Two different mechanisms of metal cluster formation in the

Article Cite This: Cryst. Growth Des. 2018, 18, 3433−3440

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Two Different Mechanisms of Metal Cluster Formation in the Rhombohedral Spinel Structures: AlV2O4 and CuZr1.86(1)S4 Mikhail V. Talanov* Southern Federal University, Rostov-on-Don 344090, Russia

Crystal Growth & Design 2018.18:3433-3440. Downloaded from pubs.acs.org by UNIV OF KANSAS on 01/22/19. For personal use only.

S Supporting Information *

ABSTRACT: An unusual case of the structural mechanisms of inorganic crystals formation is considered theoretically on the example of the AlV2O4 and CuZr1.86(1)S4 isosymmetric rhombohedral spinels with the same character of the atom distribution on the Wyckoff positions. It is shown that the difference in structures is due to the critical influence of the order parameter value that describes the real (in the case of AlV2O4) and virtual (in the case of CuZr1.86(1)S4) structural phase transitions. Owing to the low value of the order parameter, the phase transition to the rhombohedral AlV2O4 structure is associated with relatively small atomic displacements and the formation of heptamer structural blocks in the V sublattices. In contrast, the virtual phase transition to the rhombohedral CuZr1.86(1)S4 structure is associated with high value of the order parameter and giant atomic displacements. As a result, the Cu and Zr atoms form metal nanoclustersa “bunch” of dimers. The CuZr1.86(1)S4 is a second substance (the first example is CuTi2S4) with a new type of self-organization of atoms, in which the metal clusters are formed by the atoms from two (but not one, as in the known materials) geometrically frustrating spinel sublattices.

1. INTRODUCTION Geometrical frustration is an important feature of the electronic and structural organization of spinels giving rise to a rich variety of exotic condensed phases (such as spin liquid,1,2 spiral spin liquid,3 valence bond crystal,4,5 orbital liquid,6 orbital glass7 states, and others) and some novel phenomena (such as heavy Fermion behavior,8,9 anomalous Hall effect induced by the chiral order,10 large negative magnetoresistance,11 superconductivity,12,13 and so forth). Frustration is caused by competition between different interactions, and the energy cannot be simultaneously minimized on all interactions. The most important feature of the frustrated structures is the existence of highly degenerate ground states. The lifting of the degeneracy may be accompanied by structural deformation, atom-, charge-orbital-, spin-ordering. As results of such processes, unique states of material with unusual physical behavior may appear.1,14,15 One of the most widespread type of crystals with a geometrically frustrated sublattices is the AB2X4 spinel. The A atoms occupy Wyckoff position 8a (site symmetry 4̅3m), B − Wyckoff position 16d (site symmetry 3̅m), X − Wyckoff position 32e (site symmetry 3m) in the cubic Fd3m ̅ phase of a normal spinel. The spinel structure can be viewed as interpenetrating sublattices with B atoms, forming a threedimensional network of tetrahedra (the so-called pyrochlore sublattice), while the A atoms constitute a diamond lattice (Figure 1a). The pyrochlore sublattice (Figure 1a) can be imagined by two types of planes with alternate stackings: one © 2018 American Chemical Society

type is a two-dimensional triangular lattice (Figure 1b,c), consisting of the apical B ions of the tetrahedra, while the other type is a two-dimensional kagome lattice, consisting of the B triangles (Figure 1b).1,15,16 It has been recently demonstrated that the tetrahedral site (A site) of the normal spinel structure, containing transition element atoms, is also strongly frustrated in relation both to spins6 and orbitals.17 One of the characteristic structural phenomena, particularly observed in geometrically frustrated spinel systems, is the spontaneous formation of high-order atom structuresunusual molecular metal clusters or orbital molecules.18 Well known examples of such atom clusters in ordered spinel structures are Ti2 dimers in MgTi2O4,19,20 decagons in Na4Ir3O821,22 (such decagons are also in Na3Ir3O823 and in other ordered inorganic structures24), trimers in LiVO2,25 “trimerons” in Fe3O4,26−28 heptamers29 or trimers + tetramers30 in AlV2O4, octamers in CuIr2S4,31 tetrahedra in LiXY4O8 (X = Ga, Fe, In; Y = Cr, Rh), Cr4GaLiO8, Cr4InLiO8, Rh4InLiO8, Ag0.5In0.5Cr2S4, and Cu0.5In0.5Cr2S4,32−37 and in others. The formation of metal clusters is accompanied by the charge and/or spin and orbital orderings. All of this is a result of structural (and/or magnetostructural) transition in spinels. It is important to note that clustering in all these materials occurs only in B sublattices of the ordered spinels. Received: January 27, 2018 Revised: April 4, 2018 Published: April 25, 2018 3433

