Introduction

Recently, superconducting phases have been discovered both in Bernal bilayer graphene (BBG) and in rhombohedral trilayer graphene (RTG)1,2,3. These discoveries follow an intense investigation into superconducting states of twisted bilayer and trilayer graphene, where the superconducting mechanism has been ascribed to either electron–phonon interactions or the enhanced electron–electron correlations arising in flat bands4,5,6. Similar explanations have been proposed for RTG7,8,9,10,11,12, where a high density of states at the Fermi energy can be obtained via gate tuning in a perpendicular displacement field, leading to van Hove singularities at small but finite doping. More recently, the proximity to WSe2 has been shown to increase the critical temperature of BBG by a factor of ten13,14, which may be explained by the suppression of order parameter fluctuations due to an induced Ising spin-orbit coupling15.

The similar phenomenology across these material platforms indicates a common mechanism underlying their superconductivity. However, in all these platforms some superconducting regions violate the Pauli limit, a strong signal of unconventional superconductivity that naively would rule out conventional phonon-mediated pairing as the underlying mechanism (by an unconventional superconducting state, we here mean a state whose gap transforms under rotations according to a finite angular momentum representation). However, since the details of the band structure, the level of correlation, and the presence or absence of layer twisting seem to be relatively unimportant for the superconducting phenomenology, phonon-mediated pairing provides an attractive unifying principle. The question then becomes if it is possible to reconcile unconventional superconductivity with a conventional phonon-mediated pairing?

To gain further insight into the superconducting mechanism of rhombohedral multilayer graphene, we have performed extensive first principles calculations of RTG and rhombohedral hexalayer graphene (RHG) to evaluate the phonon contribution to the superconducting pairing within Eliashberg theory. In agreement with experimental findings in RTG1,2, our calculations predict two superconducting regions with critical temperatures Tc ~100 mK whose gap symmetries depend on the parent normal state. In particular, in regions where electron correlations stabilize a spin- and valley-polarized (SVP) parent state, we find a gap with triplet f-wave symmetry stabilized purely through electron–phonon interactions. The quantitative improvement over previous studies of phonon-mediated pairing11,12 can be assigned to the additional retardation effects included in the Eliashberg function, which localize the gap function to the electronic Fermi surface. We analyze the symmetry of the superconducting gap and find an extended s-wave pairing domain arising from inter-valley scattering and originating from a spin- and valley-unpolarized normal state. In addition, we find a smaller superconducting region with f-wave symmetry at lower doping levels, which is due to intra-valley scattering and arises out of an SVP normal state. This is in good agreement with the superconducting regions identified in recent experiments1,2. Compared to RTG, we find RHG shows a slightly increased critical temperature.

We also discover a superconducting region in both RTG and RHG at higher hole doping densities. This region coincides with a second set of van Hove singularities further below the Dirac cone, arising from the top of the split-off valence bands. We investigate the symmetry of this superconducting region, and again find a dominant s- or f-wave pairing depending on the spin- and valley-polarization of the parent normal state. As a mean to reach the high-doping regime we consider a heterostructure consisting of RTG and monolayer α-RuCl3, where the large work function mismatch leads to significant doping of both structures. We finally discuss the interplay of α-RuCl3 and RTG superconductivity in the high-doping regime.

Results

Trilayer and hexalayer rhombohedral graphene in a finite displacement field

The first principles electronic structure of multilayer rhombohedral stacked graphene, as obtained from density functional theory (DFT) calculations, is well-captured around the Dirac cones by a tight-binding description of the C pz-orbitals including the hopping processes illustrated in Fig. 116,17. Depending on the stacking order, the bands around the Fermi level have different characteristics: For rhombohedral stacking the highest valence band is approximately flat for k ≈ Ks, leading to a sharp van Hove singularity in the density of states (DOS). This is in contrast to Bernal stacked graphene layers, where the highest valence band approximately retains the linear dispersion of an isolated graphene sheet. The band structure and density of states (DOS) of RTG and RHG are shown in Fig. 1.

Fig. 1: Electronic structure of rhombohedral stacked trilayer and hexalayer graphene.
figure 1

a Side view of the unit cell of rhombohedral stacked trilayer and hexalayer graphene. The inter-layer hopping amplitudes ti included in the tight-binding description of the system are indicated. b Top view of rhombohedral stacked multilayer graphene. The intra-layer hopping amplitude t0 is indicated. c Low-energy band structure and density of states (DOS) of rhombohedral trilayer graphene (RTG) around the K+ point. d Low-energy band structure and DOS of rhombohedral hexalayer graphene (RHG) around the K+ point. e Electronic Fermi surface of RTG for zero displacement field at Fermi levels ϵF = −7.4 meV, ϵF = −7.8 meV, and ϵF = −10.2 meV. f Electronic Fermi surface of RHG for zero displacement field at Fermi levels ϵF = −3.9 meV, ϵF = −4.5 meV, and ϵF = −10.7 meV. The Fermi level is measured from the Dirac cone and the DOS is calculated with a Gaussian smearing of width σ = 2 meV.

