Modifying the barriers for oxygen-vacancy migration in fluorite-structured CeO₂ electrolytes through strain: a computer simulation study

Abstract

Static lattice simulation techniques were used to examine the effect of strain on oxygen-vacancy migration in the fluorite-structured oxygen-ion conducting electrolyte CeO₂. Activation energies for vacancy migration, ΔE_mig, were calculated as a function of isotropic and biaxial strain. In both cases, significant modification of the energetic barriers for oxygen-vacancy migration was found. Analysis of the data yields the activation volumes, ΔV_mig, and activation enthalpies, ΔH_mig. Simple comparisons based on the calculated data suggest that a biaxial, tensile strain of 4% may increase the in-plane conductivity at T = 500 K by close to four orders of magnitude. Enhancement of the oxygen-ion conductivity of an oxide heterostructure through space-charge effects is also discussed.

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Broader context

Researchers have over decades optimised the compositions of AO₂–M₂O₃ fluorite-structured solid solutions for their application as oxygen-ion conducting electrolytes in Solid Oxide Fuel Cells. Solid solutions based on CeO₂, in particular, are of great interest for intermediate temperature SOFC technology (operating at ca. 600 °C), which offers several benefits over the standard technology (operating at ca. 900 °C), including cheaper fabrication, improved durability and more robust construction. Recently, mechanical strain has been proposed as a new means of enhancing the ionic conductivity of such materials, and not by the factors of 2–5 typical of composition optimisation but by many orders of magnitude. There is, however, much debate as to whether the reported enhancement of 10⁸ is a real effect or an experimental artefact. Our atomistic simulations suggest that strain can increase the ionic conductivity by 3–4 orders of magnitude, but they militate against the reported enhancement of 10⁸. Strained AO₂–M₂O₃ electrolytes may be employed in micro-SOFC technology.

Introduction

Despite decades of searching for superior alternatives, fluorite-structured solid solutions, such as ZrO₂–M₂O₃ and CeO₂–M₂O₃, remain the best available oxygen-ion conducting electrolytes for Solid Oxide Fuel Cell (SOFC) applications. Such materials combine high and exclusively ionic conductivity, over a suitably wide range of oxygen partial pressures, not only with chemical and mechanical stability, but also with thermal and chemical compatibility with other SOFC components.^1–11

Concurrent with the search for superior electrolytes, the past decades have also seen the optimisation of the AO₂–M₂O₃ fluorite materials' compositions, not only with regard to the type and amount of the M₂O₃ substituent, in order to maximise both the concentration and mobility of the ionic charge carriers (oxygen vacancies),^4–11 but also with regard to lowering the level of SiO₂ and/or the addition of silica scavengers, in order to avoid the formation of highly resistive, wetting grain-boundary phases.¹²

In an attempt to increase the conductivity beyond this composition-optimised level, researchers have turned their attention to the manipulation of the microstructure. Primarily two strategies are being pursued: (a) increasing the density of homo-interfaces (grain boundaries) drastically, by moving from microcrystalline to nanocrystalline samples; and (b) creating specific hetero-interfaces, by employing epitaxial thin-film geometries. The rationale behind such work is that the ionic conductivity can be raised either through enhanced charge-carrier mobility in the interface core (a structural effect), through charge-carrier accumulation in the bulk regions adjacent to the interface core (a space-charge effect), or through a reduced activation enthalpy for charge-carrier migration in the bulk phase (a strain effect). In general, regardless of the strategy pursued, the results obtained have been rather modest:^13–24 in some cases a small increase was observed; in other cases, a small decrease. One major exception is the report by Garcia-Barriocanal et al.²⁵ of colossal ionic conductivity in ultrathin (1 nm) YSZ layers (ZrO₂ with 8 mol% Y₂O₃), sandwiched epitaxially between thicker layers of SrTiO₃. Such heterostructures were found to exhibit conductivities that were higher than bulk YSZ samples by a factor of 10⁸. This enormous increase was ascribed by the authors to the huge (7%) tensile strain in the YSZ films and disorder of oxygen ions at the interface.

