Classical spin model of the relaxation dynamics of rare-earth doped permalloy

In this paper, the ultrafast dynamic behavior of rare-earth doped permalloy is investigated using an atomistic spin model with Langevin dynamics. In line with experimental work, the effective Gilbert damping is calculated from transverse relaxation simulations, which shows that rare-earth doping causes an increase in the damping. Analytic theory suggests that this increase in damping would lead to a decrease in the demagnetization time. However, longitudinal relaxation calculations show an increase with doping concentration instead. The simulations are in a good agreement with previous experimental work of Radu et al. [Radu et al., Phys. Rev. Lett. 102, 117201 (2009)]. The longitudinal relaxation time of the magnetization is shown to be driven by the interaction between the transition metal and the laser-excited conduction electrons, whereas the effective damping is predominantly determined by the slower interaction between the rare-earth elements and the phonon heat bath. We conclude that for complex materials, it is evidently important not to expect a single damping parameter but to consider the energy transfer channel relevant to the technique and time scale of the measurement.

Figure 1

A schematic representing energy transfer channels between the thermal reservoirs of the two-temperature model (TTM) and the spin system, the dynamics of which is described by the Landau-Lifshitz-Gilbert (LLG) equation. The Gaussian profile represents the effect of laser heating, which is coupled to the electron temperature. The electron reservoir is coupled to the lattice reservoir via the coupling constant $G_{\mathrm{el}}$. The transition metal (TM) spins, which in our case are Ni and Fe, are then coupled to the electron temperature. The rare-earth (RE) element, Ho or Gd, is coupled to either the electron or lattice temperature. The TM and RE spins are coupled via antiferromagnetic exchange giving rise to an antiparallel ground state.

Figure 2

(a) The $z$ component of the total (dot-dashed black line), TM (solid blue line), and RE (red dashed line) magnetization as a function of time for Ni${}_{72}$Fe${}_{20}$Ho${}_8$ after a change in temperature from 0 to 300 K. The TM spins are coupled to the electron temperature, while the RE spins are coupled to the phonon temperature. The TM spins respond faster to the increase in temperature while the RE spins takes much longer, these different time scales cause the total magnetization to first demagnetize following the TM behavior before the demagnetization of the RE occurs. The behavior of the electron and phonon temperatures, after a step change in the electron temperature, are shown in (b). By comparing the lattice temperatures dynamics to the RE, we can see the RE spins demagnetize at a similar rate.

Figure 3

Integral relaxation times for the TM (blue squares) and RE (red circles) sublattices in Ni${}_{75}$Fe${}_{20}$Ho${}_5$ for different temperature step sizes. The spins begin in the ground state at 0 K. A Heaviside step function is then applied to the electron temperature changing it to the required value. The TM behaves as expected with a gradual increase until diverging at the Curie temperature. The RE sublattice relaxation time generally decreases but exhibits a peak at the Curie temperature consistent with the coupling to the TM by the interlattice exchange. The solid lines are a guide to the eye showing the divergent behavior.

Figure 4

Transverse ($x$) component of the reduced total magnetization as a function of time after excitation of the system by coherent rotation of the spins to an angle of 30 degrees. Different concentrations are shown highlighting the effect of doping as the concentration is increased from 0$\%$ (a) to 6$\%$ (c) at 1K. The strength of the applied field was $h=$ 0.25 T in all cases, applied along the $z$ axis. The dashed lines represent a fitted $\cos (t/{\tau }_p)/\cosh (t/{\tau }_r+d)$ function and the shaded area shows the envelope of the damping function from which the Gilbert damping factor can be extracted. The fitting parameters ${\tau }_p$ and ${\tau }_r$ are the precession and transverse relaxation times, respectively, and for a simple single macrospin, have the values ${\tau }_p=(1+{\alpha }_{\mathrm{eff}}^2)/\gamma h$, ${\tau }_r={\tau }_p/{\alpha }_{\mathrm{eff}}$. and so the ratio of the times give the damping factor.

Figure 5

The relation between the RE concentration and the Gilbert damping parameter for Ho (red squares) and Gd (blue circles). The analytic form (lines) for both Ho and Gd from Eq. (17) agrees well with the calculated damping from the model (points) at $T=1$ K. The damping diverges at the magnetization compensation point where the total $M_s$ is zero as the RE sublattice magnetization completely cancels the TM sublattice magnetization. At low doping concentrations, the change in the damping is almost linear, which agrees well with the results of Ref. 13.

Figure 6

Variation of the longitudinal relaxation time of permalloy (light blue squares), 60$\%$ Fe (dark blue triangles), and 40$\%$ Fe (black diamonds) as a function of thermal bath coupling calculated using the atomistic model. The solid lines are fitted functions using ${\tau }_{\mathrm{IRT}}=(C_0/\lambda )+d$. The variation of Fe percentage aims to vary the effective magnetic moment per atom (${\mu }_{\mathrm{eff}}$) as no structural change is considered. Initially, the system is in an ordered state until a Heaviside step function is applied to $T_e=300$ K. The relationship matches well with Eqs. (1) and (2). The lines are a fitted function to each data set, for which the constant $C_0$ is proportional to the magnetic moment as shown in the inset.

Figure 7

The change in the integral relaxation time of the TM sublattice for a range of concentrations of Ho (red squares and triangles) and Gd (blue circles). Results are shown for Ho coupled to either the lattice (squares) or the electrons (triangles); this shows that the increase of the demagnetization time occurs due to the spin-lattice interaction of Ho. This agrees well with the experimental work of Radu et al. apart from a different scale. We find the integral relaxation time is longer than the values of the demagnetization time from the experiments. The behavior is consistent but in disagreement with predictions from Eqs. (1) and (2).