The possibility of a superconductivity state in metal decorated graphene has been suggested theoretically by a few groups9,12,15. Some have suggested phonon-mediated superconductivity in single layer graphene. Most prominently, Profeta et al.15 calculated on the basis of density functional theory for superconductors that decoration by electron donating atoms such as Ca and Li will make single layer graphene superconducting, up to 8 K for the case of Li. The ab initio anisotropic Migdal-Eliashberg formalism was used by Zheng and Margine24, who predicted a single anisotropic superconducting gap with critical temperature T c = 5.1–7.6 K, in surprisingly good agreement with experimental reported superconductivity around 6 K in LiC 6 5.

Using a phenomenological microscopic Hamiltonian in a nearest-neighbor tight-binding approximation, possible superconducting phases of pristine graphene have been discussed by Uchoa and Castro-Neto9 and also by Black-Schaffer and Doniach12. The possibility of a singlet p + ip phase pairing near the Dirac points between nearest neighbors subsites were suggested by Uchoa and Castro-Neto9. They worked in terms of a plasmon mediated mechanism for metal coated graphene, and discussed the conditions under which attractive electron-electron interaction can be mediated by plasmons.

Singlet superconducting gap phases of pristine graphene have been proposed and discussed by Black-Schaffer and Doniach12. For the nearest neighbors pairing amplitudes \({{\rm{\Delta }}}_{\langle iAjB\rangle }={{\rm{\Delta }}}_{iA,iA+{\overrightarrow{\delta }}_{j}}\) where \({\overrightarrow{\delta }}_{j}\) are the vectors that connects the iA site to its three nearest neighbors, it was observed that there are three states that minimize the free energy in various regimes of the parameters, which here have been denoted by V s = (1, 1, 1)T, \({V}_{{d}_{{x}^{2}-{y}^{2}}}={(2,-1,-1)}^{T}\), and \({V}_{{d}_{xy}}={(0,-1,1)}^{T}\). Pairing symmetries d xy and \({d}_{{x}^{2}-{y}^{2}}\) are degenerate, and only the linear combination of \({d}_{{x}^{2}-{y}^{2}}+i{d}_{xy}\equiv d+id\) preserves the graphene band symmetry. Depending on the position of the Fermi energy with respect to Dirac points, d + id or s states tend to dominate. Their numerical calculation showed that d-wave solutions will always be favored for electron or hole doping in the regime \(0 < {\bar{n}}_{c} < 0.4\) where doping is defined by \({\bar{n}}_{\alpha }=\langle {\hat{c}}_{i\alpha }^{\dagger }{\hat{c}}_{i\alpha }\rangle -1\). In this regime, superconductivity can emerge from electronic correlation effects. Near the van Hove singularity at the saddle point M corresponding to 3/8 and 5/8 fillings i.e. \({\bar{n}}_{c}=0.25\), it was suggested that chiral d + id superconductivity, which breaks time-reversal symmetry, can be stabilized. In this regime d-wave superconductivity may arise from repulsive electron-electron interaction11.

Although doping by a gate voltage is normally considered to change only the chemical potential but not the band structure, gating cannot be expected to push the Fermi energy to the van Hove singularity without altering the band dispersion. The most likely way to do this is by decoration with electropositive atoms, which has been our focus. We note that doping is essential, when graphene decorated, in addition to the expected charge migration from the decorating atoms to the graphene sheet, it is then necessary the interlayer state is partially occupied to induce superconductivity as happens in GICs. Hybridization of interlayer s-band and graphene π bands changes the graphene band structure. The s orbitals of Ca have more overlap with C orbitals than Li and lead to stronger and longer range interactions as well as increasing the doping level, effects that become detrimental to superconductivity. For this reason our emphasis here is on the Li decorated graphene.

We review some of our main points. When graphene is decorated by Li, electron transfer from Li atoms to C contracts the Li-C distance and reduces the C-C bond lengths in the Li-centered hexagon. In this kekulé -type structure, hopping amplitude symmetries of all C-C neighbors are broken (our “shrunken graphene”). This model allows study of multiband effects on the superconducting phase diagram. To gain insight into our model, solutions of superconducting gap equation in both cases of folded bands otherwise pristine C 6 and the usual two band model of C 2 were compared. These two viewpoints coincide if the same pairing paradigms are considered. For pristine graphene with its two site cell, in real space picture electrons can pair with near neighbors in three inequivalent directions, \({{\rm{\Delta }}}_{i,i+\overrightarrow{\delta }}={V}_{sy}={({{\rm{\Delta }}}_{1}{{\rm{\Delta }}}_{2}{{\rm{\Delta }}}_{3})}^{T}\) which must respect honeycomb symmetries. The V sy quantities are the three vectors that belong to the irreducible representation of crystal point group D 6h i.e. \({V}_{sy}^{T}\) = (1, 1, 1), (−1, 1, 0) and (2, −1, −1) for which the sy subscript stands for symmetries s, d xy and \({d}_{{x}^{2}-{y}^{2}}\). Permutation of s-wave solution (1, 1, 1) along three different bonds constructs just one state while permutation of d xy solution (−1, 1, 0) up to a minus sign constructs two nonorthogonal linear independent states viz. (−1, 1, 0) and (−1, 0, 1) which orthogonal linear combination of them are \({{\rm{d}}}_{xy}^{T}\) = (−1, 1, 0) and \({d}_{{x}^{2}-{y}^{2}}^{T}\) = (2 −1 −1).

