Abstract
Strong manybody interactions in solids yield a host of fascinating and potentially useful physical properties. Here, from angleresolved photoemission experiments and ab initio manybody calculations, we demonstrate how a strong coupling of conduction electrons with collective plasmon excitations of their own Fermi sea leads to the formation of plasmonic polarons in the doped ferromagnetic semiconductor EuO. We observe how these exhibit a significant tunability with charge carrier doping, leading to a polaronic liquid that is qualitatively distinct from its more conventional latticedominated analogue. Our study thus suggests powerful opportunities for tailoring quantum manybody interactions in solids via dilute charge carrier doping.
Introduction
A pronounced electron−phonon coupling in solids is known to mediate the formation of polarons—composite quasiparticles of an electron dressed with a phonon cloud^{1}. Polarons exhibit significantly enhanced quasiparticle masses that modify charge carrier transport and are proposed to play a key role in unconventional superconducting and colossal magnetoresistive states in compounds including highT_{c} cuprates^{2,3}, SrTiO_{ 3 }based electron gases^{4,5,6,7,8}, manganites^{9,10} and superconducting monolayer FeSe^{11}. Developing control over polaronic states in solids therefore holds exciting potential for manipulating the collective states of quantum materials. Phonons, however, are typically only weakly modified for experimentally accessible tuning parameters. In contrast, we demonstrate in this work that polarons which are formed via a coupling to collective plasmon excitations of an electron gas provide a highly tuneable system.
We investigate this in the doped ferromagnet EuO. Stoichiometric EuO is insulating, with a Curie temperature of T_{c} = 69 K. In an ionic picture, Eu^{2+} has seven electrons halffilling the Eu 4f shell. These unpaired electrons align according to Hund’s rule, producing a large energetic splitting between occupied and unoccupied Eu 4f states via strong onsite Coulomb interactions, U (Fig. 1a). The temperaturedependent magnetisation follows an almost perfect Brillouin function^{12}, reaching a saturation magnetisation of 7 μ_{B}/Eu. EuO is thus often described as an almost ideal manifestation of a Heisenberg ferromagnet^{13}. Oxygen reduction or atomic substitution of trivalent ions, for example Gd^{3+}, for divalent Eu^{2+} dopes electrons into an Euderived 5d conduction band. This has a dramatic effect on its physical properties, increasing T_{c}^{14}, stabilising a giant temperaturedependent metal−insulator transition with up to 13 orders of magnitude change in conductivity^{15}, inducing giant magnetoresistance^{16}, and realising a halfmetallic phase, enabling almost 100% spin injection into Si and GaN^{17}.
We synthesise epitaxial films of Eu_{1−x}Gd_{ x }O via oxide molecularbeam epitaxy (MBE), allowing fine control over its charge carrier concentration, and transfer these in situ to a synchrotronbased angleresolved photoemission spectroscopy (ARPES) system (see Methods). This provides a powerful opportunity to probe the electronic structure of this threedimensional and airsensitive compound. Our corresponding ARPES measurements, together with ab initio manybody calculations^{18,19}, reveal how charge carrier doping fundamentally changes the nature of the underlying electronic liquid in EuO. In particular, we directly image the emergence of welldefined satellites in the spectral function whose energy separation, unusually, grows as \(\sqrt n\), where n is the free carrier density. This points to an intriguing polaronic state, arising due to the coupling of the induced charge carriers to the conductionelectron plasmon excitations.
Results
Electronic structure of lightly doped EuO
Figure 1 summarises the electronic structure of our Gddoped EuO films. After incorporating an onsite Coulomb repulsion, densityfunctional theory calculations (DFT, see Methods) reproduce the generic features of the electronic structure described above, including its ferromagnetic nature. A majorityspin Eu 4f level is fully occupied, sitting above an O 2pderived valence band that inherits a much weaker exchange splitting. Our ARPES measurements of the valence band dispersions from a lightly doped sample (x = 0.023, Fig. 1c) are in good general agreement with these DFT calculations, as well as with prior ARPES studies of the valence band electronic structure of EuO^{20,21}. For a weakly interacting semiconductor, charge carrier doping would simply induce a rigid shift of the Fermi level into the majorityspin Eu 5d conduction band, whose minimum is located at the Xpoint of the Brillouin zone (see also Supplementary Fig. 1). Consistent with previous experiment^{21}, we indeed find that conductionband states become populated at X with electron doping (Fig. 1d). Our measured spectra (evident in Fig. 1d and shown magnified in Fig. 2a), however, are not consistent with a simple rigidband filling.