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The DOS at the van Hove singularity can further be tuned by applying a displacement field perpendicular to the graphene layers1. To the lowest order in the electronic screening16, such a displacement field can be treated as a symmetric potential Δ applied across the layers. For RTG the displacement field leads to an increase in the DOS due to a gap opening at Ks and the highest valence band bending into a double-well shape. In contrast, for RHG, the DOS decreases with increasing displacement field, since the highest valence band is approximately flat already at Δ = 0.

Recent experiments have found two superconducting regions in rhombohedral trilayer graphene as a function of displacement field and doping1, denoted SC1 and SC2. These superconducting phases were found to be associated with a change in the Fermi surface topology, which for finite displacement field and as a function of doping evolves from three well-separated hole-pockets, via an annular Fermi surface, to an approximately circular Fermi surface (see Fig. 1). Similarly, the Fermi surface of RHG undergoes a transition from three isolated and strongly warped hole-pockets into first an annular Fermi surface, and subsequently into a single large hole pocket. Compared to RTG, the trigonal warping of the Fermi surface in RHG is more pronounced, and the total Fermi surface area is larger leading, to a larger DOS.

Eliashberg's theory of phonon-mediated superconductivity

To obtain the superconducting critical temperature in RTG and RHG resulting from phonon-mediated pairing, we use Eliashberg theory. Crucially, this approach takes into account the frequency dependence of the retarded electron–phonon interaction, which is found to have important consequences for the present systems. The key quantity of this approach is the Eliashberg function α2F(ω), which is obtained from the spectral function of the electron–phonon self-energy Πνq(ω, T) (see Methods for a detailed discussion). To evaluate the self-energy it is sufficient to calculate the electronic dispersion ϵnk, the phonon frequencies ωνq, and the electron–phonon couplings gmnν(k, q).

Due to the structure of the Eliashberg equations, electronic states contributing to the formation of a low-temperature superconducting instability are highly restricted to the Fermi surface. In both RTG and RHG, this constitutes a very small region of the Brillouin zone (see Fig. 1), and therefore only phonons with q = 0 or q = K± contribute significantly to the superconducting pairing. The Eliashberg function can then be represented as a series of peaks, such that the effective electron–phonon coupling λ is given by λ = ∑νqλνq. Here \({\lambda }_{\nu {{{\bf{q}}}}}={\gamma }_{\nu {{{\bf{q}}}}}/(\pi {\rho }_{F}{\omega }_{\nu {{{\bf{q}}}}}^{2})\) is the contribution to λ from the phonon mode in branch ν and with momentum q, γνq is the phonon linewidth, and ρF the density of states at the Fermi level. The superconducting critical temperature follows from the McMillan equation18, where the screened electron–electron interaction is accounted for by the effective parameter μ*18,19,20.

Similarly to λ, the parameter μ* gives a dimensionless measure of the strength of electron–electron repulsion, such that superconductivity is exponentially suppressed for λ < μ*. We have checked the dependence of our results on μ* in the typical range [0.1, 0.2] (see Supplementary Fig. 1), and find a suppression of Tc by about a factor two between μ* = 0.1 and μ* = 0.2. To not overestimate Tc in the following, we have used the value μ* = 0.2 in all results presented below. All quantities needed to evaluate Tc have been calculated from first principles as discussed in the Methods section.

Critical temperature

First, we calculate the critical temperature Tc of the superconducting transition in RTG (RHG) assuming an unpolarized normal state as shown in Fig. 2. The results for RTG are in good agreement with the experiments of ref. 1, although the critical temperature is overestimated by about a factor of four. This discrepancy is expected due to the shortcomings of Eliashberg theory to incorporate the effect of strong quantum fluctuations in two-dimensional systems, which tend to suppress the critical temperature. A further reduction of Tc could come from non-adiabatic effects, since the accuracy of Eliashberg theory relies on the adiabatic assumption vF/vs ≫ 1, where \({v}_{F}=2{(\hslash \rho )}^{-1}\sqrt{{n}_{e}/\pi }\) is the electronic Fermi velocity and vs the phonon velocity. Close to the VHS, the large density of states leads to a suppression of vF, such that vF/vs ~0.1. For such velocities, the effect of non-adiabatic corrections is to modify the Eliashberg relation Tc ≈ 0.25ωe−1/λ to \({T}_{c}\approx 0.12\sqrt{\omega {\epsilon }_{F}}{e}^{-1/\lambda }\)21. Estimating ϵF ≈ ℏvFk this leads to a suppression of Tc by about a factor of four. Due to the higher DOS of RHG as compared to RTG, the critical temperature in this system is enhanced by a factor of 2−3.