Doubts have been raised, however, as to the nature of the charge carriers in these samples:^26–29 Is the measured conductivity due to ionic charge carriers in YSZ or due to electronic charge carriers in SrTiO₃? At present, the balance of evidence is strongly in favour of the latter. In particular Cavallaro et al.²⁹ measured the pO₂ dependence of the total conductivity of a SrTiO₃|YSZ|SrTiO₃ heterostructure and found σ ∝ pO₂^0.20. Such an exponent is not consistent with ionic transport in YSZ, and in fact is characteristic of electron-hole conduction in Fe-doped SrTiO₃.³⁰

Nevertheless, the fundamental issue remains as to how much the ionic conductivity of a fluorite-structured oxide can be enhanced through lattice strain. Recent simulation studies^28,31 suggest that biaxial strain can increase the ionic conductivity by a factor of 10⁴–10⁶. But, as large as these predicted enhancement factors are, they are still only a small fraction of the reported 10⁸ enhancement; thus, if these simulation studies are to be believed, they provide additional evidence that the experimentally measured conductivity is electronic.

The questions hanging over the recent simulation studies^28,31 concern both the simulation methodology and the simulated data. First, being based on computationally expensive density-functional-theory (DFT) calculations, the simulations were restricted to small simulation cells, and one may question if lattice relaxation was accurately captured in the migration calculations. Second, the calculations refer to constant volume and not constant pressure; as will be shown in this study, there are significant differences between the two for the conditions of interest. Third, in neither study it is shown that the simulations can reproduce experimental transport data, such as the vacancy migration enthalpy in unstrained YSZ; this casts doubts on the accuracy of the reported degrees of enhancement.‡

In the present study, static atomistic simulations, based on empirical pair-potentials (EPP), were used to determine the effect of strain on the energetic barriers for oxygen-vacancy migration in a fluorite-structured lattice. By explicitly relaxing several hundreds of ions surrounding the migrating ion, such simulation methods are able to model long-range relaxations effectively. Besides, by carrying out such calculations one can ascertain how well methods that employ empirical pair-potentials perform at large deviations from the equilibrium lattice parameter. We took CeO₂ as a representative fluorite-structured electrolyte because the (unstrained) material, in contrast to ZrO₂, adopts cubic symmetry for M₂O₃ substituent levels from zero upwards. As a first step towards a deeper and thorough understanding of strain effects, we focus on bulk material (as opposed to an interface) and we do not take account of defect–defect interactions, that is, we assume that the oxygen vacancies introduced by the M₂O₃ substituent are mobile and non-interacting. Another issue we do not address directly is that of lattice stability. We assume that fluorite-structured ceria is mechanically and thermodynamically stable throughout the entire range of strains examined. Araki and Arai²⁴ predict on the basis on classical molecular dynamics simulations that YSZ fractures at a uniaxial tensile strain of 3.7%. From experiment^32,33 it is known that CeO₂ transforms from a cubic fluorite-type structure to an orthorhombic cotunnite-type (PbCl₂) structure at a pressure p_tr ≈ 31 GPa. In our static atomistic simulations, the fluorite lattice is prevented from exhibiting fracture and from undergoing phase transitions. Our principal concerns are the systematic investigation of the effect of strain on the migration energetics and the consequences for the ionic conductivity.

Simulation methods

The computational methodologies used in this work are well established.^34–36 The constituent ions of the solid are treated as classical particles that bear an electrical charge corresponding to their formal oxidation number Z. The ions interact with each other through long-range Coulombic forces and short-range forces that account for electron cloud overlap (Pauli repulsion) and dispersion (van der Waals) interactions:r is the spatial coordinate, and A_ij, ρ_ij and C_ij are empirically determined parameters. Polarizable ions are treated by the shell model,³⁷ in which a mass-less shell (of charge Y) is connected to a massive core (of charge Z–Y) by a harmonic spring (of force constant k).

