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Publicly Available Published by De Gruyter April 22, 2016

The coloring problem in the solid-state metal boride carbide ScB2C2: a theoretical analysis

Souheila Lassoued, Benoît Boucher, Ahmed Boutarfaia, Régis Gautier and Jean-François Halet


The electronic properties of the layered ternary metal boride carbide ScB2C2, the structure of which consists of B/C layers made of fused five- and seven-membered rings alternating with scandium sheets, are analyzed. In particular, the respective positions of the B and C atoms (the so-called coloring problem) are tackled using density functional theory, quantum theory of atoms in molecules, and electron localizability indicator calculations. Results reveal that (i) the most stable coloring minimizes the number of B–B and C–C contacts and maximizes the number of boron atoms in the heptagons, (ii) the compound is metallic in character, and (iii) rather important covalent bonding occurs between the metallic sheets and the boron–carbon network.

1 Introduction

Ternary solid-state rare-earth metal boride carbides of formula RExByCz (RE = rare-earth or actinoid metal) can crudely be classified into three families, depending upon the arrangement of the boron and carbon atoms. They can either form infinite, planar two-dimensional (2D) nets, or can be bonded in one-dimensional (1D) zig-zag boron chains with carbon atoms attached, or can assemble in finite chains of different lengths (0D) [13]. It has been established a while ago that the dimensionality of the boron–carbon network in these compounds is related to the average valence electron concentration (VEC) per main group atom [1, 4], assuming in a first approximation a Zintl–Klemm ionic bonding scheme [58] between the metal and the non-metal atoms.

Layered compounds of formulae REB2C2 (LaB2C2 and ScB2C2 types of structures, VEC = 4.25) [917], REB2C (YB2C, ThB2C, α-UB2C types, VEC = 4.33) [10, 1821], RE2B3C2 (Gd2B3C2, VEC = 4.60) [22] as well as Sc2B1.1C2.3 (VEC = 5.13) [2325] belong to the first family [1, 3]. In the tetragonal LaB2C2-type structure (also known with alkaline-earth metals), the B and C atoms form planar nets of four- and eight-membered rings stacked directly above one another with the metal atoms occupying positions between the eight-membered rings [916]. Because of the smaller size of the metal atoms, ScB2C2 adopts an orthorhombic structure (a = 5.175(5), b = 10.075(7), c = 3.440(5) Å, space group Pbam) where the B/C layers consist of fused five- and seven-membered rings with the scandium atoms occupying positions between the larger rings (Fig. 1) [17]. In the boron–carbon sheets, each pentagon contains three C and two B atoms, whereas each heptagon contains three C and four B atoms. The unit cell contains two crystallographically independent three-connected boron and carbon atoms. B1 is surrounded by one boron and two carbon atoms, whereas B2 has three carbon neighbors. Two boron atoms and one carbon atom are found around C1. On the other hand, C2 is encircled by three boron atoms. Simple geometric considerations indicate that the polygons forming these planar boron–carbon sheets cannot be regular. Indeed, the B–B and most of the B–C separations are close to 1.6 Å, but two B–C contacts are notably shorter (1.518 and 1.543 Å). As expected, the C–C bond is shorter at 1.447 Å (see Table 1). Bond angles vary from 123° to 139° in the heptagons and from 103° to 110° in the pentagons. Finally, scandium–boron and scandium–carbon distances are close to 2.50 Å overall, somewhat comparable to the Sc–B distances of 2.528 Å in the boride ScB2 [26]. Sc–Sc distances in the ab planes of approximately 3.30 Å are only 3% longer than those measured in metallic scandium (hcp structure; 3.212 Å) [27]. Longer Sc–Sc distances (3.440 Å) are measured between the planes along the c axis. The latter distance is somewhat larger than that observed in the layered ScNiB4 compound (3.272 Å) [28].

Fig. 1: (001) (left) and (100) (right) projections of the ScB2C2 structure. Sc, B, and C atoms are represented by yellow, gray, and black spheres, respectively. Atom numbering is given on the left.

Fig. 1:

(001) (left) and (100) (right) projections of the ScB2C2 structure. Sc, B, and C atoms are represented by yellow, gray, and black spheres, respectively. Atom numbering is given on the left.

Table 1:

Comparison of the pertinent metrical parameters for the experimental (Exp.) and computed (Opt.) ScB2C2 structure.

