LaBH_{8}: the first high-T_{c} low-pressure superhydride
Simone Di Cataldo, Christoph Heil, Wolfgang von der Linden, Lilia Boeri
LLaBH : the first high- T c low-pressure superhydride Simone Di Cataldo,
1, 2, ∗ Christoph Heil, Wolfgang von der Linden, and Lilia Boeri † Institute of Theoretical and Computational Physics,Graz University of Technology, NAWI Graz, 8010 Graz, Austria Dipartimento di Fisica, Sapienza Università di Roma, 00185 Roma, Italy
In the last five years a large number of new high-temperature superconductors have been predictedand experimentally discovered among hydrogen-rich crystals, at pressures which are way too highto meet any practical application. In this work, we report the computational prediction of a hydridesuperconductor, LaBH , with a T c of 126 K at a pressure of 50 GPa, thermodynamically stableabove 100 GPa, and dynamically stable down to 40 GPa, an unprecedentedly low pressure for high- T c hydrides. LaBH can be seen as a ternary sodalite-like hydride, in which a metallic hydrogensublattice is stabilized by the chemical pressure exerted by the guest elements. The combination oftwo elements with different atomic sizes in LaBH realizes a more efficient packing of atoms thanin binary sodalite hydrides. A suitable choice of elements may be exploited to further reduce thestabilization pressure to ambient conditions. The discovery of high-temperature superconductivity(HTSC) at 203 K in sulfur hydride at a pressure of 200GPa rekindled the dream of achieving room-temperaturesuperconductivity [1–3], triggering a hydride rush [4–11]which culminated in the report of superconductivity witha critical temperature (T c ) of 287 K (15 ◦ C) at 267 GPain a carbonaceous sulfur hydride [12]. The first discoveryof a room-temperature superconductor set a major mile-stone in the history of condensed matter physics, but theexceptional pressure required to stabilize the supercon-ducting phase thwarts any practical application.Obviously, the next challenge for materials research isto find materials exhibiting comparable superconductingproperties at, or close to, ambient pressures. Also in thiscase hydrides, which realize the requirements for conven-tional HTSC, are a promising hunting ground.In the five years following the SH discovery, all pos-sible combinations of X H n binary hydrides have beencomputationally explored in an effort to achieve room-temperature superconductivity; these studies revealedthat the formation, stability and superconducting prop-erties of these high-pressure (HP) hydrides strongly de-pend on the ionic size, electronegativity and electronicconfiguration of the X elements. High-T c superconduc-tors are found either among covalent hydrides, in which X and H form directional bonds driven metallic by pres-sure, [13–15] or among alkali, alkaline-earth and rare-earth hydrides, which form sodalite structures, in whichthe X atoms do not directly bind to hydrogen but pro-vide a scaffold, that stabilizes a dense sponge-like hy-drogen lattice [16–20]. Most HTSC binary hydrides arepredicted to be stable above 150-200 GPa; a few are pre-dicted to survive down to 70 GPa, where they are onthe verge of a dynamical instability [18, 21]; uranium hy-drides are stable above 35 GPa, but do not exhibit HTSC[6, 22].Having exhausted all possible combinations of binaryhydrides, it is natural to extend the search for HTSCto multinary hydrides, where the addition of a third element other than hydrogen, enormously expands thephase space [23, 24]. A few works have already tried toexploit this additional flexibility to to raise the T c of high-pressure hydrides well beyond room temperature. [25, 26]In this