Abstract
Femtosecond laser pulses can be used to induce ultrafast changes of the magnetization in magnetic materials. However, one of the unsolved questions is that of conservation of the total angular momentum during the ultrafast demagnetization. Here we report the ultrafast transfer of angular momentum during the first hundred femtoseconds in ferrimagnetic Co_{0.8}Gd_{0.2} and Co_{0.74}Tb_{0.26} films. Using timeresolved Xray magnetic circular dichroism allowed for timeresolved determination of spin and orbital momenta for each element. We report an ultrafast quenching of the magnetocrystalline anisotropy and show that at early times the demagnetization in ferrimagnetic alloys is driven by the local transfer of angular momenta between the two exchangecoupled sublattices while the total angular momentum stays constant. In Co_{0.74}Tb_{0.26} we have observed a transfer of the total angular momentum to an external bath, which is delayed by ~150 fs.
Introduction
The mechanism of ultrafast quenching of the magnetization triggered by a femtosecond laser pulse is currently still hotly debated. Despite the large amount of work devoted to the characterization of femtosecond demagnetization in various metallic systems^{1,2,3,4,5,6,7,8,9,10,11}, so far no consensus over the subpicosecond demagnetization mechanisms has emerged. The basic phenomena involved, especially regarding the dissipation of angular momenta, are still disputed^{2,3,4,5,6,7,8,9,12}. Different other aspects are still discussed controversially, especially the timescales characterizing energy and angular momentum transfer between various degrees of freedom, such as electrons (e), spins (s) and phonons (ph). However, the literature documents critical differences between rare earth (RE) elements^{4} and transition metals (TMs)^{3,5,7,8,10,11}. The disparities lie in the couplings between e, s, and ph, and are related to the localization and hybridization of the electron orbitals carrying the spin momentum. As the loss of magnetization requires transfer of energy into the spin system and of angular momentum out of it, one of the important and unsolved questions is related to the description of the fundamental processes involving the transfer of angular momentum out of s.
In earlier work, it has been proposed that phonon or defectmediated spinflip scattering^{5} or electron–magnon spinflip scattering^{11} can account for the transfer of angular momentum on the femtosecond timescale but recent quantitative ab inito calculations show that the contribution of electron–phonon spinflip scattering is too small to describe the experimental femtosecond demagnetization^{13}. Alternatively, ultrafast quenching of the magnetocrystalline anisotropy (MCA) has been proposed by Boeglin et al.^{10} as a new mechanism of demagnetization in TM and has been supported by microscopic theoretical work^{6}. Recently, a substitute for the transfer of angular momentum during the ultrafast demagnetization has been proposed^{14,15,16} involving ultrafast superdiffusive spinpolarized transport. In this process, no angular momentum transfer to the lattice is required. However, the relative weights of each precited contribution to the total demagnetization process are still unclear^{17,18,19,20}.
In the case of multilattice magnets, the situation is more complex^{21} since additional mechanisms may emerge. For instance, transfer of angular momentum between magnetic lattices has been predicted by Mentink et al.^{22} and ab initio density functional theory (DFT) calculations performed by Wienholdt et al.^{12} have demonstrated that the transfer of angular momentum between both sublattices is a key phenomenon to establish a transient ferromagneticlike state in FeCoGd alloys^{23}. An experimental attempt to demonstrate the efficiency of such a process has been made by Graves et al.^{24} using timeresolved resonant Xray scattering in FeCoGd alloys. However, the scenario they propose for the transfer of angular momenta strongly relies on the microscopic inhomogeneity of the sample studied and cannot be viewed as a generic mechanism.
