Ultrafast and Distinct Spin Dynamics in Magnetic Alloys

Controlling magnetic order on ultrashort timescales is crucial for engineering the next-generation magnetic devices that combine ultrafast data processing with ultrahigh-density data storage. An appealing scenario in this context is the use of femtosecond (fs) laser pulses as an ultrafast, external stimulus to fully set the orientation and the magnetization Achieving such control on ultrashort timescales, e.g., comparable to the excitation event itself, remains however a challenge due to the lack of understanding the dynamical behavior of the key parameters governing magnetism: The elemental magnetic moments and the exchange interaction. Here, we investigate the fs laser-induced spin dynamics in a variety of multi-com-ponent alloys and reveal a dissimilar dynamics of the constituent magnetic moments on ultrashort timescales. Moreover, we show that such distinct dynamics is a general phenomenon that can be exploited to engineer new magnetic media with tailor-made, optimized dynamic properties. Using phenomenological considerations, atomistic modeling and time-resolved X-ray magnetic circular dichroism (XMCD), we demonstrate demagnetization of the constituent sub-lattices on signi¯-cantly di®erent timescales that depend on their magnetic moments and the sign of the exchange interaction. These results can be used as a \recipe" for manipulation and control of magnetization dynamics in a large class of magnetic materials.


Introduction
Femtosecond (fs) laser-induced controlling of ferromagnetic order 1 has intrigued researchers since the pioneering work of Beaurepaire et al., 2 who found that fs optical excitation of ferromagnetic Ni can demagnetize the sample on a sub-picosecond time-scale. This demagnetization event was much faster than one might expect supposing that the demagnetization process is de¯ned by the characteristic time of the spin-lattice relaxation in the ground state. Later, it was demonstrated that circularly polarized laser pulses could act as equally short and relatively strong e®ective magnetic¯eld pulses, 3 even leading to fs laser-induced switching of the magnetization. 4 These intriguing observations have triggered intense experimental and theoretical e®orts to understand the fundamental processes underpinning such ultrafast laser-induced dynamics. 1 Although a few models of laser-induced demagnetization of simple ferromagnets such as Ni, Fe, Co and Gd have been suggested, 5-7 the physics of ultrafast magnetic phenomena in systems with two or more magnetic sublattices [8][9][10][11][12][13] still challenges our understanding. Yet many magnetization dynamics studies were performed with magnetic compounds containing at least two magnetic elements, like the well-known Permalloy, [14][15][16] orthoferrites, 3 manganites 17 and yttrium-iron-garnets. 18 In the conventional macrospin approximation, 1 the constituent magnetic sublattices of an alloy are represented by a single macrospin [see Fig. 1(a)], which is used to describe their static and dynamic properties. This implies that two exchange-coupled magnetic sublattices are dynamically indistinguishable and thus would lose their magnetization on the very same time-scale upon ultrafast (fs) laser excitation. However, recent experiments using fs X-ray magnetic circular dichroism (XMCD) have revealed distinctly di®erent magnetization dynamics of the constituent magnetic moments in a Gd(FeCo) alloy excited by a fs optical pulse. 10 Moreover, it was subsequently shown 19 that these distinct dynamics in combination with the exchange interaction between the sublattices, can lead to a deterministic switching of this ferrimagnetic Gd(FeCo) by an ultrafast heat pulse alone. 20 These results lead to the intriguing question of whether such element-speci¯c spin dynamics is a general phenomenon or something strictly related to a narrow class of GdFe-like ferrimagnetic materials?
