Abstract
Using the ab initio anisotropic Eliashberg theory including Coulomb interactions, we investigate the electronphonon interaction and the pairing mechanism in the recentlyreported superconducting Caintercalated bilayer graphene. We find that C_{6}CaC_{6} can support phononmediated superconductivity with a critical temperature T_{c} = 6.8–8.1 K, in good agreement with experimental data. Our calculations indicate that the lowenergy Ca_{xy} vibrations are critical to the pairing and that it should be possible to resolve two distinct superconducting gaps on the electron and hole Fermi surface pockets.
Introduction
Recent progress in the fabrication of metalintercalated bilayer graphene^{1,2,3}, the thinnest limit of graphite intercalation compounds (GICs), opened promising new avenues for studying exotic quantum phenomena in two dimensions. Recently two independent studies reported the observation of superconductivity in Caintercalated bilayer graphene at 4 K^{4} and in Caintercalated graphene laminates around 6.4 K^{5}, while a third study presented evidence for superconductivity in Lidecorated monolayer graphene around 5.9 K^{6}. The electronphonon interaction is expected to play a central role in these observations, hence it is important to develop a detailed understanding of electronphonon physics in these newlydiscovered superconductors.
The strength of the electronphonon coupling (EPC) in graphite and graphene has been widely investigated using angleresolved photoelectron spectroscopy (ARPES), however the interpretation of the results is not always straightforward^{7,8,9,10,11,12,13}. For example the anisotropy of the EPC in alkalimetal decorated graphene generated significant debate in relation to the role of van Hove singularities^{10,11,12,14}. Furthermore the relative importance of the π^{*} bands and of the interlayer (IL) band in the pairing mechanism of bulk CaC_{6}^{15} has been the subject of an intense debate. Indeed several studies suggested that superconductivity arises from the EPC of both bands^{13,16,17,18,19,20}, while ARPES studies proposed that superconductivity sets in either the π^{*} bands or the IL band alone^{21,22}. This debate was reignited by the experimental observation of a freeelectron IL band in Ca and Rbintercalated bilayer graphene^{1,2,3}. The possibility of superconductivity in C_{6}CaC_{6} was suggested theoretically based on the analogy with bulk CaC_{6}^{23,24}, however predictive ab initio calculations of the critical temperature have not yet been reported.
In this work we elucidate the role of the electronphonon interaction in the normal and in the superconducting state of C_{6}CaC_{6} by performing stateoftheart ab initio calculations powered by electronphonon Wannier interpolation^{25,26}. For the normal state we study the electron selfenergy and spectral function in the Migdal approximation; for the superconducting state we solve the anisotropic MigdalEliashberg equations including Coulomb interactions from first principles. Our main findings are: (i) superconductivity in C_{6}CaC_{6} can be explained by a phononmediated pairing mechanism; (ii) in contrast to bulk CaC_{6}, lowenergy Ca vibrations are responsible for the majority of the EPC in the superconducting state; (iii) unlike bulk CaC_{6}, C_{6}CaC_{6} should exhibit two superconducting gaps. For clarity all technical details are described in the Methods.
Figure 1(a) shows a ballandstick model of Caintercalated bilayer graphene, while Fig. 1(b–d) show the corresponding band structure, Brillouin zone and Fermi surface, respectively. Two sets of bands cross the Fermi level around the Γ point. The bands labeled as α^{*} and β^{*} in Fig. 1(b) represent π^{*} states and are obtained by folding the Dirac cone of graphene from K to Γ, following the superstructure induced by Ca. The band labeled as IL is the Caderived nearlyfree electron band, which disperses upwards from about 0.5 eV below the Fermi energy at Γ^{1,23,24}. In Fig. 1(d) we see two sets of Fermi surface sheets: triangular hole pockets around the K′ points, corresponding to the states; and a bundle of small electron pockets around Γ, which arise from the other π^{*} states (, and ) and from the IL band.
