Abstract
Strong spin–orbit coupling fosters exotic electronic states such as topological insulators and superconductors, but the combination of strong spin–orbit and strong electron–electron interactions is just beginning to be understood. Central to this emerging area are the 5d transition metal iridium oxides. Here, in the pyrochlore iridate Pr_{2}Ir_{2}O_{7}, we identify a nontrivial state with a singlepoint Fermi node protected by cubic and timereversal symmetries, using a combination of angleresolved photoemission spectroscopy and firstprinciples calculations. Owing to its quadratic dispersion, the unique coincidence of four degenerate states at the Fermi energy, and strong Coulomb interactions, nonFermi liquid behaviour is predicted, for which we observe some evidence. Our discovery implies that Pr_{2}Ir_{2}O_{7} is a parent state that can be manipulated to produce other strongly correlated topological phases, such as topological Mott insulator, Weyl semimetal, and quantum spin and anomalous Hall states.
Introduction
Following the discovery of topological insulators^{1,2,3,4}, the next frontier is the regime in which both spin–orbit coupling and correlation effects are strong^{5,6,7,9,10,11,12}. Theory has suggested that the pyrochlore iridates, a family of cubic 5d transition metal oxides^{13,14}, may realize both band inversion, the essential ingredient of topological insulators and strong correlations^{9,12}. Empirical evidence for the latter is plentiful. Notably, Pr_{2}Ir_{2}O_{7} appears to be proximate to an interactiondriven antiferromagnetic quantum critical point tuned by the Asite ionic radius, which is located between the two ions with largest radii, A=Pr and A=Nd^{14,15}. It is the only compound among the series in which the iridium electrons remain paramagnetic and itinerant down to the lowest measured temperatures. It displays bad metallic behaviour and nontrivial spontaneous Hall transport, suggesting strong correlations^{6,15,16}. Moreover, recent thermodynamic measurements have revealed zerofield quantum criticality without tuning^{17}.
The phenomenological suggestion of Moon et al.^{11}, whose implications are summarized in Fig. 1, is that the Fermi surface of Pr_{2}Ir_{2}O_{7} contains a single Fermi node at the Γ point, which emerges as the touching point of two quadratically dispersing conduction and valence bands^{18}. The presence of this touching is actually required by symmetry and group theory (the quadruplet at the zone centre lies in the Γ_{8} representation of the double group of O_{h}), but its location directly at the Fermi energy was an ad hoc theoretical assumption. If the assumption is correct, Pr_{2}Ir_{2}O_{7} becomes a strongly correlated analogue of HgTe^{19,20}, which has a mathematically identical quadratic node at the Fermi energy, and implies that Pr_{2}Ir_{2}O_{7} should be tunable into various topological states (see Fig. 1). Furthermore, theory has predicted that the quadratic nodal semimetal itself is fundamentally altered by longrange Coulomb interactions (negligible in HgTe due to the large dielectric constant, but not so here), becoming a nonFermi liquid state. Thus the Fermi node, if correct, means that Pr_{2}Ir_{2}O_{7} is a natural parent material for strongly interacting topological phases and nonFermi liquid states.
In this paper, we focus on the phenomena of band inversion, and present theoretical calculations and experimental angleresolved photoemission spectroscopy (ARPES) spectra, which support it in the form of electronic structure with a node at the Fermi energy. We also observe strongly temperaturedependent singleparticle spectral weight and lineshape structure in ARPES, which suggest electronic correlations and coupling to collective modes.
Results
Fermi node state expected by band calculations
In support of this proposition, we first present ab initio electronic structure calculations in the paramagnetic state. As detailed in the Supplementary Fig. 1, we show that the quadratic band touching systematically approaches the Fermi level with increasing Asite ionic radius, reaching it for A=Pr. The corresponding band dispersion is shown in Fig. 2g along the highsymmetry lines. The nodal Fermi point and quadratic band touching at Γ is clearly visible at the Fermi energy. The bandwidth is narrower in energy than that of HgTe^{21} by one order of magnitude, reflecting the localized nature of the 5d orbitals in Pr_{2}Ir_{2}O_{7}. Theoretical uncertainty remains, however, as discussed in the Supplementary Note 1 (also see Supplementary Figs 1 and 2), so direct experimental evidence for the unique Fermi node state is strongly desired.
