The nitridoborate nitrides Mg₃[BN₂]N and Ca₃[BN₂]N – electronic structure and chemical bonding

1 Introduction

Nitridoborates form with the alkali, alkaline earth and rare earth metals [1], [2], [3], [4]. They can be synthesized from the binary nitrides and hexagonal boron nitride [5] or via metathesis reactions [6], [7]. Among all ternary and quaternary nitridoborates, the linear or slightly bent [BN₂]³⁻ anion is the most frequent anionic motif. The structure-property relationships of these materials have recently been reviewed [8].

Among the 40 different [BN₂]³⁻ containing phases [8] only few nitridoborate nitrides have been reported: two modifications of Mg₃[BN₂]N [9], [10], [11], [12], Mg₆Al₂[BN₂]₂N₄ and Mg₆Ga₂[BN₂]₂N₄ [13], [14] and Ca₃[BN₂]N [15]. These phases are electron-precise and show complete anion ordering between [BN₂]³⁻ and N³⁻. The profound interest in these nitridoborate nitrides lies in the field of catalysis for the high-pressure high-temperature transformation of hexagonal to cubic boron nitride. Especially Mg₃[BN₂]N has repeatedly been discussed as an important intermediate product [16], [17], [18], [19], [20], [21], [22], [23], although the complete mechanism in not yet fully understood. The second latest research topic concerns luminescent materials. Various Eu²⁺-doped nitridoborate nitrides are promising candidates for red-emitting phosphors [12], [13], [14].

The dimorphism and the electronic structure of Mg₃[BN₂]N have not yet been studied in detail. Only the density of states of the normal-pressure modification has been reported [24], substantiating a direct band gap of ca. 0.88 eV at the Γ point. The theoretically obtained transformation pressure of ca. 2 GPa is close to the experimentally used pressure of 4 GPa at 1573 K [10].

Herein the theoretical study is extended to include both modifications of Mg₃[BN₂]N along with data for tetragonal Ca₃[BN₂]N [15], presenting energy-volume curves, density of states and the band structures. Also, an analysis of chemical bonding on the basis of overlap populations is performed.

2 Crystal chemistry

The crystal structures of the normal-pressure (NP) and high-pressure (HP) modifications of Mg₃[BN₂]N are presented in Fig. 1. The nitride anions in hexagonal NP-Mg₃[BN₂]N, space group P6₃/mmc, have trigonal bi-pyramidal magnesium coordination. These N@Mg₅ units are condensed via common corners, leading to layers, which extend in the ab plane. The layers are stacked in ABAB sequence (a consequence of the 6₃ axis) and separated by the [BN₂]³⁻ units. The latter are coordinated by six magnesium cations in a strongly distorted octahedral manner.

Fig. 1:

The crystal structures of NP- and HP-Mg₃[BN₂]N. Magnesium, boron and nitrogen atoms are drawn as blue, green and medium grey circles, respectively. The magnesium coordination of the isolated nitride anions and the B–N distances (Å) are emphasized.

The transformation to HP-Mg₃[BN₂]N is a reconstructive phase transition and one observes different coordination modes. For the nitride anions the coordination number increases from five to six and the resulting octahedra are condensed to layers via common edges. According to the pressure-distance paradoxon, the high-pressure phase exhibits longer Mg–N distances (2.056–2.202 Å) as compared to NP-Mg₃[BN₂]N (2.040–2.046 Å). In the orthorhombic high-pressure phase we observe an increasing coordination number for the [BN₂]³⁻ units: 10 Mg²⁺ in an orthorhombically distorted bi-capped, square prismatic arrangement. The corresponding interatomic distances are listed in Table 1.

