Multiscale modeling of ultrafast element-specific magnetization dynamics of ferromagnetic alloys

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.

Figure 1

Ab initio calculated exchange constants $J_{ij}$ for the ${\mathrm{Fe}}_{20}{\mathrm{Ni}}_{80}$ alloy as a function of the distance $r_{ij}$ between atoms $i$ and $j$. Results are given for the three different possible sublattice interactions $(J_{\mathrm{Fe}\text{−}\mathrm{Fe}},J_{\mathrm{Ni}\text{−}\mathrm{Ni}}$, and $J_{\mathrm{Fe}\text{−}\mathrm{Ni}})$. Note our hyperbolic scaling. In our atomistic spin simulations the exchange constants are taken into account up to a distance $r_{ij}$ (cutoff) where they are finally small enough to be neglected.

Figure 2

Element-specific zero-field equilibrium magnetizations $m^{\epsilon }$ of either Fe or Ni as a function of temperature calculated by a rescaled mean-field approximation (MFA) (lines) and by the atomistic spin dynamics simulation (open symbols). In the MFA the exchange parameters are renormalized by equalizing the Curie temperatures $T_{\mathrm{C}}$ computed with atomistic simulations with those obtained from the rescaled MFA. System size: $128\times 128\times 128$ atoms, damping parameter: $\lambda =1.0$.

Figure 3

Schematics of the magnetic unit cell used in the mean-field approximation for the FeNi alloys. The unit cell shown by the box contains two spins, one Fe and one Ni. The only interaction among spins located at the same unit cell $\mathbf{r}$ is defined by $J_0^{\mathrm{Ni}\text{−}\mathrm{Fe}}$. The self-interactions are neglected, $J_0^{\mathrm{Ni}\text{−}\mathrm{Ni}}(\mathbf{r},\mathbf{r})=J_0^{\mathrm{Fe}\text{−}\mathrm{Fe}}(\mathbf{r},\mathbf{r})=0$. The rest of the interactions are among spins located in neighboring unit cells $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$.

Figure 4

Calculated $z$ component of the normalized element-specific magnetization $m_z^{\epsilon }$ vs time for Py (top panel) and ${\mathrm{Py}}_{60}{\mathrm{Cu}}_{40}$ (bottom panel). In both cases the quenching of the element-specific magnetizations for Fe and Ni due to a temperature step of $T_{\mathrm{pulse}}=0.8T_{\mathrm{C}}$ are shown, computed with atomistic Langevin spin dynamics (open symbols) as well as LLB simulations, considering only the longitudinal part of the equation of motion [see Eq. (7)] (lines). System size: $64\times 64\times 64$, damping parameter: $\lambda =0.02$.

Figure 5

Top panel: Normalized magnetization dynamics of Fe and Ni sublattices after the application of a heat pulse $T=0.8T_{\mathrm{C}}$ as computed with the atomistic spin model. The ratio between the Fe and Ni demagnetization times is 1.8. The intersection of the linear fit to the abscissa gives the relaxation time for each sublattice. Bottom panel: Plot of the unnormalized magnetization dynamics which shows that after the first 0.2 ps the element-specific demagnetization proceeds at the same rate.

Figure 6

Relaxation times of the dynamical system obtained by the LLB equation as a function of temperature. Inset: The ratio between the relaxation times.

Figure 7

(a) Temperature dependence of the individual longitudinal damping parameters for Fe and Ni. (b) Matrix elements of the dynamical system defining the magnetization dynamics. (c) Ratio between the individual damping parameters. (d) Ratio between the “effective” susceptibilities ${\mathrm{\Lambda }}^{\epsilon \epsilon }$ and the actual susceptibilities ${\tilde{\chi }}_{\parallel }^{\epsilon }$.

Figure 8

Ratio between the relaxation times $\tau$ of the Fe and Ni sublattices in Py after the application of a heat pulse of temperature $T_{\mathrm{pulse}}$ for a range of values of the ratio of intrinsic damping parameters ${\lambda }^{\mathrm{Fe}}/{\lambda }^{\mathrm{Fe}}$. Black lines represent ${\lambda }^{\mathrm{Fe}}/{\lambda }^{\mathrm{Ni}}$ values where the ratio between relaxation times ${\tau }^{\mathrm{Ni}}$ and ${\tau }^{\mathrm{Fe}}$ is constant with the value given by the label.