Possible high-T C superconductivity at 50 GPa in sodium hydride with clathrate structure

Ambient-pressure room-temperature superconductivity is one ultimate goal of science, for it will bring worldwide revolutionary changes in all kinds of technology. Several room temperature and near room temperature hydride superconductors at ultra high pressure (≳100 GPa) have been predicted theoretically. In particular, the hydrogen sulfide (H3S) with T C ≃ 203 K at 200 GPa has soon been confirmed experimentally, establishing a milestone toward room temperature superconductivity. However, high-T C superconductors at lower pressure (≲100 GPa) have not been reported before. In this work, we present high-T C superconductivity of 180 K at a relatively low pressure of 50 GPa in sodium hydride clathrate structure NaH6. The T C can be raised up to 206 K at 100 GPa, similar to the T C of H3S but at a much lower pressure. At 200 GPa, it reaches the highest T C of 210 K, slightly higher than that of H3S. The strong electron–phonon coupling strength given by the T 2g phonon mode at Γ point plays the key role in superconductivity. Our work demonstrates theoretically that hydrides could stabilize at a relatively low pressure and host high-T C superconductivity.

and phonon frequency. Moreover, the high density of state at the Fermi level under high pressure can induce strong electron-phonon coupling [35]. These two conditions play crucial roles in BCS theory toward room temperature superconductivity.
Unfortunately the ultra high pressure required for metallic hydrogen is very difficult to achieve in experiments. A report claims the metallic hydrogen has been synthesized, however more details need further confirmations [38]. Therefore, many hydrogen-based compounds with high-T C have been proposed because the additional ions can play the role to stabilize the lattice structure at relatively lower pressure. H 2 S is firstly proposed to be metalized and stabilized under 160 GPa [28]. Then H 3 S [26][27][28] and LaH 10 (including the rare-earth series REH 6 , REH 9 , REH 10 ) [29][30][31][32][33]39] are predicted to be electron-phonon superconductors. Soon after the theoretical predictions, high-T C superconductivity of H 3 S and LaH 10 are realized by experiments, demonstrating the powerful ability to predict conventional superconductors by BCS-Eliashberg theory. In addition, more accurate calculations such as solving the Eliashberg equation [40], including the anharmonic correction [41], and so on (see the method section S-I in supplementary information (https://stacks.iop.org/NJP/23/093007/mmedia)) have also been reported. On the other hand, superconductivity of T C up to 100 K [42][43][44] are discovered experimentally in H 3 P under 100-250 GPa first, and soon followed by theoretical calculations of T C = 70-80 K [45][46][47] compatible with previous experimental data.
In this letter, we demonstrate theoretically that sodium hydride NaH 6 in the truncated cubic cellulation structure exhibits high-T C of 180 K at relatively low pressure of 50 GPa with respect to the extremely high pressure of hundreds of GPa required for the aforementioned high-T C hydrides. Moreover, the T C of NaH 6 can be further enhanced to 206 K by raising the pressure to 100 GPa, which is similar to T C = 203 K of H 3 S at a much higher pressure of 200 GPa. At 200 GPa, NaH 6 reaches the maximum T C of 210 K, slightly higher than that of H 3 S. The relatively low pressure of 50 GPa is experimentally much more achievable than H 3 S at 200 GPa [27], thus serves as an important step toward ambient-pressure room-temperature superconductivity.
Sodium, the amplest element of group I alkali metals, is the sixth most abundant species in Earth's crust and widely exists in numerous minerals such as rock salt. Because of the low electronegativity, the single valent s-electron can readily donate to the HCS. To search for high-T C superconductors with high-abundance highly-donating metal ions in the HCS, we study the electronic structure and phonon spectrum of NaH 6 through first-principles calculations based on density functional theory (DFT) and density functional perturbation theory (DFPT) (supplementary information S-I presents the computational details and introduces some advanced electron-phonon-coupling methods [21,41,43,[53][54][55][56][57]). The crystal structure of sodium hydride NaH 6 in the cubic HCS with high crystal symmetry (Pm-3m) is shown in figures 1(a) and (b). This structure, which is the same as that of the topological Kondo insulator SmB 6 [58][59][60][61][62][63][64], has been reported in a previous work [65]. However, the dynamical stability, electron-phonon coupling, and superconductivity have not been studied through DFPT yet [66,67]. On the other hand, a previous study based on neural network research [68] has shed light on possible T C higher than 100 K for the Na-H system, being compatible with our ab initio study.
Under 50 GPa, the optimized lattice constant of NaH 6 is a = 2.98Å with the long (short) hydrogen bond length of 1.50Å (0.85Å) longer than 0.74Å of H 2 gas molecules. Owing to sodium's low electronegativity, the HCS receives additional electrons from Na to stabilize the stretched H-H bonds. Even though the extra electrons occupying the hydrogen anti-sigma (σ * ) bond stabilize the stretched H-H bonds, they are actually weaker than the shorter H-H bonds in the molecular phase. Consequently these weaker H-H bonds in the HCS are easier to vibrate and hence can enhance the electron-phonon coupling strength [39] in NaH 6 . With the pressure increased to 100 GPa (200 GPa), the lattice parameter decreases to a = 2.78Å (2.56Å); meanwhile, the short and long H-H bonds become 0.84Å (0.81Å) and 1.37Å (1.24Å), respectively.
