Structural, Electronic, and Thermodynamic Properties of Tetragonal t-SixGe3−xN4

The structural, mechanical, anisotropic, electronic, and thermal properties of t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 in the tetragonal phase are systematically investigated in the present work. The mechanical stability is proved by the elastic constants of t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4. Moreover, they all demonstrate brittleness, because B/G < 1.75, and v < 0.26. The elastic anisotropy of t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 is characterized by Poisson’s ratio, Young’s modulus, the percentage of elastic anisotropy for bulk modulus AB, the percentage of elastic anisotropy for shear modulus AG, and the universal anisotropic index AU. The electronic structures of t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 are all wide band gap semiconductor materials, with band gaps of 4.26 eV, 3.94 eV, 3.83 eV, and 3.25 eV, respectively, when using the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional. Moreover, t-Ge3N4 is a quasi-direct gap semiconductor material. The thermodynamic properties of t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 are investigated utilizing the quasi-harmonic Debye model. The effects of temperature and pressure on the thermal expansion coefficient, heat capacity, Debye temperature, and Grüneisen parameters are discussed in detail.


Introduction
IV A Group nitrides have attracted considerable interest as high-performance ceramics due to their outstanding physical properties [1]. Silicon nitride has many advantages, such as high strength, wear resistance, a high decomposition temperature, oxidation resistance, outstanding thermal shock resistance, a low friction coefficient, and low corrosion resistance, thus making it the ideal material for use in engineering ceramics [2]. Germanium nitride also has advantages, such as corrosion resistance and an adjustable band gap [3,4].
Si 3 N 4 under normal temperature and pressure has two polymorphs: α-Si 3 N 4 and β-Si 3 N 4 . It is generally accepted that α-Si 3 N 4 is a metastable state, and β-Si 3 N 4 is the low-temperature phase of Si 3 N 4 crystal [5]. In 1999, Zerr et al. synthesized the third polymorph, γ-Si 3 N 4 (cubic spinel structure, also named c-Si 3 N 4 [6]) [7]. Since then, more researchers have studied the transitions among the three phases of Si 3 N 4 . Togo et al. [8] found that the phase transition pressure of β → γ was 12.5 GPa at 300 K. Through X-ray diffraction experiments, Kruger et al. [9] found that the α-Si 3 N 4 could remain stable at room temperature and in the pressure range of 0-48 GPa. The phase transition pressure of α → γ was 15.22 GPa at 300 K, as reported by Yu and Chen [6]; they also found that the stability order of the three types of polymorphs was β-Si 3 N 4 > α-Si 3 N 4 > γ-Si 3 N 4 . Kroll predicted two new phases of Si 3 N 4 : post-spinel and wII- [10,11]. Yu et al. [12] investigated the structural and elastic properties of post-spinel and wIIof Si 3 N 4 . They found that post-spinel and wII-Si 3 N 4 were stable at 160 GPa and

Structural Properties
The crystal structures of t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 are displayed in Figure 1. The t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 are obtained when germanium atoms replace the Si atom of t-Si 3 N 4 with the smallest energy. The equilibrium lattice parameters of t-Si x Ge 3−x N 4 and c-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys with the related reference data are listed in Table 1. As seen, our calculated lattice parameters for t-Si 3 N 4 and c-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys are in excellent agreement with the existing results. The lattice parameters of t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys as a function with the Ge component are shown in Figure 2a. From Figure 2a, the lattice parameter a increases with an increase in the percentage of the germanium composition, except for t-Ge 3 N 4 . The lattice parameter a of t-Ge 3 N 4 is slightly smaller than t-SiGe 2 N 4 ; the possible cause of this phenomenon is that the angle of ∠Si-N-Ge (113.95 • ) and ∠Ge-N-Ge (112.16 • ) in t-SiGe 2 N 4 decreases in t-Ge 3 N 4 (∠Si-N-Ge: 112.86 • ; ∠Ge-N-Ge: 111.75 • ) parallel to the ac plane. Moreover, the lattice parameter c of t-Si 