High Pressure X-ray Diffraction as a Tool for Designing Doped Ceria Thin Films Electrolytes

Rare earth-doped ceria thin films are currently thoroughly studied to be used in miniaturized solid oxide cells, memristive devices and gas sensors. The employment in such different application fields derives from the most remarkable property of this material, namely ionic conductivity, occurring through the mobility of oxygen ions above a certain threshold temperature. This feature is in turn limited by the association of defects, which hinders the movement of ions through the lattice. In addition to these issues, ionic conductivity in thin films is dominated by the presence of the film/substrate interface, where a strain can arise as a consequence of lattice mismatch. A tensile strain, in particular, when not released through the occurrence of dislocations, enhances ionic conduction through the reduction of activation energy. Within this complex framework, high pressure X-ray diffraction investigations performed on the bulk material are of great help in estimating the bulk modulus of the material, and hence its compressibility, namely its tolerance toward the application of a compressive/tensile stress. In this review, an overview is given about the correlation between structure and transport properties in rare earth-doped ceria films, and the role of high pressure X-ray diffraction studies in the selection of the most proper compositions for the design of thin films.


Introduction
Ceria doped by trivalent rare earth (RE) ions forms a wide family of mixed oxides of major relevance for their remarkable properties of ionic conductivity [1]. The partial substitution of Ce 4+ by RE 3+ induces in fact the formation of oxygen vacancies which at a sufficiently high temperature are able to migrate through the lattice by hopping from one to another oxygen site [2]. In the presence of an oxygen gradient, such as the one existing between the air and the fuel electrode in a solid oxide cell, the described movement gives rise to an organized and net flow of oxygen ions, thus making RE-doped ceria an ideal class of electrolytes for solid oxide cells. Nevertheless, this is not the only employment of the studied material, being it also used as an ionic conductivity enhancer in mixed ionic electronic conductors (MIECs), namely the perovskite-based materials used for both air and fuel electrodes in solid oxide cells [3]. Moreover, RE-doped ceria is also the active material in many gas sensors [4,5] and memristive systems [6].
The current and increasing need for portable devices makes miniaturization a forefront topic in the materials science research; in this respect, the deposition of thin films having the same or even better properties than the bulk material is one of the main goals connected with the realization of small-scale devices. The issue related to the thin films vs. bulk properties is a crucial point, since the 2D dimensionality, as well as the presence of the film/substrate interface, make the former constitutionally different from the latter, and the properties of thin films unpredictable on the sole basis of the bulk properties.

Rare Earth-Doped Ceria Thin Films: A Versatile Material to Be Used in Many Fields
As aforementioned, ionic conductivity is the most relevant physical property of REdoped ceria systems, and the most exploitable one in terms of applications; its maximization, which depends on several different issues, such as composition, RE chemical nature, microstructure, and also extrinsic factors, like the synthesis conditions, plays a central role in driving the research on this material [14]. The occurrence of significant values of ionic conduction, generally ranging between 0.01 and 0.1 S cm −1 at 873 K [15], makes doped ceria more desirable in solid oxide cells than yttria-stabilized zirconia (YSZ), which effectively works only at higher temperatures. The employment of doped ceria allows in fact the reduction of the operating temperature of the cell, thus favoring a lengthening of the device lifetime.
The first and foremost employment of RE-doped ceria (in particular for RE ≡ Gd and Sm) is thus in solid oxide cells, either fuel (SOFCs) or electrolysis (SOECs) cells, where it is used mainly as a solid electrolyte [16,17]. To this purpose, it is generally fabricated as a highly dense and thin pellet, in order to favor the movement of oxygen ions through the material. In addition, it is also often mixed, either by co-electrospinning [18] or by infiltration [3,19], to the electrode material with the aim to improve the electrochemical activity of the latter by enhancing its ionic conductivity. In SOFCs, for instance, the presence of Gd-or Sm-doped ceria within the electrode material allows the oxygen reduction reaction (ORR) to take place not only at the triple phase electrolyte/oxygen/air electrode boundary (TPB), but also in the electrode interior [20]. While, in fact, the electrode material is characterized by a high electric conductivity, it has mostly a negligible ionic conductivity, such as in the case of lanthanum strontium manganite (LSM), which is one of the most studied and used cathode materials in SOFCs [21]. Even in the case of lanthanum, strontium, cobalt ferrite (LSCF), presenting a higher ionic conductivity, the addition of some doped ceria significantly improves the material performance [20]. Again, in SOFCs it is also sometimes added to the perovskite fuel electrode material, with the aim to enhance charge transfer under reducing conditions, and in general to improve the electrode morphological stability [3,22]. In Figure 1, taken from [18], the arrangement of a SOFC cathode is shown: the Gd-doped ceria (GDC) electrolyte is in contact with the GDC/LSCF co-electrospun cathode. The EAT (electrochemical active thickness) is the thin layer of electrode where the ORR takes place. The so-formed electrons move through the GDC fibers toward the GDC electrolyte. The gold mesh acts as a current collector, and Pt ensures the contact between gold and the cell. For sake of completeness, it cannot be overlooked that ce its gas sensing properties, thanks to its ability to release or in of changes in the ambient oxygen concentration [4,5], and to volatile organic compounds (VOCs) [23]. The partial substitut such as Sm 3+ , was proven to induce an augmented sensitivit sensor [24]; the deposition of a Pr-doped ceria film on the ytt gas sensor, for instance, is reported to be a valid enhancer of ties of the device [25].
A further and more recent application of RE-doped c memories employing ions in place of electrons as carriers, nam These devices are resistive switches consisting in a switching two electrodes; within the cited material a redox reaction take resistivity variation. The application of a voltage promotes t the device. Among switching materials, CeO2−x [26] and Ce1− studied.

