  We present electronic structure calculations of twisted double bilayer
graphene (TDBG): A tetralayer graphene structure composed of two AB-stacked
graphene bilayers with a relative rotation angle between them. Using
first-principles calculations, we find that TDBG is semiconducting with a band
gap that depends on the twist angle, that can be tuned by an external electric
field. The gap is consistent with TDBG symmetry and its magnitude is related to
surface effects, driving electron transfer from outer to inner layers. The
surface effect competes with an energy upshift of localized states at inner
layers, giving rise to the peculiar angle dependence of the band gap, which
reduces at low angles. For these low twist angles, the TDBG develops flat
bands, in which electrons in the inner layers are localized at the AA regions,
as in twisted bilayer graphene.
