Room-Temperature Superconductivity in Yb/Lu Substituted Clathrate Hexahydrides under Moderate Pressure

Room temperature superconductivity is a dream that mankind has been chasing for a century. In recent years, the synthesis of H3S, LaH10, and C-S-H compounds under high pressures has gradually made that dream become a reality. But the extreme high pressure required for stabilization of hydrogen-based superconductors limit their applications. So, the next challenge is to achieve room-temperature superconductivity at significantly low pressures, even ambient pressure. In this work, we design a series of high temperature superconductors that can be stable at moderate pressures by incorporating heavy rare earth elements Yb/Lu into sodalite-like clathrate hexahydrides. In particular, the critical temperatures (Tc) of Y3LuH24, YLuH12, and YLu3H24 can reach 283 K at 120 GPa, 275 K at 140 GPa, and 288 K at 110 GPa, respectively. Their critical temperatures are close to or have reached room temperature, and minimum stable pressures are significantly lower than that of reported room temperature superconductors. Our work provides an effective method for the rational design of low-pressure stabilized hydrogen-based superconductors with room-temperature superconductivity simultaneously and will stimulate further experimental exploration.

Among these hydrogen-based superconductors, clathrate hydrides are one type that exists widely, including the well-known CaH 6 with H 24 cage [15], YH 9 with H 29 cage [16], and LaH 10 with H 32 cage [11,12]. The clathrate hexahydrides Im-3m-XH 6 (X = Mg, Ca, Sc, Y, La, Tm, Yb, and Lu) are prevalent in alkaline earth and rare earth metal hydride [12,15,[17][18][19][20], in which, the metal atoms form a body centered cubic (bcc) lattice, and hydrogen atoms occupy all the tetrahedral voids of the bcc lattice, forming a H 24 cage. CaH 6 and YH 6 have been experimentally synthesized and exhibit high T c of 215 K at 172 GPa and 227 K at 166 GPa, respectively [21,22]. Theoretical predicted T c s of MgH 6 , ScH 6 , and LaH 6 are 260 K at 300 GPa, 147 K at 285 GPa, and 174 K at 100 GPa, respectively. It is reported that the large chemical precompression of H-rich clathrate structures can be attained in rare earth hydrides with 4f electrons [23]. With the filling of the f orbitals of metal atoms, the structure is more easily stabilized at low pressure, but the unfilled f electrons can negatively affect superconductivity. For example, the CeH 9 with unfilled 4f orbitals were experimentally synthesized at low pressure of 88 GPa with low T c of 57 K [24]. TmH 6 , with unfilled 4f orbitals, is predicted to stable at 50 GPa and has a lower T c of 25 K. YbH 6 and LuH 6 , with filled f -shelled, are predicted to exhibit high-T c superconductivity of 145 K and 273 K at relatively low pressures of 70 GPa and 100 GPa, respectively [20]. These results suggest that heavy rare earth metals Yb/ Lu are suitable elements to reduce the pressure of stability and keep high T c simultaneously.
Incorporating a new element into binary hydrides to form ternary hydrides is an important way to improve the superconducting transition temperature or reduce the superconducting phase stable pressure. In 2019, Li 2 MgH 16 with the highest T c to date (473 K at 250 GPa) was designed by introducing extra electrons (Li element) to fill the antibonding orbital of the H 2 molecular units of MgH 16 [25]. In 2020, a compound of hydrogen, carbon, and sulfur showed a superconducting transition at 288 K [26] and 267 GPa. However, the stoichiometry and crystal structure of this compound have not yet been determined. This experiment is still subject to many controversies [16,27,28]. Although the T c s of Li 2 MgH 16 and C-S-H compounds can reach room temperature, the extreme pressure above 250 GPa also makes them difficult in practical application. Recently, a new class of fluorite-type clathrate ternary hydrides AXH 8 (A = Ca, Sr, Y, and La, X = B, Be, and Al) with hydrogen alloy backbone were predicted [29]. The most outstanding one, LaBeH 8 , is dynamically stable down to 20 GPa with a high T c ∼185 K. It is inspiring that the cubic clathrate superhydrides La x Y 1-x H 6,10 via laser heating of yttrium-lanthanum alloys have been experimental synthesized exhibiting a maximum critical temperature T c of 253 K without increasing pressure [30]. This experiment demonstrates that selecting a suitable central metal element to substitute sodalite-like clathrate hydrides is feasible.
