ABSTRACT

Comparison of well-determined single crystal data for stoichiometric, or near-stoichiometric, metal hexaborides confirms previously identified lattice parameter trends using powder diffraction. Trends for both divalent and trivalent forms suggest that potential new forms for synthesis include Sc and Mn hexaborides. Density Functional Theory (DFT) calculations for KB₆, CaB₆, YB₆, LaB₆, boron octahedral clusters and Sc and Mn forms show that the shapes of bonding orbitals are defined by the boron framework. Inclusion of metal into the boron framework induces a reduction in energy ranging from 1 eV to 6 eV increasing with ionic charge. For metals with d₁ character, such a shift in energy brings a doubly degenerate band section along with the G-M reciprocal space direction within the conduction bands tangential to the Fermi surface. ScB₆ band structure and density of states calculations show directions and gap characteristics similar to those of YB₆ and LaB₆. These calculations for ScB₆ suggest that it may be possible to realize superconductivity in this compound if synthesized.

1. Introduction

Metal hexaborides, MB₆ (where M includes alkali, transition and rare-earth metals) show simple cubic symmetry (Pmm) in which the metal is in 24-fold coordination with a framework of boron octahedra at the vertices of a cube. This boron octahedral framework is a key contributor to the physical properties ascribed to many hexaborides. The isostructural forms of MB₆ are superconductors (e.g. M = Y, Th) [1,2], metallic conductors (e.g. M = La, Gd, Dy) [3,4] and semi-conductors (e.g. M = Ca, Sr and Yb) [5,6] while others display intermediate or unusual behavior (e.g. M = Sm, Ce, Nd) [7,8] or have notable thermal stability and hardness [9,10].

In general, metal ions are divalent or trivalent except for KB₆ [11,12] and ThB₆ [13]. The smaller size metal ion generally does not favor formation of hexaborides for multi-valent elements (e.g. Pr³⁺ is preferred over Pr⁴⁺; Ce³⁺ over Ce⁴⁺). Formation of actinide hexaborides can be enhanced through substitution, for example, in HoB₆ using Ca or La as minor substitutents [14]. In general, mixed metal forms of hexaboride such as (Nd_1−xGd_x)B₆, (Ce_1−xNd_x)B₆ or (Ce_1−xLa_x)B₆ [15-17] follow Vegard’s law [18] with the exception of Sm_xM_1−xB₆ structures. Other exceptions to Vegard’s law across a full substitution range include (Eu_xCa_1−x)B₆ for which phase separation occurs at x = 0.27 [19].

Seminal work on hexaborides undertaken prior to the 1970’s is well documented in the volume by Matkovich [20]. In this work, general inferences relating to structures and properties including band structure calculations are presented for CaB₆, LaB₆ and BaB₆ [21]. More recently, high resolution crystal structure determinations of additional hexaboride compositions have been completed (e.g. Sm, Ce, Sr, Ca, Nd, Eu). These carefully defined structures allow further insight into bond length variations and their influence on electronic properties within the hexaboride suite. In this work, we compile and compare these structure determinations of end-member metal hexaborides with other hexaborides that may be formed at stoichiometric, or near stoichiometric composition given the development of facile and chemically precise synthesis methods. While electronic structure calculations have been undertaken for a range of compositions [5,12,21], theoretical studies of Sc, Ti or Mn hexaborides are not extant in the literature and are provided in this work.

2. Experimental Data and Calculation Methods

Data from single crystal refinements and high resolution powder diffraction studies are compiled from the literature. Data from refinements with quality goodness of fit measures (e.g. R_w ≤ 5%) are listed in Table 1 along with reference sources.

Band structure calculations are undertaken using the CASTEP module of the Materials Studio 6.1 software package. Details of this software and the CASTEP module are well described [22]. Most calculations have been performed using the Local Density Approximation (LDA). For comparison, some calculations have also been performed in the Generalized Gradient Approximation (GGA). The LDA is exact in the uniform electron density limit and the GGA includes information on the spatial variations in electron density, using functions which are only valid for slowly varying densities [23]. A well established outcome of previous work is that LDA calculations yield bonds shorter than experimental values and GGA models estimate longer bonds than experimental. Further details on model assumptions and outcomes are available at Milman et al. [24].

