Disorder and defects are not intrinsic to boron carbide

A unique combination of useful properties in boron-carbide, such as extreme hardness, excellent fracture toughness, a low density, a high melting point, thermoelectricity, semi-conducting behavior, catalytic activity and a remarkably good chemical stability, makes it an ideal material for a wide range of technological applications. Explaining these properties in terms of chemical bonding has remained a major challenge in boron chemistry. Here we report the synthesis of fully ordered, stoichiometric boron-carbide B13C2 by high-pressure–high-temperature techniques. Our experimental electron-density study using high-resolution single-crystal synchrotron X-ray diffraction data conclusively demonstrates that disorder and defects are not intrinsic to boron carbide, contrary to what was hitherto supposed. A detailed analysis of the electron density distribution reveals charge transfer between structural units in B13C2 and a new type of electron-deficient bond with formally unpaired electrons on the C–B–C group in B13C2. Unprecedented bonding features contribute to the fundamental chemistry and materials science of boron compounds that is of great interest for understanding structure-property relationships and development of novel functional materials.

A unique combination of useful properties in boron-carbide, such as extreme hardness, excellent fracture toughness, a low density, a high melting point, thermoelectricity, semi-conducting behavior, catalytic activity and a remarkably good chemical stability, makes it an ideal material for a wide range of technological applications. Explaining these properties in terms of chemical bonding has remained a major challenge in boron chemistry. Here we report the synthesis of fully ordered, stoichiometric boron-carbide B 13 C 2 by high-pressure-high-temperature techniques. Our experimental electron-density study using high-resolution single-crystal synchrotron X-ray diffraction data conclusively demonstrates that disorder and defects are not intrinsic to boron carbide, contrary to what was hitherto supposed. A detailed analysis of the electron density distribution reveals charge transfer between structural units in B 13 C 2 and a new type of electron-deficient bond with formally unpaired electrons on the C-B-C group in B 13 C 2 . Unprecedented bonding features contribute to the fundamental chemistry and materials science of boron compounds that is of great interest for understanding structure-property relationships and development of novel functional materials.
Boron carbide is one of the hardest substances, surpassed only by diamond and boron nitride 1 . The high mechanical and thermal stability, low density and low costs of fabrication have made boron carbide the prime choice in a series of technological applications [1][2][3][4][5][6][7] . Boron carbide preserves the same structure for a range of compositions, and details of this crystal structure have been discussed in terms of chemical disorder of boron and carbon atoms as well as the presence of vacancies 1,[8][9][10][11] . Electronic-structure calculations suggest that the properties of boron carbide depend on the stoichiometry and the details of the disorder 2,7,12,13 .
Experimentally, chemical bonding can be accessed through single-crystal x-ray diffraction. Reliable information on the distribution of the electron density in the unit cell can be obtained only for good-quality single crystals with minimal structural disorder 14 . Synthesis of defect-free material is the most challenging task in boron carbide chemistry. We have succeeded in growing small single crystals of the stoichiometric composition B 13 C 2 by high-pressure-high-temperature techniques (see Methods). The material is transparent with a dark red or maroon color, indicating an insulator or a large-band-gap semiconductor. This is in agreement with some literature data 15 , but it is inconsistent with the relatively high electrical conductivity reported for boron carbide 1 . To the best or our knowledge, dark red transparent boron carbide has not been reported before.
A multipole (MP) model has been obtained for the crystal structure of B 13 C 2 by refinement against accurately measured intensities of Bragg reflections (see Methods and Supplementary Information Section S1) 14 . The excellent fit to the diffraction data with R 1 = 0.0197 provides strong evidence for the stoichiometry of B 13 C 2 , in agreement with the composition obtained by Energy-dispersive x-ray (EDX) analysis (see Methods). The excellent fit furthermore indicates the absence of disorder: B 13 C 2 is composed of B 12 icosahedral clusters and CBC linear chains ( Fig. 1 and Supplementary Information Section S2). Lattice parameters and values of atomic displacement parameters (ADPs) fall within a range previously assigned to the composition B 12 C 3 1,8-10 . The possibility of different compositions was investigated by additional MP refinements with small amounts of carbon at the B P site, corresponding to B 12 + x C 3 − x stoichiometries with x = 0.44 and x = − 0.11, respectively (see Supplementary Information Section S1 for details). Both models gave a slightly worse fit to the diffraction data than the B 13 C 2 model. More importantly, the number of valence electrons of C at the B P site refined to zero, thus showing that the MP refinement has effectively removed carbon from the B P site, providing further support for the ordered stoichiometric character of the investigated crystal. Interestingly, a refinement of the independent atom model (IAM) including the site occupancy factors of C at the B P and B E sites resulted in 19% occupancy of the B P site by carbon (x = − 0.11). Contrary to the MP model (R 1 = 0.0197), the IAM with disorder (R 1 = 0.0287) leads to only a small improvement of the fit to the data (Table S4). These results suggest that the charge transfer towards B P in the MP model is mimicked in the disordered IAM by a fractional occupancy of the B P site by C.
