Combined valley and layer control in twisted double bilayer graphene

We show full control over the valley polarization and interlayer coupling in twisted double bilayer graphene with a twist angle of 2.37$^\circ$. This intermediate angle is small enough for the layers to couple and large enough such that they can be tuned individually. Using a dual-gated geometry we identify and control all possible combinations of valley polarization and population of the two bilayers. In addition, we control the coupling of the bilayers by tuning the Fermi energy through a Lifshitz transition, where the Fermi surface topology changes. By modifying the Lifshitz transition with the displacement field, we identify combinations of Fermi surfaces relevant for correlated physics phenomena, possibly allowing to study the interplay between correlated states and valley polarization in the future.