The stability, electronic properties, and hardness of SiN₂ under high pressure

Abstract

The structures of silicon nitride (SiN₂) under high pressures have been predicted using the developed particle swarm optimization algorithm for crystal structure prediction. The pyrite SiN₂ can be synthesized above 17 GPa, and the current thermodynamical calculations reveal that the pyrite-type structure is the most stable up to 100 GPa. No imaginary phonon frequencies in the whole Brillouin zone indicate the pyrite SiN₂ is dynamically stable. Strikingly, the mechanical analyses show the pyrite SiN₂ displays a behavior very similar to isotropy and high bulk modulus, shear modulus, and high simulated hardness indicate its very incompressible and superhard nature (63 GPa), which was driven by the strong covalent Si–N bonds and localized nonbonding.

---

Introduction

Superhard materials are of great importance due to their science and technology applications such as scratch-resistant coatings to polishing and cutting tools, etc.^1–3 Extensive experiments and theoretical work have been performed with a focus on searching novel superhard materials and understanding the properties of those. Light elements e.g., boron, carbon, nitrogen, and oxygen can form strong covalent compounds^4–6 which can withstand both elastic and plastic deformations. Moreover, pressure can also be seen as an effective tool for increasing the density of materials and altering the electronic bonding states to induce the formation of new superhard materials or modify their physical properties.⁷ With these ideas, designing novel superhard materials under high pressure is a scientifically effective approach.

The triple N≡N bond in nitrogen is the strongest bond known, however, the triple bond collapses to form the N–N bond at moderate pressures,^8–10 which can help to form new nitrides with other elements, moreover, these nitrides, for example silicon nitride, have novel physical properties. Si₃N₄ is an important ceramic with many diverse applications, especially, at high temperature and under corrosive environments. To date, there are three reported stable phases of Si₃N₄: the trigonal (α-Si₃N₄), hexagonal (β-Si₃N₄), and cubic (γ-Si₃N₄) phases.^11–13 Theoretical calculations have shown that the β-Si₃N₄ phase was more stable than the α-Si₃N₄ phase at 0 K. Under high pressure and high temperature, α-Si₃N₄ and β-Si₃N₄ can transform into γ-Si₃N₄. As for silicon nitrides, the other form is SiN₂, which is considered as a pyrite-type structure with a space group of Pa3̄ that is consistent with CN₂ and GeN₂.¹⁴ Recently, first principles studies¹⁵ have showed that bct-CN₂ is energetically much superior to the previously proposed pyrite structure under high pressures and bct-CN₂ has high calculated ideal strength, bulk modulus, shear modulus, and simulated hardness. However, the structures of SiN₂ under high pressure are unclear, which are well worth exploring, moreover, the stability, electronic properties, and hardness of SiN₂ under high pressure are the subjects of our research.

In this paper, we have predicted the structures of SiN₂ under high pressures by recently developed particle swarm optimization (PSO) algorithm^16–23 on crystal structural prediction. The pyrite SiN₂ can be synthesized above 17 GPa, and the current thermodynamical calculations reveal that the pyrite-type structure is the most stable up to 100 GPa. The electronic calculations indicate the pyrite SiN₂ is insulator by the energy band gap 5.13 eV, and the strong covalent bonds are formed by the hybridizations between Si and N electrons. For further understanding the mechanical properties, the anisotropy, high bulk modulus, high shear modulus, and high simulated hardness are uncovered, which indicate its very incompressible and superhard nature (63 GPa), which was driven by the strong covalent Si–N bonds and localized nonbonding.

