## Abstract

The discovery of superconductivity at 200 kelvin in the hydrogen sulfide system at high pressures^{1} demonstrated the potential of hydrogen-rich materials as high-temperature superconductors. Recent theoretical predictions of rare-earth hydrides with hydrogen cages^{2,3} and the subsequent synthesis of LaH_{10} with a superconducting critical temperature (*T*_{c}) of 250 kelvin^{4,5} have placed these materials on the verge of achieving the long-standing goal of room-temperature superconductivity. Electrical and X-ray diffraction measurements have revealed a weakly pressure-dependent *T*_{c} for LaH_{10} between 137 and 218 gigapascals in a structure that has a face-centred cubic arrangement of lanthanum atoms^{5}. Here we show that quantum atomic fluctuations stabilize a highly symmetrical \({Fm}\overline{3}{m}\) crystal structure over this pressure range. The structure is consistent with experimental findings and has a very large electron–phonon coupling constant of 3.5. Although ab initio classical calculations predict that this \({Fm}\overline{3}{m}\) structure undergoes distortion at pressures below 230 gigapascals^{2,3}, yielding a complex energy landscape, the inclusion of quantum effects suggests that it is the true ground-state structure. The agreement between the calculated and experimental *T*_{c} values further indicates that this phase is responsible for the superconductivity observed at 250 kelvin. The relevance of quantum fluctuations calls into question many of the crystal structure predictions that have been made for hydrides within a classical approach and that currently guide the experimental quest for room-temperature superconductivity^{6,7,8}. Furthermore, we find that quantum effects are crucial for the stabilization of solids with high electron–phonon coupling constants that could otherwise be destabilized by the large electron–phonon interaction^{9}, thus reducing the pressures required for their synthesis.

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## Data availability

All the data generated in this work is available upon request from I.E. and J.A.F.-L.

## Code availability

Quantum ESPRESSO is an open-source suite of computational tools available at https://www.quantum-espresso.org. VASP is a proprietary program. The SSCHA and the SCDFT codes are private codes developed by some of the authors, and are being prepared for distribution as an open-source code.

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## Acknowledgements

This research was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 802533); the Spanish Ministry of Economy and Competitiveness (FIS2016-76617-P); Grant-in-Aid for Scientific Research (number 16H06345, 18K03442 and 19H05825) from the Ministry of Education, Culture, Sports, Science and Technology, Japan; and NCCR MARVEL funded by the Swiss National Science Foundation. Computational resources were provided by the Barcelona Superconducting Center (project FI-2019-1-0031) and the Swiss National Supercomputing Center (CSCS) with project s970.

## Author information

### Affiliations

### Contributions

The project was conceived by I.E. and J.A.F.-L. The SSCHA was developed by I.E., L.M., R.B., M.C. and F.M. In particular, R.B. developed the method to compute the quantum energy Hessian and the anharmonic phonon dispersions, and L.M. developed the method to perform a quantum relaxation of the lattice parameters. I.E. and F.B. performed the SSCHA calculations. A.S., T.K., T.T., R.A. and J.A.F.-L. conducted studies on structure prediction and superconductivity. All authors contributed to the editing of the manuscript.

### Corresponding author

Correspondence to José A. Flores-Livas.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Additional information

**Peer review information** *Nature* thanks Yanming Ma and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

**Publisher’s note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Convex hull of enthalpy formation and diffraction pattern of candidate LaH_{10} phases.

Top, classical calculations of enthalpy (without zero-point energy) at different hydrogen contents at 100, 150 and 200 GPa. At low pressure (100 GPa), LaH_{10} is not stable and only develops as stable point in the convex hull of enthalpy formation at pressures above about 175 GPa. Bottom, diffraction patterns of different structures at 150 GPa (classical pressure), compared to the experimental data reported in ref. ^{5} for LaH_{10} in the \(Fm\bar{3}m\) phase at 150 GPa. The pattern is shown in the vicinity of the (111) peak of the \(Fm\bar{3}m\) phase. This peak is clearly split in the distorted *C*2 and \(R\bar{3}m\) phases predicted classically. The figure provides confirmation that the experimental resolution in ref. ^{5} would have been sufficient to distinguish between these phases.

### Extended Data Fig. 2 Convergence of SSCHA-phonon supercells and different anharmonic phonon calculations for LaH_{10} at 163 GPa.

