Abstract
Magnetization manipulation is essential for basic research and applications. A fundamental question is, how fast can the magnetization be reversed in nanoscale magnetic storage media. When subject to an ultrafast laser pulse, the speed of the magnetization dynamics depends on the nature of the energy transfer pathway. The order of the spin system can be effectively influenced through spinflip processes mediated by hot electrons. It has been predicted that as electrons drive spins into the regime close to almost total demagnetization, characterized by a loss of ferromagnetic correlations near criticality, a second slower demagnetization process takes place after the initial fast drop of magnetization. By studying FePt, we unravel the fundamental role of the electronic structure. As the ferromagnet Fe becomes more noble in the FePt compound, the electronic structure is changed and the density of states around the Fermi level is reduced, thereby driving the spin correlations into the limit of critical fluctuations. We demonstrate the impact of the electrons and the ferromagnetic interactions, which allows a general insight into the mechanisms of spin dynamics when the ferromagnetic state is highly excited and identifies possible recording speed limits in heatassisted magnetization reversal.
Introduction
The FePt L1_{0} alloy represents the most important material for novel concepts in magnetic recording due to its high magnetic anisotropy, which ensures longtime thermal stability of nanometer sized bits^{1}. Thin films of FePt with perpendicular anisotropy and small grain sizes are the most promising candidates for heatassisted 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 and a deeper insight into the processes involved. However, nondeterministic fractioning in ultrafast magnetization reversal can limit the switching speed in recording schemes and thus has inspired fundamental research for more than a decade^{3,4}. Recently, a new concept of ultrafast alloptical magnetic recording with an unprecedented switching timescale below 1 ps was suggested^{5}. This opened new possibilities to reduce the speed limit established by the spinorbit coupling timescale to that governed by the much stronger exchange interaction. Here we show that in FePt the intrinsic dynamics surprisingly is connected with the electronic structure: the alloying of Pt, because of its low lying dbands, reduces the density of states significantly^{6,7} and increases the electron temperature significantly above the Curie temperature. We identify the large electron energies of the heated electron system in FePt L1_{0}, arising from its reduced density of states at around the Fermi level, as its origin driving the spin fluctuations^{8,9}.
Even though CoPt_{3} was among the first thin film systems investigated^{10} since the discovery of ultrafast demagnetization in 1996 by Beaurepaire et al.^{11}, most investigations were centered on samples with inplane anisotropy. So little is known about the behavior of materials with high perpendicular anisotropy^{12,13,14}. A notable exception of materials with perpendicular anisotropy is the ferrimagnetic CoFeGd, which was studied in alloptical ultrafast switching triggered by a single laser pulse^{5,15}. The modeling of this mechanism involves the exchange interaction of the two spin subsystems^{16}. Different models have been proposed^{17,18,19,20,21}. On the other hand, recent work of Koopmans et al.^{22} 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 subpicosecond fast demagnetization. However, in the first case the fast femtosecond demagnetization is followed by a magnetization recovery (as in Ni^{23}), 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^{24}). 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^{25}. Thus, the question of whether the high anisotropy material 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^{26,27,28} that can be directly detected by the emission of THz radiation^{29}. We use insulating substrates and cap layers to minimize these effects. A transition from type I to type II demagnetization is found, triggered by the laser fluence. Theoretically this is analyzed by mapping the properties of the electron system onto a Heisenberglike spin system in a thermal ensemble that determines the timescales of the ultrafast fluctuations^{30}. A coupling parameter λ, determined by the microscopic spinflip transitions, couples them^{31}. Via two scenarios, we calculate a different response of the electronic system to the laser pulse. We can pinpoint that the quasiequilibrium values of the electron temperature above the Curie temperature is determining the response of the spin system. This is leading to the critical fluctuations and a slowing down, a fundamental limit to recording speeds in heatassisted reversal on the ultrafast timescales. To enable progress in highspeed and highcapacity magnetic storage devices, a fundamental understanding of these dynamic processes is required in the future.
