Abstract
Many properties of real materials can be modeled using ab initio methods within a singleparticle picture. However, for an accurate theoretical treatment of excited states, it is necessary to describe electronelectron correlations including interactions with bosons: phonons, plasmons, or magnons. In this work, by comparing spin and momentumresolved photoemission spectroscopy measurements to manybody calculations carried out with a newly developed firstprinciples method, we show that a kink in the electronic band dispersion of a ferromagnetic material can occur at much deeper binding energies than expected (E_{b} = 1.5 eV). We demonstrate that the observed spectral signature reflects the formation of a manybody state that includes a photohole bound to a coherent superposition of renormalized spinflip excitations. The existence of such a manybody state sheds new light on the physics of the electronmagnon interaction which is essential in fields such as spintronics and Febased superconductivity.
Introduction
Spinflip excitations, including singleparticle Stoner and collective spinwave excitations (magnons), schematically shown in Fig. 1a, are fundamental for the description of ferromagnetic materials^{1,2,3,4}. The interaction between conduction electrons and magnons is critical for fundamental physical properties, such as the temperature dependence of the resistivity^{5} and magnetotransport^{6}. It also plays an essential role in models that describe the laserinduced ultrafast demagnetization^{7}. On the more applied side, electronmagnon interactions are the basis of the field of magnonics, which offers prospects of faster and more energyefficient computation^{8}. While magnons have a well defined dispersion relation with excitation energies up to a few hundred meV, the Stoner excitations form a quasicontinuum in the magnetic excitation spectrum and their excitation energies are typically in the order of a few eV (Fig. 1b). Although the spin of majority electrons is more likely flipped than that of minority electrons (Fig. 1b), a minority spin flip can have a strong effect on the electronic dispersions, as this article reveals.
Electron dispersion anomalies, such as kinks, are regarded as signatures of an electronboson interaction, expected to occur at the scale of the boson energy (typically up to few hundred meV)^{9,10,11,12,13}. In the case of superconducting materials, the appearance of kinks is a priceless clue pointing to the origin of the electronelectron coupling^{14,15,16,17}. In ferromagnetic materials, kinks observed by photoemission at binding energies of 100–300 meV were interpreted as originating from the electronmagnon interaction because the involved energy scale was regarded as too large to reflect electronphonon interaction^{12,13,18}. Up to now, this interpretation was merely a suggestion, as no ab initio method has been able so far to reproduce magnoninduced kinks.
In this work, we have experimentally mapped the electronic band structure of an Fe(001) thin film and identified a characteristic kink located 1.5 eV below the Fermi level, which can be reproduced by ab initio calculations based on a diagrammatic expansion of the selfenergy, a quantity that describes the deviation of the quasiparticle spectrum from the ‘undressed’ electron picture. This GT selfenergy (Fig. 1c) accounts for the coupling of electrons or holes (Green function G) to the correlated manybody system through the creation and absorption of spin excitations (T matrix), taking into account the full nonlocal excitation spectrum with magnons and Stoner excitations treated on an equal footing. The T matrix, which describes the correlated motion of an electronhole pair with opposite spins, is a mathematically complex quantity because it depends on four points in space (two incoming and two outgoing particles) and time (or frequency). The method is a firstprinciples approach, therefore, apart from the atomic composition, no additional parameters are used. It naturally takes into account nonlocal electron correlations (momentum dependence of selfenergy), which were recently experimentally shown to be important for 3d ferromagnets^{19}. Details of the theory are presented in refs. ^{20,21}.
Results
Momentumresolved photoemission
To get experimental access to the bulk electronic structure of Fe, we have used a thin Fe film (38 ML) deposited on a Au(001) single crystal. The photoemission measurements have been performed at the NanoESCA beamline of Elettra, the Italian synchrotron radiation facility, using a modified FOCUS NanoESCA photoemission electron microscope (PEEM) in the kspace mapping mode^{22}. The experiment is shown schematically in Fig. 1d. We will discuss here the results obtained using spolarized light of hν = 70 eV, which according to the freeelectron final state model induces transitions from the initial states located close to the Γ point of the bulk Brillouin zone. Complementary results of the measurements with ppolarized light, as well as spinresolved measurements are shown in the Supplementary Figs. 1 and 2 and discussed in the Supplementary Notes 1 and 2.