In order to quantify the strength of EPCs for each of these bands we calculate the spectral function in the normal state using the Migdal approximation^{27}. Figure 1(e–g) show that the EPC induces sudden changes of slope in the bands, which are referred to as ‘kinks’ in highresolution ARPES experiments. A pronounced kink is seen in all bands at a binding energy of 180 meV. This feature can be assigned to the coupling with the inplane C_{xy} stretching modes^{28} (see Supplementary Fig. S1 for the phonon dispersion relations). The occurrence of this highenergy kink in the π^{*} bands is consistent with the observed broadening of the quasiparticle peaks in the ARPES spectra of bulk CaC_{6} in the same energy range^{13}. This feature results from a sharp peak at 180 meV in the real part of the electron selfenergy of the π^{*} electron pockets, as shown in Supplementary Fig. S2. A second kink is clearly seen at a binding energy of 70 meV and is mostly visible for the π^{*} bands defining the hole pockets [magenta line in Fig. 1(e)] and for the interlayer band [cyan line in Fig. 1(g)]. This lowenergy kink corresponds to a second, smaller peak at the same energy in the real part of the electron selfenergy (Supplementary Fig. S2) and arises from the coupling with the outofplane C_{z} modes of the graphene sheets. Closer inspection reveals also a third kink around 12 meV, however this feature is hardly discernible and is unlikely to be observed in ARPES experiments. This faint structure arises from the coupling to the inplane Ca_{xy} vibrations and can be seen more clearly in the real and the imaginary parts of the electron selfenergy in Supplementary Fig. S2.
From the calculated electron selfenergy we can extract the electronphonon mass enhancement parameter λ_{F} for each band using the ratio between the bare and the renormalized Fermi velocities. Along the ΓK′ direction we obtain λ_{F} = 0.53 (), 0.48 (), 0.30 ( and ) for the π^{*} bands and λ_{F} = 0.68 for the IL band. Along the other two directions ΓM′ and M′K′ the mass enhancement parameters are up to a factor of three smaller than the corresponding values along ΓK′, suggesting a rather anisotropic EPC. Our findings are in agreement with ARPES studies on bulk KC_{6} and CaC_{6} superconductors^{10,22} and also with recent work on Rbintercalated bilayer graphene^{3}.
We now move from the normal state to the superconducting state and consider an electronphonon pairing mechanism in analogy with bulk CaC_{6}. Figure 2(a) shows a comparison between the isotropic Eliashberg spectral functions, α^{2}F(ω) and the cumulative total EPC, λ(ω), calculated for C_{6}CaC_{6} and for bulk CaC_{6}. In both cases we can distinguish three main contributions to the EPC, to be associated with the Ca_{xy} vibrations (~10 meV), the outofplane C_{z} vibrations (~70 meV) and the inplane C_{xy} modes (~180 meV). The Eliashberg functions of C_{6}CaC_{6} and CaC_{6} look similar in shape and the total EPC is λ = 0.71 in both cases. However the relative contributions of each set of modes differ considerably. The lowenergy Ca modes are slightly softer in C_{6}CaC_{6}, leading to a larger contribution to the EPC than in CaC_{6}. These modes account for 60% of the total coupling in the bilayer, while they only account for less than 30% of the coupling in bulk CaC_{6}. In both cases the EPC strength associated with the inplane CC stretching modes is too weak (15% of the total) to make a sizable contribution to the superconducting pairing. This is somewhat counterintuitive, given that the C_{xy} modes lead to the most pronounced kinks in the spectral function in Fig. 1.
The smaller contribution of the outof plane C_{z} modes to the EPC and the softening of the inplane Ca_{xy} vibrations, obtained when going from the bulk to the bilayer, are similar to the results found for Li and Cadecorated monolayer graphene^{29}. In the monolayer case, the removal of quantum confinement causes a shift of the IL wave function farther away from the graphene layer as compared to bulk, and, therefore, the EPC coupling between π^{*} and IL states mediated by C_{z} vibrations is reduced. Although in the bilayer case the IL state is strongly localized around the Ca atom, the fact that the interlayer charge density is only present between the graphene layers gives rise to a weaker coupling of the outofplane C_{z} modes with the interlayer electrons, and, therefore, a lower contribution to the global EPC.
