Ba Solid Solution Crystals as an Effective Solid Electrolyte: Growth and Properties

: A series of nonstoichiometric La 1– y Ba y F 3– y (0 ≤ y ≤ 0.12) single crystals with a tysonite-type structure (sp. gr. P -3 c 1) was grown from the melt by the directional crystallization method in a ﬂuorinating atmosphere, and some physical properties were characterized. The concentration dependence of electrical conductivity σ dc ( y ) La 1– y Ba y F 3– y crystals was studied. The composition of the ionic conductivity maximum for this solid electrolyte was reﬁned. It was conﬁrmed that the maximum conductivity σ max = 8.5 × 10 –5 S/cm (295 K) was observed at the composition y max = 0.05 ± 0.01. Analysis of the electrophysical data for the group of tysonite-type solid electrolytes R 1– y M y F 3– y ( M = Ca, Sr, Ba, Eu 2+ and R = La, Ce, Pr, Nd) showed that the compositions of the maxima of their conductivity were close and amount to y = 0.03 − 0.05. This fact indicates a weak inﬂuence of the size effect (ionic radii R 3+ and M 2+ ) on the value of y max for R 1– y M y F 3– y solid electrolytes.


Introduction
One of the paths in the search for new materials with specified properties in modern materials science is the complication of chemical composition, i.e., transition from onecomponent to multicomponent crystals. Nonstoichiometric phases M 1-x R x F 2+x with a fluorite-type structure and R 1-y M y F 3-y with tysonite-type one are formed in all binary systems MF 2 -RF 3 (M are alkaline-earth elements; R are rare-earth elements). With a change in the component concentration of such phases, the physical properties of crystals vary in wide ranges due to heterovalent isomorphic substitutions. The ability to smoothly change the chemical composition of crystals allows effectively controlling the physical properties and determining compositions with an optimal combination of parameters.
Wide areas of solid solutions with a fluorite-type (sp. gr. Fm-3m) structure Ba 1-x La x F 2+x (0 ≤ x ≤ 0.52) and with a tysonite-type (sp. gr. P-3c1) structure La 1-y Ba y F 3-y (0 ≤ y ≤ 0.14) are formed in a condensed system BaF 2 -LaF 3 ( Figure 1) [1]. These solid solutions demonstrate high stability to degradation; therefore, there is a possibility of growing both fluorite and tysonite single crystals from a melt in a wide concentration range. Heterovalent solid solutions Ba 1-x La x F 2+x [2] and La 1-y Ba y F 3-y are promising optical materials with variable properties due to the changeable composition. Their high fluorine-ionic conductivity σ dc is especially attractive. [3][4][5][6][7]. They have very low electronic conductivity [8,9] and therefore are effective fluorine-conducting solid electrolytes. The method of growing fluorite-type crystals Ba 1-x La x F 2+x is well developed, and their electrophysical properties have been studied in detail [3][4][5][6][7]10,11]. Tysonite La 1-y Ba y F 3-y crystals have higher conductometric characteristics than Ba 1-x La x F 2+x , but the conditions for their preparation and properties have been studied much less. At the present time, the solid electrolyte La 1-y Ba y F 3-y in terms of the set of performance characteristics has been proposed for use in the designs of the prototypes of fluorine-ion current sources of a new generation [12][13][14][15][16].
solid electrolyte La1-yBayF3-y in terms of the set of performance characteristics has been proposed for use in the designs of the prototypes of fluorine-ion current sources of a new generation [12][13][14][15][16].