Figure 2a presents a comparison between experiment and a meanfield band structure of bulk Fe obtained with the localspindensity approximation (LSDA)^{23} of density functional theory (DFT). The blue (red) labels: Δ_{1}, Δ_{2}, etc. identify the symmetry of the orbital part of the wavefunctions for minority (majority) states along Fe(001) direction^{24}. Here, we neglect the spinorbit coupling (SOC), as we are interested in the general shape of the bands within a wide binding energy range. The effect of the SOC on the Fe(001) electronic states close to the Fermi level was discussed earlier^{25}. In the Supplementary Fig. 1, we also compare the experimental spectrum to GW calculations^{26,27,28}, which include quasiparticle renormalization effects beyond DFT. The identification of the experimentally observed electronic states is possible thanks to the results of the spinpolarized measurements (Supplementary Fig. 2) and the consideration of the dipole selection rules that depend on the photon polarization (Supplementary Note 1). Specifically, we identify a minority band of Δ_{2} symmetry, which is particularly sharp, especially in contrast to the majority bands (e.g., Δ_{1}), which become broad and diffuse directly below the Fermi level (Fig. 2a). Importantly, in contrast to the prediction of LSDA and GW calculations, the experimentally observed minority band Δ_{2} exhibits a peculiar anomaly near a binding energy of E_{b} = 1.5 eV marked by arrows in Fig. 2a.
Ab initio calculations
Figure 2b shows the theoretical spectral function as obtained from the GT calculation summed over the spins on the left and only for the minority spin on the right. We observe a strong renormalization and lifetime broadening of the band structure, in particular, for the majority bands. For example, the majority Δ_{2} band loses its quasiparticle character completely below E_{b} = 1 eV, which explains why this band is not visible in the experiment (Fig. 2a) despite favorable dipole selection rules. Such spin dependence of the electronelectron correlation effects is in line with earlier theoretical reports^{29,30,31} and experimental findings^{19,32}. In the minority channel, the calculated band dispersions remain relatively sharp. However, the minority Δ_{2} band exhibits an anomaly that seems to coincide with the kink observed in the photoemission experiment.
Experiment vs. theory
In order to compare the theoretical prediction with the experimental measurement, we have fitted experimental momentum distribution curves (MDC) with a Lorentzian function on a linear background (Fig. 2e) for binding energies between 0.8 and 2.0 eV and superimposed the fitted peak positions on the experimental and theoretical spectral functions in Fig. 2c, d. Both curves, the experimental and the theoretical one, strikingly show the band anomaly at roughly the same energy and momentum. For further analysis, we compare in Fig. 2f the experimental (circles) and theoretical spectral functions (lines) taken at k = 1.3 Å^{−1}. We observe a characteristic doublepeak structure in both, which indicates a transfer of spectral weight from one branch of the quasiparticle band to another resulting in the appearance of a kink. The Lorentzians fitted to the experimental dispersion can be used to derive the experimental selfenergy, which compares remarkably well with the calculated selfenergy (see Supplementary Note 3 for the discussion of the selfenergy and Supplementary Fig. 3 for the comparison between theoretical and experimental selfenergy).
