Phase Stability Analysis of Ternary Alkaline-Earth Hexaborides

Article Cite This: ACS Appl. Electron. Mater. XXXX, XXX, XXX−XXX

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Phase Stability Analysis of Ternary Alkaline-Earth Hexaborides: Insights from DFT Calculations Kevin M. Schmidt,† Robert J. Koch,‡ Scott T. Misture,‡ Olivia A. Graeve,¶ and Victor R. Vasquez*,† †

Chemical and Materials Engineering Department, University of Nevada, Reno, Reno, Nevada 89557, United States Kazuo Inamori School of Engineering, Alfred University, 2 Pine Street, Alfred, New York 14802, United States ¶ Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California 92093, United States Downloaded via UNIV OF SOUTH DAKOTA on December 22, 2018 at 00:07:01 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

ABSTRACT: We present a phase stability analysis of ternary mixtures of alkaline-earth hexaborides using insights from DFT calculations to determine the effect of homogeneity on properties including stability and domain formation. We find that linear mixing rules and Vegard’s law type, describe very well octahedral volumes, inter and intraoctahedral bond lengths, lattice constants, bulk moduli, cation−facial 2 distances for the mixtures of M1xM1−x B6, where Mk can be Ca, Ba, or Sr, and composition (x) ranges from 0 to 1. In general, variations are less than 5% of the calculated mixture properties with positive deviations, except the interoctahedral boron bond lengths in systems with Ba. We find that doping with lighter cations may be an effective means of strengthening MB6 materials. Electronic structure calculations predict that the lattice constants and interoctahedral bond lengths with a mixture are identical, as the degree of homogeneity increases, indicative of formation of a single phase. However, bond lengths within the boron framework are found to be heavily dependent upon the local cation environment, and energies taken at absolute zero suggest phase splitting as a general tendency for certain stoichiometric ratios, in particular, for compositions at the 50% levels, which is in agreement with experimental evidence. The phase shifted regions are likely the product of the slight mismatch between interoctahedral bonds. These nanoregimes are interpreted as regions that have not yet fully mixed. KEYWORDS: DFT, hexaboride, ternary mixtures, electronic structure, stability

1. INTRODUCTION

Metal hexaborides have potential for wear and corrosionresistant coatings,28 boron ion beam sources,29 light filters,17,30 protective layers in plasma displays,31 n-type thermoelectric materials,32 H2O2 sensors,33 lightweight armor,34 and nearinfrared triggered medical devices.35,36 There is also evidence suggesting that MB6 can be used in catalytic applications. LaB6 surfaces equilibrate mixtures of hydrogen and deuterium at low temperatures, most likely on terminal boron atoms.37 LaB6 can be potentially used as an industrial catalyst in dissociative adsorption of water at room temperature and carbon monoxide above 400 K.38,39 Metal hexaboride materials have been studied extensively from both experimentally and modeling standpoints,40−43 with most of work performed on pure MB6 materials. An interesting feature of MB6 materials is that they readily form solid solutions containing two different metal cation species. The resulting ternary hexaborides will generally display intermediate properties dependent upon stoichiometry, suggesting a method to tune certain features for practical applications.

Metal hexaborides (MB6) are refractory crystals with relatively low work functions commonly used as cathodes in devices for electron emission.1 An assortment of metals can be used to synthesize these materials,2−4 and each choice will generally produce different electronic properties in the crystal such as ferromagnetism, paramagnetism, superconductivity, semiconductivity, and electrical conduction.5 For example, as a result of its thermionic characteristics, lanthanum hexaboride (LaB6) has been widely used as an electron emission source. One of the major applications as an electron emitter is in the construction of single crystal and polycrystalline cathodes for electron beam instruments,6,7 mass spectrometry,8 and space applications.9 More recently, these materials are finding applications as field emitters at nanoscale levels10−12 and because of their nanostructured nature are also excellent as light absorbing materials in the near-infrared region.13−17 Other properties and potential applications include their use for improving the abrasive resistance in convector applications;18,19 these materials also have high chemical inertness, low density, low thermal expansion coefficients, low volatility, high melting points, and high hardness values.1,20−27 © XXXX American Chemical Society

