arXiv:1707.08491v2 [cond-mat.supr-con] 29 May 2018 inlter,i aaersdb two-orbital a by parametrised func- struc- is ( density electronic theory, with This tional consistent observation qualitatively structure. direct ture, two-band provide a that La- of on cuprates angle-resolved experiments based report we spectroscopy Here photoelectron multi- no exists structures. supporting there band evidence because experimental used strong copper-oxygen widely the are of orbitals models low- single-band Effective energy debated. heavily (cuprates) remains copper-oxide materials layered of physics sential sacuilse oad eeln h einprinciples design the mechanism revealing pairing towards the step underlying crucial a is ature superconductivity. for element sation urtsadw ocueta yrdsto,re- hybridisation, that unification conclude straining single-layer a we for to and parameters cuprates leads hopping analysis the electronic Our to of and singularity van-Hove level. topology the Fermi surface of an Fermi proximity provides the the which for hybridisation explanation orbital the tify hretase a ewe h xgnadcpe site copper and oxygen the between gap transfer charge w ieetwy.Frt h distance the First, ways. different two e 3 per aes h pcloye oiinmypa nimpor- temperature an transition play the may CuO defining in position role oxygen copper-oxide tant apical the the within promoted layers, certainly of is variation ductivity compounds dramatic single-layer known the the for explanation an 3 tance rpit odsrb h o nrybn tutr.It ap- structure. that band most predicted energy been are low also models the has two-band describe to or propriate single whether fines a ate / d 4 x dniyn h atr htlmttetasto temper- transition the limit that factors the Identifying -filled 2 h iia nrdet oepantees- the explain to ingredients minimal The − 6 y d T 1 d 2 caernlfstedgnrc ftenn cop- nine the of degeneracy the lifts octahedron eetosadgnrtsflyoccupied fully generates and -electrons / c A 2 ietOsraino ria yrdsto naCpaeSu Cuprate a in Hybridisation Orbital of Observation Direct and -filled 5 e fhg-eprtr urt superconductivity cuprate high-temperature of nttt fPhysics, of Institute oteCuO the to g .Horio, M. .Schmitt, T. d .E Matt, E. C. states wv arn,i nipratoptimi- important an is pairing, -wave 10 d .M Hayden, M. S. 4 z T oa nttt fTcnlg,MtrasPyis SE-1 Physics, Materials Technology, of Institute Royal KTH 1 d 8 eateto ple cecs uoa nttt fTechn of Institute Muroran Sciences, Applied of Department 2 hskIsiu,Universit¨at Z¨urich, Winterthurerstr Physik-Institut, .H il hsc aoaoy nvriyo rso,Bris Bristol, of University Laboratory, Physics Wills H. H. 3 x ih-idn oe.W quan- We model. tight-binding ) eateto hsc n srnm,UpaaUiest,S University, Uppsala Astronomy, and Physics of Department 2 8 − 7 ihicesn pcloye dis- oxygen apical increasing With . eateto dacdMtras nvriyo oy,Kas Tokyo, of University Materials, Advanced of Department y 1 2 2 2 ws ih ore alShre nttt H53 Villig CH-5232 Institut, Scherrer Paul Source, Light Swiss .Destraz, D. 2 ad h distance The band. ln,the plane, 9 ,2 1, .Dudin, P. eateto hsc,Hkad nvriy-Spoo060-0 Sapporo - University Hokkaido Physics, of Department 6 imn ih ore awl aps ictO1 D,UK. 0DE, OX11 Didcot Campus, Harwell Source, Light Diamond .Sutter, D. 8 d cl oyehiu e´rl eLuan EF) Lausann (EPFL), Lausanne Fed´erale de Polytechnique Ecole ´ A .Kurosawa, T. influences 1 6 2 1 e lhuhsupercon- Although . tmyas provide also may It . .G Fatuzzo, G. C. g .Hoesch, M. ttssltt cre- to split states 1 .M Cook, M. A. d A T otosthe controls d 9 c A na least at in .Momono, N. T 6 3 hsde- thus c – t .Pyon, S. 2g 7 5 across The . .Hauser, K. 1 and .Sassa, Y. 7 ,10 9, .Takayama, T. ttsi La in states hc,i un upesssuperconductivity suppresses turn, in which, XS xeiet,sniiet h ncuidstates, of unoccupied hybridisation spectroscopy marginal the absorption only to concluded X-ray sensitive experiments, revealed. (XAS) been not has above ue h oprbysotaia xgndistance oxygen apical short struc- band comparably cuprate The d the of character ture. orbital the reveal rprisi ervcnt oteFrilevel Fermi self-energy the pseudogap, to quasiparticle the vicinity on near and in focused properties gap have majority studies superconducting large ARPES A of experiments. photoelectron angle-resolved (ARPES) by spectroscopy directly identified been between eemn h ria hrce ftesae erthe near states the to ( of challenging energy character is Fermi orbital it however, the Experimentally, determine study. a a enage htfrLC the LSCO for that argued been has hrfr h opigbtenthe between coupling the therefore of layer larger single with Generally, materials directions. opposite in points ever, level Fermi td fthe of study orbital between excitations probe d to bands used these d of scattering been x-ray character inelastic has Resonant orbital addressed. hence never was and origin the the below structures band xeto oti rn r tde fteso-called the of studies are trend structure this waterfall to exception anti- an suggest cuprates between Bi-based correlation of studies STM ever, h arn strength pairing the enobtained been A lvl.I hsfsin nih bu h oiinof position the about insight fashion, z this In -levels. 2 eew rvd ietutavoe n soft-xray and ultra-violet direct provide we Here relation causal these disentangle to quest the In d 1 .Oda, M. , 3 z ae La makes se10 H85 Z¨urich, Switzerland CH-8057 190, asse .Shi, M. 2 .M˚ansson,M. d hbiiainrmisambiguous remains -hybridisation xz d xz , d A , d 9 yz 44 it,Sokom Sweden Stockholm, Kista, 40 64 d 2 o S T,Uie Kingdom United 1TL, BS8 tol lg,Mrrn0088,Japan 050-8585, Muroran ology, 2 d and z 7 − d .Neupert, T. yz 2 , .Kobayashi, M. 2 18 z x − .Takagi, H. hbiiain eedn on depending -hybridisation, 2 E -52 psl,Sweden Uppsala, E-75121 , Sr d x lhuhdffiutt ietnl,it disentangle, to difficult Although . nPI Switzerland PSI, en xy ia2786,Japan 277-8561, hiwa Sr F oetmdpnec smsigand missing is dependence momentum T d x 4 .I at the fact, In ). 13 c xy CuO x ti meaiet experimentally to imperative is it , .Tjernberg, O. d – ttswt epc to respect with states CuO 6 19 17 A , d 10 1,Japan 810, A , 20 and 4 xeietleiec,how- evidence, Experimental . htla oteosrainof observation the to lead that (LSCO) aeide larger a indeed have 4 H11,Switzerland CH-1015, e 7 odt,acomprehensive a date, To . 1 .J Lipscombe, J. O. n .Chang J. and nielcniaefrsuch for candidate ideal an d T 2 x c perconductor 2 .N Strocov, N. V. 11 − . y 21 2 4 hrfr,terole the Therefore, . d band .Das, L. d z z 2 2 d and z 22 adhsnever has band 2 14 1 d . , ee sfound is level x 16 1 d 2 x 5 8 − d However, . 2 d 2 , A 9 y T x − Second, . 2 2 reduces , c y 12 − 2 and 2 How- . y An . bands 2 has d z 2 2

FIG. 1: ARPES spectra showing eg-bands of overdoped La2−xSrxCuO4 x = 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 dx2−y2 (dz2 ) bands. The curve thickness in b and e is scaled to the contribution of the dz2 orbital. Semitransparent lines are guides to the eye. (f)-(g), EDMs along cut 1 recorded withσ ¯ andπ ¯ light, (f) sensitive to the low-energy dx2−y2 and dxz/dyz bands and (g) the dz2 and dxy-derived bands. All data has been recorded with hν = 160 eV.

ARPES measurements of the dz2 band in La-based overdoped compound of La1.8−xEu0.2SrxCuO4 with single layer compounds. The dz2 band is located x =0.21 have been studied. These compounds represent about 1 eV below the Fermi level at the Brillouin zone different crystal structures: low-temperature orthorhom- (BZ) corners. From these corners, the dz2 band is bic (LTO), low-temperature tetragonal (LTT) and the dispersing downwards along the nodal and anti-nodal high-temperature tetragonal (HTT). Our results are directions, consistent with density functional theory very similar across all crystal structures and dopings (DFT) calculations. The experimental and DFT band (see Supplementary Fig. 1). To keep the comparison to structure, including only dx2−y2 and dz2 orbitals, is band structure calculations simple, this paper focuses parametrised using a two-orbital tight-binding model23. on results obtained in the tetragonal phase of overdoped The presence of the dz2 band close to the Fermi level La2−xSrxCuO4 with x =0.23. allows to describe the Fermi surface topology for all single layer compounds (including HgBa2CuO4+x and Tl2Ba2CuO6+x) with similar hopping parameters for the Electronic band structure: A raw ARPES energy dx2−y2 orbital. This unification of electronic parameters distribution map (EDM), along the nodal direction, is implies that the main difference between single layer displayed in Fig. 1a. Near EF, the widely studied nodal cuprates originates from the hybridisation between quasiparticle dispersion with predominately dx2−y2 char- 12 dx2−y2 and dz2 orbitals. The significantly increased acter is observed . This band reveals the previously hybridisation in La-based cuprates pushes the van-Hove reported electron-like Fermi surface of La2−xSrxCuO4, singularity close to the Fermi level. This explains x =0.2324,25 (Fig. 1b), the universal nodal Fermi veloc- 26 why the Fermi surface differs from other single layer ity vF ≈ 1.5 eVA˚ and a band dispersion kink around compounds. We directly quantify the orbital hybridi- 70 meV26. The main observation reported here is the sation that plays a sabotaging role for superconductivity. second band dispersion at ∼ 1 eV below the Fermi level EF (see Fig. 1 and 2) and a hybridisation gap splitting the two (see Fig. 3). This second band – visible in both Results raw momentum distribution curves (MDC) and constant Material choices: Different dopings of La2−xSrxCuO4 energy maps (CEM) – disperses downwards away from spanning from x = 0.12 to 0.23 in addition to an the BZ-corners. Since a pronounced kz-dependence is 3

FIG. 2: Comparison of observed and calculated band structure. (a)-(d), Background subtracted (see methods section) π ′ soft-xray ARPES EDMs recorded on La2−xSrxCuO4, x = 0.23 along in-plane high-symmetry directions for kz = 0 and kz = /c as indicated in g. White lines represent the two-orbital (dz2 and dx2−y2 ) tight-binding model as described in the text. The line width in b and d indicates the orbital weight of the dz2 orbital. (e),(f), Corresponding in-plane DFT band structure at π ′ kz = 0 and kz = /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 dx2−y2 orbitals is found. In contrast, symmetry prevents any hybridisation along the nodal lines (Γ–M or Z–A). (g), Sketch of the 3D BZ of LSCO with high symmetry lines and points as indicated. observed for this band structure (see Figs. 2 and 4) a this plane the electromagnetic light field has odd (even) trivial surface state can be excluded . Subtracting a parity forσ ¯ (¯π) polarisation (see Supplementary Fig. 4). background intensity profile (see Supplementary Fig. 2) Orienting the mirror plane along the nodal direction (cut is a standard method that enhances visualisation of 1 in Fig. 1), the dz2 and dxy (dx2−y2 ) orbitals have even this second band structure. In fact, using soft x-rays (odd) parity. For a final-state with even parity, selec- 12 (160 eV - 600 eV), at least two additional bands (β tion rules dictate that the dz2 and dxy-derived bands and γ) are found below the dx2−y2 dominated band should appear (vanish) in theπ ¯ (¯σ) polarisation chan- crossing the Fermi level. Here, focus is set entirely on nel and vice versa for dx2−y2 . Due to their orientation the β band dispersion closest to the dx2−y2 dominated in real-space, the dxz and dyz orbitals are not expected band. This band is clearly observed at the BZ corners to show a strict switching behavior along the nodal di- 27 (see Figs. 1–3). The complete in-plane (kx, ky) and rection . As shown in Fig. 1f-g, two bands (α and γ) out-of-plane (kz) band dispersion is presented in Fig. 4. appear withσ ¯-polarised light while forπ ¯-polarised light ′ bands β and γ are observed. Band α which crosses EF is assigned to dx2−y2 while band γ has to originate from Orbital band characters: To gain insight into the or- dxz/dyz orbitals as dz2 and dxy-derived states are fully bital character of these bands, a comparison with a suppressed forσ ¯-polarised light. In the EDM, recorded DFT band structure calculation (see methods section) of withπ ¯-polarised light, band (β) at ∼ 1 eV binding en- ergy and again a band (γ′) at ∼ 1.6 eV is observed. From La2CuO4 is shown in Fig. 2. The eg states (dx2−y2 and the orbital shape, a smaller kz dispersion is expected for dz2 ) are generally found above the t2g bands (dxy, dxz, d 2− 2 and d -derived bands than for those from d 2 or- and dyz). The overall agreement between the experiment x y xy z and the DFT calculation (see Supplementary Fig. 3) thus bitals. As the β band exhibits a significant kz dispersion (see Fig. 4), much larger than observed for the dx2−y2 suggests that the two bands nearest to the Fermi level ′ band, we conclude that it is of d 2 character. The γ are composed predominately of dx2−y2 and dz2 orbitals. z This conclusion can also be reached by pure experimen- band which is very close to the γ band is therefore of tal arguments. Photoemission matrix element selection dxy character. Interestingly, this dz2 -derived band has rules contain information about the orbital band char- stronger in-plane than out-of-plane dispersion, suggest- acter. They can be probed in a particular experimen- ing that there is a significant hopping to in-plane px and tal setup where a mirror-plane is defined by the incident py oxygen orbitals. Therefore the assumption that the light and the electron analyser slit12. With respect to dz2 states are probed uniquely through the apical oxy- 4

correlation28,29. Thus the single-band structure applied broadly to all single-layer cuprates lead to conclusions that challenge conventional theoretical approaches.

The observation of the dz2 band calls for a re- evaluation of the electronic structure in La-based cuprates using a two-orbital tight-binding model (see methods section). Crucially, there is a hybridisation term Ψ(k)=2tαβ[cos(kx) − cos(ky)] between the dx2−y2 and dz2 orbitals, where tαβ is a hopping parameter that char- acterises the strength of orbital hybridisation. In princi- ple, one may attempt to describe the two observed bands independently by taking tαβ = 0. However, the prob- lem then returns to the single band description with the above mentioned contradictions. Furthermore, tαβ = 0 implies a band crossing in the antinodal direction that FIG. 3: Avoided band crossing. Left panel: Ultravio- is not observed experimentally (see Fig. 3). In fact, let ARPES data recorded along the antinodal direction using from the avoided band crossing one can directly estimate 160 eV linear horizontal polarised photons. Solid white lines tαβ ≈ 200 meV. As dictated by the different eigenval- are the same tight-binding model as shown in Fig. 2. Right ues of the orbitals under mirror symmetry, the hybridi- 2 2 2 panel: Tight-binding model of the dx −y and dz bands along sation term Ψ(k) vanishes on the nodal lines kx = ±ky the anti-nodal direction. Gray lines are the model prediction (see inset of Fig. 3). Hence the pure dx2−y2 and dz2 in absence of inter-orbital hopping (tαβ = 0) between dx2−y2 orbital band character is expected along these nodal and dz2 . In this case, the bands are crossing near the Γ-point. lines. The hybridisation Ψ(k) is largest in the anti-nodal This degeneracy is lifted once a finite inter-orbital hopping pa- region, pushing the van-Hove singularity of the upper rameter is considered. For solid black lines tαβ = 210 meV and other hopping parameters have been adjusted accord- band close to the Fermi energy and in case of overdoped ingly. Inset indicates the Fermi surface (green line) and the La2−xSrxCuO4 across the Fermi level. − Γ X cut directions. coloured background displays the am- In addition to the hybridisation parameter tαβ and plitude of the hybridisation term Ψ(k) that vanishes on the the chemical potential µ, six free parameters enter the nodal lines. tight-binding model that yields the entire band structure (white lines in Figs. 2 and 4). Nearest and next-nearest ′ in-plane hopping parameters between dx2−y2 (tα, tα) and 21 ′ gen pz orbital has to be taken with caution. dz2 (tβ,tβ) orbitals are introduced to capture the Fermi surface topology and in-plane dz2 band dispersion (see Supplementary Fig. 4). The kz dispersion is described by Discussion ′ Most minimal models aiming to describe the cuprate nearest and next-nearest out-of-plane hoppings (tβz, tβz) physics start with an approximately half-filled single of the dz2 orbital. The four dz2 hopping parameters and the chemical potential µ are determined from the exper- dx2−y2 band on a two-dimensional square lattice. Exper- imentally, different band structures have been observed imental band structure along the nodal direction where across single-layer cuprate compounds. The Fermi Ψ(k) = 0. Furthermore, the α and β band dispersion in the anti-nodal region and the Fermi surface topology surface topology of La2−xSrxCuO4 is, for example, ′ provide the parameters tα, t and tαβ. Our analysis re- less rounded compared to (Bi,Pb)2(Sr,La)2CuO6+x α veals a finite band coupling −tαβ =0.21 eV resulting in (Bi2201), Tl2Ba2CuO6+x (Tl2201), and HgBa2CuO4+x a strong anti-nodal orbital hybridisation (see Fig. 2 and (Hg1201). Within a single-band tight-binding model 24 the rounded Fermi surface shape of the single layer Table I). Compared to the single-band parametrisation a significantly larger value r ∼−0.32 is found and hence compounds Hg1201 and Tl2201 is described by setting ′ ′ ′′ ′ ′′ 6 a unification of tα/tα ratios for all single-layer compounds r = (|tα| + |tα|)/tα ∼ 0.4 , where tα,tα, and tα are nearest, next nearest and next-next nearest neighbour is achieved. hopping parameters (see Table I and Supplementary Finally, we discuss the implication of orbital hy- Fig. 4). For La2−xSrxCuO4 with more flat Fermi bridisation for superconductivity and pseudogap physics. surface sections, significantly lower values of r have been First, we notice that a pronounced pseudogap is found reported. For example, for overdoped La1.78Sr0.22CuO4, in the anti-nodal region of La1.8−xEu0.2SrxCuO4 with r ∼ 0.2 was found24,25. The single-band premise thus x = 0.21 – consistent with transport experiments30 leads to varying hopping parameters across the cuprate (see Supplementary Fig. 5). The fact that tαβ of families, stimulating the empirical observation that La1.59Eu0.2Sr0.21CuO4 is similar to tαβ of LSCO sug- max ′ 2 Tc roughly scales with tα . This, however, is in gests that the pseudogap is not suppressed by the dz2 - direct contrast to t-J models that predict the opposite hybridisation. To this end, a comparison to the 1/4-filled 5

FIG. 4: Three-dimensional band dispersion: (a) kz dispersion recorded along the diagonal (π,π) direction of the dx2−y2 and dz2 bands (along gray plane in b). Whereas the dx2−y2 band display 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 π ′ 1 at EF is largest in the antinodal region of the kz = /c plane where the dz2 admixture at kF amounts to ∼ /3.

31,32 eg system Eu2−xSrxNiO4 with x =1.1 is interesting . to beamline I05 (proposal SI10550-1) that contributed This material has the same two-orbital band structure to the results presented here and thank all the beamline with protection against hybridisation along the nodal staff for technical support. lines. Both the dx2−y2 and dz2 bands are crossing the 31 Fermi level, producing two Fermi surface sheets . De- Authors contributions: SP, TT, HT, TK, NM, spite an even stronger dz2 -admixture of the dx2−y2 de- MO, OJL, and SMH grew and prepared single crystals. 32 rived band a d-wave-like pseudogap has been reported . CEM, DS, LD, MH, DD, CGF, KH, JC, NP, MS, OT, The pseudogap physics thus seems to be unaffected by MK, VS, TS, PD, MH, MM, and YS prepared and the orbital hybridisation. carried out the ARPES experiment. CEM, KH, JC It has been argued that orbital hybridisation – of the performed the data analysis. CEM carried out the kind reported here – is unfavourable for superconducting DFT calculations and AMC, CEM, TN developed the pairing6,10. It thus provides an explanation for the tight-binding model. All authors contributed to the max varying Tc across single layer cuprate materials. manuscript. Although other mechanisms, controlled by the apical oxygen distance, (e.g. variation of the copper-oxygen Methods charge transfer gap4) are not excluded our results Sample characterisation: High-quality single crys- demonstrate that orbital hybridisation exists and is an tals of La2−xSrxCuO4, x = 0.12, 0.23, and important control parameter for superconductivity. La1.8−xEu0.2SrxCuO4, x = 0.21, were grown by the floating-zone technique. The samples were characterised by SQUID magnetisation35 to determine superconduct- Acknowledgements: D.S., D.D., L.D., and J.C. ing transition temperatures (Tc = 27 K, 24 K and acknowledge support by the Swiss National Science 14 K). For the crystal structure, the experimental lattice Foundation. Further, Y.S. and M.M. are supported by parameters are a = b =3.78 A˚ and c =2c′ = 13.2 A˚36. the Swedish Research Council (VR) through a project (BIFROST, dnr.2016-06955). This work was performed at the SIS33, ADRESS34 and I05 beamlines at the Swiss ARPES experiments: Ultraviolet and soft x-ray Light Source and at the Diamond Light Source. A.M.C. ARPES experiments were carried out at the SIS and wishes to thank the Aspen Center for Physics, which ADRESS beam-lines at the Swiss Light Source and at is supported by National Science Foundation grant the I05 beamline at Diamond Light Source. Samples PHY-1066293, for hosting during some stages of this were pre-aligned ex situ using a x-ray LAUE instrument work. We acknowledge Diamond Light Source for access and cleaved in situ – at base temperature (10 - 20 K) 6

−11 and ultra high vacuum (≤ 5 · 10 mbar) – employing which describe the intra-orbital hopping for dx2−y2 and 37 a top-post technique or cleaving device . Ultraviolet dz2 orbitals, respectively. The inter-orbital nearest- (soft x-ray38) ARPES spectra were recorded using a neighbour hopping term is given by SCIENTA R4000 (SPECS PHOIBOS-150) electron analyser with horizontal (vertical) slit setting. All data Ψ (k)= 2tαβ cos kxa − cos kyb . (5) was recorded at the cleaving temperature 10-20 K. h

i To visualize the dz2 -dominated band, we subtracted In the above, µ determines the chemical potential. The in Figs. 1f,g and Figs. 2-4 the background that was ′ ′′ hopping parameters tα, tα and tα characterise nearest obtained by taking the minimum intensity of the MDC neighbor (NN), next-nearest neighbour (NNN) and next- at each binding energy. next-nearest neighbour (NNNN) intra-orbital in-plane ′ hopping between dx2−y2 orbitals. tβ and tβ characterise NN and NNN intra-orbital in-plane hopping between dz2 Tight-binding model: A two-orbital tight-binding ′ model Hamiltonian with symmetry-allowed hopping orbitals, while tβz and tβz characterise NN and NNN terms is employed to isolate and characterise the extent intra-orbital out-of-plane hopping between dz2 orbitals, of orbital hybridisation of the observed band structure23. respectively (see Supplementary Fig. 3). Finally, the For compactness of the momentum-space Hamiltonian hopping parameter tαβ characterises NN inter-orbital in- matrix representation, we introduce the vectors plane hopping. Note that in our model, dx2−y2 intraor- bital hopping terms described by the vectors (Eqs. 1) T Qκ = (a, κ b, 0) , (1) are neglected as these are expected to be weak com- pared to those of the d 2 orbital. This is due to the κ1,κ2 T z R = (κ1a,κ1κ2b,c) /2, fact that the inter-plane hopping is mostly mediated by κ1,κ2 T hopping between apical oxygen p orbitals, which in turn T1 = (3κ1a,κ1κ2b,c) /2, z 2 2 2 κ1,κ2 T only hybridize with the dz orbitals, not with the dx −y T2 = (κ1a, 3κ1κ2b,c) /2, orbitals. Such an argument highlights that the tight- binding model is not written in atomic orbital degrees of where κ, κ1, and κ2 take values ±1 as defined by sums T freedom, but in Wannier orbitals, which are formed from in the Hamiltonian and denotes vector transposition. the Cu d orbitals and the ligand oxygen p orbitals. As Neglecting the electron spin (spin orbit coupling is not follows from symmetry considerations and is discussed considered) the momentum-space tight-binding Hamilto- in Ref. 10, the Cu dz2 orbital together with the api- nian, H (k), at a particular momentum k = (kx, ky, kz) cal oxygen pz orbital forms a Wannier orbital with dz2 is then given by symmetry, while the Cu dx2−y2 orbital together with the 2 2 four neighboring p orbitals of the in-plane oxygen forms x −y σ M (k) Ψ (k) 2 2 H (k)= 2 (2) a Wannier orbital with dx −y symmetry. One should  Ψ (k) M z (k)  thus think of this tight-binding model as written in terms of these Wannier orbitals, thus implicitly containing su- ⊤ in the basis ck,x2−y2 ,ck,z2 , where the operator ck,α perexchange hopping via the ligand oxygen p orbitals. annihilates an electron with
momentum k in an eg-orbital Additionally we stress that all hopping parameters effec- 2 2 2 dα, with α ∈{x − y ,z }. The diagonal matrix entries tively include the oxygen orbitals. Diagonalising Hamil- are given by tonian (2), we find two bands

2− 2 1 x2−y2 z2 M x y (k)=2t cos k a + cos k b + µ ε± (k)= M (k)+ M (k) α x y 2 h i h i (6) ′
κ
2 + 2t cos(Q · k) 1 2 2 2 α (3) ± M x −y (k) − M z (k) + 4Ψ2(k) κX=±1 2rh i ′′ +2tα cos 2kxa + cos 2kyb , and make the following observations: along the kx = ±ky h

i lines, Ψ (k) vanishes and hence no orbital mixing appears and in the nodal directions. The reason for this absence of mixing lies in the different mirror eigenvalues of the two z2 M (k)=2tβ cos(kxa) + cos(kyb) − µ orbitals involved. Hence it is not an artifact of the finite h i range of hopping processes included in our model. The ′ κ + 2tβ cos(Q · k) parameters of the tight-binding model are determined by κX=±1 fitting the experimental bandstructure and are provided in Table I. κ1,κ2 (4) + 2tβz cos(R · k)  κ1X,2=±1 DFT calculations: Density functional theory (DFT) ′ κ1,κ2 + 2t cos(T · k) , calculations were performed for La2CuO4 in the tetrag- βz i  iX=1,2 onal space group I4/mmm, No. 139, found in the 7

the calculation, the Kohn-Sham equation is solved self- Compound LSCO Hg1201 Tl2201 LSCO consistently by using a full-potential linear augmented Doping p 0.22 0.16 0.26 0.23 plane wave (LAPW) method. The self consistent field − Tight Binding Parameters in units of tα= 1.21 eV calculation converged properly for a uniform k-space −µ 0.88 1.27 1.35 0.96 ′ grid in the irreducible BZ. The exchange-correlation −tα 0.13 0.47 0.42 0.32 ′′ term is treated within the generalized gradient ap- tα 0.065 0.02 0.02 0.0 proximation (GGA) in the parametrization of Perdew, −t 0 0 0 0.175 αβ Burke and Enzerhof (PBE)44. The plane wave cut- tβ - - - 0.062 ′ off condition was set to RKmax = 7 where R is the tβ - - - 0.017 tβz - - - 0.017 radius of the smallest LAPW sphere (i.e., 1.63 times − ′ tβz - - - 0.0017 the Bohr radius) and Kmax denotes the plane wave cutoff. Ref. 24 39,40 41,42 This work

TABLE I: Tight-binding parameters for single-layer Data availability. cuprate materials. Comparison of tight-binding hopping All experimental data are available upon request to the parameters obtained from single-orbital and two-orbital mod- corresponding authors. els. Once a coupling tαβ between the dx2−y2 and dz2 band is introduced for La2−xSrxCuO4, the dx2−y2 hopping parame- ters become comparable to those of Hg1201 and Tl2201. Competing financial interest. The authors declare no competing financial interests.

Additional information. Correspondence to: overdoped regime of La2−xSrxCuO4 using the WIEN2K J. Chang ([email protected]) and C. Matt package43. Atomic positions are those inferred from ([email protected]). neutron diffraction measurements36 for x = 0.225. In

1 Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott (2012). insulator: Physics of high-temperature superconductivity. 11 Slezak, J. A. et al. Imaging the impact on cuprate super- Rev. Mod. Phys. 78, 17–85 (2006). conductivity of varying the interatomic distances within 2 Pavarini, E. et al. Band-Structure Trend in Hole-Doped individual crystal unit cells. Proceedings of the National Cuprates and Correlation with Tcmax. Phys. Rev. Lett. Academy of Sciences of the United States of America 105, 87, 047003 (2001). 3203–3208 (2008). 3 Ohta, Y., Tohyama, T. & Maekawa, S. Apex oxygen and 12 Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved critical temperature in copper oxide superconductors: Uni- photoemission studies of the cuprate superconductors. versal correlation with the stability of local singlets. Phys. Rev. Mod. Phys. 75, 473–541 (2003). Rev. B 43, 2968–2982 (1991). 13 Graf, J. et al. Universal high energy anomaly in the 4 Weber, C. et al. Orbital currents in extended hubbard angle-resolved photoemission spectra of high temperature models of high-Tc cuprate superconductors. Phys. Rev. superconductors: Possible evidence of spinon and holon Lett. 102, 017005 (2009). branches. Phys. Rev. Lett. 98, 067004 (2007). 5 Weber, C., Haule, K. & Kotliar, G. Apical oxygens and 14 Xie, B. P. et al. High-energy scale revival and giant kink correlation strength in electron- and hole-doped copper ox- in the dispersion of a cuprate superconductor. Phys. Rev. ides. Phys. Rev. B 82, 125107 (2010). Lett. 98, 147001 (2007). 6 Sakakibara, H. et al. Two-Orbital Model Explains the 15 Valla, T. et al. High-energy kink observed in the electron Higher Transition Temperature of the Single-Layer Hg- dispersion of high-temperature cuprate superconductors. Cuprate Superconductor Compared to That of the La- Phys. Rev. Lett. 98, 167003 (2007). Cuprate Superconductor. Phys. Rev. Lett. 105, 057003 16 Meevasana, W. et al. Hierarchy of multiple many-body (2010). interaction scales in high-temperature superconductors. 7 Raimondi, R., Jefferson, J. H. & Feiner, L. F. Effective Phys. Rev. B 75, 174506 (2007). 17 single-band models for the high-Tc cuprates. II. Role of Chang, J. et al. When low- and high-energy electronic apical oxygen. Phys. Rev. B 53, 8774–8788 (1996). responses meet in cuprate superconductors. Phys. Rev. B 8 Fink, J. et al. Electronic structure studies of high-tc su- 75, 224508 (2007). perconductors by high-energy pectroscopies. IBM J. Res. 18 Sala, M. M. et al. Energy and symmetry of dd excita- Dev. 33, 372 (1989). tions in undoped layered cuprates measured by cu l 3 res- 9 Ruan, W. et al. Relationship between the parent charge onant inelastic x-ray scattering. New Journal of Physics transfer gap and maximum transition temperature in 13, 043026 (2011). cuprates. Science Bulletin 61, 1826–1832 (2016). 19 Peng, Y. Y. et al. Influence of apical oxygen on the extent 10 Sakakibara, H. et al. Origin of the material dependence of of in-plane exchange interaction in cuprate superconduc- Tc in the single-layered cuprates. Phys. Rev. B 85, 064501 tors. Nature Physics EP – (2017). 8

20 Ivashko, O. et al. Damped spin excitations in a doped 41 Plat´e, M. et al. Fermi surface and quasiparticle excitations cuprate superconductor with orbital hybridization. Phys. of overdoped Tl2Ba2CuO6+δ. Phys. Rev. Lett. 95, 077001 Rev. B 95, 214508 (2017). (2005). 21 42 Chen, C. T. et al. Out-of-plane orbital characters of intrin- Peets, D. C. et al. Tl2Ba2CuO6+δ brings spectroscopic sic and doped holes in La2−x Srx CuO4. Phys. Rev. Lett. 68, probes deep into the overdoped regime of the high-Tc 2543–2546 (1992). cuprates. New Journal of Physics 9, 28 (2007). 22 Hozoi, L. et al. Ab initio determination of cu 3d orbital 43 Blaha, P. et al. Wien2k: An Augmented Plane Wave + energies in layered copper oxides. Scientific Reports 1, 65 Local Orbitals Program for Calculating Crystal Properties (2011). (Vienna University of Technology, Vienna, 2001). Wien2k: 23 Bishop, C. B. et al. On-site attractive multiorbital hamilto- An Augmented Plane Wave + Local Orbitals Program for nian for d-wave superconductors. Phys. Rev. B 93, 224519 Calculating Crystal Properties (Vienna University of Tech- (2016). nology, Vienna, 2001) (2001). 24 Yoshida, T. et al. Systematic doping evolution of the un- 44 Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized derlying Fermi surface of La2−xSrxCuO4. Phys. Rev. B gradient approximation made simple. Phys. Rev. Lett. 77, 74, 224510 (2006). 3865–3868 (1996). 25 Chang J. et al. Anisotropic breakdown of Fermi liq- uid quasiparticle excitations in overdoped La2−xSrxCuO4. Nat. Commun. 4, 2559 (2013). 26 Zhou, X. J. et al. High-temperature superconductors: Uni- versal nodal fermi velocity. Nature 423, 398–398 (2003). 27 Zhang, Y. et al. Orbital characters of bands in the iron- based superconductor BaFe1.85Co0.15As2. Phys. Rev. B 83, 054510 (2011). 28 White, S. R. & Scalapino, D. J. Competition between ′ stripes and pairing in a t − t − J model. Phys. Rev. B 60, R753–R756 (1999). 29 Maier, T. et al. d-wave superconductivity in the hubbard model. Phys. Rev. Lett. 85, 1524–1527 (2000). 30 F. Lalibert´e et al. Fermi-surface reconstruction by stripe order in cuprate superconductors. Nat. Commun. 2, 432 (2011). 31 Uchida, M. et al. Orbital characters of three-dimensional fermi surfaces in Eu2−xSrxNiO4 as probed by soft-x-ray angle-resolved photoemission spectroscopy. Phys. Rev. B 84, 241109 (2011). 32 Uchida, M. et al. Pseudogap of metallic layered nickelate R2−xSxNiO4 (R = Nd, Eu) crystals measured using angle- resolved photoemission spectroscopy. Phys. Rev. Lett. 106, 027001 (2011). 33 Flechsig, U., Patthey, L. & Schmidt, T. Performance Mea- surements at the SLS Spectroscopy Beamline. AIP Conf. Proc. 705, 316 (2004). 34 Strocov, V. N. et al. High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectro- scopies. J. Synchrotron Radiat. 17, 631–643 (2010). 35 Lipscombe, O. J. et al. Persistence of high-frequency spin fluctuations in overdoped superconducting La2−xSrxCuO4 (x = 0.22). Phys. Rev. Lett. 99, 067002 (2007). 36 Radaelli, P. G. et al. Structural and superconducting prop- erties of La2−x Srx CuO4 as a function of Sr content. Phys. Rev. B 49, 4163–4175 (1994). 37 M˚ansson, M. et al. On-board sample cleaver. Rev. Sci. Instrum. 78, 076103 (2007). 38 Strocov, V. N. et al. Soft-X-ray ARPES facility at the ADRESS beamline of the SLS: Concepts, technical reali- sation and scientific applications. Journal of Synchrotron Radiation 21, 32–44 (2014). 39 Wang, S. et al. Strain derivatives of Tc in HgBa2CuO4+δ: The cuo2 plane alone is not enough. Phys. Rev. B 89, 024515 (2014). 40 Vishik, I. M. et al. Angle-resolved photoemission spec- troscopy study of HgBa2CuO4+δ. Phys. Rev. B 89, 195141 (2014).