The calculation of defect energies employed the standard procedure of partitioning the crystal into two regions: a spherical inner region with the defect at its centre (region I) and an outer region that extends to infinity (region II). In region I, which contained between 300 and 400 ions, the positions of the individual ions are relaxed explicitly, under the perturbation of the defect, to zero net force. For the remainder of the crystal (region II), the forces due to the defect are relatively weak, and therefore the response can be treated by a quasi-continuum approximation (Mott–Littleton³⁸). An interfacial region is also introduced to provide a smooth transition between the two regions.

Various sets of empirical parameters for CeO₂ are available in the literature.^39–41 In this study, the parameter set reported by Balducci et al.⁴¹ was used, since it yields a migration energy of an oxygen vacancy in cubic unstrained, undoped CeO₂, ΔE_mig = 0.6 eV, close to that found experimentally.^42,43 The parameters for these empirical pair-potentials (EPP) are summarised in Table 1. All calculations were carried out with the GULP code.⁴⁴

Table 1 Parameters for the empirical pair-potentials (EPP) used in this study⁴¹

Interaction ij	A ij /eV	ρij/Å	C ij /eV Å^6
O^2−⋯O^2−	22764.30	0.1490	27.89
Ce^4+⋯O^2−	1986.83	0.3511	20.40

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Ion	Y/e	k/eV Å^−2
O^2−	−2.077	27.290
Ce^4+	7.7	291.75

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Results and discussion

Vacancy migration in isotropically strained CeO₂

The lattice energy of fluorite-structured ceria (corresponding to a unit cell of Ce₄O₈) is plotted in Fig. 1(a) as a function of unit cell volume. The data are described well by a 3rd order Birch–Murnaghan equation of state,⁴⁵ with bulk modulus B_EPP = 261 GPa and bulk modulus derivative B′_EPP = 3.78. The calculated bulk modulus is ca. 15% higher than experimental values^33,46 (B_expt = 220–230 GPa); DFT calculations with standard functionals yield values ca. 15% lower than experiment (B_DFT ≈ 180 GPa).^46–49 From the data in Fig. 1(a) we calculated the equivalent applied pressure according to p = −∂ E_latt/∂ V_latt; and plotting normalised cell volume against pressure in Fig. 1(b), we find good agreement between our theoretical data and experimental data obtained by Duclos et al.³³

Fig. 1 Fluorite-structured CeO₂: (a) Calculated lattice energy E_latt as a function of lattice volume V_latt (open circles); fit to a 3rd order Birch–Murnaghan equation of state (solid line). (b) Normalised lattice volume V_latt/V⁰_latt as a function of pressure p, calculated from the data shown in (a) (solid line), compared with experimental data from Duclos et al.³³ (open diamonds).

On the other hand, preliminary calculations (not shown), based solely on consideration of lattice enthalpies, H_latt = E_latt(p = 0) + pV_latt, yield a pressure p_tr ≈ 95 GPa for the fluorite to cotunnite transition, significantly higher than experiment (p_tr ≈ 31 GPa).^32,33 Fortunately the calculated transition pressure is close to the highest pressure examined in this study (see Fig. 1b), and hence the migration energies obtained below refer to a hypothetical cubic structure above the transition pressure. The discrepancy is probably due in part to the use of empirical pair-potentials and in part to the simplification we made, when varying the volume of the orthorhombic structure, of maintaining constant a/b and b/c ratios. It is an open question whether there are other phase transitions, either to be found experimentally or to be predicted theoretically with this set of empirical pair-potentials, in particular at negative p. (Negative pressures correspond to isotropic tensile strain—a situation that is of course difficult to achieve experimentally.) As previously noted we assume in the following that fluorite-structured CeO₂ neither fractures nor undergoes any phase transitions within the range of strains/pressures investigated.

Oxygen transport in cubic fluorite-structured oxides takes place by oxygen ions jumping along the six equivalent <100> directions of the cubic structure into adjacent vacant sites. The variation in the migration energy of an oxygen vacancy in cubic CeO₂ with isotropic strain is shown in Fig. 2. Compressing the lattice increases ΔE_mig, whilst dilating the lattice has the opposite effect. The magnitude of the effect is surprisingly large. If the lattice is subjected to sufficient tensile strain (ε ≈ +0.05, in this case), the migration energy decreases in fact to zero. Further lattice dilation (not shown) yields negative migration energies, i.e., the migrating ion prefers the saddle-point configuration to the initial configuration. This is indicative of a lattice instability.

