Existence of Majorana bound states near impurities in the case of a small superconducting gap

We consider the edge states of a 2D topological insulator in the presence of the Zeeman field and in proximity to a s-wave superconductor. We analytically show that two linearly independent Majorana bound states (MBSs) can appear near the impurity located in a small region in which both the pairing parameter Δ and the Zeeman field M may be changed. We find two conditions for the existence of the MBSs: firstly, |Δ|≈|M|, ie, the superconducting gap in the spectrum should be sufficiently small; secondly, the absolute value of the average w of the impurity potential should have a certain value; the last condition is necessary. The equation |Δ|=|M| determines the boundary of the topological phase of the system, thus the system as a whole must be close to this boundary in relation to the parameters. If the same is true for the impurity region, then the second condition has the form w≈±v/2 where v is the edge states velocity. In this case, the electron transmission probability is equal to 1 for energies close to zero.