First-principles investigation on elastic and thermodynamic properties of Pnnm-CN under high pressure

The elastic anisotropy and thermodynamic properties of the recently synthesized Pnnm-CN have been investigated using first-principles calculations under high temperature and high pressure. The calculated equilibrium crystal parameters and normalized volume dependence of the resulting pressure agree with available experimental and theoretical results. Within the considered pressure range of 0–90 GPa, the dependences of the bulk modulus, Young’s modulus, and shear modulus on the crystal orientation for Pnnm-CN have been systematically studied. The results show that the Pnnm-CN exhibits a well-pronounced elastic anisotropy. The incompressibility is largest along the c-axis. For tension or compression loading, the Pnnm-CN is stiffest along [001] and the most obedient along [100] direction. On the basis of the quasi-harmonic Debye model, we have explored the Debye temperature, heat capacity, thermal expansion coefficient, and Grüneisen parameters within the pressure range of 0–90 GPa and temperature range of 0–1600K.


I. INTRODUCTION
In material science, the superhard materials have attracted considerable interest owing to their farranging applications in cutting, polishing tools, and wear-resistant coatings. 1 Because the compounds consisting of light elements display relatively short and strong covalent bonds, they usually possess low-compressibility and high hardness. Therefore, scientists made a great effort to synthesize or theoretically predict new covalent compounds. In 1989, Liu and Cohen 2 suggested a new compound β-C 3 N 4 . which to be ultra-incompressible with a considerable bulk modulus (427 GPa) as that (442 GPa) of diamond. 3 Subsequently, numerous studies searching for new CN compounds were carried out. [4][5][6][7][8][9][10][11] Experiments selected different carbon-nitrogen rich compounds as precursors to synthesize CN phases, such as cyanamide, melamine, and other related triazine-based compounds. [12][13][14] However, it is a faced challenge to determine the crystal structures, chemical compositions and internal atomic arrangement. To solve this problem, several theoretical approaches were employed to predict new CN materials.
One of the typical CN phase is the predicted Pnnm-CN, which has been successful synthesized by recent experiment. In 2012, Wang 15 predicted an orthorhombic Pnnm-CN as the energetically most stable structure for carbon mononitride below 100 GPa. The theoretical results shown that the Pnnm-CN possesses the highest hardness (62.3 GPa) and can be synthesized using graphite and nitrogen as precursors at pressure of 10.9 GPa. Further interest is the experimental synthesis of Pnnm-CN under the condition of 55 GPa and 7000 K, as reported by Stavrou et al.. 16 In their work, the pressure dependence of the lattice parameters of Pnnm-CN was discussed. Moreover, they repoted that the anisotropic behavior of Pnnm-CN was the high compressibility of a-axis. More recently, Tang et al. examined its mechanical properties by simulating the strain-stress relations at large strains. They pointed out that the weakest peak tensile stress of 41 GPa in the <100> direction and strongest peak tensile stress of 94 GPa in the <001> direction for Pnnm-CN. 17 From above discussion, it is remarkable that the Pnnm-CN shows anisotropic behavior. Furthermore, few systematic studies on the elastic anisotropy and thermodynamic properties of the Pnnm-CN have been carried out until now.
In present work, the elastic properties of Pnnm-CN under pressure up to 90 GPa are studied, from which the elastic anisotropy is also found. Then the quasi-harmonic Debye model have been employed to explore the thermodynamic properties of the Pnnm-CN.

II. COMPUTATIONAL METHODS
All first-principles calculations have been performed with the VASP package 18 using the Perdew-Burke-Ernzrehof (PBE) generalized gradient approximation (GGA). 19 The all electron projector augmented wave (PAW) method is adopted valence electrons of 2s 2 p 2 and 2s 2 p 3 for C and N atoms, respectively. The calculations of total energy and stress selected the energy cutoff of 800 eV and appropriate Monkhorst-Pack k meshes 20 of 8×10×16. The elastic constants of the Pnnm-CN under different pressure have been obtained via strain-stress approach. In the light of the Voigt-Reuss-Hill approximation, 21 one can calculate the bulk modulus, shear modulus, Young's modulus, and Poisson's ratio. Furthermore, the thermodynamics properties of the Pnnm-CN are also investigated according to the quasi-harmonic Debye model.

