Superconducting Chevrel phase PbMo6S8 from first principles Giovanni Marini,1, 2 Antonio Sanna,3 Camilla Pellegrini,4 Christophe Bersier,5 Erio Tosatti,6, 7, 8 and Gianni Profeta9, 10 1Department of Physical and Chemical Sciences, University of L'Aquila, Via Vetoio 10, I-67100 L'Aquila Italy 2SPIN-CNR, University of L'Aquila, Via Vetoio 10, I-67100 L'Aquila Italy 3Max-Planck Institut f¨urMikrostrukturPhysik, Weinberg 2, 06120 Halle, Germany 4Department of Physics and Geology, University of Perugia, Via Pascoli 33, 06123 Perugia, Italy 5Max-Planck Institut f¨urMikrostrukturPhysik, Weinberg 2, 06120 Halle, (Germany) 6International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy 7CNR-IOM Democritos National Simulation Center, Via Bonomea 265, 34136 Trieste, Italy 8The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy 9Department of Physical and Chemical Sciences, University of L'Aquila, Via Vetoio 10, I-67100 L'Aquila (Italy) 10SPIN-CNR, University of L'Aquila, Via Vetoio 10, I-67100 L'Aquila (Italy) Chevrel ternary superconductors show an intriguing coexistence of molecular aspects, large electron-phonon and electron-electron correlations, which to some extent still impedes their quan- titative understanding. We present a first principles study on the prototypical Chevrel compound PbMo6S8, including electronic, structural and vibrational properties at zero and high pressure. We confirm the presence of an extremely strong electron-phonon coupling, linked to the proximity to a R3-P1 structural phase transition, which weakens as the system, upon applied pressures, is driven away from the phase boundary. A detailed description of the superconducting state is obtained by means of fully ab initio superconducting density functional theory (SCDFT). SCDFT accounts for the role of phase instability, electron-phonon coupling with different intra- and inter-molecular phonon modes, and without any empirical parameter, and accurately reproduces the experimental critical temperature and gap. This study provides the conclusive confirmation that Chevrel phases are phonon driven superconductors mitigated, however, by an uncommonly strong Coulomb repul- sion. The latter is generated by the combined effect of repulsive Mo states at the Fermi energy and a band gap in close proximity to the Fermi level. This is crucial to rationalize why Chevrel phases, in spite of their extreme electron-phonon coupling, have critical temperatures below 15 K. In addition, we predict the evolution of the superconducting critical temperature as a function of the external pressure, showing an excellent agreement with available experimental data. I. INTRODUCTION tive to the A15 SC class for the fabrication of SC mag- nets14,15. Molybdenum chalcogenides, better known as Chevrel From a fundamental point of view, this family repre- phases, were discovered in 1971 by Roger Chevrel and sents a unique platform to study the interplay between collaborators1 as the first known ternary superconduc- superconductivity and magnetism, when the cations are tors (SC)2. To date, this family counts more than magnetic atoms. For example, REMo6S8 and REMo6Se8 one hundred examples, and has been extensively inves- (RE = rare-earth) show coexistence between supercon- 16 tigated3. The general stoichiometry of Chevrel phases ductivity and long-range magnetic ordering . Moreover, is MyMo6X8 ( M = metal and X = chalcogen), but re-entrant SC transitions and ferromagnetic phases have they can often deviate from the stoichiometric formu- been reported for ErRh4B4 and HoMo6S8 (many details 3 lae in experimental observations. Their peculiar crystal on the subject can be found in the reviews of Pe~na and 13 structure, characterized by molecular-like interpenetrat- Fischer ). ing Mo6 octahedra and X8 cubes, allows for a wide chem- Research on Chevrel phases has thrived in the 70s ical variability as different chemical species can be inter- and 80s, until the discovery of cuprate superconductors17 calated into the structure, a fact which is at the origin shifted the attention away, leaving behind an incomplete arXiv:2104.07982v1 [cond-mat.supr-con] 16 Apr 2021 of the broad spectrum of interesting physical properties characterization with several unsolved problems, espe- they display. Owing to the peculiar cage-like structure, cially concerning the nature and properties of the su- Chevrel phases have been considered as promising ther- perconducting pairing. In the Uemura plot18,19, which 4{7 moelectric materials , as cathode materials in battery relates the SC critical temperature Tc to the ratio be- design8,9 and, when X=S, as catalysts in desulfurization tween carrier density and effective mass, the Chevrel processes10{12. As superconductors, the