Structural Principles of Ion-Conducting Mineral-like Crystals with Tetrahedral, Octahedral, and Mixed Frameworks

Abstract

Materials with high ion mobility are widely used in many fields of modern science and technology. Over the last 40 years, they have thoroughly changed our world. The paper characterizes the structural features of minerals and their synthetic analogs possessing this property. Special attention is paid to the ionic conductors with tetrahedral (zincite- and wurtzite-like), octahedral (ilmenite-like), and mixed (NASICON-like) frameworks. It is emphasized that the main conditions for fast ionic transport are related to the size and positions occupied by a mobile ion, their activation energy, the presence and diameter of conduction channels running inside the structure, isomorphic impurities, and other structural peculiarities. The results of the studies of solid electrolytes are dispersed in different editions, and the overview of new ideas related to their crystal structures was the focus of this paper.

1. Introduction

The first references to the high conductivity of ionic crystals date back to the early 19th century [1]. In 1833, M. Faraday noted [2] the anomalously high electrical conductivity of PbF₂ and Ag₂S, comparable to metals. Both compounds are characterized by existence of polymorphic modifications. Fluorocronite, PbF₂, is a cubic fluorite-type natural analog of β-PbF₂, which is stable at environmental conditions and shows a conductivity 2.4 S·cm⁻¹ at 1000 K. Orthorhombic α-PbF₂ with cotunnite (PbCl₂) structure is more stable at higher pressures. Ag₂S has three polymorphs: low-temperature monoclinic (space group P2₁/c) α-Ag₂S (natural analog acanthite) exists at temperatures below 450 K, body-centered cubic (space group Im-3m) superionic β-Ag₂S (argentite) exists in the temperature range of 452–860 K, and the high-temperature face-centered cubic (space group Fm3m) γ-Ag₂S phase is stable at T > 860 K [3]. The conductivity of β-Ag₂S at room temperature is (1.3–1.6) × 103 S·cm⁻¹, and at elevated temperature range it exceeds 5 S·cm⁻¹ [4,5]. A similar effect in oxide materials was discovered by W. Nernst in 1897, who used ceramics based on zirconium oxide ZrO₂ doped with yttrium (Y₂O₃) as a material for incandescent lamps [6].

In the early 20th century, scientists proved that the high conductivity of such substances was due to the movement of different ions rather than electrons, as observed in liquid electrolytes. Under normal conditions, the ion transport in ordinary solids—both crystalline and amorphous—is not very significant, and at room temperature, the specific conductivity does not exceed 10⁻¹⁰–10⁻¹² S·cm⁻¹ (the siemens (symbol: S) is the unit of electric conductance in the International System of Units (SI); one siemens is equal to the reciprocal of one ohm (Ω⁻¹)). There exist many solids with high ionic conductivity > 10⁻⁴ S·cm⁻¹ [7]. There are two main groups of crystals that exhibit this physical property. The first group includes superionic conductors, also called fast ion conductors or solid electrolytes, whose conductivity is several orders higher and lies in the range of 10⁻³ S·cm⁻¹ to 10 S·cm⁻¹. These values are very high for crystalline ionic solids but are still lower than those of many electronic conductors, such as metals, which have typical values ranging from 10 to 10⁵ S·cm⁻¹ [8].

Superionic conductors have high ionic conductivity because of strong disorder in the sublattice of conducting ions. There are crystals with so-called impurity-induced structure disorder. In these solids, the conductivity is due to a high concentration of impurity ions, which promote the disordering of the structure.

The main general requirements for the practical use of solid electrolytes are as follows: (1) High ionic conductivity; (2) Negligible electronic conductivity; (3) Stability with respect to adjacent phases and thermal and electrochemical decomposition [9].

High ionic diffusion is observed in the second group of crystalline solids, the so-called intercalation compounds. Such compounds are formed by the injection of guest ions from the electrolyte and their charge-compensating electrons into a solid framework. The materials are mixed, ionic, and electronic conductors, so they can be used as electrode material for metal-ion batteries. The different aspects of mineralogical crystallography, composition, and physical properties of these compounds were considered in [10]. The ionic conductivity of certain crystals is now widely used in metal-ion power supplies in the military, medical, household, and industrial electronic devices. The structural types of a number of minerals have been found to be intrinsic to most materials possessing ionic conductivity [11,12]. The aim of this paper is devoted to the structure peculiarities and diffusion mechanism in three types of mineral-like fast ion conductors with the tetrahedral, octahedral, and mixed frameworks.

