1 Introduction

Si3N4 is widely used as an important ceramic with many industrial applications, such as under high temperature and corrosive environments. Three polymorphs for Si3N4 exist: α-Si3N4, β-Si3N4, and γ-Si3N4. Both α-Si3N4 and β-Si3N4 have hexagonal structures P31/c and P63/m and can be produced at ambient conditions [1, 2]. γ-Si3N4 has a cubic structure. It can be produced at temperatures above 2000 K and pressures above 15 GPa [3, 4]. It is found that β-Si3N4 is a more stable phase [5]. Recently, Cui et al. have predicted three potential hard metastable phases of t-Si3N4, m-Si3N4, and o-Si3N4 using a recently developed particle swarm optimization method within the CALYPSO software package [6]. In this paper, we focus on the β-Si3N4. In recent years, frequent experimental and theoretical attempts have been used to study the β-Si3N4 [7–14]. On the experimental side, Flammini et al. have carried out the thermal oxidation of the β-Si3N4 (0001)-8 × 8 surface [10]. On the theoretical side, Ching et al. have optimized the experimentally observed phases of Si3N4 crystal structures by employing first principles calculation [11]. Kuwabara et al. have calculated the lattice dynamics of α-Si3N4, β-Si3N4, and γ-Si3N4 phases by interatomic force constants in the real space method combined with first-principles calculations [12]. Lu et al. have performed to clarify the shear mechanical property of β-Si3N4 nano-thin layer in the basal plane with different extreme strain rates and loading temperatures via molecular dynamics simulations [13]. Jiang et al. have also investigated the different roles of Lu and La in the intergranular film in β-Si3N4 on the growth morphology investigated via molecular dynamics simulations [14].

Although there are some studies, the mechanical anisotropy properties of β-Si3N4 are still limited both experimentally and theoretically as far as we are aware. Therefore, we here have extended the mechanical properties and presented the Young’s modulus of β-Si3N4 in detail.

2 Methods of Calculation

β-Si3N4 crystallize in the hexagonal structure belonging to the space group of P63/m (No. 176). The atomic positions are N 2c (0.3333, 6667, and 0.25), 6h (0.3302, 0.0301, and 0.25), and Si 6h (0.1751, 0.7689, and 0.25). The experimental lattice parameter a and c were determined to be 7.6069 and 2.9091 Å from the experimental data [15].

The structure and lattice dynamics of β-Si3N4 calculations are carried out by using the Cambridge Serial Total Energy (CASTEP) code [16], based on density functional theory (DFT) by using Vanderbilt-type ultrasoft pseudopotentials [17] and density functional perturbation theory (DFPT) [18] by using norm-conserving pseudopotential [19], respectively. The effects of exchange-correlation interaction are descried with the local density approximation (LDA) of the Ceperley-Alder data as parameterized by Perdew-Zunger (CA-PZ) [20]. Considering accuracy, we choose the cutoff energy to be in 520 eV, and the Brillouin-zone sampling mesh parameters for the k-point set to be 7 × 7 × 8 in this work for every calculation. For calculation of the elastic constants, we adopted the same cutoff energy and Monkhorst-Pack mesh as the structure. Through the method described above, Moreira et al. have successfully investigated the elastic properties of some materials [21]. It suggests that the simulation method adopted in this work is reasonable and reliable, giving us confidence in the following calculations.

The thermal properties calculations of β-Si3N4 are carried out by using the quasi-harmonic Debye model [22, 23].

3 Results and Discussions

3.1 Structural Properties

By performing ab initio density functional calculations using LDA functional within CA-PZ type, we obtained the equilibrium lattice constants for a = 7.5152 Å and c = 2.8702 Å, which are very close to the experimental works: a = 7.6069 Å and c = 2.9091 Å [15, 24] and other theoretical results [8, 11, 25–27]. The above optimized lattice constants are listed in Table 1 together with the available experimental data and theoretical results. Then we further obtained the lattice constants a, c and the relative lattice constants a/a 0, c/c 0, and relative volume V/V 0 as a function of the applied pressure are given in Figs. 1 and 2, respectively, along with Li et al. XRD experimental data [11]. We found that present calculated values are consistent with Li et al. XRD experimental data. It is also worth seeing that the lattice constants a of β-Si3N4 decreases more with applied pressure than the lattice constants c in Fig. 1.

