Fig. 1: Nonlocal measurement scheme and electric field tunability in TDBG.

a Nonlocal measurement scheme of bulk valley current. A net valley current is generated by the charge current through the local probes in its transverse direction due to nonzero Berry curvature via valley Hall effect (VHE). In the nonlocal probes, the valley current generates a voltage drop by inverse VHE. b A schematic depicting the electric field tunable moiré bands in twisted double bilayer graphene (TDBG). Two low-energy flat bands—the conduction band (CB in blue) and the valence band (VB in red) are separated from high energy dispersing bands (gray) by two electric field tunable moiré gaps. CNP gap opens up between two flat bands as electric field is increased. c Schematic of a band structure of TDBG at a finite electric field with locations of the Berry curvature hotspots encircled. d A map of calculated Berry curvature Ω in k-space for K valley of the conduction flat band at a finite electric field corresponding to interlayer potential difference Δ = 10 meV. Here a is the lattice constant of graphene. This map shows the hotspots located at the symmetry points \(\tilde{\Gamma }\), \(\tilde{{\rm{K}}}\), and \({\tilde{{\rm{K}}}}^{\prime}\) in the moiré Brillouin zone. The Berry curvature for \({\rm{K}}^{\prime}\) valley has same magnitude but opposite sign.