The binary carbide consisting of metal or nonmetal, has various intriguing properties in physics and chemistry, and thus a large number of promising structures have been studied. The well-known nonmetal carbides, silicon carbides,^[1–7] which are regarded as the third-generation semiconductor material, have been employed in the high temperature and voltage power devices and optical sensors, owing to the indirect wide band gap. Furthermore, the nano-SiC^[2] also has obtained a wide application in superconductors and ceramics to enhance the critical current density or to utilize its refractory nature and low neutron absorption.^[3–5] Recently, the superhard and superconductive boron carbide material (e.g. BC₃,^[8,9] BC₅,^[10,11] BC₇,^[12,13] and B₂C,^[14,15] etc.) have received increasing attention, and both experiments and computational simulations show that they are more thermally and chemically stable than pure diamond. In addition, a diversity of metal carbides, such as LiC_x,^[16] TiC_x,^[17,18] MgC_x,^[19,20] and CaC_x^[21–25] are reported. The spontaneous intercalation of cations in the two-dimensional Ti₃C₂ MXene layers, which was studied by Lukatskaya et al.,^[17] provides a basis for the further study of the high volumetric capacitance. The magnesium methanide Mg₂C was first synthesized by Kurakevych et al.,^[19] and the rare C⁴⁻, which is observed, allows easy hydrolysis towards the alkaline-earthmetal carbide. Furthermore, the high pressure polymorph β-Mg₂C₃,^[20] with the linear chains nearly aligned along the crystallographic c axis, was synthesized by Strobel et al. For the calcium carbides, the well-known CaC₆^[21,22] and CaC₂^[23] were also investigated widely to show an unexpected pressure-composition property, and even the pressure-induced superconductivity. All of these studies enrich the study of binary carbides, and provide guidance for applications. Recently, Li et al.^[26] reported a series of novel stable and metastable calcium carbide phases (Ca₅C₂, Ca₂C, Ca₃C₂, CaC, and Ca₂C₃), and successfully synthesized the Pnma-Ca₂C and C2/m-Ca₂C₃ by using the in situ synchrotron powder x-ray diffraction measurements. In the present work, we systematically study the C2/m-Ca₂C₃ phase in the mechanical and dynamical stability, elastic anisotropy and electronic band structure and the partial density of states (PDOSs) under the action of pressure, to reveal the mechanical, elastic and electronic properties of Ca₂C₃.

All of the calculations are performed based on the first-principles calculations. The exchange and correlation functional is in the generalized gradient approximation (GGA) which is parameterized by Perdew, Burke, and Ernzerhof (PBE)^[27] in the Cambridge Serial Total Energy Package (CASTEP) code,^[28] and the density functional theory (DFT)^[29,30] uses the Vanderbilt ultrasoft pseudopotentials.^[31] The Broyden–Fletcher–Goldfarb–Shanno (BFGS)^[32] minimization scheme is used in geometry optimization, and the total energy convergence tests are within 1 meV/atom. For the C2/m-Ca₂C₃, the energy cutoff is 420 eV, and the k-point mesh is 11×9×9 in the Brillouin zone. The self-consistent convergence of the total energy is 5.0×10⁻⁶ eV/atom, the maximum ionic Hellmann–Feynman force is 0.01 eV/Å, the maximum stress is 0.02 GPa, and the maximum ionic displacement is 5.0×10⁻⁴ Å. Furthermore, the electronic properties are calculated by the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional.^[33]

As shown in Fig. 1, C2/m-Ca₂C₃ has two types of nonequivalent carbon atoms, i.e., 4i (0.04, 0, 0.2283) and 2a (0, 0, 0), and one type of calcium atom 4i (0.4017, 0, 0.7208). The Ca atoms are isolated cations, and the C atoms constitute a carbon trimer chain which is along the c-axis and has a length of 1.336-Å C–C bond at 0 GPa. The optimized lattice parameters at 0, 18.1, and 24 GPa are listed in Table 1. To make a comparison, the known values are also given. As can be seen, these values are in good agreement with each other at 18.1 GPa.

Fig. 1.

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	Fig. 1. (color online) 1×2×2 supercell of C2/m-Ca2C3. The black and blue spheres represent C and Ca atoms, respectively.

Table 1.

Table 1.

Table 1. Calculated lattice parameters, volume, and density. . a/Å b/Å c/Å β /(°) V /(Å3/atom) ρ (g/cm3) Ca2C3 0 GPa 5.3909 5.3022 6.5661 128.4 14.72 2.622 18.1 GPa 5.1441 4.9562 6.2979 128.81 12.51 3.084 5.1511[26] 4.9620[26] 6.3059[26] 128.81[26] 24 GPa 5.0897 4.8829 6.2382 129.03 12.05 3.204

Table 1. Calculated lattice parameters, volume, and density. .

