Abstract
The minimal ingredients to explain the essential physics of layered copperoxide (cuprates) materials remains heavily debated. Effective lowenergy singleband models of the copper–oxygen orbitals are widely used because there exists no strong experimental evidence supporting multiband structures. Here, we report angleresolved photoelectron spectroscopy experiments on Labased cuprates that provide direct observation of a twoband structure. This electronic structure, qualitatively consistent with density functional theory, is parametrised by a twoorbital (\(d_{x^2  y^2}\) and \(d_{z^2}\)) tightbinding model. We quantify the orbital hybridisation which provides an explanation for the Fermi surface topology and the proximity of the vanHove singularity to the Fermi level. Our analysis leads to a unification of electronic hopping parameters for singlelayer cuprates and we conclude that hybridisation, restraining dwave pairing, is an important optimisation element for superconductivity.
Introduction
Identifying the factors that limit the transition temperature T_{c} of hightemperature cuprate superconductivity is a crucial step towards revealing the design principles underlying the pairing mechanism^{1}. It may also provide an explanation for the dramatic variation of T_{c} across the known singlelayer compounds^{2}. Although superconductivity is certainly promoted within the copperoxide layers, the apical oxygen position may play an important role in defining the transition temperature^{3,4,5,6,7}. The CuO_{6} octahedron lifts the degeneracy of the nine copper 3delectrons and generates fully occupied t_{2g} and 3/4filled e_{g} states^{8}. With increasing apical oxygen distance d_{A} to the CuO_{2} plane, the e_{g} states split to create a 1/2filled \(d_{x^2  y^2}\) band. The distance d_{A} thus defines whether single or twoband models are most appropriate to describe the lowenergy band structure. It has also been predicted that d_{A} influences T_{c} in at least two different ways. First, the distance d_{A} controls the charge transfer gap between the oxygen and copper site which, in turn, suppresses superconductivity^{5,9}. Second, Fermilevel \(d_{z^2}\) hybridisation, depending on d_{A}, reduces the pairing strength^{6,10}. Experimental evidence, however, points in opposite directions. Generally, singlelayer materials with larger d_{A} have indeed a larger T_{c}^{2}. However, scanning tunneling microscopy (STM) studies of Bibased cuprates suggest an anticorrelation between d_{A} and T_{c}^{11}.
In the quest to disentangle these causal relation between d_{A} and T_{c}, it is imperative to experimentally reveal the orbital character of the cuprate band structure. The comparably short apical oxygen distance d_{A} makes La_{2−x}Sr_{ x }CuO_{4} (LSCO) an ideal candidate for such a study. Experimentally, however, it is challenging to determine the orbital character of the states near the Fermi energy (E_{F}). In fact, the \(d_{z^2}\) band has never been identified directly by angleresolved photoelectron spectroscopy (ARPES) experiments. A large majority of ARPES studies have focused on the pseudogap, superconducting gap and quasiparticle selfenergy properties in near vicinity to the Fermi level^{12}. An exception to this trend are studies of the socalled waterfall structure^{13,14,15,16,17} that lead to the observation of band structures below the \(d_{x^2  y^2}\) band^{14,16}. However, the origin and hence orbital character of these bands was never addressed. Resonant inelastic Xray scattering has been used to probe excitations between orbital dlevels. In this fashion, insight about the position of \(d_{z^2}\), d_{ xz }, d_{ yz } and d_{ xy } states with respect to \(d_{x^2  y^2}\) has been obtained^{18}. Although difficult to disentangle, it has been argued that for LSCO the \(d_{z^2}\) level is found above d_{ xz }, d_{ yz } and d_{ xy }^{19,20}. To date, a comprehensive study of the \(d_{z^2}\) momentum dependence is missing and therefore the coupling between the \(d_{z^2}\) and \(d_{x^2  y^2}\) bands has not been revealed. Xray absorption spectroscopy (XAS) experiments, sensitive to the unoccupied states, concluded only marginal hybridisation of \(d_{x^2  y^2}\) and \(d_{z^2}\) states in LSCO^{21}. Therefore, the role of \(d_{z^2}\) hybridisation remains ambiguous^{22}.
