Thermodynamic Ground States of Multifunctional Metal

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Cite This: Chem. Mater. XXXX, XXX, XXX−XXX

Thermodynamic Ground States of Multifunctional Metal Dodecaborides Yongcheng Liang,*,† Yubo Zhang,‡ Haitao Jiang,† Liangcai Wu,*,† Wenqing Zhang,*,‡ Kilian Heckenberger,§ Kathrin Hofmann,§ Andreas Reitz,§ Frederick C. Stober,§ and Barbara Albert*,§

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College of Science and State Key Laboratory for Modification of Chemical Fibers and Polymer Materials, Donghua University, Shanghai 201620, China ‡ Department of Physics and Shenzhen Institute for Quantum Science & Technology, Southern University of Science and Technology, Guangdong 518055, China § Eduard-Zintl-Institute of Inorganic and Physical Chemistry, Technische Universität Darmstadt, 64287 Darmstadt, Germany ABSTRACT: A large class of metal dodecaborides (MB12) is currently raising great expectations as multifunctional materials, but their refined structures are not fully resolved, which severely limits the understanding of structure−property relationships. Here, we report that the tetragonal tI26 structure is the thermodynamic ground state of ScB12, and we predict the tetragonal YB12, ZrB12, and HfB12 to be metastable, whereas the cubic cF52 structure is the high-temperature phase of ScB12 and represents the thermodynamic ground state of YB12, ZrB12, and HfB12. Crystal structures based on experimental synchrotron data are reported for tetragonal ScB12 and cubic YB12, and hightemperature X-ray data prove the phase transformation into cubic ScB12. In both types of crystal structures, the most prominent feature is that the boron atoms are linked into a rigid three-dimensional network of interconnected, empty B12cuboctahedra with metal atoms in large cages in form truncated octahedra consisting of 24 boron atoms. It is the uniqueness of these configurations that causes unusual functionalities, i.e., the coexistence of high hardness, low density, and good electrical conductivity. Furthermore, we elucidate that these physical properties are of electronic origins. These findings not only resolve the longstanding structural puzzle of this family of MB12 but also provide crucial insights into the underlying nature of their remarkable properties.

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INTRODUCTION The crystal chemistry of borides is rich not only because boron can combine with most of the metals to form a wide variety of phases with a metal-to-boron ratio ranging from 4:1 to 1:66 but also because they crystallize in complex polyhedra such as octahedra, bipyramids, dodecahedra, antiprisms, cuboctahedra, and icosahedra.1−4 Among these, metal dodecaborides (MB12) have recently attracted particular interest in the development of multifunctional materials. In conventional superhard materials like diamond and cubic boron nitride (cBN), strong covalent bonds are favorable for high hardness but unfavorable for electron transport, leading to band gaps that characterized these materials as semiconductors or insulators. In contrast, MB12 (e.g., ZrB12, ScB12, YB12, GdB12, SmB12, NdB12, PrB12, and their solid solutions)5−8 are a class of materials tailored to be a new generation of superhard compounds with high electrical conductivities. Implicit in such efforts is an idea that heavy-metal atoms provide high valence-electron densities to contribute to incompressibility and metallicity, and boron atoms form strong covalent networks to enhance hardness.9−14 As illustrated in some superconductors (e.g., MgB 2 , FeB4),15−17 metallicity and covalency also are key factors for © XXXX American Chemical Society

phonon-mediated superconductivity. The quest for new superconducting materials led to the (re)discovery of superconductivity in MB12 (e.g., ScB12, YB12, ZrB12, LaB12, LuB12).18−20 In addition, MB12-type phases have been found to show the Kondo effect (e.g., YbB12)21 and anomalous magnetic, thermal, and transport properties (e.g., HoB12, ErB12, TmB12).22−24 These intriguing characteristics confer exciting technological prospects to this large family of MB12. It should be mentioned that the cuboctahedron as an entity in boronrich solids is rare compared to structures containing icosahedra (e.g., Na2B29, Li∼1B13C2)25,26 and different from those with body-centered tetragonal carbon (bct-C4) structures that have also been discussed to be exceptionally hard (e.g., CrB4, MnB4, FeB4).16,17,27−31 A definitive determination of crystal structures is an essential prerequisite for understanding their multiple functionalities. However, until now, the agreement has been far from satisfactory on crystal structures of MB12. Dodecaborides are Received: November 15, 2018 Revised: December 19, 2018 Published: January 7, 2019 A

