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 systems1,2,3,4,5,6,7,8,9,10,11, so far no consensus over the sub-picosecond demagnetization mechanisms has emerged. The basic phenomena involved, especially regarding the dissipation of angular momenta, are still disputed2,3,4,5,6,7,8,9,12. 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) elements4 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 defect-mediated spin-flip scattering5 or electron–magnon spin-flip scattering11 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 spin-flip scattering is too small to describe the experimental femtosecond demagnetization13. 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 work6. Recently, a substitute for the transfer of angular momentum during the ultrafast demagnetization has been proposed14,15,16 involving ultrafast superdiffusive spin-polarized transport. In this process, no angular momentum transfer to the lattice is required. However, the relative weights of each pre-cited contribution to the total demagnetization process are still unclear17,18,19,20.

In the case of multi-lattice magnets, the situation is more complex21 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 ferromagnetic-like state in FeCoGd alloys23. An experimental attempt to demonstrate the efficiency of such a process has been made by Graves et al.24 using time-resolved resonant X-ray 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 time-resolved X-ray magnetic circular dichroism (XMCD) at the TM L2,3 and RE M4,5 edges at the BESSY II femtoslicing source of the Helmholtz-Zentrum Berlin7. This tool combines element, spin and orbital sensitivity9,25,26 with femtosecond time resolution resolving the ultrafast dynamics of the orbital (Li(t)), spin (Si(t)) and total angular momentum (Ji(t)) (where i stands for Co, Gd or Tb) in ferrimagnetic Co0.8Gd0.2 and Co0.74Tb0.26 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 LCo(t)/SCo(t), our results reveal an ultrafast quenching of the MCA in both ferrimagnetic films10. 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 LGd(t)=0 at all times.

Results

Experimental details

The magnetic configuration at the thermodynamic equilibrium of the two different ferrimagnetic alloy films Co0.8Gd0.2 and Co0.74Tb0.26 have been analysed by XMCD measurements; see Methods. Thanks to the chemical sensitivity of XMCD, our previous static measurements27 are used to quantify the ultrafast dynamics of the angular momenta Szi(t), Lzi(t) and Jzi (t) in Co and in the RE elements. To test the possible ultrafast increase in Li(t) starting from an initial value Li(t<0)=0, we study one of the most convenient cases, Gd in Co0.8Gd0.2. 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 momenta27. According to element-resolved hysteresis loops obtained for both alloys27, we conclude that our films are homogenous and are fully saturated under the applied magnetic field of 5 kOe. Therefore, we have L(t)~Lz(t) and S(t)~Sz(t) for Co, Gd and Tb. Thus, L(t) and S(t) are aligned parallel, so that we may define the time-dependent angular momentum Ji(t)=Li(t)+Si(t) per element i.

The X-ray transmission experiment, using an infrared laser pump and X-ray probe configuration, is schematically shown in Fig. 1. The external magnetic field is applied along the propagation direction of the X-rays. The lower part sketches the time evolution of the pump-induced atomic demagnetization on the exchange-coupled Co and RE in the multisublattice ferrimagnet. The time t0=0 is defined by the temporal overlap between the laser pulse and the X-ray pulse. A film thickness of 15 nm ensures a homogenous infrared laser excitation. The incident X-rays are circularly polarized and time-resolved XMCD is measured using the difference of intensities measured in transmission between two opposite magnetic fields +H and −H applied parallel to the X-rays (see Methods).

Figure 1: Experimental set-up and the microscopic model.
figure 1

(a) Scheme of the X-ray transmission experiment using an infrared laser pump and X-ray probe configuration. The external magnetic field is applied parallel to the incoming X-rays. (b) Sketch of the time evolution of the pump-induced atomic magnetization dynamics on the exchange-coupled Co and RE in the ferrimagnetic multisublattice. Ultrafast dynamics of Ji(t) for each element is illustrated and start at the temporal overlap t0=0. At any time, both elements are exchange coupled and show antiferromagnetic order, attesting that we do not cross the compensation temperature Tcomp. During the demagnetization, a coupling between the two angular momenta may exist in the ferrimagnet.

