Pressure induced superconductivity and electronic structure properties of scandium hydrides using first principles calculations

Abstract

The electronic, vibrational and superconducting properties of scandium hydrides (ScH₂ and ScH₃) under pressure were studied using first-principles calculations. The results indicate that ScH₂ and ScH₃ are dynamically stable in the pressure ranges of 0–85 GPa and 46–80 GPa, respectively. The superconducting properties of ScH₂ and ScH₃ were investigated by employing Bardeen–Cooper–Schrieffer (BCS) theory, and this shows that the superconducting temperature of ScH₂ initially increases exponentially and then reaches a maximum value of about 38.11 K at 30 GPa, while it remains constant under further compression. However, the superconducting behavior of ScH₃ is not obvious under low pressure (P < 46 GPa), and it almost disappears under higher pressure, in agreement with experimental observations. Analysis of the energy band structures demonstrates that the distinct superconducting behaviors of ScH₂ and ScH₃ are related to the hybridization between the s-state of the H atom and the d-state of the Sc atom. The superconducting behavior of ScH₂ follows the variation of the hybridization between the H_O-s state and Sc-d state, while for ScH₃, it is found that there is no density of states observed for H_T or H_O when the pressure is above 46 GPa. Analysis of the electronic structure of ScH₂ was also performed to allow for further comprehension of the metallic behavior of ScH₂ under pressure. This work may offer help to understand the mechanism of pressure-induced superconductivity in metal–hydride systems.

---

I. Introduction

The superconductivity of hydrides has raised tremendous interest since experiments found that the superconducting temperature (T_c) in a sulfur hydride system can reach as high as 203 K in the pressure range of 200 GPa. Recently, various metal-hydrides, such as CrH_x,¹ LuH₂,² GeH₄,³ RuH_x,⁴ SnH₈,⁵ PtH⁶ and GaH₃,⁷ etc.,^8–11 have been investigated in detail. However, some hydrides may decompose under compression, as has been confirmed by calculations and experiments,^12–24 resulting in enormous difficulty in exploring the underlying mechanism that induces superconductivity in hydrides. Using common hydrides which can exist stably in experiments, even under high pressures, is an advisable way to study the mechanism of superconductivity. Generally, rare-earth metals exhibit successive hydride forms from a solid solution of MH_x (x < 2, M = Sc, Y, La, etc.) to trihydride MH₃ via dihydride (MH₂), as the concentration of hydrogen increases, which is accompanied by a structural change from face-centered-cubic (fcc) MH₂ to a hexagonal-close-packed (hcp) MH₃ lattice before the onset of the fcc-structure of MH₃. This progression is interpreted as a metallic–insulator–metallic transition from dihydride to trihydride. Simultaneously, a contraction of the lattice can also be observed during the transition from MH₂ to MH₃.

Experiments on H₂S have demonstrated that the superconducting properties of the sulfur hydride system may originate from other hydrides such as SH₃, produced by the decomposition or synthesis of H₂S.¹³ The stoichiometry of hydrides, consisting of hydrogen and other elements, has an extraordinary influence on the superconducting behavior of these compounds. Therefore, it is necessary to make a detailed investigation into the superconducting properties of metal–hydride systems of different chemical formulae. Scandium is known as the lightest element with the smallest radius among transition metals and displays interesting changes in electronic structure induced by different concentrations of hydrogen, making it a perfect candidate to obtain a detailed comprehension of the superconducting properties of hydrides. Commonly, scandium hydrides also follow the above transition from ScH₂ to ScH₃, i.e. the Sc lattice retains its fcc structure and hydrogen atoms fill the tetrahedral and octahedral sites. Raman and IR measurements²⁵ have confirmed that ScH₃ comes into being when the pressure is above 50 GPa, as the hydrogen content increases and the closing of the optical band gap can be observed. So far, Ohmura et al.²⁶ have studied the pressure-induced phase transition of ScH₃ using X-ray diffraction measurements, finding a structural transition from the hcp to fcc lattice which began at 30 GPa and was completed at 46 GPa. The static ground-state structures and pressure-induced phase transition of three scandium hydrides (ScH, ScH₂, ScH₃) have been calculated by Ye et al.²⁷ Peterman et al.²⁸ have also previously carried out a detailed investigation on the electronic structure of metal hydrides. Up to now, studies on the lattice dynamical and superconducting properties of scandium hydrides remain scare, as well as those on the underlying mechanism of superconductivity in these metal hydrides.

