Crystal Structures and Mechanical Properties of Ca2C at High Pressure

Recently, a new high-pressure semiconductor phase of Ca2C (space group Pnma) was successfully synthesized, it has a low-pressure metallic phase (space group C2/m). In this paper, a systematic investigation of the pressure-induced phase transition of Ca2C is studied on the basis of first-principles calculations. The calculated enthalpy reveals that the phase transition which transforms from C2/m-Ca2C to Pnma-Ca2C occurs at 7.8 GPa, and it is a first-order phase transition with a volume drop of 26.7%. The calculated elastic constants show that C2/m-Ca2C is mechanically unstable above 6.4 GPa, indicating that the structural phase transition is due to mechanical instability. Both of the two phases exhibit the elastic anisotropy. The semiconductivity of Pnma-Ca2C and the metallicity of C2/m-Ca2C have been demonstrated by the electronic band structure calculations. The quasi-direct band gap of Pnma-Ca2C at 0 GPa is 0.86 eV. Furthermore, the detailed analysis of the total and partial density of states is performed to show the specific contribution to the Fermi level.


Introduction
Hitherto, the pressure-composition (P-x) phases of binary systems have gained increasing interest and been extensively researched. Among these predicted compounds, some of them have been successfully synthesized [1,2], but the others still need further experiments to confirm their theoretical predictions [3,4]. For the Ca-C system, there are many works that have been done and obtained remarkable achievements [5][6][7][8][9][10][11][12][13][14][15][16]. Gauzzi et al. [5] found the superconductivity will be enhanced in the intercalated graphite CaC 6 at high pressure. It performs the structural instability and leads to a structural transition with pressure. Nylen et al. [6] studied the structural behavior of CaC 2 at high pressure via the first-principles calculations. Their results suggest an irreversible amorphization, corroborating the structural peculiarities of acetylide carbides, which persists at high pressure conditions. Li et al. analyzed the pressure-induced superconductivity of CaC 2 [7]. They uncovered that it is calcium that contributes to the superconducting behavior, and it is capable of stabilizing carbon sp 2 hybridization at a larger range of pressure. Nourbakhsh et al. [8] investigated the magnetism in CaC ionic compound and observed a perfect Fermi level spin polarization and a half-metallic behavior.
Recently, Li et al. [9] systematically explored all the stable calcium carbides at pressures from 0 to 100 GPa. This resulted in five newly predicted stable stoichiometries (Ca 5 C 2 , Ca 2 C, Ca 3 C 2 , CaC and Ca 2 C 3 ). Using in situ synchrotron powder X-ray diffraction measurements, they successfully synthesized the Ca 2 C and Ca 2 C 3 . The Ca 2 C has two phases: the semiconducting phase Pnma-Ca 2 C at high pressure and the metallic metastable phase C2/m-Ca 2 C at low pressure. The Pnma-Ca 2 C exists in the pressure range of 7.5-100 GPa and possesses the isolated C anions. Carbon atoms polymerize to

Results and Discussion
The 2 × 1 × 2 supercell structures of Ca2C are illustrated in Figure 1. The black and blue spheres represent C and Ca atoms, respectively. At zero pressure, the optimized lattice parameters of Pnma-Ca2C are a = 6.677 Å, b = 4.384 Å, c = 7.979 Å with two inequivalent Ca atoms occupying 4c To determine the phase transition pressure of Ca2C, the enthalpy differences between two structures are plotted as a function of pressure up to 100 GPa in Figure 2a. There is an intersection between the two enthalpy curves, indicating that the C2/m-Ca2C phase transforms to the Pnma-Ca2C phase at 7.8 GPa and the Pnma-Ca2C is more stable than the C2/m-Ca2C above this pressure point. The known transition pressure data is 7.5 GPa [9], and it is in a good agreement with our result. Meanwhile, the dependence of volume on pressure is presented in Figure 2b. The C2/m-Ca2C is larger than the Pnma-Ca2C in volume. The change of volume at 7.8 GPa shows