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We investigate the effect of Rashba splitting on the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in a topological-insulator (TI) thin film both at finite and zero chemical potential. We show that the spin susceptibility of the TI thin film depends strongly on the direction of the distance vector between impurities. In addition to the well-known Heisenberg-, Ising-, and Dzyaloshinskii-Moria (DM)-like terms reported before in TIs, we find another term in the off-diagonal part of the spin-susceptibility tensor which is symmetric in contrast to the DM term. Furthermore, we show how one can tune the RKKY interaction by using electric field applied perpendicularly to the surface plane of the TI, where in the presence of such a field the RKKY interaction can be enhanced drastically for small chemical doping. We present our results for two different situations, namely intersurface pairing of magnetic impurities as well as intrasurface pairing. The behavior of these two situations is completely different, which we describe by mapping the density of states of each surface on the band dispersion.
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