Ionic Conductivity in Bulk Doped Ceria
Ionic conductivity in ceria doped by a trivalent RE ion oxygen ions toward oxygen unoccupied crystallographic sites the movement of oxygen vacancies from one to another oxy For sake of completeness, it cannot be overlooked that ceria is also widely studied for its gas sensing properties, thanks to its ability to release or incorporate oxygen as a result of changes in the ambient oxygen concentration [4,5], and to respond to the presence of volatile organic compounds (VOCs) [23]. The partial substitution of Ce 4+ by a trivalent ion, such as Sm 3+ , was proven to induce an augmented sensitivity to oxygen of the resulting sensor [24]; the deposition of a Pr-doped ceria film on the yttria-stabilized zirconia-based gas sensor, for instance, is reported to be a valid enhancer of the toluene-sensing properties of the device [25].
A further and more recent application of RE-doped ceria applies to non-volatile memories employing ions in place of electrons as carriers, namely memristive systems [6]. These devices are resistive switches consisting in a switching material interposed between two electrodes; within the cited material a redox reaction takes place, thus giving rise to a resistivity variation. The application of a voltage promotes the flow of O 2− ions through the device. Among switching materials, CeO 2−x [26] and Ce 1−x Gd x O 2−x/2 [27,28] are widely studied.

Ionic Conductivity in Bulk Doped Ceria
Ionic conductivity in ceria doped by a trivalent RE ion occurs through hopping of oxygen ions toward oxygen unoccupied crystallographic sites; or, in other words, through the movement of oxygen vacancies from one to another oxygen site. The occurrence of empty oxygen positions takes place to ensure electroneutrality as a consequence of the partial replacement of Ce 4+ by RE 3+ . The described phenomenon becomes significant above a certain temperature: Sm-and Gd-doped ceria, namely two of the most effective systems in terms of ionic conduction, are commonly used in solid oxide cells working in the intermediate temperature range, i.e., between 673 and 973 K (IT-SOC). The dependence of ionic conduction (σ) on temperature (T) is ruled by the Arrhenius Equation: where A is the pre-exponential factor, E a is the activation energy to ionic conductivity, and R the gas constant. E a can be thought as the sum of association energy (E ass ) and migration energy (E m ), being the former the energy needed to separate an oxygen vacancy from a defect aggregate, and the latter the energy needed by vacancies to move through the lattice. Vacancies are in fact free to move only until RE 3+ ions are incorporated into the CeO 2 crystal structure as isolated RE Ce defects and oxygen vacancies randomly distributed within the matrix, both acting as guests of a CeO 2 -based solid solution [14]. Above a certain RE amount vacancies are partly blocked within defect aggregates. The reasons behind this behavior can be found in the structural arrangement of both the CeO 2 -based solid solution and defect aggregates. CeO 2 crystallizes in a fluorite-like cubic structure (called F) belonging to the Fm3m space group [29] containing four formula units per cell; Ce and O are hosted at the two atomic positions 4a (0,0,0) and 8c (1/4, 1/4, 1/4), respectively, and Ce is eight-coordinated to O. As aforementioned, up to a certain RE 3+ amount, RE Ce defects and oxygen vacancies occupy the 4a and 8c sites respectively, with vacancies being independent of their ionic counterpart and free to move at sufficiently high temperature. With increasing the RE 3+ content, doping ions and vacancies are stabilized by assuming the cubic atomic arrangement (called C) of the RE 2 O 3 sesquioxides of the heaviest rare earths (RE ≡ Gd-Lu) (space group: Ia3), where RE is six-coordinated to O [30]; in this configuration, vacancies cannot flow through the lattice, unless E ass is provided to the system in order to dislodge them from defect clusters. In Figure 2 the atomic arrangements of the F and the C phase are represented. CeO2 crystal structure as isolated ′ defects and oxygen vacancies randomly distrib uted within the matrix, both acting as guests of a CeO2-based solid solution [14]. Above certain RE amount vacancies are partly blocked within defect aggregates.
The reasons behind this behavior can be found in the structural arrangement of bot the CeO2-based solid solution and defect aggregates. CeO2 crystallizes in a fluorite-lik cubic structure (called F) belonging to the 3 ̅ space group [29] containing four fo mula units per cell; Ce and O are hosted at the two atomic positions 4 (0,0,0) and 8 (1/4, 1/4, 1/4), respectively, and Ce is eight-coordinated to O. As aforementioned, up to certain RE 3+ amount, ′ defects and oxygen vacancies occupy the 4 and 8 sites re spectively, with vacancies being independent of their ionic counterpart and free to mov at sufficiently high temperature. With increasing the RE 3+ content, doping ions and vacan cies are stabilized by assuming the cubic atomic arrangement (called C) of the RE2O3 se quioxides of the heaviest rare earths (RE ≡ Gd-Lu) (space group: 3 ̅ ), where RE is six coordinated to O [30]; in this configuration, vacancies cannot flow through the lattice, un less is provided to the system in order to dislodge them from defect clusters. In Fig  ure 2 the atomic arrangements of the F and the C phase are represented. From this perspective, it is possible to understand why in many doped ceria system two different values of activation energy can be recognized from the Arrhenius plot, wit the lower one at higher temperature [32][33][34][35]. This is quite a common feature in doped ceri systems, and it is ascribable to the occurrence of at least two different defect aggregate namely 1 •• ′ positively charged dimers and 1 •• 2 ′ neutral trimers [36]. In gen eral, a higher binding energy is calculated for dimers rather than for trimers [37]: as consequence, the latter, due to their lower configurational entropy, become progressivel less stable with increasing temperature, and their dissociation at a sufficiently high tem perature induces the release of oxygen vacancies which become able to migrate, thus re From this perspective, it is possible to understand why in many doped ceria systems two different values of activation energy can be recognized from the Arrhenius plot, with