that the phase transition is first-order with a volume drop of 26.7%. To interpret this large volume collapse, we estimated the ionic radii of the C and Ca within these two structures at 7.8 GPa through Bader charge analysis. The obtained results are listed in Table 1. The calculated charges of the two Ca2C phase show increasing trends from C2/m-Ca2 0.928 C −0.928 to Pnma-Ca2 2.348 C −2.348 at phase transition pressure point. Compared to To determine the phase transition pressure of Ca 2 C, the enthalpy differences between two structures are plotted as a function of pressure up to 100 GPa in Figure 2a. There is an intersection between the two enthalpy curves, indicating that the C2/m-Ca 2 C phase transforms to the Pnma-Ca 2 C phase at 7.8 GPa and the Pnma-Ca 2 C is more stable than the C2/m-Ca 2 C above this pressure point. The known transition pressure data is 7.5 GPa [9], and it is in a good agreement with our result. Meanwhile, the dependence of volume on pressure is presented in Figure 2b. The C2/m-Ca 2 C is larger than the Pnma-Ca 2 C in volume. The change of volume at 7.8 GPa shows that the phase transition is first-order with a volume drop of 26.7%. To interpret this large volume collapse, we estimated the ionic radii of the C and Ca within these two structures at 7.8 GPa through Bader charge analysis. The obtained results are listed in Table 1   The lattice parameters of Ca2C at different pressures are listed in Table 2. In Table 2, an excellent agreement with the previous theoretical and experimental values is shown [9]. The calculated lattice parameters decrease with pressure. To get more details, the variations of lattice parameters X/X0 of the two Ca2C phases with pressure are shown in Figure 3. For Pnma-Ca2C (see Figure 3a), along the b-and c-axis, the degrees of anti-compression along these two directions are almost the same. At low pressure range (P < 23 GPa), the incompressibility along a-axis is larger than that along b-and c-axis, which is contrary to the case at high pressure range (P > 23 GPa). In Figure 3b, the changes of lattice parameters along the a-, b-and c-axis are similar for C2/m-Ca2C when below 6 GPa, suggesting the same incompressibility along these three directions. a Calculated data in Ref. [9]; b Experimental results in Ref. [9].   The lattice parameters of Ca 2 C at different pressures are listed in Table 2. In Table 2, an excellent agreement with the previous theoretical and experimental values is shown [9]. The calculated lattice parameters decrease with pressure. To get more details, the variations of lattice parameters X/X 0 of the two Ca 2 C phases with pressure are shown in Figure 3. For Pnma-Ca 2 C (see Figure 3a), along the band c-axis, the degrees of anti-compression along these two directions are almost the same. At low pressure range (P < 23 GPa), the incompressibility along a-axis is larger than that along band c-axis, which is contrary to the case at high pressure range (P > 23 GPa). In Figure 3b, the changes of lattice parameters along the a-, band c-axis are similar for C2/m-Ca 2 C when below 6 GPa, suggesting the same incompressibility along these three directions.   The calculated elastic constants and moduli of Ca2C at 0 GPa and high pressures are shown in Table 3. The strain-stress method was used to calculate the single crystal elastic constants. A small finite strain was applied on the optimized structure and the atomic position was fully optimized. Then, the elastic constants were obtained from the stress of the strained structure. The generalized Born's mechanical stability criteria of orthorhombic phase at 0 GPa are given by [17,18]: The stability criteria of monoclinic phase at 0 GPa are given by [17,18]:   The calculated elastic constants and moduli of Ca 2 C at 0 GPa and high pressures are shown in Table 3. The strain-stress method was used to calculate the single crystal elastic constants. A small finite strain was applied on the optimized structure and the atomic position was fully optimized. Then, the elastic constants were obtained from the