Coulomb-driven band unflattening suppresses $K$-phonon pairing in moiré graphene

It is a matter of current debate whether the gate-tunable superconductivity in twisted bilayer graphene is phonon mediated or arises from electron-electron interactions. The recent observation of the strong coupling of electrons to so-called $K$-phonon modes in angle-resolved photoemission spectroscopy experiments has resuscitated early proposals that $K$ phonons drive superconductivity. We show that the bandwidth-enhancing effect of interactions drastically weakens both the intrinsic susceptibility towards pairing as well as the screening of Coulomb repulsion that is essential for the phonon attraction to dominate at low temperature. This rules out purely $K$-phonon-mediated superconductivity with the observed transition temperature of $\sim 1$ K. We conclude that the unflattening of bands by Coulomb interactions challenges any purely phonon-driven pairing mechanism, and must be addressed by a successful theory of superconductivity in moiré graphene.

Figure 1

Suppression of pairing by Coulomb interactions. Column (a) shows BM model results as a reference; columns [(b)–(d)] show results for unstrained HF-renormalized bands that respectively preserve both flavor and $C_3$ symmetry, only flavor, and break both; (e) and (f) show results for strained HF-renormalized bands with and without flavor symmetry (cf. Table 1). Top row: Density of states at the Fermi level with a broadening corresponding to temperature $T=2$ K. Middle row: Energy scales of the Thomas-Fermi screened Coulomb interaction $U=V(q=0)$, and of the phonon-mediated attraction $U_{\text{ph}}=\frac{g}{A_M}=0.5$ meV, where $A_M$ is the moiré unit cell area. Even for the highest level of screening, $U\gg U_{\text{ph}}$, suppressing SC. Bottom row: Gap equation eigenvalue $\lambda$ without and with Coulomb interactions at temperature $T=2$ K, where $\lambda =-1$ signifies an SC instability. At the HF level, interactions reduce $\lambda$ by more than an order of magnitude, reducing $T_c$ well below the experimentally observed scale.

Figure 2

Interaction dependence of $T_c$ and symmetry of the gap function. (a) As the Coulomb interaction is reduced by increasing ${\epsilon }_r$, $T_c$ increases to around 2 K. $T_c$ only reaches the experimental value of around 2 K for an unrealistically weak Coulomb interaction with ${\epsilon }_r>100$. (b), (c) mBZ-even and mBZ-odd components of the gap function. All results are for filling $\nu =2.48$.