From Trivial Kondo Insulator Ce{}_{3}Pt{}_{3}Bi{}_{4} to Topological Nodal-line Semimetal Ce{}_{3}Pd{}_{3}Bi{}_{4}

From Trivial Kondo Insulator CePtBi to Topological Nodal-line Semimetal CePdBi

Chao Cao Condensed Matter Group, Department of Physics, Hangzhou Normal University, Hangzhou 310036, P. R. China Center of Correlated Materials, Zhejiang University, Hangzhou 310058, China    Guo-Xiang Zhi Department of Physics, Zhejiang University, Hangzhou 310013, P. R. China    Jian-Xin Zhu Theoretical Division and Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
July 26, 2019
preprint: 1

The electronic structure of strongly correlated systems is one of the most intriguing problems in condensed matter physics. Taking heavy fermion systems as an example, its electronic structure exhibits strong temperature dependence and Kondo effect, due to the subtle interplay between the hybridization of -electrons with the itinerant band electrons and strong Coulomb repulsion among the -electrons. Recently, there are renewed interests in these systems due to the discovery of electron band topology Hasan and Kane (2010); Qi and Zhang (2011). In addition to possible topological Kondo insulators (TKIs) Kim et al. (2014); Chang et al. (2017), the possibility of Weyl Kondo semimetals WKSs has been brought up Lai et al. (2018); Guo et al. (2017); Chang and Coleman (2018); Guo et al. (2018); Dzsaber et al. (2017). In particular, the proposal of a truly heavy-fermion WKS Lai et al. (2018) has stimulated intensive theoretical and experimental studies Chang and Coleman (2018); Guo et al. (2018); Dzsaber et al. (2017). Experimentally, it has been argued that a TKI-WKS phase transition could be realized from CePtBi to CePdBi Dzsaber et al. (2017, 2018). Here, using state-of-art first-principles method based on density functional theory (DFT) and its combination with dynamical mean-field theory (DMFT) Georges et al. (1996); Kotliar et al. (2006), we show that the CePtBi is a topologically trivial Kondo insulator; while CePdBi is a topological nodal-line Kondo semimetal, which is protected by the non-symmorphic symmetry of the crystal.

Figure 1: Electronic structure of CePtBi. (a) The DFT+DMFT band structure at 18 K. (b) The DFT band structure with itinerant Ce-4 states. The contributions from Ce-4, Bi-6 and Pt-5 states are represented by green, blue and red circles, respectively. (c) The DFT+DMFT band structures at 290 K, 72 K, 36 K and 18 K. (d) The DFT+DMFT density of states at 290 K, 72 K, 36 K and 18 K. The solid black line is total density of states, while the green solid line, red dashed line, and blue dotted line represent Ce-4, Ce-4, and Ce-4 contributions, respectively.

Both CePtBi and CePdBi compounds are body-centered cubic crystals with symmetry group (No. 220), which contains 6 gliding mirror symmetry operations. Our main results of CePtBi is shown in Fig. 1. At high temperatures (=290 K), the Ce-4 electrons are localized, leading to incoherent -bands near the Fermi energy , while coherent conduction bands can still be identified between -H, H-N, and P-H (Fig. 1c), thus the system is metallic. Density of states (DOS) shows two small humps near the Fermi level, formed by the Ce-4 at and Ce-4 380 meV above the Fermi energy, respectively. As the temperature is lowered to 72 K, the flat Ce-4 band begin to form around 10 meV above , which strongly hybridizes with the conduction bands. The DOS shows sharper Ce-4 peak, but without any trace of hybridization gap at 72 K. At 36 K, the hybridization gap is already present, as the DOS shows a clear dip around the Fermi level . Therefore, the coherence temperature is expected to be between 36K and 72K from our calculation. Experimentally, the hybridization gap starts to form below 100 K, and its maximum value is measured to be approximately 50 K (or 4.3 meV) Bucher et al. (1994); Hundley et al. (1990); Severing et al. (1991); Cooley et al. (1997); Wakeham et al. (2016). In addition, the band structure is more coherent near at 36 K, clearly showing an energy gap at . The band structure is completely coherent below 18 K, and an indirect gap of 6 meV can be identified from both DOS and band spectrum plots. In all these DFT+DMFT calculations, the Ce-4 occupation ranges from around 1.00 at 290 K to 0.98 at 18 K, suggesting negligible valence fluctuation. Therefore, the system is consistent with the Kondo lattice model, where the -electrons can be treated as localized spins coupled to conduction electrons via a Kondo exchange interaction.