Here we provide direct ultraviolet and softXray ARPES measurements of the \(d_{z^2}\) band in Labased singlelayer compounds. The \(d_{z^2}\) band is located about 1 eV below the Fermi level at the Brillouin zone (BZ) corners. From these corners, the \(d_{z^2}\) band is dispersing downwards along the nodal and antinodal directions, consistent with density functional theory (DFT) calculations. The experimental and DFT band structure, including only \(d_{x^2  y^2}\) and \(d_{z^2}\) orbitals, is parametrised using a twoorbital tightbinding model^{23}. The presence of the \(d_{z^2}\) band close to the Fermi level allows to describe the Fermi surface topology for all singlelayer compounds (including HgBa_{2}CuO_{4+x} and Tl_{2}Ba_{2}CuO_{6+x}) with similar hopping parameters for the \(d_{x^2  y^2}\) orbital. This unification of electronic parameters implies that the main difference between singlelayer cuprates originates from the hybridisation between \(d_{x^2  y^2}\) and \(d_{z^2}\) orbitals. The significantly increased hybridisation in Labased cuprates pushes the vanHove singularity close to the Fermi level. This explains why the Fermi surface differs from other singlelayer compounds. We directly quantify the orbital hybridisation that plays a sabotaging role for superconductivity.
Results
Material choices
Different dopings of LSCO spanning from x = 0.12 to 0.23 in addition to an overdoped compound of La_{1.8−x}Eu_{0.2}Sr_{ x }CuO_{4} with x = 0.21 have been studied. These compounds represent different crystal structures: lowtemperature orthorhombic, lowtemperature tetragonal and the hightemperature tetragonal. Our results are very similar across all crystal structures and dopings (Supplementary Fig. 1). To keep the comparison to band structure calculations simple, this paper focuses on results obtained in the tetragonal phase of overdoped LSCO with x = 0.23.
Electronic band structure
A raw ARPES energy distribution map (EDM), along the nodal direction, is displayed in Fig. 1a. Near E_{F}, the widely studied nodal quasiparticle dispersion with predominately \(d_{x^2  y^2}\) character is observed^{12}. This band reveals the previously reported electronlike Fermi surface of LSCO, x = 0.23^{24,25} (Fig. 1b), the universal nodal Fermi velocity v_{F} ≈ 1.5 eVÅ^{26} and a band dispersion kink around 70 meV^{26}. The main observation reported here is the second band dispersion at ~1 eV below the Fermi level E_{F} (Figs. 1 and 2) and a hybridisation gap splitting the two (Fig. 3). This second band—visible in both raw momentum distribution curves (MDC) and constant energy maps—disperses downwards away from the BZ corners. Since a pronounced k_{ z } dependence is observed for this band structure (Figs. 2 and 4) a trivial surface state can be excluded. Subtracting a background intensity profile (Supplementary Fig. 2) is a standard method that enhances visualisation of this second band structure. In fact, using soft Xrays (160–600 eV), at least two additional bands (β and γ) are found below the \(d_{x^2  y^2}\) dominated band crossing the Fermi level. Here, focus is set entirely on the β band dispersion closest to the \(d_{x^2  y^2}\) dominated band. This band is clearly observed at the BZ corners (Figs. 1–3). The complete inplane (k_{ x }, k_{ y }) and outofplane (k_{ z }) band dispersion is presented in Fig. 4.
