Atomic order in the spinel structure – a group-theoretical analysis

Missing Piece in the Crystal Chemistry of Zn-Sb Secondary Phases in ZnO-Sb2O3-Bi2O3 Varistor Ceramics: Orthorhombic β-Zn7Sb2O12. An Experimental and Theoretical Study of the Crystal Structure and Its Thermal and Vibrational Spectroscopic Characterization

The ternary Zn-Sb oxide β-Zn₇Sb₂O₁₂ was prepared by solid state synthesis from ZnO and Sb₂O₃ at 1250 °C and a reaction time of 168 h. The crystal structure of β-Zn₇Sb₂O₁₂ was solved from powder diffraction data by direct-space methods and was refined by the Rietveld technique. The title compound β-Zn₇Sb₂O₁₂ crystallizes in the orthorhombic space group Cmme with a = 1210.6(1) pm, b = 1856.7(1) pm, c = 852.2(1) pm, V = 1.9154(1) nm³, and eight formula units per unit cell. The structure can be described as a distorted cubic closest packing of oxide ions with Zn²⁺ and Sb⁵⁺ in the tetrahedral and octahedral interstitial sites. According to group-subgroup relations, the anion packing is directly derived from the spinel structure of α-Zn₇Sb₂O₁₂; however, the occupation pattern of the interstitials is completely different than in the spinel structure type. The most prominent structural feature in β-Zn₇Sb₂O₁₂ is the clustering of all [ZnO₄] polyhedra into a [Zn₇O₂₀] moiety representing an excerpt from the sphalerite structure. The structural model is corroborated by vibrational spectroscopy as well as density functional theory calculations. Thermal analysis reveals an irreversible phase transition of β-Zn₇Sb₂O₁₂ into the α-polymorph at 1330 °C.

A Pressure-Induced Inverse Order-Disorder Transition in Double Perovskites

Given the consensus that pressure improves cation ordering in most of known materials, a discovery of pressure-induced disordering could require recognition of an order-disorder transition in solid-state physics/chemistry and geophysics. Double perovskites Y₂ CoIrO₆ and Y₂ CoRuO₆ polymorphs synthesized at 0, 6, and 15 GPa show B-site ordering, partial ordering, and disordering, respectively, accompanied by lattice compression and crystal structure alteration from monoclinic to orthorhombic symmetry. Correspondingly, the long-range ferrimagnetic ordering in the B-site ordered samples are gradually overwhelmed by B-site disorder. Theoretical calculations suggest that unusual unit-cell compressions under external pressures unexpectedly stabilize the disordered phases of Y₂ CoIrO₆ and Y₂ CoRuO₆ .

Formation of breathing pyrochlore lattices: structural, thermodynamic and crystal chemical aspects

The structural diversity of breathing pyrochlore lattices was investigated on the example of ordered pyrochlores in terms of group-theoretical analysis, Landau thermodynamics and crystal chemistry.

Effect of pressure on the order-disorder phase transitions of B cations in AB'1/2B''1/2O3 perovskites

Perovskite-like oxides AB'_1/2B''_1/2O₃ may experience different degrees of ordering of the B cations that can be varied by suitable synthesis conditions or post-synthesis treatment. In this work the earlier proposed statistical model of order-disorder phase transitions of B cations is extended to account for the effect of pressure. Depending on the composition, pressure is found to either increase or decrease the order-disorder phase transition temperature. The change in transition temperature due to pressure in many cases reaches several hundred kelvin at pressures accessible in the laboratory, which may significantly change the degree of atomic ordering. The work is intended to help in determining how pressure influences the degree of atomic ordering and to stimulate research into the effect of pressure on atomic order-disorder phase transitions in perovskites.

