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The Hall conductivity $\sigma_{xy}$ of many condensed matter systems presentsa step structure when a uniform perpendicular magnetic field is applied. Wereport the quantum Hall effect in buckled AB-bottom-top bilayer silicene andits robust dependence on the electronic valley and spin-orbit coupling. Withthe unique multi-valley electronic structure and the lack of spin degeneracy,the quantization of the Hall conductivity in this system is unlike theconventional sequence as reported for graphene. Furthermore, the conductivityplateaux take different step values for conduction ($2e^2/h$) and valence($6e^2/h$) bands since their originating valleys present inequivalentdegeneracy. We also report the emergence of fractions under significant effectof a uniform external electric field on the quantum Hall step structure by theseparation of orbital distributions and the mixing of Landau levels fromdistinct valleys. The valley- and spin-dependent quantum Hall conductivityarises from the interplay of lattice geometry, atomic interaction, spin-orbitcoupling, and external magnetic and electric fields.

Valley- and spin-dependent quantum Hall states in bilayer silicene