First-Principles Study of the Nonlinear Elasticity of Rare-Earth Hexaborides REB₆ (RE = La, Ce)

Abstract

The complete set of independent second- and third-order elastic constants of rare-earth hexaborides LaB${}_6$ and CeB${}_6$ are determined by the combination method of first-principles calculations and homogeneous deformation theory. The ground-state lattice parameters, second-order elastic constants, and bulk modulus are in reasonable agreement with the available experimental data. The third-order elastic constant of longitudinal mode $C_{111}$ has a larger absolute value than other shear modes, showing the contribution to lattice vibrations from longitudinal modes to be greater. The pressure derivatives of the second-order elastic constants related to the third-order elastic constants are calculated to be positive for the two hexaborides, which are consistent with those of their polycrystalline bulk modulus and shear modulus. Furthermore, the effect of pressure on the structural stability, mechanical property, and elastic anisotropy of the two hexaborides are investigated, showing a reduction in mechanical stability and an increase in ductility and anisotropy with increasing pressure.

1. Introduction

Rare-earth hexaborides ($RE$B${}_6$) have been widely used in various field-electron emitter devices and high-energy optical systems due to their attractive properties of high melting point, high mechanical strength, low work function, low volatility at high temperatures, conductivity, chemical resistance, brightness, small optical size, long service life, and the monoenergetic character of their electrons [1]. These attractive properties are closely related to their crystal structure, which is a simple cubic CsCl (space group Pm$\overline{3}$m) structure with boron octahedra at the cube corners and a rare-earth atom occupying the body-center position. In this structure, each rare-earth atom is surrounded by eight boron octahedra, and the boron octahedra are linked together into a three-dimensional network. It is generally accepted that the structural peculiarity should be taken as one of the basic starting points to explain some of the anomalies in the $RE$B${}_6$ systems. Currently, research focuses on the developments and applications of the $RE$B${}_6$ nanostructures. Understanding the mechanical behavior of these structures is very important in the development and application stages. It is well-known that nonlinear effects become significant in nanostructural materials. Nonlinear elastic properties are important for describing nonlinear effects in mechanical behavior. Thus, it is very necessary to study nonlinear effects in the elasticity of the hexaborides.

In general, the second-order elastic constants (SOECs) $C_{ij}$ describe the linear elastic stress–strain response for single crystals. Third-order elastic constants (TOECs) $C_{ijk}$ are important quantities to characterize the nonlinear elasticity of the crystals. Both SOECs and TOECs are important parameters to model the mechanical response of crystals under high pressure. Tanaka et al. [2] have made measurements of the transit times of pulses of longitudinal and transverse ultrasonic waves propagating in single crystal LaB${}_6$ at room temperature, and have determined its SOECs from the resultant velocities. Baranovskiy et al. [3] measured the sound velocities along the principal crystallographic axes of LaB${}_6$ at 78 K by employing the phase-frequency method, and evaluated its SOECs and bulk modulus from the resultant sound velocities. Nakamura et al. [4,5] performed sound velocity measurements with an ultrasonic apparatus based on the phase comparison method to investigate the temperature dependence of the SOCEs in LaB${}_6$ and CeB${}_6$. Goto et al. [6] also studied the temperature dependence of the SOECs in CeB${}_6$. Lüthi et al. [7] took the ultrasonic and Brillouin scattering measurements to redetermine the SOECs of CeB${}_6$ at room temperature. Moreover, Gürel and Eryiǧit [8] used a first-principles calculations method to study structural, elastic, lattice-dynamical, and thermodynamical properties of LaB${}_6$ and CeB${}_6$. Tang et al. [9] performed a first-principles study of structural, elastic, and electronic properties of CeB${}_6$ under pressure. Besides, high-pressure phase transitions in LaB${}_6$ and CeB${}_6$ have been investigated experimentally. By using Raman and X-ray diffraction measurements on LaB${}_6$, Teredesai et al. [10] proposed that a phase transition from cubic to orthorhombic crystal structure occurs around 10 GPa, while Godwal et al. [11] stated that there is no structural phase transitions up to at least 25 GPa. By using X-ray diffraction measurements on CeB${}_6$, Leger et al. [12] revealed no structural phase transition at ambient temperature up to 20 GPa. Foroozani et al. [13] failed to detect any change in crystal structure up to 85 GPa, but they did not exclude the possibility of structural changes in the pressure range of 85−122 GPa. Nevertheless, the TOECs and related elastic properties of LaB${}_6$ and CeB${}_6$ have not yet been investigated either experimentally or theoretically to the best knowledge of the authors.

