Theoretical Investigations of the Hexagonal Germanium Carbonitride

The structural, mechanical, elastic anisotropic, and electronic properties of hexagonal germanium carbonitride (h-GeCN) are systematically investigated using the first-principle calculations method with the ultrasoft pseudopotential scheme in the frame of generalized gradient approximation in the present work. The h-GeCN are mechanically and dynamically stable, as proved by the elastic constants and phonon spectra, respectively. The h-GeCN is brittle because the ratio B/G and Poisson’s ratio v of the h-GeCN are less than 1.75 and 0.26, respectively. For h-GeCN, from brittleness to ductility, the transformation pressures are 5.56 GPa and 5.63 GPa for B/G and Poisson’s ratio v, respectively. The h-GeCN exhibits the greater elastic anisotropy in Young’s modulus and the sound velocities. In addition, the calculated band structure of h-GeCN reveals that there is no band gap for h-GeCN with the HSE06 hybrid functional, so the h-GeCN is metallic.


Introduction
Ternary compounds have attracted more and more attention, such as B-C-N [1][2][3], B-C-O [4][5][6][7] superhard materials, and Si-Ge-N [8,9], Si-C-N [10][11][12][13][14], Ge-C-N [15,16], and so on. Si-Ge-N is an alloy of silicon nitride and germanium nitride. The structural, elastic anisotropic, and electronic properties of m-Si 2 GeN 4 and m-SiGe 2 N 4 were investigated using density functional theory calculations by Ma et al. [8], where m-Si 2 GeN 4 and m-SiGe 2 N 4 are alloys of m-Si 3 N 4 and m-Ge 3 N 4 . They found that the m-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) series exhibit larger anisotropy and that the anisotropy of m-SiGe 2 N 4 is largest among the m-Si x Ge 3−x N 4 (x = 0, 1, 2, 3). The calculated band structures show that both m-Si 2 GeN 4 and m-SiGe 2 N 4 are direct semiconductors with band gaps of 4.76 eV and 4.81 eV, respectively. Very recently, the structural, mechanical, anisotropic, electronic, and thermal properties of t-Si 2 GeN 4 and t-SiGe 2 N 4 in the tetragonal phase were systematically investigated by Han et al. [9]. They found that both t-Si 2 GeN 4 and t-SiGe 2 N 4 demonstrate brittleness, and that t-Si 2 GeN 4 and t-SiGe 2 N 4 exhibit larger elastic anisotropy than that of c-Si 2 GeN 4 and c-SiGe 2 N 4 characterized by Young's modulus, Poisson's ratio, the percentage of elastic anisotropy for shear modulus A G , the percentage of elastic anisotropy for bulk modulus A B , and the universal anisotropic index A U . The electronic structures of t-Si 2 GeN 4 and t-SiGe 2 N 4 are both wide-bandgap semiconductor materials, with band gaps of 3.94 eV and 3.83 eV using the HSE06 hybrid functional, respectively. In addition, the effects of temperature and pressure on the Debye temperature, thermal expansion coefficient, heat capacity, and Grüneisen parameters were discussed in detail utilizing the quasi-harmonic Debye model. In addition, other III-V group compounds have also been studied extensively, including previous values [15] by 0.07% and 0.13%, while lattice parameters a and c of the LDA (local density approximation) deviate from the corresponding previous values [15] by 0.26% and 1.47%; that is to say, the values of the GGA deviate from other previous values less than do those of the LDA. Therefore, in this paper, all the results are based on the GGA. In addition, the lattice parameters are a = 3.621 Å in this work, and a = 3.622 Å [2] with GGA for c-BN, while the lattice parameters are a = 3.582 Å in this work, and a = 3.576 Å [2] with LDA for c-BN; the experimental value of c-BN is 3.620 Å, so the result of GGA is very close to the experiment value for c-BN. Therefore, in this work, all the results are based on the crystal structure from GGA. The crystal structure of the hexagonal representation and rhombohedral representation for h-GeCN are shown in Figure 1. The red, black, and blue spheres represent Ge, C, and N atoms, respectively. For h-GeCN, the bond lengths of the C-N, C-Ge, and N-Ge bonds are 1.362 Å, 2.091 Å, and 2.180 Å, respectively. The C-N and C-Ge bond lengths are slightly greater than the N-Si (1.895 Å) and C-Si (1.875 Å) bond lengths in h-SiCN, while the C-N bond length is slightly smaller than that of C-N (1.373 Å) in h-SiCN. Compared with t-GeCN, the C-N bond length in h-GeCN is slightly smaller than that (1.445 Å) in t-GeCN, while the C-Ge and N-Ge bond lengths in h-GeCN are slightly greater than the C-Ge (2.015 Å) and N-Ge (1.884 Å) bond lengths in t-GeCN. In addition, there are C-C bond lengths (1.619 Å) in t-GeCN. The lattice constants and conventional cell volumes of h-GeCN and t-GeCN are shown in Figure 2. From Figure 2a, the compression along the lattice constants' a-axis and c-axis for h-GeCN is slightly larger than that of t-GeCN when the pressure increases. In addition, it is clear that the compression of h-GeCN is slightly larger than that of t-GeCN; that is to say, the bulk modulus of t-GeCN is slightly larger than that of h-GeCN. 1.47%; that is to say, the values of the GGA deviate from other previous values less than do those of the LDA. Therefore, in this paper, all the results are based on the GGA. In addition, the lattice parameters are a = 3.621 Å in this work, and a = 3.622 Å [2] with GGA for c-BN, while the lattice parameters are a = 3.582 Å in this work, and a = 3.576 Å [2] with LDA for c-BN; the experimental value of c-BN is 3.620 Å, so the result of GGA is very close to the experiment value for c-BN. Therefore, in this work, all the results are based on the crystal structure from GGA. The crystal structure of the hexagonal representation and rhombohedral representation for h-GeCN are shown in Figure 1. The red, black, and blue spheres represent Ge, C, and N atoms, respectively.  Figure 2. From Figure 2a, the compression along the lattice constants' a-axis and c-axis for h-GeCN is slightly larger than that of t-GeCN when the pressure increases. In addition, it is clear that the compression of h-GeCN is slightly larger than that of t-GeCN; that is to say, the bulk modulus of t-GeCN is slightly larger than that of h-GeCN.  [15]. GGA, generalized gradient approximation. LDA, local density approximation.

