Electronic and topological properties of bilayer graphene nanostructures

This thesis is devoted to studying the electronic properties of bilayer graphene (BLG) by focusing on the confined states, especially the topological states, and the study of the transport in this material, addressing charge transport for electrons with and without magnetic field. Electrostatic confinement in BLG is achieved by applying top and bottom microelectrodes acting with reversed signs on the two graphene sheets. We discuss in this thesis two types of electrostatic confinement: trivial and topological. Trivial electrostatic confinement in BLG is characterized by all micro-electrodes on the top side having the same potential sign, which is opposed to the sign of all micro-electrodes on the bottom side of the graphene sheets. Topological electrostatic confinement in BLG is characterized by different microelectrodes of the top side having sign inversion of the potential. This creates boundaries separating regions of opposite directions of the inter-layer electric field. The boundary with a straight line shape is known as a kink, and the low-energy electronic states propagate along the kink. Chapter 1 is a general introduction, discussing the fundamental aspects and theoretical background of monolayer and bilayer graphene, quantum transport, and its paradigmatic systems (quantum dots and quantum points contacts). In Chapter 2, we discuss the trivial and topological confinement in bilayer graphene wires, comparing the two types of confinement depending on the potential applied to the bilayer graphene sheets. We discuss the behavior of the confined states in both cases. We found that for trivial confinement the spectrum opens a gap, and the states are confined in a region with a low energy gap. Otherwise, in the topological confinement, in the middle of the gap, we found states propagating in opposite directions for each valley. This phenomenon is known as momentum-valley locking in bilayer graphene. To investigate and know more about these states and their behavior, Chapter 3 describes a system where we can study and control the back-scattering of those topological states under kink-antikink potentials. We demonstrate that a kink-antikink constriction can modulate the transmission electrostatically. If we change the geometry of bilayer graphene what will happen to the topological states? The fourth Chapter answers this question when discussing the geometry dependence on the bilayer graphene. We take four shapes from higher to lower symmetry (circle, square, rectangle, and polygon). Our study shows that for small sizes the spectrum depends on the loop shape. The magnetic field induces a valley splitting and asymmetry in the spectrum. We have also done a comparative study between the trivial and topological confinement in the case of circular geometry (ring and dot), as discussed in Chapter 5. The study discusses the trivial confinement where it shows the bunching of levels into degenerate Landau bands, with an energy asymmetric gap, while topological confinement shows no field-induced gap and a sequence of state branches always crossing zero energy. Finally, a summary of our results is included in Chapter 6. In this Chapter, we also give a perspective and future works that would either treat systems not considered in this work or extend the applicability range of our theoretical formalism.