Figure 1.
Phase diagram of the BaF2-LaF3 system [1] and the appearance of the grown crystals of the La0.93Ba0.07F2.93 tysonite (T) phase. As an example, the previously grown single crystal of the fluorite (F) phase Ba0.69La0.31F2.31 [2] is shown for comparison.
Bulk R1-yMyF3-y (R = La-Lu, Y, and M = Ca, Sr, Ba) crystals are usually grown by melt crystallization methods, predominantly the Bridgman-Stockbarger method [17,18] or Czochralski method. [19,20]. The Bridgman-Stockbarger method is technologically simple and allows, due to the use of multi-cell crucibles, a concentration series of solid solution crystals to grow under identical preparative and thermal conditions.
Bulk single crystals are of particular interest for studying the fundamental characteristics of ionic transfer since only the single-crystal form gives fundamental conductivity values that are not distorted by the influence of the grain boundaries and pores. Investigating single crystals can accurately identify the composition La1-yBayF3-y solid solution, which corresponds to the maximum conductivity level. It is known that electrophysical studies of La1-yBayF3-y single crystals were carried out in [5] only. The authors used samples with a large step in composition y, which made it impossible to determine the coordinate of the σmax exactly. A large scatter in the experimental data on the value ymax = 0.04-0.06 [21,24,28], 0.07-0.09 [5,22], 0.10 [25,27,30], and 0.15 [26] was observed for La1-yBayF3-y solid solution crystals.
Until now, there are few studies on growing concentrated tysonite-type fluoride solid solutions. Only the possibility of growing crystals of R1-ySryF3-y (R = La-Nd (0 ≤ y ≤ 0.16)) tysonite solid solutions, which are isostructural to La1-yBayF3-y crystals, have been studied [17,32]. These SrF2-containing solid solutions have an important advantage for crystallization from a melt-congruent compositions corresponding to a singular point (temperature maximum) on the liquidus curve. The crystals with a homogeneous axial  [1] and the appearance of the grown crystals of the La 0.93 Ba 0.07 F 2.93 tysonite (T) phase. As an example, the previously grown single crystal of the fluorite (F) phase Ba 0.69 La 0.31 F 2.31 [2] is shown for comparison.
Bulk R 1-y M y F 3-y (R = La-Lu, Y, and M = Ca, Sr, Ba) crystals are usually grown by melt crystallization methods, predominantly the Bridgman-Stockbarger method [17,18] or Czochralski method. [19,20]. The Bridgman-Stockbarger method is technologically simple and allows, due to the use of multi-cell crucibles, a concentration series of solid solution crystals to grow under identical preparative and thermal conditions.
Bulk single crystals are of particular interest for studying the fundamental characteristics of ionic transfer since only the single-crystal form gives fundamental conductivity values that are not distorted by the influence of the grain boundaries and pores. Investigating single crystals can accurately identify the composition La 1-y Ba y F 3-y solid solution, which corresponds to the maximum conductivity level. It is known that electrophysical studies of La 1-y Ba y F 3-y single crystals were carried out in [5] only. The authors used samples with a large step in composition y, which made it impossible to determine the coordinate of the σ max exactly. A large scatter in the experimental data on the value y max = 0.04-0.06 [21,24,28], 0.07-0.09 [5,22], 0.10 [25,27,30], and 0.15 [26] was observed for La 1-y Ba y F 3-y solid solution crystals.
Until now, there are few studies on growing concentrated tysonite-type fluoride solid solutions. Only the possibility of growing crystals of R 1-y Sr y F 3-y (R = La-Nd (0 ≤ y ≤ 0.16)) tysonite solid solutions, which are isostructural to La 1-y Ba y F 3-y crystals, have been studied [17,32]. These SrF 2 -containing solid solutions have an important advantage for crystallization from a melt-congruent compositions corresponding to a singular point (temperature maximum) on the liquidus curve. The crystals with a homogeneous axial and radial distribution of the components can be grown in the vicinity of this chemical composition. Unfortunately, La 1-y Ba y F 3-y solid solution does not have compositions with congruent melting; a large difference in the liquidus and solidus temperatures leads to a significant inhomogeneous distribution of the components along the length of the crystals during directional crystallization from the melt.
The aims of the present work are: -Growing from the melt of concentration series of La 1-y Ba y F 3-y single crystals under identical conditions and its some physical properties characterization; -Measurement of their ionic conductivity to refine y max on the dependence σ dc (y); -Comparative analysis of conductometric data for tysonite-type solid electrolytes in systems MF 2 -RF 3 with M = Ca, Sr, Ba, Pb, Eu 2+ and R = La, Ce, Pr, Nd.

