Role of longitudinal fluctuations in $\mathrm{L}{1}_0$ FePt

$\mathrm{L}{1}_0$ FePt is a technologically important material for a range of novel data storage applications. In the ordered FePt structure the normally nonmagnetic Pt ion acquires a magnetic moment, which depends on the local field originating from the neighboring Fe atoms. In this work a model of FePt is constructed in which the induced Pt moment is simulated by using combined longitudinal and rotational spin dynamics. The model is parameterized to include a linear variation of the moment with the exchange field, so that at the Pt site the magnetic moment depends on the Fe ordering. The Curie temperature of FePt is calculated and agrees well with similar models that incorporate the Pt dynamics through an effective Fe-only Hamiltonian. By computing the dynamic correlation function the anisotropy field and the Gilbert damping are extracted over a range of temperatures. The anisotropy exhibits a power-law dependence on the magnetization with exponent $n\approx 2.1$. This agrees well with what was observed experimentally, and it is obtained without including a two-ion anisotropy term as in other approaches. Our work shows that incorporating longitudinal fluctuations into spin dynamics calculations is crucial for understanding the properties of materials with induced moments.

Figure 1

Energy as a function of the reduced spin length in FePt. Fe possesses a spontaneous moment, which is modeled by having an energy minimum at $\left|S\right|=1$, while for Pt the moment is induced by the Fe exchange interaction. As such the longitudinal energy is modeled by a quadratic function. The arrows indicate the relevant energy scale. The left-hand (right-hand) scale is for Fe (Pt).

Figure 2

The relaxation of the atomic magnetic moments to equilibrium at $T=0\mathrm{K}$ for two cases: ferromagnetic (FM; open symbols) and anti-ferromagnetic (AFM; solid symbols) ordering of the Fe atoms. In both cases the Fe moments start fully saturated (i.e., $\left|S\right|=1$), while the Pt atoms start with $\left|S\right|=0$ for the FM case and $\left|S\right|=1$ for the AFM case to highlight the relaxation dynamics. The left-hand (right-hand) scale is for Fe (Pt).

Figure 3

The average magnetization as a function of temperature in ${\mathrm{L}1}_0$ FePt calculated by using (a) the GSE, (b) the LLG, and (c) the ES model. Lines show results obtained by using the spin dynamics models, while the symbols are for Monte Carlo simulations. Each model gives slightly different Curie temperatures, but they all are close to the experimental value. In (a) and (b) the average sublattice magnetizations of Fe and Pt are plotted separately. In (a) the average of the Pt magnetization unit vector $n_{\text{Pt}}$ is shown as a dotted line as a comparison with (b).

Figure 4

Probability distribution of finding (a) an Fe or (b) a Pt moment with a certain spin length at different temperatures. For Fe the probability is peaked at around $\left|S\right|=1$ for all temperatures due to the form of the longitudinal energy function, while for Pt it decreases with temperature due to the increasing noncollinearity of the neighboring Fe atoms.

Figure 5

Dynamics structure factor for the (a) GSE, (b) LLG, and (c) ES model. The color intensity encodes the logarithm of the power spectral density normalized to the peak values along each $k$ vector. Since the GSE and LLG models contain two atoms in the primitive cell, both acoustic and optical branches are present. Only the acoustic branch is observed for the ESM.

Figure 6

The temperature dependence of the average Hamiltonian energies in FePt separated into the three main contributions, (a) exchange ($E_J$), (b) anisotropy ($E_k$), and (c) longitudinal ($E_l$), for each element. Solid (open) symbols correspond to the LLG (GSE) model. The Fe energies are given on the left-hand axis, and the Pt ones are on the right-hand side. In (c) $E_l$ is given relative to the $T=0\mathrm{K}$ values. The exchange energy shows a consistent decrease (in magnitude) with temperature. For Fe the GSE and LLG models are similar, while for Pt there is a change in the power scaling from $<1$ for GSE to $>1$ for LLG. Likewise, the anisotropy energy for Fe behaves similarly for the two models, while that of Pt decreases with $T$ for GSE and increases for LLG. The longitudinal energy in the GSE model increases with $T$ for both Fe and Pt but on a different scale.

Figure 7

Fourier transform of the $m_x$ correlation function at 10, 300, and 550 K. The solid lines show a fit using Eq. (12). With increasing temperature the resonance field drops, and so does the magnitude of the background white noise.

Figure 8

Anisotropy field as a function of temperature calculated from the FMR for each model. The ESM and GSE models drop slowly until reaching the Curie temperature, while the LLG model shows a fast decay with temperature even well below $T_{\mathrm{C}}$. The inset shows the scaling of the anisotropy with the total magnetization, showing a power of $n=2.17$ for the GSE and ES models, while for the LLG model we find $n=8.90$. For the LLG model we also present the case where the anisotropy is scaled with the Pt sublattice magnetization (solid orange circles). In this case the scaling follows a $n=3.1$ power law, closer to the $n=3$ expected for pure uniaxial anisotropy.

Figure 9

Temperature dependence of the Gilbert damping extracted from fitting the FMR line shape. At low temperature the Gilbert damping is close to the atomistic damping value ($\lambda =0.1$), while it diverges as we approach the Curie temperature.