ARPES is a powerful technique to directly observe the electronic structure of matter^{22,23}. One incident photon energy corresponds to one k_{z} value in solid, thus the momentum space observed at a fixed photon energy is limited to a k_{x}–k_{y} plane at a fixed k_{z}. To locate the Γpoint, therefore, sweeping the photon energy is required. Figure 2a,b shows the photos of the cleavage surface with the (111) plane measured by ARPES and the highquality single crystal we used, respectively. While the cleaved surface looks rough in the photo, the scanning electron microscope image of it (Fig. 2c,d) exhibits very flat parts in the multiple locations, which are large enough compared with the photon beam size (∼150 μm, marked with a yellow circle in Fig. 2d). Several k_{x}–k_{y} sheets perpendicular to k_{(111)} measured at different photon energies are coloured in Fig. 2f. The band dispersions obtained by the ab initio calculation for these sheets are plotted in Fig. 2h–j. The experimentally obtained energy dispersion is expected to have a large gap when the k_{(111)} location is far from the Γ. As k_{(111)} is reduced, the gap decreases and eventually vanishes at Γ, where the two parabolic dispersions touch at E_{F} (Fig. 2j).
Observation of Fermi node in Pr_{2}Ir_{2}O_{7} by ARPES
To observe the Fermi node, we measured the ARPES spectra at various photon energies: hν=7 eV (a laser source) and 21.2 eV (a He lamp) in the lab system, and hv=8∼18 eV and 39∼60 eV from two different synchrotron facilities. The results for the 1st Brillouin zone (BZ) are shown in Fig. 3d, where the energy distribution curves (EDCs) along k_{x} (defined in Fig. 3a) are plotted. The observed momentum cut shifts with photon energy along the k_{(111)} axis within a momentum sheet crossing Γ, L and K points (coloured dashed lines in Fig. 3a). The spectra are symmetrized to remove the effect of the Fermi cutoff: the EDC is flipped about E_{F} and added to the original one. This method is widely accepted as a means of determining the presence of a gap (or a gap node). Small but clear quasiparticle peaks are seen for all the spectra, which allows us to determine the energy dispersion, as marked by bars and dotted curves. At hν=7 eV (see Fig. 3d), a large gap of ∼20 meV is opened at k_{x}=0 (green curve), and a parabolic dispersion is obtained. The most significant finding is that the parabolic dispersion moves towards the E_{F} with increasing incident photon energy (or increasing k_{(111)}), and it eventually touches E_{F} at hν=10 eV. This behaviour is more clearly demonstrated in Fig. 3b,c, where the band dispersion determined from the peak energies of spectra at 7, 8, 9 and 10 eV are plotted. As the photon energy is further increased, the dispersion moves away from E_{F} again, following the quadratic dispersion along k_{(111)} as shown in Fig. 3e (also see Supplementary Figs 3 and 4; Supplementary Note 2). We have also examined the dispersion along a different momentum sheet crossing Γ, L and W points (see Supplementary Fig. 5 and Supplementary Note 2) for a different piece of sample. This time the photon energy was swept by a finer step (≤0.5 eV), and the Fermi node was again detected at hν=10.5 eV. While it is not possible to eliminate a slight uncertainty in the exact photon energy, which yields k_{(111)}=0, owing to the broad shape of the spectral peak, our data show that the threedimensional band structure of Pr_{2}Ir_{2}O_{7} has the theoretically predicted Fermi point at the momentum reached by hv∼10 eV, which is thus assigned to be Γ. Other scans of different k_{(111)} values up to the L point in the 1st BZ, which is reached at hν=18 eV, revealed no other states touching or crossing E_{F} (see Supplementary Fig. 3). This absence of other bands crossing E_{F} is another consistency condition on the Fermi node model, which requires this situation by state counting and charge neutrality.
Here we validate our conclusion by making several further checks. First, we investigate another BZ to verify the required repetition of the Fermi nodal state along k_{(111)}. We used higher photon energies, corresponding to the 3rd BZ, and found the Fermi node at the expected Γ point (hν=52 eV) (see Supplementary Fig. 6). In the k_{x}–k_{y} sheet at hν=52 eV (Fig. 4a,b), the ARPES intensities at E_{F} becomes strongest at the zone centre, and the band touching at E_{F} is confirmed in the dispersion maps along k_{x} and k_{y} (Fig. 3d) and the corresponding symmetrized EDCs (Fig. 4f, also see the raw EDCs in Supplementary Fig. 7). In contrast, these features are missing in the k_{x}–k_{y} sheet across L point (hν=39 eV) (see Fig. 4c), where a rather flat, gapped dispersion is observed (Fig. 4e,g).