At this point it is interesting to compare the structure of HP-Mg₃[BN₂]N with that of the higher congener Ca₃[BN₂]N. Both compounds follow the empirical pressure-homologue rule: HP-Mg₃[BN₂]N crystallizes approximately with the structure of the higher congener. The striking difference is the space group symmetry, Pmmm for HP-Mg₃[BN₂]N (the orthorhombic distortion has experimentally clearly been proven on the basis of powder X-ray diffraction in [10]) and P4/mmm for Ca₃[BN₂]N. Thus, HP-Mg₃[BN₂]N crystallizes with a translationengleiche subgroup of P4/mmm. The corresponding group-subgroup scheme is presented in the Bärnighausen formalism [26], [27] in Fig. 2. From this scheme it is readily evident that the distortion only concerns the course of the lattice parameters: a=b=3.5494 Å for tetragonal Ca₃[BN₂]N as compared to a=3.0933 and b=3.1336 Å for HP-Mg₃[BN₂]N. There is no need for other free positional parameters. These structural peculiarities are addressed based on ab initio calculations in the following.

Fig. 2:

Group-subgroup scheme in the Bärnighausen formalism [26, 27] for the structures of Ca₃[BN₂]N [15] and HP-Mg₃[BN₂]N [9]. The index for the translationengleiche symmetry reduction (t) and the evolution of the atomic parameters is given.

3 Computational details

Within the accurate quantum mechanical density functional theory DFT [28], [29] two methods have been used complementarily.

The first calculations are performed using the Vienna ab initio simulation package (VASP) code [30], [31] which allows geometry optimization, total energy calculations, and establishing the energy-volume equations of state (EOS). This method is based on the projector augmented wave (PAW) [31], [32] which is executed for atomic potentials built within the generalized gradient approximation (GGA) scheme following Perdew, Burke and Ernzerhof (PBE) [33]. This exchange-correlation (XC) scheme is preferred to the local density approximation LDA [34] known to underestimate interatomic distances and energy band gaps. Relaxing atoms of the different crystal setups is based on the conjugate-gradient algorithm [35] and the tetrahedron method with Blöchl corrections [36] as well as a Methfessel-Paxton [37] scheme are applied for both geometry relaxation and total energy calculations. Brillouin zone (BZ) integrals are approximated using a special k-point sampling of Monkhorst and Pack [38]. Optimization of structural parameters has been performed until the forces on the atoms were less than 0.02 eV Å⁻¹ and all stress components less than 0.003 eV Å⁻³. Calculations are converged at an energy cut-off of 350 eV for the plane-wave basis set with respect to the k-point integration with a starting mesh of 6×6×6 up to 12×12×12 for best convergence and relaxation to zero strains.

The second method fully accounts for the electronic structure, electronic band structure, site projected density of states (PDOS) and properties of chemical bonding based on the overlap matrix (S_ij) with the COOP criterion [39]. It is based on the full potential augmented spherical wave (ASW) method [40], [41] and the generalized gradient approximation GGA [33] for exchange-correlation effects. In the minimal ASW basis set, outermost shells are used for representation of valence states and the matrix elements are constructed using partial waves up to l_max+1=3 for Ca, Mg and l_max+1=2 for B and N. Self-consistency is achieved when charge transfers and energy changes between two successive cycles are ΔQ<10⁻⁸ and ΔE<10⁻⁶ eV, respectively. The BZ integrations are performed using the linear tetrahedron method within the irreducible hexagonal and orthorhombic wedges following Blöchl [36].

4 Electronic structure and chemical bonding

4.1 Energy-volume curves

Table 2 shows the starting experimental and calculated atomic positions and structure parameters of the two compounds. There is a reasonably good agreement between experiment and calculations as well as with the computations by Medvedeva et al. [24] and Wang et al. [25]. Further, following the introduction of the orthorhombic form of Mg₃[BN₂]N crystallizing with a translationengleiche subgroup of the tetragonal group P4/mmm of Ca₃[BN₂]N with group-subgroup correspondence on one hand and on the other hand in view of the computational results by Wang et al. [25] where Mg₃[BN₂]N is computed in the tetragonal Ca₃[BN₂]N type structure, additional calculations were carried out in order to clearly evaluate the energetic consequences of the closeness of the two structures (orthorhombic versus tetragonal). Such energy discriminations between the three forms of Mg₃[BN₂]N can be assessed from the energy (E)-volume (V) values calculated for the optimized structure parameters (Table 2). The E(V) data, which then present a quadratic evolution, are fitted with a Birch EOS [42] up to the third order following:

Table 2:

Experimental and (calculated) crystal data.