To check whether the softening of the low energy Ca_{xy} modes is due to the difference in the structural parameters between the bilayer and the bulk, we recalculate the vibrational spectrum of bilayer C_{6}CaC_{6} after setting the inplane lattice constant and the interlayer distance to the values of the bulk. As shown in Supplementary Fig. S3, the inplane C_{xy} modes (ω > 105 meV) soften due to the increase in the inplane lattice constant and the outofplane C_{z} and Ca_{z} modes (20 < ω < 45 meV) harden due to the decrease in the interlayer distance. On the other hand, the two lowestlying modes involving mainly inplane Ca_{xy} vibrations are considerably less affected, although a closer inspection reveals an overall slight hardening of approximately 1 meV, particularly along the ΓM′ and M′K′ directions. This suggests that the observed softening of the lowenergy Ca_{xy} modes from bulk to bilayer is most likely caused by changes in the electronic structure which is consistent with an increased density of states at the Fermi level.
In order to determine the superconducting critical temperature of C_{6}CaC_{6} we solve the anisotropic Eliashberg equations^{26,30,31}, with the Coulomb pseudopotential μ^{*} calculated from first principles (the calculation of μ^{*} is discussed below). In Fig. 2(b,c) we plot the energydependent distribution of the superconducting gap, separated into contributions corresponding to the two sets of Fermi surfaces centered around the Γ and K′ points. We see that two distinct gaps open on the Γcentered electron pockets, with average values Δ_{1} = 0.55 meV and Δ_{2} = 1.29 meV at zero temperature. The Δ_{1} gap is characterized by a very narrow energy profile and the EPC on these pockets is essentially isotropic, resulting primarily from the coupling with the Ca_{xy} phonons [Fig. 2(d)]. The Δ_{2} gap exhibits a much broader energy profile (0.82 < Δ_{2} < 1.75 meV) and originates mainly from the coupling to Ca modes and outofplane C_{z} phonons [Fig. 2(e)]. In between the two Γcentered gaps, a third gap with an average value Δ_{3} = 0.99 meV opens on the triangular hole pockets around K′ [ states] [Fig. 2(f)]. These states couple primarily to C_{xy} phonons and the gap has an anisotropic character with a large spread in energy (0.73 < Δ_{3} < 1.25 meV). In Supplementary Fig. S4 we show the anisotropic EPC parameters λ_{k} leading to this superconducting gap structure. For completeness, we compare our results with the gap structure of bulk CaC_{6}. In the latter case, only one superconducting gap is predicted (see Supplementary Fig. S5), in agreement with previous theoretical studies^{20,32} and experiments^{33,34}. Although multiple sheets of the Fermi surface contribute to the superconducting gap, there is no separation into distinct gaps, giving rise to a smeared multigap structure^{20,32}. This situation is similar to the Δ_{2} and Δ_{3} gaps in the bilayer case. Based on our results we suggest that in C_{6}CaC_{6} highresolution ARPES experiments might be able to resolve two distinct gaps on the electron pockets, corresponding to Δ_{1} and Δ_{2}, but only one gap on the triangular hole pockets, corresponding to Δ_{3}.
The calculations of the gap function and the superconducting critical temperature in Fig. 2(b,c) require the knowledge of the Coulomb pseudopotential μ^{*}. In order to determine this parameter we first calculate the dimensionless electronelectron interaction strength within the randomphase approximation, obtaining μ = 0.254 (see Methods). Then we renormalize this interaction using μ^{*} = μ/[1 + μlog(ω_{pl}/ω_{ph})]^{35}, where ω_{pl} and ω_{ph} are characteristic electron and phonon energy scales, respectively. We set ω_{pl} = 2.5 eV, corresponding to the lowest plasmon resonance in GICs^{36,37} and ω_{ph} = 200 meV, corresponding to the highest phonon energy in C_{6}CaC_{6}. Figure 2(g) shows the variation of the Coulomb pseudopotential μ_{k} across the Fermi surface. The repulsive interaction is strongest on the electron pockets and weakest on the hole pockets. Since the EPC exhibits a very similar anisotropy, the net coupling strength is only moderately anisotropic and can be replaced by a single, average μ^{*} in the Eliashberg equations. From Fig. 2(g) we obtain μ^{*} = 0.155 and this is the value employed in Fig. 2(b,c). As we can see in Fig. 2(b,c), our Eliashberg calculations yield a superconducting critical temperature T_{c} = 8.1 K, which is only slightly higher than the experimental value K reported in ref.^{4}.
