Abstract
The quantum nature of the hydrogen lattice in superconducting hydrides can have crucial effects on the materialâs properties. Taking a detailed look at the dynamic stability of the recently predicted BaSiH8 phase, we find that the inclusion of anharmonic quantum ionic effects leads to an increase in the critical dynamical pressure to 20 GPa as compared to 5 GPa within the harmonic approximation. We identify the change in the crystal structure due to quantum ionic effects to be the main driving force for this increase and demonstrate that this can already be understood at the harmonic level by considering zero-point energy corrections to the total electronic energy. In fact, the previously determined critical pressure of kinetic stability pkin = 30 GPa still poses a stricter bound for the synthesizability of BaSiH8 and similar hydride materials than the dynamical stability and therefore constitutes a more rigorous and accurate estimate for the experimental realizability of these structures.
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Introduction
The discovery of high-temperature superconductivity in H3S at extreme pressures1 stimulated an intense hunt for novel hydride compounds with even higher critical temperatures (Tc), spearheaded by computational material discovery2,3,4,5,6. One prominent example is LaH10, which has been shown to superconduct up to temperatures of 265 K at pressures of ~190 GPa7,8.
While it is very tempting to continue searching for materials with record-breaking Tcâs9,10,11, lowering the required stabilization pressures is even more important in view of technological applications11,12,13,14,15,16,17,18,19,20.
In a recent paper15 some of us proposed a strategy to bring the stabilization pressures of high-Tc hydrides closer to ambient pressure, based on the concept of an optimized chemical precompression. In fact, we identified LaBH8, a hydride superconductor with a Tcâ>â100âK, dynamically stable at an unprecedentedly low pressure. In a follow-up work, we showed that other hydride superconductors with the same \(Fm\bar{3}m\) XYH8 structural template can be identified with even lower critical pressures of stability, such as SrSiH8 and BaSiH819. The latter is particularly interesting, as it remains dynamically stable down to 3 GPa. We note in passing that these estimates of dynamical stability were based on anharmonic frozen-phonon calculations.
All of the mentioned \(Fm\bar{3}m\) XYH8 compounds are thermodynamically stable only at pressures aboveâ~100âGPa, and metastable below. A conceivable route to synthesize these materials would hence be to obtain them at high pressures where they are thermodynamically stable, and quench them to lower pressures. Such a procedure has recently been successfully employed to realize the \(Fm\bar{3}m\) LaBeH8 at around 110 GPa with subsequent quenching down to a pressure of ~80 GPa21.
The standard criterion employed in literature to estimate how far a metastable phase can be quenched down in pressure is dynamical (phonon) stability. However, dynamical stability indicates only that a structure is in a local minimum of the potential energy surface. To estimate its actual lifetime (kinetic stability) one needs also to estimate the height of the barriers that separate the current minimum from other minima. In a previous work19, some of us introduced a rigorous method to assess the kinetic stability pressure pkin by explicitly calculating the energy barrier protecting the metastable \(Fm\bar{3}m\) structure from decomposition as a function of pressure, using the variable-cell nudged elastic band method22. For BaSiH8, for example, we found a pkin ofâ~30âGPa, significantly higher than the dynamical value pdynâ=â3âGPa.
It was argued in a recent work23 that quantum lattice effects treated within the stochastic self-consistent harmonic approximation (SSCHA) drastically increase the dynamical stabilization pressure pdyn for LaBH8 and it was further suggested that a similar increase in pdyn should be expected for other \(Fm\bar{3}m\) XYH8 hydrides.
To investigate this, we apply the SSCHA formalism to BaSiH8, which, so far, has the lowest pdyn among all \(Fm\bar{3}m\) XYH8 hydrides. In SSCHA, a major bottleneck is represented by the need to use large supercells and large numbers of randomly displaced structures if one wants to fully converge the calculation. We overcome this problem by employing machine-learned moment tensor potentials (MTP)24,25 that allow us to obtain total energies, forces, and stresses with density-functional-theory (DFT) accuracy but at a fraction of the computational cost26,27,28. To our knowledge, this work represents the first combination of MTPs with the SSCHA method. In addition, we introduce a method to discern the contribution of quantum ionic (QI) effects from those of anharmonic (anh) and phonon-phonon (ph-ph) effects.
We find that pdyn increases from 3 GPa to about 20 GPa within the SSCHA and that this rise can almost entirely be attributed to QI effects, with actual anharmonic and ph-ph effects playing only a subordinate role. In fact, the same crystal structure that minimizes the free energy within SSCHA can already be obtained at the harmonic level using DFT by including zero-point energies (ZPE).
Particularly, we demonstrate that even after including QI, anharmonic, and ph-ph effects within the framework of SSCHA, the actual limit of stability is still set by pkin (~30 GPa), as stated in our previous work19.
