Direct Observation of Orbital Hybridisation in a Cuprate Superconductor

Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch

Year: 2018

Direct observation of orbital hybridisation in a cuprate superconductor

Matt, C E ; Sutter, D ; Cook, A M ; et al

Abstract: The minimal ingredients to explain the essential physics of layered copper-oxide (cuprates) materials remains heavily debated. Effective low-energy single-band models of the copper–oxygen orbitals are widely used because there exists no strong experimental evidence supporting multi-band structures. Here, we report angle-resolved photoelectron spectroscopy experiments on La-based cuprates that provide direct observation of a two-band structure. This electronic structure, qualitatively consistent with density functional theory, is parametrised by a two-orbital (dx2−y2 and dz2) tight-binding model. We quantify the orbital hybridisation which provides an explanation for the Fermi surface topology and the proximity of the van-Hove singularity to the Fermi level. Our analysis leads to a unification of electronic hopping parameters for single-layer cuprates and we conclude that hybridisation, restraining d-wave pairing, is an important optimisation element for superconductivity.

DOI: https://doi.org/10.1038/s41467-018-03266-0

Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-157585 Journal Article Published Version

The following work is licensed under a Creative Commons: Attribution 4.0 International (CC BY 4.0) License.

Originally published at: Matt, C E; Sutter, D; Cook, A M; et al (2018). Direct observation of orbital hybridisation in a cuprate superconductor. Nature Communications, 9:972. DOI: https://doi.org/10.1038/s41467-018-03266-0 ARTICLE

DOI: 10.1038/s41467-018-03266-0 OPEN Direct observation of orbital hybridisation in a cuprate superconductor

C.E. Matt 1,2, D. Sutter 1, A.M. Cook1, Y. Sassa3, M. Månsson 4, O. Tjernberg 4, L. Das1, M. Horio 1, D. Destraz1, C.G. Fatuzzo5, K. Hauser1, M. Shi2, M. Kobayashi2, V.N. Strocov2, T. Schmitt2, P. Dudin6, M. Hoesch6, S. Pyon7, T. Takayama7, H. Takagi 7, O.J. Lipscombe8, S.M. Hayden8, T. Kurosawa9, N. Momono9,10, M. Oda9, T. Neupert1 & J. Chang 1

1234567890():,; The minimal ingredients to explain the essential physics of layered copper-oxide (cuprates) materials remains heavily debated. Effective low-energy single-band models of the copper–oxygen orbitals are widely used because there exists no strong experimental evidence supporting multi-band structures. Here, we report angle-resolved photoelectron spectroscopy experiments on La-based cuprates that provide direct observation of a two- band structure. This electronic structure, qualitatively consistent with density functional

theory, is parametrised by a two-orbital (dx2Ày2 and dz2 ) tight-binding model. We quantify the orbital hybridisation which provides an explanation for the Fermi surface topology and the proximity of the van-Hove singularity to the Fermi level. Our analysis leads to a uniﬁcation of electronic hopping parameters for single-layer cuprates and we conclude that hybridisation, restraining d-wave pairing, is an important optimisation element for superconductivity.

1 Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. 2 Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland. 3 Department of Physics and Astronomy, Uppsala University, SE-75121 Uppsala, Sweden. 4 Materials Physics, KTH Royal Institute of Technology, SE-164 40 Kista, Stockholm, Sweden. 5 Institute of Physics, École Polytechnique Fedérale de Lausanne (EPFL), Lausanne CH-1015, Switzerland. 6 Diamond Light Source, Harwell Campus, Didcot OX11 0DE, UK. 7 Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, Japan. 8 H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK. 9 Department of Physics, Hokkaido University, Sapporo 060-0810, Japan. 10 Department of Applied Sciences, Muroran Institute of Technology, Muroran 050-8585, Japan. Correspondence and requests for materials should be addressed to C.E.M. (email: [email protected]) or to J.C. (email: [email protected])