Unified View of the Local Cation-Ordered State in Inverse Spinel Oxides

Cation ordering/disordering in spinel oxides plays an essential role in the rich physical and chemical properties which are hallmarks of the structural archetype. A variety of cation-ordering motifs have been reported for spinel oxides with multiple cations residing on the octahedral site (or B-site). This has attracted tremendous attention from both experimental and theoretical communities in the last few decades. However, no unified view has been reached, presumably due to the richness of cation species and corresponding complex arrangements emergent in this large family of compounds. In this report, local cation-ordered ground states of (inverse) spinel oxides with two different cations on the octahedral site have been thoroughly investigated using neutron and X-ray total scattering, and a comprehensive theory has been proposed to explain the commonly observed cation-ordered polymorphs. It is found that a cation-zigzag-ordered structure (space group P4₁22) is the ground state for inverse spinel oxides with a pure or strong ionic lattice, while a cation-linear-ordered arrangement (space group Imma) emerges when one of the B-site cations forms very strong directional covalent bonds with lattice oxygen. The degree and length scale of cation ordering is strongly correlated with the charge and ionic radius difference between the two octahedral site cations. More complicated cation ordering schemes can be formed when there is a concomitant charge and orbital ordering which fall on a similar energy scale. This can lead to the formation of orbital-driven cation clusters or the broad concept of "molecules" in solid- state compounds. It is expected these findings will help to better understand the observed physical properties of spinel oxides and thus facilitate design strategies for improved functional materials.

Group-theoretical analysis of 1:3 A-site-ordered perovskite formation

The quadruple perovskites AA'3B4X12 are characterized by an extremely wide variety of intriguing physical properties, which makes them attractive candidates for various applications. Using group-theoretical analysis, possible 1:3 A-site-ordered low-symmetry phases have been found. They can be formed from a parent Pm{\bar 3}m perovskite structure (archetype) as a result of real or hypothetical (virtual) phase transitions due to different structural mechanisms (orderings and displacements of atoms, tilts of octahedra). For each type of low-symmetry phase, the full set of order parameters (proper and improper order parameters), the calculated structure, including the space group, the primitive cell multiplication, splitting of the Wyckoff positions and the structural formula were determined. All ordered phases were classified according to the irreducible representations of the space group of the parent phase (archetype) and systematized according to the types of structural mechanisms responsible for their formation. Special attention is paid to the structural mechanisms of formation of the low-symmetry phase of the compounds known from experimental data, such as: CaCu3Ti4O12, CaCu3Ga2Sn2O12, CaMn3Mn4O12, Ce1/2Cu3Ti4O12, LaMn3Mn4O12, BiMn3Mn4O12 and others. For the first time, the phenomenon of variability in the choice of the proper order parameters, which allows one to obtain the same structure by different group-theoretical paths, is established. This phenomenon emphasizes the fundamental importance of considering the full set of order parameters in describing phase transitions. Possible transition paths from the archetype with space group Pm{\bar 3}m to all 1:3 A-site-ordered perovskites are illustrated using the Bärnighausen tree formalism. These results may be used to identify new phases and interpret experimental results, determine the structural mechanisms responsible for the formation of low-symmetry phases as well as to understand the structural genesis of the perovskite-like phases. The obtained non-model group-theoretical results in combination with crystal chemical data and first-principles calculations may be a starting point for the design of new functional materials with a perovskite structure.

Vanadium clusters formation in geometrically frustrated spinel oxide AlV2O4

The spinel oxide AlV₂O₄ is a unique material, in which the formation of clusters is accompanied by atomic, charge and orbital ordering and a rhombohedral lattice distortion. In this work a theory of the structural phase transition in AlV₂O₄ is proposed. This theory is based on the study of the order-parameter symmetry, thermodynamics, electron density distribution, crystal chemistry and mechanisms of formation of the atomic and orbital structures of the rhombohedral phase. It is established that the critical order parameter is transformed according to irreducible representation k₉(τ₄) (in Kovalev notation) of the Fd \bar{3}m space group. Knowledge of the order-parameter symmetry allows us to show that the derived AlV₂O₄ rhombohedral structure is a result of displacements of all atom types and the ordering of Al atoms (1:1 order type in tetrahedral spinel sites), V atoms (1:1:6 order type in octahedral sites) and O atoms (1:1:3:3 order type), and the ordering of d_xy, d_xz and d_yz orbitals. Application of the density functional theory showed that V atoms in the Kagomé sublattice formed separate trimers. Also, no sign of metallic bonding between separate vanadium trimers in the heptamer structure was found. The density functional theory study and the crystal chemical analysis of V-O bond lengths allowed us to assume the existence of dimers and trimers as main clusters in the structure of the AlV₂O₄ rhombohedral modification. The trimer model of the low-symmetry AlV₂O₄ structure is proposed. Within the Landau theory of phase transitions, typical diagrams of possible phase states are built. It is shown that phase states can be changed as a first-order phase transition close to the second order in the vicinity of tricritical points of the phase diagrams.