Results
Ab-initio machine-learned interatomic potentials
In the self-consistent harmonic approximation, the system of fully anharmonic and interacting lattice vibrations is mapped onto an auxiliary harmonic system and the free energy \({{{{{{{\mathcal{F}}}}}}}}\) of the full system is approximated by the minimum of the free energy of the auxiliary harmonic system29,30,31,32. In the SSCHA, this minimization is performed stochastically via Monte Carlo summation and importance sampling over several consecutive ensembles (populations) of a large number of individuals. Each individual here corresponds to a supercell structure with displaced atomic positions, where the supercell size determines the density of phonon wave vectors in the Brillouin zone33. More details are provided in the Method section, in Supplementary Method 2, and in refs. 34,35,36,37,38,39,40.
In practice, to calculate accurate phonon frequencies within SSCHA, in particular for slow-converging soft modes, one needs to consider population sizes of several ten or hundred thousands individuals. In addition, one also needs to converge the supercell size.
Doing this fully at a DFT level is computationally prohibitive, which is why we made use of MTPs in this work. For every pressure, MTPs were trained on DFT results of 50 structures randomly chosen out of the SSCHA random-displacement individuals in 2âÃâ2âÃâ2 supercells. We then validated the trained MTPs for all other individuals by comparing the total energies, forces and stress components. This validation is shown in Fig. 1, demonstrating the exceptional accuracy of the used MTPs (see Supplementary Fig. 2 for other pressures, as well as forces and stresses). As can be appreciated in this figure, the root-mean-squared error (RMSE) is below 1 meV/atom, i.e., at the same level as the error in DFT. The inset also shows that the potential energy surface of the slow-converging, T2g mode at Î is reproduced very nicely with the MTPs.
As a final validation, we compare the SSCHA phonon dispersions obtained using only DFT with those employing only MTPs (see Supplementary Fig. 3) and find very good agreement, with only minor differences in the T2g mode at Î and the Eg at X. To fully converge these modes within 1 meV, we increased the populations sizes within MTP-SSCHA up to 100,000 individuals compared to 10,000 for the DFT-SSCHA calculations.
The use of MTPs does not only substantially speed up the calculations (we found MTPs to be a factor of about 20,000 faster than DFT in the case of 2âÃâ2âÃâ2 supercells of BaSiH8), but also gives access to larger supercells. In this work, we performed additional SSCHA calculations using MTPs on nâÃânâÃân supercells with nâ=â1,â2,â3,â4 at all studied pressures. The convergence of the free energy, the structural parameters, and the phonon dispersions with respect to the supercell size is provided in Supplementary Figs. 4â6. An overview of all performed SSCHA runs is given in Supplementary Tab. 1. Unless stated otherwise, all SSCHA results presented in the following have been obtained with MTPs for 100,000 individuals in 4âÃâ4âÃâ4 supercells at 0 K.
Structural parameters and electronic dispersion
The \(Fm\bar{3}m\) phase of BaSiH8 has a face-centered cubic unit cell with 10 atoms in the primitive cell, where Ba and Si occupy Wyckoff 4a/b sites and the H occupy 32f sites. The eight H atoms form rhombicuboctahedral cages around the Ba atoms and cubic cages around the Si atoms. The structure has only two free parameters, namely the lattice constant a and the Wyckoff coordinate of the 32f sites x, defining the H-H distance dH-Hâ=â2aââ âx (side length of the cubic cage) and the H-Si distance \({d}_{{{{{{{{\rm{H-Si}}}}}}}}}=a\sqrt{3}\cdot x\) (half the space diagonal of the cubic cage).
Relaxing the structure within DFT to target pressures of 10, 20, 25, and 30 GPa, we obtained lattice constants between 6.5 Ã and 6.2 Ã , and H-Si distances of about 1.6 Ã , as shown in Table 1. An extensive list of the structural parameters from ambient pressure up to 100 GPa, as well as the fit to the BirchâMurnaghan equation of state can be found in Supplementary Fig. 1 and Supplementary Note 1.
Starting from the atomic positions obtained in DFT and the harmonic dynamical matrices obtained in density-functional perturbation theory (DFPT) calculations at each pressure, we performed constant-volume SSCHA relaxation calculations. The corresponding parameters, indicated by \(\tilde{x},\tilde{d}\), and \(\tilde{p}\), are reported in Table 1.
We observe an elongation of dH-H/Si of about 30âmà (2%) for all pressures and an increase in pressure of about 2 to 3 GPa, i.e., ~20% at 10 GPa and ~10% at 30 GPa. The change in atomic positions introduces only small changes in the electronic structure, as demonstrated in Fig. 2, where we compare the electronic bands and densities of states (DOS) for x and \(\tilde{x}\). The largest differences are found above and below the Fermi energy, whereas electronic bands and DOS at the Fermi energy, and hence the Fermi surface, remain essentially unchanged.