Fig. 2 Activation energy of oxygen-vacancy migration, ΔE_mig, in fluorite-structured CeO₂ as a function of isotropic strain ε. (a⁰ is the lattice constant at zero strain.)

In executing a jump, a migrating oxygen ion has to push past two cations, as shown in Fig. 3. The radius of a sphere that just passes through this aperture without disturbing the lattice, r_crit, is far smaller than the radius of an oxygen ion, r_O. And the smaller the critical radius, the more the lattice has to be perturbed in order for the oxygen ion to migrate, and hence the higher the migration energy. The critical radius r_crit for an AO₂ fluorite-structured oxide can be expressed⁵⁰ in terms of the cubic lattice parameter a and the ionic radius of the cation r_A

Fig. 3 (a) Cubic fluorite-structured lattice of CeO₂; the <100> migration paths within the unit cell are shown in grey. (b) View from an oxygen vacancy towards a neighbouring oxygen ion along the migration path (the ions are drawn proportional to their Shannon ionic radii⁵¹). The two cations present a steric hindrance to the migration of the oxygen ion, as r_O > r_crit.

Although the behaviour shown in Fig. 2 is qualitatively consistent with the critical-radius model—lattice compression, for instance, leads to smaller r_crit, and thus to higher ΔE_mig—, one cannot make a quantitative comparison between ΔE_mig and r_crit because r_A in eqn (2) [and r_O of the migrating ion] will also vary with a. Furthermore it is unclear how to apportion a change in a to changes in r_A and r_O in an unambiguous and physically reasonable fashion.

The activation volume for vacancy migration, ΔV_mig, describes the change in lattice volume, as the migrating ion passes through the saddle-point configuration. It is defined as the first derivative of the Gibbs energy of migration, ΔG_mig, with respect to pressure p,

The data shown in Fig. 2, however, refer to the change in migration energy ΔE_mig with lattice volume V_latt. By means of thermodynamic identities,⁵² it can be shown thatif one assumes the change in migration entropy ΔS_mig with V_latt to be negligible. We obtained ΔV_mig by fitting a 5th order polynomial to the data of ΔE_migvs. V_latt; differentiating the polynomial; and subsequently evaluating eqn (4). Results are shown in Fig. 4(a) as a function of pressure. Identical results are obtained by differentiating ΔE_mig with respect to p. One finds that ΔV_mig is not constant but varies with pressure. Enormous pressures (of the order of tens of GPa) are required, though, to observe a significant effect.

Fig. 4 Fluorite-structured CeO₂ subjected to isotropic strain: (a)activation volume for oxygen-vacancy migration, ΔV_mig, as a function of pressure p; (b) the activation enthalpy of migration, ΔH_mig, (symbols) and the activation energy of migration, ΔE_mig, (dashed line) as a function of isotropic strain ε.

In Fig. 4(b) we plot the migration enthalpy, calculated from ΔH_mig = ΔE_mig(p = 0) + pΔV_mig, against isotropic strain. For small strains, there is little difference between the activation energy of migration and the activation enthalpy of migration. For larger strains, significant differences appear; in particular the same decrease in activation energy can be achieved at smaller dilatative strains, e.g., ΔH_mig goes to zero at ε ≈ +0.035, whereas ΔE_mig goes to zero at ε ≈ +0.05.

Vacancy migration in biaxially strained CeO₂

Upon subjecting cubic fluorite-structured CeO₂ to a biaxial strain along [100] and [010], the unit cell becomes tetragonal, a = b ≠ c. For an isotropic medium, c is given bywhere ν is Poisson's ratio (ν_EPP = 0.22). Values of c calculated with eqn (5) are consistent with values obtained by minimising E_latt with respect to c for fixed a and b. Once again it is assumed that the system, in this case (tetragonally distorted) fluorite-structured ceria, does not exhibit material failure nor undergo any phase transitions over the entire range of biaxial strains examined.

