Two Novel C3N4 Phases: Structural, Mechanical and Electronic Properties

We systematically studied the physical properties of a novel superhard (t-C3N4) and a novel hard (m-C3N4) C3N4 allotrope. Detailed theoretical studies of the structural properties, elastic properties, density of states, and mechanical properties of these two C3N4 phases were carried out using first-principles calculations. The calculated elastic constants and the hardness revealed that t-C3N4 is ultra-incompressible and superhard, with a high bulk modulus of 375 GPa and a high hardness of 80 GPa. m-C3N4 and t-C3N4 both exhibit large anisotropy with respect to Poisson’s ratio, shear modulus, and Young’s modulus. Moreover, m-C3N4 is a quasi-direct-bandgap semiconductor, with a band gap of 4.522 eV, and t-C3N4 is also a quasi-direct-band-gap semiconductor, with a band gap of 4.210 eV, with the HSE06 functional.


Introduction
Studies on light element-based materials trace back to the middle of the last century. Since Lavoisier found that diamond was isostructural to carbon and much denser than graphite, many studies have been devoted to its synthesis under high pressure [1][2][3][4]. More and more researchers have begun to investigate the carbon allotropes [5][6][7][8][9][10][11][12][13][14][15]. The second light element-based material to be evidenced was boron nitride. It includes three different structures: blende-, wurtzite-and graphitic-type structures. Cubic boron nitride (c-BN) was first elaborated upon in 1957 by Wentorf, who performed direct conversion using graphitic boron nitride (at 7 GPa and 1500˝C) [16]. Many boron nitride allotropes have been investigated by researchers, such as O-BN, Pbca-BN, Z-BN, W-BN, h-BN, bct-BN, P-BN, and cT8-BN. Interest in carbon nitrides has been initiated by studying materials that exhibit mechanical properties comparable with those of diamond. A fullerene is a molecule of carbon in the form of a hollow sphere, ellipsoid, tube, and many other shapes. Gueorguiev et al. [17,18] studied the formation mechanisms and structural features of fullerene-like carbon nitride (FL CNx), utilizing first-principles calculations.

Materials
This The calculated pressure-volume relationships of m-C 3 N 4 and t-C 3 N 4 , together with diamond, c-BN, and other C 3 N 4 allotropes, are shown in Figure 2. The highest incompressibility along the c-axis is due to m-C 3 N 4 in the C 3 N 4 allotropes, while along the c-axis, m-C 3 N 4 yields the lowest incompressibility at pressures from 0 to 87 GPa; along the b-axis, m-C 3 N 4 yields the lowest incompressibility at pressures from 87 to 100 GPa. For the crystal structure, pseudocubic-C 3 N 4 has the greatest incompressibility in the C 3 N 4 allotropes discussed above, while m-C 3 N 4 has the weakest incompressibility. However, the incompressibility of t-C 3 N 4 is greater than that of c-BN and the incompressibility of m-C 3 N 4 is weaker than that of c-BN. The calculated pressure-volume relationships of m-C3N4 and t-C3N4, together with diamond, c-BN, and other C3N4 allotropes, are shown in Figure 2. The highest incompressibility along the c-axis is due to m-C3N4 in the C3N4 allotropes, while along the c-axis, m-C3N4 yields the lowest incompressibility at pressures from 0 to 87 GPa; along the b-axis, m-C3N4 yields the lowest incompressibility at pressures from 87 to 100 GPa. For the crystal structure, pseudocubic-C3N4 has the greatest incompressibility in the C3N4 allotropes discussed above, while m-C3N4 has the weakest incompressibility. However, the incompressibility of t-C3N4 is greater than that of c-BN and the incompressibility of m-C3N4 is weaker than that of c-BN.

Purple Cyan Olive
Lattice parameters ratio

Elastic Properties and Hardness
In an effort to assess the thermodynamic stability of two novel C3N4 allotropes, enthalpy change curves with pressure for various structures were calculated, as presented in Figure 3. The dashed line represents the enthalpy of the summary of diamond and α-N2. It can be clearly seen that g-C3N4 has the lowest minimum value of enthalpy, which is in good agreement with previous reports and supports the reliability of our calculations [43]. Pseudocubic-C3N4 has the greatest minimum value of enthalpy. The minimum value of total energy per formula unit of t-C3N4 is slightly larger than that of g-C3N4, α-C3N4, m-C3N4, and β-C3N4 but much smaller than those of pseudocubic-C3N4 and cubic-C3N4, indicating that t-C3N4 and m-C3N4 should be thermodynamically metastable phases.

