Introduction

Superconductivity at near room temperature has been discovered in clathrate polyhydrides at megabar pressures1,2,3,4,5,6,7. Determining the crystal structure responsible for the superconductivity is of critical importance, yet a great challenge8,9. Experimentally, difficulties in probing lighter elements arise in XRD, impeding a direct determination of the complete lattice symmetry. The measured Tc and its pressure dependence (dTc/dp) tighten constraints on structural models; however, calculations on the candidate structures based on Bardeen–Cooper–Schrieffer (BCS) theory10 unequivocally suggest a negative dTc/dp11,12,13,14,15,16,17,18,19,20, which contradicts the experimentally observed positive dTc/dp in some pressure regimes (Supplementary Fig. 1). This phenomenon was initially observed in the 250-kelvin superconducting lanthanum polyhydride1, and later in polyhydrides of yttrium6,7 and calcium21 with Tc reaching 257 K and 212 K, respectively. This ‘positive dTc/dp contradiction’ presents a major obstacle to our full understanding of the crystal structures of these superconducting polyhydrides.

As the first superconductor with a Tc above 250 K, lanthanum polyhydride has been studied by several groups independently1,2,3,4,5. In addition to the high Tc, a positive dTc/dp was observed between 137 and 150 GPa by measurements on four samples (samples 1–4 in Ref. 1) synthesized at different pressures with laser heating. XRD measurements determine that the lanthanum atoms form a face-centered cubic (fcc) lattice at pressures of 137–218 GPa, while the hydrogen atoms have undetermined locations within the lattice. Based on the measured crystal volume, the hydrogen-to-lanthanum (H/La) ratio around the maximum Tc was estimated to be 9.6 (150 GPa)1 or 9–11 (180–200 GPa)2,3 in the two studies, respectively. This trend was recently confirmed by new experiments on the fcc → C2/m phase transformation at pc = 135 GPa, which shows an even steeper decrease of Tc below pc5.

With a moderate synthetic pressure and high symmetry, ‘fcc’ lanthanum polyhydride provides a model system for exploring the structural nature of high-Tc hydrides in both experiment and theory. In fact, ab initio calculations have guided the experimental discovery of lanthanum polyhydrides, and predicted the appearance of high Tc superconductivity, i.e., an estimated Tc of 280-kelvin for fcc-LaH10 at 210 GPa11,12. This prediction turned out to be very close to the fcc lanthanum polyhydride later synthesized, with some differences in Tc, synthetic pressure, and hydrogen content. Recently, it has been theorized that the inclusion of quantum atomic fluctuation is essential for a correct calculation of Tc and the pressure boundary of fcc-LaH10, which is dictated by the quantum nature of lanthanum polyhydride structures16. However, a negative dTc/dp predicted at 137–150 GPa16 remains in contradiction to the experimental observations (Supplementary Fig. 1). This disagreement suggests that critical factors have been overlooked in previous calculations of superconductivity. In particular, the stoichiometric defect observed by experimental studies may play an important role in determining the structure and underlying superconductivity.

Results and discussion

Although minor in amount, defects in solid materials can strongly affect their properties. The thermodynamic stability of a defect in high-pressure solids as well as the relative stabilities of different defect structures can be evaluated by the defect formation enthalpy (Hf). As shown in Fig. 1a, the crystal structure of fcc-LaH10 is represented as the insertion of an ‘H cube’ into octahedral interstices of the fluorite-LaH2 structure. In this model, we calculated the Hf for a vacancy defect either at a corner of the H cube (VC) or at a tetrahedral interstice of the La lattice (VT), and found that the configurationally averaged Hf becomes negative below 158 GPa. This suggests that fcc-LaH10 is prone to vacancy defects, which agrees well with the experimental finding of hypostoichiometric LaH10-δ at 150 GPa specifically LaH9.61. Notably, the calculated ‘vacancy occurring region’ coincides the region where the Tc of the fcc lanthanum polyhydride depends positively on the pressure (which previous calculations have failed to predict), indicating that the vacancy structure has a key role to play in the superconductivity.

