Probing surface states of topological insulators: Kondo effect and Friedel oscillations in a magnetic field

Abstract

We address three issues on the role of magnetic impurities in the surface state of a three dimensional topological insulator. First, we prove that the Kondo effect of the topological surface is essentially the same as that of the graphene surface, demonstrating that an effective impurity action of the topological surface coincides with that of the graphene surface. Second, we study the role of the $z$-directional magnetic field ($h$) in the Kondo effect, and show that the peak splitting in the impurity local density of states does not follow the $h$-linear behavior, the typical physics in the soft-gap Kondo model. We discuss that the origin is spin locking in the helical surface. Third, we examine the Friedel oscillation around the magnetic impurity. It turns out that the pattern of Friedel oscillation in the helical surface is identical to that of the graphene surface, displaying the inverse-square behavior $\sim r^{-2}$ if the inter-valley scattering is not introduced in the graphene case. However, introduction of the magnetic field leads the electron density of states from the inverse-square physics to the inverse behavior $\sim r^{-1}$ in the topological insulator's surface while it still remains as $\sim r^{-2}$ in the graphene Kondo effect. We discuss that this originates from spin flipping induced by magnetic field.