Graphene1 is the physical realization of many fundamental concepts and phenomena in solid-state physics 2 . However, in the list of graphene's many remarkable properties 3-6 , superconductivity is notably absent. If it were possible to find a way to induce superconductivity, it could improve the performance and enable more efficient integration of a variety of promising device concepts including nanoscale superconducting quantum interference devices, single-electron superconductor-quantum dot devices 7,8 , nanometre-scale superconducting transistors 9 and cryogenic solid-state coolers 10 . To this end, we explore the possibility of inducing superconductivity in a graphene sheet by doping its surface with alkaline metal adatoms, in a manner analogous to which superconductivity is induced in graphite intercalated compounds 11,12 (GICs). As for GICs, we find that the electrical characteristics of graphene are sensitive to the species of adatom used. However, contrary to what happens in GICs, Li-covered graphene is superconducting at a much higher temperature with respect to Ca-covered graphene.As graphene itself is not superconducting, phonon-mediated superconductivity must be induced by an enhancement of the electron-phonon coupling (λ),In equation (1), N (0) is the electronic density of states per spin (DOS) at the Fermi level, D is the deformation potential, and M and ω ph are the effective atomic mass and phonon frequency that in metallic alloys reflect the role of the different atomic species and phonon vibrations involved in superconductivity. In undoped graphene λ is small and phonon-mediated superconductivity does not occur as the small number of carriers, intrinsic to a semimetal, leads to a vanishingly small N (0). In this respect the situation is similar to the bulk graphite case, where, without intercalation of foreign atoms superconductivity is not stabilized.A first guess to induce superconductivity could then be to fill by rigid-band doping the carbon π -states to have enough carriers. However, besides the fact that the π -DOS grows very slowly with doping, its impact will be hindered by two major difficulties. First, even if the deformation potential related to coupling between π -bands and in-plane phonon vibrations is large and leads to Kohn anomalies 13 , these vibrations are highly energetic (ω ph ≈ 0.16 eV) and λ is small owing to the ω 2 ph factor in the denominator. Second, symmetry forbids the coupling between π-states and the softer out-of-planes vibrations.To promote coupling to carbon out-of-plane vibrations, it is then necessary to promote new electronic states at the Fermi level as happens in GICs. Indeed, in superconducting GICs, an intercalant band (interlayer state) occurs at the Fermi

Two hydrogen-rich materials: H 3 S and LaH 10 , synthesized at megabar pressures have revolutionized the field of superconductivity by providing a first glimpse into the solution for the hundred-year-old problem of room-temperature superconductivity. The mechanism governing these exceptional superconductors is the conventional electron-phonon coupling. Here, we describe recent advances in experimental techniques, superconductivity theory and first-principles computational methods, which made this discovery possible. The aim of this work is to provide an up-to-date compendium of the available results on superconducting hydrides and explain the synergy of different methodologies that led to the extraordinary discoveries in the field. Furthermore, in an attempt to evidence empirical rules governing superconductivity in binary hydrides under pressure, we discuss general trends in electronic structure and chemical bonding. The last part of the Review introduces possible strategies to optimize pressure and transition temperatures in conventional superconducting materials. Directions for future research in areas of theory, computational methods and high-pressure experiments are proposed, in order to advance our understanding of superconductivity.

The density functional theory for superconductors developed in the preceding article is applied to the calculation of superconducting properties of several elemental metals. In particular, we present results for the transition temperature, for the gap at zero temperature, and for thermodynamic properties like the specific heat. We obtain an unprecedented agreement with experimental results. Superconductors both with strong and weak electron-phonon coupling are equally well described. This demonstrates that, as far as conventional superconductivity is concerned, the first-principles prediction of superconducting properties is feasible.

A novel approach to the description of superconductors in thermal equilibrium is developed within a formally exact density-functional framework. The theory is formulated in terms of three "densities": the ordinary electron density, the superconducting order parameter, and the diagonal of the nuclear N -body density matrix. The electron density and the order parameter are determined by Kohn-Sham equations that resemble the Bogoliubov-de Gennes equations. The nuclear density matrix follows from a Schrödinger equation with an effective N -body interaction. These equations are coupled to each other via exchange-correlation potentials which are universal functionals of the three densities. Approximations of these exchange-correlation functionals are derived using the diagrammatic techniques of many-body perturbation theory. The bare Coulomb repulsion between the electrons and the electron-phonon interaction enter this perturbative treatment on the same footing. In this way, a truly ab-initio description is achieved which does not contain any empirical parameters.