Fig. 1: Vacancy formation enthalpy and pressure-volume relation.
figure 1

a The formation enthalpy (Hf) of a single vacancy in fcc-LaH10 at two inequivalent lattice sites in the clathrate hydrogen framework, VT and VC, and their configurational average calculated at different pressures using 2 × 2 × 2 extension of conventional unit cell of fcc-LaH10. The locations of VT and VC are illustrated in a conventional unit cell of the fcc-LaH10 structure by red and black balls, respectively. b The experimental pressure-volume relations of fcc-LaH9.6 and fcc-LaD10 measured by Drozdov et al.1 compared to the theoretical values of fcc-LaH10, fcc-LaH9 and fcc-LaD10 derived from quantum simulations at constant pressure and temperature of 300 K. The volumes selected for subsequent quantum simulation (VCMD) for fcc-LaH9.63 are thereby linked to the ‘quantum’ pressure. Formula unit is abbreviated as f.u.

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A low concentration of vacancies in hydrogen sites does not change the XRD pattern of the fcc lanthanum polyhydride, which is determined primarily by the La sublattice. However, these vacancies significantly affect the dynamics of hydrogen in the crystal (Supplementary Fig. 2). The distortion in the hydrogen sublattice spreads out away from the vacancy sites in both the VC and VT models of LaH10 with an H/La ratio of 9.97, and even results in a ‘liquid-like’ H framework in the former at 150 GPa. For LaH10-δ, the presence of vacancies changes the potential energy surface, while quantum nuclear fluctuation is expected to affect the transport behavior of vacancies. The calculated zero-point energy (ZPE) of ~150 meV/LaH10 is comparable to the vacancy migration energy of 160/200 meV for the migration path of VC → VC’/VC → VT at 150 GPa. This suggests that nuclear zero-point motion can promote proton hopping to neighboring vacancies and cause the LaH10-δ ground state to not be a single structure around which the atoms vibrate, but instead one where the protons dynamically explore different vacancies in a fixed fcc-La framework.

We refer to this behavior as being ‘fluxional’ in the sense used by Goncharov et al. for phase IV of solid hydrogen22. It is well-established by experiment and theory that thermal effects dominate the fluxionality in this structure: A classically ordered phase III replaces the fluxional phase IV at low temperature23,24,25. Here, we demonstrate that, in the presence of vacancies, hydrogen NQEs are strong enough—and indeed essential—to stabilize a fluxional framework in the fcc lanthanum polyhydride. Therefore, we are dealing with a quantum fluxional structure (QFS).

Taking LaH9.63 as an example, we investigate the NQE and thermal effects on the structural fluxion at 150 GPa and low temperatures up to 240 K. The stoichiometry is selected according to the experimental estimation (LaH9.6). Our theoretical simulations suggest a H/La ratio of 9.54 (or 9.71) for the experimental samples synthesized at 150 GPa (or 137 GPa), in good agreement with experiments1 (Fig. 1b). We adopt ab initio centroid molecular dynamics (CMD)26 to treat the nuclei quantum mechanically. As shown by the mean square displacement (MSD) curves in Fig. 2a, the simulations reveal appreciable proton migration, with equivalent mean migration distances of vacancies reaching 1.0 to 2.0 Å between 60 and 240 K (Supplementary Fig. 3). These distances are almost as large as or larger than the shortest H-H separation around 1.2 Å. This is consistent with the ‘network’-like density distribution patterns obtained, for instance, at 60 K (Fig. 2b). In contrast, the diffusivity is much smaller in ab initio molecular dynamics (MD), where the nuclei are treated classically (Fig. 2a, c). This confirms that the structural fluxion in LaH9.63 is dominated by the NQE, rather than thermal effects. The crucial role of vacancies in this intriguing quantum phenomenon is demonstrated by a comparison to the dynamics of stoichiometric fcc-LaH10 at the same pressure, which exhibits no diffusion below 700 K27. Experimentally, the occurrence of ‘quenched-in’ vacancies in polyhydrides is inevitable, as a result of the thermal treatment of the samples. This finding therefore suggests that vacancy effects interact with the strong NQE to result in strong quantum structural fluxion in polyhydrides.

Fig. 2: Structural fluxion in LaH9.63 at 150 GPa.
figure 2

a The proton MSD derived from centroid trajectories of the CMD simulations and those of MD simulations. b The [100] view of the quantum nuclear density distribution at 60 K extracted from a CMD simulation, with full consideration of the 16 beads. Neighboring protons are illustrated in different colors. c the [100] view of the classical nuclear density distribution in 16 MD simulation runs of 4 ps distinguished by various proton colors. The 16 runs were initialized from different centroid configurations of the 4-picosecond CMD trajectory with a sampling interval of 0.25 ps.

