Pyrochlore-type lanthanide titanates and zirconates: Synthesis, structural peculiarities, and properties Open Access

Multicomponent oxides are an interesting class of inorganic solids, exhibiting a broad range of atomic structures, bonding characteristics, and functionalities, which are behind many advances of modern technology. Among them, A₂B₂O₇ ternary oxides crystallizing with the pyrochlore structure, form a relatively large family of phases with a rich variety of chemical and physical properties of fundamental, and practical interest. This review is concerned with two specific, but fascinating series of pyrochlore oxides, i.e., lanthanide titanates, and zirconates: Ln₂Ti₂O₇ and Ln₂Zr₂O₇ (Ln = Ln³⁺ and Y³⁺). The burgeoning research interest in this limited group of compounds (only 18 ternary phases) and their solid solutions, is well-evidenced by the considerable number of papers published every year, dealing with different aspects of their synthesis, structural peculiarities, and physicochemical properties.

The pyrochlore structure is named after a relatively common, and chemically diverse family of minerals, which was first described by Wöhler¹ in 1826, and is best represented by the general formula A_2−mB₂X_6−nY_1−p·wH₂O, where A and B represent cations, and X and Y anions,^2,3 respectively. Typically, A is a large (r ∼ 0.85–1.55 Å),⁴ and naturally abundant alkali or alkaline earth cation (e.g., Na⁺ or Ca²⁺), which can be partially replaced by minor amounts of various additional ions, including lanthanides and actinides (e.g., Ce³⁺ or U⁴⁺). By contrast, B is usually a smaller (∼0.40–0.78 Å),⁴ and higher-valence cation, most commonly Nb⁵⁺, Ta⁵⁺ or Ti⁴⁺, and less frequently, W⁶⁺ or Sb⁵⁺; in fact, pyrochlore minerals are primary ores for niobium and tantalum. As for the anion sublattice, X is generally oxygen, and Y may be O²⁻, OH⁻ or F⁻. The m, n, and p symbols in the formula, indicate partial occupancy of the A, X, and Y sites, respectively, whereas vacancies at the B-site are rare in natural samples. The total water content in pyrochlore minerals (w) may be as high as 10 to 15 wt. %.

The pyrochlore structure is also widely adopted by synthetic multicomponent oxides because it can accommodate a broad range of cation combinations, mixed site populations, and vacancies. The A-site in synthetic A₂B₂O₆O′ (where O and O′ denote two crystallographically nonequivalent oxygen ions) or, more commonly, A₂B₂O₇ pyrochlore oxides, can be occupied not only by the same elements found in natural pyrochlore, but also by a diverse range of large cations in various oxidation states such as Bi³⁺, Cs⁺, Tl⁺, and Pb²⁺, among others. The B-site might host almost any positively charged ion capable of adopting a sixfold coordination, such as Sn⁴⁺, Zr⁴⁺, Ru^4+,5+, Mo^4+,5+, Hf⁴⁺, Te^4+,6+, Os⁶⁺, or Ir⁴⁺. Surprisingly, even small ions such as Si⁴⁺, and Ge⁴⁺ can also form pyrochlore-type phases, when subjected to high-pressure and high temperature.^5,6 Some oxygen- and cation-deficient pyrochlore oxides can even show considerable deviations from the ideal A₂B₂O₇ stoichiometry and yet, remain thermodynamically stable, even without a significant distortion of the ideal isometric unit cell. In addition to structural and compositional flexibility, pyrochlore oxides display a plethora of unique, exciting, and potentially useful physical and chemical properties. Moreover, pyrochlore represents an important structure type in view of magnetic geometrical frustration due to ordering of the A, and B cations into separate interpenetrating lattices of corner sharing tetrahedra. An almost endless list of low-, and room-temperature exotic magnetic states (e.g., spin-glass, spin-liquid, spin-ice) are possible depending on whether the A-, the B-site or both, are occupied by magnetic cations.^7–15 Pyrochlore-structured oxides also exhibit a multitude of different electrical and electrochemical behaviors; some members show conductivity values typical of metal-like or even, superconducting materials, whereas some others are fast ion or mixed ionic-electronic conductors, semiconductors, and insulators.^16–25 Therefore, pyrochlore oxides find application in energy conversion and storage devices, oxygen separation membranes, and sensing and combustion control systems. Some pyrochlore oxides also exhibit very low thermal conductivities (κ), even close to the glass-like lower limit value of highly disordered solids. Consequently, they are being applied as thermally insulating coatings (Thermal Barrier Coatings, TBCs), to protect exposed metal components of gas turbines, and diesel engines, and as supports for high temperature catalysts.^26–32 Furthermore, pyrochlore oxides are remarkably effective in accommodating actinides (e.g., Pu, Np, Cm, Am), some compounds sustaining prolonged exposure to radiation due to alpha-decay events, all the while preserving the integrity of the periodic, atomic structure; they have been proposed as major components in actinide-bearing ceramics, for the long-term immobilization, and disposal of long-lived waste radionuclides from nuclear programs.^33–38 Some of these properties may be displayed simultaneously allowing for new and promising applications. Furthermore, properties are mediated by subtle electronic differences, or by the presence of structural defects and disorder, which occur easily in this family of oxides. Moreover, the pyrochlore structure offers a unique “playground” to examine the interplay between chemical composition, defect chemistry, and properties. Despite decades of intensive research, the understanding of the fundamental aspects that control the ordering/disordering (and vice versa) processes taking place in pyrochlore-type materials remains far from satisfactory. Moreover, the relative importance of the electronic structure, bonding characteristics, or even processing conditions upon disorder, is still the subject of an active debate.

This contribution addresses the singularities of the pyrochlore structure and provides an overview of pyrochlore-type lanthanide titanates and zirconates (Ln₂Ti₂O₇, and Ln₂Zr₂O₇), their most interesting properties (i.e., electrical, magnetic, and thermal), and their behavior under extreme conditions (pressure and irradiation). Brief sections on the crystal chemistry of the elemental oxides, the relevant phase diagrams, and the energetics of defect formation, which are crucial to fully understand the properties of multicomponent oxides, have been included. The most comprehensive review on pyrochlore oxides was published 40 years ago,³⁹ but this paper did not incorporate for example data on phases with multiple A and B populations or address differences between short- and long-range structures.

The pyrochlore structure was first determined by von Gaertner for a naturally occurring (Ca,Na)₂Nb₂O₆(F,OH) mineral.⁴⁰ There are different possible visualizations of the structure, which are mainly based on the shape and topology of the cation coordination polyhedra; although some are notably straightforward, they prove immensely beneficial for discussing specific aspects of pyrochlore crystal chemistry and physicochemical properties. Therefore, this review begins by outlining the most important characteristics of some of these simple structure models. For a detailed description of the pyrochlore structure, interested readers are referred to alternative comprehensive sources such as Subramanian et al.³⁹ and Chakoumakos.⁴¹ 

The most popular depiction of this structure considers pyrochlore to be a fluorite-related superstructure, with a 1/8th anion deficiency; in fact, many pyrochlore-type oxides undergo a solid-state phase transition to an anion deficient fluorite structure at high temperatures. As illustrated in Fig. 1, the similarity between both structure-types is apparent, albeit with some significant differences. The well-known AX₂ fluorite structure [S.G. = Fm $3 ¯$m (225)] is characterized by a face-centered cubic (fcc) array of cations (A), with all available tetrahedral interstices filled with anions in a simple cubic array, having a repeat unit half of the cation array. Cations and anions fully occupy the 4a (0,0,0) and 8c (¼,¼,¼) Wyckoff sites, and the structure is completely described by specifying the cell constant. In contrast, cations in the A₂B₂X₆Y pyrochlore structure [S.G. = Fd $3 ¯$m (227)] are ordered on two nonequivalent sites while retaining the fcc array. Choosing the inversion center for the origin of centrosymmetric space group 227 (origin choice 2), the smaller B cation is generally confined to the 16c Wyckoff site (0,0,0), whereas the A cation is located at the 16d position (½,½,½). Moreover, 1/8th of the anions are missing compared to the fluorite structure, lowering the coordination environment of the 16c-site to sixfold. B-cations are located at the center of a trigonal antiprism of $3 ¯$m symmetry (D_3d), with all six anions (X in A₂B₂X₆Y) at equal distances from the metal atom (i.e., metal coordination polyhedron is BX₆). On the other hand, A-cations are surrounded by eight anions, forming an axially compressed scalenohedron also of $3 ¯$m symmetry, with two metal-anion bonds (A–Y), significantly shorter than the other six (i.e., metal coordination polyhedron is AX₆Y₂).

Like in fluorites, anions in the pyrochlore structure occupy the tetrahedral interstices of the cation framework, though not all are equivalent. They fully occupy two symmetry-independent positions in the unit cell, the 48f (x,⅛,⅛) and 8b (⅜,⅜,⅜) Wyckoff sites. Anions (X) at the 48f-site [site symmetry mm (C_2v)] are surrounded by two A and two B nearest neighbors, forming a distorted tetrahedron (i.e., coordination polyhedron is XA₂B₂). Anions (Y) on the 8b-site $[ 4 ¯$3m (T_d)], are surrounded by four A cations, forming a regular tetrahedron (i.e., YA₄). Therefore, the two anion sites have different Pauling electrostatic valences:⁴² anions on the 48f site receive charge from two A-cations, each subdivided eight ways, and charge from two B-cations, each subdivided six ways. In contrast, the anions on the 8b site receive charge from four A-cations, each subdivided eight ways. This results in slight overbonding at the 48f anion site and considerable underbonding at the 8b site, which cancels out over the entire unit cell. Compared to the fluorite structure, the ordered distribution of cations on the one hand, and of anions and vacancies on the other, results in a different space group and a doubling of the lattice constant (a_pyrochlore ≈ 2a_fluorite). The pyrochlore structure features a large (typically, a ≈ 1 nm), and rather populated unit cell (Z = 8), although all the atoms, except for the anions at the 48f-site, are in special positions. Therefore, the structure is fully described by only two independent structural parameters: i.e., the lattice constant (a), and the fractional x-coordinate of the anions in 48f (x_48f). A distinctive peculiarity of the pyrochlore structure is the existence of an additional interstitial site in the unit cell, Wyckoff site 8a (⅛,⅛,⅛), easily accessible for anions, but nominally empty in ideal (ordered) pyrochlore structures. This 8a-site (⁠ $4 ¯$3m symmetry) is also tetrahedrally coordinated as the 48f and 8b positions, though by four B cations (□B₄). This constitutional vacancy, which would complete a fluorite-type anion sublattice, provides the pyrochlore structure with a remarkable ability to tolerate structural distortions. The absence of electrostatic repulsions allows the surrounding anions (especially the nearest neighbors in 48f), to relax away from their ideal positions and toward the vacant site. The value of the variable positional parameter x_48f, shows a strong correlation with both, A and B absolute sizes, and their size difference. Furthermore, x_48f determines the shape of the cation coordination polyhedra, and might be used as a measure of the structural deviation from the ideal fluorite atomic arrangement.^43,44 With the origin still fixed at the 16c Wyckoff site, an x_48f -value of 0.3125 (i.e., ^5/₁₆) would render a regular octahedron around B, and a distorted trigonal scalenohedron around A. On the other hand, anions would form a perfect cubic arrangement when x_48f = 0.375 (i.e., ^6/₁₆), with the coordination polyhedra around B and A, becoming a trigonally flattened octahedron, and a regular cube respectively. The former atomic arrangement corresponds to the ideal pyrochlore structure, whereas the anion sublattice in the latter matches the topology of its analog in an ideal, anion-deficient fluorite structure. As the space group imposes the same symmetry for both cation sites, forming pyrochlore materials at ambient pressure necessarily requires^45,46 x_48f values between these two limits, 0.375 and 0.3125. Most pyrochlore structures, however, present x_48f values closer to 0.3125 than to 0.375, suggesting that the B-cation prefers adopting closer to regular octahedral coordination more than the A-cation tends to adopt regular cubic coordination.⁴⁷ 

The ordering process that transforms a disordered fluorite structure into a fully ordered pyrochlore structure, can be described using the tools of representation theory. This approach offers a powerful insight into the system's Hamiltonian, which presides over the long-range order process and is a bridge between structure and properties. The symmetry breaking produced by the compositional order characteristic of the pyrochlore structure, involves zone boundary irreducible representations at the L and X points of the fluorite Brillouin zone. More precisely, the compositional order parameter describing the characteristic cation order observed in a pyrochlore structure only requires a four-dimensional order parameter involving all the branches of the star of the L¹⁺ representation. Simultaneously, the complete O vacancy localization also demands a three-dimensional order parameter involving all the branches of the star of the X⁴⁺ irreducible representation. The same four branches of the L¹⁺ irreducible representation also provide a complete description of the observed displacive component of the O atom (at the 48f site) along the [100] directions, a displacement characteristic shared by all pyrochlore structures.

As depicted in Fig. 2, the pyrochlore structure can be alternatively represented⁴⁸ as two interpenetrating yet largely independent and loosely bound networks. The first network is a fairly rigid B₂X₆ framework, composed of corner-sharing BX₆ octahedra organized in three- and six-membered rings, reminiscent of the hexagonal tungsten bronze configuration. In contrast, the second network is an A₂Y linear network, forming zigzag chains along the largest channels of the former. Accordingly, the characteristic pyrochlore A₂B₂X₆Y stoichiometry, can be conveniently expressed as A₂Y·B₂X₆. This structural model proves especially useful when accounting for the many structural derivatives of the pyrochlore structure. While the B₂X₆ network is typically preserved in any derivative, the A₂Y substructure surprisingly is not required for pyrochlore structural integrity. Accordingly, a high level of deficiency can exist at both the A and Y sites (16d and 8b Wyckoff sites, respectively) resulting in numerous rather complex stoichiometries. Therefore, a range of defect A_xB₂X₆Y_z structures (0 ≤ x ≤ 2; 0 ≤ z ≤ 1) has been documented in the literature, where the A₂Y network is either partially present or completely absent. Although the A and Y site deficiencies are generally coupled, stoichiometries such as A₂B₂X₆ (i.e., ABX₃), AB₂X₆ or even B₂X₆ (i.e., BX₃) as in pyrochlore-type WO₃, are also feasible.^49–53 Moreover, the 8b-site, typically occupied by anions in ideal pyrochlore structures, may host cations as seen in the so-called “inverse” (rattling) □₂B₂X₆A pyrochlore⁵⁴ (A = large monovalent cation, such as K⁺, Rb⁺ or Tl⁺; □ = vacancy). This group of complex oxides garnered significant interest after the initial reports of superconductivity in pyrochlore-type AOs₂O₆^55,56 (A = alkali metal). In this structural variant, A-cations are partially delocalized (rattling motion) toward the nearby 32e (x,x,x) general position, considered collinear and intermediate between the 16d and 8b sites. The magnitude of such displacements (0 ≤ x ≤ ⅜) depends on the charge, size, and polarizability of each specific metal ion; smaller ions exhibit larger displacements. Missing anions reduce the A-cation coordination to sixfold. Interestingly, these changes do not necessarily compromise the original pyrochlore isometric symmetry. Furthermore, some unit cell expansion has been observed upon water uptake in otherwise easily hydrated potassium osmates: certain K⁺ ions occupy the 16d site to accommodate water molecules, which, in turn, occupy a position centered around the 32e site.

A third common description of the structure, particularly beneficial when discussing the magnetic properties of pyrochlore oxides, conceptualizes two rigid and interpenetrating arrays of B₄ and A₄ corner-sharing tetrahedra.⁴⁸ Illustrated in Fig. 3, in this representation, B-cations occupy the corners of regular tetrahedra with an empty center (the 8a-site), while A-cations form analogous tetrahedra, but with the center occupied by anions in 8b (Y). Therefore, the B₄ tetrahedra network forms a 3D configuration of Fd $3 ¯m$ space group, akin to that observed in cristobalite, SiO₂. However, in this case, the oxygen positions of the [SiO₄]⁴⁻ tetrahedra would be occupied by the B-cations, and the silicon vacant site corresponds to the pyrochlore 8a position. Anions at the 48f-site (X) reside in variable positions outside the B₄ tetrahedra, close to the middle point of the edges and at a distance from the center dependent on the positional parameter x_48f. This three-dimensional tetrahedral network can be adequately represented by B_4/2□(X₆). Likewise, the A₄ tetrahedra form a 3D anticristobalite-type network and can be represented by A_4/2Y.

A fourth description of the pyrochlore structure (Fig. 4), developed by Sickafus et al.,⁴⁷ and derived from concepts introduced by Iida,⁵⁷ is in a layered form along its cubic unit cell body diagonal (i.e., along ⟨111⟩). This same arrangement can be obtained by transforming the conventional Fd $3 ¯$m unit cell, to a primitive trigonal cell with rhombohedral axes [S.G. = R $3 ¯$ (148)], in which the stacking along ⟨111⟩_Fd-3m, is instead along ⟨001⟩_R-3. In this representation, the pyrochlore structure is composed of pure layers of either cations or anions arranged in triangular, or Ising, nets.⁵⁸ 

Consistent with the general stoichiometry of a fluorite-derived structure, there are two anion (A) layers for every cation (C) layer, arranged in the sequence: ACAACA. More specifically, to maintain the pyrochlore stoichiometry A₂B₂O₇, these six layers are composed as follows: (1) a fully dense triangular net of anions, (2) a fully dense triangular net of cations, (3) a fully dense triangular net of anions, (4) a ¾ dense triangular net of anions, (5) a fully dense triangular net of cations, and (6) a ¾ dense triangular net of anions. In the ¾ dense anion layers (numbers 4 and 6), the anion vacancies are distributed such that the resulting layer of anions, form a Kagome sublattice. In cation layer (2), sandwiched between the fully dense anion layers (1 and 3), forms an A₃B Kagome arrangement, while in cation layer (5), it is a AB₃ Kagome arrangement. Alternatively, taking layers (1–3) together as a compound layer, the stoichiometry is A₃BO₈, and the compound layer formed by layers (4–6) has the stoichiometry AB₃O₆; thus, the cation coordination scheme of layers (1–3) has been called “8–8–8–6” and “6–6–6–8” for layers (4–6).⁵⁹ Because the cations are in special positions, their layers are composed of ideal triangular nets. However, due to the deviation of oxygen from an ideal fluorite position, the anion layers are not ideal triangular nets.

Component oxides play a major role in determining the stability of complex oxides. The following section will provide a concise overview of the crystal structures, polymorphism and metal coordination of TiO₂, ZrO₂, and Ln₂O₃, as well as key features of the BO₂-Ln₂O₃ phase diagrams (Ln = lanthanides and Y; B = Ti and Zr). Additionally, the structural peculiarities, non-stoichiometry and defect formation in pyrochlore-type lanthanide titanates and zirconates, will be also discussed in detail.

The most common polymorphs of titanium (IV) oxide are anatase and rutile (Fig. 5). Anatase-TiO₂ is metastable and undergoes an irreversible transformation into rutile between 400 and 1200 °C,⁶⁰ depending on factors such as sample preparation conditions (activation energy ∼377 kJ·mol⁻¹). However, the majority of TiO₂ nanoscale materials with a particle size below ∼15 nm exhibits the anatase structure.⁶¹ Both crystal structures share common features including (i) a tetragonal unit cell (though different space groups: P4₂/mnm (136) in rutile vs I4₁/amd (141) in anatase) (ii) Ti⁴⁺ ions in a distorted octahedral coordination with slightly longer apical Ti–O bonds (see Fig. 8), and (iii) O²⁻ ions surrounded by three metal atoms. However, anatase and rutile-TiO₂ differ in the way the TiO₆ octahedra are interconnected.

In rutile, the TiO₆ octahedra form edge-sharing chains along the z direction (i.e., each TiO₆ octahedron shares two edges with neighboring octahedra), and corners in the x-y plane. In contrast, the TiO₆ octahedra are significantly more distorted in anatase-TiO₂, forming zigzag chains along the x and y directions with each octahedron sharing four edges with neighboring octahedra. Accordingly, the anatase-to-rutile phase transition is accompanied by ∼8% decrease in the unit cell volume; each unit cell contains four formula units in anatase and two in rutile, which also has higher density (3.895 g·cm⁻³ vs 4.274 g·cm⁻³). Therefore, the anatase-to-rutile transformation is considered reconstructive. Ab initio calculations have shown that the ⟨Ti–O⟩ bonds have a dominant covalent character.⁶² The melting point of TiO₂ (∼1830 °C) is significantly lower than that of ZrO₂ or any of the lanthanide oxides Ln₂O₃.

