Mechanical, Anisotropic, and Electronic Properties of XN (X = C, Si, Ge): Theoretical Investigations

The structural, mechanical, elastic anisotropic, and electronic properties of Pbca-XN (X = C, Si, Ge) are investigated in this work using the Perdew–Burke–Ernzerhof (PBE) functional, Perdew–Burke–Ernzerhof for solids (PBEsol) functional, and Ceperly and Alder, parameterized by Perdew and Zunger (CA–PZ) functional in the framework of density functional theory. The achieved results for the lattice parameters and band gap of Pbca-CN with the PBE functional in this research are in good accordance with other theoretical results. The band structures of Pbca-XN (X = C, Si, Ge) show that Pbca-SiN and Pbca-GeN are both direct band gap semiconductor materials with a band gap of 3.39 eV and 2.22 eV, respectively. Pbca-XN (X = C, Si, Ge) exhibits varying degrees of mechanical anisotropic properties with respect to the Poisson’s ratio, bulk modulus, shear modulus, Young’s modulus, and universal anisotropic index. The (001) plane and (010) plane of Pbca-CN/SiN/GeN both exhibit greater elastic anisotropy in the bulk modulus and Young’s modulus than the (100) plane.


Introduction
In the last few decades, nitride-based ceramics such as silicon nitride (Si 3 N 4 ) have attracted increasing attention from researchers in the ceramics, mechanical, and aerospace industries, as well as in fields such as solar cells, as they have a wide range of applications [1][2][3][4][5][6][7][8]. This is due to their significant chemical stability, good compression resistance, corrosion resistance, high hardness, good mechanical properties, and good optical performance characteristics. Other stoichiometries like Si 3 N 4 , SiN 2 , and Si 2 N 2 (NH) have also been proposed to exist [9][10][11][12][13][14]. Silicon and germanium-based compounds and alloys such as the Si/Ge-group-III and Si/Ge-group-V compounds have been widely investigated [15][16][17].
CxNy with different stoichiometries is often used as a potential superhard material [18][19][20][21]. Li et al. [18] have reported a novel body-centered tetragonal CN 2 named bct-CN 2 , using the newly-developed particle swarm optimization algorithm for crystal structure prediction. They found that the hardness of bct-CN 2 is 77.4 GPa, and it is an indirect wide gap semiconductor material with a band gap of 3.6 eV. Wang et al. [19] suggested a new carbon nitride phase consisting of sp 3 hybridized bonds, with cubic symmetry and a P2 1 3 space group (i.e., cg-CN). Unlike most of the other superhard materials that are insulators or semiconductors, it is a metallic compound, and its Vickers hardness is 82.56 GPa. They found that cg-CN is the most favorable stable crystal structure, with carbon nitride with 1:1 stoichiometry. Using the particle swarm optimization technique, Wei et al. [20] proposed a cubic superhard phase of C 3 N (c-C 3 N) with a Vickers hardness of 65 GPa, which is more energetically favorable than the recently proposed o-C 3 N [21]. o-C 3 N was proposed by Hao et al. [21]. It has a C222 1 phase, and its Vickers hardness is 76 GPa.

Structural Properties
In the newly-formed (Pbca phase) solid, all the nitrogen atoms have sp 2 hybridizations and all the carbon/silicon/germanium atoms have sp 3 hybridization with their nearest neighboring N and C/Si/Ge atoms. The crystal structures of CN/SiN/GeN in Pbca phase are shown in Figure 1a. The C/Si/Ge atoms and N atoms consist of zigzag six-membered rings and eight-membered rings. The C/Si/Ge and N atoms are located at Wyckoff 8c (0. 1396 Figure 1b,c, respectively. The eight-membered rings are normal to the [001] direction in the structure of Pbca-CN/SiN/GeN, and the six-membered rings are normal to the [010] direction. The optimal lattice parameters of Pbca-CN/SiN/GeN, together with the previous results [27,39] of Pbca-CN are listed in Table 1. The optimized lattice parameters are a = 5.504 Å, b = 4.395 Å, and c = 4.041Å, which are in excellent agreement with [27,39]. In addition, taking into account the van der Waals forces, we also calculated the lattice parameters of Pbca-CN/SiN/GeN and diamond, c-BN using the dispersion-corrected Perdew-Burke-Ernzerhof (PBE + D) [40]. For diamond and c-BN, the theoretical results obtained by the GGA-PBE level (diamond: 3.566 Å for PBE level, 3.526 Å for CA-PZ [41], experimental value 3.567 Å [42]; c-BN: 3.626 Å for PBE level, 3.569 Å for CA-PZ [43], experimental value 3.620 Å [44]) are closer to the experimental values; the obtained results of c-BN and diamond using PBE + D are not much different from those obtained by PBE functional compared to corresponding experimental values, so the results obtained by the GGA-PBE level are all used in our paper. The lattice parameters of Pbca-XN with X changing from C to Ge are illustrated in Figure 2a. It is clear that the lattice parameters of Pbca-XN increase with X changing from C to Ge. From CN to SiN, the lattice parameters increase 31.4%, 21.52%, and 20.7% for a, b, and c of SiN compared to CN, while the lattice parameters increase 8.25%, 5.69%, and 5.84% for a, b, and c of GeN compared to SiN, respectively. This is because the average bond length of Si-N (1.751 Å) is much greater than that of the C-N bond (1.452 Å), and the average bond length of Ge-N (1.871 Å) is slightly longer than that of the Si-N bond.