Phenomenological thermodynamics and the structure formation mechanism of the CuTi₂S₄ rhombohedral phase

The theory of structural phase transition in CuTi2S4 is proposed. The symmetry of order parameters, thermodynamics and the mechanism of the atomic structure formation of the rhombohedral Cu-Ti-thiospinel have been studied. The critical order parameter inducing the phase transition has been found. Within the Landau theory of phase transitions, it is shown that the phase state may change from the high-symmetry cubic disordered Fd3[combining macron]m phase to the low-symmetry ordered rhombohedral R3[combining macron]m phase as a result of phase transition of the first order close to the second order. It is shown that the rhombohedral structure of CuTi2S4 is formed as a result of the displacements of all types of atoms and the ordering of Cu-atoms (1 : 1 order type in tetrahedral spinel sites), Ti-atoms (1 : 1 : 6 order type in octahedral spinel sites), and S-atoms (1 : 1 : 3 : 3 order type). The Cu- and Ti-atoms form metal nanoclusters which are named a "bunch" of dimers. The "bunch" of dimers in CuTi2S4 is a new type of self-organization of atoms in frustrated spinel-like structures. It is shown that Ti-atoms also form other types of metal nanoclusters: trimers and tetrahedra.

Anion order in perovskites: a group-theoretical analysis

Anion ordering in the structure of cubic perovskite has been investigated by the group-theoretical method. The possibility of the existence of 261 ordered low-symmetry structures, each with a unique space-group symmetry, is established. These results include five binary and 14 ternary anion superstructures. The 261 idealized anion-ordered perovskite structures are considered as aristotypes, giving rise to different derivatives. The structures of these derivatives are formed by tilting of BO6 octahedra, distortions caused by the cooperative Jahn-Teller effect and other physical effects. Some derivatives of aristotypes exist as real substances, and some as virtual ones. A classification of aristotypes of anion superstructures in perovskite is proposed: the AX class (the simultaneous ordering of A cations and anions in cubic perovskite structure), the BX class (the simultaneous ordering of B cations and anions) and the X class (the ordering of anions only in cubic perovskite structure). In most perovskites anion ordering is accompanied by cation ordering. Therefore, the main classes of anion order in perovskites are the AX and BX classes. The calculated structures of some anion superstructures are reported. Comparison of predictions and experimentally investigated anion superstructures shows coherency of theoretical and experimental results.

Unique atom hyper-kagome order in Na4Ir3O8and in low-symmetry spinel modifications

Group-theoretical and thermodynamic methods of the Landau theory of phase transitions are used to investigate the hyper-kagome atomic order in structures of ordered spinels and a spinel-like Na4Ir3O8crystal. The formation of an atom hyper-kagome sublattice in Na4Ir3O8is described theoretically on the basis of the archetype (hypothetical parent structure/phase) concept. The archetype structure of Na4Ir3O8has a spinel-like structure (space group Fd\bar 3m) and composition [Na1/2Ir3/2]16d[Na3/2]16cO32e4. The critical order parameter which induces hypothetical phase transition has been stated. It is shown that the derived structure of Na4Ir3O8is formed as a result of the displacements of Na, Ir and O atoms, and ordering of Na, Ir and O atoms, orderingdxy,dxz,dyzorbitals as well. Ordering of all atoms takes place according to the type 1:3. Ir and Na atoms form an intriguing atom order: a network of corner-shared Ir triangles called a hyper-kagome lattice. The Ir atoms form nanoclusters which are named decagons. The existence of hyper-kagome lattices in six types of ordered spinel structures is predicted theoretically. The structure mechanisms of the formation of the predicted hyper-kagome atom order in some ordered spinel phases are established. For a number of cases typical diagrams of possible crystal phase states are built in the framework of the Landau theory of phase transitions. Thermodynamical conditions of hyper-kagome order formation are discussed by means of these diagrams. The proposed theory is in accordance with experimental data.