DOI: 10.1021/acs.cgd.8b00151 Cryst. Growth Des. 2018, 18, 3433−3440

Crystal Growth & Design

Article

Figure 1. Cubic spinel structure fragments. A and B cation tetrahedra in the spinel structure (a). A atoms and B atoms are shown by green and deep blue color, respectively; X anions are not shown. In the spinel structure, B sublattices are formed by a network of corner-sharing B tetrahedra (deep blue color). The B sublattice can be visualized as alternating kagome (b) and triangular (b, c) layers of the structure along a [111] direction as designated in (b). Kagome layers are shown by deep blue; triangular layers are shown by dark yellow color.

Recently, we have first found the existence of a new exotic type of high-order clusters in the CuTi2S4 rhombohedral modification with R3̅m space group. Each of these clusters is formed not only by B atoms but also by A atoms spinel also. These metal clusters were called a “bunch” of dimers.38 One can suppose that the “bunch” of dimers should exist in the CuZr1.86(1)S4 structure, since this substance is isostructural with CuTi2S4.39 The structure of the AlV2O4 rhombohedral phase is characterized by the same space group as the CuZr1.86(1)S4 structure.40 The atom distribution on the Wyckoff positions is the same for both phases, but the structures of these phases differ noticeably. This unusual and intriguing situation needs theoretical analysis. This research has two aims. One goal is a new interpretation of the clustering in the CuZr1.86(1)S4 structure. Within this interpretation, we first prove the existence of (Cu-Zr) clusters similar to clusters in the CuTi2S4 structure. The other purpose of this paper is to study the reasons for the substantial differences between isosymmetric spinel-like structures (CuZr1.86(1)S4 and AlV2O4). To realize the latter, we carry out a detailed group-theoretical and structural study of the rhombohedral phases formation mechanisms from a cubic spinel structure as well as clustering in CuZr1.86(1)S4 and AlV2O4 structures. As a result, we expect to determine two fundamentally different mechanisms of the metallic clusters formation in rhombohedral spinel modifications. In our study, we use a combination of ISOTROPY41/ ISODISTORT42 and AMPLIMODES43,44 computer software to analyze the structural phase transition from the hightemperature spinel phase with the Fd3m ̅ space group to the low-temperature phase with the R3̅m space group. In the case of AlV2O4, this structural phase transition is real and was detected by synchrotron X-ray and electron diffraction experiments at T < 70029,45 or at P > 20 GPa on hydrostatic46 and nonhydrostatic conditions,47 and also with an increase of the Cr concentration in the B sublattice at x > 0.1.48 Note that the temperature-induced phase transition Fd3̅m ↔ R3̅m accompanied by charge ordering in AlV2O4 was observed by Matsuno et al. for the first time,49 but in more recent works, the distribution of the atoms on the Wyckoff positions of R3m ̅ phase was revised. On the other hand, there are not any phase transitions in rhombohedral CuZr1.86(1)S440 and the structural phase transition from parent phase with spinel structure is virtual. By using the concepts of one critical irreducible representation (irrep)50 and the parent phase (praphase, archetype), we derive the theoretical structure (or more precisely, motif of crystal structure) of low-symmetry R3̅m structure from highsymmetry spinel structure (parent Fd3̅m phase) and find out

the different role of critical and noncritical (improper) order parameters in the rhombohedral spinel modifications formation.

2. RESULTS AND DISCUSSION 2.1. Order Parameter Symmetry. The spinel structure has a face-centered cubic lattice. For this lattice, the first Brillouin zone represents a body-centered lattice and contains four points of high symmetry, namely, k11(Γ), k10(X), k9(L), and k8(W).51 For these points, there are stars of the following wavevectors: k11(Γ) = 0; k10(X) = =

