Crystallographically amorphous ferrimagnetic alloys: Comparing a localized atomistic spin model with experiments

We present a computational model of crystallographically amorphous ferrimagnetic alloys using a stochastic Landau-Lifshitz-Gilbert equation of motion for atomistic spins and an atomistic spin Hamiltonian with Heisenberg exchange. The spontaneous equilibrium magnetization is calculated and a comparison with a mean field model is made. The simulations show excellent agreement with experiments on GdFeCo using x-ray magnetic circular dichroism to determine the individual sublattice magnetizations. The calculated temperature dependence of the magnetization shows a polarization of the Gd sublattice leading to a common Curie temperature, in agreement with the experimental data. The intersublattice exchange is shown to be an important energy transfer channel for ultrafast dynamics.

Figure 1

Element- and temperature-dependent hysteresis loops measured for the Gd and Fe spins in a Gd${}_{23.4}$Fe${}_{73.3}$Co${}_{3.3}$ sample using XMCD, clearly showing the antiferromagnetic coupling between the sublattices.

Figure 2

The variation of the coercive field $H_{\mathrm{c}}$ and saturation magnetization $M_{\mathrm{s}}$ with temperature for a Gd${}_{23.4}$Fe${}_{73.3}$Co${}_{3.3}$ sample as deduced from element-specific hysteresis measured at the Fe and Gd absorption edges. The divergence in the coercivity indicates the magnetization compensation point. The solid lines are fits according to $M(T)$ power law (see text). Dashed lines are guides to the eye.

Figure 3

Experimental data showing temperature-dependent coercivity for a range of GdFeCo compositions. The divergence in the coercivity indicates the magnetization compensation point. The solid lines are guides to the eye.

Figure 4

Numerically calculated magnetization curves for the TM-RE ferrimagnet as a function of temperature. Results are shown for a range of RE content. The compensation point exists for concentration in the range 18%–44%, consistent with experiment (Ref. 16). Note that the sign of $M_{\mathrm{s}}$ takes into account the fact that the TM sublattice is pointing opposite to the RE sublattice. In the negative region, therefore, the RE sublattice is dominant.

Figure 5

Numerically calculated magnetization curves for the TM-RE ferrimagnet as a function of temperature. Results are shown for a range of RE content. This figure shows that the compensation point does not exist above 44% RE. Note that the sign of $M_{\mathrm{s}}$ takes into account the fact that the TM sublattice is pointing opposite to the RE sublattice. All of the data are plotted as the absolute value of the magnetization as there does not exist a compensation point in this range.

Figure 6

Magnetization curves as a function of temperature (normalized to the Curie temperature $T_{\mathrm{c}}$ for each composition) for different concentrations of RE. The data is calculated via the MFA approach (solid lines) and direct integration of the Langevin dynamics equation (2) (symbols).

Figure 7

Compositional dependence of the Curie temperature ($T_{\mathrm{c}}$) and magnetization compensation point ($T_{\mathrm{M}}$). The mean-field approximation (MFA) with renormalized exchange parameters is shown to agree very well (lines) with the Landau-Lifshitz-Gilbert (LLG) model (points). The compensation temperatures deduced from temperature-dependent hysteresis curves show excellent agreement with the mean field and LLG models.

Figure 8

Numerical values of the coercive field (points) for various RE amounts. Data shown was calculated using a sweep rate of 1 T/ns for a range of compositions. Results show good qualitative agreement with the experimental results (Fig. 3), with the divergence representing the magnetization compensation point. Lines are guides to the eye. Values are reduced to the zero-temperature, pure TM coercivity value with the same sweep rate.

Figure 9

Reduced magnetization of the Fe and Gd sublattices as a function of the intersublattice exchange ($J_{\mathrm{TM}-\mathrm{RE}}$). Magnetization is normalized to the sublattice magnetization. The exchange leads to a polarization effect between the sublattices. Here $J_{\max }=-2.18\times {10}^{-21}$ J.

Figure 10

Numerical result of the effective RE spin temperature ($T_{\mathrm{eff}}$) using the Langevin dynamics model of the TM-RE ferrimagnet. The system is initially at 0 K, and at time zero a step in temperature to 300 K is applied. The effective spin temperature is extracted from the equilibrium magnetization. The result shows that the higher exchange leads to an increase in the rate of change of the effective spin temperature in the RE system.