First-principles study of structural, elastic, and electronic properties of CeB{}_{\text{6}} under pressure

Abstract

We performed a first-principles study of the electronic, elastic, and thermal properties of the rare-earth hexaboride CeB₆ using the local density approximation (LDA) in consideration of the effective onsite Coulomb parameter U_{eff}. To systemically evaluate the effect of U_{eff} on the structure of the material, the dependences of the lattice parameter a₀ and bulk modulus B on U_{eff} were examined in the framework of the LDA+U and GGA(PBE)+U scheme. We obtained a lattice constant a₀, elastic constants C_{ij}, and a bulk modulus B at 0 K and 0 GPa that were in good agreement with the experimental results and other theoretical findings. We focused on the electronic structure by analyzing the variation of the density of states with different U_{eff} values and pressures, which indicates the metallic characteristic of CeB₆. Interestingly, the effect of high pressure was similar to that of increasing U_{eff}, as the peaks at the bottom of the conduction band moved to the high-energy region in both cases. The elastic constants C_{ij}, bulk modulus B, shear modulus G, Young’s modulus E, shear-sound velocity V_S, and longitudinal-sound velocity V_L were calculated from 0 to 120 GPa. Additionally, the Debye temperature {\Theta }_D and elastic Debye temperature {\Theta }_E were systematically calculated using the thermodynamic methods in the range of 0–100 GPa. This research may provide a comprehensive understanding of the Kondo compound CeB₆ and similar rare-earth hexaborides.

Graphical abstract

Keywords

CeB₆ / structural properties / elastic properties / thermal properties / electronic structure

1 Introduction

Cerium compounds, in which additional electron is a strongly correlated f electron on the Ce center upon the reduction of Ce(IV) to Ce(III) [1], attract considerable research interest. The pressure dependences of the structural and elastic properties of Ce₂O₃ [2] and the electronic and magnetic structures of Ce₂O₂FeSe₂ [3] have been investigated recently. CeB₆ has also been intensively studied because of its interesting low-temperature magnetic phases. Experimentally, CeB₆ exhibits an antiferromagnetic order below T_N = 2.3 K [4], in advance of another phase transition at T_Q = 3.2 K, in which the order parameter remains unknown from standard experimental probes such as neutron diffraction [5, 6]. No structural phase transition occurs when the applied pressure is up to 85 GPa, but there is a possibility of structural changes in the pressure range of 85–122 GPa [7]. Remarkably, significantly different values were reported for the elastic constants; for example, the reported experimental values for C_{12} include 16, 19, 53, and 93 GPa [8–10].

The structural properties of CeB₆ were theoretically investigated using the plane-wave pseudopotential approach [11]. Sikora et al. [12] performed a symmetry analysis of the antiferroquadrupolar order in this substance. Recently, special attention has been directed to the magnetic moment from the 4f electron [13, 14], thermodynamic properties [15], and electronic structure [16]. To further understand CeB₆, the effective onsite Coulomb parameter U_{eff} must be introduced, which yields a drastic improvement in the description of the reduced ceria [17, 18]. Because U_{eff} strongly influences the calculated observables, its optimal value remains under debate. Recently, researchers investigated the Kondo effect in CeB₆, including the observation of a Ce-resonance and its origin [19, 20].

The main objective of this work was to study the electronic structure of CeB₆ by analyzing the density of states (DOS) and elastic constants from 0 to 120 GPa under different U_{eff} and pressure values in order to understand the effects of the pressure. Furthermore, we investigated the electronic structure and thermal properties of CeB₆ from the viewpoint of the U_{eff} effect. The remaining parts of the paper are arranged as follows. In Section 2, we provide the computational details. Section 3 presents our results and discussion, and we conclude the paper in Section 4.