1 (b1 + b2); k 9(L) 2

1 (b1 + b2 + b3); k 8(W) = b1/4 − b2/4 + b3/2 2

There are 10 irreps of the wavevector k11(Γ) (four onedimensional, two two-dimensional, and four three-dimensional); four six-dimensional irreps of the wavevector k10(X), four four- and two eight-dimensional irreps of the wavevector k9(L); and two 12-dimensional irreps of the wavevector k8(W). Thus, totally there are 22 irreps. All irreps of the wavevector k8(W) and also two irreps of the wavevector k10(X) (τ3 and τ4) do not satisfy the Lifshitz criterion; i.e., they induce incommensurate phases.50 Since the considered rhombohedral structure is commensurate phase, we exclude these irreps from our analysis. On the basis of the results of the group-theoretical analysis of the phase transitions occurring according to one critical irrep in 50 the group Fd3m ̅ , which satisfies the Lifshitz criterion, we find that the phase with the R3̅m space group may be generated by three-dimensional irrep k11(τ7), as well as by two sixdimensional irreps k10(τ1) and k10(τ3), two four-dimensional irreps k9(τ1) and k9(τ4). The expression kj(τi) means the star of wavevectors kj, where i is the number of corresponding irrep τ for a given star j (according to Kovalev51). The correspondence of notation by Kovalev51 and Miller and Love52 is as follows.53 The analysis shows that, only in the case of phase, generated by irrep k9(τ4), the calculated distribution of the atoms on the Wyckoff positions of R3̅m phase is consistent with experimental data on AlV2O429 and CuZr1.86(1)S4.40 As a result of group-theoretical calculations, the theoretical structure formula of the R3̅m low-symmetry rhombohedral spinel modification should be A2c1/2(1)A2c1/2(2)B1a1/4(1)B1b1/4(2)B6h3/2(3)X2c1/2(1)X2c1/2(2)X6h3/2(3)X6h3/2(4) (the rhombohedral presentation) or A6c1/2(1)A6c1/2(2)B3a1/4(1)B3b1/4(2)B18h3/2(3)X6c1/2(1)X6c1/2(2)X18h3/2(3)X18h3/2 (the hexahedral presentation). The theoretical formula agrees with experimental data.29,40 3434



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Table 1. Role of Critical and Improper Order Parameters in the AlV2O4 and CuZr1.86(1)S4 Rhombohedral Phases Formationa amplitudes of global distortions (Å)

a

irrep

dimensions

order parameter direction

k wavevectors

isotropy subgroups

AlV2O4

CuZr1.86(1)S4

k11(τ1) k11(τ7) k9(τ4) total

1 3 4

(a) (a a a) (a 0 0 0)

(0 0 0) (0 0 0) (1/2 1/2 1/2)

Fd3m ̅ (227) R3̅m (166) R3̅m (166) R3̅m (166)

0.06090 0.07323 0.37003 0.38209

2.25228 3.87712 11.49193 12.33569

The amplitudes normalized with respect to the primitive unit cell of the parent phase.

Figure 2. Transformations of pyrochlore (a−c) and diamond (d, e) sublattices of the Fd3̅m cubic spinel structure into sublattices of R3̅mrhombohedral spinel modification. As a result of the phase transition B sublattice of spinels (pyrochlore sublattice) splits into layers that are mutually interconnected by B(1) and B(2) atoms (only in AlV2O4) (a). In this case, [B(1)B(3)6] and [B(2)B(3)6]-heptamer structural blocks are formed too (only in AlV2O4). Kagome sublattices in low-symmetry R3̅m phase have two types of [B(3)2] dimers (dark blue and red) and two types of [B(3)3] trimers (dark blue and red) with different B(3)−B(3) interatomic distances (red, long; blue, short) (b). Triangular sublattices, formed by the apical B atoms in a cubic spinel structure, are transformed into two types of triangular sublattices in a rhombohedral structure (c). One type of sublattice is formed by B(2) atoms (blue), and another type is formed by B(1) atoms (deep blue). Triangular diamond sublattice, formed by tetrahedral A atoms in a cubic spinel structure, splits in the R3̅m-rhombohedral structure into two types of triangular sublattices too; namely, one type of triangular sublattice is formed by A(1) atoms (dark green) and another type is formed by A(2) atoms (yellow) (d, e). The disposition of the layers formed by A(1) and A(2) atoms is shown in (e).