Elastic Properties and Hardness
In an effort to assess the thermodynamic stability of two novel C 3 N 4 allotropes, enthalpy change curves with pressure for various structures were calculated, as presented in Figure 3. The dashed line represents the enthalpy of the summary of diamond and α-N 2 . It can be clearly seen that g-C 3 N 4 has the lowest minimum value of enthalpy, which is in good agreement with previous reports and supports the reliability of our calculations [43]. Pseudocubic-C 3 N 4 has the greatest minimum value of enthalpy. The minimum value of total energy per formula unit of t-C 3 N 4 is slightly larger than that of g-C 3 N 4 , α-C 3 N 4 , m-C 3 N 4 , and β-C 3 N 4 but much smaller than those of pseudocubic-C 3 N 4 and cubic-C 3 N 4 , indicating that t-C 3 N 4 and m-C 3 N 4 should be thermodynamically metastable phases. For monoclinic symmetry and tetragonal symmetry, there are different independent elastic constants. Monoclinic symmetry has thirty independent elastic constants (C11, C22, C33, C44, C55, C66, C12, C13, C23, C15, C25, C35 and C46), while tetragonal symmetry has six independent elastic constants (C11, C33, C44, C66, C12 and C13). The mechanical stability criteria of monoclinic symmetry are given by [44,45]:  For monoclinic symmetry and tetragonal symmetry, there are different independent elastic constants. Monoclinic symmetry has thirty independent elastic constants (C 11 , C 22 , C 33 , C 44 , C 55 , C 66 , C 12 , C 13 , C 23 , C 15 , C 25 , C 35 and C 46 ), while tetragonal symmetry has six independent elastic constants (C 11 , C 33 , C 44 , C 66 , C 12 and C 13 ). The mechanical stability criteria of monoclinic symmetry are given by [44,45]: The criteria for the mechanical stability of tetragonal symmetry are given by [44]: pC 11´C12 q ą 0 (10) The calculated elastic constants of α-C 3 N 4 , β-C 3 N 4 , t-C 3 N 4 , d-ZB-C 3 N 4 , pseudocubic-C 3 N 4 , cubic-C 3 N 4 and m-C 3 N 4 are listed in Table 2. Elastic constants under high pressure were also studied. The elastic constants under ambient pressure and high pressure of t-C 3 N 4 and m-C 3 N 4 satisfied the mechanical stability criteria of monoclinic symmetry and tetragonal symmetry. Namely, t-C 3 N 4 and m-C 3 N 4 are mechanically stable. To confirm the stability of t-C 3 N 4 and m-C 3 N 4 , their dynamical stabilities should also be studied under ambient pressure and high pressures. Thus, we calculated the phonon spectra for m-C 3 N 4 and t-C 3 N 4 at 0 and 100 GPa, as shown in Figure 4. No imaginary frequencies are observed throughout the whole Brillouin zone, signaling dynamically the stabilities of m-C 3 N 4 and t-C 3 N 4 . The calculated elastic modulus of α- Table 3. Bulk modulus B and shear modulus G were calculated by using the Voigt-Reuss-Hill approximation [46][47][48]. The Voigt and Reuss approximation of monoclinic symmetry is calculated using the following equations [44]: a " pC 33 C 55´C Frequency (TH z)  Table 3. The bulk modulus of t-C3N4 is 375 GPa, which is slightly larger than that of c-BN, while the bulk modulus of m-C3N4 is slightly smaller than that of c-BN. The hardness of m-C3N4 is only 37 GPa, which is approximately half of that of α-C3N4, β-C3N4, d-ZB-C3N4, pseudocubic-C3N4 and t-C3N4. The reason for this phenomenon is that the mechanical properties of m-C3N4 are not excellent compared with the other C3N4 allotropes The Voigt and Reuss approximation of tetragonal symmetry is calculated using the following equations [44]: The Hill approximation of monoclinic symmetry and tetragonal symmetry is calculated using the following equation: Young's modulus and Poisson's ratio can be calculated using the following formulas, respectively: [49]. The relationships between elastic constants and pressures are shown in Figure 5a 1350  The elastic anisotropy of a solid is closely related to the possibility of inducing microcracks in materials and can be expressed by the universal anisotropic index (A U ) [58]. The universal anisotropic index is defined as A U = 5GV/GR + BV/BR-6. The calculated results of universal anisotropic index are also shown in Table 3. The universal anisotropic index of α-C3N4 is only 0.073, which is approximately one-third that of β-C3N4, approximately one-sixth that of t-C3N4, and approximately one-sixteenth that of m-C3N4. Namely, α-C3N4 and m-C3N4 exhibit the smallest and largest elastic    Table 3. The calculated elastic modulus (GPa), B/G, hardness Hv (GPa) and the universal anisotropic index of m-C 3 N 4 , t-C 3 N 4 , d-ZB-C 3 N 4 , cubic-C 3 N 4 , pseudocubic-C 3 N 4 and cubic-BN. The dependence of bulk modulus, shear modulus and Young's modulus on pressure of m-C 3 N 4 and t-C 3 N 4 is 3.49, 0.59, and 2.15 and 3.41, 1.59, and 4.35, respectively. Young's modulus of t-C 3 N 4 increases faster than other elastic modulus, while the increase in the shear modulus of m-C 3 N 4 is the slowest. At ambient pressure, the bulk modulus of α-C 3 N 4 , β-C 3 N 4 , m-C 3 N 4 and t-C 3 N 4 are 387 GPa, 406 GPa, 327 GPa and 375 GPa, respectively. The calculated hardness of α-C 3 N 4 , β-C 3 N 4 , m-C 3 N 4 , t-C 3 N 4 , Cubic-C 3 N 4 , d-ZB-C 3 N 4 , Pseudocubic-C 3 N 4 and c-BN are shown in Table 3. The bulk modulus of t-C 3 N 4 is 375 GPa, which is slightly larger than that of c-BN, while the bulk modulus of m-C 3 N 4 is slightly smaller than that of c-BN. The hardness of m-C 3 N 4 is only 37 GPa, which is approximately half of that of α-C 3 N 4 , β-C 3 N 4 , d-ZB-C 3 N 4 , pseudocubic-C 3 N 4 and t-C 3 N 4 . The reason for this phenomenon is that the mechanical properties of m-C 3 N 4 are not excellent compared with the other C 3 N 4 allotropes and the bulk modulus, shear modulus and Young's modulus are all smaller than those of other C 3 N 4 allotropes.

