Theoretical Prediction on the Structural, Electronic, Mechanical, and Thermodynamic Properties of TaSi₂ with a C40 Structure Under Pressure

3.1 Structural Parameters

The hexagonal structure (space group of P6₂22) of TaSi₂ was displayed in Figure 1. In this hexagonal structure, Ta and Si atoms are located in the Wyckoff 3d (0.5, 0, 0.5) and 6j (0.1593, 0.3185, 0.5) sites, respectively.

Figure 2:

Band structures of TaSi₂ at (a) 0 and (b) 50 GPa.

By performing PAW calculations, we obtained the lattice constants for a = 4.80 Å, c = 6.60 Å, c/a = 1.375, and volume V = 131.69 ų (Table 1), which is well consistent with experimental findings: a = 4.784 Å, c = 6.568 Å, c/a = 1.373, V = 130.18 ų, as well as previous experimental findings [7]. This indicates the accuracy of our PAW calculations.

3.2 Electronic Properties

Using the optimized lattice constants, a systematic study on band structures and density of states were carried out using the generalized gradient approximation (Perdew–Burke–Ernzerhof) functional at pressure P = 0 and 50 GPa, respectively. Representative graphs of the band structures are plotted in Figure 2 with the Fermi level (E_F) set to zero. A shift of position under pressure at P = 0 and 50 GPa can be observed in Figure 2. We observe that some bands cross the E_F, confirming that TaSi₂ is metallic. The total densities of states together with partial densities of states of TaSi₂ at P = 0 GPa are plotted in Figure 3. Figure 3 shows that the band structure of TaSi₂ can be divided into two regions. The conduction bands around 0–5 eV for TaSi₂ are dominated by Ta d orbital, whereas the valence bands from −15 to 0 eV is mainly Ta d orbital and Si s hybridized with the Si p orbital.

Figure 3:

Total and partial densities of states of TaSi₂ at 0 GPa.

3.3 Elastic Properties

The obtained single crystal elastic constants C_ij (C₁₁, C₁₂, C₁₃, C₃₃, and C₄₄) under various pressures (0–50 GPa) are shown in Table 2, as well as the independent constants of elastic compliance matrix calculated from elastic constants. Additionally, to further confirm the mechanical stability of the hexagonal structure, we predicted the mechanical stability via the Born–Huang stability criterion. For the hexagonal crystal system, the calculated elastic constants need to satisfy the following stability conditions [20]:

Figure 4:

The evolution of calculated elastic constants of TaSi₂ as a function of pressure.

From Table 2, we find that the calculated elastic constants satisfy the above conditions. As mentioned above, it is indicated that the hexagonal structure is a mechanically stable structure. The calculated elastic constants at zero temperature and under some fixed pressures compared with available previous experimental findings (at room temperature) are presented in Table 2, being in consistency with the experimental findings (at room temperature) [7] and other theoretical results [15]. We also explored the pressure evolution of elastic constants of TaSi₂ (Fig. 4). We can find that the elastic constants increase with the applied pressure. With the increase of the pressure from 0 to 50 GPa, the C₁₁, C₁₂, C₁₃, C₃₃, and C₄₄ of TaSi₂ at zero temperature is increased by 43.44 %, 53.63 %, 41.84 %, 40.20 %, and 23.71 %, respectively.

On the basis of the known C_ij values, the bulk modulus B and shear modulus G can be computed by the Voigt approximation [21]:

and the Reuss approximation, which happens to be formally equivalent to the single crystal bulk modulus in this case [22]:

Figure 5:

The calculated B, G, and E of TaSi₂ as a function of pressure.

Therefore, shear modulus upper bounded by can be presented as [21]

and lower bounded by [22]

Voigt and Reuss limits can be averaged, as proposed by Hill [23], to produce a single estimation of the bulk modulus and shear modulus of the TaSi₂:

B_V, B_R, G_V, and G_R are the bulk moduli and shear moduli referring to the Voigt approximations and Ruess approximations, respectively.

