Elastic and Thermal Properties of Silicon Compounds from First-Principles Calculations

3.1 Structural Properties

In this article, the initial crystal structures are based on the experimental crystallographic data of V-Si compounds, and the lattice parameters of these compounds are optimised by using first-principle calculations. The crystal structures of V-Si compounds are shown in Figure 1. The optimised lattice parameters are listed in Table 1, where the available experimental data and other theoretical results are included. For V₃Si, the calculated lattice parameters by the GGA-PBE method are in good agreement with the previous results (4.7001 Å) and other theoretical results [13, 15, 16]. For VSi₂, the calculated lattice parameters are found to be a=b=4.5597 Å, c=6.3602 Å, respectively, showing that they are in good agreement with the experimental values [17]. For V₅Si₃, the calculated lattice parameters are found to be a=b=9.3997 Å, c=4.7252 Å, respectively, are in good agreement with previous results [3, 7, 16]. For V₆Si₅, the calculated lattice parameters in this work are well consistent with the previous results [16, 18]. These agreements with the previous results provide a confirmation that this work is reliable.

Figure 1:

The crystal structure of V-Si compounds (a) V₃Si, (b) VSi₂, (c) V₅Si₃, and (d) V₆Si₅ (The V atoms are shown as gray spheres and the Si atoms as yellow spheres.).

The formation enthalpy E_f per atom of the V_xSi_y compounds can be expressed as

where x and y are indices that give the amount of atoms of each atomic species within the unit cell of a structure V_xSi_y. E_tot is the total energy of atom of the elements or compound. The calculated formation enthalpies of V₃Si, VSi₂, V₅Si₃, and V₆Si₅ are also listed in Table 1, together with available experimental data and other theoretical results. For V₃Si, the formation enthalpy E_f is in good agreement with previous results [13, 14, 16]. The formation enthalpy of VSi₂ is also predicted in this work and in good agreement with theoretical results [5]. Compared to the experimental data [13], the present calculated formation enthalpy of the V₅Si₃ is also in good agreement and is coherent with other theoretical results [3, 5, 16]. For V₆Si₅, the calculated formation enthalpy E_f (–54.27 kJ/mol) in this work is in good agreement with the value –53 kJ/mol obtained by the GGA-PBE method [16] and other previous measurement results [19–21]. The deviation is only 6.57%.

3.2 Elastic Constants

The number of independent elastic constants is different for various crystal structures. For a cubic crystal, it has three independent elastic constants (C₁₁, C₁₂, and C₄₄). For a hexagonal crystal, it has five independent elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, and C₄₄). For a tetragonal chalcopyrite crystal, it has six independent elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, and C₆₆). For an orthorhombic crystal, it has nine elastic constants (C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃, and C₂₃). The mechanical stability criteria are given by [22–25].

As shown in Table 2, all of the elastic constants for V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds satisfy the respective mechanical stability criteria. It indicates that all above structures are mechanically stable. The elastic constants for V₃Si are in agreement with the experimental data (at T=77 K) [4]. It is worth pointing out that our theoretical calculations of the elastic constants for VSi₂ with space group P6₄22 are in agreement with the experimental data [17]. The C_ij values calculated by the GGA-PBE method of V₅Si₃ are also provided and compared. The calculated C_ij values in this work are well consistent with the previous results [3]. Unfortunately, there are no other theoretical and experimental results for comparison with our elastic results for V₆Si₅.

It is known that the mechanical properties are determined by the elastic modulus. The polycrystalline elastic properties, including bulk modulus B, shear modulus G, Young modulus E, and Poisson’s ratio σ, can be obtained by Voigt–Reuss–Hill (VRH) approximation. The B, G, and E are listed in Table 3. As can be seen from Table 3, the bulk modulus B of for V-Si compounds follows the order V₃Si>V₅Si₃>V₆Si₅>VSi₂. V₃Si has the largest bulk modulus (193.64 GPa). The Young’s modulus of for the V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds follows the order VSi₂>V₅Si₃>V₆Si₅>V₃Si.

According to Pugh’s criterion [26], the ductility or brittleness of a solid material is judged by B/G ratio. The critical value which separates ductile and brittle material is B/G=1.75. If B/G>1.75, behaves in a ductile manner. Otherwise, behaves in a brittle. Moreover, the Poisson’s ratio σ is consistent with B/G, which refers to a ductile compound has a large σ>0.26 [27]. The values of B/G are larger than 1.75 and values of σ are larger than 0.26 for V₃Si in Table 3, which testify that they are ductile. The values of B/G and σ for VSi₂, V₅Si₃, and V₆Si₅ are less than 1.75 and 0.26, respectively. It shows that they are all brittle.

The values of the compressional velocity wave V_p and the shear wave velocity V_S can be obtained using the Navier’s equation [28], where ρ is density of the compound.

