  We investigate here how the current flows over a bilayer graphene in the
presence of an external electric field perpendicularly applied (biased
bilayer). Charge density polarization between layers in these systems is known
to create a layer pseudospin, which can be manipulated by the electric field.
Our results show that current does not necessarily flow over regions of the
system with higher charge density. Charge can be predominantly concentrated
over one layer, while current flows over the other layer. We find that this
phenomenon occurs when the charge density becomes highly concentrated over only
one of the sublattices, as the electric field breaks layer and sublattice
symmetries for a Bernal-stacked bilayer. For bilayer nanoribbons, the situation
is even more complex, with a competition between edge and bulk effects for the
definition of the current flow. We show that, in spite of not flowing trough
the layer where charge is polarized to, the current in these systems also
defines a controllable layer pseudospin.