At ambient pressure, three phases exist between room temperature, and the melting point (∼2700 °C) for pure ZrO₂ (Fig. 6). The stable form at room temperature adopts the monoclinic baddeleyite structure-type [m-ZrO₂; S.G. = P2₁/c (14); Z = 4], which transforms into a tetragonal t-ZrO₂ phase on heating at ∼1100 °C [S.G. = P4₂/nmc (137); Z = 2], and into a cubic c-ZrO₂ form at ∼2350 °C [S.G. = Fm $3 ¯$m (225); Z = 1]. The m-ZrO₂ form contains sevenfold coordinated Zr⁴⁺ atoms (see Fig. 8) with a large bond-length dispersion, ranging from 0.205 to 0.228 nm.⁶³ The structure also includes two nonequivalent oxygen ions in three- and fourfold environments. Zr⁴⁺ coordination increases to 8 in the higher symmetry polymorphs, although four ⟨Zr-O⟩ bonds in t-ZrO₂, are considerably shorter than the remaining four (∼0.2065 vs ∼0.2455 nm); whereas, all are identical in c-ZrO₂. Density is also higher in t- and c-ZrO₂ than in m-ZrO₂ (∼6.10 g·cm⁻³ vs 5.89 g·cm⁻³). The m → t transformation is commonly seen as semi-reconstructive due to a change in coordination, not only for Zr⁴⁺ but also for one of the two nonequivalent oxygen ions of m-ZrO₂, which adopts a distorted tetrahedral coordination geometry. The transformation also entails a significant reduction in molar volume (∼3.5%) and exhibits appreciable temperature hysteresis on cooling: the t → m reversed transformation usually occurs between 100 and 200 °C below the m → t. Stabilization of the higher symmetry ZrO₂ forms under ambient conditions is facilitated by reducing the particle size below a critical value (∼30 nm for t-ZrO₂) because of the surface energy contribution to the oxide's free energy.^64,65 The ⟨Zr-O⟩ bonds in ZrO₂ are mainly ionic, although covalency increases with decreasing Zr coordination.

Furthermore, the lower coordination number and higher covalency of the ⟨Zr–O⟩ bonds account for the higher stability of the low symmetry m-ZrO₂ at room temperature. As with the pyrochlore structure, all three ZrO₂ polymorphs can be treated as local distortions of the fluorite structure; interestingly, none of them features sixfold coordinated Zr⁴⁺ atoms, as found in the pyrochlore structure.

Despite the relatively small variation in ionic radii along the lanthanide series, Ln₂O₃ sesquioxides (Ln = Ln³⁺ and Y³⁺) exhibit a rich crystal chemistry with at least five different structures referred to as A-, B-, C-, H-, and X-Ln₂O₃, existing at different combinations of temperature and pressure.^66,67 Under ambient conditions, depending on the Ln³⁺ ionic radius, Ln₂O₃ sesquioxides will adopt⁶⁸ one of the three structures depicted in Fig. 7: A-, B-, or C-Ln₂O₃ of hexagonal, monoclinic, and cubic symmetry, respectively. The hexagonal H- and cubic X-Ln₂O₃ forms are only observed at very high temperatures and are not typically quenchable although, the latter has been identified for example, in some Ln₂O₃ samples after being subjected to energetic heavy ion irradiation.⁶⁹ Available thermodynamic information for transitions and fusion of Ln₂O₃ phases have been critically reviewed and summarized.^70,71 The hexagonal A-form [S.G. = P $3 ¯$m1 (164); Z = 1] is common at room temperature for the lighter lanthanides (La–Nd). The stability of the cubic (C-) form increases with decreasing Ln³⁺ radii. Therefore, the heavier members of the series (Tb–Lu) exist at room temperature as the C-Ln₂O₃ form [S.G. = Ia $3 ¯$ (206); Z = 16], whereas C-La₂O₃ and C-Nd₂O₃ are considered metastable. Most cubic Ln₂O₃ sesquioxides transform to B- and A-Ln₂O₃ with increasing temperature. Furthermore, the stability temperature interval of the B- and A-Ln₂O₃ forms narrows rapidly from Tb³⁺ to Lu³⁺, with decreasing ionic radius. Moreover, the C → B transformation in Tb₂O₃ and Dy₂O₃ requires temperatures of ∼1200 °C, and ∼1800 °C respectively (r_Tb = 0.0923 nm vs r_Dy = 0.0912 nm, both in octahedral coordination); whereas, cubic C-Lu₂O₃ (r_Lu = 0.0861 nm) is stable up to the melting point (∼2480 °C). The reversed B → C transformation mechanism is sufficiently slow for Sm³⁺, Eu³⁺ and Gd³⁺ sesquioxides, allowing the existence of the monoclinic B-form at room-temperature [S.G. = C2/m (12); Z = 6]. Promethium (Pm) is the only member of the lanthanide series not naturally occurring, formed exclusively through nuclear reactions (all its isotopes are unstable). Consequently, only limited information about its properties and chemistry is available in the literature. Nevertheless, it has been established⁷² that the room temperature stable form of Pm₂O₃ is the cubic C polymorph, transforming irreversibly upon firing to monoclinic B-Pm₂O₃. Subsequently, at higher temperatures, it undergoes a reversible transformation to the A-form. This crystal chemistry aligns more closely with that of the Sm-Gd subgroup than with the lighter lanthanides La–Nd.

From the structural point of view, the C-form is isostructural to the mineral bixbyite [(Mn,Fe)₂O₃]; however, it can also be appropriately described in terms of a simple fluorite cell with two vacancies paired along the body diagonal of the anion cube. Interestingly, whereas the anion vacancies in pyrochlore are arranged in ¾ dense triangular nets forming Kagome lattices along ⟨111⟩, in the C-Ln₂O₃ form, the anion vacancies cluster in an unusual “wishbone-shaped” pattern, referred to as “the wishbone-Kagome lattice.”⁷³ Furthermore, anions are all in tetrahedral coordination, as in the fluorite structure, whereas Ln³⁺ ions are in a heavily distorted octahedral environment. In contrast, Ln³⁺ ions form a perfect hexagonal close packed array (hcp) in A-Ln₂O₃ with alternating layers of tetrahedral, and octahedral interstices filled with anions. The density and Ln³⁺ coordination increase from C- to B-, and A-Ln₂O₃. Therefore, Ln³⁺ ions are seven coordinated in A-Ln₂O₃ (Fig. 8), with anions arranged in a distorted mono-capped octahedral geometry (four oxygen atoms are closer to the central metal atom than the rest). By contrast, up to three nonequivalent Ln³⁺ ions, and five oxygen ions, exist in B-Ln₂O₃ (Figs. 7 and 8); some Ln³⁺ ions are sevenfold coordinated, as in A-Ln₂O₃, but some others have a distorted octahedral coordination, as in C-Ln₂O₃. Nevertheless, the B-Ln₂O₃ structure is considered more closely related to A- than to C-Ln₂O₃. Accordingly, the B → A transition is considered displacive, whereas the C → B and C → A, are considered both reconstructive. Ln₂O₃ show very high thermal stability with melting points in the range of ∼2300–2500 °C. ⟨Ln–O⟩ bonds are primarily ionic because the inner 4f electrons are relatively well screened from the chemical surroundings and do not typically participate in bonding. Calculations predict increasing covalency for the ⟨Ln–O⟩ bonds as the size of Ln³⁺ increases, and La₂O₃ would feature mixed ionic-covalent bonding since La³⁺ has no 4f electrons.

The cubic C-form is the most stable polymorph of Y₂O₃ with the hexagonal, and monoclinic variants obtained only at high temperature⁷⁴ (>2300 °C) and high pressure, respectively; some authors have also suggested a cubic fluorite-type structure for the high temperature form.^75,76 The presence of any of the Ln₂O₃ oxides affects the polymorphic transformations of both TiO₂ and ZrO₂. Thus, phase transitions to the t- and c-forms of ZrO₂ are possible at lower temperatures, by the addition of Ln₂O₃; the stabilization effect is more pronounced as the Ln³⁺ ionic radius decreases. On the other hand, the presence of small amounts of Ln₂O₃, inhibits the anatase to rutile-TiO₂ transformation;⁷⁷ this effect seems to be independent of the Ln³⁺ involved, and may be related to the ease of forming lanthanide titanates at relatively low temperatures, as it will be shown in the next section.

The ZrO₂–Ln₂O₃ and TiO₂–Ln₂O₃ phase diagrams show some distinct peculiarities. Several comprehensive reviews of available experimental data and computational studies of the Ln₂O₃–ZrO₂ systems, have been published.^78,79 Therefore, only a brief discussion will be provided here. Figure 9 illustrates a typical ZrO₂–Ln₂O₃ phase diagram, using the example of the ZrO₂–Gd₂O₃ system. While there is little doubt about the number, and stoichiometry of existing intermediate phases, the refractory nature of the component oxides and the high thermal stability of the reaction products generate substantial discrepancies regarding homogeneity ranges, and transformations at high temperatures. As a function of the size of the Ln³⁺ ion, two ordered intermediate compounds occur: pyrochlore-type Ln₂Zr₂O₇ (Ln₂O₃:ZrO₂ molar ratio = 1:2) or the so-called δ-phase Ln₄Zr₃O₁₂ (Ln₂O₃:ZrO₂ molar ratio = 2:3). The pyrochlore is the only mixed oxide found for the lighter lanthanides (Ln = La–Gd); whereas the δ-phase is stable from Ho³⁺ to Yb³⁺. Dysprosium appears to be on the boundary between both phases, but neither of the two stoichiometric mixed oxides has been prepared in the Dy₂O₃–ZrO₂ system. Therefore, the stability of the pyrochlore structure decreases in the lanthanide series with increasing atomic number (decreasing Ln³⁺ size), whereas that of the δ-phase follows the opposite trend. Similar to the pyrochlore structure, the δ-phase is also a derivative of the anion-deficient fluorite structure with ordered oxygen vacancies, and cations distributed between two nonequivalent sites. However, in the rhombohedral δ-phase, cations occupy the 3a and 18f Wyckoff positions, which exhibit six- and sevenfold coordination, respectively. Despite the small change in the anion/cation ratio compared to pyrochlore (i.e., 12/7 = 1.71 vs 7/4 = 1.75), eightfold cation sites are absent in the δ-phase structure.

The specific cation distribution between the 3a and 18f sites is still the subject of some debate. Monte Carlo energy-minimization calculations have identified three possible low-energy cation distributions, depending not only on the size difference, and processing conditions of each sample but also on electronic effects.⁸⁰ Similarly to pyrochlore, most Ln₄Zr₃O₁₂ phases undergo order-disorder (O-D) transformations to an anion-deficient fluorite-structure at high temperatures. Figure 9 illustrates the ZrO₂–Gd₂O₃ phase diagram, demonstrating extensive solid solution regions on either side of the line compounds, Ln₂Zr₂O₇ or Ln₄Zr₃O₁₂. The specific crystal structure of each solid solution depends on factors such as Ln₂O₃ content, thermal history or even particle size. Furthermore, metastable states easily form. Nonetheless, fluorite-type solid solutions of either tetragonal or cubic symmetry, are commonly found for ZrO₂-rich compositions, whereas structures based on the corresponding high-temperature Ln₂O₃ forms exist for high lanthanide contents.

The phase chemistry of the Ln₂O₃–TiO₂ systems is notably more complex, especially for the larger lanthanides (Ln = La-Nd), with at least five intermediate compounds with different stoichiometries described in the literature. In contrast, as illustrated in Fig. 9 with the TiO₂–Gd₂O₃ system, only two such mixed Ln³⁺/Ti⁴⁺ phases have been reported for the remaining members of the series (Ln = Sm–Lu). On increasing lanthanide content, these lanthanide titanates include the following Ln₂O₃:TiO₂ molar ratios: 2:9 (Ln₄Ti₉O₂₄), 1:3 (Ln₂Ti₃O₉), 1:2 (Ln₂Ti₂O₇), 2:3 (La₄Ti₃O₁₂), and 1:1 (Ln₂TiO₅). Concerning the larger lanthanides, comprehensive phase diagram studies for the Ce₂O₃–TiO₂, Pr₂O₃–TiO₂, and Pm₂O₃–TiO₂ systems are lacking in the literature. The La₂O₃–TiO₂ and Nd₂O₃–TiO₂ systems have received more attention, likely due to practical considerations.^81–83 Synthesizing Ce³⁺ or Pr³⁺ titanates is challenging due to these elements strong preference for the 4+ oxidation state, with CeO₂ and Pr₆O₁₁ being their most stable oxides under ambient conditions. Nevertheless, some Ce³⁺ or Pr³⁺ containing titanates have been successfully prepared and characterized, as described below. Considering the limited availability and radioactive nature of promethium, studies on the Pm₂O₃-TiO₂ (or ZrO₂) system are not available and unlikely to be pursued. The Ln₄Ti₉O₂₄ phases (Ln = La–Nd) crystallize in the orthorhombic Fddd group (S.G. = 70). Their structure consists of a complex network of distorted TiO₆ octahedra connected in a 3D framework, with distorted LnO₈ square antiprisms and LnO₆ octahedra.^84–86 The Ln₂Ti₃O₉ phases, also represented as Ln_2/3TiO₃ (Ln = La and Nd), exhibit an A cation-deficient perovskite layered structure where one-third of the A-sites are vacant.^87–89 Vacancies are typically ordered, resulting in a doubled unit cell compared to an idealized ABO₃ perovskite cubic unit cell. These phases are slightly oxygen deficient, and the oxygen loss is compensated by the reduction of some Ti⁴⁺ atoms to Ti³⁺. The symmetry of the crystal structure increases with the oxygen deficiency level, from orthorhombic for slightly deficient phases to tetragonal and eventually cubic for higher oxygen deficiency levels. The layers are composed of three perovskite slabs stacked with a 1/2 gliding along two directions, with Ln occupying a 12-fold coordinated intralayer position and no interlayer cation. The Ln₄Ti₃O₁₂ phases (Ln = La–Nd) belong to the general homologous series of B cation-deficient perovskite-related structures with the general formula A_nB_{n − 1}O_3n (n > 3). These phases consist of a framework of mixed cubic and hexagonal stacking of close-packed LnO₃ (and Ti) layers along the z-axis.^90–92 Identical perovskite-like slabs, with a thickness of four LnO₃ layers or three corner-sharing TiO₆ octahedra, are separated by titanium vacant layers and the Bravais lattice is rhombohedral (12R polytypoid) with space group R $3 ¯$ (S.G. = 148). The Ln₂Ti₂O₇ mixed oxides crystallize on two different structures depending on the A to B size mismatch. When r_A/r_B > 1.78 (Ln = La–Nd), these titanates exhibit also a layered perovskite-type of structure belonging to the homologous series A_nB_nO_{3n + 2} (n = 4) and consist of slabs of corner-sharing TiO₆ octahedra layers with 12- and 7-coordinated A cations within the perovskite-like layers.^85,93–96 These layers are separated by oxygen-rich gaps, which are linked by A cations at their boundaries (coordination number is 10), where n is the number of octahedral layers in the perovskite layers. The symmetry is monoclinic and the space group is P2₁ (S.G. = 4). For the remaining lanthanides Ln = Sm–Lu (r_A/r_B ≤ 1.78), the Ln₂Ti₂O₇ materials crystallize in the pyrochlore-type of structure already described here. At least four different Ln₂TiO₅ polymorphs have been described in the literature, depending on temperature, pressure, and the size of the Ln³⁺ ion.^97–103 An orthorhombic α-Ln₂TiO₅ phase [S.G. = Pnma (62)] is the stable form at room temperature for the larger lanthanides (Ln = La-Sm); whereas cubic fluorite and disordered pyrochlore structures [S.G. commonly Fm $3 ¯$m (225), and Fd $3 ¯$m (227)] would be the stable forms for the smaller members of the series (Ln = Er-Lu). An additional polymorph of hexagonal symmetry, β-Ln₂TiO₅ [S.G. = P6₃/mmc (194)], has been described for the intermediate lanthanides (Eu-Ho and Y) as well as a Dy₂TiO₅ monoclinic form, which was obtained when using wet chemistry preparation routes. Accordingly, Ln₂TiO₅ oxides show temperature dependent phase changes on annealing: α → β → F. The Ln³⁺ and Ti⁴⁺ coordination environments in Ln₂TiO₅ compounds are rather complex. Lanthanide ions occupy two sevenfold coordinated sites in α-Ln₂TiO₅; whereas Ti⁴⁺ ions are in fivefold coordination, within a square pyramid polyhedron of oxygen ions. By contrast, Ln³⁺ and Ti⁴⁺ ions are distributed between six-, and fivefold coordinated sites in β-Ln₂TiO₅; whereas the octahedral site is solely occupied by Ln³⁺, and the lower coordinated site features mixed occupancy, with cations (all Ti⁴⁺ and the remaining Ln³⁺ ions), found inside a trigonal bi-pyramid polyhedron of oxygen ions. Finally, and despite the large size difference, both cations share a single common site in F-Ln₂TiO₅. Interestingly, Ln₂TiO₅ and Ln₂Ti₂O₇ show significant miscibility, which will be discussed in Sec. III F. Large differences in ionic radii account for the very limited solubility between TiO₂ and Ln₂O₃. Only very narrow solid solution regions based on both simple oxides exist in the TiO₂–Ln₂O₃ phase diagram.

Like many other crystal structures, different attempts have been made over the years to rationalize the formation and stability of the pyrochlore structure in terms of simple geometrical considerations, such as the A and B size factor and the radius ratio rule (r_A/r_B). One noteworthy effort was carried out by Chakoumakos,⁴¹ who, by analyzing the empirical structural data from compounds known in the early eighties to adopt the pyrochlore structure, developed a relatively simple yet comprehensive model. This model predicts both the unit cell parameter, a, and the fractional coordinate of anions in the 48f-site (x_48f). Based on empirical sets of effective ionic radii, the model yields values in good agreement with experimental determinations, underscoring the importance of cations size ratio in the stability of the pyrochlore structure. Subsequent improvements were achieved with a similar model that incorporates the cation:anion electronegativity difference,¹⁰⁴ particularly for oxides with multiple site occupancy. Additionally, literature includes attempts to introduce a simplified parameter, akin to the tolerance factor proposed by Goldschmidt for perovskite-type oxides.^105–107 However, due to the presence of two crystallographically nonequivalent anions in a single unit cell and the propensity for cation-antisite defects, such calculations for pyrochlore become far more complex. For instance, two different tolerance factors were suggested to measure the likelihood of a specific cation combination (A²⁺, B⁵⁺) adopting the pyrochlore structure.¹⁰⁶ Yet, these factors depend either on the cell size (a) or on the unknown x_48f positional parameter, limiting their predictive ability. The criterion of cation size mismatch still appears to be the most effective choice for this purpose.¹⁰⁸ Therefore, the radius ratio constraint for A₂B₂O₇ lanthanide titanates and zirconates results in the formation of the pyrochlore structure only when 1.46 ≤ r_A/r_B ≤ 1.78. Accordingly, Ln₂Ti₂O₇ pyrochlore phases are stable at ambient conditions only from Sm to Lu (see Table I and Fig. 10). Pyrochlore-type Ln₂Zr₂O₇, on the other hand, would form for lanthanides from La to Gd. Notably, Sm³⁺, Eu³⁺, and Gd³⁺ are the only three lanthanide ions forming pyrochlore-type oxides with both Ti⁴⁺ and Zr⁴⁺.