Here, we report experimental results evidencing the initial conservation and the following dissipation of the total angular momentum during the demagnetization time by measuring the angular momentum in a quantitative way in ferrimagnetic TM–RE alloys. We have used timeresolved Xray magnetic circular dichroism (XMCD) at the TM L_{2,3} and RE M_{4,5} edges at the BESSY II femtoslicing source of the HelmholtzZentrum Berlin^{7}. This tool combines element, spin and orbital sensitivity^{9,25,26} with femtosecond time resolution resolving the ultrafast dynamics of the orbital (L^{i}(t)), spin (S^{i}(t)) and total angular momentum (J^{i}(t)) (where i stands for Co, Gd or Tb) in ferrimagnetic Co_{0.8}Gd_{0.2} and Co_{0.74}Tb_{0.26} alloys. Our results show on what timescales and through which microscopic mechanism the total angular momentum is conserved in the case of a multisublattice magnet. They reveal that, to conserve the total angular momentum, the ferrimagnetic system involves two compensating angular momenta that flow in opposite directions, allowing for the loss of magnetization in each subsystem during the first 140±60 fs of demagnetization and resulting in a net loss of the magnetization in the system. We have determined at what delay after the laser excitation the systems start to transfer angular momentum to the external reservoir. Backed by the ultrafast dynamics of the ratio L^{Co}(t)/S^{Co}(t), our results reveal an ultrafast quenching of the MCA in both ferrimagnetic films^{10}. Finally, we show that the atomic description is still valid for Gd in the first hundred femtoseconds after the laser excitation, because the orbital momentum L^{Gd}(t)=0 at all times.
Results
Experimental details
The magnetic configuration at the thermodynamic equilibrium of the two different ferrimagnetic alloy films Co_{0.8}Gd_{0.2} and Co_{0.74}Tb_{0.26} have been analysed by XMCD measurements; see Methods. Thanks to the chemical sensitivity of XMCD, our previous static measurements^{27} are used to quantify the ultrafast dynamics of the angular momenta S_{z}^{i}(t), L_{z}^{i}(t) and J_{z}^{i} (t) in Co and in the RE elements. To test the possible ultrafast increase in L^{i}(t) starting from an initial value L^{i}(t<0)=0, we study one of the most convenient cases, Gd in Co_{0.8}Gd_{0.2}. In contrast, the RE Tb has a large orbital momentum (L=3) and is therefore well suited to study the demagnetization dynamics of the orbital momenta^{27}. According to elementresolved hysteresis loops obtained for both alloys^{27}, we conclude that our films are homogenous and are fully saturated under the applied magnetic field of 5 kOe. Therefore, we have L(t)~L_{z}(t) and S(t)~S_{z}(t) for Co, Gd and Tb. Thus, L(t) and S(t) are aligned parallel, so that we may define the timedependent angular momentum J^{i}(t)=L^{i}(t)+S^{i}(t) per element i.
The Xray transmission experiment, using an infrared laser pump and Xray probe configuration, is schematically shown in Fig. 1. The external magnetic field is applied along the propagation direction of the Xrays. The lower part sketches the time evolution of the pumpinduced atomic demagnetization on the exchangecoupled Co and RE in the multisublattice ferrimagnet. The time t_{0}=0 is defined by the temporal overlap between the laser pulse and the Xray pulse. A film thickness of 15 nm ensures a homogenous infrared laser excitation. The incident Xrays are circularly polarized and timeresolved XMCD is measured using the difference of intensities measured in transmission between two opposite magnetic fields +H and −H applied parallel to the Xrays (see Methods).
Ultrafast demagnetization in CoGd
In Fig. 2a we show the pumpprobe results obtained at the Co L_{2,3} and Gd M_{4,5} edges for the Co_{0.8}Gd_{0.2} alloy film. The normalized XMCD values at negative time delays match the XMCD magnitudes recorded at the Co L_{3}, Co L_{2}, Gd M_{5} and Gd M_{4} edges during quantitative static XMCD measurements^{27}. Applying the sum rules for the Co L_{2,3} and for the Gd M_{4,5} edges^{9,25,26}, we extract the ultrafast dynamics of the spin momentum S^{i}(t) (black circles) and the orbital momentum L^{i}(t) (red circles) for Co and for Gd. Quantitative data for the timedependent values of S^{i}(t) and L^{i}(t) are displayed in units of ħ per atom in Fig. 2b,c (red and black symbols). The values at negative delays are normalized to the values measured at thermodynamic equilibrium^{27}. The continuous lines are the results of the simulations of S^{i}(t) (black line) and L^{i}(t) (blue line; see Methods), whereas the green line is the ratio of these results.