Initial structures are normally optimized for geometry and all calculations make use of plane-wave (PW) basis, linear response functions, norm-conserving pseudo potentials and interpolation methods. The PW basis set cut-off is typically >290 eV and the energy cut off is 10 eV. For the same metal hexaboride, some calculations are undertaken with different optimized cell parameters in order to explore the limits of structure stability.

3. Crystal Structure

Metal hexaborides are cubic structures with an ambiguous bonding character that is, on the one hand, considered ionic and, on the other, covalent or hybrid bonded, particularly within the boron framework. Indeed, recent ab initio calculations on the LaB₆ electronic structure show that covalent, metallic and ionic bonding co-exist in this specific structure [25]. Ionic character seems predominantly related to M²⁺ valences with the third electron in M³⁺ ions participating in metallic conduction bands. Structure refinements of alkali, alkali-earth, rare earth and actinide hexaborides support the concept of a boron framework within which metal ions are vibrating freely in the interstitial positions [6,8,13,17,26]. In general, the refined temperature factors for boron are anisotropic and reflect the strengths of the inter-octahedral bond relative to the intra-octahedral bond [17]. For the rare earth hexaborides, structure refinements confirm that vacancies occur on the boron site [8,13,17,27]. Boron isotopic effects are also manifest in the symmetry and frequency of Raman vibrations [9].

MB₆ structures have two different boron-boron bond lengths: the inter-octahedral and the intra-octahedral as shown in Figure 1. The intra-octahedral B-B bond (BB_in in this study) is longer than the inter-octahedral bond (B-B_out in this study). The shorter length of the in-

ter-octahedral bond influences thermal stability and hardness of hexaborides among a number of properties [10].

3.1. Cell Dimensions

Of the nineteen end-member hexaborides synthesized to date, eleven have been subjected to high resolution powder diffraction or single crystal structure refinements. Data for these structure determinations have been collected at ambient or room temperature. The detailed reports on SmB₆ and CeB₆ single crystals at a range of temperatures (100 K, 165 K, 230 K and 298 K) are exceptions [28,29] and provide key evidence for donation of 4f electrons to the B-B bonds connecting B₆ octahedra in these structures. XRD studies on slightly K-deficient KB₆ have shown indications of lattice instabilities at 60K [12]. The lack of low temperature structural data for a broader range of hexaborides indicates that key phenomena are likely waiting to be discovered.

Data on end-member hexaborides for which compositions have been independently assessed or estimated from structure refinements are listed in Table 1. Table 1 lists data with a weighted residual (R_w) based on least squares refinement of structure factors that provides a measure of refinement precision. The high symmetry of MB₆ provides few crystallographic parameters to describe, or refine, a structure. Two primary parameters are the cell dimension, a, and z, the 6f position for boron at the (1/2, 1/2, z) site.

Ionic radii listed in Table 1 are for eight-fold coordination of the lowest charge form of the metal. The calculated parameter, a-2r_R [30], is listed in Table 1 because for hexaborides the mean square displacements of rare earth metal ions appear to correlate with a-2r_R [31], where r_R is the ionic radius for the respective metal ion. The data by Chernyshov et al. [31] when recalculated, reflect ionic radii for the metal ion in six-fold coordination. As noted below, experimental data show that specific metal ions prefer certain geometric configuration(s)

and, in order to correct the record, Table 1 lists calculated values for a-2r_R for metal ions in eight-fold coordination. Eight-fold coordination is a more reasonable expectation for metal ion coordination given eight vertices in a cube.

For nineteen refined cell dimensions evaluated, including those in Table 1, the variation in values is less than 5% while the variation in ionic radii for metals of the same structures is many times at 30% - 40%. Yahia et al. [9] suggest this feature is a clear indication of the rigidity of the boron lattice. However, the boron framework accommodates a wide range of valence electron distributions despite the constraints of high symmetry.

