Pressure-induced phase transitions and superconductivity in magnesium carbides

Crystal structure prediction and in silico physical property observations guide experimental synthesis in high-pressure research. Here, we used magnesium carbides as a representative example of computational high-pressure studies. We predicted various compositions of Mg–C compounds up to 150 GPa and successfully reproduced previous experimental results. Interestingly, our proposed MgC2 at high pressure >7 GPa consists of extended carbon bonds, one-dimensional graphene layers, and Mg atomic layers, which provides a good platform to study superconductivity of metal intercalated graphene nano-ribbons. We found that this new phase of MgC2 could be recovered to ambient pressure and exhibited a strong electron-phonon coupling (EPC) strength of 0.6 whose corresponding superconductivity transition temperature reached 15 K. The EPC originated from the cooperation of the out-of-plane and the in-plane phonon modes. The geometry confinement and the hybridization between the Mg s and C pz orbitals significantly affect the coupling of phonon modes and electrons. These results show the importance of the high-pressure route to the synthesis of novel functional materials, which can promote the search for new phases of carbon-based superconductors.

. Srepusharawoot et al. proposed a new phase of MgC 2 containing the pentagon structure using ab initio random searching 22 . The pentagon structure is energetically more stable than the dumbbell structure according to their calculations, which implies that the dumbbell structure can be a meta-stable structure of MgC 2 . The dumbbell and pentagon structures are both an insulating phase 22,23 . Furthermore, Wang et al., reported the structural evolution and the superconductivity of MgC 2 under pressure using first principles calculations 24 . The calculated maximum T c for orthorhombic Cmcm and monoclinic C2/m structures are 11.3 K at 4 GPa and 7.1 K at 9.6 GPa, respectively. Therefore, it is worth studying the possibility of superconductivity at ambient pressure and the origin of the high T c further.
In this paper, we systematically study Mg-C compounds under high-pressure using first-principles calculations. First, we perform crystal structure searching to find unprecedented compounds, and to verify existing experimentally-known compounds. Then, we investigate phases of MgC 2 under pressure up to 150 GPa. The metallic phase of MgC 2 is obtained and stable even at ambient pressure, which exhibits superconductivity with a T c of 15 K. The large λ originates from the cooperation of both couplings of the out-of-plane and in-plane mode to charge carriers. Geometry confinement plays an essential role in enhancing the in-plane vibration contribution to the EPC.

Results
Crystal structure searching under pressure. Figure 1 shows computational structure-searching results of Mg-C compounds at various pressures. At ambient pressure in Fig. 1a, the compounds are energetically unstable while two experimentally meta-stable compounds of MgC 2 and Mg 2 C 3 are close to the stability line (horizontal dashed lines at zero). At 15 GPa, Mg 2 C is stable with an anti-fluorite type structure, which has been reported experimentally 20 . Kurakevych et al. stated that their synthesis condition was 15 GPa 20 . This is in excellent agreement with our computational prediction. As the pressure increases, the formation enthalpy of MgC 2 decreases from ~0.3 eV/atom at ambient pressure to ~0.11 eV/atom at 15 GPa. At a high pressure of 150 GPa, MgC 2 can be one of the stable stoichiometries, as shown in Fig. 1b. This stable structure at high pressure will be discussed later. The enhanced stability of Mg-C compounds under high pressure is consistent with a previous report 25 . Figure 2a shows the formation enthalpy plot of MgC 2 with a reference line exhibiting the previously observed dumbbell structure in the space group P4 2 /mnm. It is worth noting that some of our predicted phases show the lower enthalpy than that of the experimental structure. At ambient pressure, the Mg-intercalated graphite compound (Pmn2 1 ) is predicted to be the ground state. However, when we tested its dynamical stability using ab initio lattice dynamics, it exhibits an imaginary acoustic phonon. Thus, it is unstable against decomposition. As it is a combination of graphene and Mg layers, it would be dissociated into individual graphene and Mg, which is manifested by the negligible charge transfer between C and Mg. Under pressure, we predict two structures; one is a-I4/mmm at low pressure <7 GPa, and another is C2/m structure at high pressure >7 GPa. The a-I4/mmm phase contains a C4 chain, which is more stable than the experimentally known phase at low pressure. Interestingly, C2/m consists of a single zigzag graphene nano-ribbon (zGNR) and Mg array. Hereafter, we refer to this C2/m phase as MgGNR. Figure 2b,c illustrates the top view and side view of the primitive cell of the C2/m structure. This structure has dynamic stability not only at high pressure but also at ambient pressure (See Fig. 3a), implying www.nature.com/scientificreports www.nature.com/scientificreports/ that the MgGNR phase can exist as a meta-stable structure at ambient pressure. The distances of C1-C2 and C2-C2 at ambient pressure are 1.53 Å and 1.45 Å, respectively. Figure 3 shows the phonon dispersion curve and EPC properties of MgGNR at ambient pressure. The phonon dispersion curve and total phonon DOS without phonon softening in Fig. 3a indicates the dynamic stability of the structure. This phase is theoretically predicted to exhibit superconductivity. The superconducting parameters have been calculated using the Eliashberg EPC theory and the Allen-Dynes formula 26,27 ;