DOI: 10.1021/acs.chemmater.8b04776 Chem. Mater. XXXX, XXX, XXX−XXX

Article

Chemistry of Materials

Figure 1. Crystal structures of MB12 of the cubic cF52 phase (a), the simplified cF52 phase (b), the tetragonal tI26 phase (c), and the simplified tI26 phase (d). In (a) and (b), the large and small spheres represent the metal and boron atoms, respectively. In (a), adjoining large B24 and small B12 cages are highlighted in red. In (b) and (c), the red plus signs mark the center positions of the B12 cages that are hidden to simplify the structures. The cubic cF52 structure can be regarded as an array of metal atoms and B12 cages in the NaCl-type arrangement. The tetragonal tI26 structure can be derived from the cubic structure by symmetry reduction.

of this class of multifunctional materials. Hence, it is highly urgent to resolve the structural issue of MB12, and it is likely that a hidden tetragonal phase is common in this family of MB12. In this article, we present a comprehensive study of structural stability and related physical properties of four representatives of the MB12 family (M = Sc, Y, Zr, and Hf) using first-principles calculations in combination with structural data for tetragonal ScB12 and cubic YB12 refined from high-resolution synchrotron data and for cubic ScB12 refined from laboratory data. We identify the tetragonal tI26 structure as the thermodynamic ground state of ScB12 and as a metastable state of YB12, ZrB12, and HfB12, whereas the cubic cF52 structure is in fact that of the high-temperature phase of ScB12 and represents the thermodynamic ground state of YB12, ZrB12, and HfB12. The tI26 structure is obtained from the cF52 one by slightly distorting the regular B12-cuboctahedron and reducing the high symmetry. In both types of crystal structures, a common feature is that the boron atoms form a strong threedimensional (3D) network, with the metal atoms in the large cages in form of truncated octahedra consisting of 24 boron atoms. This configurational peculiarity gives rise to multiple functionalities, i.e., the coexistence of high hardness, low density, and good electrical conductivity. Finally, we reveal the

known to exist for a series of metals: transition metals (e.g., Sc, Y, Zr, Hf), lanthanides (e.g., Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu), and actinides (e.g., Th, U, Np, Pu). Among them, UB12 was synthesized first and found to crystallize in the cubic cF52 structure (space group Fm3̅m).32 Since, nearly all of the MB12-type compounds have been reported to be isomorphous with UB12.33 As such, ScB12 was originally assumed to be of cF52 type,34 but it was later argued to adopt a tetragonal symmetry (space group I4/mmm).35,36 Unfortunately, the refined structure of this tetragonal phase has never been published due to some difficulties caused by the interference of other coexisting phases, the unavailability of crystals, and the insufficient resolution of the powder diffraction experiments. Subsequently, a number of experiments6,36−38 have confirmed the existence of this elusive tetragonal phase. At the same time, it was observed that the cubic cF52 phase can also be stabilized under certain conditions (e.g., doped Sc1−xMxB12 with M = Zr, Y, Tm, and Lu). Just when the tetragonal form was believed to exist only for ScB12, a recent experiment39 surprisingly revealed that LuB12 has a cubic symmetry at room temperature but apparently distorts to a tetragonal symmetry at lower temperatures. A dislocation of the Lu atom from the center of the cage was proposed,40 and no refined structure data have been available for this phase to date. These inconsistent reports greatly hinder in-depth understanding and further exploration B