Analysing the timedependent spin and orbital momenta for Co (Fig. 2b) we observe that, similar to CoPd films^{10}, the magnitude of the demagnetization observed for S^{Co}(t) and L^{Co}(t) are different leading to a large quenching of the ratio L^{Co}(t)/S^{Co}(t) before t=1 ps (Fig. 2a, green symbols). The thermalization times of S^{Co}(t) and L^{Co}(t) are identical (τ_{therm}=200±20 fs) within the error bars. For Gd (Fig. 2c), we notice a large decrease of S^{Gd}(t) while L^{Gd}(t)~0. The dynamics of S^{Gd}(t) show that the thermalization time is much longer, τ_{th}=480±40 fs (ref. 27). Interestingly, the value L^{Gd}(t)=0 stays constant over this time, although the orbital momentum was previously assumed to be an angular momentum sink to explain the loss of spin momentum on the subpicosecond timescale. The excess of angular momentum stored in the electron orbit was assumed to be transferred to the lattice only at longer times (~1 ps)^{28}. A transfer to the lattice was also proposed by Stamm et al.^{7} in the case of Ni, although on the faster subpicosecond timescale. The quantitative ultrafast dynamics of S^{Gd}(t) and L^{Gd}(t) reported here excludes any ultrafast transfer from the 4f spin angular momentum towards the orbital angular momentum in Gd. In addition, we rule out any ultrafast transfer from Co towards the Gd orbital momentum.
Ultrafast demagnetization in CoTb
For the Co_{0.74}Tb_{0.26} alloy the excitedstate temperature is below T_{comp}, thus leading to faster demagnetization rates for the RE (τ_{therm}=280±40 fs) as compared with other experiments where the excitedstate temperature is in the vicinity of T_{C} (refs 23, 27). The pumpprobe results measured at the Co L_{2,3} and Tb M_{4,5} edges are shown in Fig. 3a and transformed into ultrafast dynamics of S^{i}(t) and L^{i}(t) for Co and Tb as shown in Fig. 3b,c (red and black symbols). The normalized XMCD values at negative time delays match the XMCD magnitude recorded at the Co L_{3}, Co L_{2}, Tb M_{5} and Tb M_{4} edges during quantitative static XMCD measurements^{27}. Applying the sum rules, we can extract the ultrafast dynamics of the spin and orbital momenta S^{i}(t) (black circles) and L^{i}(t) (red circles) for Co and for Tb. For Co and Gd we can neglect the magnetic dipole term T_{z} (t) so that the spin momentum is then defined by S(t)=S^{eff}(t)^{27}. For Tb, according to atomic calculations done by Teramura et al.^{29}, we take T_{z}/S=−0.08. We further correct the sum ruleextracted value S^{eff}(t) by a value T_{z} (t)^{27} representing a time constant proportion of 8% of S(t). This assumption leads to an underestimation of 8% of S^{Tb}(t) for t>0 assuming a virtual and complete quenching of T_{z} (t) by the pump laser at t>0. The values at negative delays are normalized to the values measured at thermodynamic equilibrium^{27}. The continuous lines correspond to the simulations of S^{i}(t) (black line) and L^{i}(t) (blue line; see Methods) while the green lines are the ratios between these numerical results. In Fig. 3b,c we observe an ultrafast quenching of S^{i}(t) and L^{i}(t) for Co and Tb. For each element, the thermalization times of S^{i}(t) and L^{i}(t) are identical (Co τ_{th}=180±40 fs and Tb τ_{th}=280±30 fs). Computing the ratio L^{i}(t)/S^{i}(t) reveals an ultrafast quenching for Co but for Tb any change is smaller than the error bars (Fig. 3c, green symbols).