In this work, first principles calculations were carried out on the lattice dynamical and superconducting properties of the stoichiometric forms of scandium di- and tri-hydrides, to find the root cause of superconductivity obtained in these metal hydrides. The calculations demonstrate that ScH₂ exhibits a relatively high T_c under a low pressure range while almost no superconducting behavior is exhibited by ScH₃ when the pressure is above 46 GPa. Based on the analysis of the energy band structures, it was found that this phenomenon is attributed to the hybrid intensity between the 1s-state of the H (H_T for ScH₂) atom and the d-state of the Sc atom under pressure. In this work, the computational details are introduced in Section II. Some results and analyses of the energy band structures, lattice dynamics, electronic structures and superconducting properties of the scandium hydrides are presented in Section III. The main conclusions are given in Section IV.

II. Materials and methods

In this work, convergence tests of the kinetic energy cutoff as 60 Ry and 16 × 16 × 16 Monkhorst–Pack k-point sampling mesh²⁹ was applied for the Brillouin zone integration, using a smearing parameter of 50 mRy. Ultrasoft plane-wave pseudopotentials³⁰ for Sc and H were applied to calculate the relative properties of ScH₂ and ScH₃, and 3p⁶3d¹4s² for Sc as well as 1s¹ for H were used as electronic configurations to obtain a self-consistent atomic charge density. Phonon dispersion and electron–phonon calculations were obtained by applying linear response theory. The dynamical matrices were calculated firstly at 10 wave (q) vectors using a 4 × 4 × 4 q grid in the irreducible wedge of the Brillouin zone, and then the dynamical matrices were Fourier-transformed to real space, accompanied by the construction of force constant matrices, which were used to obtain phonon frequencies at arbitrary points. The Monkhorst–Pack (MP) mesh was set as 12 × 12 × 12 and the Fermi–Dirac smearing width was 0.02 Ry. The zone-center phonon mode Eigen displacements, which represent the infrared and Raman activity of the cubic phase scandium hydrides (ScH₂ and ScH₃) were obtained using an isotropy code developed by Stokes and Hatch.³¹ The energy band structures of fcc-ScH₂ and ScH₃ were calculated in the irreducible fcc Brillouin zone (BZ) using the generalized gradient approximation of the Perdew, Burke and Ernzerhof (GGA-PBE) method.^32,33 All of our work was carried out using the Quantum-Espresso (QE) package.³⁴