Table 1 Calculated lattice constants a (Å), c (Å), and c/a of β-Si3N4, together with experimental data and theoretical results
Full size table
Fig. 1
figure 1

The lattice constants a and c as a function of pressure at zero temperature

Full size image
Fig. 2
figure 2

The relative lattice constants a/a 0, c/c 0, and relative volume V/V 0 as a function of pressure at zero temperature

Full size image

3.2 Elastic Properties

For hexagonal P63/m structure, the number of the independent elastic constants reduces to five C 11, C 12, C 13, C 33, and C 44. In order to check the influence of these elastic constants (C11, C 12, C 13, C 33 and C 44) on the mechanical stability, we referred to the Born stability criteria [28]. The calculated elastic constants of β-Si3N4 under 0 pressure and high pressure up to 35 GPa are given in Table 2. From Table 2, we found C11, C 12, C 13, C 33, and C 44 still obey the Born stability criteria up to 35 GPa, indicating that β-Si3N4 is still mechanically stable up to 35 GPa. The calculated elastic constants of β-Si3N4 under 0 pressure are given in Table 2, and we compared them with experimental data by using Brillouin scattering experimental [9] and other similar theoretical calculations [29–32]. From it, we can find that the present calculated elastic constants are more close to the experimental values [9]. However, our obtained results are also quite consistent with other available calculations obtained by using a force field molecular dynamics simulation, based on ab initio calculations [29]. In Fig. 3, we also plotted the variation of elastic constants of β-Si3N4 with respect to the variation of pressure. It is found from Fig. 3 that the elastic constants C11, C 12, C 13, and C 33 linearly increase monotonously when pressure is enhanced. Moreover, C 44 decreases very slowly with the elevated pressure.

Table 2 Calculated elastic constants C ij (GPa) of β-Si3N4 under 0 pressure up to 35 GPa, together with experimental data and theoretical results
Full size table
Fig. 3
figure 3

The present LDA calculated elastic constants of β-Si3N4 as a function of pressure at zero temperature

Full size image

Base on the obtained elastic constants of β-Si3N4, we also investigated its polycrystalline elastic properties, such as Voigt bulk modulus (B V), the Reuss bulk modulus (B R), the Voigt shear modulus (G V), and the Reuss shear modulus (G R). By using the Voigt-Reuss-Hill average method [33, 34], the bulk modulus B = (B R + B V)/2 and the shear modulus G = (G R + G V)/2 are derived, then the Young’s modulus E is also obtained: E = 9BG/(3B + G) using the above B and G. The calculated results are summarized in Table 3. At T = 0 K and P = 0 GPa, the calculated bulk modulus B is 257.84 GPa, which is much accords with the experimental data of 259 GPa by using Brillouin scattering [9].

Table 3 The present LDA calculated bulk modulus B (GPa), shear modulus G (GPa), Young Modulus E (GPa), Poisson’s ratio σ, the ratio of the shear modulus G to the bulk modulus B (B/G), and the compressional wave velocities V l (km/s), the shear wave velocities V t (km/s), the average wave velocity V m (km/s), and Debye temperature Θ (K) of the β-Si3N4 under high pressures at 0 K
Full size table
Table 4 The values of heat capacity of constant volume C V (J/mol*K), heat capacity of constant pressure C P (J/mol* K), Entropy S (J/mol*K), Grüneisen parameter γ, and thermal expansion coefficient α (10−5/K) for β-Si3N4 at 300 K under different pressures
Full size table

The ratio B/G has been proposed to evaluate the ductile or brittle character of materials by Pugh [35]. The critical ratio B/G value is 1.75. If B/G > 1.75, the material behaves in a ductile manner, otherwise, B/G < 1.75 in a brittle manner. From Table 3, it can be found that the ratio of B/G is above 1.75 from 1.91 of 0 GPa to 3.19 of 35 GPa, however, the values increase with increasing pressure gradually. Thus, it becomes more and more ductile with increasing applied pressure. The Poisson’s ratios of β-Si3N4 are calculated using the formula: σ = (3B −2G)/[2(3B + G)], which are listed in Table 3.

