Equilibrium oxygen storage capacity of ultrathin CeO2-δ depends non-monotonically on large biaxial strain

Elastic strain is being increasingly employed to enhance the catalytic properties of mixed ion–electron conducting oxides. However, its effect on oxygen storage capacity is not well established. Here, we fabricate ultrathin, coherently strained films of CeO2-δ between 5.6% biaxial compression and 2.1% tension. In situ ambient pressure X-ray photoelectron spectroscopy reveals up to a fourfold enhancement in equilibrium oxygen storage capacity under both compression and tension. This non-monotonic variation with strain departs from the conventional wisdom based on a chemical expansion dominated behaviour. Through depth profiling, film thickness variations and a coupled photoemission–thermodynamic analysis of space-charge effects, we show that the enhanced reducibility is not dominated by interfacial effects. On the basis of ab initio calculations of oxygen vacancy formation incorporating defect interactions and vibrational contributions, we suggest that the non-monotonicity arises from the tetragonal distortion under large biaxial strain. These results may guide the rational engineering of multilayer and core–shell oxide nanomaterials.


Supplementary Note 1 Critical thickness for coherent films
The equilibrium theory of dislocations 1 expresses the critical thickness for a commensurate film, h c , as a function of the strain energy and the barrier for nucleating misfit dislocations. The critical thickness is given by where α is a dimensionless constant typically between 1 and 10 (related to the strength of the dislocation core), b is the magnitude of the Burger's vector, ε is the misfit strain between the film and the substrate and ν is the Poisson's ratio for the film. Solving equation (1) iteratively, it can be shown that h c is approximately 0.8 nm for α=10. Smaller α values give even smaller critical thicknesses. As a consequence of the large mismatch between YSZ and ceria, equilibrium theory predicts that a coherent film is unlikely. conditions. The ratio of this quantity, normalized by that of the oxidized reference is related to the fractional oxygen content near the surface:

Normalization of O 2p feature in the valence band spectra
Unlike the Ce 4d spectra, the O 2p feature lacks an internal normalization factor. Also, there is partial hybridization between the O 2p and Ce 4f / 5d states. As a result, we note that these ratios are not a fully quantitative measure of oxygen concentration. We also performed a similar determination using the lattice O 1s peak, which yielded a similar result as that shown in Figure 3.
where A(Ce 4+ ) represents the area under the higher BE peaks, and A(Ce 4d) represents the integrated area of the entire Ce 4d spectra. Subscript 'ref' denotes the oxidized reference.
Additionally, the Ce 4f peak in the valence band spectra (VB) is also proportional to the concentration of Ce 3+ . The area of Ce 4f peak normalized by the corresponding Ce 4d intensity is a linear function of [Ce 3+ ] (equation (2)). The slope provides a scaling factor to directly calculate [Ce 3+ ] from the VB spectra. In this work, we use Ce 4f spectra for quantifying [Ce 3+ ] due to the smaller scatter in the data.

Relating equilibrium oxygen vacancy concentration and vacancy formation energy
For the reduction of ceria at a fixed temperature, the redox reaction is given by The oxygen chemical potential in the gas phase, fixed by pO 2 Second, we approximate ΔH r 0 as invariant with δ (in reality, bulk values vary by 20 % with δ 5 ). Finally, we assume non-configurational contributions to the solid-state entropy is not dependent on strain. Using subscripts "str" and "rel" to denote the strained and relaxed oxides, and eliminating pO 2 from equation (15), we get, In this model, we assume that defect formation energies change abruptly at the interface of a thin surface region ("surface core") and the rest of the solid (position x = 0, Supplementary Figure 5). We denote the segregation energies for ions (oxygen vacancies) and electrons as Δh i = h core,i -h bulk,i and Δh e = h core,e -h bulk,e , respectively. In this model, we do not consider other sources of charge, such as those due to adsorbates and the additional SC zones near the film/substrate interface. Equilibration of electrochemical potential between the bulk and core requires that denotes the potential at x = 0 referenced to the bulk ("SC potential"). N is taken to be identical in the bulk and core. The configurational entropy terms assume an ideal, non-dilute solid solution. For simplicity, we also assume that the thickness of the core is sufficiently thin such that we can ignore the electrostatic potential variation within the core.
For non-zero defect segregation energies, a SC potential (Φ 0 ) is generated at the surface core. In the SC region adjacent to the core, charge carriers redistribute to screen the potential build-up at the core, and the electrostatic potential gradually decays to the bulk value away from the core. The electrostatic potential and concentrations are related by the Poisson equation as: The charge density is given by The individual defect concentrations can be written in a similar fashion as Eq. (4), giving The boundary conditions are: a. Zero electrostatic potential gradient far from the x= 0 interface: b. Matching the core charge (Q c,eq ) calculated by Gauss law to the core charge determined from the segregation thermodynamics: Using Eq. (4), RHS can be recast in terms of c bulk , Δh and Φ 0 from the previous iteration.
The core width, w c is taken to be 1 nm (i.e., 2 unit cell lengths of CeO 2 ). This value likely overestimates the core width (typically predicted to be less than one atomic layer by DFT).
Nevertheless, choosing a thicker core gives a higher core charge, which favors formation of a SC zone.
Using Eq. (9), ϕ(x = 0) at j+1 iteration can be calculated as : To solve the Poisson equation numerically, we use finite differences to compute derivatives, The value of ϕ(x) at iteration j+1 can be calculated based on values from iteration j as Thus, Eq. (8) and (10) are used to calculate the potential at the boundaries, while Eq. (12) updates the interior points. The system of equations is iterated until convergence is achieved.

Results of SC analysis
Supplementary Figure 6a In order to compare the simulation to experimental XPS results, we consider two limits in which the XPS probes (1) only the core region, and (2)

XPS information depth
The IMFP was obtained using the NIST Standard Reference Database 82 9 and the TPP-2M equation 10 . The parameters that were input for this calculation are summarized in Supplementary Table 5 and have been used in previous publications 2, 3 . For inorganic compounds, the IMFP is most sensitive to valence electron density, with an average uncertainty of 11 % 10 .

Treatment of electron localization in DFT +U calculations
For the bulk unit cells with an O vacancy, the structures with the lowest energy have the reduced Ce 3+ ions along the xy plane, while the unit cells with 2O vacancies have all the Ce atoms in the 3+ oxidation state. All the 2 × 2 × 2 supercell structures have the Ce 3+ ions in the nearest-neighbor positions from the oxygen vacancy. According to a previous work, the energy difference between this electron localization and the most stable one (i.e. nextnearest-neighbor) is less than 0.05 eV, which is within the intrinsic error of DFT 11 . In the case of the CeO 2 (100) surfaces, the structures with the lowest energy have the excess of charge localized at the Ce atoms of the first and second layers in an antiferromagnetic (AFM) ground state. For comparison, we also investigated the defective surface under compressive strain in a ferromagnetic (FM) spin state, obtaining an energy difference between the AFM and FM structures of less than 5 meV. Hence, the spin ground state of the reduced surfaces has a very small effect, in agreement with previous theoretical studies 12 . Our findings for the unstrained CeO 2 (100) surface are also in agreement with previous hybrid DFT calculations 13 .