The focus of hydrogen-based superconductor research is not simply the pursuit of high-temperature superconductivity or low stable pressure. A good superconductor should achieve a good balance between the pressure required for stability and the critical temperature. The next challenge is to achieve room-temperature superconductivity at significantly low pressures, even ambient pressure. In the present work, we chose heavy rare earth element Yb/Lu-doped clathrate hexahydrides to achieve this goal. According to the aforementioned introduction, the heavy rare earth metals Yb/Lu show outstanding properties in sodalite-like clathrate hexahydrides YbH 6 and LuH 6 . With full-filled f -shells, Yb/Lu can eliminate the negative impact on T c of f electrons, meanwhile satisfying the purpose of reducing the pressure.
Here, a serial of ternary hydrides A 1-x B x H 6 (A = Y, Ca, Sc, B=Yb, Lu, and A = Yb, B = Lu, x = 0:25, 0.33, 0.5, 0.67, 0.75) with high T c at moderate pressure were found, in which the atomic radii of A and B are similar. Among them, the critical temperatures of Y 3 LuH 24 , YLuH 12 , and YLu 3 H 24 are 283 K at 120 GPa, 275 K at 140 GPa, and 288 K at 110 GPa, respectively, which are close to or have reached room temperature. Recently, it is reported that pressureinduced high-temperature superconducting FeSe retained without pressure via pressure quenching (PQ) [31]. This PQ technique offers the possibility of these high T c hydrides for practical application in the future.
In order to extend the study to more ternary hydride systems, we substitute the metal elements in these six structures. The properties of clathrate structure can be further improved by choosing a suitable "precompressor" element. As discussed above, at least one of them is heavy rare earth element Yb/Lu, and the other element has similar radius with Yb/Lu, including K, Mg, Ca, Sr, Sc, Y, and La. We calculated the phonon dispersion for all possible components in the pressure range of 50-200 GPa and finally determined that 36 sodalite-like clathrate hexahydrides can be dynamically stable in seven ternary hydride systems, including Y-Lu-H, Ca-Lu-H, Sc-Lu-H, Y-Yb-H, Ca-Yb-H, Sc-Yb-H, and Yb-Lu-H (see Figures S9-S15). To determine the thermodynamic stability of these structures in Y-Yb-H, Ca-Yb-H, Sc-Lu-H, and Sc-Yb-H system, we also performed the fixed composition structure searches of A x B 1-x H 6 and constructed the convex hull, shown in Figures S5-S8. It is reported that the YH 6 , CaH 6 , and YbH 6 are always thermodynamically stable in the pressure range of 150-300 GPa. Therefore, the energetic stabilities of the Yb-containing hexahydrides in Y-Yb-H and Ca-Yb-H system are evaluated using their formation enthalpies (ΔH) with respect to binary hexahydrides. Compared with Lu-containing hexahydrides, many Ybcontaining hexahydrides can be thermodynamically stable at 200 GPa, including CaYbH 12 , Ca 3 YbH 24 , CaYb 3 H 24 , and YYbH 12 . All Sc-containing sodalite-like clathrate hexahydrides we studied are metastable phases in our studied pressure range (see Figures S7 and S8).
