Pressure-induced superconductivity in H2-containing hydride PbH4(H2)2

High pressure structure, stability, metallization, and superconductivity of PbH4(H2)2, a H2-containing compound combining one of the heaviest elements with the lightest element, are investigated by the first-principles calculations. The metallic character is found over the whole studied pressure range, although PbH4(H2)2 is metastable and easily decompose at low pressure. The decomposition pressure point of 133 GPa is predicted above which PbH4(H2)2 is stable both thermodynamically and dynamically with the C2/m symmetry. Interestedly, all hydrogen atoms pairwise couple into H2 quasi-molecules and remain this style up to 400 GPa in the C2/m structure. At high-pressure, PbH4(H2)2 tends to form the Pb-H2 alloy. The superconductivity of Tc firstly rising and then falling is observed in the C2/m PbH4(H2)2. The maximum of Tc is about 107 K at 230 GPa. The softening of intermediate-frequency phonon induced by more inserted H2 molecules is the main origin of the high Tc. The results obtained represent a significant step toward the understanding of the high pressure behavior of metallic hydrogen and hydrogen-rich materials, which is helpful for obtaining the higher Tc.

In recent decades, many scientists have devote to searching for the high-temperature superconducting materials. For the lightest element, hydrogen (H), Ashcroft applied the BCS theory to propose that the metallic hydrogen will be a room-temperature superconductor under high pressure 1 . This suggestion has motivated considerable experimental and theoretical activities. However, solid hydrogen remains insulating character at extremely high pressure, at least up to 342 GPa 2 . Due to the extremely high and experiment unreachable pressure, as a alternative, Ashcroft proposed that the hydrogen-rich alloys shall transform into metal under relatively lower pressure due to the chemical precompressions from the comparable weight elements 3 . Thus, hydrogen-rich group-IV hydrides have been extensively explored, such as CH 4 , SiH 4 , GeH 4 , SnH 4 , and PbH 4 . All of them show up interesting new structures and novel properties under pressure. CH 4 is still an insulator up to the pressure of 520 GPa 4 . Although Eremets et al. experimentally reported the metallization and superconductivity of SiH 4 above 60 GPa 5 , for the controversial result it might be understood as superconductivity of amorphous silicon, silicon hydrides, or platinum hydrides 6,7 . And theoretical prediction indicates that the stable SiH 4 can behave as metal and exhibit superconductivity above 220 GPa with the superconducting transition temperature (T c ) of about 20 K (The Coulomb parameter µ = . * 0 1, the below is same.) 8 . GeH 4 has lower metallization pressure than silane 9,10 , and the highest T c reaches to 73 K at 220 GPa 11 . Furthermore, the metallization pressure of SnH 4 decreases, the highest T c is close to 83 K at 120 GPa 12 .
It is clearly that the metallization pressure of group-IV hydrides decreases with increase of atomic number of heavy element, which is obviously less than that of solid H 2 . Unfortunately, the T c of group-IV hydrides is also greatly decreased. By analyzing the crystal feature, we find that the quasi-molecular H 2 units exist in the high-pressure structures of GeH 4 and SnH 4 . And these H 2 units have been found to contribute significantly to the superconductivity. Then, whether can the T c be improved by intercalating H 2 into group-IV hydrides? H 2 -containing compounds of CH 4 -H 2 have been fabricated up to 30 GPa, such as CH 4 (H 2 ) 2 , (CH 4 ) 2 H 2 , CH 4 (H 2 ) 4 , CH 4 H 2 13 . But both metallization and superconductivity are still lack. For the SiH 4 -H 2 system, the crystal structure, phase diagram, and metallization under pressure of SiH 4 (H 2 ) 2 were extensively explored [14][15][16][17][18][19][20][21][22] . The T c of SiH 4 (H 2 ) 2 is as high as 107 K at 250 GPa 23 , which is visibly higher than that of SiH 4 . Following the experimental observation 24 , we have also theoretically investigated the structural, phase transition, metallization, and superconductivity of GeH 4 (H 2 ) 2 under pressure 25,26 . The predicted T c of GeH 4 (H 2 ) 2 is close to 100 K at 250 GPa, higher than that of GeH 4 . These results inevitably encourage us further to seek for high-temperature superconductors and study the superconductivity in these H 2 -containing compounds. However, it is necessary to decrease the work pressure of superconducting. For examples, the decomposition pressures are as high as 248 GPa for SiH 4 (H 2 ) 2 and 220 GPa for GeH 4 (H 2 ) 2 , respectively, above which they are stable superconducting materials.