Left, the phonon spectra shown are calculated by directly Fourier-interpolating the force constants obtained from the Hessian of \(E({\boldsymbol{ {\mathcal R} }})\) in a real space 2 × 2 × 2 and a 3 × 3 × 3 supercell. The similarity of both phonon spectra obtained by Fourier interpolation indicates that these SSCHA force constants are short-ranged and can be Fourier-interpolated. Right, phonon spectra obtained from the SSCHA energy Hessian of equation (1), making different level of approximations. The purple solid line is the phonon spectrum calculated with the full-energy Hessian without any approximation. In the blue dotted spectrum we set \(\mathop{\varPhi }\limits^{(4)}\) = 0 in the equation. For the orange dash-dotted line we set \(\mathop{\varPhi }\limits^{(3)}=\mathop{\varPhi }\limits^{(4)}=0\), so that the phonon spectra correspond to that arising directly from the SSCHA variational force constants *Φ*. These results clearly show that whereas the effect of \(\mathop{\varPhi }\limits^{(3)}\) is important, setting \(\mathop{\varPhi }\limits^{(4)}\) = 0 has minimal effect. All phonon spectra are obtained by directly Fourier-interpolating the real space anharmonic force constants in a 2 × 2 × 2 supercell.

### Extended Data Fig. 3 *α*^{2}*F*(*ω*) values for the \({\boldsymbol{Fm}}\overline{3}{\boldsymbol{m}}\) phase of LaH_{10} and LaD_{10}.

Calculated *α*^{2}*F*(*ω*) values for different pressures together with the integrated electron–phonon coupling constant, which is defined as \(\lambda (\omega )=2{\int }_{0}^{\omega }{\rm{d}}\varOmega {\alpha }^{2}F(\varOmega )/\varOmega \). The results show that high frequency, optical modes of hydrogen are responsible for the large value of the electron–phonon coupling constant *λ*. It is worth noting that acoustic modes with La character contribute between 0.2 and 0.5 to *λ* and cannot be neglected when aiming to estimate an accurate value of *T*_{c}.

### Extended Data Fig. 4 Details of the \({\boldsymbol{R}}\overline{3}{\boldsymbol{m}}\) LaH_{10} cell relaxation, including quantum effects.

The initial point for the relaxation is the output from the previous internal relaxation with fixed angle presented in Extended Data Fig. 5. The \(R\bar{3}m\) phase in the rhombohedral description is described by three vectors of the same length (**a** = **b** = **c**) and by the angles between them (α = *β* = *γ*). The top left panel shows the evolution of the rhombohedral angle and the top right panel shows the evolution of the rhombohedral lattice parameter (*a* = *b* = *c*). The progression of the stress tensor in the quantum SSCHA minimization is shown in the bottom left panel. It is clear that at the end of the minimization the structure has an angle of 60°, which matches the angle of an fcc lattice and, in this case, the stress is isotropic. In the bottom right panel, we show the evolution of the Wyckoff positions in the minimization and we compare it with that of the \(Fm\bar{3}m\)phase. The occupied Wyckoff positions for both \(R\bar{3}m\) LaH_{10} and \(Fm\bar{3}m\) LaH_{10} are summarized in Extended Data Table 2. Here, the evolution of *ε*_{a}, *ε*_{b}, *ε*_{x} and *ε*_{y} parameters in the minimization can be seen. The atoms in the first set of 6*c* positions approach the 8*c* Wyckoff site of the \(Fm\bar{3}m\)phase, whereas the atoms in the second set of 6*c* positions and those in 18*h* sites approach the atoms in the 32*f* Wyckoff site of the \(Fm\bar{3}m\) phase, where *ε* = 0.12053.

### Extended Data Fig. 5 Phonon dispersion in \({\boldsymbol{R}}\overline{3}{\boldsymbol{m}}\)-phase LaH_{10} and the anisotropic pressure created in a fixed-cell quantum relaxation.