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Table 1. Calculated Lattice Constants (a, c) and Wyckoff Sites (x, y, z) of the cF52 and tI26 Structures of ScB12, YB12, ZrB12, and HfB12 cF52

a (Å) ScB12 YB12 ZrB12 HfB12

7.4106 7.4947 7.4015 7.3834

tI26

metal site

boron site

4a: x, y, z

48i: x, y, z

0, 0, 0, 0,

0, 0, 0, 0,

0 0 0 0

0.5, 0.5, 0.5, 0.5,

0.1692, 0.1688, 0.1696, 0.1697,

0.1692 0.1688 0.1696 0.1697

metal site a, c (Å) 5.2399, 5.2995, 5.2320, 5.2190,

0.5, 0.5, 0.5, 0.5,

0.5, 0.5, 0.5, 0.5,

0 0 0 0

boron site 16m: x, y, z 0.1692, 0.1688, 0.1695, 0.1697,

0.1692, 0.1688, 0.1695, 0.1697,

0.1693 0.1689 0.1696 0.1697

8i: x, y, z 0.3385, 0.3376, 0.3391, 0.3393,

0, 0, 0, 0,

0 0 0 0

Poisson’s ratio) were determined by efficient strain-energy method,47 whereas the Vickers hardness values were estimated by Chen’s model.48

electronic origins of the relative structural stability and related physical properties of these compounds.

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7.4110 7.4954 7.4065 7.3885

2b: x, y, z

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EXPERIMENTAL AND COMPUTATIONAL METHODS

RESULTS AND DISCUSSION As shown in Figure 1a, a cubic cell of the cF52 structure contains 4 formula units of MB12, in which 4 metal atoms and 48 boron atoms occupy the 4a(0, 0, 0) and 48i(1/2, x, x) Wyckoff sites, respectively, with x ∼ 1/6. Twelve boron atoms form a cuboctahedral cage (B12), and these small B12 cages are interconnected so that additional cages (B24) result that consist of 24 boron atoms and accommodate the metal atoms at their centers. These large B24 cages are Archimedean polyhedra, called truncated octahedra. The linked polyhedra form a rigid 3D network of boron atoms. Because the hole radius of the B24 cage limits which atoms can fit inside, the formation of MB12 is believed to depend on the effective size of metal atoms, with the largest atom being yttrium and the smallest being zirconium.8,33 To simplify this structure, we visualize it as the NaCl-type arrangement. As illustrated in Figure 1b, the metal atoms are located at the sodium-atom positions and the centers of the small cuboctahedral B12 cages coincide with the chloride-ion sites. With symmetry reduction from cubic to tetragonal, a body-centered setting is chosen with half the size of the unit cell volume and space group I4/mmm, called tI26. In Figure 1c,d, the basis vectors (a, b, c) of the tI26 lattice correspond to the vectors [1/2, −1/2, 0], [1/2, 1/2, 0], and [0, 0, 1] of the cF52 lattice, respectively. The tetragonal unit cell then contains two formula units of MB12. To determine the exact structures, we chose four transitionmetal dodecaborides MB12 (M = Sc, Y, Zr, and Hf) as prototypes to optimize their lattice parameters and atomic coordinates by first-principles calculations. This selection of the four systems was based on the following factors: (i) As mentioned above, the structure of ScB12 has been a controversial subject for decades. Also, there are some interesting experimental observations on the structures of YB12, ZrB12, and HfB12. (ii) Conventional DFT calculations generally provide more accurate results for d-electron elements of transition metals than for f-electron elements of lanthanides and actinides. (iii) Because Sc (Zr) is isovalent with Y (Hf), whether the corresponding borides ScB12 and YB12 (ZrB12 and HfB12) follow similar trends is worth studying. Table 1 lists our optimized lattice constants and atom positions for the cF52 and tI26 structures of MB12 (M = Sc, Y, Zr, and Hf). For the four cubic cF52 phases, our calculated results are well consistent with the available experimental data,8,33,34 indicating the reliability of the present calculations. For the tetragonal tI26 cell, 2 metal atoms occupy the 2b site of the space group I4/mmm; 24 boron atoms are located at the 16m and 8i sites. Compared with the cubic cF52 structure, the tetragonal tI26

