The debate on the superconducting mechanism of CaC6 has centred on the respective roles of the π* and interlayer (IL) bands1,2,3,4,5,6,7,8,9,10,11,12,13,14,15. Some theoretical studies proposed that strong phonon-mediated interactions between these two bands are required to explain superconductivity1,4,5,6, and would lead to both bands being superconducting and having superconducting gaps. This fully gapped picture was consistent with a number of experiments7,8,9,10, but challenged by angle-resolved photoelectron spectroscopy (ARPES)11,12,13. Valla et al.11 claimed that electron–phonon coupling (EPC) involving the π* band alone can explain superconductivity, and found no evidence for the existence of the IL band. In contrast, Sugawara et al.12 reported that the π* band is gapless, which suggested that superconductivity exists on the IL band only. Clearly, no consensus has been reached regarding which bands participate in superconductivity.

As we show in this Article, our ARPES measurement resolves the π* and IL bands with unprecedented clarity. Our analysis of the superconducting gaps and EPC on both bands yields compelling evidence for the picture in which the π*-IL interband interaction contributes >50% to the total EPC strength to enable superconductivity1,4,5,6.


Electronic band structure

We begin by presenting the Fermi surface map taken with 9 eV photons in Fig. 1a. The most pronounced features are a warped hexagonal pocket and a near-circular pocket. The hexagonal pocket can be understood as a folding of the triangular graphene π* pockets16,17 owing to the calcium superlattice illustrated in Fig. 1b. At the same time, this band-folding cannot account for the near-circular pocket. Its near-circular shape, parabolic band dispersion (Fig. 1c) and three-dimensionality (Supplementary Fig. 1) all agree with the predicted free-electron-like character of the IL band1,2,3,4,5,6, and therefore permit its confident assignment. We note that the spectral feature identified as the IL band by Sugawara et al.12 spanned a region of momentum-space which contains the folded π* bands. Their inability to resolve these separate contributions undermined their claim of observing superconductivity associated with only the IL band. In contrast, the radius of our observed IL pocket is approximately twice as large as that measured by Sugawara et al.12, and is 50% larger than the calculated value at kz=0 using density functional theory1,2,3,4,5,6. While the former difference can be understood as our photon energy probing a kz value closer to zero (Supplementary Fig. 1), the latter demonstrates that density functional theory underestimates the actual IL pocket size. Importantly, these differences allow us to clearly distinguish between the two Fermi surfaces near Γ, and therefore study the superconducting gaps and EPC on the individual bands to address the superconducting mechanism.

Figure 1: ARPES measurement revealing the CaC6 band structure near the zone centre.
figure 1

(a) Fermi surface map plotted in both the C2 (solid) Brillouin zone (BZ) and the Ca-folded (dashed) BZ. The solid blue triangles are schematic representations of the heavily electron-doped graphene π* pockets11. The dashed blue triangles illustrate the superlattice-folded π* pockets near Γ. The dashed red circle marks the IL pocket. (b) CaC6 crystal structure viewed along the c-axis. The black honeycomb lattice represents the graphene planes. The yellow, red and purple balls stand for the calcium atoms. Each colour corresponds to each calcium layer. The solid and dashed arrows are the two-dimensional lattice vectors of the graphene lattice and the calcium superlattice, respectively. (c) Electronic band dispersions along the K′-Γ-M′ trajectory. The blue- and red-guiding lines correspond to the π* and IL bands, respectively.

