Abstract
Relativistic massless Dirac fermions can be probed with highenergy physics experiments, but appear also as lowenergy quasiparticle excitations in electronic band structures. In condensed matter systems, their massless nature can be protected by crystal symmetries. Classification of such symmetryprotected relativistic band degeneracies has been fruitful, although many of the predicted quasiparticles still await their experimental discovery. Here we reveal, using angleresolved photoemission spectroscopy, the existence of twodimensional typeII Dirac fermions in the hightemperature superconductor La_{1.77}Sr_{0.23}CuO_{4}. The Dirac point, constituting the crossing of \(d_{x^2  y^2}\) and \(d_{z^2}\) bands, is found approximately one electronvolt below the Fermi level (E_{F}) and is protected by mirror symmetry. If spinorbit coupling is considered, the Dirac point degeneracy is lifted and the bands acquire a topologically nontrivial character. In certain nickelate systems, band structure calculations suggest that the same typeII Dirac fermions can be realised near E_{F}.
Introduction
Dirac fermions are classified into typeI and typeII according to the degree of Lorentzinvariance breaking^{1,2}. TypeI Dirac fermions are degeneracy points between an electronlike and a holelike band that are energetically located above and below the energy of the touching point, respectively. In contrast, typeII Dirac fermions manifest themselves as strongly tilted Dirac cones, where an electron and a holelike Fermi sheet touch at the energy of the Dirac point. Assuming that the Dirac point is in the vicinity of the Fermi level (E_{F}), typeI and typeII Dirac fermions display distinct physical properties. Many of them originate from the fact that the density of states at the Dirac node is vanishing and finite for typeI and typeII Dirac fermions, respectively. In the past decade, twodimensional and threedimensional typeI Dirac fermions near E_{F} have been identified in a variety of different systems, e.g., graphene^{3}, topological insulators^{4}, and semimetals such as Na_{3}Bi^{5}, Cd_{3}As_{2}^{6,7}, and black phosphorus^{8}. The concept of topologically protected Dirac fermions has also been applied to the band structure found in hightemperature ironbased superconductors^{9,10}. TypeII Dirac fermions seem to be much less common. Their existence has been predicted theoretically in transitionmetal icosagenides^{11}, dichalcogenide semimetals^{12}, and photonic crystals^{13}. Only recently, threedimensional typeII Dirac fermions have been identified experimentally in PtTe_{2}^{14,15} and PdTe_{2}^{16}. In these materials, the Dirac cone is tilted along the direction perpendicular to the cleavage plane making it observable only through photon energydependent angleresolved photoemission spectroscopy (ARPES) measurements. Even more recently, it has been reported that this typeII cone can be tuned to E_{F} by chemical substitution of Ir_{1−x}Pt_{x}Te_{2}^{17}.
Here we report twodimensional typeII Dirac fermions in the hightemperature cuprate superconductor La_{1.77}Sr_{0.23}CuO_{4}. The cone is found approximately 1 eV below E_{F}. There are three important characteristics. First, the typeII Dirac cone reported here is quasi twodimensional in nature, and can be viewed as a nodal line, if the band structure is considered threedimensional. Second, the tilt is along the nodal inplane direction. Third, just as in graphene, the Dirac node degeneracy is lifted when spinorbit coupling (SOC) is considered, while the Dirac electrons in PtTe_{2}^{14,15} are robust against SOC. We show theoretically that this degeneracy lifting endows the bands with a topological character, namely, a nonvanishing spinChern number. As known from graphene, SOC is, however, negligibly small for light elements such as copper and oxygen. Guided by band structure calculations, we suggest that the position of the Dirac cone can be tuned through chemical substitution. In Eu_{0.9}Sr_{1.1}NiO_{4}, the cone is expected above E_{F}. It is thus demonstrated how oxides are a promising platform for creation of twodimensional typeII Dirac fermions near E_{F}, where their topological properties become relevant for linear response and interacting instabilities.