• binary tetrahedral cation ordering (the type of order is 1:1). • ternary octahedral cation ordering (the type of order is 1:1:6). • quarternary anion ordering (the type of order is 1:1:3:3). The contribution of a critical irrep in crystal structure formation defines the symmetry of the low-symmetry phase completely. However, when the low-symmetry structure is far from the phase transition temperature, then the contribution of improper representations can become essential.54 Here, the interpretation of some experimental data cannot be carried out only by means of a critical irrep. A method of finding improper atomic displacements and ordering is proposed.54 In the case of the Fd3̅m → R3̅m phase transition, besides the critical order parameter k9(τ4), the total condensate of the

2.2. Structural Mechanisms of Rhombohedral Modifications Formation. The physical meaning of the order parameters that are transformed according to the corresponding critical and improper irreps is determined by their entry into the mechanical and permutation reducible representations of the high-symmetry structure of the parent Fd3̅m phase. The critical irrep k9(τ4) enters into both the mechanical representation and the permutation representation of the spinel structure on the Wyckoff positions 8a, 16d, and 32e.22 This means that the low-symmetry R3̅m phase formation is the result of the tetrahedral and octahedral cations and anions displacements as well as the ordering of all atom types. Grouptheoretical analysis showed that the formation of the R3̅mrhombohedral phase is due to the following types of atom ordering: 3435



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2a,b). A two-dimensional triangular sublattice, consisting of the apical octahedral B ions in cubic phase, split into two sublattices, formed by B(1) and B(2) ions (Figure 2c). The diamond spinel A sublattices are transformed into two type A(1) and A(2) sublattices in the rhombohedral structure (Figure 2d,e). In the kagome sublattice, there are two kinds of B(3)−B(3) bonds (B dimers: [B(3)2]) with different bond lengths: the shorter (dark blue; 2.610 Å for AlV2O4 and 3.579 Å for CuZr1.86(1)S4) and the longer (red; 3.141 Å for AlV2O4 and 3.773 Å for CuZr1.86(1)S4) (Figure 2a,b). These dimers alternate in the kagome sublattice (Figure 2a,b). It is important to underline that the B−B distances in the cubic form of AlV2O4 and CuZr 1.86(1) S 4 are equal to 2.896 and 3.670 Å, respectively.29,40 The kagome layers, between which the B(2) atoms are located, are shifted relative to each other in CuZr1.86(1)S4 (Figure 2a). As a result, the formation of heptamers (as structural blocks, but not as orbital molecules) linking kagome layers in CuZr1.86(1)S4 structure is observed only with B(1) atoms, while, in the AlV2O4 structure, the formation of heptamers with both B(1)- and B(2)-binding atoms is possible (Figure 2a). In addition, the triangle layers A(1) and A(2) atoms are shifted relative to each other in the CuZr1.86(1)S4 structure (Figure 2d,e). Table 2 shows the absolute atom displacements at the forming of the low-symmetry phases AlV2O4 and CuZr1.86(1)S4.

order parameters includes also two improper order parameters with the wavevector k11 = 0 from the center of the Brillouin zone: k11(τ1) and k11(τ7). The improper order parameters are due to nonlinear interactions between different degrees of freedom in the crystal. They cause additional distortions of the crystal structure, which can be significant in magnitude far from the phase transition temperature. Improper irrep k11(τ7) enters into the mechanical representation on the Wyckoff positions 8a and 32e and into the permutation representation on the Wyckoff positions 16d and 32e. This means that we can expect additional displacements of tetrahedral cations and anions and ordering of octahedral cations (the type of order is 1:3) and anions (the type of order is 1:3). Improper irrep k11(τ1) enters into the mechanical representation on the Wyckoff position 32e and into the permutation representation on the Wyckoff positions 8a, 16d, and 32e. This means that additional ordering of all types of atoms is possible, but additional displacements can be expected for anions only. 2.3. Exploring the Role of Critical and Improper Order Parameters to the Rhombohedral Spinel Modifications Formation. We calculated the contribution of the critical and improper order parameters to the formation of rhombohedral phases (Table 1). The contributions of critical and improper order parameters to the low-symmetry crystal structures formation of AlV2O4 and CuZr1.86(1)S4 differ significantly. In the case of AlV2O4, the typical picture for second order phase transitions is observed; i.e., the contribution from the critical order parameter to the displacement amplitude is dominant, and the atom displacements are relatively small: global distortion (total displacement of all atomic positions) is 0.38209 Å. The absolutely different situation is in CuZr1.86(1)S4. Indeed, there are giant atomic displacements (global distortion is 12.33559 Å) from the parent phase with a large contribution from the improper order parameters k11(τ7) and k11(τ1) at the hypothetical/virtual Fd3̅m → R3̅m phase transition. Such giant displacements practically do not occur during phase transitions in inorganic compounds. Therefore, it is no wonder that the cubic phase has never been obtained for CuZr1.86(1)S4 (the rhombohedral phase of CuTi2S4 is detected in the compound prepared by using a KCl/KI flux at low temperatures, but an irreversible phase transition to the cubic phase occurs at above 450 °C).39 One should emphasize that the choice of a cubic spinel Fd3̅m phase as a parent phase for describing the transition to a CuZr1.86(1)S4 rhombohedral structure is important for comparing the mechanisms of the low-symmetry phases formation in the two title compounds under study. Therefore, the absolute values of the theoretical displacements in the formation of the CuZr1.86(1)S4 rhombohedral phase have an exclusively conditional value. Thus, in the formation of rhombohedral R3̅m modifications of spinels, two different mechanisms compete, namely, with small (AlV2O4) and giant (CuZr1.86(1)S4) atomic displacements. These two mechanisms cause the existence of two types of rhombohedral phases with different metal clusters. 2.4. Differences in the AlV 2 O 4 and CuZr 1.86(1) S 4 Rhombohedral Structures. Let us consider the changes in the pyrochlore and diamond spinel sublattices resulting in R3m ̅ spinel modifications formation (Figure 2). A kagome sublattice in the rhombohedral structure, formed by B(3) ions, has become trimerized or breathing kagome sublattice (Figure