Orbital band characters
To gain insight into the orbital character of these bands, a comparison with a DFT band structure calculation (see Methods section) of La_{2}CuO_{4} is shown in Fig. 2. The e_{g} states (\(d_{x^2  y^2}\) and \(d_{z^2}\)) are generally found above the t_{2g} bands (d_{ xy }, d_{ xz } and d_{ yz }). The overall agreement between the experiment and the DFT calculation (Supplementary Fig. 3) thus suggests that the two bands nearest to the Fermi level are composed predominately of \(d_{x^2  y^2}\) and \(d_{z^2}\) orbitals. This conclusion can also be reached by pure experimental arguments. Photoemission matrix element selection rules contain information about the orbital band character. They can be probed in a particular experimental setup where a mirrorplane is defined by the incident light and the electron analyser slit^{12}. With respect to this plane the electromagnetic light field has odd (even) parity for \(\bar \sigma\) (\(\bar \pi\)) polarisation (Supplementary Fig. 4). Orienting the mirror plane along the nodal direction (cut 1 in Fig. 1), the \(d_{z^2}\) and d_{ xy } (\(d_{x^2  y^2}\)) orbitals have even (odd) parity. For a finalstate with even parity, selection rules^{12} dictate that the \(d_{z^2}\) and d_{ xy }derived bands should appear (vanish) in the \(\bar \pi\) (\(\bar \sigma\)) polarisation channel and vice versa for \(d_{x^2  y^2}\). Due to their orientation in realspace, the d_{ xz } and d_{ yz } orbitals are not expected to show a strict switching behaviour along the nodal direction^{27}. As shown in Fig. 1f, g, two bands (α and γ) appear with \(\bar \sigma\)polarised light while for \(\bar \pi\)polarised light bands β and γ′ are observed. Band α which crosses E_{F} is assigned to \(d_{x^2  y^2}\) while band γ has to originate from d_{ xz }/d_{ yz } orbitals as \(d_{z^2}\) and d_{ xy }derived states are fully suppressed for \(\bar \sigma\)polarised light. In the EDM, recorded with \(\bar \pi\)polarised light, band (β) at ~1 eV binding energy and again a band (γ′) at ~1.6 eV is observed. From the orbital shape, a smaller k_{ z } dispersion is expected for \(d_{x^2  y^2}\) and d_{ xy }derived bands than for those from \(d_{z^2}\) orbitals. As the β band exhibits a significant k_{ z } dispersion (Fig. 4), much larger than observed for the \(d_{x^2  y^2}\) band, we conclude that it is of \(d_{z^2}\) character. The γ′ band which is very close to the γ band is therefore of d_{ xy } character. Interestingly, this \(d_{z^2}\)derived band has stronger inplane than outofplane dispersion, suggesting that there is a significant hopping to inplane p_{ x } and p_{ y } oxygen orbitals. Therefore the assumption that the \(d_{z^2}\) states are probed uniquely through the apical oxygen p_{ z } orbital^{21} has to be taken with caution.
Discussion
Most minimal models aiming to describe the cuprate physics start with an approximately halffilled single \(d_{x^2  y^2}\) band on a twodimensional square lattice. Experimentally, different band structures have been observed across singlelayer cuprate compounds. The Fermi surface topology of LSCO is, for example, less rounded compared to (Bi,Pb)_{2}(Sr,La)_{2}CuO_{6+x} (Bi2201), Tl_{2}Ba_{2}CuO_{6+x} (Tl2201) and HgBa_{2}CuO_{4+x} (Hg1201). Within a singleband tightbinding model the rounded Fermi surface shape of the singlelayer compounds Hg1201 and Tl2201 is described by setting \(r = \left( {\left {t_\alpha ^\prime } \right + \left {t_\alpha ^{\prime\prime} } \right} \right){\mathrm{/}}t_\alpha \sim 0.4\)^{6}, where t_{ α }, \(t_\alpha ^\prime\) and \(t_\alpha ^{\prime\prime}\) are nearest neighbour (NN), next–nearest neighbour (NNN) and nextnext–nearest neighbour (NNNN) hopping parameters (Table 1 and Supplementary Fig. 4). For LSCO with more flat Fermi surface sections, significantly lower values of r have been reported. For example, for overdoped La_{1.78}Sr_{0.22}CuO_{4}, r ~ 0.2 was found^{24,25}. The singleband premise thus leads to varying hopping parameters across the cuprate families, stimulating the empirical observation that \(T_{\mathrm{c}}^{{\mathrm{max}}}\) roughly scales with \(t_\alpha ^\prime\)^{2}. This, however, is in direct contrast to t–J models that predict the opposite correlation^{28,29}. Thus the singleband structure applied broadly to all singlelayer cuprates lead to conclusions that challenge conventional theoretical approaches.
