Rare-earth hexaborides (REB₆) are commonly used in various high-energy optical devices and field electron emitter systems because of their superior properties such as high chemical stability, high melting point, high mechanical strength, high brightness, low work function, low volatility, conductibility, small visual dimensions and long lifetimes [1]. REB₆ is commonly used as cathode material. REB₆ has a cubic CsCl-type structure with a space group of Pm-3m symmetry, in which a rare-earth (RE) ion occupies the Cs site, and the B₆ octahedron is located on the Cl site. REB₆ compounds include LaB₆, CeB₆, PrB₆, NdB₆, PmB₆, SmB₆, EuB₆, GdB₆, TbB₆, DyB₆, HoB₆, ErB₆, TmB₆, YbB₆, LuB₆, ScB₆ and YB₆. LaB₆ has low volatility, CeB₆ indicates a typical dense Kondo behavior, PrB₆ shows high density, NdB₆ has low magnification, SmB₆ is a typical valence semiconductor and GdB₆ has the lowest work function among REB₆ compounds [2].

The electronic structures of the doped and binary REB₆ were calculated using density functional theory (DFT). The position of the Fermi energy level and DOS were adjusted by doping with REB₆ to improve the electron emission characteristics. The high-density d-orbital electrons play a crucial role in considerably decreasing the work function of REB₆ and contributing to the electronic states of electron emission near the Fermi level. This ensures excellent emission characteristics [3]. The second-order elastic constants (SOECs) and third-order elastic constants (TOECs) of LaB₆ and CeB₆ were studied by first-principles calculations. The effect of increased pressure on the elastic anisotropy, mechanical characteristics and structural stability of LaB₆ and CeB₆ has attracted considerable attention. When the pressure increases, the mechanical stability decreases and the ductility and anisotropy increase [1]. Lanthanum hexaborides (LaB₆) are superb thermionic and field electron emission cathode materials in the field of electron emission. LaB₆ has several applications in high-power electronics owing to its long lifetime and high luminosity. LaB₆ attracts attention by its low work function between 2.6 and 2.8 eV, its high melting point of 2715˚C, and its stable chemical and physical characteristics. Compared to polycrystalline and single-crystal applications of LaB₆, it has better potential for single-crystal applications [4]. LaB₆ works well as a thermal-field emitter. It is easily degradable and stable in air. LaB₆ was reactive at 2715˚C. LaB₆ is a violet-colored metal and its electron conductivity is approximately 1/5 that of copper [5]. Lu et al. [6] successfully fabricated LaB₆ nanocubes with an average dimension of 94.7 nm using a low- temperature molten salt technique at 800˚C. LaB₆ nanocubes exhibited high near-infrared (NIR) adsorption. As mentioned in [7], LaB₆ nanocrystalline preparation routes include many synthesis routes, such as the floating zone method, aluminum flux, molten salt, high-temperature reaction, chemical vapor deposition (CVD), direct solid-phase reaction and carbothermal reduction. It is because of its wonderful characteristics that LaB₆ is commonly used in some electrical devices, including free-electron laser, thermionic electron cathode, electron microscope, vacuum, and electron beam welder [8].

Neodymium hexaboride (NdB₆) is black solid with good chemical stability, magnetic properties, electrical conductivity, and thermal conductivity characteristic. NdB₆ is insoluble in hydrofluoric acid (HF) and hydrochloric acids (HCl). However, it can be dissolved in molten alkali, sulfuric acid (H₂SO₄) and nitric acid (HNO₃). In addition, it exhibits very high antioxidant capability [9]. NdB₆ crystallizes in a CsCl-type structure with a space group of Pm-3m symmetry, where the neodymium (Nd) occupies the Cs site and octahedral B₆ molecules are located at the Cl site. NdB₆ has a low work function (1.6 eV) [10]. NdB₆ are an efficient field-emission cathode material. These excellent properties make NdB₆ nanomaterials promising materials for use in vacuum electronic devices [11]. Thus, NdB₆ is antiferromagnetical at T_N = 7.74 K [12]. Ding et al. [13] successfully synthesized NdB₆ nanowires (NWs) by a self-catalyst method. Nanowires with diameters of approximately 80 nm and lengths spanning several micrometers have monocrystalline structures. Xu et al. [14] successfully produced NdB₆ nanostructures using a free-CVD process. The NdB₆ nanostructures exhibited a good stability. The effect of temperature on NdB₆ is important. When the temperature was increased, the turn-on and threshold electric fields decreased. The work function of NdB₆ nanostructures is considerably decreased as the temperature increases, leading to much enhanced field emission characteristics.