Mechanical Properties
The calculated elastic constants and elastic moduli of CN/SiN/GeN in the Pbca phase are listed in Table 2. The calculated elastic constants and elastic modulus of Pbca-CN are excellent agreement with the previous report [26]. For an orthorhombic phase, the criteria of mechanical stability are [45]: where the Cij is elastic constant of the material. The mechanical stability of a phase can be confirmed by using the elastic constants. The SiN/GeN in the Pbca phase both satisfy the above mechanical stability criteria. The SiN/GeN in the Pbca phase show mechanical stability under ambient pressure. The phonon dispersion curve can show dynamic stability; the phonon dispersion curves of SiN/GeN in the Pbca phase are illustrated in Figure 3. There is no imaginary frequency in the Brillouin zone, which means SiN/GeN in the Pbca phase can be dynamically stable under ambient pressure. The elastic moduli of Pbca-XN with X changing from C to Ge are illustrated in Figure 2b. It is clear that the elastic moduli of Pbca-XN decrease with X changing from C to Ge. The elastic constants and elastic moduli of other SixNy compounds [22,46,47] are also listed in Table 2. The bulk modulus B of Pbca-SiN is slightly smaller than that of SiN2, o-Si3N4, and t-Si3N4, while it is slightly larger than Si3N2 and t-Si3N4. The shear modulus G and Young's modulus E of Pbca-SiN are similar to the bulk modulus of Pbca-SiN. For Pbca-GeN, its bulk modulus is as large as that of GeN2. However, its shear modulus and Young's modulus are slightly smaller than that of GeN2.
Brittleness and ductility of materials are important properties in crystal physics and engineering sciences. Pugh [48] proposed the ratio of bulk to shear modulus (B/G) as an indication of ductile verses brittle characters. If B/G > 1.75, the material is characterized by a ductile manner; otherwise, the material has a brittle character. The Poisson's ratio v is consistent with B/G, but refers to brittle compounds, usually with a small v (less than 0.26) [49]. The B/G ratio of Pbca-CN/SiN/GeN is 1.12 (1.11 [26]), 1.63, and 1.70; it is revealed that Pbca-CN/SiN/GeN are all brittle materials, and Pbca-CN has the most brittleness. For Poisson's ratio v, we obtained the same conclusion.

Mechanical Properties
The calculated elastic constants and elastic moduli of CN/SiN/GeN in the Pbca phase are listed in Table 2. The calculated elastic constants and elastic modulus of Pbca-CN are excellent agreement with the previous report [26]. For an orthorhombic phase, the criteria of mechanical stability are [45]: C ii > 0, i = 1-6; C 11 C 22 − C 2 12 > 0; C 11 C 22 C 33 + 2C 12 C 13 C 23 − C 11 C 2 23 − C 22 C 2 13 − C 33 C 2 13 > 0, where the C ij is elastic constant of the material. The mechanical stability of a phase can be confirmed by using the elastic constants. The SiN/GeN in the Pbca phase both satisfy the above mechanical stability criteria. The SiN/GeN in the Pbca phase show mechanical stability under ambient pressure. The phonon dispersion curve can show dynamic stability; the phonon dispersion curves of SiN/GeN in the Pbca phase are illustrated in Figure 3. There is no imaginary frequency in the Brillouin zone, which means SiN/GeN in the Pbca phase can be dynamically stable under ambient pressure. The elastic moduli of Pbca-XN with X changing from C to Ge are illustrated in Figure 2b. It is clear that the elastic moduli of Pbca-XN decrease with X changing from C to Ge. The elastic constants and elastic moduli of other Si x N y compounds [22,46,47] are also listed in Table 2. The bulk modulus B of Pbca-SiN is slightly smaller than that of SiN 2 , o-Si 3 N 4 , and t-Si 3 N 4 , while it is slightly larger than Si 3 N 2 and t-Si 3 N 4 . The shear modulus G and Young's modulus E of Pbca-SiN are similar to the bulk modulus of Pbca-SiN. For Pbca-GeN, its bulk modulus is as large as that of GeN 2 . However, its shear modulus and Young's modulus are slightly smaller than that of GeN 2 .