In recent years, first-principles calculations based on density functional theory (DFT) have been employed to successfully determine the SOECs, TOECs, and higher-order elastic constants of single crystals by utilizing a series of homogeneous deformation strains applied to a crystalline system to obtain the energy–strain relations [14,15,16]. To study the nonlinear effects in the elasticity of LaB${}_6$ and CeB${}_6$, we shall use the same method to determine the SOECs and TOECs of both hexaborides. Subsequently, the pressure derivatives of the effective SOECs have been estimated from the obtained values of SOECs and TOECs. The polycrystalline bulk, shear, and Young’s moduli, Poisson’s ratio, and their pressure dependence in LaB${}_6$ and CeB${}_6$ have also been studied along with the elastic anisotropy of the two hexaborides. The paper is organized as follows. In Section 2, a brief description of computational methodology is given. In Section 3, the results we have obtained are presented with available experimental and theoretical values for comparison. Finally, the conclusions are drawn in Section 4.

2. Computational Methods

2.1. First-Principles Total-Energy Calculations

First-principles calculations have been performed by means of the Vienna Ab initio simulation package (VASP) code based on density functional theory (DFT) [17,18,19]. The projector augmented wave (PAW) method was used for describing the ion–electron interaction [20,21]. The generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) was used for evaluating the exchange-correlation energy [22,23]. The standard PAW potentials were used for La, Ce, and B elements. A cutoff energy of 600 eV was chosen for the plane wave basis. A threshold of 10${}^{-6}$ eV per atom on total energy was set for the convergence of electronic self-consistency. A 15 × 15 × 15 Monkhorst–Pack grid of k-point was adopted for sampling the Brillouin zone [24]. Before calculating the elastic constants, the structures of LaB${}_6$ and CeB${}_6$ were fully relaxed with respect to the volume, shape, and internal atomic position until the atomic forces were less than 0.01 eV/Å. To accurately calculate the elastic constants, the linear tetrahedron method was used for the final self-consistent calculations of total-energies.

2.2. SOECs and TOECs of Single Crystal

In this paper, the method of the finite-strain continuum elasticity theory is employed to calculate the SOECs and TOECs. Here we discuss the theory briefly, the details of which have been given in References [25,26,27,28,29]. Let $a_i$ be the initial Cartesian coordinates of a material point in the unstrained state. A finite homogeneous deformation carries the material point to the final position with the coordinates $x_i$ in the strained state. After introducing the Jacobian deformation gradient

$$J_{ij}=\frac{\partial x_i}{\partial a_j}(i,j=1,2,3),$$

(1)

the Lagrangian strain tensor in the stressed state may be defined as

$${\eta }_{ij}=\frac{1}{2}\sum _{r=1}^3(J_{ri}{J}_{rj}-{\delta }_{ij}).$$

(2)

The elastic strain energy ($\Delta E$) can be expanded in a Taylor series in terms of the strain tensor as

$$\Delta E=\frac{V}{2!}\sum _{ij}{C}_{ijkl}{\eta }_{ij}{\eta }_{kl}+\frac{V}{3!}\sum _{ijklmn}{C}_{ijklmn}{\eta }_{ij}{\eta }_{kl}{\eta }_{mn}+O\left({\eta }^4\right),$$

(3)

where V is the volume of the unstrained lattice. After applying the Voigt notation (11→1, 22→2, 33→3, 23→4, 13→5, and 12→6) to denote the strain tensor, the strain energy can be rewritten as

$$\Delta E=\frac{V}{2!}\sum _{ij}{C}_{ij}{\eta }_i{\eta }_j+\frac{V}{3!}\sum _{ijk}{C}_{ijk}{\eta }_i{\eta }_j{\eta }_k+O\left({\eta }^4\right).$$

(4)

For cubic systems, there are three independent SOECs ($C_{11}$, $C_{12}$, $C_{44}$) and six dependent TOECs ($C_{111}$, $C_{112}$, $C_{123}$, $C_{144}$, $C_{155}$, $C_{456}$). To obtain the complete set of SOECs and TOECs of LaB${}_6$ and CeB${}_6$, we used six Lagrangian strain tensors in terms of a single strain parameter $\xi$.

Table 1. The coefficients $A_2$ and $A_3$ in Equation (5) of the corresponding selected strain tensors as the linear combinations of the second- and third-order elastic constants for cubic crystal [14].