The observation of the \(d_{z^2}\) band calls for a reevaluation of the electronic structure in Labased cuprates using a twoorbital tightbinding model (see Methods section). Crucially, there is a hybridisation term \({\mathrm{\Psi }}\left( {\bf{k}} \right) = 2t_{\alpha \beta }\left[ {{\mathrm{cos}}\left( {k_xa} \right)  {\mathrm{cos}}\left( {k_yb} \right)} \right]\) between the \(d_{x^2  y^2}\) and \(d_{z^2}\) orbitals, where t_{ αβ } is a hopping parameter that characterises the strength of orbital hybridisation. In principle, one may attempt to describe the two observed bands independently by taking t_{ αβ } = 0. However, the problem then returns to the singleband description with the above mentioned contradictions. Furthermore, t_{ αβ } = 0 implies a band crossing in the antinodal direction that is not observed experimentally (Fig. 3). In fact, from the avoided band crossing one can directly estimate t_{ αβ } ≈ −200 meV. As dictated by the different eigenvalues of the orbitals under mirror symmetry, the hybridisation term Ψ(k) vanishes on the nodal lines k_{ x } = ±k_{ y } (see inset of Fig. 3). Hence the pure \(d_{x^2  y^2}\) and \(d_{z^2}\) orbital band character is expected along these nodal lines. The hybridisation Ψ(k) is largest in the antinodal region, pushing the vanHove singularity of the upper band close to the Fermi energy and in case of overdoped LSCO across the Fermi level.
In addition to the hybridisation parameter t_{ αβ } and the chemical potential μ, six free parameters enter the tightbinding model that yields the entire band structure (white lines in Figs. 2 and 4). Nearest and nextnearest inplane hopping parameters between \(d_{x^2  y^2}\) (t_{ α }, \(t_\alpha ^\prime\)) and \(d_{z^2}\) \(( {t_\beta ,t_\beta ^\prime } )\) orbitals are introduced to capture the Fermi surface topology and inplane \(d_{z^2}\) band dispersion (Supplementary Fig. 4). The k_{ z } dispersion is described by nearest and nextnearest outofplane hoppings (t_{ βz }, \(t_{\beta z}^\prime\)) of the \(d_{z^2}\) orbital. The four \(d_{z^2}\) hopping parameters and the chemical potential μ are determined from the experimental band structure along the nodal direction where Ψ(k) = 0. Furthermore, the α and β band dispersion in the antinodal region and the Fermi surface topology provide the parameters t_{ α }, \(t_\alpha ^\prime\) and t_{ αβ }. Our analysis reveals a finite band coupling t_{ αβ } = −0.21 eV resulting in a strong antinodal orbital hybridisation (Fig. 2 and Table 1). Compared to the singleband parametrisation^{24} a significantly larger value r ~ −0.32 is found and hence a unification of \(t_\alpha ^\prime {\mathrm{/}}t_\alpha\) ratios for all singlelayer compounds is achieved.
Finally, we discuss the implication of orbital hybridisation for superconductivity and pseudogap physics. First, we notice that a pronounced pseudogap is found in the antinodal region of La_{1.8−x}Eu_{0.2}Sr_{ x }CuO_{4} with x = 0.21—consistent with transport experiments^{30} (Supplementary Fig. 5). The fact that t_{ αβ } of La_{1.59}Eu_{0.2}Sr_{0.21}CuO_{4} is similar to t_{ αβ } of LSCO suggests that the pseudogap is not suppressed by the \(d_{z^2}\) hybridisation. To this end, a comparison to the 1/4filled e_{g} system Eu_{2−x}Sr_{ x }NiO_{4} with x = 1.1 is interesting^{31,32}. This material has the same twoorbital band structure with protection against hybridisation along the nodal lines. Both the \(d_{x^2  y^2}\) and \(d_{z^2}\) bands are crossing the Fermi level, producing two Fermi surface sheets^{31}. Despite an even stronger \(d_{z^2}\) admixture of the \(d_{x^2  y^2}\) derived band a dwavelike pseudogap has been reported^{32}. The pseudogap physics thus seems to be unaffected by the orbital hybridisation.
