Reduced Ceria Nanofilms from Structure Prediction

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Cerium oxide (ceria) may gradually change its stoichiometry between CeO 2 and Ce 2 O 3 depending on the environment.This makes it a key reducible oxide in numerous technological applications (e.g.5][6] The latter results in the increased catalytic activity of noble metals supported on CeO 2 and, in particular, on nanostructured CeO 2 .1 ,7 Since stoichiometric cerium dioxide is present only at oxidizing conditions and/or moderate temperatures, there is a growing interest in (partially) reduced forms of ceria.Recently, ultrathin supported nanoscale films of crystalline stoichiometric cerium sesquioxide Ce 2 O 3 have been prepared on various substrates. 8,9Such nanofilms provide well-defined model systems for studying the reactivity of ceria in extremely reducing conditions8 and have potential applications as high-k transistor gate dielectrics. 9Curiously, many of these nanofilms do not possess the hexagonal A-type structure, which is generally thought to be the thermodynamically stable bulk Ce 2 O 3 polymorph.
Generally, for many oxide materials reduction of their thickness to only a few monolayers has opened up a wealth of new technological opportunities in diverse application areas. 10In only a few cases, however, are supported oxide nanofilms found to possess wellordered atomic structures unlike that of the corresponding most stable bulk crystalline phase (e.g.Al 2 O 3 , 11 SiO 2 , 12,13 MgO, 14 ZnO 15 ).These nanofilms can be divided into two types: (i) non-stoichiometric films (e.g.oxides of Al, 11 Si 12 ), where chemical bonds of a noticeable strength form with a strongly interacting support, or (ii) stoichiometric films, essentially without chemical bonds with the support (e.g.MgO, 14 ZnO, 15 SiO 2 13 ).In all these cases ab initio calculations have been indispensable in confirming, 11,12,16 and even predicting 17 the atomic structure of the nanofilms.Although, in a real experimental set-up, oxide nanofilms are almost always grown on a support, computationally, via modelling free-standing sheets, one can enquire into the inherent stability of different nanofilm structures independently of a specific support.7][18] Note that even for nanofilms that weakly interact with the support epitaxial matching between the nanofilm and the support is generally observed.Comparison of freestanding models with experimental data can help to determine to what extent the observed polymorph is the result of: (i) intrinsic nanoscale structural/energetic tendencies of the material or (ii) experimental conditions (e.g.epitaxy with a specific support, metastability of obtained structures, etc.).
Herein we use a powerful structure search method and accurate electronic structure calculations to systematically explore the stabilities and structures of a range of free-standing stoichiometric Ce 2 O 3 nanofilms in order to understand the experimental observations.Specifically, we address the issue of thermodynamic versus kinetic stability in experimentally prepared Ce 2 O 3 nanofilms.Moreover, we predict new low energy Ce 2 O 3 nanofilms that may be prepared in the future.
Diminution of inorganic materials to the nanoscale often induces one or more alternative atomic orderings relative to the most stable bulk crystal. 19In order to test this possibility for reduced ceria we explored the space of stable Ce 2 O 3 nanofilm structures with ~1 nm thickness, i.e. containing four monolayers (MLs).Here, we define monolayers based on the number of cerium atoms, i.e. the O-Ce-O-Ce-O unit found in the vertical stacking of atomic layers in A-type Ce 2 O 3 (001) is counted as 2 ML.We employed the simulated mechanical annealing (SMA) technique [20][21][22] for searching the space of low energy film structures.
Following the experimental observation of structural relaxation via application of mechanical stress (termed mechanical annealing 23 ) in submicrometre atomic systems, the SMA method consists of cyclically gradually compressing and stretching the simulated Ce 2 O 3 nanofilms laterally (by up to ±30%) in a step-wise fashion.After each application of stress/strain to the nanofilm structure (achieved through systematically varying the cell parameters) all atomic positions are optimised.Upon these optimisations the atomic positions sometimes relaxed to give a new polymorph.We repeatedly applied the SMA stretching and compressing procedure to the Ce 2 O 3 nanofilms for every new polymorphic structure found until no further structural changes occurred.To reduce the bias on the choice of initial nanofilm structure, we repeated the above process starting from three distinct archetypal A 2 O 3 sesquioxide polymorphs: corundum, A-type, and bixbyite.Due to high computational cost of following this protocol directly with ab initio methods we initially performed the SMA search with suitable classical interatomic potentials (IPs) [24][25][26] using the GULP 27 code.From this search, ten of the resulting lowest energy nanofilm structures were then optimised using density functional theory (DFT) calculations.As detailed below (see also Supporting Information, Figures S1, S2) both the lattice parameters and the relative energies calculated using IPs and DFT schemes correlate with one another very well.This excellent correspondence is in line with our previous experience in modelling stoichiometric 4,28 and reduced ceria nanoparticles, 26 giving us confidence in using the IPs for our SMA searches.
In all reported periodic DFT calculations for both nanofilm and bulk structures, the unit cell parameters and all atomic positions were locally optimized (forces <0.2 eV nm -1 ) with