Electron-phonon coupling and superconductivity.
, and μ * is the effective Coulomb repulsion parameter. The predicted T c of 15 K in Table 1 is close to the maximum T c of 15.1 K of existing GICs.
To explain this high T c , we analyzed the λ qν and corresponding phonon modes. Interestingly, two regions give a large contribution to the total λ and Eliashberg function, as shown in Fig. 3a,b. The first region is around 30-60 meV at relatively low frequency, and the second one is around 140 meV. Two representative phonon modes at the Γ point produce large contribution to λ. The frequencies of the first mode, q 1 and the second mode, q 2 , are 45.8 meV with λ qν = 0.53 and 142.9 meV with λ qν = 0.36, respectively. Figure 3c represents the lattice displacements by q 1 , which is the out-of-plane vibration mode by Mg and carbon displacements. This is consistent with existing GICs whose out-of-plane mode is usually chosen to explain their superconductivity. The in-plane vibration mode, q 2 , corresponds to the in-plane carbon displacement mode, as shown in Fig. 3d. The in-plane phonon mode is very similar to the phonon mode that mediates the superconductivity in MgB 2 16 . Interestingly, MgGNR exhibits superconductivity induced by not only the out-of-plane mode but also the in-plane mode. These multiple phonon contributions to λ can produce the high T c in MgGNR.

