Two-dimensional type-II Dirac fermions in layered oxides
Relativistic massless Dirac fermions can be probed with high-energy physics experiments, but appear also as low-energy quasi-particle excitations in electronic band structures. In condensed matter systems, their massless nature can be protected by crystal symmetries. Classification of such symmetry protected relativistic band degeneracies has been fruitful, although many of the predicted quasi-particles still await their experimental discovery. Here we reveal, using angle-resolved photoemission spectroscopy, the existence of two-dimensional type-II Dirac fermions in the high-temperature superconductor LaSrCuO. The Dirac point, constituting the crossing of and bands, is found approximately one electronvolt below the Fermi level () and is protected by mirror symmetry. If spin-orbit 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 type-II Dirac fermions can be realised near .
Dirac fermions are classified into type-I and type-II according to the degree of Lorentz-invariance breaking XuPRL2015 (); Soluyanov2015 (). Type-I Dirac fermions are degeneracy points between an electron-like and a hole-like band that are energetically located above and below the energy of the touching point, respectively. In contrast, type-II Dirac fermions manifest themselves as strongly tilted Dirac cones, where an electron and a hole-like Fermi sheet touch at the energy of the Dirac point. Assuming that the Dirac point is in the vicinity of the Fermi level (), type-I and type-II 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 type-I and type-II Dirac fermions, respectively. In the past decade, two- and three-dimensional type-I Dirac fermions near have been identified in a variety of different systems e.g. graphene NovoselovNat2005 (), topological insulators HasanRMP2010 (), and semimetals such as NaBi LiuScience2014 (), CdAs NeupaneNatCommun2014 (); BorisenkoPRL2014 () and black-phosphorus KimScience2015 (). The concept of topologically protected Dirac fermions has also been applied to the band structure found in high-temperature iron-based superconductors RichardPRL2010 (); TanPRB2016 (). Type-II Dirac fermions seem to be much less common. Their existence has been predicted theoretically in transition-metal icosagenides ChangPRL2017 (), dichalcogenide semimetals MuechlerPRX2016 () and photonic crystals WangNatQMAT2017 (). Only recently, three-dimensional type-II Dirac fermions have been identified experimentally in PtTe YanNatCommun2017 (); ZhangPRB2017 () and PdTe NohPRL2017 (). In these materials, the Dirac cone is tilted along the direction perpendicular to the cleavage plane making it observable only through photon-energy dependent angle-resolved photoemission spectroscopy (ARPES) measurements. Even more recently, it has been reported that this type-II cone can be tuned to by chemical substitution of IrPtTe FeiarXiv2017 ().
Here we report two-dimensional type-II Dirac fermions in the high-temperature cuprate superconductor LaSrCuO.
The cone is found approximately below . There are three important characteristics.
First, the type-II Dirac cone reported here is quasi two-dimensional in nature, and can be viewed as a nodal line, if the band structure is considered three-dimensional.
Second, the tilt is along the nodal in-plane direction.
Third, just as in graphene, the Dirac node degeneracy is lifted when spin-orbit coupling is considered, while the Dirac electrons in PtTe YanNatCommun2017 (); ZhangPRB2017 () are robust against spin-orbit coupling (SOC). We show theoretically that this degeneracy lifting endows the bands with a topological character, namely, a nonvanishing spin-Chern 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 EuSrNiO, the cone is expected above . It is thus demonstrated how oxides are a promising platform for creation of two-dimensional type-II Dirac fermions near where their topological properties become relevant for linear response and interacting instabilities.
Density-functional-theory predictions: Single-layer transition metal oxides often crystallise in the body-centred tetragonal structure. These systems can be doped by chemical substitution on the rare-earth site. Strontium substitution, for example, drives the Mott insulator LaCuO into a superconducting ground state ImadaRMP1998 (). Density functional theory (DFT) calculations (see Methods section) of the LaSrCuO (LSCO) and EuSrNiO(ESNO, ) band structure are displayed in Fig. 1. The shared crystal structure and partially filled bands lead to a similar band structure TaoarXiv2018 (). In particular, both systems display a type-II Dirac cone that is protected by mirror symmetry preventing hybridisation between the and bands along the –M direction in the Brillouin zone CBishopPRB2016 (); MattNatCommun2018 (). For LSCO the cone is found well below CBishopPRB2016 (); HirofumiPRL10 (); HirofumiPRB12 (), whereas for ESNO it is found above. In the case of ESNO, the Dirac cone is thus inaccessible to ARPES experiments UchidaPRL11 (); UchidaPRB11 ().
Enormous efforts have been made to explore the electronic structure of cuprate superconductors DamascelliRMP2003 (). As the quasi-particles responsible for superconductivity are strongly correlated, DFT has widely been considered too simplistic LeeRMP06 (). It has for example been argued using more sophisticated methods that DFT places the band too close to HozoiSREP11 (). Only very recently, the band was directly observed by ARPES MattNatCommun2018 (). Although DFT indeed underestimates the overall 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 two-band tight-binding model (see Methods section) to parametrically describe the observed band structure.
