week ending PRL 105, 037002 (2010) PHYSICAL REVIEW LETTERS 16 JULY 2010
First-Principles Prediction of Doped Graphane as a High-Temperature Electron-Phonon Superconductor
G. Savini,1,2 A. C. Ferrari,1,* and F. Giustino3 1Department of Engineering, University of Cambridge, Cambridge, CB3 0FA, United Kingdom 2Institute of Protein Research, University of Osaka, Osaka, 565-0871, Japan 3Department of Materials, University of Oxford, Oxford, OX1 3PH, United Kingdom (Received 11 March 2010; published 14 July 2010) We predict by first-principles calculations that p-doped graphane is an electron-phonon superconductor with a critical temperature above the boiling point of liquid nitrogen. The unique strength of the chemical bonds between carbon atoms and the large density of electronic states at the Fermi energy arising from the reduced dimensionality give rise to a giant Kohn anomaly in the optical phonon dispersions and push the superconducting critical temperature above 90 K. As evidence of graphane was recently reported, and doping of related materials such as graphene, diamond, and carbon nanostructures is well established, superconducting graphane may be feasible.
DOI: 10.1103/PhysRevLett.105.037002 PACS numbers: 74.10.+v, 63.22.Rc, 73.22.Pr, 74.20.Fg
The discovery of superconductors such as MgB2 [1] and BCS formula for Tc is now replaced by more refined iron pnictides [2] intensified the search for high- expressions such as the modified McMillan equation [9], temperature superconductivity in materials other than cop- Eq. (1) still proves useful for discussing trends: One could per oxides. The critical temperature Tc reflects the energy maximize Tc by increasing the materials parameters !0, scale of the interactions driving the condensation into the NF, and V. However, these are strongly intertwined, mak- superconducting state. In high-Tc copper oxides the nature ing such optimization extremely complex. We now discuss of the interaction is still under debate, yet it is generally a simple procedure, based on Eq. (1) and on recent advan- accepted that Coulomb exchange and correlation, with ces in electron-phonon superconductivity, to design a novel energy scales around few hundred meVs, play an important high-Tc superconductor. role [3,4]. In contrast, in conventional superconductors the Let us first consider the conventional superconductor ¼ pairing is driven by the electron-phonon interaction, with with the highest Tc, MgB2 (Tc 39 K)[1]. For simplicity, an energy scale of only a few ten meVs [5]. Because of the we neglect multiband and anisotropy effects [10,11]. In order-of-magnitude difference between such energy scales, MgB2 the EPC contribution to V is large ( 1:4eV, from it is assumed that conventional superconductors cannot ¼ NFV, using NF ¼ 0:7 states=cell=eV, and 1 [11]) exhibit Tc as high as copper oxides. because the states with energy close to EF (those which Here we show by density-functional theory (DFT) cal- condensate in the superconducting state [5]) are of char- culations that p-doped graphane may be a conventional acter, i.e., derive from bonding combinations of planar B 2 superconductor with Tc well above the boiling point of sp hybrids localized around the middle of B—B bonds liquid nitrogen. As evidence of graphane was recently [10,11]. These are significantly affected by the B—B bond reported [6,7], and doping of related materials such as length variation associated with bond-stretching E2g pho- graphene, diamond, and nanocarbons is well established, nons [11], resulting in a large EPC contribution to V. superconducting graphane may be feasible. Here we iden- At the same time, the energy of the E2g phonons is large tify a conventional phonon-mediated superconducting ( 60 meV [11]), due to the small B mass, leading to a pairing in close analogy to B-doped diamond [8]. large !0 in Eq. (1). Furthermore, MgB2 is a metal with a The Bardeen-Cooper-Schrieffer (BCS) theory [5] de- significant EDOS at EF ( 0:7 states=cell=eV [11]). These fines the basic framework to understand conventional three factors cooperate to establish a superconducting state superconductivity. Its generalization, incorporating lattice with Tc ¼ 39 K [10,11]. Experimental and theoretical at- dynamics, the Migdal-Eliashberg theory [9], provides a tempts to improve upon MgB2 by investigating related predictive computational tool. Within BCS, Tc is [5] materials met limited success [12], with the experimental k T ¼ 1:14@! expð 1=N VÞ; (1) Tc never higher than MgB2. B c 0 F We thus search for an alternative material having at least @ where kB is the Boltzmann constant, !0 a characteristic some of the desirable features of MgB2, i.e., (i) electrons phonon energy, NF the electronic density of states (EDOS) at the Fermi surface, (ii) large bond-stretching phonon at the Fermi energy EF, and V an effective pairing potential frequencies, and (iii) large EDOS at EF. The first two resulting from the net balance between the attractive requirements are met by B-doped diamond, a conventional electron-phonon coupling (EPC) and the repulsive BCS superconductor with Tc ¼ 4K[8,13], where a small electron-electron interaction [5]. Even though the original holelike Fermi surface appears around the top of the va-
