#
Structural, Electronic, and Thermodynamic Properties of Tetragonal t-Si_{x}Ge_{3−x}N_{4}

^{1}

^{2}

^{*}

## Abstract

**:**

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}in the tetragonal phase are systematically investigated in the present work. The mechanical stability is proved by the elastic constants of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}. Moreover, they all demonstrate brittleness, because B/G < 1.75, and v < 0.26. The elastic anisotropy of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}is characterized by Poisson’s ratio, Young’s modulus, the percentage of elastic anisotropy for bulk modulus A

_{B}, the percentage of elastic anisotropy for shear modulus A

_{G}, and the universal anisotropic index A

^{U}. The electronic structures of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}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-Ge

_{3}N

_{4}is a quasi-direct gap semiconductor material. The thermodynamic properties of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}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.

## 1. Introduction

_{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 wII- of Si

_{3}N

_{4}. They found that post-spinel and wII-Si

_{3}N

_{4}were stable at 160 GPa and 0 GPa, respectively. The post-spinel Si

_{3}N

_{4}could be obtained when γ-Si

_{3}N

_{4}was at 152.5 GPa and 0 K, while β → wII occurred at 20 GPa and 300 K.

_{3}N

_{4}was first identified in 1930, when β-Ge

_{3}N

_{4}was first synthesized [13]. Ruddlesden and Popper [14] found that the α phase was closely related to β-Ge

_{3}N

_{4}; however, it is generally believed to be a metastable phase under ambient conditions. Ge

_{3}N

_{4}is considered to be a better candidate for obtaining the wII phase than Si

_{3}N

_{4}, because the transition to higher coordination states could be easier; it also may occur at lower pressures in germanium nitride than in silicon nitride because of the larger ionic size of Ge [15]. McMillan et al. [16] observed the first-order phase transition between the β and σ (space group: P3) phases at 23 GPa and 298 K. They also predicted the β → P$\overline{6}$ → σ phase transitions at 20 GPa and 28 GPa, respectively. Wang et al. [17] found that γ-Ge

_{3}N

_{4}could retain its stability up to 69.2 GPa at room temperature. The critical pressure of the β → wII phase transition was 10.7 GPa (at 300 K), and the β → wII transformation occurred at 13.5 GPa and 1100 K; further compression led to the wII → γ transition at 35.7 GPa, as reported by Luo et al. [18] In addition to silicon nitride and germanium nitride, IV A Group and V A Group elements compounds have been investigated, such as Si

_{3}P

_{4}and Ge

_{3}P

_{4}[19,20,21], W

_{3}N

_{4}[22,23,24], BN [25,26,27,28,29,30] and C

_{3}N

_{4}[31,32,33,34,35].

_{3}N

_{4}: the tetragonal phase t-Si

_{3}N

_{4}, the monoclinic phase m-Si

_{3}N

_{4}, and the orthorhombic phase o-Si

_{3}N

_{4}[36]. They found that ΔH for m-Si

_{3}N

_{4}and t-Si

_{3}N

_{4}were smaller than γ-Si

_{3}N

_{4}below 2.9 GPa and 2.5 GPa, respectively. o-Si

_{3}N

_{4}is a high pressure phase, with P

_{t}= 198 GPa from γ-Si

_{3}N

_{4}. Subsequently, other researchers studied the physical properties of t-Si

_{3}N

_{4}, m-Si

_{3}N

_{4}, and o-Si

_{3}N

_{4}[37,38,39]. Yao and Chen reported on the structural properties, mechanical properties, Vickers hardnesses, and electronic properties of t-Ge

_{3}N

_{4}, m-Ge

_{3}N

_{4}, and o-Ge

_{3}N

_{4}[37]. When the pressure was below 20 GPa, the formation enthalpies of t-Ge

_{3}N

_{4}, m-Ge

_{3}N

_{4}, and o-Ge

_{3}N

_{4}were, negative indicating they are stable. Fan et al. [38] investigated the elastic anisotropic and electronic properties of m-Si

_{3}N

_{4}, o-Si

_{3}N

_{4}, and t-Si

_{3}N

_{4}under high pressure. They found that the m-Si

_{3}N

_{4}transition from the direct band gap to the indirect band gap state occurred at ~32 GPa. Most of the Si