Synthesis and Characterization of the Samples. The starting materials, elemental metals or metal oxides and boron in form of small pieces (ScB12) or powders (YB12), were mixed loosely (ScB12) or ground together using an agate mortar and pestle (YB12) in the respective stoichiometric ratio of M/B = 1:12 (Sc: Smart Elements, 99.9%; Y2O3: Smart Elements, 99.99%; B: H.C. Starck GmbH, >98%, powdered, or Chempur Feinchemikalien und Forschungsbedarf GmbH, 99.95%, crystalline pieces). The reaction mixtures were then melted in an electric arc several times for 5−10 s each (ScB12) or pressed into pellets and heated in an induction furnace for 2−4 h at 1700−1800 K (YB12) to yield dark gray powders of excellent crystallinity. Those were first characterized by in-house X-ray diffraction (XRD, Stoe powder diffractometer, STADI P, transmission mode, flat plate sample holder, λ = 1.5406 Å, Ge[111]monochromator) to ensure phase purity and then sent to Advanced Photon Source (Argonne National Laboratory) to be measured at the 11-BM-B beamline with λ = 0.41267 Å at T = 100, 295, and 473 K (±1 K). Structural refinement was performed with the GSAS-II program using a Chebyschev for the background and a pseudo-Voigt profile function.41 Temperature-dependent phase transitions were investigated by differential scanning calorimetry (DSC, Netzsch, STA 449, F3 Jupiter) and high-temperature X-ray diffraction (ht-XRD, Stoe powder diffractometer, STADI P, Debye−Scherrer mode, quartz capillaries, λ = 0.7093 Å, Ge[111]-monochromator). First-Principles Calculations. The calculations on MB12 (M = Sc, Y, Zr, and Hf) were carried out using spin-polarized density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) code.42 The all-electron projectoraugmented-wave method43 was adopted with 2s22p1, 3s23p64s23d1, 4s24p65s24d1, 3s23p64s23d2, and 4s24p65s24d2 treated as valence electrons for B, Sc, Y, Zr, and Hf atoms, respectively. A plane-wave basis set with a large cutoff energy of 600 eV and dense k meshes were employed for all the considered phases to ensure that the numerical accuracy can resolve an energy difference of less than 1 meV/atom. Forces on the ions were calculated through the Hellmann−Feynman theorem, allowing a full geometry optimization of different structures (i.e., tI26, cF52) of MB12. To reveal possible temperature-induced phase transitions, we calculated the Gibbs free energies of different phases of MB12 (M = Sc, Y, Zr, and Hf) over a wide range of temperatures. Phonon calculations were performed using the Phonopy package44 with the force-constant matrices calculated from VASP. We have carefully checked the sensitivity of the calculated Gibbs energies to different energy functionals including the generalized gradient approximation (GGA)45 and the strongly constrained and appropriately normed (SCAN) metal GGA.46 Our results show the GGA gives a reasonable description of the structural parameters, relative energies, mechanical properties, and electronic structures of different phases of MB12, although SCAN also reproduces the main conclusions. In the following, we shall restrict our discussion to the GGA results for simplicity, unless otherwise specified. The mechanical properties (i.e., elastic constants, bulk modulus, shear modulus, Young’s modulus, and C


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Figure 2. Calculated phonon dispersion curves of the cubic cF52 phases (top row) and the tetragonal tI26 phase (bottom row) for ScB12, YB12, ZrB12, and HfB12 (from left to right). It can be clearly seen that these eight phases are all dynamically stable because no imaginary frequencies are observed throughout their whole Brillouin zones.

