Introduction

The experimental determination of the microscopic spin-flip scattering rate in solids is of fundamental importance in order to better understand their macroscopic properties such as the femtosecond demagnetization process1. However, while more than two decades of experimental work have been devoted to quantify the ultrafast demagnetization time constants using mainly pump-probe strategies, the experimental quantification of the spin-flip rates appears to be more challenging and therefore, scarcely investigated. More specifically, the femtosecond demagnetization of ferromagnets2, the transient states in ferrimagnets3 or the modifications of antiferromagnetic order4 have common microscopic physical drivers. Among these drivers are the atomic electron-phonon5,6,7,8,9,10 and electron-magnon11 mediated spin-flip scattering, non-collinear momenta reordening4, different velocities of minority and majority spin electrons in superdiffusive spin transport2,12 or intersite spin-selective charge transfer13. All these drivers satisfy the boundary condition of angular momentum conservation within accessible spin, electron orbital and lattice degrees of freedom of a material.

Here, we present a method to determine the temperature-dependent atomic electron-phonon induced spin-flip scattering rate. We exploit the quantifiable change in the decay peak intensities in static x-ray emission spectroscopy (XES) spectra when changing the temperature, i.e. when changing the phonon population. We apply this method to nickel and copper as test model systems. For nickel we observe a decrease of the intensity of the emission peak corresponding to the spin-polarized 3d valence band to the created 2p3/2 core hole. We interpret this decrease as a result of the Elliott-Yafet type spin-flip scattering of valence electrons with phonons, which reduces the decay probability. Accordingly to our interpretation, the diamagnetic counterexample copper presents no temperature dependance of the decay peak.

The basic underlying idea of the method is illustrated in the simplified schematics of Fig. 1, which depicts the radiative decay from a valence band electron to a created core-hole. At low temperature this decay occurs during the core-hole lifetime from electrons having the same spin. At high temperature, electron-phonon scattering-driven angular momentum transfer events can flip the electron spin and lead to a lower radiative decay rate, visible as a lower corresponding peak intensity in XES spectra.

Figure 1
figure 1

Schematic principle of the XES process in the presence or absence of spin-flip scattering in nickel after the creation of a core-hole. (a) Low temperature case: radiative decay from the filling of a core hole by a valence band electron. (b) High temperature case: spin-flip processes induced by electron-phonon scattering events reduce the radiative decay probability of the core-hole.

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Results and Discussion

Temperature-dependent x-ray emission spectroscopy

We apply this method to the 3d-ferromagnetic model system nickel. The experimental data is presented in Fig. 2(a), which shows XES spectra taken with an incident energy of hν = 865 eV. At this chosen incident photon energy, selectively a Ni 2p3/2 core level vacancy is created, which decays within the natural Ni 2p3/2 core level life time of τcore−hole = 1.04 fs14. The incident energy is chosen well above the Ni L3 edge in order to excite the core electron to the continuum and to be in the non-resonant regime. This allows probing the weakly perturbed valence band when measuring the radiative decay. The radiative decay of the Ni 2p3/2 core vacancy through valence electrons within XES obeys the atomic dipole selection rules of Δl = ±1 and Δs = 0. Thus, we detect within the 700 eV to 900 eV photon emission energy range of our X-ray spectrometer simultaneously the 3s → 2p3/2, and the 3d → 2p3/2 transitions. Note that the 3s → 2p1/2, and the 3d → 2p1/2 transitions are also slightly visible. A 3p → 2p non-dipole x-ray emission arising due to resonant Raman scattering was reported previously15. However, its spectral signature is not visible in our data.

Figure 2
figure 2

Temperature dependence of the XES spectra of nickel and copper. (a) Temperature-dependent XES spectra of nickel. An increase in the temperature leads to a decrease in the 3d → 2p3/2 peak intensity. (b) The room temperature NEXAFS spectrum of nickel recorded during the experiments shows no indication of an oxide. (c) Schematically illustrated nickel valence density of states (DOS) with the 2p core levels. The magnetic properties arise from the half-filled spin minority 3d band. (d) XES spectra of copper. Here, the 3d → 2p3/2 peak intensity is temperature-independent. (e) The room temperature NEXAFS spectrum of copper recorded during the experiments shows no indication of an oxide. (f) Schematically illustrated copper valence DOS with the 2p core levels. Due to the fully occupied bands, copper is diamagnetic.