the lower one at higher temperature [32][33][34][35]. This is quite a common feature in doped ceria systems, and it is ascribable to the occurrence of at least two different defect aggregates, namely 1V •• O RE Ce positively charged dimers and 1V •• O 2RE Ce neutral trimers [36]. In general, a higher binding energy is calculated for dimers rather than for trimers [37]: as a consequence, the latter, due to their lower configurational entropy, become progressively less stable with increasing temperature, and their dissociation at a sufficiently high temperature induces the release of oxygen vacancies which become able to migrate, thus reducing the activation energy value. The crossover temperature results to be roughly the same for all the systems, and it was observed around 750 K [34]; in Figure 3, reproduced from [34], the trend of the activation energy below and above 750 K, as well as the preexponential factor A appearing in Equation (1), are reported as a function of the ionic size of the doping ion. In spite of the large differences in ionic conductivity existing betw with different RE 3+ [34], the maximum in ( ) occurs within a na interval, namely at x (in Ce1−xRExO2−x/2) ranging between 0.10 and 0.2 the studied systems, such as for RE ≡ Gd [2], Sm [38], Lu [39], as well higher x, ionic conductivity significantly drops, and its absolute valu significant for any application; nevertheless, even heavily RE-doped interest in fundamental research, since they allow to correlate subtle transport properties [31,32,[40][41][42]. The substantial coincidence of the different systems suggests that the appearance in diffraction patterns o to defect clusters at largely different x values for different RE ions, do F phase has a different extent according to the RE identity; on the co gregates are present and stable even at very low RE concentration, bu ent spatial correlation, they can be identified by X-ray diffraction at di In addition to the numerous studies performed on singly-doped a large number of doubly-and multiply-doped systems has been in Gd/Y- [43], Gd/Sm- [44], La/Sm- [35], Sm/Nd- [45], Nd/Gd- [46], La/ [48]. A general lowering of activation energy [33] and an improveme In spite of the large differences in ionic conductivity existing between systems doped with different RE 3+ [34], the maximum in σ (σ max ) occurs within a narrow compositional interval, namely at x (in Ce 1−x RE x O 2−x/2 ) ranging between 0.10 and 0.25 for the majority of the studied systems, such as for RE ≡ Gd [2], Sm [38], Lu [39], as well as (Nd,Tm) [33]. At higher x, ionic conductivity significantly drops, and its absolute value becomes no more significant for any application; nevertheless, even heavily RE-doped systems are of great interest in fundamental research, since they allow to correlate subtle structural details to transport properties [31,32,[40][41][42]. The substantial coincidence of the position of σ max in different systems suggests that the appearance in diffraction patterns of peaks attributable to defect clusters at largely different x values for different RE ions, does not mean that the F phase has a different extent according to the RE identity; on the contrary, C defect aggregates are present and stable even at very low RE concentration, but due to their different spatial correlation, they can be identified by X-ray diffraction at different x values [33].

Ionic Conductivity in Doped Ceria Thin Films
When approaching thin films, the first and foremost character to be taken into account is their 2D nature, which strongly affects all their properties. Going into detail of the particular case of doped ceria, ionic conductivity results to be specifically driven, in addition to the factors mentioned in the previous paragraph, also valid for the material in bulk form, by the strain arising at the film/substrate interface, and by the dislocations content.
The unavoidable lattice mismatch occurring at the film/substrate interface is responsible for the former item, namely for the occurrence of strain. Tensile strain, in particular, is believed to operate a decrease in the activation energy to ionic conduction, and therefore an improvement of this property, while the opposite occurs when compressive strain arises [51,52]. This happens because a migrating oxygen ion, while hopping from one to another vacancy site, has to move through a narrow aperture, namely the space between two cations, which is generally much smaller than the oxygen ion radius. A tensile strain contributes to widen this aperture, thus favoring the movement of oxygen ions. As the activation energy to ionic conduction is directly correlated to the lattice perturbation needed to let the O 2− ion through, a tensile strain helps to reduce this quantity. This evidence is further corroborated by the increase in the activation energy observed in buckled freestanding doped ceria membranes, which are subjected to a compressive stress due to the buckling itself [53]. For instance, an increase of 0.13 eV was observed in the activation energy of a Ce 0.8 Gd 0.2 O 1.9 free-standing membrane as a response to a 1.78% compressive strain with respect to the flat substrate-supported thin film with the same composition [54]. Similarly, a multilayer oxide-based device consisting in alternating layers of Ce 0.9 Gd 0.1 O 1.95 and Er 2 O 3 , where the number of individual layers was changed from 1 to 60 in order to vary the strain-activated volume, presents an increase in activation energy by 0.31 eV with enhancing the compressive strain by 1.16% [54].
The effect of strain on ionic conductivity is comparable to that of introducing doping ions into the ceria lattice: inducing strain by using a doping ion with a significantly different size than Ce 4+ causes in fact a change in ionic conduction, due to the variation of internal pressure caused by the foreign ion [34]. Computational simulations performed on Ce 1−x Y x O 2−x/2 , for example, suggest that the application of isotropic strain to the bulk material induces an increase in ionic conductivity by three times at 1273 K for x = 0.18 [8]. It can be thus concluded that both doping and the existence of a lattice mismatch have to be carefully considered when designing doped ceria thin films. Therefore, the desired transport properties can be engineered as a combined result of internal and external strain.