Received: November 6, 2018 Accepted: December 17, 2018 Published: December 17, 2018 A

DOI: 10.1021/acsaelm.8b00047 ACS Appl. Electron. Mater. XXXX, XXX, XXX−XXX

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ACS Applied Electronic Materials

configurations of 3s 2 3p 6 3d 0 4s 2 , 4s 2 4p 6 4d 1 5s 1 5p 0 , and 5s25p65d06s26p0 for atoms Ca, Sr, and Ba, respectively, and the boron pseudopotential has three valence electrons in a 2s22p1 configuration. Kinetic energy cut-offs for the plane wave basis are 30 Ry for the wave function and 400 Ry for the charge density for all systems, and these values are converged to 1 mRy/atom using a bulk unit cell calculation. Smearing widths of σ = 5 mRy are shown to give accurate and reproducible results. In terms of k-points, a 16 × 16 × 16 mesh is used per unit cell. However, many of the calculations require supercells, and the k-point mesh is modified accordingly; for example, the mesh is reduced by a factor of 2 in the x-direction if the corresponding real-space length is doubled. Bulk lattice parameters are calculated by relaxing the ionic positions and volume of a single unit cell until total energy and forces are converged to within 10−6 Ry and 10−5 Ry/bohr, respectively. In the case of supercells, the three orthogonal vectors spanning the unit cell are individually optimized. To determine the bulk moduli, atomic positions are relaxed for a sequence of cell volumes ranging between ±5% of the equilibrium cell size using identical convergence thresholds as for bulk lattice parameter calculations. The total energy and volume resulting from these relaxations are then fit to a thirdorder Birch−Murnaghan equation of state.60 2.2. Supercell Constructions. Hexaboride compounds having M1xM21−xB6 stoichiometry are studied at substitutions in 25% increments, providing three ternary solutions (x = 0.25, 0.50, 0.75) and two pure binary crystals (x = 0.00, 1.00) for each set (M1, M2). In addition, three types of atomic segregation patterns for ternary mixtures are considered. Representative supercells for each stoichiometry and heterogeneity are shown in Figure 1. Well-mixed solutions refer to

Indeed, several studies have investigated the use of hexaboride mixtures to optimize or adjust thermoelectric properties,44 magnetism,45 thermionic emission,46 thermal conductivity,17 and optical properties30,47 with promising results. Common to all metal hexaborides is the covalently linked cage of boron atoms providing structural stability and a unique placement for each of the metal atoms, located at the 1a (0, 0, 0) Wyckoff site of the simple cubic unit cell. An octahedral unit is formed by the six boron atoms with vertices at the 6f (1 2 , 1 2 , z ) Wyckoff position, each of which has additional connections to boron atoms from neighboring unit cells through interoctahedral bonds. Structural features of the hexaboride framework are fairly insensitive to cation substitution, and relatively minor adjustments in the boron bond lengths between different binary hexaborides48 suggest mixtures can form with minimal strain. Although a number of features can be adjusted according to the stoichiometry of ternary hexaborides, problems arise during synthesis that affect the ability to form into a single stable phase. Several features can significantly affect the homogeneity of the crystal, including the synthesis approach, metal vapor pressures, and chemical form of the reactant species.49,50 Processing crystals with methods such as spark plasma sintering51 can help to reduce the heterogeneity, but isolated regions containing separate phases persist.47 Lattice dislocations formed at the intersection of these domains have been observed using high-resolution transmission electron microscopy (HRTEM) in multiple cases.47,52 While several computational studies have been performed to understand effects related to ternary hexaborides, issues concerning phase stability have not yet been investigated. Luo et al.53 used density functional methods to determine the effect of cation substitution on the mechanical properties of CaB6 crystals, finding a linear increase in the lattice constant and correlations between d-orbital electrons and the shear modulus. Other electronic structure calculations have predicted the emergence of magnetism in La-doped SrB6.54 In this work, we examine the effect of homogeneity on specific structural features for the alkaline-earth hexaborides containing Ca, Sr, and Ba metals to help clarify these issues. By considering several geometries for each stoichiometric composition, we find that each boron bond is significantly influenced by its nearby cation environment. Subtle changes in the local ion concentration, which lead to increased strain in the material, will favor domain formation. Additionally, equimolar cation concentrations are found to be least favorable when there is a large difference in the cation sizes.