stress of the strained structure. The generalized Born's mechanical stability criteria of orthorhombic phase at 0 GPa are given by [17,18]: The stability criteria of monoclinic phase at 0 GPa are given by [17,18]: The mechanical stability in crystals under isotropic pressure is provided by Ref. [19]. This requires the symmetric matrixĜ has a positive determinant. InĜ matrix, r C αα " C αα´P , α " 1, 2, . . . , 6 r C 12 " C 12`P , r C 13 " C 13`P , r C 23 " C 23`P (15) where P is the isotropic pressure. If the elastic constants satisfy these stability criteria, it means the structure is mechanically stable. From Table 3, one can see that orthorhombic Pnma-Ca 2 C is mechanical stable up to at least 100 GPa. For monoclinic C2/m-Ca 2 C, the criteria r C 44 r C 66´C 2 46 ą 0, which is similar to the Equation (9), is obeyed only up to 6.4 GPa, as seen in Figure 4, showing that it has mechanical stability below 6.4 GPa. Furthermore, the phonon spectra are presented in Figure 5 to ensure the dynamical stability. As observed, there is no imaginary frequency in the whole Brillouin zone, indicating that Pnma-Ca 2 C is dynamically stable up to at least 100 GPa and that the C2/m-Ca 2 C is dynamically stable below 6.4 GPa. The elastic constants as a function of pressure are displayed in Figure 6 with an approximately upward tendency. We noticed that, for Pnma-Ca 2 C, C 11 is larger than C 22 or C 33 at 0 GPa, whereas it is less than C 22 or C 33 at high pressures, which is in consistent with our previous analyses on the incompressibility along the a-, b-, and c-axis. The elastic constants as a function of pressure are displayed in Figure 6 with an approximately upward tendency. We noticed that, for Pnma-Ca2C, C11 is larger than C22 or C33 at 0 GPa, whereas it is less than C22 or C33 at high pressures, which is in consistent with our previous analyses on the incompressibility along the a-, b-, and c-axis.   The elastic constants as a function of pressure are displayed in Figure 6 with an approximately upward tendency. We noticed that, for Pnma-Ca2C, C11 is larger than C22 or C33 at 0 GPa, whereas it is less than C22 or C33 at high pressures, which is in consistent with our previous analyses on the incompressibility along the a-, b-, and c-axis.   In Table 3, the bulk modulus B and shear modulus G are calculated by Voigt-Reuss-Hill approximations [20][21][22]. The Young's modulus E and Poisson's ratio, υ are given by the following equations [22]: The Pnma-Ca2C is larger than C2/m-Ca2C in bulk modulus, shear modulus and Young's modulus at 0 GPa, as listed in Table 3. All the elastic modulus increase with pressure for Pnma-Ca2C. According to Pugh [23], the brittle material has a small B/G ratio (B/G < 1.75), whereas, the ductile material has a larger ratio (B/G > 1.75). It is interesting that Pnma-Ca2C and C2/m-Ca2C show the brittle manner at 0 GPa and transform to ductile manner at 9.3 GPa and 2.0 GPa, respectively.
Calculating the elastic anisotropy of crystal is of great importance to further study the physical and chemical properties. The calculated universal elastic anisotropy index (A U ), shear anisotropic factors (A1, A2 and A3) and percentage of anisotropy in compressibility and shear (AB and AG) are listed in Table 4. For arbitrary symmetry, the universal elastic anisotropy index A U is obtained by [24,25]:  Pressure (GPa) Pressure (GPa) In Table 3, the bulk modulus B and shear modulus G are calculated by Voigt-Reuss-Hill approximations [20][21][22]. The Young's modulus E and Poisson's ratio, υ are given by the following equations [22]: The Pnma-Ca 2 C is larger than C2/m-Ca 2 C in bulk modulus, shear modulus and Young's modulus at 0 GPa, as listed in Table 3. All the elastic modulus increase with pressure for Pnma-Ca 2 C. According to Pugh [23], the brittle material has a small B/G ratio (B/G < 1.75), whereas, the ductile material has a larger ratio (B/G > 1.75). It is interesting that Pnma-Ca 2 C and C2/m-Ca 2 C show the brittle manner at 0 GPa and transform to ductile manner at 9.3 GPa and 2.0 GPa, respectively.