III. Results and discussion

A. Vibrational modes and lattice dynamical properties

For scandium dihydride, the scandium atom (Sc) occupies the fcc site (Wyckoff position: 4a (0, 0, 0)) and the hydrogen atom (H_T) fills in the tetrahedral site (Wyckoff position: 8c (0.25, 0.25, 0.25)). For scandium trihydride, ScH₃, it has a BiF₃ structure with the same space group as ScH₂, and the additional hydrogen atom (H_O) is located at the 4b (0.5, 0.5, 0.5) octahedral Wyckoff position. Based on group theory,³¹ ScH₂ and ScH₃ belong to the Fm3̄m space group and the point group at Γ is O_h. Therefore the irreducible representations for fcc-ScH₂ and fcc-ScH₃ can be expressed as Γ = T_2g (R) ⊕ 2T_1u (IR) and Γ = T_2g (R) ⊕ 3T_1u (IR),²⁵ respectively. Thus, for the fcc structures of ScH₂ and ScH₃, the H atom at the O site as well as that at the metal site occupied by a Sc atom are infrared active, while that at the T site is Raman active, indicating that the H atom of the T site (H_T) is not located at the center of the tetrahedral site. If the H atom is centered at the T site, the vibrations become Raman inactive. Compared with ScH₂, the octahedral hydrogen of ScH₃ adds a T_1u mode to the zone center mode decomposition. The Eigen displacement of the zone-center modes for ScH₂ and ScH₃ are given in Fig. 1. It can be seen that the T_2g mode in both fcc-ScH₂ and fcc-ScH₃ (Fig. 1b and e) involves the anti-symmetric breathing motion of H_T, while the T_1u mode of ScH₂ (Fig. 1a), as well as the T¹_1u mode of ScH₃ (Fig. 1c), present a reverse motion towards the opposite direction of each layer (for the ScH₂ containing Sc-plane and H_T-plane, and for ScH₃ involving the Sc ⊕ H_O-plane and H_T-plane). For another infrared active mode, T²_1u in ScH₃, this means completely opposite directions between the motions of the Sc atoms and H atoms (both H_T and H_O). This mode, located at the Γ point, reflects the move of the H (H_T and H_O) lattice in the opposite direction of the metal lattice, and therefore, is at a much lower frequency compared with T¹_1u in the phonon dispersion curves.

Fig. 1 Schematic diagrams showing atom displacement of Raman active modes (R) and infrared active modes (IR) in ScH₂ (a and b) and ScH₃ (c–e). The scandium and hydrogen atoms are shown by the large navy and small red spheres, respectively. The H_T and H_O (for fcc-ScH₃) atoms are located at tetragonal and octahedral sites, which are surrounded by four and eight scandium atoms, respectively.

The calculated phonon dispersion curves along W–L–Γ–X–W–K high symmetry directions of the Brillouin zone for ScH₂ and ScH₃ under various pressures are displayed in Fig. 2 and 3, respectively. Based on the calculated phonon dispersion curves under various pressures, it is found that ScH₂ is dynamically stable in the pressure range of 0–85 GPa, while ScH₃ is stable at 46–80 GPa. Our calculations are in good agreement with the Raman and infrared experiments performed by Kume,²⁵ which found that the hcp-intermediate-fcc phase transition of ScH₃ occurs at 25 and 46 GPa, respectively. In Fig. 2 and 3, each band of ScH₂ and ScH₃ moves to a higher frequency with elevated pressure. The most striking feature in Fig. 2 is the existence of a broad gap between the optical and acoustical branches, and this gap becomes enlarged under higher pressure, while the ranges of the optical modes shrink inversely. This finding originates from the huge mass difference between the hydrogen and scandium atoms. Since the low frequencies (below 10 THz) of the phonon dispersion curves correspond to the vibration of Sc atoms, and the high frequencies (in the range of about 20–45 THz) are mainly related to H atoms (the contribution of the Sc atoms is very slight), the above phenomenon can be interpreted as a pressure-induced continual compression of the lattice structure. The behavior of the phonon dispersion curves of ScH₃ under pressure is interesting, aside from the fact that the frequency of each mode increases with pressure. It can be seen that the phonon band structures are constituted mainly by three parts, i.e. the lowest frequencies below about 10 THz (Part-I) corresponding to Sc atoms, the intermediate frequencies (Part-II) switched by the other two parts representing the vibration of H atoms at T sites, and the high frequency part (Part-III), dominated by the H atoms at O sites. Initially, the frequencies of Part-I and Part-II adjoin each other. Then Part-II moves to higher frequencies and departs from Part-I gradually with elevated pressure, and simultaneously becomes closer to Part-III.

Fig. 2 Phonon band structures of ScH₂ under various pressures (a–d) compared with phonon density states. The Raman active mode (T_2g) and infrared active mode (T_1u) at the BZ center are also labeled.

Fig. 3 Phonon band structures of ScH₃ under various pressures (a–d) compared with phonon density states. The Raman active mode (T_2g) and infrared active modes (T¹_1u, and T²_1u) at the BZ center are also given.