ARPES evidence of type-II Dirac fermions: The low-energy quasi-particle structure of LSCO with is well documented YoshidaPRB06 (); ChangNATC13 (). Its electron-like Fermi surface is shown in Fig. 2a. Figures 2–4 focus on the crossing of the and bands that constitutes the type-II Dirac cone. The band dispersion along the nodal direction carries the most direct experimental signature of the Dirac cone. Along this direction, the two bands with and orbital character are not hybridising and cross at a binding energy of approximately (Fig. 2b). The light polarisation dependence presented in Figs. 2e and 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 self-energy effects are visible. Most noticeable is the waterfall feature, indicated by the energy scales and in Fig. 2f. We stress that this self-energy structure is consistent with previous reports on cuprates GrafPRL2007 (); XiePRL2007 (); MeevasanaPRB2007 (); ChangPRB2007 () and other oxides AizakiPRL2012 (); IwasawaPRL2012 ().
As previously reported in Ref. MattNatCommun2018, and shown in Fig. 3, the band has a weak but clearly detectable dispersion near
the in-plane zone center. This effect translates into a weak dispersion of the Dirac point from
1.4 eV near to 1.2 eV around Z.
As LaSrCuO has body-centred tetragonal structure, the and Z points can be probed simultaneously in constant energy maps that cover first and second in-plane zones (Fig. 4j).
The 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 band disperses faster towards than to Z (see Fig. 4k).
As the binding energy increases, this pocket grows and eventually crosses the dominated band on the nodal line (i.e., the line of Dirac points extended in -direction in momentum space). This happens first at in the second zone near Z (Fig. 4f) and next in the first zone in vicinity to at (Fig. 4g). The type-II Dirac cone thus forms a weakly dispersing
line along the direction. Note that the bands appearing below 1.5 eV around the M point (Figs. 2–4) are of origin MattNatCommun2018 () and irrelevant in this discussion.
Dirac fermions are classified by their dimensionality and the degree to which they break Lorentz invariance (see Table 1). Type-I Dirac fermions break Lorentz invariance such that it is still possible for 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 Type-II Dirac fermions, this is not possible.
Three-dimensional Dirac points are characterised by linearly dispersing bands (around the Dirac point) along all reciprocal directions (,,). For type-I, such Dirac points have been identified in NaBi LiuScience2014 () and CdAs NeupaneNatCommun2014 (); BorisenkoPRL2014 (). Two-dimensional Dirac fermions, by contrast, have linear dispersion in two reciprocal directions. Graphene, being a monolayer of graphite has a perfect two-dimensional band structure. The Dirac cones found in graphene are therefore purely two-dimensional. A three-dimensional version of type-II cones has recently been uncovered in PtTe YanNatCommun2017 (). There, the Dirac cone is defined around a single point in (,,) space. The type-II Dirac cone in LSCO is different since it is found along a line (,) running along the direction. The topological characteristics of the nodal line and a strictly two-dimensional Dirac cone are very similar, for instance, both carry a Berry phase of with respect to any path enclosing them. The observations reported here are, to the best of our knowledge, thus the first demonstration of two-dimensional type-II Dirac fermions. We stress that possible topological boundary modes of the type-II Dirac fermions are obscured by the projections of the bulk bands in the boundary Brillouin zone (see supplementary Fig. 3).
Given the quasi-two-dimensionality, it is imperative to compare our results with graphene. Although SOC is generically 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 KanePRL2005 () demonstrated that graphene, in the presence of SOC, is turned into a topological insulator with spin-Hall conductivity . We stress that this conclusion is independent of the microscopic details. If graphene’s crystal symmetries and time-reversal 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 pre-formed 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 and bands change their order from the to the M point. (This is a precise statement since mirror symmetry , mapping , implies a well-defined orbital character of the bands along the lines .) This is a band inversion of rotation eigenvalues of the lower band between and M, being at and at M (for the spinless case). It has previously been shown that the Chern number of a band can be determined from the rotation eigenvalues of a -symmetric system (mod 4) by the formula FangPRB2012 (), where and are the and eigenvalues of the band in question, respectively, at the indicated high-symmetry points. In the presence of and inversion symmetry, the -component of spin is conserved and we can generalise this formula to the spin Chern number Sheng2006 () of our time-reversal symmetric system. The band inversion then implies that (mod 4) if the degeneracy of the Dirac points is lifted by SOC. (The eigenvalue is irrelevant for this discussion, as it is for both of the orbitals involved.)
|GrapheneNovoselovNat2005 ()||LSCO (This work)|
|3D||CdAsNeupaneNatCommun2014 (); BorisenkoPRL2014 (), NaBiLiuScience2014 ()||PtTeYanNatCommun2017 ()|
In order for this nonvanishing spin Chern number (see also Supplementary Fig. 4 and Supplementary Note 2) to have measurable consequences, the type-II Dirac point should be tuned to . The type-II Dirac node reported here resides 1.3 eV below . This is similar to the Dirac point found in PtTe YanNatCommun2017 () and PdTe NohPRL2017 (). In case of PtTe, chemical substitution of Ir for Pt has been used to position the Dirac point near FeiarXiv2017 (). In a similar fashion, we envision different experimental routes to control the Dirac point position. The position of the band is controlled by the distance between apical oxygen and the CuO plane HirofumiPRL10 (); HirofumiPRB12 (). A smaller -axis lattice parameter is thus pushing the band, and hence the Dirac point, closer to . Uni-axial pressure along the -axis on bulk crystals or substrate induced tensile strain on films are hence useful external tuning parameters. Chemical pressure is yet another possibility. Partial substitution of La for Eu reduces the -axis 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. MattNatCommun2018, , a 20% substitution pushes the band about 200 meV closer to .