Table 2. Atomic Displacements in the Transition Fd3̅m → R3̅ma displacements (Å)

a

atoms

Wyckoff position

AlV2O4

CuZr1.86(1)S4

Al(1)/Cu(1) Al(2)/Cu(2) V(1)/Zr(1) V(2)/Zr(2) V(3)/Zr(3) O(1)/S(1) O(2)/S(2) O(3)/S(3) O(4)/S(4)

6c 6c 3a 3b 18h 6c 6c 18h 18h

0.13056 0.09214 0.00000 0.00000 0.15721 0.03755 0.14587 0.08197 0.04048

0.03941 1.54078 0.00000 0.00000 6.08456 0.09047 3.13621 0.04341 3.04844

In bold type are the groups of atoms with the largest displacements.

The formation of the rhombohedral phase of CuZr1.86(1)S4 is associated with much more noticeable changes than in AlV2O4 compared to the parent phase structure. The largest displacements occur with the atoms A(2), B(3), X(2), and X(4). The amplitudes of their absolute displacements in thiospinels exceed 1 Å. An important feature of the phase transition Fd3̅m → R3̅m in the above compounds is the absence of displacements of the B(1)3a and B(2)3b atoms, as well as the displacement of cations and anions in the Wyckoff position 6c along only the z axis (Supporting Information). Note the giant displacements of Zr(3) atoms from neighboring kagome layers toward each other. As a result, the kagome layers shift relative to each other, and [Zr(2)Zr(3)6] structural blocks are not formed, while [V(2)V(3)6] in AlV2O4 are formed (Figure 3a). Cu(2) and Zr(3) atoms form a “bunch” of dimers (Figure 3a). Each “bunch” consists of three [Cu(2)− Zr(3)] dimers which are joined by a common Cu(2) atom. Each Cu(2) atom is surrounded by three Zr(3) atoms, resulting in three Cu(2)−Zr(3) bonds with interatomic distances of 2.937 Å. The shortest Cu−Zr distance in the cubic spinel is 3436