Strain	${\mathit{A}}_2$	${\mathit{A}}_3$
${\eta }_1=(\xi ,0,0,0,0,0)$	$C_{11}$	$C_{111}$
${\eta }_2=(\xi ,\xi ,0,0,0,0)$	2$C_{11}$ + 2$C_{12}$	2$C_{111}$ + 6$C_{112}$
${\eta }_3=(\xi ,\xi ,\xi ,0,0,0)$	3$C_{11}$ + 6$C_{12}$	3$C_{111}$ + 18$C_{112}$ + 6$C_{123}$
${\eta }_4=(\xi ,0,0,2\xi ,0,0)$	$C_{11}$ + 4$C_{44}$	$C_{111}$ + 12$C_{144}$
${\eta }_5=(\xi ,0,0,0,2\xi ,0)$	$C_{11}$ + 4$C_{44}$	$C_{111}$ + 12$C_{155}$
${\eta }_6=(0,0,0,2\xi ,2\xi ,2\xi )$	12$C_{44}$	$48C_{456}$

gives the relationship between the coefficients $A_2$ and $A_3$ and SOECs and TOECs for the six selected strain tensors. For each strain tensor, the strain parameter $\xi$ varied from −0.08 to 0.08 in steps of 0.01. Inserting these strain tensors into Equation (4), the strain energy density $\mathsf{\Phi }$ can be written as an expansion in the strain parameter $\xi$ as

$$\mathsf{\Phi }=\frac{\Delta E}{V}=\frac{1}{2}{A}_2{\xi }^2+\frac{1}{6}{A}_3{\xi }^3+O\left({\xi }^4\right),$$

(5)

where $A_2$ and $A_3$ are the combinations of SOECs and TOECs, respectively. For every deformed configuration, the atomic positions were optimized, and the total-energy was calculated by using first-principles method based on DFT. The strain energy is defined as the total-energy difference between the deformed and the perfect crystals. In this way, the dependencies of the strain energy $\Delta E$ on the strain parameter $\xi$ were obtained for each homogeneous deformation. By comparing with the expressions from the finite-strain elasticity theory given in Table 1, the elastic constants were extracted from a polynomial fit to the strain energy versus strain parameter curves under the various strains.

2.3. Pressure Derivatives of the Effective SOECs

When an external hydrostatic pressure is applied to a crystal, the effective SOECs are very useful to describe the nonlinear elastic properties of the crystal. Usually, the effective SOECs under hydrostatic pressure P ($C_{ij}\left(P\right)$) can be expanded by a Taylor expansion as [15]

$$C_{ij}\left(P\right)\approx C_{ij}+\frac{d{C}_{ij}\left(P\right)}{dP}P=C_{ij}+C_{ij}^{\prime }P,$$

(6)

where $C_{ij}^{\prime }$ is the first-order pressure derivative, which can be determined from SOECs and TOECs, and can be expressed for cubic systems as [15,25]

$$\begin{array}{c}{C}_{11}^{\prime }=-\frac{C_{111}+2C_{112}+2C_{11}+2C_{12}}{C_{11}+2C_{12}},\\ C_{12}^{\prime }=-\frac{2C_{112}+C_{123}-C_{11}-C_{12}}{C_{11}+2C_{12}},\\ C_{44}^{\prime }=-\frac{2C_{155}+C_{144}+C_{11}+2C_{12}+C_{44}}{C_{11}+2C_{12}}.\end{array}$$

(7)

2.4. Pressure Derivatives of Polycrystalline Elastic Moduli

On the basis of the effective elastic constants, the bulk modulus B and shear modulus G for LaB${}_6$ and CeB${}_6$ under different pressure were obtained using the Voigt, Reuss, and Hill approximations [30,31,32]. For the specific case of cubic structures, the Voigt’s and Reuss’s bulk moduli can be expressed as

$$B_V=B_R=\frac{C_{11}+2C_{12}}{3},$$

(8)

and the Voigt’s and Reuss’s shear moduli are defined as

$$\begin{array}{c}{G}_V=\frac{C_{11}-C_{12}+3C_{44}}{5},\\ G_R=\frac{5(C_{11}-C_{12})C_{44}}{3(C_{11}-C_{12})+4C_{44}}.\end{array}$$

(9)

Hill proposed that the effective elastic moduli are the arithmetic averages of the Voigt and Reuss moduli, and thus obtained by

$$\begin{array}{c}{B}_H=\frac{B_V+B_R}{2},\\ G_H=\frac{G_V+G_R}{2},\end{array}$$

(10)

where the subscripts “V” and “R” correspond to the Voigt and Reuss bounds, and the subscript “H” represents the Hill averaging method. The Young’s modulus (E) and Poisson’s ratio ($\nu$) are given by

$$\begin{array}{c}{E}_X=\frac{9B_X{G}_X}{3B_X+G_X},\\ {\nu }_X=\frac{3B_X-2G_X}{2(3B_X+G_X)},\end{array}$$