work, we will demonstrate a strategy to bringthe stabilization pressure of high- T c superconducting hy-dride phases close to ambient pressure in a ternary hy-dride. In short, it consists of identifying a suitable combi-nation of elements with different sizes and electronegativ-ity. Our strategy is demonstrated by the prediction of anew ternary high-temperature superconductor, identifiedthrough a evolutionary search of the lanthanum-boron-hydrogen (La-B-H) phase diagram [27, 28]. This phase,with LaBH composition, exhibits a superconducting T c of 126 K at 50 GPa. It is a remarkable example of a ternary sodalite hydride , in which a cubic La-B scaffold-ing confines a highly-symmetric, dense hydrogen sublat-tice, and makes it stable down to moderate pressures.Ternary sodalite hydride phases have been predicted onlyat pressures above 170 GPa [25, 29], while LaBH appearson the ternary La-B-H hull at 110 GPa, and is dynami-cally stable down to 40 GPa.The new LaBH phase was identified through a struc-tural search at 100 GPa using variable-composition evo-lutionary crystal structure prediction, as implemented inUSPEX [27, 28], by sampling a total of over twelve thou-sand structures [30]. The ternary La-B-H convex hullwas then constructed, including also the zero-point lat-tice contribution to the total energy. At 100 GPa fourstable compositions lie on the convex hull: La(BH ) ,La(BH ) , LaBH , and LaBH . The first two exhibitcrystal structures analogous to those observed in othermetal borohydrides, i.e. molecular structures with BH − and BH − anions interspaced by La cations [24, 31–34],and are insulating. LaBH and LaBH are characterizedby the same La-B rocksalt sublattice. In LaBH , boronand hydrogen form a BH − tetrahedral anion, and an ad-ditional H atom is trapped at the center of a La tetra-hedron. For LaBH we predict a F m ¯3 m sodalite-like a r X i v : . [ c ond - m a t . s up r- c on ] F e b Figure 1. Crystal structure of the
F m ¯3 m phase of LaBH (conventional unit cell). La, B, and H atoms are shown asgreen, orange, and blue spheres, respectively. The ElectronLocalization Function (ELF) is projected onto the ¯100 plane. structure. Both LaBH and LaBH are metallic. Prelim-inary T c calculations showed that the LaBH structureexhibits a T c of 53 K at 50 GPa, whereas LaBH exhibitsa T c of 126 K at the same pressure [35]. These prelimi-nary results led us to focus on the much more promisingLaBH F m ¯3 m sodalite-like structure. F m ¯3 m LaBH lies only 23 meV/atom above the hull at100 GPa, becomes thermodynamically stable above 110GPa, and is dynamically stable down to 40 GPa. Thisimplies that this phase can be realistically synthesized bylaser heating at 110 GPa, and quenched at low pressuresdown to a minimum of 40 GPa. The crystal structure of F m ¯3 m -LaBH is shown in Fig. 1: La and B occupy b and a Wyckoff positions, respectively, while H atoms siton f sites with x = 0 . . The hydrogen sites occupythe vertices of a rhombicuboctahedron centered aroundLa atoms, and vertices of cubes around B atoms. In-terestingly, a structure with an identical M -B sublatticewas observed with neutron diffraction on M BH ( M =K, Rb, Cs), which only differs by the / occupancy ofthe f site by hydrogen [36].The Electron Localization Function (ELF) for LaBH ,shown in Fig. 1 along the ¯100 plane, has maxima aroundthe atoms, and along the H-H bonds, but not between Laand H or B and H, indicating that neither La nor B formbonds with H, but both act as spacers. The absence ofa B-H covalent bond is quite unusual for a borohydride,and implies that this structure is not a covalent hydride,like SH . Rather, it is reminiscent sodalite