Topological Phase and Strong Correlation in Rare-Earth Hexaborides XB6 (X = La, Ce, Pr, Nd, Pm, Sm, Eu)

Abstract

The rare-earth hexaboride SmB₆, known as the topological Kondo insulator, has attracted tremendous attention in recent years. It was revealed that the topological phase of SmB₆ is insensitive to the value of on-site Coulomb interactions (Hubbard U), indicating that the topological phase in SmB₆ is robust against strong correlations. On the contrary, the isostructural YbB₆ displays a sensitivity to the Hubbard U value. As U increases, YbB₆ transforms from topological Kondo insulator to trivial insulator, showing the weak robustness of the topological phase of YbB₆ against U. Consequently, the dependence of the topological phase on Hubbard U is a crucial issue in the rare-earth hexaboride family. In this work, we investigate the structural and electronic properties of rare-earth hexaboride compounds through first-principles calculations based on density functional theory. By taking the strong correlations into consideration using a wide range of on-site U values, we study the evolution of the topological phases in rare-earth hexaboride (XB₆, X = La, Ce, Pr, Nd, Pm, Sm, Eu). Unlike YbB₆, the topological trends in all the examples of XB₆ studied in this work are insensitive to the U values. We conclude that in addition to the well-known SmB₆, PmB₆, NdB₆ and EuB₆ are also topologically nontrivial compounds, whereas LaB₆, CeB₆ and PrB₆ are topologically trivial metal.

1. Introduction

The discovery of the topological phase in condensed matter paved the way to classify electronic states [1,2]. Topological insulators have been attracting world-wide extensive attention in recent research [3,4,5,6]. Three dimensional materials with time reversal symmetry and inversion symmetry may harbor a topologically nontrivial phase if a band gap and band inversion emerge owing to spin–orbit interaction (SOI) [7].

The rare earth hexaboride XB₆ crystallizes in the CaB₆-structure, as shown in Figure 1. Its lattice structure is similar to a body-centered cubic such as the CsCl-type lattice with Cs replaced by rare earth ions, and with Cl substituted by B₆ octahedra. The variety of the physical properties observed in these compounds is intriguing. For example, the application of LaB₆ has been paid attention due to its low work function, which is suitable for thermionic emission. LaB₆ is metallic and becomes superconducting at T_C = 0.45 K [8]. CeB₆ is considered as a Kondo system. CeB₆ presents an antiferro-quadrupolar ordering in the paramagnetic phase between T_q = 3.3 K (quadrupolar ordering temperature) and T_N = 2.4 K (Neel’s Temperature) [9,10]. PrB₆ has been confirmed that negative quadrupolar pair interactions exist in the paramagnetic phase (T_N = 6.9 K) [11]. NdB₆ is a localized 4f system that orders ferro-magnetically at low temperatures [12]. SmB₆ is a well-known topological Kondo insulator [1,13,14]. EuB₆ orders ferro-magnetically below 15.1 K with a huge decrease of resistivity and a significant blue shift of the reflectivity plasma edge [15,16,17]. At 12.7 K, another phase transition takes place, which is observed as a broad peak in the specific heat or an anomaly in the resistivity [18]. GdB₆ is a localized 4f system with a ferromagnetic order at low temperatures [19]. YbB₆ is a topology Kondo insulator at low temperatures, and is a classical mixed valence narrow band gap semiconductor [1,20,21]. Structural studies are also presented in Ref. [22,23]. As reported in Ref. [1], the topological phase of YbB₆ is sensitive to the Hubbard U value. As U increases, YbB₆ transforms from topological Kondo insulator to trivial insulator, showing the weak robustness of the topological phase of YbB₆ against U.

In this study, the lattice structures of rare-earth hexaboride (XB₆, X = La, Ce, Pr, Nd, Pm, Sm, Eu) are fully optimized through first-principles calculations. We then perform self-consistent field electronic structure calculations with and without SOI. To reveal the topological phases, we analyze if SOI would open up a continuous energy gap at the Fermi level with band inversion around the energy gap. To examine the robustness of the topological phase upon the strong correlation in XB₆, we trace the evolution of its electronic structure by tuning the on-site U of the f electrons. We demonstrate that besides SmB₆, PmB₆, NdB₆ and EuB₆ are also topologically nontrivial compounds, while the others are topologically trivial normal metals.