The ultrafast quenching of L^{Co}(t)/S^{Co}(t) in Co_{0.8}Gd_{0.2} and Co_{0.74}Tb_{0.26} leads to several conclusions. Interestingly, one notices that in this experiment we quenched L^{Co}(t)/S^{Co}(t) towards the value of 0.12, which is the value of hexagonal close packed Co (ref. 9). This reflects a transition from the highly anisotropic electronic distribution in Co–RE alloys towards the more isotropic distribution in hexagonal close packed structures. The quenching of L^{Co}(t)/S^{Co}(t) is in agreement with the model of ultrafast quenching of MCA proposed by Boeglin et al.^{10}, in the framework of Bruno’s model^{30}. This model predicts a direct relationship between the anisotropy of the orbital momentum and the magnetic anisotropy energy in 3d TMs. In the case of 4flocalized moments, the simple relationship given by Bruno^{30} is no longer valid. Considering this limitation for RE 4f moments, the ultrafast dynamics of L(t)/S(t) does not reflect the quenching of the MCA. However, the ultrafast dynamics of L^{Tb}(t)/S^{Tb}(t) observed in Fig. 3c (green line) is consistent with the fact that the 4f moments are not directly pumped by the infrared laser but are exchange coupled with the laser pumped 3d–5d states, explaining a quasisimultaneous demagnetization of L^{Tb}(t) and S^{Tb}(t), whereas the 3d moments show an ultrafast quenching of L^{Co}(t)/S^{Co}(t). Considering the low infrared pump energy, we suggest that the exchange coupling between the 4f and the pumped 5d3d moments prevent large ultrafast quenching of the magnetocrystalline and dipolar anisotropy of the 4f moments. It follows that under the given pump conditions, the proposed correction value of T_{z} (t) as a constant proportion of S(t) is justified for Tb.
Total angular momentum
We calculate the angular momentum J^{i}(t)=L^{i}(t)+S^{i}(t) by adding the elementresolved L^{i}(t) and S^{i}(t). We multiply J^{i}(t) by the elemental concentration in the alloy for each i=Co, Gd or Tb in Co_{0.8}Gd_{0.2} (Fig. 4a) and Co_{0.74}Tb_{0.26} (Fig. 4b). All individual angular momenta J^{i}(t) are quenched on laser excitation at t=0 ps. For each alloy, we then derive the total angular momentum J(t)=J^{Co}(t)+J^{Gd}(t) (Fig. 4a, blue open symbols) and J(t)=J^{Co}(t)+J^{Tb}(t) (Fig. 4b, blue open symbols). To account for the time dependence of T_{z}(t) in Tb, we estimated the variation for J(t) to be ~0.020 ħ at^{−1}%, less than the given error bars of ±0.035 ħ at^{−1}%. In Co_{0.8}Gd_{0.2}, where the working temperature is close to but above T_{comp}, we have J^{Co}≈−J^{Gd} and the total angular momentum is mostly compensated. The remaining J(t) in CoGd is too small compared with the experimental noise level to reliably detect changes within the first few hundred of femtoseconds. The small negative values of J(t) observed for t>1 ps illustrates the nonequilibrium state of the system determined by the working temperature and pump conditions^{27}.
For Co_{0.74}Tb_{0.26} where the working temperature is below T_{comp}, we have J^{Co}<−J^{Tb} and a sizable value of J(t)=J^{Co}(t)+J^{Tb}(t) is obtained that allows us to follow unambiguously the ultrafast dynamics of J(t). The validity of the determination of J(t) is supported by the limited error introduced by neglecting a virtual quenching of T_{z} (t) at t>0. In such a case, we estimate that for J(t) we make an error of 0.025 ħ at^{−1}, well below the experimental error bars of 0.035 ħ at^{−1} given for the experimental values (Fig. 4b). Analysing the dynamics of J(t), we observe a loss of the total angular momentum J(t) towards an external bath until the value J(t)≈0 is reached at t=1.5 ps. This dissipation happens with a characteristic time of 300±50 fs (Fig. 4b). At t=1.5 ps, the system is at quasiequilibrium at a temperature close to T_{comp}^{27} and the value J(t)=0 shows that the CoTb alloy is now magnetically compensated. The appealing feature is that the quenching of J(t) is delayed by δt=140±60 fs, whereas both J^{Co}(t) and J^{Tb}(t) have already started to be quenched (Fig. 4b). The value δt is estimated by performing simulations using a double exponential function for J(t) (Fig. 4b, blue line; see Methods). In Fig. 4b we superpose the weighted sum of the twofit functions obtained for J^{Co}(t) and J^{Tb}(t) as the magenta dotted line. The twofit functions for J(t) agree perfectly. The delay of δt=140±60 fs is attributed to an ultrafast transfer of angular momenta between the coupled Co and Tb sublattices.