hydrides likeLaH , where a dense, metallic hydrogen sublattice isstabilized at pressures lower than the pure hydrogen at metallization pressure, due to the chemical pressure ex-erted by the host atoms. In a very simplified picture,one could see LaBH as a chemically-precompressed ver-sion of LaH (for a visual impression see Fig. S3 of theSM [37]). In fact, the two structures share the same La-La sublattice, with almost identical lattice parameters atall pressures; LaH is stable at the harmonic level onlyabove 200 GPa; in LaBH , boron atoms fill the voidsbetween the second-nearest La atoms and provide addi-tional chemical pressure, making the metallic hydrogensublattice stable down to 40 GPa. The analogy of LaBH with binary sodalite structures, which is confirmed bythe analysis of the electronic structure and vibrationalproperties, is ultimately at the heart of the HTSC at lowpressure. We also observe that at all pressures H-H in-teratomic distances are 13% larger than sodalite LaH ,and 20% larger than LaH [18] (see Fig. S5 of the SM[37]). In short, both the geometric and bonding proper-ties indicate that this structure is a natural extension ofthe concept of sodalite hydrides, to the case of a ternaryhydride [16, 17].In Fig. 2 we show the electronic band structure, alongwith the atom-projected density of states, calculated at50 GPa. Here and in the following, we will focus onthis pressure, which is sufficiently close to the moderate-pressure regime, but is about 10 GPa higher than thedynamical instability pressure, so that predictions arestill not dramatically affected by anharmonic effects. Aband structure formation analysis (see SM, Fig. S6 [37])reveals that the eight bands in the -15 to 2 eV range fromthe Fermi level derive from the eight quasi-free-electron-like bands of the empty H sublattice, which are onlyweakly perturbed by hybridization with the s − p boronstates, and more strongly by hybridization with the threeLa semi-core bands from -20 to -15 eV. A Bader chargeanalysis [38] predicts a net charge of +1.46 for La, +0.88for B, and -0.29 for each H, indicating that both boronand lanthanum donate charge to the hydrogen sublattice.Hence, the band structure analysis confirms the bond-ing picture observed in real space: H forms a dense sub-lattice, stabilized by the La-B scaffolding with which hy-drogen forms only weak bonds. The absence of a covalentB-H bond here is crucial, and explains the free-electron-like behavior of the hydrogen-derived electronic states.In fact, as a result of this weak hybridization electronicbands at the Fermi level are highly-dispersive and havean almost purely (70%) hydrogen character, exactly likebinary sodalite hydrides [18].In the left panel of Fig. 4 we show the Fermi surfacedecorated with H character. The Fermi surface is char-acterized by three sheets i) a large electron-like spherearound the Γ point, which has the greatest weight in thereciprocal space, ii) a cross-shaped sheet and a small holepocket around the X point, mostly of H character, and iii)a small hole pocket around the L point with mixed B andH character. Overall, the whole Fermi surface exhibits astrong hydrogen character, i.e. the partial H contributionto the DOS is never less than 50 %, on average around70%. Figure 2. Left panel: electronic band structure of
F m ¯3 m LaBH at 50 GPa, decorated with hydrogen character (blue)vs non-hydrogen character (gray). Right panel: atom-projected density of states in units of eV − spin − . Projectiononto La, B, and H is shown in green, orange, and blue, re-spectively. The zero of the energy corresponds to the Fermilevel. In order to compute the superconducting propertiesof the