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Ultrafast demagnetization in CoGd

In Fig. 2a we show the pump-probe results obtained at the Co L2,3 and Gd M4,5 edges for the Co0.8Gd0.2 alloy film. The normalized XMCD values at negative time delays match the XMCD magnitudes recorded at the Co L3, Co L2, Gd M5 and Gd M4 edges during quantitative static XMCD measurements27. Applying the sum rules for the Co L2,3 and for the Gd M4,5 edges9,25,26, we extract the ultrafast dynamics of the spin momentum Si(t) (black circles) and the orbital momentum Li(t) (red circles) for Co and for Gd. Quantitative data for the time-dependent values of Si(t) and Li(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 equilibrium27. The continuous lines are the results of the simulations of Si(t) (black line) and Li(t) (blue line; see Methods), whereas the green line is the ratio of these results.

Figure 2: Dynamics of spin and orbital momenta for Co and Gd in Co0,8Gd0.2.
figure 2

(a) Time-resolved XMCD at the Co L3 (filled red circles), Co L2 (empty red circles), Gd M5 (filled black circles) and Gd M4 edges (empty black circles) measured on a 15-nm Co0,8Gd0.2 alloy film. The XMCD values at negative time delays match the XMCD magnitude recorded at the Co L3, Co L2, Gd M5 and Gd M4 edges during quantitative static XMCD measurements27. The pump fluence used during our experiment was adjusted to 8 mJ cm−2. (b) Ultrafast dynamics of the spin and orbital momenta −SCo(t) (black circles) and −LCo(t) (red circles) for Co as extracted by applying the sum rules. The values of LCo(t) and SCo(t) at negative delays are normalized to the static values LCo and SCo at thermodynamic equilibrium27. The continuous lines are the simulations of SCo(t) (black line) and LCo(t) (blue line; see Methods) with characteristic thermalization times of τCotherm=200±20 fs for both momenta. The ratio LCo(t)/SCo(t) (green circle) obtained as a function of the delay time shows that the orbital momentum reduces more than the spin momentum during the ultrafast demagnetization. (c) Ultrafast dynamics of the spin and orbital momenta SGd(t) (black circles) and LGd(t) (red circles) for Gd. The values at negative delays are normalized to the static values of LGd and SGd provided by the static XMCD measurements27. A constant value of LGd(t)=0 is evidenced during the ultrafast demagnetization process. The continuous line is a simulation of SGd(t) (black line) with a characteristic thermalization time of τGdtherm=480±40 fs. The simulations of Li(t) and Si(t) (where i stands for Co, Gd or Tb) have been obtained using a nonlinear least square fitting procedure. The extracted error bar on τitherm is the s.d., which is given as an output parameter of the fitting procedure. The error bars obtained for Li(t) and Si(t) as shown in b and c are given by the s.d. of the experimental data with respect to the fitting functions (see Methods).