Tsuji et al. [15], studied the magnetoresistance, magnetization and specific heat of Nd_xLa_1-xB₆ (x = 0. 9, 0.8, 0.7, 0) by a FZM method. The magnetoresistance, magnetization and specific heat are affected by temperature. As the temperature increased the others increase. Chaolong et al. [16] successfully investigated Nd_xLa_1-xB₆ bulks using spark plasma sintering (SPS) method. The work function of Nd_xLa_1-xB₆ was 2.72 eV. The Nd content enhanced thermionic emission characteristic and decreased the work function.

Li et al. [17] fabricated successfully high-quality, uniform La_xNd_1-xB₆ nanowires by catalyst-free CVD technique. La_xNd_1-xB₆ nanowires exhibit a superb field emission performance. Nanowires are used in optoelectronic devices such as nanoelectronic building blocks and flat panel displays.

Ab initio material modelling based on DFT was performed quantum espresso software (QE) packages based on modelling the material at nanoscales or on an atomic scale [18]. First-principles calculations were performed using the VASP. [19]. The projector augmented wave (PAW) method and the functional form of the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) were preferred for exchange. The kinetic energy cut-off of the plane-wave basis set was 500 Ry. The Brillouin zone integration was performed at 3 × 3 × 3 k mesh points using methfessel-paxton smearing with a width of 0.02 Ry. A k-mesh 3 × 3 × 3 was used in the brillouin zone integration with Methfessel-Paxton smearing width of 0.02 Ry. Both LaB₆ and NdB₆ have cubic CsCl-type structures with a space group of Pm-3m symmetry [20].

The bulk unit cell of REB₆ is simple cubic and is found in the symmetry of the space group Pm-3m. The lattice of NdB₆ can be entirely defined using merely the lattice constant, a, and the positional parameter, z, as indicated in Figure 1. The lattice constant, a was 1, intra-octahedral boron bond was 2, and inter-octahedral boron bond was 3.

Table 1 lists the positional parameters of the given REB₆. The lattice constant of LaB₆ (Pm-3m space group) was calculated as a value of 4.157 Å. This was consistent with the experimental consequences indicated in Table 1. Furthermore, parameters 2 and 3 were 1.766 and 1.660 Å, respectively, with a boron positional parameter of approximately z = 0.226 Å. Chen et al. [28] conducted a study on the structural refinement and thermal expansion of hexaborides. In this study, based on the X-ray powder diffraction technique, the intra-octahedral and inter-octahedral boron–boron distances were calculated as 1.766 and 1.659 Å, respectively. Xiao et al. investigated the optical features of LaB₆ using first-principles DFT calculations. They calculated the LaB₆ parameter to be 4.154 Å [33]. Hasan et al. [34] synthesized LaB₆ via carbothermal reduction. The calculated value of the lattice parameter was 4.157 Å. Furthermore, other experimental studies [35], [36] are consistent with those of presidential study. Mackinnon et al. [37] calculated the lattice constant of LaB₆ using DFT calculations. The calculated boron parameter (z) was 0.225 Å.

The doping of Nd instead of one La atom led to a slight increase in the lattice constant to 0.502 Å. In addition, 2 and 3 parameters were found to be 1.857 and 1.801 Å, respectively, which indicates an increase in these parameters while the z parameter remains nearly the same as that of LaB₆. In a study related to La_xGd_1-xBd₆ synthesized by the SPS technique, the doping of Gd into LaB₆ strengthened the lattice parameters of the structure [38]. In a similar study, Chao et al. [39] fabricated La₇Sm₁B₆ by solid-state technique. Sm doping led to a decrease in the La₇Sm₁B₆ lattice.

