Spin-valley Hall conductivity of doped ferromagnetic silicene under strain

The spin-valley Hall conductivity (SHC-VHC) of two-dimensional material ferromagnetic graphene's silicon analog, silicene, is investigated in the presence of strain within the Kubo formalism in the context of the Kane–Mele Hamiltonian. The Dirac cone approximation has been used to investigate the dynamics of carriers under the strain along the armchair (AC) direction. In particular, we study the effect of external static electric field on these conductivities under the strain. In the presence of the strain, the carriers have a larger effective mass and the transport decreases. Our findings show that SHC changes with respect to the direction of the applied electric field symmetrically while VHC increases independently. Furthermore, the reflection symmetry of the structure has been broken with the electric field and a phase transition occurs to topological insulator for strained ferromagnetic silicene. A critical strain is found in the presence of the electric field around $45\%$. SHC (VHC) decreases (increases) for strains smaller than this value symmetrically while it increases (decreases) for strains larger than one. (paper)