Beyond the above-mentioned range, the size mismatch favors a structural change to different atomic arrangements. As already mentioned in Sec. III B, a perovskite-related monoclinic layered structure forms for size ratios higher than 1.78 (Ln₂Ti₂O₇, Ln = La–Nd), while an anion-deficient fluorite structure exists when r_A/r_B < 1.46 (Ln₂Zr₂O₇, Ln = Tb–Lu and Y). These stability field limits of these compounds can undergo significant alterations based on processing conditions. For instance, Sm₂Ti₂O₇ and Eu₂Ti₂O₇ adopt the layered perovskite structure instead of the pyrochlore-type, when prepared under high pressure.¹⁰⁹ Similarly, soft chemistry routes typically yield fluorite-type Eu₂Zr₂O₇, Sm₂Zr₂O₇, and Gd₂Zr₂O₇.^110–112 A systematic study of the enthalpy of formation of pyrochlore-type titanates, conducted on single crystals,¹¹³ reveals that all Ln₂Ti₂O₇ (Ln = Sm-Lu) are stable compared to their oxides (see Table I). Generally, the formation enthalpies become more exothermic with an increasing cation size ratio (r_Ln/r_Ti), indicating greater stability of the pyrochlore structure compared to the oxides. Consequently, Lu₂Ti₂O₇ would be the least stable among pyrochlore-type titanates, while the Gd and Sm-containing analogues, with similar formation enthalpies, are the most stable. A similar trend is observed for the Ln₂Zr₂O₇ series,^110,114,115 and pyrochlore stability decreases with the decreasing size of the lanthanide ion. As anticipated, the unit cell constant decreases along both series, exhibiting an approximately linear correlation with the size of the lanthanide ion. However, the relation between the Ln³⁺ ionic radii and the positional parameter x_48f is less straightforward. Some studies suggest an almost linear relationship in titanates, while others propose that this trend is not followed in the case of the larger lanthanides (Sm-Gd). Structural refinement of synchrotron x-ray data for single crystals¹¹⁶ has revealed that Gd₂Ti₂O₇ instead of Sm₂Ti₂O₇, has the lowest x_48f value of all pyrochlore titanates. This deviation is attributed to the Gd³⁺ electronic structure with half-filled f-shell ([Xe]4f⁷), and the higher ionicity of the ⟨Gd-O⟩ bond. In any case, x_48f values for lanthanide zirconates, and titanates vary widely, spanning from ∼0.320 to 0.350. Ti–O_48f bond lengths undergo slight changes along the series, falling within the range known for TiO₆ octahedra in most Ti-containing oxides (∼0.194–0.200 nm). Interestingly, the decrease in the unit cell parameter from Sm₂Ti₂O₇ to Lu₂Ti₂O₇, with decreasing Ln³⁺ size, is compensated by greater trigonal distortion of the TiO₆ octahedra, keeping the Ti–O_48f bond length almost constant.⁴⁶ The Zr–O_48f bond lengths are, on average, slightly larger but also remain almost constant throughout the series. Thus, Zr–O_48f distances vary from ∼0.210 for La₂Zr₂O₇, to ∼0.213 nm for the Gd end-member.^117,118

The evolution of the Ln–O_8b and Ln–O_48f bond lengths along the series is not straightforward, but both decrease with decreasing lanthanide size, more than the ⟨Ti–O_48f⟩ or ⟨Zr–O_48f⟩ bonds. The ⟨Ln–O_8b⟩ bond is among the shortest known for lanthanide oxides (∼0.210–0.230 nm), depending only on the cell parameter (⁠ $a 3 / 8$⁠). In contrast, Ln–O_48f bond-lengths are significantly larger (between ∼0.240 and ∼0.265 nm), and depend on both x_48f and a. Furthermore, ⟨Ln–O_48f⟩ bonds exhibit a more rapid expansion with increasing Ln³⁺ size compared to ⟨Ln–O_8b⟩. Since all cations occupy special positions, metal-metal distances (i.e., Ln–Ln, Ln–Ti, Ti–Ti, and Zr–Zr) are consistently equal to $a 2 / 4$⁠. Similarly, the O_8b–O_8b bond length is fixed by symmetry to $a 3 / 4,$ maintaining the Ln–O_8b–Ln angle at a constant 109°28′ (the tetrahedral angle). In contrast, the O_8b–Ln–O_8b angle is 180°, while the B-O_48f-B angle falls within a narrow range, typically from about 127° to 134°, with little influence from the A-cation. According to electronegativity differences (i.e., 1.54 vs 1.33 for Ti and Zr, respectively, based on the Pauling scale), ⟨Ti-O⟩ bonds in pyrochlore are inherently more covalent than ⟨Zr–O⟩. Although lanthanide electronegativities decrease along the series from 1.10 for La to 1.27 for Lu, it's important to note that the ionic character of the ⟨Ln–O⟩ bond increases with decreasing ionic radii.

While geometric and electrostatic criteria provide valuable insights, they alone may not be sufficient to explain the formation and stability of pyrochlore-type phases. Additional factors such as bonding character, processing conditions, and thermal history play crucial roles in either stabilizing or destabilizing the pyrochlore structure.

Pyrochlore-type titanates and zirconates tend to exhibit increased disorder as the A to B size difference, represented by the r_A/r_B size ratio, decreases. This structural disorder plays a crucial role in determining the physical and chemical properties of multicomponent oxides. The scientific community has invested considerable effort in identifying and understanding the prevailing order-disorder (O-D) mechanisms within the pyrochlore structure. Interestingly, disordering in pyrochlore does not require solutes or impurities, as observed in many other oxide systems; instead, external factors such as temperature, pressure, processing conditions or ion irradiation, can induce disorder.

The enthalpy differences between the ordered pyrochlore structure, and the fully disordered anion-deficient fluorite structure in zirconates [Ln₂Zr₂O₇ vs (Ln_0.5Zr_0.5)O_1.75], are relatively small, for instance, approximately 10 and 38 kJ·mol⁻¹ at 25 °C for Gd₂Zr₂O₇ and Eu₂Zr₂O₇, respectively.^110,119 Moreover, most Ln₂Zr₂O₇ pyrochlores experience a thermally induced O-D transformation to the defect fluorite structure. The transition temperatures decrease rapidly with decreasing Ln ion size (i.e., r_A/r_Zr ≈ 1.46),^{115,120–122} dropping from around 2300 °C for Nd₂Zr₂O₇, to ∼2000 °C for Sm₂Zr₂O₇, and ∼1530 °C for Gd₂Zr₂O₇. However, the kinetics of these O-D processes are sluggish, and most zirconates retain a certain degree of structural disorder, even at room temperature.¹¹⁷ In contrast, disordering of pyrochlore Ln₂Ti₂O₇ titanates has been reported only for the smaller lanthanides, i.e., Ln = Tm–Lu.¹²³ This disordering occurs at temperatures near their melting points (∼1700 °C), with the dominant mechanism controlled by the anion substructure.

Various theoretical models, employing different mathematical approximations have been used to evaluate the energetics of defect formation and disordering in the pyrochlore structure.^{34,35,124–132} While these models may differ in their calculated energy costs, there is a consensus that the most stable point defect predicted for the pyrochlore structure is the cation antisite defect (A and B swap sites around O_48f). The energy needed to create this defect decreases as the A to B size mismatch decreases. Additionally, anion Frenkel pairs, involving a 48f-site anion vacancy plus an oxygen interstitial at the 8a-site (V_48f + O_8a), are predicted to be the next most stable defect. The formation of isolated anion Frenkel pairs would require higher energy, but the presence of cation disorder may lower this energy penalty, increasing the likelihood of coupled cation antisite/Frenkel pairs and the higher the formation energy of these coupled defects, the lower the pyrochlore tendency to disorder.³⁵ Notably, disordering energies for the zirconate series are predicted to be lower compared to their titanate counterparts, though increasing a near linear fashion with increasing Ln³⁺ size. However, debates in the literature persist and some studies have suggested that defect formation energies in the Ln₂Ti₂O₇ series may not be simple functions of the cation size mismatch,^130,133 but exhibit a maximum at Gd₂Ti₂O₇. The electronic structure and bonding character of lanthanides also influences the disordering process, with Gd₂Ti₂O₇, and La₂Zr₂O₇ featuring the highest defect formation energies from among the pyrochlore-type titanates and zirconates, respectively. Finally, within a specific family of pyrochlore oxides (e.g., same B-cation), models agree in considering the O_48f positional parameter as a good metric of the tendency to disorder of the oxygen sublattice, though it is less reliable when comparing trends between different series.

Reconciling theoretical predictions of defect formation energies with experimental data are challenging given the inherent complexity of defect migration in multicomponent oxides.¹³⁴ Different studies employing different experimental probes, have arrived at apparently contradictory conclusions. Much research in this context has focused on chemically induced disordering using A_2-yA′_yB_2-xB′_xO₇ solid solution systems, where increasing the average size of cations at the B-site and/or decreasing that of cations at the A-site, is expected to increase structural disorder.^135–152 Moreover, the progressive substitution of Zr⁴⁺ for Ti⁴⁺ in Ln₂Ti₂O₇ pyrochlore (r_Zr = 0.0720 vs r_Ti = 0.0605 nm, both in sixfold coordination) increases the average B cation size (i.e., r_A/r_B decreases), leading to a higher likelihood of defective structures. Some lanthanides (Tb–Lu and Y), experience a phase transition to fluorite structure with critical Zr-content specific to each Ln₂Ti_2−xZr_xO₇ system. The debate whether O-D processes in the pyrochlore structure are guided by anions, cations, or both simultaneously continues, with differences in emphasis on chemical composition vs size difference. The advent of new and very powerful analytical techniques sensitive to local atomic environments and bonding characteristics at different length scales, have further fueled these discussions. Adding to the complexity, cation and anion disordering appears to occur to varying degrees and is highly sensitive to sample processing conditions. Different research groups using the same experimental probe on identical solid solutions often infer different mechanisms. According to some studies, disordering of the anion and cation sublattices in the Y₂Ti_2−xZr_xO₇ system, proceeds rather independently, with the disordering of the anion sublattice starting at low Zr⁴⁺ levels (x > 0.6) levels, with the filling of the initially empty interstitial 8a-site at the expense of the neighboring 48f-site.¹³⁶ Moreover, disordering at the two nonequivalent oxygen sites, O_48f and O_8b, proceeds at different rates. Oxygens at O_8b participate in the process only at higher Zr concentrations (x > 0.9), almost simultaneously with the onset of disordering of the cation sublattice. Finally, complete mixing of cations and anions in their corresponding sublattices, and the transition to a fluorite structure, occurs when x ≥ 1.8. Other studies proposed a different disordering mechanism,¹⁴⁰ indicating a two-phase region and evolving on increasing Zr addition, from a pyrochlore-type solid solution (0 ≤ x ≤∼1.1) to a fluorite-like region (∼1.4 ≤ x ≤ 2). The disordering mechanism suggested involves mostly the B₂O₆ network (Ti, Zr, O_48f and the 8a-site) with the A₂O′ substructure (Y and O_8b) remaining almost unaffected during the process.

Discrepancies also extend to the Ln_2-yLn′_yZr₂O₇ systems, showing almost unlimited solid solubility, where disorder is expected to increase with decreasing average Ln size. Separate studies of the Nd_2−yY_yZr₂O₇ solid solution confirm the existence of a two-phase region, although with different compositional limits, and a different mechanism for the gradual occupancy of the 8a-site with increasing Y-content: occupancy of the 8a-site may occur via O_48f¹⁴⁷ or O_8b migration.¹⁴⁸ Increasing disorder of the anion sublattice changes the local coordination environment around Zr⁴⁺; whereas, that of Y³⁺ remains essentially as in the pyrochlore structure. A similar disordering mechanism, involving mostly the O_8b ions, were suggested for other Ln_2−yLn′_yZr₂O₇ systems.^139,152 The importance of bonding characteristics in the O–D process is highlighted by experimental data.¹³⁷ Despite of having almost the same cation ionic radius ratios, the Gd₂Ti_2−xSn_xO₇ and Gd₂Ti_2−xZr_xO₇ systems (r_Sn = 0.069 vs r_Zr = 0.072 nm, both in sixfold coordination) show different evolution patterns upon Ti-substitution; whereas, no significant change in the degree of structural disorder is observed on increasing tin content, a high number of defects evolve in the other system, with increasing Zr content. The higher covalency of the ⟨Sn–O⟩ bonds as compared with ⟨Ti–O⟩ and ⟨Zr–O⟩ accounts for such differences.

Unraveling the details of the O-D mechanisms occurring in pyrochlore-type oxides has proven to be an exceptionally complex task. Likewise, interpreting the disordered state across different length scales remains an ongoing challenge. Two primary scenarios leading to disordering and eventually, triggering a phase transformation to the fluorite structure across the long-range can be identified:¹⁵³ (i) preservation of the pyrochlore long-range periodicity with the accumulation of defects (cation antisite/Frenkel pairs) over time, leading to the fluorite structure; or (ii) coexistence of ordered pyrochlore-type domains with antiphase domains where cation antisite/Frenkel pairs predominate. Early experimental evidence generally supported the microdomain concept, portraying disordered pyrochlore materials as typically comprising fluorite-type microdomains embedded in an ordered, pyrochlore-like matrix.^16,154–157 Microdomain dimensions would depend on stoichiometry, thermal history, processing conditions, and the specific lanthanide ion involved. This conceptual model of microdomains aligned with early studies on local atomic structure of tetragonal and cubic lanthanide-stabilized zirconia solid solutions, Ln_xZr_1−xO_2−x/2. These studies postulated different coordination environments around Ln³⁺ ions and the Zr⁴⁺ host, suggesting that charge-compensating vacancies would be preferentially located next to Zr⁴⁺, resulting in cation coordination numbers closer to 7 than 8.^158,159 Ln³⁺ ions were believed to be surrounded by 8 oxygen ions. Moreover, these studies indicated that the distortion of the Zr⁴⁺ network is so severe that the long-range fluorite-like structure implied by the cubic symmetry, is not evident in the local atomic structure. In parallel with the advancement of experimental techniques probing the atomic-scale structure of crystalline materials, there has recently been renewed interest in elucidating the local structure of disordered crystalline materials.¹⁶⁰ This is crucial since the actual local configuration of atoms governs the expression of physical properties,¹⁶¹ such as electron, phonon, and thermal transport.^162–165 

Conventionally, the average, long-range structure (i.e., deduced from diffraction experiments) of disordered pyrochlore and anion-deficient fluorite oxides, was assumed to be comprised of randomized distributions of cations and/or anions (including vacancies) on their respective sublattices. The random assemblage of atoms leads to an averaging of coordination numbers and occupation of identical polyhedral, regardless of atomic species. Recent studies using advanced structural characterization techniques, however, have demonstrated that the short-range structure of these disordered oxides is distinct from the long-range structure. For instance, it was shown, using x-ray absorption^151,153 and neutron scattering¹⁴⁵ data, that the stoichiometric oxygen vacancies in the disordered structures are preferentially coordinated with the B-site cation, given the preference for a higher coordination number for the A-site cation. However, it was not clearly understood how this local order manifests within and is related to the observed long-range structure and why it occurs in the first place.

Much recent work has been dedicated to answering these questions and has centered around neutron total scattering as an experimental approach. This technique is ideal as neutrons are sensitive to oxygen, and total scattering collects information about local deviations from long-range symmetry, which can be analyzed in real space. Using this method, Shamblin et al.¹⁶⁶ have shown that within the anion deficient, disordered fluorite, Ho₂Zr₂O₇, short-range atomic ordering (and associated structural relaxations) necessarily creates a local atomic configuration that is no longer modeled well, by the conventional disordered fluorite (Fm $3 ¯$m) model. Therefore, the randomization of Ho³⁺, Zr⁴⁺, and anion vacancies, which is observed over long length-scales, is not manifested over the sub-nanoscale. Instead, using neutron pair distribution function analysis (PDF), they showed that the configuration is actually well-modeled by a different fluorite superstructure, the orthorhombic, weberite-type (Ccmm) arrangement.^167,168 This atomic configuration is characterized by three unique cation polyhedra: (i) an eight-coordinated site occupied by Ho³⁺, (ii) a six-coordinated site occupied by Zr⁴⁺, and (iii) two seven-coordinated sites randomly occupied by Ho³⁺ and Zr⁴⁺. This cation distribution requires an ordering of anion vacancies. Thus, the disordered pyrochlore structure is actually comprised of two different material length scales: an ordered short-range arrangement modeled by weberite-type domains, and the long-range disordered fluorite structure.¹⁶⁹ This same heterogeneous disorder was also identified in the AB₂O₄ spinel family of oxides,¹⁷⁰ suggesting that this phenomenon is perhaps general to disordered materials.

In a more recent work, Drey et al.¹⁷¹ used a solid solution between an ordered pyrochlore titanate (Ho₂Ti₂O₇) and an anion deficient fluorite zirconate (Ho₂Zr₂O₇), to focus on the disordering process between the two structures. It was found, in agreement with previous studies on compositionally related solid solution series,^136,145 that a transformation between the long-range pyrochlore and the fluorite structure, occurs in Ho₂Ti_2−xZr_xO₇ over a relatively narrow compositional range around x = 1.2. In contrast, any substitution for Ti⁴⁺ with Zr⁴⁺ induces a local weberite-type configuration, which increases linearly across the entire series with x. This suggests that disordering across both length scales is decoupled and proceeds at different rates with respect to composition. The origin of this behavior is not clear but is probably related to distinct mesoscale processes. Indeed, di Biase et al.,¹⁷² also probed this phenomenon in the Gd₂Ti_2−xZr_xO₇ series with x-ray total scattering and demonstrated that disordered pyrochlore exhibits symmetry breaking at shorter length scales; modeling of these very local structures was accomplished with a weberite-type model but at higher values of real space, density functional theory calculations yielded extended clusters about anti-Frenkel pairs. Drey et al.¹⁷³ were able to show that a similar disordering process with respect to length scale, is also present across a non-stoichiometric pyrochlore series Nd_xZr_1−xO_2−0.5x; however, details such as the relative level of order/disorder at the short-range vs long-range differed with the Ho₂Ti_2−xZr_xO₇ series, which was explained by differences in chemical composition and excess oxygen. By using neutron PDF, it was shown that the atomic-scale ordering is in both pyrochlore series better represented by the C222₁ space group, vs the Ccmm space group proposed by Shamblin et al.¹⁶⁶ This space group, which features a splitting of one oxygen site in Ccmm into two oxygen sites, is also the correct symmetry for the ordered weberite-type oxide, Y₃TaO₇.¹⁷⁴ While the weberite-type local domains have been identified in materials for which a long-range disordered, defect fluorite structure is observed, an interesting local phenomenon was identified in nano-domains of La₂Zr₂O₇, a compound which according to the cation's size difference criterion, should exhibit the ordered pyrochlore structure. Simeone et al.¹⁶⁵ used a combination of diffraction and spectroscopic approaches to study polycrystalline La₂Zr₂O₇ samples prepared with different constituent grain sizes. The sample with the smallest grain size was shown to exhibit a defect fluorite pattern with electron microscopy, though all samples were shown to have highly similar local coordination environments with Electron Energy Loss Spectroscopy. Thus, Simeone et al. concluded that in La₂Zr₂O₇, a long-range defect fluorite structure can be observed in materials that consists of a random distribution of pyrochlore nano-domains. This finding showed that the local structure of all A₂B₂O₇ defect fluorites may not always be the weberite-type ordering, but may exhibit different short-range configurations dependent on the cationic radius ratio.

What remained to be understood was a comprehensive explanation of why this distinct heterogeneous ordering is present in disordered complex oxides. If the short-range structure of disordered pyrochlore oxides is to be manipulated, then the underlying chemical rules that govern the phenomena must be known to predict and engineer the desired outcomes. Recently, O'Quinn et al.⁵⁹ have shown that heterogeneous disorder in complex oxides can be explained and predicted simply by applying Pauling's rules.⁴² Pauling's rules were originally formulated based on the chemical bond, a short-range interaction, so it should not be entirely surprising that they can be used to predict the short-range structural details also of disordered materials. For instance, as mentioned before, it was not entirely understood how an ordered pyrochlore structure undergoes the O-D transformation; do cation anti-site defects precede anion sublattice disorder or vice versa? Do the two phenomena occur simultaneously? Analysis with Pauling's first and second rules showed that anion disorder must precede cation disorder, specifically the movement of an O_48f to the nearest vacant 8a site. The motion of the 48f oxygen (Fig. 11) disrupts the aforementioned “8–8–8–6” and “6–8–8–8” cation coordination scheme and creates new seven-coordinated polyhedra which can, in terms of size, be occupied by both the larger A-cation and smaller B-cation.⁵⁹ 

The resulting polyhedral configuration was shown to be nearly indistinguishable from the weberite-type motif consisting of an “8–7–7–6” coordination scheme in each of the compound layers. Once a weberite-type building block is created, the movement of nearby oxygens is severely limited, as certain correlated movement would destroy the newly created building block. Thus, accumulation of weberite-type units must be diluted and maintain a certain critical distance. This would qualitatively be consistent with the very recent findings of di Biase et al.,¹⁷² which find that modeling of disordered pyrochlore compositions in higher values of real space necessitates the presence of extended clusters about anti-Frenkel pairs. Eventually, saturation occurs and triggers a global phase transformation to an observed disordered fluorite structure. Evaluation using a crystallographic approach introduced by Neder et al.,¹⁷⁵ demonstrated that an assemblage of randomly oriented variants of this weberite-type motif will yield, indeed, the observed fluorite diffraction pattern with no observable Bragg reflections associated with the C222₁ symmetry. This finding also may help explaining the pyrochlore-type local ordering identified in La₂Zr₂O₇ by Simeone et al.¹⁶⁵ In the case of this specific compound, the high level of cation size mismatch may preclude the formation of the weberite-type order, but long-range structural disorder still manifests as a long-range defect fluorite structure. The formation of locally ordered building blocks and the atomic arrangement in disordered AB₂O₄ spinel oxides were similarly explained⁵⁹ by straightforward application of Pauling's rules, which highlights the general applicability of this concept to explain local order in disordered oxides.