Discussion
Interestingly, the binding energy at which the anomaly appears is much higher than what one would normally expect for electronmagnon scattering. Specifically, it is larger than typical magnon energies. The reason for this is twofold. First, selfenergy resonances appear not at the boson energy, but rather at the sum of two energies: the boson (e.g., magnon) energy (from T) and a singleparticle energy (from G). Second, we consider a coupling of a propagating minority spin hole to excitations that, due to spin conservation, would have to carry a spin of +1, which is just opposite to the respective spin transfer of magnon excitations. The coupling is thus predominantly with renormalized Stoner excitations, whose energies are typically larger than magnon energies. In fact, Fig. 1b (left panel) shows a particularly strong resonance around 0.7 eV close to the H point, which, together with a peak in the majority density of states of bulk iron (Supplementary Fig. 4 and Supplementary Note 4) at 0.8 eV, produces a selfenergy resonance at around 1.5 eV. This resonance is a manifestation of a broadened manybody state that consists of a majority hole and a superposition of correlated (electronhole) Stoner excitations and that, by interaction with the minority band, is ultimately responsible for the appearance of the band anomaly. In other words, a minority photohole is created in the photoemission process, which becomes dressed with electronhole pairs of opposite spins (Stoner excitations), and flips its spin in the process. This broadened manybody scattering state has a resonance at around E_{b} = 1.5 eV in bcc iron. It bears similarities to the spin polaron^{33} in halfmetallic ferromagnets and to the Fermi polaron^{34} in ultracold fermion gases.
By examining other kspace directions, we find that the appearance of the highenergy kink is very sensitive to the value of the selfenergy, and therefore strongly dependent on the k direction (see Supplementary Note 5 and Supplementary Fig. 5).
The highenergy kink identified in our work seems similar to the results of F. Mazzola et al.^{35,36}, who found a kink in the σ band of graphene close to E_{b} = 3 eV. In this case, however, the authors attribute the kink to the strong electronphonon coupling near the top of the σ band, which effectively places the kink exactly at the boson energy. It is also important to note that some highenergy anomalies, observed especially for cuprates^{37,38,39}, were later interpreted to be the result of the photoemission matrix elements^{40,41,42}. We can rule out such an explanation in our experiment, as it is not related to the suppressed photoelectron intensity near a highsymmetry direction^{40,41,42}. It is interesting to note, however, that we observe a similar suppressed intensity for the majority band Δ_{1} near the Γ point (Fig. 2a).
While the experimental position and shape of the kink match the prediction by the GT renormalization very well, it should be mentioned that the calculation has been carried out without SOC. The SOC gives rise to an avoided crossing (a spinorbit gap) between the minority Δ_{2} band and both the majority Δ_{5} and \({\mathrm{\Delta }}_2^\prime\) bands of the size equal to 60 meV and 100 meV, respectively^{25}. However, experimentally, we observe only one kink along the Δ_{2} minority band, with the separation in the doublepeak structure as large as 600 meV (shown in Fig. 2f), which is why we can rule out that SOC is responsible for the observed band anomaly. Furthermore, the surface states that could potentially interfere with the bulk electronic dispersion are not visible in our experiment (also when measured with other photon energies)^{25}. However, to unambiguously prove that the observed kink is not a result of an anticrossing with the surface state, we have analyzed the orbital character of the Fe(001) majority surface state based on relativistic DFT slab calculations (Supplementary Fig. 6). The details of this analysis can be found in the Supplementary Note 6.
The manybody scattering state observed in our experiment can be compared to the ‘plasmaron’, a bound state of an electron (or hole) and a plasmon^{43}, which can appear as satellite resonances in photoemission spectra. However, there are important differences, too. First, from a formal point of view, the plasmon propagator is a twopoint function (in space and time), while the T matrix is a fourpoint function obtained from a solution of a BetheSalpeter equation. Second, the plasmon energy is quite large (typically around 20 eV), and the plasmaron peaks therefore appear at large binding energies well separated from the quasiparticle bands. In our case, the selfenergy resonances are energetically so close to the quasiparticle bands that they strongly interact with each other, potentially leading to anomalous band dispersions like the one discussed in this work.