Distance (Å)Exp.Opt.Distance (Å)Exp.Opt.
Angles (deg)Exp.Opt.Angles (deg)Exp.Opt.

It is noteworthy that the boron–carbon arrangement is reminiscent of the boron part encountered in ScNiB4 which belongs to the family of the ternary rare-earth metal-transition metal borides REMB4 [29, 30] some of which have been considered as potential thermoelectric materials at high temperatures [31, 32].

Looking at this structure, two questions come to mind: (i) What are the formal oxidation states of Sc and B2C2? In other words what are the electronic properties of ScB2C2 and what is the nature of the electron transfer from the metals to the nonmetals? (ii) Is the proposed atomic B vs. C distribution on the available crystallographic sites valid, knowing that X-ray diffraction studies encounter some difficulty to differentiate B from C in the presence of the strongly scattering scandium atoms?

2 Computational details

Density functional theory (DFT) band structure calculations were performed on 3D structures with the Castep 8.0 code [33]. Lattice parameters and the free atomic position parameters were optimized using the Perdew-Burke-Ernzerhof (PBE) functional [34]. Pseudo-potentials were generated using the on-the-fly (OTF) ultrasoft pseudo-potential generator included in the program. A cut-off energy of 500 eV and a Monkhorst–Pack k-point grid [35] of 5 × 3 × 8 were used to achieve convergence. Density of states (DOS) and band structures of the optimized structural arrangements were obtained via the scalar relativistic tight-binding linear muffin-tin orbital method in the atomic sphere approximation including the combined correction (LMTO) using the program TB-LMTO-ASA [3641]. Exchange and correlation were treated in the local density approximation using the von Barth–Hedin local exchange correlation potential [42]. Within the LMTO formalism, interatomic spaces are filled with interstitial spheres. The optimal positions and radii (rES) of these additional “empty spheres” (ES) were determined by the procedure described in [43]. One non-symmetry-related ES with rES = 1.15 Å was introduced in these calculations. The full LMTO basis set consisted of 4s, 4p, and 3d functions for Sc spheres, 2s, 2p, and 3d for B and C spheres, and s, p, and d functions for ES. The eigenvalue problem was solved using the following minimal basis set obtained from the Löwdin downfolding technique: Sc 4s, 3d, C 2s, 2p, B 2s, 2p; and interstitial 1s LMTOs. The k-space integration was performed using the tetrahedron method [44]. Charge self-consistency and average properties were obtained from 125 irreducible k points. DOS and crystal orbital Hamiltonian population (COHP) curves were shifted so that the Fermi level lies at 0 eV. A measure of the magnitude of the bonding was obtained by computing the COHP corresponding to the Hamiltonian population weighted DOS [45]. As recommended, a reduced basis set (in which ES LMTOs were downfolded) was used for the COHP calculations.

QTAIM (quantum theory of atoms in molecules) [46] and ELI (electron localizability indicator) [47, 48] analyses were realized using a module [49] implemented in the full-potential local orbital method [50]. Calculations were performed on the six colorings previously optimized with a 16 × 8 × 32 k-point mesh and using the PBE functional [34]. The physical-space analysis was realized with the program Dgrid [51] with an equidistant assembly of points separated by 0.05 Å in all the three directions. The topology of the electron density divides the space into non-overlapping and space filling atomic regions, called atomic basins, which contain exclusively one nucleus and possess well-defined electronic total energies. The integral of the electron density over an atomic basin yields its effective charge,


with Z being the charge of the nucleus [46]. The topology of ELI divides the space into atomic core regions and valence regions (bonds, lone pairs). It allows the determination of the amount of electrons, which participate to each bond, and provides direct information about chemical bonding. The ELI distribution was plotted with the Paraview program [52, 53].

3 Results and discussion

3.1 Electronic structure

Some years ago, Burdett et al. proposed that ScB2C2 could formally be described as being made of (B2C2)3– sheets sandwiching fully oxidized Sc3+ ions [54, 55]. However, the sp2-hybridized atoms of boron and carbon obey the octet rule for the charge of 2– only with either single or double B–B, C–C, and B–C bonds (see Fig. 2), requiring partially oxidized Sc2+ atoms. This is in agreement with extended-Hückel-tight binding (EH-TB) calculations performed on an isolated B2C2 layer extracted from ScB2C2 which show a hole at the Fermi level for such a charge, i.e. isoelectronic to graphene [56].

