Two-gap superconductivity in heavily $n$-doped graphene: Ab initio Migdal-Eliashberg theory

Graphene is the only member of the carbon family from zero- to three-dimensional materials for which superconductivity has not been observed yet. At this time, it is not clear whether the quest for superconducting graphene is hindered by technical challenges, or else by the fluctuation of the order parameter in two dimensions. In this area, ab initio calculations are useful to guide experimental efforts by narrowing down the search space. In this spirit, we investigate from first principles the possibility of inducing superconductivity in doped graphene using the fully anisotropic Migdal-Eliashberg theory powered by Wannier-Fourier interpolation. To address a best-case scenario, we consider both electron and hole doping at high carrier densities so as to align the Fermi level to a van Hove singularity. In these conditions, we find superconducting gaps of $s$-wave symmetry, with a slight anisotropy induced by the trigonal warping, and, in the case of $n$-doped graphene, an unexpected two-gap structure reminiscent of ${\mathrm{MgB}}_2$. Our Migdal-Eliashberg calculations suggest that the observation of superconductivity at low temperature should be possible for $n$-doped graphene at carrier densities exceeding ${10}^{15}{\text{cm}}^{-2}$.

Figure 1

Electronic-structure and phonon-dispersion relations of pristine and doped graphene. (a) Electronic density of states and (b) band structures of pristine graphene (black solid line), $p$-doped graphene (dashed red line), and $n$-doped graphene (dotted blue line). (c),(d) Fermi surfaces of $p$-doped and $n$-doped graphene, respectively. In both cases, the Fermi surface corresponds to the green region of the energy isosurface, as indicated by the small arrow in the color bar. The long black arrows indicate the dominant electron-phonon scattering mechanisms at the Fermi surface. (e) Phonon-dispersion relations of pristine, $p$-doped, and $n$-doped graphene, using the same color code as in (a). The vibrational modes discussed in the text are indicated. (f) Phonon density of states corresponding to the dispersions in (e).

Figure 2

Electron-phonon coupling and superconducting gap function in doped graphene. (a),(b) Eliashberg spectral function and cumulative EPC calculated for $p$-doped and $n$-doped graphene, respectively. ${\alpha }^{2}F$ is the black solid line (left scale), $\lambda \left(\omega \right)$ is the blue dashed line (right scale). In the case of $n$-doped graphene, we also show the decomposition of $\lambda \left(\omega \right)$ into contributions from the intraband ${\pi }^{*}$ scattering (red dotted line) and interband ${\pi }^{*}$-FEL scattering (green dash-dotted line). (c),(d) Momentum-resolved EPC ${\lambda }_{\mathbf{k}}$ for electrons on the Fermi surface, for $p$-doped and $n$-doped graphene, respectively. (e)–(h) Anisotropic superconducting gap of $p$-doped graphene and $n$-doped graphene calculated using the Migdal-Eliashberg equations and ${\mu }_c^{*}=0$. Panels (e) and (f) represent the energy distribution of the gap across the Fermi surface at various temperatures (shown in black) for $p$-doped and $n$-doped graphene, respectively. The isotropic superconducting gap is shown as blue filled dots. The dotted lines are BCS fits to the calculated data. Panels (g) and (h) show the corresponding superconducting gap at zero temperature on the Fermi surface (in meV). In the case of $n$-doped graphene, there are two superconducting gaps, one for the ${\pi }^{*}$ band and one for the FEL band (h).

Figure 3

Superconducting critical temperature in doped graphene vs Coulomb parameter. (a),(b) Calculated superconducting critical temperature as a function of the Coulomb pseudopotential ${\mu }_c^{*}$ for $p$-doped graphene and $n$-doped graphene, respectively. The filled black squares are from anisotropic Migdal-Eliashberg calculations, the filled blue circles correspond to the isotropic approximation, and the filled green triangles are estimates based on the Allen-Dynes equation. The lines are guides to the eye. The vertical red line in (b) indicates the Coulomb parameter estimated here using the model dielectric function of Ref. [60], and the small arrow indicates our best estimate for the $T_c$ of $n$-doped graphene. The schematic band-structure plots illustrate the two doping scenarios considered in this work.