Ultrahigh-Density Superhard Hexagonal BN and SiC with Quartz Topology from Crystal Chemistry and First Principles

Abstract

Based on superdense C₆ with a quartz (qtz) topology, new ultrahigh-density hexagonal binary phases, qtz BN and qtz SiC, were identified via full geometry structure relaxations and ground state energies using calculations based on the quantum density functional theory (DFT) with a gradient GGA exchange–correlation XC functional. Like qtz C₆, with respect to diamond, the resulting binary qtz BN and qtz SiC were found to be less cohesive than cubic BN and cubic SiC, respectively, but were confirmed to be mechanically (elastic constants) and dynamically (phonon band structures) stable. Higher densities of the new phases correlate with higher hardness values compared to cubic BN and cubic SiC. In contrast to the regular tetrahedra that characterize the cubic BN and SiC phases, the corner-sharing tetrahedra in the new phases are distorted, which accounts for their exceptional density and hardness. All three qtz phases were found to be semiconducting to insulators, with reduced band gaps compared to diamond, cubic BN, and cubic SiC.

1. Introduction

Diamond is the hardest known material and has a Vickers hardness (H_V) of about 100 GPa and a density of ρ = 3.635 g/cm³ [1]. Its structure (space group Fm-3m) is formed by corner-sharing tetrahedra of sp³-hybridized carbon atoms with ∠C-C-C = 109.47° and is characterized by the highest atomic density (i.e., the number of atoms per unit cell volume) and highest density per valence electron of any material [2]. It has been claimed that a rare hexagonal form of diamond (lonsdaleite) (space group P6₃/mmc) with virtually the same density is stronger and stiffer than diamond [3].

The network topologies of diamond and lonsdaleite are dia and lon, respectively, and many theoretically predicted carbon allotropes have been identified using these topologies (see [4] and references therein). The topology determination for the new phases is now made easy with the TopCryst program [5]. Information on all carbon allotropes extracted from the literature is indexed in the “SACADA” database [6], which currently contains 703 allotropes.

Diamond (both cubic and hexagonal) is still considered to have superior atomic density, elastic moduli, and hardness [1]. Recently, however, several superdense (ρ > 3.635 g/cm³) carbon allotropes have been predicted from the first-principle studies [7,8], and hexagonal (space group P₆222) C₃ with assigned quartz topology (SACADA qtz #11) has even been claimed to have a hardness of 113 GPa [8], i.e., 15% harder than diamond. Also, very recently, we proposed a superdense (ρ = 3.666 g/cm³) ultrahard (H_V ≈ 102 GPa) hexagonal C₆ (space group P6₅22) allotrope with a qtz topology [9].

The quartz topology is also observed in binary (ZnTe) and ternary (GaAsO₄, FePO₄) compounds. In particular, under pressure, in addition to the cubic sphalerite structure (space group F-43m) characteristic of zinc chalcogenides, ZnTe also exhibits a trigonal α-HgS-type structure (space group P3₁21) [10] with a qtz topology. This may be because the Pauling electronegativity (χ) of the chalcogen decreases from oxygen (χ(O) = 3.44) to tellurium (χ(Te) = 2.10). With the knowledge that χ(Zn) = 1.65, ZnO is an ionic compound (Δχ|Zn-O|=1.79), while ZnTe is already a covalent compound (Δχ|Zn-Te| = 0.45).

In this regard, the aim of the present study is to address the challenging issue of the existence of ultrahigh-density polymorphs of compounds that are of great practical importance. We aimed to prove the existence of two known binary compounds, BN and SiC, in a quartz topology based on the previously studied qtz C₆ [9]. For this purpose, we conducted accurate quantum mechanical calculations of crystal structures and physical properties in the framework of density functional Theory (DFT) [11,12].

2. Computational Framework

The structures of the binary compounds were all subjected to geometry relaxations of atomic positions and lattice constants down to the respective ground states characterized by minimum energies. The protocol consists of iterative calculations performed using the DFT-based plane-wave Vienna Ab initio Simulation Package (VASP) [13,14]. For atomic potentials, the projector augmented wave (PAW) method was used [14,15]. The exchange and correlation effects were treated using a generalized gradient approximation (GGA) scheme [16]. The relaxation of the atoms to the ground state geometry was carried out by applying a conjugate gradient algorithm [17]. A tetrahedron method [18], with corrections made according to the Methfessel–Paxton scheme [19], was used for geometry optimization and energy calculations. A special k-point sampling [20] was applied to approximate the Brillouin zone (BZ) integrals in the reciprocal space. For better reliability, the optimization of the structural parameters was carried out along with successive self-consistent cycles with increasing k-mesh until the forces on atoms were less than 0.02 eV/Å and the stress components had values below 0.003 eV/ų. The plane waves energy cutoff was 400 eV.