2 Computational details

The calculations were performed using the Cambridge Sequential Total Energy Package [21, 22]. The local density approximation (LDA) proposed by Vosko et al. [23] for the exchange-correlation potential was used in the LDA+U variant, where the orbital-dependent LDA+U functional form is given as

\begin{array}{}\text{(1)}& E_{LDA+U}=E_{LDA}+{\displaystyle \frac{U-J}{2}}{\displaystyle \sum _{\sigma }}[\mathrm{tr}{\rho }^{\sigma }-\mathrm{tr}({\rho }^{\sigma }{\rho }^{\sigma })].\end{array}

Here, {\rho }^{\sigma } is the density matrix of the f states, and U and J are the Coulomb energy and the exchange energy, respectively. As described in this paper, the two energy parameters integrate into a single meaningful parameter U_{eff} (U_{eff}=UJ). This method can be regarded as the introduction of a penalty function that disapproves of non-integer occupation numbers of the onsite density matrix. More information about this functional is found in Refs. [24, 25].

Plane-wave basis sets were used with an energetic cut-off of 420 eV. For integration within the Brillouin zone, a 9 × 9 × 9 Monckhorst-Pack grid [26] was used. We employed the on-the-fly pseudopotential for the interactions of the electrons with the ion cores and regarded the atomic levels of 4{\mathrm{f}}^{1}5{\mathrm{s}}^{2}5{\mathrm{p}}^{6}5{\mathrm{d}}^{1}6{\mathrm{s}}^2 and 2{\mathrm{s}}^{2}2{\mathrm{p}}^1 as the valence electron states for the Ce and B atoms, respectively. In the structure optimization, the criteria for convergence were set as 1 × 10^-5 eV/atom for the energy, 0.03 eV/Å for the force, 0.05 GPa for the stress, and 0.001 Å for the ionic displacement. To efficiently obtain self-consistency, the density-mixing scheme reported by Kresse and Furthmüller [27] is recommended for metallic systems. The spin-orbit interaction apparently has little influence on the structural properties, leading to the speculation that it is strongly atomic in nature and that transformations with geometry are non-essential with the accuracy of current DFT methods, as indicated by Pickard et al. [11]. Thus, we neglected the spin-orbit coupling in our study. All of the above-mentioned parameters were carefully tested and leading to a well-converged total energy. We fixed the equilibrium lattice constant for the obtained value within the traditional LDA scheme and then optimized the atomic position with an ideal U_{eff} value of 4 eV for the structure optimization. Therefore, we only considered the U_{eff} effect on the atomic interaction, ignoring its effect on the lattice parameter for the electronic-structure and thermal calculations.

3 Results and discussion

3.1 Structural properties of CeB₆

CeB₆ is one of the strongly correlated systems sharing the Pm3m structure of CaB₆ with the latest experimentally determined lattice parameter a_0\approx 4.132 Å [7]. The conventional unit cell of the ClCs-type structure of CeB₆ is presented in Fig. 1. The Ce ions and B₆ octahedra are located at the corners and body center of the cubic lattice, respectively. The lattice parameter a₀ and bulk modulus B were obtained by fitting our energy-volume data to the BirchMurnaghan equation for the state [29]. Table 1 presents the lattice constant of CeB₆ according to a popular theoretical calculation and experiment. The previous theoretical results underestimate or overestimate a₀, especially compared with the latest experimental result [7]. However, our calculated lattice parameter a₀ was 4.121 Å (U_{eff} = 0 eV), which has discrepancies of approximately 0.27% and 0.46% compared with the experimental values of 4.132 [7] and 4.140 Å [28], respectively. Our result is in good agreement with both of the experimental results and agrees far more closely with the latest one. Figure 2 illustrates the dependence of the lattice parameter and bulk modulus on U_{eff} in the range of 0–9 eV. Clearly, the GGA calculations exhibit a somewhat longer a₀ and smaller bulk modulus B, causing a slightly less accurate description of the structural properties. Overall, the LDA yields a better agreement with the lattice parameter and bulk modulus. There is a stable increase in the lattice parameter with U_{eff} for both the LDA and GGA functionals. The lattice parameter a₀ was overestimated at the simply GGA level (U_{eff} = 0 eV); thus, considering that U_{eff}> 0 increases the discrepancy with respect to the experiment. However, the LDA+U method behaves contrarily: the calculation ignoring U_{eff} underestimates a₀ by \sim0.011 Å compared with the experiment. Thus, it is practical to introduce U_{eff} into the calculation of the lattice constant for the LDA functional.