_{3}N

_{4}and Ge

_{3}N

_{4}semiconductors are wide band gap semiconductors; however, most of them are also indirect band gap semiconductors, such as α-Si

_{3}N

_{4}, β-Si

_{3}N

_{4}, o-Si

_{3}N

_{4}, t-Si

_{3}N

_{4}, post-spinel Si

_{3}N

_{4}, and α-Ge

_{3}N

_{4}[7,36,40,41]. It is reported that the band gaps of (Si

_{1−x}Ge

_{x})

_{3}N

_{4}could be adjusted by the ratio of Si:Ge [42]. The cubic spinel c-Si

_{3}N

_{4}, c-SiGe

_{2}N

_{4}, and c-Ge

_{3}N

_{4}were theoretically predicted to have wide and direct band gaps of 3.45 eV, 1.85 eV, and 2.22 eV, respectively, and c-GeSi

_{2}N

_{4}had an indirect band gap of 2.56 eV and a direct band gap of 2.64 eV [43]. The cubic spinel SiGe

_{2}N

_{4}has been shown to be of particular interest, because it is a stable compound with a direct band gap [44]. Ma et al. [45] studied the structural, mechanical, elastic, anisotropic, and electronic properties of the monoclinic phase of m-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys. Their results indicated that the m-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys are all direct and wide band gap semiconductor materials, and that the band gaps could be adjusted from 3.34–5.08 eV. From Ref. [43,44,45], it could be envisioned that the proper mixing of (Si, Ge)

_{3}N

_{4}in the cubic spinel and monoclinic phases may lead to compounds with appropriate properties, which we needed. For example, the band gaps of (Si, Ge)

_{3}N

_{4}could be tuned by changing the Si:Ge ratio. These tunable band gap materials are particularly useful in applications such as light-emitting diodes (LEDs), hybrid solar cells, sensors, and photocatalysts.

_{3}N

_{4}in the tetragonal phase, we propose two new double nitrides, t-Si

_{2}GeN

_{4}and t-SiGe

_{2}N

_{4}, in the I-42m space group. In the present work, we investigate and discuss the structural, elastic, electronic, and thermodynamic properties for t-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys, which would be helpful for future experiments and theoretical explorations.

## 2. Calculation Methods

^{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-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}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-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys were investigated utilizing the quasi-harmonic Debye model [53,54].

## 3. Results and Discussion

#### 3.1. Structural Properties

_{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.

_{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}.

#### 3.2. Mechanical and Anisotropic Properties

_{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. The elastic constant is an index that reflects the ability of materials to resist elastic deformation. From Table 2, the elastic constants of the t-Si

_{2}GeN

_{4}and t-SiGe

_{2}N

_{4}satisfy Born’s mechanical stability criteria of tetragonal symmetry [57]; as a result, t-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys are mechanically stable. In order to confirm the stability of t-Si

_{2}GeN

_{4}and t-SiGe

_{2}N

_{4}, the phonon spectra are calculated at ambient pressure, which are shown in Figure 3. There is no imaginary frequency, proving that the alloys are stable.

_{H}and shear modulus G

_{H}. The Young’s modulus E and Poisson’s ratio v could be obtained as: E = 9B

_{H}G

_{H}/(3B

_{H}+ G

_{H}) 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.

_{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, all of the v are less than 0.26. Thus, t-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys all demonstrate brittleness; this is consistent with the conclusion based on B

_{H}/G

_{H}. Moreover, c-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys are quite brittle because of the small ratio of B

_{H}/G

_{H}; m-SiGe

_{2}N

_{4}is ductile; and m-Si

_{3}N

_{4}, m-Si

_{2}GeN

_{4}, and m-Ge

_{3}N

_{4}are brittle in nature.

_{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, φ = 0.00, respectively.