2. METHODOLOGY 2.1. Density Functional Theory. First-principles calculations with density functional theory (DFT) are performed using the plane-wave electronic structure codes from Quantum ESPRESSO.55 The Perdew−Burke−Ernzerhof (PBE) functional56 within the generalized gradient approximation (GGA) is used to describe exchange and correlation effects. Integration over the Brillouin zone uses a Monkhorst−Pack set of special k-points57 and Marzari−Vanderbilt cold smearing58 for electron occupations. Vanderbilt59 ultrasoft pseudopotentials (USPP) with nonlinear core corrections are used to represent the interactions between the ionic core and valence electrons for all atoms in this work.55 Each of the metals have 10 valence electrons with

Figure 1. Supercell examples of ternary M1xM21−xB6 systems for each of the stoichiometries used in this work. Left: Pure phases of the binary hexaborides. Right, top: Well-mixed systems with highest homogeneity. Right, middle: Cubic ordered system displaying phase separation for 2 × 2 × 2 supercells. Right, bottom: Tetragonal ordered system showing planar segregation in 2 × 2 × 4 supercells.

both the pure phases and mixtures with the highest degree of homogeneity, and these can be seen in the left and top portions of Figure 1, respectively. Segregation within the mixture is simulated through ordering schemes to partition the metals within the supercell. The cubic ordered system (Figure 1, second row) represents a slight deviation from complete homogeneity, while the tetragonal ordered system (Figure 1, B


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consider include electrochemical differences, Brillouin zone effects, and valence electron densities.62 It is interesting to observe that although many of these properties vary in a roughly linear fashion with composition, the computed positional parameter obeys this relationship exactly for the Ca−Ba system. Figure 2 displays these trends

third row) starts to resemble phase separation within the mixture. It should be noted that although each of the lattice parameters for the supercells is allowed to change during the optimization, the periodicity imposed may artificially constrain certain systems depending on the atomic arrangements in each layer. For example, the tetragonal ordered system at x = 0.50 must have identical lattice vectors in the x- and y-directions, even though the bottom and top two layers would be expected to produce slightly different values. Lattice vectors normal to stacked planes of atoms are less problematic (e.g., z-direction for any tetragonal ordered supercell).

3. RESULTS AND DISCUSSION 3.1. Well-Mixed Solutions. Equilibrium lattice parameters and bulk moduli are calculated for each of the well-mixed solutions, including the pure metal hexaborides. These values are compared with experimental results in Table 1 with very Table 1. Calculated Lattice Parameters and Bulk Moduli for M1xM21−xB6 Compared with Experimental Results a (Å) boride BaB6 CaxBa1−xB6 CaxBa1−xB6 CaxBa1−xB6 CaB6 SrxCa1−xB6 SrxCa1−xB6 SrxCa1−xB6 SrB6 BaxSr1−xB6 BaxSr1−xB6 BaxSr1−xB6

x 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75

B0 (GPa)

z

calc

expt

calc

expt

calc

4.244 4.220 4.196 4.171 4.145 4.158 4.170 4.182 4.194 4.206 4.219 4.231

4.272 4.247 4.213 4.181 4.154 4.169 4.177 4.186 4.201 4.221 4.238 4.256

0.2046 0.2039 0.2032 0.2025 0.2018 0.2022 0.2025 0.2029 0.2032 0.2035 0.2039 0.2043

0.2019 0.1999 0.2021 0.2019 0.2011 0.1994 0.1976 0.1965 0.1935 0.1956 0.2027 0.2052

