Nitrides consisting of III A and IV A group elements can be used as high temperature materials, catalysts, light-emitting diodes and refractory ceramics because of their excellent mechanical and thermodynamic properties.^[1–4] Silicon nitride is an important ceramic material. It is a superhard material and atomic crystal which has good lubricity and abrasion resistance. It resists oxidation at high temperature, and it can resist cold and hot shocks, it can be heated to over 1000 K in the air and will not be broken by rapid cooling and rapid heating. Because silicon nitride ceramic has such excellent characteristics, it is often used in bearings, gas turbine blades, mechanical seals, permanent die, and other mechanical components. If the silicon nitride ceramic with high temperature resistance and high heat transfer resistance is used to make the heating surfaces of the engine parts, it can not only improve the quality of diesel engine, save fuel, but also enhance the thermal efficiency. In fact, germanium is adopted because Ge atoms have higher carrier mobility than Si atoms. The semiconductor field effect transistor has superior performances. The germanium atom radius is bigger, so the critical pressure of the phase transition of Ge₃N₄ is less than that of Si₃N₄. Germanium nitride also has the advantages of corrosion resistance, high hardness, and its band gap can be adjusted. Many new structures of silicon nitride and germanium nitride have been reported in many researches.^[5–8] It has been known for several decades that silicon nitride exists in two stable phases (namely α- and β-Si₃N₄) with very similar structures,^[5] they both have hexagonal symmetry. The phase boundaries of the transitions are investigated using first-principles calculations within the quasi-harmonic approximation by Yu and Chen.^[6] Using the plane-wave pseudopotential method, Yu and Chen^[7] have investigated the structural properties, mechanical properties and thermodynamic properties of β-, γ-, wII-, and post-spinel Si₃N₄ and also the phase transition characters among these polymorphs. Wang et al. have conducted a comparative structural and electronic study of a new class of material, olivine-Si₃N₄ and olivine-Ge₃N₄ by using the first principles approach.^[8] They found that olivine-Si₃N₄ has a large bulk modulus (262.9 GPa), and olivine-Si₃N₄ and olivine-Ge₃N₄ are both direct bang gap semiconductors with the band gaps of 2.99 eV and 1.55 eV, respectively.

The structural, electronic, and optical properties of the new cubic spinel nitrides c-Si₃N₄, c-SiGe₂N₄, c-Si₂GeN₄, and c-Ge₃N₄ were studied by a first-principles method.^[9] The c-SiGe₂N₄ is a direct band gap semiconductor with a band gap of 1.85 eV, and the c-Si₂GeN₄ is an indirect band gap semiconductor with a band gap of 2.56 eV. In addition, the bulk moduli of c-SiGe₂N₄ and c-Si₂GeN₄ are 277.1 GPa and 258.3 GPa, respectively. Using first-principles calculations, the structural, electronic, elastic, and optical properties of cubic spinel SiGe₂N₄ under pressure were investigated by Moakafi et al.^[10] Bouhemadou et al.^[11] have investigated the structural, elastic, electronic, optical and thermal properties of c-SiGe₂N₄ by using the ultrasoft pseudopotential density functional method in the generalized gradient approximation under pressure. The structural and mechanical properties of the cubic spinel c-CSi₂N₄ were studied by first-principles total energy calculations based on the density-functional theory.^[12] The Vickers hardness of c-CSi₂N₄ is 52.07 GPa, so it is a potential superhard material. Cui et al. explored three Si₃N₄ metastable phases, i.e., tetragonal t-Si₃N₄, monoclinic m-Si₃N₄ and orthorhombic o-Si₃N₄, by analyzing the crystal structure obtained through using the particle swarm optimization (CALYPSO) code.^[13] The m-Si₃N₄ belongs to the space group of Cm in a monoclinic system and it is mechanically and dynamically stable under ambient pressure based on their elastic constants and phonon dispersions calculations. Cang et al. predicted the lattice structures, densities of states, phonon dispersion curves of m-, t-, and o-Ge₃N₄.^[14] The stabilities of these Ge₃N₄ structures were verified by the proving of negative formation enthalpy, the satisfying of Born’s stability criteria and no imaginary frequency in the phonon dispersion curves in a pressure range of 0 GPa–20 GPa. Fan et al. investigated the electronic structures, elastic, anisotropic, and electronic properties of m-Si₃N₄, t-Si₃N₄, and o-Si₃N₄ under pressure,^[15] and it is interesting that the shift of band gap from direct to indirect occurs for m-Si₃N₄ when the pressure rises up to 50 GPa. Chen et al.^[16] predicted the structural, electronic, optical and thermodynamic properties of m-, t-, and o-M₃N₄ (M = Si, Ge, Sn).

