Eur. Phys. J. B (2018) 91: 177 https://doi.org/10.1140/epjb/e2018-90168-7 THE EUROPEAN PHYSICAL JOURNAL B Regular Article Superconductivity in hydrogenated carbon nanostructures? Antonio Sanna1, a, Arkadiy Davydov2, John Kay Dewhurst1, Sangeeta Sharma1, and Jos´eA. Flores-Livas3 1 Max-Planck Institute f¨urMikrostrukturPhysik, Weinberg 2, 06120 Hale, Germany 2 Department of Physics, Kings College London, London WC2R 2LS, UK 3 Department of Physics, Universit¨atBasel, Klingelbergstr. 82, 4056 Basel, Switzerland Received 13 March 2018 / Received in final form 29 May 2018 Published online 6 August 2018 c The Author(s) 2018. This article is published with open access at Springerlink.com Abstract. We present an application of density functional theory for superconductors to superconductivity in hydrogenated carbon nanotubes and fullerane (hydrogenated fullerene). We show that these systems are chemically similar to graphane (hydrogenated graphene) and like graphane, upon hole doping, develop a strong electron phonon coupling. This could lead to superconducting states with critical temperatures approaching 100 K, however this possibility depends crucially on if and how metallization is achieved. 1 Introduction gating or by a chemical substitution. So far the doped configuration of graphane has not been synthesized and The development of accurate ab-initio many body meth- the prediction is not yet confirmed. Still this material ods for superconductivity, such as Eliashberg´ theory [1,2] is of great importance for several reasons: first graphane and density functional theory for superconductors [3], as is stable at ambient conditions of pressure, making it of well as efficient numerical methods to compute electronic technological relevance; second it has the highest critical interactions [4{6] allows prediction of superconducting temperature ('100 K) predicted and known at ambient properties and the critical temperature (Tc) of materi- conditions of pressure in phononic superconductors; third als without need of empirical parameters or experimental graphane, in spite of being an hydrogen compound, does input. This, combined with the fact that computational not owe its high-Tc to the high energy phonon modes of approaches are nowadays also able to scan the configura- hydrogen (i.e. Ashcroft mechanism [33]) but rather to the tion space of stable materials [7{11] in a faster and efficient carbon phonon modes and carbon electronic states. This fashion (as compared to cumbersome trial-and-error pro- opens the possibility of other high temperature phononic cess of synthesis and characterization), has revived hope superconductors may exist in a broader set of materials, of finding a, long dreamed, room temperature supercon- since at room pressure hydrogen states are usually fully ductor. compensated while carbon related states are more likely In recent years many theoretical predictions ([12{24] to to be pinned at the Fermi level where they can contribute cite a few) have been made. One of which was SH3 at high to superconductivity. pressure (200 GPa) [25], this was then confirmed through In this work we will show that the excellent super- diamond-anvil cell experiments [26] SH3 breaking the then conducting properties of graphane also occur in related record for the highest-known critical temperature and the structures of lower dimensionality such as fullerane, the prejudice that high-Tc superconductivity is impossible to hydrogenated buckyball (C60H60), and hydrogenated car- predict. bon nanotubes (CH-NT). While in graphane hydrogena- Another very interesting prediction was that of super- tion occurs on both sides of the layer, for these systems conductivity in graphane [27]. Graphane [28{31] is a fully we consider only exohedral hydrogenation as shown in hydrogenated graphite layer where hydrogen atoms are Figure1. bonded on the two sides of the layer attached to alternate The superconducting state is described by means of den- carbons [32]. Unlike SH3, graphane does not superconduct sity functional theory for superconductors [3] (SCDFT) in its stoichiometric structure, it has to be hole doped and an extension used to compute the order parameter and this could be in principle achieved by electrostatic in real space [34]. This provides a natural description for large systems with low periodicity as nanotubes and ? Contribution to the Topical Issue \Special issue in honor buckyballs and is a strength of SCDFT over the more con- of Hardy Gross", edited by