Results
Density functional theory (DFT) predictions
Singlelayer transition metal oxides often crystallise in the bodycentred tetragonal structure. These systems can be doped by chemical substitution on the rareearth site. Strontium substitution, for example, drives the Mott insulator La_{2}CuO_{4} into a superconducting ground state^{18}. DFT calculations (see Methods section) of the La_{2−x}Sr_{x}CuO_{4} (LSCO) and Eu_{2−x}Sr_{x}NiO_{4} (ESNO, x = 1.1) band structure are displayed in Fig. 1. The shared crystal structure and partially filled e_{g} bands lead to a similar band structure^{19}. In particular, both systems display a typeII Dirac cone that is protected by mirror symmetry preventing hybridisation between the \(d_{z^2}\) and \(d_{x^2  y^2}\) bands along the Γ–M direction in the Brillouin zone^{20,21}. For LSCO the cone is found well below E_{F}^{20,22,23}, whereas for ESNO it is found above. In the case of ESNO, the Dirac cone is thus inaccessible to ARPES experiments^{24,25}.
Enormous efforts have been made to explore the electronic structure of cuprate superconductors^{26}. As the quasiparticles responsible for superconductivity are strongly correlated, DFT has widely been considered too simplistic^{27}. It has, for example, been argued using more sophisticated methods that DFT places the \(d_{z^2}\) band too close to E_{F}^{28}. Only very recently, the \(d_{z^2}\) band was directly observed by ARPES^{21}. Although DFT indeed underestimates the overall \(d_{z^2}\) band position, it captures the observed band structure in essence, in particular as far as qualitative features protected by symmetry or topology go. To account for differences between the DFT calculation and the experiment, we use a twoband tightbinding model (see Methods section) to parametrically describe the observed band structure.
ARPES evidence of typeII Dirac fermions
The lowenergy quasiparticle structure of LSCO with x = 0.23 is well documented^{29,30}. Its electronlike Fermi surface is shown in Fig. 2a. Figures 2–4 focus on the crossing of the \(d_{z^2}\) and \(d_{x^2  y^2}\) bands that constitutes the typeII Dirac cone. The band dispersion along the nodal (k_{x} = k_{y}, 0) direction carries the most direct experimental signature of the Dirac cone. Along this direction, the two bands with \(d_{x^2  y^2}\) and \(d_{z^2}\) orbital character are not hybridising and cross at a binding energy of approximately 1.4 eV (Fig. 2b). The light polarisation dependence presented in Figs. 2e, f indicates the opposite mirror symmetry of the two bands and hence that the crossing is indeed protected by the crystal symmetry. A perpendicular cut through this Dirac point is shown in Fig. 2c. Along both cuts, significant selfenergy effects are visible. Most noticeable is the waterfall feature, indicated by the energy scales E_{1} and E_{2} in Fig. 2f. We stress that this selfenergy structure is consistent with previous reports on cuprates^{31,32,33,34} and other oxides^{35,36}.
As previously reported in ref. ^{21} and shown in Fig. 3, the \(d_{z^2}\) band has a weak but clearly detectable k_{z} dispersion near the inplane zone centre. This effect translates into a weak k_{z} dispersion of the Dirac point from 1.4 eV near Γ to 1.2 eV around Z. As La_{1.77}Sr_{0.23}CuO_{4} has bodycentred tetragonal structure, the Γ and Z points can be probed simultaneously in constantenergy maps that cover first and second inplane zones (Fig. 4j). The \(d_{z^2}\) dominated band enters for binding energies of approximately 1 eV (Fig. 4d) as an elongated pocket centred around the zone corner. This “cigar” contour stems from the fact that the \(d_{z^2}\) band disperses faster towards Γ = (0, 0, 0) than to Z = (0, 0, 2π/c) (see Fig. 4k). As the binding energy increases, this pocket grows and eventually crosses the \(d_{x^2  y^2}\) dominated band on the nodal line (i.e., the line of Dirac points extended in k_{z}direction in momentum space). This happens first at 1.2 eV in the second zone near Z (Fig. 4f) and next in the first zone in vicinity to Γ at 1.4 eV (Fig. 4g). The typeII Dirac cone thus forms a weakly dispersing line along the k_{z} direction. Note that the bands appearing below 1.5 eV around the M point (Figs. 2–4) are of d_{xz/yz} origin^{21} and irrelevant in this discussion.