In light of the previous considerations, the aforementioned film/substrate lattice mismatch comes as an ideal item, acting as a strain source; coherently grown films, in particular, are especially affected. Nonetheless, provided that the lattice mismatch, if exerting a tensile strain, plays a positive role in improving ionic conductivity, it should never overcome a certain threshold, which primarily depends on the elastic properties of the material; otherwise, the excess strain is released through the formation of dislocations. The segregation of both trivalent cations and oxygen vacancies in the vicinity of the dislocation satisfies the natural tendency to reduction of elastic energy around these defects, and it necessarily reduces ionic conductivity [55]. For this reason, a direct growth of doped ceria on a MgO substrate is not recommended: the 8.62% lattice misfit ε (ε = a ceria −a substrate a ceria ·100, with a the lattice parameter) is too large to ensure a proper substrate/film match [56]. If, on the contrary, a SrTiO 3 buffer layer is interposed between MgO and the film, in the particular case of Sm-doped ceria ε is reduced to 1.62% if the cubic axis of the growing film is rotated by 45 • with respect to the one of the substrate, thus allowing to take advantage of a significant ionic conductivity increase. Moreover, the lattice mismatch lies at the root of the often observed columnar growth of the film, which takes place as an attempt to accommodate interfacial strain [57].
That being said, the literature provides a large amount of even not entirely agreeing data; nevertheless, even if the colossal ionic conductivity increase by a factor 10 8 observed in YSZ grown on SrTiO 3 and ascribed to the very large tensile strain [58] does not take place in doped ceria, most studies point at an increase in ionic conductivity of strained films with respect to their bulk unstrained counterpart. This evidence is generally deemed as a direct consequence of the increase of mobile ions at the strained interface [59]. Ce 0.8 Gd 0.2 O 1.9 thin films grown on a MgO crystal, for instance, are reported to present very similar values of ionic conductivity and activation energies as the corresponding bulk material, perhaps due to the very large lattice mismatch [60], while Ce 0.8 Sm 0.2 O 1.9 films grown on SrTiO 3 -buffered MgO present at 973 K a σ value of 0.07 S·cm −1 , to be compared to σ = 0.02 S·cm −1 deriving from dense polycrystalline pellets [57] an even larger ionic conductivity enhancement takes place in Sm-doped ceria/YSZ films deposited on the same substrate [60]. A clear increase in ionic conductivity with increasing strain was observed at each temperature considered in the range 723-1123 K in epitaxial Ce 0.9 Gd 0.1 O 0.95 thin films grown on SrTiO 3 -buffered MgO [7].
It is the interplay between two main parameters which rules the dependence of ionic conductivity on strain: film thickness and grain size. The former can be understood in terms of volume fraction of strained material: the thinner the film, the larger the strained volume fraction. A strong increase in ionic conductivity for instance was observed in epitaxial grown Sm-doped ceria/YSZ films with reducing film thickness [61]; moreover, an ideal film thickness of~80 nm was suggested to achieve the maximum ionic conductivity in Ce 0.8 Gd 0.2 O 0.9 films deposited by pulsed laser deposition on single crystal (0001) Al 2 O 3 substrates, as the best compromise between the strain induced by lattice misfit and the grain boundary conduction associated to grain size [62]. Nevertheless, even the thickness of the buffer layer plays a role: a controlled reduction of the SrTiO 3 layer implies a weaker compensation of the ceria/MgO lattice mismatch, thus leading to an enhancement of ionic conductivity, as experimentally observed in epitaxial Ce 0.9 Gd 0.1 O 0.95 thin films [63]. Finally, it is noteworthy that an increase in ionic conductivity is also observed in doubly-layered Sm-doped ceria/YSZ films with increasing the amount of bilayers and keeping constant the overall film thickness [62].
As aforementioned, even the grain size is a relevant item when discussing the ionic conductivity of doped ceria films. Generally speaking, the reduction of the grain size always implies an increase in the grain boundary volume and, as a consequence, a progressively larger contribution to transport properties of the latter. This normally translates into a conductivity decrease, at least in bulk materials. In ionic conducting thin films, something different happens. As already observed and discussed by Christie and Berkel [63], in such films a grain size decrease induces an increase in the grain boundary conductivity. An increase in σ, for example, was observed in Ce 0.8 Gd 0.2 O 0.9 films deposited by spin-coating on a sapphire substrate with decreasing the grain size, most probably due to the reduction of the activation energy as a consequence of the segregation of impurities within the grain boundary volume [64,65]. This is a general achievement; despite the scatter among absolute conductivity values reported in different papers [66][67][68], it seems to be correlated to the decrease in activation energy, occurring irrespective of the different deposition techniques. Anyway, since films containing smaller grains display larger strain [66], the described evidence is a further clear indication toward a direct correlation between strain and ionic conductivity.

Thin Films Deposition Processes
Deposition techniques useful to prepare doped ceria thin films can be divided into two main groups, namely the vacuum-and the precipitation-based ones. To the former belong all techniques making use of a target consisting in a sintered pellet; the latter, on the contrary, rely on the employment of solvents.
To obtain dense thin films of doped ceria, sputtering and pulsed laser deposition (PLD) are widely used, since they allow to obtain thin doped ceria layers, starting from the precursors powders. For instance, Liu et al. [26] used the rf magnetron sputtering method to deposit a~16 ± 2 nm film on a Pt/Ti/SiO2/Si substrate for the use in memristive devices, working at room temperature, and starting from a CeO 2−x ceramic target, under ultra-fine Ar atmosphere. Recently, Kumar et al. [16] employed pulsed laser deposition to deposit Sm-Gd co-doped ceria films on Si (111), starting from a dense sintered pellet of Gd 2 O 3 , Sm 2 O 3 and CeO 2 in the molar ratio 1:1:8; the deposition process was performed under different oxygen pressure conditions, to determine the effect of pO 2 on the structural and morphological properties of the grown films.