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Analysing the time-dependent spin and orbital momenta for Co (Fig. 2b) we observe that, similar to CoPd films10, the magnitude of the demagnetization observed for SCo(t) and LCo(t) are different leading to a large quenching of the ratio LCo(t)/SCo(t) before t=1 ps (Fig. 2a, green symbols). The thermalization times of SCo(t) and LCo(t) are identical (τtherm=200±20 fs) within the error bars. For Gd (Fig. 2c), we notice a large decrease of SGd(t) while LGd(t)~0. The dynamics of SGd(t) show that the thermalization time is much longer, τth=480±40 fs (ref. 27). Interestingly, the value LGd(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 sub-picosecond 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 sub-picosecond timescale. The quantitative ultrafast dynamics of SGd(t) and LGd(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 Co0.74Tb0.26 alloy the excited-state temperature is below Tcomp, thus leading to faster demagnetization rates for the RE (τtherm=280±40 fs) as compared with other experiments where the excited-state temperature is in the vicinity of TC (refs 23, 27). The pump-probe results measured at the Co L2,3 and Tb M4,5 edges are shown in Fig. 3a and transformed into ultrafast dynamics of Si(t) and Li(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 L3, Co L2, Tb M5 and Tb M4 edges during quantitative static XMCD measurements27. Applying the sum rules, we can extract the ultrafast dynamics of the spin and orbital momenta Si(t) (black circles) and Li(t) (red circles) for Co and for Tb. For Co and Gd we can neglect the magnetic dipole term Tz (t) so that the spin momentum is then defined by S(t)=Seff(t)27. For Tb, according to atomic calculations done by Teramura et al.29, we take Tz/S=−0.08. We further correct the sum rule-extracted value Seff(t) by a value Tz (t)27 representing a time constant proportion of 8% of S(t). This assumption leads to an underestimation of 8% of STb(t) for t>0 assuming a virtual and complete quenching of Tz (t) by the pump laser at t>0. The values at negative delays are normalized to the values measured at thermodynamic equilibrium27. The continuous lines correspond to the simulations of Si(t) (black line) and Li(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 Si(t) and Li(t) for Co and Tb. For each element, the thermalization times of Si(t) and Li(t) are identical (Co τth=180±40 fs and Tb τth=280±30 fs). Computing the ratio Li(t)/Si(t) reveals an ultrafast quenching for Co but for Tb any change is smaller than the error bars (Fig. 3c, green symbols).

Figure 3: Dynamics of spin and orbital momenta for Co and Tb in Co0,74Tb0,26.
figure 3

(a) Time-resolved XMCD at the Co L3 (filled red circles), Co L2 (empty red circles), Tb M5 (filled black circles) and Tb M4 edges (empty black circles) measured on a 15-nm Co0,74Tb0,26 alloy film. The XMCD values at negative time delays match the XMCD magnitude recorded at the Co L3, Co L2, Tb M5 and Tb M4 edges during quantitative static XMCD measurements27. The pump fluence used during our experiment was adjusted to 12 mJ cm−2. (b) Ultrafast dynamics of the spin and orbital momenta SCo(t) (black circles) and LCo(t) (red circles) for Co extracted by applying the sum rules. The continuous lines are the simulations of SCo(t) (black line) and LCo(t) (blue line; see Methods) with characteristic thermalization times of τCotherm=180±40 fs for both momenta. The dynamics of the ratio LCo (t)/SCo(t) (green circle) shows that the orbital momentum decays more than the spin momentum during the ultrafast demagnetization. (c) Ultrafast dynamics of the spin and orbital momenta −STb(t) (black circles) and −LTb(t) (red circles) for Tb. The continuous lines are the simulations of STb(t) (black line) and LTb(t) (red line; see Methods) with characteristic thermalization times of τTbtherm=280±40 fs for both momenta. The simulations of Li(t) and Si(t) have been obtained using a nonlinear least square fitting procedure. The extracted error bar on τitherm is the s.d., which is given as an output parameter of the fitting procedure. The error bars obtained for Li(t) and Si(t) as shown in b and c are given by the s.d. of the experimental data with respect to the fitting functions (see Methods).

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The ultrafast quenching of LCo(t)/SCo(t) in Co0.8Gd0.2 and Co0.74Tb0.26 leads to several conclusions. Interestingly, one notices that in this experiment we quenched LCo(t)/SCo(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 LCo(t)/SCo(t) is in agreement with the model of ultrafast quenching of MCA proposed by Boeglin et al.10, in the framework of Bruno’s model30. 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 4f-localized moments, the simple relationship given by Bruno30 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 LTb(t)/STb(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 quasi-simultaneous demagnetization of LTb(t) and STb(t), whereas the 3d moments show an ultrafast quenching of LCo(t)/SCo(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 Tz (t) as a constant proportion of S(t) is justified for Tb.

Total angular momentum

We calculate the angular momentum Ji(t)=Li(t)+Si(t) by adding the element-resolved Li(t) and Si(t). We multiply Ji(t) by the elemental concentration in the alloy for each i=Co, Gd or Tb in Co0.8Gd0.2 (Fig. 4a) and Co0.74Tb0.26 (Fig. 4b). All individual angular momenta Ji(t) are quenched on laser excitation at t=0 ps. For each alloy, we then derive the total angular momentum J(t)=JCo(t)+JGd(t) (Fig. 4a, blue open symbols) and J(t)=JCo(t)+JTb(t) (Fig. 4b, blue open symbols). To account for the time dependence of Tz(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 Co0.8Gd0.2, where the working temperature is close to but above Tcomp, we have JCo≈−JGd 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 non-equilibrium state of the system determined by the working temperature and pump conditions27.