Left, harmonic and anharmonic phonon spectrum, maintaining a 62.3° rhombohedral angle. The harmonic calculation is performed with the internal atomic positions that yield classical vanishing forces. The anharmonic calculation is performed after relaxing (with the SSCHA) the internal degrees of freedom but maintaining the 62.3° rhombohedral angle. At the harmonic level there are unstable phonon modes even at Γ. Symmetry prevents the relaxation of this structure according to the unstable phonon mode at Γ. The harmonic phonons are calculated at a classic pressure of 150 GPa. Quantum effects add around an extra 10 GPa to the pressure. To the right of the graph is shown the behaviour of *λ*(*ω*) and *α*^{2}*F*(*ω*) for the anharmonic calculation. Right, pressure along the different Cartesian directions during the SSCHA relaxation of the internal parameters, keeping the rhombohedral angle fixed at 62.3°. At step 0 the pressure reported is obtained directly from *V*(**R**), neglecting quantum effects. It is isotropic within 1 GPa of difference between the *x–y* and *z* directions. At each of the other steps it is calculated from the quantum \(E({\boldsymbol{ {\mathcal R} }})\) and along the minimization it becomes anisotropic. When the minimization stops at step 12—that is, the internal coordinates are at the minimum of the \(E({\boldsymbol{ {\mathcal R} }})\) for this lattice—the stress anisotropy between the *z* and the *x–y* directions is about 6%. This clearly indicates that quantum effects act to relax the crystal lattice—in particular, because *P*_{z} is larger—by reducing the rhombohedral angle. It is worth noting that quantum effects increase the total pressure by approximately 10 GPa, which is calculated as *P* = (*P*_{x} + *P*_{y} + *P*_{z})/3. The initial cell parameters before the minimization are *a* = 3.5473398 Å and *α* = 62.34158°. The initial values of the free Wyckoff parameters, which yield classical vanishing forces and a 150-GPa isotropic stress, are *ε*_{a} = 0.26043, *ε*_{b} = 0.09950, *ε*_{x} = 0.10746 and *ε*_{y} = 0.12810. See Extended Data Table 2 for more details.

### Extended Data Fig. 6 Anisotropic pressure of the *C*2 phase of LaH_{10} in a cell quantum relaxation.

Pressure along the different Cartesian directions is plotted during the SSCHA cell minimization. The target pressure for this minimization is 160 GPa. At the end of the minimization the isotropy of the stress tensor is recovered. A symmetry analysis performed on the structure at the end of the minimization confirms that the *C*2 LaH_{10} evolves in the \(Fm\bar{3}m\)-phase LaH_{10}. The initial values *P*_{x} = 163.2 GPa, *P*_{y} = 159.7 GPa, *P*_{z} = 155.0 GPa are obtained by an atomic internal relaxation performed using the SSCHA with a fixed cell.

### Extended Data Fig. 7 SSCHA minimization on LaH_{10} and DOS.

Top left and top right, two initial structures (*C*2 and \(R\bar{3}m\)) of low enthalpy that were considered in our SSCHA simulations. When considering quantum effects, both structures evolve towards the \(Fm\bar{3}m\) structure. The corresponding total electronic DOS at different pressures is plotted for each structure (for comparison, at the same energy scale). The highly symmetric motif \((Fm\bar{3}m)\) maximizes \({N}_{{E}_{F}}\), whereas in distorted structures (\(R\bar{3}m\) and *C*2) the occupation at the Fermi level is reduced by more than 23% for *C*2 and by 11% for \(R\bar{3}m\) (with respect to \(Fm\bar{3}m\) at 150 GPa). Values at classical pressures are shown for comparison. Note that the shape of the DOS plot is also strongly modified at different pressures.

### Extended Data Fig. 8 Details of LaH_{11}.

Left, crystal structure of the *P*4/*nmm* phase of LaH_{11} at 100 GPa, which is thermodynamically stable in the convex hull. Top right, dispersion of harmonic phonons along the momentum space for LaH_{11}: it is dynamically stable. Bottom right, superconducting Eliashberg spectrum function (*α*^{2}*F*(*ω*)) calculated for LaH_{11} at the pressure indicated with harmonic phonons. The *T*_{c} estimated using the Allen–Dynes formula (*μ** = 0.1) is around 7 K at 100 GPa (harmonic phonons).

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Errea, I., Belli, F., Monacelli, L. *et al.* Quantum crystal structure in the 250-kelvin superconducting lanthanum hydride.
*Nature* **578, **66–69 (2020). https://doi.org/10.1038/s41586-020-1955-z

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