Here, we investigate the dynamics of multi-sublattice magnetic materials with both ferromagnetic and antiferromagnetic coupling between their sublattices. Using phenomenological modeling and atomistic spin simulations, we show that their demagnetization dynamics is distinct and element-speci¯c. These simulations are fully supported by element-speci¯c fs time-resolved XMCD experiments on various NiFe and rare earth-transition metal (RE-TM) alloys, which provide evidence for a demagnetization time that scales with the elemental magnetic moment and varies with the sign of the exchange interaction. As such, one can tune the speed of magnetization processes in multi-sublattice alloys by properly choosing the magnitude of the constituent magnetic moments and the sign of the exchange interaction that couples them. These results may lead to new design principles of magnetic media for fast and energy-e±cient magnetic data manipulation and storage. 12,13,23

Phenomenological Model of Ultrafast Magnetization Dynamics in Ferromagnetic Alloys
To gain a qualitative understanding of the ultrafast demagnetization dynamics in ferromagnetically ordered two-sublattice magnets, we employ a macroscopic theory developed originally by Baryakhtar 21 for the description of the dynamics and relaxation of the macroscopic (sublattice) magnetization of ferromagnets and antiferromagnets. It was recently shown that this approach can qualitatively describe the spatially averaged and dominantly longitudinal dynamics obtained from atomistic spin dynamics simulations. 8 Clearly, the Baryakhtar theory is not restricted to longitudinal dynamics, and it has recently been demonstrated 22 for antiferromagnetic coupling in the exchange approximation, that in addition to the purely longitudinal solutions, the theory exhibits a second class of solutions, which for speci¯c conditions allows for the appearance of transverse time evolution with the same timescale as the longitudinal exchange relaxation. For the present work, we are interested in a di®erent regime, where both the relativistic relaxation as well as the exchange relaxation play an important role (regime 3 according to Ref. 8), and we will restrict ourselves to the class of longitudinal solutions only. This approach is supported by atomistic simulations, 23 which show that even when the system exhibits a small canting before laser excitation, the fast initial relaxation directly after excitation, i.e., the ultrafast demagnetization, is dominantly longitudinal. Furthermore, all experiments that operate at macroscopic length scales, including the X-ray experiments we describe below, demonstrate a longitudinal evolution of the (sublattice) magnetization on the sub-picosecond timescale. Hence on macroscopic length scales, the longitudinal time-evolution seems to be the relevant one. Under these conditions, the equations of motion are: where S i denotes the macroscopic spin angular momentum of sublattice i ¼ 1; 2, and H i ¼ ÀW =S i is the e®ective magnetic¯eld for sublattice i, which is derived from the magnetic (Landau) free energy W. i are the relativistic relaxation parameters describing the transfer of angular momentum with the environment, while e describes the exchange relaxation responsible for transfer of angular momentum between the sublattices. Previous work 8 considered both i;e and the free energy W as phenomenological parameters. As a next step, to assess the in°uence of the concentration, we derive the macroscopic free energy from a microscopic Heisenberg model, which is also the basis for atomistic spin dynamics simulations. Hence, similar to the latter, we explicitly assume that the longitudinal time-evolution on the macroscopic scale originates from spatial averaging over the transverse dynamics of the atomistic spins. Using the methods described in e.g., Refs. 24-26, we obtain in the mean¯eld approximation for classical spins and for small S i values: For a binary alloy, we can use x is the concentration of sublattice 2, z the average number of nearest neighbors in the alloy, and N the total number of spins per unit volume. In addition, i indicates the magnitude of the spin angular momentum in sublattice i and J ij are exchange parameters between spins in sublattice i and j that we take for nearest neighbors only. Further, k B T is the energy of the temperature°uctuations experienced by the spins, which we assume to be solely determined by the electron temperature. In principle, it is possible to extract the relaxation parameters from microscopic theory in a similar way as¯rst-principle calculations of the transverse Gilbert relaxation parameter. 27,28 This will generally yield parameters that depend on the temperature, the exchange interactions and the concentration. Below, we will consider the relaxation parameters i;e as purely phenomenological, in order to gain qualitative understanding in the limits of small and large concentration.