Mechanical properties were derived from elastic constants calculations [21,22].

The phonons were calculated to verify the dynamic stability of the new phases. The phonon modes were computed following a protocol that considers harmonic approximation by operating finite displacements of the atoms around their equilibrium positions in different input files (POSCAR-YYY, where YYY is the displacement number), corresponding to configurations generated using the interface code “Phonopy” [23]. Calculations were then performed for all individual POSCAR-YYY. Then, the summation of the different resulting forces could be obtained for the different configurations. Finally, the phonon dispersion curves (bands) along the main directions of the hexagonal Brillouin zone were plotted.

Crystal information files (CIF); structure sketches, including tetrahedral representations; and illustrations of the charge density plots were generated using VESTA graphics software (version 3.5.7) [24]. The electronic band structures and densities of states were obtained using the full-potential augmented spherical wave ASW method based on DFT using the same GGA scheme as above [25].

3. Crystal Chemistry

The structure of qtz C₆ is shown in Figure 1a. A differentiation of carbon into two different sites was highlighted by resolving the initial C₆ structure (space group P6₅22, No. 179) with unique (6a) sites [9] into carbon atoms with (3c) and (3d) Wyckoff positions within space group P6₄22, No. 181 (as shown in

Table 1. Crystal structure parameters of phases with quartz topology.

P6422 (No. 181)	C6 (C13C23)	qtz BN	qtz SiC
a, Å	2.598	2.595	3.079
c, Å	5.586	5.799	7.054
Vcell, Å3	32.65	33.83	57.90
<Vatom> Å3	5.44	5.64	9.65
Shortest bond, Å	1.598	1.64	1.936
Angles (deg.)	108.73/90.52	106.64/92.0	105.26/93.01
Atomic positions	C1 (3c) ½, 0, 0	B (3c) ½, 0, 0	Si (3c) ½, 0, 0
	C2 (3d) ½, 0, ½	N (3d) ½, 0, ½	C (3d) ½, 0, ½
Etotal, eV	−47.87	−46.65	−41.79
Ecoh/atom, eV	−1.38	−1.73	−1.15

). Such an elementary modification allowed us to consider the binary compounds, B₃N₃ and Si₃C₃. The lattice parameters of the ground state structures are given in columns 3 to 5 of Table 1. The corresponding crystal structures are shown in Figure 1b–c with ball-and-stick and tetrahedral representations, the latter being characterized by corner-sharing irregular tetrahedra (vide infra). The atoms are described using general Wyckoff positions at (3c) ½, 0, 0 and at (3d) ½, 0, ½.

Differences are observed in the volumes (total and atom-averaged) and in the interatomic distances, which increase along the series due to the increase in the respective atomic radii. If we look at the angles related to the constituent tetrahedra, they significantly differ from the regular tetrahedral angle (∠109.47°), thus indicating the specificity of the qtz topology.

All three qtz phases were found to be cohesive with negative E_coh/atom values (see Table 1), which are systematically lower than the corresponding values of diamond (−2.43 eV), cubic BN (−2.63 eV) and cubic SiC (−1.72 eV).

4. Mechanical Properties from the Elastic Constants

The analysis of the mechanical behavior was carried out using the elastic properties by performing finite distortions of the lattice. The phase is then described by the bulk (B) and the shear (G) moduli obtained by averaging the elastic constants. Here, we used Voigt’s method (cf. original [21] and modern [22] studies) based on a uniform strain. The calculated sets of elastic constants C_ij (i and j corresponding to directions) are given in

Table 2. Elastic constants (C_ij) of phases with quartz topology. Bulk (B_V) and shear (G_V) moduli calculated via Voight averaging. All values are in GPa.

	C11	C12	C13	C33	C44	C66	BV	GV
qtz C6 [9]	1184	88	63	1161	548	539	440	546
qtz BN	892	164	109	1013	364	342	396	370
qtz SiC	448	87	97	598	180	201	228	197

. All C_ij values are positive. The elastic constants of qtz C₆ have the largest values, close to diamond [1], and smaller magnitudes were obtained for qtz BN and qtz SiC. The bulk (B_V) and shear (G_V) moduli (see the last two columns of Table 2) were calculated using the equations for the hexagonal system [22]:

B_V = 1/9 {2(C₁₁ + C₁₂) + 4C₁₃ + C₃₃}

(1)

G_V = 1/30 {C₁₁ +C₁₂ + 2C₃₃ − 4C₁₃ + 12C₄₄ + 12C₆₆}

(2)

qtz C₆ has the largest moduli, close to the accepted values for diamond (B_V =445 GPa and G_V = 550 GPa [1]), while the binary phases, especially qtz SiC, have much smaller values.