Stability and Mechanical Properties
The stability of h-GeCN can be characterized by phonon spectra and Born stability conditions. The phonon spectra of h-GeCN are displayed in Figure 3. The phonon spectra show that all the lattice vibrations in the Brillouin region are positive, indicating that the h-GeCN is dynamically stable. The elastic constants of h-GeCN under different pressures are listed in Table 2. The criteria for mechanical stability of hexagonal symmetry are [50] C44 > 0, C 2 11 > C 2 12 , and (C11 + 2C12)C33 > 2 C 2 12 . From Table 2, we note that all the elastic constants of h-GeCN under different pressures satisfy the Born stability conditions of hexagonal symmetry.  The elastic moduli of h-GeCN under different pressures are listed in Table 3. According to our previous prediction, the bulk modulus of t-GeCN (183 GPa) is indeed larger than that of h-GeCN (130 GPa). Similarly, the shear modulus and the Young's modulus are the same as the bulk modulus. However, the bulk modulus, shear modulus, and Young's modulus of h-GeCN are all slightly larger than those of m-Ge3N4 [8]. A kind of material showing brittleness or ductility is usually characterized

Stability and Mechanical Properties
The stability of h-GeCN can be characterized by phonon spectra and Born stability conditions. The phonon spectra of h-GeCN are displayed in Figure 3. The phonon spectra show that all the lattice vibrations in the Brillouin region are positive, indicating that the h-GeCN is dynamically stable. The elastic constants of h-GeCN under different pressures are listed in Table 2. The criteria for mechanical stability of hexagonal symmetry are [50] C 44 > 0, C 2 11 > C 2 12 , and (C 11 + 2C 12 )C 33 > 2 C 2 12 . From Table 2, we note that all the elastic constants of h-GeCN under different pressures satisfy the Born stability conditions of hexagonal symmetry.