F m ¯3 m -LaBH phase, we calculated the phonondispersions and the electron-phonon coupling using lin-ear response theory within the harmonic approximation,using Wannier interpolation on very fine (cid:126)k and (cid:126)q grids,as implemented in the EPW code [39–41]. In Fig. 3 weshow the phonon dispersions decorated with the par-tial electron-phonon ( e-ph ) coupling coefficients λ ν(cid:126)q ,together with the atom-projected Eliashberg function α F ( ω ) and the phonon density of states. The e-ph cou-pling is spread rather evenly on all optical branches, andis stronger for modes which involve vibrations of the hy-drogen sublattice, again in close analogy with other bi-nary sodalite hydrides [18, 42]. The high peak in theEliashberg function at 50 meV corresponds to a flat re-gion of the dispersion around the Γ point, which experi-ences a particularly strong e-ph coupling. This phononmode, named T ∗ g in the following, is characterized by a T g symmetry at the Γ point and corresponds to a dis-tortion of the tetrahedra formed by nearest-neighbours Hatoms, and carries around 15 % of the total e-ph coupling(See SM Fig. S13 for more details [37]). In addition, atriply degenerate branch with T u symmetry at Γ , acci-dentally quasi-degenerate with the T ∗ g mode at 50 GPa,is also notable. This branch, in fact, corresponds to arattling mode of the boron atoms inside the cubic hy-drogen cages surrounding it, and is mostly dispersionlessthroughout the Brillouin zone, coherently with the de-scription of boron as passively pressurizing the metallichydrogen sublattice, without bonding to it.Integrating the Eliashberg function we obtain the two moments: [43, 44] λ = 2 (cid:82) α F ( ω ) ω − dω = 1 . and ω log = exp (cid:2) λ − (cid:82) dωα F ( ω ) ω − log( ω ) (cid:3) meV. TableI reports the critical temperature obtained by a directsolution of ab-initio Migdal-Eliashberg equations, as im-plemented in the EPW code [41]. Coulomb effects areincluded via the Morel-Anderson pseudopotential µ ∗ = µ/ [1 + µ log( ω el /ω ph )] [45], with ω el and ω ph being char-acteristic energies for electrons (band-width of the Fermisurface electrons) and phonons (highest phonon energy),respectively. The double Fermi-surface average of thescreened Coulomb interaction µ was evaluated within therandom phase approximation [46–49]. We find a value of µ ∗ =0.09 at pressures of 50 and 100 GPa, close to the stan-dard values (0.10-0.14) assumed for most conventionalsuperconductors [18]. This rules out possible anomalouseffects of Coulomb repulsion which were suggested foryttrium sodalite hydride [5].The T c obtained from the fully anisotropic solution( T c =126 K) is extremely close to the isotropic one( T c =122 K). The anisotropy of the superconducting gapis in fact very limited [50] The distribution of the calcu-lated superconducting gap on the Fermi surface is shownin the right panel of Fig. 4. Indeed, with the exceptionof a hotspot around the X point, which has a negligi-ble weigth in reciprocal space, the gap is rather uniform,with a deviation of ±
15% around its mean value ∆ avg =26 meV. The mean value differs from the isotropic aver-age ∆ iso = iso (0) /T c is 4.3, confirming the strong-couplingnature of superconductivity in LaBH . Figure 3. Left panel: phonon dispersions of LaBH at 50GPa (black thin lines), decorated with the e-ph coupling(red thick lines). Center panel: atom-projected (colored filledlines) and total (black line) Eliashberg function, and its firstinverse moment λ ( ω ) (dashed black line). Right panel: atom-projected (colored filled lines) and total (black line) phonondensity of states. Projection onto La, B, and H is shown ingreen, orange, and blue, respectively. P N ( E F ) λ ω log T AME c T IME c ∆ iso ∆ H(GPa) ( eV − ) (meV) (K) (K) (meV) (meV/at)50 0.62 1.54 71 126 122 23.5 12575 0.60 1.06 91 101 96 16.8 71100 0.56 0.64 88 42 32 5.6 23Table I. Summary of the main superconducting properties ofLaBH at 50 and 100 GPa. The DOS at the Fermi level N ( E F in the second column is in units of eV − spin − . AME and