Fig. 2: An example of a Lewis formula for the B2C2 sub-network in ScB2C2.

Fig. 2:

An example of a Lewis formula for the B2C2 sub-network in ScB2C2.

DFT calculations were conducted on ScB2C2 in order to gain insight regarding its structural and electronic properties. The structure was first optimized using the Castep program (see above for the computational details). The optimized lattice parameters (a = 5.268, b = 10.162, c = 3.495 Å) and cell volume (V = 187.1 Å3) deviate from the experimental ones [17] by approximately 1% (below the overestimation that is often computed with the PBE functional). Good agreement is observed overall between the computed and experimentally measured atomic separations (see Table 1). Within the metalloid sheets, some computed distances compare very well (C1–B2, C1–C1, C2–B1, C2–B2) whereas others (C1–B1 and B1–B1) are slightly overestimated with respect to the experiment. The computed bond angles are reproduced at remarkably high accuracy with deviations of 1°–3° overall from the experimental values. The computed metal–nonmetal and metal–metal distances are up to 0.04 Å larger than the observed ones (see Table 1).

The partial and total density of states (PDOS and DOS) and the band structure of ScB2C2 are depicted in Figs. 3 and 4, respectively (similar results are obtained within the TB-LMTO-ASA formalism on the experimental structure, see Fig. S1, Supplementary Information). The Fermi level εF (arbitrarily set at 0 eV) crosses a substantial peak of DOS indicating a metallic character. This is in contrast with the semiconducting properties of YCrB4, which adopts the same 2D nonmetal arrangement [5659]. The total DOS divides basically into two main regions. The first one located below εF derives mainly from the nonmetal orbitals. The large DOS above the Fermi level shows the dominance of the Sc orbitals, which are combined to a small extent with C and B orbitals. The metal orbital participation in the valence band in the vicinity of εF, coupled with the boron/carbon orbital contribution to the conduction band which otherwise predominantly comprises metal states, reflects the rather strong metal–nonmetal covalent interactions. There is a high-energy dispersion, reflecting significant covalent interactions between the metal atoms and lighter atoms. This is confirmed by the PDOS of the scandium atoms.

Fig. 3: Total and atom projected DOS for ScB2C2.

Fig. 3:

Total and atom projected DOS for ScB2C2.

Fig. 4: Band structure of ScB2C2 (see the Brillouin zone above for symmetry lines).

Fig. 4:

Band structure of ScB2C2 (see the Brillouin zone above for symmetry lines).

The metallic behavior of ScB2C2 is confirmed in Fig. 4, where some calculated electron bands are illustrated along high symmetry lines. The bands around εF are rather dispersive in all directions, particularly along the lines ΓZ, ΓY, and ΓX, reflecting rather important covalent character not only in the boron/carbon and scandium sheets, but also between the sheets. Metal–nonmetal covalency is also evidenced by the atomic net charges which show that charge transfer from the scandium atoms to the more electronegative boron and carbon atoms is far from complete (Sc1.68+(B0.27–)2(C0.57–)2).

COHP indicating energetic contributions of crystal orbitals between atoms were computed for several contacts in the structure. The resulting curves are sketched in Fig. 5 and their corresponding integrated COHP (ICOHP) values are given in Table 2. Their inspection shows that the C–C COHP curve shows some antibonding states below the Fermi level indicating that C–C bonding is not maximized in this compound. An ICOHP value of –0.775 Ry per cell is computed for the C–C bonds. Oppositely, looking at the B–B and B–C curves, many B–B and B–C bonding states remain unoccupied which indicates that both of these contacts are not at their maximum value (ICOHP = –0.432 and –0.561 Ry per cell for B–B and B–C bonds, respectively). Interestingly, the widely dispersed Sc–B and Sc–C curves show nearly a maximum for the metal–metalloid bonding interactions (ICOHP = –0.219 and –0.092 Ry per cell for Sc–B and Sc–C bonds, respectively). Finally, weak but noticeable Sc–Sc interactions are observed in the metallic sheets (ICOHP = –0.018 and –0.024 Ry per cell). On the other hand, hardly any direct metal–metal interactions occur along the c direction (ICOHP = –0.002 Ry per cell).

Fig. 5: Averaged COHP curves for C–C, B–B, B–C, Sc–B, Sc–C, and Sc–Sc contacts in ScB2C2.

