Pure and doped Ceria are among the most studied nonstoichiometric oxides, as these have been used as commercial catalysts for a long time.¹ The catalytic capability of ceria owes to the reversible transition between Ce⁴⁺ and Ce³⁺. This phenomenon also has promising usability outside its conventional catalytic application. In a recent experimental work, Korobko et al.,² showed that the nonstoichiometric Gd-doped ceria (GDC) can be a new class of electrostrictor exhibiting anomalously large electrostriction effect. It was shown, GDC thin film can generate up to a 500 MPa stress on the application of electric-field, three times more than the stress generated by commercially available piezoelectric or electrostrictor materials. This electrostriction effect in GDC is a good example of the electro-chemo-mechanical coupling concept, where the application of electric field on a nonstoichiometric GDC thin film orients the local chemical strain fields (with polaron) producing observable mechanical stress and paraelectricity. Korobko et al.² hypothesized this effect was related to the anisotropic chemical strain, but most computational studies only focused on the average deformation due to the oxygen vacancy.¹ In this paper, the anisotropic chemical strain of oxygen vacancy in CeO_2-δwas calculated from density functional theory (DFT) calculations. The anisotropic nature of the chemical strain can explain the reason behind electrostriction effect.

Computational Methods

Fully spin-polarized calculations with generalized gradient approximation (GGA) method along with Hubbard-U correction implemented in VASP was used for the energy calculations. The Hubbard-U ( = 4.5 eV) parameter was required to treat the localized electron in Ce 4f orbital.³ A 2x2x2 ceria cubic-supercell (Figure 1) was used for the oxygen vacancy formation calculation (δ = 0.03) to avoid any vacancy-vacancy interaction due to periodic boundary condition. The anisotropic chemical strain coefficient tensor was calculated following the work of James et al.⁴, by balancing the local deformation due to oxygen vacancy formation and the long-range elastic strain. The average chemical strain was calculated from the principle strain components in the anisotropic chemical strain tensor.

Results and Discussion

In CeO₂ Fluorite structure, Ce-O₈ forms cubic coordination but O-Ce₄ forms tetrahedral coordination (Figure 1). As Ce atom has an electronic configuration of [Xe]4f¹5d¹6s², so in CeO₂, Ce⁺⁴ will have [Xe] electronic configuration. Therefore on the formation of the neutral oxygen vacancy (Equation 1), two excess electrons will be distributed among two out of the four neighboring Ce atoms. The GGA+U calculation of the oxygen vacancy formation was performed with two different structures. In the first case, four Ce atoms were equidistant from the oxygen vacancy site (V_o) and the charge on them was +3.5. This is because DFT tends to equally distribute two excess electrons to the 4f orbital of four Ce atoms. However after moving two Ce atoms away from the V_obefore relaxing the structure, a low energy structure with a localized electron on two Ce atoms was found. The two moved-Ce maintained +4 charge and the other two Ce became +3 (Figure 2). Thus, the structure with charge disproportionation was energetically favorable.

The calculated Ce-O bond distance in a perfect lattice was 2.38 Å. After the formation of the oxygen vacancy Ce⁺⁴ (2.56 Å) moved further away from the V_o than Ce⁺³ (2.52 Å). According to the bond distance calculations, the oxygen atoms (O-Ce-V_o) opposite to V_o of Ce, moved closer to the Ce in [111] (Figure 2). This observation is opposite to the hypothesis of Korobko et al.² The chemical strain tensor (Figure 3, left image) was not able to show the anisotropic effect in principle directions and it was known that the magnitude of chemical strain was higher in [111] direction, so the <001> direction was mapped to <111> and the strain tensor was rotated into a new coordinate system (Figure 3, right image). The rotated chemical strain tensor clearly shows 1% extensional strain in <111> but 0.6% contraction in <0-11> (at δ = 0.03), indicating the anisotropic effect.

Conclusion

The presence of highly localized 4fvalance orbital in Ce causes charge disproportionation on the formation of neutral oxygen vacancy producing anisotropic chemical strain in crystal with cubic symmetry. On the application of electric field, O-Ce-Vo orients perpendicular to the thin film producing paraelectricity.

References

1. Er et al., J. Electrochem. Soc., 161, F3060–F3064 (2014).

2. Korobko et al., Adv. Mater., 24, 5857–5861 (2012).

3. Chen et al., J. Power Sources, 234, 69–81 (2013).

4. James et al., 1, 1037–1042 (2016).