Phonon dispersions and lattice instability
Moving on, we evaluate and compare the DFPT and SSCHA phonon dispersions at all studied pressures, as shown in Fig. 3. Similar to the results for LaBH8 reported by Belli et al.23, we find that the high-energy optical modes are strongly renormalized to lower frequencies. In particular, a significant softening occurs for the threefold degenerate T2g mode at Î (harmonic values around 50 meV in Fig. 3), which becomes imaginary and indicates a (dynamic) lattice instability for lattice constants aâ>â6.323 Ã , corresponding to pâ=â20 GPa and \(\tilde{p}\) = 22.6 GPa. Thus, the inclusion of quantum lattice effects within the SSCHA shifts the dynamical stability pressure from the anharmonic frozen-phonon value pdynâ=â3 GPa to \({\tilde{p}}_{{{{{{{{\rm{dyn}}}}}}}}}\)â=â20 GPa (see Supplementary Fig. 7). This ~17 GPa difference is substantial, but considerably smaller than the ~40âGPa shift reported for LaBH823.
We want to note in passing that we also calculated the fourth order corrections to the phonon frequencies in SSCHA (cf. Supplementary Method 2) and find, in contrast to the work on LaBH823, but in accordance with other works employing SSCHA35,36,38,39,40, only minor differences to the results obtained only up to third order. The maximum phonon energy differences are on the order of about 1 meV for all pressures. The phonon dispersions in the 2âÃâ2âÃâ2 supercells with second (auxiliary), third, and fourth order terms for all studied pressures are shown in Supplementary Fig. 8.
Different effects contributing to frequency shifts
The observed changes in the phonon dispersions when employing SSCHA and the resulting different dynamical stabilization pressures result from a combination of several effects that are not included at the level of standard DFT and DFPT. These are most importantly the vibrational contributions of the ions to the free energy, phonon anharmonicity, and ph-ph interactions. In the following, we will present an attempt to disentangle and determine the importance of each of these effects for BaSiH8. Before doing so, however, we want to briefly touch upon terminologies around phonon anharmonicity. Anharmonicity gives rise to ph-ph interactions, but also to the phonon self-interaction of a single mode; effects that are sometimes collectively referred to as anharmonic effects. In a frozen-phonon approach, single-mode potentials are usually obtained without incorporating ph-ph interactions, and there anharmonicity then refers to any deviation from the idealized parabolic potential. For the sake of clarity, we mention both aspects explicitly in the subsequent discussion.
QI effects
First, we want to look at the contributions to the total energy originating from the so-called zero-point vibrations, i.e, vibrations of the ions around their equilibrium positions due to the quantum mechanical treatment of the nuclei, absent in the classical, clamped-nuclei picture41. In the BornâOppenheimer approximation, the total energy Etot[R] (at Tâ=â0âK) for ionic positions R is given by the sum of the internal electronic energy Eel[R] and the ZPE contributions of the nuclei EZP[R]. In most solids, EZP is much smaller than Eel and can be safely neglected. However, due to the small mass of H and the resulting high phonon frequencies in hydrides, the ZPE can become substantial and thus cause a modification of the equilibrium crystal structure.
At the harmonic level, the true ZPE can be approximated via \({E}_{{{{{{{{\rm{ZP}}}}}}}}}[{{{{{{{\bf{R}}}}}}}}]\approx {E}_{{{{{{{{\rm{ZP}}}}}}}}}^{{{{{{{{\rm{harm}}}}}}}}}[{{{{{{{\bf{R}}}}}}}}]=\int\nolimits_{0}^{\infty }{{{{{{{\rm{d}}}}}}}}\omega {\rho }_{{{{{{{{\bf{R}}}}}}}}}(\omega )\hslash \omega /2\), where ÏR(Ï) is the DFPT phonon density of states and âÏ/2 the ZPE of a quantum harmonic oscillator. At constant volume, the only free parameter in the \(Fm\bar{3}m\) structure is the Wyckoff parameter x, defining the H-Si distance, for which we have plotted Etot, Eel, and \({E}_{{{{{{{{\rm{ZP}}}}}}}}}^{{{{{{{{\rm{harm}}}}}}}}}\) relative to their respective values at the DFT minimum x in Fig. 4 for a lattice constant of a = 6.242 à . The results for other lattice constants, i.e., pressures, are provided in Supplementary Fig. 9. We also want to note at this point that for structures not in the DFT minimum, non-vanishing forces occur, at odds with the underlying harmonic approximation. In all the cases considered here, however, the individual atomic force components are still small enough within our computational setup to result in purely real frequencies and sufficiently accurate estimates for the ZPE (see Supplementary Figs. 11 and 12 for further details).
As can be appreciated in this figure, the inclusion of the ZPE, even at the harmonic level, shifts the position of the minimum of the total energy considerably and puts it almost exactly at the minimum position \(\tilde{x}\) predicted by the SSCHA. The differences in dH-Si between the SSCHA calculations and the ZPE analysis are, in fact, of the order of 1 mà , i.e., well within the observed stochastic noise in SSCHA. We want to note that the same is true for LaBH8 (see Supplementary Fig. 10). Furthermore, inclusion of \({E}_{{{{{{{{\rm{ZP}}}}}}}}}^{{{{{{{{\rm{harm}}}}}}}}}[{{{{{{{\bf{R}}}}}}}}]\) reduces the total energy at its minimum byâ~30 meV/uc compared to its value at the DFT minimum, agreeing very nicely with the result from SSCHA (~27 meV/uc).