Fig. 2: Superconducting critical temperature of rhombohedral tri- and hexalayer graphene.
figure 2

a, b Superconducting critical temperature Tc of rhombohedral trilayer graphene (RTG, a) and rhombohedral hexalayer graphene (RHG, b) as a function of electron doping density ne and displacement potential Δ. c, d Electronic density of states (DOS, blue) and superconducting critical temperature Tc (red) of RTG (c) and RHG (d) as a function of doping density in the high-doping regime. Here Δ = 0 since the bands far below the Dirac cone are insensitive to the displacement field, and the DOS is calculated with a Gaussian smearing of width σ = 2 meV.

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To account for the superconducting region SC2, arising out of an SVP state1, we assume that the effective DOS in the active channel is increased by a factor of four (since all electrons are accumulated in one spin and valley sector), and that inter-valley scattering is negligible. This gives a superconducting region at lower doping with slightly lower Tc compared to the phase SC1. For both RTG and RHG, the critical temperature is found to closely follow the electronic DOS, which is a consequence of the self-energy scaling like \(\Pi \sim {\rho }_{F}^{2}\), such that α2F(ω)~ρF. Since the DOS increases (decreases) with increasing displacement field in RTG (RHG) (see Supplementary Fig. 2), the critical temperature shows the same trend. We note that compared to conventional BCS theory11,12, the main consequence of including phonon retardation effects is an overall reduction of the critical temperature by a factor of 5−10 and the range of dopings over which superconductivity is found.

The DOS and critical temperature in the high-doping regime are also shown in Fig. 2. The density of states is similar to that found in the low-doping regime, and the corresponding critical temperature is therefore of the same order. Since the active bands in the high-doping regime arise from the intermediate layers of the RTG and RHG stacks, their energies are largely insensitive to the displacement field.

Linearized gap equation

Next, we analyze the symmetry of the leading superconducting order parameter in both doping regimes. We focus on the regime T ≲ Tc, where the gap function can be obtained by solving the linearized gap equation22,23

$${\Delta }_{m{{{\bf{k}}}}}=\pi \sum\limits_{n{{{\bf{p}}}}}{\chi }_{mn{{{\bf{k}}}}{{{\bf{p}}}}}\frac{\tanh (\beta {\xi }_{n{{{\bf{p}}}}})}{\beta {\xi }_{n{{{\bf{p}}}}}}{\Delta }_{n{{{\bf{p}}}}},$$
(1)

where β is the inverse temperature, ξnk = ϵnk − ϵF and the susceptibility is

$${\chi }_{mn{{{\bf{k}}}}{{{\bf{p}}}}}=\sum\limits_{\nu }\frac{2| {g}_{mn\nu }({{{\bf{k}}}},{{{\bf{q}}}}){| }^{2}}{{N}_{{{{\bf{k}}}}}{\rho }_{F}{\omega }_{{{{\bf{q}}}}\nu }}\delta ({\xi }_{m{{{\bf{k}}}}})\delta ({\xi }_{n{{{\bf{p}}}}}).$$
(2)

Here it is implicitly assumed that the phonon momentum satisfies q = p − k. This equation is of the same form as the standard BCS gap equation, however with the interaction restricted to the Fermi surface. The main difference between BCS and Eliashberg theory is that the latter includes retardation effects from the electron–phonon interaction, which give rise to a non-trivial frequency dependence in the effective electron–electron interaction (see Methods). This frequency dependence significantly improves the temperature dependence of the theory and, in the low-temperature limit, localizes the susceptibility to the Fermi surface22,23. For many systems, where the Fermi surface occupies a significant portion of the Brillouin zone, this effect is rather small. For RTG and RHG however, where the Fermi surface occupies a tiny portion of the full Brillouin zone, the quantitative difference between the two approaches is quite dramatic. In general, Eliashberg theory is expected to be more accurate than BCS theory, since it captures the dynamical aspects of the electron–phonon interaction.

We note that an accurate estimate of the magnitude of the zero temperature gap requires the solution of the full (as compared to the linearized) gap equation, which is a highly complex numerical task. We therefore focus on the region close to the superconducting phase transition, where the critical temperature and symmetry of the gap are expected to be well captured by the linearized equation. To obtain an estimate for the magnitude of the zero temperature gap, we use the BCS relation Δ(0) = πe−γkBTc ≈ 0.16Tc meV K−1. For critical temperatures Tc ~ 0.1−1 K this gives a superconducting gap of Δ(0) = 0.016−0.16 meV.

Spin structure of the gap function

The gap function can, in principle, depend on the spin as well as on band and momentum indexes. However, since the electron–phonon interaction is independent of spin, this is true also for the superconducting susceptibility χ. The gap equation can therefore be solved in each spin sector separately. Due to this fact, the spin symmetry of the superconducting state will be sensitive to the symmetries of the parent state. For an unpolarized normal state, the singlet sector is expected to dominate. In contrast, singlet pairing will be strongly suppressed by an SVP parent state, where the only available Cooper pairs have spin S = ±1. The normal state symmetry is modeled here by suppressing inter-valley phonon scattering, since superconductivity resulting from a valley-polarized state is dominated by intra-valley scattering. Since such a polarized normal state is believed to arise from the spontaneous breaking of spin and valley symmetry24, spin-orbit interactions are not needed to stabilize a triplet state. They are, however, expected to increase the critical temperature of the f-wave state as discussed below. In practice, the spin symmetry of the gap is found from its orbital symmetry: when the gap is even under the momentum exchange k to −k, the gap will be a spin singlet, while when the gap is an odd function of momentum, the gap will be a spin triplet (see Methods). In the following discussion, the triplet f-wave state will refer to the S = 1 state.