It has been argued that orbital hybridisation—of the kind reported here—is unfavourable for superconducting pairing^{6,10}. It thus provides an explanation for the varying \(T_{\mathrm{c}}^{{\mathrm{max}}}\) across singlelayer cuprate materials. Although other mechanisms, controlled by the apical oxygen distance, (e.g. variation of the copper–oxygen charge transfer gap^{4}) are not excluded our results demonstrate that orbital hybridisation exists and is an important control parameter for superconductivity.
Methods
Sample characterisation
Highquality single crystals of LSCO, x = 0.12, 0.23, and La_{1.8−x}Eu_{0.2}Sr_{ x }CuO_{4}, x = 0.21, were grown by the floatingzone technique. The samples were characterised by SQUID magnetisation^{33} to determine superconducting transition temperatures (T_{c} = 27, 24 and 14 K). For the crystal structure, the experimental lattice parameters are a = b = 3.78 Å and c = 2c′ = 13.2 Å^{34}.
ARPES experiments
Ultraviolet and softXray ARPES experiments were carried out at the SIS^{43} and ADRESS^{44} beamlines at the Swiss Light Source and at the I05 beamline at Diamond Light Source. Samples were prealigned ex situ using a Xray LAUE instrument and cleaved in situ—at base temperature (10–20 K) and ultra high vacuum (≤5 × 10^{−11} mbar)—employing a toppost technique or cleaving device^{35}. Ultraviolet (soft Xray^{36}) ARPES spectra were recorded using a SCIENTA R4000 (SPECS PHOIBOS150) electron analyser with horizontal (vertical) slit setting. All data was recorded at the cleaving temperature 10–20 K. To visualise the \(d_{z^2}\)dominated band, we subtracted in Fig. 1f, g and Figs. 2–4 the background that was obtained by taking the minimum intensity of the MDC at each binding energy.
Tightbinding model
A twoorbital tightbinding model Hamiltonian with symmetryallowed hopping terms is employed to isolate and characterise the extent of orbital hybridisation of the observed band structure^{23}. For compactness of the momentumspace Hamiltonian matrix representation, we introduce the vectors
where κ, κ_{1} and κ_{2} take values ±1 as defined by sums in the Hamiltonian and ⊤ denotes vector transposition.
Neglecting the electron spin (spin–orbit coupling is not considered) the momentumspace tightbinding Hamiltonian, \({\cal H}\)(k), at a particular momentum k = (k_{ x }, k_{ y }, k_{ z }) is then given by
in the basis \(\left( {c_{{\bf{k}},x^2  y^2},c_{{\bf{k}},z^2}} \right)^ \top\), where the operator c_{k,α} annihilates an electron with momentum k in an e_{g}orbital d_{ α }, with α ∈ {x^{2} − y^{2}, z^{2}}. The diagonal matrix entries are given by
and
which describe the intraorbital hopping for \(d_{x^2  y^2}\) and \(d_{z^2}\) orbitals, respectively. The interorbital nearestneighbour hopping term is given by
In the above, μ determines the chemical potential. The hopping parameters t_{ α }, \(t_\alpha ^\prime\) and \(t_\alpha ^{\prime\prime}\) characterise NN, NNN and NNNN intraorbital inplane hopping between \(d_{x^2  y^2}\) orbitals. t_{ β } and \(t_\beta ^\prime\) characterise NN and NNN intraorbital inplane hopping between \(d_{z^2}\) orbitals, while t_{ βz } and \(t_{\beta z}^\prime\) characterise NN and NNN intraorbital outofplane hopping between \(d_{z^2}\) orbitals, respectively (Supplementary Fig. 3). Finally, the hopping parameter t_{ αβ } characterises NN interorbital inplane hopping. Note that in our model, \(d_{x^2  y^2}\) intraorbital hopping terms described by the vectors (Eq. (1)) are neglected as these are expected to be weak compared to those of the \(d_{z^2}\) orbital. This is due to the fact that the interplane hopping is mostly mediated by hopping between apical oxygen p_{ z } orbitals, which in turn only hybridise with the \(d_{z^2}\) orbitals, not with the \(d_{x^2  y^2}\) orbitals. Such an argument highlights that the tightbinding model is not