Figure 2 shows a plot of cell parameter, a, against ionic radius [32] for a range of MB₆ structures containing divalent or trivalent metal ions listed in Table 1. The calculated systematic errors for each lattice parameter are contained within the symbols in Figure 2. These data show an expected linear relationship between cell dimension and ionic radius for six-fold and for eight-fold coordinated divalent or trivalent metal ions in the MB₆ structure. Both trends—using only structure determinations for divalent or trivalent metal hexaborides (blue filled symbols in Figure 2)—show a strong linear correlation for six-fold and eight-fold coordination (r² = 0.97 and 0.98, respectively) with slightly better alignment of the Th lattice parameter in eight-fold coordination. Both K and Th are not included in the least squares linear fits shown in Figure 2. Superconducting hexaborides are denoted with a red-outline to the symbol in Figure 2. Unfilled symbols in Figure 2 are for potential structures for which a cell dimension has been estimated using the respective linear trend.

However, KB₆ does not lie on this linear trend for either coordination as indicated in Figure 2. Note that the value for K in six-fold coordination plots close to the trend line for eight-fold coordination. Similarly, if twel-

ve-fold coordinated ionic radii are plotted for the respecttive hexaborides, the eight-fold coordination K value lies closest to the twelve-fold trend line. Whilst circumstantial, this notional fit for KB₆ cell parameter with ionic radius values suggests that potassium may favor a higher coordination in this structure type than normally attributed to the ion.

3.2. Bond Lengths

Bond lengths are reliably determined by least squares refinement of structure factors which take account of peak positions and peak intensities (moderated by absorption and other factors). The datasets that allow for calculation of bond lengths in metal hexaborides are limited and are listed in Table 1. Figure 3 provides detailed data on intra-octahedral (B-B_in) and inter-octahedral (BB_out) bond lengths using MB₆ structure determinations.

Trend lines have been constructed for divalent metals (e.g. Ba, Sr, Eu, Ca, Yb) and for trivalent metals (e.g. La, Nd, Y) for both types of bonds. SmB₆, CeB₆, ThB₆ and KB₆ data, although plotted in Figure 3, have not been included in the trend line calculations as they represent unusual structure types (e.g. Kondo compounds or valence fluctuation) or different valence state(s) for the metal ion. Data for KB₆ do not lie on either trend line and ThB₆ is divergent from the trend.

In Figure 3, filled symbols represent data collected from sources listed in Table 1 with +3 metal valence symbols designated as circles and triangles for interand intra-octahedral bonds, respectively. Similarly, +2 metal valences are designated as square and diamond symbols for interand intra-octahedral bonds, respectively.

Unfilled symbols in Figure 3 represent inferred values for possible hexaboride structures as discussed below. The tendency for bond-lengths of Sm and Ce compounds to lie on, or close to, the divalent trend line in Figure 3 is noteworthy, given their predominantly trivalent charge in the MB₆ structure [7], and may reflect the unusual character of these two dense Kondo structures.

Symbols for known superconductors in this suite of hexaborides are designated with a red border. Each superconducting compound has a significantly greater bond length difference between the interand intra-octahedral bonds than other MB₆ structures of similar cell dimension. Thus, for a particular electron charge, the bonds between boron octahedra are shorter and the bonds within the octahedra are longer for superconducting than for non-superconducting MB₆ structures. Note that while YB₆ and LaB₆ show superconducting properties, the LaB₆ transition is at a very low temperature [3] and electronic properties are highly subject to small substitutions (e.g. <0.01 mole% Eu shifts from semiconducting to metallic [4]) and may, in effect, be considered a “weak superconductor”.

Table 1 lists the metal-boron bond length for structure refinements of metal hexaborides with estimated values for some structures indicated in italics. Where the ex plicit M-B value is not tabulated in an earlier literature reference, the formula: a (0.5 + z²)^1/2 is used to calculate the metal-boron value [6] based on the other critical crystallographic parameter for refinement, z, the position of the B atom as noted above.