B. Superconducting properties and energy band structures

Pressure-induced superconducting behaviors of metal hydrides have attracted much interest in recent years. Here, we calculated the superconducting properties of ScH₂ and ScH₃ under various pressures, which result from the electron–phonon interaction. The calculated parameters associated with T_c for ScH₂ are listed in Table 1. Initially, T_c increases exponentially with pressure in the low pressure range and reaches a maximum value of about 38.11 K at 30 GPa, and then remains anomalously constant up to the highest achieved pressure of 80 GPa (Fig. 4a). Unlike ScH₂, the T_c of ScH₃ approaches 0 K (0.113 K at 50 GPa, and 0.199 K under 60 GPa) and stays almost unchanged under further pressures based on our calculations. Thus, in the following sections, the emphasis of our studies is mainly focused on the superconducting properties of the fcc structure of ScH₂. The results obtained for ScH₃ are consistent with the prediction made by Kim et al.,³⁵ who found that almost no superconductivity is observed when the pressure becomes higher than 30 GPa. Here it should be mentioned that in the above reports, the highest value of T_c for ScH₃ can reach as high as 18 K at around 19 GPa and decreases exponentially in the pressure range of 19–30 GPa. Based on the calculated phonon band structures, our calculations indicate that ScH₃ is dynamically stable when P > 46 GPa, and the Raman and infrared measurements also confirm that the intermediate-fcc phase transition occurs at 46 GPa.^25,26 However, a superconducting state of ScH₃ under a low pressure range below 50 GPa may exist, because experiments have found that for many transition metal hydrides, such as ErH₃, GdH₃, HoH₃, LuH₃ and SmH₃,^36–38 structure hysteresis indeed exists below the transition point during the decline of pressure. Our previous work on cubic phase AlH₃ also concluded that when finite electronic temperature effects are included, dynamical instabilities below 73 GPa can be removed in the phonon dispersion curves, rendering the metastability of this phase in the range of 63–73 GPa, while T_c (15.4 K) can become remarkably high under the lowest possible pressure (63 GPa) compared with that observed under 73 GPa (8.5 K).³⁹ In fact, the distinct behavior of T_c in metal hydrides, especially for transition metal hydrides, is reasonable since the electronic bandwidth is sensitive to volume change in these materials. The pressure-dependent evolutions of T_c are caused by the implicit dependence of T_c on volume through the phonon cutoff frequency (θ_D or ω_c) and the electronic density of states [N(E_F)], as given by the BCS relationship or the McMillan strong-coupling formalism.^40,41 Therefore, it is possible that these two mechanisms produce a distinct T_c under pressure for different stoichiometric compounds. The calculated spectral function α²F(ω) of ScH₂ obtained under 60 GPa and the corresponding the dimensionless integral of the electron–phonon parameter λ with frequency are given in Fig. 4b. As can be seen for ScH₂, the low frequency part (below 15 THz) contributes about 5% to the integrated λ, while the high frequency part, corresponding to the vibration between Sc and H_T, contributes almost the remaining 95% of the total EPC parameter. To have a clear comparison, the calculations of the el-ph information for ScH₃ under 60 GPa are also presented in Fig. 4c. As can be seen, the intensity of the coupling for both the scandium and hydrogen modes is much weaker than that in ScH₂, and lambda equals about 0.309, while the T_c value of ScH₃ under this pressure is about 0.199 K. We can see that the T_c value of ScH₃ is much smaller compared with ScH₂.