According to the previous work, we predict two new double nitrides m-Si₂GeN₄ and m-SiGe₂N₄ in the Cm space group. In the present work, the structural, elastic anisotropic, and electronic properties of m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ are investigated using density functional theory calculations. The Debye temperature, the ductile or brittle characters and elastic anisotropy of m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3.) are discussed in detail.

The structural optimizations and property predictions of the monoclinic structures of m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) were performed based on density functional theory (DFT)^[17,18] as implemented in the Cambridge Serial Total Energy Package (CASTEP) code.^[19] The exchange–correlation potential was treated with the generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE),^[20] PBE for solid (PBEsol),^[21] and the local density approximation (LDA) in the form of Ceperley and Alder data parameterized by Perdew and Zunger (CA-PZ).^[22,23] The Broyden–Fletcher–Goldfarb–Shanno (BFGS)^[24] minimization scheme was used in geometry optimization. Vanderbilt type ultrasoft pseudopotentials^[25] were employed to describe the interactions between ionic core and valence electrons. The considered valence atomic configurations had been set to be Si-3s²3p², Ge-4s²4p², and N-2s²2p³. In the structure calculation, a plane-wave basis set with energy cut-off 500 eV was used for all cases. The k-points of 6×14×7, 6×13×7, 6×13×7, and 5×13×7 were adopted for the structures of m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ by using Monkhorst–Pack mesh,^[26] respectively. The system reached the ground state via self-consistent calculation when the total energy, the maximum displacement of ion, the maximum ionic Hellmann–Feynman force within and the maximum stress were less than 5×10⁻⁶ eV/atom, 5×10⁻⁴ Å, 0.01 eV/Å, and 0.02 GPa, respectively. In order to calculate electronic structures, the HSE06 hybrid functional^[27] was used.

The crystal structures of m-Si₃N₄ and the double nitrides m-Si₂GeN₄ and m-SiGe₂N₄ are shown in Fig. 1. In a conventional cell, m-Si₃N₄ (m-Ge₃N₄) has 8 N atoms and 6 Si (Ge) atoms. For m-Si₂GeN₄ and m-SiGe₂N₄, the Ge atoms replace Si atoms with the smallest energy. The equilibrium lattice constants are determined by the minimum total energy with respect to the variation of the cell volume through utilizing both GGA and LDA. The optimized lattice parameters of m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ at zero pressure are listed in Table 1.

Fig. 1.

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	Fig. 1. (color online) Crystal structures of (a) m-Si3N4, (b) m-Si2GeN4, and (c) m-Ge3N4 under ambient pressure. The silicon, germanium, and nitrogen atoms are represented as black, red, and blue spheres, respectively.

Table 1.

Table 1.

Table 1. Calculated lattice parameters (a, b, c in unit Å and β in unit degree) of m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4 compared with previous theoretical results. . Materials PBE PBEsol CA-PZ a b c β a b c β a b c β m-Si3N4 9.606 2.932 7.419 133.3 9.590 2.928 7.409 133.6 9.442 2.888 7.308 133.6 9.512a 2.904 7.353 133.4 9.605b 2.933 7.417 133.4 9.603c 2.932 7.419 m-Si2GeN4 9.754 2.989 7.569 132.8 9.735 2.983 7.551 132.8 9.558 2.933 7.422 133.0 m-SiGe2N4 9.896 3.043 7.787 132.8 9.876 3.036 7.770 132.8 9.672 2.975 7.603 133.1 m-Ge3N4 10.218 3.104 7.931 133.6 10.188 3.095 7.913 133.6 9.924 3.021 7.714 133.7 10.215d 3.103 7.928 a Ref. [13] b Ref. [15] c Ref. [16] d Ref. [14].

Table 1. Calculated lattice parameters (a, b, c in unit Å and β in unit degree) of m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4 compared with previous theoretical results. .