Discussion
Compared with previous work by Medapalli et al.^{31} and Graves et al.^{24}, we provide quantitative timeresolved and elementselective angular momenta J^{i}(t) in ferrimagnetic TM–RE alloys. Medapalli et al.^{31} argued that a direct transfer of angular momentum occurs between the TM to the RE sublattices when the temperature of the sample is below the temperature of magnetic compensation. Although a model developed by Mentink et al.^{22} supported their claim, no direct proof of such a mechanism has been provided so far. Graves et al.^{24} discussed a nonlocal transfer of the angular momentum in FeCoGd films between chemically different nanograins. More recently, Wienholdt et al.^{12} performed ab initio calculations using DFT to develop an orbitalresolved model for spin dynamics in RE–TM alloys. From their model, they concluded that the ferromagneticlike state observed by Radu et al.^{23} in FeCoGd is a consequence of ‘dissipationless spin dynamics’ during the first picoseconds after the excitation during which the energy and angular momentum is redistributed between the RE and TM elements. The experimental results presented here support this model.
In our work we measure L^{i}(t), S^{i}(t) and J^{i}(t) for each element and attribute the constant value of J(t) in Co_{0.74}Tb_{0.26} during the first 140 fs to the angular momentum flowing between J^{Co}(t) and J^{Tb}(t) (Fig. 4b, blue symbols). This process is hidden in Co_{0.8}Gd_{0.2} because of the weak J(t). This mechanism allows an ultrafast and local demagnetization in both sublattices, whereas it conserves the total angular momentum J(t). We would like to point out that phonon or defectinduced spinflip scattering or superdiffusion of hot electrons would result in a decrease of J(t) if measured by XMCD. According to our analysis, these mechanisms do not show a major contribution up to 140±60 fs. After t=300 fs, the magnetization of Co reaches a minimum while the magnetization of Tb keeps decreasing, accompanied by a decrease of J(t). In this case, the transfer channel of angular momenta between sublattices is no more efficient and angular momentum is transferred from the Tb sublattice towards an external reservoir (phonons, hot electrons and so on). The transfer rate towards the external reservoir is now set by the transfer rate from the 4f electrons, which is usually slower than that for the 3d electrons in TM elements. We thus have clearly demonstrated that an efficient transfer channel for angular momenta exists in multisublattice ferrimagnetic alloys before any transfer to the external bath. This twostep transfer mechanism agrees perfectly with the theoretical calculations performed by Wienholdt et al.^{12}. Furthermore, our results show that this transfer results from a direct exchange of angular momenta between Co and Tb as long as both sublattices are demagnetizing (t<300 fs). In previous work by Radu et al.^{23}, uncorrelated demagnetization between the two sublattices was detected in the subpicosecond timescale. Going one step further and considering the quantitative angular momenta J^{i}(t), we show here that the two sublattices are strongly linked between t=0 and 140±60 fs corresponding to the timescale of the exchange interaction^{9}.
To conclude, our findings demonstrate the local transfer of angular momenta between the two antiferromagnetically exchangecoupled sublattices. This transfer channel induces ultrafast demagnetization at the atomic scale, whereas the total angular momentum J(t) initially stays constant. In Co_{0.74}Tb_{0.26} we observed a delayed transfer of J(t) to an external bath. These results evidence a new ultrafast mechanism, determine the related timescale for angular momentum transfer during the demagnetization in ferrimagnetic systems and are supported by recent ab initio DFT calculations^{12}. In addition, we have uncovered an ultrafast quenching of the L^{Co}(t)/S^{Co}(t) ratio in ferrimagnetic Co_{0.8}Gd_{0.2} and Co_{0.74}Tb_{0.26} alloys and attributed it to the ultrafast quenching of MCA in agreement with Bruno’s model and previous results in CoPd^{10,30}. For the RE elements, a different behaviour is observed for L(t)/S(t) that could be linked to the localized character of the 4f moments.