IME correspond to solutions of the anisotropic and isotropicMigdal-Eliashberg equations, respectively. The T c is calcu-lated with µ ∗ = 0 . . ∆ iso represents the isotropic averageof the superconducting gap. The last column describes theenthalpy per atom (including zero-point energy) above theconvex hull for the LaBH F m ¯3 m phase.Figure 4. Fermi surface of LaBH at 50 GPa. Left: decoratedwith hydrogen character, right: decorated with the value ofthe superconducting gap. The color scale goes from zero tothe maximum value of the H character (0 to 0.75), and thegap (0 to 42 meV), respectively. Having established that the superconducting proper-ties of LaBH at 50 GPa are extremely promising, we fur-ther studied their behavior as a function of pressure, com-puting the electron-phonon spectra at 75 and 100 GPa,and solving the corresponding Eliashberg equations. Themain results are summarized in Tab. I, and more detailsare provided in SM Fig. S8 [37]. The main effect of anincrease in pressure is a rather uniform shift of all phononfrequencies to higher values, which causes a decrease of λ and an increase of ω log . The T ∗ g mode around Γ , whichat 50 GPa has a frequency of 50 meV, responds morestrongly to pressure than the rest of the spectrum, caus-ing a small, counterintuitive decrease of ω log between 75and 100 GPa. The same mode drives the system towardsa dynamical instability when pressure is decreased below50 GPa.At the harmonic level, the instability occurs at 35 GPa.Vibrations involving hydrogen and, in general, light el-ements, exhibit strong anharmonic and quantum effects[42, 51], which may severely affect the dynamical sta-bility and/or superconducting properties of hydrides. In LaBH , the soft T ∗ g mode is also the only strongly anhar-monic one. Hence, in order to estimate the importanceof anharmonic effects in F m ¯3 m -LaBH , We recomputedthe frequency of the T ∗ g mode, solving the Schrödingerequation numerically as a function of pressure, as de-scribed in Ref. [49]. We estimated that the differencebetween the harmonic and anharmonic frequencies is al-most constant with pressure, and equal to ∼ meV.This causes a ∼ GPa shift of the stability pressure to40 GPa (See Figs. S11 and S12 of the SM for more details[37]), and a negligible effect on the critical temperature(See Fig. S13 of the SM [37]). The stabilization pressureof LaBH represents a new minimum at which a high- T c superhydride is predicted to be stable, beating the pre-vious record of 70 GPa in YbH [21]. We believe thatthe main reason behind the low stabilization pressure ofLaBH is chemical pressure .In conclusion, using a evolutionary crystal structureprediction and ab-initio Migdal Eliashberg theory we pre-dicted a new ternary hydride phase with LaBH stoi-chiometry and F m ¯3 m space group, which is a conven-tional HTSC at moderate pressures, with a T c of 126K at 50 GPa. According to our estimate, this struc-ture could be synthesized by means of laser heating at apressure around 100 GPa, and remains dynamically sta-ble down to 40 GPa, where a single zone-center phononmode drives a structural instability.LaBH is the first conventional superconductor with T c above liquid nitrogen boiling point that can be stabi-lized down to 50 GPa. Its