Fig. 5:

Averaged COHP curves for C–C, B–B, B–C, Sc–B, Sc–C, and Sc–Sc contacts in ScB2C2.

Table 2:

ICOHP values for pertinent contacts in ScB2C2.

ContactDistance (Å)–ICOHP (Ry/cell)

aIn plane.

bBetween planes.

3.2 The coloring problem

The second question, i.e. the B vs. C distribution in the boron–carbon layers, deals with the so-called coloring problem. In other words, is the proposed B/C arrangement based on experiments valid? Thirty year ago, Burdett et al. introduced it with the following question: “Given a molecular or extended network and two different types of atoms, we may wonder what is the best way to distribute them in the network for a fixed stoichiometry” [60, 61]. Indeed, the way the atoms are distributed can influence the nature of the electronic structure, and therefore the physical and chemical properties. Initially addressed by Burdett et al. [62, 63] on the basis of theoretical calculations of extended Hückel type, the coloring problem was definitely solved a few years later by some of us for LaB2C2 and YB2C structure types combining 11B solid-state NMR experiments and DFT calculations [6466].

Similarly, several colorings can be envisioned for ScB2C2 [56]. Six, the most simple, are shown in Fig. 6. Coloring I proposed based on experiments [17] and coloring VI favor the minimum of contacts between atoms of the same kind. Colorings II and V are based on the assembly of B4 and C4 entities reminiscent of trans-butadiene. They can alternatively be described as (B4C4) U chains linked to each other via B–C bonds. Finally, networks III and IV result from the assembly of zig-zag and U chains of carbon and boron for the first, and boron and carbon for the second, respectively. This reminds us of polyacetylene with its zig-zag or U forms. Indeed, colorings I and VI, II and V, and III and IV only differ by the B vs. C occupation sites.

Fig. 6: Some B/C colorings for ScB2C2 (coloring I is that experimentally proposed). Sc, B, and C atoms are represented by yellow, gray, and black spheres, respectively.

Fig. 6:

Some B/C colorings for ScB2C2 (coloring I is that experimentally proposed). Sc, B, and C atoms are represented by yellow, gray, and black spheres, respectively.

These different colorings were geometrically optimized. Cell parameters and energies are summarized and compared in Table 3 (see Table S1, Supplementary Information, for the atomic coordinates). Interestingly, despite the fact that the cell parameters change somewhat moving from one coloring to the other, the cell volumes hardly change and overall are all computed approximately 1% larger than the experimental one. This does not allow us to discriminate one arrangement from another. On the other hand, relative energies for the different colorings indicate that coloring I is thermodynamically more stable than the others by an appreciable margin (see Table 3), inspiring confidence for the experimental X-ray structure determination. Colorings III, V, and VI are computed to be substantially less stable (ca. 0.4–0.8 eV per formula unit (f.u.)), whereas colorings II and IV seem thermodynamically unstable with respect to the others, lying 1.35 and 1.83 eV per f.u. higher in energy. It is clear that the most stable colorings are those that minimize the number of B–B and C–C contacts and maximize the number of boron atoms in the heptagons.

Table 3:

Optimized cell parameters (Å), cell volumes (Å3), and relative energies (Erel, eV) for different colorings.

Cell parametersIIIIIIIVVVIExp.
Volume187.16 (1.04)a189.02 (1.05)187.98 (1.05)187.09 (1.04)187.97 (1.05)186.13 (1.04)179.36

aDeviation percentage vs. experimental cell volume.

bRelative energy per formula unit.

The total DOS of the different colorings were computed and examined in order to gain insight regarding their electronic properties (see Fig. S2, Supplementary Information). They are overall similar at first sight and all colorings show conducting properties with substantial DOS in the vicinity of εF.

To shed additional light on the chemical bonding governing the ScB2C2 structure, a QTAIM analysis [46] was performed for all the colorings. The effective atomic charges Qeff defined by the integration of the electron density within the QTAIM atoms [46] are compiled in Table 4. The values obtained for Sc, approximately 2, are nearly equal in all colorings and qualitatively reflect the expected trend: partially oxidized scandium atoms having transferred part of their valence electrons to the boron–carbon network. B atoms are positively charged or slightly negative [from –0.3 (coloring III) to 1.4 (coloring I)], whereas C atoms are substantially negatively charged [from –2.5 (colorings I and VI) to –0.9 (coloring III)]. Note that due to their different environments, B1 and B2 as well as C1 and C2 do not bear the same effective charge. It turns out that the most positively charged boron atoms and the most negatively charged carbon atoms are observed in colorings I and VI. Since coloring I is that observed experimentally, it could be suggested that some ionic character between boron and carbon atoms brings some additional stabilization to the structure. Correlation between ionicity and stabilization has also been observed in AlB2, which contain graphite-like boron sheets [67].