Methods
Sample preparation and magnetic properties of the films
Fifteen nanometres thick Co–RE alloys have been grown by magnetron sputtering on Si_{3}N_{4} membranes. Codeposition with convergent Co and Gd (or Tb) flux was used to get amorphous Co_{0.8}Gd_{0.2} and Co_{0.74}Tb_{0.26} alloy films. The concentration in the films were optimized to obtain moderate saturation fields of 5 kOe or less, compatible with the magnet used in the timeresolved experiments. For Co_{0.8}Gd_{0.2} and Co_{0.74}Tb_{0.26} we have verified experimentally that at working temperatures the saturation fields are 4 and 3 kOe, respectively. Both alloys have been characterized by static XMCD^{27}. As the samples were fully saturated by the external magnetic field of 5 kOe during the pumpprobe experiments, we can define L_{z} (resp. S_{z})=L (resp. S) so that it follows that J^{i}(t)=L^{i}(t)+S^{i}(t) (Fig. 4a, black and red symbols and Fig. 4b, black and red symbols) as the angular momentum of the element i (Co, Gd or Tb). The Co_{0.8}Gd_{0.2} and Co_{0.74}Tb_{0.26} films show magnetic compensation temperatures (T_{comp}) at which both magnetizations of the Co and the RE sublattices compensate (resp. T_{comp}=150 and 550 K) and Curie temperatures (T_{C}) where the magnetic order is lost (resp. T_{C}=450 and 650 K).
The Co_{74}Tb_{26} alloy film presents an outofplane magnetic anisotropy, which is measured along the normal of the film plane. We emphasize that, in contrast to CoPd and Co_{0.74}Tb_{0.26} alloys^{10,27}, Co_{0.80}Gd_{0.20} has no outofplane magnetic anisotropy. For this sample we performed our XMCD experiment at 30° from the normal of the film plane. In the case of Co_{0.80}Gd_{0.20} an inplane uniaxial anisotropy may be induced during growth^{32}. This point is confirmed by our recent results where static XMCD characterization was performed^{27}. In both alloys we found large ratios L/S of 0.21–0.29 at the Co L_{2,3} edges, whereas bulk Co shows a ratio of only 0.13. This is a strong indication for the presence of large MCA energies in all our Co–RE compounds. However, in our alloys, the quantitative value of the spin momentum (for instance, S^{Tb}=1.35±0.2 at^{−1}) is lower than expected from Hund’s rules (S^{Tb}=3 at^{−1} ) and is related to the structural disorder and finite temperature effects^{27}.
Timeresolved XMCD
Timeresolved XMCD was performed at the femtoslicing beam line of the BESSY II synchrotron radiation source of the Helmholtz–Zentrum Berlin^{7,10}. The magnetization dynamics have been measured by monitoring the transmission signal of circularly polarized Xrays, tuned to specific corelevel absorption edges as a function of a pumpprobe delay. The dynamic XMCD contrast is obtained by subtracting the gated signals obtained with and without pump beam. The energy was set to the different Co L_{2,3}, Tb M_{4,5} and Gd M_{4,5} edges using the Bragg Fresnel reflection zone plate monochromator. The experiments have been performed with a pumpprobe setup where the short Xray pulses are synchronized with a femtosecond pump laser working at 790 nm, 3 kHz repetition rate with pulses of 60 fs. The Xray pulse duration of about 100 fs in the femtoslicing operation mode ensures a global time resolution of ~130 fs (see refs 7, 10 for details). The pump fluences used during our experiments were adjusted to 8 mJ cm^{−2} for Co_{0.8}Gd_{0.2} and to 12 mJ cm^{−2} for Co_{0.74}Tb_{0.26} to reach large demagnetization magnitudes of about 60% at the Co L_{3} edge without altering the sample properties (alloy concentration, atomic diffusion and large heating).
Fitting procedure
The physical quantities L^{i}(t), S^{i}(t) and J^{i}(t) (i=Co, Gd, Tb) were adjusted using the rate equation of the twotemperature model with two exponential functions (equation (1)):
where G(t) is the Gaussian function defining the total time resolution of the experiment (130 fs), τ_{th} and τ_{s−ph} are the thermalization time and the relaxation time from the spin system to other systems (lattice, external bath), t_{0} is the delay at which the temporal overlap of the pump and the probe is achieved and H(t−t_{0}) is the Heaviside function (H(t−t_{0})=0 if t<t_{0} and H(t−t_{0})=1 if t>t_{0}) describing the energy transfer from the laser.