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Raising the temperature of the nickel crystal leads to a waning of spectral intensity of the Ni 3d → 2p3/2 transition, as highlighted in the inset of Fig. 2(a). The partially occupied Ni 3d valence states can undergo changes in orbital and spin character, due to low energy scattering events with phonons. Thus, the initially atomically prepared state of the Ni 2p3/2 core vacancy that is radiatively filled by the sub-set of dipole allowed Ni 3d electrons, is sensitive to Ni 3d electron-phonon scattering, which in particular changes the spin state of the valence electrons. However, for the Ni 3s inner valence state, that is fully occupied, no change of angular momentum and spin state can occur and a constant spectral intensity vs. the temperature for the radiative transition into the Ni 2p3/2 is expected. Therefore, the peak area of this transition is assumed to be constant and is used to normalize spectra. More precisely, the entire spectra are multipied by a factor in order to keep this peak area, after subtracting the background area under it, constant for all temperatures16. The background comes mainly from the glowing filament and the warm parts around the samples, when measuring at high temperature during several hours. What is shown in Fig. 2 is the normalized spectra including the total background. Figure 2(b) shows a nickel Near-Edge X-ray Absorption Spectroscopy (NEXAFS) spectrum measured in the total electron yield mode and acquired during our experiments. The energy range includes the L2 and L3 edges at 871.9 eV and 854.7 eV, respectively17,18. Our NEXAFS data correspond to those expected for clean nickel19. In particular, the satellite peak at 859 eV, known as the 6-eV feature, which arises from strong electronic correlation effects, is visible20.

To elucidate the aspect of spin-flip scattering further, we performed similar experiments on copper, where we can create the analogous Cu 2p3/2 core-vacancy. In contrast to nickel, the Cu 3d-band as well as the Cu 3s inner valence state are fully occupied. Thus, no spin-flip scattering is possible. Figure 2(d) shows the temperature-dependent XES spectra of copper for an incident energy of 945 eV, i.e. between the L2 and L3 edges. Here, both the radiative decay of the fully occupied Cu 3s inner valence and the Cu 3d-band into the atomic Cu 2p3/2 core-vacancy leads to no detectable changes in spectral intensities with the temperature. Since the Cu 2p3/2 core hole life time is with τcore−hole = 0.56 fs14 rather similar to the one of nickel, this cannot be attributed to a shorter scattering duration time for copper than for nickel. And again, the NEXAFS spectrum of copper (Fig. 2(e)), which presents the L2,3 peaks at 952.3 eV and 932.7 eV in addition to two distinct satellite peaks at 937.6 eV and 941.5 eV, is characteristic of clean copper with no indication of the presence of oxide or other contaminants21.

Discussion

The evolution of the 3d → 2p3/2 peak with the temperature is the consequence of a reduction of the density of 3d electrons available for the decay to the created core-holes. This reduction can be the result of either electrons with a 3d symmetry being excited to a 4s or 4p symmetry or of a spin-flip of the 3d electrons. Both scenarios can originate from an electron-phonon angular transfer. The first scenario is unlikely since (i) 4s and 4p density of states (DOS) are more than an order of magnitude smaller than the 3d DOS and would not explain a visible change in the XES peak intensity and (ii) it is not consistent with an absence of temperature dependence of the XES spectra for copper, where such excitations could also be considered. Therefore, the XES peak evolution is more likely the result of the spin-flip of 3d-electrons, which is allowed for nickel but not for copper. Spin-flip transitions of localized 3d electrons in nickel driven by spin-orbit coupling have been recently proposed as a microscopic mechanism of the demagnetization dynamics22.

To test this interpretation, we performed Density Functional Theory (DFT) calculations to simulate the decay peak intensity. Our calculations show a reduction of the peak area of 5.5 % and 5.4 % for nickel and copper, respectively, due to the temperature-induced lattice expansion and the Fermi-Dirac smearing (see16). The facts that the value for nickel is smaller than the experimentally observed waning and that we observe very similar values for nickel and copper indicate that the lattice expansion and the Fermi-Dirac smearing only, i.e. without scattering, are not the main contributions to our observations. Simulated 3d → 2p3/2 emission peaks are presented in Fig. 3. The plots show the difference in the emission peak when allowing or prohibiting the decay from the 3d bands crossing the Fermi surface. This would be the consequence of an electron scattering within the 3d band, reducing the decay probability. We find a reduction of the peak area for both nickel and copper and for low (300 K) and high (1200 K) temperatures. This reduction is in the order of 12 % for nickel and 16 % for copper. This matches only the observed peak reduction of 11 % for nickel at high temperature, consistently with our interpretation. Indeed, the fact that this reduction is not experimentally observed at low temperature indicates electron-phonon scattering. In addition, since the calculated peak reduction for copper is not observed experimentally, this speaks consistently against the possibility of a spin-flip scattering process in copper. Copper has a full 3d band, which prevents spin-flip scattering events, in contrast to nickel, which has a partially filled 3d band, below and above the Curie temperature, and for which spin-flip scattering is allowed.

Figure 3
figure 3

Simulated 3d → 2p3/2 emission peak of (a) nickel and (b) copper at 300 K and 1200 K. The plots show the change in the emission peak due to the presence or absence of electron decay from the 3d-bands crossing the Fermi surface, leaving electrons from all other bands free to decay. This change is shown for 300 K and 1200 K. For clarity, the center of the peak is set to 852 eV for nickel and 928 eV for copper and the plots are shifted vertically for different temperatures.