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}are shown in Figure 5. Clearly, t-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys exhibit large anisotropy in Young’s modulus in all of the planes except the (111) plane. The Young’s modulus of the materials in the (111) plane are approximately circular in shape, indicating that the materials exhibited the smallest anisotropy in the (111) plane. In addition, the shapes of the four lines are similar; thus, the anisotropy of Young’s modulus should change little with an increasing percentage of Ge composition. The maximum (minimum) values of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}are 421 GPa (179 GPa), 373 GPa (159 GPa), 347 GPa (149 GPa), and 306 GPa (111 GPa), respectively. The positions of the maximum values of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}are θ = 2.19, φ = 3.93; θ = 0.95, φ = 3.93; θ = 2.19, φ = 3.93; and θ = 2.17, φ = 3.93, respectively. The minimum value of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, and t-Ge

_{3}N

_{4}appears at θ = 1.57, φ = 0.00, and the minimum value of t-SiGe

_{2}N

_{4}appears at θ = 0.00, φ = 0.00. The values of E

_{max}/E

_{min}for t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}are 2.352, 2.346, 2.329, and 2.757, respectively. From these results, t-Ge

_{3}N

_{4}has the largest anisotropy, and those of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, and t-SiGe

_{2}N

_{4}are similar.

_{B}, the percentage of elastic anisotropy for the shear modulus A

_{G}, and the universal anisotropic index A

^{U}: A

^{U}= 5G

_{V}/G

_{R}+ B

_{V}/B

_{R}− 6, A

_{B}= (B

_{V}− B

_{R})/(B

_{V}+ B

_{R}), and A

_{G}= (G

_{V}− G

_{R})/(G

_{V}+ G

_{R}) [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}.

#### 3.3. Electronic Properties

_{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.

_{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.

#### 3.4. Thermodynamic Properties

_{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.

_{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}.

_{D}of t-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys with the temperature and pressure. From 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] (Θ

_{D}= (h/k

_{B})[(3n/4π)(N

_{A}ρ/M)]

^{1/3}v

_{m}, v

_{m}= [(2/${v}_{s}^{3}$ + 1/${v}_{p}^{3}$)/3]

^{−1/3}, v

_{p}= [(B + 4G/3)/ρ]

^{1/2}, v

_{s}= (G/ρ)

^{1/2}; 937 K, 767 K, 666 K, and 571 K) at 0 K and 0 GPa.

## 4. Conclusions

_{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.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Crystal structures of t-Si

_{3}N

_{4}(

**a**); t-Si

_{2}GeN

_{4}(

**b**); 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.

**Figure 2.**Calculated lattice parameters a and c (

**a**) of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}by PBE, PBEsol, and CA–PZ methods; the volume V/V

_{0}compression as functions of temperature (

**b**) and pressure (

**c**); the elastic moduli for t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}(

**d**).

**Figure 4.**Two-dimensional (2D) representations of Poisson’s ratio for t-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys in the (001) plane (

**a**); (010) plane (

**b**); (100) plane (

**c**); (101) plane (

**d**); (110) plane (

**e**); and (111) plane (

**f**). The dash dot and solid lines represent the maximum and minimum values, respectively. The black, red, blue, and cyan lines represent the Poisson’s ratios of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}, respectively.

**Figure 5.**2D representation of Young’s modulus for t-Si

_{x}Ge

_{3−x}N

_{4}(x = 0, 1, 2, 3) alloys in the (001) plane (

**a**); (010) plane (

**b**); (100) plane (

**c**); (101) plane (

**d**); (110) plane (

**e**); and (111) plane (

**f**). The black, red, blue, and green lines represent the Poisson’s ratios of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}, respectively. All units are in GPa.

**Figure 6.**Electronic band structures of 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**) with the HSE06 hybrid functional.

**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.

**Figure 8.**Temperature (

**a**) and pressure (

**b**) dependence of the thermal expansion coefficients for t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}.

**Figure 9.**Calculated specific volume C

_{V}as a function of temperature (

**a**) and as a function of pressure (

**b**); the pressure heat capacity C

_{P}as a function of temperature (

**c**) and as a function of temperature pressure (

**d**) for t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}.

**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}.

**Table 1.**The lattice parameters a and c (in Å) and band gaps Eg (in eV) of t-Si

_{3}N

_{4}, t-Si

_{2}GeN

_{4}, t-SiGe

_{2}N

_{4}, and t-Ge

_{3}N

_{4}and c-Si

_{3}N

_{4}, c-Si

_{2}GeN

_{4}, c-SiGe

_{2}N

_{4}, and c-Ge

_{3}N

_{4.}PBE: Perdew–Burke–Ernzerhof functional; PBEsol: Perdew–Burke–Ernzerhof functional for solids; CA–PZ: Perdew and Zunger.