2. Computational Details

First-principles calculations were performed using the Vienna Ab initio Simulation Package (VASP) with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional used in the generalized gradient approximation (GGA) as well as the GGA plus Hubbard U (GGA + U) schemes [24,25,26,27] based on density functional theory (DFT). The cut-off energy of 500 eV was adopted for the plane-wave basis. A Γ-centered 15 × 15 × 15 k-mesh was used in geometry optimization and self-consistent field calculations. The geometry optimization converged until all residual forces remained below 0.01 eV/Å. Table 1 compares the experimental lattice parameters of rare-earth hexaboride with our geometrically optimized ones. Good agreement between experimental and theoretical results can be found with deviations, in general, of less than 1%.

3. Results and Discussion

3.1. Topologically Trivial Normal Metal LaB₆, CeB₆, and PrB₆

Figure 2a,b show the PBE band structures of LaB₆ without and with SOI, respectively. The atom-orbital decomposition demonstrates that the valence bands below −1.5 eV are mainly composed of B-p orbital, while the conduction La-f bands (blue curves) are located mainly from 0.5 eV to 2.5 eV above the Fermi level (E_f). In between, there is a dispersive band composed of La-d orbital connecting the valence and conduction bands, resulting in an overall semimetal character. This in-gap La-d band also gives an electron pocket at E_f along ΓM. A comparison of band structures without SOI (a) and with SOI (b) shows that the SOI in LaB₆ is weak and has no significant effect on band structure. Consequently, the semimetal character remains as SOI is included. Without any continuous gap, LaB₆ is therefore a topologically trivial normal metal. On the other hand, the band structures remain more or less the same when the on-site Coulomb repulsion U is taken into account for the strong correlation in f orbitals, as can be seen in Figure 3. This constitutes preassembly, owing to the empty f states that have no effect near E_f.

With one more electron than La, the Fermi level of CeB₆ is thus raised up to the bottom of Ce-f bands, as shown in Figure 4. The flat Ce-f conduction bands are located around E_f from 0.6 eV below to 1.2 eV above E_f. As shown in Figure 5, for all the four cases with U = 2, 4, 6, 8 eV studied, there are no significant changes in band structures. Similar to LaB₆, CeB₆ is also insensitive to the on-site U values. Although the SOI is included in the calculations and the degeneracy at M is lifted by SOI, there is no continuous gap in all cases, leading to topologically trivial normal metal ground state for CeB₆.

Elementary Pr has three electrons occupying the f-orbitals in the ground state. Therefore, in PrB₆ the Pr-f conduction band is occupied by one more f electron than CeB₆ through the rigid-band shift, as shown in Figure 6. The band dispersions remain similar with different on-site U values. However, because the Fermi level is raised to the middle of the Pr-f conduction band, on-site U affects the bandwidth more significantly than that in the previous two species. With U = 8.0 eV, the f bandwidth is enhanced by about 0.5 eV. On the other hand, gapless ground state remains in PrB₆ even when the SOI is taken into consideration. Consequently, the same as LaB₆ and CeB₆, PrB₆ is also a topologically trivial normal metal.

3.2. Topologically Nontrivial Kondo Insulator SmB₆, PmB₆, NdB₆ and EuB₆

Figure 7 shows our calculated band structures of the well-known topological Kondo insulator. The relatively flat La-f bands locate around Ef, with a much more dispersive La-d band crossing all these f bands. The spin–orbit interaction splits the f bands and opens up a continuous energy gap (see Figure 8) with band inversion between Sm-f/d characters flipping around the SOI-induced gap. These results agree well with those presented in previous works [1]. Band structures of SmB₆ with SOI and on-site U ranging from 2 to 8 eV are shown in Figure 9. There are no significant changes in band structures due to all the different U values used. Similar to previous study, the SOI-induced band gap and the band inversion behavior remain, indicating the robust topological phase against strong correlations in SmB₆.