(11)

where $X=V,R,H$. The pressure derivative of the bulk and shear moduli can be given by

$$\begin{array}{c}{B}_H^{\prime }=B_V^{\prime }=B_R^{\prime }=\frac{C_{11}^{\prime }+2C_{12}^{\prime }}{3},\\ G_V^{\prime }=\frac{C_{11}^{\prime }-C_{12}^{\prime }+3C_{44}^{\prime }}{5},\\ G_R^{\prime }=\frac{5[3{(C_{11}-C_{12})}^2{C}_{44}^{\prime }+4(C_{11}^{\prime }-C_{12}^{\prime })C_{44}^2]}{{[3(C_{11}-C_{12})+4C_{44}]}^2},\\ G_H^{\prime }=\frac{G_V^{\prime }+G_R^{\prime }}{2}.\end{array}$$

(12)

The pressure derivative of the Young’s modulus $E_X^{\prime }$ and the Poisson’s ratio ${\nu }_X^{\prime }$ can be given by

$$\begin{array}{c}{E}_X^{\prime }=\frac{9(3B_X^2{G}_X^{\prime }+B_X^{\prime }{G}_X^2)}{{(3B_X+G_X)}^2},\\ {\nu }_X^{\prime }=\frac{9(B_X^{\prime }{G}_X-B_X{G}_X^{\prime })}{2{(3B_X+G_X)}^2}.\end{array}$$

(13)

Under the external hydrostatic pressure P, the polycrystalline elastic moduli $Y_X\left(P\right)$ ($Y=B,G,E,\nu$) can be given by

$$Y_X\left(P\right)=Y_X+\frac{d{Y}_X\left(P\right)}{dP}P=Y_X+Y_X^{\prime }P.$$

(14)

3. Results and Discussion

3.1. Second-Order and Third-Order Elastic Constants of $RE$B${}_6$ ($RE=$ La, Ce)

The calculated results of the lattice parameters and the SOECs of LaB${}_6$ and CeB${}_6$ at ground-state are listed in

Table 2. Lattice parameter a (in Å) and elastic constants $C_{ij}$ (in GPa) of LaB${}_6$ and CeB${}_6$.

Crystal	Method	a	${\mathit{C}}_{11}$	${\mathit{C}}_{12}$	${\mathit{C}}_{44}$
LaB${}_6$	Present	4.154	474.1	24.3	90.1
	Exp. [2]	4.156	453.3	18.2	90.1
	Exp. [4,33]	4.1569	478	43	84
	Exp. [3]	4.1565	463	45	89
	The. [8]	4.1277	466	37	88
CeB${}_6$	Present	4.114	488.2	17.9	74.8
	Exp. [5,13]	4.132	473	16	81
	Exp. [7,34]	4.1407	508	19	79
	Exp. [7]		472	53	78
	Exp. [6]		406	$-93$	78
	The. [9]	4.121	483	10	75
	The. [8]	4.154	452	34	98

along with the published experimental values and other calculated results [2,3,4,5,6,7,8,9,13,33,34]. Comparison of the lattice constants shows that the calculated quantities of the present work are in excellent agreement with the previous experimental and theoretical values. The maximum relative error between our calculated result and the experimental data is $0.07\%$ ($0.65\%$) for LaB${}_6$ (CeB${}_6$), and that between our and other calculated results correspond to $0.63\%$ ($0.96\%$) for LaB${}_6$ (CeB${}_6$). The room-temperature lattice constant of LaB${}_6$ was measured as 4.156 Å from the X-ray power diffraction patterns [2], which is larger than that (4.1407 Å) of CeB${}_6$ [34]. It is the same with the results obtained in this study. These can be explained well by the larger atomic radius of La (2.74 Å) compared with Ce (2.70 Å) [35].

