Theory of superconductivity for Dirac electrons in graphene
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SORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM
Theory of Superconductivity for Dirac Electrons in Graphene Yu. E. Lozovika, S. L. Ogarkovb, and A. A. Sokolika a
Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia b Moscow Institute of Engineering Physics, Kashirskoe sh. 31, Moscow, 115409 Russia email: [email protected] Received June 28, 2009
Abstract—Phononexchangeinduced superconducting pairing of effectively ultrarelativistic electrons in graphene is investigated. The Eliashberg equation obtained for describing pairing in the Cooper channel with allowance for delayed interaction are matrix equations with indices corresponding to the valence and con duction bands. The equations are solved in the high doping limit, in which pairing is effectively a singleband process, and in the vicinity of a critical quantum point of underdoped graphene for a value of the coupling constant for which pairing is an essentially multiband process. For such cases, analytic estimates are obtained for the superconducting transition temperature of the system. It is shown that the inclusion of dynamic effects makes it possible to determine the superconducting transition temperature, as well as the critical coupling constant for underdoped graphene, more accurately than in the static approximation of the BCS type. Esti mates of the constants of electron interaction with the scalar optical phonon mode in graphene indicate that an appreciable superconducting transition temperature can be attained under a high chemical doping level of graphene. DOI: 10.1134/S1063776110010073
and the substrate can shift the Fermi level to any of these bands, thus producing electrons or holes with a concentration of up to 1013 cm–2 [9]. If the chemical potential in graphene is sufficiently close to the Dirac point, superconducting pairing may involve both the conduction band and the valence band in view of the absence of a gap in the spectrum. In the opposite case of high doping, pairing involves only one energy band containing the Fermi surface. It was proposed that possible sources of intrinsic superconductivity of graphene can be the phonon and plasmon mechanisms in graphene coated with a layer of a metal [10], electron correlations [11–13], and the essentially anisotropic electron–electron interaction in the vicinity of Van Hove singularities of the electron subsystem [14]. Apart from intrinsic superconductiv ity, the multiband nature of pairing in graphene can manifest itself in the proximity effect in contact with a superconductor (see, for example, [15–17]) and in electron–hole pairing in a graphene bilayer in the tight binding mode [18] (in contrast to electron–hole pair ing in the weak binding regime considered for graphene [19, 20], coupled quantum wells [21], and the 3D exciton dielectric [22]). Pairing of electrons in graphene, which involves the two bands simulta neously, was considered in the model of the contact pairing potential in [23] and later in [24] by extending the BCS equations
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