In comparison with the well-known topological Kondo insulator SmB₆, the overall band dispersion of PmB₆ as shown in Figure 10 is similar to those of SmB₆ (Figure 7, Figure 8 and Figure 9). Since PmB₆ has one less valence electron than SmB₆, the Fermi level of PmB₆ is relatively lower than that of SmB₆. With SOI taken into consideration, PmB₆ opens up a continuous gap around Ef, as shown in Figure 10b,d. In addition, there is a band inversion around X point with B-p and Pm-d components exchanged near E_f. Therefore, PmB₆ can host topological nontrivial state, giving rise to the topological Kondo insulator similar to SmB₆. The electronic structure of PmB₆ around E_f is not sensitive to various U values, as shown in Figure 11. On-site U only affects the highest empty f-band without influencing the overall topological properties, indicating the topological phase is robust in PmB₆ against strong correlations.

Band structures of NdB₆ as shown in Figure 12 also demonstrate topologically nontrivial phase. The SOI not only opens up a continuous energy gap around E_f but also gives rise to band inversion around X point. Similar to PmB₆, the electronic structure and topological behavior of NdB₆ near the Fermi level are insensitive to on-site U value, as can be seen in Figure 13. Only the highest unoccupied f band is noticeably modified by U, which is irrelevant to its topology. Consequently, NdB₆ is also a topological Kondo insulator.

Figure 14 shows PBE (U = 0 eV) band structures of EuB₆ without and with SOI as well as PBE + U band structures with U = 2 eV and 6 eV. As can be seen in Figure 14a, the f bands are located at E_f with a localized flat band character. In the periodic table, Eu is the neighbor of Sm with one more electron. The additional electron raises the Fermi level of EuB₆ near the half-filling metallic regime. When SOI is included, the f bands separate themselves into two groups with an SOI-induced continuous gap in between. Furthermore, band inversion emerges around the high symmetry point X. As a result, EuB₆ exhibits nontrivial topological phase similar to SmB₆. The band structure of EuB₆ is not sensitive to U, as shown in Figure 14b–d, with U = 0–6 eV, leading EuB₆ to robust topological Kondo insulator against strong correlations.

4. Conclusions

We have systematically analyzed the electronic structures of rare-earth hexaborides to investigate their topological properties and examine the robustness of the topological phase against strong correlations by varying the Coulomb repulsion U. SmB₆ is a topological Kondo insulator due to the hybridization gap, and it will not experience topological phase transition by tuning the Coulomb interaction. YbB₆, which has a hybridization gap, on the contrary, will experience a topology phase transition from a topological Kondo insulator to a topological insulator, and finally become a trivial insulator. Our results of SmB₆ and YbB₆ are in good agreement with previous results [1]. Our study also shows that PmB₆, NdB₆, EuB₆ and SmB₆ exhibit SOI-induced continuous gaps with band inversion, revealing nontrivial topological properties. On the other hand, the weaker SOI in relatively lighter. Lanthanides La, Ce and Pr fail to open up a continuous gap in LaB₆, CeB₆ and PrB₆. Thus LaB₆, CeB₆ and PrB₆ are topologically trivial normal metals with correlated conduction electrons.

Author Contributions

S.-H.H. performed first-principles calculations and wrote the manuscript. H.-T.J. conducted the project and reviewed the manuscript. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Acknowledgments

This work was funded by the Ministry of Science and Technology, Taiwan (grant number MOST 106-2112-M-007-012-MY3). The authors also thank support from NCHC, CINCNTU, AS-iMATE-109-13 and CQT-NTHU-MOE, Taiwan.

Conflicts of Interest

There are no conflicts of interest to declare.

Figures and Table

Figure 1. CsCl-type cubic crystal structure and Brillouin Zone of rare-earth hexaboride XB6. (a) Side view. (b) Oblique view. (c) Brillouin Zone and high symmetry k-points.

Figure 2. Atom-orbital decomposed band structure of LaB6 calculated using Perdew–Burke–Ernzerhof (PBE) functional without spin–orbit interaction (SOI) (a) and with SOI (b). The size of blue, green and yellow circles indicates components from La-f, La-d and B-p orbitals, respectively.