As mentioned above, the combination the lightest H with one of the heaviest Pb seems to be a good way to improve the T c and decrease the work pressure. Chemically, PbH 4 still remains the most elusive of group-IV tetrahydrides. The pioneering theoretical work of Desclaux and Pyykkö predicted the structure and stability of PbH 4 27,28 . The theoretically predicted tetrahedral structure of an isolated molecule, with an equilibrium Pb-H distance of approximately 1.73 Å, was eventually confirmed by experiments 29,30 . But, Krivtsun et al. 30 observed that the PbH 4 molecules were kinetically unstable and readily decompose to Pb atomic layer and H 2 in approximately 10 s. Recently, Zaleski-Ejgierd et al. theoretically investigated the structure and the stability of PbH 4 under high pressure 31 . They found that PbH 4 is stable thermodynamically above 132 GPa, in forms of Imma (132-296 GPa) and Ibam (> 296 GPa) space groups. And PbH 4 even keeps the metallic character covering the whole range of pressure 31 . However, the superconductivity is indeterminate, since the dynamic stable phase of PbH 4 has been not discovered from experimental and theoretical aspects yet. By intercalating H 2 units into PbH 4 molecular crystal, e.g. PbH 4 (H 2 ) 2 , how about the structure, stability, and superconductivity? It is just the purpose of our study. In this work, we found out the stable phase of PbH 4 (H 2 ) 2 thermodynamically and dynamically and investigated its desired superconductivity. The decomposition pressure of 133 GPa is much lower than the metallization pressure of solid hydrogen, which is easily reached in experiments by diamond-anvil techniques. And the H 2 -H 2 coupling under high pressure figures out the different superconducting mechanism.

Results
Covering the wide pressure range of 0-400 GPa, variable-cell structure prediction simulations with 1 to 4 PbH 4 (H 2 ) 2 formula units per cell (f.u./cell) were performed. We have calculated the enthalpies of searched structures of PbH 4 (H 2 ) 2 to examine the thermodynamical stability induced by pressure. For several competitive structures of PbH 4 (H 2 ) 2 , the enthalpies (relative to the P-1 structure) as function of pressure are shown in Fig. 1. It is found that Pnnm phase is the most stablest below 40 GPa with the lowest enthalpy value. Starting from 40 GPa up to 135 GPa, PbH 4 (H 2 ) 2 transfers into P-1 phase. Upon further compression, the C2/m becomes to the most stablest phase above 135 GPa. As a result, there are two structural phase transitions existing in the range of 0-400 GPa. Three low-enthalpy structures were obtained, orthorhombic Pnnm (4 f.u./cell), triclinic P-1 (2 f.u./cell), and monoclinic C2/m (2 f.u./cell), respectively, as shown in Supplementary Fig. S1 online. The lattice parameters of these three structures at Enthalpy difference versus pressure for competitive structures of PbH 4 (H 2 ) 2 , referenced to the P-1 phase. The decomposition enthalpies into PbH 4 + 2H 2 , PbH 3 + 5/2H 2 , PbH 2 + 3H 2 , PbH + 7/2H 2 , and Pb + 4H 2 were also plotted. The inset exhibits the change of enthalpies induced by ZPE correction, which indicates that the decomposition pressure of the C2/m structure decreases as 133 GPa.
Scientific RepoRts | 5:16475 | DOI: 10.1038/srep16475 different pressures are also listed in Table S1 of the supplementary information online. From the crystal configurations at different pressures, PbH 4 tetrahedral molecule does not exist in PbH 4 (H 2 ) 2 , and all of hydrogen atoms construct the H 2 quasi-molecules separating from Pb atoms.