Instead, we observe a series of replica bands offset in energy from the main quasiparticle band which intersects the Fermi level. This is a characteristic spectroscopic signature of Fröhlich polaron formation^{22,23}, whereby strong electron−phonon interactions give rise to shakeoff excitations involving small q scattering processes. These yield satellite features in the spectral function, shifted to successively higher binding energy and evenly spaced by the corresponding phonon mode frequency. From our measured energydistribution curves (EDCs, Fig. 2c), we observe three such distinct replica bands which, from fitted peak positions, are separated by a constant value of \(\Delta E \approx (56\,\pm\, 3)\,{\mathrm {meV}}\). This agrees well with the longitudinal optical phonon mode frequency measured from EuO single crystals^{24} as well as with the optical phonon branch obtained from our ab initio calculations (Supplementary Fig. 2). We thus attribute these observed spectral features in very lightly doped EuO as polaronic satellites arising from a strong electron−phonon coupling.
To confirm that this is an intrinsic property of the spectral function, we perform manybody ab initio calculations within the cumulant expansion method^{25,26}, thereby including the effects of electron−phonon coupling from first principles (see Methods). Apart from a small overall energetic shift, our calculations, performed for the same carrier doping as in our experiments, yield a spectral function in excellent agreement with the one measured by ARPES (Fig. 2c, d), including the spacing and approximate spectral weights of the replica features. Indeed, this level of agreement is remarkable given that the calculations are performed fully ab initio and there are no tuning parameters employed. They reveal a pronounced quasiparticle mass renormalisation, m^{*}/m_{0} = 2.1, where m_{0} is the bare band mass, pointing to a strong electron−phonon coupling, and supporting that dilutely doped EuO is in the polaronic limit. We note that similar spectral features and electron−phonon coupling strengths have been observed recently in other lightly doped oxides including TiO_{2}, Sr_{2−x}La_{ x }TiO_{4}, as well as ZnO and SrTiO_{3}based twodimensional electron gases^{4,5,6,8,27,28,29}. Their observation here, within the markedly different system of the bulkdoped threedimensional and spinpolarised electron pocket of EuO, suggests that polaron formation is likely universal to lightly doped polar oxides.
Dopingdependent spectral function
We show in Fig. 3 how the spectral function evolves with increasing carrier doping. By increasing the density of the Gd^{3+} dopants, the band filling can be controllably increased, as evidenced by the increased quasiparticle bandwidth as well as the larger Fermi surface volume, shown inset in Fig. 3a–d. With even a small increase in carrier density, however, the pronounced multipeak satellite structure observed in the lowestdoped sample is no longer apparent. This points to a rapid reduction in the electron−phonon coupling strength, a phenomenon which we return to below. Nonetheless, a broadened satellite peak is still observed below the quasiparticle band (Fig. 3b, c), evident as a hump in EDCs (Fig. 3e) which persists over at least two orders of magnitude increase in carrier density. To demonstrate this more clearly, we show in Fig. 4a the residual of the measured EDC intensity after subtraction of a background function accounting for the quasiparticle peak intensity (see also Supplementary Fig. 3).
The satellite peak broadens with increasing charge carrier doping, but remains clearly resolved up to a carrier density n ≈ 10^{20} cm^{−3}. At the same time, the satellite exhibits a pronounced shift to higher binding energy with increasing doping. The shift is much faster than the increase in filling of the conduction band. Indeed, from fits to the measured data, we find that the separation of this hump feature from the band bottom of the quasiparticle band grows with a \(\sqrt n\) dependence, where n is the threedimensional electron density (Fig. 4b). This indicates electron−boson coupling to a mode which hardens with increasing carrier density. This is in striking contrast to the expectations for a phonon mode, which should be nearly carrier density independent. Instead, it agrees well with the functional form of the mode energy expected for a plasmon (red line in Fig. 4b).
The satellite features we observe here for our higherdensity samples therefore point to the formation of plasmonic polarons, where the conduction electrons become dressed by chargedensity fluctuations of their own electron gas^{30}. This interpretation is confirmed by our ab initio calculations, where we are able to treat electron−phonon and electron−plasmon coupling on an equal footing. Our obtained spectral functions (Fig. 3f–i) reproduce the general trends observed experimentally, also yielding plasmonic polaron satellites shifted below the quasiparticle band by the conduction electron plasmon energy. Given that EuO is a halfmetal for the levels of doping investigated here, these plasmon−polarons must necessarily also be spinpolarised. Indeed, the spinpolarised conduction band of EuO has led to significant interest in using this material for spininjection in spintronics applications^{17}. The polaronic nature of the spinpolarised charge carriers in EuO, and consequent limited intrinsic carrier mobilities that would be expected, should be carefully considered for such applications. More generally, the excellent agreement that we find between our experimental and ab initio spectral functions for a real, complex, multiorbital and magnetic system such as EuO suggests opportunities to exploit such advanced calculation schemes for not only understanding, but increasingly predicting, the interacting electronic states and properties of functional materials.