This demonstrates the importance of QI effects of the light H ions on the dynamic stability of the hydride materials, and shows that the minimum structure from SSCHA can already be obtained at the level of harmonic ZPE corrections, at least for this class of materials.
Having established that the ZPE has a crucial effect on the structure, we investigate the effect of the changed structure on the phonon dispersions. In Fig. 5a, we present the harmonic dispersions for atomic positions defined by x and \(\tilde{x}\). We observe large differences for the high-energy optical modes above 150 meV, but also for the low T2g mode at Î. The energy shifts for these modes are between 15 and 25 meV.
Anharmonicity and ph-ph interaction effects
Having established the QI effects on the structure and the phonon dispersions, we now want to assess the contributions of phonon anharmonicity (anh) and ph-ph interactions. To do that, we perform a SSCHA calculation while keeping the ions fixed at the DFT equilibrium positions, thus qualitatively removing structural effects on the phonon frequencies. In that case, we obtain non-vanishing forces within SSCHA (cf. Supplementary Method 2). As the individual force components on the H atoms are still small (~150 meV/Ã in the cell with a = 6.242 Ã , for example), a fixed-ion calculation seems applicable for a qualitative insight. The phonon dispersions obtained from this calculation are presented in Fig. 5b, where we find that all H-dominated optical modes in the whole BZ experience a sizable frequency renormalization. It is worth noting that fixed-ions SSCHA calculations indicate the onset of dynamical instability just between 5 and 10 GPa (8.4 and 12.2 GPa, respectively, for the SSCHA pressure \(\tilde{p}\), see Supplementary Fig. 13), which is reasonably close to the frozen-phonon anharmonic (harmonic) pdyn = 3(5) GPa19.
Combining QI, anh, and ph-ph effects
As we made the attempt of separating the different contributions from QI effects, anharmonicity, and ph-ph interactions, it is tempting to combine the individual contributions to the phonon dispersion as a simple sum and compare it to the full SSCHA calculation. We approach this task in real space via adding the force constants Csumâ=âCharmâ+âÎCQIâ+âÎCanh/phph, where Charm is the force constant matrix for harmonic phonons of structure x, ÎCQI are the force constant contributions due to the structural changes based on including the ZPE (QI effects), and ÎCanh/phph are the force constant contributions to the anh and ph-ph interaction effects (see Supplementary Note 3).
The dispersions obtained from Csum are shown in Fig. 5c, where we also present as reference the phonon dispersions from a full SSCHA calculation. Again, we find very good agreement between these results, providing further, a posteriori justification and support for the qualitatively introduced separation Ansatz. In Table 2, we summarize the considered effects, methods, and the corresponding physical description of the ions.
Superconductivity
As BaSiH8 is potentially a very promising high-Tc superconductor, we also want to assess the implications of the above mentioned effects on its superconducting (SC) properties. To do that, we solved the anisotropic MigdalâEliashberg (ME) equations as implemented in EPW42 for the four cases in Table 2. Details about the calculation within EPW are provided in the Method section and in Supplementary Method 3, at this point we only want to highlight that for each case we used the corresponding force constants to compute the dynamical matrices, and computed the electronâphonon (ep) coupling matrix elements as the self-consistent first-order variation of the potential using the equilibrium positions as defined in Table 2. In Table 3, we summarize the obtained values for quantities characterizing the SC state, i.e., the ep coupling strength λ, the logarithmic average of the phonon frequencies \({\omega }_{\log }\), and the SC critical temperature Tc.
The corresponding Eliashberg spectral functions α2F(Ï) and the cumulative coupling strengths λ(Ï) are shown in Supplementary Fig. 14. We want to stress that the provided values for Tc are obtained by the solution of the full ME equations. The distribution of the SC gap function Îk indicates no change in the distinct two-gap shape calculated for the pure harmonic case19. The differences in \({\omega }_{\log }\), λ, and Tc are in the order of 10-15% except for the full SSCHA calculation, where we see a considerable increase in λ to almost double the harmonic value, but also a decrease in \({\omega }_{\log }\), compensating the enhancement of λ. The resulting Tc is increased from 84 to 94 K, showing that the full inclusion of all discussed effects results only inâ~10-15% change in Tc for BaSiH8.
Discussion
In this work, we study the effects of quantum lattice dynamics within the SSCHA framework on the structure and the dynamical stability of the \(Fm\bar{3}m\) phase of BaSiH8. The SSCHA structure relaxation suggests a 2% elongation of the H-H and H-Si bonds for the studied pressure range of 10 to 30 GPa (~30âmà ).