Opening of superconducting gaps

We identify the existence of superconducting gaps on the hexagonal π* and IL bands in Fig. 2a and b, respectively. Below the superconducting transition temperature (Tc=11.5 K), the leading edges of the CaC6 spectra for both the π* and IL bands are clearly shifted to higher binding energies with respect to the spectra acquired on polycrystalline gold. Above Tc, the leading edges of the CaC6 and gold spectra coincide. We also show in Supplementary Fig. 2 that the leading edge shifts below Tc are not confined to particular momenta. As demonstrated by previous ARPES experiments, leading edge shifts (ΔE) below Tc signal the existence of superconducting gaps12,18,19,20. By fitting the CaC6 spectra to a phenomenological model based on the Fermi–Dirac distribution21, we obtain ΔEπ*=0.5±0.1 meV and ΔEIL=0.4±0.1 meV. We emphasize that our energy resolution contributes significant broadening (Methods). In this situation, leading edge shifts have been known to substantially underestimate the actual gap values18,19,20, and are not suitable for quantitative comparison to theory5 or other experiments9,10. Nevertheless, our analysis of the leading edge shifts clearly demonstrates the opening of superconducting gaps on both the π* and IL bands below Tc.

Figure 2: Opening of superconducting gaps.
figure 2

(a) Constant-momentum spectra acquired on the hexagonal π* band, both below (blue triangles) and above (red diamonds) Tc. The spectra acquired on polycrystalline gold (dashed curves) are used to determine EF. All spectra are normalized by the values at 10 meV below EF to account for the difference in photon flux and photoemission matrix element. The CaC6 spectra are fitted using a phenomenological model (solid curves)21. The model is built on a Fermi–Dirac distribution, multiplied by a linear function and convolved with the Gaussian resolution function. The inset shows the momentum for the spectral analysis. Owing to substantial resolution broadening (Methods), the leading edge shifts below Tc are significant underestimates of the actual gap values18,19,20, and are only used to indicate the opening of the gaps. (b) Spectra acquired on the IL band and reference gold. Notations and analysis are the same as those for a.

Importantly, our leading edge measurement provides the first experimental evidence with clear momentum-space resolution for the existence of a superconducting gap on the π* band of any graphitic material. Tunnelling experiments have measured the superconducting gap in CaC6 and yielded a fully gapped picture, which is consistent with our results, but they were unable to directly resolve the separate contributions from the individual bands in momentum-space9,10. On the other hand, the ARPES study by Sugawara et al.12 attributed their measured gap near Γ to the IL band alone and reported no gap on the π* band. By resolving the distinct contributions from both bands, our measurement corrects this oversimplified assignment, and establishes the critical participation of both bands in superconductivity.


Connecting to the debate on CaC6’s superconducting mechanism, our data resolve the controversy about which bands are superconducting, and agrees well with theory in this respect5. We now seek evidence for the phonon modes, which facilitate the superconducting transition1,4,5,6. The intense debate in the literature focused on whether the main contribution comes from the carbon in-plane mode (Cxy)13, which couples only to the π* electrons, or the carbon out-of-plane (Cz) and calcium in-plane (Caxy) modes1,4,5,6, which mediate interactions between the graphene sheets and intercalant atoms. The key to revealing the superconducting mechanism is to separately characterize these different phonon modes and compare their respective EPC strengths.

In ARPES experiments, EPC is manifested in sudden changes in the electroic band dispersions and linewidths11,22,23,24, which, respectively, correspond to discontinuous features in the real and imaginary parts of the electronic self-energy (Re[Σ], Im[Σ]). In Fig. 3a,d, we present the ARPES cuts on the same momenta as for the leading edge measurement in Fig. 2. The momentum-distribution curves are analysed by fitting to Lorentzian distributions to extract the band dispersions and linewidths. The π* dispersion, shown in Fig. 3b, displays two kinks which are, respectively, near 160 meV and 50 meV below the Fermi level (EF). In contrast, Fig. 3e shows that the IL dispersion has only one pronounced kink near 70 meV. These kinks appear for all momenta along the respective Fermi surfaces (Supplementary Fig. 3). Based on an experimental phonon spectrum25 and theoretical calculations1,4,5,6, we assign the 160-meV kink to the coupling with the Cxy modes which span the range of 150–200 meV, and assign the 50 and 70 meV kinks to the coupling with the Cz modes which span 30–80 meV. We do not resolve any coupling with the Caxy modes 10 meV4, yet its existence cannot be ruled out owing to technical limitations such as the finite signal-to-noise ratio. Notably these results agree with theory1,4,5,6 on the existence of the Cz modes on both the IL and π* bands, along with the non-existence of the Cxy modes on the IL band. In Fig. 3c,f we extract the electronic self-energies on the π* and IL bands using the method described in the Methods section. At the corresponding mode energies, the peak features in Re[Σ] are one-to-one matched to the step features in Im[Σ], which corroborates our observation of the kinks in the band dispersions. Notably, both the Re[Σ] and Im[Σ] features have finite widths. The overlap in energy between the Cz features on the π* and IL bands indicates that there are common Cz modes which couple to both bands (Supplementary Note 1).