The 1, 2, 3 and z parameters; 1 of La₁Nd₇B₆ were found as 4.449, 1.812, 1.801, and 0.223 Å respectively. Compared to LaB₆, the ratio of Nd/La increased in parameters 1 2, and 3 but the parameter z didn’t change considerably. On the other hand, Li et al. [17] conducted a study of single-crystal La_xNd_1-xB₆ nanowires to investigate the field emission performance and characterization. In this study, when Nd-doped into LaB₆, the lattice parameter of La_xNd_1−xB₆ was decreased. In a similar study, Chao et al. [39] produced La_xSm_1−xB₆ by solid-state reaction. They employed DFT to describe the characteristic of Sm-doped LaB₆. They obtained the lattice parameters of La_0.2Sm_0.8B₆ and La_0.4Sm_0.6B₆ as 4.123 and

4.128, respectively. Their results showed that doping LaB₆ with Sm decreases the lattice parameter of La_xSm_1−xB₆. When the La content was doped into BaB₆, the calculated value of the lattice constant of La₁Ba₇B₆ was increased [40]. Luo et al. [41] studied La₁Ca₇B₆ by first-principles calculations. Ca doping provides increases in the lattice strength of La₁Ca₇B₆.

The lattice constant of NdB₆ was calculated as a value of 4.118. Ali et al. [12] studied the thermoelectric power of NdB₆ by using the floating zone method. They measured the lattice constant was 4.126 Å. In other studies, Ping et al. [42] conducted a study of NdB₆ by using the first principle method. The lattice parameter of NdB₆ was calculated as 4.069 Å. Sandeep et al. [35] calculated the lattice parameter of NdB₆ (4.157 Å) using the full-potential linearized augmented plane wave (FP-LAPW) technique. Furthermore, parameters 2 and 3 were found to be 1.750 and 1.643 Å, respectively, and the z parameter was 1.643 Å. Mackinnon et al. [37] determined the z parameter as 0.226 Å.

Figure 2 illustrated PDOS of La_xNd_8-xB₆ (x = 0, 1, 7, 8). As explicated in Figure 2(a) and Figure 2(c) PDOS curves. La d-electron bands show typical conductive behavior as they pass fermi energy. The lowermost conduction bands (CBs) consisted of B s and the uppermost valence bands (VBs) have consisted of B p are indicated in Figure 2(b). La d-electron band passing Fermi energies are shown in Figure 2(c). The calculation converged with great intensity to a metallic ground state at the E_F, at the Fermi level, as shown in the figure. The zone near E_F is contributed mostly by La d states as explicit in Figure 2(b) and Figure 2(c). Besides that, E_F is contributed generally by Nd d states as shown in Figure 2(e) and Figure 2(g). This is obvious that the energy division of the La d states and Nd d states additives look alike to B 2p additives, which is a signature of hybridization between La d-Nd d - B 2p states.

Figure 3 shows the charge density of LaB₆, La₇Nd₁B₆, La₁Nd₇B₆ and NdB₆. Light and dark colors show high and low-intensity zones respectively. Dark and light colours indicate low and high-density regions, respectively. La₁Nd₇B₆ has a low-density region and LaB₆ has a high-density region. There are six boron atoms on the plane. The center of the figure was seen the strong B-B bonds. The LaB₆ midpoint has strong charge density peaks. La₁Nd₇B₆ has weak peaks.

We comprehensively studied the mechanical and electronic properties of LaB₆, NdB₆, Nd-doped LaB₆ and La-doped NdB₆ using the density functional theory. The lattice constant of LaB₆ was lower than that of Nd-doped LaB₆. In addition, La doping increased the lattice constant of La-doped NdB₆. We calculated the PDOS of LaB₆, B for LaB₆, La₇Nd₁B₆, B for La₇Nd₁B₆, La₁Nd₇B₆, B of La₁Nd₇B₆, NdB₆, and B of NdB₆. We found that the La d-electron bands pass the Fermi energy as shown in Figure 2(c). The light color in the charge density indicates that LaB₆ has a high-density region. Similarly, dark color in the charge density shows that La₁Nd₇B₆ has a low-density region.

The author declares no conflicts of interest regarding the publication of this paper.

Bozada, C. (2022) The Mechanical and Electronic Properties of Ternary Rare-Earth Hexaboride La_xNd_8-xB₆ (x = 0, 1, 7, 8) Materials. Modeling and Numerical Simulation of Material Science, 12, 1-11. https://doi.org/10.4236/mnsms.2022.121001