Superconducting symmetry in the low-doping regime

Figure 3 shows the superconducting gap of RTG for a displacement field potential Δ = 20 meV and the Fermi level at ϵF = −26 meV, corresponding to a hole doping density of nh ≈ 0.4 × 1012 cm−2. As can be clearly seen the gap is symmetric under inversion, Δ−k = Δk, indicating a singlet pairing. Further, in each valley, the gap has threefold rotational symmetry with a non-trivial nodal structure. Together these observations are consistent with an extended s-wave symmetry. A similar gap structure is found for RHG, again consistent with an extended s-wave symmetry (see Supplementary Fig. 3). This dominant pairing is found to arise from inter-valley scattering, corresponding to the exchange of virtual phonons with momenta q ≈ K, as expected from a symmetry analysis (see Methods for an extended discussion).

Fig. 3: Superconducting gap of rhombohedral trilayer graphene.
figure 3

a, b Superconducting gap Δk of rhombohedral stacked trilayer graphene for a hole doping density nh ≈ 0.4 × 1012 cm−2, including the attractive interaction from both inter- and intra-valley phonon scattering (a) or only intra-valley scattering (b). The dominant inter-valley scattering favors s-wave pairing (a), while the sub-dominant intra-valley scattering favors f-wave pairing (b). c, d Superconducting gap Δk of rhombohedral stacked trilayer graphene for a hole doping density nh ≈ 3.9 × 1012 cm−2, including the attractive interaction from both inter- and intra-valley phonon scattering (c), or only intra-valley scattering (d). The dominant inter-valley scattering favors s-wave pairing (c), while the sub-dominant intra-valley scattering favors f-wave pairing (d). In all panels, the momentum runs over a region ka ∈ [−0.1, 0.1] around the Ks point.

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Artificially suppressing the inter-valley scattering, to account for the superconducting state arising out of an SVP normal state, we find an odd gap Δ−k = −Δk. This indicates an unconventional triplet f-wave pairing, and that superconductivity in RTG arises from a competition between inter- and intra-valley scatterings favoring different superconducting symmetries. Again, the results for RHG are qualitatively similar (see Supplementary Fig. 3). These findings agree with the experimental results of ref. 1, where the superconducting region SC2 was found to violate the Pauli limit by more than an order of magnitude, strongly indicating a triplet superconducting pairing.

The f-wave pairing is likely stabilized by a combination of Fermi surface topology and the spin and valley polarization of the parent state. In fact, for an unpolarized parent state with momentum-independent electron–phonon scattering, the s- and f-wave pairings are found to be degenerate (see Supplementary Note 3). This indicates a highly non-trivial interplay of electronic correlations and phonon-mediated pairing, where the former stabilizes the parent state and the latter the f-wave pairing. The triplet state is, therefore, expected to be stabilized in a regime where electronic correlations lead to a spin- and valley-polarized normal state.

Superconducting symmetry in the high-doping regime

Similar results are found for both RTG and RHG in the high-doping regime, where inter-valley (intra-valley) scattering is found to favor an extended s-wave (f-wave) pairing. The superconducting gap for RTG at zero displacement field and a Fermi level of ϵF = −350 meV, corresponding to a hole doping density of nh ≈ 3.9 × 1012 cm−2, is shown in Fig. 3. We note that the critical temperature in the high-doping regime is comparable to but slightly lower than that of the low-doping regime, as expected from the respective DOS. These results indicate a new and so far unexplored region of superconductivity in RTG, arising from the van Hove singularities of the lower valence bands.

Reaching the high-doping regime

To reach the high-doping regime, we consider a heterostructure consisting of RTG and a monolayer of the Mott-Slater insulator α-RuCl3. This heterostructure has recently been found to realize a heavily hole-doped regime of graphene, with a Fermi level ~0.6 eV below the Dirac cone25. More specifically, due to the large work function mismatch of about 1.6 eV, there is a significant charge transfer from the graphene multilayer into α-RuCl3 resulting in an overall doping of ~−0.07e per Ru atom25,26. Since the charge transfer is mainly localized to the interface, the charge transfer corresponds to a hole doping of 0.01e per C atom, or equivalently a doping density of 4.2 × 1012 cm−2, in the layer adjacent to α-RuCl3.