After determining the stability of all structures, we further calculated their superconducting properties. The superconductive transition temperatures of these structures are estimated through the Allen−Dynes-modified McMillan equation with correction factors and self-consistent solution of the Eliashberg equation (see Table S1). We also calculated the critical temperatures of CaH 6 and YH 6 in the same way and compared them with the experimentally measured values (see Table S2). The results show that the critical temperature T c using the self-consistent solution of the Eliashberg equation is in good agreement with the experimental measurements. Figure 2 shows the variation of critical temperature T c and minimum dynamically stable pressure with the concentration of heavy rare earth element Lu/Yb in sodalite-like clathrate hexahydrides. YH 6 has more excellent properties 2 Research than CaH 6 and ScH 6 , and its properties are further improved after incorporation of Lu. The results show that YLu 2 H 18 and Y 2 LuH 18 with the P-3m1 space group have similar properties, with high T c of 242 K and 240 K at 100 GPa, respectively (see Figure 2(a)). And YLu 3 H 24 and Y 3 LuH 24 with the Fm-3m space group also have very close properties, exhibiting room temperature of 288 K at 110 GPa and 283 K at 120 GPa, respectively. The critical temperature of YLuH 12 with space group Pm-3m is 275 K at 140 GPa, reaching ice point temperature. The critical temperatures of Y 3 LuH 24 , YLuH 12 , and YLu 3 H 24 are close to or have reached room temperature. Surprisingly, the images of T c and minimum pressure of Y (1-x) Lu x H 6 exhibit mirror symmetry as a function of Lu doping concentration due to their similar properties (see Figure 2(a)). For Y (1-x) Lu x H 6 , the structure appears to have a greater effect on the T c and minimum pressure than the doping concentration. For CaH 6 , both T c and minimum pressure show a trend of first increasing and then decreasing with the Lu doping concentration (see Figure 2(b)). Ca 3 LuH 24 has a T c of 221 K at 170 GPa, which is close to CaH 6 . At 170 GPa, the critical temperature of CaLuH 12 is 282 K, near room temperature. And CaLu 2 H 18 can be dynamically stable at 140 GPa, and its T c is as high as 299 K at this pressure, which is the highest T c in this work. It is only 6 meV/atom above the convex hull at 300 GPa, implying the possibility of experimental synthesis. For ScH 6 , doping with Lu can significantly reduce the minimum stable pressure and increase T c . With the increase of Lu doping concentration in Sc (1-x) Lu x H 6 , T c showed an upward trend, and the minimum pressure is basically maintained at about 100 GPa (see Figure 2(c)). The effect of Lu on reducing the minimum pressure is obvious. The most prominent is ScLu 3 H 24 , which reaches a T c of 271 K at 100 GPa, close to ice point temperature. In short, doping Lu in binary hexahydrides achieves the goal of reducing the dynamical stable pressure and increasing T c , such as YLuH 12 (275 K at 140 GPa), YLu 3 H 24 (288 K at 110 GPa), Y 3 LuH 24 (283 K at 120 GPa), CaLuH 12 (282 K at 170 GPa), and CaLu 2 H 18 (299 K at 140 GPa).
The T c s of Yb-containing structures are all significantly lower than those of Lu-containing structures and show a decreasing trend with the increase of doping concentration of Yb. There are only five low Yb concentration hydrides have T c over 200 K, such as Y 3 YbH 24 (222 K at 100 GPa), Y 2 YbH 18 (221 K at 100 GPa), Sc 3 YbH 24 (203 K at 100 GPa), YbLu 2 H 18 (212 K at 150 GPa), and YbLu 3 H 24 (222 K at 100 GPa). The effect of Yb on reducing the dynamical stable pressure is obvious. The minimum dynamically stable pressure of Y (1-x) Yb x H 6 is basically maintained around 100 GPa (see Figure 2(d)). But for Ca (1-x) Yb x H 6 , the introduction of heavy rare earth element cannot always reduce the dynamically stable pressure (see Figure 2(e)). The minimum dynamically stable pressure of CaYbH 12 and Ca 3 YbH 24 is 150 GPa like that of CaH 6 , and the minimum dynamically stable pressure of Ca 2 YbH 18 is even higher than that of CaH 6 , reaching 170 GPa. Only CaYb 2 H 18 and CaYb 3 H 24 with Yb concentrations over 50% exhibit lower pressures than CaH 6 , with minimum dynamically stable pressures of 130 GPa and 100 GPa, respectively. This is similar to Ca (1-x) Lu x H 6 discussed above. Doping less than 50% Yb/Lu in CaH 6 cannot reduce the dynamically stable pressure. This is due to the large difference between the radius of Ca and Yb/Lu. The minimum The light green and dark green balls represent "precompressor" metal atoms A and B, respectively. The small pink balls represent H atoms.   Figure S15). The calculated phonon spectrum exhibits imaginary frequencies in the whole Brillouin zone, indicating that YbLuH 12 is dynamically unstable. This result suggests that 4f electrons play an important role in stabilizing these clathrate structures.