Second, we demonstrate that our conclusion is insensitive to the different analytic schemes. This is significant especially because the symmetrization technique is relevant for the particle–hole symmetric state, which is unknown in Pr_{2}Ir_{2}O_{7}. Accordingly, we have tested another widely used measure of dividing the ARPES spectra by the FermiDirac function (FD). Figure 5c plots the EDCs along a momentum cut crossing Γ (lightblue arrow in the inset of Fig. 5d) measured at various temperatures. Instead of symmetrization, the curves are divided by the energyresolution convoluted Fermi function at the measurement temperatures to remove the effect of the Fermi cutoff. It is clearly seen that the quadratic dispersion touches E_{F}, in agreement with the earlier analysis. In Fig. 5e, we compare the dispersions determined by the symmetrization and the FDdivision methods, and confirm the consistency between the two (see Supplementary Fig. 8 and Supplementary Note 3 for more details).
Comparison between ARPES results and band calculations
In Fig. 3b, we compare our experimental results near Γ with the ab initio dispersion (grey curve), which has a purely quadratic shape of the electronic structure given by ɛ(k)∝k^{2}. Close to E_{F}, we find an agreement between the two, demonstrating that the quadratic curve (lightblue dotted line) fits well to our data with almost the same effective mass, m_{eff}=6.3m_{0} (m_{0} is the mass of a free electron). On the other hand, for energy below −0.012 eV, the measured dispersion deviates remarkably from the parabolic shape. This contrasts to the calculation, which matches with ɛ(k)∝k^{2} up to much higher binding energies. It is possible that the deviation is due to correlation effects beyond the band calculation. Intriguingly, the total bandwidth in the occupied side is estimated to be ∼40 meV (arrows in Fig. 4f,g), which is actually much narrower than that (>100 meV) of the band calculation (see Fig. 2g and Supplementary Fig. 9e,h). Band narrowing relative to the density functional theory is indeed a wellknown characteristic of correlated electrons. If that is the case, however, the agreement of the effective mass around Γ between the data and calculation would be a coincidence. This is understandable, considering that the band shape of Pr_{2}Ir_{2}O_{7} with comparable energy scales between the spin–orbit interaction and the electron correlations is sensitive to different calculation methods (see Supplementary Note 4; Supplementary Figs 1 and 9).
Another possible cause of the discrepancy between the data and calculations is the strong coupling of electrons to the bosonic modes, which also could significantly renormalize the band shape. Indeed, the peak–dip–hump shape, which is a characteristic signature of strong mode coupling, is seen in EDCs (Fig. 3e), being consistent with this scenario. One of candidates for the bosonic modes is the phonons, which are usually coupled to the correlated systems very strongly. Another candidate is suggested by the similarity of the slightly distorted band shape to measurements in graphene, where it was attributed to electron–plasmon coupling^{24}. Namely, the origin could be the same in the two cases: emission of collective modes through vertical interband transitions becomes possible above a threshold energy, modifying the spectrum.
Correlationdriven anomalous temperature evolution
The strongly correlated feature of 5d electrons of Pr_{2}Ir_{2}O_{7} should be observed in the ARPES spectra. We find such a signature in the temperature evolution of the spectral shape. Figure 5a,b plots Fermifunctiondivided EDCs at Γ (hν=10.5 eV) and off Γ (k_{(111)}=0.31 Å^{−1} reached at hν=21.2 eV). The sharp peak clearly seen in the lowtemperature spectra is strongly suppressed at elevated temperatures. The behaviour is clearly more marked than the thermal broadening effects observed in the other strongly collated systems such as the wellstudied cuprates, which have the ‘marginal’ Fermi liquid state^{25,26}. We note that the peak suppression in the present data is accelerated across ∼100 K, differently from the typical thermal broadening with a gradual increase over a wider temperature range. We find that, associated with the suppression, a large portion of spectral weight above E≃−0.1 eV is transferred to higher binding energies in Pr_{2}Ir_{2}O_{7}. A plausible origin for it is the polaronic effects, which could become crucial in the purely screened electronic systems, as also proposed for the other strongly correlated systems such as the perovskite iridates, manganites and lightly doped cuprates^{27,28,29}. These features are compatible with the nonFermi liquid behaviour predicted theoretically for the Fermi node phase in refs 11, 30, and may also be related to recent observations of quantum critical behaviour in thermodynamic measurements of Pr_{2}Ir_{2}O_{7} (ref. 17). However, a fuller identification of the nonFermi liquid state and explication of its physics in Pr_{2}Ir_{2}O_{7} requires higher resolution data and more elaborate analysis, beyond the scope of this paper.
