Formation of Hubbard-like bands as a fingerprint of strong electron-electron interactions in FeSe

We use angle-resolved photo-emission spectroscopy (ARPES) to explore the electronic structure of single crystals of FeSe over a wide range of binding energies and study the effects of strong electron-electron correlations. We provide evidence for the existence of"Hubbard-like bands"at high binding energies consisting of incoherent many-body excitations originating from Fe $3d$ states in addition to the renormalized quasiparticle bands near the Fermi level. Many high energy features of the observed ARPES data can be accounted for when incorporating effects of strong local Coulomb interactions in calculations of the spectral function via dynamical mean-field theory, including the formation of a Hubbard-like band. This shows that over the energy scale of several eV, local correlations arising from the on-site Coulomb repulsion and Hund's coupling are essential for a proper understanding of the electronic structure of FeSe and other related iron based superconductors.

Introduction.-Understanding the role of electron-electron correlations in materials exhibiting high-T c unconventional superconductivity is one of the central problems within the field of strongly correlated electron systems. Unlike the cuprates, the parent compounds of the Fe-based superconductors (e.g. LaFeAsO) are not Mott insulators but antiferromagnetic metals at low temperatures, away from half-filling. Nevertheless, local electron-electron interactions on the Fe site do play an important role, although in this case it has been shown that it is the Hund's coupling J H rather than the Coulomb repulsion U which is most important both for the magnetic ordering [1] and for the degree of band renormalization [2][3][4][5][6][7][8][9][10][11]. From an experimental point of view, clear manifestations of the effect of strong correlations in Fe-based superconductors are found in enhancements of quasiparticle effective masses deduced from specific heat [12] and quantum oscillations measurements [13], and from band renormalisations observed in Angle-Resolved Photo-Emission Spectroscopy (ARPES) [14][15][16]. These measurements indicate that the low-energy electronic structure broadly resembles that predicted by Density Functional Theory (DFT) calculations, at least, at temperatures above any magnetic or orbital orderings, but with the experimental band dispersions being renormalised by a factor typically of ∼3 [14,15], although this varies substantially between systems, and is orbital-dependent [5]. However, while general considerations of many-body theory would suggest that this band renormalisation must be accompanied by the transfer of spectral weight into incoherent excitations at higher binding energies [17], the high energy spectral weight has only rarely been experimentally investigated in Fe-based superconductors [15,18].
FeSe provides an ideal case to study the effect of strong correlations in Fe-based superconductors. The recent availability of high-quality single crystals [19,20] and thin films [21] of FeSe has led to a surge of experimental work, including recent ARPES studies with a focus on the origin of the nematic phase [20,[22][23][24][25]. ARPES [16,20], quantum oscil-lations [20,26,27] and specific heat measurements of FeSe [19] have previously reported significant orbital-dependent effective mass renormalisations. Theoretically, a significant effect of correlations in FeSe has been found in combined Density Functional Theory with Dynamical Mean Field Theory (DFT+DMFT) calculations [5,28,29], in which local Coulomb repulsion U and Hund's coupling J H on the Fe site are accounted for.
In this paper, we present systematic ARPES studies of the spectral function of FeSe to high binding energies. In addition to the renormalised quasiparticle bands near the Fermi level, we find much broader features lying in a range of 1-2.5 eV binding energy, well separated from the quasiparticle structure and the Se 4p bands at ∼3-6 eV. A "peak-dip-hump" structure on such an energy scale is usually a trademark of strong electron-electron interactions, which reduce the spectral weight of the quasiparticle peak and give rise to Hubbard bands at higher and lower binding energies [30]. Our DFT+DMFT calculations are able to reproduce many of the qualitative features of the experimental electronic structure at high binding energies, including the formation of Hubbardlike bands of incoherent spectral weight. While accounting for local electron-electron interactions within DFT+DMFT alone is not sufficient for a perfect description of the experimental Fermi surface, we show that the strong interactions are responsible for the overall form of the spectral function of FeSe over an energy scale of several eV.
Methods.-Single crystals of FeSe were grown by the vaportransport method [20]. ARPES measurements were performed at the I05 beamline at Diamond Light Source at temperatures below 10 K. ARPES measurements are a probe of the one-particle spectral function A(ω, k) [17], multiplied by the Fermi occupation function and the matrix elements for photo-emission [17], with some additional background. This spectral function is commonly expressed as: arXiv:1612.02676v1 [cond-mat.supr-con] 8 Dec 2016 k is the bare non-interacting dispersion, µ the chemical potential and Σ and Σ are the real and imaginary parts of the self-energy, which in general is orbital-, frequency-and momentum-dependent. In many materials where electronic correlations are weak and do not play a significant role, Σ is small and sharp dispersions can be observed in ARPES measurements to binding energies of several eV, usually in good agreement with the DFT dispersions. On the other hand, in FeSe, electron-electron interactions on the Fe 3d site do give a significant contribution to the self-energy [28,29], while the system remains metallic. Therefore, the observed dispersions close to the Fermi level at low temperatures can be interpreted as coherent quasiparticles with renormalised dispersions q k = b k + Σ , and a scattering rate Σ that introduces a finite lifetime for quasiparticle excitations. Depending on the form of Σ(ω, k) there may be apparent "kinks" or "waterfalls" [31] in the spectral function where the observed states transform from the renormalized quasiparticle peak close to the Fermi level into incoherent excitations at higher or lower binding energies. Generally speaking, at higher binding energy, features can become very broad and incoherent when Σ becomes large, and in particular the formation of Hubbardlike bands is possible [32,33]. While experimental evidence of Hubbard bands has been largely reported for effective oneband systems [17,34], results for multiorbital systems are scarce with only a few well-studied exceptions like transition metal oxides [35][36][37][38].
