1 Low Temperature Laboratory, Aalto University School of Science and Technology, PO Box 15100, FI-00076 AALTO, Finland 2 

# Supercurrent in superconducting graphene 

L.D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia 

$^{3}$The Racah Institute of Physics, Hebrew University of Jerusalem, Israel 

N. B. Kopnin 1, 2 and E. B. Sonin 3 

(Dated: October 14, 2018) 

The problem of supercurrent in superconducting graphene is revisited and the supercurrent is cal- culated within the mean-field model employing the two-component wave functions on a honeycomb lattice with pairing between different valleys in the Brillouin zone. We show that the supercurrent within the linear approximation in the order-parameter-phase gradient is always finite even if the doping level is exactly zero. 

Recent exciting developments in transport experiments on graphene 1 have stimulated theoretical and experi- mental studies of possible superconductivity phenom- ena in this material. Experimentally, there are both hints towards intrinsic superconductivity 2 and observa- tions of proximity-induced superconductivity in graphene layers $^{3}$. Intrinsic superconductivity has been discussed theoretically in the frameworks of phonon and plas- mon mediated mechanisms 4,5 whereas resonating valence bond and density wave lattice models were proposed in Refs. 6–8. It was shown within the BCS model 7,9,10 that the superconducting transition in the undoped graphene possesses a quantum critical point at a finite interaction strength below which the critical temperature vanishes. However, electrons in graphene may become unstable to- wards formation of Cooper pairs for any finite pairing interaction if doping shifts the Fermi level by an amount µ away from the Dirac point $^{4,7,8,10}$. The effect of fluc- tuations on the critical temperature of superconducting transition in graphene has been studied in Ref. 11. A number of unusual features of superconducting state have been predicted, which are closely related to the Dirac- like spectrum of normal state excitations. In particu- lar, the unconventional normal electron dispersion has been shown to result in a nontrivial modification of An- dreev reflection 12 and Andreev bound states in Josephson junctions 13 and vortex cores (see Ref. 14 and references therein). 

Nevertheless, there still remains a controversy regard- ing the most fundamental property of superconducting graphene, i.e., the supercurrent , no matter what the mechanism, intrinsic or extrinsic, of the superconductiv- ity is. In Ref. 7 the supercurrent has been calculated within the framework of the mean field model of super- conducting graphene 7,12 that assumes the Cooper pair- ing between electrons belonging to the same sublattice in the configurational space. According to Ref. 7 the supercurrent calculated as a linear response to the phase gradient of the order parameter disappears in undoped 

PACS numbers: 73.63.-b,74.78.Na,74.25.Jb 

## I. INTRODUCTION 

graphene (i.e., zero shift of the chemical potential, µ = 0) at zero temperature even if the order parameter Δ itself is finite. However, a simpler model based on an effective Dirac type spectrum of normal electrons 10 demonstrates that the supercurrent is always finite as long as supercon- ductivity exists, Δ ̸ = 0. Though the surprising result 7 of “superconductivity without supercurrent” is an alarming indication by itself, the question may be raised, to which extent this difference between the supercurrents is model- dependent $^{15}$, or, if not, what is then the correct behavior of the supercurrent in the low doping limit, µ → 0. In the present paper we revisit this problem and cal- 

culate the supercurrent again using the two-component mean field model of superconductivity in graphene as for- mulated in Refs. 7,12. We show that the supercurrent in fact is always finite . Its value in the low doping limit µ ≪ | Δ | is independent of whether the doping level is exactly zero or not, in contrast to the claim of Ref. 7. This statement qualitatively agrees with the conclusion drawn from the simple model suggested in Ref. 10. 

The paper is organized as follows. In the next Section we outline the model of superconductivity in graphene as formulated in Refs. 7,12 and introduce the basic quan- tities relevant for further calculations. In Section III we calculate the supercurrent within the linear approxima- tion in the order-parameter phase gradient for finite dop- ing levels. The last Section III B deals with the case of low doping µ ≪ | Δ | . Details of calculations are presented in Appendix A and Appendix B. 

## II. BOGOLIUBOV–DE GENNES–DIRAC EQUATIONS 

Transport properties of graphene associated with en- ergies much smaller than the band width are conve- niently described by equations of the Dirac type for two- component wave function whose two components are en- velopes of the true wave functions for two sublattices in the configurational space, Fig. 1(a), near the so called Dirac points K or K ′ in the Brillouin zone of the recip- rocal lattice, Fig. 1(b) (for more details see, for example, 

