Owing to the the fact that topological insulators (TIs) provide a promising background for novel physics arising from their nontrivial bulk electronic structure with gapless surface states on the boundaries, they have aroused the great interest of researchers in recent years. Very recently, the rare-earth hexaborides containing f-electrons in their electronic configurations in this field were found. The compound LnB₆ may exhibit the electronic correlation effects from foreign materials,^[1–3] such as topological Mott insulators,^[4,5] topological crystalline insulators,^[3,6] and topological Kondo insulators (TKIs).^[7–9] For example, SmB₆ hopefully becomes a “topological Kondo insulator” theoretically and was confirmed recently by transport,^[7,10] photo emission,^[11–13] and STM experiments.^[14] The prediction of SmB₆ renews the scientific attention to 4f-contained hexaborides. However, the surface states of SmB₆ in the TKI phase only exist at very low temperatures, which restricts its further application in devices.^[3,13] Recently, electronic structure and thermal properties of topological insulator of SmB₆ were studied by first-principles calculations.^[15,16] And then, plutonium-based 5f compound PuB₆ and the isostructural ytterbium-based 4f compound YbB₆ are frequently considered to be TKIs theoretically when onsite Coulomb interaction is included in the computations to take into account the strong electron correlation effects.^[2,17]

However, there are some discrepancies between the experimental result and theoretical prediction of YbB₆. Theoretically, YbB₆ was proposed to be a TKI with the mixed-valence state of Yb based on the calculations from the DFT + Gutzwiller method.^[3,9,18] Besides, a new band calculation with an adjustable correlation parameter (Hubbard U = 4) revealed that the topological state is derived from band inversion between the d and p bands^[17] and the binding energy (BE) of the Yb 4f_7/2 is 0.3 eV. But the ARPES experiments^[17,19,20] manifested that the Yb-4f states are located at around 1 eV and 2.3 eV below the E_F, which is contradictory to previous theoretical prediction. Moreover, Min et al. obtained an insulating gap of YbB₆ about 100 meV in the ARPES experiment by adopting the modified Becke–Johnson (mBJ) exchange potential and the on-site Coulomb interaction U (U = 7).^[19] Unfortunately, the resistivity of YbB₆ exhibits a bulk metallic behavior,^[21] which is inconsistent with the prediction. At present, the published calculations^[3,15,22] indicate the existence of band gap almost without exception based on the experimental crystal structure.^[23] However, the lattice constant a for full relaxation and experimental result are quite similar to each other. On the other hand, the DFT+U calculations with increasing U give only a small gradual decrease in the p–d overlap energy without opening of a semiconducting p–d gap even for an unphysically large U value as high as 15 eV. Based on the above mentioned, we guess that the onsite Coulomb interaction (U) and the internal parameters x are the cause of the difference between experimental and theoretical results. Therefore, exploring the inconsistencies between the theoretical predictions and experimental observation in YbB₆ systems would be meaningful and hopeful in gaining an insight into the relation of crystal structure with electronic properties for further analyzing its topological properties.

Therefore, the effect of internal structural parameter x and U on the band gap of YbB₆ is firstly investigated by the first-principles calculation. Then, the relation of the internal structural parameter x with electronic structure near the Fermi level (E_F) with fixed U is discussed. The rest of this paper is organized as follows. In Section 2, a brief introduction of theoretical methods is presented. The results aregiven and the discussion is made in Section 3. Finally, some conclusions are drawn in Section 4.

The first-principles study of electronic structure is conducted based on the density functional theory (DFT) in conjunction with projector augmented-wave potential (PAW)^[24,25] within the GGA + U^[26–30] schemes as implemented in the Vienna ab initio simulation package (VASP).^[31–34] The GGA + U approach is an alternative method to take into account the correlation effects with less computational effort and is frequently used to correct the band gap. It combines the explicit treatment of electronic correlation with a Hubbard-like model^[35] for a subset of states in the system. Pseudopotentials with valence states 2s²2p for B, 5s²5p⁶4f¹⁴6s² for Yb are used. The wavefunctions are expanded in a plane-wave basis set with a kinetic-energy cut-off of 550 eV. The 11 × 11 × 11 k-points for YbB₆ unit cell with the Monkhorst–Pack scheme are adopted for the Brillouin zone sampling. In all cases, the total energy of self-consistent convergence is 10⁻⁵ eV/cell and the process is terminated when the atomic force is less than 10⁻³ eV/Å. All the parameters are carefully tested to give satisfactory converged results. The spin–orbit coupling (SOC) is included self-consistently in all calculations. To produce more accurate densities of states, a dense k-point mesh of 19 × 19 × 19 is used, and the DOS is computed by the Gaussian smearing method.^[31–34]

YbB₆ crystallizes into a cubic CsCl-type structure with a space group of .^[36,37] The Yb and B atoms are respectively located at the 1a (0, 0, 0) and 6f (1/2, 1/2, x) Wyckoff position, where x is the positional parameter. Figure 1 shows the crystal structure of YbB₆.

Fig. 1.

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	Fig. 1. Crystal structure of YbB6, where Yb and B atoms are located at 1a (0, 0, 0) and 6f (1/2, 1/2, x) Wyckoff positions, respectively.