In order to check the role of anisotropic Coulomb interactions we repeat the calculations by considering a Coulomb pseudopotential resolved over the electron and hole pockets. In this case we find the decrease in T_{c} to be very small, ΔT_{c} = 0.3 K. Furthermore we explore the sensitivity of the calculated T_{c} to the choice of the characteristic phonon energy ω_{ph}. To this aim we solve the Eliashberg equations again, this time by setting ω_{ph} equal to the Matsubara frequency cutoff (5 × 200 meV, see Methods). This alternative choice leads to μ^{*} = 0.207 and T_{c} = 6.8 K (Supplementary Fig. S6), which is in even better agreement with experiment. For completeness in Supplementary Fig. S7 we also show the dependence of T_{c} on the characteristic energy ω_{c} as obtained using the standard McMillan formula^{38}. Consistent with our Eliashberg calculations, we find that large variations of ω_{c} only change T_{c} by a few K’s. These additional tests show that our results are solid, therefore we can safely claim that the ab initio Eliashberg theory yields T_{c} = 6.8–8.1 K for C_{6}CaC_{6}. The close agreement between these values and experiment supports the notion that Caintercalated bilayer graphene is a conventional phononmediated superconductor.
In conclusion, we studied entirely from first principles the electronphonon interaction and the possibility of phononmediated pairing in the newlydiscovered superconducting C_{6}CaC_{6}. We showed that the Ca vibrations play an important role in the pairing but do not carry a sharp signature in the normalstate band structure; conversely the highfrequency inplane Cxy vibrations lead to pronounced photoemission kinks but have a small contribution to the pairing. The good agreement between the critical temperature calculated here and the recent experiments of ref. 4 indicate that Caintercalated bilayer graphene is an electronphonon superconductor. The present work calls for highresolution spectroscopic investigations, as well as for calculations based on alternative ab initio methods^{39,40}, in order to test our prediction of two distinct superconducting gaps in C_{6}CaC_{6}.
Methods
The calculations are performed within the local density approximation to densityfunctional theory^{41,42} using planewaves and normconserving pseudopotentials^{43,44}, as implemented in the QuantumESPRESSO suite^{45}. The planewaves kinetic energy cutoff is 60 Ry and the structural optimization is performed using a threshold of 10 meV/Å for the forces. C_{6}CaC_{6} is described using the R30° supercell of graphene with one Ca atom per unit cell and periodic images are separated by 15 Å. The optimized inplane lattice constant and interlayer separation are a = 4.24 Å and d = 4.50 Å, respectively. Bulk CaC_{6} is described using the rhombohedral lattice and the optimized lattice constant and rhombohedral angle are a = 5.04 Å and α = 50.23°, respectively. The electronic charge density is calculated using an unshifted Brillouin zone mesh with 24^{2} and 8^{3} kpoints for C_{6}CaC_{6} and CaC_{6}, respectively and a MethfesselPaxton smearing of 0.02 Ry. The dynamical matrices and the linear variation of the selfconsistent potential are calculated within densityfunctional perturbation theory^{46} on the irreducible set of regular 6^{2} (C_{6}CaC_{6}) and 4^{3} (CaC_{6}) qpoint grids.