To understand qualitatively how the demagnetization time changes by going from the pure element to the alloy, we consider¯rst the limiting case of small x. In this limit, we simplify the theory using e ðxÞ ( i for x ( 1 and the transfer of angular momentum between the sublattices will in general be a small e®ect. Furthermore, this transfer is generally less e®ective for ferromagnetic coupling than for antiferromagnetic coupling, since the terms in H 1 À H 2 partly cancel each other. Therefore, we can write the demagnetization rate after a (fs laserinduced) step-like change of the electron temperature as _ As in the pure material, we¯nd a demagnetization time that depends on the magnetic moment, and hence the Ni and Fe sublattices generally have distinct dynamics after laser excitation due to the difference in their magnetic moments. In addition, the concentration determines the sublattice magnetic moments and exchange coupling strength, which can be di®erent from the pure material. This in turn leads to a change in demagnetization times compared to the pure material. Applying this to the NiFe alloy at small Fe concentration, we have J NiNi < J NiFe . 29 Since also Ni ð0Þ < Ni ðxÞ 30 (see also Supplementary Sec. 6), we anticipate the demagnetization time of Ni increases upon moderate doping with Fe. Next, we investigate the limit of large concentration (x % 0:5). As before, the relativistic contributions generally yield distinct dynamics, however the transfer of angular momentum between the sublattices may not be negligible anymore. For example, at high-temperature (k B T ) J ij ) we have The sign of this term depends on the di®erence

Atomistic Spin Simulations of Ultrafast Demagnetization
Following the model predictions above, we have performed atomistic simulations of the response of 10 6 ferromagnetically exchange-coupled NiFe as well as antiferromagnetically coupled GdFe spins to an ultrashort 60 fs heat pulse, as explained in the Supplementary Information. The dynamics of each magnetic sublattice were obtained by solving a system of coupled stochastic Landau-Lifshitz-Gilbert equations for individual atomic moments, which describe the transient dynamics of the system upon excitation with a fs laser pulse. 34 For multi-component magnetic alloys one derives 34,35 a characteristic longitudinal relaxation time a (i.e., demagnetization time) for each constituent component which scales with: where i is the moment associated with site i; i is the coupling constant to the thermal bath, i is the gyromagnetic ratio at site i, k B is Boltzmann's constant and T is the temperature of the bath. Consequently, if an alloy consists of two sublattices and 16 ¼ 2 , a rapid heating of the system to high temperatures will lead to di®erent demagnetization dynamics of the two sublattices, despite the strong exchange interaction that couples them. The e®ect of i and i on the demagnetization time is discussed in Supplementary Sec. 7.
We have simulated both ferromagnetically coupled NiFe alloys as well antiferromagnetically coupled GdFe (ferrimagnetic) systems. Following Ref. 34, we coupled the spin system to the electron temperature that is calculated using the twotemperature model, 36 which describes the transient variation of the electron and phonon temperatures upon femtosecond laser excitation. The strength of the thermal°uctuations then scales with this timevarying temperature. For simplicity, the model assumes the same coupling constants for the di®erent sublattices ( i ¼ 0:01), and the magnetic moments were derived from XMCD measurements (see Supplementary Information). The NiFe exchange interactions were determined from the measured Curie temperature of the NiFe alloy samples as reported in Ref. 37. For instance, in the case of Ni 50 Fe 50 , this yields a total exchange per spin of 260 meV. The results of the Ni 50 Fe 50 simulations are displayed in Fig. 2(a), showing clearly a faster demagnetization for Ni than for Fe. This conclusion holds for the full stoichiometry range of the NiFe alloys, as detailed in the Supplementary Sec. 6.
To simulate the antiferromagnetically coupled GdFe system, we used the exchange constants obtained by¯tting the temperature-dependent XMCD data measured for the Fe and Gd sublattices. 35 Taking into account the number and type of neighbors, the total e®ective Fe-Gd exchange is 30 meV. The results of the GdFe simulations are displayed in Fig. 2(b), showing a dissimilar demagnetization dynamics of Fe and Gd with Fe being much faster than Gd.
Hence, distinct timescales for the demagnetization of magnetic sublattices have been computed for both ferromagnetic and antiferromagnetic coupling between the sublattices. Thus, this appears to be a general phenomenon for multi-sublattice magnets.