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Nanoscale Accepted Manuscript the PW91 29 form of the generalized gradient approximation (GGA) functional using the VASP code. 30An onsite Coulombic correction (U eff = U -J) 31,32 was applied to obtain a localized description of Ce 4f-electrons, resulting in a GGA+U corrected functional.
Following previous studies, 4, 5 a U eff value of 4 eV was used.The suggestion that a LDA+U description of the relative stabilities of Ce 2 O 3 polymorphs may provide a better match to experiment than a GGA+U approach 33 is briefly discussed below.The projector augmented wave approach 34,35 was used to describe the effect of core electrons on valence states, with the latter represented by a plane wave basis with a 600 eV cut-off.Nanofilms were separated by over 1 nm in the c-stacking direction to avoid spurious periodic interactions.Reciprocal space k-point sampling was achieved through appropriate Monkhorst-Pack grids 36 (see Table 1).Tests showed that all nanofilm energies were converged to <0.5 kJ mol -1 per Ce 2 O 3 with respect to k-point sets and completeness of the plane wave basis.

Bulk calculations.
The hexagonal A-type phase is generally thought to be the most thermodynamically stable bulk phase of Ce 2 O 3 . 37Our GGA+U calculations, however, predict the A-type structure to be higher in energy than the cubic bixbyite structure (E rel = 19.9kJ mol -1 per Ce 2 O 3, see Table 1).Using a similar calculation set-up, an apparent improvement in the treatment of reduced ceria via the use of an LDA+U approach with respect to GGA+U one has been noted previously. 33Our DFT calculations using the Local Density Approximation with a Hubbard U correction (LDA+U, with U = 6 eV) bring the energies of the two phases closer whereby the A-type phase becomes only 0.5 kJ mol -1 per Ce 2 O 3 less stable than bixbyite.We note that the lower relative energetic stability of bixbyite in ref 33 can be probably ascribed to the known problem of the presence of many meta-stable selfconsistent electronic solutions to Kohn-Sham equations for reduced cerium oxide.These solutions differ by the shape and symmetry of occupied f-orbitals of Ce 3+ cations, which may be sub-optimal in the electrostatic Madelung potential of the crystal. 38We found that, for bixbyite especially, occupied f-orbitals would often converge to be ݂ ௭ య -like, instead of more stable ݂ ௫௬௭ -like ones, significantly affecting the calculated total energy of the system. 39In general, due to their more refined account of electron density variations, GGA functionals have proven to be superior to LDA functionals for calculating the relative stability of different oxide polymorphs when the coordination environment of the constituent atoms varies (e.g.SiO 2 , 40 HfO 2 41 ).Specifically, GGA functionals help to correct the tendency of LDA functionals to overstabilise polymorphic structures that have more bonds per atom.In the present study when going from bixbyite to A-type, the average bonding coordination environment of Ce increases from six to seven; this may rationalize the increased relative stabilization of A-type in LDA+U calculations with respect to GGA+U treatments.The advantage of a GGA-based approach over LDA is expected to be more pronounced for structures with less homogeneous electron densities, in situations where bonds are being stretched, or for terminated structures.
Although in the case of the relative bulk energetics of bixbyite versus A-type polymorphs, GGA+U appears to overcompensate the failings of LDA+U, in principle GGA+U should provide an improved description of Ce 2 O 3 systems.One way to assess this assertion is to compare the GGA+U results with those from computationally intensive calculations employing hybrid functionals; the current DFT benchmark standard for periodic systems like ceria.3Using the hybrid HSE06 functional 42 we find bixbyite to be more stable than A-type by 25 kJ mol -1 per Ce 2 O 3 unit, confirming the energetic ordering calculated using GGA+U approach.
Considering the above mentioned arguments, in this work where we report the calculated properties of strained surface-terminated nanostructures, which possess novel polymorphic structures with variable bonding coordination, we preferred GGA+U over LDA+U.