The Peculiarities of the Growth Process
On the one hand, the proximity of the melting temperatures of LaF 3 (T fus = 1773 ± 10 K) and BaF 2 (T fus = 1627 ± 5 K) and their low volatility contribute to the production of La 1-y Ba y F 3-y single crystals from the melt by the direction crystallization method; on the other hand, this is hindered by the incongruent melting of the solid solution, as mentioned earlier. For tysonite-type structure, solid solutions, in comparison with fluoritetype structure solutions, a tendency to melt supercooling and spontaneous nucleation are characteristic [11], which prevents the growth of the high-quality bulk crystals. These problems are solved by increasing the temperature gradient at the growth interface and reducing the crucible pulling rate [10].
The La 1-y Ba y F 3-y solid solution compositions were chosen in the range 0 ≤ y ≤ 0.12 for the investigation. The crystals were grown by the vertical directional crystallization method in a two-section growth facility with resistive heating in graphite crucibles using non-oriented seeds. BaF 2 (purity 99.99%, Sigma-Aldrich, Darmstadt, Germany) and LaF 3 (99.99%, Lanhit Ltd., Moscow, Russia) powders were used as starting reagents. The initial powders were preliminarily calcined in a vacuum (~10 -2 Pa) at a temperature of 450 K and remelted to remove oxygen-containing impurities. The temperature gradient in the growth zone was~100 K/cm and the crucible pulling rate was 3 mm/h. It should be noted that 5 wt.% PbF 2 was added to the charge as an oxygen scavenger in [32,33]. This is not the best choice because PbF 2 is not completely removed from the RF 3 and MF 2 melts and obviously modifies the physical properties of the crystals. Probably, at low concentrations of PbF 2 in some systems, azeotropic mixtures are formed [33,34]. Therefore, in our work, a mixture of high-purity He + CF 4 was used to create a fluorinating atmosphere in growth experiments. The evaporation loss during crystallization was no more than 2 wt. %. The La 1-y Ba y F 3-y tysonite-type crystals with a diameter of 12-14 mm and a length of up to 30 mm with the compositions y = 0; 0.005; 0.025; 0.035; 0.050; 0.060; 0.080; 0.090; 0.100; 0.120 (by charge) were successfully grown. The Ba 1-x La x F 2+x fluorite-type crystals in the range 0 ≤ x ≤ 0.50 and eutectic composite (BaF 2 /LaF 3 = 0.68/0.32 by charge) were obtained by the above method for comparative analysis of properties.
Crystal samples were cut perpendicular to the growth axis from the ingots in the form of plane-parallel disks and were optically polished.

Density Measurement
The density ρ(y) of the samples was measured by the hydrostatic method in distilled water at room temperature. The density measurement error was ∆ρ = ±5 × 10 -3 g/cm 3 .

Refractive Indices
Refractive indices n D for La 1-y Ba y F 3-y single crystals were measured using the refractometric technique with an accuracy of ±5 × 10 -4 . The light source was Na-lamp (λ = 0.589 µm).

X-ray Diffraction (XRD) Analysis
The XRD analysis of the crystals was carried out by X-ray powder diffractometer Rigaku MiniFlex 600 (CuKα radiation). The diffraction peaks were recorded within the angle range 2θ from 10 to 140 • . The calculation of the unit cell parameters a(y) and c(y) for sp. gr. P-3c1 was carried out by full-profile Rietveld analysis using the HighScore Plus software (PANanalytical, Almelo, The Netherlands).
Since a concentration gradient of the components along the crystals' length was observed, the chemical composition of each sample was refined in terms of the unit cell parameters according to the equations for the concentration dependence of the unit cell parameters [11] and density [35] for the La 1-y Ba y F 3-y solid solution.

The Electrical Conductivity Measurements
The static electrical conductivity σ dc at a direct current of the La 1-y Ba y F 3-y crystals was determined by impedance spectroscopy [36]. The impedance measurements were carried out in the frequency range of 5 to 5 × 10 5 Hz using a Tesla B-507 impedance tester at temperatures of 294-800 K in a vacuum~1 Pa. The relative measurement error did not exceed 5%.