Results
We have studied a 3 nmthick continuous FePt thin layer and a 7 nmthick AgCuFePtC granular recording media, shown in Fig. 1 a) and b). In the granular media carbon intercalates the magnetic 5–8 nm grains to separate them magnetically^{32,33}. In Fig. 1 c) the timeresolved magnetooptical Kerr effect (Δθ_{K}/θ_{K}_{,0}) is shown for the granular FePt recording medium as well as the data for the continuous FePt layer for a small demagnetization of below 3–7 % M_{S}, which was set to +/− M_{S} beforehand. Besides the absolute scale, both can be described using identical parameters of the analytical solution of a rate equation model, shown as a continuous line: the microscopic mechanisms on the nanometer length scale dominate the dynamics on a femtosecond timescale. This leads to identical spectra for the continuous film and granular media. Moreover for the case of the recording medium having granular structure, the larger H_{S} and different K_{u} does not alter the magnetization dynamics. 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 exchange interaction. Simultaneously to the magnetooptical Kerr rotation of the probe beam, the timeresolved reflectivity is determined. The decay of the reflectivity signal is fitted to an exponential function before characteristic stress waves set in (Fig. 1 d)), according to R(t) ~ exp (−t/τ_{E}). The results are presented in Fig. 1 f) showing the evolution of the characteristic timescale τ_{E} that represents the relaxation time of the electron temperature. Experimentally, the value for τ_{E} ~ 1 ps is a typical value expected for transition metals and consistent with previous results^{35}. While for oxides discrepancies of the Kerr rotation and the genuine magnetization can be found even on long timescales^{36}, this is meanwhile accepted not to be the case for metals for time scales longer than the electron equilibration time (about 50–100 fs).
For the continuous FePt layer, using moderate Bfields of 200 mT saturates the magnetization and we can extract the microscopic parameters of the ultrafast magnetization dynamics of FePt in polar geometry for a strong demagnetization. In the experiment, the fluence of the pump beam is varied from 5 to 35 mJ/cm^{2}. The Kerr rotation is extracted from that of opposite external field direction^{34,37} to get the absolute degree of demagnetization, presented in Fig. 2. The first rapid demagnetization, τ_{M}, occurs at a timescale of 0.15–0.38 ps, increasing with the laser pump fluence. A “kink” is observed here on the short timescales at zero delay known from preceding experiments, for example by Bigot et al.^{38}. The remagnetization slows down as a function of the incident fluence, until finally a second, slower demagnetization with the absence of recovery is found above a fluence of 30 mJ/cm^{2}. Thus, a transition from type I to type II as classified in Ref. 22 is observed for FePt.
To understand the behavior, we model the ultrafast magnetization dynamics under external laser excitation by a thermal process of electronic origin^{23}. The model considers that within a timescale of the order of 10 fs the electrons are thermalized and can be described by a quasiequilibrium electron temperature T = T_{e}(t), coupled to the spin system via the the fluctuationdissipation theorem. We implemented the modeling within a stochastic micromagnetic LandauLifshitzBloch (LLB) model. The LLB equation for the spin S represents a thermodynamic extension of the LandauLifshitzGilbert (LLG) equation and has been derived by modeling distribution functions for thermal ensembles^{39}. The thermal bath is the electron temperature T = T_{e}(t) and the couplingtothe bath parameter λ is related to microscopic scattering mechanisms. For the spin quantum number S = 1/2 this versatile equation is equivalent to the microscopic three temperature model (M3TM) proposed for the description of ultrafast processes^{35}. Using the LLB with spin S = 3/2 it has been shown by comparison to an atomistic model for FePt that this describes its thermal equilibrium properties, e.g. damping and magnetization versus temperature, as well as the ultrafast magnetization dynamics^{30,40}. Note that in the classical case, e.g. for S → ∞, the LLB equation can be derived as a coarsegrained approach from a classical FokkerPlanck equation for a thermal ensemble of spins, whose individual motion follows the LandauLifshitzGilbert equation, coupled to a thermal source via stochastic fluctuations in a similar way. The LLB's advantage, as compared to atomistic simulations, is that it is far less computationally intensive and thus represents a thermodynamically consistent extension of classical micromagnetism. Therefore, to extend its capabilities describing lateral inhomogenities (e.g. lateral thermal gradients or domains), here the LLB is extended to a thermal micromagnetic LLB. Combining individual thermal macrospins in a discrete array allows to model systems up to the micronsize. Thereby, each cell represents a thermodynamic average over atomistic magnetic moments. Thus, thermal ultrafast heating is taken into account within each single cell by the longitudinal relaxation, equal to τ_{M}. However, this micromagnetic LLB allows describing locally a different thermal demagnetization, as well as a rotation against each other. In the model 900 thermal macrospins with a lateral cubic discretization of Δ = 3 nm with periodic boundary conditions are combined (Fig. 3). In detail, every single macrospin within the LLB formalism m_{i} = M_{i}/M_{e}(0) is described by the following formalism derived by Garanin^{39,41,42}:
M_{i} is the magnetization (thermal average of atomistic spins over the volume V at temperature T), M_{e}(T) is its thermal equilibrium value and M_{e}(0) is the maximum at T = 0 K. The temperature dependence of M_{e}(T) is evaluated via the Brillouin function with spin quantum number S = 3/2^{43}. The effective field is comprised of applied, anisotropy, micromagnetic exchange and internal exchange fields. The micromagnetic anisotropy field is determined by the perpendicular susceptibility . Experimental^{44} and theoretical^{45} results report that in FePt the anisotropy K scales with magnetization as ~ m(T)^{2.1}. Thus, we use . The micromagnetic exchange field is defined as^{30}
where j goes over neighboring elements and A_{i}(T) is the micromagnetic exchange stiffness. For FePt, we derived its temperature dependence from atomistic calculations using a FePt Hamiltonian with parameters taken from local spin density approximation (LSDA)^{46}. It scales with the magnetization as ~ m(T)^{1.76}. Here we use , where A(0) = 2.2 · 10^{−6} erg/cm. The internal exchange field H_{i}_{,J} results from the thermal average of atomistic spins. At low temperatures, it is responsible for keeping the magnetization magnitude constant. At the same time this term is responsible for the critical behavior approaching ~ T_{C}, which can be seen in the following expression:
In this expression a_{s} = 2(S + 1)^{2}/([S +1]^{2} + S^{2}) and represents the longitudinal susceptibility. The latter follows from
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. 46) and T_{C} is given by T_{C} = S(S + 1)J_{0}/3k_{B} where k_{B} is the Boltzmann's constant. The stochastic fields ζ_{i}_{,⊥} and ζ_{i,ad} are given by^{47}
where i and j denote the macrospin number and k and l denote its Cartesian components x, y and z. Finally, the longitudinal and transverse relaxation parameters are^{42}
where λ is the microscopic relaxation parameter that couples the spin dynamics to the electron temperature T = T_{e}(t), defined by the microscopic spin scattering rate λ and q_{s} = 3T_{C}m_{e}/[2(S + 1)T].
Via microscopic spin scattering rate λ and the relaxation dynamics, the magnetization dynamics is coupled to the electron temperature T = T_{e}(t) as described above. However in turn, the electron temperature within the two temperature model (2T) is coupled to the lattice temperature T_{ph}(t) via rate equations, which determine its evolution with time t after the pump pulse arrived:
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} = 340 ps 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}, where γ_{e} is determined by the electron density of states. 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 timeresolved reflectivity reveals that the change of electron temperature depends linearly on the change in reflectivity^{48}. Thus, we assume that the lacking parameters of the 2T model can be extracted from the experimental data. The thermal diffusion timescale to the substrate τ_{ph} is obtained from the longterm magnetization behavior. The proportionality constant I_{0} is estimated by fitting of the experimental demagnetization value at 30 ps and the couplingtothe bath parameter λ via matching the maximum demagnetization value. The microscopic parameter λ is related to the spinflip probability. Assuming that the main spinflip scattering mechanism is the same that is responsible for the energy dissipation processes on a longer timescale, λ is connected to the macroscopic Gilbert damping.We used 0.01 to 0.1 as a starting point for FePt^{30,49,50,51}. This value scales with the spinorbit interaction^{51}. But generally the values of the Gilbert damping are found to be larger than the electronic couplingtothe bath parameter λ. Additional contributions to the Gilbert damping are magnonmagnon scattering and inhomogeneous line broadening of the ferromagnetic resonance^{34,52}.
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 a low electron temperature profile. To model the profile for the FePt, first G_{e–ph} = 1.5 · 10^{17} W/m^{3}K is assumed in the range of Cu, Mo and Pt^{53}. Then, the analysis of the reflectivity relaxation rate gives γ_{e} = 110 J/m^{3}K^{2}. This value is of the order reported for Au and Cu^{53}, but is much smaller than the corresponding value for Ni and hence produces a high electron temperature^{23}. In the second case of a hypothetical material, a coupling constant G_{e–ph} = 1.8 · 10^{18} W/m^{3}K, similar to Ni^{23}, is assumed which gives γ_{e} = 1700 J/m^{3}K^{2}. This γ_{e} is the proper value for transition metals with a large density of states at the Fermi level due to crossing dbands. As a consequence, a lower electron temperature is reached (Fig. 3 d)).