Materials | PBE | PBEsol | CA–PZ | ||||||
---|---|---|---|---|---|---|---|---|---|

a | c | Eg | a | c | Eg | a | c | Eg | |

t-Si_{3}N_{4} | 4.167 | 8.253 | 3.05 | 4.155 | 8.241 | 2.89 | 4.099 | 8.115 | 2.99 |

- | 4.131 ^{1} | 8.168 | 3.15 | - | - | - | - | - | - |

- | 4.166 ^{2} | 8.249 | - | - | - | - | - | - | - |

t-Si_{2}GeN_{4} | 4.292 | 8.287 | 2.74 | 4.280 | 8.269 | 2.60 | 4.196 | 8.143 | 2.71 |

t-SiGe_{2}N_{4} | 4.417 | 8.312 | 2.31 | 4.406 | 8.283 | 2.33 | 4.295 | 8.156 | 2.59 |

t-Ge_{3}N_{4} | 4.410 | 8.838 | 1.79 | 4.393 | 8.86 | 1.80 | 4.285 | 8.582 | 2.67 |

c-Si_{3}N_{4} | 7.763 | - | - | 7.751 | - | - | 7.639 | - | - |

- | 7.773 ^{3} | - | - | - | - | - | - | - | |

- | 7.770 ^{4} | - | - | - | - | - | - | - | |

c-Si_{2}GeN_{4} | 7.934 | - | - | 7.918 | - | - | 7.767 | - | - |

- | 8.001 ^{5} | - | - | - | - | - | - | - | |

c-SiGe_{2}N_{4} | 8.111 | - | - | 8.095 | - | - | 7.909 | - | - |

- | 8.182 ^{6} | - | - | - | - | - | - | - | |

c-Ge_{3}N_{4} | 8.296 | - | - | 8.289 | - | - | 8.040 | - | - |

- | 8.288 ^{7} | - | - | - | - | - | - | - | |

- | 8.300 ^{8} | - | - | - | - | - | - | - |

**Table 2.**Calculated elastic constants C

_{ij}(in GPa), bulk modulus B

_{H}, shear modulus G

_{H}, and Young’s modulus E (in GPa), the ratio of B

_{H}/G

_{H}and Poisson’s ratio 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 compared with other calculated results.

Materials | C_{11} | C_{12} | C_{13} | C_{33} | C_{44} | C_{66} | B_{H} | G_{H} | E | B_{H}/G_{H} | v |
---|---|---|---|---|---|---|---|---|---|---|---|

t-Si_{3}N_{4} | 277 | 148 | 144 | 314 | 176 | 206 | 193 | 127 | 312 | 1.52 | 0.230 |

- | 277 ^{1} | 152 | 145 | 312 | 178 | 207 | 194 | 126 | 311 | 1.54 | - |

t-Si_{2}GeN_{4} | 254 | 138 | 137 | 278 | 158 | 171 | 179 | 110 | 274 | 1.63 | 0.245 |

t-SiGe_{2}N_{4} | 241 | 127 | 131 | 243 | 150 | 151 | 167 | 101 | 252 | 1.65 | 0.248 |

t-Ge_{3}N_{4} | 200 | 126 | 110 | 233 | 127 | 148 | 147 | 87 | 218 | 1.69 | 0.253 |

c-Si_{3}N_{4} | 524 | 177 | - | - | 333 | - | 293 | 256 | 595 | 1.14 | 0.161 |

- | 512 ^{2} | 177 | - | - | 331 | - | 289 | - | - | - | - |

- | - | - | - | - | - | - | 290 ^{7} | - | - | - | - |

c-Si_{2}GeN_{4} | 453 | 167 | - | - | 298 | - | 263 | 222 | 520 | 1.18 | 0.170 |

- | - | - | - | 258 ^{8} | - | - | - | - | |||

c-SiGe_{2}N_{4} | 440 | 151 | - | - | 251 | - | 247 | 201 | 474 | 1.229 | 0.179 |