Figure 4: Ultrafast dynamics of the individual and total angular momenta.
figure 4

(a) Angular momenta JCo(t) (red circles), JGd(t) (black circles) and J(t)=JCo(t)+JGd(t) (blue circles) normalized by the composition (that is, JCo(t)=(LCo(t)+SCo(t)) × 0.80 and JGd(t)=(LGd(t)+SGd(t)) × 0.2) in the Co0,8Gd0.2 film. The continuous lines are simulations of JCo(t) (red line), JGd(t) (black line), whereas the magenta line is the sum of the two simulations JCo(t) (red line) and JGd(t) (black line). (b) Angular momenta JCo(t) (red circles), JTb(t) (black circles) and J(t)=JCo(t)+JTb(t) (blue circles) normalized by the composition (that is, JCo(t)=(LCo(t)+SCo(t)) × 0.74 and JTb(t)=(LTb(t)+STb(t)) × 0.26) measured in Co74Tb26 alloy film. The continuous lines are simulations of JCo(t) (red line), JTb(t) (black line) and J(t) (blue line). The sum of both simulations (JCo(t)+JTb(t)—in magenta) cannot be distinguished from the simulation of J(t) (blue line). The loss of the total angular momentum J(t) is delayed by δt=140±60 fs compared with the time at which the demagnetization of both sublattices starts. The simulations of Ji(t) and J(t) have been obtained using a nonlinear least square fitting procedure. Error bars on Ji(t) and J(t) displayed in a and b correspond to the s.d. of experimental data with respect to the fitting functions (see Methods).

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For Co0.74Tb0.26 where the working temperature is below Tcomp, we have JCo<−JTb and a sizable value of J(t)=JCo(t)+JTb(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 Tz (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 quasi-equilibrium at a temperature close to Tcomp27 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 JCo(t) and JTb(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 two-fit functions obtained for JCo(t) and JTb(t) as the magenta dotted line. The two-fit 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 time-resolved and element-selective angular momenta Ji(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 non-local 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 orbital-resolved model for spin dynamics in RE–TM alloys. From their model, they concluded that the ferromagnetic-like 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 Li(t), Si(t) and Ji(t) for each element and attribute the constant value of J(t) in Co0.74Tb0.26 during the first 140 fs to the angular momentum flowing between JCo(t) and JTb(t) (Fig. 4b, blue symbols). This process is hidden in Co0.8Gd0.2 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 defect-induced spin-flip 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 two-step 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 sub-picosecond timescale. Going one step further and considering the quantitative angular momenta Ji(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 interaction9.

To conclude, our findings demonstrate the local transfer of angular momenta between the two antiferromagnetically exchange-coupled sublattices. This transfer channel induces ultrafast demagnetization at the atomic scale, whereas the total angular momentum J(t) initially stays constant. In Co0.74Tb0.26 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 calculations12. In addition, we have uncovered an ultrafast quenching of the LCo(t)/SCo(t) ratio in ferrimagnetic Co0.8Gd0.2 and Co0.74Tb0.26 alloys and attributed it to the ultrafast quenching of MCA in agreement with Bruno’s model and previous results in CoPd10,30. 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 Si3N4 membranes. Co-deposition with convergent Co and Gd (or Tb) flux was used to get amorphous Co0.8Gd0.2 and Co0.74Tb0.26 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 time-resolved experiments. For Co0.8Gd0.2 and Co0.74Tb0.26 we have verified experimentally that at working temperatures the saturation fields are 4 and 3 kOe, respectively. Both alloys have been characterized by static XMCD27. As the samples were fully saturated by the external magnetic field of 5 kOe during the pump-probe experiments, we can define Lz (resp. Sz)=L (resp. S) so that it follows that Ji(t)=Li(t)+Si(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 Co0.8Gd0.2 and Co0.74Tb0.26 films show magnetic compensation temperatures (Tcomp) at which both magnetizations of the Co and the RE sublattices compensate (resp. Tcomp=150 and 550 K) and Curie temperatures (TC) where the magnetic order is lost (resp. TC=450 and 650 K).

