Introduction

The long-sought goal is to find materials with sufficiently high superconducting critical temperatures (Tcs) to facilitate experimental exploration and practical implementation, ultimately allowing ambient-environment applications. Enormous experimental efforts have been made on metallic hydrogen due to its predicted the high-Tc superconductivity, potentially above room temperature1. But the external high pressure to metallize hydrogen is very hard to achieve, and experimental observations of the metallic hydrogen phase remain controversial. At the beginning of this century, after Ashcroft2 proposed that hydrogen-rich compounds is an alternative to pursue high Tc superconductivity driven by “chemically compressed” hydrogen in hydrides at lower pressure. Therefore, a large number of researches are focused on searching for high-Tc superconducting hydrides3,4,5. The employment of well-established investigative techniques in the study of superconducting hydrides has yielded notable results, such as SiH3 with a transition temperature Tc of 139 K at 275 GPa6, TeH4 with Tc = 104 K at 170 GPa7, YH6 with Tc = 264 K at 120 GPa8, YH10 with Tc = 326 K at 250 GPa9, HfH10 with Tc = 234 K at 250 GPa. And the long-known compound CaH6 has recently been successfully synthesized10. The most significant achievements are the discovery of superconductivity in sulfur hydride11,12,13 and lanthanum hydride8,14 with Tc of 203 and 250 K, respectively. In sulfur hydride, the novel phenomenon has been observed in the pressure-driven disproportionation of H2S to the covalent sixfold cubic structure H3S. In addition, hydrogen-rich sodalite-like lanthanum hydrides LaH10 with H32 cage also exhibit high Tc.

Despite achievements in binary hydrides about superconductivity3,4,5,15,16,17, ternary hydrides offer considerable potential for the development of new superconducting materials18,19,20,21,22,23,24,25. In recent years, there has been a growing number of studies exploring the properties of ternary hydrides, such as studies focused on the formation routes and superconductivity of specific ternary hydrides MgSiH619 and MgGeH618; by doping lithium in phosphorus hydrides under pressure to stabilized LiPH6, which exhibiting Tc of 167 K at 200 GPa22; Li2MgH16 exhibit the highest Tc of 473 K at 250 GPa in the theoretical prediction. Additionally, experimental studies on ternary hydrides are a rapidly expanding field of research. Experimental studies have revealed that the ternary hydride BaReH9 is a superconductor with Tc about 7 K at 100 GPa20. Li5MoH11 can be synthesized under high pressure and showing unexpected low Tc < 10 K26. The recent discovery of an anomalously high Tc in LaBeH8 showed that high-temperature superconductivity can be achieved at more modest pressures in ternary hydrides27. Under the guidance of theoretical prediction, the synthesis of ternary LaBeH8 was successfully obtained via compression in a diamond anvil cell under high pressure. with critical temperature Tc up to 110 K at 80 GPa28.

Research on ternary hydrides remains challenging, the structure space of ternary hydrides is much larger than that of binary hydrogen. Therefore, the organized research strategies are necessary to explore ternary hydride superconductors. And high-throughput calculations became more and more popular in materials science, and recently reached great achievements in superconducting hydrides: around 50 hydrides with Tc from 20 K to 80 K were discovered from dataset comprising over 150,000 hydrides at moderate pressure, and most of them are combined by alkali or alkali-earth elements and hydrogen-and-transition-metal (HTM) units29,30,31. For example, hydride Mg2IrH632,33,34, consisting of cation Mg2+ and HTM unit [IrH6]4-, is isostructural to known hydride Mg2FeH6 and predicted to exhibit Tc more than 80 K at ambient pressure. Therefore, the features of proposed hydrides we learnt from high-throughput calculations would trigger further theoretical design of hydride superconductors based on HTM units. Moreover, origin of superconductivity in hydrides with HTM units remains unclear. Clarification of the relationship between superconductivity and coordination structures of HTM units not only contribute to hunting for hydride superconductors, but also developments of theory of high-pressure superconductivity and coordination chemistry.