Tuneable plasmon polarons
We focus below on the origin, and unique properties, of the plasmon−polarons discovered here. For a threedimensional electron gas, the plasmon dispersion, ω(q), remains gapped in the longwavelength limit (ω(q→0) = ω_{p}, where ω_{p} is the plasma frequency), while the electron−plasmon coupling strength goes as 1/q. Our direct observation of satellite structures spaced by the plasma energy here indicates that this is sufficient to generate welldefined replica bands, similar to those generated by the Fröhlich electron−phonon interaction. We note that this is different to the occurrence of sharp plasmonmediated features in the spectral function of graphene, which rely upon pseudospin conservation and phasespace restrictions from matching the group velocity of plasmon and band dispersions^{31} which would not be expected in the current system. Instead, the plasmon–polarons observed here can be expected as a generic feature in the low to mediumdoping limit of a doped threedimensional semiconductor. Moreover, we note that the polarons observed here have markedly different characteristics to plasmonic polaron band structures that have been predicted to occur via excitation of highenergy valence plasmons^{30,32,33}, with experimental signatures recently observed in silicon^{33} and graphite^{34}. In such systems, the plasmon energy scales are ~3 orders of magnitude larger than the other excitations in the system. In contrast, the plasmons considered here have comparable energy to the Fermi energy, and so can be expected to have a much more dramatic influence on the lowenergy properties of the system, such as enhancing the quasiparticle mass and limiting charge carrier mobilities.
Moreover, the conductionelectron plasmons here are highly tuneable via charge carrier doping, with a characteristic mode energy that can be driven into resonance with, for example, phonon modes of the system as shown in Fig. 4. When close in energy, the two bosonic modes will in general couple to each other, leading to hybrid phonon−plasmon polaritons. Such a mode coupling is not explicitly considered in our calculations, although would be consistent with our experimentally determined satellite structure (Supplementary Fig. 4). Our study thus motivates the development of theoretical approaches and targeted experiments to investigate the polaronic signatures that might be expected in this intriguing regime.
Even without considering such mode hybridisation, our ab initio calculations shown in Fig. 4c already reveal a rich dopingdependent interplay of the coupling strength of charge carriers to different bosonic modes in the system. For the lowest carrier density investigated, a large electron−phonon coupling of λ_{e−ph} > 1 is obtained, which is the dominant coupling in the system. Given this, and that the plasma frequency is very small for this level of charge carrier doping, the multipeak satellite structure observed experimentally thus predominantly arises due to electron−phonon interactions (cf. Figs. 2b and 3f). In this regime (plasma energy much smaller than the phonon mode energy), the system hosts Fröhlich polarons^{35}.
With increasing carrier density, the plasma frequency becomes larger than the phonon frequency (Fig. 4b). The electron−phonon interaction therefore becomes efficiently screened^{23}, and so the electron−phonon coupling strength shows a rapid dropoff with increasing charge carrier doping. Nonetheless, a satellite peak remains visible in both our calculations (Fig. 3g) and in fits to our measured experimental data (Fig. 4b) until the Fermi energy becomes comparable to the phonon mode frequency (approximately at the dashed line in Fig. 4b). Beyond this point, the system moves into a Fermi liquid regime^{23}, and the electron−phonon interaction instead leads to a more conventional kink in the band dispersion near to the Fermi level. Weak signatures of this are visible in our experimental data (Fig. 3c, d), although they are somewhat obscured by broadening due to a poor k_{ z } resolution resulting from the inherent surface sensitivity of photoemission.