Along with previous theoretical results for comparison, it can be seen that the calculated lattice parameters of m-Si₃N₄ and m-Ge₃N₄ are in excellent agreement with previous results.^[13,15,16] The variations of the lattice parameters (a, b, c) with increasing the composition of germanium are shown in Fig. 2. On the whole, the lattice parameters (a, b, c) increase with increasing the composition of germanium. The augment tendency of lattice parameter b is nearly linear while the lattice parameter a and c increases nonlinearly with increasing the composition of germanium. The curves of the PBE and PBEsol methods almost coincide as shown in Fig. 2. From m-Si₃N₄ to m-Ge₃N₄, the lattice parameters a, b, and c increase by 6.34% (5.10%), 5.87% (4.61%), and 6.90% (5.56%) obtained from the PBE (CA-PZ) method, respectively.

Fig. 2.

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	Fig. 2. (color online) Calculated lattice parameters ((a) a, (b) b, and (c) c) of m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4 by PBE, PBEsol, and CA-PZ methods.

The elastic constants are closely related to various fundamental mechanical properties of a crystal, which can provide, particularly, the information about the stability and stiffness of materials. For a monoclinic crystal, there are thirteen independent elastic constants, namely C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃, C₁₅, C₂₃, C₂₅, C₃₅, and C₄₆. The calculated elastic constants of m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) are listed in Table 2. The elastic constants of m-Si₃N₄ and m-Ge₃N₄ are well consistent with pervious theoretical values.^[13,14] For a stable crystal of monoclinic structure, it must obey the following mechanical stability according to the Born’s criteria:^[28]1 2 3 4 5 6 7 8

Table 2.

Table 2.

Table 2. Calculated elastic constants (in unit GPa) of m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4 compared with other calculated results. . C11 C22 C33 C44 C55 C66 C12 C13 C15 C23 C25 C35 C46 m-Si3N4 257 455 353 93 132 92 36 131 –27 58 22 16 30 254a 478 369 70 130 95 40 142 57 23 18 34 m-Si2GeN4 242 417 316 76 112 85 37 126 −18 53 16 15 22 m-SiGe2N4 235 436 363 60 110 42 42 164 −30 68 20 21 19 m-Ge3N4 180 344 274 57 90 61 34 110 –18 40 9 19 22 175b 338 277 55 90 63 31 107 41 9.8 20 22 a Ref. [13] b Ref. [14].

Table 2. Calculated elastic constants (in unit GPa) of m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4 compared with other calculated results. .

It can be easily verified that the elastic constants of m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) satisfy their stability requirements, which indicates that they are mechanical stable. And the dynamic stabilities of m-Si₃N₄ and m-Ge₃N₄ have been verified by the phonon spectrum calculations in Refs. [13] and [14]. So it is revealed that m-Si₃N₄ and m-Ge₃N₄ are mechanically and dynamically stable via the elastic constants and phonon calculations. The elastic constants C₁₁, C₂₂, C₃₃ represent the stiffness of the material when applying stresses along the x, y, z directions,^[29] respectively. The obtained result obviously shows that all the m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ follow this ordering relationship: , which means that the resistance against deformation along the z axis is stronger than along the x axis but weaker than along the y axis. Simultaneously, with increasing the components of Ge atoms, the elastic constants C_ii decrease, and thus the compression property increases.

Furthermore, according to the Voigt–Reuss–Hill (VRH) average scheme,^[30,31] the bulk modulus B and shear modulus G can be estimated using calculated elastic constants via the following equations:9 10 11 12 13 14 15 16 17 18 19 20

The Young’s modulus E and Poisson’s ratio v are obtained from the following equations: and , respectively. Comparing with the previous results,^[13,14,16] the calculated values of bulk modulus B, shear modulus G, and Young’s modulus E are presented in Table 3. The bulk modulus B represents the resistance to fracture while the shear modulus G represents the resistance to plastic deformation.^[32] The values of shear modulus are much larger than those of bulk modulus for m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3), which means that the resistance to shape change is better than that to volume change. Generally, Young’s modulus E provides a measure of material stiffness. The larger the value of E, the stiffer the material will be.^[33] The values of B, G, and E decrease with the increase of the Ge atoms except m-SiGe₂N₄, which suggests that the resistance to deformation becomes weaker and the stiffness decreases.

Table 3.

Table 3.