2. Basic Structural Requirements for Ionic Conductivity

The existence of superionic conductivity depends largely on the structural features of the material. Here are just the basic ones [13]:

• For ions to be able to move in the unit cell, the number of crystallographic positions energetically close to their sites must be greater than the number of ions themselves.
• The energy of disorder of ions in the crystal lattice and the energy spent on their movement must be small.
• The ease with which an ion can move to a neighboring site is controlled by the activation energy. The activation energy is the main factor influencing ionic conductivity.
• In the crystal structure, the channels for ion motion must have commensurate size with their ionic radii.

The above requirements are satisfied only by special crystals, in the structure of which atoms of one or several varieties partially occupy their positions. This opens up the possibility for their movement under changing external physical and chemical conditions. Materials with high ion mobility are used in many fields of science and technology. It can be said that over the last 40 years, they have changed our world significantly, occupying the most important place in the development of compact current sources and batteries, which were first used in miniature headphones and now in electric vehicles [14].

A few examples which confirm the aforementioned statements are reported below. Synthetic analogs of Na₂TiSiO₅ natisite, a typical mineral of ultraagpaites and hydrothermalites, with the compositions Na₂TiGeO₅ and Li₂TiGeO₅ [15,16] can serve as an illustration of the structural condition of ionic conductivity. The crystal structure of Na₂TiGeO₅ is shown in Figure 1. Perpendicular to the [001] axis, layers of tightly bound Ge-tetrahedra [GeO₄] and square pyramids [TiO₅] are clearly visible. In the space between the layers, mobile Na ions (pink spheres) can move freely. As can be seen in Figure 1b, the conductivity, parallel to the a-axis, is 10³–10⁴ times higher compared to the movement of alkali cations along the c-axis.

The pearceite-polybasite group of minerals with general formula (Ag,Cu)₁₆M₂S₁₁, where M = Sb and As, can also be considered as a homogeneous series with a fast ion conducting form at high temperature [17,18]. Their structures are described as a regular succession of two module layers stacked along the c-axis: a first module layer (labeled A), with general composition [(Ag,Cu)(As,Sb)₂S₇]²⁻, and a second module layer (labeled B), with general composition [Ag₉CuS₄]²⁺ (Figure 2). The observed structural disorder in the B layer is strongly related to the fast ion conduction character exhibited by these minerals. In this module layer, Ag ions can adopt linear to tetrahedral coordination, and they have numerous available sites. Those sites are very close to each other, making a diffusion path in the plane possible (001). Thus, the conductivity in polybasite 222 increases from 5.3·10⁻⁶ S/cm at 250 K to 1.8·10⁻² S/cm at 385 K. A similar trend is observed in pearceite, where the values of conductivity increase from 4.5·10⁻⁵ S/cm to 8.2·10⁻³ at 250 K and 385 K, respectively [17].

As shown in Figure 2, there are two individual Ag⁺ layers forming a coupled Ag⁺ bi-layer hexagonal [Ag^I₉]⁹⁺ sub-structure in these mineral solid electrolyte systems within each B module with [Cu^IS₄]⁷⁻ slab located in between [19]. The motion of any one Ag⁺ ion necessarily impacts the neighboring Ag⁺ ions, suggesting that they must somehow move in a distinctly correlated fashion. However, the diffusion of Ag⁺ ions results in the mutual incommensurability of the two-component sub-structures. Regarding the use of electron diffraction, two possible modes of incommensurability were discussed: (i) The hexagonal layers with Ag⁺ ions should, as far as possible, relate to one another as close-packed layers; (ii) To maintain physically reasonable separation distances between Ag⁺ layers across the bi-layer is to slightly displace them along the c-direction, a feature that was seen in the average structure refinement [18,19].

Another extremely interesting example of a direct connection between structural features and ionic conductivity is the Na₅YSi₄O₁₂ crystal. The characteristic features of this structure are 12-membered (Si,O)-rings of silicon-oxygen tetrahedra (Figure 3) [20].