Nanofilm calculations.The strain versus total energy curves resulting from the SMA searches for low energy Ce 2 O 3 four ML nanofilms using IPs are shown in Fig. 1a.These searches revealed more than 30 distinct nanofilm structures of which ten with the lowest energy were further optimized using DFT calculations.Results for six of them, A-type, bixbyite and nanofilms 1 to 4 (NF1-NF4), as well as for A-type and bixbyite bulks are presented in Table 1.Bixbyite, as a four ML nanofilm, is still predicted by our GGA+U calculation to be more stable than the corresponding A-type nanofilm.However, its stability with respect to A-type decreases to 11.1 kJ mol -1 from 19.9 kJ mol -1 per Ce 2 O 3 in the bulk.
This reduction in polymorphic energy differences when going from bulk to nanofilm appears to be a general phenomenon that has been predicted to occur for a number of materials. 43The new nanofilms NF1-NF4 found in our SMA searches all have energies slightly higher than the bixbyite nanofilm by 5.5 -26.2 kJ mol -1 per Ce 2 O 3 .It is of note that the four nanofilms, NF1-NF4, have structures which do not correspond to any known bulk crystalline A 2 O 3 polymorph.The 4 ML NF1 nanofilm is particularly interesting as it is the only new film that is predicted to be more energetically stable than the A-type 4 ML nanofilm.We note that this prediction in also confirmed by our calculations using the hybrid HSE06 functional.In Figure 2 we show the structures of the NF1, bixbyite and A-type 4 ML nanofilms.For these three nanofilms we have performed GGA+U calculations under externally applied stress or strain Page 6 of 14 Nanoscale

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(within the plane of each nanofilm) and generated three characteristic curves of relative energy versus the in-plane a lattice parameter per Ce 2 O 3 unit (see Figure 1b).The shapes and relative positions of the three curves in Figure 1b match quite well with the corresponding IPbased curves (see highlighted curves in Figure 1a).Although the DFT-calculated energetic ordering of the nanofilms is generally well reproduced by the IP calculations, the latter results span a twice larger energy range.This finding is fully in line with a combined IP and GGA+U study of partially reduced ceria nanoclusters. 26We note that according to the IP data bixbyite films are more stable than A-type and NF1 films even at their points of minimum energy.For the DFT calculations, however, the NF1 and A-type energy minima lay outside of the energy versus strain curve of bixbyite.This finding suggests that by using substrates with different lattice parameters one could favour the epitaxial growth of a particular nanofilm structure.
Experimentally, a few Ce 2 O 3 nanofilm structures have already been produced on different substrates.In Figure 1b we include the in-plane lattice parameters of a selection of surfaces that have been employed to grow supported Ce 2 O 3 nanofilms, as calculated using GGA-based DFT.For the Cu(111) surface, 2.5 ML fluorite CeO 2 (111) nanofilms were grown with a 2:3 epitaxy.Upon heating to 1070 K these nanofilms could be transformed into Ce 2 O 3 nanofilms with the A-type structure while retaining a very similar epitaxial matching. 44From a thermodynamical perspective, such a transition is in agreement with our calculations (Figure 1b) where the Cu(111) surface and A-type films have closely matching lattice parameters (after multiplying the lattice parameter of the A-type film by 3/2).Using metallic Ce as a reducing agent, and annealing under slightly milder thermal conditions (900 K), similar Cu(111)-supported 4 ML CeO 2 films could be reduced to Ce 2 O 3 nanofilms exhibiting the bixbyite structure.9Here, assuming no structural relaxation of the Cu(111) surface and perfect 3:2 epitaxy, we predict that a suitably contracted free-standing 4 ML bixbyite nanofilm would be moderately metastable (+6 kJ mol -1 per Ce 2 O 3 ) relative to an A-type nanofilm with the same lattice parameter (see Figure 1b).We thus suggest that the observation of bixbyite films grown at relatively moderate temperature on Cu(111) does not necessarily require their preferential energetic stability on the support.Rather, it can be due to kinetics whereby the preparation retains much of the original fluorite structure of the CeO 2 precursor.Bixbyite Ce 2 O 3 nanofilms of 2-5 ML have also been grown on Cl-passivated Si(111) surfaces by Flege et al.8For such a situation we predict an even smaller metastability of 4 ML bixbyite films (+2 kJ mol -1 per Ce 2 O 3 ) with respect to A-type.This very small calculated energy difference points again to kinetic stabilization of these experimentally Page 7 of 14 Nanoscale