Crystal Characterization
The grown La 1-y Ba y F 3-y crystalline boules were transparent and did not have cracks and light-scattering inclusions. The La 0.93 Ba 0.07 F 2.93 crystal is shown in Figure 1 as a typical sample.
According to XRD analysis, the La 1-y Ba y F 3-y samples were a single-phase solid solution and belonged to the tysonite (sp. gr. P-3c1) structural type (Figure 2a). The presence of impurity phases was not detected.

X-ray Diffraction (XRD) Analysis
The XRD analysis of the crystals was carried out by X-ray powder diffractometer Rigaku MiniFlex 600 (CuKα radiation). The diffraction peaks were recorded within the angle range 2θ from 10 to 140°. The calculation of the unit cell parameters a(y) and c(y) for sp. gr. P-3c1 was carried out by full-profile Rietveld analysis using the HighScore Plus software (PANanalytical, Almelo, The Netherlands).
Since a concentration gradient of the components along the crystals' length was observed, the chemical composition of each sample was refined in terms of the unit cell parameters according to the equations for the concentration dependence of the unit cell parameters [11] and density [35] for the La1-yBayF3-y solid solution.

The Electrical Conductivity Measurements
The static electrical conductivity σdc at a direct current of the La1-yBayF3-y crystals was determined by impedance spectroscopy [36]. The impedance measurements were carried out in the frequency range of 5 to 5 × 10 5 Hz using a Tesla B-507 impedance tester at temperatures of 294-800 K in a vacuum ∼1 Pa. The relative measurement error did not exceed 5%.

Crystal Characterization
The grown La1-yBayF3-y crystalline boules were transparent and did not have cracks and light-scattering inclusions. The La0.93Ba0.07F2.93 crystal is shown in Figure 1 as a typical sample.
The low-temperature tysonite modification is described in sp. gr. P-3c1 [37]. Purely anionic and cation-anionic layers alternated in the trigonal motif along the c axis (Figure 2b). The Fanions were located in three non-equivalent structure positions. Substitution of La 3+ cations on heterovalent M 2+ ones led to the formation of vacancies in the fluorine sublattice (to maintain electroneutrality) and significantly affected the magnitude of the ionic conductivity of such tysonite solid solutions (see Section 3.2).
The La 1-y Ba y F 3-y crystals studied were uniaxial, optically negative, and were characterized by two refractive indices. The ordinary refractive index n o was measured for La 1-y Ba y F 3-y crystals. The refractive index n o (λ = 0.589 µm) of La 1-y Ba y F 3-y crystals decreased in a weakly square dependence from 1.5982(5) to 1.5841 (5) with an increase in the fraction of y from 0 to 0.12 (Figure 3a). These crystals were highly refractive, exceeding the refractive indices of Ba 1-x La x F 2+x and other crystals with a fluorite-type structure. The known data for some other tysonite-type R 1-y M y F 3-y crystals are shown for comparison. The La 1-y Ba y F 3-y crystals were similar in refractive properties to other isostructural La 1-y M y F 3-y crystals (M = Ca, Sr, Ba) [17].
Crystals 2021, 11, 629 5 of 11 a = 6.0378(2) Å for the limiting composition x = 0.50. The observed concentration dependences (Figure 2a, insert) coincided with the data of [1,11]. Eutectic composite contained both the saturated solid solution (F + T) which crystallized from the melt simultaneously. The low-temperature tysonite modification is described in sp. gr. P-3c1 [37]. Purely anionic and cation-anionic layers alternated in the trigonal motif along the c axis ( Figure  2b). The Fanions were located in three non-equivalent structure positions. Substitution of La 3+ cations on heterovalent M 2+ ones led to the formation of vacancies in the fluorine sublattice (to maintain electroneutrality) and significantly affected the magnitude of the ionic conductivity of such tysonite solid solutions (see Section 3.2).
The La1-yBayF3-y crystals studied were uniaxial, optically negative, and were characterized by two refractive indices. The ordinary refractive index no was measured for La1-yBayF3-y crystals. The refractive index no (λ = 0.589 µm) of La1-yBayF3-y crystals decreased in a weakly square dependence from 1.5982(5) to 1.5841 (5) with an increase in the fraction of y from 0 to 0.12 (Figure 3a). These crystals were highly refractive, exceeding the refractive indices of Ba1-xLaxF2+x and other crystals with a fluorite-type structure. The known data for some other tysonite-type R1-yMyF3-y crystals are shown for comparison. The La1-yBayF3y crystals were similar in refractive properties to other isostructural La1-yMyF3-y crystals (М= Ca, Sr, Ba) [17]. The density of La1-yBayF3-y samples quadratically decreased in the range from 5.917(5) to 5.741(5) g/cm 3 with an increase in the fraction of y from 0 to 0.08 (Figure 3b) and exceeded the density of fluorite-type Ba1-xLaxF2+x single crystals.