3 N 4 , t-Si 2 GeN 4 , and t-SiGe 2 N 4 is almost the same; however, the lattice parameter c of t-Ge 3 N 4 is far greater than that of the others. The bond length of the silicon nitrogen bond is certainly smaller than that of the germanium nitrogen bond in the same structure. From t-Si 3 N 4 and t-Ge 3 N 4 , as the substitution of germanium atoms for silicon atoms increases and the substitution occurs primarily along the c axis, the growth of germanium nitrogen bonds is mainly manifested in the c axis direction. In Figure 1c, the silicon or germanium atoms occupy five of the nine layers of atoms along the c axis. From t-Si 3 N 4 to t-Si 2 GeN 4 , the germanium atoms only replace two of the silicon atoms; thus, at this time, the increase along the c axis is very small. In addition, from t-Si 2 GeN 4 to t-SiGe 2 N 4 , the germanium atoms are replaced on the same layers, so the increase along the c axis is very small. From t-SiGe 2 N 4 to t-Ge 3 N 4 , all of the germanium atoms replace the silicon atoms, so the lattice parameter c suddenly increases. the local density approximation (LDA) in the form of Ceperley and Alder data, as parameterized by the Perdew and Zunger (CA-PZ) [46] exchange correlation potential. The valence electron configurations of Si, Ge, and N atoms were Si-3s 2 3p 2 , Ge-4s 2 4p 2 , and N-2s 2 2p 3 , respectively. The cut-off energy was set as 500 eV. The Brillouin zone of t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 could be described by the k-points of 10 × 10 × 5, 10 × 10 × 5, 9 × 9 × 5, and 9 × 9 × 5 using the Monkhorst-Pack mesh, [51] respectively. The crystal structures were optimized by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [52]. The self-consistent convergence of the total energy was 5 × 10 −6 eV/atom; the maximum force on the atom was 0.01 eV/Å, the maximum ionic displacement was within 5 × 10 −4 Å, and the maximum stress was within 0.02 GPa. The thermodynamic properties of t-SixGe3−xN4 (x = 0, 1, 2, 3) alloys were investigated utilizing the quasi-harmonic Debye model [53,54].

Structural Properties
The crystal structures of t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 are displayed in Figure 1. The t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 are obtained when germanium atoms replace the Si atom of t-Si3N4 with the smallest energy. The equilibrium lattice parameters of t-SixGe3−xN4 and c-SixGe3−xN4 (x = 0, 1, 2, 3) alloys with the related reference data are listed in Table 1. As seen, our calculated lattice parameters for t-Si3N4 and c-SixGe3−xN4 (x = 0, 1, 2, 3) alloys are in excellent agreement with the existing results. The lattice parameters of t-SixGe3−xN4 (x = 0, 1, 2, 3) alloys as a function with the Ge component are shown in Figure 2a. From Figure 2a, the lattice parameter a increases with an increase in the percentage of the germanium composition, except for t-Ge3N4. The lattice parameter a of t-Ge3N4 is slightly smaller than t-SiGe2N4; the possible cause of this phenomenon is that the angle of ∠Si-N-Ge (113.95°) and ∠Ge-N-Ge (112.16°) in t-SiGe2N4 decreases in t-Ge3N4 (∠Si-N-Ge: 112.86°; ∠Ge-N-Ge: 111.75°) parallel to the ac plane. Moreover, the lattice parameter c of t-Si3N4, t-Si2GeN4, and t-SiGe2N4 is almost the same; however, the lattice parameter c of t-Ge3N4 is far greater than that of the others. The bond length of the silicon nitrogen bond is certainly smaller than that of the germanium nitrogen bond in the same structure. From t-Si3N4 and t-Ge3N4, as the substitution of germanium atoms for silicon atoms increases and the substitution occurs primarily along the c axis, the growth of germanium nitrogen bonds is mainly manifested in the c axis direction. In Figure 1c, the silicon or germanium atoms occupy five of the nine layers of atoms along the c axis. From t-Si3N4 to t-Si2GeN4, the germanium atoms only replace two of the silicon atoms; thus, at this time, the increase along the c axis is very small. In addition, from t-Si2GeN4 to t-SiGe2N4, the germanium atoms are replaced on the same layers, so the increase along the c axis is very small. From t-SiGe2N4 to t-Ge3N4, all of the germanium atoms replace the silicon atoms, so the lattice parameter c suddenly increases.  ; and t-SiGe 2 N 4 (c) at ambient pressure. The silicon, germanium, and nitrogen atoms are represented as black, red, and blue spheres, respectively.