Here, we propose a strategy to construct hydride superconductors by inducing hydrogen into known HTM units. P63/mmc-BaReH9 crystallizes with ionic bonding between Ba2+ and face-capped trigonal prismatic [ReH9]2- units. BaReH9 is well characterized at ambient pressure35 and high pressure20, due to the hydrogen to metal ratio, it is one of the high-hydrogen-content hydrides. We explored high-pressure phase diagram of the ternary Ba-Re-H system, especially focused hydrides BaReH9+x (x = 1–9). We found that only one compound BaReH12, exhibits [ReH12]2- unit with hydrogen content higher than [ReH9]2-, is stable with space group \({Pm}\bar{3}\) at 100 GPa. As it is expected that hydride BaReH12 exhibits remarkable high Tc around 128 K driven by a 12-coordination HTM units, where 12 hydrogen atoms take quasi-regular icosahedron arrangements. And pressure provide an effective approach which controls the coordination number of HTM units in stable hydrides BaReH9+x: at 100 GPa, coordination number of Re in HTM unit could increase from 9 (P63/mmc-BaReH9), to 12 (\({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\)), and as pressure further increasing, the coordination number achieve the maximum of 14 (P2/m-BaReH12).

In the first transition of coordination number from 9 to 12, BaReH9 transform to metallic hydride BaReH12 with remarkably Tc. Because units [ReH9]2- and [ReH12]2- bears 18 and 21 electrons, respectively. In hydrides with HTM unit, the odd number electrons occupied orbital generally drives metallicity, and the occupation of antibonding orbitals may cause the superconductivity30. Accordingly, [IrH6]4- in superconducting hydride Mg2IrH6 also bearing odd numbers of electrons 19. In contrast, hydrides with even numbered electrons HTM units typically exhibit insulating or weaker metallic properties20,31, requiring higher pressure to induce superconductivity, such as BaReH9 (18e of HTM units), Mg2IrH5 (18e), Mg2IrH7 (20e), K2SiH6 (12e), et al. In the second transition of coordination number from 12 to 14, the high symmetry of 12-coordination unit [ReH12]2-(21e) in \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) was broken in P2/m-BaReH12, as well as weak bonding of hydrogens inter HTM units, all these lead to a significant decrease in the strength of electron–phonon coupling (the EPC λ decreases from 1.9 to 0.8) and the Tc decreases from 128 K to 42 K. Besides BaReH9+x in Ba-Re-H phase diagram, we also discover a stable hydrides Cmmm-Ba2ReH8 with [ReH8]4-(19e). Because HTM unit [ReH8]4- also bears odd numbers of electrons, so it is metallic and exhibits superconductivity about 19 K. Comparing with other hydrides, the additional Ba cation increases the distance between the HTM units and suppresses coupling of hydrogens inter HTM units, so the Tc of Cmmm-Ba2ReH8 is much lower than those of \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) and P2/m-BaReH12. Our results indicate the following features may exhibit in high-temperature superconductivity hydrides with HTM units. The metallization of hydrides is attributed to the odd-numbered electrons of HTM unit. Additionally, a high symmetrical structure of these HTM units, along with weak bonding of hydrogens are favorable to the superconductivity of the hydrides36. Our results will guide further discovery of high-temperature superconductivity hydrides with HTM units.

Results and discussion

Ternary phase diagram stable compounds

We first constructed the ternary phase diagram of Ba-Re-H system at 100 200 and 300 GPa (see Fig. 1 and Supplementary Fig. S1 in the Supplemental Material). First, as we known, Ba-H binary compounds are systematically investigated, BaHn (n = 1, 2, 6, 8, 12) can fall on the Ba-H side and our results are consistent with previous literatures37,38. While, the other two sides (Re-H and Ba-Re) of the ternary phase diagram are inevitably investigated in this paper. In Ba-Re system, there is no thermodynamic stable compound among the pressure range of our investigation. In Re-H system, there are two thermodynamic stable compound P63/mmc-ReH and \(P\bar{3}m1-{\mathrm{Re}}_{2}{\rm{H}}\), the maximum proportion of hydrogen is 50% in rhenium hydride (ReH) and its Tc is almost zero. At 100 GPa, there are three thermodynamic stable ternary hydrides, P63/mmc-BaReH9, Cmmm-Ba2ReH8 and \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\). Except for elemental solids, binary compounds and known BaReH9, we predicted two new stable ternary hydrides Ba2ReH8 and BaReH12(see Fig. 1 and Supplementary Fig. S1). There is a thermodynamic metastable hydride BaReH6, with a distance off the convex hull ~22 meV/atom. In general, this metastability does not preclude BaReH6 from experimental synthesis.