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We approximate the proton diffusion coefficient D in LaH9.63 from the slope of the MSD in simulations at 240 K and between 137 to 176 GPa. The temperature is chosen to be close to the measured Tc of the fcc lanthanum polyhydride. As shown in Fig. 3a, the D value falls in the range of 10−6 to 10−7 cm2/s in our quantum simulation, whereas it is significantly lower in the classical simulation. The D value is 2–3 orders of magnitude below the threshold criterion for classical superionicity (~10−4 cm2/s) for freely diffusing protons28. This notwithstanding, it reaches or indeed exceeds the upper bound diffusivities observed in interstitial hydrides at room temperature, such as 3.8 × 10−7 cm2/s in fcc-Cu2H29, and approaches that of phase IV hydrogen30. With increasing pressure, D decreases in interstitial hydrides (e.g. Cu2H29 and FeH31), due to the contraction of the metal lattice and the resulting increase of the activation energy for proton hopping. In phase IV hydrogen, on the other hand, D increases with pressure30, probably owing to the drastic increase of the proton hopping rate with the shortened of the distance. Our preliminary results suggest that the unique ‘host-guest’ structure of LaH9.63 likely induces a significant competition between the two mechanisms mentioned, resulting in a nonmonotonic pressure trend of the coefficient D.

Fig. 3: Pressure effects on the diffusivity and structural distortions.
figure 3

a The proton diffusion coefficient D derived from the MSD in CMD simulations (run1) of 4 ps, and from the average MSD of four MD simulations of 24 ps. The D in three additional CMD simulations of 4 ps at 150 GPa are also shown. These are two simulations at 120 K and 60 K, and a simulation using higher total-energy convergence criteria (run2). The MSD in the CMD simulations is derived from the centroid trajectories. The measured D in Cu2H29 and FeH31, and calculated D in phase IV hydrogen30 are shown for comparison. b The configurational distance ξ (with error bar indicating the standard deviation) of QFS from the crystal lattices of static fcc and C2m phases, and a triclinic P1 structure for H and La substructure. The P1 structure is built by scaling the lattice parameter a of a QFS sampled from the CMD simulation of LaH9.63 at 176 GPa.

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In LaH9.63, proton diffusion breaks the balance of internal Coulomb repulsions in the ‘H cube’ located in octahedral interstices of La sublattice, which therefore distorts the H sublattice. Lattice expansion promotes the distortion, as indicated by the heightening of the second coordination shell (peaked at ~1.85 Å) in the average radial distribution function [RDF, g(r), Supplementary Fig. 4], which is correlated to a smearing of the inter- and inner-cube H-H separations. Based on a configurational distance (denoted as ξ) of crystal fingerprint matrices32, which is sensitive to local structural changes, we parametrized the structural difference of the QFS relative to the crystal lattices of static fcc and C2/m phases, and a triclinic P1 structure mimicking the QFS, as shown in Fig. 3b by ξqf, ξqc and ξqp respectively. For the H substructure, we find ξqf > ξqc > ξqp, with the ξqf much beyond the other two at larger volumes, revealing an increasing distortion of the H sublattice in QFS from fcc to lower symmetry upon decompression. For the La substructure, ξqc > ξqf ≈ ξqp, with ξqc much larger than ξqf and ξqp. This agrees with the XRD measurements that determine an fcc sublattice for La atoms above 137 GPa1. The tiny distortions related to ξqf and ξqp are both within the uncertainty of refinements for the fcc phase in XRD studies (Supplementary Fig. 4). The results suggest a premature low-symmetry distortion of the H substructure upon volume expansion relative to the La substructure in the quantum fluxional LaH9.63.