The electronic band structures and density of states (DOS) of NaH 6 are shown in figures 1(c) and (d), respectively. There is a three-fold degenerate electron band around 2.16 eV above the Fermi level at Γ-point. This degeneracy gives rise to spikes in DOS. Driven by the T 2g phonon mode, these spikes march toward the Fermi level and offer strong electron-phonon coupling owing to the severely changing electronic properties when this electron pocket passes through the Fermi level.
To date, all the available high-T C hydrides require high pressure from 100 GPa to 450 GPa. In strong contrast, the phonon spectra of NaH 6 at 50 GPa as shown in figures 2(a)-(c) are well-behaved without imaginary (negative) mode, demonstrating that NaH 6 keeps stable at this relatively low pressure. Furthermore, the T C remains surprisingly high of 180 K. Such a high-T C at relatively low pressure, which is an important breakthrough toward ambient-pressure room-temperature superconductor, has never been reported before.
As the pressure raises, the overall band width of the phonon spectra increases from 90 to 115 THz as shown in figures 2(a)-(f), which helps enhance T C . Under 200 GPa, the Eliashberg function α 2 F(ω) exhibits a large optical band at low frequency region with the peak value around 40 THz (≈1900 K). The integrated electron-phonon coupling strength λ(ω) increases strongly in this region. The average phonon frequency ω ln ∼ 1700 K is nearly equal to ω ln ∼ 2000 K of the metallic hydrogen [36]. This is one important reason that NaH 6 yields high-T C . The strongest electron-phonon couplings originate from a special T 2g mode at Γ with frequency of 33 THz. These strong electron-phonon couplings concentrated in few phonon modes at Γ are the same as those in other HCS such as YH 10 [39]. The atom-decomposed phonon bands illustrate that vibrations among H ions, i.e. over the HCS, is the main cause of the strong electron-phonon coupling. While Na ions contribute negligibly on the low frequency phonons, serving as electron donors and cornerstones of HCS only. Because of the additional constraint that the size of the embedded cation should well match the cage-like HCS, replacing Na by Li, K, Mg, and Ca all results in unstable structures with negative phonon bands.  induced changes in electronic structures through frozen-phonon simulations are illustrated in supplementary information S-II. Figure 3 compares the electron-phonon coupling strength λ (see supplementary information S-III for convergence test), logarithm average phonon frequency ω ln , superconducting T C , and DOS at the Fermi level of NaH 6 and H 3 S [26,27]. The superconducting T C of NaH 6 (figure 3(a)) increases significantly from 180 K at 50 GPa up to 206 K at 100 GPa, and then increases gently up to 210 K at 200 GPa. Except at 50 GPa, all the obtained T C above 100 GPa are slightly higher than that of H 3 S. Nevertheless, the  [26,27]. (b) The electron-phonon coupling λ (blue) and the logarithm average phonon frequency ω ln (red) of NaH 6 as functions of pressure. The blue (red) horizontal dashed line denotes the electron-phonon coupling (logarithm average phonon frequency) of H 3 S at 200 GPa [26,27]. The grey color (below 50 GPa) indicates the unstable NaH 6 lattice region.
T C = 180 K of NaH 6 at 50 GPa remains surprisingly high. This high-T C at low pressure is important toward room-temperature superconductivity at ambient-pressure. Meanwhile DOS at the Fermi level decreases monotonically with increasing pressure, indicating that the DOS factor is actually harmful to T C . On the other hand, λ and ω ln (figure 3(b)) are respectively suppressed and strengthened by raising the applied pressure. These opposite trends in λ and ω ln is the reason why T C increases only slightly above 100 GPa. Furthermore, in comparison with H 3 S at 200 GPa, ω ln (λ) of NaH 6 is larger (smaller) than that of H 3 S. These different behaviors are given by different lattice and electric natures between NaH 6 and H 3 S. The H 3 S lattice is composed of interpenetrating cubes with H-S bonds, whereas the NaH 6 crystal is of the HCS nature with weak H-H bonds enhancing the hydrogen vibration ω ln .
The NaH 6 hydride exhibits several properties similar to those of other high-T C hydrides, including the high symmetry crystal, cubic lattice, HCS, and metallic ions. To compare with previous cases, we present the T C -pressure phase diagram of available high-T C hydrides in figure 4(a). It can be clearly seen that NaH 6 is the only superconductor presenting high-T C over 200 K at pressure below 100 GPa. Another Na hydride NaH 8 studied in this work also exhibit high-T C of 130 K at 100 GPa (supplementary information S-IV). Except the metal hydrogen with 350 K over 500 GPa, all the other superconductors with T C > 200 K are in the cubic structure. Moreover, for hydrides with T C > 200 K (green region), except H 3 S [26], all the others (CaH 6 [48], MgH 6 [50], YH 6 [39,49], YH 9 [39], YH 10 [29,39], LaH 10 [29,39], ThH 10 [78], AcH 10 [78], AcH 16 [79]) are in the HCS and the degenerate phonon modes at Γ plays the crucial role in high-T C superconductivity. This kind of enhanced superconductivity has been observed in other types of clathrate-like structures such as the potassium-doped carbon Fullerenes K 3 C 60 [84] and β-pyrochlore K x Os 2 O 6 oxide [85]. The above trends imply that simple cube, BCC, and FCC sublattice of metal ions situated at the center of cubic-cellulation hydrogen cages may pave a possible route toward ambient-pressure room-temperature superconductivity. Figure 4(b) shows the pressure-hydrogen bond lengths diagram for available high-T C hydrides in HCS. Interestingly, all the H-H bonds, even though in different kinds of hydrides, show the similar trend that The red solid (empty) spheres are for NaH 6 (NaH 8 ) from our calculations. The blue circles are given from previous studies [26, 27, 29, 35-37, 39, 42, 46, 48-50, 53, 69-80]. The yellow crosses are experimental results [27,[30][31][32][33][81][82][83]. The light-grey region indicates pressure below 100 GPa. The green region denotes hydrides in HCS with T C higher than 200 K. The blue region is for other hydrides without HCS with T C below 200 K. (b) The pressure-hydrogen bond lengths diagram for HCS. The blue circle indicates the H-H bond length of available hydrides. The red solid circle (square) indicates the NaH 6 long (short) H-H bond length at different pressure. The long H-H bonds, rather than the short ones, take part in the T C mechanism. The bond lengths of NaH 8 are also presented in red empty circle. (c) The relation between the bond length and T C . they approximately lie on a straight line, demonstrating the close relation between the pressure and H-H bond lengths. The long H-H bonds of NaH 6 , which dominate the high-T C mechanism as discussed previously, do follow this simple rule with relatively long bond length at low pressure. The only exception is the short H-H bonds in NaH 6 that they locate far away from all the others because they do not take part in the high-T C mechanism of NaH 6 . Combining the bond length-pressure relation in figure 4(b) with the bond length-T C relation in figure 4(c), the implication for designing high-T C hydrides at lower pressure is clear: to search for a HCS with suitable metal ions matching the size of the H-cage yielding longer H-H bonds. Here we note that both the short and long H-H bonds of the HCS are of the covalent-bond character. Lower pressure would lead to longer H-H bonds with suppressed covalent charge over the H-H long bonds and eventually break the covalent behavior over the H-H long bonds, resulting in lattice instability below 50 GPa. On the other hand, one should look for shorter H-H bonds if higher T C is the only concern without the lower-pressure constraint. There is usually a competition between stability, electron-phonon coupling strength, and crystalline symmetry in hydrides under pressure. It could be fortuitous that NaH 6 remains stable at relatively low pressure of 50 GPa while with strong electron-phonon coupling and crystalline symmetry. Future studies may resolve if such good behavior can also be discovered in other hydrides.
Inspired by the McMillan formula (equation (2), S1) that both the electron-electron interaction μ * and electron-phonon coupling strength λ play important roles in enhancing T C through the exponential function, we assume that the complicated argument in the exponential function may have simple correlation with the H-H bond length L. We thus fit the available hydrogen bond length-T C data of high-T C hydrides in HCS shown in figure 4(c) by the simple relation: T C = A exp(−B · L). Surprisingly, the fitting is good as depicted in the grey dashed curve in figure 4(c) with A = 1160 K and B = 1.37Å −1 . All the relevant data points locate closely to the fitted curve. Here by relevant we mean the H-H bonds formed by electrons around the Fermi level that contribute significantly to high-T C . The irrelevant short H-H bonds of NaH 6 (solid red square) and long H-H bonds of YH 9 [39] (not shown here) indeed lie away from this curve. As such, this simple relation between H-H bond length and T C can give a quick hint on T C of hydrides in HCS.
In conclusion, by using DFT and DFPT calculations, we theoretically demonstrate that NaH 6 in HCS exhibits the highest T C of 210 K at 200 GPa. It remains stable at a relatively low pressure of 50 GPa while hosts a surprisingly high T C of 180 K. In comparison with the extremely high pressure required for other hydrides, such a high-T C at low pressure is a significant achievement toward normal conditions. The strong electron-phonon coupling is dominated by the three-fold degenerate T 2g phonon at Γ, which also play the key role of enhancing T C in previous studies. More importantly, our study shed light on guidelines for designing high-T C hydrogen-based superconductors at low pressure: to search for a HCS with suitable metal ions matching the size of the H-cage yielding longer H-H bonds, which paves a possible route toward ambient-pressure room-temperature superconductivity.