Compound	Space group	a, b, c (Å)	z (Ca/Mg); z (N1)
Ca3[BN2]N	P4/mmm	3.5494 (3.52); 8.2136 (8.31)	0.2778 (0.28); 0.3354 (0.33)
Mg3[BN2]N	Pmmm	3.0933 (3.10); 3.1336 (3.12); 7.7005 (7.84)	0.2670 (0.25); 0.3262 (0.32)
Mg3[BN2]N	P63/mmc	3.5445 (3.52); 16.0354 (16.2)	0.1228 (0.13); 0.0851 (0.09)
Mg3[BN2]N	P4/mmm	(3.122); (7.91)	(0.258); (0.33)

1. Orthorhombic/tetragonal: Ca1/Mg1(0 0 0), Ca2/Mg2(1/2 1/2 z), N1(0 0 z), N2(1/2 1/2 0), B(0 0 1/2); hexagonal: Mg1(0 0 1/4), Mg2(1/3 2/3 z), N1(1/3 2/3 1/4), N2(0 0 z), B(0 0 0).

E(V)=E0(V0)+98V0B0[(V0/V)2/3−1]2+916B0(B′−4)V0[(V0/V)2/3−1]3

The latter provides equilibrium parameters for the energy, the volume, the zero pressure bulk modulus and its pressure derivative: E₀, V₀, B₀ and B′, respectively. The obtained values with accurate goodness of fit χ² ~10⁻⁵/10⁻⁴ magnitudes are displayed in the insert of Fig. 3 for tetragonal Ca₃[BN₂]N and the three forms of Mg₃[BN₂]N. For the former the volume comes close to the experiment. This is opposed to the smaller volume of the Mg compounds which come close to ~76 ų per formula unit (FU) for the orthorhombic and tetragonal forms and larger (V~88 ų per FU) for the hexagonal normal-pressure modification. The consequence is a large change of the magnitudes of the corresponding bulk moduli, i.e. the smallest one occurs for hexagonal Mg₃[BN₂]N: this follows from the fact that the larger the volume the larger the compressibility or the smaller the bulk module. The relevant result for the Mg₃[BN₂]N structural models is the proposition of the hexagonal form as the ground state structure which exhibits by far the lowest energy. The orthorhombic modification is very close to the tetragonal one, although the latter is found at slightly lower energy and smaller volume (i.e. a slightly larger bulk modulus of 140 GPa). One might suggest that the orthorhombic and tetragonal modification form under different high-pressure high-temperature conditions.

Fig. 3:

Energy-volume curves and fit values from Birch EOS calculations for Ca₃[BN₂]N and the NP, HP and hypothetic Ca₃[BN₂]N-type modifications of Mg₃[BN₂]N.

4.2 Electronic band structure and density of states

Figure 4 provides the band structure and the site projected density of states (PDOS) of Ca₃[BN₂]N. The compound is characterized by a semi-conducting behavior with a direct gap at the Γ point, i.e. at the Brillouin zone (BZ) center with Γ_V→Γ_C of ~1 eV magnitude. The zero energy is with respect to the top of the valence band (VB). At the VB bottom two distinct types of bands are identified corresponding to s-like states whereas p-like states occupy the energy range from –6 eV to –E_V. The site projected DOS clearly show different (chemical) characters of N1 versus N2, i.e. with less electronegative behavior for the latter and the presence of its p-PDOS at the top of the valence band (VB). Note the energetically low-lying s states of N1 at –15 eV whereas the s-N2 states are at approximately –10 eV.

Fig. 4:

Ca₃[BN₂]N: Electronic band structure along major lines of the tetragonal Brillouin zone (top) and site-projected density-of-states (DOS) (bottom).