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observed bixbyite nanofilms.For the significantly larger lattice parameter of Rh(111), supported CeO 2 nanofilms with 1-6 ML thicknesses have been shown to decompose at temperatures 700-800 °C to give a reduced ceria islands and a (4×4) Low-Energy Electron Diffraction (LEED) pattern. 45Although in ref 45 this LEED pattern is ascribed to Ce-Rh alloy formation, with hindsight, another interpretation of such a measurement may be the emergence of the bixbyite structure.In Figure 1b we see that such an interpretation is consistent with the calculated small energetic preference for 3:2 epitaxial 4 ML bixbyite nanofilms on Rh(111).
Although we are aware of no reports directly identifying our predicted NF1 nanofilm we can see from Figure 1b that supports with a larger lattice parameter than those cited above for ultrathin films would be required to produce NF1.For instance, Re(0001) or Pt(111) with calculated a 0 of 278 and 282 pm, respectively.In fact, reduced ceria films have been prepared on Re(0001), 46 but, as far as we are aware, only with relatively large thicknesses (>20 ML) of limited relevance to the present study.On the Pt(111) surface, reduction of 1-2 ML CeO 2 nanofilms with 4:3 epitaxy has led to novel nanofilms with, as yet, undetermined structures. 47,48Assuming a 3:2 epitaxy, our calculations indicate that the Pt(111) surface should thermodynamically favour the formation of the 4 ML NF1 nanofilm relative to bixbyite and A-type.In ref. 47 a strongly reduced 2 ML CeO 2 nanofilm is found to exhibit an unresolved structure with a 9/4(√3×√3)R30° periodicity (with respect to Pt) which is consistent with that of NF1 (see Figure 2).Similarly to the structure of NF1, the 1 ML Ce 2 O 3 nanofilm reported in ref.48 has a hexagonal unit cell with a lattice constant that is approximately twice that of A-type (see Table 1).Additionally, scanning tunneling microscopy of this latter nanofilm shows protruding add-atoms at three-fold coordinated sites covering.This observation is in line with the curious structure of NF1, which displays protruding oxygen atoms at three-fold coordinated sites, albeit with a higher density than that observed in experiment.The finding that the adatoms in the experimentally prepared 1 ML nanofilm are disordered whereas those in NF1 are ordered may be a reflection of experimental conditions (e.g.finite temperatures, 1 ML versus 4ML) or again kinetic limitations.
In order to encourage further experimental work to better characterize such reduced ceria nanofilms, in Table 2 we present some calculated properties of NF1 to help distinguish it from A-type and bixbyite nanofilms.