Calculating the elastic anisotropy of crystal is of great importance to further study the physical and chemical properties. The calculated universal elastic anisotropy index (A U ), shear anisotropic factors (A 1 , A 2 and A 3 ) and percentage of anisotropy in compressibility and shear (A B and A G ) are listed in Table 4. For arbitrary symmetry, the universal elastic anisotropy index A U is obtained by [24,25]: When A U is 0, it means the solid is isotropic, otherwise the solid is anisotropic. The results of the bonding between atoms in different planes. The shear anisotropic factor for the {100} shear plane between the <011> and <010> directions is [26,27]: 4C 44 C 11`C33´2 C 13 (18) For the {010} shear plane between the <101> and <001> directions, it is: For the {001} shear plane between the <110> and <010> directions, it is: The factors A 1 , A 2 and A 3 are 1.0 for any isotropic crystals. As observed in Table 3, all the calculated shear anisotropic factors are not 1.0, presenting the elastic anisotropy. The percentage anisotropy in compressibility and shear are defined as [26]: where B and G are the bulk and shear modulus, and the subscripts V and R represent the Voigt and Reuss bounds. The values of isotropic crystal are 0.0. In Table 3, the values of A B and A G suggest that these two structures of Ca 2 C are anisotropic in compressibility and shear.
To intuitively illustrate the elastic anisotropy, the directional dependence of elastic anisotropy was calculated by the ELAM code [28], which shows the 2D figures of the differences in each direction. The calculated Young's modulus along different directions as well as the projections in different planes are demonstrated in Figure 7. The ratios of E max /E min are 1.76 (1.19) and 2.32 (1.85) for Pnma-Ca 2 C at 0 (100) GPa and C2/m-Ca 2 C at 0 (6.0) GPa, respectively, which means C2/m-Ca 2 C has greater anisotropy. The anisotropy in yz plane is the greatest for Pnma-Ca 2 C at both 0 and 100 GPa (see Figure 7a,b). In Figure 7c,d, the C2/m-Ca 2 C also has the largest anisotropy in yz plane at both 0 and 6 GPa. The 2D representations of Poisson's ratio are revealed in Figure 8. All of them show the elastic anisotropy. From Figure 8a,b, it can be found that the Pnma-Ca 2 C has the greatest anisotropy in yz plane at 0 GPa and in xz plane at 100 GPa. However, the greatest anisotropy of C2/m-Ca 2 C is in yz plane at both 0 and 6 GPa (see Figure 8c,d). The C2/m-Ca 2 C is more anisotropic than the Pnma-Ca 2 C in Poisson's ratio. As far as the 2D projections of shear modulus in xy, yz, and xz planes shown in Figure 9, both C2/m-Ca 2 C and Pnma-Ca 2 C exhibit the obvious elastic anisotropy. From Figure 9a,b, it is seen that the 2D projections of shear modulus in xz plane at 0 GPa and in yz plane at 100 GPa are almost a perfect circle, showing a slight anisotropy character in these two cases. The anisotropy of Pnma-Ca 2 C at high pressure is smaller than that at 0 GPa. The same case occurred for C2/m-Ca 2 C, as seen in Figure 9c,d. Similar to the anisotropy of Poisson's ratio, the shear modulus of Pnma-Ca 2 C has the greatest anisotropy in yz plane at 0 GPa and in xz plane at 100 GPa, and that of C2/m-Ca 2 C is the most anisotropic in yz plane at both 0 and 6 GPa.     [9]. It is known that the calculated band gap with DFT is usually underestimated by 30%-50%, so the ideal band gap is larger than this calculated result. The DOS of Pnma-Ca2C near Fermi level is mainly originated from the contributions of C-p orbital electrons. In Figure 10b, the calculated electronic band structure crosses the Fermi level along many directions in the Brillouin zone, showing the metallic character. And the DOS near Fermi level is mainly characterized by the Ca-d orbital electrons.  