High-temperature superconductor of sodalite-like clathrate hafnium hexahydride

Hafnium hydrogen compounds have recently become the vibrant materials for structural prediction at high pressure, from their high potential candidate for high-temperature superconductors. In this work, we predict \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {HfH}_{6}$$\end{document}HfH6 by exploiting the evolutionary searching. A high-pressure phase adopts a sodalite-like clathrate structure, showing the body-centered cubic structure with a space group of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Im\bar{3}m$$\end{document}Im3¯m. The first-principles calculations have been used, including the zero-point energy, to investigate the probable structures up to 600 GPa, and find that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Im\bar{3}m$$\end{document}Im3¯m structure is thermodynamically and dynamically stable. This remarkable result of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Im\bar{3}m$$\end{document}Im3¯m structure shows the van Hove singularity at the Fermi level by determining the density of states. We calculate a superconducting transition temperature (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{c}$$\end{document}Tc) using Allen-Dynes equation and demonstrated that it exhibits superconductivity under high pressure with relatively high-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{c}$$\end{document}Tc of 132 K.

www.nature.com/scientificreports/ that a value of T c is around 213-234 K at a pressure of 250 GPa. The solution of this novel structure opened the door to the exploration of a new class of structure. Interestingly, it is worth to note that this work reported an energy difference between HfH 6 and HfH 10 which is closed by approximately 1-2 meV at a pressure of 300 GPa. The high pressure phase of HfH 6 is predicted to be a Cmc2 1 structure 30,31 and found that it is stable structure among a convex hull diagram. Moreover, the Cmc2 1 structure is reported to be dynamically stable at a pressure of 300 GPa 31 because it does not indicate any imaginary frequency. Also, the value of T c of HfH 6 is estimated to be 45.2 K to 55 K. However, there are neither experimentally nor theoretically studies under high-pressure above 300 GPa. It is interesting to note that transition metal hexahydride is thermodynamically and dynamically stable, as being in accordance with the high-T c such as ScH 6 , YH 6 , and ZrH 6 , respectively. Among the predicted the value of T c , based on the Allen-Dynes equation 32 . In 2017, ScH 6 was predicted the high-T c above 100 K from 300 to 400 GPa 33 . In the same year, ScH 6 was investigated by using the first-principles calculations, carried out the McMillan formula with Allen-Dynes corrections 32,34 . As result of this, ScH 6 displayed superconductivity with T c of 130 K at 285 GPa. Then, in 2018, ZrH 6 was explored the T c , resulting in the estimation T c to be 114 K at 295 GPa 35 . Recently, in 2019, YH 6 was determined by using fully anisotropic Migdal-Eliashberg theory. The results on superconducting properties of YH 6 manifested the T c reads 290 K at 300 GPa 23 . Motivated by the prediction of T c of transition metal hexahydride, it is worthy to further explore HfH 6 at very high compressed conditions.
In this work, we provide a potential high pressure candidate structure of HfH 6 , leading to scientific leap frog of high pressure superconductivity. We explore the high-pressure phase of HfH 6 under pressure from 300 GPa to 600 GPa by first-principles evolutionary techniques. Moreover, we aim to predict the value of T c by performing a candidate structure of HfH 6 . Regarding its potential for superconductivity, the electronic properties shown to propound a possibility of the value of T c such as a band structure, a density of states, and a nature of chemical bonding. Particularly, the electronic properties play an important role in support the value of T c .

Methods
The searching for the structures of the clathrate hafnium hexahydride HfH 6 was performed by the Universal Structure Predictor: Evolutionary Xtallography (USPEX) 36 . In all subsequent generations, the random symmetric algorithm employed 40% heredity, 20% random symmetric, 20% soft mutation, and 20% transmutation operators in the pressure range from 200 to 600 GPa with structures containing up to four formula units. A plane-wave basis set up to cutoff energy of 700 eV and an initial Brillouin-zone (BZ) sampling grid of spacing 2π × 0.02Å −1 were used for this calculation as well as a plane-wave basis set up to cutoff is guaranteed to be converged within an accuracy of 3 meV per atom. All structures were fully relaxed using the generalized gradient approximation of the Perdew-Burke-Ernzerhof (GGA-PBE) functional 37 for the exchange-correlation functional. We used the projector augmented wave (PAW) method 38 and the conjugate gradient scheme, as implemented in the Vienna ab initio simulation package (VASP) 39 . For electron-phonon and the spectral function calculations, a plane-wave energy cutoff of 80 Ry was used. The dense k-points mesh contained all k and k+q grid points were used. The subsequent electron-phonon and spectral function calculations depended on the k-point part due to it covered the grid of q-point. The calculations were computed in the first BZ on 24 × 24 × 24 k-points mesh and 2 × 2 × 2 q-meshes, showing that it is sufficient to produce accurate electron-phonon coupling. Computational details of the electron-phonon and spectral function calculations were successfully reported in the theoretical studies 16,17 The Allen-Dynes equation 32 was exploited with the effective Coulomb pseudopotential parameter, µ * = 0.10.
as follows: where ω log is the logarithmic average of the spectral function. is the total electron-phonon coupling strength.
The projected crystal orbital Hamilton population 40 (pCOHP) used to explain the chemical bonding of the sodalite-like clathrate hafnium hexahydride structure, as implemented in LOBSTER code 41 .