No one at that time paid attention to another feature of this structure. According to the symmetry of the cell, it should contain 90 Na atoms. However, regarding the occupancy of the found positions, only 48 atoms were localized. Later, in 1978, a group of researchers led by R.D. Shannon suggested [22] that this discrepancy should be associated with the possible movement of Na atoms within the structure, i.e., with ionic conductivity. The positions of these mobile Na atoms are shown in Figure 3b. As a result, it turned out that static Na atoms are located inside the (Si,O)-rings, while mobile ones are located between the columns formed by them inside the channels, the width of which is determined by the size of the octahedra of rare-earth cations. Furthermore, it was shown that the conductivity increases with increasing radius of R³⁺ cation, as well as with increasing temperature (Figure 4).

At the same time, in Na₅MSi₄O₁₂ crystals, where M = TR, Fe, In, Sc, Y, the high values of conductivity were obtained. They range from 2·10⁻³ S·cm⁻¹ in Na₅ScSi₄O₁₂ (r Sc = 0.885 Ǻ) to 3·10⁻¹ S·cm⁻¹ in Na₅SmSi₄O_I2 (r Sm = 0.958 Ǻ) at 300 °C.

The results obtained using X-ray diffraction methods are extremely important for understanding the structural requirements of ionic conductivity in crystals. The structure of K₃NdSi₆O₁₅, determined in 1977 [23] (Figure 5), confirms this statement.

The first result of this work was the discovery of the original dimeta-silicate layer [Si₆O₁₅] containing four-, six- and eight-membered silicon-oxygen rings, similar to those found in rare mineral dalyite K₂ZrSi₆O₁₅ [24]. At the same time, rather high values of isotropic thermal parameters of alkali K⁺ cations in the range of 3.90–5.46 Ų did not provoke a special interest.

After 16 years, this very circumstance allowed the authors of [25] to conclude that this material, based on structural considerations, was a likely fast ion conductor. All features of this interesting structure were confirmed, but its refinement in the anisotropic approximation [25] allowed to establish that K1 thermal displacement along one of the three coordinate axes was twice as large as the other two (×10⁻²) Ų: U11 = 5.9, U22 = 5.4, U33 = 10.4. The projection of this structure along [001] clearly indicates the presence of channels in which K1 cations are located and which are considered as probable directions for ionic conduction. Accordingly, K1 cations are most mobile along [001], which is confirmed by the values of the principal elements of the specific electrical conductivity tensor (10⁵ S·cm⁻¹) at 600 °C [26] σ11 = 3.4, σ22 = 6.9, σ33 = 17. Thus, the above-mentioned examples indicate the main crystallochemical criteria inherent in ion-conducting crystals.

3. Ion-Conducting Materials with Tetrahedral Framework of Zincite-Wurzite Type. Li₃PO₄-Polymorphs and Their Relationship with Mineral Lithiophosphate Li₃PO₄

Despite its simple composition, the structural chemistry of the compound Li₃PO₄ is quite complex. It forms three modifications—α, β, and γ [27,28]. The transformation of the β-modification into the higher-temperature γ-form occurs at 500 °C. Further γ-modification at 1170 °C is transformed into α-phase, which then experiences melting at 1220 °C. Thus, α-Li₃PO₄ is stable in a very narrow temperature range.

The main difference between the structures of β-Li₃PO₄ and γ-Li₃PO₄ is presented in Figure 6. In the structure of β-Li₃PO₄ with space group Pmn2₁ (structural type of zincite ZnO and wurtzite ZnS), all tetrahedra are oriented equally, whereas in the γ-modification (space group Pmnb), both types of tetrahedra have opposite orientation along [001]. This is the reason why the tetrahedra in the γ-Li₃PO₄ structure are interconnected not only by vertices but also by edges (Figure 6b), and the unit cell parameter b doubles with respect to β-Li₃PO₄. Lithiophosphate (Li₃PO₄, space group Pmn2₁) can be supposed to be a natural form of low-temperature β-lithium orthophosphate. Besides the same space group both phases have similar cell dimensions. However, their similarity must be validated by the results of the structural study of litiophosphate. Because of the reversibility of the γ→α transition, attempts to fix α-Li₃PO₄ by quenching have not been successful. Therefore, the question of its structure remains open.

