  Valleytronic materials, characterized by local extrema (valley) in their
bands, and topological insulators have separately attracted great interest
recently. However, the interplay between valleytronic and topological
properties in one single system, likely to enable important unexplored
phenomena and applications, has been largely overlooked so far. Here, by
combining a tight-binding model with first-principles calculations, we find the
large-band-gap quantum spin Hall effects (QSHEs) and valley Hall effects (VHEs)
appear simultaneously in the Bi monolayers decorated with halogen elements,
denoted as Bi2XY (X, Y = H, F, Cl, Br, or I). A staggered exchange field is
introduced into the Bi2XY monolayers by transition metal atom (Cr, Mo, or W)
doping or LaFeO3 magnetic substrates, which together with the strong SOC of Bi
atoms generates a time-reversal-symmetry-broken QSHE and a huge valley
splitting (up to 513 meV) in the system. With gate control, QSHE and anomalous
charge, spin, valley Hall effects can be observed in the single system. These
predicted multiple and exotic Hall effects, associated with various degrees of
freedom of electrons, could enable applications of the functionalized Bi
monolayers in electronics, spintronics, and valleytronics.