Results and discussion
Regarding ground-state structure in HfH 6 , we aimed to identify the unknown structure of HfH 6 above 300 GPa due to theoretical predictions is a crucial key to the exploration of a candidate structures at high pressure. First of all, we predicted the high-pressure phase using USPEX code, it shows that our main structural prediction revealed low-enthalpy structures, showing an orthorhombic structure with a space group of Cmc2 1 and a body centered cubic with a space group of Im3m.
For the first step in the structural predictions, a structural sequence showed that the Cmc2 1 structure transformed into the Im3m structure at a pressure of 543 GPa. Under higher pressure, it found that the Im3m structure declined steadily up to 600 GPa, as showed in Fig. 1a. Moreover, we analyzed the further stabled structure of HfH 6 with respect to the elemental hafnium (the Im3m structure) and hydrogen (the Cmca-12 structure). Considering the relative enthlapy, one can see that the Cmc2 1 structure is thermodynamically stable favored over the Im3m structure at a pressure of 300 GPa. On further compression to 600 GPa, the Im3m structure is apparently stable (Fig. 1b). Following this, we furture our investigation to the structural stability by the incorporation of the zeropoint energy (ZPE) of the nuclei estimation, indicating that the Im3m structure is thermodynamically stable throughout the whole studied pressure range, as showed in Fig. 1c. It should be mentioned that our calculations performed the DFT at 0 K, we therefore investigated by considering at elevated temperatures. As a result, the Im3m structure is thermodynamically more stable than the Cmc2 1 structure with increasing temperature up to at least 300 K, depicting in the convex hull envelopes at a pressure of 600 GPa of Fig. S1 in the Supplemental Material. This further implies the Im3m structure probably occurs at room temperature. Furthermore, we www.nature.com/scientificreports/ investigated further study of the stable structure of HfH 6 at a pressure of 300 GPa. As a result, we pointed out that the Cmc2 1 structure is a potential candidate. Our calculations are in good agreement with those recently reported in the theoretical works 30,31 .
Here, we introduce sodalite-like clathrate at extremely high-pressure, showing the stabled bcc with the Im3m space group. To further describe this structure, the H atoms which is in the form of a sodalite-like cage, composing of eight H-hexagons and six H-squares, and Hf atoms crystallize into a lattice site of body-centered cube. The structural morphology showed in Fig. 1d, which resembles the structures of MgH 6 9 , CaH 6 8 and YH 6 23 . For the electronic property in the Im3m structure, it is clearly demonstrated in Fig. 2a. The band structure manifested a metallic state because a conduction band and a valence band crossed at the Fermi level. Besides, we found that the density of states (DOS) exhibited van Hove singularities (vHs) at the Fermi level, indicating a large electron-phonon coupling (EPC). Interestingly, it is worth noting that the vHs is dominated by a d-electron of Hf. As depicted in the DOS, the characteristics of the vHs play an important role in superconductivity. For example, H 3 S 42,43 , YH 6 23 , and LaH 10 44 systems, leading to the possibility of achieving high values of T c . To further explore the electronic structure, the Fermi surface is described, as shown in Fig. 2b. It can see that the Fermi surfaces around the P-point exhibited the Fermi nesting because several Fermi surfaces are parallel to each other. It can thus enhance the EPC and the value of T c .
According to Fig. 1, we computed phonon dispersions and phonon density of states (PhDOS) of the Im3m structure at a pressure of 600 GPa. As a result, we found that the Im3m structure is dynamically stable because it does not exhibit the imaginary frequency. Also, the phonon dispersions displayed acoustic modes and optical modes, as can be seen from Fig. 3, where the acoustic modes are the vibrations of the Hf atom and the optical modes are the vibrations of the H atoms. Moreover, the optical branches showed that there was an abundantly spread, showing the stretch and bent modes. These vibrations associated with the electron-phonon interaction and it yielded the high-T c . Also, these characteristics corresponded with the PhDOS. It is interesting to note that the H atoms exhibited large vibrations by approximately from 1138 to 2918 THz. A remarkable solution is shown to propound a possibility of the high-T c ,as will be discussed