Lower valent cation substitution of P in Li₃PO₄ by Si or Ge requires charge balance through the introduction of additional Li⁺-ions, which are located in the interstitial octahedral sites. These derivative compounds, Li₄SiO₄ and Li₄GeO₄, have high ionic conductivity due to the peculiarity of their tetrahedral framework structures. The closest hexagonal package of anions O²⁻ is in their basis. Li⁺ ion has a small radius of 0.59 Å, which allows it to move in the relatively narrow voids of crystal structures. The choice of lithium as a charge carrier is also based on its other properties. Lithium is the lightest metal: it is twice as light as water! Its density of 0.534 g/cm³ is about the same as that of wood [30]. Thus, its properties, charge, high electrochemical potential, and ability to move easily in crystal structures are associated with the use of this element in lithium-ion batteries, which provide the highest gravimetric and volumetric energy densities. Therefore, the isomorphic replacements within the γ-Li₃PO₄ structure type are the focus of many research works, and the corresponding group of compounds is named LISICON (Some authors incorporate into LISICON group all materials with Li-conductivity but not only those with tetrahedral frameworks). For example, substitutions of Si⁴⁺ and Li⁺ ions in Li₄SiO₄ with other cations such as Ge⁴⁺, P⁵⁺, or Zn²⁺ could lead to significant improvement in conductivity, and Li₁₄ZnGe₄O₁₆ has the highest value of about 10⁻¹ S·cm⁻¹ at 300 °C.

In lithium superionic conductors (LISICONs) with γ-Li₃PO₄-type structure, Li atoms occupy two non-equivalent positions. However, in crystals with multivalent cations isomorphically substituting P⁵⁺ in tetrahedra, such as Li_3.68(Ge_0.6V_0.36Ga_0.04)O₄ [31], Li-sites are split, and a degree of disorder of Li⁺ cations increases. Consequently, they can move more easily within the structure, and the conductivity increases by an order of magnitude.

In total, there are 6 Li positions in the above-mentioned structure (Figure 7a). Two of them with small occupancy are near the two main ones, and the other two are located in channels, acquiring an octahedral coordination that is too large for them. Thus, Li_3.68(Ge_0.6V_0.36Ga_0.04)O₄ exhibited the highest ionic conductivity (1.5 × 10⁻⁴ S·cm⁻¹) at 298 K, with extremely low activation energy (~0.26 eV).

In the superionic conductor Li_3+xP_1−xGe_xO₄ (x = 0.34), two basic Li sites are partially filled with occupancies 0.85 and 0.83. In addition to these, there are four positions of Li cations with relatively low occupancies. Two of these Li⁺ cations are located near the main positions at distances of 0.86 and 0.81 Å, respectively, whereas two other Li atoms occupy octahedral cavities in the framework. Evidently, these cavities are too large for small Li cations. One of them has a large value of the atomic displacement parameter, and another one shows the replacement of octahedral coordination by a pseudo-tetragonal pyramidal one [32]. In this compound, all six Li atoms have population coefficients less than unity, which opens the possibility for their transport, determining high cationic conductivity (Figure 7b) with an increase in temperature. The values of specific conductivity for Li_3.34P_0.66Ge _0.34O₄ are 1.8·10⁻⁶ and 3.7·10^–2 S·cm⁻¹ at 40° and 400 °C, respectively [32].

The number of additional Li positions depends on the variety of cations substituting for P. In the structure Li_3.17(P_0.69Ge_0.24Mo_0.07)O₄ [32], due to the presence of highly covalent Mo as an isomorphic impurity, the Li content decreases, and the number of positions occupied by it decreases to 5 in the γ-Li₃PO₄-type structure (Figure 7c).

The similarity of the structure of γ-Li₃PO₄ with mineral-like electrode materials for metal-ion batteries of the triphylite LiFePO₄ group was noted in [33]. Both structures contain hexagonal closest packing of O²⁻ anions, and in both cases, phosphorus atoms occupy 1/8 of the tetrahedral voids (Figure 8) with their very close spatial arrangement.

It is no coincidence that Li₃PO₄ was used as a template at the final stage of the multistep synthesis of LiFePO₄ [33]. Thus, taking into account the structural similarity of both compounds, the formalized reaction scheme allows isomorphic substitution of 2 Li in Li₃PO₄ for 1 Fe in LiFePO₄.

4. Ion-Conducting Materials with Octahedral Ilmenite-like Framework

The structure of a synthetic analog of brizzite NaSbO₃ (Figure 9) with the structural type of ilmenite also shows a number of features characteristic of ion-conducting crystals [34].