The focus of hydrogen-based superconductor research is not simply the pursuit of high-temperature superconductivity, after the realization of room-temperature superconductivity. A good superconductor should achieve a good balance between the pressure required for stability and the critical temperature. Therefore, we use a figure of merit S [38] to evaluate the significance of all thermodynamically stable sodalite-like clathrate hexahydrides and metastable phase CaLu 2 H 18 in this work. S is obtained from the critical temperature T c and the pressure required for stabilization P: Y 3 LuH 24 and YLu 3 H 24 with S > 2 and YLuH 12 and Yb 2 LuH 18 with S close to 2 are better than the well-known hydrogen-based superconductor H 3 S, LaH 10 , and C-S-H compounds. Most importantly, it can be seen from Figure 3 that Y 3 LuH 24 is the room temperature superconductor with the lowest pressure required for stability (110 GPa), which is much lower than that of C-S-H, YH 10 and CaBeH 8 with room-temperature superconducting. It also means that room-temperature superconductivity at moderate pressure is promising in hydrogen-based superconductors.
It seems difficult to summarize the law of T c change only from the perspective of doping concentration for Lucontaining structures. So, we evaluated the distribution of H-H bond lengths for all hexahydrides in this work. Figure 4 shows the variation of superconducting critical temperature T c and H-H bond lengths with the concentration of heavy rare earth element Lu/Yb in sodalite-like clathrate hexahydrides. There are nine hexahydrides with T c over 250 K, including LuH 6 , YLuH 12 Figure 5 shows the charge transfer between two "precompressor" metal elements and hydrogen in all sodalite-like clathrate hexahydrides we studied. It can be clearly seen that Lu element is an extremely good electron donor. Each Lu atom can donate 3.07 electrons at most, which is far more than other metal elements. The electrons obtained by H atom in the Lu-containing structures increase with increasing the Lu doping concentration (black curves). In the structures without Lu, the number of electrons obtained by H atom is basically the same level (about 0.23 |e|). Although Y atom is also ideal candidates for "precompressor" metal element, it can only donate up to 1.81 electrons. Ca, Sc, and Yb atoms cannot donate more than 1.5 electrons. Both YH 6 and CaH 6 exhibit high-temperature superconductivity, but they are still not qualified for room temperature superconductor. The introduction of Lu makes it possible to achieve room-temperature superconductivity in sodalite-like clathrate hexahydrides.

Discussion
According to the aforementioned findings, the critical temperatures of Y 3 LuH 24 , YLuH 12 , and YLu 3 H 24 have reached room temperature under moderate pressure. In addition, although the T c of Ca 3 LuH 24 is only 221 K at 170 GPa, it has a lower thermodynamically stable pressure (below 200 GPa), meaning that it is easier to be synthesized experimentally. To gain insight into the origin of roomtemperature superconductivity for these sodalite-like clathrate hexahydrides, we calculated their electronic band structures and projected density of electronic states (PDOS), as shown in Figure 6. For hydrides, the contribution of electronic states of H to the Fermi level is an important basis for judging whether it is an excellent superconductor. In this work, for two structures of the same space group and the same "precompressor" metal elements A and B, their contributions of electronic states of H near the Fermi surface are basically the same. For example, the contributions of electronic states of H near the Fermi surface in YLu 3 H 24 and Y 3 LuH 24 are both 1.5 states/eV/f.u. (see Figures 6(a) and 6(b)), which is the essential reason for their close T c .