As shown in Figure 10, a research of the electronic band structure and density of state (DOS) of Ca 2 C at 0 GPa was also made. The dashed line represents the Fermi level (E F ). From Figure 10a, one can see that Pnma-Ca 2 C is a semiconductor characterized by a quasi-direct band gap of 0.86 eV (the direct band gap at Γ point is 0.87 eV). The conduction band minimum (CBM) is just at Γ point, and the valence band maximum (VBM) locates at (0, 0, 0.378) along the Γ-Z direction. The calculated band gap of Pnma-Ca 2 C at 14 GPa is direct band gap with 0.65 eV, which is close to the previous value of 0.64 eV [9]. It is known that the calculated band gap with DFT is usually underestimated by 30%-50%, so the ideal band gap is larger than this calculated result. The DOS of Pnma-Ca 2 C near Fermi level is mainly originated from the contributions of C-p orbital electrons. In Figure 10b, the calculated electronic band structure crosses the Fermi level along many directions in the Brillouin zone, showing the metallic character. And the DOS near Fermi level is mainly characterized by the Ca-d orbital electrons. As shown in Figure 10, a research of the electronic band structure and density of state (DOS) of Ca2C at 0 GPa was also made. The dashed line represents the Fermi level (EF). From Figure 10a, one can see that Pnma-Ca2C is a semiconductor characterized by a quasi-direct band gap of 0.86 eV (the direct band gap at Γ point is 0.87 eV). The conduction band minimum (CBM) is just at Γ point, and the valence band maximum (VBM) locates at (0, 0, 0.378) along the Γ-Z direction. The calculated band gap of Pnma-Ca2C at 14 GPa is direct band gap with 0.65 eV, which is close to the previous value of 0.64 eV [9]. It is known that the calculated band gap with DFT is usually underestimated by 30%-50%, so the ideal band gap is larger than this calculated result. The DOS of Pnma-Ca2C near Fermi level is mainly originated from the contributions of C-p orbital electrons. In Figure 10b, the calculated electronic band structure crosses the Fermi level along many directions in the Brillouin zone, showing the metallic character. And the DOS near Fermi level is mainly characterized by the Ca-d orbital electrons.

Computational Methods
Our calculations are performed via the generalized gradient approximation (GGA) parameterized by Perdew-Burke-Eruzerhof (PBE) [29] in the Cambridge Serial Total Energy Package (CASTEP) code [30], which is based on the density functional theory (DFT) [31,32]. For the two Ca 2 C phases, the ultrasoft pseudo-potential [33] which describes the interactions between the ionic core and valence electrons is used with the cutoff energy of 420 eV. The k-points of Pnma-Ca 2 C (7ˆ11ˆ6) and C2/m-Ca 2 C (6ˆ9ˆ3) in the first irreducible Brillouin zone are generated using Monkhorst-Pack mesh scheme [34]. Furthermore, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme [35] is used in geometry optimization. The convergence is within 1 meV/atom in the total energy convergence tests for all calculation parameters. The self-consistent convergence of the total energy is 5ˆ10´6 eV/atom, the maximum force on the atom is 0.01 eV/Å, the maximum stress is 0.02 GPa and the maximum ionic displacement is 5ˆ10´4 Å.

Conclusions
A systematic analysis of the pressure-induced phase transition of Ca 2 C is made by first-principles calculations. The enthalpy and dependence of volume on pressure of Ca 2 C are performed. We found that there is a phase transition which occurs at 7.8 GPa transforming from C2/m-Ca 2 C to Pnma-Ca 2 C with a volume drop of 26.7%. The Pnma-Ca 2 C is larger than C2/m-Ca 2 C in the calculated bulk modulus, shear modulus, Young's modulus and Poisson's ratio at 0 GPa. Both of them exhibit the elastic anisotropy. The low-pressure phase C2/m-Ca 2 C, which is mechanically stable up to 6.4 GPa, has the greater anisotropy over the Pnma-Ca 2 C. The electronic band structures reveal the semiconductivity of Pnma-Ca 2 C and the metallicity of C2/m-Ca 2 C. The quasi-direct band gap of Pnma-Ca 2 C at 0 GPa is 0.86 eV. Furthermore, the total and partial density of states is provided to study the specific contribution to Fermi level.