# Potential high-*T*_{c} superconducting lanthanum and yttrium hydrides at high pressure

_{c}

^{a}Geophysical Laboratory, Carnegie Institution of Washington, Washington, DC 20015;^{b}Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY 14853;^{c}Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853;^{d}Department of Civil and Environmental Engineering, The George Washington University, Washington, DC 20052;^{e}School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853

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Contributed by Russell J. Hemley, May 5, 2017 (sent for review March 20, 2017; reviewed by Panchapakesan Ganesh, Jeffrey M. McMahon, and Dimitrios Papaconstantopoulos)

## Significance

Theoretical predictions and subsequent experimental observations of high-temperature superconductivity in dense hydrogen-rich compounds have reinvigorated the field of superconductivity. A systematic computational study of the hydrides of lanthanum and yttrium over a wide composition range reveals hydrogen-rich structures with intriguing electronic properties under pressure. Electron–phonon coupling calculations predict the existence of new superconducting phases, some exhibiting superconductivity in the range of room temperature. Moreover, the calculated stabilities indicate the materials could be synthesized at pressures that are currently accessible in the laboratory. The results open the prospect for the design, synthesis, and recovery of new high-temperature superconductors with potential practical applications.

## Abstract

A systematic structure search in the La–H and Y–H systems under pressure reveals some hydrogen-rich structures with intriguing electronic properties. For example, LaH_{10} is found to adopt a sodalite-like face-centered cubic (fcc) structure, stable above 200 GPa, and LaH_{8} a *C*2/*m* space group structure. Phonon calculations indicate both are dynamically stable; electron phonon calculations coupled to Bardeen–Cooper–Schrieffer (BCS) arguments indicate they might be high-*T*_{c} superconductors. In particular, the superconducting transition temperature *T*_{c} calculated for LaH_{10} is 274–286 K at 210 GPa. Similar calculations for the Y–H system predict stability of the sodalite-like fcc YH_{10} and a *T*_{c} above room temperature, reaching 305–326 K at 250 GPa. The study suggests that dense hydrides consisting of these and related hydrogen polyhedral networks may represent new classes of potential very high-temperature superconductors.

Extending his original predictions of very high-temperature superconductivity of high-pressure metallic hydrogen (1), Ashcroft later proposed that hydrogen-rich materials containing main group elements might exhibit superconductivity at lower pressures, as the hydrogen in these structures may be considered “chemically precompressed” (2). These proposals, which were based on the Bardeen–Cooper–Schrieffer (BCS) (3) phonon-mediated theory of superconductivity, have motivated many theoretical and experimental efforts in the search for high-temperature superconductivity in hydrides at elevated pressures (4⇓⇓⇓⇓⇓–10). Theory has predicted the stability of a variety of dense hydride structures for which BCS arguments give superconducting transition temperatures, *T*_{c}s, that are very high (11⇓⇓⇓⇓⇓⇓–18).

In recent times, compression of hydrogen sulfides has provided a new incentive in hydride superconductivity, one in which theory played an important role. First, theoretical calculations predicted H_{2}S to have a *T*_{c} of ∼80 K at pressures above 100 GPa (19). Compression of H_{2}S led to the striking discovery of a superconducting material with a *T*_{c} of 203 K at 200 GPa (20). Moreover, the critical temperature exhibits a pronounced isotope shift consistent with BCS theory. It was proposed that the superconducting phase is not stoichiometric H_{2}S but SH_{3}, with a calculated *T*_{c} of 194 K at 200 GPa including anharmonic effects (21, 22). A subsequent experiment (23) suggested that the superconducting phase is cubic SH_{3}, in agreement with a theoretical study that gave *T*_{c} = 204 K within the harmonic approximation (24). Compression of another hydride, PH_{3}, was reported to reach a *T*_{c} of ∼100 K at high pressures (25). Subsequent theoretical calculations predicted possible structures and calculated *T*_{c}s close to the experimental results (26⇓–28). The experimental picture for these materials remains not entirely clear, and the synthesis of the superconducting phases, which has been reproduced for hydrogen sulfide, is path-dependent (23).