Fig. 1: Convex hull of Ba-Re-H at 100 GPa.
Fig. 1: Convex hull of Ba-Re-H at 100 GPa.
Full size image

Formation enthalpies of various compounds are scaled using three mixed colors to show the depth of the convex hull. Big colored circles and half circles represent thermodynamically stable and metastable compounds, respectively.

As for BaReH9+x, three stable hydrides were discovered with different coordination number of HTM units as shown in Fig. 2, they are P63/mmc-BaReH9, \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) and P2/m-BaReH12. As is well known, BaReH9 exhibits nona-hydro-rhenate dianions HTM units [ReH9]2-(18e). High-pressure hydride \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) exhibits CsCl-structure with cation Ba2+ and [ReH12]2-(21e), which is a 12-coordination HTM units, where 12 hydrogen atoms arrange in quasi-regular icosahedron. Notably, distance between hydrogens inter HTM units [ReH12]2- is about 1.0 Å, similar to that of clathrate hydrides. Therefore, we investigate the ELF of BaReH12 and found weak bonding of hydrogens inter HTM units [ReH12]2-. As pressure further increasing, \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) transforms to P2/m-BaReH12, which the coordination number of Re is 14 and four hydrogens was share by adjacent units. Therefore, the HTM units of P2/m-BaReH12 was twisted and lead to low symmetry.

Fig. 2: Crystal structures for the Ba-Re-H system.
Fig. 2: Crystal structures for the Ba-Re-H system.
Full size image

a P63/mmc-BaReH9 at 0 GPa, b \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) at 100 GPa, c P2/m-BaReH12 at 300 GPa. The green, gray and pink balls represent barium (Ba), rhenium (Re) and hydrogen atoms, respectively. d The enthalpy difference curves with respect to \({Pm}\bar{{\boldsymbol{3}}}-{{\rm{BaReH}}}_{12}\).

For Ba2ReH8, the thermodynamical favorable phase is Cmmm, depicted in Supplementary Fig. S2, and there is an energetically competing phase with space group Pbam. The crystal structures of Ba2ReH8 are depicted in Supplementary Fig. S3. Under high pressure, we found octa-hydro-rhenate tetra-anions ([ReH8]4-) in Ba2ReH8. Each rhenium atom is enclosed by 10 hydrogen atoms in Ba2ReH8, 4 of which are shared by the adjacent rhenium atoms, together construct octa-hydro-rhenate anion [ReH8]4-(19e). In the Cmmm phase the [ReH8]4- is regularly arranged with the same direction, while in the Pbam phase the [ReH8]4- is distorted, and the position of Ba also slightly changed.

Electronic properties about Ba-Re-H system

As it is known that BaReH9 is insulator, so we investigate the band structures of \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) and P2/m-BaReH12. Figure 3c shows the band structure of \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) at 100 GPa, the highly degenerate bands, which are related to high-symmetry structures, across the Fermi surface near the high symmetry point Γ is relatively flat, and there are steep band across the Fermi surface near the high symmetry points X, M and R. There are both flat and steep bands across the Fermi surface indicating a large difference in the Fermi velocity. Especially, the high symmetry of \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) structure leads to the several degenerated bands along Γ to R (see Supplementary Fig. S6). At 300 GPa, the band structure of P2/m-BaReH12 is relatively complex (see Fig. 3d). Compared with the partial electronic state density of Ba2ReH8, the contribution of electrons from H to the total DOS at the Fermi level is significantly increased. Especially for \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) at 100 GPa, the electronic state density of H above and below the Fermi level is comparable to the which of Re and much greater than that of Ba. The increased electronic state density of H at the Fermi level is beneficial to its superconductivity. In addition, we analyze the bonding characteristics through the three-dimensional electron localization function (ELF) of \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) at 100 GPa, combining with the atomic distances, as shown in Supplementary Fig. S5 and Supplementary Table S2. For the H2 unit on the face, there is electronic localization between H-H, and the H-H distance is ~1 Å, which is much larger than the H-H distance (0.73 Å) in molecule H2 under the same pressure, indicating that H-H forms a weak covalent bond.

Fig. 3: The band structures and partial electronic density of states (PDOS) for Ba-Re-H hydrides.
Fig. 3: The band structures and partial electronic density of states (PDOS) for Ba-Re-H hydrides.
Full size image

a, b Cmmm-Ba2ReH8 at 100 GPa and 300 GPa; c \({Pm}\bar{3}\)-BaReH12 at 100 GPa, d P2/m-BaReH12 at 300 GPa.