[56]-experimental.   In addition, we further investigate the influence of pressure and temperature for t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4. The ratio of volume V/V0 affected by temperature and pressure is shown in Figure 2b,c. As shown in Figure 2b,c, the volume of t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 increases as the temperature increases. When T < 300 K, the growth rate is very small; however, when T > 300 K, the volume is found to increase linearly. Furthermore, the ratio of V/V0 clearly decreases as the pressure increases, approaching a linear decrease. Moreover, Figure 2b,c indicates that the effect In addition, we further investigate the influence of pressure and temperature for t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 . The ratio of volume V/V 0 affected by temperature and pressure is shown in Figure 2b,c. As shown in Figure 2b,c, the volume of t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 increases as the temperature increases. When T < 300 K, the growth rate is very small; however, when T > 300 K, the volume is found to increase linearly. Furthermore, the ratio of V/V 0 clearly decreases as the pressure increases, approaching a linear decrease. Moreover, Figure 2b,c indicates that the effect of the pressure on the V/V 0 is more significant than that of the temperature in the pressure and temperature ranges that are considered in our study. In addition, the volume of t-Ge 3 N 4 changes more than do those of t-Si 3 N 4 , t-Si 2 GeN 4 , and t-SiGe 2 N 4 .
[59]-experimental. of the pressure on the V/V0 is more significant than that of the temperature in the pressure and temperature ranges that are considered in our study. In addition, the volume of t-Ge3N4 changes more than do those of t-Si3N4, t-Si2GeN4, and t-SiGe2N4.
[59]experimental. The Voigt-Reuss-Hill approximation [60] is typically used to calculate bulk modulus BH and shear modulus GH. The Young's modulus E and Poisson's ratio v could be obtained as: E = 9BHGH/(3BH + GH) and v = (3BH − 2GH)/[2(3BH + GH)], respectively. The bulk modulus BH represents the resistance and v = (3B H − 2G H )/[2(3B H + G H )], respectively. The bulk modulus B H represents the resistance to compressibility, and the shear modulus G H represents the resistance to plastic deformation. The calculated bulk modulus B H and shear modulus G H are listed in Table 2. For t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys, the bulk modulus are larger than the shear modulus, which indicates that it is more difficult for volume deformation to occur in alloys than shape deformation. The bulk modulus B H , shear modulus G H , and Young's modulus E all decrease with an increase in the Ge component. Comparing t-Si x Ge 3−x N 4 , c-Si x Ge 3−x N 4 , and m-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys, the elastic modulus for t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys is slightly larger than that for m-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys, whereas it is much smaller than that for c-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys.
Pugh [61] proposed that the ratio of B H /G H could accurately distinguish the brittleness or ductility of a material. If B H /G H > 1.75, then the material exhibits a ductile property; otherwise, it exhibits a brittle property. The ratio B H /G H of t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys as a function with a Ge component are shown in Figure 2d. From Figure 2d, although the value of B H /G H increases with the increasing percentage of the Ge composition, they are all less than 1.75. This suggests that t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys all exhibit brittleness. Poisson's ratio v can also be used to distinguish the brittleness or ductility of a material [62]. If v >0.26, then the material will behave in a ductile manner; otherwise, the material demonstrates brittleness. The calculated Poisson's ratios v of t-Si x Ge 3−x N 4 , c-Si x Ge 3−x N 4 and m-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys are all listed in Table 2. From Table 2 It is well known that elastic anisotropy has important implications in engineering science and crystal physics. The directional dependence of the anisotropy is calculated by the elastic anisotropy measures (ELAM) [63,64] code. The two-dimensional (2D) representation of Poisson's ratio v in (001), (010), (100), (101), (110), and (111) planes for t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys are shown in Figure 4. Clearly, the 2D figures of Poisson's