exceptional superconductingproperties can be understood as deriving from a metallichydrogen lattice, which is stabilized at low pressures bya boron and lanthanum scaffolding. The combination oftwo elements with different atomic sizes turns out to bea very effective way to boost chemical pressure on theinterstitial hydrogen sublattice. In general, our resultsdemonstrate an effective new strategy to lower the sta-bilization pressure of binary hydrides. It is likely thatthe XY H F m ¯3 m structure may be tuned to attain evenbetter performances, through a careful choice of the X , Y elements. The possibility of stabilizing a superhydrideto this pressure represents a giant leap towards hydride-based superconductivity at room pressure. ACKNOWLEDGMENTS
This work was supported by the Austrian ScienceFund (FWF) Projects No. P 30269-N36 (Superhy-dra), the dCluster of the Graz University of Technologyand the VSC3 of the Vienna University of Technology.L.B. acknowledges support from Fondo Ateneo- Sapienza2017,2018 and 2019. C. H. acknowledges support fromthe Austrian Science Fund (FWF) Project No. P 32144-N36 and the VSC4 of the Vienna University of Technol-ogy. The authors would like to thank Antonio Sanna forthe useful suggestions and for kindly sharing the code tosolve the isotropic Migdal-Eliashberg equations. ∗ [email protected] † [email protected][1] M. Einaga, M. Sakata, T. Ishikawa, K. Shimizu,M. Eremets, A. P. Drodzov, I. A. Troyan, N. Hirao, andY. Ohishi, Nature Physics , 835 (2016).[2] A. P. Drodzov, M. I. Eremets, I. A. Troyan, V. Kseno-fontov, and S. I. Shylin, Nature , 73 (2015).[3] D. Duan, Y. Liu, F. Tian, D. Li, X. Huang, Z. Zhao,H. Yu, B. Liu, W. Tian, and T. Cui, Scientific Reports , 6968 (2014).[4] P. P. Kong, V. S. Minkov, M. A. Kuzonikov, S. P.Besedin, A. P. Drodzov, S. Mozaffari, L. Balicas, F. F.Balakirev, V. B. Prakapenka, E. Greenberg, D. A.Knyazev, and M. I. Eremets, arXiv:1909.10482 (2019).[5] I. A. Troyan, D. V. Semenok, A. G. Kvashin, A. V.Sadakov, O. A. Sobolevskiy, V. M. Pudalov, A. G.Ivanova, V. B. Prakapenka, E. Greenberg, A. G. Gavril-iuk, V. V. Struzhkin, A. Bergara, I. Errea, R. Bianco,M. Calandra, F. Mauri, L. Monacelli, R. Akashi, andA. R. Oganov, arXiv:1908.01534 (2019).[6] I. A. Kruglov, A. G. Kvashin, A. F. Goncharov, A. R.Oganov, S. S. Lobanov, N. Holtgrewe, S. Jiang, V. B.Prakapenka, E. Greenberg, and A. V. Yanilkin, ScienceAdvances (2018).[7] D. V. Semenok, A. G. Kvashin, A. G. Ivanova, V. Svit-lyk, V. Y. Fominski, A. V. Sadakov, O. A. Sobolevskiy,V. M. Pudalov, I. A. Troyan, and A. R. Oganov, Mate-rials Today , 36 (2020).[8] A. P. Drodzov, M. I. Eremets, and I. A. Troyan,arXiv:1508.06224 (2015).[9] A. P. Drodzov, P. P. Kong, S. P. Besedin, M. A. Ku-zonikov, S. Mozaffari, L. Balicas, F. F. Balakirev, D. E.Graf, V. B. Prakapenka, E. Greenberg, D. A. Knyazev,M. Tkacz, and M. I. Eremets, Nature , 528 (2019).[10] M. Somayazulu, M. Ahart, A. K. Mishra, Z. M. Geballe,M. Baldini, Y. Meng, V. V. Struzhkin, and R. J. Hemley,Phys. Rev. Lett. , 027001 (2019).[11] J. A. Flores-Livas, L. Boeri, A. Sanna, G. Profeta,R. Arita, and M. Eremets, Physics Reports , 1 (2020).[12] E. Snider, N. Dasenbrock-Gammon, R. McBride,M. Debessai, H. Vindana, K. Vencatasamy, K. V. Lawler,A. Salamat, and R. P. Dias, Nature , 373 (2020).[13] J. A. Flores-Livas, M. Amsler, C. Heil, A. Sanna,L. Boeri, G. Profeta, C. Wolverton, S. Goedecker, andE. K. U. Gross, Phys. Rev. B , 020508 (2016).[14] N. Bernstein, C. S. Hellberg, M. D. Johannes, and I. I.Mazin, Phys. Rev. B (2015).[15] C. Heil and L. Boeri, Phys. Rev. B , 060508(R) (2015).[16] H. Wang, J. S. Tse, K. Tanaka, T. Iitaka, and Y. Ma,PNAS , 6463 (2012).