144.3 145.6 146.8 147.9 148.9 148.7 148.3 147.9 147.5 146.8 146.0 145.2

Figure 2. Variation of lattice constant (circles) and bulk modulus (squares) with respect to ternary hexaboride composition for wellmixed MB6 solutions.

graphically for the lattice constants and bulk moduli computed from our DFT calculations. The change in lattice sizing (top of Figure 2) with composition is in accord with Vegard’s Law.61 The bulk moduli also vary linearly, increasing with decreasing cation size and interoctahedral bond lengths. These calculations suggest that the size and strength of these materials can be tuned according to cation composition in a fairly predictable manner for homogeneous mixtures. It is useful to define some of the mechanical properties considered in this study to examine the nature of phases formedsee Figure 3. The lattice constant as well as

a

Calculated structural characteristics for the well-mixed solutions compared with experimental measurements from Gürsoy et al.44

good agreement. The positional parameter, z, can be found from B−BinterOh z= (1) 2·a where B−BinterOh is the bond length between boron atoms in neighboring octahedra, and a is the lattice constant of the unit cell. Since the crystal structures are the same for all metal hexaborides, we assume that the ternary hexaboride compounds will form ideal mixtures in the solid phase as a first approximation. One common metric used to describe the effect of composition on lattice size for crystals of the same structure was proposed by Vegard61 in 1921. For a solution containing components A and B, Vegard’s Law estimates the lattice parameter with a linear interpolation between the lattice constants of the two pure systems in thermal equilibrium, aA and aB, as aA,B, x = aA (x) + aB(1 − x)

Figure 3. Characteristic distances used in analysis of ternary structures. (a) Two unit cells displaying the lattice constant as well as interoctahedral bond and interoctahedral bond lengths. (b) The cation−facial distance is the magnitude of the vector between the centroid of the nearest octahedral face containing three boron atoms and the cation.

interoctahedral and intraoctahedral boron bonds are each labeled in Figure 3a. The cation−facial distance, shown in Figure 3b, is defined as the length between the cation and the geometric center of the triangular face created by the three nearest boron atoms located within a single octahedron, and the angle formed between a vector normal to the octahedral plane and the cation−facial vector describes the facial angle. As the name implies, the octahedral volume refers to the amount of space enclosed by the six boron atoms forming the vertices

(2)

where x is the mole fraction of component A within the mixture. The cation size is expected to be the main factor affecting mechanical properties, though other influences to C


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ACS Applied Electronic Materials of the octahedral structure. In certain cases, these shapes are found to be severely distorted from the typical regular polyhedra. All of the well-mixed solutions are considered to form a single phase, a conclusion supported by the identical metal− metal distances, interoctahedral bond lengths, and octahedral volumes found within the supercells at each particular composition. Interoctahedral distances (octahedral volumes) seem to be determined by the four (eight) cations directly surrounding the two (six) boron atoms in these configurations; these surrounding atoms are identical for all octahedra by construction. Equimolar cation ratios also yield a single intraoctahedral bond length, cation−facial distance, and facial angle. For inequivalent metal ratios at x = (0.25,0.75), two distinct values are produced for both the intraoctahedral bond lengths and cation−facial distances. The intraoctahedral bond has two nearest cation neighbors in MB6 systems, and its length is directly dependent upon the identities of these two metals for homogeneous systems. Although the octahedral volumes are identical within the 2 × 2 × 2 supercells for each of the wellmixed solutions, their shape is generally skewed from a regular octahedron. This is entirely due to their cationic environment. Every octahedral unit has its intraoctahedral bonds enclosed by an identical set of eight metal pairs, but the shape produced by the six boron atoms rotates in space accordingly. The cation− facial distances and facial angles also depend upon the type of cation, and displacements are skewed in favor of the majority component and its pure phase. In cases where one cation is found in minority, the octahedra adjust such that facial angles are perpendicular to the solute. Analogous to Vegard’s Law, certain other mechanical properties, ψ, can be described by a simple mixing rule ψA,B, x = ψA(x) + ψB(1 − x)