The electron selfenergy, spectral function and superconducting gap are evaluated using the EPW code^{25,26,47}. The electronic wavefunctions required for the WannierFourier interpolation^{48,49} in EPW are calculated on uniform unshifted Brillouinzone grids of size 12^{2} (C_{6}CaC_{6}) and 8^{3} (CaC_{6}). The normalstate selfenergy is calculated on fine meshes consisting of 100,000 inequivalent qpoints, using a broadening parameter of 10 meV and a temperature T = 20 K. For the anisotropic Eliashberg equations we use 120 × 120 (40 × 40 × 40) kpoint grids and 60 × 60 (20 × 20 × 20) qpoint grids for C_{6}CaC_{6} (CaC_{6}). The Matsubara frequency cutoff is set to five times the largest phonon frequency (5 × 200 meV) and the Dirac delta functions are replaced by Lorentzians of widths 100 meV and 0.5 meV for electrons and phonons, respectively. The technical details of the Eliashberg calculations are described extensively in ref. 26.
The electronelectron interaction strength is obtained as μ = N_{F}〈〈V_{k,k′}〉〉_{FS}, where N_{F} is the density of states at the Fermi energy, and W is the screened Coulomb interaction within the random phase approximation^{50}. Here 〈〈⋅〉〉_{FS} indicates a double Fermisurface average and k stands for both momentum and band index. The screened Coulomb interaction is calculated using the Sternheimer approach^{51,52}. Linearresponse equations are solved using 26 × 26 Brillouinzone grids, corresponding to 70 inequivalent points. The energy cutoff of the dielectric matrix is 10 Ry (815 planewaves).
Additional Information
How to cite this article: Margine, E. R. et al. Electronphonon interaction and pairing mechanism in superconducting Caintercalated bilayer graphene. Sci. Rep. 6, 21414; doi: 10.1038/srep21414 (2016).
References
Kanetani, K. et al. Ca intercalated bilayer graphene as a thinnest limit of superconducting C6Ca. Proc. Natl. Acad. Sci. USA. 109, 19610–19613 (2012).
Kleeman, J., Sugawara, K., Sato, T. & Takahashi, T. Direct evidence for a metallic interlayer band in Rbintercalated bilayer graphene. Phys. Rev. B 87, 195401 (2013).
Kleeman, J., Sugawara, K., Sato, T. & Takahashi, T. Anisotropic electronphonon coupling in Rbintercalated bilayer graphene. Journal of the Physical Society of Japan 83, 124715 (2014).
Ichinokura, S., Sugawara, K., Takayama, A., Takahashi, T. & Hasegawa, S. Superconducting calcium intercalated bilayer graphene. DOI: 10.1021/acsnano.5b07848 (2016).
Chapman, J. et al. Superconductivity in Cadoped graphene. arXiv:1508.06931 (2015).
Ludbrook, B. M. et al. Evidence for superconductivity in Lidecorated monolayer graphene. Proc. Natl. Acad. Sci. USA. 112, 11795–11799 (2015).
Bianchi, M. et al. Electronphonon coupling in potassiumdoped graphene: Angleresolved photoemission spectroscopy. Phys. Rev. B 81, 041403 (2010).
Siegel, D. A., Hwang, C., Fedorov, A. V. & Lanzara, A. Electronphonon coupling and intrinsic bandgap in highlyscreened graphene. New Journal of Physics 14, 095006 (2012).
Haberer, D. et al. Anisotropic Eliashberg function and electronphonon coupling in doped graphene. Phys. Rev. B 88, 081401 (2013).
Grüneis, A. et al. Electronic structure and electronphonon coupling of doped graphene layers in KC8 . Phys. Rev. B 79, 205106 (2009).
McChesney, J. L. et al. Extended van Hove singularity and superconducting instability in doped graphene. Phys. Rev. Lett. 104, 136803 (2010).
Fedorov, A. V. et al. Observation of a universal donordependent vibrational mode in graphene. Nat. Commun. 5, 3257 (2014).
Yang, S. L. et al. Superconducting graphene sheets in CaC6 enabled by phononmediated interband interactions. Nat. Commun. 5, 3493 (2014).