Comparison of B-B and M-B bond lengths shows that B-B_out and B-B_in vary by about 8% and 3.4% in value, respectively while the M-B bond varies by 4.3%. This latter range of values for M-B bonds is similar to the cell dimension change (~4% overall). These differences in bond lengths, particularly the larger relative change for B-B_out, suggest that the boron framework cooperatively adjusts both the B₆ octahedra and, predominantly, the bonds between B₆ octahedra. While there is some level of rigidity to the overall structure, this framework flexibility is an important contributor to phase stability for these hexaborides. Given the symmetry-defined formula that relates B-B_out and B-B_in (B-B_out = a – 2^1/2B-B_in), larger changes in B-B_out are required to accommodate changes in B-B_in.

3.3. Cell Dimension Variations and Limits

As shown in Figure 3, calculated bond lengths based on structure refinements, particularly B-B bonds, reveal more detail than the linear trends for ionic radius. For this structure type, there will be a limit at which incorporation of a particular metal ion will result in a change of symmetry or phase relation. The trend lines in Figure 3 for both types of boron bonds in divalent and trivalent structures converge at smaller cell dimension. The convergence of these trends define a geometric limit for a cubic hexaboride structure and thus, sets a limit on likely ions that may be included in new forms of stoichiometric metal hexaborides. The trends in Figure 3 imply that the lower limit of cell dimension will range between 4.01A and 4.03A (dotted vertical lines in Figure 3). With reference to Figure 2, this suggests that ionic radii 0.80 < r_R < 0.85 is a lower limit for an eight-fold coordinated metal. This lower limit is also consistent with published theoretical values of lattice constants for hexaboride structures [33].

An upper limit for the cell dimension of hexaborides is more difficult to predict as there are no divalent metal ions with ionic radius larger than Ba²⁺. Nevertheless, a single crystal study [34] of BaB₆ showed that the structure remained stable to 400 K with cell dimension expansion to approximately 4.284A. This work suggests that larger metal ions may be incorporated into a hexaboride structure albeit these would be univalent (e.g. K, Cs, Rb and Fr). To date, only KB₆ has been synthesized [11].

Ogita et al. [30] suggest that the term, a-2r_R, describes the “cage space” [30,31] or the amount of space in the boron framework within which the M ion may move. A plot of a-2r_R against cell dimension for divalent and trivalent metal ions gives an expected linear trend that appears counter-intuitive because as cell dimension decreases, the available cage space increases (data not shown). The cage space calculation [30] assumes the boron framework is rigid and that the predominant motion/vibration within the MB₆ structure is related to the metal ion. According to Chernyshov et al. [31], the atomic thermal displacements of the metal ion increase with reduced cell dimension. This trend appears to correlate with softening of the acoustic modes for phonon spectra in the rare earth hexaborides [31] and to relative peak energy of Raman spectra [30] at <200 cm⁻¹. However, a geometrical analysis of the hexaboride structure suggests these displacements would reduce with smaller lattice parameter.

Inspection of the region around the M ion shows that as B-B_in and B-B_out adjust dimensions (in six symmetry-related directions equally), the volumetric re-arrangement is that of an irregular truncated cube. To simplify our estimate of the change(s) in volume as bonds adjust to accommodate specific metal ions, we have calculated the volume bound by the boron octahedral edges defined as the “truncated cube volume” (V_tc) which is the truncated cube bound by the surfaces formed by B-B_in and B-B_out bonds surrounding the metal ion.

Figure 4 shows values for this truncated cube volume

(V_tc) plotted against the ionic radii for the relevant metal ion. Note that V_tc is not the volume contained within each boron B₆ octahedra within, or at, the vertices of the unit cell (depending on the choice of origin)—but the volume surrounding the M ion in six symmetry-related axial directions.

There is a strong positive relationship between V_tc and metal ion radius for all refined structures. While the value for KB₆ does not directly align with the trend line in Figure 4, the general relationship—a larger value of V_tc related to a larger ionic radius—applies. The trend in Figure 4 shows that within the MB₆ structure, the overall space—bounded by boron atoms—surrounding the metal ion increases with ionic radius of the metal ion (and with cell dimension). This seems to indicate that as the cage contracts, the cage itself contributes more to its own stability, probably through repulsion between boron octahedra, but also with the octahedra acting as complex ligands bound to the central metal.

