Electronic structure for a topological insulator with a smoothly varying step on surface

We study the penetration depth (PD) of surface state into the bulk, the local density of states (LDOS) and the spin-polarized STM tip tunneling conductance (TG) for a smoothly varying step described by an asymptotic model, on the surface of a three-dimensional (3D) topological insulator (TI). Using curvilinear coordinates for calculation, we find that the PD shows two peaks near step edges with maximum surface curvature and a dip value of flat surface at step middle with zero curvature. In contrast, the LDOS exhibits a double valley (dip) pattern, in which it rapidly declines from the highest value for flat surface to the dip value near step edges, and then shows a small peak at step middle. This interesting small peak is in good agreement with experimentally observed LDOS peak within a step of definite span on a 3D TI surface. Moreover, the calculated surface spin-polarized STM tip TG also confirms the LDOS qualitatively. The findings here may provide a further understanding of 3D TI surface with step line defects.