Prediction of New Low Compressibility Solids

An empirical model and an ab initio calculation of the bulk moduli for covalent solids are used to suggest possible new hard materials. The empirical model indicates that hypothetical covalent solids formed between carbon and nitrogen are good candidates for extreme hardness. A prototype system is chosen and a first principles pseudopotential total energy calculation on the system is performed. The results are consistent with the empirical model and show that materials like the prototype can have bulk moduli comparable to or greater than diamond. It may be possible to synthesize such materials in the laboratory.

Lacey, M. M. Oprysko, and T. M. Shaw for support. Part of this work was supported by a grant of the Bundesministerium fir Forschung und Technology (Projekt 13N5482).
3 May 1989; accepted 20 June 1989 Prediction of New Low Compressibility Solids AMY Y. Liu AND MARVIN L. COHEN An empirical model and an ab initio calculation ofthe bulk moduli for covalent solids are used to suggest possible new hard materials. The empirical model indicates that hypothetical covalent solids formed between carbon and nitrogen are good candidates for extreme hardness. A prototype system is chosen and a first principles pseudopotential total energy calculation on the system is performed. The results are consistent with the empirical model and show that materials like the prototype can have bulk moduli comparable to or greater than diamond. It may be possible to synthesize such materials in the laboratory.
HA ARDNESS (1) IS A COMPLEX PROPerty related to the extent to which solids resist both elastic and plastic deformation. For materials with defects, hardness can be limited by many factors including point defects, dislocations, and macroscopic defects. On the microscopic level, for ideal systems, hardness is determined by the bulk modulus, which in turn depends on the nature ofthe chemical bonding. It is the strength and compressibility of the bond that plays the primary role in a solid's ability to resist deformation. The largest bulk moduli are found in covalently bonded materials. Diamond, which has the largest bulk modulus (4.43 Mbar), is also the hardest known solid. A theory of the bulk moduli of covalent solids is therefore useful for suggesting schemes to increase hardness.
An empirical model for the bulk moduli of covalent solids was developed (2) with scaling arguments based on the Phillips-Van Vechten scheme (3) for characterizing the covalent and ionic nature of tetrahedral solids by means of their spectral properties. For purely covalent solids such as C and Si, the bulk modulus scales as the average homopolar energy gap in the reflectivity spectrum divided by the volume of the bond charge. This leads to the bulk modulus varying as d3 5, where d is the bond length. For the zinc blendes, an additional empirical term is added to account for the depletion of bond charge with increasing ionicity. The resulting scaling relation is B=(19.71 -2.20X) (1 d3. 5 (1) where B is the bulk modulus in megabars, d is in angstroms, and X is a measure of the 25 AUGUST I989 ionicity of the compound. For the homopolar semiconductors (group IV), A = 0, while for the heteropolar III-V and II-VI zinc blende solids, X = 1 and 2, respectively. Similar analytic relations between the bulk modulus and lattice constant have been obtained for the group IV and III-V semiconductors by means offirst principles methods based on a pseudopotential scheme (4). The scaling law given by Eq. 1 successfully describes the diamond and zinc blende semiconductors, giving results for the bulk moduli within a few percent. This is comparable in accuracy to first principles calculations (5) that require extensive supercomputer time. Equation 1 is also useful for illustrating trends in these materials. For example, to achieve a large bulk modulus, two properties are needed: low ionicity and short bond length. In the past, there was speculation that cubic BN might be harder than diamond. Interpolation of empirical relations for the elastic constants (6) led to an estimate of the bulk modulus of BN in  the range of4.65 Mbar. In contrast, because both the bond length and ionicity are larger in BN than in diamond, Eq. 1 predicts that BN should have a smaller bulk modulus than diamond. With X = 1 and d = 1.56 A, the scaling law yields B = 3.67 Mbar for BN. A recent measurement (7) of the bulk modulus of BN yields 3.69 Mbar. The good quantitative agreement of the results of the empirical model and experiment demonstrates the predictive power of the model. Based on this model, it was suggested (2) that a covalent solid formed between C and N could have a larger bulk modulus than diamond. In this work, we investigate the structural and electronic properties of a solid of this type using first principles pseudopotential calculations (5). In choosing a candidate structure that can be used as a prototype for a covalent C-N solid, the simple zinc blende structure is ruled out because in a hypothetical zinc blende compound the first antibonding band would be occupied. This would distribute charge in antibonding regions and it is doubtful that such a structure would be stable. Instead, a more complex structure for a C-N solid, based on the known P-Si3N4 structure with C substituted for Si, is considered. The octet rule for covalent bonding is satisfied in this structure, and no antibonding states are occupied. In Fig. 1, the structure is shown to consist of buckled layers stacked in an AAA.. . sequence (8). The unit cell is hexagonal and contains two formula units (14 atoms) with local order such that C atoms occupy slightly distorted tetrahedral sites while N atoms sit in nearly planar triply coordinated sites. This structure can be thought of as a complex network of CN4 tetrahedra that are linked at the corners. The atomic coordination suggests sp3 hybrids on the C atoms and Sp2 hybrids on the N atoms.