Our DFT calculations (Fig. 1) suggest that the sister compound ESNO provides an even better starting point. ESNO is iso-structural to LSCO, and undergoes a metal-insulator-transition at UchidaPRL11 (). The configuration of Ni in combination with considerable Sr doping leads to the filling of orbitals less than in ESNO (), shifting both and bands toward . A soft x-ray ARPES study conducted on ESNO () UchidaPRB11 () has reported an almost unoccupied band and partially filled band. Although the crossing between and bands was not observed in the previous ARPES study, our DFT band calculation displayed in Fig. 1 predicts their crossing slightly above . To bring the Dirac line node down to , chemical substitution of Eu for La is a possibility. Adjusting chemical and external pressure is thus a promising path for realisation of type-II Dirac fermions at . 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 element can be a way to include spin-orbit coupling in the system. Oxides and related compounds are thus promising candidates for type-II Dirac fermions at . The present work demonstrates how oxides, through material design, can be used to realise novel topological protected states.
Experimental specifications: High-quality single crystals of LSCO () were grown by the floating-zone method. The samples with superconducting transition temperature K have previously been used for transport CooperScience2009 (), neutron LipscombePRL07 (); ChangPRB12 () and ARPES ChangNATC13 (); FatuzzoPRB14 (); MattNatCommun2018 () experiments. ARPES experiments were carried out at the Surface/Interface Spectroscopy (SIS) beamline at the Swiss Light Source SIS (). Samples were cleaved in situ at K under ultra high vacuum ( Torr) by employing a top-post technique or by using a cleaving device cleaver (). Ultraviolet ARPES spectra were recorded using a SCIENTA R4000 electron analyser with horizontal slit setting. All the data were recorded at the cleaving temperature 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).
Density functional theory (DFT) calculations were performed for LSCO () and ESNO () in the tetragonal space group I4/mmm using the WIEN2k package. Crystal lattice parameters and atomic positions of LSCO () RadaelliPRB94 () and ESNO () HuiJSSC1998 () were used for the calculation. In order to avoid the generation of unphysically high density of states of 4 electrons near , on-site Coulomb repulsion eV was introduced to Eu 4 orbitals. Calculated band dispersions of ESNO were shifted upwards by 350 meV to reproduce the experimental band structure of ESNO () previously reported in an ARPES study UchidaPRB11 ().
Tight-binding model: We utilise a two-orbital tight-binding model Hamiltonian with symmetry-allowed hopping terms constructed in Ref. MattNatCommun2018, . The momentum-space tight-binding Hamiltonian, , at a particular momentum and for electrons with spin is given by
in the basis , where the operator annihilates an electron with momentum and spin in an -orbital , with . The right-hand side of (1) being independent of indicates that we have neglected spin-orbit coupling initially.
For compactness of the Hamiltonian matrix entries, the following vectors are defined,
where , , and take values as defined by sums in the Hamiltonian and denotes vector transposition. In terms of these, we can write
which describe the intra-orbital hopping for and orbitals, respectively. The inter-orbital nearest-neighbour hopping term is given by
In the above, represents the chemical potential. The hopping parameters and characterise nearest neighbour (NN) and next-nearest neighbour (NNN) intra-orbital in-plane hopping between orbitals. and characterise NN and NNN intra-orbital in-plane hopping between orbitals, while and characterise NN and NNN intra-orbital out-of-plane hopping between orbitals, respectively. Finally, the hopping parameter characterises NN inter-orbital in-plane hopping.
In closing, we discuss the inclusion of spin-orbit coupling in the tight-binding model. The lowest-order spin-orbit coupling term (in terms of in-plane hopping processes) that preserves inversion symmetry, mirror symmetry, rotation symmetry, and time-reversal symmetry is given by
with , and the parameter representing the strength of spin-orbit coupling. Such spin-orbit coupling gaps out the Dirac points and equips the two pairs of spin-degenerate bands with a nonvanishing spin Chern number as discussed in the main text.
The parameters of the tight-binding model, except for the spin-orbit coupling, were determined from fitting to the experimental band structure and are given by , , , , , , , all expressed in units of eV.
All experimental data are available upon request to the corresponding authors.
M.H., 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) through a project (BIFROST, dnr.2016-06955). 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: 200021-169061) and from the European Union’s Horizon 2020 research and innovation program (ERC-StG-Neupert-757867-PARATOP).
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 tight-binding model. M.H., C.E.M., T.N., and J.C. conceived the project. All authors contributed to the manuscript.
The authors declare no competing interests.
Correspondence to: J. Chang (email@example.com) and M. Horio (firstname.lastname@example.org).
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