In addition to H, f electrons also have an important impact on superconductivity and dynamical stability, especially in Yb-containing structures. Note that the 4f orbitals associated with the Yb atom form a set of localized bands that appear about 1.5 eV below the Fermi level (see Figures S19-S22). As the Yb doping concentration increases, the position of PDOS peak corresponding to the f electrons is always 1.5 eV below the Fermi level and does not shift, but the contribution of electronic states of the f electrons at the Fermi surface increases. The 4f -orbital electrons are good for stabilizing the structure, but excessive f electrons at the Fermi surface will negatively affect superconductivity. Therefore, the T c s of Ybcontaining structures in Table S1 mostly does not exceed 200 K. Although lowering the concentration of Yb can increase T c , it also makes Yb lose its role in reducing the pressure required for stabilization. For Lu-containing structures, extra electron in the 5d orbitals leads to the 4f electrons moving away from the Fermi surface in the band structure (see Figures S19-S22). Thus, Lu-containing structures are not negatively affected by the 4f -orbital electrons, and exhibits a rather high T c . Meanwhile, YLu 3 H 24 can exhibit 288 K room temperature superconductivity which is also related to the flat band near the Fermi surface along the W-K direction, as shown in Figures 6(a) and 6(b). Such flat bands exist in all sodalite-like clathrate hexahydrides with Fm-3m space group in this work, but not all the flat bands can be near the Fermi surface. The energy bands of hydrides are influenced by the precompression element. Compared to CaH 6 , YH 6 has a higher Fermi energy due to more valence electrons [19]. Ca 3 LuH 24 also has the same flat band, but at 1.7 eV above the Fermi energy (see Figure 6(d)). This results in that the contribution of electronic states of H in Ca 3 LuH 24 at the Fermi level is not as high as that in YLu 3 H 24 . It may be one of the reasons why the T c of Ca 3 LuH 24 is not as high as that of YLu 3 H 24 . Due to the lower Lu content, the flat band of Y 3 LuH 24 is slightly above the Fermi surface compared to YLu 3 H 24 (see Figure 6(b)). Among all the "precompressor" metals, Y and Lu are more favorable for hightemperaturesuperconductivity of H 24 cage. In ternary 7 Research clathrate hexahydrides A 1-x B x H 6 , by selecting the appropriate "precompressor" elements A and B, adjust their proportions, maximizing T c can be achieved.
The high T c s of hydrogen-based superconductors are largely due to strong electron-phonon coupling (EPC) from high frequency optical phonons. We calculated phonon spectrum, phonon density of state (PHDOS) and integral EPC parameter λ of all sodalite-like clathrate hexahydrides we studied, to explore the source of this strong coupling from optical phonons. As shown in Figure 7, the phonon spectra of these ternary clathrate structures are similar to those of the binary clathrate hexahydrides [15,18]. The low frequency region is mainly the vibration of metal atoms (red and blue peak in PHDOS), and the high frequency region comes from the vibration of hydrogen (black peak in PHDOS). The maximum vibrational frequency of the hydrogen atom is related to the length of the H-H bond. The shorter the H-H bond, the higher the corresponding vibrational frequency. The maximum vibrational frequency of H 2 or H 3 units is generally above 2000 cm -1 [39]. The absence of spectral lines over 2000 cm -1 means that there are no H 2 or H 3 units. It can be seen from Figure 7 that the integral curve of λ (red curve) grows rapidly in 500-1500 cm -1 , while λ grows slowly above 1500 cm -1 (see Figure 7(d)). It means that the vibration in frequency range of 500-1500 cm -1 is the most important source of electron-   Figure 4) of all sodalite-like clathrate hexahydrides, we find that the H-H bond length corresponding to this vibrational frequency range is about 1.25-1.30 Å, which is consistent with the most suitable H-H bond length summarized in Figure 4. Elongating the H-H bond to 1.25-1.30 Å can reduce the vibrational frequency of hydrogen from high frequency (above 1500 cm -1 ) to a suitable frequency of 500-1500 cm -1 , thereby increasing T c . In addition to the high frequency optical phonons, "soft mode" also plays an important role in enhancement of superconductivity [40,41]. It can be seen from YLu 3 H 24 (Figure 7(a)) that there are some soft phonon modes near the Gamma point in frequency range of 100-500 cm -1 . The integral curve of λ (red curve) grows rapidly in this frequency range which can reach 3 up to 500 cm -1 . The Y 3 LuH 24 and YLuH 12 also have soft phonon modes (see Figures 7(b) and 7(c)). The integral curve of λ corresponding to the frequency at which the phonon softening is also rising rapidly. These results indicate that in YLu 3 H 24 , Y 3 LuH 24 , and YLuH 12 near lattice instability, the phonon softening strengthens the electron-phonon coupling λ, which in turn leads to high T c .