The plots of ratio X/X₀ as a function of pressure are presented in Fig. 2, and the degree of reducing along b axis is larger than those along the other two axes, indicating a lower anti-compressibility. Furthermore, since the C–C bond is along the c axis, Ca₂C₃ has the best ability to resist the compression along c axis.

Fig. 2.

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	Fig. 2. (color online) Plots of the lattice parameter X/X0 versus pressures.

To study the elastic property, the elastic constants and moduli of Ca₂C₃ are calculated and shown in Table 2. For the monoclinic phase, the generalized Born’s mechanical stability criteria at 0 GPa are given as follows:^[34,35]

Table 2.

Table 2.

Table 2. Calculated values of elastic constant Cij, bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν, B/G ratio, and universal anisotropy index AU at various pressures. . Pressure/GPa Ca2C3 0 12 24 25 C11 171 232 292 303 C22 142 245 342 352 C33 212 281 336 346 C44 30 29 28 27 C55 41 57 69 82 C66 52 54 55 55 C12 42 50 60 66 C13 53 107 162 162 C23 20 39 53 52 C15 −8 −22 −27 −29 C25 −5 −8 −7 −4 C35 24 22 21 31 C46 −9 −10 −11 −10 B 82 126 167 G 48 58 64 E 121 151 170 v 0.255 0.301 0.33 B/G 1.72 2.18 2.61 AU 0.978 1.561 2.399

Table 2. Calculated values of elastic constant Cij, bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν, B/G ratio, and universal anisotropy index AU at various pressures. .

Based on the mechanical stability in crystal under isotropic pressure,^[36] the mechanical stability criteria of the monoclinic phase at high pressure can be expressed as

where , i = 1, …, 6, , , and P is the isotropic pressure. That the calculated results satisfy the criteria indicates that the structure is mechanically stable.

As listed in Table 2, the structure satisfies the above criteria in a pressure range from ambient pressure to 24 GPa. At 25 GPa and higher pressures, the criteria in Eq. (12) cannot be satisfied. This demonstrates that Ca₂C₃ has the mechanical stability only at pressure up to 24 GPa (see Table 2). The elastic constants under pressure are also presented in Fig. 3. From Fig. 3 it follows that C₁₁, C₃₃, and C₁₃, have similar increased tendencies, and the tendency of C₂₂ is sharper than the others, indicating that C₂₂ is the most sensitive to the pressure effect in all of the elastic constants. The phonon spectra are calculated to ensure the dynamical stability of structure. As shown in Fig. 4, because of no imaginary frequency in the whole Brillouin zone, Ca₂C₃ is dynamically stable at pressure up to 24 GPa. With Voigt–Reuss–Hill approximation,^[37–39] the bulk modulus B and shear modulus G are calculated, and the Young’s modulus E and Poisson’s ratio v are given by^[39,40] E = 9BG/(3B + G) and ν = (3B − 2G)/[2(3B + G)]. The calculated values of B, G, and E increase with pressure rising. According to Pugh,^[41] the ratio of bulk to shear modulus shows the brittle and ductile nature. When the B/G of material is less than 1.75, it shows a brittle manner. Conversely, it shows a ductile manner. For Ca₂C₃, except the ratio of 1.72 at 0 GPa, other ratios are larger than 1.75, and make this material ductile. The universal elastic anisotropy index^[42] is A^U = 5G_V/G_R + B_V/B_R − 6, and the nonzero values in Table 2 suggest the elastic anisotropy. As can be seen from this table, these values are 0.978, 1.561, and 2.399 at 0, 12, and 24 GPa, respectively, indicating that the anisotropy increases with pressure rising.

Fig. 3.

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	Fig. 3. (color online) Plots of elastic constant versus pressure.

Fig. 4.

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	Fig. 4. Phonon spectra for C2/m-Ca2C3 at 0 GPa (a) and 24 GPa (b).

To study the elastic anisotropy in detail, the directional dependence of elastic anisotropy, calculated by the ElAM code,^[43] is shown in Fig. 5. The solid and dashed lines represent the maximal and minimal positive values, respectively. The black, red and blue lines represent the xy, xz, and yz planes, respectively. In Figs. 5(a) and 5(b), the maxima of shear modulus at 0 GPa and 24 GPa are 77 GPa and 131 GPa, respectively, and the minima are 27 GPa and 24 GPa, respectively. As a result, the anisotropy of shear modulus increases with pressure rising, conforming to the results of the universal elastic anisotropy index. For each plane, the anisotropy in yz plane is the highest at both 0 GPa and 24 GPa, whereas the lowest is in xy plane at 0 GPa and in xz plane at 24 GPa. The Poisson’s ratios of Ca₂C₃ in Figs. 5(c) and 5(d) have similar anisotropies at different pressures (0.563/0.035 ≈ 6.1 at 0 GPa and 0.733/0.046 ≈ 5.9 at 24 GPa) and the lowest anisotropies at 0 GPa and 24 GPa are both in xy plane, respectively. The Poisson’s ratios have the highest anisotropy in yz plane at 0 GPa, and in xz plane at 24 GPa, respectively.