Much attention has been paid to the heterogeneous nature of disordered materials, but there is evidence that the local structure in even ordered pyrochlore may be more complex than previously believed. For instance, Trump et al.¹⁷⁶ recently studied three pyrochlore compounds: Pr₂Zr₂O₇, La₂Zr₂O₇, Yb₂Ti₂O₇ which all exhibit ideal cation ordering and stoichiometry. It would be assumed, based on known literature of these ordered compounds, that any local atomic displacements away from their ideal pyrochlore lattice positions would be dynamic and driven by temperature (varying such that their time-averaged mean position is on the pyrochlore lattice position). Using neutron PDF and High-Angle Annular Dark Field Scanning Transmission Electron Microscopy (HAADF STEM), they showed that there is perhaps a persistent, static displacement that cannot be modeled with the Fd $3 ¯$m space group, but instead with tetragonal P4₃2₁2. These static displacements mimic the structure of the mineral β-cristobalite, in which the O-tetrahedra cooperatively tilt. However, in pyrochlore, these displacements must create geometric frustration, and not be cooperative, as the observed long-range structure is Fd $3 ¯$m. They conclude that if these static displacements are the natural consequence of the A-to-B cation size mismatch, then they must be present in all pyrochlore oxides. Geometrically frustrated static displacements in ordered pyrochlore may explain why they are not ferroelectric materials but excellent dielectric materials.

Furthermore, while it was shown that the weberite-type structural motif is present in A₂B₂O₇ oxides that are disordered at equilibrium (e.g., by changes in chemical composition), the same structural behavior has been also identified in samples processed by far-from-equilibrium conditions. Swift heavy ion exposure has been utilized in many studies^{166,177–179} as a method of obtaining large volumes of highly disordered materials suitable for characterization with neutron scattering.^180,181 In contrast with Ho₂Zr₂O₇ which is a disordered fluorite at ambient conditions, it was shown that fully ordered Er₂Sn₂O₇ pyrochlore can be transformed into a disordered fluorite when exposed to swift heavy gold ions¹⁷⁹ with the same weberite-type local configuration. Interestingly, Dy₂Sn₂O₇, which was amorphized by the same ion exposure conditions, also exhibits a weberite-type configuration, despite the lack of any long-range coherent structure.¹⁷⁹ The same outcome was confirmed in irradiation studies on other amorphizable pyrochlores Dy₂Ti₂O₇¹⁷⁷ and Er₂Ti₂O₇.¹⁷⁸ Chung et al.¹⁷⁸ and O'Quinn et al.¹⁸² demonstrated that the local weberite-type configuration in swift heavy ion irradiated Er₂Ti₂O₇ pyrochlore, is nearly indistinguishable from the atomic configuration identified in Er₂Ti₂O₇ prepared by mechanochemical synthesis.

While the long- and short-range structure of disordered pyrochlores appear to be decoupled over different length scales, there is additional evidence that their structural response to thermal treatments, and associated energetics, are also decoupled, not just over different length scales, but also over different temperature regimes. An ion-induced amorphous Dy₂Ti₂O₇ sample examined by Chung et al.¹⁷⁷ was shown to anneal back to a long-range ordered pyrochlore structure in a temperature range between 600 and 900 °C. In contrast, the short-range weberite-type structure observed in the irradiated sample was shown to persist up at least 1200 °C. Thus, neutron scattering data reveal the presence of two separate structural processes: (i) an abrupt long-range crystallization event back to a long-range pyrochlore structure and (ii) a sluggish, subtle re-arrangement from a weberite-type to a pyrochlore-type configuration. This finding was corroborated with differential scanning calorimetry data, which revealed that the heat release profile of irradiated Dy₂Ti₂O₇ can be decomposed into a sharp heat release event at ∼850 °C and a broad, continuous heat release beginning ∼800 °C and continuing beyond 1200 °C. A similar multi-scale structural response was observed in Er₂Ti₂O₇ prepared by mechanochemical synthesis¹⁸² with the local weberite-type configuration identified even after heating up to 1200 °C, despite diffraction data showing a long-range fully ordered pyrochlore structure. These complementary measurements demonstrate that the structure-energetics relationship of disordered or amorphized pyrochlore oxides is much more complex than previously thought.

While the previously mentioned disordering mechanisms do not alter the ideal A₂B₂O₇ stoichiometry of the pyrochlore structure, lanthanide titanates and zirconates may exhibit slight deviations from stoichiometry, even in the absence of impurities. Sections III F 1 and III F 2 review the topic of non-stoichiometry in Ln₂Ti₂O₇ and Ln₂Zr₂O₇ pyrochlore oxides, arising either due to the presence of BO₂ (B = Zr, Ti) or Ln₂O₃ excess, or through changes in the typical Ti⁴⁺ or Ln³⁺ oxidation states.

Solution energy calculations¹⁸³ suggest that deviations from stoichiometry are more likely to occur in pyrochlore-type zirconates compared to their titanate counterparts, particularly in the presence of excess BO₂ (B = Ti or Zr) or Ln₂O₃. These calculations also indicate a higher energy penalty for accommodating A- than B-site excesses creating an asymmetric extent of non-stoichiometry around the line compound (Ln₂Zr₂O₇ or Ln₂Ti₂O₇). In cases of BO₂ excess, it is presumed that the non-stoichiometry is compensated by Ln³⁺ cation vacancies, contrasting with fluorite-type solid solutions where B-site excess is charge-balanced by oxygen interstitials. Oxygen interstitials in pyrochlore play a role mainly in phases close to the pyrochlore/fluorite phase boundary, and B-site vacancies are not considered a favorable compensating mechanism. On the other hand, lanthanide excess is expected to be consistently charge-balanced by oxygen vacancies, similar to fluorite structures.

Contrary to these expectations, different studies completed on Ln₂Ti₂O₇ pyrochlore oxides have not revealed significant TiO₂ excess but instead, an excess of Ln₂O₃. Accordingly, pyrochlore-type solid solutions have been obtained in the Ln₂Ti₂O₇–Ln₂TiO₅ subregion (Ln = Ho³⁺–Lu³⁺) of the corresponding Ln₂O₃-TiO₂ phase diagram, after firing at very high temperatures (∼1700 °C) and quenching, mixtures of simple oxides.¹⁸⁴ A general formula of Ln₂(Ti_2−xLn_x)O_7−x/2 (0 ≤ x ≤ 0.67) has been proposed for such stuffed pyrochlore-type solid solutions. Cation disorder increases with rising Ln-content, with an average defect fluorite structure observed by XRD at x = 0.67 (Ln₂TiO₅). For low levels of substitution (x < 0.3), extra lanthanide ions in Ln₂Ti₂O₇ occupy mainly the Ti⁴⁺ site, with minimal disorder on the original Ln³⁺ site. Only at high Ln-content is there mixed occupancy at both sites. Neutron and electron diffraction studies on Ho³⁺ and Yb³⁺ containing samples revealed nanodomains (∼5 nm) with short-range pyrochlore order even in the fluorite-like Ln₂TiO₅.¹⁸⁵ The size of these ordered domains increases as the quench rate decreases, and single crystals of Ho₂TiO₅ grown by the floating zone technique exhibit pyrochlore-like long-range ordering.¹⁸⁶ Tb- and Dy-containing samples required faster cooling rates to stabilize the pyrochlore solid solution while attempts with larger lanthanides (Sm³⁺–Gd³⁺) were unsuccessful, consistent with expectations based on cations size: Ln₂Ti₂O₇ (Sm³⁺–Gd³⁺) should be considered line compounds. A similar explanation for the smaller lanthanides has been also proposed for the TiO₂–Y₂O₃ system.¹⁸⁷ Thus, only a small excess of TiO₂ and Y₂O₃ are possible in the Y₂Ti₂O₇ pyrochlore below 1450 °C; whereas yttria solubility increases significantly above this temperature and the structure of the solid solution gradually transforms into a fluorite-type phase with increasing Y₂O₃.

In the Ln₂O₃–ZrO₂ systems (La³⁺–Gd³⁺), both Ln- and Zr-rich pyrochlore-like solid solutions have been also obtained at high temperature (>1400 °C).^173,188,189 The homogeneity range decreases with increasing lanthanide size, estimated to be less than 2 mol. % for La₂Zr₂O₇, even at 1700 °C.

An alternative method to achieve nonstoichiometric pyrochlore oxides based on Ti and Zr involves altering the oxidation state of either Ln³⁺ or Ti⁴⁺. For instance, an oxygen-rich Ce₂Zr₂O_7+x (0 ≤ x ≤ 0.36) solid solution was prepared at room temperature by exposing the pyrochlore-type parent Ce₂Zr₂O₇ compound to the oxidizing agent sodium hypobromite NaOBr.¹⁹⁰ Excess oxygen ions entered the initially vacant 8a-site, accompanied by additional oxygen ions migrating from the originally fully occupied 8b-site. The limit of the solid solution was expanded through low-temperature thermal oxidation of the parent pyrochlore.¹⁹¹ An isolated compound with a nominal composition of Ce₂Zr₂O_7.97 maintained the characteristic cation ordering of pyrochlore. In this case, oxygen excess was accommodated by populating the 32e interstitial site of the pyrochlore structure and increasing the coordination around Zr⁴⁺ atoms to sevenfold. Less than 5% of all Ce atoms were estimated to remain as Ce³⁺ after oxidation.

Oxygen-deficient pyrochlore titanates, known as reduced pyrochlore, are possible with small Ln ions (Lu³⁺ and Yb³⁺).¹⁹² The degree of reduction varies with the reaction temperature reaching 90% of Ti³⁺ at ≥ 600 °C in Lu₂Ti₂O_6.1. Due to Ti atoms not being bound to O_8b, anion vacancies are preferentially formed on the 48f-site. The 32e-site of the pyrochlore structure also plays a crucial role, becoming partially populated as Ti⁴⁺ reduction progresses; migrating O_48f ions create a suitable fivefold coordination environment for Ti³⁺. This redistribution of oxide ions introduces additional antisite disorder in the cation sublattice. Non-stoichiometric pyrochlore-type compounds, such as Y₂Ti₂O_5.9 and Y₂Ti₂O_6.5, have also been synthesized by reducing the parent Y₂Ti₂O₇ compound using calcium hydride.¹⁹³ 

As previously noted, the extent of structural disorder in pyrochlore is highly influenced by temperature and processing conditions. Hence, this section offers a brief overview of the typical methods employed for synthesizing pyrochlore-type materials, along with specific insights into the features of the resultant reaction products.

Just like many other multicomponent oxides, the pyrochlore-type compounds Ln₂Ti₂O₇ and Ln₂Zr₂O₇ are commonly prepared through conventional solid-state reactions (SSR) at very high temperatures, utilizing high-purity elemental oxides^45,194 or freshly prepared hydroxides^188,195,196 as initial reactants. When oxides (Ln₂O₃ and ZrO₂ or TiO₂) are employed, powders are blended in the appropriate stoichiometric ratio and typically fired in air using an electric furnace. Formation temperatures are lower for Ln₂Ti₂O₇ pyrochlore (typically ranging between 1200 and 1400 °C) than for their zirconate counterparts (>1500 °C). To overcome slow diffusion kinetics in the solid state and obtain single-phase products, the process involves repeated grinding, pressing and long firing cycles lasting several days. Some Ln cations (e.g., Ce³⁺, Pr³⁺, and Tb³⁺) are also stable at higher oxidation states than trivalent (e.g., 4+), making it preferable to carry out the pyrochlore synthesis under inert or slightly reducing atmospheres to prevent the formation of non-stochiometric phases. Similarly, an oxygen-rich atmosphere should be chosen for Eu-containing phases because of its propensity to become divalent, while Sm and Yb, which may also be divalent, rapidly oxidize to trivalent species in air or the presence of H₃O⁺ ions. Since most Ln₂O₃ are highly reactive toward atmospheric H₂O and CO₂, firing before weighing is strongly recommended to ensure accurate stoichiometries. Reactivity is more pronounced for lighter lanthanides (La³⁺–Sm³⁺), and although carbonation is limited to the particle's outer layers, hydration cannot be considered a purely surface effect.¹⁹⁷ Consequently, very high firing temperatures (≥900 °C) are necessary to eliminate any traces of hydroxides, hydroxycarbonates, and hydrated carbonates formed in the process. Ti⁴⁺ and Zr⁴⁺ oxides do not require any pretreatment and are commonly used as anatase or rutile-TiO₂ and m-ZrO₂. Synthesizing pyrochlore at such elevated temperatures yields coarse grains of the thermodynamically most stable form, with disorder primarily determined by intrinsic characteristics such as chemical composition and bonding.

Alternatively, pyrochlore oxides can be obtained at lower temperatures by SSR using freshly prepared, and intimately mixed hydroxides as starting materials. However, challenges arise with precipitation methods due to the limited availability of stable, water soluble Zr- and Ti-containing inorganic salts. Only a few commercially available chemicals, such as ZrO(NO₃)₂·xH₂O, ZrOCl₂·8H₂O and TiCl₄, are suitable. Moreover, the aqueous solution chemistry of Zr⁴⁺ and Ti⁴⁺ is rather complex, with both cations easily hydrolyzing and forming various species in solution, some of which may contain more than a one metal atom and are easily polymerized by condensation reactions; oligomeric cations and colloidal particles are easily formed, which might agglomerate and precipitate from solution. Strict control of experimental parameters such as pH, is crucial to avoid microscopic inhomogeneity and ensure the simultaneous quantitative precipitation of all the desired species. Ln³⁺ are typically added to solution as hydrated metal nitrates or, less frequently, as chlorides, and ammonia or dilute alkalis are common precipitating agents. The resulting precipitates, collected through filtration or centrifugation, are generally XRD-amorphous and consist of mixtures of the corresponding hydroxides. Although direct precipitation of pyrochlore materials has been also observed in some cases,¹⁹⁵ the precipitates are dehydrated, and fired typically between 800, and 900 °C, to promote pyrochlore formation. Additional heat treatments at higher temperatures are often required to obtain fully crystalline, single-phase products. Notably, fluorite-type Ln₂Ti₂O₇ (Ln = Tm³⁺, Yb³⁺ and Lu³⁺) have been prepared after freeze-drying and firing the precursor material obtained by the co-precipitation method.¹²³ However, these fully disordered phases exhibit limited thermal stability (600–800 °C) and are accompanied by large volumes of amorphous phase. Firing the as-prepared materials between 800, and 1100 °C yields pyrochlore oxides with a high degree of structural disorder, with the thermodynamically stable ordered phases obtained at ∼1200 °C. Similarly, fluorite structures are commonly obtained first, when firing zirconate precursors, with a phase transition to the stable pyrochlore structures occurring at increasing temperatures as the size of the Ln ion decreases: ∼1000 °C for Nd₂Zr₂O₇¹⁹⁸ and above 1400 °C for Eu₂Zr₂O₇.¹⁹⁹ Co-precipitation also enables the preparation of Ce₂Zr₂O₇; bulk and single phase Ce₂Zr₂O₇ was obtained by firing a precipitate obtained from a Ce(NO₃)₃ and ZrO(NO₃)₂ containing solution under reducing and oxidizing conditions.²⁰⁰ 

Pyrochlore-type titanates, and zirconates can be successfully prepared using sol-gel methods, which generally involve several fundamental steps:^201,202

• Dissolving the required metal salts in an appropriate solvent.

• Transforming these solutions through hydrolysis reactions into a stable dispersion of colloidal particles (sol) and, thorough condensation, into an interconnected and rigid network (gel).

• Converting the gel into a solid mass through aging.

• Drying and firing the resulting precursor at appropriate temperatures to induce the formation and crystallization of the target materials.

The preferred sources of Ln cations are hydrated metal nitrates, while Ti and Zr are commonly introduced as alkoxides M(OR)₄. The chemistry of the precursor chemicals used for the sol-gel processing of TiO₂, and ZrO₂-based compounds is complex, and was extensively reviewed by Livage et al.²⁰³ Most alkoxides, especially Ti and Zr chemicals, react vigorously with water, and caution is required in handling them in dry environments: when brought into contact with H₂O, they readily tend to form precipitates instead of gels. Clear gels can be challenging to obtain, and the use of chemical additives, such as stabilizing agents, non-aqueous solvents or acidic or basic catalysts, is recommended to control the gelation process, prevent phase separation, and avoid premature precipitation of single Zr⁴⁺ and/or Ti⁴⁺ hydroxides or hydrous oxides.²⁰⁴ The choice of alkyl group (-OR) influences the hydrolysis and condensation rates of metal alkoxides, as well as the characteristics of the final product (particle size, surface area, and morphology). Generally, the rate of hydrolysis decreases with increasing -OR size. Popular alkoxides for the synthesis of pyrochlore-type titanates, and zirconates include Ti[OCH(CH₃)₂]₄ (Ti (IV) isopropoxide, TTIP), and Zr[OCH₂CH₂CH₃]₄ (Zr (IV) propoxide). Numerous experimental parameters must be controlled for good reproducibility of the preparation protocol, as changes on the order of chemicals addition or in the heating schedule can affect the composition, and morphology of the final product. Similar to the co-precipitation method, fluorite-type lanthanide titanates, and zirconates with limited thermal stability are the initial phases to crystallize when firing the amorphous precursors. For instance, a poorly crystalline fluorite-type Er₂Ti₂O₇ is obtained initially when firing the corresponding precursor below 700 °C, gradually transforming into pyrochlore above this temperature.²⁰⁵ 

A widely used sol-gel method for obtaining pyrochlore-type oxides is the Pechini method (liquid-mix).²⁰⁶ This variant relies on the formation of metal complexes with an organic chelating agent, such as citric acid. A polyalcohol, like ethylene glycol, is then added under continuous stirring, and the solution is heated to evaporate the solvent, promoting the formation of cross-linked polymeric networks through a polyesterification reaction. The gel is subsequently dried, charred at ∼500 °C to eliminate the organic fraction, and fired at higher temperatures (750–1000 °C) to produce the target pyrochlore. Numerous examples in the literature showcase the use of this method for pyrochlore synthesis, often with slight modifications such as substituting citric acid with alternative chelating agents like ethylenediaminetetraacetic acid or oxalic acid.^207–210 As-obtained powders are typically XRD-amorphous, requiring firing in air to achieve the final products. Interestingly, the stability of the amorphous precursor phase is higher for Ti-containing materials than for those with Zr, necessitating lower crystallization temperatures (typically ∼800 °C vs ∼500 °C). Lanthanide zirconates including La₂Zr₂O₇, crystallize generally as fluorite-like materials, with the transformation to the pyrochlore structure occurring during firing at temperatures between ∼1000 and ∼1100 °C.

Recently, mechanochemical processing has gained prominence in materials science for the synthesis of various compounds due to its simplicity, solvent-free nature, and ability to provide a sufficient volume of the desired material in an economically viable manner.^211–216 As reactants undergo processing under non-equilibrium conditions, it often results in uncommon and highly defective metastable phases. Moreover, the conversion to the “true” equilibrium phases induced by further processing, such as post-milling annealing, offers the opportunity to isolate intermediate states that are inaccessible through more conventional powder processing techniques. Key factors influencing the dynamics of mechanochemical processing are the molecular mass of the reactants, their hardness, and the enthalpy of the chemical reactions involved:²¹⁷ optimal conditions for high yield mechanochemical reactions, involve large molecular weight, a small difference in hardness, and large reaction enthalpies. Dry milling has been successfully employed for the room-temperature synthesis of pyrochlore-type titanates and zirconates, as well as their solid solutions, using the corresponding elemental oxides (anatase-TiO₂, m-ZrO₂ and Ln₂O₃) as reactants.^{143,182,218–221} The formation of the target materials involves a multi-step process, often starting with the C → B-Ln₂O₃ polymorphic transformation when the lanthanide source is the former. The milling time needed for this phase transition increases as the size of the lanthanide ion decreases. Tetragonal and/or cubic-ZrO₂ are also formed during the initial stages of the milling process, stabilized in situ by Ln³⁺ incorporation. Pyrochlore formation is gradual, with starting oxides and reaction products coexisting in the Zr containing mixtures, while some lanthanide titanates crystallize only after complete amorphization of the reactants. Under the same experimental settings and milling equipment, shorter milling times are required to obtain single-phase titanates than zirconates, aligning with the higher Mohs hardness of ZrO₂ compared to TiO₂ (6–6.5 for anatase and rutile vs 6.5–8.5 for m-, t- and c-ZrO₂). Both titanates and zirconates are obtained as highly disordered phases,^{108,153,178,182,222} and thermally induced transitions to more ordered states in titanates occur through at least two different exothermic steps. These steps occur primarily between 750, and 950 °C,²²² with one associated with an ordering process affecting mostly the oxygen sublattice and the other covering a broader temperature range, related to cation ordering, grain growth, and the release of strain energy. Remarkably, defects induced by milling persist even after firing at very high temperatures, as evidenced by differences in the short-range structures between mechanochemically prepared Er₂Ti₂O₇ powders fired at 1200 °C, and those prepared by SSR.¹⁸² Despite both samples presenting identical long-range structures, their short-range structures are different, with a fraction of weberite-type domains remaining in the mechanochemically prepared Er₂Ti₂O₇.