We calculated the band structures of Cmmm-Ba2ReH8 at 100 and 300 GPa and their partial electronic density of states (PDOS), as shown in Fig. 3a, b. As the pressure increases, the band structure of Cmmm-Ba2ReH8 changes slightly overall. At 100 GPa, there are two relatively flat bands across the Fermi level at the high symmetry point Z, which is beneficial to increasing the electronic density of states at the Fermi level. But at 300 GPa, flat bands near high symmetry point Z shift up upward of the Fermi level. The disappearance of the flat region is not conducive to increasing the DOS at the Fermi level. From the Fig. 3a, b, it can be seen that the total electronic density of states at the Fermi surface of Cmmm-Ba2ReH8 are 1.50 and 1.16 states/eV/f.u. at 100 and 300 GPa, respectively. And from the PDOS, it can be seen that the contributions from Ba and Re at the Fermi level are relatively large, and the electrons of H contribute a smaller proportion at the Fermi level, which are 7.4% and 11.8% at 100 and 300 GPa respectively.

Phonon dispersion and superconductivity properties about Ba-Re-H system

In order to study the dynamic stability of each structure, we provide the phonon dispersion, projected density of states (PHDOS) of Cmmm-Ba2ReH8 P2/m-BaReH12 and \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) in Supplementary Figs. S7–10 and Fig. 4a. No imaginary frequency modes appear in all structures, indicating that these structures are dynamically stable under corresponding pressures. As can be seen from Supplementary Fig. S7, Cmmm-Ba2ReH8 has imaginary frequency mode in the phonon spectrum, indicating that Ba2ReH8 is dynamically unstable at 50 GPa. Due to the large difference in atomic mass (\({M}_{{\rm{Ba}}},{M}_{\mathrm{Re}}\)≫\({M}_{{\rm{H}}}\)) and the existence form of H atoms, obvious partition of regions appears in the phonon spectrum. The phonon spectrum of Ba2ReH8 and P2/m-BaReH12 can be divided into low-frequency, mid-frequency and high-frequency regions. The low-frequency region is mainly contributed by the vibration of Ba and Re, and the mid-frequency and high-frequency regions are mainly contributed by the vibration of H atoms. The phonon frequency range in the mid-frequency and high-frequency regions are larger, only the frequency below 500 cm−1 should be defined as the low-frequency region.

Fig. 4: The dynamic stability and superconducting properties of BaReH12.
Fig. 4: The dynamic stability and superconducting properties of BaReH12.
Full size image

a Phonon dispersion, projected density of states (PHDOS), and Eliashberg spectral function for Pm\(\bar{3}\)-BaReH12 at 100 GPa, b superconducting gap of \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) at 100 GPa.

Figure 4 also shows the spectral function α2F(ω) and the integrated electron–phonon coupling constant λ of \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) at 100 GPa. We found that the low-frequency, mid-frequency and high-frequency regions contributed 37.2%, 59.2% and 3.6% of the entire electron–phonon coupling strength respectively. The mid-frequency and high-frequency regions mainly composed of H2 unit vibration contributed 37.2%, 59.2% and 3.6% of the entire electron–phonon coupling strength. 62.8%. The contribution of the vibration of Ba and Re in the low-frequency region to electron–phonon coupling cannot be ignored. Further by solving the Allen-Dynes modified McMillan equation39, the superconducting transition temperature Tc of BaReH12 is 78.0–83.7 K (μ* = 0.1–0.13). Since the electron–phonon coupling constant λ = 1.94 > 1.50, we also obtained the Tc value of 117–128 K by self-consistently solving the Eliashberg equation(scE)40. For the case of λ < 1.5, It can give a more accurate superconducting transition temperature Tc. Table 1 lists the superconducting transition temperature Tc of Ba2ReH8 and BaReH12 under different pressures. The electron–phonon coupling constant λ of Ba2ReH8 at 100 GPa is 0.59, the logarithmic average value of phonon frequency ωlog is 885 K, and the superconducting transition temperature Tc is 19 K. As the pressure increases, the logarithmic average phonon frequency ωlog becomes larger, but the electron–phonon coupling constant λ decreases to 0.41, and the electronic state density at the Fermi surface also becomes smaller. Therefore, as the pressure increases, the superconducting transition temperature of Ba2ReH8 decreases. The electron–phonon coupling constant λ of P2/m-BaReH12 at 200 GPa is 0.76, and the ωlog is 662 K, result in a superconducting transition temperature Tc of 28.2 K. As the pressure increases, the ωlog and the electron–phonon coupling constant λ both become larger. Therefore, the superconducting transition temperature increases to 41.6 K at 300 GPa.