Four modern theoretical models [26,27,28,29] have been used to predict the Vickers hardness (H_V) of new phases. The thermodynamic (T) model [26], which is based on thermodynamic properties and crystal structures, generally shows good agreement with experiments, and is therefore recommended for the hardness evaluation of superhard and ultrahard phases [30]. The Vickers hardness and bulk modulus values calculated using this model are summarized in

Table 3. Vickers hardness (H_V) and bulk moduli (B₀) of dense carbon, BN, and SiC phases calculated using the thermodynamic model of hardness [26].

	Space Group	a = b (Å)	c (Å)	ρ (g/cm3)	HV (GPa)	B0 (GPa)
Diamond	Fd-3m	3.56661 [31]		3.517	98	445 [1]
Lonsdaleite	P63/mmc	2.5221 [32]	4.1186 [32]	3.516	97	443
qtz C3 #180 [7]	P6222	2.605 [7]	2.801 [7]	3.635	101	460
qtz C3 #180 [8]	P6222	2.613 [8]	2.811 [8]	3.600	100	456
qtz C6 #181	P6222	2.5970	5.5865	3.667	102	464
c-BN #216	F-43m	3.6160 [33]		3.487	55	381 [34]
w-BN #186	P63/mc	2.5505 [35]	4.210 [35]	3.475	54	375 [36]
qtz BN #181	P6422	2.5954	5.7993	3.654	58	395
SiC (3C) #216	F-43m	4.3581 [37]		3.218	34	238
qtz SiC #180	P6222	3.0787	7.0537	3.450	36	255

. The Lyakhov–Oganov (LO) model [27] considers the topology of the crystal structure, the strength of covalent bonding, the degree of ionicity and directionality; and the empirical models, Mazhnik–Oganov (MO) [28] and Chen–Niu (CN) [29], are based on elastic properties, namely bulk and shear moduli. As previously shown [30], in the case of superhard (H_V ≥ 40 GPa) compounds of light elements, the Lyakhov–Oganov model gives slightly underestimated values of hardness, while empirical models are not reliable. Fracture toughness (K_Ic) was evaluated using the Mazhnik–Oganov model [28].

Table 4. Mechanical properties of the dense carbon, BN and SiC phases: Vickers hardness (H_V), bulk modulus (B), shear modulus (G), Young’s modulus €, Poisson’s ratio (ν), and fracture toughness (K_Ic).

	HV				B		GV	E **	ν **	KIc ‡
	T *	LO †	MO ‡	CN §	B0 *	BV
	GPa									MPa·m½
Diamond	98	90	100	93	445 [1]		530 [1]	1138	0.074	6.4
Lonsdaleite	97	90	99	94	443	432	521	1115	0.070	6.2
qtz C3 #180 [7]	101	90	–	88 [7]	460	433 [7]	–	–	–	–
qtz C3 #180 [8]	100	88	113 [8]	110	456	452 [8]	587 [8]	1229	0.047	6.8
qtz C6 #181	102	90	104	100	464	440	546	1159	0.061	6.4
c-BN #216	55	50	73	67	381 [34]		399 [34]	887	0.112	4.8
w-BN #186	54	50	72	65	375 [36]	390	397	889	0.120	5.1
qtz BN #181	58	51	63	56	395	396	370	847	0.143	6.4
SiC (3C) #216	34	30	32	35	238	225 ††	199 ††	461 ††	0.153 ††	3.3
qtz SiC #180	36	31	31	34	255	228	197	460	0.165	3.3

* Ref. [26]; ^† Ref. [27]; ^‡ Ref. [28]; ^§ Ref. [29]; ** E and ν values calculated using isotropic approximation. ^†† Calculated via Voigt averaging of literature data on elastic constants [38].

shows the hardness and other mechanical properties of the dense carbon, BN, and SiC phases that are calculated using all four models.

Notably, the density of all three phases with a quartz topology is higher than the density of the corresponding cubic phases (see Table 3); qtz C₆, qtz BN and qtz SiC are ultrahigh-density phases.

The hardness and mechanical properties of qtz C₆ are, as expected, close to those of diamond and lonsdaleite. The corresponding values for qtz BN and especially for qtz SiC are significantly lower and are at the level of the mechanical properties of cubic boron nitride and cubic silicon carbide, respectively. It should be noted, however, that the hardness of all three phases with quartz topology is about 5% higher than that of the corresponding cubic phases. Such increased hardness is likely related to the ultrahigh densities of the qtz phases resulting from the distorted tetrahedron building blocks.

5. Dynamic and Thermal Properties from the Phonons

5.1. Phonon Band Structures

Following the protocol presented in the “Computational framework” section, the phonon band structures were plotted along the hexagonal Brillouin zone in the reciprocal space. The corresponding band structures for qtz C₆, qtz BN, and qtz SiC are shown in Figure 2 (red lines).

