Revealing the Coulomb interaction strength in a cuprate superconductor

Revealing the Coulomb interaction strength in a cuprate superconductor

S.-L. Yang [ Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA Geballe Laboratory for Advanced Materials, Departments of Physics and Applied Physics, Stanford University, Stanford, CA 94305, USA    J. A. Sobota Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA    Y. He    Y. Wang    D. Leuenberger Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA Geballe Laboratory for Advanced Materials, Departments of Physics and Applied Physics, Stanford University, Stanford, CA 94305, USA    H. Soifer Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA    M. Hashimoto    D. H. Lu Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA    H. Eisaki Electronics and Photonics Research Institute, National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8558, Japan    B. Moritz    T. P. Devereaux    P. S. Kirchmann Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA    Z.-X. Shen Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA Geballe Laboratory for Advanced Materials, Departments of Physics and Applied Physics, Stanford University, Stanford, CA 94305, USA
July 11, 2019

We study optimally doped BiSrCaYCuO (Bi2212) using angle-resolved two-photon photoemission spectroscopy. Three spectral features are resolved near , , and  eV above the Fermi level. By tuning the photon energy, we determine that the  eV feature arises predominantly from unoccupied states. The and  eV features reflect unoccupied states whose spectral intensities are strongly modulated by the corresponding occupied states. These unoccupied states are consistent with the prediction from a cluster perturbation theory based on the single-band Hubbard model. Through this comparison, a Coulomb interaction strength of  eV is extracted. Our study complements equilibrium photoemission spectroscopy and provides a direct spectroscopic measurement of the unoccupied states in cuprates. The determined Coulomb indicates that the charge-transfer gap of optimally doped Bi2212 is  eV.

74.72.-h, 78.47.J-, 71.27.+a

current affiliations: ]Kavli Institute at Cornell for Nanoscale Science; Laboratory of Atomic and Solid State Physics, Department of Physics; Department of Materials Science and Engineering. Cornell University, Ithaca, New York 14853, USA

I Introduction

Governed by Fermi-Dirac statistics, electronic states above the Fermi level are unoccupied at zero temperature Ashcroft and Mermin (1976). Studies of unoccupied states yield critical information about topological properties Sobota et al. (2013) and symmetry-breaking orders Hashimoto et al. (2010); Yang et al. (2008). In particular, knowledge of unoccupied states is essential for determining the symmetry of a spectral gap, which encodes the origin of the corresponding order Hashimoto et al. (2010); Yang et al. (2008). For cuprate superconductors which host a complex interplay of competing orders Keimer et al. (2015), the ability to resolve unoccupied electronic states is particularly important.

A Mott insulating phase is a manifestation of strong correlation physics Imada et al. (1998). Due to Coulomb repulsions, half-filled electronic states are localized resulting in an insulating phase Imada et al. (1998). The hallmark of the Mott physics is the formation of lower Hubbard band (LHB) and upper Hubbard band (UHB), separated by the Coulomb interaction strength . As the UHB is above and unoccupied, an energy- and momentum-resolved characterization of UHB in cuprates has remained challenging.

Angle-resolved photoemission spectroscopy (ARPES) enables a direct measurement of the single-particle spectral function, which contains the information of electronic band structures and the underlying interactions Damascelli et al. (2003); Shen et al. (1993); Lanzara et al. (2001). However, the application of ARPES has been typically limited to the occupied part of the spectral function. Numerical techniques such as division by the Fermi-Dirac distribution have been used to reveal the states slightly above  Lee et al. (2007), yet this method is confined to an energy range on the order of the sample temperature. A recent ARPES study on Bi-based cuprates identified features contributed by unbound states at  eV above  Miller et al. (2015). However, the key quantities of the strong correlation physics in cuprates - the energy scale of the UHB and the Coulomb interaction strength - remain underexplored.

Several techniques have studied the unoccupied electronic states in cuprates. Inverse photoemission spectroscopy (IPES) revealed unoccupied states from to  eV above  Wagener et al. (1989, 1990); Claessen et al. (1989); Drube et al. (1989); Bernhoff et al. (1990); Watanabe et al. (1991). However, IPES experiments are challenging due to the -lower efficiency compared to ARPES Johnson and Davenport (1985) and the  eV energy resolution Claessen et al. (1989); Drube et al. (1989); Bernhoff et al. (1990). X-ray absorption spectroscopy Himpsel et al. (1988); Bianconi et al. (1992); Saini et al. (1996) and scanning tunneling spectroscopy (STS) Ye et al. (2013) are also capable of characterizing the unoccupied states. Yet, these studies measure momentum-integrated density of states instead of momentum-resolved band structures. Two-photon photoemission (2PPE) enables the measurement of momentum-resolved unoccupied band structures with  meV energy resolution Petek and Ogawa (1997); Weinelt (2002); Sobota et al. (2013); Sonoda and Munakata (2002, 2004); Gilbertson et al. (2014). Pioneer 2PPE works on cuprates by Sonoda and Munakata revealed unoccupied states at the Brillouin zone center Sonoda and Munakata (2002, 2004). To further study the unoccupied band structure and the strong correlation physics, a momentum-resolved 2PPE study with a detailed comparison to theoretical calculations is needed.

