We consider core-shell nanowires with conductive shell and insulating core, and with polygonal cross section. We investigate the implications of this geometry on Majorana states expected in the presence of proximity-induced superconductivity and an external magnetic field. A typical prismatic nanowire has a hexagonal profile, but square and triangular shapes can also be obtained. The lowenergy states are localized at the corners of the cross section, i. e. along the prism edges, and are separated by a gap from higher energy states localized on the sides. The corner localization depends on the details of the shell geometry, i. e. thickness, diameter, and sharpness of the corners. We study systematically the low-energy spectrum of prismatic shells using numerical methods and derive the topological phase diagram as a function of magnetic field and chemical potential for triangular, square, and hexagonal geometries. A strong corner localization enhances the stability of Majorana modes to various perturbations, including the orbital effect of the magnetic field, whereas a weaker localization favorizes orbital effects and reduces the critical magnetic field. The prismatic geometry allows the Majorana zero-energy modes to be accompanied by low-energy states, which we call pseudo Majorana, and which converge to real Majoranas in the limit of small shell thickness. We include the Rashba spin-orbit coupling in a phenomenological manner, assuming a radial electric field across the shell.

The Majorana phase boundaries in planar 2D hybrid (semiconductor-superconductor) nanowires are modified by orbital effects due to off plane magnetic components. We show that Majorana zero modes survive sizable vertical field tiltings, uncovering a remarkable phase diagram. Analytical expressions of the phase boundaries are given for the strong orbital limit. These phase boundaries can be fulfilled with attainable setups, such as an InAs nanowire of $150\, {\rm nm}$ in transverse width.Comment: 9 pages, 6 figures. ( Only acknowledgement update

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.

We review applications of complex band structure theory to describe Majorana states in nanowires and nanowire junctions. The dimensionality of the considered wires is gradually increased, from strictly 1D to quasi-1D with one and two transverse dimensions. The complex wave number analysis is applied to two main types of Majorana modes: end states in hybrid semiconductor wires and chiral edge states in hybrid wires of quantum-anomalous Hall insulators. In topological NS and NSN junctions with Majorana modes conductance oscillations and spatial distributions of density and current are described. Finally, the optical absorption in finite systems with both types of Majorana modes is considered.