ratio all have a great deviation degree from a circular shape, indicating that the t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys exhibit anisotropy in Poisson's ratio. Moreover, the four alloys present the different degree of anisotropy on Poisson's ratio in different planes. The maximum Poisson's ratio for t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 are 0.59, 0.60, 0.62, and 0.70, respectively, following the sequence t-Si 3 N 4 > t-Si 2 GeN 4 > t-SiGe 2 N 4 > t-Ge 3 N 4 . The following directional dependence of anisotropy is usually described by vector direction (θ, ϕ), where θ (ϕ) represents the angle between the vector and the x-axis (z-axis) positive direction, and they are expressed in radians [63,64]. The positions of the maximum Poisson's ratio for t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 appear at θ = 0.79, ϕ = 0.00; θ = 2.36, ϕ = 0.00; θ = 1.57, ϕ = 3.93; and θ = 2.33, The 2D figures of Young's modulus for t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 are shown in manner; otherwise, the material demonstrates brittleness. The calculated Poisson's ratios v of t-SixGe3−xN4, c-SixGe3−xN4 and m-SixGe3−xN4 (x = 0, 1, 2, 3) alloys are all listed in Table 2. From Table 2, all of the v are less than 0.26. Thus, t-SixGe3−xN4 (x = 0, 1, 2, 3) alloys all demonstrate brittleness; this is consistent with the conclusion based on BH/GH. Moreover, c-SixGe3−xN4 (x = 0, 1, 2, 3) alloys are quite brittle because of the small ratio of BH/GH; m-SiGe2N4 is ductile; and m-Si3N4, m-Si2GeN4, and m-Ge3N4 are brittle in nature.   0.70, respectively, following the sequence t-Si3N4 > t-Si2GeN4 > t-SiGe2N4 > t-Ge3N4. The following directional dependence of anisotropy is usually described by vector direction (θ, φ), where θ (φ) represents the angle between the vector and the x-axis (z-axis) positive direction, and they are expressed in radians [63,64]. The positions of the maximum Poisson's ratio for t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 appear at θ = 0.79, φ = 0.00; θ = 2.36, φ = 0.00; θ = 1.57, φ = 3.93; and θ = 2.33, φ = 0.00, respectively. The 2D figures of Young's modulus for t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 are shown in    Table 3. From Table 3, t-SixGe3−xN4 (x = 0, 1, 2, 3) alloys show a slight degree in anisotropy in the bulk modulus, because AB is very close to zero. For Moreover, the elastic anisotropic of the material could be shown by the percentage of elastic anisotropy for the bulk modulus A B , the percentage of elastic anisotropy for the shear modulus A G , and the universal anisotropic index A U : [65]. The calculated values of A B , A G and A U of t-Si x Ge 3−x N 4 , c-Si x Ge 3−x N 4 , and m-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys are all listed in Table 3. From Table 3, t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys show a slight degree in anisotropy in the bulk modulus, because A B is very close to zero. For t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys, the percentage of elastic anisotropy for shear modulus A G is the smallest for t-Si 2 GeN 4 , and is slightly smaller than that of t-Si 3 N 4 ; meanwhile, t-Ge 3 N 4 has the largest A G . For t-Si 3 N 4 , c-Si 3 N 4 , and m-Si 3 N 4 , the A G of c-Si 3 N 4 is approximately half that of t-Si 3 N 4 . The t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys exhibit the largest anisotropy compared with c-Si x Ge 3−x N 4 and m-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys, except for m-SiGe 2 N 4 . At the same time, m-SiGe 2 N 4 exhibits the largest A G among the t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys, c-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys, and m-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys. The greater the value of A U , the greater the anisotropy. The values of the universal anisotropic index A U of the t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys are 1.231, 1.261, 1.283, and 1.637, respectively. It was also found that t-Ge 3 N 4 has the greatest A U , indicating that t-Ge 3 N 4 has the largest anisotropy among t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys, which is consistent with the results of A B and A G . Table 3. Calculated bulk modulus B V , B R and shear modulus G V , G R by the Voigt and Reuss method respectively; percent compressibility of bulk modulus and shear modulus factors (A B , A G ) % and universal anisotropic indexes A U for t-Si x Ge 3−x N 4 , c-Si x Ge 3−x N 4 , and m-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys.

