Charge, Spin and Valley Hall Effects in Disordered Graphene

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.