[17] F. Peng, Y. Sun, C. J. Pickard, R. J. Needs, Q. Wu, andY. Ma, Phys. Rev. Lett. , 107001 (2017).[18] C. Heil, S. Di Cataldo, G. B. Bachelet, and L. Boeri,Phys. Rev. B , 220502(R) (2019).[19] Y. Sun and M. Miao, Preprint available (v1) at ResearchSquare 10.21203/rs.3.rs-130093/v1 (2021). [20] S. Yi, C. Wang, H. Jeon, and J.-H. Cho, Phys. Rev. M , 024801 (2021).[21] H. Song, Z. Zhang, T. Cui2, C. J. Pickard, V. Z. Kresin,and D. Duan, arXiv:2010.12225 (2020).[22] B. Guigue, A. Marizy, and P. Loubeyre, Phys. Rev. B , 014107 (2020).[23] C. Kokail, W. von der Linden, and L. Boeri, Phys. Rev.M , 074803 (2017).[24] S. Di Cataldo, W. von der Linden, and L. Boeri, Phys.Rev. B , 014516 (2020).[25] Y. Sun, J. Lv, Y. Xie, H. Liu, and Y. Ma, Phys. Rev.Lett. , 097001 (2019).[26] D. V. Semenok, I. A. Troyan, A. G. Kvashin, A. G.Ivanova, M. Hanfland, A. V. Sadakov, O. A. Sobolevskiy,K. S. Pervakov, A. G. Gavriliuk, I. S. Lyubutin,K. V. Glazyrin, N. Giordano, D. N. Karimov, A. B.Vasiliev, R. Akashi, V. M. Pudalov, and A. R. Oganov,arXiv:2012.04787 (2020).[27] C. W. Glass, A. R. Oganov, and N. Hansen, ComputerPhysics Communication , 713 (2006).[28] A. O. Lyakhov, A. R. Oganov, H. T. Stokes, and Q. Zhu,Computer Physics Communication , 1172 (2013).[29] X. Liang, A. Bergara, L. Wang, B. Wen, Z. Zhao, X.-F. Zhou, J. He, G. Gao, and Y. Tian, Phys. Rev. B ,100505(R) (2019).[30] In addition, we re-sampled particularly promising compo-sitions. Further details are provided in the SupplementalMaterial [37].[31] M. Paskevicius, L. H. Jepsen, P. Schouwink, R. Cerný,D. B. Ravnsbaek, Y. Filinchuk, M. Dornheim, F. Be-senbacher, and T. R. Jensen, Chem. Soc. Rev. , 1565(2017).[32] Y. Zhang, E. Majzoub., V. Ozolins, and C. Wolverton,Phys. Rev. B , 174107 (2010).[33] J. B. Grinderslev, M. B. Ley, Y.-S. Lee, L. H. Jepsen,M. J. rgensen, Y. W. Cho, J. rgen Skibsted, and T. R.Jensen, Inorganic Chemistry , 7768 (2020).[34] L. H. Rude, T. K. Nielsen, D. B. Ravnsbaek, U. Bösen-berg, M. B. Ley, B. Richter, L. M. Arnbjerg, M. Dorn-heim, Y. Filinchuk, F. Besenbacher, and T. R. Jensen,Phys. Status Solidi , 1754 (2011).[35] Additional information on the crystal structures can befound in the form of Crystallographic Information File inthe Supplemental Material [37].[36] G. Renaudin, S. Gomes, H. Hagemann, L. Keller, andK. Yvon, Journal of Alloys and Compounds , 98(2004).[37] The Supplemental Material is available at..[38] E. Sanville, S. D. Kenny, R. Smith, and G. Henkelmann,J. Comp. Chem. , 899 (2007).[39] S. Baroni, S. de Gironcoli, A. D. Corso, and P. Giannozzi,Rev. Mod. Phys , 515 (2001).[40] S. Y. Savrasov and D. Y. Savrasov, Phys. Rev. B ,16487 (1996).[41] S. P. and. E. R. Margine, C. Verdi, and F. Giustino,Comp. Phys. Communications , 116 (2016).[42] I. Errea, F. Belli, L. Monacelli, A. Sanna, T. Koretsune,T. Tadano, R. Bianco, M. Calandra, R. Arita, F. Mauri,and J. A. Flores-Livas, Nature , 66 (2020).[43] W. L. McMillan, Physical Review , 331 (1968).[44] P. B. Allen and R. C. Dynes, Phys. Rev. B , 905 (1975).[45] P. Morel and P. W. Anderson, Physical Review , 1263(1962). [46] K.-H. Lee, K. J. Chang, and M. L. Cohen, Phys. Rev. B , 1425 (1995).[47] F. Giustino, M. L. Cohen, and S. G. Louie, Phys. Rev.B , 115105 (2010).[48] H. Lambert and F. Giustino, Phys. Rev. B , 075117(2013). [49] C. Heil, S. Poncé, H. Lambert, M. Schlipf, E. R. Margine,and F. Giustino, Phys. Rev. Lett. , 087003 (2017).[50] We also checked the dependence of µ ∗ on Tc and foundthat a variation of 0.01 in µ ∗ changes Tc only by 2-3 K(See SM Fig. S9 for more details [37])[51] I. Errea, M. Calandra, C. J. Pickard, J. R. Nelson, R. J.Needs, Y. Li, H. Liu, Y. Zhang, Y. Ma, and F. Mauri,Nature532