However, the sputtering and PLD methods are generally not so cost-effective, since they require very complex apparatuses, therefore making the scaling-up of the thin films production process very difficult to achieve. Schlupp et al. [69] proposed an efficient and cheap way to obtain Gd-doped ceria films by aereosol assisted chemical vapor deposition, starting from stoichiometrically mixed precursors solutions of cerium (IV) and gadolinium (III) 2,2,6,6-tetramethylheptanedionate, and performing the deposition on Si (100) and sapphire single crystals. Lair et al. [70] carried out a study on the structural and morphological properties of electrodeposited samaria-doped ceria films, to determine their possible use in µ-SOFCs. In the mentioned experiment, thin films were deposited on a stainless steel substrate at room temperature (303 K), from aqueous electrolytic solutions in which a supporting electrolyte (NaNO 3 0.1 M) and stoichiometric amounts of Ce(NO 3 ) 3 ·6H 2 O and Sm(NO 3 ) 3 ·6H 2 O were dissolved.
In this section, only a few possible techniques for the ceria-based thin films preparation are presented, but several others can be found in literature [71]: a more detailed discussion is beyond the aim of the present paper.

Why High Pressure X-ray Diffraction?
X-ray diffraction is well known to be a powerful experimental technique, which provides a fundamental contribution to the comprehension of structure [41], microstructure [72], occurrence of defects [13], and crystallinity [73]. In the particular case of doped ceria, it is evident that strain contributes in a fundamental way to ionic conduction of thin films. As aforementioned, tensile strain has to be high enough to enable a significant reduction of the activation energy to ionic conduction, but at the same time it should not exceed a certain threshold, in order to avoid the appearance of dislocations. In this respect, thin film compositions able to accommodate a large strain are potential promising candidates ionic conductors. The ability of a material to accommodate strain is expressed by the compressibility of the material (k), namely its ability to modify its volume as a response to a pressure change. Its reciprocal, the isothermal bulk modulus B 0 , is defined as: where V 0 is the cell volume at ambient conditions. If, as it is desirable, k is large, B 0 obviously assumes a small value. The latter condition is also advantageous because it implies a small volume elastic energy, and a substantial suppression of the dislocations generation. The investigation of strain and strain-related phenomena in thin films is experimentally quite difficult due to the need to exclude the contribution of the substrate from the measurement of any property. Within this framework, in-situ high pressure X-ray diffraction applied to bulk samples of doped ceria comes to aid, allowing to determine for each composition the isothermal bulk modulus B 0 by properly treating the refined cell volumes, and hence k. In this work, the calculation is conveniently limited to the F phase, since this is the atomic arrangement where ionic conduction takes place, but it could be in principle extended even to C. In order to gain an idea of the correspondence between applied pressure and strain, it is interesting to notice that applying a pressure of~20 GPa to a Gd-doped ceria bulk sample causes a compressive strain of around 5%, while the application of 5 GPa roughly corresponds to a 1% compressive strain [7].

Fundamentals on the Experimental Method
Diamond anvil cells (DACs) are generally used to apply pressure on a selected sample by pressing it between two perfectly aligned diamond culets that slowly move closer to each other thanks to a gear-driven mechanism: the maximum achievable pressures can vary a lot, mainly depending on the DAC model. The present devices can be used to apply pressure on a given sample either anisotropically or isotropically. In the former case, uniaxial pressure is applied to the sample simply pressing it between the diamonds; in the latter, sample powders are loaded inside the pressure chamber of a specific metallic sampleholder (the so-called gasket), and a pressure transmitting medium (PTM) is required to uniformly distribute pressure within the chamber. Silicon oil, NaCl powders and Ar (in membrane-DACs) are examples of widely used PTM: to ensure the hydrostatic distribution of pressure inside the chamber, an ideal sample:PTM volume ratio of 20:80 is generally needed. In any case, in order to determine the effective pressure exerted on the sample, an internal standard is required: generally, a material with a well known high-pressure structural behaviour (e.g., Cu).
Electromagnetic waves enter the cell passing through the diamonds, and interact with the pressed sample: therefore, the described device is potentially compatible with many experimental techniques, mainly depending on the DAC model and characteristics (e.g., DAC with ultra low fluorescence diamonds for Raman spectroscopy).
As aforementioned, the maximum achievable pressure is strongly influenced by the diamond culet size employed: the lower the culet diameter, the higher the maximum pressure. For instance, a cell with Boehler-Almax designed diamonds with a Ø > 1 mm culet allows to reach a maximum pressure of 5 GPa, meanwhile in principle, with a Ø = 200 µm diamond, pressures higher than 100 GPa can be reached. However, also the gasket preparation process must be performed with the utmost care. In fact, the sample chamber consists in a hole drilled throughout a metallic disc that has been previously thinned by being pressed between the diamonds of the cell: an excessive gasket thickness, a non-circular hole or a hole which is not drilled at the centre of the diamond culets impression, or even an overfilled sample chamber can affect the maximum achievable pressure. Moreover, the hydrostatic limit of the selected PTM is also very important: for instance, a 4:1 mixture by volume of methanol and ethanol, which is one of the most commonly used liquid PTM, allows to work in hydrostatic conditions up to 10 GPa, while argon allows to reach higher pressures, since it causes a very small pressure gradient (<1.5%) at 80 GPa [74]. The silicon oil employed in the experiments by the present research group ensured the hydrostatic working conditions up to 10 GPa, therefore covering the whole considered pressure range.