The strain energy versus strain parameter curves under the various strains are fitted with a suitable polynomial to obtain the coefficients $A_2$ and $A_3$ in the Equation (5) for determining the SOECs and TOECs, as illustrated in Figure 1. The discrete points and the solid lines represent the results obtained from the first principles calculations and fitted polynomials, respectively. Obviously, these dependent curves of the strain energy on the strain have the characteristics of the asymmetry, which is the expected behavior under finite-strain elastic deformation. The strain energy with negative strains is always larger than that with positive strains, and thus the TOECs are typically negative. The fitted curves match well with the first-principles calculations results. For the values of SOECs, it is possible that some constants may be determined from a few polynomial fits (e.g., $C_{44}$ from coefficients in $f_4\left(\eta \right)$, $f_5\left(\eta \right)$, and $f_6\left(\eta \right)$), together with obtaining slightly different results (e.g., for LaB${}_6$, $C_{44}$ = 90.2, 89.1, and 91.0 GPa from $f_4\left(\eta \right)$, $f_5\left(\eta \right)$, and $f_6\left(\eta \right)$). In such cases, the average value of all results is given in Table 2. Measurements of the SOECs for LaB${}_6$ and CeB${}_6$ exhibit large discrepancies among themselves, partly because of different techniques to obtain them. Especially, the SOEC $C_{12}$ of CeB${}_6$ was measured to be large negative in reference [6] while positive in references [5,7]. For LaB${}_6$, the calculated SOECs of the present study are in reasonable accordance with the previous experimental and theoretical results [2,3,4,8]. For CeB${}_6$, a negative value of $C_{12}$ reported in reference [6] was also not found in the present study. The calculated results of the SOECs are also in reasonable agreement with other experiments [5,7] and previous calculations [8,9]. For the two hexaborides, the SOEC $C_{11}$ measures the resistance to linear compression along the uniaxial axes, and the others are mainly related to the non-axial sound propagation. The values of $C_{11}$ are significantly larger than the others in both systems, implying that they are difficult to compress under uniaxial stress. In addition, the ground-state SOECs of LaB${}_6$ and CeB${}_6$ can satisfy the three Born stability criteria for the cubic system [36]: $C_{11}+2C_{12}>0$, $C_{11}-C_{12}>0$ and $C_{44}>0$, and thus their cubic structures are both mechanically stable.

The TOECs allow the determination of anharmonic properties of crystals, such as thermal expansion, interactions of acoustic and thermal phonons, and temperature and pressure dependence of elastic constants. The evaluation of the TOECs is of general interest. The calculated TOECs of LaB${}_6$ and CeB${}_6$ are given in

Table 3. Third-order elastic constants $C_{ijk}$ (in GPa) of LaB${}_6$ and CeB${}_6$.

Crystal	${\mathit{C}}_{111}$	${\mathit{C}}_{112}$	${\mathit{C}}_{123}$	${\mathit{C}}_{144}$	${\mathit{C}}_{155}$	${\mathit{C}}_{456}$
LaB${}_6$	$-2647.1$	$-373.2$	254.3	$-167.6$	$-340.4$	$-403.8$
CeB${}_6$	$-2568.4$	$-402.8$	304.8	$-69.6$	$-286.9$	$-408.4$

. All the TOECs of the two hexaborides are negative except for the values of $C_{123}$ (=254.3 GPa (304.8 GPa) for LaB${}_6$ (CeB${}_6$)). The absolute values of their longitudinal mode $C_{111}$ (=$-2647.1$ GPa ($-2568.4$ GPa) for LaB${}_6$ (CeB${}_6$)) were found to be much greater than the corresponding shear modes $C_{112}$, $C_{123}$, $C_{144}$, $C_{155}$, and $C_{456}$, implying the contribution to lattice vibrations from their longitudinal modes is much greater. The highest absolute values of $C_{111}$ indicate a pronounced anisotropy in both materials. Unfortunately, no measurements or calculations on the TOECs of LaB${}_6$ and CeB${}_6$ are available for comparison.

3.2. Pressure Derivatives of the Effective Second-Order Elastic Constants of $RE$B${}_6$ ($RE=$ La, Ce)

The TOECs above were further used to evaluate the first-order pressure derivatives of LaB${}_6$ and CeB${}_6$. These values are summarized in

Table 4. Pressure derivatives $C_{ij}^{\prime }$ of the second-order elastic constants of LaB${}_6$ and CeB${}_6$.

Crystal	${\mathit{C}}_{11}^{\prime }$	${\mathit{C}}_{12}^{\prime }$	${\mathit{C}}_{44}^{\prime }$
LaB${}_6$	4.586	1.895	0.451
CeB${}_6$	4.507	1.921	0.085

. A linear increase with pressure was observed for $C_{11}$, $C_{12}$, $C_{44}$ with pressure derivative $C_{ij}^{\prime }$ of 4.586, 1.895, 0.451 and 4.507, 1.921, 0.085 for LaB${}_6$ and CeB${}_6$. The pressure-induced variation in the longitudinal mode $C_{11}$ was the largest, followed by the shear mode $C_{12}$, and the smallest for the pure shear mode $C_{44}$. For the two hexaborides, the effect of the pressure on the $C_{12}$ was obviously smaller than that on the $C_{11}$, but markedly greater than that on the $C_{44}$. Under hydrostatic pressure, the Born stability criteria for the cubic system are given by [9]

$$\begin{array}{c}{K}_1={\tilde{C}}_{11}^P+2{\tilde{C}}_{12}^P>0,\\ K_2={\tilde{C}}_{11}^P-{\tilde{C}}_{12}^P>0,\\ K_3={\tilde{C}}_{44}^P>0,\end{array}$$