Brittleness and ductility of materials are important properties in crystal physics and engineering sciences. Pugh [48] proposed the ratio of bulk to shear modulus (B/G) as an indication of ductile verses brittle characters. If B/G > 1.75, the material is characterized by a ductile manner; otherwise, the material has a brittle character. The Poisson's ratio v is consistent with B/G, but refers to brittle compounds, usually with a small v (less than 0.26) [49]. The B/G ratio of Pbca-CN/SiN/GeN is 1.12 (1.11 [26]), 1.63, and 1.70; it is revealed that Pbca-CN/SiN/GeN are all brittle materials, and Pbca-CN has the most brittleness. For Poisson's ratio v, we obtained the same conclusion.   The Debye temperature (ΘD) is a fundamental physical property, and correlates with many physical properties of solids (e.g., specific heat and the thermal coefficient) [50]. Debye temperature ΘD can be estimated by elastic moduli. The Debye temperature can be estimated from the average sound velocity by the following equation based on elastic constant evaluations [51]:  [49,52].
The calculated results of Debye temperature, longitudinal sound velocity, and transverse sound velocity of Pbca-XN (X = C, Si, Ge) are all listed in Table 3. The densities of Pbca-XN (X = C, Si, Ge) are also listed in Table 3.   Table 2. The calculated elastic constants C ij (in GPa) and bulk moduli B (in GPa), shear moduli G (in GPa), Young's moduli E (in GPa), and Poisson's ratio v of Pbca-CN/SiN/GeN and other C x N y , Si x N y , and Ge x N y compounds with PBE level. The Debye temperature (Θ D ) is a fundamental physical property, and correlates with many physical properties of solids (e.g., specific heat and the thermal coefficient) [50]. Debye temperature Θ D can be estimated by elastic moduli. The Debye temperature can be estimated from the average sound velocity by the following equation based on elastic constant evaluations [51]: (100) propagation direction, polarization direction (100)v l = (C 11 /ρ) 1/2 , (010)v t1 = (C 66 /ρ) 1/2 , and (100)v t2 = (C 55 /ρ) 1/2 [49,52].
The calculated results of Debye temperature, longitudinal sound velocity, and transverse sound velocity of Pbca-XN (X = C, Si, Ge) are all listed in Table 3. The densities of Pbca-XN (X = C, Si, Ge) are also listed in Table 3. For Pbca-XN (X = C, Si, Ge), in the (001) propagation direction, the (001) polarization direction has the largest sound velocity. The longitudinal sound velocity in the (010) propagation direction aligns with the (001) polarization direction, and the longitudinal sound velocity in the (100) propagation direction aligns with the (010) polarization direction. The longitudinal sound velocity is generally larger than the transverse sound velocity, mainly because the elastic constants that determine the longitudinal sound velocity are greater than those of the transverse sound velocity. In addition, for the same the propagation direction and polarization direction, the sound velocity decreases with X changing from C to Ge. Furthermore, the Debye temperature of Pbca-XN (X = C, Si, Ge) decreases with X changing from C to Ge. For Pbca-SiN, the Debye temperature is 863 K; it is slightly smaller than that of m-Si 3 N 4 (892 K), o-Si 3 N 4 (1107 K), and t-Si 3 N 4 (949 K) [53]. The longitudinal sound velocity and transverse sound velocity of Pbca-XN (X = C, Si, Ge) are different along different directions; this shows that the sound velocity of Pbca-XN (X = C, Si, Ge) is also anisotropic.