Computational methods

Searches for candidate high-pressure structures (0–100 GPa) of SiN₂ are performed using the variable-cell PSO simulations^16–23 with the CALYPSO code²⁰ developed for crystal-structure prediction and up to 12 atoms in the unit cell. The underlying ab initio structure relaxations are performed using density functional theory^24,25 within the Perdew–Burke–Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA)²⁶ as implemented in the Vienna ab initio simulation package VASP code.²⁷ The all-electron projector-augmented wave (PAW)²⁸ method is adopted with the choices of 3s²3p² and 2s²2p³ cores for Si and N atoms, respectively. The plane-wave kinetic energy cutoff of 800 eV and a k-mesh of 0.03 Å⁻¹ grid spacing within the Monkhorst–Pack scheme²⁹ for sample the Brillouin zone are used, which give an excellent convergence of the total energies, energy differences, and structural parameters. In order to obtain the elastic properties, ab initio pseudopotential plane-wave density functional method implemented in the CASTEP code³⁰ has been used. Exchange and correlation effects are described in the scheme of Perdew–Burke–Ernzerhof of GGA.²⁶ Vanderbilt ultrasoft pseudopotentials³¹ were generated for Si and N with the valence configuration of 3s²3p² and 2s²2p³, respectively. The cutoff energy and k-mesh are same as the setting in VASP calculations. The phonon frequencies are calculated using direct supercell method³² with the PHONOPY code.³³

Results and discussion

Searches of the structures within 2, 3 and 4 formula units (f.u.) are performed in the pressure range of 0–100 GPa by using the pso method. Analysis of the predicted structures give us a shortlist of candidate structures with space groups Pa3̄ (4 f.u. per cell), C2m (6 f.u. per cell), Cmc2₁ (4 f.u. per cell), Pnma (4 f.u. per cell), P6₃/mmc (2 f.u. per cell), and R3̄m (3 f.u. per cell), as schematically shown in Fig. 1. Pa3̄ structure (pyrite structure) predicted here is the same as the structure proposed by Weihrich et al.¹⁴ using a substitution method with the knowledge of isoelectronic SiP₂ structure. Si atoms occupy the center of the octahedral and one Si atom bonds to six N atoms forming SiN₆ cluster. The other five structures reported here for the first time have N–N bonds that are different from pyrite-type structure. The structural details of the lattice parameters and the atomic positions are listed in Table 1 For further investigating the stability of the structures, the thermodynamic stability of the various predicted structures need to calculate the formation enthalpies at a large pressure range from 0 to 100 GPa according to the reaction:

Fig. 1 Crystal structures of competing SiN₂ phases at 0–100 GPa.

Table 1 Predicted lattice constants and atomic coordinates, as referred to the conventional unit cells, of the predicted structures at 0 GPa

Space group	Lattice parameter (Å)	Atomic coordinates (fractional)
Pa3̄	a = 4.455	Si 4b (0.5, 0, 0)
		N 8c (0.4049, 0.9048, 0.5951)
Pnma	a = 7.181, b = 2.823, c = 4.491	Si 4c (0.3731, 0.25, 0.3174)
		N 4c (0.7942, 0.25, 0.5788)
		N 4c (0.9547, 0.25, 0.7797)
C2/m	a = 10.254, b = 2.824, c = 9.051, β = 144.8	Si 2c (0, 0, 0.5)
		Si 4i (0.5543, 0, 8261)
		N 4i (0.3365, 0, 0.4424)
		N 4i (0.8870, 0, 0.7461)
		N 4i (0.8173, 0, 0.1688)
P63/mmc	a = 2.809, c = 6.764	Si 2a (0, 0, 0)
		N 4f (0.6667, 0.3331, 0.6418)
Cmc21	a = 2.906, b = 7.487, c = 4.548	Si 4a (0, 0.8566, 0.1686)
		N 4a (0, 0.9327, 0.5367)
		N 4a (0, 0.7839, 0.7410)
R3̄m (rhombohedral representation)	a = 4.716, α = 108.8	Si 3d (0.5, 0, 0)
		N 6h (0.6563, 0.6563, 0.0391)

---

To get a more realistic prediction, both the experimental³⁴ and computational^35,36 high-pressure (low temperature) phase orders of nitrogen were adopted. The α-Si₃N₄, and β-Si₃N₄ was chosen as the reference phases in our calculations. Fig. 2 shows the curves of the formation enthalpy for the predicted structures at high pressures. Pyrite SiN₂ can be formed at 17 GPa and becomes the most stable phase until 100 GPa.