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T 13–17 dentifying the factors that limit the transition temperature c waterfall structure that lead to the observation of band d 14,16 Iof high-temperature cuprate superconductivity is a crucial step structures below the x2Ày2 band .However,theoriginand towards revealing the design principles underlying the pairing hence orbital character of these bands was never addressed. mechanism1. It may also provide an explanation for the dramatic Resonant inelastic X-ray scattering has been used to probe 2 variation of Tc across the known single-layer compounds . excitations between orbital d-levels. In this fashion, insight Although superconductivity is certainly promoted within the about the position of dz2 , dxz, dyz and dxy states with respect to 18 copper-oxide layers, the apical oxygen position may play an dx2Ày2 has been obtained .Althoughdifficult to disentangle, it 3–7 important role in defining the transition temperature . The has been argued that for LSCO the dz2 level is found above dxz, 19,20 CuO6 octahedron lifts the degeneracy of the nine copper 3d- dyz and dxy . To date, a comprehensive study of the dz2 t fi e 8 electrons and generates fully occupied 2g and 3/4- lled g states . momentum dependence is missing and therefore the coupling With increasing apical oxygen distance dA to the CuO2 plane, the between the dz2 and dx2Ày2 bands has not been revealed. X-ray e fi d d g states split to create a 1/2- lled x2Ày2 band. The distance A absorption spectroscopy (XAS) experiments, sensitive to the thus defines whether single or two-band models are most unoccupied states, concluded only marginal hybridisation of 21 appropriate to describe the low-energy band structure. It has also dx2Ày2 and dz2 states in LSCO .Therefore,theroleofdz2 22 been predicted that dA influences Tc in at least two different ways. hybridisation remains ambiguous . d First, the distance A controls the charge transfer gap between the Here we provide direct ultraviolet and soft-X-ray ARPES oxygen and copper site which, in turn, suppresses measurements of the dz2 band in La-based single-layer com- 5,9 superconductivity . Second, Fermi-level dz2 hybridisation, pounds. The dz2 band is located about 1 eV below the Fermi level d 6,10 d depending on A, reduces the pairing strength . Experimental at the Brillouin zone (BZ) corners. From these corners, the z2 evidence, however, points in opposite directions. Generally, band is dispersing downwards along the nodal and anti-nodal d T 2 single-layer materials with larger A have indeed a larger c . directions, consistent with density functional theory (DFT) cal- However, scanning tunneling microscopy (STM) studies of Bi- culations. The experimental and DFT band structure, including d T 11 d d based cuprates suggest an anti-correlation between A and c . only x2Ày2 and z2 orbitals, is parametrised using a two-orbital d 23 d In the quest to disentangle these causal relation between A tight-binding model . The presence of the z2 band close to the T and c, it is imperative to experimentally reveal the orbital Fermi level allows to describe the Fermi surface topology for all character of the cuprate band structure. The comparably short single-layer compounds (including HgBa2CuO4+x and Tl2Ba2- d d apical oxygen distance A makes La2−xSrxCuO4 (LSCO) an ideal CuO6+x) with similar hopping parameters for the x2Ày2 orbital. candidate for such a study. Experimentally, however, it is This unification of electronic parameters implies that the main challenging to determine the orbital character of the states near difference between single-layer cuprates originates from the E d d d fi the Fermi energy ( F). In fact, the z2 band has never been hybridisation between x2Ày2 and z2 orbitals. The signi cantly identified directly by angle-resolved photoelectron spectroscopy increased hybridisation in La-based cuprates pushes the van- (ARPES) experiments. A large majority of ARPES studies have Hove singularity close to the Fermi level. This explains why the focused on the pseudogap, superconducting gap and quasi- Fermi surface differs from other single-layer compounds. We particle self-energy properties in near vicinity to the Fermi directly quantify the orbital hybridisation that plays a sabotaging level12. An exception to this trend are studies of the so-called role for superconductivity.

cut1 – - pol a bc f 0 Max M cut1

1 X E –

0 ] a / F

–1 [eV] [

y Γ k 0 Max X Min

–0.5 –2 –1 E=EF –0.45 eV

[eV] –- pol F g de 0 Min E –

1 E – –1.0

] a / F

–1 [eV] [ y

k 0

–1.5 ' –1 –2 –0.9 eV –1.3 eV –2 –1 0 1 2 –1 0 1 –1 0 1 –2 –10 1 2 kll [ /a] kx [ /a] kx [ /a] kll [ /a]

Fig. 1 ARPES spectra showing eg-bands of overdoped La2−xSrxCuO4x = 0.23. a Raw ARPES energy distribution map (EDM) along cut 1 as indicated in (c). Dashed green line indicates the position of MDC displayed on top by turquoise circles. A linear background has been subtracted from the

MDC which is ﬁtted (blue line) by four Lorentzians (red lines). b–e Constant binding energy maps at EF (b) and at higher binding energies (c–e)as indicated. The photoemission intensity, shown in false colour scale, is integrated over ± 10 meV. Black (red) lines indicate the position of dx2Ày2 ðÞdz2 bands. The curve thickness in b, e is scaled to the contribution of the dz2 orbital. Semitransparent lines are guides to the eye. f, g EDMs along cut 1 recorded with σ and π light, f sensitive to the low-energy dx2Ày2 and dxz/dyz bands and g the dz2 and dxy-derived bands. All data have been recorded with hν = 160 eV

2 NATURE COMMUNICATIONS | (2018) 9:972 | DOI: 10.1038/s41467-018-03266-0 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03266-0 ARTICLE

kz = 0 - plane 2 ab e d d 0 xy xz/yz d 2 d 2 2 1 z x – y g [eV] [eV]

F 0 –1 F E E – – Z E

E –1 R A –2 /' –2

Γ Γ Γ MMXX MXX M X Γ X Min Max dz 2 -weight TB kz = - plane M k 2 z cd f k 0 y 1 kx [eV] F

[eV] 0 E F –1 E – – E –1 E

–2 –2

XARAARRZ RZ A R

Fig. 2 Comparison of observed and calculated band structure. a–d Background subtracted (see Methods section) soft-X-ray ARPES EDMs recorded on

La2−xSrxCuO4, x = 0.23 along in-plane high-symmetry directions for kz = 0 and kz = π/c′ as indicated in g. White lines represent the two-orbital (dz2 and dx2Ày2 ) tight-binding model as described in the text. The line width in b, d indicates the orbital weight of the dz2 orbital. e, f Corresponding in-plane DFT band structure at kz = 0 and kz = π/c′, calculated for La2CuO4 (see Methods section). The colour code indicates the orbital character of the bands. Around the anti-nodal points (X or R), strong hybridisation of dz2 and dx2Ày2 orbitals is found. In contrast, symmetry prevents any hybridisation along the nodal lines (Γ–MorZ–A). g Sketch of the 3D BZ of LSCO with high symmetry lines and points as indicated

Results Material choices. Different dopings of LSCO spanning from Min Max –4t 4t x = 0.12 to 0.23 in addition to an overdoped compound of 0.0 x = – - pol La1.8−xEu0.2SrxCuO4 with 0.21 have been studied. These compounds represent different crystal structures: low- X temperature orthorhombic, low-temperature tetragonal and the –0.5 Γ t = 0 eV high-temperature tetragonal. Our results are very similar across all crystal structures and dopings (Supplementary Fig. 1). To keep t = –0.21 eV

[eV] –1.0 M the comparison to band structure calculations simple, this paper F focuses on results obtained in the tetragonal phase of overdoped LSCO with x = 0.23. E – –1.5

Electronic band structure. A raw ARPES energy distribution –2.0 map (EDM), along the nodal direction, is displayed in Fig. 1a. ~2t Near EF, the widely studied nodal quasiparticle dispersion with 12 Γ predominately dx2Ày2 character is observed . This band reveals XX the previously reported electron-like Fermi surface of LSCO, x = 24,25 v ≈ Fig. 3 Avoided band crossing. Left panel: ultraviolet ARPES data recorded 0.23 (Fig. 1b), the universal nodal Fermi velocity F 1.5 26 26 along the ant-inodal direction using 160 eV linear horizontal polarised eVÅ and a band dispersion kink around 70 meV . The main photons. Solid white lines are the same tight-binding model as shown in observation reported here is the second band dispersion at ~1 eV d d E Fig. 2. Right panel: tight-binding model of the x2Ày2 and z2 bands along the below the Fermi level F (Figs. 1 and 2) and a hybridisation gap anti-nodal direction. Grey lines are the model prediction in absence of inter- splitting the two (Fig. 3). This second band—visible in both raw orbital hopping (tαβ = 0) between dx2Ày2 and dz2 . In this case, the bands are momentum distribution curves (MDC) and constant energy Γ ﬁ — crossing near the -point. This degeneracy is lifted once a nite inter-orbital maps disperses downwards away from the BZ corners. Since a t = − k hopping parameter is considered. For solid black lines αβ 210 meV and pronounced z dependence is observed for this band structure other hopping parameters have been adjusted accordingly. Inset indicates (Figs. 2 and 4) a trivial surface state can be excluded. Subtracting Γ − ﬁ the Fermi surface (green line) and the X cut directions. Coloured a background intensity pro le (Supplementary Fig. 2) is a stan- background displays the amplitude of the hybridisation term Ψ(k) that dard method that enhances visualisation of this second band vanishes on the nodal lines structure. In fact, using soft X-rays (160–600 eV), at least two additional bands (β and γ) are found below the dx2Ày2 dominated band crossing the Fermi level. Here, focus is set entirely on the β d calculation (see Methods section) of La2CuO4 is shown in Fig. 2. band dispersion closest to the x2Ày2 dominated band. This band e d d t The g states ( x2Ày2 and z2 ) are generally found above the 2g is clearly observed at the BZ corners (Figs. 1–3). The complete in- d d d k k k bands ( xy, xz and yz). The overall agreement between the plane ( x, y) and out-of-plane ( z) band dispersion is presented experiment and the DFT calculation (Supplementary Fig. 3) thus in Fig. 4. suggests that the two bands nearest to the Fermi level are com- posed predominately of dx2Ày2 and dz2 orbitals. This conclusion Orbital band characters. To gain insight into the orbital char- can also be reached by pure experimental arguments. Photo- acter of these bands, a comparison with a DFT band structure emission matrix element selection rules contain information

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a hv= bc 26 600 eV /c'] E = EF EB = –1.3 eV [ 25 k z 550 eV 24 500 eV

–0.5

[eV] –1.0 F E

– –1.5 E

–2.0 kz

X ky Max M E B = –1.3 eV k –2 x –1.5 Γ –1.0 –0.5 Min k 0 2 2 2 || [/a] 0.5 dx -y dz

Fig. 4 Three-dimensional band dispersion. a kz dispersion recorded along the diagonal (π, π) direction of the dx2Ày2 and dz2 bands (along grey plane in b).

Whereas the dx2Ày2 band displays no kz dependence beyond matrix element effects, the dz2 band displays a discernible kz dispersion. The iso-energy map below the cube has binding energy E − EF = −1.3 eV. White lines represent the tight-binding model. b, c Tight-binding representation of the Fermi surface (α band) and iso-energy surface (−1.3 eV) of the β band. The colour code indicates the k-dependent orbital hybridisation. The orbital hybridisation at EF is largest in the anti-nodal region of the kz = π/c′ plane where the dz2 admixture at kF amounts to ~1/3

is observed. From the orbital shape, a smaller kz dispersion is Table 1 Tight-binding parameters for single-layer cuprate expected for dx2Ày2 and dxy-derived bands than for those from dz2 materials orbitals. As the β band exhibits a significant kz dispersion (Fig. 4), much larger than observed for the dx2Ày2 band, we conclude that Compound LSCO Hg1201 Tl2201 LSCO it is of dz2 character. The γ′ band which is very close to the γ band Doping p 0.22 0.16 0.26 0.23 is therefore of dxy character. Interestingly, this dz2 -derived band Tight binding parameters in units of tα = −1.21 eV has stronger in-plane than out-of-plane dispersion, suggesting −μ 0.88 1.27 1.35 0.96 that there is a significant hopping to in-plane px and py oxygen ′ −tα 0.13 0.47 0.42 0.32 orbitals. Therefore the assumption that the dz2 states are probed ′′ 21 tα 0.065 0.02 0.02 0.0 uniquely through the apical oxygen pz orbital has to be taken tαβ 0 0 0 0.175 with caution. tβ –– – 0.062 t′ –– – 0.017 β Discussion tβz –– – 0.017 ′ Most minimal models aiming to describe the cuprate physics −tβz –– – 0.0017 Ref. 24 39,40 41,42 This work start with an approximately half-filled single dx2Ày2 band on a two-dimensional square lattice. Experimentally, different Comparison of tight-binding hopping parameters obtained from single-orbital and two-orbital band structures have been observed across single-layer cuprate models. Once a coupling tαβ between the dx2 Ày2 and dz2 band is introduced for La2−xSrxCuO4, the dx2 Ày2 hopping parameters become comparable to those of Hg1201 and Tl2201 compounds. The Fermi surface topology of LSCO is, for example, less rounded compared to (Bi,Pb)2(Sr,La)2CuO6+x (Bi2201), Tl2Ba2CuO6+x (Tl2201) and HgBa2CuO4+x (Hg1201). about the orbital band character. They can be probed in a par- Within a single-band tight-binding model the rounded Fermi ticular experimental setup where a mirror-plane is defined by the surface shape of the single-layerÀÁ compounds Hg1201 and 12 ′ ′′ 6 incident light and the electron analyser slit . With respect to this Tl2201 is described by setting r ¼ tα þ tα =tα 0:4 ,where ′ ′′ plane the electromagnetic light field has odd (even) parity for tα, tα and tα are nearest neighbour (NN), next–nearest neigh- σ (π) polarisation (Supplementary Fig. 4). Orienting the mirror bour (NNN) and next-next–nearest neighbour (NNNN) hop- plane along the nodal direction (cut 1 in Fig. 1), the dz2 and ping parameters (Table 1 and Supplementary Fig. 4). For LSCO dxy (dx2Ày2 ) orbitals have even (odd) parity. For a final-state with with more flat Fermi surface sections, significantly lower values 12 even parity, selection rules dictate that the dz2 and dxy-derived of r have been reported. For example, for overdoped π σ r 24,25 bands should appear (vanish) in the ( ) polarisation channel La1.78Sr0.22CuO4, ~0.2wasfound . The single-band pre- and vice versa for dx2Ày2 . Due to their orientation in real-space, mise thus leads to varying hopping parameters across the the dxz and dyz orbitals are not expected to show a strict switching cuprate families, stimulating the empirical observation that 27 Tmax t′ 2 behaviour along the nodal direction . As shown in Fig. 1f, g, two c roughly scales with α . This, however, is in direct contrast bands (α and γ) appear with σ-polarised light while for to t–J models that predict the opposite correlation28,29.Thus π-polarised light bands β and γ′ are observed. Band α which the single-band structure applied broadly to all single-layer E d γ crosses F is assigned to x2Ày2 while band has to originate from cuprates lead to conclusions that challenge conventional theo- dxz/dyz orbitals as dz2 and dxy-derived states are fully suppressed retical approaches. for σ-polarised light. In the EDM, recorded with π-polarised light, The observation of the dz2 band calls for a re-evaluation of the band (β) at ~1 eV binding energy and again a band (γ′) at ~1.6 eV electronic structure in La-based cuprates using a two-orbital

4 NATURE COMMUNICATIONS | (2018) 9:972 | DOI: 10.1038/s41467-018-03266-0 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03266-0 ARTICLE tight-binding model (see MethodsÂÃ section). Crucially,ÀÁ there is a ARPES experiments. Ultraviolet and soft-X-ray ARPES experiments were carried 43 44 hybridisation term ΨðÞ¼k 2tαβ cosðÞÀkxa cos kyb between outattheSIS and ADRESS beam-lines at the Swiss Light Source and at the I05 beamline at Diamond Light Source. Samples were pre-aligned ex situ using a the dx2Ày2 and dz2 orbitals, where tαβ is a hopping parameter that X-ray LAUE instrument and cleaved in situ—at base temperature (10–20 K) and − characterises the strength of orbital hybridisation. In principle, ultra high vacuum (≤5×10 11 mbar)—employing a top-post technique or one may attempt to describe the two observed bands indepen- cleaving device35. Ultraviolet (soft X-ray36) ARPES spectra were recorded using a dently by taking tαβ = 0. However, the problem then returns to SCIENTA R4000 (SPECS PHOIBOS-150) electron analyser with horizontal (vertical) slit setting. All data was recorded at the cleaving temperature 10–20 K. the single-band description with the above mentioned contra- d – t = To visualise the z2 -dominated band, we subtracted in Fig. 1f, g and Figs. 2 4 the dictions. Furthermore, αβ 0 implies a band crossing in the anti- background that was obtained by taking the minimum intensity of the MDC at nodal direction that is not observed experimentally (Fig. 3). In each binding energy. fact, from the avoided band crossing one can directly estimate tαβ ≈ −200 meV. As dictated by the different eigenvalues of the Tight-binding model. A two-orbital tight-binding model Hamiltonian with orbitals under mirror symmetry, the hybridisation term Ψ(k) symmetry-allowed hopping terms is employed to isolate and characterise the extent of orbital hybridisation of the observed band structure23. For compactness of the vanishes on the nodal lines kx = ±ky (see inset of Fig. 3). Hence momentum-space Hamiltonian matrix representation, we introduce the vectors the pure dx2Ày2 and dz2 orbital band character is expected along > these nodal lines. The hybridisation Ψ(k) is largest in the anti- Qκ ¼ða; κb; 0Þ ; κ ;κ > 1 2 ¼ðκ a; κ κ b; cÞ = ; nodal region, pushing the van-Hove singularity of the upper band R 1 1 2 2 ð Þ κ ;κ > 1 1 2 ¼ð κ a; κ κ b; cÞ = ; close to the Fermi energy and in case of overdoped LSCO across T1 3 1 1 2 2 κ ;κ > the Fermi level. 1 2 ¼ðκ a; κ κ b; cÞ = ; T2 1 3 1 2 2 In addition to the hybridisation parameter tαβ and the chemical μ κ κ κ fi ⊤ potential , six free parameters enter the tight-binding model that where , 1 and 2 take values ±1 as de ned by sums in the Hamiltonian and yields the entire band structure (white lines in Figs. 2 and 4). denotes vector transposition. Nearest and next-nearest in-plane hopping parameters between Neglecting the electron spin (spin–orbit coupling is not considered) the ′ ′ momentum-space tight-binding Hamiltonian, H(k), at a particular momentum dx2Ày2 (tα, tα) and dz2 ðtβ; t Þ orbitals are introduced to capture the β k = (kx, ky, kz) is then given by Fermi surface topology and in-plane dz2 band dispersion (Sup- "# k Mx2Ày2 ðÞ ΨðÞ plementary Fig. 4). The z dispersion is described by nearest and HðÞ¼ k k ; ð Þ ′ k z2 2 next-nearest out-of-plane hoppings (tβz, tβz) of the dz2 orbital. ΨðÞk M ðÞk d μ The four z2 hopping parameters and the chemical potential are ÀÁ c ; c > c determined from the experimental band structure along the nodal in the basis k;x2Ày2 k;z2 , where the operator k,α annihilates an electron with e d α ∈ x2 − y2 z2 direction where Ψ(k) = 0. Furthermore, the α and β band dis- momentum k in an g-orbital α, with { , }. The diagonal matrix entries persion in the anti-nodal region and the Fermi surface topology are given by ′ ÂÃÀÁ t t t fi x2Ày2 provide the parameters α, α and αβ. Our analysis reveals a nite M ðÞ¼k tα ðÞþkxa kyb þ μ 2 cosP cos band coupling tαβ = −0.21 eV resulting in a strong anti-nodal þ t′ ðÞκ Á 2 αcos Q k ð3Þ orbital hybridisation (Fig. 2 and Table 1). Compared to the single- κ¼ ±1ÂÃÀÁ 24 þ t′′ ðÞþk a k b ; band parametrisation a significantly larger value r ~ −0.32 is 2 α cos 2 x cos 2 y ′ found and hence a unification of tα=tα ratios for all single-layer compounds is achieved. and ÂÃÀÁ z2 Finally, we discuss the implication of orbital hybridisation for M ðkÞ¼ tβ ðÞþkxa kyb À μ 2 cosP cos superconductivity and pseudogap physics. First, we notice that a ′ κ þ 2tβ cosðÞQ Á k pronounced pseudogap is found in the anti-nodal region of κ¼P±1 Â κ ;κ þ t ðÞ1 2 Á La −xEu SrxCuO with x = 0.21—consistent with transport 2 βz cos R k ð4Þ 1.8 0.2 4 κ ¼ 30 1;2 ±1 experiments (Supplementary Fig. 5). The fact that tαβ of # P κ ;κ tαβ ′ 1 2 La1.59Eu0.2Sr0.21CuO4 is similar to of LSCO suggests that the þ 2tβz cosðÞTi Á k ; i¼ ; pseudogap is not suppressed by the dz2 hybridisation. To this end, 1 2 a comparison to the 1/4-filled eg system Eu2−xSrxNiO4 with x = 31,32 d d 1.1 is interesting . This material has the same two-orbital band which describe the intra-orbital hopping for x2 Ày2 and z2 orbitals, respectively. The inter-orbital nearest-neighbour hopping term is given by structure with protection against hybridisation along the nodal ÂÃÀÁ lines. Both the dx2Ày2 and dz2 bands are crossing the Fermi level, ΨðÞ¼k 2tαβ cosðÞÀkxa cos kyb : ð5Þ producing two Fermi surface sheets31. Despite an even stronger dz2 admixture of the dx2Ày2 derived band a d-wave-like pseudogap 32 In the above, μ determines the chemical potential. The hopping parameters tα, has been reported . The pseudogap physics thus seems to be ′ ′′ tα and tα characterise NN, NNN and NNNN intra-orbital in-plane hopping unaffected by the orbital hybridisation. ′ between dx2 Ày2 orbitals. tβ and tβ characterise NN and NNN intra-orbital in-plane — ′ It has been argued that orbital hybridisation of the kind hopping between dz2 orbitals, while tβz and t characterise NN and NNN intra- 6,10 βz reported here—is unfavourable for superconducting pairing .It orbital out-of-plane hopping between dz2 orbitals, respectively (Supplementary max t thus provides an explanation for the varying T across single- Fig. 3). Finally, the hopping parameter αβ characterises NN inter-orbital in-plane c d layer cuprate materials. Although other mechanisms, controlled hopping. Note that in our model, x2Ày2 intraorbital hopping terms described by the vectors (Eq. (1)) are neglected as these are expected to be weak compared to by the apical oxygen distance, (e.g. variation of the those of the dz2 orbital. This is due to the fact that the inter-plane hopping is mostly – 4 copper oxygen charge transfer gap ) are not excluded our results mediated by hopping between apical oxygen pz orbitals, which in turn only d d demonstrate that orbital hybridisation exists and is an important hybridise with the z2 orbitals, not with the x2Ày2 orbitals. Such an argument control parameter for superconductivity. highlights that the tight-binding 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 10 in ref. , the Cu dz2 orbital together with the apical oxygen pz orbital forms a Methods Wannier orbital with dz2 symmetry, while the Cu dx2Ày2 orbital together with the Sample characterisation. High-quality single crystals of LSCO, x = 0.12, 0.23, and four neighbouring pσ orbitals of the in-plane oxygen forms a Wannier orbital with x = fl d La1.8−xEu0.2SrxCuO4, 0.21, were grown by the oating-zone technique. The x2 Ày2 symmetry. One should thus think of this tight-binding model as written in samples were characterised by SQUID magnetisation33 to determine super- terms of these Wannier orbitals, thus implicitly containing superexchange hopping T = p conducting transition temperatures ( c 27, 24 and 14 K). For the crystal struc- via the ligand oxygen orbitals. Additionally we stress that all hopping parameters ture, the experimental lattice parameters are a = b = 3.78 Å and c = 2c′ = 13.2 Å34. effectively include the oxygen orbitals. Diagonalising Hamiltonian (2), we find two

NATURE COMMUNICATIONS | (2018) 9:972 | DOI: 10.1038/s41467-018-03266-0 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03266-0 bands 18. Sala, M. M. et al. Energy and symmetry of dd excitations in undoped layered ÂÃ ε ðÞ¼1 Mx2 Ày2 ð ÞþMz2 ð Þ cuprates measured by cu l 3 resonant inelastic x-ray scattering. New J. Phys. ± k 2 k k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð6Þ 13, 043026 (2011). 1 ½Mx2Ày2 ð ÞÀMz2 ð Þ 2þ Ψ2ð Þ 19. Peng, Y. Y. et al. Influence of apical oxygen on the extent of in-plane exchange ± 2 k k 4 k interaction in cuprate superconductors. Nat. Phys. 13, 1201 (2017). 20. Ivashko, O. et al. Damped spin excitations in a doped cuprate superconductor and make the following observations: along the kx = ±ky lines, Ψ(k) vanishes and Phys. Rev. B hence no orbital mixing appears in the nodal directions. The reason for this with orbital hybridization. 95, 214508 (2017). absence of mixing lies in the different mirror eigenvalues of the two orbitals 21. Chen, C. T. et al. Out-of-plane orbital characters of intrinsic and doped holes Phys. Rev. Lett. – involved. Hence it is not an artifact of the finite range of hopping processes in La2−xSrxCuO4. 68, 2543 2546 (1992). included in our model. The parameters of the tight-binding model are 22. Hozoi, L. et al. Ab initio determination of Cu 3d orbital energies in layered determined by fitting the experimental band structure and are provided in copper oxides. Sci. Rep. 1, 65 (2011). Table 1. 23. Bishop, C. B. et al. On-site attractive multiorbital hamiltonian for d-wave superconductors. Phys. Rev. B 93, 224519 (2016). 24. Yoshida, T. et al. Systematic doping evolution of the underlying Fermi surface DFT calculations. DFT calculations were performed for La2CuO4 in the tetra- of La −xSrxCuO . Phys. Rev. B 74, 224510 (2006). gonal space group I4/mmm, No. 139, found in the overdoped regime of 2 4 25. Chang, J. et al. Anisotropic breakdown of Fermi liquid quasiparticle LSCO using the WIEN2K package37. Atomic positions are those inferred from excitations in overdoped La −xSrxCuO . Nat. Commun. 4, 2559 (2013). neutron diffraction measurements34 for x = 0.225. In the calculation, the 2 4 – 26. Zhou, X. J. et al. High-temperature superconductors: universal nodal fermi Kohn Sham equation is solved self-consistently by using a full-potential linear Nature – augmented plane wave (LAPW) method. The self consistent field calculation velocity. 423, 398 398 (2003). 27. Zhang, Y. et al. Orbital characters of bands in the iron-based superconductor converged properly for a uniform k-space grid in the irreducible BZ. The Phys. Rev. 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B 84, 241109 (2011). R = Received: 21 August 2017 Accepted: 1 February 2018 32. Uchida, M. et al. Pseudogap of metallic layered nickelate R2−xSxNiO4 ( Nd,Eu) crystals measured using angle-resolved photoemission spectroscopy. Phys. Rev. Lett. 106, 027001 (2011). 33. Lipscombe, O. J. et al. Persistence of high-frequency spin fluctuations in x = Phys. Rev. Lett. overdoped superconducting La2−xSrxCuO4 ( 0.22). 99, 067002 (2007). References 34. Radaelli, P. G. et al. Structural and superconducting properties of La2 −xSrxCuO as a function of Sr content. Phys. Rev. B 49, 4163–4175 (1994). 1. Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of 4 35. Månsson, M. et al. On-board sample cleaver. Rev. Sci. Instrum. 78, 076103 high-temperature superconductivity. Rev. Mod. Phys. 78,17–85 (2006). (2007). 2. Pavarini, E. et al. Band-structure trend in hole-doped cuprates and correlation 36. Strocov, V. N. et al. Soft-X-ray ARPES facility at the ADRESS beamline of the with Tc . 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Phys. 9, 28 (2007). energy pectroscopies. IBM J. Res. Dev. 33, 372 (1989). 43. Flechsig, U., Patthey, L. & Schmidt, T. Performance measurements at the SLS 9. Ruan, W. et al. Relationship between the parent charge transfer gap and spectroscopy beamline. AIP Conf. Proc. 705, 316 (2004). maximum transition temperature in cuprates. Sci. Bull. 61, 1826–1832 (2016). 44. Strocov, V. N. et al. High-resolution soft X-ray beamline ADRESS at the 10. Sakakibara, H. et al. Origin of the material dependence of Tc in the single- Swiss Light Source for resonant inelastic X-ray scattering and angle- layered cuprates. Phys. Rev. B 85, 064501 (2012). resolved photoelectron spectroscopies. J. Synchrotron Radiat. 17,631–643 11. Slezak, J. A. et al. Imaging the impact on cuprate superconductivity of varying (2010). the interatomic distances within individual crystal unit cells. Proc. Natl. Acad. Sci. USA 105, 3203–3208 (2008). 12. Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission Rev. Mod. Phys. – studies of the cuprate superconductors. 75, 473 541 (2003). Acknowledgements 13. Graf, J. et al. Universal high energy anomaly in the angle-resolved photoemission spectra of high temperature superconductors: possible D.S., D.D., L.D., T.N., C.E.M, C.G.F. and J.C. acknowledge support by the Swiss National evidence of spinon and holon branches. Phys. Rev. Lett. 98, 067004 (2007). Science Foundation. Further, Y.S. and M.M. are supported by the Swedish Research 14. Xie, B. P. et al. High-energy scale revival and giant kink in the dispersion of a Council (VR) through a project (BIFROST, dnr.2016-06955). O.T. acknowledges support cuprate superconductor. Phys. Rev. Lett. 98, 147001 (2007). from the Swedish Research Council as well as the Knut and Alice Wallenberg foundation. 15. Valla, T. et al. High-energy kink observed in the electron dispersion of high- This work was performed at the SIS, ADRESS and I05 beamlines at the Swiss Light temperature cuprate superconductors. Phys. 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Author contributions 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. Open Access This article is licensed under a Creative Commons performed the data analysis. C.E.M. carried out the DFT calculations and A.M.C., C.E.M. Attribution 4.0 International License, which permits use, sharing, and T.N. developed the tight-binding model. All authors contributed to the manuscript. 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 Additional information material in this article are included in the article’s Creative Commons license, unless Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- indicated otherwise in a credit line to the material. If material is not included in the 018-03266-0. 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 Competing interests: The authors declare no competing interests. the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/. Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ © The Author(s) 2018 Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.

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