A similar procedure again can be applied to pristine graphene but now in enlarged six site unit cell. Unit cell of C 6 includes six carbon subsites and nine different bonds that support nine possible nearest neighbor bond pairing amplitudes as illustrated in Fig. 4 and denoted them by \({{\rm{\Phi }}}_{sy}^{T}=[({{\rm{\Delta }}}_{1}^{^{\prime\prime} },{{\rm{\Delta }}}_{2}^{^{\prime\prime} }\,{{\rm{\Delta }}}_{3}^{^{\prime\prime} })\,({{\rm{\Delta }}}^{1},{{\rm{\Delta }}}^{2},{{\rm{\Delta }}}^{3})\,({{\rm{\Delta }}}_{1}^{^{\prime} },{{\rm{\Delta }}}_{2}^{^{\prime} }\,{{\rm{\Delta }}}_{3}^{^{\prime} })]\). The gap equation is a 9 × 9 matrix equation given by Eq. 14. The folded bands supercell include three vertices numbered 5, 6, 7, and nine bonds as shown in Fig. 4(a). There are nine orthogonal solutions that preserve symmetries of this supercell. One of these configurations has s-wave symmetry (1, 1, 1, 1, 1, 1, 1, 1, 1) the other eight solutions are constructed by all possible permutations of (−1, 1, 0) along these bonds that preserve our supercell symmetry. There are only three solutions which can preserve symmetry of both two and six atoms cells simultaneously which they are of the form \({{\rm{\Phi }}}_{sy}^{+}={({V}_{sy}{V}_{sy}{V}_{sy})}^{T}\) as illustrated in Fig. 4. For these solutions, the folded 9 × 9 gap equation reduces to 3 × 3 gap equations of ordinary pristine graphene. The Cooper pair formation energy for these three modes are significantly less than the other six phases which are not reducible to the two band model.

In fact reduction of symmetry leads to increasing of the system free energy. After the orthogonalization procedure, one obtains three solutions Φ f , \({{\rm{\Phi }}}_{{p}_{x}}\) and \({{\rm{\Phi }}}_{{p}_{y}}\), of the form \({{\rm{\Phi }}}_{sy}^{0}={\mathrm{(0}{V}_{sy}-{V}_{sy})}^{T}\). These phases have been designated as island phases, as illustrated in Fig. 5(b) for Φ f , within which a pairing amplitude is localized within island hexagons and cannot propagate. For these island phases, numerical calculation of the electron pair potential energy g 0 shows that g 0 is large. This kind of solutions is a consequence of the six atom basis and does not appear for the two atom basis. Also, there are three solutions of the form \({{\rm{\Phi }}}_{sy}^{-}=(\,-\,2{V}_{sy}\,{V}_{sy}\,{V}_{sy})\) which also break symmetry of two atom cell. For these reasons, in association with the normal state band structure of graphene, we concentrate on superconductivity in the three \({{\rm{\Phi }}}_{sy}^{+}\) symmetry phases.

For pristine graphene C 2 , two normal bands are E ± = ±t 1 |η 0 | which fold to six branches in mini-BZ of C 6 i.e. \({E}_{\gamma }^{\pm }=\pm \,{t}_{1}|{\eta }_{0}|\), \({E}_{\beta }^{\pm }=\pm \,{t}_{1}|{\eta }_{1}|\) and \({E}_{\alpha }^{\pm }=\pm \,{t}_{1}|{\eta }_{2}|\) as shown in figure:eta-k, also Bloch-wave symmetry character of each branch has been distinguished. The Bloch coefficients of the branch labeled by γ are of s-wave character, \({C}_{{A}_{i}}=(1,1,1)\) and for those labeled as α and β are of the form d ± id type, i.e. \({C}_{{A}_{i}}=(1,{e}^{\pm i\frac{i2\pi }{3}},{e}^{\pm i\frac{i4\pi }{3}})\). Based on Bloch wave character of these branches one can obtain the dominant superconducting phases of pristine graphene in various doping regimes. d-wave pairing emerges from the d-wave branches of the folded band structure \({E}_{\alpha }^{\pm }\) and \({E}_{\beta }^{\pm }\), while s-wave pairing arises from the s-wave branch \({E}_{\gamma }^{\pm }\). For folded but otherwise pristine graphene, Fig. 2 illustrates that the lowest conduction band, weakly dispersive along Γ → M, is responsible for dominant singlet superconductivity in chiral d ± id symmetry. Upon electron doping to the critical vHs at \({\bar{n}}_{c}=0.25\), the pairing potential g 0 in the d ± id phase decreases, beyond which density of states decreases. g 0 increases until a second critical value of doping \({\bar{n}}_{c}=0.4\) at which a phase transition to s-wave pairing occurs. Bloch states in higher conduction bands include combinations of s and f symmetries that favor extended s wave pairing. The multiband character is responsible for stabilizing singlet s superconductivity at high electron or hole doping.

