Theory of Phonon-Mediated Superconductivity in Twisted Bilayer Graphene

We present a theory of phonon-mediated superconductivity in near magic angle twisted bilayer graphene. Using a microscopic model for phonon coupling to moiré band electrons, we find that phonons generate attractive interactions in both $s$- and $d$-wave pairing channels and that the attraction is strong enough to explain the experimental superconducting transition temperatures. Before including Coulomb repulsion, the $s$-wave channel is more favorable; however, on-site Coulomb repulsion can suppress $s$-wave pairing relative to $d$ wave. The pair amplitude varies spatially with the moiré period, and is identical in the two layers in the $s$-wave channel but phase shifted by $\pi$ in the $d$-wave channel. We discuss experiments that can distinguish the two pairing states.

Figure 1

(a) Real- and (b) momentum-space structure of twisted bilayer graphene. Low-energy electrons are located in $\pm K$ valleys. (c) Moiré bands in $+K$ valley along high-symmetry lines at a twist close to the first magic angle. The spectra along $\nu \gamma$ and $\overline{\nu }\gamma$ lines are different, reflecting the absence of time-reversal symmetry within one valley.

Figure 2

Illustration of the atomic displacements in the four phonon modes. Each blue dot represents a carbon atom.

Figure 3

(a) Critical temperature $T_c$ in $s$ wave (blue lines) and $d$ wave channels (green lines) as a function of chemical potential (upper panel), and the DOS per spin and per valley as a function of energy (lower panel) for the twist angle 1.05°. The vertical dashed lines indicate the chemical potential at which the upper or lower flatband is half filled. (b) $T_c$ in $s$ wave (upper panel) and $d$ wave (lower panels) channels as a function of reduced attractive interaction strength for $\theta =1.05°$ and $\mu =-0.3\mathrm{meV}$ (half filling of the lower flatband).

Figure 4

Real-space map of pair amplitudes $\mathrm{\Delta }(\mathit{r})$ that satisfy (a) $s$-wave and (b) $d$-wave linearized gap equations for $\theta =1.05°$ and $\mu =-0.3\mathrm{meV}$. $\mathrm{\Delta }(\mathit{r})$ is peaked in the $AA$ regions in both cases. The inset in (b) illustrates $d_{+}$ pairing at the atomic scale, where electrons in the same layer but on different sublattices are paired with the indicated bond-dependent phase factors.