C.A. Ullrich, F.M.S. Nogueira, ventional Eliashberg´ approach in reciprocal space [35{38]. A. Rubio, and M.A.L. Marques. This work is dedicated to Hardy Gross who, not a e-mail: [email protected] only jointly invented SCDFT [3], but also devoted a Page 2 of 10 Eur. Phys. J. B (2018) 91: 177 where are electronic field operators and µ the chemical potential. Nuclei need to be considered explicitly (not just as source of an external potential like in conventional DFT [59]) because in most known superconductors the ion dynamics provides an essential part of the supercon- ducting coupling: Z r2 H = − d3RΦy (R) Φ (R) n 2M 1 Z + d3Rd3R0Φy (R) Φy (R0) 2 Z2 × Φ (R0) Φ (R) (3) jR − R0j X Z Z Fig. 1. View of the crystal structure of CH-NT the hydro- H = − d3Rd3r y (r) Φy (R) Φ (R) (r); en σ jR − rj σ genated (6,0)-nanotube (a { front view; b { top view) and σ C60H60 the hyrogenated fullerene (c). The CH{NT unit cell (4) (gray lines) contains 24 C atoms and 24 H atoms. In violet are indicated the lattice parameters of the two crystals. Atomic where Φ are ionic creation operators, M the mass and Z distances (in A)˚ are collected in the following table: the atomic number. CH{NT C60H60 The Hamiltonian needs to includes an external symme- try breaking field [56] that for singlet superconductivity C{C C{H C{C C{C C{H can be chosen as: 1.552 1.097 1.548 1.561 1.091 Z 3 3 0 ∗ 0 0 H∆ext = d rd r ∆ext (r; r ) " (r) # (r ) + h:c: (5) large effort to develop the theoretical framework into In addition to this one should also add an external field a fully functioning method, investigating functionals coupling with the electronic density: [39{42], extensions [43{46] and transforming it into a useful and predictive tool in material science [23,47{54]. Z 3 X y Hvext = d rvext (r) σ (r) σ (r) (6) σ 2 Basics of SCDFT and an external field that couples with the nuclei: SCDFT is an extension of DFT to account for the very Z peculiar symmetry breaking that occurs in a superconduc- Y 3 y HWext = Wext (fRig) d RjΦ (Rj) Φ (Rj) : (7) tor [55,56]. Proposed in 1988 [3] by Oliveira, Gross and j Kohn was later revisited and extended in Hardy Gross' group [41,42,57] to merge with the multi-component DFT In its modern form [41,42] SCDFT is based on the three of Kreibich and Gross [58] to include nuclear motion. densities: The starting point of SCDFT is the non relativistic Hamiltonian for interacting electrons and nuclei: " # X y ρ (r) = Tr %0 σ (r) σ (r) (8) σ H = He + Hen + Hn + Hext; (1) 0 0 χ (r; r ) = Tr [%0 " (r) # (r )] (9) where e stands for electrons, n for nuclei and ext for 2 3 Y y external fields. Γ (fRig) = Tr 4%0 Φ (Rj) Φ (Rj)5 ; (10) j Z X 3 y 1 2 He = d r σ (r) − r − µ σ (r) 2 where %0 is the grand canonical density matrix. σ Z The SCDFT generalization of the Hohenberg{Kohn 1 X 3 3 0 y y 0 + d rd r (r) 0 (r ) theorem [59] (at finite temperature [60]) states: 2 σ σ σσ0 1. There is a one-to-one mapping between the set of 1 0 0 0 densities ρ (r), χ (r; r ), Γ (fRig) onto the set of × 0 σ (r ) σ (r) ; (2) 0 jr − r j external potentials vext (r), ∆ext (r; r ), Wext (fRig) Eur. Phys. J. B (2018) 91: 177 Page 3 of 10 2. There is a variational principle so that it exists a that are the electronic Kohn{Sham equation for SCDFT. functional Ω that: Their mathematical form is well known in super- conductivity literature as Bogoljubov-deGennes (BdG) equations [56] which are mostly used, within the BCS Ω [ρ0; χ0;Γ0] = Ω0 model, to describe superconducting structures in real Ω [ρ, χ, Γ ] > Ω0 for ρ, χ, Γ 6= ρ0; χ0;Γ0; (11) space. In SCDFT these equations become exact for the calculation of the total energy and the densities: where ρ0; χ0;Γ0 are the ground state densities and Ω0 the X h 2 2 i grand canonical potential. ρ (r) = 2 ju (r)j f (E ) + jv (r)j f (−E ) (16) The fact that all observables are functionals of the i i i i densities and that H is the sum of internal interactions i 0 X ∗ 0 (Eq. (1)) and couplings with external fields (Eq. (5) + χ (r; r ) = ui (r) vi (r ) f (−Ei) Eq. (6) + Eq. (7)) allows Ω [ρ, χ, Γ ] to be written as: i ∗ 0 −vi (r) ui (r ) f (Ei) : (17) Z 3 Ω [ρ, χ, Γ ] = F [ρ, χ, Γ ] + d rvext (r) ρ (r) In absence of superconductivity both χ and ∆ are zero Z and the Kohn{Sham equations (15) and (16) become the Y 3 + Γ (fRig) Wext (fRig) d Rj usual Kohn{Sham equations of conventional DFT: j Z r2 3 3 0 ∗ 0 0 − + vs (r) − µ 'nk (r) = ξnk'nk (r) : (18) + d rd r ∆ext (r; r ) χ (r; r ) + c:c: (12) 2 This form is slightly more general because it would still that defines the universal functional F [ρ, χ, Γ ].