Discussion
Dirac fermions are classified by their dimensionality and the degree to which they break Lorentz invariance (see Table 1). TypeI Dirac fermions break Lorentz invariance such that it is still possible for E_{F} to intersect the bands forming the Dirac point at only the Dirac point when considering sufficiently small regions of momentum space about the point. For typeII Dirac fermions, this is not possible.
Threedimensional Dirac points are characterised by linearly dispersing bands (around the Dirac point) along all reciprocal directions (k_{x}, k_{y}, k_{z}). For typeI, such Dirac points have been identified in Na_{3}Bi^{5} and Cd_{3}As_{2}^{6,7}. Twodimensional Dirac fermions, by contrast, have linear dispersion in two reciprocal directions. Graphene, being a monolayer of graphite, has a perfect twodimensional band structure. The Dirac cones found in graphene are therefore purely twodimensional. A threedimensional version of typeII cones has recently been uncovered in PtTe_{2}^{14}. There, the Dirac cone is defined around a single point in (k_{x}, k_{y}, k_{z}) space. The typeII Dirac cone in LSCO is different since it is found along a line (k_{x} = k_{y} ≈ 0.25, k_{z}) running along the k_{z} direction. The topological characteristics of the nodal line and a strictly twodimensional Dirac cone are very similar, for instance, both carry a Berry phase of π with respect to any path enclosing them. The observations reported here are, to the best of our knowledge, thus the first demonstration of twodimensional typeII Dirac fermions. We stress that possible topological boundary modes of the typeII Dirac fermions are obscured by the projections of the bulk bands in the boundary Brillouin zone (see Supplementary Fig. 3).
Given the quasitwodimensionality, it is imperative to compare our results with graphene. Although SOC is generally small in graphene and the cuprates, it is of conceptual importance to understand the fate of the Dirac electrons if SOC is considered. The seminal work of Kane and Mele^{37} demonstrated that graphene, in the presence of SOC, is turned into a topological insulator with spinHall conductivity \(\sigma _{xy}^{\mathrm{s}} = e/2\pi\). We stress that this conclusion is independent of the microscopic details. If graphene’s crystal symmetries and timereversal symmetry are to be preserved by SOC, the only perturbative way to open a gap leads to a topological band structure. The reason for this is a preformed band inversion at the M points in the band structure of graphene, away from the nodal Dirac points. Turning to the Dirac cones discussed in this work, a very similar analysis can be made. Already without SOC, the \(d_{x^2  y^2}\) and \(d_{z^2}\) bands change their order from the Γ to the M point (this is a precise statement since mirror symmetry M_{xy}, mapping \((x,y)\,\mapsto\,(y,x)\), implies a welldefined orbital character of the bands along the lines k_{x} = k_{y}). This is a band inversion of C_{4} rotation eigenvalues of the lower band between Γ and M, being −1 at Γ and +1 at M (for the spinless case). It has previously been shown that the Chern number C of a band can be determined from the rotation eigenvalues of a C_{4}symmetric system (mod 4) by the formula i^{C} = ξ(Γ)ξ(M)ζ(X)^{38}, where ξ and ζ are the C_{4} and C_{2} eigenvalues of the band in question, respectively, at the indicated highsymmetry points. In the presence of M_{xy} and inversion symmetry, the zcomponent of spin is conserved and we can generalise this formula to the spin Chern number C_{s}^{39} of our timereversal symmetric system. The band inversion then implies that C_{s} = 2 (mod 4) if the degeneracy of the Dirac points is lifted by SOC. (The C_{2} eigenvalue ζ(X) is irrelevant for this discussion, as it is +1 for both of the orbitals involved.)