Table 3. Calculated values of elastic modulus (in unit GPa), Poisson’s ratio v, percent compressibility of bulk modulus and shear modulus factors , , and anisotropic index for m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4, compared with other calculated results. . Materials B G E B/G v m-Si3N4 168 163 119 101 165 110 270 1.505 0.228 1.663 8.545 0.968 165a 113 276 1.457 0.220 165b 104 259 1.586 0.239 m-Si2GeN4 156 152 105 89 154 97 241 1.586 0.239 1.204 8.097 0.905 m-SiGe2N4 176 165 93 62 170 78 202 2.200 0.303 3.131 19.949 2.557 m-Ge3N4 129 124 82 64 126 73 184 1.732 0.258 2.287 12.788 1.513 124c 73 a Ref. [13]. b Ref. [16]. c Ref. [14].

Table 3. Calculated values of elastic modulus (in unit GPa), Poisson’s ratio v, percent compressibility of bulk modulus and shear modulus factors , , and anisotropic index for m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4, compared with other calculated results. .

Poisson’s ratio v can be used as an indicator for ductility or brittleness. According to the Frantsevich’s rule,^[34] the critical value that distinguishes between the ductile and brittle nature of material is 0.26. If , the material will behave in a ductile manner, otherwise the material demonstrates brittleness in nature. Additionally, the ratio of bulk modules to shear modulus (B/G) can also judge whether a material would be a ductile or brittle manner according to the Pugh’s criterion.^[35] The critical value of this ratio which distinguishes between the ductility and brittle is 1.75. For ductile materials, the ratio of B/G is larger than 1.75. Here the calculation Poisson’s ratio v and ratio of B/G of m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) are listed in Table 3. Only the Poisson’s ratio v (B/G) of m-SiGe₂N₄ is larger than 0.26 (1.75), meaning that m-SiGe₂N₄ is ductile and m-Si₃N₄, m-Si₂GeN₄, and m-Ge₃N₄ are brittle in nature. The brittle materials are sensitive to thermal shocks, as the material cannot efficiently dissipate thermal stresses via plastic deformations. Thus, a brittle material can suffer limited thermal shocks before its strength drops dramatically. Hence, the capability of resisting thermal shocks for m-SiGe₂N₄ is stronger than for m-Si₃N₄, m-Si₂GeN₄, and m-Ge₃N₄.

The Debye temperature ( is one of fundamental parameters for solid materials, which is related to many physical properties, such as thermal expansion and specific heat. It is the Debye temperature that reflects the temperature of a crystal’s normal mode of vibration. The Debye temperature can be estimated from averaged sound velocity ( ), compressional sound velocity ( and shear sound velocity ( through the following formulas:^[36,37]21 22 23 where h is the Planck’s constant, is the Boltzmann constant, is the Avogadro’s constant, M is the molecular weight, n is the number of atoms in the molecule, and ρ is the density of crystal. The calculated results of density, the Debye temperature and the sound velocities for m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) are listed in Table 4 at 0 GPa and 0 K. Following the sequence , the densities increase whereas the Debye temperature and the sound velocities decrease. According to Refs. [38]–[40], a larger suggests a higher normal vibration, which is associated with a better thermal conductivity. Meanwhile, the Debye temperature can characterize the strength of covalent bond for solid. The larger the , the stronger the covalent bond is.^[41] Therefore, thermal conductivities and covalent bonds of the m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) become weaker following the above sequence.

Table 4.

Table 4.

Table 4. Values of density (in units g/cm3), compressional, shear, and average sound velocities (in m/s) as well as Debye temperature (in K) for m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4.. . Materials ρ m-Si3N4 3.070 5984 10082 6627 892 m-Si2GeN4 3.791 5066 8656 5618 744 m-SiGe2N4 4.442 4178 7854 4668 602 m-Ge3N4 4.994 3824 6694 4249 539

Table 4. Values of density (in units g/cm3), compressional, shear, and average sound velocities (in m/s) as well as Debye temperature (in K) for m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4.. .