Table 4:

Effective charges Qeff of QTAIM atoms in ScB2C2.


Further insight into the nature of the bonding in ScB2C2 can be provided by the analysis of the ELI [47, 48]. The topology of its distribution offers the possibility of distinguishing regions of covalent interactions. Its sketch in the unit cell for an ELI isovalue of 0.7 (Fig. 7) reveals that besides the core basins of Sc, B, and C, 24 bonding basins (attractors) per unit cell are all located at the midpoint or close to the midpoint between neighboring B and/or C atoms, indicating strong covalent bonding. For this ELI isovalue, the distribution is 2D delocalized and nearly identical in both directions in the sheet for colorings I and VI, four-atom “localized” for colorings II and V, and chain-like delocalized for colorings III and IV.

Fig. 7: ELI distribution for colorings I to VI (ELI = 0.7).

Fig. 7:

ELI distribution for colorings I to VI (ELI = 0.7).

The electronic population of common homoatomic edges between two pentagons or between two heptagons is given in Table 5. The C–C bonding basin contains roughly two electrons irrespective of the carbon positions. The B–B bonding basin possesses a population of approximately two electrons for the pentagon–pentagon shared edge and three electrons for the heptagon–heptagon shared edge, suggesting on average the formation of single and partial double bonds, respectively. Real-space topological analysis has been performed on structures such as YRhB4 and YbRhB4 [68], TmAlB4 [69], or ScNiB4 [28] that exhibit the YCrB4 structure type related to ScB2C2 (see above). Our results compare well with these previous analyses which suggest on average the formation of two-electron two-center B–B bonds and boron electronic charges nearly identical as in ScB2C2. In the latter, it can be added that C–C bonds between two pentagons together with B–B bonds between two heptagons characterize the most energetically stable colorings (I and V, see Table 3). Thus, colorings which offer the maximum electronic population for edges shared between heptagons are the most energetically stable. Finally, it should be added that the population of the remaining B–B, B–C, and C–C bonding basins shows for all colorings intermediate values between 2.0 and 3.5 electrons.

Table 5:

Populations of some homoatomic valence (ELI) basins.


aPentagon–pentagon shared edge.

bHeptagon–heptagon shared edge.

4 Conclusion

The electronic properties of the layered ternary metal boride carbide ScB2C2, the structure of which consists of boron–carbon layers made of fused five- and seven-membered rings alternating with scandium sheets, have been computed and analyzed. Results reveal that (i) contrary to YCrB4, the nonmetal framework of which adopts the same topology, ScB2C2 is metallic in character, and (ii) rather important covalent bonding occurs between the atoms of the metal sheets and the boron–carbon network. Importantly, the study of the respective positions of the B and C atoms (the so-called “coloring problem”) in the nonmetal sheet has indicated that the most stable coloring minimizes the number of B–B and C–C contacts and maximizes the number of boron atoms in the heptagons, confirming the coloring proposed experimentally.

5 Supporting information

LMTO-ASA DOS and band structure for the experimental structure of ScB2C2 (Fig. S1), optimized atomic coordinates for the different colorings in ScB2C2 (Table S1), and total density of states for the different colorings (Fig. S2) are given as Supporting Information available online (DOI: 10.1515/znb-2016-0056).

Dedicated to: Professor Wolfgang Jeitschko on the occasion of his 80th birthday and in recognition of his outstanding contributions to solid-state chemistry.


SL is grateful to the University of Ouargla, Algeria, for providing her a travel grant. BB thanks the Région Bretagne, France, and the Max-Planck-Gesellschaft, Germany, for a PhD grant.


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Supplemental Material:

The online version of this article (DOI: 10.1515/znb-2016-0056) offers supplementary material, available to authorized users.

Received: 2016-3-4
Accepted: 2016-3-18
Published Online: 2016-4-22
Published in Print: 2016-5-1

©2016 by De Gruyter

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