  We study the orbital Hall effect (OHE) in the AC regime using bilayer
graphene (BLG) as a prototypical material platform. While the unbiased BLG has
gapless electronic spectra, applying a perpendicular electric field creates an
energy band gap that can be continuously tuned from zero to high values. By
exploiting this flexibility, we demonstrate the ability to control the behavior
of AC orbital Hall conductivity. Particularly, we demonstrate that the orbital
Hall conductivity at the neutrality point changes its signal at a critical
frequency, the value of which is proportional to the perpendicular electric
field. For BLG with narrow band gaps, the active frequency region for the AC
OHE may extend to a few terahertz, which is experimentally accessible with
current technologies. We also consider the introduction of a perpendicular
magnetic field in the weak coupling regime using first-order perturbation
theory to illustrate how the breaking of time-reversal symmetry enables the
emergence of AC charge Hall effect in the charge-doped situation and modifies
the AC orbital Hall conductivity. Our calculations suggest that BLG with narrow
bandgaps is a practical candidate for investigating time-dependent orbital
angular momentum transport.