Since biaxial strain breaks the fluorite structure's cubic symmetry, the six equivalent, orthogonal migration paths for an oxygen vacancy in the cubic structure become four equivalent orthogonal migration directions in the ab plane of the tetragonal cell (in-plane migration), and two anti-parallel migration directions perpendicular to the ab plane (out-of-plane migration). The activation energies for vacancy migration along these two non-equivalent paths, ΔE^out_mig and ΔEⁱⁿ_mig, are shown in Fig. 5(a).

Fig. 5 (a) Activation energies for vacancy migration in CeO₂ biaxially strained along [100] and [010]. (b) Inverse separation of Ce ions in biaxially strained fluorite-structured CeO₂, d⁻¹_Ce–Ce, as a function of biaxial strain ε.

For biaxial strains |ε| < 0.02, the behaviour of ΔE^out_mig and ΔEⁱⁿ_mig can easily be rationalised within the critical-radius model. As shown in Fig. 5(b), compressive biaxial strain in the ab plane closes the out-of-plane migration aperture, and to a lesser extent, the in-plane migration aperture (the inverse Ce–Ce separation increases), and the migration energies increase correspondingly. Small tensile biaxial strains have the opposite effect, as they open the migration apertures, albeit to differing degrees. For larger biaxial strains, there are significant deviations from the expected behaviour: in particular, at large tensile strains [rhs of Fig. 5(a)], ΔE^out_mig goes through a minimum and ΔEⁱⁿ_mig decreases to zero rather rapidly. The reasons for the unexpected behaviour are unclear. It may represent the limits of the critical radius model (a simple static picture!) in describing the dynamics of the migration process. It may simply be due to the use of empirical pair potentials, i.e., the possible failure of pair potentials to capture accurately the atomic interactions at large deviations from equilibrium values. It may also be evidence of lattice instability at large ε.

For a system characterised by the stress tensor σ, the Gibbs free energy of migration can be written as⁵³

ΔG_mig = ΔE_mig − T ΔS_mig − σ·ΔV_mig, (6)

where ΔV_mig is the activation volume tensor. Here we consider for simplicity the scalar activation volume of migration (which is the trace of ΔV_mig). In this case, and neglecting activation entropy of migration, one can reduce eqn (6) to⁵³

The biaxial stress σ_xx is given by

and combining this data with the data plotted in Fig. 5(a) yields the scalar activation volumes of migration, ΔV^out_mig and ΔVⁱⁿ_mig. Results are shown as a function of biaxial stress in Fig. 6(a).

Fig. 6 Fluorite-structured CeO₂ subjected to biaxial strain along [100] and [010]: (a)activation volume for oxygen-vacancy migration, ΔV^out_mig and ΔVⁱⁿ_mig, as a function of biaxial stress σ_xx; (b) the activation enthalpies of migration, ΔH^out_mig and ΔHⁱⁿ_mig, (symbols) and the activation energies of migration, ΔE^out_mig and ΔEⁱⁿ_mig, (dashed lines) as a function of biaxial strain ε.

The activation volumes are different for the two paths, and show different dependences on biaxial stress (with huge stresses needed to produce a significant effect); more importantly, one or other of the migration volumes becomes negative at the extreme biaxial stresses, which is highly suggestive of lattice instabilities.

Fig. 6(b) is a plot of the migration enthalpies calculated from eqn (7) against biaxial strain. As seen for the isotropic case in Fig. 4(b), there is no significant difference between ΔE_mig and ΔH_mig for small strains, but deviations do appear for larger strains.