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Determination of the scattering rate

Following our interpretation, we quantify our experimental findings in Fig. 4, where the angular momentum transfer rate of nickel (a) and copper (b) as a function of the temperature are shown in direct comparison. An important remark must be made here about the analysis of the peak area, which consists in a normalization against the 3s → 2p3/2 peak area, as discussed above and which gives the spectra shown in Fig. 2, in addition to a background subtraction. Indeed, after normalization and especially for nickel, we still observe a slight difference in the background signal (see Fig. 2). For the data shown in Fig. 4, in addition to the normalization, we estimated carefully and subtracted this background area below the 3s → 2p3/2 and the one below the 3d → 2p3/2 peaks. Further details about the background subtraction are given in the Supplementary Material16.

Figure 4
figure 4

Angular momentum transfer rate. (a) nickel. (b) copper. Points are experimental data obtained from the XES spectra. Lines are fits. (c) Momentum transfer lifetime deduced from the fitted rates in (a).

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We established previously in semiconductors how the angular momentum transfer scattering rate R(T) can be deduced from the evolution of valence to core-hole decay peaks with temperature23,24 as:

$$R(T)=\frac{1}{{\tau }_{core-hole}}\cdot \frac{{A}_{inc}}{{A}_{coh}}=\frac{1}{{\tau }_{core-hole}}\cdot \frac{{A}_{cold}-{A}_{hot}}{{A}_{cold}}$$
(1)

where τcore−hole is the core-hole lifetime of the excited state, Ainc = Acold − Ahot (purple hatched area in Fig. 2(a)) is the fraction of decay modified by electron-phonon scattering and Acoh = Acold is the fraction not affected by this. This rate can be decomposed in a temperature-independent and a temperature-dependent contribution. The former is caused by lattice distortions due to the core excited state. The latter is proportional to the phonon population and thus, to the Bose-Einstein distribution24. Therefore, the evolution of the electron-phonon transfer rate with temperature can be written as:

$$R(T)={C}_{indep}+\frac{1}{{e}^{\frac{\langle {E}_{ph}\rangle }{kT}}}\cdot {C}_{dep}$$
(2)

where Cindep and Cdep correspond to the temperature independent and the temperature dependent contribution, respectively, and are used as fitting parameters. 〈Eph〉 is the average phonon energy.

From the peak areas, we deduce the electron-phonon spin-flip scattering rate vs. temperature and show it in Fig. 4. The error bars are determined by analyzing the fluctuation in the intensity when iterating data acquisition in similar conditions16. For nickel, the fit of our experimental data using Eq. (2), where 〈Eph〉 = 24 meV25, shows an almost linear increase of the momentum transfer rate from close to zero up to 0.15 fs−1 within our 300 K – 1200 K temperature range. For copper, where 〈Eph〉 = 20 meV25, no detectable spectral evolution with temperature is seen. As shown in Fig. 4(c), our method leads to an angular momentum transfer time scale at room temperature in the order of 50 fs for nickel. Even though this quantity, which refers to a process at the atomic scale, cannot be directly compared to the macroscopic demagnetization time measured using pump-probe experiments, it unambiguously demonstrate the importance of the Elliott-Yafet contribution in the demagnetization mechanism in nickel.

Conclusion

To conclude, we present here a unique approach to measure the Elliott-Yafet contribution in the demagnetization process in nickel. It is based on static measurements and can therefore be applied in all synchrotron based facilities. It is also general to a broad range of magnetic materials9,26. Finally, our method can easily be applied for a better understanding of electron-phonon interactions in systems like (high-TC) superconductors27,28, graphene29, topological insulators30,31 or Weyl semimetals32.

Methods

XES experiments were performed with the SolidFlexRIXS endstation on the high flux U49-2 PGM-1 beamline at BESSY II in the multibunch operating mode. Temperature dependent measurements were performed from room temperature up to almost the melting point of nickel and copper, reached by electron bombardment from a Tungsten filament. The base pressure was in the low 10−8 mbar range but rose up to the low 10−6 mbar range for the highest temperatures. Spectra were acquired with a GRAZE IV – type spectrometer equipped with a single photon counting microchannel plate (MCP) detector from Scienta. Samples, purchased at Matek, were placed on a tungsten sample plate. The temperature was measured using both a thermocouple on the sample plate and a pyrometer.

Density Functional Theory (DFT) calculations of the temperature effects on the X-ray emission spectra of nickel and copper were done using the linearized augmented plane-wave elk code (elk.sourceforge.net). The effect of thermal expansion on the emission spectrum was simulated by expanding the room temperature lattice parameters of Ni (3.52 Å) and Cu (3.58 Å) by 1.8 %. The product of the smallest muffin-tin radius and the largest G vector of the plane wave basis RMT,min × Gk,max was set to 7. The ground state and spectrum calculations were performed on 40 × 40 × 40 k-point grids. The emission spectra were calculated in the random phase approximation. Effects of the initial (final) state core (valence) hole were neglected. Fermi surface smearing effects were accounted for by using the physical temperatures of 300 K and 1200 K in the calculation of the spectra. The calculations were performed with and without contributions from the bands crossing the Fermi level. The latter case simulates the effect of a spin-flip near the Fermi surface on the emission intensity.