  We study exchange interaction in zigzag bilayer phosphorene nanoribbons
(ZBLPNRs) under a perpendicular electric field. We evaluate the spatial and
electric field dependency of static spin susceptibility in real space in
various configurations of magnetic impurities at zero temperature. The
electronic properties of ZBLPNR in the presence of a gate voltage is also
obtained. In comparison to the other two-dimensional (2D) materials such as
graphene and silicene, ZBLPNR has two nearly degenerated quasiflat edge modes
at the Fermi level, isolated from the bulk states. The band gap modulation of
ZBLPNRs by the ribbon width and perpendicular electric field is investigated.
Due to the existence of these quasiflat bands at the Fermi level, in the
absence of the electric field, a sharp peak in the RKKY interaction is seen. As
shown, the signatures of these quasiflat edge modes in ZBLPNRs could be
explored by using the RKKY interaction. In the presence of a large bias
potential, a beating pattern of the RKKY oscillations occurs when two
impurities located inside the ZBLPNR. The electrically tunable RKKY coupling of
ZBLPNRs is expected to have important consequences on the spintronic
application of biased ZBLPNR.