Irrespective of their r_Ln/r_Zr size ratio, mechanochemically prepared lanthanide zirconates are obtained as fluorite-like materials with the transformation temperature to the thermodynamically stable pyrochlore phase, decreasing from Gd³⁺ to La³⁺: the larger the size mismatch (i.e., the higher the stability of the pyrochlore phase), the lower is the expected O-D transition temperature. Notably, mechanochemically prepared Gd₂Zr₂O₇ maintains the fluorite structure even after annealing at 1200 °C. Similar effects are observed in mechanochemically prepared Ln₂Ti_2−xZr_xO₇ powders (Ln = Gd and Dy),^143,221 where the temperature needed to trigger the transformation decreases with increasing Ti-content. Intermediate and partially ordered phases emerge during annealing, displaying unusual cation distributions, such as Gd³⁺ atoms distributed between the 16c, and 16d positions, and Zr⁴⁺ atoms relegated to the larger and eightfold 16d-site.

Pyrochlore oxides have also been successfully obtained through the combustion method, harnessing the heat generated from highly exothermic side chemical reactions to prepare the ceramic of interest.^223–231 Self-sustained combustion reactions occur when stoichiometric mixtures of corresponding metal precursors (e.g., nitrates), and urea/hydrazine/glycine/citric acid-based fuels are fired between 300 and 500 °C in the presence of air.^232–236 This approach rapidly produces fine particles of the target materials. Due to the intense heat generated, the method results in the formation of the high-temperature, most stable phase. However, the energetics of the reaction can be controlled by altering the fuel or metal precursors to achieve metastable materials. For instance, the use of zirconyl nitrate [ZrO(NO₃)₂] or zirconium nitrate [Zr(NO₃)₄] as metal sources allows the preparation of fluorite- or pyrochlore-type Ln₂Zr₂O₇. The crystallization of the fluorite phase is attributed to the lower heat of combustion of ZrO(NO₃)₂, leading to a cooler flame, and lower synthesis temperature.²²³ The selection of a suitable fuel is also a critical parameter in combustion reactions because it may affect the mechanism and kinetics of the process and thus, the purity, particle size, morphology, and properties of the reaction product.

Hydrothermal and solvothermal methods, conducted under high pressures and mild temperatures, have been also employed for synthesizing pyrochlore titanates and zirconates. For instance, La₂Zr₂O₇ was prepared in a stainless-steel autoclave using lanthanum chloride and zirconium oxychloride as metal sources.²³⁷ Slightly basic conditions, lower reaction temperatures, and shorter reaction times (180 °C/60 h), favored the formation of the fluorite-type phase, while increasing basicity, higher reaction temperature, and longer reaction times (240 °C/72 h), resulted in pyrochlore type La₂Zr₂O₇. Similarly, mesoporous Er₂Ti₂O₇ spherical particles were obtained by firing the amorphous precursor precipitated in an autoclave using TTIP and erbium acetate as metal precursors, and absolute ethanol as the solvent, at 800 °C.²³⁸ 

Finally, large single crystals of Ln₂Ti₂O₇ and Ln₂Zr₂O₇ have been grown using the floating zone technique, starting from polycrystalline samples.^239–244 While obtaining lanthanide zirconate single crystals has historically been challenging due to their high melting points, recent success has been achieved by using Xe arc lamps.^243,244

In contrast, anion vacancies are inherently present in pyrochlore oxides, and oxygen conduction does not require dopant incorporation. Pyrochlore-type zirconates, for example, are intrinsic oxygen ion conductors where ion conduction occurs through sequential jumps of oxygen ions into neighboring vacant sites, akin to the fluorite structure. However, ideal (ordered) stoichiometric pyrochlore oxides are poor ionic conductors because oxygen ions fully occupy the 48f, and 8b positions, and anion vacancies are ordered at the 8a-site. Migration becomes possible only when these positions are partially filled, leading to disordering of the anion sublattice. Accordingly, highly ordered Ln₂Ti₂O₇ pyrochlore oxides have lower ionic conductivity than more disordered Ln₂Zr₂O₇. However, migration activation energies are higher in zirconates indicating that conductivity is controlled by the formation energy of structural defects facilitating ion hopping. Computer simulations have pointed to the important role of cation disordering on oxygen diffusion.^{124,127,245–250} A defect structure consisting of “split vacancies,” a 48f vacancy pair in the ⟨110⟩ direction, with an interstitial ion in between, has been proposed for highly conductive pyrochlore oxides.¹²⁴ This configuration becomes more favored than a single V_48f with increasing disorder (i.e., when r_A/r_B approaches 1.46), offering greater geometric flexibility, and longer effective jump distances.²⁴⁷ Oxygen diffusion would take place basically, by sequential jumps, either between (i) nearest neighbor sites (along ⟨100⟩) and/or (ii) next nearest neighbor sites (along the ⟨110⟩ direction), involving the “split vacancy” structure. In contrast, highly ordered pyrochlore structures with no “split vacancy” configuration, exhibit a simpler oxygen migration process consisting of a 48f → 48f hopping mechanism, which depends mainly on the B-O_48f bond strength.²⁵⁰ Theoretical models also suggest the significant role of O_8b in vacancy migration, providing low-energy mechanisms for diffusion in highly ordered pyrochlore. Oxygen ions that move to the initially empty 8a site with increasing structural disorder do not significantly contribute to overall conductivity. Calculations further indicate that the activation energies for migration in pyrochlore oxides are mostly B-site dependent, even though O_48f has two B⁴⁺ and two A³⁺ nearest neighbors. Additionally, intermediate-sized lanthanides (Sm³⁺ to Er³⁺) at the A-site are expected to reduce the E_a compared to larger lanthanide ions (La³⁺ to Nd³⁺).

Interest in the electrical properties of pyrochlore-type titanates and zirconates emerged following initial reports of high ionic conductivity at elevated temperatures in some Ln₂Zr₂O₇ materials.^16,251–253 For instance, Gd₂Zr₂O₇ exhibits conductivity values close to those characteristics of YSZ (σ ≈ 10⁻² S·cm⁻¹ at 800 °C) maintaining purely ionic conductivity even at very high temperatures. Studies conducted in Ln₂O₃–ZrO₂ systems, where no pyrochlore phase is stable (Ln = Tb–Lu), reveal an ion conductivity maximum for relatively low dopant levels in the defect-fluorite type phase region. In contrast, systems with a stable and disordered pyrochlore-type phase (Ln = Sm–Gd), show an additional maximum in conductivity at the pyrochlore stoichiometry.^251,254 Moreover, conductivity in the Gd₂O₃–ZrO₂ system increases by order of magnitude with the fluorite-to-pyrochlore phase transition. This enhanced conductivity is attributed to lower activation energies in pyrochlore and the existence of a preferential and connected pathway for oxygen-ion migration through the 48f vacancy hopping mechanism. For instance, E_a dropped from ∼1.2 eV for the fluorite type Gd₂Zr₂O₇ to ∼0.85 eV for the disordered pyrochlore phase. Studies also confirm that oxygen ion conductivity in Ln₂Zr₂O₇ (Ln = Yb, Y, Gd, Eu, Sm, Nd, La), is strongly influenced by the size of the Ln³⁺ ion and the r_Ln/r_Zr ratio, exhibiting a sharp maximum close to the boundary between the pyrochlore, and fluorite-type phases.²⁵⁵ Different mechanisms explain lower conductivities on either side of the maxima. Decreased unit-cell free volume (difference between the total cell volume, and the sum of volumes occupied by individual ions in the unit cell) accounts for conductivity losses in fluorite-type phases (Ln = Yb, Y, Gd). In contrast, diminishing conductivity in the pyrochlore side (Ln = Eu, Sm, Nd, La) may be explained by a decreasing number of mobile charge carriers as the Ln³⁺ size increases (increasing ordering of oxygen vacancies). Similar effects of decreasing E_a, and increasing conductivity with higher structural ordering, were observed in Ln_2-yLn′_yZr₂O₇ solid solutions.^256–263 Moreover, increasingly ordered pyrochlore zirconates obtained by replacing Gd³⁺ in Gd₂Zr₂O₇ with a larger lanthanide (e.g., La³⁺, Nd³⁺, or Sm³⁺) show enhanced conductivity. In contrast, anion-deficient fluorites obtained by replacing Gd³⁺ with smaller Ln³⁺ ions exhibit lower conductivity than the pristine Gd₂Zr₂O₇ phase. Stronger disorder leads to increased correlations (ion–ion interactions) between oxygen ion jumps and higher E_a, with partially disordered materials yielding higher conductivity. High oxygen ion mobility is also found in the Gd_2-yLa_yZr₂O₇ solid solution.^262,263 Highly resistive La₂Zr₂O₇ layers frequently form at the interface between YSZ and LaCoO₃ or LaMnO₃-based electrode materials in-state-of-the-art solid oxide fuel cell (SOFC) technology. Both La₂Zr₂O₇ and SrZrO₃ appear during co-sintering, and prolonged exposure to high temperatures during operation, adversely affecting long-term cell performance. Interestingly, pyrochlore-type Gd₂Zr₂O₇ exhibits high tolerance to La³⁺ substitution without any significant reduction in oxide ion mobility at intermediate or high temperatures.²⁶² Bulk conductivity values reach σ ∼ 5·10⁻³ S·cm⁻¹ at 800 °C when La³⁺ contents are 0 ≤ y ≤ 0.8, despite increasing ordering, with the E_a dropping from ∼1.1 eV in Gd₂Zr₂O₇ to ∼0.85 eV in the Gd_1.2La_0.8Zr₂O₇. Conductivity is also lower when La³⁺ replaces Gd³⁺ in fluorite-type Gd₂Zr₂O₇, as compared with pyrochlore.²⁶³ 

Attempts to enhance the conductivity in zirconate pyrochlore by acceptor doping (e.g., Ca²⁺ substitution for Gd³⁺ and/or Sc³⁺ for Zr⁴⁺ on Gd₂Zr₂O₇²⁶⁴) have generally been unsuccessful. The small number of additional anion vacancies created by extrinsic Gd₂Zr₂O₇ doping does not have any significant impact on oxygen transport properties.

Ideal Ln₂Ti₂O₇ pyrochlore are, in general, poor ionic conductors. Only those containing the heavier lanthanides (Ln = Tm³⁺, Yb³⁺ and Lu³⁺) show significant oxide ion mobility following annealing at very high temperatures (≥1600 °C).^265–267 For instance, Yb₂Ti₂O₇ shows σ values of ∼10⁻² S·cm⁻¹ at 740 °C with E_a ≈ 0.90 eV. The enhanced oxygen transport properties in these compounds result from an O-D transformation occurring at extremely high temperatures and its effect on the anion sublattice. One of the compelling attributes of pyrochlore-type titanates is their ability to support ionic and electronic charge carriers, essential for mixed ionic-electronic conduction. This characteristic opens avenues for constructing monolithic solid-state electrochemical cells, based on the pyrochlore structure, offering significant advantages in mechanical, thermal, and chemical stability compared to more heterogeneous devices.²⁶⁸ Consequently, extensive efforts have been dedicated to studying the impact of different dopants on the electrical properties of lanthanide titanates. Notably, there is an important opportunity to modulate the electrical properties of Gd₂Ti₂O₇-based solid solutions, ranging from those typical of solid electrolytes, and mixed conductors with high levels of electronic and ionic conduction.^269–276 A substantial increase in ionic conductivity has been reported in acceptor-doped Gd₂Ti₂O₇, whether in the Gd³⁺ or Ti⁴⁺ sites.^272,273 Similarly, to fluorite-type oxides, the best results were obtained with Ca²⁺ as the A-site dopant due to its close size to Gd³⁺. Conductivity increased by more than two orders of magnitude on doping, with a shallow maximum (∼5 × 10⁻² S·cm⁻¹ at 1000 °C) achieved in 8–10 mole% Ca-doped Gd₂Ti₂O₇. The incorporation of Ca led to a significant reduction in E_a, transforming the material into an extrinsic oxygen ion conductor with E_a decreasing from 0.94 eV in the un-doped material, to 0.63 eV in the 5 mol. % Ca-containing titanate. However, higher doping levels increased the overall activation energy for migration and had a detrimental effect on oxygen transport. In contrast to pyrochlore-type zirconates, these highly conductive titanates show negligible cation disordering due to the large size difference between Ca²⁺/Gd³⁺, and Ti⁴⁺ (r_Ca = 0.112 nm, r_Gd = 0.1053 nm in eightfold coordination vs r_Ti = 0.0605 nm, sixfold environment). Solubility limits are reduced to about 1 mole% when replacing Ti⁴⁺ with Al³⁺ (r_Al = 0.0535 nm, sixfold coordination). Measurements under different oxygen partial pressures confirmed a dominant ionic contribution to conductivity in both Ca-, and Al-doped Gd₂Ti₂O₇. High ionic conductivity has been also observed in acceptor-doped Ln₂Ti₂O₇ (Ln = Dy, Ho and Yb):^277,278 σ ∼ 0.2 S·cm⁻¹ for Yb_1.8Ca_0.2Ti₂O₇ at 1000 °C with an E_a ∼0.70 eV. Similar to Ca-doped Gd₂Ti₂O₇, neutron diffraction data showed no significant cation disordering in these systems on doping.

Electronic conduction in lanthanide titanates can be significantly enhanced by partially replacing Ti⁴⁺ with polyvalent cations such as Ru, Mo or Mn.^{269,270,279,280} Various Gd₂Ti₂O₇-based pyrochlore phases have demonstrated high levels of electronic conduction while retaining acceptable values of ionic conductivity under oxidizing and/or reducing conditions in an electrochemical cell. Highly conductive pyrochlore phases have been identified in the Gd₂Ti₂O₇–Gd₂Mo₂O₇ system. Molybdenum's ability to change oxidation states and larger ionic radii as compared with Ti⁴⁺ (r_Mo = 0.065 nm, sixfold coordination) increases disorder and the number of mobile charge carriers, leading to enhanced ionic conductivity. Furthermore, Gd₂Mo₂O₇ pyrochlore is an excellent electronic conductor even at room temperature. The solid solution's total conductivity increases by over two orders of magnitude with increasing Mo content, with Gd₂TiMoO₇ showing a total conductivity of ∼25 S·cm⁻¹ at 1000 °C, with an ionic contribution comparable to that of Ca-doped Gd₂Ti₂O₇. Some of these materials have exhibited remarkable tolerance to sulfur-containing fuels while maintaining acceptable cell performance.²⁷⁵ 

High oxide ion conductivity has also been reported in Ln₂Ti_2-xZr_xO₇ solid solutions, with a remarkable increase in over two orders of magnitude observed in the Gd₂Ti_2−xZr_xO₇ system for x ≥ 0.8.^135,137,281 The E_a remains almost composition-independent at approximately 0.8 eV. This enhanced conductivity is attributed to the development of higher structural disorder with increasing Zr-content, rather than improved vacancy mobility to lower E_a. It is noteworthy, however, that conductivity does not exhibit a steady increase across all compositions. Similar to the Ln_2−yLn′_yZr₂O₇ solid solutions, conductivity maxima at intermediate temperatures are typically observed in partially disordered materials (e.g., Gd₂Ti_0.4Zr_1.6O₇ or Dy₂Ti_0.9Zr_1.1O₇), rather than the most disordered phases (e.g., Gd₂Zr₂O₇ or Dy₂Zr₂O₇). These findings further support the idea that conductivity is linked to an optimum degree of disorder. Additionally, acceptor doping can further enhance as exemplified by the monotonic increase in σ in Gd₂Ti_1.4Zr_0.6O₇ when Ca²⁺ substitutes for Gd³⁺ up to 2 mole%.²⁸² Furthermore, high oxide ion conductivity has been found in solid solutions of Ln_2+xTi_2−xO_7−x/2 [or Ln₂(Ti_2−xLn_x)O_7−x/2] (Ln = Dy–Lu).^283–289 Generally, the highest conductivity values are obtained for the stoichiometric or nearly stoichiometric pyrochlore. Nevertheless, a recent investigation in the Tm₂(Ti_2−xTm_x)O_7−x/2 system pinpointed the conductivity peak at x = 0.268, within a region of coexistence of two pyrochlore-type phases (0.134 ≤ x ≤ 0.563), one likely enriched in Tm and the other Tm-deficient.²⁸⁹ 

Remarkably, in a humid environment, notable proton conduction is observed in some pyrochlore-type zirconates and titanates, particularly in materials based on La₂Zr₂O₇.^290–296 Contrary to expectations, protons do not occupy the initially empty 8a-site; instead, they replace additional vacancies generated by acceptor doping in the 48f and 8b positions. Consequently, La₂Zr₂O₇ has limited capability to accommodate a substantial number of protons, whereas its Ca-doped derivative emerges as promising proton-conducting material. The conductivity has been found to depend on both the level of Ca²⁺ doping and the doping site (La³⁺ or Zr⁴⁺), with the highest value measured (∼6.8 × 10⁻² Sm·cm⁻¹ at 600 °C; E_a = 0.68 eV) achieved in (La_1.95Ca_0.05)Zr₂O_7-δ.²⁹² Migration paths for proton are different from those of mobile oxygen vacancies, and density functional calculations have pinpointed²⁹⁶ two preferential sites within the pyrochlore structure for proton diffusion: the 96g (x,x,y), and 192i (x,y,z) Wyckoff sites.

Lanthanide titanates Ln₂Ti₂O₇, represent the most extensive class of geometrically frustrated materials studied to date. A review paper published in 2010 by Gardner et al.⁷ presents an extensive survey on magnetic pyrochlores including the Ln₂Ti₂O₇ family. Consequently, this section aims to provide an overview of subsequent studies on the magnetic properties of Ln₂Ti₂O₇ and Ln₂Zr₂O₇.

In general, the magnetic ground states of Ln₂Ti₂O₇ and Ln₂Zr₂O₇ compounds, are determined by several factors: (i) the primary determinant is the crystal electric field (CEF) of the Ln³⁺ cations, influencing magnetic anisotropy; (ii) the second factor is the nearest neighbor exchange interaction; (iii) the third factor is the nearest neighbor ferromagnetic dipolar interaction, particularly significant for Ln³⁺ ions with substantial magnetic moments, such as Dy³⁺ and Ho³⁺; and (iv) various other factors with more subtle effects, including disorder and dipole-octupole coupling.

As outlined below, the studied members of the Ln₂Ti₂O₇ and Ln₂Zr₂O₇ family are classified into three groups based on their magnetic ground states: spin ice, quantum spin ice, and long-range ordering.

The ground states for Dy³⁺ and Ho³⁺ ions are Kramers and non-Kramers doublets, characterized by dominant terms such as |⁶H_15/2, ±15/2⟩ and |⁵I₈, ±8⟩, respectively.^297,298 Both cations exhibit strong Ising-like anisotropy, with magnetic moments aligned along the local ⟨111⟩ axis (z-axis).

The ground state of DTO is a spin ice state, representing a short-range ordered spin configuration for each magnetic tetrahedron, with two spins pointing in and two pointing out (2-in-2-out) along the ⟨111⟩ axis [Fig. 12(a)].²⁹⁹ Experimental evidence confirms that DTO hosts the same zero-point entropy as the water-ice phase transition,¹⁰ as originally predicted by L. Pauling.³⁰⁰ Upon applying a magnetic field along the ⟨111⟩ axis, DTO transitions into a Kagome spin ice state^301,302 [Fig. 12(b)] and subsequently into by a polarized state³⁰³ [Fig. 12(c)]. Additionally, when a spin between two adjacent tetrahedra flips due to local defects, thermal excitation, and/or small magnetic fields, the ice rule is violated. Previous research suggests that this violation can be approximated as the formation of a pair of monopoles^304–308 of opposite sign [Fig. 12(d)], which are deconfined and may separate, and move independently. HTO shares the same spin ice ground state as DTO and a comparable magnetic phase diagram³⁰⁹ [Fig. 12(f)].