Table 1 The calculated EPC parameter λ, logarithmic average phonon frequency ωlog (K), electronic density of states at Fermi level N(εf) (states/spin/Ry/f.u.) and superconducting transition temperatures Tc (K) with μ* = 0.1–0.13
Full size table

Conclusions

In summary, we proposed a strategy to construct hydride superconductors by inducing hydrogen into known HTM units. Based on this idea, we induce additional hydrogen in hydride BaReH9, which is known as hydrogen-rich material at moderate pressure, and systematically studied the ternary Ba-Re-H system. We identified the stable hydrides Cmmm-Ba2ReH8, \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) and thermodynamic metastable P2/m-BaReH12 under high pressure. The \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\) exhibits remarkable high Tc ~ 128 K driven by a hitherto undiscovered 12-coordination HTM units [ReH12]2-, where 12 hydrogen atoms take quasi-regular icosahedron arrangements. And the H-H distance inter HTM units [ReH12]2- is ~1 Å. In P2/m-BaReH12, four hydrogens were shared by adjacent Re, lead to the coordination number of Re increased from 12 to 14, and the superconducting transition temperature is decreasing to 41.6 K at 300 GPa. In addition, we found that all the hydrogens construct the HTM unit [ReH8]4- in Cmmm-Ba2ReH8. Additional Ba in Cmmm-Ba2ReH8 increases the H-H distance inter HTM units [ReH8]4- compared with \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\), and the superconducting transition temperature of Ba2ReH8 is only 19 K at 100 GPa. In the end, we proposed several rules to construct hydride superconductors with HTM units. The electropositive ions are crucial, they should provide sufficient electrons and stabilize the HTM units. The odd-numbered electrons of HTM unit generally makes hydrides metallic. Finally, the weak bonding of hydrogens inter high symmetry HTM unit probably causes strong electron–phonon coupling.

Methods

Variable-composition structure searches for BaxReyHz (x = 1–4, y = 1–4, z = 1–18) were conducted at 100, 200 and 300 GPa using the CALYPSO41 code, and BaxReyHz (x + y + z ≤ 20) by AIRSS42,43. Both AIRSS and CALYPSO have proven to be highly effective in predicting high-pressure compounds9,43. In total, about 400 different compositions are considered for Ba-Re-H system, ~6000 structures were generated at each pressure point in the exploratory variable-composition structure searches. Furthermore, we concentrated our efforts on H-rich hydrides BaReH9+x (x = 1–9), employing fixed composition predictions for the selected promising compositions. The structural relaxation was conducted within the framework of density functional theory as implemented in the VASP44 and CASTEP45,46,47. The interactions between electrons and ions were modeled by using projector augmented wave (PAW) method48 with 5s25p66s2, 5d56s2 and 1s1 treated as valence electrons for Ba, Re and H, respectively. The generalized gradient approximation (GGA) in the scheme of Perdew-Burke-Ernzerhof (PBE) was chosen for the exchange-correlation functional, and kinetic cut-off energy of 700 eV and Monkhorst-Pack49 k meshes with grid spacing of 2π × 0.03 Å−1 were adopted to ensure convergence of enthalpy. The calculation of electron localization function (ELF) which is used to describe and visualize chemical bonds in molecules and solids were performed with grid spacing of 2π × 0.02 Å−1.

The calculations of the electron–phonon coupling and superconducting Tc were performed using density functional perturbation theory (DFPT)50 and the Allen-Dynes modified McMillan equation39 which are implemented in the Quantum-ESPRESSO package51. Ultra-soft pseudopotentials52 were used with a kinetic energy cutoff of 90 Ry. Self-consistent electron density and EPC was evaluated by employing 18 × 18 × 18 k mesh and Γ-centered 6 × 6 × 6 q mesh for \({Pm}\bar{3}-{{\rm{BaReH}}}_{12}\), 12 × 12 × 18 k mesh and Γ-centered 4 × 4 × 6 q mesh for Cmmm-Ba2ReH8