We notice that the coherent band structure at 18K from DFT+DMFT calculation is indeed very similar to DFT results with itinerant Ce-4 states (Fig. 1b). The DFT band structure yields an insulating ground state with 140 meV indirect gap near . The highest occupied state at is the four-fold degenerate state in DFT calculation. It splits into two doubly degenerate states along -H line. At H point, these four states are joined by the doubly degenerate state from at around 0.2 eV below at , as well as another doubly degenerate states stemming from the split state at 0.4 eV below , to form an 8-fold degenerate state as the highest occupied state. This is exactly the same result we obtained for the coherent band structure at 18 K from DFT+DMFT calculations. Such analysis can be conducted along all high symmetry lines we presented, although the band energies are strongly renormalized in the DFT+DMFT calculations. This is consistent with the fact that the states close to is dominated by Ce-4 orbitals. As a result, the DFT calculation yields an insulating ground state with an indirect gap of 140 meV, which is about 20 times larger of the DFT+DMFT result at 18 K.

The great similarities between the DFT result and the DFT+DMFT result below the coherence temperature suggest that the coherent DFT+DMFT band structure can be adiabatically transformed from DFT result without closing the energy gap. Therefore, the topological properties of CePtBi coherent band structure from the DFT+DMFT is anticipated to be the same as its DFT band structure. It is worth noting that the global energy gap is already present in the calculations for CePtBi even when the spin-orbit coupling is turned off (Supplementary Information (SI) Fig. S-1), and is therefore not related to the band inversion. Therefore, the CePtBi is expected to be a Kondo insulator with trivial band topology. In fact, we have fitted the DFT band structure to a tight-binding Hamiltonian using the maximally localized Wannier function method Souza et al. (2001), and performed Wilson loop calculations Yu et al. (2011). The resulting is (0;000) for CePtBi, showing its trivial topology nature. Therefore, the experimentally observed resistivity saturation must have an origin other than topological arguments, and robust surface states are not guaranteed. Indeed, the Hall measurement suggested the conduction of CePtBi at low temperature is dominated by bulk in-gap states, unlike SmB, and is susceptible to low concentrations of disorders Wakeham et al. (2016).

Figure 2: Electronic structure of CePdBi. (a) The DFT+DMFT band structure in the energy range [-1.5,1.0] eV with respect to the Fermi energy at 4 K. (b) The DFT+DMFT band structures in a zoomed-in energy range close to the Fermi level at 290 K, 72 K, 36 K, 18 K, 9 K and 4 K.

Having established that the CePtBi is a trivial Kondo insulator, we now focus on CePdBi compound. The CePdBi compound is isostructural to CePtBi with nearly identical lattice constants and slightly different Bi atomic coordinates. It is therefore tempting to believe that the electronic structure of CePdBi is similar to CePtBi. However, our DFT+DMFT calculations show that the temperature dependence of CePdBi electronic structure is very different (Fig. 2). At 290 K, the Ce-4 states are localized, as suggested by the highly dispersive quasi-particle conduction bands in Fig. 2b. Such incoherence near persists until very low temperature, and the states near becomes coherent only below 9 K. This is most prominent from the states between -N (indicated by the red circles in Fig. 2, also in SI Fig. S-3) and -P near . At 18 K, these states are very incoherent. At 9 K, coherent quasi-particle bands due to ligand states can be identified between -P, but the states between -N remains incoherent, and shows only a blurb of dip, which becomes coherent at 4 K. During the whole temperature range, we do not observe well defined gap opening at the Fermi level down to 4 K in our calculation, thus CePdBi is most likely to be a correlated metal at low temperatures. At 4 K, the Ce-4 flat bands are 10 meV above , and do not form coherent Ce-4 bands below the Fermi level, thus the Ce-4 electrons remains localized at low temperatures for CePdBi.

We now compare the DFT+DMFT band structure for CePdBi at 4 K with DFT calculations treating Ce-4 state as core-electrons. Again, the coherent 4 K band structure for CePdBi resembles the DFT results, if the Fermi level in the DFT band structure is shifted 0.2 eV higher (marked by the gray line in Fig. 3(a)-(b)). The DFT band structure has a large band gap of nearly 500 meV at approximately 1 eV above the Fermi level, or equivalent to counting 6 more electrons per unit cell. The states near are dominated by Bi-6 and Ce-5 orbitals with a small mixture of the Pd-4 orbitals. We note that the centers of gravity of the Pd-4 orbitals are located in the energy region between [-4, -2] eV below . The highest occupied states at are doubly degenerate , quarterly degenerate and doubly degenerate (from higher energy to lower energy), respectively. Along -H line, the states splits into two doubly degenerate bands, and both and states remain doubly degenerate. The state and one pair of the split state are joined by other 2 doubly degenerate bands to form an 8-fold degenerate highest occupied state at H. It again splits into 4 doubly degenerate states along H-N, and further splits into 8 singly degenerate states along N-. Such splitting are all present in coherent 4K band structure from DFT+DMFT calculations.