In order for this nonvanishing spin Chern number (see also Supplementary Fig. 4 and Supplementary Note 2) to have measurable consequences, the typeII Dirac point should be tuned to E_{F}. The typeII Dirac node reported here resides ~1.3 eV below E_{F}. This is similar to the Dirac point found in PtTe_{2}^{14} and PdTe_{2}^{16}. In the case of PtTe_{2}, chemical substitution of Ir for Pt has been used to position the Dirac point near E_{F}^{17}. In a similar fashion, we envision different experimental routes to control the Dirac point position. The position of the \(d_{z^2}\) band is controlled by the distance between apical oxygen and the CuO_{2} plane^{22,23}. A smaller caxis lattice parameter is thus pushing the \(d_{z^2}\) band, and hence the Dirac point, closer to E_{F}. Uniaxial pressure along the caxis on bulk crystals or substrateinduced tensile strain on films are hence useful external tuning parameters. Chemical pressure is yet another possibility. Partial substitution of Eu for La reduces the caxis lattice parameter. This effect is a simple consequence of the fact that the atomic volume of Eu is smaller than La. As shown in the Supplementary Information of ref. ^{21}, a 20% substitution pushes the \(d_{z^2}\) band about 200 meV closer to E_{F}.
Our DFT calculations (Fig. 1) suggest that the sister compound ESNO provides an even better starting point. ESNO is isostructural to LSCO, and undergoes a metalinsulator transition at x ~ 1.0^{24}. The d^{8} configuration of Ni^{2+} in combination with considerable Sr doping leads to the filling of e_{g} orbitals less than 1/4 in ESNO (x > 1.0), shifting both \(d_{x^2  y^2}\) and \(d_{z^2}\) bands toward E_{F}. A soft Xray ARPES study conducted on ESNO (x = 1.1)^{25} has reported an almost unoccupied \(d_{z^2}\) band and partially filled \(d_{x^2  y^2}\) band. Although the crossing between \(d_{x^2  y^2}\) and \(d_{z^2}\) bands was not observed in the previous ARPES study, our DFT band calculation displayed in Fig. 1 predicts their crossing slightly above E_{F}. To bring the Dirac line node down to E_{F}, chemical substitution of La for Eu is a possibility. Adjusting chemical and external pressure is thus a promising path for realisation of typeII Dirac fermions at E_{F}. Most likely, the electron correlations found in the nickelate and cuprate systems will be preserved irrespective of the pressure tuning. It might thus be possible to create a strongly correlated topologically protected state. In addition, replacing the transition metal (Ni or Cu) with a 5d element can be a way to include SOC in the system. Oxides and related compounds are thus promising candidates for typeII Dirac fermions at E_{F}. The present work demonstrates how oxides, through material design, can be used to realise novel topological protected states.
Methods
Experimental specifications
Highquality single crystals of LSCO (x = 0.23) were grown by the floatingzone method. The samples with superconducting transition temperature T_{c} = 24 K have previously been used for transport^{40}, neutron^{41,42}, and ARPES^{21,30,43} experiments. ARPES experiments were carried out at the Surface/Interface Spectroscopy (SIS) beamline at the Swiss Light Source^{44}. Samples were cleaved in situ at ~20 K under ultra high vacuum (≤5 × 10^{−11} Torr) by employing a toppost technique or by using a cleaving device^{45}. Ultraviolet ARPES spectra were recorded using a SCIENTA R4000 electron analyser with horizontal slit setting. All the data were recorded at the cleaving temperature ~20 K. For better visualisation of energy distribution maps, a background defined by the minimum MDC intensity at each binding energy was subtracted (see Supplementary Figs. 1, 2, and Supplementary Note 1).
DFT calculations
DFT calculations were performed for LSCO (x = 0) and ESNO (x = 0) in the tetragonal space group I4/mmm using the WIEN2k package. Crystal lattice parameters and atomic positions of LSCO (x = 0.225)^{46} and ESNO (x = 1.0)^{47} were used for the calculation. In order to avoid the generation of unphysically high density of states of 4f electrons near E_{F}, onsite Coulomb repulsion U = 14 eV was introduced to Eu 4f orbitals. Calculated band dispersions of ESNO were shifted upwards by 350 meV to reproduce the experimental band structure of ESNO (x = 1.1) previously reported in an ARPES study^{25}.