The diffuse magnetic scattering pattern of HTO reveals distinct pinch point features⁹ [Fig. 13(a)], which have been successfully explained by a dipolar spin ice model, emphasizing the significance of ferromagnetic dipolar interactions.³¹⁰ A study conducted by Chandra et al. has revealed that a partial substitution of Ti by Ge, increases chemical pressure and weakens ferromagnetic interactions in Ho₂Ti₂O₇; neither magnetic order nor spin glass-like behavior occur in the Ho₂(Ti_1-xGe)₂O₇ system on cooling down to 2 K.³¹¹ 

Studies on spin ice over the last decade have primarily centered on monopole dynamics. AC susceptibility measurements by Quilliam et al., reveal an activation energy of about 6J_eff (effective exchange interaction) for low-temperature dynamics in HTO,³¹² while AC results from Yaraskavitch et al. suggest a 9 J_eff activation energy for DTO.³¹³ Giblin et al. reported the creation of a relaxing magnetic monopole current that lasts for several minutes in DTO.³¹⁴ Bovo et al. suggested that the monopole diffusion in DTO follows a Brownian motion.³¹⁵ Revell et al. found that extra spins inserted in DTO by stuffing Dy³⁺ on Ti⁴⁺ sites, trap magnetic monopole excitations.³¹⁶ Dusad et al. reported measurements of the frequency and temperature dependence of magnetic flux noise for magnetic monopole plasma.³¹⁷ Ruminy et al. proposed a phonon mediated spin-flipping mechanism in DTO and HTO.³¹⁸ Stöter et al. reported a decay constant of the order of 10⁴ s for the nonequilibrium monopole dynamics in DTO.³¹⁹ Wang et al.³²⁰ grew ultrapure HTO crystals and reported that their magnetic relaxation at low temperatures is associated with monopole motion through the spin-ice vacuum, while at elevated temperatures, relaxation occurs through the reorientation of increasingly spin-like monopolar bound states. Multiple thermal conductivity measurements from different groups consistently indicate that the magnetic thermal conductivity of DTO^321–325 and HTO³²⁶ is linked to monopole excitations.

Another focal point is the field effect, mainly with an applied field along the ⟨111⟩ axis to study the nature of the phase transition from spin ice to Kagome spin ice. Slobinsky et al.³²⁷ reported that the equilibrium magnetization curve of DTO could not be achieved below 600 mK. The AC susceptibility reported on DTO by Matthews et al.³²⁸ revealed a precursor at lower field before entering the Kagome spin ice phase. Jackson et al.³²⁹ concluded that the avalanches in spin ice are quite slow. Grams et al.³³⁰ reported the magnetoelectric character of the monopole excitations. Kao et al.³³¹ performed numerical studies, finding a first-order transition from a Kagome ice to a q = X sate when the field is tilted toward the ⟨11–2⟩ axis. Opherden et al.³³² reported that the magnetization of HTO exceeds the angle-dependent saturation value of the Ising limit, up to 120 T. Edberg et al.³³³ predicted the induction of ordering under uniaxial pressure, but their experimental results on HTO did not confirm such ordering under pressure. Meanwhile, studies on series of Dy(Ho)₂B₂O₇ (B = Sn, Ti, and Ge) samples by Zhou et al. indicate that increasing chemical pressure drives the system toward the antiferromagnetic phase boundary from the spin ice region¹⁴ and contributes to the stabilization of a larger fraction of monopole dimers.³³⁴ 

Interestingly, there are a few studies that challenge the notion of a spin ice ground state. The magnetization dynamics of DTO and HTO, measured using toroidal, boundary-free magnetization transport techniques, show the characteristics of a glass-forming spin liquid.^335,336 Specific heat measurements by Pomaranski et al.³³⁷ suggest an entropy recovery, or the absence of Pauling's residual entropy in DTO over extended timescales. However, this claim has recently been disputed by Giblin et al.³³⁸ with an alternative set of specific heat data. A MuSR study by Dunsiger et al.³³⁹ shows temperature-independent spin fluctuations in DTO, suggesting magnetic excitations without monopole signatures. Recently, Quémerais et al.³⁴⁰ proposed that this feature results from the quantum diffusion of muons in the material.

In quantum spin ice (QSI) materials, quantum-mechanical tunneling between different ice configurations (Fig. 14) has the potential to transform the magnetostatics of spin ice into a quantum spin liquid (QSL), realizing a fully dynamical lattice analog of quantum electromagnetism.^341–344 Several pyrochlores have been studied as QSI.

The ground state of Ce³⁺ ions in CZO is a Kramers doublet with m_j = ±3/2 and exhibits dipole-octupole character. The calculated g tensor indicates Ising-like anisotropy, with a parallel component (along the ⟨111⟩ axis) g_z = 2.57 and a perpendicular component g_⊥ = 0.³⁴⁵ The reported specific heat and AC susceptibility data show no ordering down to 50 mK. The low-energy inelastic neutron scattering (INS) spectrum reported by Gaudet et al.,³⁴⁵ resembles that of a dipolar spin ice, but the lack of intensity around Q = (000) and the ring of diffuse scattering peaks along (00 L), provide evidence for a QSI state [Fig. 13(b)]. In fact, the long-range dipole interaction in CZO is expected to be ∼64 times weaker than that in dipolar spin ices DTO and HTO, suggesting that the spin ice correlations in CZO, originate from quantum effects. Theoretical studies propose that the dipole-octupole exchange Hamiltonian on the pyrochlore lattice can be reduced to an XYZ model, allowing for the emergence of multiple phases, including QSI.^346,347 Another study by Gao et al.³⁴⁸ suggests that CZO may be a non-spin-ice pyrochlore QSL due to the absence of Pauling entropy and spin freezing.

The ground state of Pr³⁺ ions in PZO is a non-Kramers doublet with a large mixing of |³H₄, ±4⟩ and |³H₄, ±1⟩ terms and a small mixing with |¹G₄⟩ and |³H₅⟩ terms.³⁴⁹ The calculated g tensor is g_z = 5.5 and g_⊥ = 0, indicating Ising-like anisotropy.²⁴⁴ 

The diffuse neutron scattering spectrum reported by Kimura et al.,³⁵⁰ shows sharp pinch point features near (111) and (002), akin to those in spin ice. However, more than 90% of the neutron scattering is inelastic and lacks pinch points, in contrast to classical spin ice, indicating a QSI state [Fig. 13(c)]. The negative Curie temperature suggests that the exchange Hamiltonian, including antiferromagnetic transverse terms, induces the quantum dynamics, as proposed by various theoretical studies.^351–353 Simultaneously, Petit et al.²⁴⁴ have suggested that the ground state is an antiferroquadrupolar liquid with spin ice-like excitations.

Recent attention has been drawn to disorder effects in PZO. INS measurements by Wen et al.,³⁵⁴ reveal a continuum of excitations, indicating a continuous distribution of quenched transverse fields acting on the non-Kramers doublet. This disorder-induced QSL scenario is also supported by a theoretical study by Savary and Balents.³⁵⁵ Diffuse scattering measurements by Martin et al.,³⁵⁶ provide evidence for the presence of random strain, indicating a short-range antiferroquadrupolar correlations picture.

The ground state of Sm³⁺ ions in SZO is a dipolar-octupolar Kramers doublet dominated by a |⁶H_5/2, ±3/2⟩ term. The calculated g tensors are g_z = 0.5 and g_⊥ = 0, supporting Ising-like anisotropy.³⁵⁷ 

Muon spin relaxation (MuSR) and specific heat measurements exclude the possibility of long-range ordering but unveil dynamic spin fluctuations.³⁵⁷ Therefore, SZO could potentially represent another QSI state due to dipolar-octupolar doublet, similar to CZO. However, further clarification is required to ascertain whether this disordered state in SZO is attributed to site disorder.

The ground state of Tb³⁺ ions in TTO is a non-Kramers doublet, separated from the first excitation field by approximately 1.5 meV. The calculated g tensors are g_z = 10.7 and g_⊥ = 0, indicating Ising-like anisotropy.³⁵⁸ The precise nature of this ground doublet, whether it is predominantly composed of the |⁷F₆, ±4⟩ or |⁷F₆, ±5⟩ components, remains a subject of debate.^358,359

Studies indicate the absence of ordering in TTO down to the lowest measured temperature. Diffuse neutron scattering data reveals pinch-point scattering patterns,^360,361 indicating a QSI state [Fig. 13(d)]. Various explanations for this non-ordered state have been proposed, including (i) a theory based on Heisenberg exchange and the admixing of the first excited level by virtual crystal field excitations supporting a QSI state;³⁶² (ii) a study on the Tb_2+xTi_2−xO_7+y system suggests a U(1) QSL state for nominal TTO;³⁶³ (iii) an alternative theory based on a splitting of the ground doublet into two singlets by a Jahn-Teller distortion, combining the anisotropic exchange interaction, explains the spin liquid state;^364–370 (iv) the observed strong magnetoelastic excitations suggests a “magnetoelastic spin liquid”;^371,372 (v) the proposal of a magnetic moment fragmentation.³⁷³ 

Another focus of research is on field effects with two motivations. The first is the search for a magnetization plateau since the theory predicts that for a QSI, a magnetization plateau should appear when a field is applied along the ⟨111⟩ axis.³⁷⁴ While most studies show no existence of such a magnetization plateau,^375–378 ultra-low temperature AC susceptibility measurements by Yin et al.³⁷⁹ reveal a weak magnetization plateau below 50 mK and suggest the existence of a quantum Kagome spin ice state at low fields [Fig. 12(g)]. The second motivation is to study field-induced spin structure with the field along the ⟨110⟩ axis.^380,381 A double-layer monopolar order has been proposed for this field-induced order.³⁸² Recently, this order has been suggested to correlate with multiferroicity.³⁸³ 

Several studies on Tb_2+xTi_2−xO_7+y indicate that a small percentage of stuffed Tb on Ti sites induces an ordered state.^384–387 A thermal conductivity study by Li et al.³⁸⁸ shows a phonon-glass behavior in TTO, while thermal Hall measurements by Hirschberger et al.,³⁸⁹ suggest the existence of neutral spin excitation related to QSI. Jin et al.³⁹⁰ reported nonlinear phonon splitting under magnetic fields and anomalous crystal field excitations, which were attributed to the electric dipoles induced by magnetic monopoles.

The ground state of ⁴I_9/2 Nd³⁺ ions in NZO is a dipolar-octupolar doublet exhibiting a significant mixing of |⁴I_9/2, ±9/2⟩ with |⁴I_9/2, m_J ≠ ±9/2⟩ terms, and a minor mixing with 4/I_11/2. The calculated g tensor is g_z = 5.3 and g_⊥ = 0 indicating an Ising anisotropy.³⁹¹ NZO orders at T_N = 0.4 K with an “all in all out” spin structure [Fig. 15(a)]. The refined magnetic moment varies between 1.26 μ_B³⁹¹ and 0.8 μ_B,³⁹² significantly reduced from the estimated ground state moment of 2.65 μ_B. The MuSR relaxation measurements show slow spin dynamics at low temperatures,³⁹³ suggesting ordering coexisting with strong spin fluctuations. Various scenarios have been proposed to explain this ordering: (i) INS data by Petit et al.³⁹⁴ in 2016, shows the characteristic pinch-point features of a Coulombic spin liquid occurring alongside the Bragg peaks of the ordered state [Fig. 13(e)]. This observation strongly supports magnetic moment fragmentation. The dipolar-octupolar coupling, as seen in QSI in CZO, was used to explain the quantum dynamics of this fragmentation. Xu et al.³⁹⁵ recently proposed a Higgs mechanism to explain the ordering process; (ii) Benton explained the quantum origins of this fragmentation within the framework of a quantum model of nearest-neighbor exchange;³⁹⁶ (iii) Guruciaga et al.³⁹⁷ proposed a field-tuned order by disorder process.

Several other studies focus on field effects. With field along ⟨111⟩ axis, magnetic domain inversion^398,399 and a Kagome spin ice state have been observed.⁴⁰⁰ A “two-in-two-out” disordered state could be induced with field along the ⟨110⟩ axis.⁴⁰¹ 

The ground state of STO is mainly of |⁶H_5/2, ±3/2⟩, calculated from two reported INS studies.^402,403 The calculated g-tensor gives g_z = 0.857, g_⊥ = 0, corresponds to Ising-like anisotropy. Specific heat data confirms a phase transition around 0.35 K. Single crystal neutron diffraction data confirms a “two-in-two-out” spin structure in the ordered phase.³⁸⁹ The refined moment varies between 0.44 μ_B⁴⁰² and 0.17 μ_B.⁴⁰³ MuSR spin relaxation data shows persistent spin dynamics without oscillation below T_N.⁴⁰² Two possibilities have been raised to explain these results: (i) an octupolar ordering occurring below T_N; and (ii) a dipolar-octupolar coupling leading to the magnetic-moment fragmentation, in a similar fashion as observed in NZO.

GTO is an example of a Heisenberg spin since the S = 7/2 Gd³⁺ ion has zero orbital momentum with its half-filled 4f shell. Two magnetic transitions occur at T₁ = 1.1 K and T₂ = 0.7 K.⁴⁰⁴ The nature of its spin structure has been a historic puzzle whether it is a “1 − k structure” involving a single propagation vector k = [1/2,1/2,1/2]⁴⁰⁵ or a “4 − k structure” involving a superposition of the four symmetry-equivalent k⁴⁰⁴ [Figs. 15(b) and 15(c)]. The most recent theory proposed a spin fluctuation driven 4-k structure.⁴⁰⁶ However, recent single crystal and powder neutron diffraction data reveal the population of non-cubic magnetic domains, rhombohedral strains, and diffuse scattering, all supporting the 1 − k structure.⁴⁰⁷ 

Several other recent studies on GTO focus on its magnetic phase diagrams, such as the [100] phase diagram constructed by longitudinal and transverse magnetization data⁴⁰⁸ [Fig. 16(a)] and the [110] [111] phase diagrams constructed using thermal conductivity data.⁴⁰⁹ 

The specific heat⁴¹⁰ of polycrystalline GZO shows a sharp peak at 0.77 K, indicating a phase transition. Meanwhile, AC susceptibility and specific heat measurements in single crystal GZO show only a broad peak around 0.7 K.³⁵⁷ The difference could be due to the different levels of disorder in powder and single crystal samples.

The ground state of ETO is Kramers doublet consisting of a mixture of m_J = |±13/2⟩, |±7/2⟩, |±1/2⟩, |±5/2⟩, and |±11/2⟩, with anisotropic g tensor as g_z = 3.9 and g_⊥ = 6.3.⁴¹¹ Therefore, ETO is a model XY antiferromagnet with local (111) plane as the easy plane. It orders below T_N = 1.2 K with a ψ₂ spin structure,^412,413 where magnetic moments align along the ⟨211⟩ axes [Fig. 15(d)].

Recent studies focus on two aspects. The first involves understanding the origin of this ψ₂ ordering. The extraction of the Hamiltonian from INS data under polarized field,⁴¹⁴ the observation of a spin gap around 0.053 meV,⁴¹⁵ and several theoretical studies by using minimal spin model,⁴¹⁶ Hamiltonian including anisotropic spin-spin couplings,⁴¹⁷ or 1/S in spin wave theory,⁴¹⁸ all support a “quantum order by disorder” scenario. Monte Carlo simulations^419,420 and high-temperature series expansion method⁴²¹ show that thermal fluctuations also select ψ₂ phase. On the other hand, other studies including another set of INS data⁴²² and a theoretical study by using a combination of strong-coupling perturbation theory and the random-phase approximation⁴²³ propose that the selection of the ψ₂ phase may be also influenced by higher-multipolar interactions from virtual crystal field transitions.

The second aspect revolves around how this ψ₂ phase behaves under various perturbations. For instance, (a) Y-doping on Er sites, (b) applied magnetic field, and (c) chemical pressure. Theoretical studies have predicted that on Y-doping, the ψ₂ phase should be unstable to its ψ₃ partner [Fig. 15(e)] by magnetic dilution.^424,425 Experimental studies on the Er_2−xY_xTi₂O₇ system,⁴²⁶ show that the domain selection and energy gap associated with the ψ₂ phase, disappear for 10%–20% Y-doping. As for the applied magnetic field effect, Monte Carlo simulation⁴²⁷ predict a phase transition from ψ₂ to ψ₃ order, in a weak field H//⟨001⟩ axis and reentrant transitions for H//⟨111⟩ axis. Lhotel et al.⁴²⁸ reported a phase transition toward ψ₃ structure for H//⟨001⟩ axis with H_c = 0.043 T and no transitions for H//⟨111⟩ axis. On the other hand, Gaudet et al.⁴²⁹ reported a ψ₂ to ψ₃ transition for H//⟨001⟩ axis with H_c = 0.18 T and a series of domain-based transitions for H//⟨111⟩ axis with H_c = 0.15 and 0.4 T. Single crystal neutron diffraction⁴³⁰ and thermal conductivity⁴⁰⁹ also have been measured to study the magnetic phase diagram [Fig. 16(b)]. Finally, studies on Er₂X₂O₇ (X = Sn, Ti, Ge) show that under chemical pressure, the ground states vary through the enlarged XY type exchange interactions, from a Palmer-Chalker (PC) phase for Er₂Sn₂O₇ [Fig. 15(f)], to a ψ₂ phase for ETO, and then a ψ₃ phase for Er₂Ge₂O_7,^431,432 fitting the phase diagram proposed by Wong et al.⁴¹⁷ This XY anisotropy change could originate from the change of the crystal field for Er₂X₂O₇.⁴¹¹ 

The ground state of Yb³⁺ ions⁴³³ is a predominant |²F_7/2, ±1/2⟩ Kramers doublet with g tensor as g_z = 1.92 and g_⊥ = 3.69. Therefore, YTO has XY-type anisotropy. It orders with a splayed ferromagnetic spin structure at T_c = 0.27 K [Fig. 15(g)].

Early studies on YTO obtained on image furnace grown single crystals, suggested a non-ordered state with strong spin fluctuations.^434–437 It has been proposed as a QSI state, in which the anisotropic exchange interactions introduce the strong quantum fluctuations^438–441 or a QSL due to the competition between the ferromagnetic and antiferromagnetic phases.^442,443

More recently, Thompson et al.⁴⁴⁴ reevaluated the Hamiltonian and found dominating quantum exchange terms, which they proposed to be responsible for the strong spin fluctuations. Various studies have been devoted to monopole dynamics in the QSI state.^{324,445–450} On the other hand, all studies carried out on powder samples support a ferromagnetic transition around 0.27 K.^451–457 Polarized neutron scattering results on a single crystal from Chang et al.⁴⁵⁸ supports a Higgs transition from a magnetic Coulomb liquid to a ferromagnet in YTO.

The ground state of YTO seems to be sensitive to sample quality from different groups.^176,459,460 Indeed, the structural studies from Kate et al.⁴⁶¹ show that the crystal grown by image furnace is lightly stuffed as Yb₂(Ti_2-xYb_x)O_7-x/2 with x = 0.046. INS studies further reveal that the stuffed Yb ions affect the local CEF scheme.^433,462 Recent diffuse scattering studies on several crystals with different chemical stoichiometry, have shown that oxygen vacancies stabilize the spin liquid phase and Yb stuffing suppresses it, while the samples annealed in oxygen shows an intrinsic ferromagnetic transition.⁴⁶³ 

Most recently, Arpino et al. grew ultra-pure YTO in image furnace by using TiO₂/YTO as solvent, which shows 0.27 K ordering temperature.⁴⁵⁹ The magnetic phase diagram measured on this sample reveals a reentrance phase with low field along the ⟨111⟩ axis [Fig. 16(c)], which could be related to quantum spin fluctuations.^464,465 More interestingly, the newest neutron studies on this sample show a usual coexistence of ferro and antiferromagnetism in YTO,⁴⁶⁶ in which the ψ₂ antiferromagnetism is metastable within the splayed ferromagnetic ground state. Recent theoretical studies propose that such kind of coexistence could be due to magnon interactions.⁴⁶⁷ 

Another area where pyrochlore-type oxides have attracted increasing interest is that of thermally insulating coatings, or TBCs. The cutting-edge technology employed in TBCs involves a multilayer design, comprising an oxidation-resistant metallic bond coat in proximity to the metal substrate surface and a thermally insulating oxide-based top layer, typically t′-ZrO₂. The bond coat serves a dual purpose: acting as a foundation for the refractory oxide coating and ensuring a suitable thermal expansion match between the topcoat and the substrate. Commonly fabricated from Al-enriched metal alloys, these coatings promote the formation of a slow-growing, uniform α-Al₂O₃ scale, serving as an effective oxygen diffusion barrier and enhancing coating oxidation resistance. In general, TBCs are applied using either electron beam physical vapor deposition (EB-PVD) or air plasma-spraying (APS). Both techniques lead to porous microstructures with typical thicknesses ranging from 100 μm to 1 mm, which are essential for high strain tolerance, and reduced thermal conductivity.