written in atomic orbital degrees of freedom, but in Wannier orbitals, which are formed from the Cu d orbitals and the ligand oxygen p orbitals. As follows from symmetry considerations and is discussed in ref. ^{10}, the Cu \(d_{z^2}\) orbital together with the apical oxygen p_{ z } orbital forms a Wannier orbital with \(d_{z^2}\) symmetry, while the Cu \(d_{x^2  y^2}\) orbital together with the four neighbouring p_{ σ } orbitals of the inplane oxygen forms a Wannier orbital with \(d_{x^2  y^2}\) symmetry. One should thus think of this tightbinding model as written in terms of these Wannier orbitals, thus implicitly containing superexchange hopping via the ligand oxygen p orbitals. Additionally we stress that all hopping parameters effectively include the oxygen orbitals. Diagonalising Hamiltonian (2), we find two bands
and make the following observations: along the k_{ x } = ±k_{ y } lines, Ψ(k) vanishes and hence no orbital mixing appears in the nodal directions. The reason for this absence of mixing lies in the different mirror eigenvalues of the two orbitals involved. Hence it is not an artifact of the finite range of hopping processes included in our model. The parameters of the tightbinding model are determined by fitting the experimental band structure and are provided in Table 1.
DFT calculations
DFT calculations were performed for La_{2}CuO_{4} in the tetragonal space group I4/mmm, No. 139, found in the overdoped regime of LSCO using the WIEN2K package^{37}. Atomic positions are those inferred from neutron diffraction measurements^{34} for x = 0.225. In the calculation, the Kohn–Sham equation is solved selfconsistently by using a fullpotential linear augmented plane wave (LAPW) method. The self consistent field calculation converged properly for a uniform kspace grid in the irreducible BZ. The exchangecorrelation term is treated within the generalised gradient approximation in the parametrisation of Perdew, Burke and Enzerhof^{38}. The plane wave cutoff condition was set to RK_{max} = 7 where R is the radius of the smallest LAPW sphere (i.e. 1.63 times the Bohr radius) and K_{max} denotes the plane wave cutoff.
Data availability
All experimental data are available upon request to the corresponding authors.
Additional information
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Acknowledgements
D.S., D.D., L.D., T.N., C.E.M, C.G.F. and J.C. acknowledge support by the Swiss National Science Foundation. Further, Y.S. and M.M. are supported by the Swedish Research Council (VR) through a project (BIFROST, dnr.201606955). O.T. acknowledges support from the Swedish Research Council as well as the Knut and Alice Wallenberg foundation. This work was performed at the SIS, ADRESS and I05 beamlines at the Swiss Light Source and at the Diamond Light Source. A.M.C. wishes to thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY1066293, for hosting during some stages of this work. We acknowledge Diamond Light Source for access to beamline I05 (proposal SI105501) that contributed to the results presented here and thank all the beamline staff for technical support.
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S.P., T.T., H.T., T.K., N.M., M.O., O.J.L. and S.M.H. grew and prepared single crystals. C.E.M., D.S., L.D., M.H., D.D., C.G.F., K.H., J.C., M.S., O.T., M.K., V.N.S., T.S., P.D., M.H., M.M. and Y.S. prepared and carried out the ARPES experiment. C.E.M., K.H. and J.C. performed the data analysis. C.E.M. carried out the DFT calculations and A.M.C., C.E.M. and T.N. developed the tightbinding model. All authors contributed to the manuscript.
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The authors declare no competing interests.
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Correspondence to C. E. Matt or J. Chang.
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