Despite these qualitative changes in the nature of the electron−phonon interactions, the electron−plasmon coupling strength evolves more smoothly with increasing charge carrier density (Fig. 4c). It thus plays a more dominant role at somewhat higher carrier densities, where the electron−phonon interaction is more efficiently screened. It still, however, displays a pronounced dependence on charge carrier doping in the system. To investigate this over a wider carrier density range, we show in Fig. 5a the effective electron−plasmon coupling constant, α, and the plasmonic polaron radius derived from the selfenergy for a homogeneous electron gas with the same effective mass and dielectric permittivity as EuO (see Methods). The increase of α with decreasing carrier density suggests that a strong coupling regime between electrons and plasmons may be approached at low doping concentrations. In practice, the critical doping density which marks the onset of a Mott metal−insulator transition, poses a strict limit to the highest coupling that will be achievable in practice, since below this value the system becomes insulating and plasmons cannot be excited. The critical density in EuO, for example, is ~10^{17} cm^{−3} and it is marked by the vertical dashed line in Fig. 5a. At this doping, we find α ≃ 2.9 and a polaron radius of 53 Å. These values and their dependence on carrier concentrations are compatible with the results obtained from our firstprinciples calculations at the experimental doping densities (marked by dotted lines in Fig. 5a) reported in Supplementary Table 1.
Interestingly, we show in Fig. 5a that the polaron radius decreases with increasing carrier density. In fact, over the doping range considered in our experiments, the plasmonic polaron radius approximately doubles, as illustrated in Fig. 5b, c. This is in striking contrast to the expected behaviour known from phononic polarons, where the polaron radius decreases with increasing coupling strength (i.e., increases with increasing carrier density)^{36}. This unconventional behaviour results from the strong dependence of the plasmon energy on carrier concentration, and may lead to unconventional trends in dopingdependent mobilities. Moreover, this further points to the highly tuneable nature of plasmonic polaron states, whereby a broad spectrum of electron−boson coupling regimes can be explored by tuning the carrier concentrations.
Discussion
We stress that our findings should not be specific to EuO, but rather a general feature of band insulators where conductivity can be induced by dilute charge carrier doping. They suggest substantial opportunity to engineer the relative importance of different bosonic modes, and may allow triggering or controlling instabilities of the collective system via electron−plasmon as well as electron−phonon interactions. Indeed, superconductivity in SrTiO_{3}/LaAlO_{3} interface 2D electron gases has recently been argued to emerge from a phonon polaron liquid, with its superconducting dome linked to a dopingdependent strength of electron−phonon coupling^{5}. A similar superconducting instability could generically be expected to occur for the plasmon−polarons introduced here.
As shown above, the coupling strength for both phonon and plasmon polarons decreases with increasing doping. In the phononic case, the mode energy is fixed, and so this decrease in coupling strength must lead to a decrease in superconducting transition temperature as the doping is increased. In contrast, for the plasmonic case, while the coupling strength still decreases with increasing carrier doping, the influence on T_{c} should be partially offset by a hardening of the relevant mode energy. This could even lead to a dopingdependent crossover from phonon to plasmonmediated superconductivity with increasing charge carrier doping. While such considerations would not be relevant for EuO as studied here, due to its ferromagnetic nature, our results point to the intriguing possibility to stabilise unusual dopingdependent superconducting instabilities in, for example, lightly doped oxide semiconductors. Furthermore, they highlight the complexity of chargecarrier doping in oxides even in the absence of strong electronic correlations, opening routes to the targeted design of their materials properties.
Methods
Molecularbeam epitaxy
The EuO thin films were grown by MBE utilising a Createc miniMBE system^{37} installed on the I05 beamline at Diamond Light Source, UK. The films were grown on YAlO_{3} substrates in an absorptioncontrolled, or distillation^{38}, growth mode at a temperature of 425 °C, using an Eu partial pressure of p_{Eu} ≈ 2.3 × 10^{−7} mbar and a molecular oxygen partial pressure of \(p_{{\mathrm{O}}_{\mathrm{2}}} \approx 2.0 \times 10^{  8}\,{\mathrm {mbar}}\), as measured by a beam flux monitor. Gd dopants were introduced by exposing the films to a Gd partial pressure of p_{Gd} ≈ 6.3 × 10^{−9} mbar during growth, and shuttering the Gd source (for four equal length periods within each monolayer of EuO growth) to further reduce the incorporated Gd concentration. The films were monitored in situ using reflection high energy electron diffraction (see Supplementary Fig. 5), from which the inverse growth rate was determined to be ≈120 s per monolayer. The total thickness of the grown films is ≈20 nm, thick enough to ensure that their electronic structure is bulklike^{39}. Following growth, the films were transferred under ultrahigh vacuum to the HRARPES endstation (see below). After the ARPES measurements, they were further characterised by lowenergy electron diffraction (see Supplementary Fig. 5) and xray photoelectron spectroscopy (Supplementary Fig. 6), which indicated their high crystalline and chemical quality. We note that our undoped samples are highly insulating, pointing to negligible oxygen vacancy concentrations, while our xray photoelectron spectroscopy (XPS) measurements indicate an Eu^{2}+ charge state, indicative of the growth of stoichiometric EuO. The samples were then capped with 5–15 nm of amorphous silicon (p_{Si} ≈ 2.5×10^{−8} mbar), allowing these airsensitive samples to be removed from the ultrahigh vacuum environment. A subset of the films were then further probed by superconducting quantum interference device magnetometry to probe their magnetic properties and xray absorption spectroscopy to assess the Gd doping (Supplementary Fig. 7). These measurements confirmed material and magnetic properties of our grown EuO films that are in good agreement with previous studies of this compound. We also show in Supplementary Fig. 8 temperaturedependent ARPES measurements of a moderate carrier density sample. The majority band (occupied at low temperature) can be seen moving up through the Fermi level upon increasing through the Curie temperature, driving a temperaturedependent metal−insulator transition as a result of the loss of exchange splitting. This is fully consistent with the expected presence of spinpolarised exchangesplit bands at low temperature, entirely in line with our spinpolarised DFT calculations (Fig. 1b).