Park, C.H. et al. Van Hove singularity and apparent anisotropy in the electronphonon interaction in graphene. Phys. Rev. B 77, 113410 (2008).
Weller, T. E., Ellerby, M., Saxena, S. S., Smith, R. P. & Skipper, N. T. Superconductivity in the intercalated graphite compounds C6Yb and C6Ca. Nat. Phys. 1, 39–41 (2005).
Mazin, I. I. Intercalantdriven superconductivity in YbC6 and CaC6 . Phys. Rev. Lett. 95, 227001 (2005).
Calandra, M. & Mauri, F. Theoretical explanation of superconductivity in C6Ca. Phys. Rev. Lett. 95, 237002 (2005).
Kim, J. S., Kremer, R. K., Boeri, L. & Razavi, F. S. Specific heat of the Caintercalated graphite superconductor CaC6 . Phys. Rev. Lett. 96, 217002 (2006).
Boeri, L., Bachelet, G. B., Giantomassi, M. & Andersen, O. K. Electronphonon interaction in graphite intercalation compounds. Phys. Rev. B 76, 064510 (2007).
Sanna, A. et al. Anisotropic gap of superconducting CaC6: A firstprinciples density functional calculation. Phys. Rev. B 75, 020511(R) (2007).
Sugawara, K., Sato, T. & Takahashi, T. Fermisurfacedependent superconducting gap in C6Ca. Nat. Phys. 5, 40 (2009).
Valla, T. et al. Anisotropic electronphonon coupling and dynamical nesting on the graphene sheets in superconducting C6Ca using angleresolved photoemission spectroscopy. Phys. Rev. Lett. 102, 107007 (2009).
Mazin, I. I. & Balatsky, A. V. Superconductivity in Caintercalated bilayer graphene. Philos. Mag. Lett. 90, 731–738 (2010).
Jishi, R. A. & Guzman, D. M. Theoretical investigation of twodimensional superconductivity in intercalated graphene layers. Adv. Studies Theor. Phys. 5, 703–716 (2011).
Giustino, F., Cohen, M. L. & Louie, S. G. Electronphonon interaction using Wannier functions. Phys. Rev. B 76, 165108 (2007).
Margine, E. R. & Giustino, F. Anisotropic MigdalEliashberg theory using Wannier functions. Phys. Rev. B 87, 024505 (2013).
Giustino, F., Cohen, M. L. & Louie, S. G. Small phonon contribution to the photoemission kink in the copper oxide superconductors. Nature 452, 975–978 (2008).
Park, C.H., Giustino, F., Cohen, M. L. & Louie, S. G. Electronphonon interactions in graphene, bilayer graphene and graphite. Nano Letters 8, 4229–4233 (2008).
Profeta, G., Calandra, M. & Mauri, F. Phononmediated superconductivity in graphene by lithium deposition. Nat. Phys. 8, 131 (2012).
Allen, P. B. & Mitrović, B. Theory of superconducting Tc . Solid State Phys. 37, 1–92 (1982).
Margine, E. R. & Giustino, F. Twogap superconductivity in heavily ndoped graphene: Ab initio MigdalEliashberg theory. Phys. Rev. B 90, 014518 (2014).
Sanna, A. et al. Phononic selfenergy effects and superconductivity in CaC6 . Phys. Rev. B 85, 184514 (2012).
Bergeal, N. et al. Scanning tunneling spectroscopy on the novel superconductor CaC6 . Phys. Rev. Lett. 97, 077003 (2006).
Lamura, G. et al. Experimental evidence of swave superconductivity in bulk CaC6 . Phys. Rev. Lett. 96, 107008 (2006).
Morel, P. & Anderson, P. W. Calculation of the superconducting state parameters with retarded electronphonon interaction. Phys. Rev. 125, 1263–1271 (1962).
Mele, J. J., E. J. & Ritsko Dielectric response and intraband plasmon dispersion in stage1 FeCl3 intercalated graphite. Solid State Comm. 33, 937–940 (1980).