In the phonon dispersions, we find an overall softening of the high optical modes, as well as a dynamic lattice instability characterized by imaginary SSCHA phonon frequencies in the T2g mode at Î below 20 GPa, setting the estimate for the critical dynamical pressure to \({\tilde{p}}_{{{{{{{{\rm{dyn}}}}}}}}}\approx\) 20 GPa. We have further demonstrated the importance of QI effects over anharmonicity and ph-ph interactions, and found that the change in structure, and consequently in pressure, can already be understood by considering harmonic ZPE corrections to the total electronic energy of the system alone (which can be obtained much faster than performing a full SSCHA calculation, cf. Supplementary Note 4). We want to stress at this point that while the total energy ground-state can be obtained already within the harmonic theory, quantum ionic effects and anharmonicity have to be taken into account to accurately determine the corresponding dynamical critical pressure.
We are now left with the question: what is the stability boundary of \(Fm\bar{3}m\)-BaSiH8? In our previous work on BaSiH819, we challenged the common practice of assuming the range of metastability of high-pressure hydride phases to coincide with the range of (an)harmonic dynamical stability, which systematically underestimates the stabilization pressures needed to synthesize these materials in reality43,44,45,46. Dynamical stability is only a prerequisite for thermodynamic metastability, which is characterized by the existence of a distinctive enthalpy barrier that protects a metastable phase from decomposition into other phases (kinetic stability). In our previous work19, we calculated the enthalpy transition path to the thermodynamic groundstate at different pressures (corresponding to a decomposition of the \(Fm\bar{3}m\) BaSiH8 phase into BaSiH6 + H2 in molecular form), and could estimate the barrier height from the intermediate structures (cf. Supplementary Fig. 16, where we also report the minor influence of ZPE on the determined barrier heights). In combination with the calculated convex hulls for the B-S-H system, we can argue with confidence that the \(Fm\bar{3}m\) BaSiH8 phase could be synthesized above 100 GPa, and retained down toâ~30 GPa, where a distinctive enthalpy barrier still exists. At lower pressures, metastable \(Fm\bar{3}m\) BaSiH8 will decompose, even though (anharmonic) lattice dynamics calculations predict it to be stable. Hence, kinetic stability poses a stricter bound for synthesizability than dynamical stability.
In conclusion, employing ab-initio machine-learned MTPs, we were able to perform SSCHA calculations for BaSiH8 at various pressures for supercells up to 4âÃâ4âÃâ4 and more than 100,000 individuals. The inclusion of QI effects, anharmonicity, and ph-ph interactions within the SSCHA-framework increases the pressure of dynamical stability from pdynâââ3 GPa to \({\tilde{p}}_{{{{{{{{\rm{dyn}}}}}}}}}\approx\) 20âGPa. We identified the change in structure due to QI effects to be the main driving force here, something that can already be captured to good approximation at the level of harmonic zero-point energy corrections.
Most importantly, the determined \({\tilde{p}}_{{{{{{{{\rm{dyn}}}}}}}}}\approx\) 20 GPa is still below pkinâââ30 GPa posed by the concept of kinetic stability, thus the latter represents a much stricter bound for the stability and realizability in these materials.
Methods
DF(P)T calculations
All DFT and DFPT calculations of electronic and vibrational properties were carried out using the plane-wave pseudopotential code QUANTUM ESPRESSO47, scalar-relativistic optimized norm-conserving Vanderbilt pseudopotentials48, and the PBE-GGA exchange and correlation functional49. The unit cell calculations are done in the face-centered cubic primitive unit cell with 10 atoms, a 12âÃâ12âÃâ12 k-grid, and a plane-wave cutoff energy of 80 Ry. The 2âÃâ2âÃâ2 supercell calculations were done on a 6âÃâ6âÃâ6 k-grid. Further details are provided in Supplementary Method 1.
SSCHA calculations
The calculations in the SSCHA are done in the constant-volume relaxation mode, i.e., minimizing the free energy with respect to the average atomic positions \({{{{{{{\mathcal{R}}}}}}}}\) and the force constants Φ, as implemented in the SSCHA python package40. We use the DFT equilibrium atomic positions and the DFPT dynamical matrices on a 2âÃâ2âÃâ2 q-grid as initial guesses for \({{{{{{{\mathcal{R}}}}}}}}\) and Φ, respectively. The starting point for the larger supercells is obtained by interpolating the previously converged auxiliary dynamical matrices.
The total energies, forces, and stress tensors for the individuals are obtained from DFT calculations or from machine-learned interatomic potentials in the framework of MTPs, see below. At the end of a minimization run, a new population with higher number of individuals N is generated from the minimized trial density matrix until convergence. We set two stopping criteria for the minimization loops: a KongâLiu ratio for the effective sample size of 0.2, and ratio of <10â7 between the free energy gradient with respect to the auxiliary dynamical matrix and its stochastic error. In calculations based on DFT we increased N up to 104 individuals, for the MTP cases up to 105.
