Ab initio lattice dynamics and thermodynamics of rare-earth hexaborides ${\text{LaB}}_6$ and ${\text{CeB}}_6$

We have performed an ab initio study of structural, elastic, lattice-dynamical, and thermodynamical properties of rare-earth hexaborides ${\text{LaB}}_6$ and ${\text{CeB}}_6$. The calculations have been carried out within the density-functional theory and linear-response formalism using pseudopotentials and a plane-wave basis. Thermodynamical properties of ${\text{LaB}}_6$ and ${\text{CeB}}_6$ obtained from quasiharmonic approximation are in a good agreement with the available experimental data. We also present the complete phonon-dispersion curves, phonon density of states, and mode-Grüneisen parameters and compared with the experimental measurements. A sizable difference between the vibrational contribution to entropy of ${\text{LaB}}_6$ and ${\text{CeB}}_6$ is found. The thermal electronic contribution to entropy and specific heat is found to be important for ${\text{CeB}}_6$.

Figure 1

(Color online) Eigendisplacement patterns of the normal modes of hexaborides at the Brillouin-zone center. (R: Raman active, IR: infrared active, and Ac.: acoustic modes.)

Figure 2

(Color online) (a) Phonon-dispersion relations of ${\text{LaB}}_6$. Solid lines show the results of the present study and the dashed lines are the frozen-phonon calculations of Monnier and Delley (Ref. 24). Experimental values (circles) are taken from Smith et al. (Ref. 18). (b) Phonon density of states of ${\text{LaB}}_6$. Solid lines show the results of the present study and the dashed lines are obtained by Kunii (Ref. 74) using the experimental-dispersion relation of Smith et al. (Ref. 18).

Figure 3

(Color online) (a) Phonon-dispersion relations of ${\text{CeB}}_6$. Solid lines show the results of the present study and experimental values (circles) are taken from Kunii et al. (Ref. 19). (b) Phonon density of states of ${\text{CeB}}_6$ calculated in this study.

Figure 4

(Color online) The calculated mode-Grüneisen parameters of (a) ${\text{LaB}}_6$ and (b) ${\text{CeB}}_6$. Dashed lines are the acoustic modes and solid lines are the optical modes.

Figure 5

(Color online) Vibrational contribution to Helmholtz free energy for (a) ${\text{LaB}}_6$ and (b) ${\text{CeB}}_6$ and vibrational contribution to entropy for (c) ${\text{LaB}}_6$ and (d) ${\text{CeB}}_6$ together with experimental results (circles) of Refs. 77, 80 for ${\text{LaB}}_6$ and ${\text{CeB}}_6$, respectively. The solid (dotted) lines show the vibrational (vibrational plus thermal electronic) contributions.

Figure 6

(Color online) Difference between the calculated entropies of ${\text{LaB}}_6$ and ${\text{CeB}}_6$ (squares) and the difference between the experimental entropy (Ref. 80) and the calculated entropy for ${\text{CeB}}_6$ (circles).

Figure 7

(Color online) Thermal volume expansion of (a) ${\text{LaB}}_6$ and (b) ${\text{CeB}}_6$. Experimental data are taken from the studies of Sirota et al. with Ref. 65 for ${\text{LaB}}_6$ and Ref. 66 for ${\text{CeB}}_6$. The solid (dotted) lines represent the vibrational (vibrational plus thermal electronic) contributions.

Figure 8

(Color online) Temperature dependence of linear-expansion coefficient of ${\text{LaB}}_6$ and ${\text{CeB}}_6$. The experimental values are from the study of Sirota et al. (Ref. 66). The solid (dotted) lines represent the vibrational (vibrational plus thermal electronic) contributions.

Figure 9

(Color online) Temperature dependence of (a) bulk modulus and (b) its pressure derivative of ${\text{LaB}}_6$ and ${\text{CeB}}_6$. The solid (dotted) lines represent the vibrational (vibrational plus thermal electronic) contributions.

Figure 10

(Color online) Temperature dependence of constant volume and constant pressure specific heat of (a) ${\text{LaB}}_6$ and (b) ${\text{CeB}}_6$. For ${\text{LaB}}_6$, the experimental values are from Ref. 20 (circles) and Ref. 83 (diamonds), for ${\text{CeB}}_6$ values are from Ref. 20 (circles) and Ref. 79 (diamonds).

Figure 11

(Color online) Pressure dependence of unit-cell volume of (a) ${\text{LaB}}_6$ and (b) ${\text{CeB}}_6$ at several temperatures. The solid lines in figures are for $T=294\text{K}$. Dashed lines are separated by 100 from 0 to 800 K. The experimental values in (a) are from Ref. 12 (triangles), Ref. 14 (circles), and Ref. 68 (diamonds). The experimental values in (b) are from the study of Leger et al. (Ref. 16) (circles).