Electronic Properties
In solid-state physics and semiconductor physics, the band structure of a solid or a material describes the energy that is forbidden or permitted by electrons. The band structure of a material determines a variety of properties-especially its electronic and optical properties. It is known that since the calculated band gap with DFT is usually underestimated by 30-50%, the band gap should be greater than the calculated results with the PBE functional. Hence, the band structures of Pbca-CN/SiN/GeN calculated utilizing the Heyd-Scuseria-Ernzerhof (HSE06) [37,38] hybrid functional are shown in Figure 4a-c, respectively. The band gap of Pbca-CN is 5.41 eV within the HSE06 hybrid functional and 3.96 eV within the PBE functional; the results of the PBE functional of Pbca-CN are in excellent agreement with previous report [26]. The valence band maximum is located at the G point in the Brillouin zone, whereas the conduction band minimum is located at the X point.

Elastic Anisotropy Properties
The elastic anisotropy properties are an important characteristic of materials. Along the different crystallographic directions, various elastic moduli exhibit different values. In this work, we mainly investigated the anisotropy of Poisson's ratio v, shear modulus, bulk modulus, and Young's modulus in different planes and different directions. The Poisson's ratio v and shear modulus G have two unit vectors (a, b) and three angles [43,54], so they have a maximum value and a minimum value in the same direction, while the Young's modulus has only two unit vectors (a, b) and a two-angle description [43,54], so it is in the same direction with only one value. The Poisson's ratio v of Pbca-CN/SiN/GeN in the (001) plane (namely the xy or ab plane), the (010) plane (namely the xz or ac plane), and the (100) plane (namely the yz or bc plane) are displayed in Figure 5a Figure 6a,c, with X change from C to Ge, the shape of the minimum value for shear modulus is increasingly rounded in the (001) plane

Elastic Anisotropy Properties
The elastic anisotropy properties are an important characteristic of materials. Along the different crystallographic directions, various elastic moduli exhibit different values. In this work, we mainly investigated the anisotropy of Poisson's ratio v, shear modulus, bulk modulus, and Young's modulus in different planes and different directions. The Poisson's ratio v and shear modulus G have two unit vectors (a, b) and three angles [43,54], so they have a maximum value and a minimum value in the same direction, while the Young's modulus has only two unit vectors (a, b) and a two-angle description [43,54], so it is in the same direction with only one value. The Poisson's ratio v of Pbca-CN/SiN/GeN in the (001) plane (namely the xy or ab plane), the (010) plane (namely the xz or ac plane), and the (100) plane (namely the yz or bc plane) are displayed in Figure 5a Figure 6a,c, with X change from C to Ge, the shape of the minimum value for shear modulus is increasingly rounded in the (001) plane and (100) plane, while in the (010) plane the shape of the minimum is closer to a square. The ratios G max /G min of Pbca-CN/SiN/GeN are 1.24, 1.42, and 1.53; in other words, the elastic anisotropy in shear modulus becomes larger and larger with X changing from C to Ge.   Young's modulus is a measure of the stiffness of a solid material. It is a mechanical property of linear elastic solid materials. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material. To study the elastic anisotropy in more detail, a variation of Young's modulus with crystallographic direction is displayed in a three-dimensional manner. The directional dependence of Young's modulus E for orthorhombic crystal is [55]: E −1 = l 4 1 S11 + l 4 2 S22 + l 4 3 S33 + 2l 2 1 l 2 2 S12 + 2l 2 1 l 2 3 S13 + 2l 2 2 l 2 3 S23 + l 2 1 l 2 2 S66 + l 2 1 l 2 3 S55 + l 2 2 l 3 2 S44, where l1, l2, and l3 are the direct cosines of the [uvw] direction, and Sij refers to the elastic compliance constants. The three-dimensional surface representations of Young's modulus E for Pbca-CN/SiN/GeN are illustrated in Figure 7a-c. For an isotropic system, the three-dimensional directional dependence exhibits a spherical shape. If there is a deviation of degrees from the spherical shape, it reflects the material exhibiting elastic anisotropy [56]. From Figure 7a-c, it is obvious that the shape of the three-dimensional directional dependence does not exhibit a spherical shape, and the shapes of the three-dimensional directional dependence for Pbca-CN/SiN/GeN all exhibit mechanical anisotropy in Young's modulus.