Here we report a momentum-resolved 2PPE study on optimally doped BiSrCaYCuO (OP Bi2212, T =  K). Near the Brillouin zone center we resolve features near , , and  eV above , denoted as , , and , respectively. Tuning the photon energy from to eV, the binding energies of and stay unchanged, whereas feature becomes weak and unidentifiable. Comparison with the ARPES spectrum suggests that as well as correspond to unoccupied states whose spectral intensities are strongly modulated by the respective occupied states. Furthermore, we compare our results with calculations using the cluster perturbation theory (CPT), from which a Coulomb interaction strength of  eV is extracted. Our study provides an important benchmark for studying correlation physics in cuprate superconductors.

Ii Methods

Our optical setup is based on a regenerative amplifier system which typically outputs  eV photons with  kHz repetition rate,  fs pulse duration, and  J pulse energy. Two stages of nonlinear frequency conversions are employed: the first -BaBO (BBO) crystal yields the second harmonic; the second BBO sums the frequencies of the fundamental and the second harmonic. The third harmonic pulse duration is  fs. Its photon energy is tunable between and  eV. The incident fluence for our measurements is The -polarized third harmonic is focused on optimally doped Bi2212 samples to conduct monochromatic 2PPE measurements. The photon polarization is orthogonal to the analyzer slit. For occupied-state studies,  eV photons are generated by two stages of second harmonic generation from the  eV laser. The energy resolution of  eV ARPES is  meV. We also take ARPES measurements using  eV photons at the Stanford Synchrotron Radiation Lightsource, with a resolution of  meV. The Bi2212 samples are grown using the traveling-solvent floating-zone technique Eisaki et al. (2004), and cleaved in situ under ultrahigh vacuum with a pressure  Torr. The measurement temperature is set at  K.

Our theoretical calculation is based on a single-band Hubbard model solved by CPT Sénéchal et al. (2000, 2002); Wang et al. (2015). Although CPT is an approximate method, we believe it is most suitable for the comparison with experimental data due to its continuous momentum resolution evaluated in a zero-temperature many-body wavefunction. We refer readers to Ref. Wang et al. (2015) for a detailed implementation of the calculation.

Figure 1: Overview of the two-photon photoemission (2PPE) data on OP96 Bi2212 at 20 K. (a) Illustration of the ARPES and 2PPE processes. , , and are defined in the text. (b) 2PPE spectrum along the Brillouin zone diagonal. At the zone center, features , , and are identified near 1.5, 2.7, and 3.6 eV, respectively. The intensities in the energy range of 3.44.8 eV are magnified by a factor of to highlight the weak feature .

Iii Results

We present an overview of the 2PPE spectrum using  eV photons in Fig. 1. Figure 1(a) illustrates the one-photon excitation in ARPES and the two-photon excitation in 2PPE Sobota et al. (2013). For the latter, the first photon promotes electrons from occupied states below to high-lying unoccupied states. Scattering processes can occur to populate the lower-energy unoccupied states at energy . These intermediate states are subsequently promoted by the second photon to final states above the vacuum level . Throughout this work we follow the usual convention of ARPES experiments and discuss the binding energies of the intermediate states referenced to on the detector. This defines the intermediate state energy scale Sonoda and Munakata (2004), which allows a consistent comparison between the occupied and unoccupied states. In Fig. 1(b) we display the 2PPE spectrum along the - direction. At the zone center () we identify features near  eV (),  eV (), and  eV (). The observed features are consistent with previous 2PPE measurements at  Sonoda and Munakata (2002, 2004).

