the given atomistic data, representing an infinite system.

Figure 9

Temporal evolution (a) of the $z$ component of the normalized sublattice magnetizations and (b) of the angle between the sublattices of ${\mathrm{Gd}}_{25}{\left(\mathrm{FeCo}\right)}_{75}$ (see Table 2). The illustrated heat pulse with a duration of 50 fs is applied to the particle. The trajectories are computed with the derived coarse-grained LLB model and the atomistic code vampire. (c) Zoom of the switching process in (a).