Table 1 The calculated logarithmic average phonon frequency 〈ω_log〉 (K), electronic density of states per atom and spin at the Fermi level N_f (states/spin/Ry/unit cell) and electron-phonon coupling parameter λ combined with T_c (K) for the commonly chosen value of the Coulomb pseudopotential 0.1 for ScH₂ at various pressures

P (GPa)	〈ωlog〉	Nf	λ	Tc (μ* = 0.1)
0	364.2	5.057	0.696	22.826
10	433.4	4.697	0.707	27.832
20	396.7	4.466	0.878	34.347
30	389.0	4.335	0.976	38.110
40	381.4	4.064	0.849	31.661
50	342.7	4.299	0.936	32.024
60	367.5	3.998	0.872	31.550
70	361.8	4.102	0.856	30.349
80	384.9	4.374	0.869	32.903

---

Fig. 4 (a) Pressure dependence of the superconducting transition temperature T_c of ScH₂ when μ* = 0.1. Inset: the sum of hybridization (states per eV cell) between the s-state of the H_T atom and the d-state of the Sc atom varies with pressure. One can see that the variation of T_c follows the behavior of the hybridization between the s-state of the H_T atom and the d-state of the Sc atom. (b) and (c) The Eliashberg phonon spectral function α²F(ω) and the dimensionless integral of the electron–phonon parameter λ(ω) of ScH₂ and ScH₃ under 60 GPa, respectively.

To interpret the different superconducting behaviors of these two hydrides, the energy band structures have been analyzed carefully for each atom in the two hydrides. According to the calculated energy band structures of ScH₂ and ScH₃ (both at 46 GPa), as shown in Fig. 5a and b, the lowest Γ₁ point is a metal–s–hydrogen–s (H_T s for ScH₂, while both H_T s and H_O s for ScH₃, and therefore, Γ₁ for ScH₃ has lower energy than that for ScH₂) bonding state, while Γ′₂ is formed by an antibonding combination of the two hydrogen s-states located in T sites. The calculated energy band structures of ScH₂ and ScH₃ are the same as those obtained in the work of Peterman et al.,²⁸ in which the main focus of the study was on the electronic structure of metal hydrides. Based on group theory, it can also be inferred that Γ′₂₅, which has higher energy compared with Γ′₂, is composed of the t_2g part of the scandium 3d orbitals, and Γ₁₂ corresponds to the e_g part. The linear combination of different irreducible characteristics at Γ for ScH₂ and ScH₃ is listed in Table 2. Since the band at the Γ point across the Fermi energy is mainly composed of Γ′₂ and Γ′₂₅ in the fcc-ScH₂ lattice (Fig. 5a), the conductivity of ScH₂ originates considerably from the interaction of electrons in the Sc and H_T atoms. It can be deduced that the hydrogen at the T site likely interacts as an anion with the surrounding metal atoms, whereas the localization of electrons for a H atom at an O site (for ScH₃) is comparatively stronger than that for H_T. Interestingly, this bonding character can be regarded as a “cage” formed by the Sc and H_T atoms, trapping the H_O atom inside. However, for ScH₃, it is observed that Γ′₂₅ and Γ′₂ are located at the Γ point, indicating that no s-state of the H_T atom is observed at Fermi energy and therefore, no hybridization is found between the t_2g part of the scandium 3d orbitals and the s-state of the H_T atom (Fig. 5b). In fact, the superconducting behavior of ScH₂ follows the variation of the hybridization between the H_T-s state and Sc-d state (Fig. 4a). To make a rigorous certification of this conclusion, further analysis and a concrete contrast between the states of each atom in ScH₂ and ScH₃ are given in the next section.

Fig. 5 Energy band structure of ScH₂ (a) and ScH₃ (b) along the symmetry points and axes. The dotted line indicates the position of the Fermi energy.

Table 2 Linear composition of different levels at the Γ point for the fcc-ScH₂ and ScH₃ lattice. The states listed below are composed of the scandium s orbital, the d orbitals T_2g and e_g, the 1s orbitals H_T1 and H_T2 of the two tetrahedral hydrogen atoms and the 1s orbital H_O of the octahedral atom

Level	Linear combination
Γ^low1	s ⊕ HT1 + HT2 ⊕ HO
Γ′2	HT1 − HT2
Γ′25	(t2g)^3
Γ^high1	s ⊕ HT1 + HT2 ⊙ HO
Γ12	(eg)^2