As shown in Fig. 2(b), the bulk modulus distinctively increased with U_{eff} from 1 to 4 eV, but decreased slightly from 4 to 9 eV, suggesting that this variable is less sensitive to higher values of U_{eff}. The bulk modulus values from the LDA calculation with the above mentioned range of U_{eff} are primarily located within 166–191 GPa [8, 9, 34], which offers better consistency with the experiment than the GGA values. For a certain U_{eff} value around 4 eV, both a₀ and B yield a well-balanced agreement with the experiment. Hence, we used the LDA+U method to study the electronic, elastic, and thermal properties of CeB₆ with an optimal U_{eff} of 4 eV.

3.2 Elastic properties

The bulk modulus B varied over a broad range, with values including 166 [30], 191 [9], 168, and 182 GPa [8] among the three experimental results. Our results are consistent with two experimental results [8, 30] and slightly lower than another two experimental results [8, 9]. Our calculated pressure derivative of the bulk modulus B’ agrees better with the experimental result [30] than that with the recent theoretical result [15]. The elastic constant C_{11} is in agreement with the results of Refs. [8] and [9], and C_{44} is consistent with the results of Refs. [8], [9], and [10]. For the controversial elastic constant C_{12}, our results validate the experimental results of Lüthi et al., exhibiting no negative values. We predict that the most probable value of C_{12} tends to be ¿10 GPa. Overall, our calculated elastic constants agree better with the experimental results than that with the recent theoretical results obtained using the ABINIT code [31].

It seems insignificant that elastic and thermal properties are studied with respect to the effects of U_{eff}, as this method is mainly proposed for studying the electronic structure of strongly correlated systems. Thus, we examined both properties under pressure with an optimal U_{eff} value of 4 eV. The elastic properties of a material help determine its Debye temperature and mechanical stability. The calculated and experimental bulk modulus B (GPa) and elastic constants C_{ij} (GPa) at 0 K and 0 GPa are listed in Table 1. To the best of our knowledge, the computation of the bulk modulus using the stress tensor yields a slightly less precise bulk modulus because of the Pulay stress because of the incompleteness of the plane-wave basis set. However, our results for 0 K and 0 GPa agree with the experimental results and other theoretical data because of the sufficiently accurate approximation.

Figure 3 shows the pressure dependence of the elastic constants from 0 to 120 GPa. Here, C_{11} varies largely according to the applied pressure compared with C_{12} and C_{44}, indicating that C_{12} and C_{44} are less sensitive to the pressure than C_{11}. In the wide pressure range of 0–120 GPa, the elastic constants C_{11}, C_{12}, and C_{44}, the bulk modulus B, and the shear modulus Gincreased monotonically with pressure, but C_{44} was relatively stable and had a similar variation trend to the shear modulus G. A similar variation phenomenon is also observed for C_{11} and C_{12} in the comparable structure CaB₆ [32], but C_{44} has some differences. The C_{44} of CeB₆ varies from 75 to 139 GPa, whereas that of CaB₆ is relatively stable, remaining within 43–46 GPa.