Ionic Conductivity of La1-yBayF3-y Crystals
We analyzed the impedance spectra of the La1-yBayF3-y crystals with silver electrodes. The impedance spectra were characterized by a depressed semi-circular arc at high frequency and an oblique tail at low frequency. As an example, Figure 4 shows the hodograph of the impedance Z*(ω) for the La0.994Ba0.006F2.994 crystal with Ag-electrodes. The impedance spectrum Z*(ω) contained a semicircle (the center of which was displaced from the abscissa axis) simulating the electrical response from the crystal bulk and an The density of La 1-y Ba y F 3-y samples quadratically decreased in the range from 5.917(5) to 5.741(5) g/cm 3 with an increase in the fraction of y from 0 to 0.08 (Figure 3b) and exceeded the density of fluorite-type Ba 1-x La x F 2+x single crystals.

Ionic Conductivity of La 1-y Ba y F 3-y Crystals
We analyzed the impedance spectra of the La 1-y Ba y F 3-y crystals with silver electrodes. The impedance spectra were characterized by a depressed semi-circular arc at high frequency and an oblique tail at low frequency. As an example, Figure 4 shows the hodograph of the impedance Z*(ω) for the La 0.994 Ba 0.006 F 2.994 crystal with Ag-electrodes. The impedance spectrum Z*(ω) contained a semicircle (the center of which was displaced from the abscissa axis) simulating the electrical response from the crystal bulk and an oblique straight line (at low frequencies) simulating the electrical response from the crystal/electrode interface. An equivalent electrical circuit was used to describe the impedance hodographs. The circuit contained the crystal bulk resistivity R b , the geometric capaci-Crystals 2021, 11, 629 6 of 11 tance of the crystal C g , and the capacitance of the double layer of the crystal/electrode interface C dl .
Crystals 2021, 11, 629 6 of 11 oblique straight line (at low frequencies) simulating the electrical response from the crystal/electrode interface. An equivalent electrical circuit was used to describe the impedance hodographs. The circuit contained the crystal bulk resistivity Rb, the geometric capacitance of the crystal Cg, and the capacitance of the double layer of the crystal/electrode interface Cdl. The bulk resistance Rb was found from the intersection of the hodograph Z*(ω) with the abscissa axis. Specific static electrical conductivity σdc was calculated by the formula: where h-sample thickness and S-electrode area. The temperature dependences of the ionic conductivity for La1-yBayF3-y crystals with y = 0.036 and 0.050 are shown in Figure 5a.  The bulk resistance R b was found from the intersection of the hodograph Z*(ω) with the abscissa axis. Specific static electrical conductivity σ dc was calculated by the formula: where h-sample thickness and S-electrode area. The temperature dependences of the ionic conductivity for La 1-y Ba y F 3-y crystals with y = 0.036 and 0.050 are shown in Figure 5a.
Crystals 2021, 11, 629 6 of 11 oblique straight line (at low frequencies) simulating the electrical response from the crystal/electrode interface. An equivalent electrical circuit was used to describe the impedance hodographs. The circuit contained the crystal bulk resistivity Rb, the geometric capacitance of the crystal Cg, and the capacitance of the double layer of the crystal/electrode interface Cdl. The bulk resistance Rb was found from the intersection of the hodograph Z*(ω) with the abscissa axis. Specific static electrical conductivity σdc was calculated by the formula: where h-sample thickness and S-electrode area. The temperature dependences of the ionic conductivity for La1-yBayF3-y crystals with y = 0.036 and 0.050 are shown in Figure 5a.  