The anharmonic phonon dispersions are obtained from the positional free-energy Hessians without the fourth-order term, if not specified otherwise explicitly. The final atomic positions are obtained from the converged average atomic positions \({{{{{{{\mathcal{R}}}}}}}}\) and the pressure as derivative of the converged free energy with respect to a strain tensor. The free energy difference between the last two populations in the 2âÃâ2âÃâ2 cells is well below 1 meV/uc, and well below 0.1 meV/uc for higher cells. The total forces in the last population are well below 10â6âmeV/à . The physical phonon frequency differences between the last two populations are below 5 meV for the DFT cases and well below 1 meV for the MTP case (T2g and A2u converge slower, see Supplementary Fig. 6). All calculations were carried out at zero temperature. We note in passing that with these settings, we could reproduce all LaBH8 results from Belli et al.âs work23.
Further details are provided in Supplementary Method 2 and details on convergence of the free energy, its gradients, and the auxiliary frequencies in a SSCHA calculation are shown in Supplementary Fig. 15.
Moment tensor potentials
The MTPs were trained and evaluated using the MLIP package24,50. We choose a functional form of level 26, eight radial basis functions, Rcut = 5.0 à and \({R}_{\min }\) = 1.2âà , and trained on 50 structures in a 2âÃâ2âÃâ2 supercell randomly chosen out of all individuals of the DFT SSCHA calculations. We trained separate MTPs for each set of pressure to ensure the highest possible accuracy of our calculations. We validated the potentials on all individuals in the DFT SSCHA calculations and find a RMSE on the total energy of 0.5â0.6 meV/atom, 45â50 meV/à for the force components, and 0.3â0.4 GPa for the diagonal stress tensor components. We further validated the MTPs on 30 randomly chosen individuals in a 3âÃâ3âÃâ3 supercell and achieve similar RMSEs. The validations and RMSEs for each pressure are shown in Supplementary Fig. 2 and Supplementary Note 2.
ZPE and total energy
The internal electronic energy Eel[R] is obtained from DFT calculations at fixed volume by varying the H-Si distance via the Wyckoff parameter x of the H positions. The phonon density of states ÏR(Ï) is obtained using DFPT on a 2âÃâ2âÃâ2 q-grid, interpolated on a 16âÃâ16âÃâ16 q-grid. Smooth ZPE and total energy curves are obtained by second-order polynomial fits in the H-Si distance. Due to the shift out of the DFT equilibrium structure, forces on the individual H atoms arise at the DFT level. Around the total energy minimum, the force components are in the order of 150 meV/à , i.e., small enough to warrant the use of linear-response theory to gain qualitative and systematic insights. DFT diagonal stress tensor components (pressures) are decreased by about 2 GPa in the shifted structures around the total energy minimum. The small non-zero forces do not lead to imaginary phonon frequencies in any DFPT calculation for BaSiH8 with varied H-Si distance.
Note on similarities to the quasi-harmonic approximation (QHA): Within the QHA, the temperature-dependent internal vibrational energy and the vibrational entropy are commonly added to a system to study temperature effects. The standard QHA only varies external coordinates, such as the volume, but keeps atomic positions in the DFT minimum. In our approach to QI effects, we vary explicitly the internal coordinates at fixed volume, instead. Incidentally, the full optimization of the quasi-harmonic free energy with respect to all degrees of freedom within QHA has just been reported51.
ME theory
The Wannier interpolation of the ep matrix elements onto dense k- and q-grids, and the subsequent self-consistent solution of the fully anisotropic ME equations were done in EPW42,52, for all the cases in Table 3. We used coarse 6âÃâ6âÃâ6 and fine 30âÃâ30âÃâ30 k- and q-grids, a Matsubara frequency cutoff of 1 eV, and a MorelâAnderson pseudopotential μ*â=â0.10. The ep coupling strength \(\lambda =2\int\nolimits_{0}^{\infty }{{{{{{{\rm{d}}}}}}}}\omega \frac{{\alpha }^{2}F(\omega )}{\omega }\) and the logarithmic average phonon frequency \({\omega }_{\log }=\exp \left(\frac{2}{\lambda }\int\nolimits_{0}^{\infty }{{{{{{{\rm{d}}}}}}}}\omega \frac{{\alpha }^{2}F(\omega )\ln \omega }{\omega }\right)\) are obtained from the Eliashberg spectral function α2F(Ï). Further details are provided in Supplementary Method 3.
Data availability
The authors confirm that the data supporting the findings of this study are available within the article and its Supplementary Materials. Further information is available upon request.
References
Drozdov, A. P., Eremets, M. I., Troyan, I. A., Ksenofontov, V. & Shylin, S. I. Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system. Nature 525, 73 (2015).
Boeri, L. & Bachelet, G. B. Viewpoint: the road to room-temperature conventional superconductivity. J. Phys. Condens. Matter 31, 234002 (2019).
Lilia, B. et al. The 2021 room-temperature superconductivity roadmap. J. Phys. Condens. Matter 34, 183002 (2022).