Fig. 2 The enthalpy curves relative to Si₃N₄ + N₂ as a function of pressure.

It is of fundamental importance to check the phonon spectra of the pyrite SiN₂ due to its providing the information of the structural stability. The calculations of the phonon frequencies are performed at 0 GPa and 20 GPa, respectively. The phonon dispersion curves along the high symmetry direction are shown in Fig. 3. It is clear that no imaginary phonon frequency exists in the whole BZ, indicating the dynamical stability of the phase. As the pressure increases, all modes shift to higher frequencies. The results show that pyrite SiN₂ will be dynamically stable up to 100 GPa.

Fig. 3 The phonon dispersion curves for pyrite SiN₂. (a) at 0 GPa and (b) at 20 GPa.

The calculations of the elastic constants are essential, which can help to investigate the mechanical properties of the materials that are important in the industry applications, and also to be regarded as the mechanical stability criteria.³⁷ The elastic constants C_ij of the pyrite SiN₂ have been calculated using the CASTEP code and are listed as the following: C₁₁ = 704 GPa, C₄₄ = 379 GPa, C₁₂ = 127 GPa, which satisfy the mechanical criteria: C₁₁ > 0, C₄₄ > 0, C₁₁ > |C₁₂|, and (C₁₁ + 2|C₁₂|) > 0. The elastic anisotropy factors³⁸ providing the convenient measure of in-plane phonon focusing are studied for the pyrite SiN₂. C_L = C₄₄ + (C₁₁ + C₁₂)/2 = 794.5 GPa; A_100 = 2C₄₄/(C₁₁ − C₁₂) = 1.31; A_110 = C₄₄ (C_L + 2C₁₂ + C₁₁)/(C_LC₁₁ − C₁₂²) = 1.22. It is found that the elastic anisotropy for the [100] planes is close to that for the [110] planes. Poisson's ratio (ν) provides important information about the characteristics of the bonding forces; for example, it characterizes the stability of the crystal against shearing strain. Our results show that the Poisson's ratio of the pyrite SiN₂ is 0.15. Due to such a low Poisson's ratio, the pyrite-type SiN₂ would be isotropic. The elastic anisotropy can be represented by the Young's modulus on a direction in a crystal.³⁹ For the cubic materials, it can be described by the following equation:

E⁻¹ = s₁₁ − β₁(α²β² + α²γ² + β²γ²)

where α, β, and γ are the direction cosines of the tensile stress direction, β₁ = 2s₁₁ − 2s₁₂ − s₄₄ < 0, and s₁₁, s₁₂, and s₄₄, are the elastic compliance constants. The elements of the matrix representation of tensor s have the relationships with those of the matrix representation of tensor C in Voigt's contracted notation as follows: s₁₁ = (C₁₁ − C₁₂)/(C₁₁² + C₁₁C₁₂ − 2C₁₂²) = 1.044 × 10⁻³ GPa⁻¹, s₁₂ = C₁₂/(C₁₁² + C₁₁C₁₂ − 2C₁₂²) = 2.3 × 10⁻⁴ GPa⁻¹, and s₄₄ = 1/C₄₄ = 2.600 × 10⁻³ GPa⁻¹. From the elastic compliance constants above we can get β₁ = 9.7 × 10⁻³ GPa⁻¹, which is very close to zero indicating the pyrite SiN₂ display a behavior very similar to isotropy, that is consistent with the calculated Poisson's ratio.