Besides, it has well-known that quantum effects related to hydrogen atoms are very important. The hydrogen zero-point energy (ZPE) has significantly revised the structural stability as in the cases of solid hydrogen 34 and hydrogen-rich materials 9,11 . To judge the effect on stability, we also calculated the ZPEs of PbH 4 (H 2 ) 2 , PbH 4 , and H 2 in the range of 100-200 GPa using the quasiharmonic approximation 35 . As the insert shown in Fig.1, the ZPE effect does not change the order of the phase transitions but lowers the decomposition pressure of the C2/m structure into ~133 GPa. This decomposition pressure of PbH 4 (H 2 ) 2 is obviously lower than 248 GPa of SiH 4 (H 2 ) 2 23 and 220 GPa of GeH 4 (H 2 ) 2 25 , which indicates that PbH 4 (H 2 ) 2 will exist in the wider pressure range. For this stability, the subsequent crystal structural, electronic, phonon, and electron-phonon coupling (EPC) calculations are focused on the C2/m structure above 133 GPa, and typical results are presented at 200 GPa.
For C2/m structure, Pb atoms occupy the crystallographic 2a sites and four non-equivalent H atoms sit on the 4i sites under high pressure. All of H atoms pairwise coupling into two types of quasi-molecules as shown in Fig. 2a. The nearest distance between Pb and H atom is about 2 Å. In this dense structure, we can not find any plumbane molecules existing, but H 2 quasi-molecules distribute around Pb atoms and are ordering (Fig. 2). This kind of ordered arrangements of H 2 units is clearer at high pressure, while H 2 units tend to be inordering at low pressure 18,25 . A visible character of Pb and H 2 in layers is observed from (001)-plane (Fig. 2b) or (010)-plane (Fig. 2c). Noticeably, the layered feature is also a common phenomenon in some hydrogen-rich systems. With the increase of pressure, all of the lattice constants of C2/m structure in a, b, and c directions decrease. However, the H-H bond lengths in H 2 quasi-molecules marked as d1 H−H (formed by H1 and H2 sites shown in Fig. 2a) and d2 H−H (formed by H3 and H4 sites shown in Fig. 2a) firstly increase then decrease as shown in Fig. 3a. There are three kinds of intermolecular distances among H 2 molecules in the C2/m structure, all of them are monotonously decreased with the pressurizing, as shown in Fig. 3b. Reviewing the high-pressure structural character, we find that part hydrogen atoms form H 2 units with the other hydrogen atoms strongly bonding with Si in Ccca phase of SiH 4 (H 2 ) 2 23 , while all of hydrogen atoms pairwise coupling into H 2 quasi-molecules with the nearest distance of ~1.7 Å between Ge and H in P2 1 /c phase of GeH 4 (H 2 ) 2 25 . As a comparison, with the help of analysis of atomic distances, the intermolecular and intramolecular couplings of H 2 gradually strengthen, while the interaction between H and the heavy atom evidently weakens from SiH 4 (H 2 ) 2 to GeH 4 (H 2 ) 2 and then to PbH 4 (H 2 ) 2 .   2 and GeH 4 (H 2 ) 2 reported previously, they remain the characteristics of insulator under low pressure. The insulator-to-metal transition occurs at 92 GPa in SiH 4 (H 2 ) 2 and at 48 GPa in GeH 4 (H 2 ) 2 , respectively. However, we didn't find the transition point of PbH 4 (H 2 ) 2 . It seems to be metal even in ambient pressure, which consist with PbH 4 31 . So the low pressure metallization does not come from the intercalation of H 2 molecules. Comparing with Si and Ge, Pb has larger ionic radius which results in more strong itinerant property of valent electrons. Figure 4 shows the projected density of state (PDOS) at several selected pressures. According to the electronic PDOS at Fermi level we can draw a conclusion that at low pressure in Pnnm structure the Pb-p electrons make the most contribution to density of state and exhibit properties of a nearly free-electron metal (Fig. 4a,b). As the pressure increases, the strengthening of H 2 -H 2 interaction leads to the overlap of H-s wave functions. The contribution of H-s electrons to Fermi surface increases. PDOS tends to be uniform distribution, and the bandwidth further broadens from 100 GPa to 300 GPa (Fig. 4c-f). It indicates that with the increase of pressure PbH 4 (H 2 ) 2 mainly like to be Pb-H 2 alloy. The Pb interlayer interaction is connected by these H 2 molecules. To gain more insight into the bonding nature of PbH 4 (H 2 ) 2 , the electron location function (ELF) of C2/m phase at 200 GPa was calculated. ELF shown in Fig. 5 displays the electronic location around Pb and H atoms as well as the nearly free-electron-like distribution among Pb atoms. However, the high ELF values between Pb and H atoms (Fig. 5a) and of intermolecular H 2 (Fig. 5b) indicate that the electrons become delocalized, suggesting a feature of nearly free-electron metal.