Figure 3. Calculated total energy versus volume (top row) and Gibbs energy versus temperature (bottom row) of the cubic cF52 and tetragonal tI26 structures for the ScB12, YB12, ZrB12, and HfB12 phases (from left to right). All energies are rescaled for one formula unit of MB12.

structure has a slight elongation along the c-axis and a tiny shortening along the a-axis and b-axis. It is known that the phonon dispersion is a strict measure for the dynamical stability of structures. We have carefully performed phonon calculations for the cF52 and tI26 structures of MB12 (M = Sc, Y, Zr, and Hf). Figure 2 displays the calculated phonon dispersion curves of these eight phases. No imaginary frequencies are observed throughout their whole Brillouin zones and thus they are all dynamically stable. The calculated total energy as a function of volume for the cF52 and tI26 phases of the ScB12, YB12, ZrB12, and HfB12 systems are presented in Figure 3a−d, respectively. Unexpectedly, the widely accepted cF52 structure is the energetically unfavorable phase at 0 K, whereas the newly identified tI26 structure has slightly lower total energies for ScB12, YB12, ZrB12, and HfB12. Among the four systems, YB12

(HfB12) possesses the largest (smallest) equilibrium volume and the volume of ScB12 is larger than that of ZrB12. This volume order is closely related to the effective sizes of the respective metal atoms. It can be clearly seen from Figure 3a− d that there is no crossing between the energetically competitive cF52 and tI26 structures of MB12 (M = Sc, Y, Zr, and Hf), indicating that there will be no pressure-induced phase transition. To take temperature effects into account, we have further evaluated Gibbs free energies of MB12. Figure 3e−h plot our calculated Gibbs free energies as a function of temperature for the cF52 and tI26 structures of ScB12, YB12, ZrB12, and HfB12, respectively. It is interesting to note that ScB12 undergoes a temperature-induced phase transition from the ground-state tI26 phase to the high-temperature cF52 phase at about 180 K according to the calculations. This may explain why the tI26 D


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Chemistry of Materials phase of ScB12 can be observed experimentally, but its cF52 phase exists only under certain conditions.6,35−38 However, YB12, ZrB12, and HfB12 do not follow the relative stability trends of ScB12. Regardless of temperature changes, the cF52 type is always more stable than the tI26 type. That is to say, the cF52 structure represents the ground state of YB12, ZrB12, and HfB12, whereas the tI26 structure is predicted to be metastable for these phases. At the same time, we would like to mention that the differences in the Gibbs energies of the tI26 and cF52 structures are very small at low temperature, and the maximum span is as low as 3 meV for YB12. Moreover, as evidenced by the above phonon spectra, the tI26 phases also exhibit dynamical stability; thus, they may be viable under proper conditions. Therefore, our calculations conclude that the tetragonal tI26 structure is the thermodynamic ground state of ScB12 and might exist at very low temperatures for YB12, ZrB12, and HfB12, whereas the cubic cF52 structure is the hightemperature phase of ScB12 and represents the thermodynamic ground state of YB12, ZrB12, and HfB12. In the light of these results, new investigations of the crystal structures were required. Whereas ScB12 and YB12 (and ZrB12) were obtained as almost phase pure powders by solid-state reactions (small amounts of side phases: ScB27.76 and YB6), HfB12 could not be synthesized in accordance with the literature49 that describes this phase to be obtained under high pressure only. Initial room-temperature XRD experiments indicated a tetragonal symmetry for ScB12 and a cubic symmetry for YB12. Importantly, ScB12 was shown to transform reversibly into the cubic phase at 510 K by ht-XRD and DSC measurements, which substantiates the DFT prediction that it undergoes a temperature-induced phase transition from the tI26 structure to the cF52 structure. Figure 4 shows the highresolution synchrotron data, which is obtained at 295 and 473 K for the Sc compound and at 100 and 295 K for the Y compound, that has been used to refine the structures. Furthermore, the cubic structure of the ht-phase of ScB12 was refined based on the laboratory data obtained by an in situ XRD experiment (Figure 5). The refined results are listed in Table 2, and they are in perfect agreement with the calculated data in Table 1. The stability of the tetragonal structure of ScB12 was confirmed, as shown in Figure 1c. The B−B bond lengths are measured to be 1.707, 1.712, and 1.756 Å. There is no indication of a deviance from full occupancy of the metal or boron atom sites. Although cubic YB12 (and ZrB12) were synthesized, their tI26 metastable phases have not been observed. The temperatures of the phase transition between tetragonal and cubic phases of yttrium and zirconium borides could be lower than 100 K. As MB12 compounds are important multifunctional materials, Table 3 summarizes the theoretical results of their mechanical properties and densities (cF52 and tI26 phases of MB12, with M = Sc, Y, Zr, and Hf). For a crystal, the mechanical stability requires its strain energy to be positive so that the whole set of elastic constants meets the Born−Huang criterion.50 For the cubic and tetragonal lattices, there are three (C11, C12, and C44) and six independent elastic constants (C11, C12, C13, C33, C44, and C66), respectively. The calculated elastic constants have carefully been checked and all of the phases are mechanically stable. Interestingly, the eight borides share very similar mechanical behavior, even though their metal atoms and crystal structures are different. Their bulk moduli are predicted to be within the small range of 218−238 GPa, which is in good agreement with the experimental result (221 GPa)5