The band structure of the heterostructure was calculated in a 5 × 5 and 2 × 2 supercell for the RTG and α-RuCl3 subsystems respectively, and is shown in Fig. 4. The local interactions in the active Ru manifold are described by a Hubbard-Kanamori Hamiltonian27,28 treated within the unrestricted Hartree-Fock approximation. The interaction between the subsystems is treated as a position and orbital-dependent hybridization17, which largely restricts the coupling to the graphene layer adjacent to α-RuCl3 (see Supplementary Note 4). We note that the band structure displays clear avoided crossings at the band intersections, and that a gap opens at the Dirac cone of the RTG bands. This effect is in agreement with first principles calculations (see Supplementary Fig. 4), indicating that the heterostructure sets up an intrinsic displacement field through the charge transfer process.

Fig. 4: Superconductivity of a rhombohedral trilayer graphene and α-RuCl3 heterostructure.
figure 4

a Supercell of the rhombohedral trilayer graphene (RTG) and monolayer α-RuCl3 heterostructure considered, containing eight Ru atoms (orange) and 150 C atoms (gray). b Electronic band structure of the heterostructure, with the flat bands originating from the Ru t2g orbitals and the dispersive bands from the C pz orbitals. c Estimated electron–phonon coupling λest (blue) for the electron–phonon couplings gRTG = 200 meV and \({g}_{{{{\rm{RuC{l}}}_{3}}}}=20\) meV, and a phonon frequency ω = 200 meV, as well as the corresponding critical temperature Tc. The gray dashed line indicates the Fermi level of the heterostructure.

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Superconductivity of the RTG-RuCl3 heterostructure

We now use Eliashberg theory to obtain the critical temperature of the heterostructure. To estimate Tc we calculate the dimensionless variable λest = g2ρ/Ω (see Fig. 4), which is closely related to the dimensionless electron–phonon coupling λ. The electron–phonon couplings of the separate subsystems, obtained from first principles density functional perturbation theory calculations, are found to be on the order of gRTG ≈200 meV29 and \({g}_{{{{\rm{RuC{l}}}_{3}}}}\approx 20-40\) meV, and assuming a typical optical phonon energy of Ω = 200 meV as appropriate for graphene, we find λest ≈ 1 around the graphene van Hove singularities. This estimate is in good agreement with the more detailed calculations for RTG based on the Eliashberg function. The resulting critical temperature agrees well with Fig. 2 around the van Hove singularities, and is on the order of a few mK close to the heterostructure Fermi level.

Depending on the electron–phonon coupling in α-RuCl3 as well as on the sub-system hybridization strength, the effective coupling λest of the heterostructure can be modified by a few percent. However, while a stronger electron–phonon coupling in α-RuCl3 is found to enhance λest, a stronger hybridization predominantly reduces the effective electron–phonon coupling in RTG-dominated bands. This effect can be attributed to the much smaller Fermi surface of RTG as compared to α-RuCl3, which makes the contribution to λest from RTG much more sensitive to the hybridization than the contribution from α-RuCl3. A small enhancement of Tc with increasing hybridization can be observed for bands with a dominant projection on α-RuCl3.

Discussion

By combining first principles calculations with effective low-energy models, we have investigated the phenomenology of phonon-mediated superconductivity in RTG, RHG, and RTG-α-RuCl3 heterostructures. Including the retardation effects of the phonon-mediated attraction, we find a substantial reduction of both the critical temperature and the region of doping and displacement fields in which superconductivity appears. These effects lead to a substantial improvement between theory and experiments1,2, promoting phonon-mediated superconductivity as a strong contender to explain superconductivity across a wide range of graphene platforms1,2,4,5,6.

More surprisingly, we find that inter-valley and intra-valley phonon scattering favors superconductivity with different symmetry, such that phonon-mediated pairing can stabilize both extended s-wave as well as unconventional f-wave triplet pairing. More specifically, we find that s-wave pairing is dominant when superconductivity arises out of an unpolarized parent state, while f-wave pairing dominates when the parent state is SVP. If verified, this would demonstrate that phonons can, in fact, stabilize unconventional triplet superconductivity30. Further, the interplay of electronic correlations and phonon-mediated pairing underlying the observed f-wave pairing, indicates a path to realize unconventional triplet pairing in conventional phonon-mediated superconductors.

We also discover a so far unexplored superconducting region, stemming from the van Hove singularity of the lower valence bands, that can be reached in heterostructures of RTG and monolayer α-RuCl3. Although a strong inter-layer hybridization is found to suppress superconductivity in the supercell considered here, it is possible that maximizing the Fermi surface overlaps of the two subsystems17, e.g., via layer twisting, could lead to an enhancement of Tc. It might also be the case that reducing spin fluctuations in the heterostructure by breaking the spin rotational symmetry could lead to an enhancement of the critical temperature for spin-triplet pairings, as have recently been proposed for Bernal bilayer graphene in proximity to WSe22,15. In fact, our first principle calculations show a small induced Ising spin-orbit coupling of ~1 meV in the heterostructure (see Supplementary Fig. 4).