The B, G, and E under various pressures (0–50 GPa) are illustrated in Figure 5. It can be clearly seen that calculated B, G, and E of TaSi₂ increase with the applied pressure. In Table 3, we also listed the values of the B, G, and E with the pressure of TaSi₂. The B, G, and E at 0 K and 0 GPa are 191.06, 145.59, and 348.29 GPa, which are in accord with experimental results (at room temperature) of 192.5, 151, and 359.0 GPa, respectively [7]. The Poisson’s ratio ν is 0.196, which is almost equal to the experimental value of 0.1892 [7]. On the basis of the Pugh rule (B/G = 1.75) [24], the ductility or brittleness of a solid is deduced by the B/G ratio. If B/G > 1.75, the solid’s behavior is ductile. On the contrary, the solid is brittle. From Table 3, it is found that TaSi₂ under different pressures (0–50 GPa) show a brittle behavior because the obtained B/G ratio of TaSi₂ is smaller than 1.75. In this work, the microhardness of C40 TaSi₂ is calculated by the following five empirical rules [25], [26]:

We obtain that the maximum value is 29.5 GPa, and the minimum is 18.4 GPa. The results from the above rules tend to overestimate the experimental value (15.6 GPa) [27].

The anisotropy in elasticity can be characterized by universal anisotropic index (A^U), the percent anisotropy (A_B and A_G), and shear anisotropic factors (A₁, A₂, and A₃), which can be expressed from the following equations [25], [28], [29]:

Figure 6:

The 3D Young’s modulus E curved surface of TaSi₂ under (a) 0 and (c) 50 GPa. The 3D bulk modulus B curved surface under (b) 0 and (d) 50 GPa.

From Table 4, it is noted that percent anisotropy A_G is larger than A_B under different pressures (0–50 GPa) of TaSi₂. It means that the shear modulus shows stronger directional dependence than the bulk modulus. The results of A₁, A₂, and A₃ under different pressures (0–50 GPa) are also illustrated in Table 4. If A₁ = A₂ = A₃ = 1, the material is an isotropic crystal. Otherwise, it is an anisotropic crystal. The results indicate that it is an anisotropic crystal. The deviations of universal anisotropic index (A^U) under different pressures (0–50 GPa) from zero mean the highly single crystal anisotropy.

Figure 7:

The projections in different planes of Young’s modulus E of TaSi₂. (a) XY plane, (b) XZ plane and YZ plane. The units are in GPa.

Figure 8:

The projections in different planes of bulk modulus B of TaSi₂. (a) XY plane, (b) XZ plane and YZ plane. The units are in GPa.

In this work, the anisotropy of the three-dimensional (3D) surface of E and B of TaSi₂ under 0 and 50 GPa is presented in Figure 6. As seen in Figure 6, the nonspherical shape shows the extent of the elastic anisotropy, especially, and the anisotropy behavior becomes more obvious with the increasing pressure. In Figures 7 and 8, we, respectively plotted the projections of B and E at 0, 10, 20, 30, 40, and 50 GPa. As shown in Figures 7 and 8, it is found that the projections of XZ and YZ planes are the same. From Figures 7 and 8, the projections of E on the XY, XZ, and YZ planes show more anisotropic behavior than B. From the projections of E and B, it is concluded that anisotropic behavior of TaSi₂ strengthened with increasing pressure.

Figure 9:

The calculated wave velocities (a) and Debye temperature (b) of TaSi₂ as a function of pressure.

As we know, the Debye temperature θ can be observed from the average wave velocity [30], [31]. The calculated θ as well as the compressional velocity V_P, the shear wave velocity V_S, and the average wave velocity V_m under different pressures (0–50 GPa) are shown in Table 3 and Figure 9. As seen in Figure 9, the wave velocities and θ increase gradually with applied pressure. As shown in Table 3, at T = 0 and P = 0, θ is 545.54 K, which is a bit smaller than the experimental value of 552 K of Chu [7] and slightly more than the theoretical value of 526 K [15].

