Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Measuring the atomic spin-flip scattering rate by x-ray emission spectroscopy

Abstract

While extensive work has been dedicated to the measurement of the demagnetization time following an ultra-short laser pulse, experimental studies of its underlying microscopic mechanisms are still scarce. In transition metal ferromagnets, one of the main mechanism is the spin-flip of conduction electrons driven by electron-phonon scattering. Here, we present an original experimental method to monitor the electron-phonon mediated spin-flip scattering rate in nickel through the stringent atomic symmetry selection rules of x-ray emission spectroscopy. Increasing the phonon population leads to a waning of the 3d → 2p3/2 decay peak intensity, which reflects an increase of the angular momentum transfer scattering rate attributed to spin-flip. We find a spin relaxation time scale in the order of 50 fs in the 3d-band of nickel at room temperature, while consistantly, no such peak evolution is observed for the diamagnetic counterexample copper, using the same method.

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.

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 τcorehole = 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.

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 τcorehole = 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.

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).

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 τcorehole 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.

Data Availability

All data are available upon reasonable request.

References

  1. Beaurepaire, E., Merle, J.-C., Daunois, A. & Bigot, J.-Y. Ultrafast spin dynamics in ferromagnetic nickel. Phys. Rev. Lett. 76, 4250–4253 (1996).

    ADS  CAS  Article  Google Scholar 

  2. Eschenlohr, A. et al. Ultrafast spin transport as key to femtosecond demagnetization. Nat. Mater. 12, 332 (2013).

    ADS  CAS  Article  Google Scholar 

  3. Radu, I. et al. Transient ferromagnetic-like state mediating ultrafast reversal of antiferromagnetically coupled spins. Nat. 472, 205 (2011).

    ADS  CAS  Article  Google Scholar 

  4. Thielemann-Kühn, N. et al. Ultrafast and energy-efficient quenching of spin order: Antiferromagnetism beats ferromagnetism. Phys. Rev. Lett. 119, 197202 (2017).

    ADS  Article  Google Scholar 

  5. Elliott, R. J. Theory of the effect of spin-orbit coupling on magnetic resonance in some semiconductors. Phys. Rev. 96, 266–279 (1954).

    ADS  CAS  Article  Google Scholar 

  6. Yafet, Y. Solid State Phys. Vol. 14 (Eds Seitz, F. & Turnbull, D.) (Academic) (1963).

  7. Koopmans, B., Ruigrok, J. J. M., Longa, F. D. & de Jonge, W. J. M. Unifying ultrafast magnetization dynamics. Phys. Rev. Lett. 95, 267207 (2005).

    ADS  CAS  Article  Google Scholar 

  8. Cinchetti, M. et al. Spin-flip processes and ultrafast magnetization dynamics in co: Unifying the microscopic and macroscopic view of femtosecond magnetism. Phys. Rev. Lett. 97, 177201 (2006).

    ADS  CAS  Article  Google Scholar 

  9. Koopmans, B. et al. Explaining the paradoxical diversity of ultrafast laser-induced demagnetization. Nat. Mater. 9, 259 (2009).

    ADS  Article  Google Scholar 

  10. Stamm, C. et al. Femtosecond modification of electron localization and transfer of angular momentum in nickel. Nat. Mater. 6, 740 (2007).

    ADS  CAS  Article  Google Scholar 

  11. Carpene, E. et al. Dynamics of electron-magnon interaction and ultrafast demagnetization in thin iron films. Phys. Rev. B 78, 174422 (2008).

    ADS  Article  Google Scholar 

  12. Battiato, M., Carva, K. & Oppeneer, P. M. Superdiffusive spin transport as a mechanism of ultrafast demagnetization. Phys. Rev. Lett. 105, 027203 (2010).

    ADS  CAS  Article  Google Scholar 

  13. Dewhurst, J. K., Elliott, P., Shallcross, S., Gross, E. K. U. & Sharma, S. Laser-induced intersite spin transfer. Nano Lett. 18, 1842–1848 (2018).

    ADS  CAS  Article  Google Scholar 

  14. Krause, M. O. & Oliver, J. H. Natural widths of atomic K and L levels, Ka x-ray lines and several KLL Auger lines. J. Phys. Chem. Ref. Data 8, 329–338 (1979).

    ADS  CAS  Article  Google Scholar 

  15. Jiménez-Mier, J., Ederer, D. L., Schuler, T. & Callcott, T. A. Direct evidence for 3p to 2p non-dipole x-ray emission in transition metals. J. Phys. B: At. Mol. Opt. Phys. 36, L173 (2003).

    ADS  Article  Google Scholar 

  16. see Suppl. Material (online).