The bands develop along the main lines (horizontal direction) of the hexagonal Brillouin zone (reciprocal k-space). The vertical direction represents the frequencies ω, which are given in terahertz (THz). There are 3N phonon total modes with three acoustic modes starting from zero frequency (ω = 0 at the Γ point, center of the Brillouin zone), up to a few terahertz, and then 3N-3 optical modes are observed at higher frequencies. The three acoustic modes correspond to the lattice rigid translation modes with two transverse modes and one longitudinal mode. The remaining bands correspond to the optical modes. In all three subfigures, there are no negative frequencies, and the corresponding carbon allotrope and two binary phases are dynamically stable. The latter indicates that these phases, once synthesized, can exist at ambient conditions. In qtz C₆, the highest band culminates in the vicinity of ω ~ 40 THz, a value that has been observed for diamond by Raman spectroscopy [39]. Binary compounds are characterized by lower energy bands, but a similarity between the BN band structure and the phonons of the carbon allotrope can be observed. Finally, SiC phonons show the highest bands at 10 THz less than BN.

5.2. Thermodynamic Properties

The thermodynamic properties of the new phases were calculated from the phonon frequencies using the statistical thermodynamic approach [40] on a high-precision sampling mesh in the Brillouin zone. The temperature dependencies of the heat capacity at constant volume (C_v) and entropy (S) of qtz C₆, qtz BN, and qtz SiC are shown in Figure 3 in comparison with experimental C_v data for diamond [41,42], cubic BN [43] and cubic SiC [44]. Systematically, the heat capacities of all three phases with quartz topology formed by distorted tetrahedra are found to be higher than the heat capacities of the corresponding cubic phases.

6. Electronic Band Structures and Density of States

Using the crystal structure parameters given in Table 1, the electronic band structures were obtained using the all-electron DFT-based augmented spherical method (ASW) [25] and are shown in Figure 4. The bands (blue lines) develop along the main directions of the primitive hexagonal Brillouin zones to the extent that all three phases exhibit band structures characterized by energy gaps between the valence band (VB) and the empty conduction band (CB). The energy reference along the vertical energy axis is considered as the top of the VB: (E-E_V). qtz C₆ has a band gap half of size of that of diamond (~5 eV), showing a different behavior between dia and qtz topologies in the electronic structure behavior. The largest band gap is observed for qtz BN, which remains smaller than for cubic BN. The same feature of reduced band gap is observed for qtz SiC. It is also relevant to highlight that continuous VB in qtz C₆ versus two blocks in the binary compounds has a separation between low-energy lying s-like states and higher-energy lying p-like states up to E_V. In conclusion, the phases with qtz topology are provided with an enhanced covalence.

The band structure features are reflected in the site-projected electronic densities of states (DOS), as shown in Figure 5 with energy along the horizontal axis and DOS as 1/eV units along the vertical axis. The vertical red line represents the top of the valence band. qtz C₆, expressed as C1₃C2₃, is characterized by identical DOS for both sites and a continuous VB extending over ~27 eV, indicating the purely covalent nature of the carbon allotrope. The sharp DOS peak at 1 eV below E_V belongs to the carbon p-states, which are more localized than the s-states smeared in the lower part of the VB. The (empty) CB also shows structured p-DOS. Turning to the binary compounds, the VB is now divided into two parts corresponding to s-states up to −15 eV in qtz BN (−9 eV in qtz SiC), followed by a broad block up to E_V. The band gap in qtz BN is the largest (~5 eV) which is close to c-BN as stated above, while qtz SiC has the smallest band gap of about 1.5 eV.

More explanation can be found from electronegativity χ according to Pauling: χB = 2.04; χC = 2.55; χN = 3.04, and χSi = 1.80; one obtains Δχ|B-N| = 1 and Δχ|Si-C| = 0.75. These results indicate a larger covalence of SiC compared to the polar covalence of BN, hence the larger band gap in boron nitride.

7. Conclusions

This paper presents a new class of binary compounds with quartz topology using boron nitride and silicon carbide as representative compounds. The structures of qtz BN and qtz SiC were constructed from the template carbon allotrope C₆ with a quartz topology. It has been shown that the new phases are the densest among all known BN and SiC polymorphs. Accordingly, they are characterized by the highest hardness. In addition to mechanical stability, the new phases are also dynamically stable, as indicated by the phonon band structures. The heat capacities of the new phases calculated from the phonon frequencies were found to be higher than those of the corresponding cubic phases; this is also true for qtz C₆ when compared to diamond. It can be assumed that all of the above is a consequence of the presence of distorted tetrahedra in the crystal structures of the phases with quartz topology. Finally, from the analysis of electronic band structures and densities of states, it was found that the new phases are insulating (BN) to semiconducting (SiC), with band gaps between the filled valence band VB and the empty conduction band CB.