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gaps (O2p -Ce4f+5d) are rather similar, ~3.9 eV, in the considered bulk structures and Atype film.However, in bixbyite and NF1 films these band gaps are reduced to ~2.7 eV, which could be explained by the presence of five-coordinated Ce ions.In fact, under-coordinated Ce ions were already shown to reduce the band gap in CeO 2 nanoparticles and concomitantly greatly increase their reducibility. 49As these properties are amenable to measurement (e.g. via EXAFS, PES), we hope that our predicted new NF1 nanofilm will be indentified in future experimental studies.

CONCLUSIONS
To summarise, using simulated mechanical annealing searches and density functional calculations we identify a range of new low energy 4 ML Ce 2 O 3 nanofilm structures.We find that our calculations of energetic stability versus in-plane lattice parameter are consistent with stability of experimentally observed nanofilm phases depending on the substrates used to prepare them.Further, we propose a new energetically stable NF1 film structure and suggest suitable substrates that would favor its growth.We note that our predicted NF1 nanofilm appears to have some structural properties consistent with those reported for reduced ceria films on Pt(111) surfaces.Finally, we present specific calculated properties of the NF1 nanofilm that should assist in its experimental identification.
Firstly, in line with its relatively larger in-plane lattice parameter, both the Ce and O atoms in the NF1 nanofilm have lower average coordination numbers than in A-type and bixbyite nanofilms.With respect to conductivity, GGA+U band Page 8 of 14 Nanoscale Electronic supplementary information (ESI) available: Graph of IP versus DFT relative energies for nanofilms, GGA+U calculated lattice parameters and atomic coordinates of NF1-4 nanofilms.See DOI: 10.1039/XXXXX ACKNOWLEDGMENTS SMK thanks the Spanish MEDU for FPU grant AP2009-3379.This study has been supported by grants of the Spanish Ministry of Economy and Competitiveness (CTQ2012-34969; MAT2012-30924), the European Union FP7 Program under grant agreement number 310191, the Generalitat de Catalunya (2014SGR97; XRQTC) and by the COST Action CM1104.We also acknowledge use of supercomputing resources provided by the Red Española de Supercomputación.

Figure 1 .
Figure 1.Results of a) the IP-based SMA search, and b) GGA+U calculations for films with: A-type (circles), bixbyite (squares) and NF1 (triangles), NF2-4 (diamonds), corundum (black, no symbol) and other structures (brown, no symbol).Energies (relative to that of the optimized bixbyite nanofilm) and lattice parameters are given per Ce 2 O 3 unit.Solid lines in b) are parabolic fits to the data points to guide the eye.Vertical dotted lines in b) indicate GGA-calculated lattice parameters of possible supports for nanofilm growth (multiplied by 3/2 for transition metals).

Figure 2 .
Figure 2. Top and side views of A-type, bixbyite and NF1 Ce 2 O 3 nanofilms of 4 ML thickness.O atoms are displayed as red spheres and Ce 3+ ions as grey spheres.Atoms with darker colors are located in surface layers.Employed unit cells are denoted by black lines.

Table 1 .
37-plane film lattice parameter (a 0 in pm), relative energies (E rel , with respect to bixbyite, per Ce 2 O 3 unit, in kJ mol -1 ), Monkhorst-Pack k-point mesh, and thicknesses (in pm) of optimised Ce 2 O 3 bulk polymorphs and nanofilms from GGA+U calculations.The experimental value is 389 pm.37bTwo lattice parameters are given for films with distorted hexagonal structure. a

Table 2 .
Calculated GGA+U energy gap values ∆ε (in eV) between the highest occupied (HO) and the lowest unoccupied (LU) states of Ce and O and average coordination numbers of Ce, N(Ce), in bulk and 4 ML nanofilm structures.a System ∆ε(HO Ce -HO O ) ∆ε(HO Ce -LU O ) ∆ε(HO O -LU Ce ) N(Ce) b Note that the presented GGA+U absolute band gap values ∆ε(HO O -LU Ce ) are expected to be notably underestimated with respect to both those from hybrid-functional DFT calculations and experimental data.3b Average coordination numbers of O are 1.5 times smaller than N(Ce). a