All-thermal switching of amorphous Gd-Fe alloys: analysis of structural properties and magnetization dynamics
Raghuveer Chimata, Leyla Isaeva, Krisztina Kadas, Anders Bergman, Biplab Sanyal, Johan H. Mentink, Mikhail I. Katsnelson, Theo Rasing, Andrei Kirilyuk, Alexey Kimel, Olle Eriksson, Manuel Pereiro
AAll-thermal switching of amorphous Gd-Fe alloys: analysis of structural propertiesand magnetization dynamics
Raghuveer Chimata, Leyla Isaeva, Krisztina K´adas, , Anders Bergman, Biplab Sanyal, Johan H. Mentink, Mikhail I. Katsnelson, Theo Rasing, Andrei Kirilyuk, Alexey Kimel, Olle Eriksson, and Manuel Pereiro Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden Radboud University Nijmegen, Institute of Molecules and Materials,Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands and Institute of Solid State Physics and Optics, Wigner Research Centre for Physics,Hungarian Academy of Sciences, 1525 Budapest, P.O.B. 49, Hungary (Dated: April 21, 2018)In recent years, there has been an intense interest in understanding the microscopic mechanism ofthermally induced magnetization switching driven by a femtosecond laser pulse. Most of the efforthas been dedicated to periodic crystalline structures while the amorphous counterparts have beenless studied. By using a multiscale approach, i.e. first-principles density functional theory combinedwith atomistic spin dynamics, we report here on the very intricate structural and magnetic natureof amorphous Gd-Fe alloys for a wide range of Gd and Fe atomic concentrations at the nanoscalelevel. Both structural and dynamical properties of Gd-Fe alloys reported in this work are in goodagreement with previous experiments. We calculated the dynamic behavior of homogeneous andinhomogeneous amorphous Gd-Fe alloys and their response under the influence of a femtosecondlaser pulse. In the homogeneous sample, the Fe sublattice switches its magnetization before theGd one. However, the temporal sequence of the switching of the two sublattices is reversed inthe inhomogeneous sample. We propose a possible explanation based on a mechanism driven bya combination of the Dzyaloshiskii-Moriya interaction and exchange frustration, modeled by anantiferromagnetic second-neighbour exchange interaction between Gd atoms in the Gd-rich region.We also report on the influence of laser fluence and damping effects in the all-thermal switching.
I. INTRODUCTION
Switching the sublattice magnetization directions ofamorphous Gd-Fe alloys [1] (doped with small amountsof Co) by intense femtosecond laser pulses has gener-ated significant interest both experimentally and the-oretically. Amorphous Gd-Fe alloys are ferrimagnetic,with a strong antiferromagnetic (AFM) coupling betweenthe rare-earth and transition metal moments, a couplingwhich has its explanation in the hybridization of the and orbitals of the constituting elements [2]. InRef. [1], it was found that an optical excitation causedthe net magnetization of both the Gd and Fe sublat-tices to rapidly collapse. However, the time scales of thedynamics of the two sublattices were found to be quitedifferent: the net magnetic moment of the Fe sublatticewas found to vanish after 0.4 ps and then for a short pe-riod of time, up to 2 ps, be parallel to the Gd moment.The Gd sublattice, which initially is antiferromagnetic toFe, vanished after 2 ps, after which it reversed to have itsmagnetization opposite to that of Fe, hence completingthe reversal process (see Fig. 3 of Ref. [1]). The interestof these results obviously have great potential for techno-logical applications, since they open up for possibilitiesto store information in a magnetic medium without ap-plying an external magnetic or electric field. In fact, theexperimental results reported in Ref. [1] follow intenseinvestigations of magnetization dynamics, which startedin the mid ’90s [3–8].Different theoretical models have attempted to explainthese results. For instance, in Ref. [9] it was argued that two time and temperature domains were relevant, wherethe spin-relaxation was driven first by a relativistic con-tribution whereas subsequently relaxation was argued tobe governed by an exchange origin. A different explana-tion was provided in Ref. [10] where the coupling betweenGd and Fe dominated magnon modes were identified asthe most important aspect of the complex switching be-havior of the Gd-Fe system. It should also be mentionedthat in the experimental investigation of Ref. [11], it wasspeculated that angular momentum was transferred be-tween the different sublattices via spin-currents, and thiswas identified as the most important aspect of the mag-netization dynamics of amorphous Gd-Fe alloys.Although the main experimental findings of Ref. [1]have been repeated in subsequent experiments, there aredetails in a more recent work that have so far not been ad-dressed satisfactorily by theory. For instance, in Ref. [11]several hitherto unexplained experimental facts were re-ported. Moreover, in the samples measured by Graveset al. [11] concentration profiles were detected, with Gdrich/Fe poor regions and Gd poor/Fe rich regions, in thesame sample. Surprisingly, it was found that for the Gdrich regions, the Gd moment has a different dynamicalresponse compared to that of the Gd poor regions. Thisamounted to situations where in the Gd rich regions, theGd moment reversed before the Fe moment, in contrastto the result reported for the average magnetization ofGd or Fe sublattices, reported in Ref. [1]. Hence, inamorphous Gd-Fe alloys, it seems that sometimes the Femoment reverses before the Gd moment, and sometimesGd switches before Fe, depending on the concentration a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug of Gd and Fe in local regions of the sample.None of the theories presented so far has addressed therole of the amorphous structure and the chemical inho-mogeneity of the Gd-Fe system and how this influencesthe ultrafast switching behavior. In this work, we presenta multi-scale approach to address this problem, wherewe coupled first principles electronics structure theory toatomistic spin dynamics simulations [12]. After intro-ducing our methodology, we substantiate our approachby comparing equilibrium magnetization curves with ex-periment for a wide range of concentrations, elucidatingthe crucial role of the amorphous atomic arrangement onthe magnetism. Subsequently, we demonstrate that theresults for homogenous samples are in agreement withprevious theoretical analysis reported in Ref. [9]. Finallywe turn to inhomogenous samples and demonstrate thatthe switching is crucially affected by the chemical inho-mogeneity and the non-collinearity of the spins in therare-earth sublattice. In the appendix, the methods areexplained in more detail and an analysis of the role ofthe damping is also provided. II. METHODA. Details of the simulation of structuralproperties