The phonon dispersion curves for C2/m structure at 200 GPa (Fig. 6) and other selected pressure point (see Supplementary Fig. S2 online) were calculated to explore the lattice dynamics of PbH 4 (H 2 ) 2 . The absence of any imaginary frequencies implies the dynamical stability of C2/m phase under high pressure. The whole phonon spectrum can be divided into three parts. By combining with the phonon density of states (PhDOS) projected on atoms shown in Fig. 7a, in the case of 200 GPa, we find that the low-frequency vibration below 215 cm −1 mainly come from the vibrations Pb atoms. The intermolecular strong phonon coupling among H 2 molecules appear in the intermediate-frequency range of 295-1876 cm −1 . After a large gap, in high frequency area above 2695 cm −1 , the H-H vibration in H 2 formed by H3 and H4 sites mainly contributes in the range of 2695-2898 cm −1 , while the vibration in H 2 formed by H1 and H2 sites around 3220 to 3380 cm −1 . Comparing these three systems of Si-, Ge-, and Pb-based, we find a strong phonon coupling between silicon and hydrogen in SiH 4 (H 2 ) 2 23 , very weak phonon coupling between metal and hydrogen in GeH 4 (H 2 ) 2 25 as well as PbH 4 (H 2 ) 2 . The H-H vibration in H 2 molecule is the strongest in PbH 4 (H 2 ) 2 . From the Eliashberg phonon spectral function α 2 F(ω) and the integrated EPC parameter λ(ω) shown in Fig. 7b, the intermediate-frequency (295-1876 cm −1 ) vibrational modes of H 2 molecules contribute 81.5% of total λ. This percentage is larger than 66% in   Si-based and 75% in Ge-based case. This result highlights the significant role played by H 2 molecules on the electron-phonon interaction.
At 200 GPa, the calculated total EPC constant λ is 1.296 for C2/m PbH 4 (H 2 ) 2 . From Si to Ge and then to Pb case, the λ gradually decreases from 1.625 to 1.43 and then to 1.296, which implies a weak coupling between metal and hydrogen. However, the phonon frequency logarithmic average ω log rises gradually, from 871 K in SiH 4 (H 2 ) 2 to 1051 K in PbH 4 (H 2 ) 2 . This means more higher Debye temperature in PbH 4 (H 2 ) 2 . Based on the obtained α 2 F(ω) and λ(ω), we now can analyze the superconductivity using the modified McMillan equation by Allen and Dynes 36 , With the typical choice of the Coulomb pseudopotential µ = . * 0 1 3 , a remarkable large T c of 103 K was obtained for C2/m phase of PbH 4 (H 2 ) 2 , which is comparable with those of copper oxide superconductors.
To figure out the pressure effect on superconductivity in PbH 4 (H 2 ) 2 , in addition, the T c values at several typical pressure points were calculated and shown in Supplementary Fig. S3 online. An interesting phenomenon exhibits the superconductivity firstly strengthening before weakening. The T c has a maximum between 140 and 350 GPa, ~107 K for µ = .
* 0 1. Seen from the distances of among H 2 molecules shown in Fig. 3b, the monotonously decreasing makes a hint of "hardening" of intermediate-frequency phonon with the increase of pressure. The phonon spectra shown in Supplementary Fig. S2 online confirm this point. To analyze this phenomenon of T c variations, we have further calculated the Eliashberg phonon spectral function and the EPC strength at different pressures, the results are presented in Supplementary Fig. S4 online. With the increase of pressure, the calculated EPCs are 1.280, 1.296, 1.379, and 1.341 for 180 GPa, 200 GPa, 250 GPa, and 300 GPa, respectively, which shows a tendency of first increase and then decrease similar to T c . In the T c rising zone, the contribution of Pb-H coupling to the EPC strength is decreased from 14.3% at 180 GPa to 12% at 200 GPa, and the phonon vibration of H-H  in H 2 units also weakens the EPC (The contribution is from 7.3% to 6.5% corresponding pressures.). However, the contribution of H 2 -H 2 coupling to the EPC is strengthening from 78.4% at 180 GPa to 81.5% at 200 GPa. So the initial rising of T c results from the contribution increasing of H 2 -H 2 for the EPC. As shown in Fig. S4 online, from 75.8% at 250 GPa to 73.5% at 300 GPa, the decrease of contribution of H 2 -H 2 for the EPC leads to the fall of T c . The result further reveals the significance of H 2 -H 2 coupling to superconductivity in PbH 4 (H 2 ) 2 .