Figure 4. Synchrotron X-ray powder patterns of ScB12 at T = 295 and 473 K and YB12 at T = 295 and 100 K (from top to bottom). Experimental (blue) and calculated (green) traces, difference curves below, markers indicate the positions of the reflections of the main (blue) and small amounts of side (red) phases (ScB27.76, YB6).

for ZrB12. The values are a bit smaller than those calculated for the promising superhard WB3 (295 GPa)51 and ReB2 (356 GPa),52 but they still rival that of the superhard B6O (231 GPa).53 Moreover, our calculations indicate that the MB12 phases have large Young’s moduli (484−518 GPa), high shear moduli (214−229 GPa), and small Poisson’s ratios (0.124− 0.139). Therefore, these results suggest that MB12 not only are low-compressible materials but also possess strong rigidity against the linear compressibility and shear deformation involved in microhardness indentation experiments. It is now well recognized that high elastic moduli sometimes do not guarantee a high hardness of a material, and we thus estimated the theoretical hardness in comparison with the available experimental data. According to Chen’s model,48 all eight phases of MB12 were evaluated to possess superhardness (41.9−44.2 GPa), which is well compatible with the experimental data6 of tetragonal ScB12 (41.7 ± 2.2 GPa), E


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each boron atom in the B12 cage is bonded to a boron atom of the another B12 cage, and scandium atoms lie in the interstices outside of the B12 cages. The electronic structure is determined by a strong hybridization between the B 2sp and Sc 3d states and the total DOS consists of energy bands separated into four groups. The lowest 13 bands in the range of (−15, −4) eV are dominated by the B 2sp bonding states, which are responsible for the formation of intracage covalent bonds, specifically of three-centered B−B bonds in the cuboctahedral faces.54 The six bands in the range of (−4, 0) eV are also derived from the B 2sp bonding states, but they are responsible for the formation of intercage covalent bonds. Hence, these 19 valence bands may be viewed as the bonding part of a rather complex system of intracage and intercage B−B bonds, whereas the unoccupied highest lying feature above 4 eV is the corresponding antibonding part. In the region of (0, 4) eV, the conduction bands are hybrid states mainly composed of the B 2p and Sc 3d orbitals, and they possess considerable energy dispersion. Such results are consistent with the valence orbitals structures of other MB12 by Lipscomb et al.,55 who use the linear combination of atomic orbitals approach to give reasonable bonding arrangement. As mentioned above, the tetragonal tI26 phase is derived from the cubic cF52 phase by a small structural distortion. As a result of this distortion, one boron atom site splits into two nonequivalent sites. The total DOS, projected DOS, and band structures of the tetragonal tI26 phase of ScB12 are presented in Figure 6d−f, respectively. The total DOS profile of the tI26 phase shares many common features with that of the cF52 phase. However, because of the small distortion of the cuboctahedral B12 cage and the slight deviation from cubic symmetry, the 19 valence bands obviously shift downward, as can be seen from the band structures of the tI26 phase in Figure 6f. This lowers the total energy of the tI26 phase compared to the cF52 phase, as confirmed by our firstprinciples calculations in Figure 3a. These well explain why ScB12 phases should prefer the tetragonal tI26 phase over the cubic cF52 one at low temperatures. Taking into account the contribution from phonons, the free energy of the cF52 structure decreases relative to that of the tI26 one with increasing temperature. For YB12, ZrB12, and HfB12, static total energies of the tI26 phase are lower than that of the cF52

