Band structure and superconductivity in twisted trilayer graphene

We study the symmetries of twisted trilayer graphene's band structure under various extrinsic perturbations, and analyze role long-range electron-electron interactions near first magic angle. The electronic is modified by these in a similar way to bilayer graphene. electron pairing due long-wavelength charge fluctuations, which are coupled among themselves via Coulomb interaction additionally mediated longitudinal acoustic phonons. find superconducting phases with either spin-singlet/valley-triplet or spin-triplet/valley-singlet symmetry, critical temperatures up few Kelvin for realistic choices parameters.