Table 2 presents our calculated bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio \sigma, B/G values, longitudinal acoustic velocities V_L, shear acoustic velocities V_S, and Debye temperature {\Theta }_E of CeB₆ under hydrostatic pressure in the range of 0–120 GPa at 0 K. The bulk modulus and shear modulus indirectly indicate the hardness [33]. Generally, a superhard material has a high bulk modulus, high shear modulus, and high shear strength. Thus, we predict that CeB₆ has high hardness. Table 2 shows that the bulk modulus B increased gradually with the increasing pressure, demonstrating that the CsCl-type structure CeB₆ became more difficult to compress as the pressure increased. A high (low) B/G value is associated with ductility (brittleness), and the critical value that separates ductile and brittle materials is \sim1.75. Overall, the calculated B/G values increased with pressure, and the pressure at which this substance transformed from brittle to ductile was predicted as 27 GPa. The pressure can undoubtedly improve the ductility of a material. The Young’s modulus E is used to determine the stiffness of solids. A stiff material has a large E. Information about the bonding force can be obtained using Poisson’s ratio \sigma. The Young’s modulus E and Poisson’s ratio \sigma are defined as

\begin{array}{}\text{(2)}& E={\displaystyle \frac{9BG}{3B+G}},\sigma ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{3B-2G}{3B+G}}\right).\end{array}

At the ground state, E = 482.7 GPa. As the applied pressure increases, E gradually increases, demonstrating that the pressure enhances the stiffness of the substance. The Poisson’s ratio σ is 0.018 at 0 GPa, indicating that the inter-atomic forces within CeB₆ are non-central, as the low limit of Poisson’s ratio σ for central-force solids is 0.25 [34]. As the pressure increases, the remaining Poisson’s ratio σ remains less than 0.25, suggesting that the inter-atomic forces in CeB₆ are non-central at higher pressures. Additionally, many crystals present elastic anisotropies to some extent. The percentages of the compression anisotropy A_{comp} and shear anisotropy A_{shear} are defined as [35]

\begin{array}{}\text{(3)}& \begin{array}{c}{A}_{comp}={\displaystyle \frac{B_V-B_R}{B_V+B_R}}\times 100\%,\hfill \\ A_{shear}={\displaystyle \frac{G_V-G_R}{G_V+G_R}}\times 100\%.\hfill \end{array}\end{array}

The maximum and minimum values of A_{comp} and A_{shear} are 100% and 0%, respectively. A zero value indicates isotropy, whereas 100% indicates maximum anisotropy. At the ground state of CeB₆, A_{comp} is 0, and A_{shear} is 15.21%, indicating that there is isotropy in compression, but anisotropy in shear. The mechanical-stability conditions in cubic structures can be expressed with regard to the elastic constants as

\begin{array}{}\text{(4)}& {\tilde{C}}_{44}>0,{\tilde{C}}_{11}>\left|{\tilde{C}}_{12}\right|,{\tilde{C}}_{11}+2{\tilde{C}}_{12}>0,\end{array}

where {\tilde{C}}_{ii}=C_{ii}-P(i=1,4), and {\tilde{C}}_{12}=C_{12}+P. A strong increase in the temperature dependence of the resistance at 122 GPa was reported, which may signify a structural change around 85–122 GPa [7]. All the elastic constants given in Fig. 3 satisfy these stability conditions, indicating that CeB₆ remains mechanically stable up to 120 GPa.

3.3 Sound velocities and thermal properties

The Debye temperature is an important fundamental quantity that is closely related to many physical properties of solids, such as the specific heat and melting temperature. Above the Debye temperature, the quantum effects can be ignored, whereas below the Debye temperature, the quantum mechanical effects are very important for understanding the thermal properties of materials. According to the quasi-harmonic Debye model [36], we can obtain the Debye temperature {\Theta }_D of CeB₆ under different pressures and temperatures. In the quasi-harmonic Debye model, the non-equilibrium Gibbs energy under a pressure p and temperature T is written as

\begin{array}{}\text{(5)}& G^{\ast }(V;p;T)=E(V)+pV+A_{{}_{vib}}({\Theta }_D(V);T),\end{array}