To further understand the elastic anisotropy of the Young's modulus along different directions, the dependence of the Young's modulus on orientation is investigated when we take the tensile axis within a given plane.  [26]. Pbca-CN/SiN/GeN has a maximum of Emax = 380/241 GPa and a minimum of Emin = 179/120 GPa. In order to quantify the elastic anisotropy, we introduce a ratio; that is, the ratio of the maximum and minimum Young's modulus (ratio Emax/Emin). The greater the ratio Emax/Emin, the greater the maximum and minimum differences, and the greater the anisotropy of the material. Through the values of the ratio Emax/Emin = 2.31, 2.12, and 2.01, it is shown that the elastic anisotropy in Young's modulus for Pbca-XN (X = C, Si, Ge) decreases with X changing from C to Ge.  Young's modulus is a measure of the stiffness of a solid material. It is a mechanical property of linear elastic solid materials. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material. To study the elastic anisotropy in more detail, a variation of Young's modulus with crystallographic direction is displayed in a three-dimensional manner. The directional dependence of Young's modulus E for orthorhombic crystal is [55]: E −1 = l 4 1 S 11 + l 4 2 S 22 + l 4 3 S 33 + 2l 2 1 l 2 2 S 12 + 2l 2 1 l 2 3 S 13 + 2l 2 2 l 2 3 S 23 + l 2 1 l 2 2 S 66 + l 2 1 l 2 3 S 55 + l 2 2 l 3 2 S 44 , where l 1 , l 2 , and l 3 are the direct cosines of the [uvw] direction, and S ij refers to the elastic compliance constants. The three-dimensional surface representations of Young's modulus E for Pbca-CN/SiN/GeN are illustrated in Figure 7a-c. For an isotropic system, the three-dimensional directional dependence exhibits a spherical shape. If there is a deviation of degrees from the spherical shape, it reflects the material exhibiting elastic anisotropy [56]. From Figure 7a-c, it is obvious that the shape of the three-dimensional directional dependence does not exhibit a spherical shape, and the shapes of the three-dimensional directional dependence for Pbca-CN/SiN/GeN all exhibit mechanical anisotropy in Young's modulus.
To further understand the elastic anisotropy of the Young's modulus along different directions, the dependence of the Young's modulus on orientation is investigated when we take the tensile axis within a given plane. Let α be the angle of between (100) and (uv0) for the (001) plane; the Young's modulus between (100) and (uv0) for the (001) plane can be expressed as: E −1 = S 11 cos 4 α + S 22 sin 4 α + 2S 12 sin 2 αcos 2 α + S 66 sin 2 αcos 2 α. Let β be the angle of between (001) and (u0w) for the (010) plane; the Young's modulus between (001) and (u0w) for the (010) plane can be calculated as: E −1 = S 11 sin 4 β + S 33 cos 4 β + (2S 13 sin 2 2β + S 55 sin 2 2β)/4. Let γ be the angle of between (001) and (0vw) for the (001) plane, the Young's modulus between (001)  Pbca-CN has a maximum of E max = 1034 GPa and a minimum of E min = 447 GPa. The calculated results of elastic anisotropy in Young's modulus for Pbca-CN are in excellent agreement with [26]. Pbca-CN/SiN/GeN has a maximum of E max = 380/241 GPa and a minimum of E min = 179/120 GPa. In order to quantify the elastic anisotropy, we introduce a ratio; that is, the ratio of the maximum and minimum Young's modulus (ratio E max /E min ). The greater the ratio E max /E min , the greater the maximum and minimum differences, and the greater the anisotropy of the material. Through the values of the ratio E max /E min = 2.31, 2.12, and 2.01, it is shown that the elastic anisotropy in Young's modulus for Pbca-XN (X = C, Si, Ge) decreases with X changing from C to Ge. In addition, the maximum values of Pbca-CN/SiN/GeN all occupy the position θ = 0, φ = 0; that is, the maximum values of Pbca-XN (X = C, Si, Ge) all occupy the z (c) axis, while the minimum values of Pbca-CN/SiN/GeN do not occupy the same position (x (a) axis). The minimum value of Pbca-SiN is located at θ = 1.32, φ = 0, but the minimum value of Pbca-CN/GeN occupies the position of θ = π/2, φ = 0. For the orthorhombic phase, the dependence of the bulk modulus B along the crystallographic direction is expressed by: B −1 = (S 11 + S 12 + S 13 )l 1 + (S 12 + S 22 + S 23 )l 2 + (S 13 + S 23 + S 33 )l 3  position of θ = π/2, φ = 0. For the orthorhombic phase, the dependence of the bulk modulus B along the crystallographic direction is expressed by: B −1 = (S11 + S12 + S13)l1 + (S12 + S22 + S23)l2 + (S13 + S23 + S33)l3.   In addition, apart from the Poisson's ratio, shear modulus, and Young's modulus, there is another significant physical quantity which describes the elastic anisotropy of a material: the universal anisotropic index A U [57], which is defined as A U = 5GV/GR + BV/BR − 6, where G and B are the shear modulus and bulk modulus, and the subscripts V and R denote the Voigt and Reuss approximations, respectively. The calculated universal anisotropic indices of Pbca-XN (X = C, Si, Ge) are 0.717, 0.671, and 0.662, respectively. The elastic anisotropy in the universal anisotropic index A U of Pbca-XN (X = C, Si, Ge) is similar to the bulk modulus, Young's modulus, and shear modulus; it also decreases with X changing from C to Ge. Furthermore, for Pbca-CN, the universal anisotropic index is slightly smaller than that of m-C3N4 (0.798 [58]), while it is much higher than that of t-C3N4 (0.305 [58]). The universal anisotropic index of Pbca-SiN is slightly larger than that of o-Si3N4 (0.582 [49]), but it is smaller than that of m-Si3N4 (0.968) and t-Si3N4 (1.231) [49].