First-principles spin polarized calculations were per-formed by means of the density functional theory [13, 14]and projector augmented wave [15, 16] method as im-plemented in the Vienna ab initio simulation package(VASP) [17–19]. The exchange-correlation potentialwas treated using the generalized gradient approxima-tion with the Perdew, Burke, and Ernzerhof functional[20], including the valence states 5 s p d s f for Gdand 3 d s for Fe. The LDA+U method [21] was appliedto Gd with U eff =7 eV and J =1 eV.The amorphous structures were generated by meansof the stochastic quenching (SQ) method [22, 23], as de-scribed in Ref. [24]. This method is based on the single-random-valley approximation in vibration-transit (VT)theory [25, 26]. The SQ method was demonstrated toprovide reliable atomic coordinates of amorphous mate-rials [24]. In the initial structures, 200 atoms were bothspatially and chemically randomly distributed in a cubicunit cell with periodic boundary conditions and a den-sity of (cid:37) =7.87 g/cm for Gd . Fe . , (cid:37) =7.88 g/cm forGd . Fe . , and (cid:37) =7.89 g/cm for Gd . Fe . . Theatomic positions were then relaxed until the force on ev-ery atom was negligible. The calculations were performedusing the Γ point.The kinetic energy cutoff of 550 eV together withMethfessel-Paxton band smearing [27] of σ = 0.2 eV wereused for electronic structure calculations. The atomiccharges were determined from Bader analysis [28–30]. B. Details of the atomistic spin-dynamicssimulations
In our simulations, we combined the two-temperature(2T) model [31] with the atomistic spin dynamics (ASD)in the UppASD code [12] using the Landau-Lifshitz-Gilbert (LLG) equation. Model exchange parameterswere used for all simulations. At a finite temperature,the temporal evolution of individual atomic moments inan effective field is governed by Langevin dynamics, d m i dt = − γ m i × [ B i + b i ] − γ α m m i × ( m i × [ B i + b i ]) , (1)where γ is the gyromagnetic ratio, α represents the di-mensionless phenomenological Gilbert damping constantand m i stands for an individual atomic moment on site i .The “effective” magnetic field is represented by B i while b i is a time evolved stochastic magnetic field, which de-pends on the electron temperature from the 2T model.After applying a femtosecond laser pulse on the sam-ples, the electron temperature increase from the initialtemperature T to a peak temperature in less than 50fs. Then, the electron temperature slowly cools down inabout 5 · fs since the heat of the electron system istransferred to the phonon bath via electron-phonon in-teractions [31]. With this method, details of all thermalswitching are investigated in detail, and reported uponbelow. III. RESULTS AND DISCUSSIONA. Ab-initio theory and structural properties
In this section we provide structural information ofGd x Fe − x ( x = 0 . , . , .
76) magnetic alloys based on ab initio theory. We note here that the first-principlescalculations result in a metallic character of these amor-phous materials, in agreement with experimental obser-vations. The electronic properties of Gd-Fe systems aredescribed in more details in the Appendix A.The local atomic environment in amorphous Gd x Fe − x can be analyzed by using radial distribution functions(RDF) calculated for different atomic pairs (see Fig. 1).From RDFs of the SQ-generated structures, we findshort-range order up to 8 ˚A for Gd-Gd and Gd-Fe, andup to 6 ˚A for Fe-Fe atomic pairs. Gd-Gd, Gd-Fe andFe-Fe bond lengths, extracted from RDFs, are shown inTable I along with the bond lengths in selected referencesystems. The theoretical Gd-Gd bond length in amor-phous Gd x Fe − x is found to be shorter, and therefore thebonds are stronger than in hexagonal close-packed Gd.At the same time, the Gd-Gd bond length is longer thanin crystalline compounds consisting of Fe and Gd, suchas cubic GdFe , and trigonal GdFe , suggesting a weakerbonding in the amorphous matrix. We find a favorableagreement between theoretical Gd-Gd bond distance inthe Fe-rich ( x = 0 .
24) and equiatomic ( x = 0 .
50) amor-phous systems compared to the experimental values formelt-quenched amorphous Gd . Fe . and Gd . Fe . ,respectively (see Table I). The theoretical Gd-Fe bondlength in amorphous Gd x Fe − x is very close to that incrystalline GdFe and GdFe . We find the bond distancebetween Fe atoms in Gd x Fe − x to be shorter than in bccFe and cubic GdFe , but at the same time larger thanin trigonal GdFe . We also find a remarkable agree-ment between theoretical bond lengths between differ-ent pairs of atoms, such as Gd-Gd, Gd-Fe and Fe-Fe, inGd x Fe − x ( x = 0 . , .
50) obtained by SQ simulationsand the experimental ones for quench-melted Gd x Fe − x ( x = 0 . , .
56) with similar stoichiometry. This illus-trates the efficiency and accuracy of the SQ method todescribe the structural properties of amorphous materi-als.
DFT: Gd Fe DFT: Gd Fe DFT: Gd Fe MD: Gd Fe Gd-Gd
DFT: Gd Fe DFT: Gd Fe DFT: Gd Fe MD: Gd Fe Gd-Fe R ad i a l d i s t r i bu t i on f un c t i on ( a r b . un i t s ) DFT: Gd Fe DFT: Gd Fe DFT: Gd Fe MD: Gd Fe Fe-Fe
Radial distance (¯)
FIG. 1. (Color online) Radial distribution function inGd x Fe − x with three different stoichiometries. The dashedlines show data calculated with DFT, while the black solidline represents data provided by molecular dynamics calcula-tions. Next, we analyze the local environment in amorphousGd x Fe − x , as it is represented by the average coordi-nation numbers (Table II). With increasing Gd concen-tration, the Gd-Gd average coordination number, as ex- TABLE I. Bond lengths (˚A) in amorphous Gd x Fe − x system.The bond distances in selected crystalline and amorphous sys-tems are listed for comparison.System Gd-Gd Gd-Fe Fe-FeGd . Fe . . Fe . . Fe . [35] 3.21 2.97 2.37GdFe [36] 3.22 3.08 2.63am-Gd . Fe . [32] 3.47 3.11 2.57am-Gd . Fe . [32] 3.54 2.95 2.51 pected, increases from 4.6 for x = 0 .
24 to 10.9 for x = 0 .
76. Similarly, the Gd-Fe coordination number in-creases with the number of Gd atoms. In case of the Gd-Fe atomic pair, the coordination number is almost fivetimes smaller in the Fe-rich amorphous matrix comparedto the Gd-rich one. Also, while the Fe concentration de-creases within the series (from x = 0 .