There is great experimental and theoretical interest in searching for related materials with both higher *T*_{c} and potentially broader ranges of stability. To date, simple hydrides with the highest predicted superconducting *T*_{c}s are MgH_{6} (271 K at 300 GPa) (29), CaH_{6} (235 K at 150 GPa) (17), and YH_{6} (264 K at 120 GPa) (30). In searching for other high-*T*_{c} superconducting hydrides, here we investigated theoretically possible high-pressure crystal structures of La–H and Y–H. We predict the existence of new stable hydride phases of these elements, with remarkably high *T*_{c}s at attainable pressures. The search for low-energy crystalline structures of La–H was performed using particle swarm optimization methodology implemented in the CALYPSO code (31, 32). This method has been applied successfully to a wide range of crystalline systems ranging from elemental solids to binary and ternary compounds (33⇓–35) and has proven to be a powerful tool for predicting crystal structures at high pressures (36⇓⇓–39). Structure searches were performed in the pressure range of 150–300 GPa using models consisting of 1–4 formula units. In general, the structure search was terminated after the generation of 1,500 structures. Structural optimizations, enthalpies, electronic structures, and phonons were calculated using density-functional theory (DFT). Structure relaxations were performed using DFT using the Perdew–Burke–Ernzerhof (40) generalized gradient approximation. Phonon dispersion and electron–phonon coupling (EPC) calculations were performed with density functional perturbation theory. Ultrasoft pseudopotentials for La and H were used with a kinetic energy cutoff of 80 Ry. A *q* mesh of 6 × 6 × 6 and *k* mesh of 24 × 24 × 24 for fcc-LaH_{10} structure in the first Brillouin zone (BZ) was used in the EPC calculations. The superconductivity calculations were performed with the Quantum-ESPRESSO package (17)·

## Results

We began with structure searches at ambient pressure for LaH_{2}. The fcc structure is found to be stable at low pressure, in agreement with experiment. Thinking the high-pressure regime would be most productive for new stoichiometries, we moved directly to 150 and 300 GPa. We first performed structure prediction at 150 and 300 GPa for LaH_{x} (*x* = 1–12). LaH_{2}, LaH_{3}, LaH_{4}, LaH_{5}, LaH_{8}, and LaH_{10} are found to be stable at 150 GPa (Fig. 1). Note that this figure is plotted with LaH_{2} as the lanthanum-rich endpoint, because this composition has the most negative enthalpy/atom at every pressure considered. Fig. 2 shows the computed geometries for the most stable phases we found at each stoichiometry, for *n* = 2–8. Interestingly, the stable phase predicted for LaH_{6} is not sodalite-type but an *R*-3*m* structure. The convex hull shows that LaH_{6} is not stable from 150 to 300 GPa (Fig. S1); LaH_{5} in a *P*-1 structure is stable from 150 to 200 GPa. LaH_{2} adopts a *C*2/*m* structure with a H–H distance of 1.53 Å, LaH_{3} adopts a *Cmcm* structure with H–H distance of 1.42 Å, and LaH_{8} has a *C*2/*m* structure with H–H distance of 1.02 Å (all at 300 GPa).

Most interestingly, we find that LaH_{10} adopts a sodalite-like structure with the La atoms arrayed on an fcc rather than a bcc lattice (Fig. 3). To illustrate the difference between fcc LaH_{10} and previous sodalite-like bcc CaH_{6}/YH_{6} structure, Fig. 3*B* shows the sodalite-like LaH_{6} structure. The fcc-LaH_{10} structure contains [4^{6}6^{12}] polyhedra which contain hydrogen cubes (Fig. 3 *A* and *C*), whereas conventional sodalite is built up of [4^{6}6^{8}] polyhedra (Fig. 3 *B* and *D*). The H network, made of different atoms, is known in the clathrate and zeolite community as AST (41). The La atoms sit at the 4b Wyckoff position (0, 0, 0), the H atoms at the 32f position (0.12, 0.38, 0.12) and 8c position (0.25, 0.25, 0.75). The shortest H–H distance is 1.1 Å (250 GPa), which is close to H–H distance predicted for atomic metallic hydrogen near 500 GPa (1 Å) (1). In contrast, the H–H distance in sodalite-type CaH_{6} is 1.24 Å at 150 GPa.

