Influence of Dirac cone warping and tilting on the Friedel oscillations in a topological insulator

We calculate the Ruderman-Kittel-Kasuya-Yosida interaction between impurity spins, localized at the surface of an imperfect topological insulator (TI). Namely, we consider the warping and tilting of a TI Dirac cone. We show that the warping and tilting addition to the initial Rashba spin-orbit interaction (SOI) yields a highly anisotropic response to the localized spin rotation. Also, the Friedel oscillation strength of such an imperfect TI depends on the direction in its plane. Latter dependence, parametrized by Lissajous curves of spin-response components, can serve as a ``fingerprint'' of the initial Dirac cone distortion in the TI. The ensuing unusual dynamics is due to the interplay of SOI, warping and tilting of the initial Dirac cone, and the resulting indirect exchange between localized impurities. Variation of the warping and tilting strengths can be used to manipulate the impurity spin reversal, which is important for technological applications.