Using the centroid configurations of CMD simulations at 240 K, i.e. the QFS, we calculated the electronic density of states at the Fermi level \(N\left({\epsilon }_{F}\right)\) in LaH9.63, which exhibits opposite pressure trends at pressure below and above 150 GPa (Fig. 4a). The fcc-LaH10 (quantum) crystal has a monotonously negative pressure dependence of \(N\left({\epsilon }_{F}\right)\) above 100 GPa12,16, and this prediction can be extended to LaH9.63 using the rigid-band model of the electronic structure by virtue of an artificial shift of \({\epsilon }_{F}\)33. The pressure dependences of \(N\left({\epsilon }_{F}\right)\) in the C2/m phase of LaH10 and LaH9.63 (within the rigid-band approximation) are both non-monotonic, similar to quantum fluxional LaH9.63, suggesting that pressure effects on d\(N\left({\epsilon }_{F}\right)\)/dp are much more significant in distorted structures compared to the high-symmetry fcc phase. The non-monotonic pressure trend of \(N\left({\epsilon }_{F}\right)\) in LaH9.63 is a statistical consequence of the QFS having an average fcc-La sublattice with diversely distorted H substructures. This trend cannot be attributed to a single classical configuration, but can roughly be reproduced by the P1 structure mimicking the QFS (Supplementary Fig. 5), which implies that the premature distortion of the H substructure (e.g. fcc → P1 symmetry) accounts for the sign change of d\(N\left({\epsilon }_{F}\right)\)/dp in the quantum fluxional LaH9.63.

Fig. 4: Pressure effects on electronic properties and superconductivity.
figure 4

a The pressure trend of the averaged electronic density of states at the Fermi level \(N\left({\epsilon }_{F}\right)\) in quantum fluxional LaH9.63 (with distributions and standard deviation of statics shown in Supplementary Fig. 5), and the pressure trend of \(N\left({\epsilon }_{F}\right)\) in static P1-LaH9.63, fcc-LaH10, and C2/m-LaH10. The \(N\left({\epsilon }_{F}\right)\) of LaH9.63 in fcc and C2/m phase estimated using the rigid-band approximation are also shown for comparison. b The pressure trend of Tc in quantum fluxional LaH9.63 calculated based on various Gaussian smearings (σ) of the phonon spectrum \(F\left(\omega \right)\), together with the values measured for LaH9.6 by Drozdov et al.1 and those derived from AD, Migdal–Eliashberg equations (ME), and superconducting DFT (SCDFT) for fcc-LaH10 by Errea et al.16. Fitted lines are shown to guide the sight.

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The analytic McMillan34 and Allen-Dynes (AD)35 formulas (see details in Supplemental Information) have played an important role in analyzing the mechanism of pressure-dependent superconductivity in high-Tc hydrides. In these formulas, the electron-phonon coupling (EPC) constant λ depends explicitly on characteristic parameters of both electrons and phonons, in addition to the average of the electron-phonon matrix elements, \(\langle {{\rm I}}^{2}\rangle\). Specifically, λ can be expressed as the product of \(\langle {{\rm I}}^{2}\rangle /M\) (denoted as β, with \(M\) being atomic mass) and \(N\left({\epsilon }_{F}\right)/\left({\bar{\omega }}_{2}^{2}\right)\) (denoted as ζ, with \({\bar{\omega }}_{2}\) being the second frequency moment of the Eliashberg function). ζ directly describes the competition between electron and phonon contributions. Analysis of literature data suggest that ζ decreases more than two times faster than β increases upon compression (Supplementary Table 1). Furthermore, it was found that dλ/dp plays a dominating role in determining the dTc/dp, despite the fact that the Tc also depends explicitly on \({\omega }_{\log }\) and \({\bar{\omega }}_{2}\)/\({\omega }_{\log }\) (with \({\omega }_{\log }\) being the logarithmic frequency moment of the Eliashberg function). These findings suggest two rules for single crystal phases of high-Tc hydrides: (I) phonons affect dTc/dp mainly through the parameter λ; and (II) dλ/dp is primarily determined by dζ/dp, rather than dβ/dp.