later. www.nature.com/scientificreports/ The spectral function α 2 F of the Im3m structure is calculated at a pressure of 600 GPa, as shown in Fig. 3. The Allen-Dynes equation 32 carried out for the estimation T c . It showed that the Eliashberg spectral function contributed slightly by approximately 0 cm −1 to 324 cm −1 and it contributed mainly by approximately 1120 cm −1 to 2918 cm −1 . The solution of the integrating of lambda displayed that it climbed dramatically from 147 cm −1 to 249 cm −1 . After that, it remained stable between 247 cm −1 and 1149 cm −1 . Then, it increased moderately up to 2918 cm −1 , showing the integrating of lambda is 1.06. Here, we found that ω log is 1741 K and the T c is 132 K, using µ * = 0.10 . Additionally, the T c is estimated by directly solving the McMillan formula with Allen-Dynes corrections µ * = 0.13 34 . The calculated result shows that the estimated T c is 114 K. As a possible cause of this, one might think that the H atoms contributed a large frequency. Here again, we have already mentioned the DOS, it can see that an s-electron of H showed a large contribution to the DOS in comparison to s and p-electron of Hf, showing that it supported the existence of the high-T c . As a result of this, we suggested that the existence of an unforeseen the Im3m structure with remarkably high-T c can pave the way for further studies on the trend of the high-temperature superconductors.  www.nature.com/scientificreports/ To further analyse the spectral function of the Im3m structure, we calculated at a pressure of 550 GPa. Our calculations show that the character of the the spectral function is similar at a pressure of 600 GPa, as can be seen in Fig. 4. It exhibited the integrating of lambda is 1.07 and the ω log is 1692 K, showing a high-T c of 130 K. At this point, as reported above, we found that the high-T c of the Im3m structure increased with increasing pressure. As a possible, one might think of the ω log . It showed that at a pressure of 600 GPa is the maximal of the ω log , which is higher than a a pressure of 550 GPa. We thus can point out that the ω log plays an important role in the high-T c of HfH 6 .
As mentioned earlier, it is also interesting to answer the question of why the T c of the Im3m structure is the high-T c . At this point, we perform the electron localization function (ELF) and the projected crystal orbital Hamilton populations (pCOHP) solutions, the ELF method 45 calculated to investigate bonding. The characteristics of ELF have successfully explained several materials 14,46-48 . To begin with, the ELF of the Im3m structure is described a uniform electron gas of the same density in the (001) plane, as shown in Fig. 5a, it can be seen that a contribution of electrons between the H-H bonds are a weak bonding while the distribution of electrons in Hf atoms likely to be lone pairs in that region. Moving on to the pCOHP calculation, we described the character of the nature of a chemical bonding, which further supports the ELF calculation. This method can examine covalent bonding in several materials 11,49,50 . To further  www.nature.com/scientificreports/ understand the superconductivity, the influence of bonding plays an important role in considering the value of T c . The pCOHP calculation interprets the wave function into the covalent character. First of all, it can see that the H-H pairs promoted the anti-bonding. Following this, one can see that the Hf-H pairs were found to be the anti-bonding as well (Fig. 5). As a result of this, one might think that the nature of chemical bonding supported the value of T c . This because the antibonding states in the covalent system led to the way of strong coupling of the EPC, which associated with the large vibration of H-rich.

Conclusion
In this work, we identify the high-pressure phases of HfH 6 by performing an evolutionary searching. Overall, the incorporating of the zero-point energy shows that the Im3m structure is thermodynamically stable favored over the Cmc2 1 structure. The sodalite-like clathrate hafnium hexahydride is predicted to be a high-temperature superconductor with estimated T c of 132 K at a pressure of 600 GPa. The nature of the chemical bonding is associated with the electron localized function, implying that the characteristics of the chemical bonding entail the high-T c . Finally, we point out that the existence of an unexpected the Im3m structure can pave the way for further studies on the development of the high-temperature superconductors.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. www.nature.com/scientificreports/