The DFT+DMFT calculations were performed within the local density approximation in DFT and using the fullpotential linear augmented plane-wave (FLAPW) basis within the WIEN2k [39] package. Calculations were done for the orthorhombic crystal structure [40], and differences in the calculation to the tetragonal crystal structures were small (Supplemental Material, SM [41]). We used the projection method onto a local basis as described in Refs. [3,42], with a window encompassing both the iron 3d and selenium 4p states. The impurity problem for the Fe 3d orbitals was solved with the strong-coupling continuous-time quantum Monte-Carlo method [43] using the ALPS package [44]. As interaction parameters we use the established values of U =4 eV, J H =0.8 eV [28,45]. We employed the fully-localized limit [46,47] for the double counting term, and the stochastic analytic continuation method for obtaining real-frequency data [48]. Calculations were performed at a temperature of β=100 eV −1 , corresponding to T =116 K.
Results.-In Fig. 1 we present high-symmetry ARPES measurements for FeSe in the M-Γ-M direction, using linear vertical (LV) polarisation. In this geometry, strong matrix elements effects dictate that the spectral weight arises overwhelmingly from a single hole-like band with d yz character [20], which simplifies the observation. Fig. 1a) focuses on the disper- Binding Energy (eV) 7 7 k (Å −1 ) sion of this d yz hole band close to E F . The quasiparticle band dispersions undergo ∼20 meV band shifts in the nematic phase [25], but these are very small perturbations on the energy scales of a few eV as considered in this paper. Due to spin-orbit coupling there is a small mixing of spectral weight onto the outer (d xz ) hole band near the Fermi level [20]. In gies. Varying the photon energy has multiple effects. Firstly, the k z of the slice of the Brillouin zone probed varies (e.g. 37 eV and 56 eV are near Γ and Z points respectively [20]) which can affect the position and orbital character of bands. Secondly ARPES matrix elements themselves have a complex photon-energy dependence. Finally, if the photon energy passes through an Fe or Se resonance this may affect the relative intensity of Fe or Se contributions to the photoemission (this gives an enhancement of the Se bands in the 56 eV spectra in Fig. 1c)). We do not attempt to disentangle all these effects which lead to the differences between spectra presented in Figs. 1 b-d), but rather point out five common features which are observed at all photon energies, as we have represented schematically in Fig. 1e,f); (i) near the Fermi level the observed band is both shifted and renormalised with respect to DFT calculations, as has been widely reported in Fe-based superconductors [49,50], although the much smaller than expected Fermi surfaces in FeSe is a unique feature. (ii) The quasiparticle band dispersions become much sharper towards the Fermi level. (iii) There is generally a dip in intensity in the range ∼0.5-1 eV in experiments, where neither quasiparticles nor incoherent excitations are found. (iv) traces of the Se 4p bands are detected in the range 3-6 eV binding energy, as predicted by DFT. Therefore, the Se 4p bands do not experience any significant renormalisation. Finally, (v) in the range of ∼1-2.5 eV we observe an anomalous broad band of intensity which cannot be attributed to either a Fe-3d quasiparticle band or a Se 4p band. The width of this spectral feature is of the order of ∼1 eV which indicates that these excitations are very short lived. We interpret this as a "Hubbardlike band", consisting of incoherent spectral weight that is a precursor of the localized electron-removal states, the lower Hubbard band, in Mott-Hubbard-insulating systems. No significant temperature-dependence was found in the high energy features up to 150 K (SM).
In Fig. 2 we present a selection of ARPES spectra obtained in different measurement conditions, which indicate that this incoherent spectral weight in the region around ∼1-2.5 eV is a general feature of FeSe, and not specific to a particular band or geometry. Next to each experimental measurement, we also show how the high energy features of FeSe seen by ARPES can be qualitatively reproduced by calculations of the spectral function in DFT+DMFT. In order to perform a comparison to ARPES data, simple selection rules are employed to simulate the photoemission matrix elements in that geometry. They are based on both symmetry considerations and the identified orbital character of the primary quasiparticle bands in the cut [25] [51]. As presented in Fig. 2a) DFT+DMFT reproduces the observed renormalised quasiparticle d yz band and some additional high energy spectral weight around 1-2.5 eV. However, the agreement is not perfect, and the renormalisation of the effective masses in  Fig. 2a,c,d). c) Orbital-resolved spectral weight. d) Comparison of Fermi surfaces determined experimentally at 100 K in the tetragonal phase and e) as calculated by DFT+DMFT projected in the kz = π plane.