Comparison with experiment

Migration volume. Although experimental data for the activation volume of vacancy migration are available for fluorite-structured AO₂ electrolytes, comparing experimental data and theoretical predictions is problematic because the experimental data refer to two different ZrO₂ solid solutions subjected to uniaxial stress at two different (elevated) temperatures, whereas our calculations refer to (undoped) CeO₂ subjected either to isotropic or biaxial stress at room temperature. From uniaxial compression experiments in the range 20 to 55 MPa on ZrO₂ with 8 mol% Y₂O₃ at T = 1023 K, Park and Park⁵⁴ obtained ΔV_mig = 2.08 cm³ mol⁻¹. M'Peko and de Souza⁵⁵ measured the conductivity of ZrO₂ with 3 mol% Y₂O₃ at T = 443 K at two different uniaxial compressive stresses, 0 and 270 MPa: from their two data points we calculated ΔV_mig ≈ 1.2 cm³ mol⁻¹. From the static lattice calculations on CeO₂ we obtain, for a similar pressure range, ΔV_mig = 1.3 cm³ mol⁻¹ for the isotropic case and ΔVⁱⁿ_mig = 0.5 cm³ mol⁻¹ and ΔV^out_mig = 1.7 cm³ mol⁻¹ for the biaxial case. That the right order of magnitude is obtained from our calculations is strong evidence that the predicted variation in ΔE_mig with strain is physically reasonable. Furthermore, the activation volume for vacancy migration in fluorite-structured oxides at pressures |p| < 1 GPa is, evidently, 1 to 2 cm³ mol⁻¹, and dependent on the stress state but not strongly dependent on temperature or composition.

Enhancement of σ_ion through strain

Describing quantitatively the ionic conductivity of AO₂–M₂O₃ solid solutions as a function of temperature and amount of M₂O₃ requires three effects to be taken into account:^48,56oxygen vacancies interacting with the aliovalent substituent cations; oxygen vacancies interacting with one another (i.e. there is a repulsive vacancy–vacancy interaction); and the migration barriers for oxygen vacancies being modified by aliovalent substituent cations. Determining the effect of strain on all these effects is left for future study. Here, in order to estimate the degree to which the ionic conductivity of a fluorite-structured oxide can be enhanced through strain, we perform a simple calculation based on the migration of non-interacting oxygen vacancies in a fluorite solid solution. In this case, the ionic conductivity can be written aswhere x_V is the site fraction of oxygen vacancies, N_O is volume density of anion sites, β is a geometric factor, d the jump distance, ν₀ a characteristic lattice frequency, and ΔS_mig and ΔH_mig are, respectively, the entropy and enthalpy of migration. e, k and T have their usual meanings. Assuming that the only quantity in eqn (9) that varies significantly with strain is the migration enthalpy ΔH_mig, we obtain for the ratio of the ionic conductivity in the strained and unstrained states

Let us consider, then, a system comprising a thin film CeO₂-based electrolyte on a substrate with a lattice mismatch of, say, 4%. Let us assume that no misfit dislocations form at the interface; that the film is epitaxial, uniformly strained and continuous over macroscopic length scales; and that the entire mismatch between the expitaxial thin film and the substrate is taken up exclusively by the film, i.e., there is no relaxation of the substrate in the vicinity of the interface. Consequently, from the data shown in Fig. 6(b) one can predict that a biaxial strain of ε = + 0.04 will increase the in-plane ionic conductivity at T = 500 K by σ_ion/σ⁰_ion ≈ 10^3.7, that is, by several orders of magnitude. For two reasons we refrain from predicting a maximum possible enhancement through strain. First, there are various indications that CeO₂ is unstable at large biaxial strains, with ΔHⁱⁿ_mig for ε > +0.05 [see Fig. 6(b)]taking (physically unreasonable) negative values. This brings us to the second reason: since the lowest possible migration enthalpy is zero, the maximum possible enhancement depends only on the calculated migration enthalpy in the absence of strain (here, ΔH⁰_mig = 0.59 eV). If this set of empirical potentials had predicted a lower (higher) value for ΔH⁰_mig, the maximum possible enhancement would of course also be lower (higher). The main conclusion, though, is that, in line with another simulation study,²⁸ a strain of +0.04 is predicted to enhance the ionic conductivity of a fluorite-structured electrolyte at T = 500 K by around four orders of magnitude.

