Mean-field theory for superconductivity in twisted bilayer graphene

Recent experiments show how a bilayer graphene twisted around a certain magic angle becomes superconducting as it is doped into a region with approximate flat bands. We investigate the mean-field $s$-wave superconducting state in such a system and show how the state evolves as the twist angle is tuned, and as a function of the doping level. We argue that part of the experimental findings could well be understood to result from an attractive electron-electron interaction mediated by electron-phonon coupling, but the flat-band nature of the excitation spectrum also makes the superconductivity quite unusual. For example, as the flat-band states are highly localized around certain spots in the structure, also the superconducting order parameter becomes strongly inhomogeneous.

Figure 1

(a) Twisted bilayer graphene and its moiré superlattice. The upper layer is rotated by an angle $\theta$ relative to the lower layer. (b) Position dependence of the self-consistent $\mathrm{\Delta }$, shown here at $T=0$ for the magic angle $\theta =0.{96}^{\circ }$ and $\lambda =5\mathrm{eV}{a}^2$. In both figures also a line passing through high-symmetry points with AB, AA, and BA stacking is shown.

Figure 2

(a)–(c) Normal-state dispersion, (d)–(f) local, and (g)–(i) total density of states for three different angles near the magic angle $\theta =0.{96}^{\circ }$ in the normal state. The bottom row (j)–(l) shows the corresponding total density of states in the superconducting state, in the case $T=0$ and $\lambda =5\mathrm{eV}{a}^2$ and when doped to the point ${\mu }_0$ marked as a dashed line in (g)–(i).

Figure 3

Maximum of the position-dependent superconducting order parameter $\mathrm{\Delta }\left(\mathit{r}\right)$ at $T=0$ as a function of (a) the rotation angle and (b) the coupling strength for $\theta =0.{96}^{\circ }$. In (b) we also show how doping to the DOS peak affects the small-$\lambda$ behavior.

Figure 4

$max\left(\mathrm{\Delta }\right)$ as a function of temperature in the case $\theta =0.{96}^{\circ }$ for two values of $\lambda$, showing the approximate linear relation $k_B{T}_c\approx 0.25max\left[\mathrm{\Delta }(T=0)\right]$ for the critical temperature. The dots are the calculated values and the lines are a guide to the eye.

Figure 5

Effects of electrostatic doping $\mu ={\mu }_0+\delta \mu$ for $\theta =0.{96}^{\circ }$. (a) $max\left(\mathrm{\Delta }\right)$ vs chemical potential for various values of $\lambda$ at $T=0$. (b) Charge density in the normal state at $T=0$ as a function of chemical potential. The units of the charge density $n$ are $e/A_{\mathrm{moir}\acute{\mathrm{e}}}$, where $e$ is the electron charge and $A_{\mathrm{moir}\acute{\mathrm{e}}}$ is the area of the moiré unit cell. In both figures the vertical dashed lines mark the location of the DOS peaks at $\delta \mu \approx \pm 0.26\mathrm{meV}$.