Figure 5. Laboratory X-ray powder pattern of ScB12 at T = 773 K. Experimental (red) and calculated (green) traces, difference curve (pink) below, markers indicate the positions of the reflections of the main phase.

cubic YB12 (40.9 ± 1.6 GPa), and cubic ZrB12 (40.4 ± 1.8 GPa) at the load of 0.49 N. We would like to mention that the asymptotic hardness of MB12 phases may fall below the 40 GPa threshold value of superhard materials because of the large indentation size effect. For example, Ma et al.5 have reported that the determined hardness of cubic ZrB12 is 40 ± 1 GPa under the load of 0.49 N and decreases gradually to the asymptotic value of 27 GPa under the load of 4.9 N, but this asymptotic hardness still matches the corresponding values of other hard borides such as WB4 (25.5−31.8 GPa)11,12 and ReB2 (26.6−30.1 GPa).10,11 Moreover, the MB12 phases show the advantage of their light weights over other hard materials. ScB12, YB12, ZrB12, and HfB12 are calculated to have relatively low densities of 2.85, 3.45, 3.62, and 5.08 g/cm3, respectively, which are significantly lower than those of WB4 (8.40 g/cm3) and ReB2 (12.67 g/cm3).6 The combination of strong stiffness and high hardness with low density makes MB12 promising materials for applications in machining tools and lightweight protective coatings. To reveal the electronic origins of the structural stability and physical properties of MB12, ScB12 is used as an illustrative case. For the cubic cF52 phase, its total density of states (DOS), projected DOS, and band structures are plotted in Figure 6a−c, respectively. In this highly symmetrical structure,

Table 2. Experimental Lattice Constants (a, c), Refined Wyckoff Sites (x, y, z), and Isotropic Displacement Parameters (Uiso) of the Tetragonal tI26-Type ScB12 and of the Cubic cF52-Type ScB12 and YB12 Phases ScB12 (tI26) wavelength (Å) space group temperature (K) a (Å) c (Å) metal atom site x y z Uiso boron atom sites x y y Uiso

0.41267 I4/mmm (no. 139) 295 ± 1 5.235964(7) 7.35869(2) 2b 0.5 0.5 0 0.0201(2) 16m, 8i 0.1682(2), 0.3395(4) 0.1682(2), 0 0.1685(2), 0 0.0308(3), 0.0318(5)

YB12 (cF52)

473 ± 1 5.240297(5) 7.37417(1) 2b 0.5 0.5 0 0.0196(1) 16m, 8i 0.1686(2), 0.03384(3) 0.1686(2), 0 0.1686(2), 0 0.0301(3), 0.0305(4) F

0.7093 Fm3̅m (no. 225) 773 ± 1 7.44146(6)

0.41267 Fm3̅m (no. 225) 295 ± 1 7.53381(3)

100 ± 1 7.500139(3)

4a 0 0 0 0.0187(4) 48i 0.5 0.1700(2) 0.1700(2) 0.0203(5)

4a 0 0 0 0.00263(4) 48i 0.5 0.16934(7) 0.16934(7) 0.0133(2)

4a 0 0 0 0.00116(4) 48i 0.5 0.16944(7) 0.16944(7) 0.0133(2)


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Table 3. Calculated Elastic Constants Cij (GPa), Bulk Modulus B (GPa), Young’s Modulus E (GPa), Shear Modulus G (GPa), Poisson’s Ratio v, Vickers Hardness H (GPa), and Density ρ (g/cm3) of the cF52 and tI26 Structures of ScB12, YB12, ZrB12, and HfB12 cF52 C11, C12, C44 ScB12 YB12 ZrB12 HfB12

