Effects of tilting the magnetic field in one-dimensional Majorana nanowires

Abstract

We investigate the effects that a tilting of the magnetic field from the parallel direction has on the states of a 1D Majorana nanowire. Particularly, we focus on the conditions for the existence of Majorana zero modes, uncovering an analytical relation (the sine rule) between the field orientation relative to the wire, its magnitude and the superconducting parameter of the material. The study is then extended to junctions of nanowires, treated as magnetically inhomogeneous straight nanowires composed of two homogeneous arms. It is shown that their spectrum can be explained in terms of the spectra of two independent arms. Finally, we investigate how the localization of the Majorana mode is transferred from the magnetic interface at the corner of the junction to the end of the nanowire when increasing the arm length.

Tilting of the magnetic field in Majorana nanowires: Critical angle and zero-energy differential conductance

“…29. As the angle ϑ is increased through the value ϑ c , topologically trivial zero-energy states will appear with the momentum ± √ κ 1 .…”

“…This is readily seen, since this latter condition acts on the energy gap at zero-momentum, which does not depend on the direction of the field. In contrast, if the phase transition is driven by ϑ, the gap closes near the Fermi momentum 29 at √ κ 1 , cf. Fig.…”

“…If the field is tilted, on the other hand, the Majorana modes disappear at a critical angle 24,29 ϑ c , where the energy gap closes. Figure 2 illustrates the eigenenergies of the BdG Hamiltonian Eq.…”

“…It is immediately clear that the Majorana zero-modes cannot exist for arbitrary field directions. Some experiments have included a rotation of the external magnetic field, but there has been limited quantitative analysis 24,29 of the precise impact of the field direction on the Majorana zero-modes and the measured quantity, namely the differential conductance in a junction of the Majorana nanowire with a normal lead.…”

Extended Majorana zero modes in a topological superconducting-normal T-junction

“…This topological phase occurs when two conditions on the Zeeman field h are met 44,45 . Namely, the full field satisfies the topological criterion |h| > h c ≡ |∆| 2 + µ 2 (in a finite lattice model, there is an additional upper critical field due to the finite band width, but that field does not play any role in this paper).…”