Figure 5 shows the variation of V_tc with B-B bond length difference, ΔB_B, for the refined structures listed in Table 1. This plot shows that the highest values of ΔB_B occur with KB₆, YB₆, NdB₆, ThB₆, LaB₆ and CeB₆—all metallic conductors—while other forms follow a decreasing trend of ΔB_B with increased octahedral cage volume. The three known forms of superconducting hexaborides show higher values for ΔB_B as shown in Figure 5 and, implicitly, in Figure 3. Hexaboride insulators are to the lower right side of Figure 5.

4. New Hexaborides

Trends in Figures 2-4 allow an estimate of cell dimension or V_tc for an unknown MB₆ structure provided either the ionic radius or the cell dimension is known. For example, given a trivalent eight-fold coordinated ionic radius of 0.87 for Sc [32], the linear trend in Figure 2 provides an estimate of cell dimension for a putative

ScB₆ structure, a = 4.0398A. The elements Sc and Mn with +3 and +2 charge, respectively, are considered potential candidates for a metal hexaboride structure because they meet the minimum crystallographic requirements noted above. Accordingly, with determination of cell dimension from Figure 2, estimates of bond lengths, V_tc, and ΔB_B have also been calculated and are shown in Figures 3-5. Following a similar argument, the eight-fold coordinated ionic radius for Li¹⁺ is 0.92A [32] and infers that with a cell dimension of 4.06A, a LiB₆ structure may be possible. A lower symmetry, but structurally-related compound, Li₂B₆, is well characterized [35].

Known hexaborides from Row IV of the Periodic Table are KB₆ and CaB₆ each of which have been synthesized with significantly different degrees of stoichiometry [12,34]. Other forms of hexaborides from this row are not documented. For potential structures with Sc and Mn, the estimated cell dimension, a, is greater than 4.03A, and the B–B bond lengths are substantially smaller than KB₆ or CaB₆; consistent with the influence of ionic radius on the overall structure. In each case, bond length estimates for these putative structures imply a stronger inter-octahedral bond and relatively smaller B₆ octahedra compared with their Row IV counterparts. If it is possible to synthesize these MB₆ compounds (M = Sc³⁺ or Mn²⁺), a comparison of experimental data with these simple estimates of structural parameters would be informative.

Barantseva and Yu [36] also examined the state of hexaboride syntheses at the time and concluded that it is impossible to make ScB₆ except perhaps via high pressure. In their analysis [36], they argue that the difficulty of synthesis is due to the strong affinity of Sc atoms for B atoms rather than dimensional features related to ionic radii, structure and symmetry. Barantseva and Yu [36] propose a slightly larger cell dimension for ScB₆ (a = 4.08A) than estimated from Figure 2, but develop an argument based on ionic polarisability and the relative influence on the M-B bond on overlapping boron orbitals particularly along the B-B_out dimension. We shall return to the viability of a ScB₆ structure via theoretical band structure calculations below.

For Ti, only the +4 valence state allows an eight-fold coordination for which the ionic radius is estimated at 0.74A [32]. An estimated cell dimension of a possible TiB₆ structure using the equation from the trend line in Figure 2 (but for a divalent or trivalent valence state) is 3.988A. This estimated cell dimension is significantly less than the estimated lower value for metal hexaborides and thus, suggests that stoichiometric TiB₆ may be difficult to synthesize. As noted below, geometry optimizations in band structure calculations converge close to this value at a = 3.9802A even when the input value is nominally 4.08A.

5. Electronic Structures

Theoretical calculations of elastic, electronic and lattice dynamic parameters for metal hexaboride structures are ultimately constrained by geometry or crystal structure [33,34,37]. Valence state also influences these properties as divalent hexaborides are commonly insulators or semiconductors and trivalent hexaborides are commonly metallic [5,33,37]. As noted earlier, KB₆ is identified as not a “normal” metal [12] and ThB₆ is a paramagnetic metal [13]. Defects clearly influence electronic properties in this structure type and are considered below.

