Transport and STM studies of hyperbolic surface states of topological insulators

Motivated by the transmission of topological surface states through atomic scale steps, we study the transport of gapless Dirac fermions on hyperbolic surfaces. We show that the density of states of the electrons shows a dip at concave step edges which can be measured by a scanning tunneling microscope tip. We also show that the tunneling conductance measured by a polarized scanning tunneling probe exhibits an unconventional dependence on the polar and azimuthal angles of the magnetization of the tip as a function of the curvature of the surface and the sharpness of the edge. However, no backward scattering takes place and transmission of the topological surface states remains completely independent of the geometrical shape (within the hyperbolic model) of the surface.