Quadratic band touching
In passing, we point out that broad spectral weight emerges beyond E_{F} as seen in the Fermifunctiondivided image for the 75K data (arrow in Fig. 5f). The related spectral intensity is obtained off Γ, showing an upturn behaviour beyond E_{F} (arrow in Fig. 5b and Supplementary Figs 10 and 11, and see Supplementary Notes 5 and 6). While the strong suppression of quasiparticle peaks at elevated temperatures prevents us from a definitive determination of the conduction band dispersion, the observation of spectral weight above E_{F} is compatible with the predicted existence of a quadratic band touching on the unoccupied side (Fig. 5d).
Discussion
Prior measurements^{6,16,31} showing ferromagnetic spinicetype correlations among the Pr moments below 2 K may be explained due to unconventional ferromagnetic RKKY interactions arising from the pointlike Fermi surface. The Fermi node also leads to strong sensitivity to small timereversal breaking perturbations, producing Weyl points close to the Fermi energy^{12} and a gigantic anomalous Hall effect. This is also in accord with the experimental fact that Pr_{2}Ir_{2}O_{7} was the first material found to exhibit a large spontaneous Hall effect in a spin liquid state at zero field^{6,31}.
Our results suggest that tuning of a unique quantum critical point between an antiferromagnetic Weyl semimetal and the nonFermi liquid nodal phase may be possible by alloying or hydrostatic pressure. Correlated topological phases and device applications with iridate films could be accessed by controlling the straininduced breaking of the cubic crystal symmetry and size quantization (subband formation) in quantum well structures. We indeed verified theoretically the opening of a significant topological gap with uniaxial compression along the 〈111〉 axis by firstprinciples calculations (see Supplementary Fig. 12 and Supplementary Note 7). This analysis, moreover, shows the presence of three twodimensional surface Dirac cones in this TI state (see Supplementary Fig. 13; Supplementary Table 1; Supplementary Note 7). It will be exciting to investigate whether correlations, neglected in the density function theory, lead to spontaneous timereversal breaking at surfaces^{32,33} with fractional excitations^{34}, or surface topological order^{35,36,37,38}, both of which have been predicted theoretically.
Methods
Samples and ARPES experiments
Single crystals of Pr_{2}Ir_{2}O_{7} with 1 mm^{3} size were grown using a flux method. The sample surface with the (111) plane was prepared by cleaving the single crystal in situ with a top post clued on the crystal.
The ARPES experiments were performed at BL7U of Ultraviolet Synchrotron Orbital Radiation (UVSOR) facility (hv=8∼18 eV) with a MBS A1 electron analyzer^{23}, BL28A of Photon Factory, KEK (hv=39∼60 eV) with a Scienta SES2002 electron analyzer and in our laboratory using a system consisting of a Scienta R4000 electron analyzer equipped with a 6.994eV laser (the 6th harmonic of Nd:YVO_{4} quasicontinuous wave with a repetition rate of 240 MHz)^{22} and He discharge lamp (HeIα, hν=21.2 eV). The overall energy resolution in the ARPES experiment was set to ∼2 meV and ∼6 meV in the lab system with a laser and He lamp, respectively, and ∼15 meV for the synchrotron facility data. The sample orientation was determined with a Laue picture taken before the ARPES experiment.
The intrinsic k_{z} broadening, δk_{z}, is inversely proportional to the photoelectron escape depth λ (δk_{z}=1/λ). We used low photon energies to find the nodal point in the 1st BZ, which enables the bulk sensitive measurements. For example, the photon energy corresponding to the Γ point is around 10 eV, at which the λvalue is estimated to be ∼20 Å according to the ‘universal curve’. It translates into δk_{z} with ∼5 % of the BZ size, which is rather small and sufficient to validate our results. In the Supplementary Fig. 4, we demonstrate that the δk_{z} (or δk_{(111)}) is small enough to resolve the E versus k_{(111)} relation with a 1eV step of photon energy. We also used higher photon energies of ∼50 eV to investigate the 3rd BZ. In this case, the λvalue becomes small (∼5 Å), resulting in a relatively large δk_{z} (∼20 % of the BZ). Nevertheless, we note that the energy dispersion in Pr_{2}Ir_{2}O_{7} is very weak especially around the Γ point (quadratic Fermi node), thus the effects of k_{z} broadening on the quasiparticle peaks should not be critical in our study.