Further work is needed to conclusively determine the superconducting mechanism in RTG and RHG, taking into account electron–phonon as well as electron–electron interactions on an equal footing. In any case, our work shows that the exotic and unconventional superconducting phenomenology of RTG is consistent with a phonon-mediated pairing, thereby opening new directions to stabilize unconventional superconductivity in systems that combine phonon-mediated pairing with strong electronic correlations of the parent state.

Methods

Electronic structure, phonon dispersion, and electron–phonon coupling

The electronic structure of rhombohedral and Bernal stacked trilayer and hexalayer graphene were calculated as a function of displacement field using the Abinit electronic structure code31,32,33,34,35. The electronic ground state was calculated within the local density approximation (LDA) on a 72 × 72 k-point grid, using a plane wave cut-off of 20 Ha. The displacement field was included via a coupling to the polarization as described within the modern theory of polarization36. The Kohn-Sham Bloch functions were subsequently transformed to maximally localized Wannier functions using the Wannier90 code, including the lowest lying 15 (30) bands for trilayer (hexalayer) systems. The Wannier functions were used to interpolate the electronic band structure to arbitrary k-points, allowing to calculate the density of state (DOS) on a very dense k-point grid with 1200 × 1200 grid points.

The phonon band structure was obtained using density functional perturbation theory (DFPT) as implemented in the Abinit electronic structure code. The atomic structure was first relaxed to obtain maximal forces below 5 ⋅ 10−6 Ha/Bohr, after which the phonon frequencies were calculated on a 12 × 12 q-point grid. To calculate the electron–phonon coupling the electronic bands were interpolated using star functions37, allowing to densely sample the region of the Brillouin zone around the K± points (with a density corresponding to a 720 × 720 k-point grid). Similarly, the derivatives of the Kohn-Sham potential were Fourier interpolated to obtain the electron–phonon coupling on an arbitrarily dense q-point grid, here taken to consist of 72 × 72 points.

Eliashberg theory of phonon-mediated superconductivity

To obtain the superconducting critical temperature in RTG and RHG resulting from phonon-mediated pairing, we use Eliashberg theory. Crucially, this approach takes into account the frequency dependence of the retarded electron–phonon interaction, which is found to have important consequences for the current systems. The key quantity of this approach is the Eliashberg function α2F(ω), defined as

$${\alpha }^{2}F(\omega )=\frac{1}{{N}_{q}{\rho }_{F}}\sum\limits_{\nu {{{\bf{q}}}}}\frac{{\gamma }_{\nu {{{\bf{q}}}}}}{{\omega }_{\nu {{{\bf{q}}}}}}\delta (\omega -{\omega }_{\nu {{{\bf{q}}}}})$$
(3)

where Nq is the number of q-points for the phonons, ρF the electronic density of states (per spin) at the Fermi energy, and γνq and ωνq is the linewidth and frequency of phonon mode (νq). The phonon linewidth is related to the phonon self-energy by \({\gamma }_{\nu {{{\bf{q}}}}}=2\,{{{\rm{Im}}}}\,{\Pi }_{\nu {{{\bf{q}}}}}({\omega }_{\nu {{{\bf{q}}}}},T)\), where the self-energy results from electron–phonon scattering is given by

$${\Pi }_{\nu {{{\bf{q}}}}}(\omega ,T)=\frac{2}{{N}_{{{{\bf{k}}}}}}\sum\limits_{{{{\bf{k}}}}mn}\frac{| {g}_{mn\nu }({{{\bf{k}}}},{{{\bf{q}}}}){| }^{2}({f}_{n{{{\bf{k}}}}}-{f}_{m,{{{\bf{k}}}}+{{{\bf{q}}}}})}{{\epsilon }_{n{{{\bf{k}}}}}-{\epsilon }_{m,{{{\bf{k}}}}+{{{\bf{q}}}}}-\omega -i\eta }.$$
(4)

To evaluate the self-energy it is sufficient to calculate the electronic dispersion ϵnk, the phonon frequencies ωνq, and the electron–phonon couplings gmnν(k, q).

Due to the presence of the Fermi functions in Eq. (4), the sum over momenta is at low temperatures highly restricted to the electronic Fermi surface. In both RTG and RHG this constitutes a very small region of the Brillouin zone (on the order of 10−3G, see Fig. 1), and therefore only phonons with q = 0 or q = Q = K+ − K− contribute significantly to the superconducting pairing. The Eliashberg function can then be represented as a series of peaks, such that the effective electron–phonon coupling λ is given by

$$\lambda =\int\frac{{\alpha }^{2}F(\omega )}{\omega }d\omega =\sum\limits_{\nu {{{\bf{q}}}}}{\lambda }_{\nu {{{\bf{q}}}}}.$$
(5)

Here \({\lambda }_{\nu {{{\bf{q}}}}}={\gamma }_{\nu {{{\bf{q}}}}}/(\pi {\rho }_{F}{\omega }_{\nu {{{\bf{q}}}}}^{2})\) is the contribution to λ from mode (νq). The superconducting critical temperature can now be obtained from the McMillan equation18

$${T}_{c}=\frac{{\omega }_{\log }}{1.2}\exp \left[\frac{-1.04(1+\lambda )}{\lambda (1-0.62{\mu }^{* })-\mu^{*} }\right],$$
(6)

where \({\omega }_{\log }\) is a logarithmic average of the phonon frequencies, and μ* is the average screened electron–electron interaction strength (here taken to have the typical value μ* = 0.2). All the quantities needed to evaluate Tc have been calculated from the first principles as discussed below. For further details on the numerical evaluation, see Supplementary Note 1.