(15)

with

$$\begin{array}{c}{\tilde{C}}_{ii}^P=C_{ii}\left(P\right)-P(i=1,4),\\ {\tilde{C}}_{12}^P=C_{12}\left(P\right)+P.\end{array}$$

(16)

In terms of the effective SOECs, these criteria can be expressed as

$$\begin{array}{c}{K}_1=(C_{11}+2C_{12})+(C_{11}^{\prime }+2C_{12}^{\prime }+1)P>0,\\ K_2=(C_{11}-C_{12})+(C_{11}^{\prime }-C_{12}^{\prime }-2)P>0,\\ K_3=C_{44}+(C_{44}^{\prime }-1)P>0.\end{array}$$

(17)

From the pressure derivatives of the SOECs, the first-order pressure derivatives of $K_1$ and $K_2$ were calculated as 9.376 (9.349) and 0.691 (0.586) for LaB${}_6$ (CeB${}_6$), while that of the corresponding $K_3$ was calculated to be $-0.549$ ($-0.915$). Thus, the $K_1$ and $K_2$ values of both materials linearly increase while the $K_3$ values linearly decrease with increasing pressure. When the pressure applied to LaB${}_6$ (CeB${}_6$) is beyond 164.1 GPa (81.7 GPa), the $K_3$ has a negative value, as shown in Figure 2. This indicates that the cubic structure of LaB${}_6$ (CeB${}_6$) can remain mechanically stable up to 164.1 GPa (81.7 GPa). Previous high-pressure phase transitions showed that no structural phase transitions occur up to 25 GPa (85 GPa) for LaB${}_6$ (CeB${}_6$) [11,12,13]. Therefore, the results of the present work are basically consistent with those of previous experiments.

3.3. Pressure Derivatives of the Polycrystalline elastic moduli of $RE$B${}_6$ ($RE=$ La, Ce)

The polycrystalline bulk (B), shear (G), and Young’s (E) moduli and Poisson’s ratio ($\nu$) of LaB${}_6$ and CeB${}_6$ were calculated based on their single crystal SOECs, which are collected in

Table 5. Polycrystalline bulk modulus B, shear modulus G, and Young’s modulus E (in GPa), and Poisson’s ratio $\nu$ of LaB${}_6$ and CeB${}_6$.

Crystal	Method	B	G	E	$\mathit{\nu }$	$\mathit{B}/\mathit{G}$
LaB${}_6$	Voigt	174.2	144.0	338.8	0.176
	Reuss	174.2	118.5	289.8	0.223
	Hill	174.2	131.3	314.8	0.199	1.327
CeB${}_6$	Voigt	174.7	138.9	329.4	0.186
	Reuss	174.7	102.8	257.8	0.254
	Hill	174.7	120.9	294.6	0.219	1.445

. The calculated bulk modulus (174.2 GPa) of LaB${}_6$ agrees well with the reported experimental values of 163, 184, 188, $164\pm 2$, $173\pm 7$, and 172 GPa [2,3,4,11,37], and the previous theoretical results of 185 and 180 GPa [3,8]. For CeB${}_6$, the calculated bulk modulus (174.7 GPa) of the present study are also in good accordance with experimental measurements of 191, 168, 182, and 166 GPa [5,7,12] and other theoretical results of 173 and 166.8 GPa [8,9]. The calculated G value of 120.9 GPa is excellently consistent with that of 121.2 GPa reported in Ref. [9]. The pressure derivatives of the bulk modulus and shear modulus of the two hexaborides are calculated using the pressure derivatives of the single crystal SOECs, and then those of the Young’s modulus and the Poisson’s ratio are also estimated. All the obtained results are presented in

Table 6. Pressure derivatives of polycrystalline elastic modulus $B^{\prime }$ (bulk modulus), $G^{\prime }$ (shear modulus), and $E^{\prime }$ (Young’s modulus), and Poisson’s ratio ${\nu }^{\prime }$ of LaB${}_6$ and CeB${}_6$.