5.1. Stoichiometry

Shirai and Uemura [38] show that for icosahedral boron structures valence counting has a valid place in the interpretation of covalently bonded structures including borides and boron carbides. This simple analysis is particularly useful for high symmetry structures which tend to incorporate vacancies and interstitials into large unit cells and for structures with close similarity in elemental configurations (such as with boron and carbon) for which experimental quantification may be difficult [39]. In addition, when the electronic structure determined by experiment seems to be in disagreement with a band structure calculation, Shirai and Uemura [38] suggest that this is an indicator for the presence of defects.

As shown in Table 1 and other structural studies [27], the level of defects—identified by refinement techniques as stoichiometry—for these hexaboride structures is low except for KB₆ (K~0.94) and ThB₆ (B~6.15). For all other structures identified in this work, the nonstoichiometry for a particular hexaboride structure is less than 2%. Experimental and theoretical values of lattice constants and bulk moduli for defect-free structures compare well for Ca, Sr, Ba, Eu, Yb, Y, La and Cd hexaborides [34]. Experimental data from NMR analysis [37] of YB₆ is also strongly supported by theoretical calculations of band structure, density of states and valenceelectron distributions. Estimates for z, the boron 6f position shows good correlation [34,40] for YB₆, BaB₆ and ThB₆.

Compositions further from stoichiometry are observed in KB₆ (e.g. K_<0.84B₆ [12]) and ThB₆ (e.g. Th_0.78B₆ [41]) and occur in other hexaborides [42] (e.g. Sm_0.96B₆). Indeed, (K_1−x)B₆ shows a significant change of color with greater non-stoichiometry (from “Bordeaux red” to black) that is dependent on synthesis temperature [12]. This work [11,12] shows that the valence bands of KB₆ are partially empty while Katsura et al. [43] show a clear difference in calculated electronic properties between stoichiometric and hole-doped KB₆.

Very small concentrations of defects influence electronic properties in structures with small, or no apparent, band gap [37]. For example, in CaB₆ the presence of boron-related defects (the difference between 6N and 3N purity or << 1%) are closely associated with carrier doping and the formation of mid-gap states 0.18 eV below the conduction band [5]. Small amounts of La substitution for Ca are also associated with magnetic effects [44]. Vacancy defects also influence the complexity of Raman spectra—particularly the A_1g and E_g bands—and reduce the basic cubic symmetry of hexaborides [9]. Yahia et al. [45] also show that an excess of boron in the hexaboride structure influences local symmetry.

5.2. Valence Electrons

In contrast to boride structures [38], valence counting for metal hexaborides requires consideration of the B₆ octahedra which demonstrate strong covalence [28,46] while partially ionic character is evident for the M-B bond. Recent work [28,29] on electron distribution in SmB₆ and CeB₆ shows there is a high sensitivity of electron distribution to a) temperature and b) anharmonic vibrations. In CeB₆, donated electrons localize around the B-B_out bonds with reduction in temperature supporting the notion that the boron framework is not “rigid” but flexible.

Simple valence counts for metal ions imply that for charge balance to the overall structure, the charge on a boron atom, or the B₆ moiety, will accommodate a wide range of values (from −1 to −4) depending on the metal ion. This variability of charge balance or of bond populations in the boron framework is explicitly revealed [25] in theoretical calculations of the electronic structure of LaB₆. These calculations show that the B-B_out bond is almost completely covalent while other bonds within the B₆ octahedra display mixed covalent and ionic character [25].

This capacity for boron atoms to show valence ambiguity within a responsive framework of connected octahedra (i.e. within a specific structure type) engenders a high sensitivity to subtle variations in electron distribution and, as a consequence, physical properties are also highly influenced by the presence of boron-related vacancies or defects. This sensitivity is manifest in electronic and magnetic properties across a structure type which displays the gamut of physical properties.

Schmitt et al. [34] calculate the band structure for BaB₆ and CaB₆ using DFT and show that the total energy, E_tot, depends on the positional parameter, z, for the minimum energy range. In addition, the direct band gap for both structures [34] depends linearly on z while SmB₆ shows similar dependence [42]. Calculations show that lowering z below the equilibrium value results in significant overlap of valence and conduction bands [34]. In the case of BaB₆ and CaB₆ these shifts in z value, which may be less than 0.003 au, are consistent with a change from metallic to insulating properties [34].