The above structures of LaH_{10} encouraged us to explore the Y–H system at similar pressures (Fig. S2). Previously, YH_{3} was predicted to adopt an fcc structure of Y, with atomic H located in the tetrahedral and octahedral interstitial sites. The compound was predicted to be superconducting with a maximum *T*_{c} is 40 K near 18 GPa (*μ** = 0.1) (16). Another theoretical study suggested two energetically competing hydrogen-rich polymorphs YH_{4} and YH_{6} (30). At 120 GPa, both are predicted to be superconductors with maximum *T*_{c} of 95 K and 264 K for YH_{4} and YH_{6}, respectively. Our calculations indicate that YH_{6} is stable in the sodalite structure up to 300 GPa (Fig. 4 and Fig. S2). As expected, YH_{10} adopts the same structure as LaH_{10} over a range of pressures. We predict that YH_{10} is the energetically stable phase from 250 to 300 GPa, and is dynamically stable down to 220 GPa.

Fig. 5 shows computed YH_{n} geometries for the most stable phases found at each stoichiometry, for *n* = 2, 3, 4, 6, 8, and 12. The YH_{2} adopts a *P*6/*mmm* structure with H–H distances of 1.46 Å at 300 GPa. The YH_{3} is stable from 150 to 250 GPa and adopts a *Pnma* structure with H–H distances of 1.54 Å at 300 GPa (Fig. S2). The YH_{4} adopts an *I*4/*mmm* structure with H–H distances of 1.37 Å at 300 GPa. It is interesting to see that YH_{6} has the same sodalite structure as CaH_{6} from 150 to 300 GPa with H–H distances of 1.19 Å at 300 GPa and is stable from 150 to 300 GPa. We found YH_{8} is unstable from 150 to 300 GPa with a *Cc* structure (Fig. S2). The YH_{12} is stable from 50 to 250 GPa and is a *C*2/*c* structure with shortest H–H distances of 0.79 Å at 250 GPa.

Before calculating possible superconducting properties we analyze the electronic band structures of the *C*2/*m*-LaH_{8} and fcc-LaH_{10} structures. The band dispersion shows the metallic character of both structures at these pressures (Fig. 6). The fcc LaH_{10} is a good metal with several bands crossing the Fermi level along many directions. This fact manifests itself in a noticeable density of electronic states at the Fermi level: 10.0 states/Ry, which is a factor of 1.4 higher than that previously found in SH_{3} (42) at an optimal pressure of 200 GPa (Fig. S3). Remarkably, not only the *d* electrons of La and *s* electrons of H contribute to N(E_{f}) but also *f* electrons of La; moreover, the latter contribution is dominant. This is due to the fact that external pressure destabilizes La 6*s* and La 5*d* orbitals to a greater extent than La 4*f*: the first two have five and two nodes in their radial wave functions, respectively, whereas the latter has none. This is in sharp contrast with the YH_{10} system, where only the *d* electrons of Y and *s* electrons of H are the main contributors to N(E_{f}). Phonon calculations reveal no imaginary frequencies for LaH_{8} and LaH_{10} over a wide pressure range, indicating dynamic stability. Specifically, we find that the structure shown for LaH_{10} is dynamically stable down to 210 GPa. The EPC calculations for different phonon modes indicate that there are no significant contributions in particular directions (Fig. 6 *C* and *D*). Note that there are no high-frequency vibrations, consistent with the absence of molecular H_{2} entities. The highest frequency for the structure here (2,000 cm^{−1}) can be compared to the 2,600 cm^{−1} calculated for atomic metallic hydrogen in the Cs-IV structure at 500 GPa (43); the lower frequency for the former suggests still weaker H–H interactions in the hydride.

Electron-coupling calculations for LaH_{4} give a relatively small *λ* of 0.43. The estimated *T*_{c} is 5–10 K at 300 GPa with the typical choice of the Coulomb potential of *μ** = 0.1–0.13. Using the McMillan equation (44), we calculated *T*_{c} using the spectral function [*α*^{2}*F*(*ω*)], again with *μ** = 0.1–0.13, as previously used in superconductivity calculations for Y (15), La (45, 46), SH_{3} (24), CaH_{6} (17), YH_{6} (30), and other superconductors.*λ* is the first reciprocal moment of *α*^{2}*F*(*ω*),** q**) is the weight of a

**(wave vector of crystal vibrations) point in the first BZ. The EPC spectral function**

*q**α*

^{2}

*F*(

*ω*) is expressed in terms of the phonon linewidth

*γ*

_{qj}arising from EPC,

*N*

_{f}is the electronic density of electron states at the Fermi level. The linewidth

*γ*

_{qj}of a phonon mode

*j*at wave vector

**, arising from EPC is given by**

*q*_{BZ}as the volume of the BZ, and

*ξ*

_{kn}are the energies of bands measured with respect to the Fermi

*ξ*

_{F}level at point k. Here,

*g*

^{j}

_{kn,k+qm}is the electron–phonon matrix element for scattering from an electron in band n at wave vector