---

C. Electronic structure

The density of states (DOS) plays an important role in the analysis of the behavior of electrons in metal hydrides under pressure. To further comprehend the metallic behavior of ScH₂ under pressure, a detailed investigation on the changes in the DOS for fcc-ScH₂ was conducted in the pressure range of 0–80 GPa. The calculated partial density of states (PDOS) of the Sc and H atom under various pressures are illustrated in Fig. 6. It can be obtained that: (1) the conduction band is mainly composed of Sc-3d and H-1s, and Sc-3d obviously makes a crucial contribution compared with H-1s. The valence band is composed of Sc-3d, Sc-4s and H-1s, and Sc-4s makes little contribution. (2) Distinct changes are apparent for both the Sc-3d and H-1s shapes. Firstly, most of the peaks that appear in the conduction and valence bands of the Sc atom decrease under compression, and the peak at the coordinate of −5 eV splits into two visible peaks. Secondly, it can be seen that almost all peaks observed in H-1s become larger, and a new peak appears near −4.5 eV. The change in the shape of the DOS in the Sc and H atoms means the hybridization between these two atoms become more complex under pressure. (3) H-1s has a strong hybridization with both Sc-3d and Sc-4s.

Fig. 6 Calculated partial density of states of Sc (a) and H_T (b) atoms for ScH₂ under various pressures. (c) The partial density of states of Sc-3d (a) and H_T-1s and H_O-1s (b and c) atoms in ScH₃ under 60 GPa. The dotted line represents the Fermi energy level. Here we would like to declare that for ScH₃, we just give the partial density of states under 60 GPa for a simple contrast with ScH₂. In fact, the PDOS of H_T 1s and H_O 1s in ScH₃ are zero for the whole pressure range above 46 GPa. The reason for choosing this pressure is because the Eliashberg phonon spectral function α²F(ω) and the electron–phonon parameter λ(ω) of ScH₃ are given under 60 GPa in Fig. 4c.

To explore the underlying mechanism behind the different pressure-induced superconducting behaviors of ScH₂ and ScH₃, here the partial density of states of ScH₃ are also illustrated in Fig. 6c. Based on the calculations, it can be seen that there are indeed no states at the Fermi energy for both the H_T and H_O atoms in ScH₃, while the energy of Sc-3d states is considerably high. This indicates that the metallization or the superconducting behavior mainly results from Sc atoms and the contributions from H_T and H_O are very little. Based on the calculations, in fact, it can also be deduced that the superconducting temperature should be close to the T_c value of the transition element Sc. From the previous studies on the Sc element, it can be found that the T_c value of Sc is 1 K at pressures below 40 GPa, and equals about 1.5 K under 43 GPa.⁴² Since the energy of states at the Fermi level for ScH₃ is less than that of the Sc metal, it is reasonable that the T_c value of ScH₃ is smaller than that of Sc metal.

Compared with ScH₃, a very important character of ScH₂ is that the PDOS value of the Sc-3d orbital at the Fermi energy level becomes smaller as the pressure increases and it presents an enhanced trend for the H-1s orbital, indicating that H-1s plays a more and more important role in the metallization of ScH₂, which is consistent with the above analysis. To give a clearer picture of the influence of the hybridization intensity between Sc-3d and H_T-1s on the superconducting behavior of ScH₂, the variation in the sum of the hybridization between the s-state of the H_T atom and the d-state of Sc atom with pressure is shown in Fig. 4a. This inset graph in Fig. 4a corresponds to an enlargement of the intensity of the H_T-1s and Sc-3d atoms referred to in Fig. 6. One can see that the variation of T_c almost follows the behavior of the hybridization between the 1s-state of the H_T atom and the 3d state of the Sc atom.