Metastable tetragonal-prime yttria-stabilized zirconia, t′-ZrO₂ (7YSZ), contains ∼7 wt. % Y₂O₃ (∼4 mole%), and has emerged the natural choice for TBCs due to its exceptional properties.^27,468 Zr⁴⁺ aliovalent substitution creates a substantial number of oxygen vacancies, and solute cations. These elements are strong phonon scattering sources, leading to enhanced thermal resistance. Moreover, it has a high melting point (>2500 °C), a relatively high thermal expansion coefficient (TEC), low density, high chemical stability, and remarkable fracture toughness at elevated temperatures. However, the maximum operating temperature of t′-ZrO₂-based coatings is limited to ∼1200 °C. Prolonged exposure above this temperature prompts its accelerated sintering, resulting in lower porosity, and reduced strain tolerance. Additionally, metastable t′-ZrO₂ tends to decompose at high temperatures into the thermodynamically more stable c- and t-ZrO₂ polymorphs, with the latter further evolving to the m-ZrO₂ phase upon cooling. The t → m-ZrO₂ transformation, and its associated increase in molar volume, causes high compressive stresses with catastrophic consequences on TBCs durability. Various strategies have been proposed to address these challenges such as reducing the t′-ZrO₂ sintering rate or using lanthanide oxides, Ln₂O₃, instead of Y₂O₃, as ZrO₂ stabilizers. However, these efforts have encountered limited success.^27,469 Consequently, current research is heavily focused on identifying alternative refractory oxides, including pyrochlore-type zirconates, to replace t′-ZrO₂ in TBCs. In evaluating prospective materials for TBCs, while numerous thermal and mechanical requirements must be considered, the primary imperative is achieving a low thermal conductivity (κ) at the high operating temperatures.

Pyrochlore oxides garnered significant attention for TBCs mainly, primarily due to their notable chemical flexibility, intrinsic oxygen vacancy concentration, and the facile formation of cation antisite defects, offering a diverse array of strong phonon scattering centers.^{26–28,469–474} Lanthanide zirconates Ln₂Zr₂O₇ exhibit high thermal stability, surpassing that of t′-ZrO₂, along with sluggish sintering kinetics, both important attributes for TBC materials. Computer simulations predict^475,476 anisotropic thermal conductivity for most pyrochlore zirconates in the range of 1.85–2.15 W·m⁻¹·K⁻¹, displaying minimal dependence on Ln³⁺. Given that the presence of pores, along with their morphology and spatial distribution, significantly influences heat transport, a diverse range of values is evident in experimental literature data. Nevertheless, reported data consistently report lower κ values for lanthanide zirconates (within ∼1.1–1.7 W·m⁻¹·K⁻¹ in the 700–1200 °C temperature range) compared to predictions and those measured for the t′-ZrO₂ phase (∼2 W·m⁻¹·K⁻¹ obtained for bulk t′-ZrO₂ at the same operating temperatures).^{27,471,476,477} Notably, κ values exhibit minimal dependence on lanthanide size, temperature, or crystal structure (i.e., fluorite vs pyrochlore). Among pyrochlore compositions, lanthanum zirconate emerges as a promising candidate for TBC applications for several reasons.^478,479 First, La₂Zr₂O₇ undergoes no thermally induced O-D transformation, maintaining pyrochlore stability up to its high melting point (∼2300 °C). Second, it serves as a poor oxygen ion conductor, enhancing coating oxidation resistance, a critical factor for mitigating bond coat oxidation -the predominant damaging mechanisms affecting TBCs durability. When a thermally insulating oxide such as t′-ZrO₂, readily conducts oxygen ions at high temperatures, the oxidation of the bond coat can progress even after applying the top layer: i.e., oxygen reaches the substrate not only by gas permeation through pores and microcracks in the coating, but also through the t′-ZrO₂ crystal structure. Finally, the low volatility of La, among the least volatile lanthanide metals, makes it an ideal choice for thermal co-deposition with Zr. The intrinsically low thermal conductivity of La₂Zr₂O₇ (∼1.5 W·m⁻¹·K⁻¹), has been further decreased through single or multiple chemical substitutions. Doped-La₂Zr₂O₇, extensively employed to investigate the role of substitutional defects in lowering κ in pyrochlore oxides, demonstrates substantial reduction in thermal conductivity. The efficiency of substituting ions on enhancing thermal resistance depends primarily mass difference (“mass disorder”), relative ionic size difference (“bond disorder”) between the guest and host atoms, and changes in atomic bonding.⁴⁸⁰ There are some disagreements in the literature about which of these factors is more effective in ZrO₂-based ceramics or even, if substitutional defects themselves, are stronger phonon scatters at high temperature than oxygen vacancies since the latter may exhibit some degree of short-range ordering as discussed above. Nevertheless, La₂Zr₂O₇ thermal conductivity has been substantially reduced by partially replacing La³⁺ with smaller lanthanides (Nd³⁺, Eu³⁺, and Gd³⁺).⁴⁸¹ The effect increases with increasing atomic weight (decreasing size) of the substituted element, with a minimum of ∼0.9 W·m⁻¹·K⁻¹ measured for a sample containing 30% Gd³⁺. These values approach those observed in glass-like materials, which represent the lower limit for thermal conductivity of highly disordered solids.⁴⁸² The feasibility of using chemical substitutions to tailor the thermophysical properties of pyrochlore zirconates, has been further explored in different La₂Zr₂O₇-based solid solutions. Thermal conductivity in the La_2-yGd_yZr₂O₇ system reaches a minimum of ∼1.4 W·m⁻¹·K⁻¹ at 400 °C when y = 1 (LaGdZr₂O₇). The scattering effect of point defects, reaching a maximum at 50% substitution, has been suggested as responsible for the reduced κ.⁴⁷² Low κ values have been also observed⁴⁸³ in the La_2-yYb_yZr₂O₇ system, which were explained by the combined effect of localized “rattling” of Yb³⁺ ions in the cation sublattice, and the intrinsic concentration of oxygen vacancies of the anion sublattice. Yb³⁺ atoms are much smaller and heavier than La³⁺, resulting in a more loosely bound configuration and low frequency atomic vibrations (“rattle”), around their equilibrium positions. Such atomic vibrations are highly anharmonic, and their interaction with lattice phonons result in substantial scattering events that hinders thermal transport and renders low κ (∼1.2–1.3 W·m⁻¹·K⁻¹ at 700 °C). Although measurements on the effect of chemical substitutions at the B-site are less abundant, the same “rattling” effect has been proposed to explain low thermal conductivity values found in the (La_2-xY_x)(Zr_2-yY_y)O_7-y/2 system (κ ∼1.5 W·m⁻¹·K⁻¹ at 700 °C).⁴⁸⁴ Studies on Hf⁴⁺- and Ce⁴⁺-doped La₂Zr₂O₇⁴⁸⁵ emphasize the significant role of size difference in controlling phonon scattering at lower temperatures and the influence of dopant's absolute size and mass on the κ_min plateau at higher temperatures. Hf⁴⁺ is almost the same size as Zr⁴⁺ but has a substantially larger atomic mass, whereas Ce⁴⁺ has both significant size, and atomic mass contrast with Zr⁴⁺. Substitutional atoms of larger size and higher mass generally result in lower conductivities.

A similar doping effect has been observed in the remaining lanthanide zirconates confirming the viability of this approach to further diminish the inherently low thermal conductivity of pyrochlore oxides.^{473,486–488} Additionally, studies conducted on the Sm_2-yYb_yZr₂O₇ system have revealed that the O-D pyrochlore-to-fluorite phase transition, progressing with increasing Yb³⁺ content, significantly impacts the softening of acoustic modes.⁴⁷² This effect introduces additional phonon scattering, leading to a minimum in thermal conductivity at the phase boundary composition (Sm_2/3Yb_1/3)₂Zr₂O₇. Likewise, a minimum κ of 1.35–1.54 W·m⁻¹·K⁻¹ has been observed⁴⁸⁶ in the Nd_2-yYb_yZr₂O₇ system for the NdYbZr₂O₇ composition. As with pure zirconates, molecular dynamics simulations for κ in Gd_2-yLn_yZr₂O₇ solid solutions (Ln = La, Y, and Sm), predict higher values than those determined experimentally.⁴⁸⁹ Discrepancies between modeling results and experimental data are more pronounced at lower temperatures, likely attributed to microstructural effects and/or the model's limitations in adequately predicting the significance of additional point defect scattering at these temperatures. Models also suggest that “bond disorder” is more effective than “mass disorder” at decreasing κ for these materials, and lower κ values were calculated for Gd_2−yLa_yZr₂O₇ than for Gd_2−yY_yZr₂O₇.

In comparison to t′-ZrO₂, the major drawbacks for pyrochlore oxides in TBC applications include their lower thermal expansion coefficients (TECs), poor toughness, and propensity to react with α-Al₂O₃ resulting in mixed oxides (e.g., LnAlO₃ or Ln₃Al₅O₁₂). A substantial TEC mismatch related to both the metal substrate and the bond coat induces high thermal stresses during heating and cooling cycles. Typical TECs for superalloy substrates, and a NiCrAlY bond coat are⁴⁹⁰ ∼16 × 10⁻⁶ K⁻¹, and ∼17.5 × 10⁻⁶ K⁻¹, respectively, while TECs reported for the t′-ZrO₂ phase between room-temperature and 1200 °C, are ∼11 × 10⁻⁶ K⁻¹. In contrast, TECs measured in pyrochlore-type zirconates Ln₂Zr₂O₇ (Ln = La–Gd) in the same temperature range, are typically between 8 and 11 × 10⁻⁶ K⁻¹, increasing with decreasing Ln³⁺ radii.⁴⁹¹ According to molecular dynamics simulations, TECs are primarily determined by the stronger ⟨Zr–O⟩ bond, with the ⟨Ln–O⟩ bonds playing a secondary role.⁴⁹² Consequently, Zr-site doping is highly recommended to increase thermal expansion coefficients of zirconates. Simulations also predict that for identical chemical composition, fluorites would have relatively higher TECs than the corresponding pyrochlore due to the weaker ⟨Zr–O⟩ bonding strength. Additional approaches suggested to overcome this issue include doping with a divalent oxide such as MgO, inducing a significant increment in Sm₂Zr₂O₇ thermal expansion and reducing the mismatch with other TBC system components.⁴⁷³ However, MgO also induces higher sintering activity at lower temperatures. Double-layer topcoat systems have been also proposed, consisting of an inner layer containing t′-ZrO₂ with a higher TEC and high toughness, and an outer layer with lower thermal conductivity fabricated with pyrochlore-type La₂Zr₂O₇ or Gd₂Zr₂O₇.^29,493–497 These duplex coatings are capable of withstanding temperatures at least 100 °C higher than single layered t′-ZrO₂, and exhibit excellent cycling behavior.

Pyrochlore-type titanates are not considered inherently suitable for TBCs applications due to their higher thermal conductivity and enhanced sintering activity at lower temperatures. Computer simulations have predicted κ-values in the 2.30–3.05 W·m⁻¹·K⁻¹ range for Ln₂Ti₂O₇ pyrochlore,⁴⁷⁶ and these predictions have been confirmed using single crystals,⁴⁹⁸ with room-temperature values for the series falling within the 2 to 3 W·m⁻¹·K⁻¹ range. Unfortunately, there is limited experimental data available on the high-temperature thermal conductivities in pyrochlore-type titanates. Nevertheless, measurements conducted in the Gd₂Zr_2−yTi_yO₇ system (0 ≤ y ≤ 2) indicate that κ increases with increasing Ti⁴⁺, transitioning from ∼1.4 W·m⁻¹·K⁻¹ in Gd₂Zr₂O₇ to 2.1 W·m⁻¹·K⁻¹ for Gd₂Ti₂O₇ at 1000 °C.⁴⁹⁹ However, a separate study within the same system reveals an opposing trend below ≈ 400 °C, with conductivity initially decreasing as Ti⁴⁺ content increases (y ≤ 0.8) before resuming an upward trajectory for Gd₂ZrTiO₇.⁵⁰⁰ No higher Ti-contents were tested. TECs demonstrate the opposite trend, initially increasing before reaching a peak at y = 0.4 and subsequently dropping significantly with an increase in titanium content. The impact of lanthanide substitutions on the thermophysical properties of lanthanide titanates have been explored through the (Er_2−xY_x)Ti₂O₇ solid solution.⁵⁰¹ This study focused on the effect of “mass disorder” on κ, considering that Er³⁺ and Y³⁺ have nearly identical size (R_VIII = 0.1004 vs 0.1019 nm), but considerably different atomic masses (167.26 vs 88.91). Using single crystals, and very low working temperatures where the influence of dopant ions is more significant, the study yielded results consistent with those mentioned earlier: κ reaches a minimum value with the highest mixed lanthanide content (∼50 atom% each), representing the highest number of point defects.

Many modern energy technologies require materials that can reliably withstand extreme operating environments. Therefore, understanding the basic science associated with the behavior and performance of materials under extreme conditions and changes in their properties represents an area of growing research interest. This final section is concerned with the response of Ln₂Zr₂O₇, and Ln₂Ti₂O₇ pyrochlore oxides to pressure, and irradiation.

The characterization of high-pressure phases is challenging because many of them are in general, not quenchable to ambient conditions; research should be carried out in situ, while the sample is subject to compression. Furthermore, structural analysis by diffraction methods at tens of GPa is difficult because of the significant broadening and overlapping of diffraction peaks. However, modern synchrotron x-ray sources with beamlines devoted exclusively, to in situ high pressure diffraction experiments, have revealed a wealth of information about pressure induced effects on crystalline solids. This section will start by briefly, reviewing pressure-induced phase transformations in the simple ZrO₂, TiO₂, and Ln₂O₃ oxides, which is followed by the behavior of pyrochlore-type lanthanide titanates, and zirconates.

At least two high pressure ZrO₂ phases of orthorhombic symmetry, are known, i.e., “OI” (≥3 GPa), and “OII” (≥15 GPa).^502–504 The m-ZrO₂ → “OI” (S.G. = Pbca (61), Z = 4) transition is reversible; whereas, cotunnite-like “OII” is easily quenched to ambient conditions. Figure 17 presents an idealized representation of the orthorhombic [S.G. = Pnma (62), Z = 4] cotunnite-type unit cell (PbCl₂); this structure-type is very common amid high-pressure phases in AX₂ compounds (A = M²⁺ or M⁴⁺; X = halogen or oxygen anion), including the prototype phase of the fluorite structure, CaF₂.⁵⁰⁵ Pressure increases packing efficiency of crystal structures by mainly, deforming the anion coordination polyhedra, and/or increasing the coordination around metal atoms. Moreover, cotunnite-like ZrO₂, features a single cation site (Wyckoff position 4c), with a ninefold coordination environment vs sevenfold coordinated Zr⁴⁺ in m-ZrO₂.

Oxygen ions are also in 4c although in five-, and fourfold coordination. Zr⁴⁺ higher coordination implies elongated metal-oxygen bonds: the average Zr–O distance increases from ∼0.216 nm in baddeleyite-type ZrO₂, to ∼0.228 nm in the cotunnite-like phase.^506,507 In fact, ⟨Zr–O⟩ bond lengths in “OII” vary widely, from 0.210 to 0.256 nm. Rutile and anatase TiO₂ show both, similar sequence of pressure-induced phase transitions:^508–510 i.e., TiO₂ → columbite-like (α-PbO₂) TiO₂ → baddeleyite-like TiO₂ → “OI” → “OII.” Less consensus exists about the existence, and nature of post-cotunnite phases. Obtaining the cotunnite-like “OII” form for TiO₂ requires high pressure (>55 GPa) and temperature and preserving the structure to ambient conditions requires quenching at very low temperatures (<170 K). The average Ti-O distance increases from ∼0.194 nm in rutile, and anatase, to ∼0.210 nm in cotunnite-type TiO₂; this high-pressure phase presents higher density than any of the ambient pressure polymorphs. Both, cotunnite-like ZrO₂, and TiO₂ are considered promising hard materials,^509,510 with bulk modulus of ∼320 and ∼431 GPa respectively, being the latter close even to the value of ∼440 GPa characteristic of diamond.

By contrast, compression does not modify substantially, the crystal chemistry of Ln₂O₃ oxides, and the pressure-induced phase transformation sequence is similar to that thermally promoted. In general, increasing pressure tends to stabilize the hexagonal A-form over the C- and B-type Ln₂O₃ polymorphs.^511–514 Pressure needed to trigger the transformation increases with higher occupancy of the f-shell (decreasing ionic radii and increasing at. wt.). In some cases (mostly for the smaller lanthanides), the C → A phase transition involves the formation of monoclinic B-Ln₂O₃ as an intermediate step. The B-Ln₂O₃ form is commonly also, the stable form upon decompression.

The response of the pyrochlore structure to elevated pressures is ruled, mostly, by the energetics of defect formation, and the relative compressibility of the AX₆Y₂ and BX₆ polyhedra. Pyrochlore-type titanates and zirconates are both unstable to high compressive stresses, though with some distinct peculiarities. On the one hand, all Ln₂Zr₂O₇ zirconates seem to exhibit similar response, and undergo a phase transition to a distorted defect cotunnite-like structure above ∼20 GPa.^515–519 On the other hand, pyrochlore-type titanates show higher structural stability under compression and are capable to withstand pressures of up to ∼40 GPa.^{517,520–524}

Establishing absolute phase stability limits for pyrochlore-type zirconates under pressure, is hindered by sluggish transformation kinetics, i.e., the pyrochlore, and cotunnite structures coexist over a wide pressure range (e.g., ∼10–15 GPa). Furthermore, the transformation is irreversible; upon decompression, the cotunnite phase, probably monoclinic in symmetry, evolves to a fluorite structure in the case of Ln₂Zr₂O₇ zirconates. Therefore, theoretical and computational methods have been used to complement experimental studies and predict the stability of the high-pressure phases. Consistent with experimental data, calculations have determined that under compression the cotunnite structure is the most stable for Ln₂Zr₂O₇; the theoretical formation enthalpy of the cotunnite phase is lower than the corresponding values calculated for pyrochlore, fluorite, or even for amorphous phases.⁵¹⁵ Calculations also anticipate increasing transformation pressures as the r_Ln/r_Zr size ratio increases: e.g., ∼13, ∼18, ∼20, and ∼22 GPa have been estimated respectively, for Gd₂Zr₂O₇, Sm₂Zr₂O₇, Nd₂Zr₂O₇, and La₂Zr₂O₇.^515,525 The pyrochlore-to-cotunnite transformation involves complete mixing of both cations and anions, in their corresponding sublattices. Experimental data suggest that disordering proceeds at different rates, with higher pressures needed to disorder the cation sublattice. In general, polyhedra containing large and less charged ions are more compressible than those containing small ions with high charge. Correspondingly, average ⟨cation-anion⟩ bond distances in the defect-cotunnite structure (∼0.23–0.25 nm), are very close to the ⟨Ln-O⟩ bond lengths in pyrochlore-type Ln₂Zr₂O₇, but much higher than the ⟨Zr–O⟩ distance; i.e., the ⟨Zr–O⟩ bonds undergo more significant changes upon compression, and play a more active role during the transformation, than the ⟨Ln-O⟩ bonds. A similar high-pressure cotunnite phase has been obtained for fluorite-type lanthanide zirconates,^519,526 with Ho₂Zr₂O₇, Er₂Zr₂O₇ and Y₂Zr₂O₇ transforming above ∼30, ∼22, and ∼27 GPa, respectively. In this case, ambient-, and high-pressure phases coexist over an even wider pressure range, and complete transformation for the Er-containing zirconate is not achieved even at 43 GPa. The transition is reversible, and the cotunnite structure is reverted back to the fluorite structure upon decompression.