Flores-Livas, J. A. et al. A perspective on conventional high-temperature superconductors at high pressure: methods and materials. Phys. Rep. 856, 1 (2020).
Hilleke, K. P., Bi, T. & Zurek, E. Materials under high pressure: a chemical perspective. Appl. Phys. A 128, 441 (2022).
Hilleke, K. P. & Zurek, E. Tuning chemical precompression: theoretical design and crystal chemistry of novel hydrides in the quest for warm and light superconductivity at ambient pressures. J. Appl. Phys. 131, 070901 (2022).
Drozdov, A. P. et al. Superconductivity at 250 K in lanthanum hydride under high pressures. Nature 569, 528 (2019).
Somayazulu, M. et al. Evidence forsuperconductivity above 260 K in lanthanum superhydride at megabar pressures. Phys. Rev. Lett. 122, 027001 (2019).
Peng, F. et al. Hydrogen clathrate structures in rare earth hydrides at high pressures: possible route to room-temperature superconductivity. Phys. Rev. Lett. 119, 107001 (2017).
Grockowiak, A. D. et al. Hot hydride superconductivity above 550 K. Front. Electron. Mater. 2, 837651 (2022).
Di Cataldo, S., von der Linden, W. & Boeri, L. First-principles search of hot superconductivity in La-X-H ternary hydrides. npj Comput. Mater. 8, 2 (2022).
Pickard, C. J., Errea, I. & Eremets, M. I. Superconducting hydrides under pressure. Annu. Rev. Condens. Matter Phys. 11, 57 (2020).
Lv, J., Sun, Y., Liu, H. & Ma, Y. Theory-orientated discovery of high-temperature superconductors in superhydrides stabilized under high pressure. Matter Radiat. Extremes 5, 068101 (2020).
Di Cataldo, S., Von Der Linden, W. & Boeri, L. Phase diagram and superconductivity of calcium borohyrides at extreme pressures. Phys. Rev. B 102, 014516 (2020).
Di Cataldo, S., Heil, C., von der Linden, W. & Boeri, L. LaBH8: towards high-Tc low-pressure superconductivity in ternary superhydrides. Phys. Rev. B 104, L020511 (2021).
Shipley, A. M., Hutcheon, M. J., Needs, R. J. & Pickard, C. J. High-throughput discovery of high-temperature conventional superconductors. Phys. Rev. B 104, 054501 (2021).
Zhang, Z. et al. Design principles for high-temperature superconductors with a hydrogen-based alloy backbone at moderate pressure. Phys. Rev. Lett. 128, 047001 (2022).
Liang, X. et al. Prediction of high-Tc superconductivity in ternary lanthanum borohydrides. Phys. Rev. B 104, 134501 (2021).
Lucrezi, R., Di Cataldo, S., von der Linden, W., Boeri, L. & Heil, C. In-silico synthesis of lowest-pressure high-Tc ternary superhydrides. npj Comput. Mater. 8, 119 (2022).
Sun, Y., Sun, S., Zhong, X. & Liu, H. Prediction for high superconducting ternary hydrides below megabar pressure. J. Phys. Condens. Matter 34, 505404 (2022).
Song, Y. et al. Stoichiometric ternary superhydride LaBeH8 as a new template for high-temperature superconductivity at 110 K under 80 GPa. Phys. Rev. Lett. 130, 266001 (2023).
Qian, G.-R. et al. Variable cell nudged elastic band method for studying solid-solid structural phase transitions. Comput. Phys. Commun. 184, 2111 (2013).
Belli, F. & Errea, I. Impact of ionic quantum fluctuations on the thermodynamic stability and superconductivity of LaBH8. Phys. Rev. B 106, 134509 (2022).
Shapeev, A. V. Moment tensor potentials: a class of systematically improvable interatomic potentials. Multiscale Model. Simul. 14, 1153 (2016).
Gubaev, K., Podryabinkin, E. V., Hart, G. L. & Shapeev, A. V. Accelerating high-throughput searches for new alloys with active learning of interatomic potentials. Comput. Mater. Sci. 156, 148 (2019).
Ranalli, L. et al. Temperature-dependent anharmonic phonons in quantum paraelectric KTaO3 by first principles and machine-learned force fields. Adv. Quantum Technol. 6, 2200131 (2023).
Pedrielli, A. et al. Understanding anharmonic effects on hydrogen desorption characteristics of MgnH2n nanoclusters by ab initio trained deep neural network. Nanoscale 14, 5589 (2022).
Zuo, Y. et al. Performance and cost assessment of machine learning interatomic potentials. J. Phys. Chem. A 124, 731 (2020).
Born, M. in Festschrift zur Feier des Zweihundertjährigen Bestehens der Akademie der Wissenschaften in Göttingen: I. Mathematisch-Physikalische Klasse 1â16 (Springer Berlin Heidelberg, 1951).
Hooton, D. J. LI. A new treatment of anharmonicity in lattice thermodynamics: I. Lond. Edinb. Dublin Philos. Mag. J. Sci. 46, 422 (1955).