Electronic structures without/with phonon modulation.
To investigate the properties of the predicted MgC 2 and how q 1 and q 2 contribute to superconductivity, we performed electronic structure calculations without/with phonon modulations. The band structure of MgGNR in Fig. 4a shows the metallic property of the material. The p z orbitals of C1 and C2, and the Mg s orbital are mainly observed near E F . The other p x and p y orbitals of C1 and C2 do not contribute near E F . Each Mg p orbital contributes less than the Mg s orbital. The crossing Mg s orbital corresponding to the interlayer state at E F is consistent with that of superconducting GICs, which is proposed as a necessary condition for superconductivity 11,13,14 . Both the C p z and Mg s bands are dispersive and show a strong hybridization feature except along the K2-K3 path. The p z orbital of C2 is dominant and forms the www.nature.com/scientificreports www.nature.com/scientificreports/ flat band along the K2-K3 near E F . The flat band by C2 p z shows the two-dimensional (2D) feature near E F . This electron-like p z state along K2-K3 resembles the hole-like σ state along Γ-A in MgB 2 28,29 making a cylinder-shape Fermi surface from the 2D nature. Figure 4b represents the Fermi surface consisting of two sheets. The green sheet is derived from hybridization of the C p z and Mg s orbitals, while the blue one is derived from the p z orbitals of C2. Both Fermi surfaces are electron pockets. The blue Fermi surface by C2 p z has a flattened cylinder-shape indicating its 2D character. The similarity to GICs and MgB 2 suggests two different charge carriers; one is the electron in the Mg Fermi surface in green coupled with the out-of-plane phonon mode, and another is the electron in C Fermi surface in blue coupled with the in-plane phonon mode. This feature in the band structure calculations with deformed structures by each phonon mode will be discussed later. It is worth noting that the two-band superconductivity from the interband anisotropy is known to be important for MgB 2 . Compared to the isotropic calculation, considering the interband anisotropy increases the calculated electron-phonon coupling constant and predicted T c from 0.77 and 22 K to 1.01 and 40 K, respectively 30,31 . Based on the similarity of MgC 2 to MgB 2 , it would be interesting to investigate the anisotropic effect in MgC 2 as future work.
To investigate the role of the Mg in the system, we set the hypothetical structure (HS) by removing the Mg atom from the MgGNR structure. The flat band along K2-K3 of HS in Fig. 4c is almost intact compared to that of MgGNR. The deformation of the bands by Mg orbitals mainly occurs in unoccupied p z bands along Γ-K1-K2 and K3-Γ-K4. Another significant difference between MgGNR and HS is the position of E F . The band structure in HS is shifted upward by ~4 V with respect to that in MgGNR. MgGNR has the electron in the C2 p z band whereas HS    www.nature.com/scientificreports www.nature.com/scientificreports/ has the hole in the C2 p z band, indicating that the role of Mg is electron doping to unoccupied p z bands of GNR, forming the interlayer state at E F . Figure 4d shows the charge density plot near E F . Note that p z bands of C2 in zGNR are strongly deformed compared to those of graphite. The geometry confinement produces the bonding state between the p z orbitals of two C2s. The top view of charge density resembles the σ bonding observed in MgB 2 . This deformation of the orbital shape can enhance the coupling of the in-plane phonon mode to the C2 p z orbital. Therefore, the bonding state between the p z orbitals of C2s increases the deformation potential D of the in-plane phonon mode, and provides the sizable λ despite its high frequency. This result also explains the difference and similarity between MgGNR and MgB 2 . Because MgGNR has two more electron per f.u. than MgB 2 , the C p z orbital is mainly located at E F , whereas the B p x,y orbitals are at E F in MgB 2 28,29 . The geometry confinement, however, produces the out-of-plane C p z orbitals similar to the in-plane B p x,y .
To check the effect of each phonon mode on the electronic structure, we calculated the electronic structures of the deformed structures by each phonon mode. For each mode, we considered two structures corresponding to two opposite displacements. Figure 5a-h show the band structures of the deformed structure with the out-of-plane mode, q 1 and the in-plane mode, q 2 , respectively. The deformation by the out-of-plane phonon mode mainly occurs in the hybridized bands of Mg s and C p z (the green Fermi surface), whereas the blue Fermi surface from p z of C2 does not change much by the out-of-plane phonon mode. This suggests that the Mg electrons in the green Fermi surface are coupled with the out-of-plane mode and are the charge carrier for the superconductivity mediated by the out-of-plane phonon mode. It is consistent with GICs systems whose λ originates mainly from the out-of-plane mode, and the charge carriers are the intercalated-material electrons in the Fermi surface 11 .
On the other hand, the in-plane phonon mode produces a larger deformation in the whole Brillouin zone (BZ) as in Fig. 5e-h. Both green and blue Fermi surfaces are modified by the in-plane phonon mode. The blue Fermi surface by C2 p z shows a huge modification in contrast to the case of the out-of-plane mode. It qualitatively shows that the deformation potential, D, of the in-plane mode is larger than that of the out-of-plane mode. The C2 p z electron-like state also can be the charge carrier coupled with the in-plane phonon mode. Therefore, our results demonstrate that despite the high ω ph of the in-plane vibration mode, it can contribute to the EPC with a large enough D.
In addition, we checked the HS structure (where Mg atoms are removed) with respect to the atomic displacements of each phonon mode. The electronic structures of the distorted structures are almost intact with respect to the out-of-plane phonon mode, while the electronic structures of the deformed structures by the in-plane mode are modified even without Mg states. This clearly suggests that the coupling between the electrons and the www.nature.com/scientificreports www.nature.com/scientificreports/ out-of-plane mode needs the interlayer state, whereas the coupling between the C p z orbital and in-plane mode is not significantly affected by the interlayer state.
Properties of MgC 2 at 150 GPa. MgC 2 undergoes a structural transformation to an ionic compound at around 150 GPa and is energetically stable, as predicted in Fig. 1b. The high-pressure phase of MgC 2 possesses an intriguing geometry consisting of two-dimensional magnesium layer sandwiched by two buckled graphene layers, as shown in Fig. 6a,b, in good agreement with a previous theoretical prediction 24 . The carbon atoms consist of   a hexagonal honeycomb lattice whose lattice constant of 2.42 Å (C-C distance, 1.48 Å) is comparable to the lattice constant 2.46 Å for graphene. The structure is similar to the BCS-type high-T c superconductor, MgB 2 , except that the carbon atoms are not located on a plane due to a distortion along the z direction. It is a semiconductor with indirect-bandgap of 0.42 eV, as shown in Fig. 6c. The phonon dispersion curve without imaginary bands indicates its dynamical stability (See Fig. 6d). Figure 7a,b show the band structure and phonon bands of the high-pressure phase at ambient pressure, respectively. The phonon dispersion curve remains stable down to ambient pressure as in Fig. 7b, which implies its meta-stability at ambient conditions. Note that the band structure at ambient pressure in Fig. 7a shows that the conduction band minimum and valence band maximum become close to the E F near A point. We further engineered the band gap by adjusting lattice parameters. A closer look at the bandgap in Fig. 7c exhibits a small gap opening at the A point with 1% expansion of a and c lattice parameters. The band gap is closed with 5% lattice expansion in Fig. 7d.

