Majorana fermions inch closer to reality

It is then natural to ask, What is a doped topological insulator good for? While one hopes that many of the topological phenomena of the true insulating state might be manifested in some form in a doped system, many questions still remain unanswered. However, Hosur et al. have made a striking prediction that MBS can still be realized in doped topological insulators under certain mild conditions [3]. A true insulating state is important in the Fu-Kane proposal because if the bulk contains low-energy states then the MBS can tunnel away from the surface and delocalize into the bulk, which effectively destroys the MBS. Hosur et al. circumvent this delocalization by requiring that the entire doped topological insulator become superconducting. They show that as long as the doping is not too large, vortices in superconducting topological insulators will bind MBS at the places where the vortex lines intersect the material surfaces. While this might seem like a big leap in complexity, experimental evidence already shows that, indeed, copper-doped Bi2Se3 is a superconductor below 3.8 K [12]. In this context, Hosur et al. make a strong prediction that vortex lines in superconducting CuxBi2Se3 can harbor MBS.

To understand the prediction, we begin with the Fu-Kane proximity effect scenario, as mentioned above, with a vortex line stretched between two surfaces. MBS are trapped where each end of the vortex line meets the topological insulator surface (see Fig. 1). If we tune the bulk chemical potential to lie in the conduction band, as opposed to the nominal insulating gap, then the MBS on each end of the vortex line could tunnel through the bulk and hybridize with the state on the opposite end. This is prevented in Hosur et al.’s work by inducing a superconducting gap in the entire bulk so that the MBS remain trapped. If the superconducting state were homogeneous, then the MBS would be trapped on the ends of the vortex line for any doping level. However, the superconducting order parameter varies rapidly near the vortex core, which is essentially a thin tube of normal metal (doped topological insulator) containing bound states with energies that lie below the nominal superconducting gap. It is easiest for the MBS to tunnel through the “mini-gap” region in the vortex core, and in fact, Hosur et al. go on to show that there is a critical chemical potential level where a vortex-core bound state becomes gapless and the MBS can easily tunnel through the vortex line to annihilate. Beyond this critical doping, the vortex line re-enters a gapped phase, but the MBS are absent. See Fig. 1 for an illustration of this process. The critical chemical potential can be calculated solely from low-energy information about the Fermi surface, and depends on the orientation of the vortex line with respect to the crystal structure. It is estimated that vortex lines oriented along the c axis of CuxBi2Se3 are just on the trivial side of the transition, while vortices perpendicular to the c axis should be well within the nontrivial regime and should trap MBS.