Figure 3: DFT electronic structure of CePdBi. (a) The DFT band structure calculated by assuming open-core Ce-4 state. The contributions from Ce-4, Bi-6 and Pd-4 states are represented by green, blue and red circles, respectively. (b) The band structure on the gliding mirror plane -N-P. K is (0, 1/4, 1/4). Along P-H, the bands can be classified with C eigenvalues (red squares) and 1 (blue squares). In other directions, the bands can be labeled with (red circles) or (blue circles). (c) The first Brillouin zone showing the high symmetry points and the nodal rings. The green circle is the path used to calculate the Berry phase. (d) The nodal rings (blue lines) on one of the gliding-mirror planes.

Similar to our previous analysis for CePtBi, we argue that the similarity between 4 K coherent band structure and DFT band structure ensures that the topological properties of CePdBi can be analyzed using the DFT results with proper Fermi level shifting. Judging from the bands crossing the Fermi level between N- and around P, the DFT results can be compared with DFT+DMFT results (see also SI Fig. S-4(a)) if the DFT Fermi level is shifted upward by approximately 0.2 eV. In addition, the Fermi surface from open-core Ce-4 DFT calculations matches that from the DFT+DMFT calculations at 4 K after 0.2 eV Fermi level shifting, both of which consist of 4 sheets around H and 4 sheets around P points (SI Fig. S-5(a) and (c)). On the contrary, the Fermi surface from open-core Ce-4 DFT calculations without Fermi level shifting consists of 2 sheets around H and 2 sheets around P points (SI Fig. S-5(b)). From the DFT DOS (SI Fig. S-3(b)), we found that 0.2 eV Fermi level shifting is equivalent to 1 electron doping per formula (or 2e per unit cell). Therefore, we need to locate for the nodal structures between band 4 and band 5 (band indices are marked on SI Fig. S-6(b)). Notice that the crystal contains 6 gliding-mirror planes in {100} and five other equivalent directions, the states in and all equivalent k-planes can be classified using the mirror symmetry. Let us consider 1 of the 6 equivalent planes (, , ) (in crystal coordinate). First of all, there is a high symmetry line P-H within this plane, which is shared by other 2 gliding mirror planes. In addition to the gliding mirror symmetry, rotation symmetry is also preserved along this line. Therefore, all states can be labelled with eigenvalues or 1, the former is doubly degenerate. At point P, bands 2-4 and 5-7 are triply degenerate due to the local T symmetry. As it moves along P-H, they both split into a singly degenerate band (4 and 7) and a pair of doubly degenerate band (2,3 and 5,6). Band 4 then crosses doubly degenerate band 5/6 twice, creating two triply degenerate points at K (0.238, 0.254, 0.254) (all the coordinates are in direct unit, unless otherwise specified) and K (0.046, 0.318, 0.318), as well as a segment of nodal-line between them. It is also instructive to notice that bands 1-7 and 10 will join together at H and form an 8-fold degenerate state; while bands 8-9 are doubly degenerate states and will be joined by other 6 states at H to form another 8-fold degenerate state. Secondly, we take look at another K-path from to K (0.0, 0.25, 0.25) point. As all these states are in the mirror plane, they can be labeled with . Evidently, there is a band inversion between the band 4 and 5 between and K points, and therefore there must be a nodal point. In fact, such nodal point can be identified between K and any K-point along -P or -H. Therefore, we conclude that these nodal points must form a nodal ring within the triangle -H-P. Since there are 6 gliding-mirror planes, and each plane contains 4 such triangles within the first BZ, there are 24 nodal rings, forming 8 groups around 8 P points, each group contains 3 nodal rings sharing the same K-K nodal line. The Berry phase around a circular path perpendicular to P-H around the K-K nodal line is or , indicating its nontrivial topology. In the open-core Ce-4 electron DFT calculations, these nodal rings occupy energy ranges from 0.1 eV to 0.3 eV above , therefore they cross the Fermi level if the Fermi level is shifted 0.2 eV upward (1 e doping per formula). In the DFT+DMFT calculations, these nodal rings can also be identified in the coherent state. From P-H, the doubly degenerate states 5/6 are not very coherent a few meVs below . However, such crossings can still be identified at 50 meV below . In addition to the 24 nodal rings we described above, we have also identified band crossings at 6 H points, as well as at 24 points at (0.328, 0.153, 0.000) (in cartesian coordinates, Å unit) and its symmetrically equivalent positions. However, the chirality of all these crossings are 0, meaning that they are not topological.