Figure 2: Illustration of different two-photon excitation schemes. Notations are the same as in Fig. 1(a). (a) Unoccupied-state spectroscopy. The resolved binding energy is () and does not depend on the photon energy. (b) Occupied-state spectroscopy. The resolved binding energy is () and depends on the photon energy. (c) Resonant excitation of unoccupied states. The spectral intensity is enhanced with respect to the non-resonant schemes in (a) and (b).
Figure 3: Photon energy dependence of the 2PPE features. (a, b) 2PPE spectra near the zone center using (a)  eV and (b)  eV photons. Both spectra are obtained with  photons/( (c, d) Comparison of the EDCs at (c)  Å and (d)  Å. Each EDC is obtained by integrating over a momentum window of  Å. The cuts are indicated by dashed lines in panels (a) and (b).
Figure 4: Resonant 2PPE process for feature using  eV photons. (a) Illustration of the momentum-space trajectory (solid brown) for the spectra in (b). Overlaid on the graphs are the tight-binding Fermi surfaces (solid black) Markiewicz et al. (2005). (b) 2PPE spectrum of feature using  eV photons and ARPES spectrum of the occupied states near using  eV photons.

Importantly, 2PPE can be used to probe both the occupied and unoccupied states Sonoda and Munakata (2002, 2004); Sobota et al. (2013). Figure 2(a) illustrates the ideal unoccupied-state spectroscopy where the 2PPE spectrum is predominantly determined by unoccupied states. In this case, the resolved binding energy is () and does not depend on the photon energy. Meanwhile, a distinct 2PPE process in Fig. 2(b) shows that occupied states at energy can be photoemitted by a direct two-photon process. The binding energy of the virtual intermediate state increases linearly with photon energies. Moreover, a resonant excitation scheme can occur when an occupied state is projected to an unoccupied state by the first photon (Fig. 2(c)). In this case, the spectral intensity is much enhanced compared to the non-resonant cases in Fig. 2(a) and (b).

To distinguish between different excitation scenarios, we perform a photon energy dependent study on the 2PPE spectrum (Fig. 3). Spectra in Fig. 3(a) and (b) are obtained with and  eV photons, respectively. The incident beam flux is maintained at photons/( We compare energy distribution curves (EDCs) taken at constant momentum points in Fig. 3(c) and (d). At  Å, features and display negligible shifts when tuning the photon energy, which indicates that they correspond to unoccupied states. Intriguingly, using  eV photons the spectral intensity of feature at  Å is significantly higher than that using  eV photons (Fig. 3(c)). The spectral peak of feature using  eV photons becomes unidentifiable. These observations suggest that features and are substantially influenced by their corresponding initial states Sobota et al. (2013).

To examine the optical excitation for feature , we compare the 2PPE spectrum using  eV photons with the ARPES spectrum using  eV photons (Fig. 4). In Fig. 4(a) we plot the Fermi surface calculated by a tight-binding model Markiewicz et al. (2005). The momentum trajectory along - intercepts the Fermi surface, resulting in the occupied-state dispersion measured by  eV ARPES, as shown in the lower panel of Fig. 4(b). Photoexcitations promote this occupied state to  eV above , leading to the dispersive feature in the 2PPE spectrum near  Å. This resonant excitation explains the enhancement in spectral intensities of feature using  eV photons. In the ARPES spectrum we also observe band structures near the zone center induced by the incommensurate modulation of the BiO planes along the crystallographic b axis Mans et al. (2006). It is challenging to determine whether the same effect is observed in the 2PPE spectrum due to the strong diffuse background.

To investigate the optical excitation for feature , we compare the 2PPE spectrum using  eV photons with the valence-band ARPES spectrum using  eV photons at Stanford Synchrotron Radiation Lightsource. We notice that feature is almost non-dispersive across the entire Brillouin zone, which resembles the characteristics of localized non-bonding states. As shown in Fig. 5, by shifting the ARPES spectrum  eV upwards, a clear correspondence is established between feature on the ARPES spectrum and feature on the 2PPE spectrum. Previous ARPES studies have identified feature as a non-bonding oxygen state He et al. (2016), which explains the non-dispersive character of feature . Therefore, Fig. 5 demonstrates that originates mostly from the non-bonding oxygen state.

Our interpretation of feature is different from that in a previous 2PPE study Sonoda and Munakata (2004). Ref. Sonoda and Munakata (2004) attributed feature purely to the UHB, which is an unoccupied state. However, the UHB is highly dispersive across the Brillouin zone Wang et al. (2015); Moritz et al. (2009); Kusko et al. (2002), which is inconsistent with our observation on feature . We emphasize that the modulation in intensity due to initial-state dispersions is key to understanding the origin of feature .