Fig. 5.

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	Fig. 5. (color online) 2D representations of shear modulus at 0 GPa (a) and at 24 GPa (b); and 2D representations of Poisson’s ratio at 0 GPa (c) and at 24 GPa (d). The solid and dashed lines represent the maximal and minimal positive values, respectively. The black, red, and blue lines represent the projections in xy, xz, and yz planes, respectively.

The directional dependence of Young’s modulus is shown in Figs. 6(a) and 6(b). A spherical shape indicates that the structure is isotropic. It is found that the anisotropy at 0 GPa is lower than that at 24 GPa, and the average Young’s moduli in all directions are 125 GPa and 165 GPa, close to the 121 GPa and 170 GPa by Voigt–Reuss–Hill approximations, respectively. Figures 6(c) and 6(d) display the two-dimensional (2D) plots of differences in each direction of Young’s modulus. The maxima at 0 GPa and 24 GPa are 211 GPa and 327 GPa, respectively, and the minima are 83 GPa and 87 GPa, respectively. The E_max/E_min ratios at 0 GPa and 24 GPa are 2.52 and 3.76, respectively. Furthermore, the highest anisotropies at 0 GPa and 24 GPa are in xz plane and yz plane, respectively, and the lowest anisotropies at 0 GPa and 24 GPa are in xy plane and xz plane, respectively.

Fig. 6.

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	Fig. 6. (color online) Directional dependence of the Young’s modulus at 0 GPa (a), and 24 GPa (b); and 2D representations of the Young’s modulus at 0 GPa (c), and 24 GPa (d). The black, red, and blue lines represent the projections in xy, xz, and yz planes, respectively.

The calculated electronic properties of Ca₂C₃ by the HSE06 functional are shown in Fig. 7. It is found that Ca₂C₃ is a semiconductor. At 0 GPa, the conduction band minimum (CBM) is located at Γ point and the valence band maximum (VBM) is at Z point with an indirect band gap of 1.81 eV (see Fig. 7(a)). At 3 GPa, both the CBM and the VBM are at Γ point with a direct band gap of 1.82 eV, which means that there occurs an indirect–direct band gap transition. At 24 GPa, the CBM is at Γ point, the VBM is at (−0.5, 0, 0.34) along the B–D direction. For convenience, we denote this point as P point. The indirect band gap is 1.63 eV. It is known that the GGA calculations usually underestimate the gap. As a result, our calculations by the HSE06 functional are larger than the available results at GGA level.^[26]

Fig. 7.

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	Fig. 7. (color online) Electronic band structures at 0 GPa (a), 3 GPa (b), and 24 GPa (c); the gaps of Γ –Z, Γ –Γ, and Γ –P each as a function of pressure (d); partial DOSs of Ca2C3 (e). The blue and red points represent CBM and VBM, respectively.

As seen in Figs. 7(a)–7(c), Γ, Z, and P points have almost the same energies. To clarify the energies of these three points, the magnifications near Fermi level are shown at the bottom of Figs. 7(a)–7(c). It can be seen that the difference between Γ and Z points is only about 0.01 eV, thus Ca₂C₃ is a quasi-direct band gap semiconductor. To show the indirect–direct–indirect transition in detail, the gaps of Γ –Z, Γ –Γ, and Γ –P under pressures are calculated, and the results are illustrated in Fig. 7(d). From this figure it follows that the transition points are about 1.1 GPa and 4.0 GPa, respectively. The energy at Z point decreases with pressure increasing, however at P point it increases. For the Γ point, the energy increases below 4.0 GPa, whereas decreases from 4.0 GPa to 24.0 GPa. Figure 7(e) shows the total and partial DOS whose values decrease with pressure rising. The conduction band region is mainly characterized by the Ca-d states, and the valence band region is mainly contributed from the C-p states.

The mechanical, elastic, and electronic properties of Ca₂C₃ are investigated based on the first-principles calculations. We calculate the elastic constants and phonon spectra of Ca₂C₃, discovering that it has the mechanical and dynamical stability only in a pressure range from 0 GPa to 24 GPa, the instability at higher pressure is due to the mechanical instability. Furthermore, we find that this structure exhibits an increasing elastic anisotropy. As a semiconductor, the band gap of Ca₂C₃ can be regarded as quasi-direct at ambient pressure. The partial density of states is studied in detail, and the conduction band region is mainly characterized by the Ca-d states, and the valence band region is mainly contributed from the C-p states.