An important question is what leads to the difference between CePtBi and CePdBi compounds. Since Pt and Pd belong to the same element family, and both compounds are isostructural with nearly identical lattice constants, the spin-orbit coupling (SOC) change seems to be one very plausible reason. However, if we compare the band states in the energy range [-0.5,1.0] eV in the DFT calculations by assuming open-core Ce-4 electrons (SI Fig. S-2), it is clear that the CePtBi SOC splitting near is not very different from CePdBi. From the DFT calculations, the spin-orbit coupling constants are 0.453/0.659/1.051 eV for Ce/Pt/Bi atoms in CePtBi, and 0.454/0.324/1.074 eV for Ce/Pd/Bi atoms in CePdBi, respectively. More importantly, we notice that the CePdBi compound would also be an insulator with a large band gap in the DFT calculations if Ce-4 electrons are treated as being itinerant. Therefore, such difference is more likely related to a change in hybridization. In fact, the hybridization function for 4 exhibits completely different behavior for CePtBi and CePdBi. If we compare the imaginary part of in Matsubara frequency domain for both compounds at 18 K (please refer to SI Fig. S-7), it is clear that for CePtBi blows up below meV, consistent with a Kondo insulator behavior Tomczak (2018); while for CePdBi, remains finite and reduces in the same frequency range.

In conclusion, we have performed a systematic DFT and DFT+DMFT study of CePtBi and CePdBi compounds. The CePtBi compound is a trivial Kondo insulator with an indirect gap of 6 meV. The CePdBi compound is a correlated metal with topological nodal lines at low temperatures. The hybridization change is the main driving force for the above-mentioned transition.

Method The DFT calculations were performed using plane-wave projected augmented wave method as implemented in the VASP Kresse and Hafner (1993); Kresse and Joubert (1999) code, and cross-checked with both PWscf code Giannozzi et al. (2017) and the full-potential linearized augmented plane-wave (FP-LAPW) Wien2k code Schwarz et al. (2002). The plane-wave basis energy cut-off were chosen to be 480 eV in VASP calculations and 48 Ry in PWscf calculations. In FP-LAPW calculations, was set to 9. In all these calculations, a dense -centered K-mesh was used to perform Brillouin zone integration. The VASP band structures were used to fit a tight-binding Hamiltonian using 108 Wannier orbitals Souza et al. (2001), which were used to perform topology analysis by the Wilson loop method. Experimental lattice parameters were used for both compounds Vil (a, b).

The DMFT calculations were performed using DMFTF package in connection with the Wien2k code Haule et al. (2010). The continuous-time quantum Monte Carlo (CT-QMC) method was employed to solve the Anderson impurity problem Haule (2007), and charge-density self-consistency was achieved. For calculation at 4 K, the charge-density converges after 60 DMFT cycles, with up to 2.88 QMC steps in each cycle.

Acknowledgements The authors are grateful to Qimiao Si, S. K. Kushwaha, P. F. S. Rosa, N. Harrison, J. Lawrence, Huiqiu Yuan, Jianhui Dai, and Frank Steglich for stimulating discussions. C.C also acknowledges the hospitality of Los Alamos National Laboratory, where this work was initiated. This work at Los Alamos was carried out under the auspices of the U.S. Department of Energy (DOE) National Nuclear Security Administration under Contract No. 89233218CNA000001. It was support by NSFC 11874137 and 973 Project 2014CB648400 (C.C. & G.-X.Z.), and U.S. Office of Basic Energy Sciences under LANL-E3B5 (J.-X.Z.). The calculations were performed on clusters in Los Alamos National Lab, the High Performance Computing Cluster of Center of Correlated Matters at Zhejiang University, and Tianhe-2 Supercomputing Center.

Additional information Correspondence and requests for materials should be addressed to C. Cao ( or J.-X. Zhu (

Author contributions J.-X.Z. conceived the project; C.C. and G.-X.Z. performed DFT calculations with the VASP code and performed topology analysis; C.C. and J.-X.Z. performed Wien2k-based DFT and DFT+DMFT calculations. All authors participated in the discussions. C.C. and J.-X.Z. wrote the manuscript.

Competing financial interests The authors declare no competing financial interests.



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