Band calculations
Calculations were carried out using both the standard generalized gradient approximation (GGA) and the Tran–Blaha modified Becke–Johnson (TBmBJ)^{39} exchange potential as implemented in Wien2k, for a series of different Asite ions. An RK_{max} parameter 7.0 was chosen and the wavefunctions were expanded in spherical harmonics up to l_{max}^{wf}=10 inside the atomic spheres and l_{max}^{pot}=4 for nonmuffin tins. Bulk A_{2}Ir_{2}O_{7} (A=rareearth element) has a cubic crystal structure with the space group Fdm. The experimental lattice parameters were used for A=Pr, Nd, Eu and Y in both GGA and TBmBJ paramagnetic calculations, and spin–orbit coupling was applied to both the heavy rareearth element and the Ir electrons. The paramagnetic GGA and TBmBJ calculations put the 4f states of the rareearth element at the Fermi energy; to avoid this, since the 4f electrons are highly localized, their potential is shifted by a constant.
The TBmBJ method is believed to produce improved results for small band gap systems^{40}, and has been widely used in studies of topological insulators^{41}. Both methods (GGA and TBmBJ) yield almost indistinguishable results away from the Fermi energy, with a slight decrease of bandwidth in the TBmBJ calculation, and small differences near E_{F}, as shown in Supplementary Fig. 1. We observed the symmetryrequired nodal band touching at Γ for all calculations, but the Fermi energy is shifted from the nodal point by an amount that decreases with increasing A (A=Y, Eu, Nd and Pr) site ionic radius in both methods^{42}. This occurs because of accidental crossing of states near the L point, where the valence band rises and approaches the Fermi energy, especially in the smaller rare earths. In general, the GGA calculation underestimates the effects of correlations, which tend to narrow the bands and to push occupied states deeper below the Fermi energy. Hence, our expectation is that the accidental crossing near L, which is responsible for the Fermi level shift in GGA, will be suppressed by correlations. The TBmBJ method may be regarded as a crude way to do this. Indeed, in the presumed more accurate TBmBJ method, all the paramagnetic band structures for the A_{2}Ir_{2}O_{7} series show smaller shifts of the Fermi level compared with their GGA counterparts. In the case of A=Pr, we find the shift vanishes and the nodal point occurs precisely at the Fermi energy. While there is a universal trend of a smaller shift of the Fermi energy as rareearth ionic radius increases for the rareearth elements we have tested, the small differences between GGA and TBmBJ suggest that some theoretical uncertainties remain. We note, however, that we expect methods that include correlations more accurately, such as combined local density approximation and dynamical mean field theory (LDA+DMFT), are likely to further suppress the states near the Fermi energy at L even beyond TBmBJ, favouring placing the Fermi level precisely at the node.
Additional information
How to cite this article: Kondo, T. et al. Quadratic Fermi node in a 3D strongly correlated semimetal. Nat. Commun. 6:10042 doi: 10.1038/ncomms10042 (2015).
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Acknowledgements
This work was supported by JSPS KAKENHI (Nos 24740218, 25220707 and 25707030), by the Photon and Quantum Basic Research Coordinated Development Program from MEXT, by PRESTO, Japan Science and Technology Agency, GrantsinAid for Scientific Research (No. 25707030), by GrantsinAids for Scientific Research on Innovative Areas (15H05882 and15H05883), and Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers (No. R2604) from the Japanese Society for the Promotion of Science. L.B. and R.C. were supported by DOE grant DEFG0208ER46524, and E.G.M. was supported by the MRSEC Program of the NSF under Award No. DMR 1121053. We thank D. Hamane for technical assistance in the SEM measurement. The use of the facilities (the Materials Design and Characterization Laboratory and the Electronic Microscope Section) at the Institute for Solid State Physics, The University of Tokyo, is gratefully acknowledged.
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T.K., S.N., L.B. and S.S. designed the experiment. T.K., M.N., H.K., Y.N., T.Y., Y.O., W.M., Y.I., R.Y. and H.Y. carried out the ARPES experiment, and M.M., S.K., N.I., K.O. and H.K. assisted with measurements at the synchrotron radiation facilities. T.K. and M.N. performed the data analysis. J.J.I. and S.N. grew the highquality single crystals. R.C., E.G.M. and L.B. carried out the calculations. T.K., R.C. and L.B. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Kondo, T., Nakayama, M., Chen, R. et al. Quadratic Fermi node in a 3D strongly correlated semimetal. Nat Commun 6, 10042 (2015). https://doi.org/10.1038/ncomms10042
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DOI: https://doi.org/10.1038/ncomms10042
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