In conclusion, the incorporation of heavy rare earth elements Yb/Lu is an effective method to tune the superconducting transition temperature and the required pressure for stabilization of sodalite-like clathrate hydrides. In particular, the introduction of Lu element can further improve the superconductivity and keep the low-pressure stability. The three most prominent compounds Y 3 LuH 24 (283 K at 120 GPa), YLuH 12 (275 K 140 GPa), and YLu 3 H 24 (288 K at 110 GPa) exhibit room temperature superconductivity at much lower pressure than that of previously discovered   9 Research room temperature superconductors, such as C-S-H, YH 10 , and CaBeH 8 . The enhancement of T c is achieved by adjusting the H-H bond length to affect the hydrogen vibrational frequency and thereby enhance the electron-phonon coupling. Our results represent an important step towards room-temperature superconductivity at ambient pressure and will stimulate further experimental exploration.

Computational Methods
High-pressure structure searches were performed using the ab initio random structure searching (AIRSS) technique [42,43]. For Ca-Lu-H, Y-Lu-H, and Yb-Lu-H systems, we predicted more than 8000 structures using variable composition structure searches in each system for their ternary convex hulls. Furthermore, for 36 A 1-x B x H 6 (A = Ca, Y, Sc, B=Yb, Lu, and A = Yb, B = Lu, x = 0:25, 0.33, 0.5, 0.67, 0.75), we predicted about 500 structures for each composition. Structure relaxations during structure searches were performed using the ab initio calculation of the Cambridge Serial Total Energy Package (CASTEP) code [44]. The generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization [45] for the exchangecorrelation functional and ultrasoft pseudopotentials with cut-off energy of 400 eV and Brillouin zone sampling grid spacing of 2π × 0:07 Å −1 were chosen for the structure searching.
Considering the results of pseudopotential detection for Yb-H and Lu-H in previous work [20], we used CASTEP with ultrasoft pseudopotentials for structural relaxation and calculations of enthalpies and electronic properties. A cut-off energy of 1000 eV and a Brillouin zone sampling grid spacing of 2π × 0:03 Å −1 were used. All enthalpy calculations are well converged to less than 1 meV per atom, which is acceptable for density functional theoretical calculations. The charge transfer calculations are obtained using Mulliken population analysis [46]. The Quantum-ESPRESSO package [47] was used in phonon and electron−phonon calculations. Ultrasoft pseudopotentials were used with a kinetic energy cut-off of 90 Ry. The k-point and q-point meshes in the first Brillouin zone of 12 × 12 × 12 and 4 × 4 × 4 grids were adopted, respectively. The superconductive transition temperatures are estimated through the Allen−Dynes-modified McMillan equation (A-D-M) [48] with correction factors and selfconsistent solution of the Eliashberg equation (scE) [49] with the Coulomb pseudopotential μ * = 0:10 and 0.13.

Data Availability
The data supporting the findings of this study are available within the article and its Supplementary Materials files and from the corresponding author upon reasonable request.

Disclosure
This paper is dedicated to the 70th anniversary of the physics of Jilin University. Parts of calculations were performed in the High Performance Computing Center (HPCC) of Jilin University and TianHe-1 (A) at the National Supercomputer Center in Tianjin.