Discrepancies between the present values of the lattice parameters and ADPs and those reported in the literature 1,8-10 for the same composition may be the result of different degrees of disorder and defects between different samples. The single MP refinement 16 reported previously for B 13 C 2 gave a much worse fit to their XRD data (R 1 = 0.0440), which questions the reliability of that model. The single MP refinement 17 for B 12 C 3 also led to a substantially worse fit to their XRD data (R 1 = 0.0250) than we have obtained for our model against the present XRD data (R 1 = 0.0197). Thus, a highly precise MP refinement refutes recent less accurate diffraction studies 13 and theoretical electronic-structure calculations 2,12 , where a disorderly replacement by carbon of a certain fraction of the boron atoms of the B 12 clusters was considered as absolutely essential for the stability of B 13 C 2 .
The MP model extends the independent atom model (IAM) of spherical atoms by parameters describing the reorganization of electron density due to chemical bonding. Previous electron-density studies on boron carbide 18,19 have been restricted to a discussion of the qualitative features of the electron densities. Quantitative information about chemical bonding can be extracted from the static electron density of the MP model through its topological properties according to Bader's quantum theory of atoms in molecules (QTAIM) 14,20 . Critical points (CPs) are defined as the positions where the gradient of the electron density is zero [∇ρ (r) = 0] 20 . They are classified according to the number of positive eigenvalues of the Hessian matrix of second derivatives as local maxima (3 positive eigenvalues), bond critical points BCPs (2), ring critical points RCPs (1) and local minima (0 positive eigenvalues) 20 .
All atomic positions of the present MP model can be identified with local maxima in the static electron density, while additional local maxima do not exist. BCPs and RCPs have been found between the atoms of the B 12 cluster in a similar pattern as for α-boron 21 , and with comparable values for the electron densities and Laplacians (Table 1). Together, these features indicate similar bonding by molecular-type orbitals on the B 12 clusters in B 13 C 2 and α-boron 21 . According to Wade's rule 22 , this bonding involves 26 of the 36 valence electrons of the twelve boron atoms of this closo-cluster, thus leaving for each boron atom one orbital but only 5/6 electrons for exo-cluster bonding 21,23 .
The crystal structure of B 13 C 2 comprises four crystallographically independent atoms. CBC chains contain the carbon atom and a boron atom denoted as B C ; the B 12 cluster is made of six polar and six equatorial atoms, denoted as B P and B E , respectively (Fig. 1). According to the QTAIM 20 , bonding between a pair of atoms exists, if the electron density possesses a BCP between those atoms. For B 13 C 2 , we have found BCPs between pairs of B P atoms from neighboring clusters. The distance B P -B P is slightly larger and the magnitudes of the electron density, ρ BCP , and Laplacian, ∇ 2 ρ BCP , are slightly smaller than those of the corresponding inter-cluster bonds in α-boron 21 and γ -boron 24 (Table 1). The high value of ρ BCP together with a negative value of ∇ 2 ρ BCP of large magnitude indicate a strong covalent interaction between these atoms 20 . The similarities with bonding in α-boron 21 (Table 1) allow this bond to be classified as a 2-electron-2-center (2e2c) bond. Further evidence for this interpretation comes from the QTAIM theory, which assigns a charge to each atom by integration of the electron density over the atomic basins. A charge of − 0.21 electrons has been obtained by integrating the experimental static electron density over the atomic basin of B P ( Table 2). This value is in good agreement with electron counting. With 5/6 electrons per boron atom available for exo-cluster bonding, a formal charge of − 0.17 is obtained for B P involved in a 2e2c B P -B P exo-cluster bond.
Bond-critical points are also found between a B E atom and the closest C atom. Large magnitudes of ρ BCP and the negative Laplacian ∇ 2 ρ BCP indicate a strong covalent interaction and a 2e2c C-B E bond. An equal split of these cluster. Each B 12 cluster is bonded by B P -B P bonds to three B 12 clusters in each of the two neighboring closepacked planes, and to six CBC chains by C-B E bonds. (b) Perspective view highlighting the environment of the CBC chain. Each carbon atom is bonded to three B 12 clusters within a single close-packed plane. Color code: B P is blue, B E is green, B C is red, and C is grey. electrons between C and B E again gives a formal charge of − 0.17 for B E , and it would result in a (B 12 ) 2− group 2 However, carbon is more electronegative than boron and should attract most of the bonding electrons. Indeed, the integration of the electron density over the atomic basins leads to a positive atom B E and a strongly negative C atom (Table 2). A detailed analysis of the electron density shows that the positive charge of B E is the result of a strong polar-covalent character of the C-B E bond, with the BCP much closer to B E than to C ( Fig. 2; Table 1), but with a large value of ρ BCP as opposed to an expected small value for ionic bonding 20 .