The calculated bulk modulus can reach 320 GPa, which imply the pyrite SiN₂ is an ultra-incompressible material. The shear modulus is 343 GPa estimated by using the Wu's results,³⁷ that can resist to the reversible deformation under shear strain and provide the proofs for the superhardness. The simulated hardness of the pyrite SiN₂ is 63 GPa calculated with the model by Xing-Qiu Chen et al.,⁴⁰ which is closed to c-BN (66 GPa) and BC₅ (71 GPa) indicating pyrite SiN₂ is a superhard material. In order to better understand the origin of the superhardness, we examine the electronic band structure, electronic partial density of states, and electron localization functions (ELFs). The electronic band and partial density of states calculations indicate the pyrite SiN₂ is insulator by the energy band gap 5.13 eV as plotted in Fig. 4(a) and (b). The ELFs provide the information for distinguishing the ionic, metallic, and covalent bonding. High ELF (≥0.8) indicates the formation of covalent bonds. Fig. 4(c) shows the isosurface of ELF = 0.88, it is clear that one Si atom bonds to six N atoms and form the strong covalent directional bonds. The electronic partial density of states is depicted by Fig. 4(b) and characterizes two aspects: One is that the strong covalent directional bonds are formed by the hybridizations between Si s, p electrons and N s, p electrons. The other is the excess electrons of N p electrons in the energy rang of −10 eV and 0 eV cannot form the covalent bonds, but form stable and localized non-bonded states, which share the same properties as bct-CN₂. The strong covalent Si–N bonds and localized nonbonding are responsible for its high bulk shear modulus and simulated hardness at equilibrium.

Fig. 4 (a) The electronic band structure at 0 GPa (b) the electronic partial density of states at 0 GPa (c) the electron localization functions (ELFs) at 0 GPa.

Conclusion

In summary, the structures of silicon nitride (SiN₂) under high pressures have been predicted using the developed particle swarm optimization algorithm for crystal structure prediction. The pyrite SiN₂ can be synthesized above 17 GPa, and the current thermodynamical calculations reveal that the pyrite-type structure is the most stable up to 100 GPa. No imaginary phonon frequencies in the whole Brillouin zone indicate the pyrite SiN₂ is dynamically stable. The electronic calculations show the pyrite SiN₂ is insulator by the energy band gap 5.13 eV, and the strong covalent bonds are formed by the hybridizations between Si and N electrons. Strikingly, the mechanical analyses show the pyrite SiN₂ display a behavior very similar to isotropy and high bulk modulus, shear modulus, and high simulated hardness indicate its very incompressible and superhard nature, which was driven by the strong covalent Si–N bonds and localized nonbonding. We believe that the current study will contribute to the experimental synthesis and determination of silicon nitrides and understanding the novel properties of the materials.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (nos 11104019, 11104102). Parts of calculations were performed in the Scientific Computation and Numerical Simulation Center of Changchun University of Science and Technology.