24 to x = 0 . i.e. decrease through theseries. The reduction in the total coordination numberboth for Gd and Fe atoms can be referred to the changefrom a more close packed structure (Fe-rich system) to amore open one (Gd-rich system). TABLE II. Average coordination numbers for amorphousGd x Fe − x system. Coordination numbers in reference sys-tems are listed for comparison in the lower part of the table.In GdFe the coordination for Gd and Fe atoms with differentsite symmetries is different. Therefore, we show coordinationfor all inequivalent positions (specified in parentheses).System Gd-Gd Gd-Fe Fe-Gd Fe-FeGd . Fe . . Fe . . Fe . [38] 4 12 6 6GdFe [35] (3 a ) 2 (3 a ) 6-12 (3 b ) 6 (3 b ) 6(6 c ) 1-3 (6 c ) 3-6 (6 c ) 3 (6 c ) 3(18 h ) 1-2 (18 h ) 1-2am-Gd . Fe . [32] 3.0 8.8 2.5 7.9am-Gd . Fe . [32] 7.5 3.2 4.2 3.0 B. Generation of amorphous samples usingmolecular dynamics
As commented in Sec. III A, we optimised the struc-tures of Gd x Fe − x (x=0.24, 0.50, 0.75) magnetic alloysby means of ab initio methods considering a supercellof 200 atoms, but the lack of crystal periodicity in amor-phous structures led us to consider even bigger supercellsjust to be sure that the results are reliable and the physicsof the amorphous structure was fully captured. The sizeof the new supercells is beyond the limits of the presentstate of the art of the computational resources using aDFT methodology. In order to deal with bigger super-cells, we employed a molecular dynamics approach, sothat the dynamics of atomistic Fe and Gd spins shown inupcoming sections have been performed using as inputparameters the structural data provided by the molec-ular dynamics method. Consequently, we adapted atwo step procedure. Initially, we constructed a cubicunit cell (1600 atoms) with Gd and Fe atoms using adense-random-packing-of-hard-spheres (DRPHS) modeland using as input the lattice parameter ( ∼ . . Fe . . It may be observedthat the agreement is rather good between the two setsof theoretical values. C. Magnetization-dynamics and all-thermalswitching
1. Curie and compensation temperatures
To further extend the applicability of our method-ology, we performed ASD simulations using the Up-pASD method on a cell containing 1600 atomic spinswith periodic boundary conditions. We take a ferri-magnetic amorphous GdFe model system based on thestructural parameters provided by both molecular dy-namics and DFT calculations. The microscopic modelexchange parameters were taken from Ref. [42]. Weused the bulk exchange values for neighbouring TM andRE ions ( J F e − F e =0.8 mRy, J Gd − Gd =0.15 mRy) becausethey provide the correct Curie temperature for the re-spective sublattices. The value of the intersublattice ex-change coupling ( J Gd − F e =-0.25 mRy) was chosen to fitthe temperature dependence of the saturation magneti-sation of both Fe and Gd sublattices with results of x-raymagnetic circular dichroism measurements of static mag-netisation reported in Ref. [42]. For the magnetic mo-ments we take the bulk values, i. e. 7.6 µ B and 2.1 µ B for Gd and Fe, respectively. No external field was appliedin the simulations and the anisotropy of the Gd and Fesublattices was neglected in the hamiltonian. The modelexchange parameters are capable of reproducing the keystatic magnetic properties, especially the Curie temper-atures and magnetic compensation points. To illustrate this fact we show in Fig. 2 both the calculated compensa-tion and Curie temperatures for different Gd concentra-tions, in the range 20 at.% to 30 at.%. The Curie tem-peratures have been calculated using a finite size scalinganalysis as described in Ref. [43] (see Appendix B). Wefound a very good agreement with the reported experi-mental results. Moreover, we observe a general trend forthe T C to decrease as the Gd concentration increases.We attribute this magnetic softening to the addition ofmore Gd-Gd nearest neighbours, which have a smallerexchange coupling as compared to the Fe-Fe interaction.But still, since we have an amorphous structure and asupercell with limited number of atoms, this condition isnot fulfilled for every Gd concentration and this is whyat some concentrations, the T c can still increase slightly,as is the case for a concentration of Gd of about 25 at.%.In the thermodynamic limit, these smaller fluctuationsof the T C are expected to vanish. It may be seen fromFig. 2 that the simulations reproduce with good accuracythe measured compensation temperatures as well. Boththe measured trend and the absolute values of the com-pensation temperature of these alloys are reproduced bytheory, where the most noticeable feature is the increaseof the compensation temperature with increasing Gd con-centration. The reason for this trend is a competition ofmagnetic sublattices with antiparallel coupling. Too fewGd atoms result in a Gd sublattice with a net magne-tization that is smaller than that of the Fe sublattice,already at low temperatures, and there is no compensa-tion point. However, with increasing Gd concentration,the net magnetic moment of this sublattice is larger thanthat of the Fe sublattice, at low temperatures. Sincethe Fe exchange is stronger than the Gd exchange, theGd sublattice magnetization decays faster with temper-ature compared to the Fe sublattice magnetization, andat the compensation temperature, they have equal sizeand opposite direction. Increasing the Gd concentrationmakes the magnetization of this sublattice stronger rel-ative to the Fe sublattice, at low temperatures. Hencea higher temperature is needed in order to reduce theGd moment to have the same size, albeit with oppositedirection, compared to the Fe sublattice.It is rewarding that the agreement between theory andexperiment found in Fig. 2 is quite good, and that thethree parameters of exchange interactions used in oursimulations explain the compensation temperatures ofthe whole range of concentrations of Fig. 2. We alsonote that finer details of the atomic arrangement of theamorphous structure are very important in achieving theresults shown in Fig. 2, and this illustrates (as often is thecase) that atomic arrangement (structure) and magneticproperties are intimately coupled.