A. Structural properties
The crystal structure of the Pnnm-CN is displayed in Fig. 1. In Table I, we list our calculated lattice parameters as well as the previous experimental and theoretical data. It is clear that our results are completely closed to the theoretical values at 0 GPa and 10 GPa. 15,17 The mismatch of the lattice parameters at 55 GPa is within 6% in comparison with recent experimental data. 16   relationships of the normalized parameters a/a 0 , b/b 0 , c/c 0 , and V/V 0 against pressure are shown in Fig. 2, where a 0 , b 0 , c 0 , and V 0 are the equilibrium structural parameters at 0 GPa and 0 K. The fitting relationships at 0 K are to found: One can see that the incompressibility is the largest along the c-axis, whereas it is smallest along a-axis. The clear elastic anisotropy of Pnnm-CN is displayed. The low incompressibility of a-axis might originate from the tilting of the C-C dumbbells with respect to a-axis. In addition, we notice that the incompressibility of volume for Pnnm-CN is better than that of cg-CN at high pressure, although it is lower than that of c-BN.

B. Elastic properties
By strain-stress method, the elastic constants C ij of Pnnm-CN are calculated and listed in Table II. From the table, one can find that the present data accord with the values reported in Refs. 10 and 17 at 0 GPa and 0 K. In Fig. 3, the variations of elastic constants with pressure up to 90 GPa are plotted. It is found that the value of C ij against the applied pressure P increase monotonically. Relatively, the values of C 33 increase sharply against the pressure growing from 0 to 90 GPa, while those of C 44 and C 55 are slower. Up to 90 GPa, the C ij still satisfy the condition of the Born-Huang criteria, 22 illustrating that the Pnnm-CN is mechanical stable at high pressure. This is consistent with the result  reported by Dong et al. 23 According to the Voigt-Reuss-Hill approximation, we can obtain the bulk modulus B and shear modulus G, as listed in Table II 25 suggested that the Poisson's ratio ν of the brittle materials should be lower than 1/3. From the Table II, the lower values (0.139-0.242) of ν illustrates that the Pnnm-CN is a brittle material at the pressure range of 0-90 GPa.
In the crystal physics and engineering science, the elastic anisotropy is an important index for materials. Elastic anisotropy can provide an expectation of the atoms arrange, the bonding properties, and some chemical characters in different directions of materials. 26 It is well known that the shear anisotropic factor can reflect the level of anisotropy for different planes. For the (100) shear plane between the [011] and [010] directions, the shear anisotropic factor A 1 can be written as the following formula: 27 For an isotropic materials, the shear anisotropy factors must be 1.0. Any departure from 1.0 can reflect the level of elastic anisotropy. From the in which S ij represent the elastic compliance constants given by Nye. 29 The α, β, and γ represent the direction cosines of [uvw] direction. The shear modulus G on the (hkl) shear plane with the shear stress applied alone the [uvw] direction is written as 30 G −1 = 4S 11 α 2 1 α 2 2 + 4S 22 β 2 1 β 2 2 + 4S 33 γ 2 1 γ 2 2 + 8S 12 α 1 α 2 β 1 β 2 + 8S 23 β 1 β 2 γ 1 γ 2 + 8S 13 α 1 α 2 γ 1 γ 2 + S 44 ( β 1 γ 2 + β 2 γ 1 ) 2 + S 55 (α 1 γ 2 + α 2 γ 1 ) 2 ] + S 66 (α 1 β 2 + α 2 β 1 ) 2 , where α 1 , β 1 , γ 1 , α 2 , β 2 , γ 2 are the direction cosines of the [uvw] and [HKL] directions in the coordinate systems, and the [HKL] direction shows the vector normal to the (hkl) shear plane. 30 Figs. 4(a) and (c) display the 3D surface depictions of the E and B. For an isotropic crystal, the 3D surface depictions should be the spheric shape. A divergence from the spheric shape may well reflect the level of the elastic anisotropy. For the Pnnm-CN, both E and B show large divergence from the spheric shape. Hence, It is concluded that the Pnnm-CN exhibits a significant elastic anisotropy. In addition, the projections of 3D surface depictions of both E and B on the ab, ac, and bc planes have   (7) can be deduced as: 26 E −1 = s 11 cos 4 θ + s 22 sin 4 θ + 2s 12 sin 2 θ cos 2 θ + s 66 sin 2 θ cos 2 θ.