The elastic anisotropy analysis of a crystal is of great significance for understanding the mechanisms of material microcrack and durability in engineering science.^[42] There are a variety of methods to quantity the anisotropy of a crystal structure, such as the universal anisotropic index ,^[43] the percent compressibility of bulk and shear modulus factors ( and ).^[44] According to the following equations, these mechanical anisotropic indexes mentioned above can be evaluated from24 25 26

For isotropic crystals, the percentages of elastic anisotropy for bulk modulus and shear modulus , and universal anisotropic index must be zero. The deviation from zero measures the anisotropy degree of crystal. The calculated anisotropy factors ( , , and are presented in Table 3. It is obvious that the m-Si_xGe_{3 −x}N₄ (x = 0, 1, 2, 3) are anisotropic. The value of is much larger than for each crystal, indicating that the anisotropy of shear modulus is more conspicuous than that of bulk modulus. The values of the universal anisotropic index are 0.968 for m-Si₃N₄, 0.905 for m-Si₂GeN₄, 2.557 for m-SiGe₂N₄, and 1.513 for m-Ge₃N₄, respectively. The anisotropy factors ( , , and of m-SiGe₂N₄ are maxima, which indicates the anisotropy of m-SiGe₂N₄ is the largest among the m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3).

Simultaneously, the elastic anisotropic m-Si_xGe_{3 −x}N₄ (x = 0, 1, 2, 3) in Poisson’s ratio, shear modulus and Young’s modulus were investigated in the present work. The three-dimensional (3D) directional dependence and two-dimensional (2D) representation of anisotropy are calculated by utilizing the program of Elastic Anisotropy Measure (ELAM).^[45] The vector direction (θ, ϕ) in the spherical coordinate system is used to describe the directional dependence of anisotropy, where θ (ϕ) represents the angle between the vector and the x-axis (z-axis) positive direction and they are expressed in radians. For an isotropic material, the 3D directional dependence would exhibit a spherical shape, while the deviation degree from the spherical shape reflects the content of anisotropy.^[46]

The calculated 3D directional dependence of Poisson’s ratio and 2D representation in the (001), (010), and (100) planes for m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ is shown in Figs. 3 and 4, respectively. It is evident that the 3D figures of the Poisson’s ratio have a large degree of deviation in shape from the sphere, which implies that the Poisson’s ratios of m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) are greatly anisotropic. The maximal (minimal) values of the Poisson’s ratio for m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ are 0.55 (0.02), 0.55 (0.03), 0.77 (0.02), and 0.67 (0.02) at 0 GPa. Following the sequence , the maximal and minimal Poisson’s ratio directions are (3.14, 0.19), (3.14, 0.17), (0, 2.95), (3.14, 0.17) and (3.14, 0.89), (2.28, 0), (0, 2.38), (3.68, 1.35), respectively. It can be found that the maximal and minimal Poisson’s ratio directions for m-Si₃N₄ and m-SiGe₂N₄ are both in the (010) plane. From Fig. 4, it can be seen that four compounds present the different degrees of anisotropy in Poisson’s ratio in different planes. Following the above sequence, the ratios of are 0.44/0.02 = 22, 0.43/0.04 = 10.75, 0.63/0.02 = 31.5, and 0.52/0.04 = 13 in the (001) plane, 0.55/0.02 = 27.5, 0.55/0.03 = 18.33, 0.77/0.02 = 35, and 0.67/0.02 = 33.5 in the (010) plane, and 0.52/0.02 = 26, 0.53/0.04 = 13.25, 0.48/0.03 = 16.33, and 0.64/0.04 = 16 in the (100) plane, respectively. Obviously, the anisotropy of the Poisson’s ratio for m-SiGe₂N₄ is the largest one among m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3).

Fig. 3.

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	Fig. 3. (color online) 3D directional dependence of Poisson’s ratio for (a) m-Si3N4, (b) m-Si2GeN4, (c) m-SiGe2N4, and (d) m-Ge3N4 at ambient pressure.

Fig. 4.

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	Fig. 4. (color online) 2D representations of Poisson’s ratio in the (001), (010), and (100) planes for m-SixGe3−xN4 (x = 0, 1, 2, 3). The dash–dotted and solid lines represent the maximum and minimum values in the (001), (010), and (100) planes, respectively. The black, blue, green, and red lines denote the Poisson’s ratios of m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4, respectively.