Synthetic NaSbO₃ crystals form four polymorphic forms. Initially, a cubic modification with space group Pn3 became known [36]. Another cubic polymorphic form of NaSbO₃, but with sp. gr. Im3 and with a disordered arrangement of Na atoms in the unit cell, became known almost 35 years later [37]. This metastable compound was synthesized at a pressure of 2 GPa on the basis of ion exchange involving cubic modification of KSbO₃ (sp. gr. Im3) in NaNO₃ melt. The detailed analysis of the rhombohedral ilmenite-like modification of NaSbO₃ with space group R-3 was reported in [34]. The lower ionic conductivity of this ilmenite-like phase NaSbO₃ (3.0 × 10⁻⁵ S·cm⁻¹ at 400 °C) in comparison with that of the metastable cubic NaSbO₃ (5.6 × 10⁻² S·cm⁻¹ at 300 °C) has been attributed to its structural low dimensionality, strong Na⁺–oxygen bond, and high sodium occupancy factor [34]. Later on, considering that the structural characteristics, physical properties, and consequently the potential applications of NaSbO₃ compound in the industry depend on the synthesis method and reaction conditions, its rhombohedral crystals were obtained at temperatures lower than 750 °C [38]. This symmetry is also characteristic of the mineral brizzite [35]. Besides these phases, an orthorhombic form of NaSbO₃ with a distorted perovskite-like structure and with the space group Pnma is known [39].

The structures of both cubic modifications contain a framework of SbO₆ octahedra. Pairs of such octahedra are connected along the common edge form [Sb₂O₁₀] dimers, which are linked to each other along the vertices (Figure 10). Inside the octahedral framework, tunnels are formed along [111], which intersect at the vertices and in the center of the unit cell. In these tunnels, Na atoms are localized either with an ordered arrangement in two non-equivalent positions in sp. gr. Pn3 or with partial occupancy of the two main positions in sp. gr. Im3 (the corresponding occupancy coefficients are 0.82 and 0.29).

Similarly to ilmenite, gibbsite layers of edge-bound Sb-octahedra and more distorted Na-octahedra alternate in the structure of the rhombohedral modification of NaSbO₃ along [001] [34]. The data obtained on structural modifications of NaSbO₃ polymorphs allow to understand [34] the difference of ion-conducting properties of the cubic phase (Im3) and rhombohedral phase with the structural type of ilmenite. It was noted above that the rhombohedral polymorph exhibits a lower value of conductivity. Additionally, its activation energy is higher: 0.66 eV vs. 0.29 eV in the metastable cubic phase.

In the structure of cubic polymorph (Im3), Na atoms are disordered in two positions inside the 3-dimensional system of intersecting tunnels. In rhombohedral NaSbO₃ with ilmenite structure type, these atoms are ordered in octahedral positions between layers of SbO₆ octahedra. Therefore, the Na⁺ motion in the former case occurs along 3D axes, whereas in the latter case, it is restricted to 2 dimensions.

In the ilmenite-like structure, the average Na-O distance of 2.424 Å is significantly shorter compared to the cubic phase, where it is 2.70 Å. Accordingly, this difference leads to reduced conductivity in the rhombohedral polymorph. On the contrary, the Na-Na distance of the cubic structure is 2.30 Å, while in the rhombohedral structure, it is 3.175 Å. These values should affect the difference in the reduced activation energy: 0.66 eV for rhombohedral ilmenite-like crystals vs. 0.29 eV for the cubic phase. Another factor that has a significant influence on ionic conductivity is determined by the difference in the degree of occupancy of Na⁺ cations in their positions. In the cubic structure, these values are equal to 0.82 and 0.29, and in the rhombohedral structure, the occupancy coefficients of Na⁺ cations are equal to 1.

Finally, the transport of Na⁺ cations should be affected by the size of conductive channels. According to [40], their optimal width for Na⁺ cations should be 4.58 Å. In the cubic structure, it exceeds 5.0 Å, and this partly determines the low value of the activation energy of 0.29 eV. In ilmenite-like NaSbO₃, the diffusion of Na⁺ cations occurs inside the layers. The width of their conductive pathways is reduced to the smaller distance of 4.568 Å between the nearest two O atoms. Consequently, it should lead to an increase in the activation energy value of 0.66 eV.