*k*state to band m at wave vector

**+**

*k***via a phonon with wave vector**

*q***.**

*q*The calculated electron-coupling parameter for LaH_{8} at 300 GPa is relatively large (*λ* = 1.12), and the calculated *T*_{c} is 114–131 K, assuming *μ** = 0.1–0.13 (Eq. **1**). That result encouraged us to calculate the *T*_{c} of LaH_{10}. Previous theoretical studies also found that sodalite-structured hydrides (e.g., CaH_{6} and YH_{6}) were associated with calculated high-*T*_{c} superconductivity. In the LaH_{10} structure, the predicted shortest H–H distance is 1.1 Å (Fig. S4) at these pressures, indicating some H…H interactions and no clear distinction between stretching and bending vibrations. As a result, all H vibrations effectively participate in the EPC process, which appears to enhance high superconductivity. The calculated EPC is quite large (*λ* = 2.2). *T*_{c} was estimated from *α*^{2}*F*(*ω*) by numerically solving the Eliashberg equations (17, 30) with *μ** = 0.1–0.13. Coulomb repulsion is taken into account in terms of *μ** scaled to a cutoff frequency. At 250 GPa, the estimated *T*_{c} of LaH_{10} is 257–274 K with *μ** = 0.1–0.13. The calculated *T*_{c} is found to decrease with increasing pressure (Fig. 7 and Table S1). The predicted *T*_{c} for YH_{10} is very high (Fig. 7) using similar EPC calculations. At 250 GPa, the *λ* is 2.56 and gives a *T*_{c} of 305–326 K with *μ** = 0.1–0.13 based on numerically solving the Eliashberg equations. The *T*_{c} of YH_{10} increased by ∼30 K relative to YH_{6} (30) despite the fact that *λ* decreases (from 2.93 to 2.56). But, this change is offset by the higher average *ω*_{log} calculated for YH_{10} compared with YH_{6} (1,282 K versus 1,124 K), as a result of the higher hydrogen content in the former. The information of all predicted structures for La–H and Y–H is summarized in Table S2.

## Discussion

We now examine the above results, including various assumptions and comparisons with previous studies. Because zero-point energy (ZPE) has been shown to play an important role in determining the stability of hydrogen-rich materials, we consider its effect. We recomputed the formation enthalpies of different phases in La–H and Y–H systems at 300 GPa by including ZPE (Fig. S5). The phases LaH_{10} and YH_{10} are still stable; therefore, the above conclusions are not altered by considering zero-point vibrations. The authors of ref. 30 focused on the YH_{6} stoichiometry, and the authors did not consider hydrogen compositions higher than 1:8. To understand the role of the La atoms in stabilization of clathrate hydrogen structure, we computed the “formation volume” *V*_{LaH10} − *V*_{H10} − *V*_{La}, where *V*_{LaH10} is the volume of LaH_{10}, *V*_{H10} is the volume of hydrogen in the sodalite-like structure, and *V*_{La} is the volume of La at 300 GPa. We found the formation volume to be negative: −4.66 A^{3} per primitive cell. This suggests that the La atoms help to stabilize the clathrate hydrogen structure.

We also investigated the bonding in these hydrides by a crystal orbital Hamiltonian population (COHP) analysis. The COHP provides an atom-specific measure of the bonding character of states in a given energy region (47). A negative COHP indicates bonding and positive COHP indicates antibonding. As Fig. 8 shows, predictably most of the states below the Fermi level are H–H bonding for LaH_{10} and YH_{10}. The interesting difference is that some H–H antibonding states are occupied in the yttrium hydride.

We comment on the reliability of the present predictions of very high-*T*_{c} superconductivity by comparing our results with existing experimental data and additional BCS calculations for related superconductors. We calculated a *T*_{c} of 6–7 K for fcc La under ambient pressure using *μ** = 0.1–0.13, which is close to previous theoretical (46) and experimental work (*T*_{c} = 6 K) (48), and consistent with high-pressure behavior (*T*_{c} = 13 K at 20 GPa) (45, 48). Likewise, our method gives the *T*_{c} of elemental Y under pressure to be 15–16 K using *μ** = 0.1–0.13, which is also consistent with previous theoretical (15) and experimental studies (*T*_{c} = 17 K at 89 GPa) (49). All of our calculations were carried out using the harmonic approximation. It is well known that in many hydrides anharmonicity tends to lower *T*_{c} (50). For the well-studied SH_{3} system, however, such an effect is weak: *T*_{c} is lowered from its harmonic 204 K value (24) only to 194 K (21) at 250 GPa, and both are close to the reported experiment *T*_{c} at that pressure (20).