Stability and Mechanical Properties
The stability of h-GeCN can be characterized by phonon spectra and Born stability conditions. The phonon spectra of h-GeCN are displayed in Figure 3. The phonon spectra show that all the lattice vibrations in the Brillouin region are positive, indicating that the h-GeCN is dynamically stable. The elastic constants of h-GeCN under different pressures are listed in Table 2. The criteria for mechanical stability of hexagonal symmetry are [50] C44 > 0, C 2 11 > C 2 12 , and (C11 + 2C12)C33 > 2 C 2 12 . From Table 2, we note that all the elastic constants of h-GeCN under different pressures satisfy the Born stability conditions of hexagonal symmetry.  The elastic moduli of h-GeCN under different pressures are listed in Table 3. According to our previous prediction, the bulk modulus of t-GeCN (183 GPa) is indeed larger than that of h-GeCN (130 GPa). Similarly, the shear modulus and the Young's modulus are the same as the bulk modulus. However, the bulk modulus, shear modulus, and Young's modulus of h-GeCN are all slightly larger than those of m-Ge3N4 [8]. A kind of material showing brittleness or ductility is usually characterized  The elastic moduli of h-GeCN under different pressures are listed in Table 3. According to our previous prediction, the bulk modulus of t-GeCN (183 GPa) is indeed larger than that of h-GeCN (130 GPa). Similarly, the shear modulus and the Young's modulus are the same as the bulk modulus. However, the bulk modulus, shear modulus, and Young's modulus of h-GeCN are all slightly larger than those of m-Ge 3 N 4 [8]. A kind of material showing brittleness or ductility is usually characterized by two physical quantities: B/G and Poisson's ratio v. A larger B/G [51] value (B/G > 1.75) and a larger v (v > 0.26) [52] for a solid represent a ductile state, while a smaller B/G value and a smaller v usually mean that the solid is brittle. The B/G and Poisson's ratio v of h-GeCN are also presented in Table 3. From Table 3, with increasing pressure, both B/G and Poisson's ratio v increase. At ambient pressure, B/G = 1.69 and v = 0.25 of h-GeCN, indicating that the h-GeCN exhibits brittleness. As the pressure increases, the h-GeCN changes from brittle to ductile. From brittleness to ductility, we note that the transformation pressures of h-GeCN are 5.56 GPa and 5.63 GPa for B/G and Poisson's ratio v, respectively. The Debye temperature (Θ D ) is a fundamental physical property and correlates with many physical properties of solids, such as specific heat and the thermal coefficient [53].
Avogadro's number; n is the number of atoms in the molecule; M is molecular weight; ρ is the density; and v m is the mean sound velocity, v m = [(2/v 3 s + 1/v 3 p )/3] −1/3 . The v l and v t are the longitudinal and transverse sound velocities, respectively, which can be obtained from Navier's equation [54]:  Table 4. At ambient pressure, the Debye temperature of h-GeCN is 506 K-smaller than that of t-GeCN (756 K). The Debye temperature of h-GeCN increases with increasing pressure except for the situation under 20 GPa. The changes of almost all of the sound velocities for h-GeCN are consistent with the changes of the Debye temperature, except for v p . The sound velocity v p increases with increasing pressure until the pressure increases to 20 GPa. The Debye temperature of h-GeCN shows different behavior at 20 GPa because the elastic constants and elastic moduli of h-GeCN decreased quickly from 15 to 20 GPa than from 10 to 15 GPa. Therefore, the Debye temperature of h-GeCN shows different behaviors at 20 GPa.