In the phase with a body-centered unit cell, even the 6 shortest Na-O 2.65Ǻ distances in Na-octahedra exceed the sum of ionic radii, which should also contribute to the mobility of Na⁺ cations. According to [37], the mobility of Na⁺ cations in NaSbO₃ and in superionic crystals of β-aluminum oxide Na₂O·11Al₂O₃ (conductivity is of the order of 10⁻² S·cm⁻¹) must have common prerequisites. First of all, it should be recalled that the structure of β-aluminum oxide, as well as its composition, is very different from that of α-Al₂O₃ corundum. The structure of β-aluminum oxide (sp. gr. P6₃/mmc) contains mirror-bound spinel-like blocks of Al-tetrahedra and Al-octahedra. The ion-conducting layers with Na⁺ cations are located between them (Figure 11). In this phase, Na⁺ cations cross an edge common to the two octahedra during their movement, whereas in the NaSbO₃ structure, they pass through a face common to the neighboring octahedra [37]. However, both compounds have close values of ionic conductivity.

The β-alumina compounds with mono-valent cations, like sodium or potassium, have no stoichiometric composition, and their disordered structures are known as fast ion conductors. This peculiarity makes them attractive for energy storage applications [42]. The β-alumina presents the great advantage of being a very stable host lattice. It is separated into two species, namely, hexagonal β-alumina and rhombohedral β”-alumina types [43]. The incorporation of Li in sodium β-alumina by laser chemical vapor deposition (CVD) was reported in [41].

The structure of β-alumina exhibits the Frenkel defects. This type of defect forms when Al leaves its place in the structure, creating a vacancy and becoming an interstitial by lodging in a nearby location. The ionic conductivity depends on the correlations between the compensating defects and the diffusing ions. With the use of diffuse scattering measurements, the formation of Al Frenkel defects, which lock the extra bridging oxygen ions in the diffusing plane of β-alumina, was considered in [44]. Rods of diffuse scattering were observed in β-alumina grown from the flux with Ag, K, Rb, and Eu substituted for Na in the diffusing plane. They result from a strong interaction among the diffusing ions within a conducting layer and a weak correlation between layers. Consequently, three types of superstructure unit cells agree with observed XRD data for isomorphic varieties of β-alumina: (i) Those containing one diffusing ion; (ii) Those with two diffusing ions; (iii) Those with two ions plus one extra bridge oxygen.

It is worth noting that, to date, only two minerals with structures of the β-alumina type are known: kahlenbergite KAl₁₁O₁₇ and diaoyudaoite NaAl₁₁O₁₇ [45]. The structure of kahlenbergite is shown in Figure 12.

5. Mixed Frameworks in Natural and Synthetic Ionic Conducting Crystals with NASICON Structure Type

The discovery of fast-ionic conductivity in β-alumina stimulated interest in using solid electrolytes in thermoelectric generators, which convert heat into electrical energy. The ionic mobility in the layered structure of β-alumina occurs in two dimensions. Therefore, the framework structures with possible ionic transport in three dimensions attracted special attention [46,47].

Among the most promising compounds from the point of view of ionic conductivity, the representatives of the NASICON family with the general formula Na_1+xZr₂Si_xP_3−xO₁₂, 0 < x < 3 occupy an important place. It was their composition that determined the name of the whole NASICON family: NA—from Na, SI—from super ionic, and CON—from the conductor. The study of its representatives began in 1976 when the prototype of NASICON, Na_1+xZr₂Si_xP_3−xO₁₂, was reported [46,47]. The ionic conductivity of this solid solution at 300 °C achieved a high value of 0.2 S·cm⁻¹ with low activation energy 0.29 eV when x = 2 [47]. These values are comparable to Na-β-alumina above 443 K.

Later, the class of these materials expanded, and now it includes a wider set of compounds, such as AM₂(TO₄)₃, where A = Li, Na, K; M = Zr, Th, Fe, Sc, etc.; and T = P, As, Si, S, Se, etc., which are characterized by high cation mobility at relatively low temperatures and should have high hydrolytic and thermal stability. The data on the structure and ionic conductivity of NASICONs are discussed in an enormous number of papers [48]. Here, only some aspects of their correlation are considered.

The most specific feature of NASICON structure type is a mixed framework with the rhombohedral (R-3c) symmetry which is formed by corner-sharing metal octahedra and oxo-tetrahedra, as illustrated in Figure 13a.

All oxygen atoms are bridging, i.e., they are involved in sharing the links between polyhedra. Thus, each octahedron is connected to six tetrahedra and each tetrahedron—to four octahedra. Na⁺ ions occupy two positions: Na(1) ions are located in distorted [NaO₆]-octahedra and Na(2) ions lie in framework voids formed by 10 oxygen atoms. Na ions can migrate from one site to another through two triangular bottlenecks. Their transport determines the fast ion conductivity of the NASICON crystal and is shown in Figure 13b.