2. All-thermal switching of homogenous samples
In this section, we discuss the dynamic behavior ofhomogenous amorphous alloys, using the exchange pa- hola
21 23 25 27 29 31 33 350100200300400500600 T M Exp T M LLG T C LLG T e m pe r a t u r e ( K ) Gd concentration (%) T C Exp
FIG. 2. (Color online) Curie temperature ( T C ) and magneticcompensation temperature ( T M ) for amorphous Gd-Fe alloys,for different concentrations of Gd. The experimental data hasbeen taken from Ref. [44]. GdFe
GdFe M z ( B ) Time (ps)
Gd (Gd Fe ) Fe (Gd Fe )1.32 ps 2.20 ps Gd (Gd Fe ) Fe (Gd Fe )2.73 ps T e m pe r a t u r e ( K ) Time (ps)
FIG. 3. (Color online) Time evolution of the magnetiza-tion (M z ) for two different concentrations of amorphous Gd-Fe alloys under the influence of a thermal heat pulse. Thesolid and dash-dot lines represent the sample concentrationGd . Fe . , while the dash and dot lines are for Gd . Fe . .The magnetization of the two sublattices is plotted separately.In the inset we show a typical temperature profile inducedfrom the laser fluence, as given by the two temperature modelof Ref. [31]. rameters discussed above, and the magnetic response toa femtosecond laser pulse. Only temperature effects fromthe laser pulse were considered, where we adopted a two-temperature model, as described in the Appendix C. Allsimulations started with the spin system at room tem-perature, from which the heat pulse increased the tem-perature of the spin system in a very short period of time(50 fs) to a maximum value, T max , after which the sam-ple cooled down again. We observed sublattice switching for a wide range of concentrations, i.e. 21 at.% to 30at.% Gd. As an example we show in Fig. 3 the switchingbehavior of the Gd . Fe . and the Gd . Fe . alloys.We find for both concentrations that initially both Gdand Fe sublattices demagnetize fast, and that the Fe sub-lattice reverses its magnetization first, so that for a shortperiod of time both Fe and Gd sublattices have parallelmagnetic moments. Figure 3 shows that after ∼ T max = 800 K. If the spin temperature is too high (e.g. thesimulation with T max = 2000 K) both sublattices simplydemagnetize to a zero moment state, which is stable forsufficiently long time in order to make the remagnetiza-tion completely stochastic in terms of direction of eachsublattice moment of the final configuration. However,for intermediate temperatures (e.g. T max = 1000 K) theall thermal switching occurs, as is also shown in Fig. 4.This shows that it is essential to find the appropriatelaser fluence with respect to the strength of the exchangeinteractions, for all-thermal switching to occur.We end this section with a short note on the effects ofthe damping. We investigated the magnetization dynam-ics for a wide range of damping parameters, as detailedin the Appendix D, and found that the switching behav-ior reported in Fig. 3 was essentially very dependent onthe choice of damping parameter. This shows that thestrength of the intrinsic damping actually determines ifthe all-thermal switching can either occur or not. M z ( B / a t o m ) Time (ps)
Gd T max =800 K Fe T max =800 K Gd T max =1000 K Fe T max =1000 K Gd T max =2000 K Fe T max =2000 K
FIG. 4. (Color online) Time evolution of the magnetization(M z ) of amorphous Gd-Fe alloys (Gd . Fe . ) for differentpeak temperatures caused by the laser fluence.
3. All-thermal switching of inhomogeneous samples
After the initial experimental work of Ref. [1] addi-tional experimental data was reported, and in particularit was argued in Ref. [11] that amorphous Gd-Fe alloysmay have non-uniform concentration profiles, such thatsome regions are richer in Gd and some are poorer, withthe opposite trend for the Fe concentration. Interest-ingly, it was reported that in an all-thermal switchingexperiment of an amorphous Gd-Fe alloy with heteroge-nous concentration, the Gd magnetic moment reachedzero before the Fe magnetic moment in the Gd rich re-gions. After a period of parallel alignment of Fe and Gdmagnetic moments, the reversal completed with both Feand Gd moments having reversed orientations with re-spect to their original direction. Hence, Gd rich regionsexhibited similar behavior as shown here in Fig. 3, albeitwith the Gd sublattice reaching zero first.In order to investigate this experimental result andmimic the experimental samples as closely as possible,we generated simulation cells with concentration profiles,such that, some regions had enhanced (depleted) Gd (Fe)concentration with respect to the nominal concentrationwhile some other regions had a depletion (enhancement)of Gd (Fe) concentration. We considered an amorphousalloy with average concentration Gd . Fe . and the Gdrich regions had an increase of 6 at.% Gd, while the Gdpoor regions had a reduction of 6 at.% of the Gd con-centration. The Fe concentration was modified in thesame way. A schematic illustration of such a heteroge-nous sample is shown in Fig. 5, and further details onhow the inhomogeneous simulation-cells were generatedcan be found in the Appendix E. In these simulations, wehave in the initial configuration also considered different degrees of non-collinearity of the Gd moments. This issupported by our first principles calculations, presentedabove, that show an exchange driven non-collinear con-figuration as the ground state even at T=0 K. From first-principles non-collinear theory, we find that the degree ofnon-collinearity is more pronounced for calculations in-cluding spin-orbit (LS) coupling than without it. More-over, the effect is more prominent for Gd sublattices (seeTable III and Appendix A). For example, it is worthyto mention here that the current DFT calculations withspin-orbit coupling predict a maximal angle deviation ofGd atomic magnetic moments with respect to the quan-tization axis of about 35 ◦ . However, the angle deviationpredicted by DFT without spin-orbit coupling is dras-tically reduced down to less than the half of the valuewith LS couping. Since the amorphous structure lacksinversion symmetry, the Dzyaloshinskii-Moriya (DM) in-teraction, which favours canted (non-collinear) configu-rations, is non-zero for both Gd and Fe sublattices. Sincethe spin-orbit interaction is larger for Gd, it is for thissublattice we expect a larger effect of the DM interac-tion. In Table III we have collected the average and max-imum angles of the magnetic moments, with respect toa common z-axis, for various concentrations. It may beseen that exchange effects alone produce a certain degreeof non-collinearity of the moments, where in particularthe maximum deviation from collinearity is larger for Gdthan for Fe. With spin-orbit effects included, the de-gree of non-collinearity increases, in particular for theGd sublattice. In practice, we have included the de-gree of non-collinearity, from exchange as well as DMinteraction, among the Gd atomic moments via a secondnearest neighbour (SNN) exchange interaction ( J ) withanti-ferromagnetic character, and as reported below, wehave followed the simulated magnetization dynamics asa function of the degree of non-collinearity of Gd sublat-tices. TABLE III. Degree of non-collinearity predicted by the cur-rent DFT calculations with and without spin-orbit (LS) cou-pling. The collected data represent the average (A) and maxi-mal (M) angle deviation with respect to the z-axis of the mag-netic moments on Gd and Fe atoms for amorphous Gd x Fe − x structures ( x = 0 . , . , . . Fe . ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Gd . Fe . ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Gd . Fe . ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ The results from the simulation cells with inhomoge-neous concentration are shown in Fig. 6. The figureshows results for four different degrees of non-collinearityamong the Gd atoms in the initial configuration beforethe heat-pulse enters the spin system. The figure alsoshows for each panel the sublattice magnetization of theGd rich-regions and Gd poor-regions. Several things may
FIG. 5. (Color online) Schematic figure showing the inhomo-geneous samples of Gd . Fe . plotted in the right side whilein the the left side is shown a detailed figure of the amorphousstructure with Fe and Gd atoms in red and blue, respectively.Above, Gd rich regions are called Part A and Fe rich regionsare called Part B. be noted from this figure, where the most important factis that the experimental results of Ref. [11] are repro-duced in these simulations, if a non-collinear configura-tion of Gd atomic spins are considered within Gd rich-regions. Figure 6 shows that for a small degree of non-collinear moments of the initial configuration (panels aand b in Fig. 6), results in a dynamical behavior thatis similar to the homogenous results shown in Fig. 3.Nonetheless, the data shown in Fig. 3 can be consideredqualitatively similar to that in Fig. 6 when J is zero, asshown by the fact that there is all-thermal switching andthat the Fe sublattice demagnetizes before the Gd sublat-tice. Moreover, increasing the degree of non-collinearityof the initial configuration causes the Gd moment to de-magnetize faster, as show in Fig. 6, and for sufficientlylarge values of J it demagnetizes faster than the Fe sub-lattice. This finding is in agreement with the observationreported in Ref. [11].In order to further analyze the results of the heteroge-nous sample, we show in Fig. 7 the magnetization dy-namics of the Gd and Fe sublattices, from the differentregions of the sample, i.e. the Gd poor (Fe rich), Gdrich (Fe poor) and regions with an average concentration.We can observe that the Gd sublattice switches faster(0.42 ps) than the Fe one (1.97 ps) in the Gd-rich regionswhile for the Fe-rich regions the process is reversed. Ascommented above, this finding was already observed ex-perimentally in Ref. [11] for GdFeCo. The explanationproposed by the authors relies on the assumption thatthere are spin currents which transfer torque towards theGd spins in the enriched Gd nanoregions. The theoryput forth here does not involve explicitly a spin currentmechanism. Instead, we propose an alternative explana-tion based on a mechanism driven by a combination ofthe Dzyaloshiskii-Moriya interaction and exchange frus-tration that produces non-collinearity of the Gd atomsbelonging to the Gd-rich nanoregion. Using this model,the atomistic spin-dynamics simulations show that a non-collinear configuration of spins before a heat-pulse enters -0.150.000.15-0.150.000.15-0.150.000.15 0 1 2 3 4 5 6-0.150.000.15 dcb J = 0.0 mRyJ = -0.4 mRy J = -0.1 mRy Gd Part A Fe Part A Gd Part B Fe Part B a J = -0.3 mRy M z ( B ) Time (ps)
FIG. 6. (Color online) Magnetization profile (M z ) for Gd(blue) and Fe (red) sublattices for inhomogeneous samples.The magnetization of Gd rich-regions is shown as solid anddash-dot lines (Part A, Gd . Fe . ) while Gd poor-regionsare represented by dash and dot lines (Part B, Gd . Fe . ).The strength of the second-nearest neighbour exchange pa-rameter ( J ) for Gd rich-regions is different for any of thepanels (a, b, c and d) outlined in the figure. For Gd poor-regions the J values are considered as 0 mRy. the system, explains the faster switching of Gd momentsin the Gd rich region. We note that the assumption ofnon-collinear moments agrees with the non-collinearitypredicted by DFT results for the three stoichiometriesshown in Fig. 1 (see Appendix A). It is important to em-phasize here the role played by the inhomogeneity of the M z ( B ) Time (ps)
Gd Part A Fe Part A Gd Part B Fe Part B Gd average Fe average
FIG. 7. (Color online) Magnetization profile (M z ) for Gd andFe sublattices for inhomogeneous concentration profiles in asample with average concentration Gd . Fe . . The mag-netization of the Gd rich regions (Part A), Gd poor regions(Part B) and sample average is shown in black, green and redlines, respectively. The strength of the next nearest neigh-bour exchange parameter ( J ) for the Gd sublattice was -0.4in this simulation. The figure show also similarities with ex-perimental data reported in Fig. 3c of Ref. [11]. sample. Thus, as shown in Fig. 7, only the Gd sublat-tice in the Gd-rich region switches faster than Fe spinswhile for the sample average, the change of the magne-tization occurs first for the Fe sublattice. In order tomeasure, detect and use that property for technologicalapplications, the experimental techniques are required tohave at least a nanometer spatial resolution or lower,as for example, measuring nanometer-femtosecond spinscattering dynamics using X-ray lasers [11].Ultimately, and based on the DFT data collected inTable III, we observed that the spin-orbit effects con-tributes to both sublattices, but are more important inGd sublattices. Moreover, the degree of non-collinearityis not evenly distributed over the Gd and Fe atoms asshown in Table III by the substantial difference betweenthe average and maximal angle deviation of the atomicmagnetic moment. Consequently, the distribution of theDM vectors or SNN exchange