where E(V) is the total energy per formula unit, pV corresponds to the constant hydrostatic-pressure condition, A_{vib} is the vibrational Helmholtz free energy, and {\Theta }_D is the Debye temperature {\Theta }_D, which is given as

\begin{array}{}\text{(6)}& {\Theta }_D={\displaystyle \frac{\mathrm{\hslash }}{k}}{(6{\pi }^2{V}^{1/2}n)}^{1/3}f(\sigma )\sqrt{{\displaystyle \frac{B_s}{M}}},\end{array}

with the assumption of an isotropic solid having Poisson’s ratio σ. Here, M is the molecular mass per formula unit, B_s is the adiabatic bulk modulus, and f(\sigma ) is given as

\begin{array}{}\text{(7)}& f(\sigma )={\left\{3{\left[2{\left({\displaystyle \frac{2}{3}}{\displaystyle \frac{1+\sigma }{1-2\sigma }}\right)}^{3/2}+{\left({\displaystyle \frac{1}{3}}{\displaystyle \frac{1+\sigma }{1-\sigma }}\right)}^{3/2}\right]}^{-1}\right\}}^{1/3}.\end{array}

Additionally, the elastic Debye temperature [37] can be obtained using the elastic constants. The Debye temperature {\Theta }_E is estimated as

\begin{array}{}\text{(8)}& {\Theta }_E={\displaystyle \frac{h}{k}}{\left[{\displaystyle \frac{3n}{4}}\left({\displaystyle \frac{N_A\rho }{M}}\right)\right]}^{1/3}{V}_m,\end{array}

where h is Planck’s constant, k is Boltzmann’s constant, N_A is Avogadro’s number, n is the number of atoms per formula unit, Mis the molecular mass per formula unit, ρ is the density, and V_m is obtained as [38]

\begin{array}{}\text{(9)}& V_m={\left[{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{2}{V_S^3}}+{\displaystyle \frac{1}{V_L^3}}\right)\right]}^{-1/3}.\end{array}

Here, V_S and V_L are the shear and longitudinal sound velocities, respectively. The probable values of the average shear and longitudinal sound velocities can be calculated using Navier’s equations as follows [39]:

\begin{array}{}\text{(10)}& V_S=\sqrt{{\displaystyle \frac{G}{\rho }}},V_L=\sqrt{{\displaystyle \frac{B+(4/3)G}{\rho }}}.\end{array}

Here, a Cauchy solid ratio of 0.25 is used as Poisson’s ratio σ.

The obtained shear and longitudinal sound velocities are 4.95 and 8.15 km/s, respectively, at 0 K and 0 GPa (Table 2). Clearly, the shear sound velocities increase slowly, and the longitudinal sound velocities increase relatively quickly. These effects are associated with the shear anisotropy and compression isotropy, respectively. Because the sound velocities are directly proportional to the elastic modulus of the medium, the shear and longitudinal sound velocities exhibit variation trends quite similar to those of the shear modulus G and elastic modulus E, respectively.

To obtain typical and persuasive energy-volume data points for the thermal-property calculation, every energy value at the corresponding volume was obtained from the novel point of the U_{eff} effect on the atomic interaction. According to the quasi-harmonic Debye model, the bulk modulus B is 181.3 GPa, which agrees with the experimental value of 182 GPa [8], confirming the validity of our results. Here, the Debye temperature {\Theta }_D was 718 K at 0 K and 0 GPa, which agrees with the previously obtained elastic Debye temperature {\Theta }_E = 760 K for CeB₆. The Debye temperature increased monotonically with pressure up to 100 GPa (Fig. 4). With pressure below 12 GPa, the difference between the two Debye temperatures decreased. However, at higher pressures—especially above 50 GPa—this difference increased. This suggests that the anisotropy of the cubic structure decreased before 12 GPa and increased after 12 GPa, which is consistent with the fluctuation of the shear anisotropy A_{shear} under pressure. This is partially because the Debye temperature {\Theta }_D was obtained under the assumption that the material was isotropic.