Ultrafast Dynamics of the Elemental Magnetic Moments from Time-Resolved XMCD Experiments
To compare these theoretical predictions and simulations with experiments, we measured the transient demagnetization dynamics of the magnetic sublattices in NiFe and GdFe alloys b as well as their respective pure elements, employing the element-speci¯c XMCD technique in transmission at the femto-slicing facility 38,39 of the Helmholtz-Zentrum Berlin. In addition, we have also investigated a ferrimagnetic DyCo 5 (i.e., Dy 16:7 Co 83:3 ) alloy 40 that shows the largest di®erence between the magnitudes of the constituent magnetic moments (see Supplementary Fig. 3). The experiments were performed in a stroboscopic pump-and-probe mode, where the metallic samples were excited with 60 fs laser pulses of 1.55 eV photon energy. Subsequent laser-induced magnetization dynamics were probed by measuring the XMCD changes in transmission with circularly polarized 100 fs X-ray pulses 39 tuned at the L 3 edges of Ni (853 eV), Fe (707 eV), Co (778 eV) and at the M 5 edges of Gd (1189 eV) and Dy (1292 eV). At these energies, we probe the dynamics of the 3d magnetic moment of Ni, Fe, Co and the 4f moment of Gd and Dy, respectively. We point out that at these absorption edges, there is a nonambiguous relationship between the measured XMCD signal and the elemental magnetic moments 41 that remains valid even in the high nonequilibrium regime after laser excitation. 42 The XMCD measurements were performed on Fe, Ni, Ni 50 Fe 50 , Ni 80 Fe 20 , Dy 16:7 Co 83:3 and Gd 25 Fe 66 Co 9 samples at 300 K except for DyCo 5 , which were done at 100 K (under laser exposure the e®ective sample temperature of DyCo 5 rose well above its compensation temperature of 120 K due to a permanent heating e®ect, as deduced from the polarity of the XMCD signal and the elemental hysteresis). For Gd(FeCo), the magnetization compensation temperature was 250 K. The typical absorbed laser°uence was $ 6.8 mJ/cm 2 for NiFe alloys, $10.1 mJ/cm 2 for DyCo 5 while for Gd(FeCo) it was $ 4.5 mJ/cm 2 . The Fe, Ni and NiFe¯lms exhibit an in-plane magnetization orientation, while Gd(FeCo) and DyCo 5 have an out-of-plane magnetic anisotropy. The samples are poly-crystalline except for Gd(FeCo) which is amorphous.
The results of the element-speci¯c laser-induced demagnetization of ferromagnetic Ni 50 Fe 50 are shown in Fig. 3(a). Upon fs laser excitation, the alloy demagnetizes to about 50% at both the Fe and Ni edges. Although a similar degree of demagnetization is achieved for both elements, their transient demagnetization behavior is considerably di®erent: While Fe is demagnetized within $ 800 fs, it takes only $ 300 fs for Ni to demagnetize to the same extent as Fe, in qualitative agreement with the predictions from Fig. 2(a). Although less pronounced, we measure similar distinct demagnetization dynamics of the Fe and Ni moments in a Ni 80 Fe 20 sample with again Fe being slower than Ni (see Supplementary Fig. 7).
Essentially, a similar decoupled demagnetization behavior of the constituent, but antiferromagnetically coupled, magnetic moments is encountered for the ferrimagnetic Gd(FeCo) alloy, as shown in Fig. 3(b). In this case, the demagnetization of Fe takes $ 400 fs while Gd demagnetizes within $ 1.2 ps. Due to the lower T C of $ 550 K (see Ref. 35), here we totally quench the magnetic order at both Fe and Gd edges i.e., achieve a demagnetization degree of 100%. This dynamic decoupling of Fe and Gd is however also encountered for much lower demagnetization degrees down to $ 40% (see Supplementary Fig. 8).
For the ferrimagnetic alloy DyCo 5 , according to Eq. (4) and given the di®erent values of the constituent Co ($ 1.52 B ) and Dy ($ 8.91 B ) magnetic moments (see Supplementary Fig. 3), we expect a large di®erence between their dynamics. Indeed, as shown in Supplementary Fig. 9, Co demagnetizes down to $ 50% within $ 350 fs while it takes several ps for Dy to reach the same demagnetization level.
To extract the time constants of the measured demagnetization transients, we have used the following bi-exponential¯t function: where D and R are the demagnetization and relaxation time constants, g(t) is the Gaussian function describing the time resolution of the experiment, B and C are the exponential amplitudes and A is the value of the transient signal at negative delays. The demagnetization time constants obtained from the¯ts are listed in Table 1 of Supplementary Information and plotted in Fig. 4(a).