The shear moduli for all possible directions of shear strain are also calculated. The calculated 3D directional dependence of shear modulus and 2D representations in the (001), (010), and (100) planes for the m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) are plotted in Figs. 5 and 6, respectively. The 3D figures reveal that the anisotropy of shear modulus is weak since the deviation degree from the sphere shape is small. The values of ratio are as follows: for m-Si₃N₄, for m-Si₂GeN₄, for m-SiGe₂N₄, for m-Ge₃N₄ at 0 GPa. From Fig. 6, it is clear that the anisotropies of shear modulus in the (010) plane for the m-Si_xGe_{3 −x}N₄ (x = 0, 1, 2, 3) are slightly larger than those in other planes. From m-Si₃N₄ to m-Ge₃N₄, the values of are 150.31/63.24 = 2.38, 129.87/57.71 = 2.25, 118.83/29.98 = 3.96, and 100.23/36.25 = 2.76 in the (001) plane, 139.05/63.24 = 2.2, 117.46/57.71 = 2.04, 120.27/29.98 = 4.01, and 97.68/36.25 = 2.69 in the (010) plane, and 156.54/63.24 = 2.48, 139.54/57.71 = 2.42, 139.86/29.98 = 4.6, and 119.49/36.25 = 3.29 in the (100) plane, respectively. Consequently, m-SiGe₂N₄ exhibits larger anisotropy among m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3).

Fig. 5.

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	Fig. 5. (color online) 3D directional dependence of shear modulus for (a) m-Si3N4, (b) m-Si2GeN4, (c) m-SiGe2N4, and (d) m-Ge3N4 (d) at ambient pressure.

Fig. 6.

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	Fig. 6. (color online) 2D representations of shear modulus in the (001), (010) and (100) planes for m-SixGe3−xN4 (x = 0, 1, 2, 3). The dash dotted and solid lines represent the maximum and minimum values in the (001), (010), and (100) planes, respectively. The black, blue, green and red lines represent the Poisson’s ratios of m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4, respectively.

The 3D directional dependencies of Young’s modulus for m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3.) are plotted in Fig. 7. It can be seen that the 3D figures of the m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ have more or less deviations in shape from sphere along the x, y, z axes, which indicates that the Young’s moduli of these compounds possess some kind of degree anisotropy. The values of ratio are as follows: for m-Si₃N₄, for m-Si₂GeN₄, for m-SiGe₂N₄, and for m-Ge₃N₄ at 0 GPa. The calculated Young’s moduli along different directions as well as in different planes for the m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ are shown in Fig. 8. At first glance, the anisotropy of Young’s modulus in (001) plane for m-Si_xGe_3−xN₄ ( , 1, 2, 3) is similar to that in (100) plane but slightly larger than that in (010) planes. The values of for m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ are 440.27/193.73 = 2.28, 404.44/181.34 = 2.23, 418.92/106.91 = 3.92, and 334.56/123.52 = 2.71 in the (001) plane, 344.61/190.17 = 1.81, 304.96/178.67 = 1.71, 331.71/135.61 = 2.45, and 267.23/120 = 2.23 in the (010) plane, and 440.27/227.36 = 1.94, 404.44/194.14 = 2.08, 418.92/153.44 = 2.73, and 334.56/139.31 = 2.4 in the (100) plane, respectively. It can be noted that the maximum values of Young’s modulus for m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ in the (001) plane are the same as those in the (100) plane. A comparison of the values of leads to a conclusion that the anisotropy of Young’s modulus for m-SiGe₂N₄ is stronger among those of m-Si_xGe_{3 −x}N₄ (x = 0, 1, 2, 3), which is consistent with the pervious conclusion.

Fig. 7.

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	Fig. 7. (color online) 3D directional dependence of Young’s modulus for (a) m-Si3N4, (b) m-Si2GeN4, (c) m-SiGe2N4, and (d) m-Ge3N4 at ambient pressure.

Fig. 8.

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	Fig. 8. (color online) 2D representations of Young’s modulus in the (001), (010), and (100) planes for m-SixGe3−xN4 (x = 0, 1, 2, 3). The black, red, blue, and magenta lines represent the Poisson’s ratios of m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4, respectively.

Physical properties can be measured in different directions. Various crystal properties, including the elastic modulus, hardness, fracture resistance, thermal expansion coefficient, thermal conductivity, resistivity, and electric displacement vector, have different levels of anisotropy. If a physical property is closely related to orientation, it is anisotropic. Why does this reverse phenomenon appear? This is because of the different atomic arrangements and different atomic positions, and different atoms have different lengths in different directions, resulting in the fact that the physical quantities along the different directions possess different values. For m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3), they have a monoclinic symmetry that belongs to the Cm space group, the symmetry of the monoclinic system is very low. Young’s modulus in Figs. 8(a) and 8(c) are symmetric with respect to the xy plane and yz plane, respectively, but in Fig. 8(b), the symmetry axis are not along the xz plane. And both the Poisson’s ratio, and the shear modulus have the same situation as Young’s modulus. That is to say, the symmetries of Young’s modulus, Poisson’s ratio, and shear modulus in the xz plane are lower than in the xy plane and yz plane.