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significantly from the remaining atoms.47 The existence of voids where [B(2)S(3)6] octahedra are located should be taken into account when designing thermoelectric materials based on the rhombohedral thiospinels. Another important difference in the rhombohedral structures of AlV2O4 and CuZr1.86(1)S4 is the mutual arrangement of coordination polyhedrons formed around the atoms B(2) and B(3) (Figure 3c). In the AlV2O4 structure, they have common edges, and in the CuZr1.86(1)S4 structure, they have common vertices. This significant transformation in the local environment of transition metal atoms determines electronic interactions and, as a consequence, the formation of charge, orbital, and spin degrees of freedom.55 As a result, the localized ground states can differ drastically that must be taken into consideration when choosing the materials for spintronics and orbitronics. Previously, the authors40 suggested that the structural formula of CuZr1.86(1)S4 rhombohedral modification that considers atom valence states (obtained from the crystalchemical analysis of the crystal bond lengths and electronic structure calculations) should be (Cu+)4Zr(1)4+Zr(2)3+Zr(3)3.5+6(S2−)16. This formula means that the Zr(1)4+ ion has no t2g orbitals; i.e., there are no chemical bonding between Zr(1) and Zr(3) atoms. Therefore, the formation of Zr heptamers is improbable. Besides, the calculated densities of states of rhombohedral modification show that Cu(1)−Zr(3) and Zr(3)−Zr(3) interactions in “bunch” of dimers and [Zr(3)3] trimers (with interatomic distances Zr(3)−Zr(3) 3.579 Å40) have bonding character.40 We consider the formation of clusters only on the basis of the analysis of the average structure and interatomic distances obtained from the experimental diffraction data. Therefore, in this case, heptamers in AlV2O4 are only the structural units, i.e., building blocks. We believe that, in AlV2O4, only a fraction of the [V(2)V(3)6] heptamers, namely, [V(3)3] trimers with the interatomic distances V(3)−V(3) equal to 2.610 Å, can be considered as orbital metal molecules. In this orbital molecules, the lengths of the V(3)−V(3) bonds are even lower than the V−V bond length in the metal vanadium (2.62 Å). According to Goodenough,56−58 there is a critical cation−cation distance Rc in oxide, dividing the regions of localized and itinerantelectron behavior. Note that Rc for interatomic distances V−V is 2.62 Å. Therefore, we believe that the nature of the V(3)− V(3) bond in the [V(3)3] trimers is close to the metal bond. Recently, Browne et al.,30,59 based on an analysis of the pair distribution functions of the AlV2O4 and its analogue GaV2O4 structures, showed that some local distortions in the above structures do not agree with that the heptamers are orbital molecules.29 Thus, in the of rhombohedral spinels structure, two mechanisms of formation of metal clusters are observed. The above mechanisms are explained by various displacement amplitudes for a real or hypothetical phase transitions from a cubic structure. With small distortions in the structure of the cubic spinel, [B(2)B(3)6]-heptamer structural blocks are formed, giving rise to [V(3)3] trimers60 or (trimers + tetramers) local molecules30 as in AlV2O4. At large amplitudes of distortions, giant displacements of B(3) atoms occur and no [B(2)B(3)6]-heptamer structural blocks are formed, whereas a new type of clusters, “bunches” of dimers [Cu(1)Zr(3)3], is formed. As mentioned above, all known clusters in the structures of the low-symmetry phase of spinels are formed by atoms of only

Figure 3. Cations A(1) (green) and B(2) (blue) located between the kagome layers of cations B(3) (gray) (a). Anions are not shown. Kagome layers and the B(2) coordination polyhedra [B(2)X(3)6] along the [001] direction (b). The coordination polyhedra around B(2) and B(3) octahedra: O(1)/S(1) (red), O(2)/S(2) (purple), O(3)/S(3) (orange), O(4)/S(4) (dark purple) (c).

much longer and equals 4.304 Å. This distance is too large for a significant interaction, while the Cu(2)−Zr(3) interatomic distance in the rhombohedral form of CuZr1.86(1)S4 is the shortest one among all the metal−metal bond lengths. Note that Zr(1) and Zr(3) atoms also form [Zr(1)Zr(3)] dimers, but the interatomic distance 3.787 Å (on Figure 2a, they are shown blue) is much longer. Thus, the “bunches” of [Cu(2)Zr(3)] dimers are the smallest-sized clusters. It is interesting to note that only contracted [Zr(3)3] trimers (with shortest interatomic distances Zr(3)−Zr(3)) belong to the “bunch” of dimers. Thus, it has been found that CuZr1.86(1)S4 is the second compound (after CuTi2S438) in which such type of metal clusters exists. In the CuZr1.86(1)S4 rhombohedral thiospinel structure, Zr(2) atoms occupy a position being equidistant from 12 Zr(3) atoms with a considerable distance between them that practically excludes the possibility of a chemical bonding between them (for comparison, in CuTi2S4, distance Ti(2)−Ti(3) = 4.488 Å39) (Figure 3a,b). As a result, the octahedron Zr(2)S(3)6 occupies a position in the voids (Figure 3a) formed as a result of the kagome layers shift, and the Zr(3) atoms form a peculiar cage-skeleton, causing the rattling effect. Such local environment of B(2) atoms in the rhombohedral CuZr1.86(1)S4 and CuTi2S4 structures is assumed to result in the high values of isotropic displacement factors observed.39,40 In contrast, the V(2) atoms in the AlV2O4 rhombohedral structure, as assumed in ref 29, are to be a part of the [V(2)V(3)6]-heptamer structural blocks, and the [Al(2)O(3)6] octahedron is more “clamped” between the two kagome layers (Figure 3b) that leads to the limitation in their thermal vibrations. And the values of their isotropic displacement factors do not differ 3437