NASICON compounds can also form in a monoclinic symmetry (C2/c) depending on the composition and temperature, which splits the Na2 site into two different sites. However, the skeleton remains the same, and thus, the ion transport within the framework channel is similar [48].

The split of Na2-positions into three sites in monoclinic modification of NASICON-type structure Na₃FeV(PO₄)₃, which leads to the doubling of unit cell volume [49], is shown in Figure 14a in comparison with the chemically related rhombohedral structure of Na₄FeV(PO₄)₃ with almost fully occupied two Na-sites (Figure 14b).

The correlation between conductivity and composition of NASICON compounds was carefully analyzed in [48]. In general, the ionic conductivity increases as the silicate content increases in mixed SiO₄/PO₄ NASICON compounds [48]. In particular, the total conductivities of NASICON crystals with the pure phosphate group are in the range of 10⁻⁶ to 10⁻⁵ S·cm⁻¹, whereas those of phases with the (SiO₄)(PO₄)₂ groups are in the range of 10⁻⁵ to 10⁻⁴ S·cm⁻¹. The highest values of ionic conductivity of 4.4 × 10⁻⁴ S·cm⁻¹ at room temperature (~25 °C) were achieved in Na₃HfZr(SiO₄)₂(PO₄) [48] and 1.65 × 10⁻¹ S·cm^–1 at 473 K in Na_3.4Zr₂Si_2.4P_0.6O₁₂ [50]. Within each polyanion group, the ionic conductivity shows a non-monotonic trend with an average radius <r M> of M-cations. Indeed, the ionic conductivity increases with <r M> only up to an optimal value. For example, the ionic conductivity increases from 2.40 × 10⁻⁶ to 6.48 × 10⁻⁵ S·cm⁻¹ in going from Na₃HfMg(PO₄)₃ (<r M> = 0.715 Å) to Na₃ScIn(PO₄)₃ (<r M> = 0.773 Å) and from 6.17 × 10⁻⁵ S·cm⁻¹ for Na₃Hf_1.5Mg_0.5(SiO₄)(PO₄)₂ (<r M> = 0.713 Å) to 1.87 × 10⁻⁴ S·cm⁻¹ for Na₃HfSc(SiO₄)(PO₄)₂ (<r M> = 0.728 Å).

The conductivity of the solid solution Na_1+xZr₂Si_xP_3-xO₁₂ correlates with the following limits on x values: 1.8 < x < 2.4. At smaller values of x, there will be few Na⁺ cations (charge carriers) in the structure and a correspondingly smaller conductivity of the crystal. At larger values of x > 2.4, all possible positions for Na⁺ will be occupied, and they will not be able to move along the transport pathways.

According to [51], one of the representatives of NASICON—LiZr₂(PO₄)₃ has a linear dependence of conductivity on temperature (Figure 15), which is typical for many above considered fast ion conductors.

A natural analog of the NASICON family compounds is kosnarite KZr₂(PO₄)₃ [52]. Its rombohedral (space group R-3c) structure (Figure 16), like other representatives of NASICON, contains a mixed framework [Zr₂P₃O₁₂]. K atoms are located in the centers of antiprisms on the inversion axes -3 in its cavities. In addition to K mobility, kosnarite possesses ion-exchange properties, accumulating actinide atoms [53].

6. Ion Transport in the Structures of Conducting Crystals

Li⁺ superionic conductors are highly desirable for solid-state lithium batteries. However, only a few crystalline Li⁺ compounds are characterized by high ionic conductivity, particularly at room temperature. Among the materials exhibiting high ionic conductivity with relatively low transition temperatures to the superionic state, lithium-ion conductors with general formula Li₃M₂(PO₄)₃ (M = Fe, Sc, Cr, In) and with mixed framework structures are very promising. The three-dimensional framework of the Li₃Sc₂(PO₄)₃ structure is built of [ScO₆] octahedra and [PO₄] tetrahedra connected through common oxygen vertices (Figure 17). Three non-equivalent Li⁺ ions are located in the channels of this framework [54].