Discussion
We computationally studied the high-pressure phases of MgC 2 . The phases show dynamic stability not only at high pressure but also at ambient pressure. From an application perspective, this feature is intriguing because the properties of this newly-designed materials can be exploited both at high and ambient pressures. The superconductivity of MgC 2 is predicted to have a T c of 15 K, which is high among similar carbon-based superconductors. The large T c originates from the cooperation of the out-of-plane and in-plane phonon modes. The hybridization between Mg s and C p z produces an interlayer state at E F , and the coupling with the out-of-plane mode. The geometry confinement deformed the C p z orbital to the bonding state of C p z orbitals, which is similar to the σ-bonding states in MgB 2 , and produces the large deformation potential of the in-plane mode. The graphene-nanoribbon superconductor, MgC 2 , shows the coupling mechanism between charge carriers and phonon modes observed in GICs and MgB 2 . Thus, we believe that metal-intercalated graphene-nanoribbon superconductors like MgC 2 can be a model system to investigate the relationship between geometry confinement and superconductivity in low-dimensional systems. www.nature.com/scientificreports www.nature.com/scientificreports/

Methods
To investigate ground state structures of various Mg-C compounds containing up to four formula units under pressure, a structure searching algorithm USPEX 32,33 combined with VASP was employed 34 . We conducted USPEX calculations up to 12 atoms in the simulation cell and set the maximum generation number to 50 and structure number to 60. The calculations were converged within the limit. The generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE) was used to describe the exchange correlation functional 35 . PAW pseudopotential 36 was use in the USPEX-VASP calculations with energy cutoff 500 eV and 2 π × 0.01 Å −1 Brillouin zone grid of spacing to show a converged energy. After obtaining the structure at high pressure, the structure at ambient pressure was generated by the full relaxation of the atomic positions and the volume.
Harmonic phonon and Electron phonon coupling (EPC) calculations were conducted using Quantum Espresso software 37 based on density functional perturbation theory 38 . We utilized ultrasoft pseudopotential and PBE functional for EPC calculations. EPC matrix elements were computed in the first BZ on a 6 × 6 × 6 q-mesh with a 24 × 24 × 24 k-mesh and 60 Ry energy cutoff. The convergence of EPC matrix elements was checked with a 12 × 12 × 12 q-mesh with a 24 × 24 × 24 k-mesh. To evaluate the integrated EPC constant (λ), we used the Gaussian broadening of 0.025 Ry.
The band structure calculations were performed with the full-potential linearized augmented plane wave band method implemented in Wien2k package 39 . Generalized gradient approximation (GGA-PBE) is used as the exchange-correlation.