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one sublattice, while “bunches” of dimers are formed by atoms of two frustrating sublattices (kagome and triangular sublattices). Therefore, the substances, in the structures of which similar clusters exist, are expected to have unique physical properties, above all magnetic and electrical ones. In particular, one should note that the interaction between the A and B cations will lead to an unusual spiral short-range magnetic order. This magnetic order can induce ferroelectricity as proposed in CoCr2O4.61

clusters. This suggests that CuZr1.86(1)S4 and CuTi2S4 are prototypes of a new class of materials with possible exotic phases as well as unexpected intriguing physical properties. Also, our study showed for the first time that the values of critical and improper (noncritical) order parameters will be considered as a new source of structural diversity of materials.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.8b00151. Atomic representations of AlV2O4 and CuZr1.86(1)S4 structures. Calculation details (ISODISTORT: modes details for AlV2O4 and CuZr1.86(1)S4) (PDF)

3. CONCLUSIONS The main result of the work is to establish the general reasons (group-theoretical and structural) for the significant differences in AlV2O4 and CuZr1.86(1)S4 structures. It is proved that different values of the critical and improper order parameters explain the formation of various structures. In the case of the AlV2O4 rhombohedral modification, the atomic displacements are insignificant. The role of improper order parameters in the phase formation is also insignificant. In this case, Landau thermodynamics, which operates with small in magnitude order parameters, is a quite suitable tool for describing of the structural transition and the rhombohedral phase formation mechanism. The situation changes completely in the case of the CuZr1.86(1)S4 rhombohedral modification. If a phase transition from a cubic spinel to a rhombohedrally distorted spinel occurred in this substance, then the displacements of the atoms, according to our calculations, would have to be giant. At the same time, the displacements generated by improper (secondary) parameters of the order of k11(τ7) and k11(τ1) should contribute significantly to the calculated value of the displacements of Cu(2), S(2), and S(4) atoms. The various values of the critical and the improper order parameters describing the formation of rhombohedral structures are caused by the crystal-chemical features of the compositions and possibly by the methods of sample preparation. This is an interesting topic for further research. This study first proposed a structural mechanism of the lowsymmetry rhombohedral modifications formation from highsymmetry cubic spinel phase, and showed that the calculated low-symmetry rhombohedral phase is due to displacements and the ordering of all types of atoms. An important result of the work is that a “bunch” of [Cu(1)Zr(3)] dimers, two types of Zr(3) trimers [Zr(3)3] exist in CuZr1.86(1)S4. The discovery of metal nanostructures, such as [Cu(1)Zr(3)] dimers, trimers [Zr(3)3] with the shorter bond lengths, and a “bunch” of [Cu(1)Zr(3)] dimers, is of particular interest. The metallic dimers and trimers in geometrically frustrated structures (for example, dimers: MgTi2O4,19,20,62 CuTe2O5,63,64 VO2,65 CuGeO3;66 trimers: LiVO2,25 LiVS2,67 NaV6O10,68 BaV10O15,69−72 SrV8Ga4O19,73 AV13O18 (A = Ba, Sr),74 A2V13O22 (A = Ba, Sr),75 Ba4Ru3O1076) have long been known. The “bunch” of [Cu(1)−Zr(3)] dimers in CuZr1.86(1)S4 is a new type of self-organization of atoms in geometrically frustrated spinel-like structures. It is vital to stress that metal clusters in all known spinel materials are formed by the atoms with only one geometrically frustrating sublattice. CuZr1.86(1)S4 and CuTi2S4 rhombohedral phases are the only examples of the crystals in which the metal clusters are formed in two geometrically frustrating spinel sublattices. The appearance of two sublattices means new combinatorial possibilities for the selection of cations in these sublattices. Therefore, multiple scenarios of interaction between two frustrating sublattices are possible in crystals having similar

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Mikhail V. Talanov: 0000-0002-5416-9579 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The reported study was funded by RFBR, according to the research project No. 16-32-60025 mol_a_dk. REFERENCES

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Two different mechanisms of metal cluster formation in the

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