Electronic Properties
As is known, the electronic structure determines the physical and chemical properties of materials. The electronic band structures of t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 are shown in Figure 6, according to calculations with the HSE06 hybrid functional [66,67]. From Figure 6, the t-Si x Ge 3−x N 4 (x = 1, 2, 3) alloys are all indirect band gap semiconductor materials, and t-Ge 3 N 4 is a quasi-direct band gap semiconductor material. In addition, the band gaps of t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 are 4.26 eV, 3.94 eV, 3.83 eV, and 3.25 eV, respectively; the conduction band minima are all located at the M point, and the valence band maxima are all located at the G point. With the increasing percentage of the Ge composition, the band gap decreases. For t-Ge 3 N 4 , the direct gap at G point is 3.34 eV, which is slightly larger than the indirect gap of 3.25 eV; as a result, t-Ge 3 N 4 is a quasi-direct gap semiconductor. The calculated band gaps that utilize other functions, such as PBE, PBEsol, and CA-PZ, are listed in Table 1. It is known that the calculated band gaps are usually underestimated with DFT; i.e., the true band gap is larger than the calculated results. The band gaps of t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 with the HSE06 hybrid functional are found to be much larger than those calculated by other functions.
To further investigate the nature of the electronic band structure for t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 , we also investigated the partial density of states (PDOS) for t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys displayed in Figure 7. Since the t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys are all tetragonal crystal and Si and Ge belong to the IV A Group, their PDOS are similar and consist of three regions. The first region is from approximately −20 eV to −15 eV; the second and third regions are from −10 eV to the Fermi energy (E F ) and from 5 eV to 12 eV, respectively. For t-Si 3 N 4 and t-Ge 3 N 4 , the first region originates from the N-s, Si-s/p or Ge-s/p states. The second region is from the N-p states, with a mixture of Si-s/p or Ge-s/p states. The last region is primarily from Si/Ge-p states and a mixture of Si/Ge-s states and N-p states. For t-Si 2 GeN 4 and t-SiGe 2 N 4 , the first region is primarily from N-s states, with an admixture from Si-s/p and Ge-s/p states. The second region primarily originates from N-p states, with significant contributions from the Si-s and Ge-s states between −9 and −7 eV, and the Si-p and Ge-p states between −7 eV and E F . The Ge-s/p and Si-p states dominate the last region, ranging from 5 eV to approximately 10 eV. and N-p states. For t-Si2GeN4 and t-SiGe2N4, the first region is primarily from N-s states, with an admixture from Si-s/p and Ge-s/p states. The second region primarily originates from N-p states, with significant contributions from the Si-s and Ge-s states between −9 and −7 eV, and the Si-p and Ge-p states between −7 eV and EF. The Ge-s/p and Si-p states dominate the last region, ranging from 5 eV to approximately 10 eV.

Thermodynamic Properties
The thermodynamic properties of semiconductors are very important at higher temperatures and pressures. In this work, the highest temperature is 1400 K, and the highest pressure is 18 GPa. In the above conditions, the thermal expansion coefficient α, the heat capacities CV and CP, and the Debye temperature ΘD are all investigated here. The thermal expansion coefficient describes how the size of a material changes with a change in temperature. Specifically, the thermal expansion coefficient measures the fractional change in size per degree change in temperature at a constant and N-p states. For t-Si2GeN4 and t-SiGe2N4, the first region is primarily from N-s states, with an admixture from Si-s/p and Ge-s/p states. The second region primarily originates from N-p states, with significant contributions from the Si-s and Ge-s states between −9 and −7 eV, and the Si-p and Ge-p states between −7 eV and EF. The Ge-s/p and Si-p states dominate the last region, ranging from 5 eV to approximately 10 eV.

Thermodynamic Properties
The thermodynamic properties of semiconductors are very important at higher temperatures and pressures. In this work, the highest temperature is 1400 K, and the highest pressure is 18 GPa. In the above conditions, the thermal expansion coefficient α, the heat capacities CV and CP, and the Debye temperature ΘD are all investigated here. The thermal expansion coefficient describes how the size of a material changes with a change in temperature. Specifically, the thermal expansion Figure 7. Comparison of total density of states and partial density of states for t-Si 3 N 4 (a); t-Si 2 GeN 4 (b); t-SiGe 2 N 4 (c); and t-Ge 3 N 4 (d). The black and red curves represent the s states and p states, respectively, of silicon, germanium, and nitrogen atoms.