Gap equation

Within Eliashberg's theory, the gap equation is of the form22,23

$$\begin{array}{ll}{Z}_{n{{{\bf{k}}}}}(i{\omega }_{m})=1+\pi {k}_{B}T\sum\limits_{{m}^{{\prime} }{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}\frac{({\omega }_{{m}^{{\prime} }}/{\omega }_{m})}{\sqrt{{\omega }_{{m}^{{\prime} }}^{2}+{\Delta }_{{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}^{2}(i{\omega }_{{m}^{{\prime} }})}}\\\qquad\qquad\quad\, \times {\lambda }_{n{{{\bf{k}}}},{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}(i{\omega }_{m}-i{\omega }_{{m}^{{\prime} }})\end{array}$$
(7)
$$\begin{array}{ll}{Z}_{n{{{\bf{k}}}}}(i{\omega }_{m}){\Delta }_{n{{{\bf{k}}}}}({\omega }_{m})=\pi {k}_{B}T\sum\limits_{{m}^{{\prime} }{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}\frac{{\Delta }_{{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}}{\sqrt{{\omega }_{{m}^{{\prime} }}^{2}+{\Delta }_{{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}^{2}(i{\omega }_{{m}^{{\prime} }})}}\\\qquad\qquad\qquad\qquad\, \times \left[{\lambda }_{{{{\bf{k}}}}{{{{\bf{k}}}}}^{{\prime} }}^{n{n}^{{\prime} }}(i{\omega }_{m}-i{\omega }_{{m}^{{\prime} }})-{N}_{F}{U}_{{{{\bf{k}}}}{{{{\bf{k}}}}}^{{\prime} }}^{n{n}^{{\prime} }}\right]\end{array}$$
(8)

where \({U}_{{{{\bf{k}}}}{{{{\bf{k}}}}}^{{\prime} }}^{n{n}^{{\prime} }}\) is the screened Coulomb interaction between momenta k and \({{{{\bf{k}}}}}^{{\prime} }\) on the Fermi surface, and the electron–phonon coupling is

$$\begin{array}{ll}{\lambda }_{{{{\bf{k}}}}{{{{\bf{k}}}}}^{{\prime} }}^{n{n}^{{\prime} }}(i\omega )={N}_{F}^{-1}\sum\limits_{\nu }\frac{2{\omega }_{{{{\bf{q}}}}\nu }}{{\omega }_{{{{\bf{q}}}}\nu }^{2}+{\omega }^{2}}| {g}_{n{n}^{{\prime} }}({{{\bf{k}}}},{{{\bf{q}}}}){| }^{2}\\\qquad\qquad\, \times \delta ({\epsilon }_{{{{\bf{k}}}}}-{\epsilon }_{F})\delta \left({\epsilon }_{{{{\bf{k}}}}}^{{\prime} }-{\epsilon }_{F}\right).\end{array}$$
(9)

In this equation the phonon momentum has to satisfy \({{{\bf{q}}}}={{{{\bf{k}}}}}^{{\prime} }-{{{\bf{k}}}}\). Assuming Z ≈ 1 and taking the static limit, we find the gap equation

$$\begin{array}{l}{\Delta }_{n{{{\bf{k}}}}}=\pi {k}_{B}T\sum\limits_{i{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}\frac{{\Delta }_{{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}}{\sqrt{{\omega }_{i}^{2}+{\Delta }_{{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}^{2}}}\left[{\lambda }_{{{{\bf{k}}}}{{{{\bf{k}}}}}^{{\prime} }}^{n{n}^{{\prime} }}-{U}_{{{{\bf{k}}}}{{{{\bf{k}}}}}^{{\prime} }}^{n{n}^{{\prime} }}\right]\\ {\lambda }_{n{{{\bf{k}}}},{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}(i\omega )=\sum\limits_{\nu }\frac{2| {g}_{n{n}^{{\prime} }}({{{\bf{k}}}},{{{\bf{q}}}}){| }^{2}}{{N}_{F}{N}_{{{{\bf{k}}}}}{\omega }_{{{{\bf{q}}}}\nu }}\delta ({\epsilon }_{{{{\bf{k}}}}}-{\epsilon }_{F})\delta ({\epsilon }_{{{{{\bf{k}}}}}^{{\prime} }}-{\epsilon }_{F}).\end{array}$$
(10)

Linearizing this equation, we find an equation of the same form as the BCS gap equation,

$${\Delta }_{n{{{\bf{k}}}}}=\pi \sum\limits_{{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}{\lambda }_{n{{{\bf{k}}}},{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}\frac{\tanh (\beta {\xi }_{{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }})}{\beta {\xi }_{{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}}{\Delta }_{{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }},$$
(11)

although with the interaction restricted to the Fermi surface.