Discussion
The results for the integration of the set of the LLB equations, coupled to the 2T model, with the set of parameters for FePt are presented in Fig. 5 for all fluences. As in the experiment, the simulations show a transition between type I and type II behavior. A fair agreement is found for the evolution of the demagnetization values ΔM/M(300 K) = 0.05–0.7. The demagnetization time τ_{M} are matched and range in between τ_{M} = 0.2–0.35 ps and, within this fluence range, they show an increase (Fig. 1 f)). Because of the large electron temperatures, a small λ = 0.01 is sufficient to reach the demagnetization values. Contrary to this, in case of the integration of the LLB equation coupled to the 2T model with γ_{e} = 1700 J/m^{3}K^{2} for the hypothetical material, the demagnetization time τ_{M} stays at a constant value and is always followed by the remagnetization within several picoseconds (see Supplementary Materials). Here a large λ = 0.1 is necessary to reach the experimental demagnetization values. The difference can be seen in Fig. 3 d) where both scenarios are presented directly opposed to each other. We obsereve that the second scenario does not produce a transition between type I and type II behavior. The origin of the high electron temperatures is the reduced density of states of FePt by Pt alloying. Because of lower lying dbands in Pt as compared to Fe (Fig. 3 a, b), the ferromagnet Fe becomes more noble in the FePt L1_{0} compound^{6,7}. This is the underlying reason for the electron specific heat being more similar to Au and Cu with a small γ_{e}. For the hypothetical scenario with large γ_{e}, only below 1 ps a T_{e} above the Curie temperature T_{C} is found. For the full calculation for all fluences refer to the Supplementary Material. We conclude that the transition from type I to type II demagnetization is defined by a critical temperature that the electrons have reached. The microscopic picture behind is that the electron system drives via spinflips the magnetic system into a region, where the ferromagnetic correlations are strongly disturbed as schematically shown in Fig. 3 c). The role of the coupling parameter λ is only to determine the rate of spinflips^{31}. Microscopic spin fluctuations are responsible for the decrease of the micromagnetic exchange stiffness A(T) and thermal behavior of M_{e}(T) within the LLB model. The crucial effect, however, is that they slow down the magnetization response when the electron temperature is approaching the Curie temperature. In the LLB theory this is taken into account by the critical behavior of the longitudinal susceptibility, diverging at the Curie temperature. The effect is seen by looking at the shaded regions in Fig. 3 d) and Fig. 4. Only if the electron temperature T_{e} stays near the Curie temperature T_{C} for several picoseconds this transition in demagnetization behavior is found. This is a collective effect, constituting the magnetitization dynamics: from the theory of phase transitions we know that the dynamics is dominated by spin fluctuations and divergence of the correlation lengths. This leads to a slowing down of correlation times and connected to that, a longer demagnetization time τ_{M}^{40,54}. The transition to the type II behavior occurs when the timescale τ_{M}, here given in an extension valid approaching T_{C}, exceeds the timescale τ_{E}^{22}:
For the FePt case, a delicate balance between the driving electron temperature T_{e}, determined by the small specific heat of the electron system for FePt and the other parameters, drives the observation of the type II demagnetization in a small window of experimental parameters. This is in contrast to Ni, where for the same parameters only a slowing down is observed^{23}.
However also certain interesting discrepancies are observed. In comparison to the model (Fig. 5), in the experiment (Fig. 2) the transition from type I to type II demagnetization takes place for a lower fluence. To investigate this further, we plot the demagnetization dynamics for the highest fluences, now normalized to the value of M_{S} at room temperature when the pumpbeam is blocked. This value can be determined by the hysteresis curves given in Fig. 6. The demagnetization relative to M_{S} is lowered further with pump fluence and the two step behavior of the magnetization changes to a decrease within the first picosecond and stays on a plateau. This discrepancy between experiment and theory is related to the decrease of the magnetization at negative delay. The magnetization does not recover in between two laser pulses (e.g. 4 μs pulse repetition). These nonreversible effects are also mirrored in the strong reduction of the coercive fields, ideal for heatassisted writing, meaning that the reversal process is significantly altered. We do not have a full understanding at the moment. Two effects could be the origin that the magnetization dynamics slows down so strongly and stays at its value. The heat is not conducted away and the repeated laser pulses increase the basis temperature, or that the magnetization itself does not recover in between two pulses completely. But it will be most interesting to explore the origin and importance for a reliable heat assisted recording technology in future harddisc drives.
Ultrafast demagnetization dynamics experiment at highest fluences for 30 mJ/cm^{2} to 40 mJ/cm^{2}.
Here the timeresolved data is normalized to its magnetization value with blocked pump beam. The hysteresis curves for 5 mJ/cm^{2} and 40 mJ/cm^{2} are shown for blocked pumpbeam and negative delay of −5 ps. They demonstrate the irreversible parts of the laser induced dynamics present before the new laser pulse arrives (in a time scale of 4 μs pulse repetition period), becoming prominent for 40 mJ/cm^{2}. For 0.5 ps and 2 ps delay the hysteresis is scaled to the maximum demagnetization, demonstrating the size of the demagnetization in total (reversible and irreversible contributions).