The σ dc (T) dependences for all studied samples were divided into two sections, which satisfy the Arrhenius-Frenkel equation: where A-preexponential conductivity factor, ∆H σ -activation enthalpy of ion transport, k B -Boltzmann's constant, T-temperature. The Arrhenius-Frenkel equations parameters A and ∆H σ are given in Table 1. Table 1. Unit cell parameters a and c, preexponential conductivity factor A and activation enthalpy of ionic conductivity ∆H σ for single crystals of La 1-y Ba y F 3-y solid electrolyte. The dependence lg σ dc (y) at room temperature for the solid electrolyte La 1-y Ba y F 3-y , plotted based on our results and data [5,38], is shown in Figure 5b. In the range of compositions y = 0.006-0.086, the ionic conductivity varied from 1 × 10 -5 to 8.5 × 10 -5 S/cm.
The coordinates of the conductivity maximum were σ max = 8.5 × 10 -5 S/cm and y max = 0.05 ± 0.01. The minimum activation enthalpy ∆H σ = 0.33 eV (low-temperature segment σ dc (T)) corresponded to the maximum conducting composition y max .
Note that the nonstoichiometric R 1-y M y F 3-y crystals with a tysonite-type structure have an advantage in conductivity value over crystals with a fluorite-type structure [3]. Apparently, this is due to the features of defect formation in these two structures. The conductivity is equalized in value only when the fluorite-type crystals are heated to 150-200 • C, which significantly complicates their practical application.
The coordinates of the conductivity maxima (y max , σ max ) for a large family of tysonitetype solid electrolytes R 1-y M y F 3-y with M = Ca, Sr, Ba, Pb, Eu 2+ and R = La, Ce, Pr, Nd are presented in Table 2. It can be seen that the maxima of ionic conductivity were realized in a narrow concentration range of y = 0.03-0.05. This fact indicates a weak effect of the R 3+ and M 2+ cations sizes on the value of the coordinate y max for this type of solid electrolytes with heterovalent isomorphic substitutions. Nd 1-y Pb y F 3-y S 0.05 3 × 10 -5 [46] * S-single crystals, P-polycrystals.

Conclusions
The series of La 1-y Ba y F 3-y solid electrolyte crystals with 0.00 ≤ y ≤ 0.12 was successfully grown from the melt by the Bridgman technique. Temperature measurements of the ionic conductivity of La 1-y Ba y F 3-y crystals by impedance spectroscopy were carried out in the range of 294 to 800 K. The composition of single-crystal samples was refined in terms of the unit cell parameters and density for the La 1-y Ba y F 3-y tysonite-type phase. The results of conductometric data for La 1-y Ba y F 3-y single crystals indicate that the σ dc (y) dependence had a maximum σ max = 8.5×10 -5 S/cm at y max = 0.05 ± 0.01. Analysis of the published data on the compositions of the conductivity maxima for a large family of solid electrolytes R 1-y M y F 3-y (M = Ca, Sr, Ba, Eu 2+ and R = La, Ce, Pr, Nd) showed that they fall within the range of 0.03 ≤ y max ≤ 0.05 regardless of the type of R 3+ and M 2+ cations. It is shown that the size of the cation forming a solid solution practically does not affect the value of y max for R 1-y M y F 3-y solid electrolytes.
Despite the high level of room temperature conductivity of the studied LaF 3 -based materials, crystals of R 1-y M y F 3-y (R = Ce, Pr, Nd; M = Ca, Sr) should be considered promising for practical implementation as a solid electrolyte with better electrophysical characteristics. It is these crystals that will become the next step in solid-state ionics as