The authors' research group focused on RE-doped ceria systems containing RE ≡ Lu [11], Sm [12] and a mixture of Nd/Tm ions [13]. In particular, the last-mentioned system was synthesized using a Nd/Tm = 0.74/0.26 ratio, in order to reproduce the average ionic size of Sm 3+ (C.N. 8), being Sm-doped ceria one of the most promising electrolytes for solide oxides cells [75]: Ce 0.8 Sm 0.2 O 1.90 , for example, reaches the remarkable σ value of 10 −2 S cm −1 at 773 K [38]. All the samples were prepared by coprecipitation of the corresponding mixed oxalates, and by subsequent thermal treatment at 1173 K for 3 days in order to obtain a high crystallinity degree, as thoroughly described in [41,76]. Going into detail, authors studied Ce 1−x RE x O 2−x/2 systems with x ranging between 0.1 and 0.4 for RE ≡ Lu [11], 0.2 and 0.6 for RE ≡ Sm [12], and 0.1 and 0.6 RE ≡ (Nd,Tm) [13]. Moreover, regarding the Lu-doped ceria system, even the composition having x = 0.8 was studied. Samples are named according to the RE doping ion identity and to its percentage amount with respect to the total lanthanide content. For example, the name Sm20 corresponds to the composition Ce 0.8 Sm 0.2 O 1.9 .
High pressure X-ray diffraction data of the aforementioned systems were collected at the Xpress beamline of the Elettra synchrotron facility, located in Basovizza (Trieste, Italy). Pressure was applied to all the doped ceria powders up to~7 GPa, thus obtaining comparable data for all the considered systems; at least six different pressure values were taken into account for each sample. Powders were loaded in a gear-driven Boehler-Almax plate DAC equipped with diamonds with a 300, 500 or 600 µm culet size (respectively for RE ≡ Nd/Tm, Sm and Lu), and characterized by a large X-ray aperture. For instance, the 300 µm culets DAC is characterized by an angular aperture of 60 • , thus allowing to collect the image of the diffraction rings in the angular range from -30 • to 30 • . The pressure chambers were settled by drilling hole with a diameter of 100, 200 or 150 µm (respectively for RE ≡ Nd/Tm, Sm and Lu) through spark-erosion in several 200-µm-thick rhenium gaskets, which had been previously indented to reduce their thickness below 110 µm (70 for RE ≡ Lu, for which stainless steel gaskets were employed). Silicone oil was used as the pressure transmitting medium, and pressure calibration was performed by two different methods: either by placing Cu foils into the sample chamber (as depicted in Figure 4), and taking into account the displacement of the (111) diffraction peak with increasing pressure, or by using ruby chips, studying the displacement of their fluorescence lines. All the diffraction analyses were performed in hydrostatic conditions.

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taking into account the displacement of the (111) diffraction peak with increasing pressure, or by using ruby chips, studying the displacement of their fluorescence lines. All the diffraction analyses were performed in hydrostatic conditions.  The Xpress beamline is equipped with a MAR345 image plate detector allowing to collect images of the diffraction rings, which are subsequently converted into the typical 2 -intensity diffraction patterns by the Fit2D software [77]. The 2 range spanned between 5° and 24° for RE ≡ Nd/Tm, between 6° and 36° for RE ≡ Lu, and between 4° and 30° for RE ≡ Sm. The FullProf Suite [78] was used to refine by the Rietveld method [79] the structural models reported in Section 3.1, which describe the obtained diffraction patterns.

Fundamentals on Data Analysis
Diffractograms were treated through the Rietveld method by refinement of the structural models which better describe experimental data. As elucidated in Section 3.1, REdoped ceria crystallizes in the F structure up to a certain RE 3+ doping ion amount; beyond this limit, defect aggregates characterized by the C structure appear. If Ce 4+ and RE 3+ are sufficiently close in size, the cited C defect clusters act as guests of the CeO2-based solid solution. Since C is a superstructure of F, and due to the aforementioned Ce 4+ /RE 3+ close size similarity, such as in the case of RE ≡ Gd 3+ and Sm 3+ (Ce 4+ , CN:8, r = 0.97 Å ; Sm 3+ , CN:6, r = 0.958 Å ; Gd 3+ , CN:6, r = 0.938 Å [80]), diffraction patterns of C and F only differ by the The Xpress beamline is equipped with a MAR345 image plate detector allowing to collect images of the diffraction rings, which are subsequently converted into the typical 2ϑ-intensity diffraction patterns by the Fit2D software [77]. The 2ϑ range spanned between 5 • and 24 • for RE ≡ Nd/Tm, between 6 • and 36 • for RE ≡ Lu, and between 4 • and 30 • for RE ≡ Sm. The FullProf Suite [78] was used to refine by the Rietveld method [79] the structural models reported in Section 3.1, which describe the obtained diffraction patterns.

Fundamentals on Data Analysis
Diffractograms were treated through the Rietveld method by refinement of the structural models which better describe experimental data. As elucidated in Section 3.1, REdoped ceria crystallizes in the F structure up to a certain RE 3+ doping ion amount; beyond this limit, defect aggregates characterized by the C structure appear. If Ce 4+ and RE 3+ are sufficiently close in size, the cited C defect clusters act as guests of the CeO 2 -based solid solution. Since C is a superstructure of F, and due to the aforementioned Ce 4+ /RE 3+ close size similarity, such as in the case of RE ≡ Gd 3+ and Sm 3+ (Ce 4+ , CN:8, r = 0.97 Å; Sm 3+ , CN:6, r = 0.958 Å; Gd 3+ , CN:6, r = 0.938 Å [80]), diffraction patterns of C and F only differ by the superstructure peaks, while all the other ones coincide. The exactly double length of the C cell parameter with respect to the one of F is a further proof of that. As the amount of C defect aggregates gradually grows with increasing the RE 3+ content, neither the F nor the C model properly describe the real atomic arrangement; on the contrary, an intermediate or hybrid model (called H), is the best candidate. In H, structural parameters sensitive to the F/C passage, such as the x coordinate of the 24d site, occupied by Ce and RE, and the occupancy factor of the 16c site, populated by O, display a progressive transition from the values typical of F to the ones of C with increasing the RE content. If, on the contrary, Ce 4+ and RE 3+ are too different in size, namely starting from RE ≡ Tm and going toward smaller RE elements [81], a two-phase field (F + C) becomes stable beyond the stability region of the CeO 2 -based solid solution. In the light of these considerations, among the studied systems, while Sm-and (Nd,Tm)-doped ceria show the hybrid region, in Lu-doped ceria the biphasic (F + C) field appears. Relying on the occurrence of diffraction peaks, in Ce 1−x RE x O 2−x/2 the F region extends up to x = 0.3, 0.4 and 0.5 for RE ≡ Sm, Lu, and (Nd,Tm), respectively. This means for instance that in sample Lu80 a mixture of the F and C phase is present, with C largely prevailing over F [42]. Structural models were refined accordingly; for sake of clarity, the F, C and H structural models are reported in the Supplementary Materials (Table S1).