There is a good reason to think that hydrogen-rich hydrides should behave similarly to pure hydrogen. Recent first-principles calculations of metallic hydrogen in structures with space groups *I4*_{1}*/amd* and *Cmca-*4 predict anharmonic behavior that differs from calculated results for hydrides with relatively low hydrogen content (51, 52). Whereas in the *I4*_{1}*/amd* the inclusion of anharmonicity slightly lowers *T*_{c} (i.e., from 318 to 300 K at 500 GPa), the situation is drastically different in the *Cmca-4* phase, where the calculated *T*_{c} increases by a factor of 2 with anharmonicity introduced, such that the phase is predicted to be a superconductor above 200 K. In this context it is also worth noticing that anharmonic vibrations may enhance the electron–phonon matrix elements, e.g., in the case of disordered materials (53).

In summary, exploration of La–H and Y–H systems at high pressures reveals stable hydrogen-rich phases with calculated unusually high superconducting temperatures. To be realistic, we should be careful about these high *T*_{c}s. Although well-calibrated on low *T*_{c} elemental superconductors, more high-*T*_{c} compounds need to be studied––we only have SH_{3} mentioned above (with its attendant ambiguities––not in the *T*_{c}, measured, but in the structure). The predicted stabilities of the phases are in the range of current high-pressure techniques, although the megabar pressures required present challenges for combined synthesis and characterization. On the other hand, it may also be possible to synthesize these and related materials metastably at lower pressure. For instance, one might reason from the fact that Y (54) itself has a distorted fcc structure that it may be possible to produce sodalite-like YH_{10} at a metal–hydrogen interface. The exploration, theoretical and experimental, of high hydrides of heavy metals is thus a promising field (55⇓⇓–58).

## SI Text

Table S1 presents calculated *T*_{c}s of LaH_{n} (*n* = 4, 8, and 10) and YH_{10} at high pressures for different models and parameters. It is seen that the McMillan and Eliashberg equations lead to similar results for small *λ*. For larger *λ*, however, the Eliashberg equation predicts higher *T*_{c} values. Table S2 contains the lattice parameters and atomic coordinates for the predicted structures.

We compare the calculated density of states (DOS) for LaH_{10}, YH_{10}, and SH_{3} in Fig. S3. In the case of SH_{3,} we found a DOS at the Fermi level of 7.1 states/Ry which is close to the value of 6.93 states/Ry found in the previous work (42). The calculated *N*(*E*_{f}) of LaH_{10} and YH_{10} at 250 GPa is larger than that of SH_{3} at 200 GPa. For LaH_{10}, both the *d* and *f* electrons of La as well as the *s* electrons of H contribute to *N*(*E*_{f}). Indeed, the *f*-electron contribution is dominant, which is in sharp contrast with the YH_{10} system, where only the *d* electrons of Y and *s* electrons of H are the main contributors to *N*(*E*_{f}).

The distances between H atoms in LaH_{10} as a function of pressure are shown in Fig. S4. Each La atom is coordinated with 12 hydrogen atoms. H atoms are 4-coordinated, with bond distances of 1.08 and 1.15 Å (see also Fig. 3). Finally, the formation enthalpy of predicted La–H and Y–H structures at 300 GPa including ZPE is shown in Fig. S5.

## Acknowledgments

We are grateful to the reviewers for their comments on the manuscript. This research was supported by EFree, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award DE-SC0001057. The infrastructure and facilities used are supported by the US Department of Energy/National Nuclear Security Administration (Grant DE-NA-0002006, Capital/Department of Energy Alliance Center).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: rhemley{at}gwu.edu.

Author contributions: H.L. and R.J.H. designed research; H.L., I.I.N., R.H., N.W.A., and R.J.H. analyzed data; and H.L., I.I.N., R.H., N.W.A., and R.J.H. wrote the paper.

Reviewers: P.G., Oak Ridge National Laboratory; J.M.M., Washington State University; and D.P., George Mason University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1704505114/-/DCSupplemental.

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