Dynamical response and instability in ceria under lattice expansion

J. Buckeridge, D. O. Scanlon, A. Walsh, C. R. A. Catlow, and A. A. Sokol

Phys. Rev. B 87, 214304 – Published 17 June 2013

Abstract

We present results of density functional theory calculations on the phonon dispersion and elastic constants of bulk ceria (CeO $_{2}$ ) as a function of positive and negative isotropic strain, which could be induced thermally or by cationic doping. We find that, as the lattice is expanded, there is a significant softening of the $B_{1 u}$ mode at the $X$ point. This mode consists of motions of oxygens in the $[001]$ direction. At a strain of $1.6$ $%$ , corresponding to a temperature of 1600 K, the $B_{1 u}$ and $E_{u}$ modes at the $X$ point cross, with an associated high, narrow peak in the phonon density of states appearing. We infer that this crossing indicates a coupling of the modes, leading to a transition to a superionic phase, where conductivity occurs in the $[001]$ direction, mediated by anion interstitial site occupation. As the lattice is expanded further, the $B_{1 u}$ mode continues to soften, becoming imaginary at a strain of $3.4$ $%$ , corresponding to a temperature of 2500 K. Following the imaginary mode would result in a cubic to tetragonal phase transition, similar to those known to occur with reducing temperature in zirconia (ZrO $_{2}$ ) and hafnia (HfO $_{2}$ ). Our calculated elastic constants, however, indicate that the structure remains mechanically stable, even at this level of expansion. As confirmed by our semiclassical free energy calculations, the cubic phase of ceria remains the most stable, while the imaginary mode indicates a change to a thermally disordered cubic phase, with the majority of disorder occurring on the anion sublattice. Our results explain the high temperature ionic conductivity in ceria and other fluorite-structured materials in terms of the intrinsic lattice dynamics, and give insight to the stability and anionic disorder at elevated temperatures.

Received 18 March 2013

DOI:https://doi.org/10.1103/PhysRevB.87.214304

Authors & Affiliations

J. Buckeridge^1,*, D. O. Scanlon¹, A. Walsh², C. R. A. Catlow¹, and A. A. Sokol¹

¹University College London, Kathleen Lonsdale Materials Chemistry, Department of Chemistry, 20 Gordon Street, London WC1H 0AJ, United Kingdom
²Centre for Sustainable Chemical Technologies and Department of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

^*j.buckeridge@ucl.ac.uk

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Vol. 87, Iss. 21 — 1 June 2013

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Images

Figure 1
Phonon dispersion curve for CeO $_{2}$ calculated using GGA+ $U$ (solid red line) compared with measurements using Raman scattering (blue upward-pointing and orange downward-pointing triangles; Refs. 87 and 83), infrared spectroscopy (sidewards-pointing brown triangles; Ref. 88), reflectivity-derived (green diamonds; Ref. 89), and inelastic neutron scattering (purple squares; Ref. 90) methods. The modes are labeled according to their symmetry representations at the $Γ$ , $X$ , and $L$ points.Reuse & Permissions
Figure 2
Phonon dispersion curves and densities of states for CeO $_{2}$ determined using GGA+ $U$ at different values of linear strain $ɛ$ . The horizontal axis is labeled according to the convention of Fig. 1. At the $X$ point, the $B_{1 u}$ singlet is indicated by the downward-pointing (green) triangle, while the $E_{u}$ doublet is indicated by the upwards-pointing (blue) triangle. The unshaded region highlights the softening of the $B_{1 u}$ mode at the $X$ point.Reuse & Permissions
Figure 3
Calculated $B_{1 u}$ and $E_{u}$ mode frequencies at the $X$ point as a function of linear strain $ɛ$ , determined using GGA+ $U$ (black circles and red squares, respectively), and hybrid-DFT with the HSE06 functional (green diamonds and blue triangles, respectively). The lines are a guide for the eye. Negative frequency indicates an imaginary mode.Reuse & Permissions
Figure 4
Schematic of the eigenvectors associated with (a) the $B_{1 u}$ singlet mode and (b),(c) the $E_{u}$ doublet mode at the $X$ point. O (dark-gray/red spheres) atomic motion is indicated by dark-gray (green) arrows. Ce (light-gray/blue spheres) atomic motion is indicated by light-gray (orange) arrows. The eigenvectors are shown in the cubic unit cell with the simple cubic anion sublattice highlighted. The lengths of the arrows indicate the relative proportion of the magnitude of the O and Ce atomic motions, which reduces as the lattice is expanded (see text).Reuse & Permissions
Figure 5
Combination of the $B_{1 u}$ and $E_{u}$ modes at the $X$ point, shown for clarity acting on a single O chain (dark-gray/red spheres) in the [001] direction (adjacent O chains not shown). The interstitial site is represented by a black cross in the center of the cube formed by the Ce ions (light-gray/blue spheres). The motions associated with the $B_{1 u}$ and $E_{u}$ modes are indicated by thinner, darker-colored/shaded arrows, while the resulting motion from the combination of the modes is indicated by thicker, lighter-colored/shaded arrows.Reuse & Permissions
Figure 6
Calculated elastic constants $C_{11}$ , $C_{12}$ , $C_{44}$ , and high frequency dielectric constant $ε^{\infty}$ , as a function of linear strain $ɛ$ , determined using GGA+ $U$ (black circles). The lines are a guide for the eye.Reuse & Permissions

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