In summary, by means of timeresolved Kerr magnetometry we have investigated ultrafast magnetization dynamics in FePt thin films with perpendicular anisotropy. Our results indicate that the electron temperatures reached by the laser heating play 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. We discuss two scenarios with different specific heat of the electron system. Within this framework, we find that a transition to type II behavior is a consequence of the 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 responsible for this transition^{40}. This nondeterministic spin dynamics is responsible for a speed limitation of the magnetic response to the laser pulse. Note that this is not only determined by the ultrafast optical pulse, but also by the nature of the FePt's density of states at the Fermi level, which defines the increase in electron temperature. From our experiments we conclude that a careful adjustment between electron heating is essential, while the anisotropy energy of the system does not play a crucial role for the response time. We identified the nature of the electronic density of states as a tuning parameter for the strength, a spin system will react to the laser power. Our results open possibilities for ultrafast control of the demagnetization in FePt, the most promising candidate for future magnetic recording. 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.
Methods
Fabrication
The continuous FePt thin layer and AgCuFePtC granular recording media have been prepared by sputtering. The structure from bottom to top is glass/NiTa/MgO/FePt/SiO_{2} and glass/NiTa/MgO/AgCuFePtC/carbon overcoat respectively. The granular film has small amount of Ag to reduce the FePt Ll_{0} ordering temperature and a small amount of Cu to lower the T_{C}. Carbon intercalates the magnetic 5–8 nm grains in a specific heating cycle^{32}. The continuous film sample has a Curie temperature of 650 K, a saturation magnetization M_{S} = 1070 emu/cm^{3}, a maximum anisotropy constant K_{u,max} = 2.2 · 10^{7} erg/cm^{3} and an average anisotropy constant K_{u,av} = 1.4 · 10^{7} erg/cm^{3}. Hystersis curves are given in the Supplementary Materials and reveal for the granular recording media μ_{0}H_{S} = 2.4 T and for the continuous FePt film μ_{0}H_{S} = 150 mT.
Experimental setup
In the alloptical pump probe experiment the magnetooptical Kerr rotation of the probe beam is measured^{34}. The pulse has a full width at half maximum of τ_{p} = 40 fs and a wave length of λ = 800 nm (Ti:Sapphire amplifier system RegA 9040, Coherent). The repetition rate of the laser system used was 250 kHz. The Kerr rotation is extracted from that of opposite external field direction^{37}. 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 (Δθ_{K,min} ~ ΔM_{z,min}).
Thermal modelling
To describe the ultrafast magnetization dynamics, we model a thermal process of electronic origin^{23}. The model considers that within a timescale of the order of 10 fs the electrons are thermalized and can be described by a quasiequilibrium electron temperature T_{e}(t). A multimacrospin model is used with cubic discretization elements with a lateral size of Δ = 3 nm (volume V = Δ^{3}). Within each cell the LandauLifshitzBloch (LLB) micromagnetic formalism is applied^{41}. For the present simulation, a system of 30 × 30 × 1 macrospins with periodic boundary conditions in x and y directions is used.
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Acknowledgements
We thank A. McCallum for providing a sample and TEM image, O. Mosendz for a TEM image and U. Atxitia for his help with the theory. Research at Göttingen University was supported by German Research Foundation (DFG) through Research at Göttingen University was supported by German Research Foundation (DFG) through MU 1780/ 61 PhotoMagnonics, SPP 1538 SpinCaT and SFB 1073. Research in Madrid was supported by the European Community's Seventh Framework Programme under grant agreement NNP3SL2012281043 (FEMTOSPIN) and the Spanish Ministry of Science and Innovation under the grant FIS201020979C0202. We acknowledge support by the German Research Foundation and the Open Access Publication Funds of the Göttingen University.
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T.S., M.M. and O.C. designed the experiment. J.M. and J.W. carried out the experiments and analyzed experimental data. J.M. and O.C. developed the theoretical model. T.S. and S.P. prepared and characterized the samples. J.M., P.M., O.C., T.S., S.P. and M.M. discussed the data and anlysis and wrote the manuscript. All the authors reviewed the manuscript.
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Mendil, J., Nieves, P., ChubykaloFesenko, O. et al. Resolving the role of femtosecond heated electrons in ultrafast spin dynamics. Sci Rep 4, 3980 (2014). https://doi.org/10.1038/srep03980
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