Iv Discussion

Various techniques have been used to study the origins of the unoccupied states in cuprates. IPES studies in the early 1990s observed features near and  eV Wagener et al. (1989, 1990); Claessen et al. (1989); Drube et al. (1989); Bernhoff et al. (1990); Watanabe et al. (1991), which likely correspond to features and in this work. Influenced by the band structure calculations available by then Krakauer and Pickett (1988); Massidda et al. (1988), most IPES studies attributed features and to BiO bands. However, several issues have been noticed with this assignment. First, the band structure calculations Krakauer and Pickett (1988); Massidda et al. (1988) are based on the local density approximation, which is questionable for strongly correlated materials such as cuprates. Second, as pointed out by Ref. Claessen et al. (1989) the observed dispersions of features and are vastly different from the predicted dispersions of the BiO bands Krakauer and Pickett (1988); Massidda et al. (1988).

Previous 2PPE studies conducted polarization dependence study to investigate the origins of the unoccupied states Sonoda and Munakata (2002, 2004). It was shown that and disappear when photons are -polarized, yet survives for both - and -polarized photons. Accordingly, they concluded that and have out-of-plane characters consistent with the Cu orbital, and that has in-plane characters consistent with the Cu orbital. This interpretation assigns the unoccupied states to states in the CuO layers where the many-body Mott physics occurs.

Figure 5: Influence of initial states on feature . (a) ARPES spectrum of OP96 Bi2212 along the zone diagonal using  eV photons. The energy axis is offset by  eV to be compared with the 2PPE spectrum. (b) 2PPE spectrum on the same sample using  eV photons.
Figure 6: Calculated spectrum using cluster perturbation theory. (a) Theoretical single-particle spectral function calculated by CPT for optimal doping and  eV. The charge-transfer gap  eV is marked on the spectrum. (b) Theoretical unoccupied spectrum in the experimentally measured momentum and energy range. The corresponding color scales are indicated on top of panels (a) and (b). (c) Comparison of an experimental EDC taken at the zone center and the theoretical counterparts for various values. EDCs are offset on a log scale for clear demonstration. The theoretical EDCs are multiplied by a Fermi-Dirac function with the Fermi edge at  eV above , which is the cutoff energy for 2PPE. The best match is obtained for  eV.

To obtain further understanding of the Mott physics, we compare our experimental results with a CPT calculation based on the single-band Hubbard model, which exclusively captures the low-energy Mott physics in CuO planes Wang et al. (2015). The Hubbard Hamiltonian is comprised of a nearest (next nearest) neighbor hopping term parametrized by energy t (t), and a Coulomb repulsion term parametrized by the interaction strength . For cuprate superconductors, this Coulomb corresponds to the Cu-O charge transfer gap  Zaanen et al. (1985); Damascelli et al. (2003); Lee et al. (2006); Xiang et al. (2009). We include only the Zhang-Rice singlet band Zhang and Rice (1988) in the single-band Hubbard model, and solve for the spectral function . Figure 6(a) demonstrates the calculated spectrum corresponding to optimal doping and  eV. Here we adopt  eV determined from previous ARPES experiments Xie et al. (2007), and . It is worth noting that the UHB is comprised of fine features corresponding to different electron hopping mechanisms in the energy range of to  eV Wang et al. (2015).

To compare the theoretical results with the experimental data, we emphasize that the entire feature and feature at  Å are strongly modulated by the occupied states, and hence should not be compared directly to the pure unoccupied states obtained by theory. Restraining our discussion to features and near the zone center, we identify the two features on the theoretical spectrum as shown in Fig. 6(b). We further plot the experimental EDC at , and compare it to theoretical EDCs for a series of values (Fig. 6(c)). Although the spectral shapes of and depend on matrix elements and inelastic scattering processes Sobota et al. (2012), the peak positions can be utilized for a quantitative comparison. Varying between and  eV with an increment of  eV, we determine that the optimal matching between theory and experiment is achieved when  eV.

The comparison between CPT calculations and 2PPE results suggests that features and at the zone center both belong to the UHB. Specifically, these features reflect the inter- and intra-sublattice electron motions Wang et al. (2015). We emphasize that there can be additional contributions from different origins. For features and , contributions from the orbital cannot be excluded Sonoda and Munakata (2002, 2004). For feature , the binding energy with respect to is close to that of the image potential state (IPS) Höfer et al. (1997). However, it is not readily evident in the 4.8 eV data (Fig. 3(b)) that feature possesses a free-electron-like dispersion expected for an IPS. Hence a contribution of the IPS to feature is unlikely but cannot be excluded.