Tightbinding model
We utilise a twoorbital tightbinding model Hamiltonian with symmetryallowed hopping terms constructed in ref. ^{21}. The momentumspace tightbinding Hamiltonian, H_{σ}(k), at a particular momentum k = (k_{x}, k_{y}, k_{z}) and for electrons with spin σ = ↑/↓ is given by:
in the basis \(\left( {c_{\sigma ,{\mathbf{k}},x^2  y^2},c_{\sigma ,{\mathbf{k}},z^2}} \right)^{\rm T}\), where the operator c_{σ,k,α} annihilates an electron with momentum k and spin σ in an e_{g}orbital d_{α}, with α ∈ {x^{2} − y^{2}, z^{2}}. The righthand side of (1) being independent of σ indicates that we have neglected SOC initially.
For compactness of the Hamiltonian matrix entries, the following vectors are defined:
where κ, κ_{1}, and κ_{2} take values ±1 as defined by sums in the Hamiltonian and T denotes vector transposition. In terms of these, we can write
and
which describe the intraorbital hopping for \(d_{x^2  y^2}\) and \(d_{z^2}\) orbitals, respectively. The interorbital nearestneighbour hopping term is given by:
In the above, μ represents the chemical potential. The hopping parameters t_{α} and t_{α′} characterise nearestneighbour (NN) and next nearestneighbour (NNN) intraorbital inplane hopping between \(d_{x^2  y^2}\) orbitals. t_{β} and t_{β′} characterise NN and NNN intraorbital inplane hopping between \(d_{z^2}\) orbitals, while t_{βz} and \(t_{\beta z}^\prime\) characterise NN and NNN intraorbital outofplane hopping between \(d_{z^2}\) orbitals, respectively. Finally, the hopping parameter t_{αβ} characterises NN interorbital inplane hopping.
In closing, we discuss the inclusion of SOC in the tightbinding model. The lowestorder SOC term (in terms of inplane hopping processes) that preserves inversion symmetry, M_{xy} mirror symmetry, \(C_4^z\) rotation symmetry, and timereversal symmetry is given by:
with g_{↑} = +1, g_{↓} = −1 and the parameter λ representing the strength of SOC. Such SOC gaps out the Dirac points and equips the two pairs of spindegenerate bands with a nonvanishing spin Chern number as discussed in the main text.
The parameters of the tightbinding model, except for the SOC, were determined from fitting to the experimental band structure and are given by μ = −0.97, \(t_{\alpha} ^{\prime} =  0.31\), t_{αβ} = −0.175, t_{β} = 0.057, \(t_{\beta} ^{\prime} = 0.010\), t_{βz} = 0.014, \(t_{\beta {z}}^{\prime} =  0.005\), all expressed in units of t_{α} = −1.13 eV.
Data availability
All experimental data are available upon request to the corresponding authors.
Change history
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Acknowledgements
M.H., K. K., D.S., and J.C. acknowledge support by the Swiss National Science Foundation. Y.S. and M.M. are supported by the Swedish Research Council (VR) and the European Commission (projects Dnr.20146426, Dnr.201606955 and Dnr.201705078) as well as the Carl Tryggers Foundation for Scientific Research (CTS16:324). This work was performed at the SIS beamline at the Swiss Light Source, and we thank all technical beamline staffs. T.N. acknowledges support from the Swiss National Science Foundation (grant number: 200021169061) and from the European Union’s Horizon 2020 research and innovation programme (ERCStGNeupert757867PARATOP).
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O.J.L. and S.M.H. grew and prepared single crystals. M.H., C.E.M., K.K., D.S., Y.S., K.H., M.M., N.C.P., M.S., and J.C. prepared and carried out the ARPES experiment. M.H. and C.E.M. analysed the data. M.H. and K.K. carried out the DFT calculations. A.M.C., C.E.M., and T.N. developed the tightbinding model. M.H., C.E.M., T.N., and J.C. conceived the project. All authors contributed to the manuscript.
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Horio, M., Matt, C.E., Kramer, K. et al. Twodimensional typeII Dirac fermions in layered oxides. Nat Commun 9, 3252 (2018). https://doi.org/10.1038/s41467018057152
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DOI: https://doi.org/10.1038/s41467018057152
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