We now apply our treatment to the experimentally examined SrTiO₃|YSZ|SrTiO₃ heterostructure.²⁵ The measured activation enthalpy of conduction was found to decrease from ΔH⁰_mig = 1.1 eV at ε = 0 to ΔH_mig = 0.64 eV at ε = 0.07. According to eqn (10), the resulting enhancement in ionic conductivity at T = 500 K is σ_ion/σ⁰_ion ≈ 10^4.6. The enhancement can be increased to σ_ion/σ⁰_ion ≈ 10^5.6 by including a large possible variation in ΔS_mig with strain (see Appendix). Hence, the discrepancy between this predicted value and the measured enhancement of 10⁸ indicates, that the measured conductivity is electronic;^26–29 that one of the pre-exponential factors exhibits an extremely strong dependence on strain (‘extremely strong’ because σ is linearly proportional to the various pre-exponential factors, and we require changes of several orders of magnitude); or that another effect is active (space charge has been suggested,³¹ and in principle charge carrier concentrations in space-charge layers may be orders of magnitude different from bulk values). We discuss the latter two options below.

The missing orders of magnitude

Of the pre-exponential factors in eqn (9), the candidate most likely to exhibit an increase with strain is x_V (the increase in d² with ε is trivial and small, and ν₀ is likely to decrease with increasing ε). In the simple model of mobile, non-interacting vacancies generated by an M₂O₃ substituent, x_V is by definition unaffected by strain. But in a more realistic model of the ionic conductivity containing interactions between substituent cations and oxygen vacancies, interactions between oxygen vacancies themselves and modification of vacancy migration barriers by substituent cations, strain may, through changes in these defect-defect interactions, cause the effective value of x_V to increase. It remains to be seen, however, whether substantial increases are possible.

Lastly we consider whether charge carrier accumulation in space-charge layers can provide the missing orders of magnitude. Before we address this question, however, we need to confront the more fundamental question of why charge carriers should accumulate in YSZ. If we cannot provide an answer to the latter, more fundamental question, the first question becomes irrelevant.

Is there, then, a space-charge effect through which the oxygen vacancy concentration in a thin YSZ film can be increased beyond the level defined by the yttrium concentration? Yes, there is: If the standard chemical potential for oxygen vacancies (as building units), , is lower in YSZ than in SrTiO₃, and the interface does not trap oxygen vacancies (i.e. at the SrTiO₃|YSZ interface is not lower than in either SrTiO₃ or YSZ),⁵⁷ then oxygen vacancies will transfer from the SrTiO₃ substrate to the YSZ layer, generating space-charge layers in both materials. The situation is analogous to the case of bringing two semiconductors into contact: there will be transfer of charge carriers from one semiconductor to the other until the electrochemical potential of electrons (Fermi level) is equal in both systems. In the present case two oxides containing oxygen vacancies are brought into contact, and there will be transfer of oxygen vacancies from one to the other until the electrochemical potential of oxygen vacancies is equal in both systems. It must be noted that the transfer of oxygen vacancies from SrTiO₃ to YSZ is conditional upon being lower in YSZ than in SrTiO₃, and at present there is no experimental or theoretical evidence for or against this condition. Thus, although it is more or less certain that vacancy re-distribution will take place, since the values in SrTiO₃ and YSZ are unlikely to be equal, it remains to be seen whether vacancies transfer from SrTiO₃ to YSZ or vice versa.

Bearing this in mind and turning now to the original question we note that, since the YSZ solid solution used in the experiments contains 8 mol% Y₂O₃, the site fraction of oxygen vacancies is already rather high, x_V = 0.04. As a consequence, an increase in conductivity of even one order of magnitude is not possible. For the extremely simplified case of non-interacting vacancies migrating on a uniform lattice, the conductivity varies [see eqn (9)] according to x_V(1 − x_V), that is, maximum conductivity is obtained for half the sublattice being empty (i.e. occupied with vacancies). Thus, the maximum possible enhancement in conductivity through accumulation of oxygen vacancies in YSZ (with 8 mol% Y₂O₃) is

And this maximum enhancement requires the oxygen-ion sublattice of the entire 1 nm film to be half-filled, that is, the YSZ film would have the composition (Zr_0.84Y_0.16O₁)^1.84+; neither experiment nor theory provides any support for such major changes in the oxygen-ion sublattice. Upon inclusion of defect–defect interactions, the degree of enhancement through space-charge accumulation will change, but the changes are expected to be minor; furthermore they may not necessarily be beneficial. It is conceivable that, on account of the vacancy–vacancy interaction being repulsive, a transfer of vacancies from SrTiO₃ to YSZ leads to a decrease in the electrolyte's conductivity.