There are a few important differences between theory and experiment. First, the theoretical spectrum contains a sharp feature near  eV and  Å which is not resolved experimentally. In 2PPE, this sharp feature can be overwhelmed by the strong modulation due to the occupied state (Fig. 5(a)). Second, the theoretical and features in Fig. 6(b) are rather non-dispersive. To avoid complications due to occupied states, we compare the theoretical features to the 2PPE results obtained in a non-resonant excitation regime (Fig. 3(b)). Here the spectral intensities of and quickly decrease as a function of momentum away from , which makes it challenging to determine the exact band dispersions. Further investigations are needed to quantify the experimental dispersions of and .

Nevertheless, the overall agreement between the CPT calculation and our 2PPE experiment has important implications. Momentum-resolved 2PPE lets us identify the UHB at the zone center, and furthermore the Coulomb interaction strength . The Coulomb represents the energy cost forming a doubly-occupied state on a Cu site (doublon) Xiang et al. (2009), and was determined by earlier experiments which did not resolve the UHB Shen et al. (1987). Our study showcases a modern method to directly unveil the UHB and deduce the Coulomb , which provides the basis for theoretical modeling of superconductivity and magnetism based on the single-band Hubbard model.

Taking into account the quasiparticle bandwidth  Xie et al. (2007), our measurement suggests a charge-transfer gap  eV. This is a factor of two smaller than the counterparts in undoped LaCuO Falck et al. (1992), CaCuOCl Ye et al. (2013), and Bi2201 Cai et al. (2016). On the other hand, our result is consistent with gap values reported by optical spectroscopies on doped Bi2212 Itoh et al. (1999) and STS on undoped Bi2212 Ruan et al. (2016). These comparisons suggest that varies substantially between different cuprate families. A recent STS study Ruan et al. (2016) discovered an anticorrelation between in the parent compound and the maximum superconducting transition temperature upon doping. This indicates a direct connection between electronic correlations and the superconducting pairing mechanism.

Interestingly, our results provide a new perspective to understand the chemical potential puzzle in the cuprate literature, where people have found a chemical potential shift  eV when tuning from electron doping to hole doping Ikeda et al. (2010). This shift is supposed to match , yet the experimental value is much smaller than the conventional of  eV Falck et al. (1992); Ye et al. (2013); Waku et al. (2004); Cai et al. (2016). Our results show that in hole-doped Bi2212 is as small as  eV, which suggests that this apparent discrepancy in the literature is due to comparison across different material families with different magnitudes of . Notably, a careful analysis of the photoemission and optical spectroscopy data on electron-doped NdCuO yields a gap of  eV Xiang et al. (2009). These values would be consistent with a chemical potential shift  eV when tuning from electron doping to hole doping. Future 2PPE experiments on electron-doped cuprates are clearly needed to verify this picture.

V Conclusion

Our momentum-resolved 2PPE measurement characterizes the unoccupied band structure for optimally doped Bi2212. By tuning the photon energy, we identify an unoccupied state near  eV above . Two other features near and  eV reflect unoccupied states strongly modulated by occupied-state dispersions. These results are compared with the UHB spectrum calculated by CPT, which yields a Coulomb interaction strength of  eV and a charge-transfer gap of  eV. Notably, our study provides a clean method to characterize the Coulomb repulsion for doped Mott insulators. Our technique is advantageous compared to optical measurements which are complicated by the emergence of Drude peaks for finite doping Terasaki et al. (1990); Itoh et al. (1999). If the 2PPE measurement conditions are further optimized, it is conceivable that the full unoccupied band structure can be determined unambiguously. In the study of advanced materials such as cuprates Damascelli et al. (2003) or iridates Kim et al. (2014), obtaining the full unoccupied band structure can determine the gap symmetries corresponding to various symmetry-breaking orders Hashimoto et al. (2010); Yang et al. (2008), which will be key to understanding the complex phase diagrams.

Acknowledgments This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under contract DE-AC02-76SF00515. S.-L.Y. and Y.W. acknowledge support by the Stanford Graduate Fellowship. S.-L.Y. is also supported by the Kavli Postdoctoral Fellowship at Cornell University. J.A.S. is in part supported by the Gordon and Betty Moore Foundations EPiQS Initiative through Grant GBMF4546. D.L. acknowledges partial support by the Swiss National Science Foundation under fellowship P300P2151328. H.S. acknowledges support from the Fulbright Scholar Program. Stanford Synchrotron Radiation Lightsource is operated by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences.


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