Koehler, T. R. Theory of the self-consistent harmonic approximation with application to solid neon. Phys. Rev. Lett. 17, 89 (1966).
Werthamer, N. R. Self-consistent phonon formulation of anharmonic lattice dynamics. Phys. Rev. B 1, 572 (1970).
Errea, I., Calandra, M. & Mauri, F. Anharmonic free energies and phonon dispersions from the stochastic self-consistent harmonic approximation: application to platinum and palladium hydrides. Phys. Rev. B 89, 064302 (2014).
Errea, I., Calandra, M. & Mauri, F. First-principles theory of anharmonicity and the inverse isotope effect in superconducting palladium-hydride compounds. Phys. Rev. Lett. 111, 177002 (2013).
Bianco, R., Errea, I., Paulatto, L., Calandra, M. & Mauri, F. Second-order structural phase transitions, free energy curvature, and temperature-dependent anharmonic phonons in the self-consistent harmonic approximation: theory and stochastic implementation. Phys. Rev. B 96, 014111 (2017).
Ribeiro, G. A. S. et al. Strong anharmonicity in the phonon spectra of PbTe and SnTe from first principles. Phys. Rev. B 97, 014306 (2018).
Monacelli, L., Errea, I., Calandra, M. & Mauri, F. Pressure and stress tensor of complex anharmonic crystals within the stochastic self-consistent harmonic approximation. Phys. Rev. B 98, 024106 (2018).
Aseginolaza, U. et al. Phonon collapse and second-order phase transition in thermoelectric SnSe. Phys. Rev. Lett. 122, 075901 (2019).
Errea, I. et al. Quantum crystal structure in the 250-kelvin superconducting lanthanum hydride. Nature 578, 66 (2020).
Monacelli, L. et al. The stochastic self-consistent harmonic approximation: calculating vibrational properties of materials with full quantum and anharmonic effects. J. Phys. Condens. Matter 33, 363001 (2021).
Born, M. & Oppenheimer, R. Zur quantentheorie der molekeln. Ann. Phys. 389, 457 (1927).
Poncé, S., Margine, E., Verdi, C. & Giustino, F. EPW: electron-phonon coupling, transport and superconducting properties using maximally localized Wannier functions. Comput. Phys. Commun. 209, 116 (2016).
Heil, C., di Cataldo, S., Bachelet, G. B. & Boeri, L. Superconductivity in sodalite-like yttrium hydride clathrates. Phys. Rev. B 99, 220502 (2019).
Kong, P. et al. Superconductivity up to 243 K in the yttrium-hydrogen system under high pressure. Nat. Commun. 12, 5075 (2021).
Errea, I. et al. Quantum hydrogen-bond symmetrization in the superconducting hydrogen sulfide system. Nature 532, 81 (2016).
Einaga, M. et al. Crystal structure of the superconducting phase of sulfur hydride. Nat. Phys. 12, 835 (2016).
Giannozzi, P. et al. Advanced capabilities for materials modelling with Quantum ESPRESSO. J. Phys. Condens. Matter 29, 465901 (2017).
Hamann, D. R. Optimized norm-conserving Vanderbilt pseudopotentials. Phys. Rev. B 88, 085117 (2013).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Novikov, I. S., Gubaev, K., Podryabinkin, E. V. & Shapeev, A. V. The MLIP package: moment tensor potentials with MPI and active learning. Mach. Learn. Sci. Technol. 2, 025002 (2020).
Masuki, R., Nomoto, T., Arita, R. & Tadano, T. Full optimization of quasiharmonic free energy with an anharmonic lattice model: application to thermal expansion and pyroelectricity of wurtzite GaN and ZnO. Phys. Rev. B 107, 134119 (2023).
Margine, E. R. & Giustino, F. Anisotropic Migdal-Eliashberg theory using Wannier functions. Phys. Rev. B 87, 024505 (2013).
Acknowledgements
We thank Pedro Pires Ferreira, Carla Verdi, and Luigi Ranalli for fruitful discussions. This work was supported by the Austrian Science Fund (FWF), projects P30269 and P32144. Calculations have been performed at the dCluster of TU Graz, as well as on the Vienna Scientific Cluster (VSC4 and VSC5). L.B. acknowledges support from Fondo Ateneo-Sapienza 2018-2022. S.D.C. acknowledges computational resources from CINECA, proj. IsC90-HTS-TECH_C and IsC99-ACME-C.
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R.L., E.K., and S.D.C. performed the calculations, M.A. introduced the idea of MTP, and C.H. and L.B. conceived and supervised the project. All authors contributed to the discussion of the results and participated in preparing the manuscript.
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Lucrezi, R., Kogler, E., Di Cataldo, S. et al. Quantum lattice dynamics and their importance in ternary superhydride clathrates. Commun Phys 6, 298 (2023). https://doi.org/10.1038/s42005-023-01413-8
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DOI: https://doi.org/10.1038/s42005-023-01413-8
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