The parameters C_{0}, C_{1}, t_{0} and τ_{th} in equation (1) were optimized to minimize independently the χ^{2} factor for L^{i}(t), S^{i}(t) and J^{i}(t) (equation (2)).
where y is the rate equation, y_{j} are the experimental data and σ_{j} is the standard deviation (for j=1–n, and n is the number of data points).
The minimization was performed via the Levenberg–Marquardt algorithm, based on a nonlinear least square procedure. The extracted error bars for τ_{th} and t_{0} correspond to the standard deviation (s.d.), which is given as an output parameter of the fitting procedure.
The error bars appearing in Figs 2b,c and 3b,c for L^{i}(t) and S^{i}(t) as well as on Fig. 4a,b for J^{i}(t) correspond to the s.d. (σ) of experimental data with respect to the fitting functions that minimize the χ^{2} factor. The amplitude of the error bars has been calculated according to equation (3):
The final fitting parameters for J^{i}(t) are compatible with the result of the sum of the twofit functions obtained for L^{i}(t) and S^{i}(t).
The total angular momentum for each alloy was obtained by performing the weighted sum J(t)=J^{Co}(t)+J^{Tb}(t) according to the composition of the alloy. In Co_{0.8}Gd_{0.2}, the total angular momentum J^{Co}(t)+J^{Gd}(t) remains small and comparable to the noise level. For Co_{0.74}Tb_{0.26}, a sizable value of J(t)=J^{Co}(t)+J^{Tb}(t) is obtained. In this case, we used the rate equation (1) to adjust J(t) (blue solid line in Fig. 4). In parallel, we plotted the weighted sumofthefit functions obtained for J^{Co}(t) and J^{Tb}(t); (magenta dotted line in Fig. 4). The twofit functions for J(t) agree perfectly for our parameters C_{o}, C_{1}, t_{0} and τ_{th}. The error bars given in Fig. 4a,b for J(t) correspond to the s.d. of the fitting function derived using equation (3). The results extracted from the rate equation lead to the following zerotime values: t_{0}=0±40 fs for J^{Co}(t) and J^{Tb}(t) and t′_{0}=140±40 fs for J(t), which allows us to define δt=140±60 fs. The error bar for δt has been derived as follows:
Since the main source of uncertainty for δt stems from the error bars of t_{0} as extracted from the rate equation, we performed the simulations using a time axis for J^{Co}(t) shifted by an amount of±40 fs with respect to J^{Tb}(t). Doing so, we extracted the minimum and maximum delay times δt for J(t) of 80 and 200 fs. These lower and upper limits define our error bar of 60 fs for the delais δt=140 fs.
Additional information
How to cite this article: Bergeard, N. et al. Ultrafast angular momentum transfer in multisublattice ferrimagnets. Nat. Commun. 5:3466 doi: 10.1038/ncomms4466 (2014).
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Acknowledgements
We are indebted to A Eschenlohr, K Holldack, R Mitzner and T Kachel for help and support during the femtoslicing experiments, and A Boeglin for discussions and correction of the manuscript. This work was supported by the CNRS—PICS, by Université de Strasbourg and the EU Contract Integrated Infrastructure Initiative I3 in FP6Project number R II 3 CT20045060008, BESSY IASFS Access Program, by the ‘Agence Nationale de la Recherche’ in France via the project EQUIPEX UNION: number ANR10EQPX52’ and by the German Ministry of Education and Research BMBF Grant 05K10PG2 (FEMTOSPEX). V.L.F. acknowledges the Ministry of Education of Spain (Programa Nacional de Movilidad de Recursos Humanos del Plan Nacional de ID+i 20082011) for financial support.
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N.B., V.L.F., V.H., N.P., C.S, E.B. and C.B. performed the timeresolved measurements and data exploitations. M.H. grew and characterized the samples. N.B. and C.B. wrote the paper. All authors discussed and improved the manuscript.
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Bergeard, N., LópezFlores, V., Halté, V. et al. Ultrafast angular momentum transfer in multisublattice ferrimagnets. Nat Commun 5, 3466 (2014). https://doi.org/10.1038/ncomms4466
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DOI: https://doi.org/10.1038/ncomms4466
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