interactions is inhomoge-neous within the Gd-rich and Fe-rich regions. In order tomimic and study the effects of the aforementioned inho-mogeneous distribution, we performed several ASD sim-ulations with different sets of SNN exchange interactionse.g. using values of -0.3 mRy and -0.5 mRy for the Gdsublattice distributed randomly in the Gd rich region.Interestingly, we observe that an inhomogeneous distri-bution of the non-collinearity can favour a faster switch-ing of Gd sublattice if, and only if, the four exchangeinteractions are above a specific threshold (in this set of calculations the threshold was -0.3 mRy, as shown for twosets of parameters in Fig. 8b). Consequently, an inhomo-geneous distribution of non-collinear magnetic momentscan also cause Gd to switch faster than Fe only above aminimum value of the degree of non-collinearity. In linewith the results described in this section, we also stud-ied the influence of the non-collinearity in the Gd-poorregion for Gd sublattice and also for Fe sublattice in theGd-rich region. We observe in both cases that the in-crease of the degree of non-collinearity in this situationcompletely eliminate the switching behavior. We showas an example in Fig. 8a the magnetization profile of theGd sublattice in the Gd-poor region for two values ofthe SNN exchange interaction while for the Gd-rich re-gion the SNN exchange interaction is kept constant as-0.4 mRy. The results show that for J =-0.2 mRy, theswitching behavior is suppressed for both Fe and Gd sub-lattices, while if J2 = -0.1 mRy the all thermal switchingoccurs. The conclusion of all these simulations show thatall-thermal switching is determined by delicate details inthe concentration profile and the exchange interactionsof different regions of the sample. IV. CONCLUSIONS
The study of amorphous Gd-Fe alloys represent an out-standing theoretical challenge because of the lack of crys-talline periodicity and intrinsic sample inhomogeneities.The intricate structural properties clearly determine themagnetic ones, as usually is the case, and consequently,the magnetization dynamics. We have here been able toaddress the very complicated structural, magnetic anddynamical properties of several concentrations of amor-phous Gd-Fe alloys by using ab initio
DFT in conjunc-tion with an atomistic spin dynamics approach. With theaim to assess the validity of this multi-scale approach, wecompare both structural and magnetic parameters withthe experimental results and where a comparison can bemade, we find that they are in a very good agreementwith observed data. In particular, the T C and T M pre-dicted by our simulations compare quite well with theexperiment and we are able to explain the increase of thecompensation temperature with increasing Gd concen-tration. The explanation mainly resides in the competi-tion of magnetic sublattices with antiparallel coupling.Among the most conspicuous results obtained here, welay emphasis on the crucial role played by the degree ofhomogeneity and non-collinearity of atomic moments inthe Gd-Fe alloys for the thermally induced magnetizationswitching driven by a femtosecond laser pulse. For ho-mogeneous samples, the Fe sublattice reverse its magne-tization before the Gd sublattice for a Gd concentrationranging from 21 at.% to 30 at.%. We observe all-thermalswitching irrespective of whether the initial temperaturewas above or below T M , which is in clear disagreementwith previous reported results in literature [45]. In thatregard, the mechanism proposed for all-thermal switch- -0.20-0.15-0.10-0.050.000.050.100.150.20 0 1 2 3 4 5 6-0.20-0.15-0.10-0.050.000.050.100.15 Fe J (Part B)=-0.2 mRy Gd J (Part B)=-0.2 mRy b J (Part A)=-0.4 mRy M z ( B ) Fe J (Part B)=-0.1 mRy Gd J (Part B)=-0.1 mRy a Fe J (Part A)=-0.4,-0.3,-0.1 mRy Gd J (Part A)=-0.4,-0.3,-0.1 mRy Time (ps)
Fe J (Part A)=-0.3,-0.5 mRy Gd J (Part A)=-0.3,-0.5 mRy FIG. 8. (Color online) Magnetization profile (M z ) for Gd andFe sublattices for inhomogeneous concentration profiles in asample with average concentration Gd . Fe . . a) The SNNexchange interaction ( J ) in the Gd-poor region (Part B) hasbeen chosen to be -0.1 and -0.2 mRy while in the Gd-richregion (part A) J =-0.4 mRy. b) Two sets of J parametersdistributed randomly over the Gd-rich region. The values of J parameters are listed in the insets of the figure. ing put forward in Ref. [45] seems to break down foramorphous materials and makes these systems more ver-satile for spintronic applications since they are less sen-sitive to the applied initial temperatures. However, forinhomogeneous samples, we found the opposite behaviorwith respect to homogeneous case, i.e. the Gd sublat-tice reaches zero magnetization faster than Fe sublattice,at least for the regions with higher Gd concentration.Here, we propose a mechanism based on the influence ofDzyaloshinskii-Moriya interaction and the exchange frus-tration that we model by considering a second-neighbourexchange interaction between Gd atoms in the Gd-richregions. The microscopic origin of the antisymmetricDzyaloshinskii-Moriya interaction is in general known tobe coupled to spin-orbit effects and the absence of inver-sion symmetry, that clearly is present in the amorphousGd-Fe samples. The influence of the damping parameterwas also considered in this work and we observe that thisparameter plays a crucial role when dealing with ultra- fast switching experiments. Thus, the amorphous Gd-Fesample with values of α lower or in the vicinity of 0.02undergo a switching process while for higher values ofthe damping, the switching mechanism is totally absent.Furthermore, our results point out that a microscopicmechanism for all-thermal switching does not need to in-volve spin current effects. ACKNOWLEDGMENTS
We gratefully acknowledge financial support from theSwedish Research Council (VR). O.E. is in additiongrateful to the ERC (project 247062 - ASD) and theKAW foundation for support. J.H.M. acknowledgesfunding from the Nederlandse Organisatie voor Weten-schappelijk onderzoek (NWO) by a Rubicon Grant. Sup-port from eSSENCE, Stichting voor Fundamenteel On-derzoek der Materie (FOM), De Nederlandse Organisatievoor Wetenschappelijk Onderzoek(NWO), the EuropeanUnion via ERC Grant agreements No. 257280 (Femto-magnetism), No. 339813 (EXCHANGE), No. 338957(Femto/Nano) and EC FP7 No. 281043 (FEMTOSPIN)is acknowledged. We also acknowledge Swedish NationalInfrastructure for Computing (SNIC) for the allocationof time in high performance supercomputers.
