Effect of the electron-phonon interaction on the spin texture in Bi2−ySbySe3−xTex

Abstract

The electronic surface state of a three-dimensional strong topological insulator (TI) is characterized by an isotropic Dirac cone at the $\mathrm{\ensuremath{\Gamma}}$ point and an ideally perpendicular spin-momentum locking. Away from this point, the lattice anisotropy becomes effective where deviations from the ideal picture are observed. In particular, weak violations of the spin-momentum orthogonality is observed in many experiments. In parallel to these, the existence of the electron-phonon interaction has been verified by a number of experimental groups. In this article we devise an interacting theory of the surface spin. We observe that the interactions, while weakly affecting the energy bands, can yield changes in the spin texture in the presence of a Fermi surface anisotropy. In particular, using the coupling of a specific optical phonon experimentally discovered in the ${\mathrm{Bi}}_{2\ensuremath{-}y}{\mathrm{Sb}}_{y}{\mathrm{Se}}_{3\ensuremath{-}x}{\mathrm{Te}}_{x}$ family to the surface state, we obtain a sixfold nonorthogonal modulations in the in-plane spin-momentum pair. The out-of-plane spin experiments are also discussed in this context. Our results indicate that there is an additionally new mechanism contributing to the deviations in the spin texture beyond the conventional independent particle picture in topological materials where interactions are observed to be important. Recent experimental reports on the spin-momentum nonorthogonality in other symmetries, and even in nearly isotropic Fermi surfaces, indicate that the theory introduced here may be relevant to a larger set of interacting Dirac materials and deserves further exploration.