Thermodynamic Properties
The thermodynamic properties of semiconductors are very important at higher temperatures and pressures. In this work, the highest temperature is 1400 K, and the highest pressure is 18 GPa. In the above conditions, the thermal expansion coefficient α, the heat capacities C V and C P , and the Debye temperature Θ D are all investigated here. The thermal expansion coefficient describes how the size of a material changes with a change in temperature. Specifically, the thermal expansion coefficient measures the fractional change in size per degree change in temperature at a constant pressure; it is one of the important parameters to measure for determining the thermodynamic properties of materials. The values of the thermal expansion coefficient α of t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys as functions of temperature and pressure are shown in Figure 8. Figure 8a shows that α increases exponentially with an increase in temperature until 300 K at 0 GPa. When T > 300 K, the growth rate of α decreases, and α increases linearly after T > 800 K. The growth rate of thermal expansion α at a high temperature is far less than that at a low temperature; i.e., the temperature dependence of α is very small at a high temperature. In addition, at a given temperature, α decreases with an increase in pressure, and the decline rate decreases with the increase in pressure. The effect of the temperature on the thermal expansion coefficient α is found to be more significant than that of pressure. Note that the thermal expansion coefficient α is the smallest for t-Si 3 N 4 , whereas t-Ge 3 N 4 has the largest thermal expansion coefficient α in t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys.  Figure 8. Figure 8a shows that α increases exponentially with an increase in temperature until 300 K at 0 GPa. When T > 300 K, the growth rate of α decreases, and α increases linearly after T > 800 K. The growth rate of thermal expansion α at a high temperature is far less than that at a low temperature; i.e., the temperature dependence of α is very small at a high temperature. In addition, at a given temperature, α decreases with an increase in pressure, and the decline rate decreases with the increase in pressure. The effect of the temperature on the thermal expansion coefficient α is found to be more significant than that of pressure. Note that the thermal expansion coefficient α is the smallest for t-Si3N4, whereas t-Ge3N4 has the largest thermal expansion coefficient α in t-SixGe3−xN4 (x = 0, 1, 2, 3) alloys.  The variation of the heat capacity (C V ) at constant volume and the heat capacity (C p ) of t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 at a constant pressure versus temperature and pressure variations are shown in Figure 9. The curves of heat capacity for t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 have similar tendencies, as do the variations of the temperature and pressure that are shown in Figure 9. From Figure 9a, the heat capacity curves are proportional to T 3 when T < 300 K, and the growth rate of the heat capacity decreases above 300 K. For T > 800 K, the heat capacity gradually approaches the fixed value of the Dulong-Petit limit (174.54 J mol −1 K −1 ) [68]. As shown in Figure 9c, the change regulation of C P is similar to that of C V below 800 K. C P increases linearly as the temperature increases above 800 K. From Figure 9b,d, the effect of pressure and temperature on heat capacity is the opposite; i.e., the heat capacities for t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys decrease with an increase in pressure at a given temperature. Moreover, the sensitivity of heat capacity to temperature is far greater than that to pressure. Figure 9 also shows that the heat capacity for t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys follows the sequence: t-Ge 3 N 4 > t-SiGe 2 N 4 > t-Si 2 GeN 4 > t-Si 3 N 4 .  Figure 9. Calculated specific volume CV as a function of temperature (a) and as a function of pressure (b); the pressure heat capacity CP as a function of temperature (c) and as a function of temperature pressure (d) for t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4.
The variation of the heat capacity (CV) at constant volume and the heat capacity (Cp) of t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 at a constant pressure versus temperature and pressure variations are shown in Figure 9. The curves of heat capacity for t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4 have similar tendencies, as do the variations of the temperature and pressure that are shown in Figure 9.   