APPENDIX A: ELECTRONIC PROPERTIES
Total and partial densities of states (DOS) of the Gd-Fe system are shown in Fig. 9. The metallic characterof amorphous Gd-Fe is due to Fe 3 d and Gd 5 d states,which contribute to the DOS at the Fermi level. TheGd 4 f states in the LDA+U treatment are localized ina narrow energy interval around 8 eV below the Fermilevel. This is in a rather good agreement with the bind-ing energy of the occupied 4 f states obtained in XPSmeasurements (9.4 eV) [46]. Although the LDA+U treat-ment has been criticized for rare-earths in general [47],it is shown from our electronic structure calculations tobe sufficiently accurate for the purposes of the presentstudy.Since the electronic structure calculations performed inthis work have non-collinear spin densities, the spin-upor spin-down band picture is no more applicable in ourtreatment. However, the main features of the DOS, i.e.the shape of Gd 4 f and Fe 3 d states are similar to thosecalculated for ferrimagnetic amorphous Gd . Fe . al-loys within DFT+LDA theory [48]. For both approaches,the center of mass of the Fe 3 d states is located below theFermi level, while the center of mass of Gd 5 d states islocated above the Fermi level. With the increase of Gdconcentration, the Fe-Fe bond distance becomes slightlyshorter, while the Fe-Fe coordination number reduces al-most 4 times from 7.6 to 2.0. This leads to a narrowerand less intense Fe 4 d states in Gd-rich alloy. We canobserve, in the middle and lower panels of Fig. 9, that0around the Fermi level there is a strong coupling betweenFe 3 d and Gd 5 d states. These orbitals are responsiblefor the strong AFM coupling between Fe and Gd atoms.The calculated average magnetic moment is 7.40 µ B for Gd and 2.38 µ B for Fe atoms. This is in line withprevious experimental data at T =4.2 K for amorphousGd . Fe . ferrimagnetic alloy [49] and also with theo-retical values obtained for Gd . Fe . [48]. In Table III,we show the average and maximal angles between the z-axis and the magnetic moments on Gd and Fe atoms toestimate the degree of non-collinearity in the amorphousGd x Fe − x structures ( x = 0 . , . , . ◦ without LS coupling and ∼ ◦ with LS coupling.The maximal angle deviation tends to be higher as theGd concentration increases. Thus, the non-collinearity isenhanced in Gd-rich structure. Note that our first princi-ple results contrast the empirical models used to supportexperimental studies on amorphous alloys, which alwaysassume the Gd sublattice to be collinear [50]. On theother hand, similar as in our results the degree of non-collinearity of the Fe sublattice increases with increasingthe Gd concentration [51]. If the spin-orbit coupling ispresent in amorphous Fe-Gd alloys, as is indicated bythe current spin-orbit DFT calculations, and due to thefact that amorphous structures lack inversion symmetry,then these conditions create a suitable environment forthe DM interaction to be present in these alloys. Eventhough in amorphous materials it is not possible to applystraightforwardly the usual symmetry-related argumentsencompassed by Moriya rules, however it is feasible touse the rather general formulas such as the ones derivedin Refs. [52, 53].Bader analysis shows that in amorphous Gd x Fe − x ,there is a charge transfer from Gd to Fe atoms. With theincrease of Gd concentration from 24 at.% to 76 at.%,Fe atoms gain more negative charge, simply due to thefact that the probability of finding Gd atoms located innearest neighbour positions around an Fe atom increases.At the same time, the average valence electron increasesby 0.73 and 0.53 for Gd and Fe atoms, respectively. APPENDIX B: BINDER CUMULANT
The fourth order Binder cumulant was introduced inRef. [43] in the context of the finite size scaling the-ory [54]. For magnetic atoms arranged in a lattice ofsize L, the Binder cumulant is defined by: U L = 1 − < m > L < m > L (B1) Fe 3d (Gd Fe ) Fe 3d (Gd Fe ) D O S ( e V - a t o m - ) Fe 3d (Gd Fe ) Fe 3d (Gd Fe ) Energy (eV)
Gd 5d (Gd Fe ) Gd 5d (Gd Fe ) Gd 5d (Gd Fe ) Gd 4f (Gd Fe ) Gd 4f (Gd Fe ) Gd 4f (Gd Fe ) FIG. 9. (Color online) Total and orbital projected densitiesof states calculated for theoretical Gd x Fe − x structure ( x =0 . , . , . where m is the order parameter, i.e. the magnetizationand <> denotes the statistical average taken over sys-tems at equilibrium and at constant temperature. TheBinder cumulant allows to locate the critical point andthe critical exponents in a phase transition. Thus, in thethermodynamic limit where the system size of the ferro-magnet L → ∞ and consequently L is bigger than thecorrelation length, the Binder cumulant approximates tozero for temperatures higher than T C while for temper-atures lower than T C , U L → . This property of thecumulant is very useful for obtaining very good estimatesof T C which are not biased by any prerequisites aboutcritical exponents. After performing the atomistic spindynamics simulations, we have access to the magnetiza-tion which is inserted in Eq. (B1) to obtain as a result theBinder cumulant. Then, we plot the cumulant as a func-tion of the temperature for different sample sizes and T C is estimated from the intersection point of those curves.In Fig. 10 we show, as an example, the Binder cumulantsfor two Gd concentrations and the estimated T C for bothsamples. The Curie temperatures shown in Fig. 2 have1been calculated using the procedure described above. C = 518 K Gd Fe T C = 543 KGd Fe B i nde r c u m u l an t s ( U L ) Temperature (K)
FIG. 10. (Color online) Calculated Binder cumulants forGd . Fe . and Gd . Fe . samples with 1600 atoms perunit cell. The size of the samples ( x × y × z ) was rangedfrom x = y = z = 1 to 4 unit cells in steps of 1. The Curietemperature is indicated by the arrows. APPENDIX C: DETAILS OF THETWO-TEMPERATURE MODEL
In order to study the ultrafast demagnetization, thethree temperature model (3TM) was introduced by Beau-repaire et al. in 1996 [3]. In the 3TM model, the electron,spin and phonon are thermal reservoirs and coupled toeach other by coupling constants. It is difficult to definean electron temperature for the first femto-seconds of aLaser induced pump-probe experiment, but we have herefor simplicity used the 3TM model. The analytical ex-pression of the 3TM contains three differential equationsand from that equations, the three temperatures T e (elec-tron temperature), T s (spin temperature) and T latt (lat-tice or phonon temperature) are calculated. In equilib-rium the temperature of all three thermodynamic reser-voirs is equilibrated, i.e. T e = T s = T latt . In this modelwe assume that the lattice is an infinitely large thermal reservoir with constant temperature T latt . This assump-tion seems to be quite valid as it was reported in typicalpump-probe experiments [55]. Moreover, we also assumethat the electron reservoir is a thermal reservoir muchsmaller than the lattice, but still larger than the spin sys-tem. The spin temperature is explicitly passed into thestochastic LLG equation while the electron temperaturecan be expressed in a simple analytical form as, T e = T + ( T P − T ) · (1 − e ( − t/τ i ) ) · e ( − t/τ f ) (C1)+ ( T F − T ) · (1 − e ( − t/τ f ) ) Thus, we reduced the 3TM model into a simple exponen-tial function [9, 12], that captures the essential physicsof the three temperature model. In Eq. C1, T repre-sents the initial temperature of the system, T P is themaximum temperature achieved in the simulation and T F is the final temperature of the system. The functiondepends on two time parameters, such as initial time τ i and final time τ f . The initial time describes the rise-timeof the temperature to its maximum value and the finaltime represents the relaxation time of the temperaturefrom the maximum value to the final temperature of thesystem. Here we used τ i = 50 fs and τ f = 1 ps. In thismodel, the electron temperature is used as spin tempera-ture. At each time step, the calculated spin temperatureis passed explicitly into the stochastic field of the LLGequation. The values of T P =800 K, 1000 K and 2000 Kand T = T F =300 K are used in the simulations. APPENDIX D: DAMPING EFFECTS
One of the main parameters in LLG equation is theGilbert damping, α , which is mainly responsible forbringing the system into an equilibrium state. It wasexperimentally observed that the damping constant α significantly depends on the Gd content and it becomeslarge near to the compensation temperature of the sam-ples [56]. Though the g-factors of Gd and Fe sublatticemagnetic moments in our samples are slightly different,we used same g-factor for both sublattices, and in our ini-tial simulations, the static damping parameter was alsokept equal for both sublattices in the atomistic spin dy-namics simulations. With these parameters, the evolu-tion of the magnetization of Gd and Fe moments underthe influence of an intense femtosecond laser pulse showsdifferent precession and reproduce experimental observa-tions.Later on, in a second stage of our simulations, weadapted a site-dependent damping parameters in theLLG equation and we performed ultra-fast simulations onGd . Fe . , Gd . Fe . and Gd . Fe . amorphousalloys. These results are shown in Figs. 11-13. By fixingthe damping parameter of Fe species as α F e = 0 .