In summary, our combined experimental and theoretical analysis of the electronic dispersions in iron revealed the formation of a manybody spin flip scattering channel which manifests itself by a kink located at unusually high binding energy (E_{b} = 1.5 eV). This newly discovered excited state of iron is a bound state of a majority hole and a superposition of correlated electronhole pairs of opposite spins. The observed kink structure is thus of a pure electronic origin, and its prediction from first principles requires a sophisticated quantummechanical manybody treatment, in which the kdependence of the selfenergy is sufficiently taken into account.
Methods
Momentumresolved photoemission
The momentum resolved photoemission was performed using the momentum microscope at the NanoESCA beamline in Elettra synchrotron in Trieste (Italy)^{44}. The 38 ML Fe film was grown insitu on a Au(001) single crystal at low temperature (T = 140 K) using molecular beam epitaxy and gently annealed up to 300 °C. This preparation procedure was found previously to result in highquality Fe(001) films, with no Au present on the Fe surface^{25}. This was also confirmed by Xray photoelectron spectroscopy (XPS) measurements. The microscope is equipped with a W(001)based spin detector^{45}, which enables collecting constant energy spinresolved maps within the entire Brillouin zone of Fe(001). The images were obtained using photon energy of hν = 70 eV of p or s polarization. The photon beam impinges under an angle of 25 with respect to the sample surface and along the k_{x} = 0 line. According to the freeelectron final state model, such a photon energy corresponds to performing a cut through the 3D Brillouin zone close to the Γ point. An analysis of the spinresolved images was performed following the procedure described in ref. ^{46}. Before each measurement, the sample was remanently magnetized.
Ab initio calculations
The theoretical calculations were performed in the allelectron fullpotential linearized augmentedplanewave (FLAPW) formalism as implemented in the FLEUR DFT and SPEX GW code^{27}. To describe the electronmagnon interactions, an ab initio selfenergy approximation was derived from iterating the Hedin equations^{43}, resulting in a diagrammatic expansion from which we have singled out the diagrams that describe a coupling to spinflip excitations. A resummation of these ladder diagrams to all orders in the interaction yields the GT selfenergy approximation, which has a similar mathematical structure as the GW approximation as it is given by the product of the singleparticle Green function G and an effective magnon propagator T. The T matrix depends on four points in real space and its implementation involves the solution of a BetheSalpeter equation. The numerical implementation is realized using a basis set of maximally localized Wannier functions that allows an efficient truncation of the T matrix in real space. The selfenergy is calculated by the method of analytic continuation. The details of the implementation are presented in refs. ^{20,21}.
Code availability
The FLEUR code is available at http://www.flapw.de. The SPEX code (http://www.flapw.de/spex) is available from the authors upon request.
Data availability
All data generated and analyzed during the current study are available from the corresponding author on reasonable request.
References
Edwards, D. M. & Hertz, J. A. Electronmagnon interactions in itinerant ferromagnetism. II. Strong ferromagnetism. J. Phys. F. Met. Phys. 3, 2191–2205 (1973).
Capellmann, H. Ferromagnetism and strong correlations in metals. J. Phys. F. Met. Phys. 4, 1466–1476 (1974).
Tang, H., Plihal, M. & Mills, D. Theory of the spin dynamics of bulk Fe and ultrathin Fe(100) films. J. Magn. Magn. Mater. 187, 23 (1998).
Zhukov, V. P., Chulkov, E. V. & Echenique, P. M. Lifetimes of excited electrons in Fe and Ni: firstprinciples GW and the Tmatrix theory. Phys. Rev. Lett. 93, 096401 (2004).
Mannari, I. Electrical resistance of ferromagnetic metals. Prog. Theor. Phys. 22, 335–343 (1959).
Raquet, B., Viret, M., Sondergard, E., Cespedes, O. & Mamy, R. Electronmagnon scattering and magnetic resistivity in 3d ferromagnets. Phys. Rev. B 66, 024433 (2002).
Carpene, E. et al. Dynamics of electronmagnon interaction and ultrafast demagnetization in thin iron films. Phys. Rev. B 78, 174422 (2008).