Figure 3. Atom-orbital decomposed band structures of LaB6 with on-site U =2 eV (a,b), 4 eV (c,d), 6 eV (e,f), and 8 eV (g,h).

Figure 4. Band structure of CeB6 without (a) and with (b) spin–orbit interaction.

Figure 5. (a–d) CeB6 band structures given from PBE + SOI + U with U = 2, 4, 6, 8 eV, respectively. The sizes of blue, green and yellow circles indicate components from Ce-f, Ce-d and B-p orbitals, respectively.

Figure 6. PrB6 band structure without (a) and with (b) spin–orbit interaction (noted in the figures), and with spin–orbit interaction plus on-site U = 2, 4, 6, 8 eV (c–f, respectively) as noted in the figures.

Figure 7. Band structures of SmB6 without (a) and with (b) spin-orbit interaction projected by f and d electrons of Sm.

Figure 8. Band structure of SmB6 without and with spin-orbit coupling. The right panel is the zoom-in view of the middle panel around Ef.

Figure 9. Band structures of SmB6 with spin–orbit interaction using on-site U = 2.0, 4.0, 6.0, and 8.0 eV (a–d, respectively) projected by f and d electrons of Sm.

Figure 10. PBE band structure of PmB6 without SOI (a) and with SOI (b). The size of blue, green and yellow circles show contributions from Pm-f, Pm-d and B-p orbitals, respectively. (c) Zoom-in of (a). (d) Zoom-in of (b). (c,d) demonstrate SOI-induced band inversion and gap opening.

Figure 10. PBE band structure of PmB6 without SOI (a) and with SOI (b). The size of blue, green and yellow circles show contributions from Pm-f, Pm-d and B-p orbitals, respectively. (c) Zoom-in of (a). (d) Zoom-in of (b). (c,d) demonstrate SOI-induced band inversion and gap opening.

Figure 11. Band structures of PmB6 with SOI and U = 2, 4, 6, 8 eV (a–d, respectively). As U is tuned larger, the highest f band is lifted but the band property is not changed near the Fermi level.

Figure 12. Band structures of NdB6 without (a) and with (b) spin-orbit interaction.

Figure 13. Band structures of NdB6 with SOI using U = 2, 4, 6, and 8 eV (a–d, respectively). As U is tuned larger, the highest f band is lifted but the band property is not changed near the Fermi level.

Figure 14. PBE band structure of EuB6 without SOI (a) and with SOI (b). SOI opens up an energy gap around Ef and induces band inversion around X point. PBE + U band structure of EuB6 with on-site U = 2.0 eV (c) and U = 6.0 eV (d). Similar to SmB6, the Hubbard U does not change the band structure noticeably.

Table 1

Experimental (exp) and theoretical (the) lattice parameters. The rare-earth hexaboride crystalizes in a bcc-like structure with space group of $\mathrm{Pm}\overline{3}\mathrm{m}$ (No. 221), in which metal ions are located at the Wyckoff position 1a(0,0,0) and octahedral B₆ at the Wyckoff position 6f (1/2,1/2,z). The subscripts “exp” and “the” indicate experimental and theoretical results, respectively.

	aexp (Å)	athe(Å)	Error of a	B(z)exp	B(z)the	B–B Bond Lengthexp(Å)	B–B Bond Lengththe(Å)
LaB6 [15]	4.1527	4.1553	0.06%	0.1993	0.1997	1.7660	1.7647
CeB6 [16]	4.14	4.1130	−0.65%	0.1992	0.1984	1.7611	1.7543
PrB6 [22]	4.13	4.1024	−0.67%	0.2	0.1984	1.7522	1.7498
NdB6 [16]	4.127	4.1007	−0.64%	0.1989	0.1987	1.7574	1.7473
PmB6 [23]	4.128	4.1131	−0.36%	0.2	0.1990	1.7514	1.7508
SmB6 [28]	4.1346	4.1087	−0.63%	0.2018	0.1993	1.7436	1.7474
EuB6 [29]	4.1849	4.1325	−1.25%	0.2027	0.1999	1.7595	1.7539
YbB6 [29]	4.1444	4.1325	−0.29%	0.207	0.2007	1.7173	1.7492