428, 448, 460, 458,

114, 104, 116, 121,

264 258 274 273

tI26 ρ

B, E, G, v, H 219, 218, 231, 234,

484, 494, 513, 511,

214, 220, 227, 225,

0.132, 0.124, 0.129, 0.135,

41.9 44.4 43.8 42.4

C11, C12, C13, C33, C44, C66

2.85 3.45 3.62 5.08

538, 537, 569, 569,

11, 17, 19, 21,

115, 104, 120, 125,

428, 449, 465, 463,

294, 259, 275, 274,

157 173 175 172

ρ

B, E, G, v, H 235, 219, 235, 238,

518, 495, 518, 516,

229, 220, 229, 227,

0.133, 0.124, 0.133, 0.139,

43.6 44.2 43.6 42.2

2.85 3.45 3.62 5.08

Figure 6. Total density of states (DOS), Sc d and B sp projected DOS, and band structures (from left to right) of the cubic (top row) and tetragonal (bottom row) phases of ScB12. The Fermi level is set at 0 eV and indicated by a horizontal dashed line. In our calculations, the primitive cell including one formula unit of ScB12 is adopted.

conductivity is still a challenge to modern materials science, calling for more theoretical and experimental effort along this direction. Interestingly, the coexistence of good metallicity, low density, and high hardness makes MB12 distinct from those traditional superhard materials, which could open an avenue for a new class of superhard and lightweight metals.

phase, but zero-point energies make the latter become the thermodynamic ground states. The highly directional covalent 3D framework resists elastic and plastic deformations, counteracts the creation and movement of dislocations, and causes strong stiffness and high hardness of MB12. On the other hand, it is found that the Fermi level is located in the region of extended plateau in the DOS between the bonding and antibonding bands formed by the B 2sp states. The bonding states are fully occupied, whereas the corresponding antibonding states are fully unoccupied. This optimal filling of the orbitals also has an important contribution to the hardness of MB12. In Figure 6c,f, a notable feature of the electronic structure is several bands crossing the Fermi level, indicating the metallic character of ScB12. Similarly, it has been noticed by Ma et al.5 that ZrB12 exhibits superior metallic behavior with ultralow electrical resistivity (18 μΩ cm), which is comparable to that of pure metal platinum. As we know, traditional superhard materials such as diamond and cBN are insulators or semiconductors. The large electrical resistivity severely limits their diverse applications as hard conductors under extreme conditions. In this sense, the rational design and synthesis of multifunctional materials with high hardness and high electrical

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CONCLUSIONS In summary, we have carried out a systematic investigation of structural stability and physical properties of MB12, with a focus on four transition-metal elements (M = Sc, Y, Zr, and Hf), using first-principles calculations in combination with XRD experiments for the Sc and Y variants. The tetragonal tI26 structure is predicted to be the thermodynamic ground state of ScB12 and the metastable state of YB12, ZrB12, and HfB12, whereas the cubic cF52 structure is that of the hightemperature phase of ScB12 and represents the thermodynamic ground state of YB12, ZrB12, and HfB12. The tI26 phase is derived from the cF52 phase by slightly distorting the cuboctahedral B12 cage and deviating from cubic symmetry. Detailed structure data of the tetragonal phase of ScB12 derived from high-resolution, room-temperature synchrotron powder data are presented here for the first time. In both types of G


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lattice structures, the most prominent feature is that the boron atoms are linked into a rigid 3D covalent network with empty cuboctahedra and metal atoms in the cavities in the shape of truncated octahedra. This peculiar configuration results in multiple functionalities, i.e., the coexistence of high hardness, low density, and good electrical conductivity. Furthermore, we explain the electronic origins of relative structural stability and related physical behaviors of these MB12 systems.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: *E-mail: *E-mail: *E-mail:

[email protected] (Y.L.). [email protected] (L.W.). [email protected] (W.Z.). [email protected] (B.A.).

ORCID

Yongcheng Liang: 0000-0002-3708-2076 Barbara Albert: 0000-0002-2696-521X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Ping Qin for his helpful calculations. We acknowledge the financial support from the National Natural Science Foundation of China (Nos 51671126 and 61874151). The help of Dr. Maren Lepple and Dr. Christina Birkel with various aspects of the experimental work is gratefully acknowledged.

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REFERENCES

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Thermodynamic Ground States of Multifunctional Metal

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