With the interpretation of B P -B P and C-B E bonds as 2e2c bonds, only three electrons are left for the two C-B C bonds of the CBC group (see Supplementary Material Section S3). These bonds can therefore be described as a three-electron-three-center (3e3c) bond or as resonance between two equivalent combinations of one 2e2c and one 1e2c bond (Fig. 3). The large values of the electron density along the bond path (Fig. 2a) correlate with the short bond length, which is explained by the internal pressure on the CBC group 2 . Large magnitudes of ρ BCP and ∇ 2 ρ BCP indicate a covalent interaction. The electron deficient character of this bond is in complete agreement with the ionic charge of + 2.30 of B C . The latter value is the result of the extremely small volume of the atomic basin of this atom, which demonstrates that the internal pressure has squeezed out most of the electrons of B C , reminiscent of the effect of pressure on the electrons in lithium metal 25 .
A 3e3c C-B C -C bond contains one unpaired electron per formula unit B 13 C 2 . Experimentally, unpaired spins have been observed at much lower concentrations in boron carbides of different compositions 2,4,5,26,27 . One explanation lies in chemical disorder and vacancies, which are necessarily present for other compositions than stoichiometric B 13 C 2 , and which reduce the number of unpaired spins. On the other hand, the itinerant character of the electron states or localization as bipolarons may be in agreement with low concentrations of unpaired spins 2,5,12 . The presence of an unsaturated bond on the CBC chains should result in a high chemical reactivity of this bond. However, we have found that B C is extremely small ( Table 2) and well shielded from the outside by C atoms and bulky B 12 clusters. Steric effects hindering access to reactive sites is known to stabilize radicals 28,29 . High temperatures can overcome these barriers, and a high reactivity at elevated temperatures towards oxidizing agents has been described for boron carbide 30 . Recently, amorphisation 6,31 of boron carbide B 12 C 3 has been explained on the basis of the presence of carbon atoms at a small fraction of the B P sites 32 . Stoichiometric B 13 C 2 is a form of boron carbide that lacks this detrimental property of technical boron carbide with compositions on the carbon-rich side of B 13 C 2 .
In summary, we have synthesized stoichiometric boron carbide B 13 C 2 , which is free of intrinsic disorder, and is built of B 12 icosahedral clusters and C-B C -C chains. Unlike band-structure calculations 2,12 on fully ordered B 13 C 2 , the ordered stoichiometric compound is an insulator or large band-gap semiconductor. An experimental electron-density study by X-ray diffraction conclusively determines that B 13 C 2 is an electron-precise material. The electron-deficient character is explained by B C being stripped of two of its valence electrons and the existence of a unique, electron deficient 3e3c bond on the C-B C -C chains. The low chemical reactivity follows from the extremely small volume of B C .  Table 1. Geometries and topological properties of the experimental static electron density for 2-center exo-cluster bonds in B 13 C 2 . d is the bond-length and d BCP is the distance between a BCP and each of the two constituent atoms of that bond. ρ BCP is the electron density at the BCP and ∇ 2 ρ BCP is its Laplacian. Topological properties for the inter-cluster B-B bonds in α -boron 21 and in γ -boron 24 are also given.  Table 2. Atomic basins (volume V Basin ) and ionic charges for the four crystallographically independent atoms in B 13 C 2 along with their multiplicity in the unit cell.

Methods summary
Crystal growth. Single crystals of boron-carbide were grown at high pressures of 8.5-9 GPa and high temperatures of 1873-2073 K using a 1200-ton (Sumitomo) multi-anvil hydraulic press at the Bayerisches  Geoinstitut. Energy-dispersive x-ray (EDX) analysis has been employed to determine the composition as B 6.51 (12) C, in agreement with stoichiometric B 13 C 2 . The presence of other elements could be excluded.

X-ray diffraction experiment for/and electron density analysis.
A single crystal of boron-carbide of dimensions 0.09 × 0.08 × 0.05 mm 3 was chosen for an x-ray diffraction experiment with synchrotron radiation at beamline F1 of Hasylab, DESY in Hamburg, Germany. The sample was kept at a temperature of 100 K, while a complete data set of accurate intensities was measured for Bragg reflections up to sin(θ )/λ = 1.116 Å −1 . The diffraction data were integrated using the computer program EVAL 33 . Structure refinements have been performed with the software XD2006 34 . A topological analysis of the static electron density has been performed by the modules TOPXD and XDPROP of the computer program XD2006. Two-dimensional density maps have been generated by the module XDGRAPH. See the Supplementary Information for details on procedures and the MP model.