Figure 3: EPC in CaC6 at 5 K.
figure 3

(a,d) Typical ARPES cuts for the π* and IL bands, respectively. These cuts are taken at the same momenta as for the leading edge measurement in Fig. 2. The extracted band dispersions are overlaid on the raw data. (b,e) Identification of electron–phonon modes in the dispersions. The insets illustrate the momenta for the kink analysis. (c,f) Extracted real and imaginary parts of the electronic self-energies of the π* (c) and IL (f) bands.

Importantly, our observation of the Cz modes on both the π* and IL bands evidences the interaction between the two bands1,4,6. To understand this point, consider the electron–phonon perturbation operator corresponding to the Cz modes. This operator has odd parity with respect to the z axis6, and can therefore only couple electronic states with opposite z-parities. This implies that intraband coupling to the Cz modes within the π* band is symmetry-forbidden, as the π* band has a universally odd z-parity1,4,6. On the other hand, the IL band has a strong contribution from the calcium 4s and orbitals5, both of which have even z-parities. Therefore, coupling to the Cz modes on the π* band is enabled by the existence of both bands1,4,6. The observation of the Cz modes in both bands provides direct evidence for this interband interaction.

We evaluate the EPC strengths (λ) to demonstrate the significant role of the interband interaction in superconductivity. Figure 4 presents the extracted Eliashberg functions, which can be subsequently used to deduce , and based on the Eliashberg theory as explained in the Methods section26,27. These values demonstrate that without the existence of the IL band and its associated interband interaction, the EPC strength on the π* band would be only 0.20, which is insufficient to facilitate the superconducting transition at 11.5 K (ref. 4). The contribution from the Cz-mediated interband interaction to the total EPC strength on the π* band is >50%, which is crucial to enable superconductivity1,4,6. The same analysis performed at different momenta along the π* and IL bands also supports this conclusion (Supplementary Fig. 4).

Figure 4: Evaluation of EPC strengths in CaC6.
figure 4

(a,c) Extracted real parts of the self-energies (solid circles) on the π* (a) and IL (c) bands based on the Eliashberg theory (Methods)11,26,27. (b,d) Extracted Eliashberg functions (α2F, solid) and EPC functions (λ, dashed) of the π* (b) and IL (d) bands. Here ω stands for the phonon frequency. The contributions to the total electron–phonon-coupling strengths are deduced: , , and .


Our analysis of the superconducting gaps and EPC strengths reveals that superconductivity of CaC6 cannot be attributed to either the π* or IL band alone. Instead, our discoveries demonstrate the critical role of the interaction between the two bands.

An important implication of this understanding is that the substantial contribution to the total EPC strength from the interband interaction can be readily explored in monolayer graphene. In this case, the IL band can be created by forming an array of surface adatoms with the resulting interband interaction then enabling superconductivity. This idea was recently utilized by Profeta et al.1 to predict that the monolayer LiC6 system may superconduct up to Tc 8.1 K. If this material can be fabricated, it will allow an efficient integration of various nanoscale technologies, such as superconducting quantum interference devices and single-electron superconductor-quantum dot devices1. Importantly, in this proposal the IL pocket size and the EPC strength contributed by the interband interaction are the keys to realizing the superconducting phase transition. Our work demonstrates the significant role ARPES will have in the implementation of this proposal, as it allows for direct measurement of not only the size of the IL band, but also the strength of the interband electron–phonon interaction. These results set a solid foundation for future exploratory activities in the pursuit of fabricating superconducting graphene devices.