Conclusions
The structural, mechanical, electronic, and elastic anisotropy properties of CN, SiN, and GeN in orthorhombic phase were performed using DFT calculations in this work. SiN and GeN are mechanically and dynamically stable, fulfilling the Born stability criteria for an orthorhombic phase and phonon spectra, respectively. PBE function predicts lattice parameters that agree well with the previous report. From band gap calculations with the HSE06 function, SiN and GeN are direct band gap semiconductor materials with band gap of 3.39 eV and 2.22 eV, while CN has an indirect band gap with band gap of 5.41 eV. The elastic moduli of Pbca-XN (X = C, Si, Ge) such as Young's moduli, bulk moduli, shear moduli, Poisson's ratio, and sound velocities have also been reported in this work. The Debye temperature, longitudinal sound velocities, and transverse sound velocities are also estimated using the elastic constants. The elastic anisotropy calculations showed that Pbca-XN (X = C, Si, Ge) exhibited anisotropy in bulk modulus, shear modulus, Poisson's ratio, Young's modulus, and A U . Besides, the elastic anisotropy in bulk modulus, shear modulus, Poisson's ratio, Young's modulus, and A U for Pbca-XN (X = C, Si, Ge) decreases with X changing from C to Ge. In addition, apart from the Poisson's ratio, shear modulus, and Young's modulus, there is another significant physical quantity which describes the elastic anisotropy of a material: the universal anisotropic index A U [57], which is defined as A U = 5G V /G R + B V /B R − 6, where G and B are the shear modulus and bulk modulus, and the subscripts V and R denote the Voigt and Reuss approximations, respectively. The calculated universal anisotropic indices of Pbca-XN (X = C, Si, Ge) are 0.717, 0.671, and 0.662, respectively. The elastic anisotropy in the universal anisotropic index A U of Pbca-XN (X = C, Si, Ge) is similar to the bulk modulus, Young's modulus, and shear modulus; it also decreases with X changing from C to Ge. Furthermore, for Pbca-CN, the universal anisotropic index is slightly smaller than that of m-C 3 N 4 (0.798 [58]), while it is much higher than that of t-C 3 N 4 (0.305 [58]). The universal anisotropic index of Pbca-SiN is slightly larger than that of o-Si 3 N 4 (0.582 [49]), but it is smaller than that of m-Si 3 N 4 (0.968) and t-Si 3 N 4 (1.231) [49].

Conclusions
The structural, mechanical, electronic, and elastic anisotropy properties of CN, SiN, and GeN in orthorhombic phase were performed using DFT calculations in this work. SiN and GeN are mechanically and dynamically stable, fulfilling the Born stability criteria for an orthorhombic phase and phonon spectra, respectively. PBE function predicts lattice parameters that agree well with the previous report. From band gap calculations with the HSE06 function, SiN and GeN are direct band gap semiconductor materials with band gap of 3.39 eV and 2.22 eV, while CN has an indirect band gap with band gap of 5.41 eV. The elastic moduli of Pbca-XN (X = C, Si, Ge) such as Young's moduli, bulk moduli, shear moduli, Poisson's ratio, and sound velocities have also been reported in this work. The Debye temperature, longitudinal sound velocities, and transverse sound velocities are also estimated using the elastic constants. The elastic anisotropy calculations showed that Pbca-XN (X = C, Si, Ge) exhibited anisotropy in bulk modulus, shear modulus, Poisson's ratio, Young's modulus, and A U . Besides, the elastic anisotropy in bulk modulus, shear modulus, Poisson's ratio, Young's modulus, and A U for Pbca-XN (X = C, Si, Ge) decreases with X changing from C to Ge.