For comparison, we also show the demagnetization time constants of the elemental Fe, Ni, Co and Gd samples. The demagnetization of pure Co and Gd are taken from Refs. 5 and 43, respectively. For demagnetization of pure Ni, please see also Ref. 44. We note that these demagnetization time constants are extracted from demagnetization transients (both the alloys and their elemental counterparts) that exhibit similar demagnetization degrees ÁM=M 0 of around 50%, and thus we can treat and compare these demagnetization times on equal footing. In Fig. 4(b) we show the deduced demagnetization constants versus magnetic moment for all measured samples. One observes a clear scaling behavior between the demagnetization time and the magnitude of the magnetic moment. A quantitative estimate for the variation of the demagnetization time per magnetic moment unit is obtained by¯tting the Another important piece of information provided by these data is the dependence of the demagnetization time on the magnetic ordering type of the samples. This is shown in Fig. 5 where the demagnetization time constants of Fe and Co are plotted for pure samples, ferromagnetic FeNi and CoPd and ferrimagnetic (i.e., antiferromagnetic coupling) Gd (FeCo) and DyCo alloys. For a ferromagnetic type of coupling, both the Fe and Co moments demagnetize slower than the pure material while the opposite is observed in the antiferromagnetically coupled GdFe and DyCo. Please note that the very same behavior is encountered when normalizing the elemental demagnetization times to the corresponding magnetic moment magnitudes.

Discussion
There are three major e®ects that can be observed in the overview of the demagnetization time constants shown in Figs. 4 and 5: (i) The distinctly di®erent demagnetization times of the elemental magnetic moments in the ferromagnetic NiFe and ferrimagnetic Gd(FeCo) and DyCo 5 alloys [the highlighted regions in Fig. 4(a)], (ii) the demagnetization times scale approximately with the magnitude of the elemental magnetic moment [see Fig. 4(b)] for the pure samples of Ni, Fe, Co and Gd as well as for their alloys and (iii) the demagnetization time of a speci¯c element (e.g., Fe or Co) changes with the exchange interaction type of the host alloy i.e., it is longer in ferromagnetic FeNi and shorter in antiferromagnetically coupled Gd(FeCo) and DyCo 5 compared to their pure Fe and Co counterparts (see Fig. 5).
Both the XMCD data and the simulations show that, despite the strong ferromagnetic exchange coupling between the Ni and Fe sublattices in NiFe, they apparently lose their net magnetization in a very dissimilar manner. The same situation is also observed in ferrimagnetic Gd(FeCo) and DyCo 5 , where two antiferromagnetically coupled Gd and Fe or Dy and Co sublattices demagnetize on substantially di®erent time-scales. Thus, we observe the same transient decoupling of the exchange-coupled magnetic moments in two material systems that are strikingly di®erent, i.e., ferromagnetic versus antiferromagnetic coupling, itinerant versus localized type of magnetic ordering and in-plane versus outof-plane magnetic anisotropy. Consequently this distinct dynamics seems to be a general property of multi-elemental magnetic compounds that are driven nonadiabatically to a highly nonequilibrium state by fs laser excitation. Moreover, in agreement with Eqs. (3) and (4), the observed characteristic demagnetization times scale with the magnetic moment of the sublattices i.e., the larger the magnetic moment the slower the demagnetization process [see Fig. 4(b)]. For the materials investigated here, we obtain a characteristic change of the demagnetization time per magnetic moment of 90 AE 20 fs/ B . Consequently, although in NiFe, Fe demagnetizes slower than Ni, in Gd(FeCo) the demagnetization of the Fe-sublattice is much faster than the demagnetization of Gd. This distinct dynamics is even more pronounced in DyCo 5 given the large di®erence (a factor of 6) between the Dy and Co magnetic moments.
Furthermore, the antiferromagnetic coupling in Gd(FeCo) and DyCo 5 yields faster demagnetization of both sublattices when compared to their pure elements. The ferromagnetic coupling in Ni 50 Fe 50 , on the other hand, yields faster demagnetization of Ni and slower demagnetization of Fe, despite the fact that the magnetic moment of Ni increases (see Supplementary Fig. 3 for variation of the magnetic moment with the sample stoichiometry). In Ni 80 Fe 20 , the exchange coupling between the sublattices is not e®ective enough (see Table 2 in Supplementary Information) to compensate for the increased magnetic moment, yielding an increase of sublattice dynamics. This allows designing new magnetic materials, for example using exchange coupled multilayers, which combine desirable properties for future recording media. 12,13,23