In these compounds, two phase transitions occur at different temperatures, with the high-temperature orthorhombic phase being the superionic one [55]. For Li₃Sc₂(PO₄)₃, the phase transition temperatures are 473 K and 573 K, whereas for Li₃Fe₂(PO₄)₃, these temperatures are 513 K and 573 K [56,57]. The lower temperature modifications (α and β) are characterized by the monoclinic space groups P2₁/n, whereas the structures of high-temperature γ-polymorphs in both cases are described in the frames of orthorhombic sp.gr. Pcan. Thus, the symmetry and the unit cell metrics of these compounds differ from the rhombohedral crystals of NASICON. However, the general configurations of mixed frameworks in both cases have many common features (Figure 17).

The main difference between α- and β-polymorphs lies in the arrangement of lithium atoms since the structure framework built by Sc(Fe)-octahedra and P-tetrahedra is only slightly more symmetrized in the β-phase than the framework of the α-phase. At room temperature, lithium atoms fully occupy (100%) three non-equivalent crystallographic positions, referred to as I, II, and III, in the voids of this structure type. Upon heating, the third Li-position becomes less favorable than an additional one called the supplementary (Is) position.

In high-temperature γ-polymorph, lithium atoms experience further redistribution in comparison with the α- and β-phases. As a result of the phase transition, twelve lithium atoms of the unit cell are redistributed over three eight-fold crystallographic positions, whereas only one of them is fully occupied. Two other positions are occupied with lithium atoms only partially—for 25%. During phase transitions, the relative coordinates of M and P atoms remain almost unchanged. Thus, a significant difference between the arrangement of Li in the low-temperature and high-temperature forms of Li₃Sc₂(PO₄)₃ was revealed [54,55] (Figure 18). The reason for the anisotropy of the conductivity of lithium ions in these compounds becomes clear: in the superionic orthorhombic phase, the movement of Li along the a-axis will meet an obstacle in the form of a fully occupied site, whereas along the c-axis, its movement will be easier (Figure 18b).

The lithium ionic conductivity in Li₃Sc₂(PO₄)₃ crystals at room temperature was found to be 1·10⁻⁵ S·cm⁻¹, and in the superionic state, it increased ~10³–10⁴ times to 2·10⁻² S·cm⁻¹ at T > 518 K [56].

Even at the early stages of the study of crystals with ionic conductivity, the question arose as to how cations, shifted from one polyhedron to the neighboring one, move within a dense packing of anions, often directly touching each other. The problem, related to the displacement of Li atoms inside the structure, formulated at the beginning of this section, is explained in Figure 18c, which shows the passage of a Li⁺ cation through a triangular face formed by three O²⁻ anions with short O-O ~2.5 Å distances.

Lithium-substituted lanthanum titanate Li_0,255La_0,582TiO₃ with the structural type of perovskite can serve as another illustration of the mysterious phenomenon associated with ion mobility [13]. At first glance, the crystal structure of Li_0.255La_0.582TiO₃ should not allow high ionic conductivity since the size of the conduction window is insufficient for unimpeded movement of Li⁺ ions through channels inside the framework of TiO₃ octahedra (Figure 19). The observed contradiction can be explained by the special role of thermal vibrations of the framework atoms, due to which the size of the “conduction window” is constantly changing—the channels “breathe”. The jump of ions to the neighboring position can occur at the moment of the greatest openness of the “window” [58].

7. Conclusions

The ever-increasing consumption of electricity, coupled with decreasing fossil fuel resources, implies the search for alternative sources of electrical energy. In this process, the most important point is associated with the improvement of prospecting and technology of solid electrolytes [59]. The materials used in them are characterized by high ionic conductivity and, as shown above, are structural analogs of various minerals. The main conditions for fast ionic transport are related to the disorder in the positions occupied by the mobile ion and the presence of conduction channels running inside the structure. In accordance with these requirements, the search for new solid electrolytes is conducted along these two trends [1]. Their main property is the presence of positions available for the conducting ion, between which it could freely jump. For this purpose, it is necessary that these sites were not too distant from each other and were separated by a sufficiently wide “conduction window”. As a rule, in an ideal crystal, all such positions are either fully occupied or completely free, which leads to the impossibility of ionic transport. That is why various heterovalent isomorphic impurities are introduced into the structure of a solid electrolyte to create partially occupied positions. The crystals obtained in this way are solid solutions from the point of view of solid-state chemistry and appear to be promising functional materials with ion-conducting properties. The results of their studies are dispersed in different editions, and the overview of new ideas related to their crystal structures was the focus of this paper.