The average wave velocity V_m can be calculated by

The calculated sound velocity of V_p, V_S, and V_m for the V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds are listed in Table 3. In comparison, the calculated values of V_p and V_S for V₃Si at zero pressure are a little lower than reported by Fleischer et al. [8]. For VSi₂, the calculated values of V_p and V_S are in good agreement with those reported by Fleischer et al. [8]. For V₅Si₃, the calculated V_p and V_S in this work are in agreement with the values by the VASP-GGA method [3] and other previous results [8]. Without experimental data and theoretical results of the V₆Si₅, we cannot make any comparisons. We expect that our theoretical results provide the valuable reference for the further research.

As a method to study the elastic anisotropic behavior of a solid material, the 3D surface constructions of the directional dependences of reciprocals of Young’s modulus E are very useful. The expressions of the reciprocals of Young’s modulus for the V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds are different for each other due to their various crystal structures [22].

In Figure 2, we show the mechanical stable structures of V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds. The surface in each graph represents the magnitude of Young’s modulus E along different orientations. From this figure, we can clearly see that the Young’s modulus shows some anisotropy at different orientations. It also should be noted that the Young’s modulus of V₃Si shows slight anisotropy and the Young’s modulus of VSi₂, V₅Si₃, and V₆Si₅ shows some anisotropy for different orientations. As for the hexagonal VSi₂, the 3D directional dependences of the Young’s modulus along z axis are more compressible than along x- and y-axes. The 3D figure of the Young’s modulus for the tetragonal V₅Si₃ is characterised by more anisotropic along the z-axis than that along the x- and y-axes. As for the orthorhombic V₆Si₅, the 3D surface of Young’s modulus along x-, y-, and z-axes is shown highly anisotropic.

Figure 2:

The directional dependence of the Young’s modulus for the V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds [(a) V₃Si, (b) VSi₂, (c) V₅Si₃, and (d) V₆Si₅]. The magnitude of Young’s modulus at different directions is represented by the contour. The units are in GPa.

3.3 Thermal Properties of V₃Si, VSi₂, V₅Si₃, and V₆Si₅ under High Temperature and High Pressure

In order to investigate the thermal properties of V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds under high pressure (0–25 GPa) and melting points temperature [6], we have used the quasi-harmonic Debye model as implemented in the Gibbs code [29]. First, we obtain a set of total energy results versus unit-cell volumes around the equilibrium geometry. Then, the above-mentioned results are fitted to the equation of state in order to obtain different parameters as a function of pressure and temperature from standard thermal relations. By the above-mentioned method, we can get the internal energy U, the heat capacity of constant volume C_V, the heat capacity at constant pressure C_P, the entropy S, the thermal expansion coefficient α, Debye temperature Θ, and Grüneisen parameter γ.

The thermal properties of the V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds are calculated at the different temperatures ranging from 0 to about melting points. The pressure effect is investigated in the range 0–25 GPa. Heat capacity belongs to one of the most important thermal properties of the materials. Figure 3 shows the calculated heat capacity at constant volume as functions of the temperature under zero GPa. It can be seen from Figure 3 that the heat capacity C_V increases exponentially with the temperature at T<400 K. At higher temperature, C_V follows the Debye model and approaches the Dulong–Petit limit indicating the thermal energy at high temperature excites all phonon modes.

Figure 3:

The heat capacity of constant volume C_V (J/mol K) for V₃Si, VSi₂, V₅Si₃, and V₆Si₅ in V-Si compounds at 0 GPa.

As an important physical quantity, the Debye temperature could distinguish the physical properties between high- and low-temperature regions. The variation of the Debye temperature Θ (K) as a function of pressure and temperature is displayed in Figure 4. With the applied pressure increasing, the Debye temperatures are almost linearly increasing. The temperature and pressure dependence of Θ (K) reveals that the thermal vibration frequency of atoms in V-Si (V₃Si, VSi₂, V₅Si₃, and V₆Si₅) compounds changes with temperature and pressure.

Figure 4:

Debye temperature versus pressure at various temperatures for V₃Si, VSi₂, V₅Si₃, and V₆Si₅.

Figure 5 shows the volume thermal expansion coefficient a of V₃Si, VSi₂, V₅Si₃, and V₆Si₅ across different pressures, from which it can be seen that the volume thermal expansion coefficient α increases quickly at a given temperature particularly at zero pressure below the temperature of 300 K. After a sharp increase, the volume thermal expansion coefficient of the V₃Si, VSi₂, V₅Si₃, and V₆Si₅ is nearly insensitive to the temperature above 300 K due to the electronic contributions. Thermal expansion coefficient α strongly decreases with pressure at a constant temperature.

Figure 5:

Thermal expansion coefficients versus temperature at various pressures for V₃Si, VSi₂, V₅Si₃, and V₆Si₅.

Finally, we have also calculated the internal energy U, entropy S, thermal expansion coefficient α, heat capacity C_V and C_P, Helmholtz free energy A, Grüneisen parameter γ, and Debye temperature Θ for the V₃Si, VSi₂, V₅Si₃, and V₆Si₅ in V-Si at 300 K under different pressures. The theoretical results are presented in Table 4. However, there has been no experimental and theoretical data available for the thermal properties of V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds so far. Therefore, our calculated results can provide support for future works on V₃Si, VSi₂, V₅Si₃, and V₆Si₅ compounds.