Materials
In materials science, ductility is a solid material's ability to deform under tensile stress. If a material is brittle, when subjected to stress, it will break without significant deformation (strain). Additionally, these material properties are dependent on pressure. Pugh [56] proposed a simple relationship to judge the plastic properties of materials based on their elastic modulus, i.e., B/G. If the ratio B/G is larger than 1.75, a material exhibits the ductile property; otherwise, the material is brittle. Moreover, Poisson's ratio v is another criterion for judging the plastic properties of materials [57]. A larger v value (v > 0.26) for a material indicates ductility, while a smaller v value (v < 0.26) usually denotes brittleness. At ambient pressure, the ratio B/G and v of α-C 3 N 4 , β-C 3 N 4 , d-ZB-C 3 N 4 , Cubic-C 3 N 4 , Pseudocubic-C 3 N 4 , m-C 3 N 4 and t-C 3 N 4 are as listed in Table 3. The ratio B/G and v of four C 3 N 4 allotropes are all less than 1.75 and 0.26, respectively, which indicates that the four C 3 N 4 allotropes are all brittle. t-C 3 N 4 has the most brittleness, while β-C 3 N 4 has the least brittleness. The pressure dependence of B/G and Poisson's ratio v are shown in Figure 5c,d, respectively. In Figure 5c,d, the B/G and v of m-C 3 N 4 and t-C 3 N 4 increase with increasing pressure. m-C 3 N 4 is found to change from brittle to ductile at 71 GPa, while t-C 3 N 4 does not change from brittle to ductile in this pressure range.
The elastic anisotropy of a solid is closely related to the possibility of inducing microcracks in materials and can be expressed by the universal anisotropic index (A U ) [58]. The universal anisotropic index is defined as A U = 5G V /G R + B V /B R -6. The calculated results of universal anisotropic index are also shown in Table 3. The universal anisotropic index of α-C 3 N 4 is only 0.073, which is approximately one-third that of β-C 3 N 4 , approximately one-sixth that of t-C 3 N 4 , and approximately one-sixteenth that of m-C 3 N 4 . Namely, α-C 3 N 4 and m-C 3 N 4 exhibit the smallest and largest elastic anisotropy in A U , respectively. The pressure dependence of the universal anisotropic index is shown in Figure 5e To analyze the anisotropy of m-C 3 N 4 and t-C 3 N 4 more systematically, we will investigate the anisotropy of m-C 3 N 4 and t-C 3 N 4 for Poisson's ratio, the shear modulus and Young's modulus by utilizing the ELAM codes [20,59]. The two-dimensional representations of Poisson's ratio in the xy plane, xz plane and yz plane for m-C 3 N 4 and t-C 3 N 4 are shown in Figure 6    The 2D representations of the shear modulus in the xy plane, xz plane and yz plane for m-C 3 N 4 and t-C 3 N 4 are illustrated in Figure 7. For m-C 3 N 4 in Figure 7a-c, the maximum value of the shear modulus occurs in the deviation from the x axis or y axis of approximately 45 degrees in the xy plane or yz plane, respectively. The maximum value of the shear modulus occurs in the deviation from the x axis or z axis of approximately 15 degrees in the xz plane. Moreover, the minimum value of the shear modulus occurs along the y axis in the xy plane and yz plane, respectively. The maximum and the minimum values of m-C 3 N 4 are 163 GPa and 455 GPa at ambient pressure, respectively, and the ratio G max /G min = 2.79. At the same time, the maximum and minimum values of m-C 3 N 4 are 153 GPa (110 GPa) and 611 GPa (738 GPa), respectively, at 50 GPa (100 GPa); the ratio G max /G min = 3.99 at p = 50 GPa, and the ratio G max /G min = 6.71 at p = 100 GPa. The anisotropy of m-C 3 N 4 increases with increasing pressure. The average shear modulus of m-C 3 N 4 is 266 GPa, 312 GPa and 328 GPa, respectively. For t-C 3 N 4 in Figure 7d-f, the maximum value of the shear modulus for t-C 3 N 4 in the xy plane, xz plane and yz plane appears along the coordinate axis, while the minimum value of the shear modulus for t-C 3 N 4 in the xy plane, xz plane and yz plane appears in the deviation from the coordinate axis of approximately 45 degrees. The maximum and minimum values of the shear modulus for t-C 3 N 4 are 245 GPa, 291 GPa, 323 GPa and 428 GPa, 559 GPa, 669 GPa at 0 GPa, 50 GPa, 100 GPa, respectively. The ratio G max /G min = 1.75 at p = 0 GPa, the ratio G max /G min = 1.92 at p = 50 GPa, and the ratio G max /G min = 2.07 at p = 100 GPa. It is clear that the ratio G max /G min for t-C 3 N 4 is much smaller than that for m-C 3 N 4 . In other words, m-C 3 N 4 exhibits greater anisotropy than t-C 3 N 4 . This agrees well with our previous prediction of anisotropy with respect to the universal anisotropic index and Poisson's ratio.
As a valid method to describe the elastic anisotropic behavior of a crystal completely, the 3D surface constructions of the directional dependences of reciprocals of Young's modulus are practically useful. The results are shown in Figure 8 for Young's modulus. For an isotropic system, the 3D directional dependence would exhibit a spherical shape, while the deviation degree from the spherical shape reflects the content of anisotropy [60]. In Figure 8a,c, the 3D shape of Young's modulus shows that m-C 3 N 4 exhibits greater anisotropy than t-C 3 N 4 . As the pressure increases, the anisotropy of Young's modulus for m-C 3 N 4 and t-C 3 N 4 increases, but m-C 3 N 4 still exhibits greater anisotropy than t-C 3 N 4 . To analyze the anisotropy of m-C 3 N 4 and t-C 3 N 4 in detail, the 2D representations of Young's modulus in the xy plane, xz plane and yz plane for m-C 3 N 4 and t-C 3 N 4 are depicted in Figure 9. From Figure 9, it is clear that m-C 3 N 4 has a larger anisotropy and that the anisotropy will become larger with increasing pressure.
Young's modulus of t-C 3 N 4 has the same value in different planes, while that of m-C 3 N 4 has different values in different planes. For example, at ambient pressure, the maximum and minimum values of Young's modulus for t-C 3 N 4 are 928 GPa and 612 GPa in the xy plane, xz plane and yz plane, while at p = 100 GPa, they are 1510 GPa and 864 GPa, respectively. However, the maximum value of Young's modulus is 996 GPa in the xy plane and yz plane for m-C 3 N 4 , but in the xz plane, it is 995 GPa, and the minimum value is always 476 GPa. At 100 GPa, the difference reaches a larger degree; in the xy plane and yz plane, the maximum value of Young's modulus for m-C 3 N 4 is 1638 GPa, while the maximum value is 1634 GPa. This also proves that m-C 3 N 4 has a larger anisotropy from the other side. As a valid method to describe the elastic anisotropic behavior of a crystal completely, the 3D surface constructions of the directional dependences of reciprocals of Young's modulus are plane, while at p = 100 GPa, they are 1510 GPa and 864 GPa, respectively. However, the maximum value of Young's modulus is 996 GPa in the xy plane and yz plane for m-C3N4, but in the xz plane, it is 995 GPa, and the minimum value is always 476 GPa. At 100 GPa, the difference reaches a larger degree; in the xy plane and yz plane, the maximum value of Young's modulus for m-C3N4 is 1638 GPa, while the maximum value is 1634 GPa. This also proves that m-C3N4 has a larger anisotropy from the other side.