Chumak, A. V., Vasyuchka, V., Serga, A. A. & Hillebrands, B. Magnon spintronics. Nat. Phys. 11, 453 (2015).
Valla, T., Fedorov, A. V., Johnson, P. D. & Hulbert, S. L. Manybody effects in angleresolved photoemission: quasiparticle energy and lifetime of a Mo(110) surface state. Phys. Rev. Lett. 83, 2085 (1999).
Higashiguchi, M. et al. Energy band and spindependent manybody interactions in ferromagnetic Ni(110): a highresolution angleresolved photoemission study. Phys. Rev. B 72, 214438 (2005).
Cui, X. Y. et al. Evaluation of the coupling parameters of manybody interactions in Fe(110). Phys. Rev. B 82, 195132 (2010).
Schäfer, J. et al. Electronic quasiparticle renormalization on the spin wave energy scale. Phys. Rev. Lett. 92, 097205 (2004).
Cui, X. Y. et al. Angleresolved photoemission spectroscopy study of Fe(110) single crystal: manybody interactions between quasiparticles at the Fermi level. Surf. Sci. 601, 4010 (2007).
Bogdanov, P. V. et al. Evidence for an energy scale for quasiparticle dispersion in Bi _{2} Sr _{2} CaCu _{2} O8. Phys. Rev. Lett. 85, 2581 (2000).
Kaminski, A. et al. Renormalization of spectral line shape and dispersion below Tc in Bi _{2} Sr _{2} CaCu _{2} O _{8+δ}. Phys. Rev. Lett. 86, 1070 (2001).
Lanzara, A. et al. Evidence for ubiquitous strong electron phonon coupling in hightemperature superconductors. Nature 412, 510 (2001).
Dahm, T. et al. Strength of the spinfluctuationmediated pairing interaction in a hightemperature superconductor. Nat. Phys. 5, 217–221 (2009).
Hofmann, A. et al. Renormalization of bulk magnetic electron states at high binding energies. Phys. Rev. Lett. 102, 187204 (2009).
Tusche, C. et al. Nonlocal electron correlations in an itinerant ferromagnet. Nat. Commun. 9, 3727 (2018).
Müller, M. C. T. D. Spinwave excitations and electronmagnon scattering in elementary ferromagnets from ab initio manybody perturbation theory. Ph.D. thesis, RWTH Aachen, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany (2017).
Müller, M. C. T. D., Blügel, S. & Friedrich, C. Electronmagnon scattering in elementary ferromagnets from first principles: lifetime broadening and kinks. Preprint at arXiv: 1809.02395 (2018).
Tusche, C., Krasyuk, A. & Kirschner, J. Spin resolved bandstructure imaging with a high resolution momentum microscope. Ultramicroscopy 159, 520–529 (2015).
Perdew, J. P. & Zunger, A. Selfinteraction correction to densityfunctional approximations for manyelectron systems. Phys. Rev. B 23, 5048–5079 (1981).
Callaway, J. & Wang, C. Energy bands in ferromagnetic iron. Phys. Rev. B. 16, 2095 (1977).
Mły n ńczak, E. et al. Fermi surface manipulation by external magnetic field demonstrated for a prototypical ferromagnet. Phys. Rev. X 6, 041048 (2016).
Hedin, L. On correlation effects in electron spectroscopies and the GW approximation. J. Phys. 11, 489 (1999).
Friedrich, C., Blügel, S. & Schindlmayr, A. Efficient implementation of the GW approximation within the allelectron FLAPW method. Phys. Rev. B 81, 125102 (2010).
Aulbur, W., Jönsson, L. & Wilkins, J. Quasiparticle calculations in solids. Vol. 54, In Solid State Physics, (eds Ehrenreich, H. & Spaepen, F.) 1–218 (Academic Press, New York, 1999).
Katsnelson, M. I. & Lichtenstein, A. I. LDA + + approach to the electronic structure of magnets: correlation effects in iron. J. Phys. Condens. Matter 11, 1037 (1999).