In view of ζ being proportional to \(N\left({\epsilon }_{F}\right)\), the pressure trend of \(N\left({\epsilon }_{F}\right)\) in LaH9.63 (Fig. 4a) suggests a sign change of dTc/dp at studied pressures. Since the quantum fluxional nature of LaH9.63 forbids a direct calculation of the Eliashberg function \({\alpha \left(\omega \right)}^{2}F\left(\omega \right)\) by density functional methods, a quantitative confirmation of this is unachievable yet. However, considering that the change is so significant that even qualitative estimations could provide insights, we study the sign of dTc/dp slope in LaH9.63 under two approximations, as the first step to access the problem: (I) calculating \({\omega }_{\log }\) and \({\bar{\omega }}_{2}\) based on the phonon spectrum \(F\left(\omega \right)\) of LaH9.63, combined with \({\alpha \left(\omega \right)}^{2}\) approximated by that of quantum fcc-LaH10; (II) approximating the β in LaH9.63 by that of the quantum fcc-LaH10. The rationale of such approximation is based on empirical rules, and the fact that quantum fluxional LaH9.63 largely retains the same local atomic environment as quantum fcc-LaH10 (Supplementary Fig. 6). The \(F\left(\omega \right)\) derived from the Fourier transform of the velocity autocorrelation functions in the CMD simulations at 240 K, as well as the approximated \({\alpha \left(\omega \right)}^{2}\) and \({\alpha \left(\omega \right)}^{2}F\left(\omega \right)\) are shown in Supplementary Fig. 7.

With this approach, we evaluated the pressure trend of Tc in LaH9.63 using the AD formula (with parameters listed in Supplementary Table 2). It does indeed exhibit a positive dTc/dp slope at 137–163 GPa, in qualitative agreement with experimental observation1 (Fig. 4b). Moreover, a negative dTc/dp slope is obtained at higher pressures in line with both experimental observations and theoretical results for fcc-LaH10 (Supplementary Fig. 1). The standard deviation of \(N\left({\epsilon }_{F}\right)\) (Supplementary Fig. 5) reveals fluctuations of the electron structure near the Fermi level, which may affect the Tc through parameters ζ and λ. However, considering that superconductivity of LaH9.6 is measured at a time scale far beyond that of the simulation, it is feasible to calculate the Tc by a averaged \(N\left({\epsilon }_{F}\right)\). Recently, the C2/m phase has been successfully prepared at 120 GPa by abrupt decompression of the fcc phase5. A positive dTc/dp slope measured in the subsequent compression was attributed to a boost in the Tc due to phonon softening. As illustrated in the literature12,16, the softening of phonons results in a negative dTc/dp slope for fcc-LaH10, in contrast to the experimental observations below 150 GPa1. However, the concept of ‘a higher Tc near structural instability’5 agrees well with our finding of a premature distortion of hydrogen substructure relative to an average fcc-La substructure in quantum fluxional LaH9.63, upon decompression to 137 GPa.

In summary, the present work illustrates in lanthanum polyhydride the coexistence of quantum proton fluxion and hydrogen-induced high-Tc superconductivity. A premature distortion of hydrogen substructure upon volume expansion and its impact on the superconductivity through modulating the electronic structure is revealed. The findings support the experimental argument concerning the hypostoichiometric nature of the superconducting samples, reveal the crucial role of vacancy defects in the structure, and provide a theoretical explanation for the positive dTc/dp slope of lanthanum polyhydride, with implications for understanding the structure and superconductivity of other hydrogen-rich materials. With significant progress of nuclear magnetic resonance techniques in diamond anvil cells, the measurement of proton mobility in metal hydrides is currently accessible at megabar pressures29,31. We anticipate that continuous experimental and theoretical studies of quantum fluxional structure of high-Tc hydrides will stimulate new theoretical tools for understanding the superconducting behavior of quantum fluxional materials that cannot be precisely described by a single underlying static structure.

Methods

The ab initio calculations were performed using the plane-wave pseudopotential method, as implemented in the Vienna ab initio simulation program (VASP)36, with the Perdew–Burke–Ernzerhof (PBE) Generalized Gradient Approximation (GGA) density functional37, and the bare ion Coulomb potential treated in the projector augmented wave (PAW) framework38. The vacancy formation enthalpy of fcc-LaH10 was calculated from vacancy structures of fcc-LaH10 and B2/n-H239, with enthalpies calculated using the third-order Birch–Murnaghan isothermal equation of state (EOS)40 from ab initio energy-volume relations, as implemented in the EOS code41. The vacancy migration energy was calculated by the climbing image nudged elastic band method (CI-NEB)42. The quantum nuclear dynamics were studied by path-integral molecular dynamics (PIMD) and centroid molecular dynamics (CMD)30, as implement in the PIMD code43. The superconductivity was analyzed using McMillan’s theory34 with the Tc estimated by the Allen-Dynes equations35. The crystal structures and MD trajectories were visualized by the VESTA44 and OVITO softwares45 respectively. More details are shown in the Supplementary Information.