Recently, there is a great deal of interest in materials known as topological insulators (TI). TI is a state of non-trivial topological matter that, like the quantum Hall effect, presents an insulating gap in the volume, but has conductive states on the surface, and that occurs in a null field, that is, without breaking the symmetry of temporal reversal, which differs from the quantum Hall state. These metallic states are analogous to edge states that characterize a material with a quantum spin-Hall effect, that is, a system in which there is a surface flow of two spin currents in opposite directions due to spin-orbit interaction (SOI). Similar to graphene, TI has a Dirac cone on its surface.Fermions of Majorana is a collective excitation that at the same time is its own antiparticle which can be detected in Josephson vortices (fluxons) trapped between two superconductors. In the case of topological superconductors, Josephson vortices are coupled with a zero energy Majorana mode and may behave as non-Abelian anions, despite the fact that they do not have a normal nucleus. Graphene based materials, in which topologically non-trivial local superconductivity occurs at elevated temperatures, may present a natural laboratory to test this theoretical prediction. One of the ways of detecting the presence of Majorana fermions was theoretically proposed by Ioselevich and Feigel'man, who considered a narrow channel crossing a topological insulator with both faces covered by an s-wave type superconductor. In the presence of a vortex in this channel, we will essentially have a ballistic nanowire connecting the superconducting surfaces, with a pair of Majorana states in it. Based on a series of calculations, the authors supposed that the energy of these Majorana states have a periodic dependence on the phase difference j between the surfaces.Another phenomenon of interest is the already century-old superconductivity, that still presents many challenges to scientists. The superconductivity and induced vortices, both in graphene and TI, open the possibility of forming and detecting fermions of Majorana, as well as the basis for their implementation in quantum computation. It is known that fermions of Majorana can be observed when the topological superconducting state is established in the material. This state can be achieved through the presence of strong SOI or changed Zeeman field. The Zeeman field can be induced in graphene through strong magnetic exchange interaction (MEI) in a graphene / magnetic insulation system. SOI can be achieved in graphene by the effect of proximity between graphene and transition metals with high SOI, through hydrogenation of graphene or doping with alkali metal atoms. It has been demonstrated that graphene nanobelts are suitable candidates for the observation of Majorana fermions through the topological superconductivity induced at their armchair-like edges, where degeneration of the valleys in the lower conductance zone is broken because of the boundary effects. This fact, together with the presence of SOI and / or MEI, leads to the formation of the topological state in this material.Some theoretical works in graphite show that the topological superconductivity could be realized in the surface of the rhombohedral phase with flat topologically protected bands in structure with hybrid stacking between the hexagonal and rhombohedral phases. Experimental work is now beginning to reveal a possible superconducting state in graphene in multiple layers or twisted bi-layer graphene. In these cases, the indication of the superconducting state was obtained on the surface by increasing the concentration of the carriers applying an electric field. Therefore, we can expect that carbon-based materials in a non-trivial topological format may show superconductivity that favors the observation of Majorana fermions.
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