The hexagonal unit cell and the planar coordination ofthe N sites raise the possibility that this structure may exhibit anisotropic elastic properties. However, on a local scale, the CN4 tetrahedra are linked in such a way that the in-plane and out-of-plane bonding are similar. The network is three dimensional and is not obviously weak in any direction. This is supported by measurements (9) of the elastic constants of 1-Si3N4 which show nearly equal in-plane and outof-plane linear compressibilities. Hence the 1-Si3N4 structure is a good prototype for this study even though the bonding is not exclusively sp3. The lack of complete tetrahedral coordination suggests that Eq. 1 may not be as precise as in the diamond and zinc blende cases, but it is expected that the trends contained in the scaling law will be followed. Our ab initio study described below is intended to test this assumption.
The structural and electronic properties of the hypothetical compound 13-C3N4 (C3N4 in the 13-Si3N4 structure) are calculated by means of the first principles pseudopotential total energy approach (5). The local density approximation, with the Wigner interpolation formula (10) for the exchange and correlation functional, is used. Because there are no p states in the core of C and N, the 2p pseudopotentials for these atoms are strongly attractive. Hence a localized orbital formalism (11) The structural properties of P-C3N4 are determined by calculating the total crystal energy as a function of volume. Since 1-C3N4 is used only as a prototypical structure, there is no reason a priori to believe that it is the lowest energy structure for the C-N system. However, at the equilibrium volume, the cohesive energy of 1-C3N4 is calculated to be 81 eV per cell or an average value of 5.8 eV per atom. This moderately large cohesive energy suggests that there is a good chance that 13-C3N4 is at least a metastable structure. At the calculated equilibrium volume, the average C-N bond length is found to be 1.47A. This is in good agreement with the sum of the tetrahedral covalent radii (3,12) of C and N. The bulk modulus of 13-C3N4 is determined by fitting the total energy as a function of volume to Murnaghan's equation ofstate (13) which yields a value of4.27 (+0.15) Mbar. For the calculated bond length and an estimated ionicity factor of X = 1/2 (discussed below), the scaling law (Eq. 1) overestimates the bulk modulus of 13-C3N4 by about 10%. A similar overestimate is found for the bulk modulus of 1-Si3N4 which has a measured (9) value of 2.5 Mbar. These overestimates are most likely caused by the structural difference between these materials and the zinc-blende systems on which Eq. 1 was based. In particular, the trigonal coordination of N in f3-C3N4 and 1-Si3N4 allows only three electrons per N site to participate in the bonding. The remaining N electrons occupy nonbonding p orbitals perpendicular to the bonding plane. This reduction in the number of bonds compared to a tetrahedral system results in elastic constants that are slightly lower than those predicted by Eq. 1.
The degree of ionicity in P-C3N4 can be seen in plots of the charge density. Figure 2 shows the total valence charge density in the [0001] plane of P-C3N4. This plane contains a central N atom surrounded by its three C neighbors. The bonds are highly directional, as expected for a covalent system. For comparison, linear plots of the charge density along a bond in diamond and BN, as well as P-C3N4, are given in Fig. 3. To facilitate comparison between the different structures, the charge densities have been normalized by the total number of electrons per cell. In diamond, there are two local maxima along the bond. The presence of the double hump rather than a single maximum near the center of the bond is a result of the strongly attractive C 2p potential. The two maxima reach a normalized height of 2.9. With increasing ionicity, the local maximum near the anion grows at the expense of the one near the cation. In 13-C3N4, the two local maxima remain, but the charge is weighted towards the N site. The height of the peaks are 2.7 and 4.6. In BN, the charge is strongly localized near the N site and the bond exhibits only a single peak of height 6.3. Thus the heteropolarity of the C-N bond in 13-C3N4 lies intermediate between purely covalent diamond and partially ionic BN.
In conclusion, we suggest that it may be possible to synthesize a covalent C-N solid such as 3-C3N4 with compressibility comparable to diamond. A possible starting material is amorphous carbon nitride, which has a polymeric structure (14). The application of high pressures and temperatures to cx-C: N may induce a phase transition to a structure with more tetrahedral bonding. This transition would be similar to the transition of graphite to diamond. Since other factors such as chemical reactivity and fracture toughness are important for applications, the prediction of hardness does not guarantee usefulness in engineering applications. Nevertheless, the fabrication of a covalent solid formed between C and N could result in an important new high-performance material that may best be used as part of a composite. From the scientific point of view, the fabrication of a low compressibility C-N solid would add support for the theoretical approach used here and provide information for further applications.