0031-9007=10=105(3)=037002(4) 037002-1 Ó 2010 The American Physical Society week ending PRL 105, 037002 (2010) PHYSICAL REVIEW LETTERS 16 JULY 2010 lence band [14]. The electronic states at EF have char- to estimate the EDOS increase in a diamond thin film, acter deriving from bonding combinations of tetrahedral we consider a parabolic band model. For 2% B doping, 3 ¼ C sp hybrids, bearing some analogy to MgB2. As these bulk diamond has NF 0:1 states=eV=cell at EF.A states are localized in the middle of the C—C bonds, they 0.5-nm-thick diamond film with the same doping would couple considerably to bond-stretching phonons [14,15], have NF 0:5 states=eV=cell. Such EDOS increase would resulting in a large EPC contribution to V, even super- significantly enhance Tc. With the electron-phonon poten- ¼ ¼ ior to MgB2 ( 3eV, from NFV, using NF tial and phonon frequency of bulk diamond, Eq. (1)gives 0:1 states=cell=eV, and 0:3 [16]). In addition, the light that a 0.5-nm film would superconduct at Tc 80 K. This C atoms have high energy optical phonons ( 130 meV, effect could be maximized by using an atomically thin even after softening induced by the large EPC [15,16]). diamond film. However, the question remains whether it However in B-doped diamond the EDOS at EF is rather is possible to synthesize such film. small ( 0:1 states=cell=eV for 2% doping [14]), compro- Recent work on graphene and its derivatives points to a mising Tc. While B-doped diamond shares some desirable positive answer. Soon after the discovery of graphene [17], features of MgB2, its 3D nature impliespffiffiffiffi the EDOS in several works considered how to functionalize and chemi- proximity of the valence band scales as E (with E from cally modify it [18–23]. In particular, it was proposed that the valence band edge); see Fig. 1(a). Thus, the carriers fully hydrogenated graphene (graphane) would be stable available for the superconducting state are relatively few [24]. The main difference between graphene and graphane even for large doping. Superconducting diamond is then a is that, while the former is fully sp2 bonded, the latter is 3 3D analogue of MgB2 [14,15]. sp , as diamond [24]. Experimental evidence of graphane The comparison between MgB2 and diamond leads to was recently reported [6,7]. Since graphane is the 2D the question of what would happen in a hypothetical counterpart of diamond, our scaling arguments point to B-doped diamond with reduced dimensionality, such as a doped graphane as a potential high-Tc superconductor. thin film or a nanowire, with the EDOS significantly en- Doping could be achieved by gating, including using an hanced by quantum confinement. The EDOS of a 2D electrolyte gate, or by charge transfer, as done in graphene semiconductor is a step function; hence, the number of [18–20]. Substitutional doping of graphene was also re- available carriers can be large even at low doping. In order ported, up to 1014 cm 2 [21]. We thus perform DFT calculations of EPC and Tc in doped graphane. By analogy with B-doped diamond, we (a) consider p doping. We use the DFT local density approxi- 0.3 Nanowire 1d (sp 3 ) mation, plane waves, and pseudopotentials [25]. Hole dop- Graphane 2d (sp 3 ) ing is simulated within the rigid-band approximation. We Diamond 3d (sp 3 ) 0.2 checked that this is adequate to describe substitutional B doping by comparing the electronic structure with that of a 0.1 supercell model explicitly including the B atoms [25]. Doping has the effect of lowering EF below the top of EDOS (states/eV/C atom) 0 the valence band and does not introduce localized states in -2 -1.5 -1.0 -0.5 0 Energy (eV) the band gap. Phonon modes and EPCs are calculated 2.0 within density-functional perturbation theory [26,27]. (b) The EPCs, , are calculated by using a 100 100 1 Pristine graphane 1.5 grid. We extensively checked the convergence of with p-doped graphane respect to Brillouin zone (BZ) sampling [25]. We calculate 1.0 Tc by using the modified McMillan equation [9], together with a Coulomb parameter ? ¼ 0:13. Other Coulomb 0.5 parameters do not affect our conclusions.
EDOS (states/eV/cell) The p-doped graphane EDOS close to the valence band 0 -3 -2 -1 0 1 234 maximum follows a steplike behavior, as expected in 2D; Energy (eV) see Fig. 1(b). At 3% doping this is 0:22 states=eV=cell, almost twice the 0:13 states=eV=cell of bulk diamond. FIG. 1 (color online). (a) EDOS per carbon atom in 1D (dia- Figure 2 plots the phonon dispersions of pristine and mond nanowire), 2D (graphane), and 3D (diamond). (b) EDOS p-doped graphane (see Ref. [25] for full dispersion); of pristine (solid black line) and 12.5% p-doped graphane Fig. 3(a) plots the phonon density of states (PDOS). Upon (dashed red line, 2 2 supercell with one substitutional B). doping, the optical zone-center modes associated with in- The top of the valence band is set as zero, and E ¼ F plane C-C stretching soften as a result of the inception of 0:96 eV (green line). The EDOS at EF is similar in the two models (0:26 states=eV=cell in rigid band; 0:27 states=eV=cell Kohn anomalies [28]. The softening of modes with large in supercell). We expect this to hold also at lower doping, where C-C stretching is similar to B-doped diamond [15,16]. The the perturbation to the pristine dispersions is smaller. The super- region of reciprocal space where the softening is observed cell calculation has no impurity states in the gap. matches the diameter 2kF of the hole Fermi surface around 037002-2 week ending PRL 105, 037002 (2010) PHYSICAL REVIEW LETTERS 16 JULY 2010