Crystal	Method	${\mathit{B}}^{\prime }$	${\mathit{G}}^{\prime }$	${\mathit{E}}^{\prime }$	${\mathit{\nu }}^{\prime }$($\times {10}^{-3}$)	${(\mathit{B}/\mathit{G})}^{\prime }$
LaB${}_6$	Voigt	2.792	0.809	2.664	2.645
	Reuss	2.792	0.618	2.089	2.445
	Hill	2.792	0.713	2.379	2.550	0.014
CeB${}_6$	Voigt	2.783	0.568	2.165	2.942
	Reuss	2.783	0.196	1.084	2.886
	Hill	2.783	0.382	1.636	2.918	0.018

. The calculated $B^{\prime }$ value of 2.792 (2.783) for LaB${}_6$ (CeB${}_6$) can compare well with the experimental value of $4.2\pm 1.5$ [11] (3.15 [12]). One can see that the $B^{\prime }$ and $G^{\prime }$ are positive for both materials, which agree well with the corresponding $C_{ij}^{\prime }$s. A similar behavior is found for the $E^{\prime }$ and ${\nu }^{\prime }$ being positive, and is in good accordance with the $B^{\prime }$ and $G^{\prime }$. The bulk modulus is a measure of the resistance of a material to volume changes. The shear modulus is a measure of the resistance of a material to shear deformation. The bulk modulus and shear modulus play an important role in determining the strength of solids [38]. Usually, a superhard material has a high bulk modulus, high shear modulus, and high shear strength. Thus, we can predict that LaB${}_6$ and CeB${}_6$ have high hardness. The positive values of $B^{\prime }$ and $G^{\prime }$ mean that the elastic moduli B and G can increase gradually with the pressure, showing that the two hexaborides with a simple cubic structure became more difficult to compress and shear as the pressure increased. The Young’s modulus is a measure of the stiffness of a material. The larger the value of E, the stiffer the material. The positive values of $E^{\prime }$ demonstrate that LaB${}_6$ and CeB${}_6$ become more and more stiff as the pressure increases.

Pugh [39] has introduced the ratio between the bulk modulus and the shear modulus ($B/G$) to assess the ductile/brittle behaviors of a material. A high (low) $B/G$ value is correlated with the ductility (brittleness) of the material. The critical value of the brittle-to-ductile transition was observed to be ∼1.75. The $B/G$ ratio was calculated as 1.327 (1.445) for LaB${}_6$ (CeB${}_6$) at ground-state, indicating the brittleness of the hexaboride. The pressure derivative ${(B/G)}^{\prime }$ can be obtained from those of the bulk modulus and the shear modulus, which is expressed as ${(B/G)}^{\prime }=(B^{\prime }G-B{G}^{\prime })/G^2$. On this basis, the ${(B/G)}^{\prime }$ value was evaluated to be 0.014 (0.018) for LaB${}_6$ (CeB${}_6$), showing that the $B/G$ ratio can increase with increasing pressure. The pressure at which a brittle-to-ductile transition happens is predicted as 30.1 GPa (16.5 GPa) for LaB${}_6$ (CeB${}_6$). Frantsevich et al. [40] distinguished the ductility/brittleness of the materials in terms of Poisson’s ratio. Generally, a brittle material has a lower Poisson’s ratio than 0.26. The ground-state value of the Poisson’s ratio is consistent with the $B/G$ ratio for LaB${}_6$ (CeB${}_6$). The very small and positive values of ${\nu }^{\prime }$ imply a very slow increase of the $\nu$ of both materials with the pressure. Pettifor [41] introduced the Cauchy pressure to describe the covalent character of atomic bonding related to the ductile/brittle characteristics of a material. He suggested that larger positive Cauchy pressure corresponds to a ductile material with more metallic bonds, whereas larger negative values indicate a brittle behavior with a more covalent character of bonds. For a cubic system, the Cauchy pressure $P_c$ is defined as $P_c=C_{12}-C_{44}$. The $P_c$ of LaB${}_6$ (CeB${}_6$) was calculated as $-65.9$ GPa ($-58.6$ GPa) in terms of the ground-state SOECs given in Table 2. We can find that the bonding of both materials is covalent with $B/G<1.75$, leading to a brittle behavior. The pressure derivative $P_c^{\prime }$ can be obtained from those of the effective SOECs, which is given by $P_c^{\prime }=C_{12}^{\prime }-C_{44}^{\prime }$. The $P_c^{\prime }$ value of LaB${}_6$ (CeB${}_6$) was estimated to be 1.444 (1.836) based on the values of $C_{12}^{\prime }$ and $C_{44}^{\prime }$ given in Table 4, showing the increase of the Cauchy pressure with the pressure. Overall, the brittleness of LaB${}_6$ (CeB${}_6$) can reduce as the pressure increases, which is consistent with the previous theoretical result [9].

