Double reflection and tunneling resonance in a topological insulator: Towards the quantification of warping strength by transport

We study the transport properties at the surface state of a topological insulator (TI) with a potential barrier. Due to the hexagonal warping effect, the number of reflected propagating states for an incident electron beam can change from one to two, corresponding to a change from normal reflection to double reflection, by controlling the Fermi energy. Associated with the change, the corresponding reflection probability shows a significant jump in its derivative. In addition, for a junction with potential barrier, the reflection probability should oscillate as the potential energy varies due to the tunneling resonance. The oscillation period in the TI junction is closely related to the warping strength. Thus, these two proposals both can be used to identify the hexagonal warping strength of the TI surface state and especially, are robust against the influence of lateral surfaces, as they depend on the relative variation of transport. Remarkably, the latter proposal is compatible with the recent experiment in graphene [Chen et al., Science 353, 1522 (2016), 10.1126/science.aaf5481], which achieves a direct measurement of the angle-resolved scattering probability.