The orientation dependence of the E are plotted in Fig. 5  To investigate the plastic deformation of Pnnm-CN, the variation of the shear modulus G with the shear stress direction has been investigated and plotted in Fig. 5(b). For the (001)
As one of the most significant thermodynamics parameters of the solids, the heat capacity not only provides available information of the vibrational properties but also is fundamental for many applications. 37 The temperature dependence of the heat capacity at various pressures for Pnnm-CN is displayed in Fig. 6. It is clear that both heat capacity at constant volume (C V ) and heat capacity at constant pressure (C P ) increase with temperature at the same pressure, while decrease with pressure at the same temperature. In more detail, both C V and C P of the Pnnm-CN follow the law of T 3 at low temperature. At high temperature, C P increases persistently, while C V increases slightly and closes to a constant of 3Nk B (≈49.9Jmol -1 K -1 ) at sufficient high temperature. The difference between C V and C P can be expressed as C P = C V + TV α 2 . 38 At low temperature, the value of α mainly lead to the departure of C V and C P . In case like this, there is small departure of C V and C P duo to small value of α. At high temperature, the behavior of the C V is obeying the law of Dulong-Petit, while the value of C P is proportional to T. Therefore, the departure of C V and C P is obvious. In addition, the dependence of both C V and C P on T are greater than that on the P. Thermal expansion coefficient α reflects the change of solid volume in response to the change in pressure P or temperature T. 33 The variations of α on P and T are illustrated in Fig. 7. It is shown that the α, under certain temperature, decreases sharply at P ≤ 40 GPa, then changes slowly at P>40 GPa. There are small influence of pressure on α at low temperature. As shown in Fig. 7(b), the α increases quickly with T especially for low temperature and 0 GPa, then it reaches to a linear increase under high temperatures. It is explained by the relation of α ∼ C V /B. 31 The bulk modulus slowly and linearly reduces with temperature. At low temperature, the quick increase of C V mainly cause the remarkable variation of α. At high temperature, the α shows a linear increase depended on B because the C V nearly approaches to Dulong-Petit limit.
As a key thermodynamic quantities: the Grüneisen parameter γ reflects the anharmonic effects in the vibrating lattice. 39 In Fig. 8 temperatures and pressures. For the given temperature, γ decreases sharply with P, especially at high temperature. Meanwhile, the variations of γ with P almost display a linear relationship in the pressure range of 40-90 GPa. For the given pressure, γ increases obviously with increasing temperature at P ≤ 40 GPa, then increases monotonously with increasing temperature at P > 40 GPa. The influences of P on γ are greater than T.

IV. CONCLUSIONS
At high temperature and high pressure, the elastic anisotropy and thermodynamic properties of the recently synthesized Pnnm-CN have been systematically investigated. The calculated equilibrium crystal parameters and normalized volume at given pressure are completely closed to previous experimental and theoretical data. To understand the elastic anisotropy of Pnnm-CN, the relationships of the Young's modulus and shear modulus against crystal orientation for Pnnm-CN are discussed. The evidence of the obvious elastic anisotropy for Pnnm-CN is obtained. Using quasi-harmonic Debye model, the thermodynamic properties, such as the Debye temperature, heat capacity, thermal expansion coefficient, and Grüneisen parameter, of Pnnm-CN have also been investigated under high pressure and high temperature.