- | 441 ^{3} | 150 | - | - | 129 | - | 247 | - | - | - | - |

- | 494 ^{4} | 172 | - | - | 288 | - | 283 | - | - | - | - |

c-Ge_{3}N_{4} | 375 | 140 | - | - | 223 | - | 218 | 172 | 409 | 1.27 | 0.187 |

- | 368 ^{5} | 145 | - | - | 223 | - | 220 | - | - | - | - |

- | - | - | - | - | - | - | 296 ^{9} | - | - | - | - |

m-Si_{3}N_{4} ^{6} | 257 | 36 | 131 | 353 | 93 | 92 | 165 | 110 | 270 | 1.51 | 0.228 |

m-Si_{2}GeN_{4} ^{6} | 242 | 37 | 126 | 316 | 76 | 85 | 154 | 97 | 241 | 1.59 | 0.239 |

m-SiGe_{2}N_{4} ^{6} | 235 | 42 | 164 | 363 | 60 | 42 | 170 | 78 | 202 | 2.20 | 0.303 |

m-Ge_{3}N_{4} ^{6} | 180 | 34 | 110 | 274 | 57 | 61 | 126 | 73 | 184 | 1.73 | 0.258 |

**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.

Materials | B_{V} | B_{R} | A_{B} | G_{V} | G_{R} | A_{G} | A^{U} |
---|---|---|---|---|---|---|---|

t-Si_{3}N_{4} | 193.22 | 193.74 | 0.123 | 140.60 | 112.87 | 10.940 | 1.231 |

t-Si_{2}GeN_{4} | 179.17 | 178.88 | 0.081 | 122.27 | 98.38 | 10.827 | 1.216 |

t-SiGe_{2}N_{4} | 167.22 | 167.20 | 0.006 | 112.57 | 89.59 | 11.367 | 1.283 |

t-Ge_{3}N_{4} | 147.15 | 146.98 | 0.058 | 99.47 | 74.95 | 14.058 | 1.637 |

c-Si_{3}N_{4} | 292.98 | 292.98 | 0 | 269.29 | 243.56 | 5.017 | 0.528 |

c-Si_{2}GeN_{4} | 262.92 | 262.92 | 0 | 236.36 | 207.84 | 6.421 | 0.686 |

c-SiGe_{2}N_{4} | 247.09 | 247.09 | 0 | 208.72 | 194.06 | 3.640 | 0.378 |

c-Ge_{3}N_{4} | 218.24 | 218.24 | 0 | 180.81 | 164.07 | 4.854 | 0.510 |

m-Si_{3}N_{4} ^{1} | 168.41 | 162.83 | 1.685 | 118.92 | 101.12 | 8.089 | 0.968 |

m-Si_{2}GeN_{4} ^{1} | 156.16 | 152.44 | 1.205 | 105.18 | 89.59 | 8.004 | 0.905 |

m-SiGe_{2}N_{4} ^{1} | 176.94 | 165.26 | 3.130 | 93.00 | 62.06 | 19.954 | 2.557 |

m-Ge_{3}N_{4} ^{1} | 129.33 | 123.55 | 2.286 | 82.36 | 63.68 | 12.791 | 1.513 |

^{1}Ref. [45].

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Han, C.; Chai, C.; Fan, Q.; Yang, J.; Yang, Y.
Structural, Electronic, and Thermodynamic Properties of Tetragonal *t*-Si* _{x}*Ge

_{3−x}N

_{4}.

*Materials*

**2018**,

*11*, 397. https://doi.org/10.3390/ma11030397

**AMA Style**

Han C, Chai C, Fan Q, Yang J, Yang Y.
Structural, Electronic, and Thermodynamic Properties of Tetragonal *t*-Si* _{x}*Ge

_{3−x}N

_{4}.

*Materials*. 2018; 11(3):397. https://doi.org/10.3390/ma11030397

**Chicago/Turabian Style**

Han, Chenxi, Changchun Chai, Qingyang Fan, Jionghao Yang, and Yintang Yang.
2018. "Structural, Electronic, and Thermodynamic Properties of Tetragonal *t*-Si* _{x}*Ge

_{3−x}N

_{4}"

*Materials*11, no. 3: 397. https://doi.org/10.3390/ma11030397