Elastic Anisotropy Properties
The sound velocities are determined by the symmetry of the crystal and the propagation direction. The pure transverse and longitudinal modes can only be found in [100] and [001] directions in a hexagonal crystal; the sound propagating modes in other directions are the quasi-transverse or quasi-longitudinal waves. In the primary directions, the sound velocities in a hexagonal crystal can be expressed by where v t1 and v t2 refer to the first transverse mode and the second transverse mode, respectively. The calculated sound velocities along the primary directions are listed in Table 5 0  3827  4682  1992  6955  1992  1992  5  4132  5074  2340  7314  2340  2340  10  4449  5404  2402  7561  2402  2402  15  4512  5592  2427  7754  2427  2427  20  4338  5675  2236  7925  2236  2236 The Young's modulus of the h-GeCN also exhibits anisotropy. The directional dependence of Young's modulus for h-GeCN and two-dimensional (2D) representations of Young's modulus in the (001), (010), (100), and (111) planes for h-GeCN are illustrated in Figure 4a,b, respectively. From Figure 4a, the shape of the three-dimensional representations of the Young's modulus for h-GeCN is similar to a gyroscope with the middle width and the two ends sharp. The two-dimensional representations of Young's modulus for h-GeCN are unfolding figures that cut along the (001), (010), (100), and (111) planes, where black, red, blue, and cyan lines represent the (001), (010), (100), and (111) planes, respectively. The figure obtained along the (001) plane is a circle, and the two figures along the (010) and (100) planes are the same, for a gyroscope plane shape, while the (111) plane is an irregular figure. What is more interesting is that the maximum value (509 GPa) of Young's modulus for h-GeCN occurred at the Z-axis, but the minimum value (130 GPa) of the Young's modulus for h-GeCN occurred at θ = 0.87, ϕ = 5.08 (more details see [55][56][57]). Regardless of the three-dimensional figure of the Young's modulus and the ratio of the maximum to the minimum (E max /E min = 509/130 = 3.92), it is more than that of t-GeCN (E max /E min = 2.49) [15], so the h-GeCN has larger anisotropy. In the (010) and (100) planes, the ratio E max /E min = 509/130 = 3.92; this is the largest ratio of elastic anisotropy in the Young's modulus among these planes. In the (001) plane, the maximal and minimal values of Young's modulus are both 206 GPa, so the ratio of Young's modulus in the (001) plane is E max /E min = 206/206 = 1.00; therefore, the Young's modulus exhibits isotropy in the (001) plane, and it is the smallest elastic anisotropy in Young's modulus among these planes.

Electronic Properties
It is well known that the electronic structure determines the fundamental physical and chemical properties of materials [35].

Conclusions
In this work, the structural, elastic, elastic anisotropic, and electronic properties of h-GeCN in the R3m space group were investigated utilizing first-principle calculations. The mechanical and dynamical stability of h-GeCN were proved by elastic constants and phonon spectra.

Electronic Properties
It is well known that the electronic structure determines the fundamental physical and chemical properties of materials [35]. The electronic structures of h-GeCN (using the rhombohedral cell) under 0 GPa and 20 GPa are shown in Figure 5. From Figure 5, we can see that h-GeCN exhibits metallicity.

Electronic Properties
It is well known that the electronic structure determines the fundamental physical and chemical properties of materials [35]. The electronic structures of h-GeCN (using the rhombohedral cell) under 0 GPa and 20 GPa are shown in Figure 5. From Figure 5, we can see that h-GeCN exhibits metallicity.

Conclusions
In this work, the structural, elastic, elastic anisotropic, and electronic properties of h-GeCN in the R3m space group were investigated utilizing first-principle calculations. The mechanical and dynamical stability of h-GeCN were proved by elastic constants and phonon spectra. The ratio B/G and Poisson's ratio v of the h-GeCN are less than 1.75 and 0.26, respectively, both of which indicate that the h-GeCN is brittle. For h-GeCN, from brittleness to ductility, the transformation pressures are 5.56 GPa and 5.63 GPa for B/G and Poisson's ratio v, respectively. At ambient pressure, the Debye temperature of h-GeCN is 506 K-smaller than that of t-GeCN. The calculated Young's modulus along all directions and in the primary planes, and the sound velocities along the primary directions of h-GeCN, exhibit greater elastic anisotropy. A three-dimensional figure of the Young's modulus

Conclusions
In this work, the structural, elastic, elastic anisotropic, and electronic properties of h-GeCN in the R3m space group were investigated utilizing first-principle calculations. The mechanical and dynamical stability of h-GeCN were proved by elastic constants and phonon spectra. The ratio B/G and Poisson's ratio v of the h-GeCN are less than 1.75 and 0.26, respectively, both of which indicate that the h-GeCN is brittle. For h-GeCN, from brittleness to ductility, the transformation pressures are 5.56 GPa and 5.63 GPa for B/G and Poisson's ratio v, respectively. At ambient pressure, the Debye temperature of h-GeCN is 506 K-smaller than that of t-GeCN. The calculated Young's modulus along all directions and in the primary planes, and the sound velocities along the primary directions of h-GeCN, exhibit greater elastic anisotropy. A three-dimensional figure of the Young's modulus was presented, and the ratio of the maximum to the minimum (E max /E min = 509/130 = 3.92) is greater than that of t-GeCN (E max /E min = 2.49). In addition, the band structure reveals that the h-GeCN is metallic.