IV. Conclusions

In this work, the lattice dynamics, superconducting properties and electronic structure of scandium hydrides have been studied using first principles calculations. The zone-center phonon mode Eigen displacements have been analyzed which is crucial to the analysis of spectral experiments, such as IR and Raman spectroscopy. Phonon dispersion curves indicate that ScH₂ and ScH₃ are dynamically stable in the pressure range of 0–85 GPa and 46–80 GPa, respectively. The superconducting transition temperature of ScH₂ increases exponentially with pressure and then reaches a maximum value of about 38.11 K at 30 GPa, while it remains anomalously constant upon the application of further pressure. In the present study, no clear superconducting behavior is found in ScH₃ above 46 GPa. From the analysis of the energy band structures and the density of states, the superconducting behavior of ScH₂ originates from the hybridization between the 1s-state of the H_T atom and the d-state of the Sc atom, while this hybridization is absent in ScH₃ when P > 46 GPa, resulting in no superconductivity observed in this material. The electronic structure was also investigated to obtain a further understanding of the metallic behavior of ScH₂ under pressure. We hope our studies may provide help for the interpretation of the underlying mechanism behind the superconducting behavior observed in hydrides.

Acknowledgements

The authors would like to thank the support from the Science and Technology Development Foundation of China Academy of Engineering Physics under Grant No. 2012A0201007 and 2013B0101002, the Doctoral Fund of Henan University of Technology under Grant No. 2016BS006, the Science and Technology Foundation of Henan province education department under Grant No. 16A140006 and 17A140016, the Fundamental Research Funds for the Henan Provincial Colleges and University of Technology under Grant No. 2016QNJH12 and 2016JJSB091, as well as the Open Project of State Key Laboratory Cultivation Base for Nonmetal Composites and Functional Materials under Grant No. 15zxfk10.