Figure 4. Vegard ratio (see Section 3.1) of various properties for the well-mixed Ca−Ba (top), Ca−Sr (middle), and Sr−Ba (bottom) ternary hexaboride mixtures.

former two are heavily related and are only expected to show differences for irregular polyhedra. In all cases, the octahedral volumes increase with increasing cation size for the pure binary hexaborides. The intraoctahedral bonds are found to resist deformation with smaller cation substitution, an effect noted by Oshima et al.63 when analyzing the surface structure of LaB6. Bulk moduli have a tendency to increase with decreasing cation size, an effect noted earlier for pure phases. The positive values of ψ̂ A,B,x obtained for the bulk modulus (shown in Figure 4) suggest that doping with lighter elements may be an effective means to strengthen these materials. 3.2. Higher-Order Solutions. We use segregation of cations within each supercell to represent the formation of nanoregimes, i.e., incomplete homogenization during the synthesis and/or heat treatment processes, and these geometries are shown schematically in Figure 1 as the cubic and tetragonal ordered systems. It is not within the scope of this work to fully study the phase boundaries and mechanical/ thermodynamic regimes in these materials but to provide initial insights if segregated configurations can potentially favor the formation of dispersed phases (nanoregimes) with lower free energies (Helmholtz or Gibbs) than that of a homogeneous mixture phase. From a classical thermodynamics point of view, the close interplay between enthalpic and entropic effects, which are a function of temperature and composition, determines the local stability of the mixture phase separation vs single mixture.64 The mechanical stability and properties of a segregated mixture are a rather complex problem to model, as it depends on the distribution of the nanoregimes within the continuous phase, domain geometry, and properties of the grain boundaries, in general. We do expect, generally speaking, that segregated phases of MB6 mixtures would not follow the linear behavior observed in

(3)

for two components, A and B. This provides a metric for comparison with DFT-calculated properties of the mixture, ψA,B,x * . To aid in correlating the effect of composition on these properties between different solid solutions, we introduce a parameter called the Vegard ratio, ψ̂ A,B,x, for which ̂ x= ψA,B,

* −ψ ψA,B, x A,B, x |ψA − ψB|

(4)

for a mixture having molar fraction x of component A. The pure component properties, ψA and ψB, are used to provide a suitable span of values to normalize the data sets. Results for ternary hexaborides using the Vegard ratio are shown in Figure 4, and each system is displayed such that the abscissa corresponds to the concentration of the smaller cation. Octahedral volumes, interoctahedral bond lengths, intraoctahedral bond lengths, lattice constants, bulk moduli, and cation−facial distances (see Figure 3) are analyzed and compared to their predicted values assuming these properties vary linearly with increasing solute concentration. For the octahedral volumes, the cubic root is taken as the variable length for ψ̂ A,B,x. In general, variations are less than 5% of the calculated ψ̂ A,B,x, and positive deviations are found for all properties except the interoctahedral boron bond lengths in systems containing barium. The three features displaying the greatest deviations with respect to cation substitution are the intraoctahedral bond, octahedral volume, and bulk modulus, though the D


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with a clear larger peak near 1.8 Å and a smaller long tail in the distribution toward shorter bond lengths. The result from our computational work (Figure 5) mimics the form of the distribution found in the experimental result, though the bond distance range is smaller (1.65 to 1.78 Å), and the peak in the distribution is smaller by 0.25 Å. This can be contrasted with the idealized, average structure, where exactly two B−B bond distances exist, rather than a distribution of bond lengths. The general agreement between the methods, even for the small sample size afforded by the DFT simulations, gives good confidence in the local structural distortions resulting from the vastly different metal cation sizes. Since the strength of a bond is dependent upon the electron density, we use this to assess the effect of each cation set. Contour plots of the electron density cutting through the interoctahedral bonds are shown in Figure 6, where the cations