The melting temperatures (T_m) of hexagonal crystals are related to their elastic constants C₁₁ and C₃₃ [32]. The melting temperature (T_m) is defined as follows:

where C₁₁ and C₃₃ is in GPa and T_m in K. We obtain that the T_m of TaSi₂ is 2146.2 K, which is slightly less than the measured T_m of 2473.15 K for TaSi₂ [7].

Figure 10:

The calculated and fitted V/V₀ as a function of pressure.

3.4 Thermodynamic Properties

In this work, the thermodynamic properties were evaluated using the quasiharmonic Debye model as implemented in the Gibbs code [33]. We obtained the thermodynamic properties under different temperatures (0–2100 K) and pressures (0–30 GPa).

The dependence of the relative volume V/V₀ on temperature/pressure is plotted in Figures 10 and 11. As shown in Figure 10, we found that present fitted values are consistent with calculated data. From Figure 11, it should be noted that V/V₀ decreases with increasing pressure. It is also obtained that V/V₀ tends to decrease with increasing temperature at a given pressure.

In Figure 12, we pictured the relationships between the B and pressure/temperature. Figure 12a displays the calculated B with pressure at different temperatures (0, 500, 1000, and 2000 K). One can see that the calculated B increases linearly with pressure at a given temperature. From Figure 12b, the magnitude of B decreases linearly with increasing temperature at different pressures (0, 10, 20, and 30 GPa). It is worth noticing that P has a significant effect on B than T. At zero pressure and T = 300 K, our calculated B value (187.54 GPa) satisfactorily agrees with the experimental value of 192.5 GPa (at room temperature) [7].

Figure 11:

Contours of relative volume V/V₀ versus pressure P (GPa) and temperature T (K).

Figure 12:

The calculated B of TaSi₂ as a function of (a) pressure and (b) temperature.

From Figure 13a, we can see at different temperatures (500, 1000, 1500, and 2000 K) that the calculated thermal expansion coefficient α decreases dramatically with increasing pressure. Figure 13b shows thermal expansion coefficient a of TaSi₂ at different pressures (0, 10, 20, 30 GPa), from which it can be seen that the thermal expansion coefficient α increases quickly at a given pressure particularly below the temperature of 500 K. After a sharp increase, the thermal expansion coefficient of the TaSi₂ is nearly insensitive to the temperature above 500 K because of the electronic contributions. The thermal expansion coefficient α decreases strongly with increasing pressure at a constant temperature. Our present calculated a is 0.76 × 10⁻⁵ K⁻¹ at P = 0 and T = 300 K and decreases by 21.74 %, 36.30 %, and 46.60 % with pressure increasing to 10, 20, and 30 GPa, respectively.

Figure 13:

The calculated α of TaSi₂ as a function of (a) temperature and (b) pressure.

Figure 14:

The calculated (a) C_V and (b) C_P of TaSi₂ at different pressures as a function of temperature.

The heat capacity at constant volume C_V and the heat capacity at constant pressure C_P of TaSi₂ as a function of temperature at some given pressure (0, 10, and 30 GPa) are plotted in Figure 14. The data of C_V and C_P are almost identical at low temperature, and the C_V and C_P of TaSi₂ under different pressures (0, 10, and 30 GPa) are proportional to T³ at low temperature. At high temperature, C_V is following the Dulong–Petit limit. C_P increases monotonically with temperature at high temperature. It is concluded that the influence of T on the C_V and C_P are more significant than P of TaSi₂. We can see that the variation of C_p as a function of temperature is similar to α. The influence of P on α is more significant than C_P. The calculated C_P is equal to 0.00265 J mol⁻¹ K⁻¹ at 5 K, which is larger than the experimental value of 0.000914 J mol⁻¹ K⁻¹ [9].

Furthermore, we have also completely explored the internal energy U, entropy S, Helmholtz free energy A, C_V, C_P, a, Grüneisen parameter γ, and θ of TaSi₂ at 300 K under different pressures in Table 5. So far as we have known, there are no experimental and theoretical findings available for comparison. So, our thermodynamic data provide an interesting insight and feasible guidelines for the potential applications of TaSi₂.