In order to elaborate each diffraction pattern, the peak profile was optimized by the pseudo-Voigt function, while the background was refined by linear interpolation of a set of 70 experimental points. In the last refinement cycle, structural parameters (cell parameters, and for the H phase the Ce/RE1 x coordinate, the O1 x, y and z coordinates, and both the occupancy factor and the x coordinate of O2), atomic displacement parameters, the scale factor, five peak parameters and the background points were allowed to vary. For each sample, high pressure data were refined using as starting parameters the optimized values of atomic positions found at ambient pressure. In compositions containing Cu as internal standard, angular regions where peaks of Cu occur were excluded from refinements. In Figure 5, the Rietveld plot of sample Sm60 at 4.11 GPa is reported as a representative example. The diffraction pattern presents the so called hybrid structure (H), resulting from the intimate intergrowth of the F and the C lattices (see Section 4.3), where C is a superstructure of F. The main peaks detectable in the diffraction pattern are the ones common to F and C; the minor peaks, on the contrary, can be ascribed to the C phase contribution.
gs 2021, 11, x FOR PEER REVIEW is present, with C largely prevailing over F [42]. Structural mode ingly; for sake of clarity, the F, C and H structural models are repo tary Materials (Table S1).
In order to elaborate each diffraction pattern, the peak profi pseudo-Voigt function, while the background was refined by line of ~70 experimental points. In the last refinement cycle, structural eters, and for the H phase the Ce/RE1 x coordinate, the O1 x, y and the occupancy factor and the x coordinate of O2), atomic displa scale factor, five peak parameters and the background points we each sample, high pressure data were refined using as starting pa values of atomic positions found at ambient pressure. In compos internal standard, angular regions where peaks of Cu occur wer ments. In Figure 5, the Rietveld plot of sample Sm60 at 4.11 GPa sentative example. The diffraction pattern presents the so called h sulting from the intimate intergrowth of the F and the C lattices (s is a superstructure of F. The main peaks detectable in the diffract common to F and C; the minor peaks, on the contrary, can be a contribution. Performing high pressure X-ray diffraction analyses on a giv termine the equation of state (EoS) by studying the evolution of t umes under the effect of an applied pressure. The isothermal EoS plex, and it is generally satisfactorily described by a refinable mod Performing high pressure X-ray diffraction analyses on a given system allows to determine the equation of state (EoS) by studying the evolution of the refined unit cell volumes under the effect of an applied pressure. The isothermal EoS of a solid is quite complex, and it is generally satisfactorily described by a refinable model; among all the possibilities, the four most common equations are Murnaghan [82], Birch-Murnaghan [83] natural strain [84], and Rose-Vinet [85]. All of them express the relation between the unit cell volume of the material at ambient conditions (V 0 ) and pressure (P), in terms of bulk modulus at zero applied pressure and its first pressure derivative (B 0 and B 0 , respectively). The Murnaghan EoS [82], in particular, relies on the assumption that the bulk modulus linearly varies with pressure; according to it, the dependence of pressure on volume can be written as: A further development of this theory is represented by the work of Birch [83], who concentrated on the application to a cubic cell of strain consisting in a hydrostatic compression plus a homogeneous infinitesimal strain. The treatment gives rise to the following form of P(V): The Natural Strain EoS, which was elaborated by Poirier and Tarantola [84], is a logarithmic equation expected to be valid for a wider pressure range than the aforementioned ones, and it can be expressed as follows: Finally, the Rose-Vinet EoS [85] was originally thought to predict the high temperature response of a material to the application of an external pressure; to accomplish this goal, the knowledge at each different temperature of B 0 , B 0 , V 0 and of the coefficient of thermal expansion (CTE) at zero pressure, are needed. Following the proposed approach, it is possible to provide an estimation of the temperature dependence of CTE, B 0 and B 0 . The expression of P(V) follows. with and Starting from the refined unit cell volumes of the F phase, the most suitable EoS was determined for each sample by using the EoSFit7_GUI software [86]. In particular, for all the considered systems, different EoS were tested, and in spite of the generally negligible discrepancies among the results deriving from different equations, the best results were always reached using the third order Rose-Vinet equation.
Refinements of the equation of state through the described Rose-Vinet model provided the results reported in Table 1. The examination of data reported in Table 1 reveals in each system a generally decreasing trend of B 0 of the F phase with increasing the RE content. This finding is in good agreement with the results presented by Rainwater et al. [10] on Sm-doped ceria: notwithstanding the slight different absolute B 0 values, a significant decrease of bulk modulus with increasing the doping ion content occurs even in their data.