Finally, we obtained the compressional wave velocities V l, the shear wave velocities V t, and the average wave velocity V m under 0 GPa up to 35 GPa, according to Navier’s equations [36]. Three wave velocities V l, V t, and V m are listed in Table 3. The Debye temperature (Θ) can be deduced from the above wave velocity and summarized in Table 3. Our calculated Θ at T = 0 K and P = 0 GPa is 964.41 K (see Table 3), which is quite consistent with experimental data 923 ± 5 K [7]. This indicates that our calculated Debye temperature values under high pressure also are trustable.

In order to display the elastic anisotropy more visually, we presented the direction-dependent Young’s modulus (E). The 3D figure of directional dependences of the reciprocal of E for the β-Si3N4 can be defined by the following equation due to crystal structure [37]. We plotted the E in Fig. 4a. It is shown clearly that the E shows a high degree of anisotropy along different crystallographic. From the projections of E in Fig. 4b, the anisotropy of the a-b plane seems alike, but that of the a-c or b-c planes is serious, which reveals the elastic anisotropy of β-Si3N4.

Fig. 4
figure 4

a The direction dependence of Young’s modulus under zero pressure and b its projections onto the a-b, b-c, and a-c plane for β-Si3N4 under zero pressure (unit, GPa)

Full size image

3.3 Lattice Dynamics

Figure 5 plot the phonon dispersion curves along with the total phonon density of states of β-Si3N4 at 0 GPa. It can be seen that the phonon dispersion curves and total phonon densities of states at 0 GPa agree well with previous calculation [12] and experimental data [38], they show similar profiles. From Figs. 5 and 6, there is a shift of position under pressure at P = 0 and 35 GPa. Additionally, a stable crystalline structure requires all phonon frequencies to be positive. Figures 5 and 6 shows the phonon dispersion curves of β-Si3N4 at 0 and 35 GPa. As what is shown in Figs. 5 and 6, no imaginary phonon frequency was found in the whole Brillouin zone direction, indicating the dynamical stabilities of β-Si3N4 up to 35 GPa.

Fig. 5
figure 5

Phonon dispersion curves (a) and Phonon density of states (b) for β-Si3N4 at T = 0 K and P = 0 GPa

Full size image
Fig. 6
figure 6

Phonon dispersion curves (a) and Phonon density of states (b) for β-Si3N4 at T = 0 K and P = 35 GPa

Full size image

3.4 Thermal Properties

By using the quasi-harmonic Debye model, we obtained the thermal properties of β-Si3N4 under 0 GPa up to 35 GPa and 0 K up to 800 K.

Debye temperature (Θ) as the function of the pressure and temperature illustrated by our calculated results is plotted in Fig. 7. It can be found that Θ is nearly constant at low temperature and decreases linearly with increasing temperature as shown in Fig. 7a. When the temperature is constant, the Θ increases almost linearly with applied pressure as shown in Fig. 7b. In the quasi-harmonic Debye model used here, the Θ of β-Si3N4 turns out to be 1093.18 K at the T = 0 and P = 0, which is a bit larger the value 964.41 K obtained in terms of our elastic constants and experimental data 923 ± 5 K [7].

Fig. 7
figure 7

Debye temperature (Θ) versus temperature at various pressures (a) and Debye temperature (Θ) versus pressure at various temperature (b). We have also calculated the heat capacity, entropy, Grüneisen parameter, and thermal expansion coefficient of β-Si3N4 at 300 K under different pressures. The theoretical results are presented in Table 4. No experimental and theoretical data of thermal parameters as a function of pressure and temperature are available in the literature for comparison

Full size image

4 Conclusions

In summary, we have presented a theoretical study the structural, elastic and thermal properties β-Si3N4 using GGA-PBE and quasi-harmonic Debye model. Our theoretical results of lattice constants a and c are found to be in good agreement with experimental data. Through calculations, the elastic constants, bulk modulus, shear modulus, Young Modulus, Poisson’s ratio, the ratio of the shear modulus to the bulk modulus, the wave velocities, and Debye temperature under high pressure and the mechanical anisotropies of β-Si3N4 are also calculated for the first time. By the elastic stability criteria and phonon calculation, it is predicted that β-Si3N4 is stable up to 35 GPa. Moreover, we predict the thermal properties with pressure and temperature.