  17. Bearden, J. A. X-ray wavelengths. Rev. Mod. Phys. 39, 78–124 (1967).

    ADS  CAS  Article  Google Scholar 

  18. Fuggle, J. C. & Mårtensson, N. Core-level binding energies in metals. J. Electron Spectrosc. Relat. Phenom. 21, 275–281 (1980).

    CAS  Article  Google Scholar 

  19. Nietubyć, R. et al. l-edge x-ray absorption fine structure study of growth and morphology of ultrathin nickel films deposited on copper. Phys. Rev. B 70, 235414 (2004).

    ADS  Article  Google Scholar 

  20. Magnuson, M. et al. Resonant auger spectroscopy at the L 2,3 shake-up thresholds as a probe of electron correlation effects in nickel. Phys. Rev. B 58, 3677–3681 (1998).

    ADS  CAS  Article  Google Scholar 

  21. Grioni, M. et al. Studies of copper valence states with cu L 3 x-ray-absorption spectroscopy. Phys. Rev. B 39, 1541–1545 (1989).

    ADS  CAS  Article  Google Scholar 

  22. Krieger, K., Dewhurst, J. K., Elliott, P., Sharma, S. & Gross, E. K. U. Laser-induced demagnetization at ultrashort time scales: Predictions of tddft. J. Chem. Theory Comput. 11, 4870–4874 (2015).

    CAS  Article  Google Scholar 

  23. Beye, M. et al. Dynamics of electron-phonon scattering: Crystal- and angular-momentum transfer probed by resonant inelastic x-ray scattering. Phys. Rev. Lett. 103, 237401 (2009).

    ADS  CAS  Article  Google Scholar 

  24. Miedema, P. S., Beye, M., Könnecke, R., Schiwietz, G. & Föhlisch, A. The angular- and crystal-momentum transfer through electron–phonon coupling in silicon and silicon-carbide: similarities and differences. New J. Phys. 16, 093056 (2014).

    ADS  Article  Google Scholar 

  25. Kresch, M. Temperature dependence of phonons in elemental cubic metals studied by inelastic scattering of neutrons and x-rays. California Inst. Technol. PhD thesis (2009).

  26. Vaterlaus, A., Beutler, T. & Meier, F. Spin-lattice relaxation time of ferromagnetic gadolinium determined with timeresolved spin-polarized photoemission. Phys. Rev. Lett. 67, 3314–3317 (1991).

    ADS  CAS  Article  Google Scholar 

  27. Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of superconductivity. Phys. Rev. 108, 1175–1204 (1957).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  28. Lanzara, A. et al. Evidence for ubiquitous strong electron-phonon coupling in high-temperature superconductors. Nat. 412, 510 (2001).

    ADS  CAS  Article  Google Scholar 

  29. Zhu, J., Badalyan, S. M. & Peeters, F. M. Electron-phonon bound states in graphene in a perpendicular magnetic field. Phys. Rev. Lett. 109, 256602 (2012).

    ADS  CAS  Article  Google Scholar 

  30. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  CAS  Article  Google Scholar 

  31. Zhu, X. et al. Electron-phonon coupling on the surface of the topological insulator Bi2Se3 determined from surface-phonon dispersion measurements. Phys. Rev. Lett. 108, 185501 (2012).

    ADS  Article  Google Scholar 

  32. Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Sci. 349, 613–617 (2015).

    ADS  CAS  Article  Google Scholar 

Download references

Acknowledgements

We acknowledge the ERC-ADG-2014 Advanced Investigator Grant No. 669531 EDAX under the Horizon 2020 EU Framework, Programme for Research and Innovation and the partial funding by the Helmholtz Virtual Institute VI419 ‘Dynamic Pathways in Multidimensional Landscapes’. We thank S. Sharma and P. Elliott for fruitful discussions and comments on the manuscript.

Author information

Authors and Affiliations

Authors

Contributions

R.D., A.B., R.B., C.S., S.N. R.H. and A.P. performed the measurements. R.D. and A.B. analyzed the experimental data. K.R. performed the DFT calculations. R.D. and A.F. wrote the manuscript. All the authors discussed the results and commented the manuscript. R.D. and A.B. contributed equally to this work.

Corresponding authors

Correspondence to Régis Decker or Alexander Föhlisch.

Ethics declarations

Competing Interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Decker, R., Born, A., Büchner, R. et al. Measuring the atomic spin-flip scattering rate by x-ray emission spectroscopy. Sci Rep 9, 8977 (2019). https://doi.org/10.1038/s41598-019-45242-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1038/s41598-019-45242-8

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing