ARTICLE

Direct observation of orbital hybridisation in acuprate superconductorC.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

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 evi-

dence 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: 10.1038/s41467-018-03266-0 OPEN

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. 4Materials 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. 6DiamondLight Source, Harwell Campus, Didcot OX11 0DE, UK. 7 Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, Japan. 8 H. H. WillsPhysics Laboratory, University of Bristol, Bristol BS8 1TL, UK. 9 Department of Physics, Hokkaido University, Sapporo 060-0810, Japan. 10 Department ofApplied Sciences, Muroran Institute of Technology, Muroran 050-8585, Japan. Correspondence and requests for materials should be addressed toC.E.M. (email: cmatt@g.harvard.edu) or to J.C. (email: johan.chang@physik.uzh.ch)

NATURE COMMUNICATIONS | (2018) 9:972 | DOI: 10.1038/s41467-018-03266-0 |www.nature.com/naturecommunications 1

1234

5678

90():,;

Identifying the factors that limit the transition temperature Tcof high-temperature cuprate superconductivity is a crucial steptowards revealing the design principles underlying the pairing

mechanism1. It may also provide an explanation for the dramaticvariation of Tc across the known single-layer compounds2.Although superconductivity is certainly promoted within thecopper-oxide layers, the apical oxygen position may play animportant role in defining the transition temperature3–7. TheCuO6 octahedron lifts the degeneracy of the nine copper 3d-electrons and generates fully occupied t2g and 3/4-filled eg states8.With increasing apical oxygen distance dA to the CuO2 plane, theeg states split to create a 1/2-filled dx2�y2 band. The distance dAthus defines whether single or two-band models are mostappropriate to describe the low-energy band structure. It has alsobeen predicted that dA influences Tc in at least two different ways.First, the distance dA controls the charge transfer gap between theoxygen and copper site which, in turn, suppressessuperconductivity5,9. Second, Fermi-level dz2 hybridisation,depending on dA, reduces the pairing strength6,10. Experimentalevidence, however, points in opposite directions. Generally,single-layer materials with larger dA have indeed a larger Tc2.However, scanning tunneling microscopy (STM) studies of Bi-based cuprates suggest an anti-correlation between dA and Tc11.

In the quest to disentangle these causal relation between dAand Tc, it is imperative to experimentally reveal the orbitalcharacter of the cuprate band structure. The comparably shortapical oxygen distance dA makes La2−xSrxCuO4 (LSCO) an idealcandidate for such a study. Experimentally, however, it ischallenging to determine the orbital character of the states nearthe Fermi energy (EF). In fact, the dz2 band has never beenidentified directly by angle-resolved photoelectron spectroscopy(ARPES) experiments. A large majority of ARPES studies havefocused on the pseudogap, superconducting gap and quasi-particle self-energy properties in near vicinity to the Fermilevel12. An exception to this trend are studies of the so-called

waterfall structure13–17 that lead to the observation of bandstructures below the dx2�y2 band14,16. However, the origin andhence orbital character of these bands was never addressed.Resonant inelastic X-ray scattering has been used to probeexcitations between orbital d-levels. In this fashion, insightabout the position of dz2 , dxz, dyz and dxy states with respect todx2�y2 has been obtained18. Although difficult to disentangle, ithas been argued that for LSCO the dz2 level is found above dxz,dyz and dxy19,20. To date, a comprehensive study of the dz2momentum dependence is missing and therefore the couplingbetween the dz2 and dx2�y2 bands has not been revealed. X-rayabsorption spectroscopy (XAS) experiments, sensitive to theunoccupied states, concluded only marginal hybridisation ofdx2�y2 and dz2 states in LSCO21. Therefore, the role of dz2hybridisation remains ambiguous22.

Here we provide direct ultraviolet and soft-X-ray ARPESmeasurements of the dz2 band in La-based single-layer com-pounds. The dz2 band is located about 1 eV below the Fermi levelat the Brillouin zone (BZ) corners. From these corners, the dz2band is dispersing downwards along the nodal and anti-nodaldirections, consistent with density functional theory (DFT) cal-culations. The experimental and DFT band structure, includingonly dx2�y2 and dz2 orbitals, is parametrised using a two-orbitaltight-binding model23. The presence of the dz2 band close to theFermi level allows to describe the Fermi surface topology for allsingle-layer compounds (including HgBa2CuO4+x and Tl2Ba2-CuO6+x) with similar hopping parameters for the dx2�y2 orbital.This unification of electronic parameters implies that the maindifference between single-layer cuprates originates from thehybridisation between dx2�y2 and dz2 orbitals. The significantlyincreased hybridisation in La-based cuprates pushes the van-Hove singularity close to the Fermi level. This explains why theFermi surface differs from other single-layer compounds. Wedirectly quantify the orbital hybridisation that plays a sabotagingrole for superconductivity.

Max

Max

Min

Min

k y [�

/a]

E –

EF [e

V]

k y [�

/a]

0

–1

1

cut1

XM

X

0

–1

–2

0

–1

–2

0

–1

1

0

–0.5

–1.0

–1.5

–1 0 1–2 –1 0 1 2

E – E

F [eV]

E – E

F [eV]

kll [�/a] kx [�/a] kll [�/a]–1 0 1 –2 –1 1 20

kx [�/a]

E=EF

–0.9 eV –1.3 eV

–0.45 eV

Γ

cut1

�'

�

�

�

�– - pol

�–- pol

a b c

d e

f

g

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 theMDC 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) asindicated. The photoemission intensity, shown in false colour scale, is integrated over ± 10 meV. Black (red) lines indicate the position of dx2�y2 dz2ð Þbands. 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 1recorded with σ and π light, f sensitive to the low-energy dx2�y2 and dxz/dyz bands and g the dz2 and dxy-derived bands. All data have been recorded withhν= 160 eV

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ResultsMaterial choices. Different dopings of LSCO spanning fromx= 0.12 to 0.23 in addition to an overdoped compound ofLa1.8−xEu0.2SrxCuO4 with x= 0.21 have been studied. Thesecompounds represent different crystal structures: low-temperature orthorhombic, low-temperature tetragonal and thehigh-temperature tetragonal. Our results are very similar acrossall crystal structures and dopings (Supplementary Fig. 1). To keepthe comparison to band structure calculations simple, this paperfocuses on results obtained in the tetragonal phase of overdopedLSCO with x= 0.23.

Electronic band structure. A raw ARPES energy distributionmap (EDM), along the nodal direction, is displayed in Fig. 1a.Near EF, the widely studied nodal quasiparticle dispersion withpredominately dx2�y2 character is observed12. This band revealsthe previously reported electron-like Fermi surface of LSCO, x=0.2324,25 (Fig. 1b), the universal nodal Fermi velocity vF ≈ 1.5eVÅ26 and a band dispersion kink around 70 meV26. The mainobservation reported here is the second band dispersion at ~1 eVbelow the Fermi level EF (Figs. 1 and 2) and a hybridisation gapsplitting the two (Fig. 3). This second band—visible in both rawmomentum distribution curves (MDC) and constant energymaps—disperses downwards away from the BZ corners. Since apronounced kz dependence is observed for this band structure(Figs. 2 and 4) a trivial surface state can be excluded. Subtractinga background intensity profile (Supplementary Fig. 2) is a stan-dard method that enhances visualisation of this second bandstructure. In fact, using soft X-rays (160–600 eV), at least twoadditional bands (β and γ) are found below the dx2�y2 dominatedband crossing the Fermi level. Here, focus is set entirely on the βband dispersion closest to the dx2�y2 dominated band. This bandis clearly observed at the BZ corners (Figs. 1–3). The complete in-plane (kx, ky) and out-of-plane (kz) band dispersion is presentedin Fig. 4.

Orbital band characters. To gain insight into the orbital char-acter of these bands, a comparison with a DFT band structure

calculation (see Methods section) of La2CuO4 is shown in Fig. 2.The eg states (dx2�y2 and dz2 ) are generally found above the t2gbands (dxy, dxz and dyz). The overall agreement between theexperiment and the DFT calculation (Supplementary Fig. 3) thussuggests that the two bands nearest to the Fermi level are com-posed predominately of dx2�y2 and dz2 orbitals. This conclusioncan also be reached by pure experimental arguments. Photo-emission matrix element selection rules contain information

XMM M

kz = � - plane

kz = 0 - plane

X XMin Max dz2 -weight TB

dxydz 2

dxz/yzdx 2– y 2

X

2

1

0

–1

–2

0

–1

–2

2

1

0

–1

–2

E –

EF [e

V]

E –

EF [e

V]

0

–1

–2

E –

EF [e

V]

E –

EF [e

V]

ZR

A

X

Mkz

ky

kx

ΓΓΓ M X

�

�

Γ

A AR RX A RZ RAR Z

�/�'

a b

c d

e

f

g

Fig. 2 Comparison of observed and calculated band structure. a–d Background subtracted (see Methods section) soft-X-ray ARPES EDMs recorded onLa2−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 anddx2�y2 ) 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 DFTband structure at kz= 0 and kz= π/c′, calculated for La2CuO4 (see Methods section). The colour code indicates the orbital character of the bands. Aroundthe 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

Min –4t��

t�� = 0 eV

t�� = –0.21 eV

~2t��

X

M

Γ

4t��Max

�– - pol

E –

EF [e

V]

0.0

–0.5

–1.0

–1.5

–2.0

X XΓ

Fig. 3 Avoided band crossing. Left panel: ultraviolet ARPES data recordedalong the ant-inodal direction using 160 eV linear horizontal polarisedphotons. Solid white lines are the same tight-binding model as shown inFig. 2. Right panel: tight-binding model of the dx2�y2 and dz2 bands along theanti-nodal direction. Grey lines are the model prediction in absence of inter-orbital hopping (tαβ= 0) between dx2�y2 and dz2 . In this case, the bands arecrossing near the Γ-point. This degeneracy is lifted once a finite inter-orbitalhopping parameter is considered. For solid black lines tαβ=−210meV andother hopping parameters have been adjusted accordingly. Inset indicatesthe Fermi surface (green line) and the Γ− X cut directions. Colouredbackground displays the amplitude of the hybridisation term Ψ(k) thatvanishes on the nodal lines

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about the orbital band character. They can be probed in a par-ticular experimental setup where a mirror-plane is defined by theincident light and the electron analyser slit12. With respect to thisplane the electromagnetic light field has odd (even) parity forσ (π) polarisation (Supplementary Fig. 4). Orienting the mirrorplane along the nodal direction (cut 1 in Fig. 1), the dz2 anddxy (dx2�y2 ) orbitals have even (odd) parity. For a final-state witheven parity, selection rules12 dictate that the dz2 and dxy-derivedbands should appear (vanish) in the π (σ) polarisation channeland vice versa for dx2�y2 . Due to their orientation in real-space,the dxz and dyz orbitals are not expected to show a strict switchingbehaviour along the nodal direction27. As shown in Fig. 1f, g, twobands (α and γ) appear with σ-polarised light while forπ-polarised light bands β and γ′ are observed. Band α whichcrosses EF is assigned to dx2�y2 while band γ has to originate fromdxz/dyz orbitals as dz2 and dxy-derived states are fully suppressedfor σ-polarised light. In the EDM, recorded with π-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 kz dispersion isexpected for dx2�y2 and dxy-derived bands than for those from dz2orbitals. As the β band exhibits a significant kz dispersion (Fig. 4),much larger than observed for the dx2�y2 band, we conclude thatit is of dz2 character. The γ′ band which is very close to the γ bandis therefore of dxy character. Interestingly, this dz2 -derived bandhas stronger in-plane than out-of-plane dispersion, suggestingthat there is a significant hopping to in-plane px and py oxygenorbitals. Therefore the assumption that the dz2 states are probeduniquely through the apical oxygen pz orbital21 has to be takenwith caution.

DiscussionMost minimal models aiming to describe the cuprate physicsstart with an approximately half-filled single dx2�y2 band ona two-dimensional square lattice. Experimentally, differentband structures have been observed across single-layer cupratecompounds. The Fermi surface topology of LSCO is, forexample, less rounded compared to (Bi,Pb)2(Sr,La)2CuO6+x

(Bi2201), Tl2Ba2CuO6+x (Tl2201) and HgBa2CuO4+x (Hg1201).Within a single-band tight-binding model the rounded Fermisurface shape of the single-layer compounds Hg1201 andTl2201 is described by setting r ¼ t′α

�� ��þ t′′α�� ��� �

=tα � 0:46, wheretα, t′α and t′′α are nearest neighbour (NN), next–nearest neigh-bour (NNN) and next-next–nearest neighbour (NNNN) hop-ping parameters (Table 1 and Supplementary Fig. 4). For LSCOwith more flat Fermi surface sections, significantly lower valuesof r have been reported. For example, for overdopedLa1.78Sr0.22CuO4, r ~ 0.2 was found24,25. The single-band pre-mise thus leads to varying hopping parameters across thecuprate families, stimulating the empirical observation thatTmaxc roughly scales with t′α

2. This, however, is in direct contrastto t–J models that predict the opposite correlation28,29. Thusthe single-band structure applied broadly to all single-layercuprates lead to conclusions that challenge conventional theo-retical approaches.

The observation of the dz2 band calls for a re-evaluation of theelectronic structure in La-based cuprates using a two-orbital

–0.5

24500 eV

550 eV

600 eVhv=

25

26

–1.0

–2.0

–2–1.5

–1.0–0.5

0.5

Min

Max

0

E –

EF

[eV

]

–1.5

k|| [�/a]

EB = –1.3 eV

k z [�

/c']

XM

Γ

E = EF EB = –1.3 eV

ky

kx

kz

dx2-y

2 dz2

b ca

Fig. 4 Three-dimensional band dispersion. a kz dispersion recorded along the diagonal (π, π) direction of the dx2�y2 and dz2 bands (along grey plane in b).Whereas the dx2�y2 band displays no kz dependence beyond matrix element effects, the dz2 band displays a discernible kz dispersion. The iso-energy mapbelow 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 islargest in the anti-nodal region of the kz= π/c′ plane where the dz2 admixture at kF amounts to ~1/3

Table 1 Tight-binding parameters for single-layer cupratematerials

CompoundDoping p

LSCO0.22

Hg12010.16

Tl22010.26

LSCO0.23

Tight binding parameters in units of tα=−1.21 eV

−μ 0.88 1.27 1.35 0.96−t′α 0.13 0.47 0.42 0.32t′′α 0.065 0.02 0.02 0.0tαβ 0 0 0 0.175tβ – – – 0.062t′β – – – 0.017tβz – – – 0.017−t′βz – – – 0.0017Ref. 24 39,40 41,42 This work

Comparison of tight-binding hopping parameters obtained from single-orbital and two-orbitalmodels. Once a coupling tαβ between the dx2�y2 and dz2 band is introduced for La2−xSrxCuO4, thedx2�y2 hopping parameters become comparable to those of Hg1201 and Tl2201

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tight-binding model (see Methods section). Crucially, there is ahybridisation term Ψ kð Þ ¼ 2tαβ cos kxað Þ � cos kyb

� �� �between

the dx2�y2 and dz2 orbitals, where tαβ is a hopping parameter thatcharacterises the strength of orbital hybridisation. In principle,one may attempt to describe the two observed bands indepen-dently by taking tαβ= 0. However, the problem then returns tothe single-band description with the above mentioned contra-dictions. Furthermore, tαβ= 0 implies a band crossing in the anti-nodal direction that is not observed experimentally (Fig. 3). Infact, from the avoided band crossing one can directly estimatetαβ ≈−200meV. As dictated by the different eigenvalues of theorbitals under mirror symmetry, the hybridisation term Ψ(k)vanishes on the nodal lines kx= ±ky (see inset of Fig. 3). Hencethe pure dx2�y2 and dz2 orbital band character is expected alongthese nodal lines. The hybridisation Ψ(k) is largest in the anti-nodal region, pushing the van-Hove singularity of the upper bandclose to the Fermi energy and in case of overdoped LSCO acrossthe Fermi level.

In addition to the hybridisation parameter tαβ and the chemicalpotential μ, six free parameters enter the tight-binding model thatyields the entire band structure (white lines in Figs. 2 and 4).Nearest and next-nearest in-plane hopping parameters betweendx2�y2 (tα, t′α) and dz2 ðtβ; t′βÞ orbitals are introduced to capture theFermi surface topology and in-plane dz2 band dispersion (Sup-plementary Fig. 4). The kz dispersion is described by nearest andnext-nearest out-of-plane hoppings (tβz, t′βz) of the dz2 orbital.The four dz2 hopping parameters and the chemical potential μ aredetermined from the experimental band structure along the nodaldirection where Ψ(k)= 0. Furthermore, the α and β band dis-persion in the anti-nodal region and the Fermi surface topologyprovide the parameters tα, t′α and tαβ. Our analysis reveals a finiteband coupling tαβ=−0.21 eV resulting in a strong anti-nodalorbital hybridisation (Fig. 2 and Table 1). Compared to the single-band parametrisation24 a significantly larger value r ~−0.32 isfound and hence a unification of t′α=tα ratios for all single-layercompounds is achieved.

Finally, we discuss the implication of orbital hybridisation forsuperconductivity and pseudogap physics. First, we notice that apronounced pseudogap is found in the anti-nodal region ofLa1.8−xEu0.2SrxCuO4 with x= 0.21—consistent with transportexperiments30 (Supplementary Fig. 5). The fact that tαβ ofLa1.59Eu0.2Sr0.21CuO4 is similar to tαβ of LSCO suggests that thepseudogap is not suppressed by the dz2 hybridisation. To this end,a comparison to the 1/4-filled eg system Eu2−xSrxNiO4 with x=1.1 is interesting31,32. This material has the same two-orbital bandstructure with protection against hybridisation along the nodallines. Both the dx2�y2 and dz2 bands are crossing the Fermi level,producing two Fermi surface sheets31. Despite an even strongerdz2 admixture of the dx2�y2 derived band a d-wave-like pseudogaphas been reported32. The pseudogap physics thus seems to beunaffected by the orbital hybridisation.

It has been argued that orbital hybridisation—of the kindreported here—is unfavourable for superconducting pairing6,10. Itthus provides an explanation for the varying Tmax

c across single-layer cuprate materials. Although other mechanisms, controlledby the apical oxygen distance, (e.g. variation of thecopper–oxygen charge transfer gap4) are not excluded our resultsdemonstrate that orbital hybridisation exists and is an importantcontrol parameter for superconductivity.

MethodsSample characterisation. High-quality single crystals of LSCO, x= 0.12, 0.23, andLa1.8−xEu0.2SrxCuO4, x= 0.21, were grown by the floating-zone technique. Thesamples were characterised by SQUID magnetisation33 to determine super-conducting transition temperatures (Tc= 27, 24 and 14 K). For the crystal struc-ture, the experimental lattice parameters are a= b= 3.78 Å and c= 2c′= 13.2 Å34.

ARPES experiments. Ultraviolet and soft-X-ray ARPES experiments were carriedout at the SIS43 and ADRESS44 beam-lines at the Swiss Light Source and at theI05 beamline at Diamond Light Source. Samples were pre-aligned ex situ using aX-ray LAUE instrument and cleaved in situ—at base temperature (10–20 K) andultra high vacuum (≤5 × 10−11 mbar)—employing a top-post technique orcleaving device35. Ultraviolet (soft X-ray36) ARPES spectra were recorded using aSCIENTA R4000 (SPECS PHOIBOS-150) electron analyser with horizontal(vertical) slit setting. All data was recorded at the cleaving temperature 10–20 K.To visualise the dz2 -dominated band, we subtracted in Fig. 1f, g and Figs. 2–4 thebackground that was obtained by taking the minimum intensity of the MDC ateach binding energy.

Tight-binding model. A two-orbital tight-binding model Hamiltonian withsymmetry-allowed hopping terms is employed to isolate and characterise the extentof orbital hybridisation of the observed band structure23. For compactness of themomentum-space Hamiltonian matrix representation, we introduce the vectors

Qκ ¼ ða; κb; 0Þ>;Rκ1 ;κ2 ¼ ðκ1a; κ1κ2b; cÞ>=2;Tκ1 ;κ21 ¼ ð3κ1a; κ1κ2b; cÞ>=2;

Tκ1 ;κ22 ¼ ðκ1a; 3κ1κ2b; cÞ>=2;

ð1Þ

where κ, κ1 and κ2 take values ±1 as defined by sums in the Hamiltonian and ⊤denotes vector transposition.

Neglecting the electron spin (spin–orbit coupling is not considered) themomentum-space tight-binding Hamiltonian, H(k), at a particular momentumk= (kx, ky, kz) is then given by

H kð Þ ¼ Mx2�y2 kð Þ Ψ kð ÞΨ kð Þ Mz2 kð Þ

" #; ð2Þ

in the basis ck;x2�y2 ; ck;z2� �>

, where the operator ck,α annihilates an electron withmomentum k in an eg-orbital dα, with α∈ {x2− y2, z2}. The diagonal matrix entriesare given by

Mx2�y2 kð Þ ¼ 2tα cos kxað Þ þ cos kyb� �� �þ μ

þ Pκ¼ ± 1

2t′αcos Qκ � kð Þ

þ2t′′α cos 2kxað Þ þ cos 2kyb� �� �

;

ð3Þ

and

Mz2 ðkÞ ¼ 2tβ cos kxað Þ þ cos kyb� �� �� μ

þ Pκ¼± 1

2t′β cos Qκ � kð Þ

þ Pκ1;2¼± 1

2tβz cos Rκ1 ;κ2 � kð Þ�

þ Pi¼1;2

2t′βz cos Tκ1 ;κ2i � kð Þ

#;

ð4Þ

which describe the intra-orbital hopping for dx2�y2 and dz2 orbitals, respectively.The inter-orbital nearest-neighbour hopping term is given by