homogeneous mixtures (Vegard’s law behavior), as the nanoeffects of dissimilar ionic sizes and nanoregimes at the domain boundaries add nonlinear contributions to the free energy of the system.65 MB6 materials are ionic solids, where Coulombic effects have to be taken into account as well for full consideration in a comprehensive thermodynamic framework or analysis of these materials using free energy models. For example, Vikrant et al.66 proposes such a framework that includes ionic effects on the chemical and mechanical equilibrium of ionic materials by taking into account contributions from the electrostatic energy, density of dipole moments, and mechanical equilibrium considerations. We find that as the cations begin to segregate, increasing the degree of ordering, anisotropies in bond lengths and geometrical parameters begin to appear. To investigate possible reasons for the formation of nanoregimes, we start by analyzing the dependence of the boron framework on the cation environment. Differences between interoctahedral bond lengths are approximately 2.5 times greater than intraoctahedral bond lengths for the pure phases; therefore, these are likely an important factor in the formation of dislocations. The cubic ordered system has three types of interoctahedral bond lengths for each composition, and these depend on the four cations directly surrounding the bond in a relationship similar to that found for the homogeneous solutions. However, only one characteristic bond length is observed in the wellmixed systems, since every interoctahedral bond has an identical quad-cation set. In the cubic ordered systems, we find lattice constants, octahedral volumes, and averaged intraoctahedral and interoctahedral bond lengths for each octahedra to be identical for every unit cell. Figure 5 shows that it is possible to derive a histogram of B− B bond lengths from the various DFT-derived configurations.

Figure 6. Electron density contour plots through the plane containing four boron atoms per octahedra. Barium (green), calcium (blue), and strontium (purple) atoms lying above the plane are shown as spheres. Electron density iso-values of ρel = 0.945 Å−3, corresponding to interoctahedral bonds, are displayed as yellow surfaces bisecting pairs of metals. Pure binary hexaborides are shown for (a) BaB6, (b) CaB6, and (c) SrB6. For the cubic ordered CaxBa1−xB6 system, calcium fractions are shown for (d) x = 0.25, (e) x = 0.50, and (f) x = 0.75, where the ratios written above the isosurfaces represent the number of Ba/Ca atoms surrounding the interoctahedral bond. Boron atoms are removed for clarity.

(colored spheres) are located above the plane of view. These contours are given for pure BaB6, CaB6, and SrB6 in Figure 6a, b, and c, respectively, where the boron atoms have been omitted for clarity. The interoctahedral bonds can be found at the midpoint between two cations along the x- and ydirections, and a yellow surface displays the electron density at a value of ρel = 0.945 Å−3, chosen to emphasize the density variations. It is apparent that the density is highest in CaB6 and decreases with increasing cation size, in agreement with the observed and calculated bond lengths from Table 1. To understand how these bonds are affected by the formation of a solid solution, a similar contour plot is displayed in Figure 6d,e for the CaxBa1−xB6 system with calcium fractions at x = 0.25, x = 0.50, and x = 0.75, respectively. The three numerical values given above interoctahedral bonds within each plot refer to the ratio of barium to calcium (Ba/Ca) atoms surrounding the boron− boron bond. Bond strengths undergo significant changes after only a single substitution of barium for calcium. This can be

Figure 5. Boron−boron distance histogram from 11 different structure configurations determined via DFT simulations. The histogram is an average over the three Ca/Ba solid solution compositions, the two end members, and the three ordering states: well-mixed, cubic ordered, and tetragonal ordered.

In order to compare to the experimental work of Koch et al.,67 we show an average over the three types of ordered states for each of the three Ca/Ba ratios and also include the end members. Thus, the histogram incorporates 11 atomic configurations and represents an ensemble average over many different local environments, including locally Ca-rich or Ba-rich configurations. Koch and co-workers67 used reverse Monte Carlo fitting of X-ray pair distribution function data to extract the various B−B bond lengths from a large supercell, and found that the B−B distances range from 1.43 to 1.97 Å, E


