Charge, Spin and Valley Hall Effects in Disordered Graphene

Abstract: The discovery of the integer quantum Hall effect in the early eighties of the last century, with highly precise quantization values for the Hall conductance in multiples of $e^2/h$, has been the first fascinating manifestation of the topological state of matter driven by magnetic field and disorder, and related to the formation of non-dissipative current flow. In 2005, several new phenomena such as the spin Hall effect and the quantum spin Hall effect were predicted in the presence of strong spin-orbit coupling and vanishing external magnetic field. More recently, the Zeeman spin Hall effect and the formation of valley Hall topological currents have been introduced for graphene-based systems, under time-reversal or inversion symmetry-breaking conditions, respectively. This review presents a comprehensive coverage of all these Hall effects in disordered graphene from the perspective of numerical simulations of quantum transport in two-dimensional bulk systems (by means of the Kubo formalism) and multiterminal nanostructures (by means of the Landauer-B\"{u}ttiker scattering and nonequilibrium Green function approaches). In contrast to usual two-dimensional electron gases, the presence of defects in graphene generates more complex electronic features such as electron-hole asymmetry, defect resonances or percolation effect between localized impurity states, which, together with extra degrees of freedom (sublattice pseudospin, valley isospin), bring a higher degree of complexity and enlarge the transport phase diagram.