  Gated heterostructures containing bilayer graphene with staggered sublattice
potentials are investigated by tight binding model with Rashba spin-orbital
coupling and Hubbard interaction. The topological phase diagrams depend on the
combinations of substrates and the Hubbard interaction. The presence of the
staggered sublattice potential favor the topological phase transition with
small Rashba spin-orbital coupling strength. The presence of the Hubbard
interaction modified the topological phase boundaries, increasing the minimal
spin-orbital coupling strength for topological phase transition. A phase space
of topological semi-metal with indirect band gap is identified in the
non-interacting systems. For the bilayer graphene with different staggered
sublattice potentials in the two layers, the conditions for the zigzag
nanoribbons to host edge polarized chiral edge states are discussed. The
conditions require moderate or vanishing Rashba spin-orbital coupling strength,
as well as proper range of the gate voltage. The conditions for the systems
with and without the Hubbard interaction are compared. The edge polarization
can be controlled by the gate voltage.