Figure 10a-d, the change regulation of Θ D for t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys has the same trend under variable temperature and variable pressure. In addition, the effect of the pressure on Θ D is found to be more significant than that of temperature. To better illustrate the effect of temperature and pressure variation on the Θ D of t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys, the Θ D values as functions of temperature or pressure are shown in Figure 10e-f. From Figure 10e, the Θ D remains nearly constant from 0 K to 200 K, and when T > 200 K, the Θ D decreases linearly. Moreover, at a given temperature, the value of Θ D almost increases monotonously with an increase in pressure, as shown in Figure 10f. From 0 K to 1400 K, Θ D decreases by 5.2%, 6.7%, 7.8%, and 8.8% for t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 , respectively, and from 0 GPa to 18 GPa, Θ D decreases by 12.2%, 13.7%, 15.3%, and 17.1%. In addition, the larger the Ge composition is, the larger the influence to Θ D is. The values of Θ D for t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 calculated by the quasi-harmonic Debye model (938 K, 761 K, 658 K, and 567 K) are in agreement with those calculated by the elastic modulus [65] , v s = (G/ρ) 1/2 ; 937 K, 767 K, 666 K, and 571 K) at 0 K and 0 GPa. From Figure 9a, the heat capacity curves are proportional to T 3 when T < 300 K, and the growth rate of the heat capacity decreases above 300 K. For T > 800 K, the heat capacity gradually approaches the fixed value of the Dulong-Petit limit (174.54 J mol −1 K −1 ) [68]. As shown in Figure 9c, the change regulation of CP is similar to that of CV below 800 K. CP increases linearly as the temperature increases above 800 K. From Figure 9b,d, the effect of pressure and temperature on heat capacity is the opposite; i.e., the heat capacities for t-SixGe3−xN4 (x = 0, 1, 2, 3) alloys decrease with an increase in pressure at a given temperature. Moreover, the sensitivity of heat capacity to temperature is far greater than that to pressure. Figure 9 also shows that the heat capacity for t-SixGe3−xN4 (x = 0, 1, 2, 3) alloys follows the sequence: t-Ge3N4 > t-SiGe2N4 > t-Si2GeN4 > t-Si3N4.    Figure 10a-d, the change regulation of ΘD for t-SixGe3−xN4 (x = 0, 1, 2, 3) alloys has the same trend under variable temperature and variable pressure. In addition, the effect of the pressure on ΘD is found to be more significant than that of temperature. To better illustrate the effect of temperature and pressure variation on the ΘD of t-SixGe3−xN4 (x = 0, 1, 2, 3) alloys, the ΘD values as functions of temperature or pressure are shown in Figures 10e-f. From Figure 10e, the ΘD remains nearly constant from 0 K to 200 K, and when T > 200 K, the ΘD decreases linearly. Moreover, at a given temperature, the value of ΘD almost increases monotonously with an increase in pressure, as shown in Figure 10f. From 0 K to 1400 K, ΘD decreases by 5.2%, 6.7%, 7.8%, and 8.8% for t-Si3N4, t-Si2GeN4, t-SiGe2N4, and t-Ge3N4, respectively, and from 0 GPa to 18 GPa, ΘD decreases by 12.2%, 13.7%, 15.3%, and 17.1%. In addition, the larger the Ge composition is, the larger the Figure 10. (a-d) Three-dimensional contour plots of the Debye temperature versus pressure and temperature for t-Si 3 N 4 (a); t-Si 2 GeN 4 (b); t-SiGe 2 N 4 (c); and t-Ge 3 N 4 (d). The Debye temperature as functions of temperature (e) and pressure (f) for t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 .

Conclusions
In the present work, the structural, mechanical, elastic anisotropic, electronic, and thermal properties of t-Si 3 N 4 , t-Si 2 GeN 4 , t-SiGe 2 N 4 , and t-Ge 3 N 4 in the tetragonal phase were investigated using density functional theory. The mechanically stable forms of t-Si 2 GeN 4 and t-SiGe 2 N 4 were proved by elastic constants. The t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys were all found to be brittle according to Poisson's ratio v and B/G. The elastic modulus was found to decrease with an increase in the proportion of Ge. Also, t-Ge 3 N 4 was found to exhibit the largest anisotropy among the t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys. It was found that t-Si 3 N 4 , t-Si 2 GeN 4 , and t-SiGe 2 N 4 were indirect band gap semiconductors, but t-Ge 3 N 4 was a quasi-direct gap semiconductor material. The band gaps of t-Si x Ge 3−x N 4 (x = 0, 1, 2, 3) alloys decreased with germanium content, which was suitable for visible light applications such as LEDs and photocatalysts. Moreover, all of the alloys considered were found to be wide band gap semiconductor materials, which indicated that transistors made from them could withstand higher temperatures and voltages, and the switching speed would be faster. In addition, their thermodynamic properties were investigated in detail utilizing the quasi-harmonic Debye model. The thermal expansion coefficient α and heat capacity were found to be more susceptible to temperature than pressure, whereas the Debye temperature was found to be more susceptible to pressure than temperature. These results would provide reference data for the experiments and make current theoretical research on these alloys more plenitude. It also could be envisioned that, by adjusting the Si:Ge ratio, the double nitrides in the tetragonal phases would lead to alloys with tailored electronic and thermodynamic properties for specific applications.