Discussion
Thus far, the stability of PbH 4 (H 2 ) 2 has been identified in the pressure range of 0-400 GPa. At low pressure it is metastable and possibly decomposes into Pb + H 2 or PbH 4 + H 2 . Above 133 GPa, it is stable not only thermodynamically but also dynamically. This high-pressure stable phase of C2/m exhibits the expected superconductivity of T c ~ 107 K at 230 GPa, which is obviously higher than those of conventional group-IV hydrides such as silane, germane, and stannane. Noticeably, the coupling between group-IV element and hydrogen reduces with the increase of atomic number. Namely, the contribution of group-IV element to total EPC decreases in hydrides from 33% in SiH 4 (H 2 ) 2  , it is clear that the intercalation of H 2 molecules results in the softening of intermediate-frequency phonon. As increasing the content of hydrogen in group-IV elements, it results in enhancing the EPC strength that is dominated by the coupling of the H 2 molecular in the AH 4 (H 2 ) 2 (A = Si, Ge, Sn, and Pb) crystals. This is just the origin of higher T c in H 2 -containing compounds. Furthermore, we infer that the higher T c may be obtained if more H 2 are inserted in group-IV hydrides. Actually, more future works are needed to advance the T c and understand the superconductivity.
As a comparison, the high-pressure structure of PbH 4 (H 2 ) 2 is visibly different from other hydrogen-rich compounds with high T c , such as CaH 6 37 and (H 2 S) 2 H 2 32 . In high-pressure structures of CaH 6 and (H 2 S) 2 H 2 , the H 2 quasi-molecules have been broken, with the strong bonds forming between metal and hydrogen atoms. Although the EPC is mainly contributed by hydrogen, the superconducting mechanism is different. It is the H-H coupling in CaH 6 and (H 2 S) 2 H 2 , while the H 2 -H 2 coupling in PbH 4 (H 2 ) 2 . It is interested that the H 2 quasi-molecule form keeps all along at thus high pressure up to 400 GPa. At the same time, Pb is one of the heaviest elements. The combination with the lightest H is one of the most important physical problems in high-pressure research. Pb metal makes the metallization pressure of hydrogen-rich compound decrease. Remarkably, the decomposition pressure point (133 GPa) of PbH 4 (H 2 ) 2 is the lowest among these H 2 -containing compounds of Si-, Ge-, and Pb-based. This value is much lower than the metallization pressure of bulk molecular hydrogen, which indicates the feasibility to experimentally observe. Hence, Pb-based hydrides are the potential candidates as high-T c superconductors. Our finding may hopefully stimulate the potential high-T c superconductors research in H 2 -containing hydrides.

Methods
The search for crystalline structures of PbH 4 (H 2 ) 2 phases was performed using particle swarm optimization methodology as implemented in the CALYPSO program 38,39 . Structural optimizations, enthalpies, and electronic structures were calculated using the Vienna ab initio simulation (VASP) program 40,41 and projector-augmented plane wave (PAW) potentials employing the Perdew-Burke-Ernzerhof (PBE) functional 42 . The 1s 1 and 6s 2 6p 2 electrons were included in the valence space for H and Pb atoms, respectively. For the plane-wave basis-set expansion, an energy cutoff of 800 eV was used. Dense k-point meshes were employed to sample the first Brillouin zone (BZ) and ensured that energies converged to within 1 meV/atom. All forces acting on atoms were converged 0.001 eV/Å or less, and the total stress tensor was reduced to the order of 0.01 GPa. With the noteworthy mass ratio 207:1 between Pb and H, we have involved the spin-orbit effect in this calculation.
Based on the optimized structures from VASP, lattice dynamics and superconducting properties were calculated using density functional perturbation theory 43 and the Troullier-Martins norm-conserving potentials 44 , as implemented in the QUANTUMESPRESSO code 45 . The cutoff energies of 60 and 400 Ry were used for wave functions and charge densities, respectively. 12 × 12 × 8 Monkhorst-Pack k-point grid with Gaussian smearing of 0.03 Ry was used for the phonon calculations at 3 × 3 × 2 q-point mesh, and double k-point grid was used in the calculation of the electron-phonon interaction matrix element.