Concluding remarks

We have employed static lattice simulations to examine the effect of isotropic and biaxial strain on the migration thermodynamics of oxygen vacancies in fluorite-structured ceria. We find that strain can alter the migration thermodynamics considerably, and we confirm the possibility of enhancing the ionic conductivity of an AO₂–M₂O₃ solid solution by orders of magnitude through mechanical strain. In addition we demonstrated that static atomistic simulations, based on empirical pair-potentials (EPP), are capable of providing physically reasonable results on strained systems. Our work emphasises the need for further experimental and theoretical studies of electrolyte heterostructures: experimental studies should take account of extraneous resistances⁵⁸ and should isolate the ionic contribution to the total measured conductivity, for example, by performing oxygen isotope diffusion experiments;⁵⁹ and theory should focus on the strained electrolyte's mechanical and thermodynamic stability and on quantitative prediction of the ionic conductivity of an AO₂–M₂O₃ solid solution at the strain-producing interface. We conclude that strain-induced enhancement of eight orders of magnitude²⁵ appears to be beyond the limit of what is physically reasonable and that space-charge accumulation is unlikely to enhance the conductivity of an AO₂–M₂O₃ solid solution, essentially because there are already so many charge carriers present.

Appendix

In deriving eqn (10) we neglected, in particular, any possible variation in the activation entropy of migration, ΔS_mig, with strain. Below we verify that, even though ΔS_mig/k appears as the argument of an exponential term, the effect provides at most a small correction.

Korte et al.²¹ used the Maxwell relation,

to argue that, because in all likelihood the migration volume displays a weak temperature dependence, the variation in migration entropy with pressure can be neglected. Here we determine a limit for an enhancement in ionic conductivity arising from the activation entropy of migration varying with pressure. We begin by approximating eqn (A1) as

Let us assume, then, that ΔV_mig can increase by 1 cm³ mol⁻¹ (i.e. an increase of the order of ΔV_mig itself) for a temperature increase of ΔT = 600 K. The change in isotropic pressure in straining biaxially a YSZ film by 7% can be calculated from data given by Korte et al.²¹ to be Δp = −12 GPa. Thus, from eqn (A2) we obtain a limit for the increase in the activation entropy of migration: Δ(ΔS_mig) ≈ 2.4 k. This yields, for YSZ at T = 500 K, an enhancement in in-plane ionic conductivity of

in comparison, neglecting the migration entropy term, as in the main text, leads to an increase of σ_ion/σ⁰_ion ≈ 10^4.6. Hence, by including a strong variation in ΔS_mig with strain one can increase the ionic conductivity at the most by one order of magnitude.

Acknowledgements

RDS thanks A. Ramadan and S. Hörner for coding most of the calculations presented here, and C. Korte and J. Janek for illuminating discussions.

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Footnotes

† Dedicated to Prof. Harry L. Tuller on the occasion of his 65th birthday

‡ The degree of enhancement reported by Pennycook et al.³¹ is particularly open to question. From their quantum molecular dynamics simulations of oxygen vacancies in strained ZrO₂ they calculated a migration energy of 0.4 eV. This value is compared, inconsistently, with the experimentally determined activation enthalpy for oxygen vacancy migration in unstrained YSZ,²⁵ to give an enhancement of 4 × 10⁶. If the experimental migration enthalpies in both strained and unstrained YSZ²⁵ had been correctly reproduced, a much lower enhancement of < 4 × 10⁴ would have been obtained (see later). The enhancement predicted by Kushima and Yildiz²⁸ is far less contentious.

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