Similar theoretical calculations performed for lanthanide titanates have predicted that the fluorite structure is energetically favored at high pressures, over the cotunnite-like phase; however, the enthalpy difference is small, and according to calculations, both defect configurations might coexist, unless kinetically limited.⁵²⁵ Experimental data show, in general, good agreement with calculations: e.g., compressibility studies have revealed the formation of a high-pressure (∼50 GPa) monoclinic phase in Y₂Ti₂O₇, Ho₂Ti₂O₇, Tb₂Ti₂O₇, and Yb₂Ti₂O₇.^522,527 At higher pressures (>60 GPa), Y₂Ti₂O₇ transforms into a mixture of a fluorite structure and an amorphous phase, with the former remaining stable upon decompression. In contrast, high-pressure experiments carried out on polycrystalline Gd₂Ti₂O₇, and Sm₂Ti₂O₇ result in a mixture of an amorphous phase and a distorted pyrochlore-like material, with a highly disordered anion sublattice; complete amorphization was only observed when the samples were quenched from 51 GPa. However, according to similar compressibility studies carried out in Sm₂Ti₂O₇ single crystals, this material remains cubic up to 50 GPa without any sign of increasing structural disorder or amorphization, upon increasing pressure.⁵²⁸ These differences between studies highlight the importance of the pressure transmitting media (two Al foils vs Ne) and the hydrostaticity in the diamond anvil cell pressure experiments. The stability difference between pyrochlore-type zirconates, and titanates under compression is best illustrated by the Gd₂Ti_2−xZr_xO₇ system; the transformation pressure to a cotunnite-like phase (i.e., the stability of the pyrochlore structure), increases with increasing Ti-content.⁵²⁹ Upon releasing the pressure, the cotunnite-like phase transforms to a closely related phase (“X-phase”), which cannot be recovered at ambient conditions unless the cotunnite-like phase is heavily irradiated in situ with energetic heavy ions. Differences in stability between lanthanide titanates, and zirconates suggest that the ⟨Zr–O⟩ bonds (i.e., ZrO₆ polyhedra) are less compressible than the ⟨Ti–O⟩ bonds (i.e., TiO₆ polyhedra). According to recent first-principles calculations, the energy penalty required to create point-defects decreases with increasing pressure for both, Gd₂Ti₂O₇ and Gd₂Zr₂O₇.⁵³⁰ However, whereas the cation antisite defect would be the dominant type of defect in Gd₂Ti₂O₇, anion Frenkel pairs are favored in Gd₂Zr₂O₇ at all pressures. Furthermore, defect-defect interactions would be weak in Gd₂Zr₂O₇; whereas clustering is favored in Gd₂Ti₂O₇.

Materials capable of enduring high-energy particle and photon fluxes are essential in technological areas such as spacecraft design and mission planning, nuclear fusion and fission reactors, solar cells and solar thermal collectors, and electronics for computers and communication. In particular, the possibility of developing pyrochlore-based inorganic matrixes for the disposal of high-level radioactive waste and separated actinides, such as excess weapons plutonium, have generated considerable interest over the past decades on irradiation effects on lanthanide titanates, and zirconates.^33,35 Initially, research involving actinide-bearing crystalline waste forms centered around the polyphase titanate ceramic, Synroc,⁵³¹ in which the principal component is zirconolite (Ca₂ZrTiO₇), a pyrochlore-derived structure. Interestingly, it was shown that doping of zirconolite with transuranic elements (e.g., Np, Pu), led to the formation of a titanate pyrochlore phase in which the actinides preferentially reside.^532,533 Among the earliest direct radiation studies of a lanthanide pyrochlore involved the ²⁴⁴ Cm-doping of Gd₂Ti₂O_7,⁵³⁴ which led to amorphization of the pyrochlore through spontaneous Cm-fission, as well as α-decay events. Thus, different research groups have used ion beam irradiations in the keV to MeV energy range, to simulate the α-decay damage (α-particles, and α-recoils) from incorporated actinides in pyrochlore-type Ln₂Ti₂O₇, and Ln₂Zr₂O₇.^36,535–541 Alpha particle energy is mostly dissipated by ionization processes (i.e., electronic energy loss); whereas, α-recoils undergo elastic collisions with the atomic nuclei in the solid (i.e., nuclear energy loss), and produce several hundreds of isolated atomic displacements. Many defects annihilate spontaneously and will not turn into irreversible structural damage; on increasing dose, however, defects will accumulate in the lattice causing microstructural changes (e.g., amorphization and disordering), and macroscopic swelling.^534,542 Therefore, the radiation resistance of a material is determined by the balance between damage accumulation and recovery, which in turn depends on chemical composition, bonding character, density, and crystal structure. Despite sharing the same crystal structure, pyrochlore-type titanates, and zirconates markedly differ in their response to irradiation: i.e., pyrochlore-type zirconates are significantly more radiation resistant than titanates. In fact, Ln₂Ti₂O₇ oxides are susceptible to amorphization even at relatively low irradiation doses; whereas, Ln₂Zr₂O₇ present a radiation-induced O-D transformation to the fluorite structure, in agreement with their enhanced ability to accommodate structural defects. These differences are best illustrated by Gd₂Ti₂O₇, and Gd₂Zr₂O₇. Gd₂Ti₂O₇ shows a high susceptibility to ion beam induced amorphization (critical dose ∼0.2 Displacements per Atom, dpa);⁵³⁵ whereas minor amorphization in Gd₂Zr₂O₇ has only been achieved recently, with the extremely high energy density delivered by MeV C₆₀-cluster projectiles.⁵⁴³ 

While all lanthanide titanates are amorphizable, their radiation response in the nuclear energy loss regime is highly compositional dependent.^116,544 Irradiation studies carried out by using Ln₂Ti₂O₇ single crystals (Ln = Sm–Lu and Y) and 1 MeV Kr⁺ projectiles, have revealed that susceptibility to amorphization increases with increasing Ln³⁺ size.¹¹⁶ Accordingly, the critical amorphization temperature, T_c, which is indicative of the energy required to facilitate the recovery of the crystal structure (i.e., lower T_c is equivalent to enhanced radiation resistance), increases from Lu₂Ti₂O₇ (∼200 °C) to Gd₂Ti₂O₇ (∼850 °C). However, the opposite trend is observed between Sm₂Ti₂O₇ and Gd₂Ti₂O₇, with samarium titanate being more radiation resistant (T_c ∼ 775 °C) than its gadolinium-containing counterpart. These results are also in agreement with studies mentioned before predicting a maximum in defect formation energies in the Ln₂Ti₂O₇ series, for Gd₂Ti₂O₇.^130,133 Recent machine learning techniques have also revealed a strong correlation in the Ln₂Ti₂O₇ series between T_c and the energy difference between the ordered and disordered states (E_{O → D}) and disordered and amorphous states (E_{D → A}).⁵⁴⁵ In contrast, all pyrochlore zirconates Ln₂Zr₂O₇ exhibit an irradiation-induced transformation to a fluorite structure under irradiation, which is in turn highly resistant to amorphization. An outlier is La₂Zr₂O₇ which becomes fully amorphous under irradiation (∼5.5 dpa)¹¹⁶ although it can be still considered a radiation tolerant material when compared to most other ceramics. There is also evidence that Nd₂Zr₂O₇ may be amorphizable at extremely high levels of radiation exposure;⁵⁴⁶ when exposed to 1 MeV Kr ions, it remained crystalline up to a fluence of 2·10¹⁵ ions·cm⁻², but evidence of diffuse scattering was observed with electron diffraction, indicative of some amorphous sample fractions. Upon irradiation, anion disorder is energetically favorable as compared with cation disordering;⁵⁴⁷ thus, an intermediate stage of anion-disordered pyrochlore has been observed prior to the formation of the fluorite suggesting that both sublattices disorder rather independently.³⁶ A recent study focused on the dependence of radiation tolerance with increasing zirconium content beyond ideal pyrochlore stoichiometry.⁵⁴⁸ Sharma et al. exposed Nd₂Zr₂O₇ and Nd_1.8Zr_2.2O_7.1 to 100 MeV I ions and showed by TEM and XRD analysis that the non-stoichiometric pyrochlore exhibited smaller amorphous ion tracks compared to the stoichiometric sample. This effect was attributed to the excess Zr yielding an excess of the disordered fluorite phase with enhanced radiation stability. This needs to be confirmed also for other ion-beam conditions; however, non-stoichiometry and Zr excess may be a promising route of increasing the radiation stability of pyrochlore oxides. However, compositions with non-ideal stoichiometry may have a complex local order that must be fully understood and linked to the radiation behavior.¹⁷³ 

The different behavior of titanates and zirconates upon irradiation is also observed in the Gd₂Ti_2−xZr_xO₇ system where a radiation-induced pyrochlore-to-fluorite phase transition occurs;^538,549 however, fluorite phases created by irradiation for Ti-rich compositions are metastable and change to a fully amorphous state at sufficiently high damage level. By contrast, fluorite structures obtained for the Zr-rich pyrochlores are stable and under identical irradiation conditions do not evolve to the amorphous state. The resistance to ion-beam-induced amorphization increases dramatically with zirconium content, which is related to the simultaneous enhanced stability of the disordered structure: the ease of rearrangement and relaxation of structural defects inhibits amorphization.

The importance of factors such as lanthanide ions electronic configuration, and/or bonding characteristics (i.e., degree of covalent bonding), are important factors to understand the susceptibility of pyrochlores to radiation-induced amorphization. For example, critical amorphization temperatures across stannate pyrochlores (Ln₂Sn₂O₇) are much higher than their titanate and zirconate counterparts despite a similar ionic radii ratio; this reduced radiation resistance of stannate pyrochlores has been associated with a higher degree of covalent bonding in ⟨Sn–O_48f⟩ when compared to ⟨Ti–O_48f⟩ or ⟨Zr–O_48f⟩.⁵⁵⁰ 

More recently, irradiation studies have been extended to include swift heavy ions (heavy ions accelerated to kinetic energies typically greater than 1 MeV/nucleon and v > 0.05c). In this regime, ions lose their energy (hundreds of MeV to GeV) almost entirely, by inelastic interactions with electrons (i.e., electronic energy loss); each individual ion might produce along their trajectory cylindrical volumes of modified material (so-called ion tracks), a few nanometers in diameter and tens of micrometers long. Track formation typically occurs above a critical energy loss value with their size and morphology depending upon the projectile velocity and the electronic energy loss.⁵⁵¹ In Gd₂Ti₂O₇, for example, continuous ion tracks only form above the threshold value of dE/dx = 11 keV/nm.⁵⁵¹ At higher values of energy loss, multiple concentric damage zones have been found within individual tracks in some pyrochlore compositions. Radiation-induced material modifications in this case include crystalline-to-amorphous transformations, generation of fluorite-type phases, defect clusters, and enormous irreversible deformations. Similar to the effects of low-energy irradiations, swift heavy projectiles induce primarily amorphization in Ti-rich pyrochlore, with a similar compositional dependence on the relative response,⁵⁴⁴ and create the fully disordered, defect-fluorite structure in Zr-rich pyrochlore. Indeed, La₂Zr₂O₇ can be also amorphized with swift heavy ions, as can Nd₂Zr₂O₇, which is resistant to amorphization with low-energy ions.⁵⁵² 

The morphology of swift heavy ion tracks in pyrochlore is complex and strongly dependents on chemical composition (see Fig. 18).^553,554 The ion energy deposition is maximum in the center of the track and decreases radially as 1/r²; accordingly, the degree of radiation damage observed also decreases with increasing distance from the center. In lanthanide pyrochlores, this energy deposition profile generally results in three concentric damage zones: (1) an amorphous core within the center of the track, (2) a disordered fluorite structured shell surrounding the track core, where the energy deposition is not high enough to amorphize (or an initial amorphous phase efficiently anneals), and (3) a defective pyrochlore halo in which the energy deposition is sufficient only to induce incoherent point defects. The morphology and size of ion tracks in the Gd₂Ti_2−xZr_xO₇ pyrochlore system was shown to strongly depend on the chemical composition with the total track size, increasing with the Ti content.^555–557 The decrease in the track size with increasing Zr content is a result of the enhanced radiation stability and amorphization resistance of Zr-rich pyrochlores. The track diameter in Gd₂Ti₂O₇ is on average more than a factor of 2 larger than in Gd₂Zr₂O₇. The enhanced resistance to swift heavy ion induced amorphization has been also related to the thermal properties (e.g., melting point, thermal conductivity, etc.) and enhanced damage recovery with increasing Zr content. The observation of a disordered fluorite shell in Gd₂Ti₂O₇ is intriguing due to its very large radius ratio (1.74), indicating a highly non-equilibrium phase. The formation mechanism and stabilization for the metastable disordered fluorite phase in Gd₂Ti₂O₇ have been attributed to dynamic recrystallization,⁵⁵¹ and high tensile strain⁵⁵⁸ at the crystalline-amorphous interface. MD simulations have shown that the concentric damage zones may be the result of partial recrystallization during rapid quenching of an initially molten ion track. The crystallization in some zirconate pyrochlore is complete and the results are fluorite structured tracks within a pyrochlore matrix. While each individual ion track consists of concentric damage zones, continued irradiation exposure leads to saturation of either a completely amorphous or disordered phase in Gd₂Ti₂O₇ and Gd₂Zr₂O₇, respectively. It has been proposed^559,560 that individual heterogeneous tracks lead to single phase saturation behavior through specific track overlap pathways; amorphous pyrochlore can be created directly in a track core or indirectly through the overlap of two disordered fluorite shell regions. Thus, in the amorphizable titanate pyrochlores, there is an initial increase in the total disordered fluorite phase from isolated shells, which is rapidly depleted by subsequent ion impacts. Interestingly, the stannate pyrochlores have been shown to be more radiation resistant to swift heavy ions than their titanate analogues, in contrast to what has been observed for low energy ions. Complex track morphology was observed in these compounds with large shells of disordered fluorite surrounding smaller amorphous cores.⁵⁶⁰ Furthermore, it was demonstrated that in the stannate pyrochlores, single phase saturation behavior is not attained. Instead, at high fluences, the fraction of amorphous and disordered fluorite phases reaches steady-state values below unity. To explain this behavior, it was proposed that amorphous regions can be transformed to disordered fluorite regions via epitaxial recrystallization of amorphous domains at crystalline-amorphous interfaces upon a second ion impact. Due to the large range of swift heavy ions, sufficiently large irradiated sample mass can be obtained for neutron total scattering experiments with high sensitivity to the oxygen sublattice and access to the local atomic structure. A direct comparison of Dy₂Sn₂O₇ and Er₂Sn₂O₇, which amorphized and disorder under swift heavy ion irradiation, respectively, showed that the local atomic arrangement is highly similar in both compounds despite the very different long-range response. The same weberite-type short range order formed after ion irradiation as it was revealed in otherwise disordered pyrochlores. Also, the complex track morphology with multiple concentric damage zones was recently assessed by neutron probes. In the specific case of Er₂Sn₂O₇, neutron pair distribution function analysis revealed⁵⁶¹ that a defective pyrochlore “halo” exists beyond the disordered fluorite shell, extending ∼1 nm into the pyrochlore matrix. This information was used to expand existing heterogeneous swift heavy track-overlap models developed for pyrochlores. Interestingly, while the long-range structure of individual tracks is characterized by a complex damage morphology (amorphous core + disordered fluorite shell + defective pyrochlore “halo”), the track structure is entirely homogeneous across the short-range with only a weberite-type phase.

While the effects induced by radiation exposure can, in many cases, be deleterious, the dense energy deposition can create otherwise inaccessible metastable material phases with desirable properties.⁵⁶² For instance, pyrochlore oxides are studied for their anion transport properties (Sec. V A); much recent attention has been paid to the use of swift heavy ions as a means of dramatically boosting oxygen conductivity. Combined experimental and computational approaches⁵⁵⁸ have revealed that within tracks created by 2 GeV Au ions in Gd₂Ti₂O₇ pyrochlore, the disordered fluorite shell (surrounding the amorphous core) is under significant tensile strain which lowers oxygen migration barriers and boosts diffusivity. Similar work⁵⁶³ indicated that such ion tracks may essentially be “nanochannels” of fast ion conduction attributed to severe deformation of the Ti-octahedra and subsequent modification of the electronic structure. The effect of irradiation-boosted conductivity was also observed in low-energy ion irradiation of Gd₂Ti₂O₇ thin films⁵⁶⁴ in which the largest conductivity boost was attributed to atomic disorder.

This review provides a comprehensive survey of the structural peculiarities exhibited by pyrochlore-type lanthanide zirconates and titanates, across various length scales. This paper also covers their processing methodologies and interesting properties, including responses to high pressure and ion irradiation. Despite the wealth of accumulated data, this review underscores the need for a better understanding of the order/disorder processes in pyrochlore-type materials and their impact on physical and chemical properties. Ongoing discussions center around the influence of electronic structure, bonding characteristics, cation-size mismatch, processing conditions, and the role of structural length scale.

In our opinion, several pivotal aspects merit further attention, with a primary focus on conducting a more in-depth analysis of short-range structures. A thorough comprehension of local order/disorder within both the anion and cation sub-lattices, is essential for a more fundamental understanding of the properties of pyrochlore-type oxides. Initially viewed as ordered derivatives of a straightforward anion-deficient fluorite structure, recent experimental evidence has revealed the complex nature of local structures. Isometric pyrochlore-type oxides are now believed to include short-range, lower symmetry, ordered domains, such as the orthorhombic weberite-type structure, which dictates to the observed long-range pyrochlore-type structure. Notably, short- and long-range structures appear decoupled across different length scales and temperature regimes, and these differences persist even in well-ordered samples. The intriguing discrepancies between both atomic configurations, explained by Pauling's rules, and the discovery that similar defect structures share short-range ordered domains, provide crucial insights into the fundamentals of oxide defect chemistry and engineering.

In the field of ionic and mixed conducting pyrochlore-type zirconates and titanates, a promising avenue for future exploration lies in optimizing the delicate equilibrium between electronic and ionic transport properties. Designing complex pyrochlore-type oxides and systematically investigating and manipulating defect chemistry are attractive and important goals in materials science. Strategic doping methodologies, defect engineering techniques, and a comprehensive understanding of the influence of short- and long-range structures on transport properties are essential. Additionally, exploring the impact of nano-scale structures on mixed/ionic conduction in pyrochlore-type materials is important, presenting opportunities for the synthesis of materials with shorter diffusion paths and/or unique electronic properties.

Decades into the study of magnetic Ti/Zr-pyrochlores, the field remains vibrant. Future research should focus on advanced methods, with a need for innovative approaches to enhancing the understanding of magnetism in pyrochlores. Recent advances, such as optimizing the Hamiltonian for Dy₂Ti₂O₇ through machine learning techniques, offer promising directions for mitigating background noise and artifacts in the analysis of raw neutron scattering data. Diversification of systems to “metallize” magnetic pyrochlores is another pivotal challenge, considering their incompatibility with electronic integrated-circuit technology (most magnetic pyrochlores are good insulators). Synthesizing heterostructures that combine magnetic pyrochlores with metals holds promise in inducing anomalous electronic behaviors or even metallization.

The limitations of t′-ZrO₂ for thermal barrier coating applications emphasize the need for a thorough exploration of alternative ZrO₂-based pyrochlores. Precise control over thermal conductivity can be achieved through advanced manufacturing techniques, innovative doping approaches, and strategic incorporation of specific defects. Enhancing the understanding of phonon scattering mechanisms, investigating order/disorder transformations, and sintering at elevated temperatures are critical aspects for developing more stable materials for TBCs. Advancements in characterization techniques also offer promising avenues for obtaining a comprehensive understanding of the microstructure, morphology, and thermal behavior of pyrochlore-type materials.

Finally, the exploration of compositionally complex and high-entropy oxides is a crucial avenue for future research. Leveraging configurational entropy contributions by introducing multiple cations in a single structure has emerged as an exciting topic in materials research. Increasing compositional complexity provides a unique opportunity to explore thermodynamics and structure-property relationships in highly disordered systems, such as the pyrochlore structure. Given the vast array of potential elemental combinations, unforeseen synergies may pave the path toward the development of revolutionary properties. Specifically, medium, or high configurational entropy within the pyrochlore structure can be achieved in two cation sites offering a new avenue for potential property engineering. Configurational entropy has the power to influence the electronic structure, ion migration pathways, phonon scattering mechanisms, magnetic ordering and susceptibility within the pyrochlore structure. Order-disorder transformations in pyrochlore oxides add an intriguing dimension to the investigation. This multifaceted influence underscores both the rich possibilities and challenges associated with compositionally complex and high-entropy pyrochlore-type oxides, promising significant contributions to the understanding of these intricate materials.

Work by E.C.O. and M.L. was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award No. DE-SC0024140. H.D.Z. was supported by the National Science Foundation through Grant No. NSF-DMR- 2003117. A.F.F. acknowledges financial support for his research by the Consejo Nacional de Ciencia y Tecnología (Conacyt) under Grant No. CB-2013-1-221701.

The authors have no conflicts to disclose.

Antonio F. Fuentes: Conceptualization (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (lead); Writing – review & editing (equal). Eric C. O'Quinn: Conceptualization (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (lead); Writing – review & editing (equal). Sagrario M. Montemayor: Conceptualization (equal); Investigation (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Haidong Zhou: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Maik Lang: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Rodney C. Ewing: Conceptualization (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.