Similar features are observed for hexagonal Mg₃[BN₂]N (normal-pressure structure) in Fig. 5 showing the band structure, with a smaller, direct Γ_V→Γ_C band gap with albeit less ionic character, i.e. a broader VB with p states over ~7 eV due to the presence of Mg. The top of the VB is with respect to E_V. More drastic changes are observed for the orthorhombic form which exhibits metallic behavior with flat localized bands (bottom panel) still with N-p states now at E_F, and the VB is largely broadened for all states. Then the transition from NP-Mg₃[BN₂]N to HP-Mg₃[BN₂]N involves metallization (higher degree of delocalization in the high-pressure phase).

Fig. 5:

Band structures of NP-Mg₃[BN₂]N (top) and HP-Mg₃[BN₂]N (bottom) along the major lines of the irreducible wedge of the hexagonal (top) and orthorhombic (bottom) Brillouin zones.

The PDOS given in Fig. 6 mirrors the band structure observations for the transition from NP-Mg₃[BN₂]N to HP-Mg₃[BN₂]N. In the middle panel (hypothetical tetragonal Mg₃[BN₂]N), besides similar features to the actual compound as for instance the relative positions of the s and p PDOS, a band gap is observed, similar to Ca₃[BN₂]N with, however, a smaller magnitude due to the less ionic character of Mg as compared to Ca.

Fig. 6:

Site-projected density-of-states (PDOS) for NP-Mg₃[BN₂]N (top) hypothetic tetragonal Mg₃[BN₂]N (middle) and HP-Mg₃[BN₂]N (bottom).

Insofar as both experimental orthorhombic HP-Mg₃[BN₂]N and hypothetical tetragonal Mg₃[BN₂]N are very close in energy (only 0.02 eV difference) with slightly more stabilization for the latter (cf. Fig. 3/EOS), and in view of the unexpected metallization in such a compound, it becomes questionable as to whether the stable high-pressure modification of Mg₃[BN₂]N should not be tetragonal. This inquiry needs further experimental investigation.

4.3 Crystal orbital overlap population analyses

Bonding is mainly operating in the p states-dominated VB part, i.e. extending over 6 eV below E_V. The plot for Ca₃[BN₂]N (Fig. 7) reveals interesting features whereby B–N1 covalent bonding within the [BN₂]³⁻ units dominates over the lower energy part of VB from –6 to –3 eV, whereas the ionic Ca–N2 bonding shows COOP at the higher energy part extending over 2 eV below E_V; two different features appear for the anions [BN₂]³⁻ and N³⁻.

Fig. 7:

Crystal orbital overlap population analyses for Ca₃[BN₂]N.

For Mg₃[BN₂]N (Fig. 8), in spite of features similar to Ca₃[BN₂]N, less energy differentiation between the B–N and Mg–N COOP due to the decrease of iconicity can be observed. These features stress further the more covalent character of the Mg–N bonds versus Ca–N.

Fig. 8:

Crystal orbital overlap population analyses for NP-Mg₃[BN₂]N (top) and HP-Mg₃[BN₂]N (bottom).

5 Conclusion

Electronic structure calculations quantify the different bonding situation of the covalently bonded [BN₂]³⁻ and isolated N³⁻ anions in the nitridoborate nitrides Ca₃[BN₂]N and Mg₃[BN₂]N. Total energy calculations reveal the hexagonal normal-pressure Mg₃[BN₂]N structure as the ground state structure. The high-pressure phase of Mg₃[BN₂]N is reported in space group Pmmm, an orthorhombically distorted variant of the Ca₃[BN₂]N type. A comparative study of orthorhombic Mg₃[BN₂]N and a hypothetic tetragonal, Ca₃[BN₂]N-type high-pressure phase show substantial metallization of the orthorhombic variant. These unusual bonding characteristics call for a reinvestigation of the HP-Mg₃[BN₂]N structure along with property studies.