Electronic Structures
Band theory is one of the most stringent tests of the physics of semiconductors. For example, silicon, calcite and copper all contain similar densities of electrons, but they have different physical properties, all inexplicable without quantum mechanics [61]. Thus, it is necessary to understand the band structure and density of states. The band structures and density of states of m-C 3 N 4 and m-C 3 N 4 at different pressures are shown in Figure 10. The band structure calculations show that m-C 3 N 4 is a quasi-direct band gap semiconductor, with a band gap of 4.52 eV (see Figure 10a), and at 100 GPa, m-C 3 N 4 remains a quasi-direct band gap semiconductor, with a band gap of 5.68 eV.   energy of CBM is 14.73 and 18.82 eV, respectively. The VBM of t-C3N4 is located at the point along the M and Γ directions; the energy is 10.52 and 14.03 eV, respectively. Moreover, the energy of the Γ point near the Fermi level of t-C3N4 is 10.51 eV and 14.01 eV at 0 and 100 GPa, respectively. Thus, t-C3N4 is a quasi-direct band gap semiconductor. Interestingly, the band gaps of m-C3N4 and t-C3N4 both increase with increasing pressure. At 100 GPa, m-C3N4 increases by 25.61%, and t-C3N4 increases by 13.66% compared with that at 0 GPa.