SánchezBarriga, J. et al. Strength of correlation effects in the electronic structure of iron. Phys. Rev. Lett. 103, 267203 (2009).
Grechnev, A. et al. Theory of bulk and surface quasiparticle spectra for Fe, Co, and Ni. Phys. Rev. B 76, 035107 (2007).
SánchezBarriga, J., Ovsyannikov, R. & Fink, J. Strong spin dependence of correlation effects in Ni due to Stoner excitations. Preprint at arXiv: 1805.09645 (2018).
Irkhin, V., Katsnelson, M. & Lichtenstein, A. Nonquasiparticle effects in halfmetallic ferromagnets. J. Phys. 19, 315201 (2007).
Massignan, P., Zaccanti, M. & Bruun, G. Polarons, dressed molecules and itinerant ferromagnetism in ultracold fermi gases. Rep. Progress. Phys. 77, 034401 (2014).
Mazzola, F. et al. Kinks in the σ band of graphene induced by electronphonon coupling. Phys. Rev. Lett. 111, 216806 (2013).
Mazzola, F. et al. Strong electronphonon coupling in the σ band of graphene. Phys. Rev. B 95, 075430 (2017).
Graf, J. et al. Universal high energy anomaly in the angleresolved photoemission spectra of high temperature superconductors: Possible evidence of spinon and holon branches. Phys. Rev. Lett. 98, 067004 (2007).
Valla, T. et al. Highenergy kink observed in the electron dispersion of hightemperature cuprate superconductors. Phys. Rev. Lett. 98, 167003 (2007).
Iwasawa, H. et al. Highenergy anomaly in the band dispersion of the ruthenate superconductor. Phys. Rev. Lett. 109, 066404 (2012).
Inosov, D. S. et al. Momentum and energy dependence of the anomalous highenergy dispersion in the electronic structure of high temperature superconductors. Phys. Rev. Lett. 99, 237002 (2007).
Rienks, E. D. L. et al. Highenergy anomaly in the angleresolved photoemission spectra of Nd _{2–x} Ce _{x} CuO _{4}: Evidence for a matrix element effect. Phys. Rev. Lett. 113, 137001 (2014).
Jung, S. W. et al. Sublattice Interference as the Origin of σ Band Kinks in Graphene. Phys. Rev. Lett. 116, 186802 (2016).
Hedin, L. New method for calculating the oneparticle Green’s function with application to the electrongas problem. Phys. Rev. 139, A796 (1965).
Schneider, C. M. et al. Expanding the view into complex material systems: from microARPES to nanoscale HAXPES. J. Electron. Spectrosc. Relat. Phenom. 185, 330 (2012).
Tusche, C. et al. Spin resolved photoelectron microscopy using a twodimensional spinpolarizing electron mirror. Appl. Phys. Lett. 99, 032505 (2011).
Tusche, C. et al. Quantitative spin polarization analysis in photoelectron emission microscopy with an imaging spin filter. Ultramicroscopy 130, 70–76 (2013).
Acknowledgements
We thank H. Ibach for valuable discussions. This work was supported by the Helmholtz Association via The Initiative and Networking Fund and by Alexander von Humboldt Foundation.
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E.M., P.G., T.H., M.G., M.J., G.Z., S.S. and V.F. performed experiments with supervision from L.P. and C.T. M.C.T.D.M. and C.F. developed the theoretical method with supervision of S.B. I.A. provided GW band structure calculations. G.B. performed the slab calculations. E.M. analyzed experimental data. E.M., M.C.T.D.M. and C.F. wrote the manuscript with contributions from all the coauthors. S.B. and C.M.S. supervised the project.
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Młyńczak, E., Müller, M.C.T.D., Gospodarič, P. et al. Kink far below the Fermi level reveals new electronmagnon scattering channel in Fe. Nat Commun 10, 505 (2019). https://doi.org/10.1038/s41467019084451
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DOI: https://doi.org/10.1038/s41467019084451
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