The electronic band structure reflects the law of the electron motion in a solid and determines the fundamental physical and chemical properties of material. The band structures of m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ along some high symmetry directions of the Brillouin zone are calculated with HSE06 functional at ambient pressure, and the results are shown in Fig. 9. The horizontal red dash line designates zero energy representing the Fermi lever ( ). At first glance, it appears that the four band structures are very similar. The major differences are in the width of band gap and conduction band minimum. From Fig. 9, it is clear that the valance band maximum (VBM) and the conduction band minimum (CBM) of m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) are both located at G point. Therefore, m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ are direct semiconductors with band gaps of 5.08 eV, 4.76 eV, 4.81 eV, and 3.34 eV, respectively. It should be noted that no experimental dataare available to verify the band gaps of these compounds. All the previous results are obtained via theoretical methods, which need to be verified experimentally in future.

Fig. 9.

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	Fig. 9. (color online) Electronic band structures of (a) m-Si3N4, (b) m-Si2GeN4, (c) m-SiGe2N4, and (d) m-Ge3N4 with the HSE06 hybrid functional.

To further investigate the natures of the electronic band structure for m-Si₃N₄, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄, the partial densities of states (PDOSs) are calculated as shown in Fig. 10. It is obvious that PDOSs of m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) look similar on a gross scale since they have the same crystal structure and Si and Ge are isoelectronic elements. The densities of electronic states are separated into three regions by the gaps. The first region is in the energy range between −20 eV and −15 eV, the second region is between −10 eV and Fermi energy ( , and the last region ranges from 5 eV to 11 eV in the conduction band. For m-Si₃N₄ (m-Ge₃N₄), the first region is mainly for N-s states with admixture from Si (Ge)-s/p states, the second region mainly involves N-p states with significant contributions from Si (Ge)-s states (between 10 eV to −7.5 eV) and Si (Ge)-p states (from −7.5 eV to , and the third region mainly contains Si (Ge)-p states and a mixture of Si (Ge)-s states and N-s/p states. For m-Si₂GeN₄ and m-SiGe₂N₄, the first region mainly contains N-s states with a mixture of Si- and Ge-s/p states, the second region mainly involes N-p states with significant contributions from Si- and Ge-s/p states, and the third region mainly covers Si/Ge-p states and a mixture of Si (Ge)-s states and N-s/p states.

Fig. 10.

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	Fig. 10. (color online) Comparisons of total density of states and partial density of states for m-Si3N4, m-Si2GeN4, m-SiGe2N4, and m-Ge3N4. The black/dash and red/solid curves represent s states and p states of silicon, germanium, and nitrogen atoms, respectively.

The structural, electronic, elastic, and anisotropic properties of the monoclinic phases of m-Si₃N4, m-Si₂GeN₄, m-SiGe₂N₄, and m-Ge₃N₄ are investigated systemically in this work. The calculated lattice constants, elastic constants and elastic moduli of m-Si₃N₄ and m-Ge₃N₄ are in good consistence with previous theoretical results. The elastic constants of m-Si_xGe_{3 −x}N₄ (x = 1, 2) satisfy the stability criteria, which indicates that they are mechanically stable. According to the analyses of the obtained values of Poisson’s ratio v and B/G, it is found that only m-SiGe₂N₄ is ductile in nature. The densities of m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) increase whereas the Debye temperature and the sound velocities decrease with increasing the composition of germanium. The detailed analyses of the anisotropy manifest that the m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3) exhibit lager anisotropy and the anisotropy of m-SiGe₂N₄ is the largest among the m-Si_xGe_3−xN₄ (x = 0, 1, 2, 3). The calculated band structures show that they are all direct semiconductors with band gaps of 5.08 eV, 4.76 eV, 4.81 eV, and 3.34 eV, respectively.

The authors thank Q Y Fan (School of Microelectronics, Xidian University) for being allowed to use the CASTEP code in Materials Studio.