Conclusions
In conclusion, we have predicted two novel C 3 N 4 allotropes, i.e., m-C 3 N 4 and t-C 3 N 4 , with space groups Cm and I-42m, which are both mechanically and dynamically stable up to at least 100 GPa. The bulk modulus of t-C 3 N 4 is 375 GPa, which is slightly larger than that of c-BN, while the bulk modulus of m-C 3 N 4 is slightly smaller than that of c-BN. The hardness of t-C 3 N 4 is larger than that of c-BN, thereby making it a superhard material with potential technological and industrial applications. The ratio B/G and v of the two novel C 3 N 4 phases are both less than 1.75 and 0.26, respectively, which indicates that the two novel C 3 N 4 allotropes are both brittle. The B/G and v of m-C 3 N 4 and t-C 3 N 4 increase with increasing pressure. m-C 3 N 4 is found to change from being brittle to ductile at 71 GPa, while t-C 3 N 4 does not change from being brittle to ductile in this pressure range. The elastic anisotropy calculations show that m-C 3 N 4 and t-C 3 N 4 both exhibit large anisotropy with respect to Poisson's ratio, the shear modulus and Young's modulus and universal anisotropic index. The band structure calculations show that m-C 3 N 4 and t-C 3 N 4 are a quasi-direct-band-gap semiconductor and a quasi-direct-band-gap semiconductor, respectively. Moreover, the band gaps of m-C 3 N 4 and t-C 3 N 4 continue to be a quasi-direct band-gap and quasi-direct band gap at 100 GPa, respectively. The band gaps of m-C 3 N 4 and t-C 3 N 4 are 4.522 and 4.210 eV, respectively, and these materials are both wide-band-gap semiconductors. Due to their quasi-direct band gaps, they are attractive for luminescent device applications.