Angleresolved photoemission
In situ ARPES was performed using the HighResolution ARPES instrument (HRARPES) of Diamond Light Source, UK. Measurements were performed at temperatures of ≈20 K or below using ppolarised synchrotron light. A Scienta R4000 hemispherical electron analyser was used, with a vertical entrance slit and the light incident in the horizontal plane. Photon energies of 48 and 137 eV was used. For an inner potential of 15 eV, these correspond to measured dispersions which cut centrally through the conduction band Fermi surface of EuO along k_{ z }, with an inplane dispersion along the short axis of the elliptical Fermi pocket as shown in Supplementary Fig. 1. To determine the carrier density of the doped films, we extracted the Luttinger volume of their measured Fermi surfaces. This is complicated by the threedimensional nature of these Fermi pockets, and the inherently poor k_{ z } resolution in ARPES arising from its surface sensitivity. We therefore simulated the measured Fermi surface including the effects of k_{ z } broadening, and compared this to our experimental data to determine the correct carrier density (Supplementary Fig. 9). This carrier density enters into the fit for the plasma frequency of a threedimensional electron gas, \(\omega _p = \sqrt {ne^2{\mathrm{/}}\varepsilon _0\varepsilon _\infty m^ \ast },\) where ε_{0} is the dielectric permittivity of free space, and ε_{∞} is the dielectric constant of EuO (4.5 ^{24}). m^{*} is the effective mass, which is treated as a fit parameter in our analysis (Fig. 4b), from which we find a value of m^{*} = 0.2 ± 0.1 m_{e} within the range of previous estimates of the effective mass of EuO^{40,41,42}.
Firstprinciples calculations
Densityfunctional theory calculations including Hubbard corrections (DFT + U)^{43} for the lowtemperature ferromagnetic phase of EuO were performed using Quantum ESPRESSO^{44}. We employed the Perdew, Burke and Ernzerhof (PBE)^{45} exchangecorrelation functional, an effective onsite Coulomb parameter U_{ f } = 6 eV for the Eu 4f states, and U_{ p } = 3 eV for the O 2p states. We used normconserving pseudopotentials, a plane wave kinetic energy cutoff of 150 Ry, and a 8 × 8 × 8 uniform kpoint mesh to sample the Brillouin zone. Maximally localised Wannier functions were constructed starting from a 4 × 4 × 4 uniform k grid. The effect of electron doping was included in the rigidband approximation. The lattice vibrational properties were calculated using the projector augmented wave (PAW) method^{46}, and effective Coulomb parameters U_{ f } = 8.3 eV and U_{ p } = 4.6 eV which yield the same band gap calculated with normconserving pseudopotentials. Convergence was ensured by using a kinetic energy cutoff of 70 Ry. The phonon dispersions were obtained by finite differences in a 6 × 6 × 6 supercell, using atomic displacements of 0.01 Å. The longitudinal opticaltransverse optical (LOTO) splitting was accounted for as in ref. ^{47}, using the calculated Born effective charges from Ref. ^{48}.