Ψ kð Þ ¼ 2tαβ cos kxað Þ � cos kyb� �� �

: ð5Þ

In the above, μ determines the chemical potential. The hopping parameters tα,t′α and t′′α characterise NN, NNN and NNNN intra-orbital in-plane hoppingbetween dx2�y2 orbitals. tβ and t′β characterise NN and NNN intra-orbital in-planehopping between dz2 orbitals, while tβz and t′βz characterise NN and NNN intra-orbital out-of-plane hopping between dz2 orbitals, respectively (SupplementaryFig. 3). Finally, the hopping parameter tαβ characterises NN inter-orbital in-planehopping. Note that in our model, dx2�y2 intraorbital hopping terms described bythe vectors (Eq. (1)) are neglected as these are expected to be weak compared tothose of the dz2 orbital. This is due to the fact that the inter-plane hopping is mostlymediated by hopping between apical oxygen pz orbitals, which in turn onlyhybridise with the dz2 orbitals, not with the dx2�y2 orbitals. Such an argumenthighlights that the tight-binding model is not written in atomic orbital degrees offreedom, but in Wannier orbitals, which are formed from the Cu d orbitals and theligand oxygen p orbitals. As follows from symmetry considerations and is discussedin ref. 10, the Cu dz2 orbital together with the apical oxygen pz orbital forms aWannier orbital with dz2 symmetry, while the Cu dx2�y2 orbital together with thefour neighbouring pσ orbitals of the in-plane oxygen forms a Wannier orbital withdx2�y2 symmetry. One should thus think of this tight-binding model as written interms of these Wannier orbitals, thus implicitly containing superexchange hoppingvia the ligand oxygen p orbitals. Additionally we stress that all hopping parameterseffectively include the oxygen orbitals. Diagonalising Hamiltonian (2), we find two

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bands

ε± kð Þ ¼ 12 Mx2�y2 ðkÞ þMz2 ðkÞ� �± 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMx2�y2 ðkÞ �Mz2 ðkÞ½ �2þ4Ψ2ðkÞ

q ; ð6Þ

and make the following observations: along the kx= ±ky lines, Ψ(k) vanishes andhence no orbital mixing appears in the nodal directions. The reason for thisabsence of mixing lies in the different mirror eigenvalues of the two orbitalsinvolved. Hence it is not an artifact of the finite range of hopping processesincluded in our model. The parameters of the tight-binding model aredetermined by fitting the experimental band structure and are provided inTable 1.

DFT calculations. DFT calculations were performed for La2CuO4 in the tetra-gonal space group I4/mmm, No. 139, found in the overdoped regime ofLSCO using the WIEN2K package37. Atomic positions are those inferred fromneutron diffraction measurements34 for x= 0.225. In the calculation, theKohn–Sham equation is solved self-consistently by using a full-potential linearaugmented plane wave (LAPW) method. The self consistent field calculationconverged properly for a uniform k-space grid in the irreducible BZ. Theexchange-correlation term is treated within the generalised gradient approx-imation in the parametrisation of Perdew, Burke and Enzerhof38. The planewave cutoff condition was set to RKmax= 7 where R is the radius of the smallestLAPW sphere (i.e. 1.63 times the Bohr radius) and Kmax denotes the plane wavecutoff.

Data availability. All experimental data are available upon request to the corre-sponding authors.

Received: 21 August 2017 Accepted: 1 February 2018

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AcknowledgementsD.S., D.D., L.D., T.N., C.E.M, C.G.F. and J.C. acknowledge support by the Swiss NationalScience Foundation. Further, Y.S. and M.M. are supported by the Swedish ResearchCouncil (VR) through a project (BIFROST, dnr.2016-06955). O.T. acknowledges supportfrom 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 LightSource and at the Diamond Light Source. A.M.C. wishes to thank the Aspen Center forPhysics, which is supported by National Science Foundation grant PHY-1066293, forhosting during some stages of this work. We acknowledge Diamond Light Source foraccess to beamline I05 (proposal SI10550-1) that contributed to the results presentedhere and thank all the beamline staff for technical support.

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03266-0

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

Author contributionsS.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 tight-binding model. All authors contributed to the manuscript.

Additional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-018-03266-0.

Competing interests: The authors declare no competing interests.

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NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03266-0 ARTICLE

NATURE COMMUNICATIONS | (2018) 9:972 | DOI: 10.1038/s41467-018-03266-0 |www.nature.com/naturecommunications 7