  In this work we have investigated the effect of Rashba splitting on the RKKY
interaction in TI thin film both at finite and zero chemical potential. We find
that the spin susceptibility of Rashba materials including TI thin film is
strongly dependent on the direction of distance vector. Moreover, we find
another term in the off-diagonal terms of the spin-susceptibility tensor which
in contrast to the well-known DM-like term is symmetric. We show how one can
tune the RKKY interaction by using electric field applied perpendicularly to
the surface plane and by small chemical doping giving rise to enhance the RKKY
term, drastically. We have presented our results for two different situations,
namely inter-surface pairing of magnetic impurities as well as intra-surface
one. The behavior of these two situations is completely different which we
describe it by mapping the density of states of each surface on the band
dispersion.
