Majorana physics in hybrid nanowires, topological phases and transport

Abstract

In general, Majoranas are a kind of particles that are their own antiparticles; there- fore two of the same kind put together annihilate each other. Majorana particles inherited their name from the notable italian physicist Ettore Majorana who pos- tulated their existence in 1937. Although no Majorana fundamental particles have been found in nature the concept has gained a new life in the realm of condensed matter physics. It is believed that it is possible to engineer devices able to hold Ma- joranas in the form of collective electronic excitations. These collective excitations would have the fundamental Majorana property of being created and annihilated in equal pairs albeit they would not be fermions like their particle physics equivalents but non-abelian anyons. The result of their interchange depends on the orientation and order of such interchange, a property that could be used to build quantum computers. From the many theoretical proposals of systems able to hold Majoranas this thesis is devoted to the study of Majorana modes in hybrid semiconductor-superconductor nanowires. A nanowire is a 1d or quasi-1d construct where Majoranas can appear if the proper ingredients are provided. In the context of this work these ingredients are superconductivity, spin orbit coupling and an external magnetic field, although this selection is not unique. We have chosen this system because of its simplicity, an advantage that allows for more realistic models and feasible proposals that can lead to new experiments.

Majoranas were originally theorized for purely 1d closed nanowires but this can not be easily achieved in real physical systems. Furthermore, Majoranas are very difficult to be detected experimentally because in this context they are chargeless, spinless, zero energy states. For this reason, experiments claiming Majorana detection are mainly based on their transport properties. In this thesis we take care of extending the 1d model to more general ones that take into consideration some realistic effects and we also study the Majorana transport properties in themselves. To do this we use a combination of different numerical and analytical methods that have allowed us to uncover a whole new set of Majorana features previously unknown. This thesis is presented as a compendium of publications where each contribution has been published or is in preparation to be published. First, we will focus on the study of smooth junctions between a Majorana nanowire and a normal lead. In this study the junctions are characterized by the potential and superconductor interface position and softness and it is discussed how this features affect the protection of the Majorana modes. Next, we investigate the effect that a tilting of the magnetic field has on the states of a purely 1d Majorana nanowire. Particularly, we are interested in the conditions of existence of the Majorana zero modes. We uncover a new analytical relation that restricts the parameter values in which a Majorana phase is possible. In the same chapter, we have also studied the Majorana physics of bent L-shaped 1d nanowires treated as if they were magnetically inhomogeneous straight nanowires composed of two homogeneous arms. The spectrum and the localization of Majoranas in this kind of nanowires are thoroughly discussed.

After studying 1d junctions and wires we move on to the study of Majorana physics of planar nanowires with a small but finite width. We demonstrate that Majo- ranas survive sizable vertical field tiltings of the external magnetic field in presence of the orbital motion caused by off plane components of this same magnetic field. Furthermore, we uncover a characteristic phase diagram for Majoranas where the analytical expressions of the phase boundaries are calculated for the limit of strong orbital effects. Additionally, we also obtain for the same kind of nanowire the local currents and the electromagnetic absorption cross section. This way we propose an alternative method of Majorana detection but complementary to transport measure- ments. We show how the presence of Majorana modes have a manifestation in the absorption spectra when the incident electromagnetic field is polarized in a trans- verse direction to the nanowire. Next, we also discuss the robustness of Majorana edge modes in the surface of finite quantum nanowires of cylindrical shape. More specifically we discuss the Majorana robustness for different external magnetic field tiltings. Furthermore, we investigate how the presence of Majorana modes depends on the radius of the nanowire cylinder. Finally, using a tight binding model and the Keldysh-Green function formalism we derive the time dependent electrical and energy currents in a circuit that contains a Majorana nanowire. Our main results for the AC transport report a singular behavior of the Majorana for the energy and electrical current admittances. Taken together, the published works that form this thesis increase our knowledge of the Majorana modes in nanowires with potential applications in experiments and also give a new insight of the Majorana transport properties. The experiments proposed can be implemented with the present technology and some of the numerical techniques can be generalized to non Majorana problems.