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all lattice constants and interoctahedral bond lengths within a mixture to be identical as the degree of homogeneity increases, indicating the formation of a single phase. Regions with nearby cation concentrations differing from the bulk will inevitably alter the bond lengths between boron atoms, forcing a slight mismatch between interoctahedral bonds and give rise to the shifted regions observed in HRTEM micrographs found from the literature. These nanoregimes are interpreted as regions that have not yet fully mixed. Kinetic processes are suggested to avoid the formation of dislocations within the mixture, since the heterogeneous systems are found to have lower energy than the well-mixed systems at 0 K based on our periodic calculations. The results of the this work warrant further studies on detail analysis of the chemical−mechanical stability of the mixtures of these materials for the development of new applications including determination of the thermodynamic phase diagrams using approaches such as phase field methods and detail characterization of mechanical properties in the various regimes of the phase diagram.

seen by the electron density differences between the 4:0 and 3:1 interoctahedral bonds in Figure 6d. Increasing the calcium content surrounding the interoctahedral bond further increases the electron density located between adjacent octahedra, forming shorter interoctahedral bonds and contributing to the smaller lattice sizing found. However, the strain formed by this mismatch is difficult to assess in these small cells in a periodic system. The use of tetragonal ordered systems helps to alleviate this issue, and specific changes in bond lengths and lattice constants are observed with deviations in the lattice constant as much as 1.5% along the z-axis. The thermodynamic stability of a system is intrinsically linked to its free energy, and we analyze energies calculated at 0 K to determine whether one phase is thermodynamically unstable with respect to another at the same composition. Using the well-mixed phase as a reference, Table 2 lists the Table 2. Zero Temperature Energy Differences from WellMixed M1xM21−xB6 Solutions

■

Δ × 100 (eV/cell) boride

x

cubic

tetragonal

CaxBa1−xB6 CaxBa1−xB6 CaxBa1−xB6 SrxCa1−xB6 SrxCa1−xB6 SrxCa1−xB6 BaxSr1−xB6 BaxSr1−xB6 BaxSr1−xB6

0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75

−0.903 −1.807 −1.061 −0.211 −0.350 −0.193 −0.304 −0.575 −0.275

−1.097 −1.103 −1.134 −0.251 −0.242 −0.246 −0.304 −0.306 −0.293

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; Phone: +775 784 6060; Fax: +775 784 4764. ORCID

Olivia A. Graeve: 0000-0003-3599-0502 Victor R. Vasquez: 0000-0002-6604-8096 Notes

The authors declare no competing financial interest.

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a

Calculated energy differences taken as (ordered −  well‐mixed) per unit cell. Cubic and tetragonal refer to the geometries shown in Figure 1.

ACKNOWLEDGMENTS We gratefully acknowledge funding support from the NSF [Award No. 1360561] (“SNM: Scalable Manufacturing of Unique Hexaboride Nanomaterials for Advanced Energy Generation and Gas Storage Applications”). A portion of the figures were generated in part using XCrySDen.68

stabilization energies of cubic and tetragonally ordered systems at each molar ratio. We find ordered structures to produce lower energies for all composition ranges, suggesting the homogenization of these materials is intrinsically dependent upon kinetic activation processes. Within each ternary set, the stability of the tetragonal system is fairly independent upon concentration, though this may be caused by the artificial constraints imposed for calculations, as discussed earlier. However, the cubic ordering displays significant variations in system energies. The equimolar cation ratios show an increased preference toward segregation, in full agreement with the experimental evidence.52 This tendency is highest for the Ca−Ba system, where the stabilization energy for an equimolar mixture is practically double the values determined for CaxBa1−xB6 with x = 0.25 or x = 0.75. Stabilization energies due to segregation are also found to increase with the size difference between the two cations, supporting the discrepancies observed between Ca−Sr and Ca−Ba XRD data.52

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REFERENCES

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4. CONCLUDING REMARKS The linear approximations (Vegard’s law type) used to describe property changes in mixtures have provided a useful tool to analyze the nonidealities determined from DFT calculations. We find doping with lighter cations may be an effective means of strengthening MB6 materials. Minor deviations from Vegard’s Law are found to be the norm, and bond lengths are generally larger than those predicted by the simple sum rules. The electronic structure calculations predict F


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Phase Stability Analysis of Ternary Alkaline-Earth Hexaborides

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