DFT+DMFT (e.g. m * /m LDA = 2.09 for d xz/yz , SM) is less than the experiments (∼2-4 for d xz/yz bands, [20]), which is to be expected due to the neglect of spin-flip and pair-hopping terms [28] and dynamical screening effects [52,53]. Still, we expect that any Hubbard band-like features are not qualitatively affected by these approximations, since their binding energy is governed by the low-energy static values of the interaction, which are accounted for in the calculation. In Fig. 2b) the DFT+DMFT calculation shows a broad band of incoherent d z 2 spectral weight in good correspondence with anomalous weight found in ARPES around 1.5-2.5 eV. In Fig. 2d), DFT+DMFT finds some incoherent spectral weight in the d xy orbital around the M point, similar to the ARPES data. Finally in Fig. 2e) the d z 2 weight through the M point is reproduced very well, showing a clear formation of a Hubbard-like band. Overall there is a qualitative good agreement between calculations and experiment, as the DFT+DMFT technique correctly captures both the renormalised quasiparticle bands which sharpen approaching the Fermi level, along with the incoherent spectral weight around 1-2.5 eV.
In Fig. 3a) we compare the integrated spectral weight from our DFT+DMFT calculation with the result from DFT. Notably the Fe 3d bandwidth develops a peak-dip-hump structure which is not present in the DFT; this arises from the separation of the quasi-particle bands and the Hubbard satellite peak around 2 eV. As expected, the DMFT treatment does not strongly affect the Se 4p bands. In Fig. 3c) we show the different orbital contributions to the total spectral weight. The Hubbard band feature appears most clearly in the d z 2 and d xz/yz orbitals but can be identified in all, similar to Ref. [28]. In Fig. 3b) we compare the total calculated spectral weight with a summation of the experimental data from Fig. 2a,c,d). Similar qualitative features are found, with good agreement on the position of the Hubbard-like peak, which supports the chosen values of the interaction parameters U, J H , which are also close to values recently determined from first-principles calculations [54]. We note that in DFT+DMFT the Hubbard-like peak shifts to higher binding energies with increasing U, J H , where the Hund's coupling J H has a stronger effect on the energy of the Hubbard band than U (SM).
Finally in Fig. 3d,e) we compare the experimental Fermi surfaces of FeSe at 100 K with the calculated ones. The measured Fermi surfaces are significantly shrunk compared to the prediction of DFT+DMFT. In order to match the experimental dispersions, the real parts of the self-energies would need to be significantly momentum-dependent in order to introduce a downward-shift for hole bands at the Gamma point and an upward-shift for the electron bands at the M point [20,50]. In DMFT, the considered interactions (U, J H ) are purely local and the self-energies Σ(ω) are independent of k, albeit orbital-dependent, so that momentum-dependent shifts of the DFT bandstructure can only result from the momentumdependent orbital characters of the bands. The limitations of DFT+DMFT at the Fermi level indicate that effects not included in the calculations such as non-local inter-site interactions [55], coupling to bosonic modes [56] or frustrated magnetism [57] are likely to be relevant to the low-energy physics. However, for the wide energy scales considered in this paper our DFT+DMFT calculation is able to satisfactorily capture many of the high energy features of our ARPES spectra, including the presence of incoherent spectral weight in the form of Hubbard-like bands at high binding energies, with specific orbital-dependent agreements. Our experiments and calculations place bulk FeSe as a significantly correlated metal, with coherent quasiparticles at the Fermi level, but also exhibiting incoherent spectral weight at high binding energies, consistent with earlier photoemission studies [58].
Conclusion.-To summarise, we have provided systematic experimental evidence, backed up by theoretical DFT+DMFT calculations, for the emergence of a Fe 3d Hubbard-like band in the spectral function of FeSe, distinct from the quasiparticle states near the Fermi level. This high-energy feature is interpreted as a fingerprint of the effect of strong electronelectron correlations. Despite the strong renormalisation and shift of spectral weight into the Hubbard-like features, a well-defined quasiparticle peak at the Fermi level is retained. Therefore FeSe provides a rare opportunity to study Hubbardband physics in a significantly correlated, metallic, multiorbital system. The unique properties of FeSe continue to provide theoretical challenges, but we have demonstrated that the DFT+DMFT technique captures the essential features of the high-energy spectral function well, highlighting the importance of local Coulomb interactions and Hund's coupling for both low and high energy features in Fe-based superconductors.