importance of phases lay in the region of unconventional superconduc- Chevrel phases is linked to the extremely high critical tors, together with cuprates, iron-based superconductors, 13 magnetic field (Hc ), up to 60 Tesla , that combines fullerenes, organics superconductors and heavy-fermions, 2 ≈ with critical temperatures (Tc) as high as 15 K. For these and far from the \conventional" BCS-like part of the figures Chevrels have been considered a valuable alterna- graph. Nevertheless, the origin of the superconducting 2 transition in Chevrel phases is attributed to the electron- II. METHODS phonon mechanism13,20,21. In addition, a recent exper- iment reports the appearance of a double SC gap in First-principles calculations of normal state proper- 22,23 PbMo6S8, SnMo6S8 and AgMo6S8 . Attempts to ties have been performed within density functional the- clarify the situation with modern theoretical methods ory (DFT). We have used norm-conserving pseudopoten- have been scarce, and few works provide ab initio results tials generated from the Troullier-Martins scheme35 to 6,7,21 21 on the topic . Only recently, Chen et al. afforded describe the electron-ion interaction, while employing a a first-principles density functional theory description of kinetic energy cutoff of 60 Ry in the plane-wave expan- Chevrel phases. They predicted an electron-phonon cou- sion of the Kohn-Sham wavefunctions, as implemented pling (EPC) parameter, λ, (2.29 for PbMo6S8), much in the Quantum Espresso package36. For the exchange- 13 larger than previous estimates blue predicting, in fact, correlation potential we have adopted the generalized SC critical temperatures systematically higher than the gradient approximation (GGA) in the Perdew, Burke and experimental values reported for (Pb, Sn, Yb, La, Y) Ernzerhof (PBE)37 formulation. A 4 4 4 Monkhorst- 2,24 Mo6S8 , therefore still leaving questions on the origin Pack wave-vector grid38 has been required× × to reach an and quantitative understanding of the superconducting accuracy in the total energy better than 1 meV/atom, transition. while the energy cutoff has been increased to 80 Ry in order to converge the stress tensor in the variable-cell cal- culations. The structural phase diagram has been con- structed by performing a variable-cell optimization of the structures at fixed pressures, and then calculating the corresponding enthalpy (H(P ) = E(V (P )) + PV (P )). The aim of the present work is to definitely shed The lattice dynamics has been simulated using den- sity functional perturbation theory (DFPT)39, with a 83 light on these unsolved issues in the characterization of 3 Chevrel phases by reinvestigating their electronic and vi- (4 ) sampling in k (q) space of the electronic (phononic) wavefunctions in the Brillouin Zone (BZ). Accurate in- brational properties, and by giving an account of their ν;mn superconducting state using density functional theory tegration of the electron-phonon matrix elements gk+q;k for superconductors (SCDFT)25{27, an ab-initio cutting- in the Eliashberg function: edge approach to phonon-mediated superconductivity. 2 1 X X ν;mn 2 α F (!) = gk+q;k Due to the accuracy of the method, a careful compar- N(EF ) j j ison of our predictions with available experimental data kqν nm 24 m n at ambient and high pressures is expected to detect δ(! !qν )δ(k+q EF )δ(k EF ); (1) × − − − anomalies in the superconducting behaviour of Chevrel has been achieved by using a random mesh of k points ac- phases. We focus our attention on the PbMo6S8 com- cumulated on the Fermi surface by means of a Metropo- pound, as the prototype system with highest Tc of a lis algorithm, and by Fourier interpolating the calcu- wide class of Chevrel phases, therefore extending and lated phonon frequencies (!qν ) and electronic eigenvalues complementing previous theoretical approaches, partic- n27,40,41 21 (k . The integrated EPC constant λ has been ob- ularly the latest by Chen et al. Incidentally, we note tained as: that a tempting analogy could initially have been drawn Z α2F (!) with alkali fullerides (doped molecular crystals with λ = 2 d!: (2) 28 29 high Tc , strong electron-phonon couplings and very ! strong electron-electron correlations30). For them it was To compare with conventional theoretical approaches, proposed31,32, and experimentally demonstrated, partic- we have estimated the superconducting critical tempera- 33 ularly for Cs3C60 , that superconductivity is strongly ture using the Allen-Dynes-modified McMillan formula 34 correlated, arising from a low-spin Mott insulator af- f1f2!ln 1:04(1 + λ) ter doping and metallization. We will show, first of all, Tc = exp − (3) that negative pressures on Chevrel phases can lead to 1:20 λ µ∗(1 + 0:62λ) − electron-phonon