Speed limit of FePt spin dynamics on femtosecond timescales

Magnetization manipulation is becoming an indispensable tool for both basic and applied research. Theory predicts two types of ultrafast demagnetization dynamics classified as type I and type II. In type II materials, a second slower process takes place after the initial fast drop of magnetization. In this letter we investigate this behavior for FePt recording materials with perpendicular anisotropy. The magnetization dynamics have been simulated using a thermal micromagnetic model based on the Landau-Lifshitz-Bloch equation. We identify a transition to type II behavior and relate it to the electron temperatures reached by the laser heating. This slowing down is a fundamental limit to reconding speeds in heat assisted reversal.

recording due to its high magnetic anisotropy, which ensures long-time thermal stability of nanometer sized bits 1 . Thin films of FePt with perpendicular anisotropy and small grain sizes are the most promising candidate for heat-assisted magnetic recording, which could reach storage densities beyond 1 Tb/inch 2 . Patterning continuous FePt into individual bits 2 can in principle extend recording densities to 100 Tb/inch 2 . The ultimate magnetic recording applications will also require faster bit switching. However, non-deterministic fractioning in ultrafast magnetization reversal can limit the switching speed in recording schemes, and thus has inspired fundamental research for nearly two decades 3 . Recently a new concept of ultrafast all-optical magnetic recording with an unprecedented switching timescale below 1 ps was suggested 4 . This opened new possibilities to reduce the speed limit established by the spin-orbit coupling timescale to that governed by the much stronger exchange interaction.
Here we show that in FePt fractioning limits the ultimate switching speed through critical fluctuations at the high electron temperatures following femtosecond laser excitation.
Even though CoPt 3 was among the first thin film systems investigated 5 since the discovery of ultrafast demagnetization in 1996 by Beaurepaire et al. 6 , most investigations were centered on samples with in-plane anisotropy, so that little is known about the behavior of materials with perpendicular anisotropy. A notable exception is ferrimagnetic CoFeGd, which was studied in all-optical ultrafast switching triggered by a single laser pulse 4,7 . The modeling of this mechanism involves the complex interaction of the two spin subsystems 8 and is still under debate. To enable progress in high-speed and high-capacity magnetic storage devices, a fundamental understanding of the ultrafast demagnetization dynamics in these materials is required.
Recent work of Koopmans et al. 9 suggests the classification of materials as "fast" (or type I) and "slow" (or type II) based on the ratio T C /µ at , where µ at is the magnetic momentum per atom and T C is the Curie temperature. In both cases, there is an initial sub-picosecond fast demagnetization. However, in the first case the fast femtosecond demagnetization is followed by a magnetization recovery (as in Ni 13 ), while in the latter a second slower demagnetization takes place. The recovery occurs on the timescale on the order of 50 ps and more (as in Gd 14 ). According to this classification FePt should be regarded as a fast magnetic material. However, more recently it has been shown that in Ni both behaviors can be observed, depending on the amount of deposited energy 15 . Thus, the question of whether d) The reflectivity dynamics from which the exponential decay τ E and e) the electron temperature T e are obtained. f) The relaxation time τ E for the electron temperature and τ M for the ultrafast demagnetization is given below for a set of pump fluences.
FePt can behave as "fast" or "slow" under specific laser excitation is an open question. In addition, for thin films and granular media the contributions of spin currents to the ultrafast demagnetization dynamics cannot be neglected [10][11][12] . In this work, we use isolating substrates and cap layers to minimize these effects. The ultrafast demagnetization dynamics of a FePt continuous film sample and a high anistropy granular recoding medium is investigated. A transition from type I to type II is found, triggered by the laser fluence. We pinpoint the electron temperature reached by the laser heating as the underlying mechanism in FePt.
We have studied a 3 nm-thick continuous FePt thin layer (H C = 200 mT) and 7 nmthick AgCuFePt-C granular recording media (H C = 2.4 T), shown in Fig. 1 a)  The energy scale of the magnetic anisotropy K u < 1 meV is too small to affect the dynamics on the timescale related to the energy scale of the of exchange interaction.
We extract the microscopic parameters of the ultrafast magnetization dynamics of FePt in polar geometry using moderate B-fields for switching the magnetization. In the experiment, the fluence of the pump beam is varied from 5 to 40 mJ/cm 2 in steps of 5 mJ/cm 2 . The magneto-optical Kerr rotation of the probe beam is measured 19 and its time delay relative to the pump beam is varied. Similarly, the time resolved reflectivity is detemined. The decay of the reflectivity signal is fitted to a simple exponential function before characteristic stress waves set in (Fig. 1 d)  The Kerr rotation is extracted from that of opposite external field direction in order to remove all non-magnetic and thus symmetric contributions 17 . To get the absolute degree of demagnetization, the Kerr signal is scaled to hysteresis measurements at two states of reference; one at negative delay (θ K ∼ M z,0 ) and the other at a time delay that shows the lowest magnetization (∆θ K,min ∼ ∆M z,min ). The pulse shape is assumed to be Gaussian  Table I. The parameters are extracted from the reflectivity dynamics (Fig. 1). Within the shaded area marked in the left panel, the electron temperature exceeds the Curie temperature.