Figure 6 shows the calculated values for z derived from refinements of experimental data shown in Table 1. These z values are plotted against B-B_in and B-B_out values from each structure refinement. Each metal ion of the hexaboride structure is identified in Figure 6. B-B_out bond length values encompass a wider range of values compared with B-B_in values across these structures. The z values, except for KB₆, show a strong linear trend with B-B_out values (r = 0.96) and show a difference in value of 0.008 au (~0.03 nm). Metallic and superconducting hexaborides have refined values for z < 0.201 and, as noted above, this implies that the difference in bond lengths, ΔB_B, is a key indicator of physical or electronic properties. Note also that B-B_out values for the superconducting hexaborides are <0.200 and furthest from the position at which both bond lengths are equivalent [34] (designated by the red dotted line at z = 0.207 in Figure 6).

KB₆ is also a metal hexaboride with unusual properties which may reflect not only non-stoichiometry (and the presence of defects) but also the presence of localized electrons [11]. We suggest that cell parameter data are important indicators of unusual electronic properties and is evident from the refined powder diffraction data [11] for K_0.97B₆. For example, cell parameters reduce systematically with lower temperature but show dramatic changes below 100 K [12]. At 60 K, the B₆ octahedra contract to a minimum size and then, with further reduction in temperature, increase rapidly to a local maximum at about 40 K. This bulk behavior appears related to a magnetic hysteresis effect below 100 K identified by measurements of magnetic susceptibility and Seebeck coefficient [11,12]. This behavior also supports the notion that the boron framework is, within limits, flexible and responsive to the electronic structure of the hexaboride.

5.3. Band Structure Calculations

For LaB₆ and YB₆, structural parameters determined from single crystal studies (Table 1) are used. Lattice parameters for ScB₆ suggested by Barantseva and Yu [36] and based on linear extrapolation of the data in Figure 2 are utilized in calculations. Given a theoretical ScB₆ structure, a range of values for the key boron position at (1/2, 1/2, z) has been included in these calculations. We have also attempted to calculate the band structures for hypothetical TiB₆ and MnB₆ using the estimated cell dimensions inferred from the compiled crystallographic trends given above in Figures 2 and 3. The reciprocal space structure for these calculations is shown in Figure 7 as a reference to lattice points and crystallographic directions.

The calculated band structure for LaB₆ in this study is similar to that reported by Hossain et al. [25] who provide a detailed interpretation of the electronic structure. The band structure for YB₆ is similar to that described by Xu et al. [37] and is shown in Figure 8(a).

For ScB₆, Figure 8(b) shows the LDA band structure calculation and density of states for a hypothetical ScB₆ with a = 4.05A and z = 0.2 using the origin as described in Figure 7. A second calculation with a = 4.08A and z = 0.2 shows similar structure largely because geometry optimization takes the final lattice parameters to similar values. The primary differences in band structure are those near the Fermi energy.

Slight changes in the cross-over of valence electrons on the X-R join near the Fermi surface are observed when comparing band structure calculations between LaB₆, YB₆ and ScB₆, and sometimes within the same compounds, when modeled under different approximations. This change in energy levels for valence electrons with minimal change to cell dimension (~0.05A) is consistent with a boron framework that is highly responsive

to subtle structural variations.

A second feature of note is the almost flat region along the G-M join to the Fermi energy (dotted outline in Figure 8). As pointed out by Hossain et al. [25], this region is the most important for electronic properties. The region encloses a small pocket of electrons just under the Fermi energy. The G-X join describes the orientation from the centre of the cubic lattice to the centre of the cube face in reciprocal space and provides a direct interconnected path from deeper regions in the valence band to the G-M join. In this region, the join presents small cusps at G that may represent hole pockets.

The height from the cusp to the Fermi energy decreases in order from LaB₆, YB₆ and ScB₆. These pockets, in close proximity to each other and to the Fermi energy, resemble conditions suited to formation of Bogoliubov quasiparticles (i.e. broken Cooper pairs). These quasiparticles are an important feature of the BCS model and high T_c superconductivity models [47]. Thus, the flatter band structure of the conduction orbitals in ScB₆ suggests that it may be possible to realize superconductivity in this hexaboride if synthesis is successful.