The firstprinciples spectral functions were obtained from the cumulant expansion method^{23,49,50} using the electron−phonon and electron−plasmon selfenergy as implemented in the EPW code^{51,52,53} as a seed:
Here, η is a positive infinitesimal, f_{mk + q} and n_{ q ν } are Fermi−Dirac and Bose−Einstein occupations, respectively, ε_{mk + q} is the electron energy, and \(\hbar \omega _{{\bf{q}}\nu }\) is the energy of a plasmon/phonon with wavevector q. The coupling matrix elements due to electron−plasmon and electron−phonon coupling were computed as in ref. ^{52} and ref. ^{54}, respectively. For the electron−phonon coupling, dynamical screening arising from the added carriers in the conduction band was taken into account by using nonadiabatic matrix elements^{23}: \(g_{mnv}^{{\mathrm{NA}}}\left( {{\bf{k}}{\mathrm{,}}{\bf{q}}} \right)\,=\,g_{mnv}\left( {{\bf{k}},{\bf{q}}} \right){\mathrm{/}} \varepsilon \left( {{\bf{q}}{\mathrm{,}}\omega _{{\bf{q}}v} + i{\mathrm{/}}\tau _{n{\bf{k}}}} \right)\). Here ε(q,ω) is the Lindhard dielectric function for a spinpolarised homogeneous electron gas with effective mass m^{*} and dielectric permittivity ε_{∞} of EuO, and \(\hbar /\tau _{n{\bf{k}}}\) is the electron lifetime near the band edge, taken to be 50 meV. Finite resolution effects were accounted for by applying two Gaussian masks of widths 20 meV and 0.015 Å^{−1}, and by integrating the spectral function along the outofplane direction k_{ z }. The temperature broadening at the Fermi level was included via a Fermi–Dirac distribution at T = 20 K. The electron−plasmon and electron−phonon coupling strengths λ were extracted from the selfenergy via \(\lambda\,=\, \hbar ^{  1}\left. {\partial {\mathrm{R}}{\mathrm e}{\kern 1pt} \Sigma ({\bf{k}}_{\mathrm{F}},\omega ){\mathrm{/}}\partial \omega } \right_{\varepsilon _{\mathrm{F}}}\)^{19}. The effective electron−plasmon coupling constants α were obtained from the mass renormalisation 1 + λ_{e−pl}^{22}, whereas the plasmonic polaron radius was estimated following ref. ^{55}: \(r_p\,\simeq\,\left( {3{\mathrm{/}}0.44\alpha } \right)^{\frac{1}{2}}(2m\omega _{{\mathrm{pl}}}{\mathrm{/}}\hbar )^{  \frac{1}{2}}\). The polaron wavefunction was calculated as the product of the latticeperiodic component of the Kohn−Sham eigenstate at the conductionband bottom and a Gaussian with isotropic width σ corresponding to the polaron radius.
Code availability
The calculations were performed using the opensource software projects Quantum ESPRESSO, EPW, and Wannier90, which can be downloaded free of charge from www.quantumespresso.org, epw.org.uk, and www.wannier.org, respectively. Input files and calculation workflows can be downloaded from the GitHub repository https://github.com/mmdgoxford/papers.
Data availability
The data that underpins the findings of this study are available at https://doi.org/10.17630/4e82a73157c64cf5b8c2841486b8dbde.
References
Landau, L. Spin exchange in excitons, the Quasicubic model and deformation potentials in IIVI compounds. Phys. Z. Sowjetunion 3, 664 (1933).
Shen, K. M. et al. Missing quasiparticles and the chemical potential puzzle in the doping evolution of the cuprate superconductors. Phys. Rev. Lett. 93, 267002 (2005).
Rösch, O. et al. Polaronic behavior of undoped highT_{ c } cuprate superconductors from angleresolved photoemission spectra. Phys. Rev. Lett. 95, 227002 (2005).
Chen, C. et al. Observation of a twodimensional liquid of Frohlich polarons at the bare SrTiO_{3} surface. Nat. Commun. 6, 8585 (2015).
Wang, Z. et al. Tailoring the nature and strength of electron−phonon interactions in the SrTiO_{3} (001) 2D electron liquid. Nat. Mat. 15, 835 (2016).
Cancellieri, C. et al. Polaronic metal state at the LaAlO_{3}/SrTiO_{3} interface. Nat. Commun. 7, 10386 (2016).
van Mechelen, J. L. M. et al. Electron−phonon interaction and charge carrier mass enhancement in SrTiO_{3}. Phys. Rev. Lett. 100, 226403 (2008).
Swartz, A. G. et al. Strong polaronic behavior in a weak coupling superconductor. Proc. Natl. Acad. Sci. USA 115, 1475 (2018).
Adams, C. P. et al. Charge ordering and polaron formation in the magnetoresistive oxide La_{0}.7Ca_{0}.3MnO_{3}. Phys. Rev. Lett. 85, 3954 (2000).
Mannella, N. et al. Nodal quasiparticle in pseudogapped colossal magnetoresistive manganites. Nature 438, 474 (2005).
