Final Report Summary - HOWTOCONTROLGRAPHENE (Search for mechanisms to control massless electrons in graphene) Graphene has brought Dirac fermions from relativistic high-energy physics into the low-energy realm of condensed matter physics. The fact that conduction electrons in graphene have no mass, makes them very efficient carriers of electricity — but it also makes them difficult to control. Normal electrostatic barriers cannot stop Dirac fermions, so other means to control the current are needed. This project aims to explore and develop methods to control massless Dirac fermions.The edges of a nanostructure, such as a quantum dot or a nanoribbon, are an effective way to control Dirac fermions. How an electron reflects from the edge of a graphene sheet depends on the crystalline orientation (zigzag or armchair edges) and on possible atomic reconstructions of the edge (notably the formation of a pentagon-heptagon pair out of two hexagons of carbon atoms). We have discovered that a large clas of boundaries can be described by a single parameter, which governs the density of states and the propagation speed along the edge. The single-parameter boundary condition provides an efficient alternative to microscopic computer calculations, and we have shown it to be highly accurate. Based on this boundary condition we could predict the absence of intervalley scattering for a broad class of edge reconstructions and edge orientations.Graphene has the unique property that the quantum Hall effect is reached in magnetic fields that are sufficiently low for superconductivity to survive. The coexistence of Andreev reflection and the quantum Hall effect leads to a new type of edge states along the graphene-superconductor interface. The edge states are charge neutral, because electrons are continuously scattered into holes and vice versa, but they do carry a spin. These neutral spin currents could have applications in spintronics. We have predicted that the Fraunhofer oscillations in the critical current as a function of magnetic flux have twice the usual periodicity, providing a unique signature for these edge states.Edges are one way to control massless electrons in graphene. Giving them a mass is another way. A mass is produced in graphene by a suitable substrate, which imposes a different electrostatic potential on the two sublattices of carbon atoms. Hydrogenation has the same effect, since the hydrogen is preferrentially attached to carbon atoms on each sublattice. It was a controversial issue in the literature whether or not a random mass would always localize the Dirac fermions, turning the graphene sheet from a metal into an insulator. This issue has obvious importance for electronics applications of hydrogenated graphene. Different models gave conflicting results, some showed metallic behavior for large energy disorder, others had only insulating behavior.We were able to settle this controversy by benefiting from the recent discovery of another class of materials that have Dirac fermions, socalled topological insulators. By comparing the two types of Dirac fermions, in graphene and in topological insulators, we could explain the model dependence of the results in the literature. We concluded, supported by computer simulations, that there is no metallic phase in graphene with a random mass.The proximity to a superconductor can convert a Dirac fermion into a pair of socalled Majorana fermions, particles that are their own antiparticles. We have proposed several signatures in trans- port properties that could provide a smoking gun evidence for these elusive particles, and we have invented a method to demonstrate their socalled “non-Abelian statistics” — a property that would make Majorana fermions ideal building blocks for a quantum computer.