All the groups well versed in topological properties adopt experimental crystallographic structure^[23] in the calculation. The experimental lattice constant a and the positional parameter x are 4.144 Å and 0.207, respectively. The equilibrium crystal structure is obtained theoretically by minimizing the total energy in the two-dimensional parameter space (a, x). The GGA+U (U = 6.5) relaxed minimum value is located at 4.140 Å and x = 0.201. The lattice constant shows that it is in good agreement with experimental result. However, the x is not in good agreement with the experimental result. On the other hand, as shown in Fig. 2, the band gap is not found by using the present equilibrium crystal structure with the different values of U but it appeared in previous researches^[15,3] based on the DFT + SOC + U (U = 4 eV) and GGA + GW + U (U = 7).

Fig. 2.

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	Fig. 2. First-principles calculation-based band structure of YbB6 with different values of internal structural parameter x and U (U = 4 eV (a) and 7 eV (c)). (b) and (d) Zoomed part of band structure in panels (a) and (c) near EF, respectively.

In view of the above, it is necessary to investigate the electronic structures of YbB₆ with three representative values of x within a range of U. As depicted in Figs. 3–5, as the values of U range from 3 to 8, the band gap between Yb-5d state and B-2p state is not discovered with x = 0.201 or 0.204. When x is equal to 0.207, the band gap between Yb-5d state and B-2p state is closed and reopened with the increase of U.^[22] Therefore, we affirm that the positional parameter x plays a significant role in forming the band gap while the U is not the main factor.

Fig. 3.

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	Fig. 3. Energy band structures along principal high-symmetry direction of YbB6 for varying U with x = 0.201.

Fig. 4.

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	Fig. 4. Energy band structures along principal high-symmetry direction of YbB6 for varying U with x = 0.204.

Fig. 5.

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	Fig. 5. Energy band structures along principal high-symmetry direction of YbB6 for varying U with x = 0.207.

To further investigate the connection between band gap and internal parameters x, another theoretical calculation is first performed within GGA + SOC + U, in which the experimental U = 6.5 is used. Figure 6 shows the band structure of YbB₆ with experimental internal parameters at U = 6.5, the non-dispersive Yb-4f bands are located at about 1 eV and 2.3 eV below Fermi energy (E_F), respectively. The energy difference between the two multiplets is 1.3 eV, which is consistent with previous experimental results.^[17,19,20] Then, figures 7(a)–7(c) show the contributions of B-2p, Yb-4d and Yb-4f orbitals as a function of x near the E_F with the different values of internal parameter x, respectively. We find that the p–d overlap decreases with x increasing, and the Yb-5d orbitals are ultimately away from the Yb-4f orbitals which remain at nearly the same binding energy. Therefore, the Yb-4f band exhibits an decreasing overlap of Yb-5d and also implies a decreasing mixed-valency. More importantly, the Yb-5d moves up and B-2p moves in the opposite direction, which is a source of small band gap.

Fig. 6.

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	Fig. 6. Energy band structure along principal high-symmetry direction of YbB6 with U = 6.5 and x = 0.207.

Fig. 7.

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	Fig. 7. Zoom-in views of band structure around the EF with different values of internal parameter x (x = 0.201 (a), 0.204 (b), and 0.207 (c)).

Furthermore, the curves of total and partial density of states (PDOS) of YbB₆ with different values of internal parameter x at U = 6.5 are presented in Fig. 8. The hybridization of 2s and 2p of B dominates the lower bonding peak located at −14 eV below the E_F. The B 2s and 2p interactions from intraoctahedron contribute mainly to the upper bonding peaks in an energy range from −10 eV to −6 eV. The upper subgroup, which is in a range from −6 eV to the Fermi level, is composed of B 2p and Yb 4f orbital. The contributions are primarily from the interoctahedral bonds except around the Fermi level where the intraoctahedral B–B bonds become important again. Nevertheless, both B-p and Yb-d states are distributed on both sides of the E_F. As illustrated in Fig. 8, the band overlap in the Fermi level appears between B-s and Yb-d state, and the Yb atom is further apart from B cage with the increase of x. And then it results in the promotion of dispersed conduction band. The intra-cage bonding states come down in energy, resulting in the opening of a small gap. From the above, we can see that the internal parameter x has an influence on the band gap and density of states of YbB₆.

Fig. 8.

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	Fig. 8. Total and angular-momentum-projected densities of states of YbB6 with different values of internal parameter x.

The effects of internal parameter x and the value of U on the band gap and density of states of YbB₆ are explored by means of first-principles generalized gradient approximation GGA + U. The electronic structures of YbB₆ for three representative values of x within a range of U are calculated, and the results indicate that the internal structural parameter x plays an important role in determining the band gap. Our results explain the discrepancy between experimental and theoretical results. Furthermore, the zoom-in views of band structure around the E_F and density of states are discussed to further illustrate the relation between the internal structural parameter x and band gap. The Yb-4f orbitals are located at around 1 eV below the E_F, p–d overlaps decrease with x increasing, and these bands around the Fermi level are unclear currently. We expect the present reserach to be able to present a basis for further experimental and theoretical studies of this promising material and other topological insulators band gap.