Lee, J. J. et al. Interfacial mode coupling as the origin of the enhancement of T_{ c } in FeSe films on SrTiO_{3}. Nature 515, 245 (2014).
Matthias, B. T., Bozorth, R. M. & Van Vleck, J. H. Ferromagnetic interaction in EuO. Phys. Rev. Lett. 7, 160 (1961).
Blundell, S. J. et al. Phase transition in the localized ferromagnet EuO probed by μSR. Phys. Rev. B 81, 092407 (2010).
Shafer, M. W. & McGuire, T. R. Studies of Curiepoint increases in EuO. J. Appl. Phys. 39, 588 (1968).
Torrance, T. B., Shafer, M. W. & McGuire, T. R. Bound magnetic polarons and the insulatormetal transition in EuO. Phys. Rev. Lett. 29, 1168 (1972).
Shapira, Y., Foner, S. & Reed, T. B. EuO. I. Resistivity and Hall effect in fields up to 150 kOe. Phys. Rev. B 8, 2299 (1973).
Schmehl, A. et al. Epitaxial integration of the highly spinpolarized ferromagnetic semiconductor EuO with silicon and GaN. Nat. Mat. 6, 882 (2007).
Giustino, F., Cohen, M. L. & Louie, S. G. Electron−phonon interaction using Wannier functions. Phys. Rev. B 76, 165108 (2007).
Giustino, F. Electron−phonon interactions from first principles. Rev. Mod. Phys. 89, 015003 (2017).
Miyzaki, H. et al. Threedimensional angleresolved photoemission spectra of EuO thin film. J. Elec. Spc. Rel. Ph. 191, 7 (2013).
Shai, D. E. et al. Temperature dependence of the electronic structure and Fermisurface reconstruction of Eu_{1−x}Gd_{ x }O through the ferromagnetic metalinsulator transition. Phys. Rev. Lett. 108, 267003 (2012).
Devreese, J. & Alexandrov, S. Fröhlich polaron and bipolaron: recent developments. Rep. Prog. Phys. 72, 066501 (2009).
Verdi, C., Caruso, F. & Giustino, F. Origin of the crossover from polarons to Fermi liquids in transition metal oxides. Nat. Commun. 8, 15769 (2017).
Güntherodt, G. Optical properties and electronic structure of Europium chalcogenides. Phys. Condens. Matter 18, 37 (1974).
Langreth, D. C. Singularities in the Xray spectra of metals. Phys. Rev. B 1, 471 (1970).
Caruso, F. & Giustino, F. The GW plus cumulant method and plasmonic polarons: application to the homogeneous electron gas. Eur. Phys. J. B 89, 238 (2016).
Moser, S. et al. Tunable polaronic conduction in anatase TiO_{2}. Phys. Rev. Lett. 110, 196403 (2013).
Yukawa, R. et al. Phonondressed twodimensional carriers on the ZnO surface. Phys. Rev. B 94, 165313 (2016).
Nie, Y. F. et al. Formation and observation of a quasitwodimensional d _{ xy } electron liquid in epitaxially stabilized Sr_{2−x}La_{ x }Tio_{4} thin films. Phys. Rev. Lett. 115, 096405 (2015).
Caruso, F., Lambert, H. & Giustino, F. Band structures of plasmonic polarons. Phys. Rev. Lett. 114, 146404 (2015).
Bostwick, A. et al. Observation of plasmarons in quasifreestanding doped graphene. Science 328, 999 (2010).
Guzzo, M. et al. Valence electron photoemission spectrum of semiconductors: ab initio description of multiple satellites. Phys. Rev. Lett. 107, 166401 (2011).
Lischner, J. et al. Satellite band structure in silicon caused by electron−plasmon coupling. Phys. Rev. B 91, 205113 (2015).
Guzzo, M. et al. Multiple satellites in materials with complex plasmon spectra: from graphite to graphene. Phys. Rev. B 89, 085425 (2014).
Alexandrov, A. S. & Yavidov, B. Y. Small adiabatic polaron with a longrange electronphonon interaction. Phys. Rev. B 69, 073101 (2004).
Mahan, G. D. ManyParticle Physics 3rd edn (Kluwer Academic, Dordrecht, The Netherlands, 2000).
Baker, A. A. et al. An ultracompact, highthroughput molecular beam epitaxy growth system. Rev. Sci. Instr. 86, 043901 (2015).
Sutarto, R. et al. Epitaxy, stoichiometry, and magnetic properties of Gddoped EuO films on YSZ (001). Phys. Rev. B 80, 085308 (2009).