Echeverry, J. P., Chulkov, E. V., Echenique, P. M. & Silkin, V. M. Lowenergy plasmonic structure in CaC6 . Phys. Rev. B 85, 205135 (2012).
McMillan, W. L. Transition temperature of strongcoupled superconductors. Phys. Rev. 167, 331–344 (1968).
Lüders, M. et al. Ab initio theory of superconductivity. I. Density functional formalism and approximate functionals. Phys. Rev. B 72, 024545 (2005).
Marques, M. A. L. et al. Ab initio theory of superconductivity. II. Application to elemental metals. Phys. Rev. B 72, 024546 (2005).
Ceperley, D. M. & Alder, B. J. Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 45, 566–569 (1980).
Perdew, J. P. & Zunger, A. Selfinteraction correction to densityfunctional approximations for manyelectron systems. Phys. Rev. B 23, 5048–5079 (1981).
Troullier, N. & Martins, J. L. Efficient pseudopotentials for planewave calculations. Phys. Rev. B 43, 1993–2006 (1991).
Fuchs, M. & Scheffler, M. Ab initio pseudopotentials for electronic structure calculations of polyatomic systems using densityfunctional theory. Computer Physics Communications 119, 67–98 (1999).
Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and opensource software project for quantum simulations of materials. Journal of Physics: Condensed Matter 21, 395502 (2009).
Baroni, S., de Gironcoli, S., Dal Corso, A. & Giannozzi, P. Phonons and related crystal properties from densityfunctional perturbation theory. Rev. Mod. Phys. 73, 515–562 (2001).
Noffsinger, J. et al. EPW: A program for calculating the electronphonon coupling using maximally localized Wannier functions. Computer Physics Communications 181, 2140–2148 (2010).
Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: Theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012).
Mostofi, A. A. et al. wannier90: A tool for obtaining maximallylocalised Wannier functions. Computer Physics Communications 178, 685–699 (2008).
Lee, K.H., Chang, K. J. & Cohen, M. L. Firstprinciples calculations of the Coulomb pseudopotential μ^{*}: Application to al. Phys. Rev. B 52, 1425–1428 (1995).
Lambert, H. & Giustino, F. Ab initio SternheimerGW method for quasiparticle calculations using plane waves. Phys. Rev. B 88, 075117 (2013).
Giustino, F., Cohen, M. L. & Louie, S. G. GW method with the selfconsistent sternheimer equation. Phys. Rev. B 81, 115105 (2010).
Emery, N. et al. Superconductivity of bulk CaC6 . Phys. Rev. Lett. 95, 087003 (2005).
Acknowledgements
H.L. and F.G. acknowledge support from the European Research Council (EU FP7/grant no. 604391 Graphene Flagship and EU FP7/ERC grant no. 239578).
Author information
Affiliations
Contributions
E.R.M. carried out the calculations of the electron selfenergy and the spectral function for the normal state and the calculations of the gap function for the superconducting state. H.L. carried out the calculations of the Coulomb pseudopotential. All authors analysed the results and contributed with writing the paper.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Electronic supplementary material
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Margine, E., Lambert, H. & Giustino, F. Electronphonon interaction and pairing mechanism in superconducting Caintercalated bilayer graphene. Sci Rep 6, 21414 (2016). https://doi.org/10.1038/srep21414
Received:
Accepted:
Published:
Further reading

Excitations of Intercalated Metal Monolayers in Transition Metal Dichalcogenides
Nano Letters (2021)

Superconductivity in alkaline earth metal doped boron hydrides
Physica B: Condensed Matter (2021)

Singlelayer polymeric tetraoxa[8]circulene modified by sblock metals: toward stable spin qubits and novel superconductors
Nanoscale (2021)

Stability and superconductivity of Caintercalated bilayer blue phosphorene
Physical Chemistry Chemical Physics (2021)

Nonadiabatic superconductivity in a Liintercalated hexagonal boron nitride bilayer
Beilstein Journal of Nanotechnology (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.