The Co74Tb26 alloy film presents an out-of-plane magnetic anisotropy, which is measured along the normal of the film plane. We emphasize that, in contrast to CoPd and Co0.74Tb0.26 alloys10,27, Co0.80Gd0.20 has no out-of-plane magnetic anisotropy. For this sample we performed our XMCD experiment at 30° from the normal of the film plane. In the case of Co0.80Gd0.20 an in-plane uniaxial anisotropy may be induced during growth32. This point is confirmed by our recent results where static XMCD characterization was performed27. In both alloys we found large ratios L/S of 0.21–0.29 at the Co L2,3 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, STb=1.35±0.2 at−1) is lower than expected from Hund’s rules (STb=3 at−1 ) and is related to the structural disorder and finite temperature effects27.

Time-resolved XMCD

Time-resolved XMCD was performed at the femtoslicing beam line of the BESSY II synchrotron radiation source of the Helmholtz–Zentrum Berlin7,10. The magnetization dynamics have been measured by monitoring the transmission signal of circularly polarized X-rays, tuned to specific core-level absorption edges as a function of a pump-probe 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 L2,3, Tb M4,5 and Gd M4,5 edges using the Bragg Fresnel reflection zone plate monochromator. The experiments have been performed with a pump-probe set-up where the short X-ray pulses are synchronized with a femtosecond pump laser working at 790 nm, 3 kHz repetition rate with pulses of 60 fs. The X-ray 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 Co0.8Gd0.2 and to 12 mJ cm−2 for Co0.74Tb0.26 to reach large demagnetization magnitudes of about 60% at the Co L3 edge without altering the sample properties (alloy concentration, atomic diffusion and large heating).

Fitting procedure

The physical quantities Li(t), Si(t) and Ji(t) (i=Co, Gd, Tb) were adjusted using the rate equation of the two-temperature model with two exponential functions (equation (1)):

where G(t) is the Gaussian function defining the total time resolution of the experiment (130 fs), τth and τs−ph are the thermalization time and the relaxation time from the spin system to other systems (lattice, external bath), t0 is the delay at which the temporal overlap of the pump and the probe is achieved and H(t−t0) is the Heaviside function (H(t−t0)=0 if t<t0 and H(t−t0)=1 if t>t0) describing the energy transfer from the laser.

The parameters C0, C1, t0 and τth in equation (1) were optimized to minimize independently the χ2 factor for Li(t), Si(t) and Ji(t) (equation (2)).

where y is the rate equation, yj are the experimental data and σj 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 τth and t0 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 Li(t) and Si(t) as well as on Fig. 4a,b for Ji(t) correspond to the s.d. (σ) of experimental data with respect to the fitting functions that minimize the χ2 factor. The amplitude of the error bars has been calculated according to equation (3):

The final fitting parameters for Ji(t) are compatible with the result of the sum of the two-fit functions obtained for Li(t) and Si(t).

The total angular momentum for each alloy was obtained by performing the weighted sum J(t)=JCo(t)+JTb(t) according to the composition of the alloy. In Co0.8Gd0.2, the total angular momentum JCo(t)+JGd(t) remains small and comparable to the noise level. For Co0.74Tb0.26, a sizable value of J(t)=JCo(t)+JTb(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 sum-of-the-fit functions obtained for JCo(t) and JTb(t); (magenta dotted line in Fig. 4). The two-fit functions for J(t) agree perfectly for our parameters Co, C1, t0 and τth. 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 zero-time values: t0=0±40 fs for JCo(t) and JTb(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 t0 as extracted from the rate equation, we performed the simulations using a time axis for JCo(t) shifted by an amount of±40 fs with respect to JTb(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).