single macrospin m i = M i /M e (0) is described using the LLB equation for a finite spin S that reads 21,22 : The micromagnetic exchange field is defined as 26 where j goes over neighboring elements and A(T ) is the micromagnetic exchange stiffness, Hereχ = ∂m/∂H || represents the longitudinal susceptibility, evaluated in the MFA as ; ξ e = 3ST C m e (S + 1)T (4) where B () stands for the derivative of the Brillouin function. The relationship between the internal exchange parameter J 0 (also related to A(T = 0 K), see Ref. 27 ) and T C is given by The stochastic fields ζ i,⊥ and ζ i,ad are given by 28 where λ is the microscopic relaxation parameter that couples the spin dynamics to the electron temperature, defined by the microscopic spin scattering rate, and q s = 3T C m e /[2(S + The magnetization dynamics is coupled to the electron temperature T e (t). In turn, the electron temperature within the two temperature model (2T) is coupled to the lattice temperature T ph (t) via rate equations: Here C e and C ph denote the specific heat of the electrons and the lattice, respectively, G e−ph is the coupling constant determining the energy exchange between the electron and lattice systems, and τ ph is the heat diffusion time to the substrate. For C e , the free electron approximation is used resulting in C e = γ e T e . C ph is set constant, since FePt has a Debye temperature well below the room temperature T room . The laser absorbed power is defined by P (t) = I 0 F exp [−(t/τ p ) 2 ] proportional to the laser pump fluence F . The time resolved reflectivity reveals that the change of electron temperature depends linearly on the change in reflectivity 29 . Thus, we assume that the lacking parameters of the 2T model can be extracted from the measured reflectivity at the lowest pump fluence F = 5 mJ/cm 2 . The long-term diffusion timescale τ ph is obtained from the long-term magnetization behavior.
The proportionality constant I 0 is estimated by fitting of the experimental demagnetization value at 30 ps, and the coupling-to-the bath parameter λ via matching the maximum demagnetization value. Note that the value obtained for λ is similar to those used previously for the simulations of FePt 30 and corresponds to an enhanced spin scattering rate at high temperatures.
The determination of the material specific constants such as γ e and G e−ph is crucial for a proper simulation of the demagnetization process. Two approaches were performed, resulting in a high and low electron temperature (see Supplementary Materials). First,  G e−ph = 1.5 · 10 17 W/m 3 K is assumed and thus in the range of Cu, Mo, and Pt 31 . Then, the fitting to the reflectivity relaxation rate, measured at F = 5 mJ/cm 2 , gives γ e = 110 J/m 3 K 2 . This value is of the order reported for Au and Cu 31 but is much smaller than the corresponding value for Ni 13 and hence produces a high electron temperature. The final set of parameters is presented in Table I. The corresponding simulated electron temperature is shown in Fig. 4. The electron temperature decay fitted to the same exponential decay function as in the experiment and the relaxation time τ E shows a reasonable agreement with the experiment for all fluences (Fig. 1 f)). In the second case, a coupling constant G e−ph = 1.8 · 10 18 W/m 3 K, similar to Ni 13 is assumed which gives γ e = 1700 J/m 3 K 2 , more proper to transition metals, and thus about one order of magnitude larger than in the first case. As a consequence, a lower electron temperature (with the maximum value up to 1000 K) is reached.
The results for the integration of the set of the LLB equations, coupled to the 2T model, with the set of parameters from Table I  between type I and type II behavior and the sub-ps demagnetization is always followed by the remagnetization within several ps (see Supplementary Materials). We conclude that the transition is defined by a critical temperature that the electrons have reached. This is illustrated by the shaded regions in Fig. 4. Only if the electron temperature T e stays near the Curie temperature T C for several picoseconds this transition is found. From the theory of phase transitions, we know that at such temperatures the dynamics will be characterized by an increased dominance of magnetization fluctuations. Particularly, the divergence of correlation lengths leads to slowing down of correlation times 32,33 . We find that the characteristics of type II materials are related to a non-deterministic fractioning into dynamic spin excitations. In the experiment (Fig. 2) the relative decay is deviating for the two highest laser fluences, in comparison to the model (Fig. 5). This discrepancy is related to the decrease of the magnetization at negative delay due to the accumulation of the high pump power, which is not taken into acount in the LLB model.
In summary, by means of the time-resolved Kerr magnetometry we have investigated ultrafast magnetization dynamics in FePt thin films with perpendicular anisotropy. Our results indicate that the amount of the absorbed energy plays a crucial role in the character of the ultrafast demagnetization in FePt. The measurements reveal a transition from type I to type II behavior. Our experimental results are modeled in terms of the micromagnetic LLB model, coupled to the 2T model. Within this framework, we find that transition to type II behavior is a consequence of high electron temperature. We identify that at large pump fluences the resulting electron temperature remains close to the Curie temperature and is leading to critical magnetization fluctuations 33 responsible for this transition. This nondeterministic spin dynamics is responsible for a speed limtitation of the magnetic response to the laser pulse. Note that this is defined not only by the laser fluence, but also by the nature of the FePt's density of states at the Fermi level defining the increase in electron temperature. Our results open possibilities for ultrafast control of the demagnetization in FePt, the most promising candidate for future magnetic recording applications. Importantly, we have shown that we are able to manipulate the degree of demagnetization and its ultrafast rates. We propose that for efficient writing the degree of heating and its speed have to be balanced by varying the amount of the energy deposited.