02, wechanged the damping parameter of Gd ( α Gd ) from 0.02 to0.1 in steps of 0.02. If α Gd = 0 .
02, the simulations for thethree sample concentrations shown in Figs. 11-13 predictthat the Fe sublattice demagnetize faster than the Gd2 -2-10123450 1 2 3 4 5-2-1012345 0 1 2 3 4 5 6 Gd Fe Gd Fe Gd Fe Gd Fe Gd Fe Gd Fe M z ( B / a t o m ) Time (ps)
Gd Fe
Gd Fe
FIG. 11. (Color online) Time dependence of ultrafast mag-netization (M z ) in Gd . Fe . for different damping param-eters ( α = 0.02, 0.04, 0.08, 0.1) with a fixed electron temper-ature profile. sublattice. Samples with compensation temperatures,i.e. with a composition of Gd ranging from 21 at.% to 30at.%, shows switching behavior (see Fig. 2). In Fig. 11,we found that for Gd . Fe . amorphous sample, thereis no ultrafast switching. This result clearly shows thatthe compensation point is a very important parameterin the spin dynamics of Gd-Fe alloy. For higher concen-trations of Gd, we observed the switching behavior, asshown in Figs. 12-13. In our calculations, we found thetransition metal demagnetize faster than the rare-earthelement and forms a ferromagnetic-like state for a shortperiod of time due to the AFM interaction between Gdand Fe atoms. This was already explained in Ref. [1].The idea is that the AFM coupling between Gd and Featoms favors the spin flipping of Fe atoms when Gd atomsare becoming reversed. Thus, the process promotes anincrease of the net Fe magnetization parallel to the re-maining Gd magnetization.In the case that α Gd > .
02, we observed that the Gdsublattice moves towards sub-picosecond times but neverbecomes FM to Fe sublattice, as shown in Figs. 11-13.The main message of these results is that the damping isa very crucial parameter in ultrafast switching process.
APPENDIX E: GENERATINGINHOMOGENEOUS SAMPLES
The inhomogenous sample was constructed from thehomogeneous unit cell of Gd . Fe . by repeating the -2-10123450 1 2 3 4 5-2-1012345 0 1 2 3 4 5 6 M z ( B / a t o m ) Time (ps) Gd Fe Gd Fe Gd Fe Gd Fe Gd Fe
Gd Fe
Gd Fe
Gd Fe
FIG. 12. (Color online) Time dependence of ultrafast magne-tization (M z ) in Gd . Fe . for four different damping pa-rameters ( α = 0.02, 0.04, 0.08, 0.1) with a fixed electron tem-perature profile. unit cell twice in x, y and z direction (12800 atoms), asshown in the right panel of Fig. 5. Then, a cube is se-lected randomly and we reduced the percentage of Fe by6 at.%. Thus, the cube turns out to be Gd rich regionand in the same way the Fe rich regions were created.The new supercell consists of an inhomogeneous envi-ronment and it mimics the original samples of GdFeCoexperimental results. After that, we study the compen-sation temperatures for rich and poor areas of Gd. Theobtained compensation temperatures are similar to ho-mogenous unit cells. As shown in Fig. 14, we obtaineda compensation temperature of about 50 K and 350 Kfor Gd poor and Gd rich areas, respectively. The re-sults shown in Appendix D clearly pinpoint an impossi-bility of Gd sublattice to switch first than the Fe one.Thus, in Sec. III C 3, we incorporate non-collinearity inthe sample by introducing an extra AFM exchange valueto Gd sublattice in the Gd rich areas. Such type of effectsare observed in the experimental samples. The origin ofthose effects can mainly reside in the concentration ofGd-Fe amorphous samples, which modify their structure-sensitive properties, such as the magnetic ones, compen-sation temperatures between the Gd rich and poor re-gions and also sperrimagnetism found in the Fe sublatticeof Gd-Fe amorphous alloys for higher Gd-concentrationwhile the collinear structure may be expected to exist forGd-poor alloys [51].3 -2-10123450 1 2 3 4 5-2-1012345 0 1 2 3 4 5 6 Time (ps) M z ( B / a t o m ) Gd Fe
Gd Fe
Gd Fe
Gd Fe Gd Fe Gd Fe Gd Fe Gd Fe FIG. 13. (Color online) Time dependence of ultrafast magne-tization (M z ) in Gd . Fe . for four different damping pa-rameters ( α = 0.02, 0.04, 0.08, 0.1) with a fixed electron tem-perature profile. T M =50 K Fe (30 at.% Gd-rich region) Gd (30 at.% Gd-rich region) Fe (22 at.% Gd-poor region) Gd (22 at.% Gd-poor region)
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