References

1. S. Y. Yu, X. J. Jia, G. Frapper, D. Li, A. R. Oganov, Q. F. Zeng and L. T. Zhang, Sci. Rep, 2015, 5, 17764.
2. J. W. Yang and L. An, Int. J. Hydrogen Energy, 2016, 41, 2720–2726.
3. H. D. Zhang, X. L. Jin, Y. Z. Lv, Q. Zhuang, Q. Q. Lv, Y. X. Liu, K. Bao, D. Li, B. B. Liu and T. Cui, Phys. Chem. Chem. Phys., 2015, 17, 27630.
4. Y. X. Liu, D. F. Duan, F. B. Tian, C. Wang, Y. B. Ma, D. Li, X. L. Huang, B. B. Liu and T. Cui, Phys. Chem. Chem. Phys., 2016, 18, 1516–1520.
5. H. D. Zhang, X. L. Jin, Y. Z. Lv, Q. Zhuang, Y. X. Liu, Q. Q. Lv, D. Li, K. Bao, B. B. Liu and T. Cui, RSC Adv., 2015, 5, 107637.
6. D. Y. Kim, R. H. Scheicher, C. J. Pickard, R. J. Needs and R. Ahuja, Phys. Rev. Lett., 2011, 107, 117002.
7. G. Y. Gao, H. Wang, A. Bergara, Y. W. Li, G. T. Liu and Y. M. Ma, Phys. Rev. B, 2011, 84, 064118.
8. D. C. He, H. X. Shao and Y. K. Wei, Can. J. Phys., 2015, 93(12), 1630–1637.
9. X. L. Feng, J. R. Zhang, G. Y. Gao, H. Y. Liu and H. Wang, RSC Adv., 2015, 5, 59292.
10. Y. W. Li, J. Hao, H. Y. Liu, J. S. Tse, Y. C. Wang and Y. M. Ma, Sci. Rep., 2015, 5, 9948.
11. D. Y. Kim, R. H. Scheicher and R. Ahuja, Phys. Rev. Lett., 2009, 103, 077002.
12. X. F. Zhao, A. R. Oganov, X. Dong, L. X. Zhang, Y. J. Tian and H. T. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 84, 054543.
13. A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov and I. Shylin, Nature, 2015, 525, 73.
14. Y. W. Li, J. Hao, H. Y. Liu, Y. L. Li and Y. M. Ma, J. Chem. Phys., 2014, 140, 174712.
15. Y. W. Li, L. Wang, H. Y. Liu, Y. W. Zhang, J. Hao, C. J. Pickard, J. R. Nelson, R. J. Needs, W. T. Li, Y. W. Huang, I. Errea, M. Calandra, F. Mauri and Y. M. Ma, Phys. Rev. B, 2016, 93, 020103.
16. M. Einaga, M. Sakata, T. Ishikawa, K. Shimizu, M. I. Eremets, A. P. Drozdov, I. A. Troyan, N. Hirao and Y. Ohishi, Nat. Phys. DOI:10.1038/NPHYS3760.
17. H. Fujihisa, H. Yamawaki, M. Sakashita, A. Nakayama, T. Yamada and K. Aoki, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 69, 214102.
18. S. Kometani, M. Eremets, K. Shimizu, M. Kobayashi and K. Amaya, J. Phys. Soc. Jpn., 1997, 66, 2564–2565.
19. M. Sakashita, H. Yamawaki, H. Fujihisa and K. Aoki, Phys. Rev. Lett., 1997, 79, 1082.
20. M. Einaga, M. Sakata, T. Ishikawa, K. Shimizu, M. Eremets, A. Drozdov, I. Troyan, N. Hirao and Y. Ohishi, Crystal structure of 200 K-superconducting phase of sulfur hydride system, arXiv:1509.03156, 2015, 1–8.
21. N. Bernstein, C. S. Hellberg, M. D. Johannes, I. I. Mazin and M. J. Mehl, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 060511.
22. D. F. Duan, X. L. Huang, F. B. Tian, D. Li, H. Y. Yu, Y. X. Liu, Y. B. Ma, B. B. Liu and T. Cui, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 180502.
23. I. Errea, M. Calandra, C. J. Pickard, J. Nelson, R. J. Needs, Y. W. Li, H. Y. Liu, Y. W. Zhang, Y. M. Ma and F. Mauri, Phys. Rev. Lett., 2015, 114, 157004.
24. T. Jarlborg and A. Bianconi, Sci. Rep., 2016, 6, 24816.
25. T. Kume, H. Ohura, T. Takeichi, A. Ohmura, A. Machida, T. Watanuki, K. Aoki, S. Sasaki, H. Shimizu and K. Takemura, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 84, 064132.
26. A. Ohmura, A. Machida, T. Watanuki, K. Aoki, S. Nakano and K. Takemura, J. Alloys Compd., 2007, 446, 598–602.
27. X. Q. Ye, R. Hoffmann and N. W. Ashcroft, J. Phys. Chem. C, 2015, 119, 5614–5625.
28. D. J. Peterman, B. N. Harmon and J. Marchiando, Phys. Rev. B: Solid State, 1979, 19, 4867–4874.
29. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188–5192.
30. D. Vanderbilt, Phys. Rev. B: Condens. Matter Mater. Phys., 1990, 41, 7892.
31. H. T. Stokes and D. M. Hatch, Isotropy Subgroups of the 230 Crystallographic Space Groups, World Scientific, Singapore, 1988.
32. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1997, 78, 1396.
33. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
34. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari and R. M. Wentzcovitch, J. Phys.: Condens. Matter, 2009, 21, 395502.
35. D. Y. Kim, R. H. Scheicher, H. K. Mao, T. W. Kang and R. Ahuja, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 2793–2796.
36. T. Palasyuk and M. Tkacz, Solid State Commun., 2004, 130, 219–221.
37. T. Palasyuk and M. Tkacz, Solid State Commun., 2005, 133, 481–486.
38. T. Palasyuk and M. Tkacz, Solid State Commun., 2007, 141, 302–305.
39. Y. K. Wei, N. N. Ge, X. R. Chen, G. F. Ji, L. C. Cai and Z. W. Gu, J. Appl. Phys., 2014, 115, 124904.
40. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev., 1957, 106, 162–164.
41. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev., 1957, 108, 1175–1204.
42. L. W. Nixon, D. A. Papaconstantopoulos and M. J. Mehl, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 134512.

---