Note added in proof:

After the acceptance of our paper, we became aware of the ARPES study by Fedorov et al.29 which reports Ca-enhanced EPC on doped monolayer graphene.


Sample synthesis

High-quality CaC6 samples were synthesized from 3–4-mm platelets of natural single-crystal graphite using the lithium-alloy method28. The phase purity of the crystals was confirmed by X-ray diffraction to be >99% with no observable peaks from LiC6 or graphite impurities. A sharp superconducting transition was observed at 11.5 K using a SQUID magnetometer.

ARPES measurement

Our ARPES measurement was performed at Beamline 5-4 of Stanford Synchrotron Radiation Light source using the Scienta R4000 electron analyser. Atomically clean surfaces were obtained by in situ cleaving in an ultra-high vacuum chamber with a pressure 2 × 10−11 Torr. The synchrotron light was incident on the sample surface with a linear polarisation perpendicular to the analyser slit in the π configuration. The measurement temperature was regulated to be stationary within 0.2 K. The energy resolution combining the broadenings from the radiation bandwidth and the analyzer slit was 5 meV. The angular (momentum) resolution was 0.2° (0.004 Å−1). Our results were repeated on multiple samples.

Self-energy analysis

The connection between the Eliashberg function (α2F) and the real part of electron self-energy (Re[Σ]) is established in equation (1)26,27.

Here f is the Fermi–Dirac (FD) distribution; ωmax represents the maximum phonon frequency. At the low-temperature limit, the FD distribution becomes a step function, leading to a simplified formula as equation (2).

There is a well-established relation between Re[Σ] and the measured dispersion ω(k) from the ARPES literature22,23,27.

In principle, equations (2) and (3) allow us to extract α2F from a measured dispersion, yet the correct bare-band dispersion ε0(k) is a priori unknown. Conventionally, an ansatz is chosen for ε0(k), which is subtracted from ω(k) to yield Re[Σ]11,22. The shortcoming of this approach is that any subsequent analysis on Re[Σ] is highly sensitive to the original choice of ε0(k). Here we overcome this problem by fitting ω(k) in a single step, allowing the parameters describing both ε0(k) and Re[Σ] to vary freely. We find that all parameters are well constrained by the shape of ω(k), thus leading to a reliable extraction of both ε0(k) and Re[Σ]. This is demonstrated in Supplementary Fig. 5, where we use a linear function to parametrize ε0(k) for a π* dispersion11, and assume two Gaussian peaks in α2F to account for the Cxy and Cz modes. We find a drastic degradation in the fitting quality when we force the bare-band velocity to deviate 10% from the value determined by a free fit.

To fit IL dispersions, we use the ansatz of a single Gaussian peak for α2F. We also find that a parabolic bare-band dispersion is necessary for high-quality fitting, which is consistent with the free-electron-like nature of the IL band1,2,3,4,5,6.

The fitting procedure determines an effective α2F. The EPC function λ(ω,k) is subsequently deduced according to equation (4)4,6,26,27.

In Fig. 4, we illustrate how to obtain the respective coupling strengths. For the IL band, we integrate over the whole range (0–250 meV) to obtain . For the π* band, we selectively obtain or by varying the integration window.

To estimate the uncertainties in the extracted EPC strengths, we consider in the α2F function a spectral peak represented by . Here A, ω0 and σ stand for the amplitude, the central energy and the peak width, respectively. If ω0σ, we can obtain an estimate of the EPC strength: . According to the principle of error propagation, we deduce the uncertainty of using equation (5).

is well determined as the quantities of the right-hand side of equation (5) can be all obtained from fitting the measured band dispersion.

Additional information

How to cite this article: Yang, S.-L. et al. Superconducting graphene sheets in CaC6 enabled by phonon-mediated interband interactions. Nat. Commun. 5:3493 doi: 10.1038/ncomms4493 (2014).