As a check on this result, the effective electron–phonon coupling can be compared to that derived from the electron–phonon self-energy. Recalling that the phonon linewidth is

$$\begin{array}{ll}{\gamma }_{mn\nu }^{{{{\bf{q}}}}}(T)=-\frac{2{\omega }_{{{{\bf{q}}}}\nu }}{{N}_{{{{\bf{k}}}}}}\sum\limits_{{{{\bf{k}}}}{{{{\bf{k}}}}}^{{\prime} }mn}| {g}_{mn\nu }({{{\bf{k}}}},{{{\bf{q}}}}){| }^{2}\delta ({\epsilon }_{{{{\bf{k}}}}n}-{\epsilon }_{F})\\\qquad\qquad\, \times \delta ({\epsilon }_{{{{{\bf{k}}}}}^{{\prime} }m}-{\epsilon }_{F}),\end{array}$$
(12)

this results in the electron–phonon coupling

$${\lambda }_{n{{{\bf{k}}}}{n}^{{\prime} }{{{{\bf{k}}}}}^{{\prime} }}=\sum\limits_{\nu }\frac{2| {g}_{n{n}^{{\prime} }}({{{\bf{k}}}},{{{\bf{q}}}}){| }^{2}}{\pi {N}_{F}{N}_{{{{\bf{k}}}}}{\omega }_{{{{\bf{q}}}}\nu }}\delta ({\epsilon }_{{{{\bf{k}}}}}-{\epsilon }_{F})\delta ({\epsilon }_{{{{{\bf{k}}}}}^{{\prime} }}-{\epsilon }_{F}).$$
(13)

From these considerations, we note that the approximate Eliashberg gap equation is obtained from the BCS gap equation by including a factor \({N}_{F}^{-1}\delta ({\epsilon }_{{{{\bf{k}}}}n}-{\epsilon }_{F})\delta ({\epsilon }_{{{{{\bf{k}}}}}^{{\prime} }{n}^{{\prime} }}-{\epsilon }_{F})\) in the susceptibility. The main difference between these approaches is, therefore, that the Eliashberg treatment localizes the susceptibility to the Fermi surface. For most metallic systems, where the Fermi surface occupies a significant portion of the Brillouin zone, this difference might not be so severe. For multilayer graphene systems, however, where the Fermi surface is a tiny portion of the full Brillouin zone, the difference between the approaches is quite dramatic.

For a further discussion on the derivation of the gap equation, see Supplementary Note 2.

Gap symmetry analysis

The symmetry of the gap is determined by the point group of the material, and rhombohedral multilayer graphene belongs to the point group D3d. The wave functions at K+ and K− therefore have to satisfy the symmetry constraint \({C}_{3}\psi ({{{\bf{r}}}}){C}_{3}^{-1}={e}^{(2\pi i/3){\tau }_{z}{\sigma }_{z}}\psi ({R}_{3}{{{\bf{r}}}})\), where τz (σz) is a Pauli matrix in valley (sublattice) space. In both RTG and RHG, intra-valley pairing is associated with finite momentum Cooper pairs and will, therefore, be strongly suppressed. This follows from the inequivalence ϵτn(k) ≠ ϵτn(−k), where τ and n are valley and band indexes, such that no nesting conditions are met (see Fig. 1). It is therefore expected that inter-valley pairing is the dominant mechanism in these systems.

For inter-valley Cooper pairs, the symmetry constraint implies that intra-sublattice pairings are invariant under C3 symmetry, while inter-sublattice pairings will acquire a net phase. The former property is expected of s- and f-wave pairings (\({C}_{3}{\Delta }_{{{{\bf{k}}}}}{C}_{3}^{-1}={\Delta }_{{{{\bf{k}}}}}\)), while the latter is expected for p- and d-wave pairings (\({C}_{3}{\Delta }_{{{{\bf{k}}}}}{C}_{3}^{-1}={e}^{i\phi }{\Delta }_{{{{\bf{k}}}}}\)). Therefore, s- and f-wave symmetries can be distinguished from p- and d-wave symmetries by looking at how the gap transforms under threefold rotations. Similarly, s-wave symmetry can be distinguished from f-wave symmetry by the behavior under inversion, since s- and d-wave gaps have to be singlets (Δ−k = Δk), while p- and f-waves gaps have to be triplets (Δ−k = −Δk).

For the low-energy bands of RTG and RHG, the wave function is mainly localized to the dangling sites A1 and BN (with N the number of layers), such that inter-sublattice pairings are strongly suppressed. Therefore, the dominant pairing in these bands is expected to be either s- or f-wave. However, for the lower valence bands at ~−0.5 eV, other pairing channels might become competitive through inter-sublattice scattering in the intermediate layers.