3.4 Electronic structures and magnetic properties

We focused on studying the electronic structure by analyzing the DOS with different values of U_{eff} and pressure. The DOS plays an important role in the analysis of the physical properties of materials. To precisely determine the energy positions of Ce-4f with respect to the Fermi level, we illustrate the calculated DOS with different approximations (LDA, LDA+U with U = 4 eV, and LDA+U with U = 8 eV) in Fig. 7(a). Obviously, the main peak of the Ce-4fstates was located at 0.5 eV in the conduction band, which is not in good agreement with the experimental result. According to Neupane et al., the peak is located around 2 eV in the conduction band [16]. The inappropriate treatment of the Ce-4f states within the LDA scheme largely causes this error. The main peak of Ce-4f transformed toward a higher level as U_{eff} increased. When the position of the Ce-4f peak was consistent with that of the experiment, as shown in Fig. 7(a), the obtained value of U_{eff} also agrees well with that reported in Ref. [13]. Thus, it is reasonable to calculate the physical properties with an optimal U_{eff} value of 4 eV.

Sandeep et al. [13] calculated the magnetic moment of CeB₆ as 1.03 u_B, which agrees with the results of Singh et al. [40]. Our results for the magnetic moment of the system, originating from the spin of the Ce atom, also indicate a value of 1.03 u_B at 0 GPa, whereas the magnetic moment decreased slightly to 0.97 u_B at 85 GPa. In magnetic materials, anti-parallel coupling between the magnetic moment and conduction electrons yields a non-magnetic Fermi liquid phase, which is known as the Kondo effect. Using neutron-diffraction experiments on CeB₆ single crystals, Rossat-Mignod et al. [41] detected that there is a homogeneous moment reduction because of the Kondo effect in low-temperature phase III. This reduction is consistent with our results. A comparison between Figs. 5(a) and (b) suggests that through the general shapes of the DOS for both spin directions, the magnetic moment did not vary significantly with pressure. There was a decrease in the intensity of the peaks with an absence of the DOS in the energy range of 7.5–17.5 eV at 85 GPa compared with that at 0 GPa. There was also a new small peak for spin-up electrons, which may cause the formation of the magnetic moment at this pressure.

Figure 6 depicts the band structures of spin-down and spin-up electrons at 0 GPa. Obviously, there is no bandgap between the conduction band minimum (CBM) and valence band maximum (VBM) for spin-up electrons, whereas there is a small gap for spin-down electrons. According to recent experiments by Neupane et al. [16], the Fermi surface of CeB₆ comprises large oval-shape pockets at the X points within the Brillouin zone, and the Ce quasi-particle states participate in the formation of hotspots at the Fermi surface, yielding an overlap between the CBM and VBM around the X points. The antibonding states, comprising the most localized Ce-4f states, have the smallest interaction with the neighboring atoms and thus have the smallest energy spread. Consequently, the un-localized Ce-5dstates and localized Ce-4f states correspond to the dispersive bands and flat bands, respectively. In Fig. 6, we observe dispersive Ce-5d bands and flat Ce-4f bands in the vicinity of the Fermi level with the Ce-5d and 4f states across the Fermi level along the high-symmetry direction. Moreover, one additional energy band appears in the valence band shown in Fig. 6(a) compared with that shown in Fig. 6(b), which agrees with the case where spin-up electrons contribute more to the DOS than spin-down electrons, as shown in Fig. 5(a). Our results show that the Ce-4f band is mainly above the Fermi level in the conduction band, whereas the ARPES data show that the 4f states are below the Fermi level. There are two reasons for this discrepancy: i) ARPES only measures the states below the Fermi level; ii) the theoretical band structure indicates that the maximum weight of the f bands are above the Fermi level [16], which agrees with our calculated DOS and band structure. One cannot exclude the possibility that the 4f states are above the Fermi level.