To understand how superconducting phases of graphene can be affected by decoration by Li, one can compare the LiC 6 gap solutions with those of folded bands C 6 at the same doping. Numerical results for pristine graphene gap equation performed in the nearest neighbor approximation in ref.12 have been extended by applying a more accurate tight binding model fit to the DFT band structure of pristine graphene23. Although a quantum critical point for zero doping reported by Black-Schaffer and Doniach12 at dimensionless coupling \(\frac{{g}_{0}}{t}=1.91\) which d- and s-wave solutions are degenerate. In the more realistic tight binding model we applied, this degeneracy is not observed at the Γ point, and the d-wave solution is dominant. This difference may be consequence of particle-hole symmetry breaking of valence and conduction bands. Also the van Hove singularity at the M point is moved from 0.25 doping for nearest neighbor hopping to 0.16 doping in the accurate model. The phase transition from d-wave to s-wave is shifted to 0.35 doping instead of the 0.4 doping reported for nearest neighbor hopping12. Numerical calculations for this more detailed model are illustrated in Fig. 7.

Figure 7 Shows cooper pair interaction g 0 in terms of doping \(\bar{n}\) for d and s-wave phases for pristine graphene at T = 0.1 K. The solid (dashed) red line indicates d- wave (s- wave) pairing interaction in first nearest neighbor hopping t 1 = 2.5 eV and similarly green line for accurate tight binding model can fit on DFT. For red line at the charge neutrality s- and d- wave are degenerate with g 0 = 4.76 while for full approximation they are not degenerate. Full size image

When graphene is decorated by Li, around 0.68 electron per lithium atom transfers to neighboring C sites, viz. \({\bar{n}}_{c}=0.11\), and the Dirac points folded to Γ move to −1.52 eV. Symmetry breaking of the hopping partially removes degeneracies of band structure of pristine graphene, which leads to creation of the small gap at Γ, with energy \({E}_{g}=2|{t}_{1}-{t}_{1}^{^{\prime} }|=0.36\,eV\). Also two of four-fold degeneracies between valence and conduction bands at the Dirac points are removed. Compression between band structure of decorated graphene and folded pristine graphene at the same doping shows that hybridization of the Li s band and C π band is small. This means nearest neighbor Li-C hopping is in the range \({t}_{1}^{LiC}\) ~ 0.3–0.5, and further hoppings are negligible.

Li decoration of graphene changes not only the band structure but also the Bloch wave coefficients from those of pristine graphene. While pristine graphene Bloch wave coefficients have pure s- or d-wave character and their magnitudes are \(\overrightarrow{k}\)-independent. In the case of LiC 6 they become mixed and vary with \(\overrightarrow{k}\), hence gap equation symmetry is reduced. Because of this symmetry reduction, for the longer C-C bonds, a new coefficient α sy appears in the pairing amplitudes. In terms of this coefficient we have classified superconducting phase symmetries into three groups. Eqs 18, 19, and 20 present all nine possible pairing phases of LiC 6 . There are three categories of solutions which have not appeared in complete form in the literature. The total of nine phases arise from spatial, and therefore hopping parameter, symmetry breaking.

In the first category Φ f , \({{\rm{\Phi }}}_{{p}_{x}}\) and \({{\rm{\Phi }}}_{{p}_{y}}\), there is α sy = 0 identical to that of folded pristine C 6 . For the second category, α sy (denoted by α−) is negative, in the case of pristine α− = −2 as discussed. These three phases break the two site cell symmetry, and numerical calculation shows that the pairing potential g 0 must be large to realize these phases. For the last category α+ is positive. Three phases which correspond to α+ > 0 include \({{\rm{\Phi }}}_{{d}_{{x}^{2}-{y}^{2}}}^{+}\), \({{\rm{\Phi }}}_{{d}_{xy}}^{+}\), and \({{\rm{\Phi }}}_{s}^{+}\), and these have the lowest pairing potentials with respect to the other six phases.