References

1. A. G. Thornton and J. Wilks, Nature, 1978, 274, 792.
2. R. B. Kaner, J. J. Gilman and S. H. Tolbert, Science, 2005, 308, 1268.
3. V. L. Solozhenko, D. Andrault, G. Fiquet, M. Mezouar and D. C. Rubie, Appl. Phys. Lett., 2001, 78, 1385.
4. X. Hao, Y. Xu, Z. Wu, D. Zhou, X. Liu and J. Meng, J. Alloys Compd., 2008, 453, 413.
5. F. Tian, J. Wang, Z. He, Y. Ma, L. Wang, T. Cui, C. Chen, B. Liu and G. Zou, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 235431.
6. Z. Wang, Y. Zhao, P. Lazor, H. Annersten and S. K. Saxena, Appl. Phys. Lett., 2005, 86, 041911.
7. Z. Zhao, K. Bao, D. Li, D. Duan, F. Tian, X. Jin, C. Chen, X. Huang, B. Liu and T. Cui, Sci. Rep., 2014, 4, 04797.
8. W. J. Nellis, N. C. Holmes, A. C. Mitchell and M. van Thiel, Phys. Rev. Lett., 1984, 53, 1661.
9. M. I. Eremets, R. J. Hemley, H.-k. Mao and E. Gregoryanz, Nature, 2001, 411, 170.
10. Y. Ma, A. R. Oganov, Z. Li, Y. Xie and J. Kotakoski, Phys. Rev. Lett., 2009, 102, 065501.
11. W.-Y. Ching, Y.-N. Xu, J. D. Gale and M. Rühle, J. Am. Ceram. Soc., 1998, 81, 3189.
12. C. Dong and Y. Ben-Hai, Chin. Phys. B, 2013, 22, 023104.
13. A. Mirgorodsky, M. Baraton and P. Quintard, Phys. Rev. B: Condens. Matter Mater. Phys., 1993, 48, 13326.
14. R. Weihrich, V. Eyert and S. F. Matar, Chem. Phys. Lett., 2003, 373, 636.
15. Q. Li, H. Liu, D. Zhou, W. Zheng, Z. Wu and Y. Ma, Phys. Chem. Chem. Phys., 2012, 14, 13081.
16. L. Zhu, Z. Wang, Y. Wang, G. Zou, H.-k. Mao and Y. Ma, Proc. Nutr. Soc., 2012, 109, 751.
17. H. Liu, Q. Li, L. Zhu and Y. Ma, Phys. Lett. A, 2011, 375, 771.
18. P. Li, G. Gao, Y. Wang and Y. Ma, J. Phys. Chem. C, 2010, 114, 21745.
19. Y. Wang, J. Lv, L. Zhu and Y. Ma, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 094116.
20. Y. Wang, M. Miao, J. Lv, L. Zhu, K. Yin, H. Liu and Y. Ma, J. Chem. Phys., 2012, 137, 224108.
21. G. Gao, H. Wang, L. Zhu and Y. Ma, J. Phys. Chem. C, 2011, 116, 1995.
22. H. Wang, J. S. Tse, K. Tanaka, T. Iitaka and Y. Ma, Proc. Natl. Acad. Sci. U. S. A., 2012, 109, 6463.
23. Y. Wang, H. Liu, J. Lv, L. Zhu, H. Wang and Y. Ma, Nat. Commun., 2011, 2, 563.
24. S. Baroni, P. Giannozzi and A. Testa, Phys. Rev. Lett., 1987, 58, 1861.
25. P. Giannozzi, S. De Gironcoli, P. Pavone and S. Baroni, Phys. Rev. B: Condens. Matter Mater. Phys., 1991, 43, 7231.
26. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
27. G. Kresse and J. Furthmüller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169.
28. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953.
29. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188.
30. M. Segall, P. J. Lindan, M. Probert, C. Pickard, P. Hasnip, S. Clark and M. Payne, J. Phys.: Condens. Matter, 2002, 14, 2717.
31. D. Vanderbilt, Phys. Rev. B: Condens. Matter Mater. Phys., 1990, 41, 7892.
32. K. Parlinski, Z. Li and Y. Kawazoe, Phys. Rev. Lett., 1997, 78, 4063.
33. A. Togo, F. Oba and I. Tanaka, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 134106.
34. E. Gregoryanz, A. F. Goncharov, C. Sanloup, M. Somayazulu, H.-k. Mao and R. J. Hemley, J. Chem. Phys., 2007, 126, 184505.
35. C. J. Pickard and R. J. Needs, Phys. Rev. Lett., 2009, 102, 125702.
36. X. Wang, Y. Wang, M. Miao, X. Zhong, J. Lv, T. Cui, J. Li, L. Chen, C. J. Pickard and Y. Ma, Phys. Rev. Lett., 2012, 109, 175502.
37. Z.-j. Wu, E.-j. Zhao, H.-p. Xiang, X.-f. Hao, X.-j. Liu and J. Meng, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 054115.
38. K. Lau and A. K. McCurdy, Phys. Rev. B: Condens. Matter Mater. Phys., 1998, 58, 8980.
39. A. Cazzani and M. Rovati, Int. J. Solids Struct., 2003, 40, 1713.
40. X.-Q. Chen, H. Niu, D. Li and Y. Li, Intermetallics, 2011, 19, 1275.

---