First-principles study of phonon anharmonicity and negative thermal expansion in ${\mathrm{ScF}}_3$

The microscopic origin of the large negative thermal expansion of cubic scandium trifluorides (${\mathrm{ScF}}_3$) is investigated by performing a set of anharmonic free-energy calculations based on density functional theory. We demonstrate that the conventional quasiharmonic approximation (QHA) completely breaks down for ${\mathrm{ScF}}_3$ and the quartic anharmonicity, treated nonperturbatively by the self-consistent phonon theory, is essential to reproduce the observed transition from negative to positive thermal expansivity and the hardening of the ${\mathrm{R}}^{4+}$ soft mode with heating. In addition, we show that the contribution from the cubic anharmonicity to the vibrational free energy, evaluated by the improved self-consistent phonon theory, is significant and as important as that from the quartic anharmonicity for robust understandings of the temperature dependence of the thermal expansion coefficient. The first-principles approach of this study enables us to compute various thermodynamic properties of solids in the thermodynamic limit with the effects of cubic and quartic anharmonicities. Therefore, it is expected to solve many known issues of the QHA-based predictions particularly noticeable at high temperature and in strongly anharmonic materials.

Figure 1

Crystal structure of cubic ${\mathrm{ScF}}_3$ with space group $Pm\overline{3}m$ (created with vesta [29]). Each scandium atom is surrounded by six fluorine atoms, which form corner-sharing octahedra. The structure is similar to the ${ABX}_3$ cubic perovskite, but the $A$ site is vacant.

Figure 2

Comparison of harmonic phonon dispersion curves calculated with the lattice constants close to the optimized values: 3.985, 4.075, and 4.030 Å for LDA, PBE, and PBEsol, respectively.

Figure 3

Volume dependence of the phonon frequency of the $R^{4+}$ soft mode. The imaginary frequencies are shown as negative values. The filled symbols indicate the volumes corresponding to the results shown in Fig. 2.

Figure 4

CTE calculated within the QHA compared with the experimental results [6] and the previous QHA results [20, 21].

Figure 5

Anharmonic free-energy correction $\mathrm{\Delta }{F}_{\mathrm{vib}}^{\left(\mathrm{SCP}\right)}(V,T)$ [Eq. (17)] and $F_{\mathrm{vib}}^{\left(\mathrm{B}\right)}(V,T)$ [Eq. (10)]; (a) volume dependence at 600 K and (b) temperature dependence for PBEsol with $a=4.025$ Å.

Figure 6

(a) CTE and (b) isothermal bulk modulus calculated within the SCP and ISC theories compared with the experimental results of Greve et al. [6] and Morelock et al. [50]. The DFT-MD result of Lazar et al. [23] obtained within PBE is also shown for comparison.

Figure 7

Upper panel: temperature dependence of the average bond length $r$ between the nearest Sc-F pair. Lower panel: temperature dependence of the lattice constant $a$ and the average distance $r$ between the nearest Sc-Sc pair. All values were calculated within LDA. The temperature dependence of the $a$ value was obtained from the ISC free energy and the perpendicular MSRD at each temperature was calculated from the SCP lattice dynamics wave function.

Figure 8

Comparison of the thermal expansion coefficients calculated by using the quantum and classical SCP theories. The quantum SCP theory [Eq. (7)] correctly accounts for the nuclear quantum effects, whereas the classical SCP theory [Eq. (B2)] neglects it.

Figure 9

Temperature dependence of squared phonon frequency of the $R^{4+}$ soft mode calculated within PBEsol in comparison with the experimental result of Handunkanda et al. [18]. (a) The calculation at each temperature is conducted with the corresponding ISC volume. (b) The calculation is conducted with the lattice constant adjusted to match the experimental lattice constant at 0 K. (c) Same as (b) but the classical approximation is made.

Figure 10

Mode-dependent contribution to the quartic renormalization of the $R^{4+}$ soft mode at 0 K. The open circle shows the contribution from phonon mode $q^{\prime }$ and the dotted line shows its cumulative value. The inset shows the displacement patterns of the phonon modes having large positive and negative quartic interaction with the $R^{4+}$ soft mode.