Calculations that include atoms with increased valence electrons in the 3d transition metal series result in a significantly increased DOS with increased d-character at the Fermi energy. This can be seen for band structure calculations of ScB₆ and MnB₆ shown in Figures 8(b) and 9. In both cases, the relatively narrow character of the 3d bands is also apparent. ScB₆ displays d character at the Fermi energy interconnected in a few reciprocal space directions to higher energy conduction bands. However, MnB₆ displays a gap of about 2.5 - 3.0 eV between orbitals at the Fermi energy and the next group of orbitals in higher energy conduction bands. In addition, the nearly tangential nature of bands to the Fermi surface for MnB₆ is more extensive over a wider range of reciprocal space directions. The possibility that this additional 3d electron doping with high DOS at the Fermi energy favors band structure modulations that assist superconductivity is worthy of experimental verification.

In order to better understand the influence of metal ions on band structure and DOS for the MB₆ structure type, we have undertaken LDA calculations for four different structures with increasing nominal metal ion charge. These structures are (a) a “metal-free” boron framework of B₆ octahedra, (b) KB₆, (c) CaB₆ and (d) LaB₆ using the parameters given in Table 1. These band structures are shown in Figure 10. The detail of conduction bands and DOS for KB₆, CaB₆ and LaB₆ are as described previously. The notable feature of this comparison is that the overall shape of the valence and conduction bands is largely the same irrespective of the metal as shown in Figure 10 and previously identified by Perkins [21] for LaB₆ and CaB₆. The relative position of the valence and conduction bands with respect to the Fermi energy depends on the type of metal ion and reflects the atomic configuration of the metal.

With respect to the boron framework, introduction of a metal primarily lowers the overall energy of the conduction orbitals, while simultaneously adjusting their relative value compared to the Fermi energy. This comparison in Figure 10 shows that the shape and density of the orbitals are primarily defined by the boron framework of B₆ octahedra alone. The energy reduction to the overall

framework by introduction of a metal ion largely represents a voltage polarization of the conduction band and ranges from 1 eV to 6 eV (see Figure 10). When the metal has d character, such as Sc, Y and La with 3d, 4d and 5d orbitals, respectively, the metal also introduces d character into the Fermi surface to an extent comparable to the 2p character contributed by the boron framework as shown in Figure 10.

Band structure calculations were also performed for TiB₆ to evaluate the nature of the Fermi surface in this putative structure. Band structure calculations for TiB₆ follow the overall trend for DOS and narrow band structure identified for 3d transition metals such as ScB₆ and MnB₆ discussed above. In general, geometry optimization for 3d transition metal elements within the hexaboride structure converge at significantly reduced lattice parameters with incrementally lower values as the number of d valence electrons increases. Under the same initial parameters, convergence for TiB₆, after allowing for unit cell optimization, occurs at cell dimension values just under 4.0A, consistent with projections from data shown in Figure 2. Thus, we suggest that MnB₆ and TiB₆ are structurally unstable forms at stoichiometric, or near stoichiometric, compositions. These compositional forms are more likely to stabilize with, for example, carbon substitution within the boron framework. This possibility is analogous to the existence of symmetrically equivalent, isotypic and isoelectronic compounds NaB₅C and KB₅C [48].

6. Conclusion

Linear extrapolations of high quality crystallographic data suggest that other types of metal hexaborides, such as ScB₆ and MnB₆, may be structurally feasible. While we have not determined the thermodynamic stability of

these hexaborides, the possibility of their existence awaits facile synthesis methods that exploit kinetic drivers rather than thermodynamic. Electronic structure calculations for these possible new forms of hexaboride suggest that TiB₆ is unlikely to form and that MnB₆ may also be unstable. On the other hand, a well-defined band structure for ScB₆, and similarity to both YB₆ and LaB₆ suggest that this hexaboride may show metallic or superconducting properties if synthesized.

7. Acknowledgements

We wish to acknowledge QUT for support of this research.

REFERENCES