In the limiting case of folded six band pristine graphene \({\alpha }_{{d}_{{x}^{2}-{y}^{2}}}^{+}\), \({\alpha }_{{d}_{xy}}^{+}\), and \({\alpha }_{s}^{+}\) are all equal to unity, which maps the results to the two-band symmetries as it should. But when Li decorated, depending on doping strength viz. w t and \({t}_{1}^{LiC}\) these coefficients \({\alpha }_{sy}^{+}\) no longer remain unity. The pairing amplitude distortion along longer C-C bonds α+, for s-wave phase is significant due to its spatial isotropic symmetry. In spite of the pristine nature this phase no longer preserves two band model symmetry. On the other hand, d-wave phases are hardly affected by doping and their superconductivity is more persistent against perturbation. The chirality or non-chirality of Cooper pairs in these phases is undetermined, however. As shown in Fig. 6(b), at low temperature \({\alpha }_{s}^{+}\approx 0.6\) for \({{\rm{\Phi }}}_{s}^{+}\), and \({\alpha }_{{d}_{{x}^{2}-{y}^{2}}}^{+}={\alpha }_{{d}_{xy}}^{+}\equiv {\alpha }_{d}^{+}\) is approximately equal to unity and varies little with temperature.

At a given critical temperature T c and chemical potential μ 0 , for each of nine possible superconducting phases, Eqs 10, 13 and 17 were evaluated numerically over the BZ of LiC 6 to find the corresponding pairing potential \({g}_{0}=\frac{1}{{J}_{sy}}\) and α sy coefficient. Smaller g 0 means less Cooper pair formation energy is required. Figure 6(a) provides the phase boundaries for T c in terms of the pairing potential g 0 for LiC 6 in which μ 0 = 0. For a given transition temperature T c , by changing the chemical potential μ 0 of LiC 6 via gating, one can engineer the pairing potential g 0 . Figure 8 gives a g 0 -μ 0 phase boundary diagram at T c = 0.1 K. As illustrated in this figure, similarly to pristine graphene, decoration with Li atoms makes it is possible to change the dominant pairing and to have a symmetry-change phase transition from d to “distorted s-wave.” Changing μ o up to μ o−v ≈ 0.22 eV so that the distance between the Fermi energy and the saddle points decreases, leads to a decrease in g 0 . Continuously increasing μ o up to 0.5 eV causes g 0 to increase for both d-wave and “distorted s-wave” pairing, and after that a smooth decrease proceeds. For both symmetries at critical μ o−c = 1.3 eV mixed state exist.

Figure 8 This diagram illustrates interaction potential g 0 in terms of chemical potential μ 0 at T c = 0.1 K. Upon electron doping to a critical chemical potential μ o−v = 0.22 eV (van Hove singularity) for symmetries \({{\rm{\Phi }}}_{{d}_{{x}^{2}-{y}^{2}}}^{+}\), \({{\rm{\Phi }}}_{{d}_{xy}}^{+}\), and \({{\rm{\Phi }}}_{s}^{+}\) the pairing potential decreases, then increases until a second critical value μ o−c = 1.3 eV at which a phase transition to \({{\rm{\Phi }}}_{s}^{+}\) occurs. Full size image

Up to μ o−c = 1.3 eV, the flat band plays a primary role in formation of Cooper pairs with lowest energy. The Bloch wave function of this band consists of d and p character, therefore Γ 12 , Γ 15 , Γ 45 and Γ 48 in Eq. 13 carry minus signs. This makes it evident from Eq. 17 that d–wave pairing is dominant. Beyond that, the uneven part of the “flat band” and also upper bands assume a major role. These bands consist of d, p, s, and f character Bloch wave functions (as defined in earlier sections) with a significantly low density of states. In this case Γ 12 , Γ 15 , Γ 45 and Γ 48 change their sign, hence s-wave pairing is favored.

Numerically we have demonstrated that electron pairing g 0 in the limit of pristine graphene is minimal for all dopings. Our calculations indicate that any perturbation of the flat band reduces T c . The flat band can be perturbed through electron hopping from decorating atoms to carbon sites (\({t}_{1}^{LiC}\)) or by hopping symmetry breaking index w t . For fixed doping at \(\bar{n}\) = 0.11 electron per carbon site and for fixed w t = 0.94 as obtained for lithium decorated, in a variety of Li-C hopping between 0.3–0.4 eV, numerical calculation doesn’t show significant altering of pair interaction potential g 0 in s- and d-wave phases. But, as one could see there is not an explicit behavior in a general coupling strength. A result is that a general aspect of superconducting pairing in LiC 6 and pristine graphene is almost the same in the \({d}_{{x}^{2}-{y}^{2}}\) and d xy phases due to robustness of the flat band against perturbation.