Figure 7 shows the calculated total DOS (TDOS) and partial DOS (PDOS) at 0 and 85 GPa. At both pressures, the DOS in the vicinity of the Fermi level is predominated by the Ce-4f and Ce-5d states. The typical characteristic of the TDOS for transition-metal borides may be what is regarded as a pseudogap (a sharp valley around the Fermi level) [42], which appear in crystalline solids [43], amorphous alloys [44], and quasi-crystals [45]. The two mechanisms were proposed for the formation of pseudogap in binary alloys. One is of ionic origin, and the other involves a hybridization effect. However, there is no noticeable pseudogap in CeB₆ and no distinctive hybridized peak between the Ce 4f+5d states and B 2p states shown in Fig. 7, which indicates the weakness of the covalent bonding between these two types of atoms. According to the atomic Mulliken population shown in Table 3, 2.34 and 2.72 electrons are transferred from Ce to B at 0 and 85 GPa, respectively, indicating that the Ce-B bonds have more distinctive ionic characteristics at 85 GPa. The electronegativity difference between Ce and B is high; hence, the ionicity strongly affects the bonding behavior. To further clarify the bonding property, the total charge densities at different pressures are shown in Fig. 8. Clearly, the electrons gradually accumulated at the B atoms and Ce atoms at 85 GPa compared with 0 GPa, suggesting that the charge transfer still dominates the bonding at a high pressure. This result agrees well with the population analysis.

It is clear that the traditional LDA method incorrectly predicts the DOS of CeB₆ in the vicinity of the Fermi level, making the peak of DOS closer to the Fermi level. According to Fig. 7, the inclusion of the Hubbard U term in the LDA functional causes the peak of the DOS to shift away from the Fermi level and moves the energy ranges of Ce 5d+4f states to the higher-energy region in the conduction band. This also occurs at 85 GPa. We can observe the difference in the DOSs of CeB₆ between 0 and 85 GPa. First, at 85 GPa, the TDOS and PDOS decrease over the entire range of their distribution. Second, the valence band and conduction band broaden at 85 GPa. Moreover, the peak at the bottom of the conduction band moves to the higher-energy region compared with that at 0 GPa. In conclusion, we find it interesting that the mechanism of high pressure is similar to that of increasing U_{eff} because the peaks at the bottom of conduction band move to the high-energy region in both cases.

4 Conclusions

We evaluated the structural, elastic, and electronic properties of the Kondo compound CeB₆ by using the LDA+U method according to density functional theory. It seems reasonable to conduct calculations of the physical structure and properties with an optimal U_{eff} value of 4 eV. Our calculated lattice constant and elastic constants are in good agreement with the experimental results and other theoretical findings. The U_{eff} dependence of the lattice constant and bulk modulus, as well as the elastic constants under pressure, were investigated for the first time. The important conclusions of our calculations are as follows. i) There was a stable increase in the lattice parameter as U_{eff} increased, whereas the bulk modulus was less sensitive to higher values of U_{eff} for both the LDA and GGA functionals. ii) The pressure at which this substance transformed from brittle to ductile was predicted as 27 GPa. In contrast to experiments speculating the possibility of a structural change in the range of 85–122 GPa, our results indicate that CeB₆ is mechanically stable up to 120 GPa. iii) We estimated a Debye temperature {\Theta }_D of 718 K using thermodynamic methods, which is close to the elastic Debye temperature {\Theta }_E of 760 K obtained with elastic constants at 0 K and 0 GPa. iv) The magnetic moment of this system was insensitive to the pressure, according to the calculations and TDOS analysis. The Ce-B bonds had a more distinctive ionic characteristic at 85 GPa than at 0 GPa. The mechanism of high pressure is similar to that of the increase in U_{eff} for the movement of the peak to the high-energy region at the bottom of the conduction band.

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