King, P. D. C. et al. Atomicscale control of competing electronic phases in ultrathin LaNiO_{3}. Nat. Nano 9, 443 (2014).
Schoenes, J. & Wachter, P. Exchange optics in Gddoped EuO. Phys. Rev. B 9, 3097 (1974).
Bebenin, N. On magnetic red shift of absorption edge in EuO. Solid State Comms 55, 823 (1985).
Patil, C. G. & Krishnamurthy, B. S. Magnetic field dependent carrier mobility in ferromagnetic semiconductors. Phys. Status Solidi 105, 391 (1981).
Cococcioni, M. & de Gironcoli, S. Linear response approach to the calculation of the effective interaction parameters in the LDA+U method. Phys. Rev. B 71, 035105 (2005).
Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and opensource software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953 (1994).
Wang, Y. et al. Mixedspace approach for calculation of vibrationinduced dipoledipole interactions. Phys. Rev. B 85, 224303 (2012).
Pradip, R. et al. Lattice dynamics of EuO: evidence for giant spinphonon coupling. Phys. Rev. Lett. 116, 185501 (2016).
Aryasetiawan, F., Hedin, L. & Karlsson, K. Multiple plasmon satellites in Na and Al spectral functions from ab initio cumulant expansion. Phys. Rev. Lett. 77, 2268 (1996).
Gumhalter, B. et al. On the combined use of GW approximation and cumulant expansion in the calculations of quasiparticle spectra: the paradigm of Si valence bands. Phys. Rev. B 94, 035103 (2016).
Giustino, F., Cohen, M. L. & Louie, S. G. Electron−phonon interaction using Wannier functions. Phys. Rev. B 76, 165108 (2007).
Caruso, F. & Giustino, F. Theory of electron−plasmon coupling in semiconductors. Phys. Rev. B 94, 115208 (2016).
Poncé, S. et al. EPW: electron−phonon coupling, transport and superconducting properties using maximally localized Wannier functions. Comput. Phys. Commun. 209, 116 (2016).
Verdi, C. & Giustino, F. Fröhlich electron−phonon vertex from first principles. Phys. Rev. Lett. 115, 176401 (2015).
Schultz, T. D. Slow electrons in polar crystals: selfenergy, mass, and mobility. Phys. Rev. 116, 526 (1959).
Acknowledgements
We are indebted to Rainer Held, Darrell Schlom and Kyle Shen for sharing their expertise on the growth of EuO with us, and for useful discussions. We further gratefully acknowledge support from the Royal Society, the Leverhulme Trust (Grants PLP2015144 and RL2012001), the Graphene Flagship (Horizon 2020 Grant No. 696656—GrapheneCore1), and the EPSRC (Grant No. EP/M020517/1). The calculations were performed using the ARCHER UK National Supercomputing Service via the AMSEC Leadership project, and the Advanced Research Computing facility of the University of Oxford (http://dx.doi.org/10.5281/zenodo.22558). C.V. is grateful to Przemek Piekarz for sharing his results on the Born effective charges. J.M.R. and L.B. acknowledge EPSRC for PhD studentship support through grant Nos. EP/L505079/1 and EP/G03673X/1. L.B.D. acknowledges studentship support from EPSRC and the Science and Technology Facilities Council (UK). We thank Diamond Light Source for access to Beamlines I05 and I10 via Proposal Nos. NT15481, SI13539 and SI16162 that contributed to the results presented here.
Author information
Authors and Affiliations
Contributions
Sample growth and characterisation was performed by J.M.R., L.B.D. and T.H. ARPES measurements were performed by J.M.R., M.D.W., L.B., K.V., M.H. and P.D.C.K. and the data were analysed by J.M.R. The calculations were performed by F.C., C.V. and F.G. XMCD measurements were performed by L.B.D., G.v.d.L. and T.H. P.D.C.K. and M.H. were responsible for overall project planning and direction. P.D.C.K., J.M.R., F.C., C.V. and F.G. wrote the paper with inputs and discussion from all coauthors.
Corresponding authors
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.
Electronic supplementary material
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/.
About this article
Cite this article
Riley, J.M., Caruso, F., Verdi, C. et al. Crossover from lattice to plasmonic polarons of a spinpolarised electron gas in ferromagnetic EuO. Nat Commun 9, 2305 (2018). https://doi.org/10.1038/s4146701804749w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4146701804749w
Further reading

Tuneable electron–magnon coupling of ferromagnetic surface states in PdCoO2
npj Quantum Materials (2022)

Bandselective Holstein polaron in Luttinger liquid material A0.3MoO3 (A = K, Rb)
Nature Communications (2021)
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.