The described behavior can be interpreted in the light of the masterly work by Anderson and Nafe [87], who in 1965 studied the correlation between bulk modulus and mean atomic volume (V at ) of covalent crystals and oxides. They moved from the observation that ln B 0 shows a linear correlation to ln(2V at ), which is obeyed by a huge variety of compounds, and finally accomplished the development of the following empirical law: Equation (9) is valid both for covalent crystals and oxides, with the only difference that m assumes the value 4/3 in the former compounds, while it ranges between 3 and 4 in the latter. It suggests that oxides are characterized by a much stronger decrease of B 0 with increasing V at with respect to covalent compounds.
In Figure 6, the trend of ln B 0 vs. ln(2V at ) is reported for all the studied systems, together with the line representing the expected trend for oxides according to Anderson and Nafe; the latter was built attributing to m the intermediate value 3.5. At first sight it can be observed that qualitatively even doped ceria follows the trend common to oxides. V at is defined as the lattice volume divided by the number of atoms therein contained; it is, therefore, noteworthy that with increasing the content of the trivalent doping ion, the oxygen amount, and hence the total number of atoms per formula unit, proportionally decreases. It results thus clear that within the considered systems, an increase in V at derives from an increase in the RE 3+ content. A closer observation of the diagram reveals that, even if the trend of doped ceria systems roughly complies with that of other oxides, the decreasing rate is higher for the former, being for instance m~6 for both Sm-and (Nd,Tm)doped ceria at low doping content. Moreover, it can also be observed that the deviation from the expected trend becomes progressively larger on going from Sm-to (Nd,Tm)to Lu-containing systems, i.e., with introducing progressively smaller doping ions. This evidence finds an explanation considering that the binding energy, and thus the stability, of C defect aggregates becomes larger with decreasing the RE 3+ size [37]; therefore, at a given x in Ce 1−x RE x O 2−x/2 , for RE ≡ Lu the amount of C defect associates is higher than for RE ≡ (Nd,Tm) and Sm. The case of doubly-doped ceria is particularly intriguing, because out of Nd 3+ and Tm 3+ , the latter preferentially and in large amount enters C defect clusters, being Tm 3+ the smaller of the two ions. This item leads the (Nd,Tm)-doped system to present lower B 0 values than the Sm-containing system, notwithstanding the identical average ionic size of doping ions in the two systems. also provided.
The significant difference in 0 between different sy in Figure 3, is also revealed in compressibility. Figure 7 d ibility as a function of the doping ion amount x. A gen preciated with increasing x for each system, but even mo smaller doping ions promotes the occurrence of larger given x. (2 ) for all the studied system while the ones of the Sm-and (Nd,Tm)-containing systems deri Lu-doped ceria are taken from [12]. Cell volumes of samples cr plied by 8 in order to make them comparable with those crystal Figure 6. Trend of lnB 0 vs. ln(2V at ) for all the studied systems. Data of CeO 2 are taken from [88], while the ones of the Sm-and (Nd,Tm)-containing systems derive from [13]; experimental points of Lu-doped ceria are taken from [12]. Cell volumes of samples crystallizing in the F phase are multiplied by 8 in order to make them comparable with those crystallizing in the C structure.
The subtraction of RE ions from the F matrix through the formation of C domains implies that the total number of atoms/unit formula of the F phase is overestimated if the nominal stoichiometry of the overall oxide is taken into consideration. This issue explains the discrepancy between the expected and the experimental trend of ln B 0 vs. ln(2V at ), and it is thoroughly discussed in [13], where an estimate of the defect clusters amount is also provided.
The significant difference in B 0 between different systems at a given x, clearly visible in Figure 3, is also revealed in compressibility. Figure 7 depicts the behavior of compressibility k as a function of the doping ion amount x. A general increasing trend can be appreciated with increasing x for each system, but even more interestingly, the presence of smaller doping ions promotes the occurrence of larger compressibility values at each given x.
These findings have some important implications in the framework of the design of thin films. As previously discussed, ionic conductivity in thin films is strictly dependent, in addition to other factors, also on the amount of tensile interfacial strain which can be accommodated without releasing it through the formation of dislocations; in other words, it strongly depends on the material compressibility. Therefore, the results presented in this work suggest that, in terms of compressibility, the employment of (a) small doping ions and (b) high doping contents are to be preferred. These guidelines need of course to be reconciled with other parameters contributing to ionic conduction, such as the effect of the RE chemical identity, and the observed drop beyond a certain x value. Nevertheless, they can be considered as a further reference point to be taken into account when designing doped ceria thin films to be used as electrolytes in solid oxide microdevices. Figure 6. Trend of 0 vs. (2 ) for all the studied systems. Da while the ones of the Sm-and (Nd,Tm)-containing systems derive fr Lu-doped ceria are taken from [12]. Cell volumes of samples crysta plied by 8 in order to make them comparable with those crystallizin

Conclusions
This work critically reviews papers recently appeared in the literature dealing with the use of in situ high pressure X-ray diffraction in the design of doped ceria thin films to be mainly used as solid electrolytes in solid oxide electrochemical microdevices; it discusses the results therein reported highlighting the role of the bulk modulus of the bulk material in the choice of the most proper compositions of films.
The tensile strain possibly arising at the film/substrate interface gives a significant contribution to ionic conduction in doped ceria thin films, provided that it is not released through the occurrence of dislocations; compressibility is therefore a measure of the strain amount which can be tolerated by the structure. Within this scenario, the evaluation of bulk modulus through the refinement of the material equation of state, and hence of compressibility, is of help. Results obtained from Sm-, (Nd,Tm)-and Lu-doped ceria show that the smaller the doping ions, the higher compressibility, as a consequence of the higher binding energy of defect clusters containing smaller rare earths.

Data Availability Statement:
No new data were created or analyzed in this study. Data sharing is not applicable to this article.