3.4. Pressure Derivatives of the Elastic Anisotropy of $RE$B${}_6$ ($RE=$ La, Ce)

The elastic anisotropy of a material has an important implication in engineering science due to its high association with the possibility of inducing microcracks in the material [42]. The elastic anisotropy factor for a cubic crystal introduced firstly by Zener [43] is expressed as

$$A_Z=\frac{2C_{44}}{(C_{11}-C_{12})}.$$

(18)

A single crystal with $A_Z=1$ is isotropic, while values smaller or greater than unity describe the degree of elastic anisotropy. The pressure derivative $A_Z^{\prime }$ can be obtained from those of the effective SOECs, which is given by

$$A_Z^{\prime }=\frac{[C_{44}^{\prime }(C_{11}-C_{12})-C_{44}(C_{11}^{\prime }-C_{12}^{\prime })]}{{(C_{11}-C_{12})}^2}.$$

(19)

Subsequently, Chung and Buessem [44] empirically improved the anisotropy factor of Zener by

$$A_C=\frac{(G_V-G_R)}{(G_V+G_R)}.$$

(20)

A single crystal with $A_C=0$ is isotropic, otherwise it is anisotropic. The pressure derivative $A_C^{\prime }$ can be obtained from those of the elastic modulus $G_V^{\prime }$ and $G_R^{\prime }$, which is given by

$$A_C^{\prime }=\frac{2(G_V^{\prime }{G}_R-G_V{G}_R^{\prime })}{{(G_V+G_R)}^2}.$$

(21)

However, it is noteworthy that $A_Z$ and $A_C$ do not account for the bulk part of the elastic stiffness tensor, and they are only effective for cubic crystals. To consider the contributions of the shear modulus and bulk modulus, Shivakumar et al. [45] proposed a universal anisotropy index that is applicable to various crystal systems, which was expressed as $A_S=B_V/B_R+5G_V/G_R-6$. A nonzero value of $A_S$ is a measure of the anisotropy. Because the $B_V$ is equal to the $B_R$ for cubic system, the $A_S$ can be simplified as

$$A_S=5(\frac{G_V}{G_R}-1).$$

(22)

The pressure derivative $A_S^{\prime }$ can be calculated in terms of those of the elastic modulus $G_V^{\prime }$ and $G_R^{\prime }$, which is given by

$$A_S^{\prime }=5\frac{(G_V^{\prime }{G}_R-G_V{G}_R^{\prime })}{G_R^2}.$$

(23)

In terms of the SOECs, the elastic moduli, and the corresponding pressure derivatives above, we predicted the values and the pressure derivatives of various anisotropy factors for LaB${}_6$ and CeB${}_6$, as presented in

Table 7. Anisotropic factors A of LaB${}_6$ and CeB${}_6$ and their pressure derivatives $A^{\prime }$ ($\times {10}^{-4}$).

Crystal	${\mathit{A}}_{\mathit{Z}}$	${\mathit{A}}_{\mathit{Z}}^{\prime }$	${\mathit{A}}_{\mathit{C}}$	${\mathit{A}}_{\mathit{C}}^{\prime }$	${\mathit{A}}_{\mathit{S}}$	${\mathit{A}}_{\mathit{S}}^{\prime }$
LaB${}_6$	0.401	$-1.956$	0.097	2.002	1.076	24.560
CeB${}_6$	0.318	$-6.900$	0.149	10.700	1.756	147.830

. The $A_Z$ values of less than one and the nonzero $A_C$ and $A_S$ values show the elastic anisotropy of the two hexaborides at ground-state. Meanwhile, the negative values of $A_Z^{\prime }$ and the positive ones of $A_C^{\prime }$ and $A_S^{\prime }$ values indicate that the corresponding anisotropy factors can be far away from unity with increasing pressure. These results show that the anisotropy of both materials can be enhanced by increasing the pressure.

4. Conclusions

The SOECs and TOECs of rare-earth hexaborides LaB${}_6$ and CeB${}_6$ have been determined by using first-principles calculations and homogeneous deformation methods. The calculated lattice parameters, SOECs, and bulk moduli are in reasonable accordance with the available experimental and theoretical values. All the TOECs of both hexaborides have negative values except for $C_{123}$. The highest absolute values of the longitudinal modes $C_{111}$ show that their contribution to lattice vibrations is the greatest, and so also their anisotropy in $C_{111}$. From the calculated elastic constants, the pressure derivatives of the effective SOECs have been investigated along with the pressure effects on the structural stability, mechanical properties, and elastic anisotropy. As the pressure increases, the mechanical stability reduces, and the ductility and anisotropy increase for both hexaborides. The full set of TOECs of both materials have been determined for the first time, and comparison could not be made as the same is not available in literature. The measurements of the TOECs are essential for the research of LaB${}_6$ and CeB${}_6$ in the future.