We develop the high frequency expansion based on the Brillouin-Wigner (B-W) perturbation theory for driven systems with spin-orbit coupling which is applicable to the cases of silicene, germanene and stanene. We compute the effective Hamiltonian in the zero photon subspace not only to order O(ω −1 ), but by keeping all the important terms to order O(ω −2 ), and obtain the photoassisted correction terms to both the hopping and the spin-orbit terms, as well as new longer ranged hopping terms. We then use the effective static Hamiltonian to compute the phase diagram in the high frequency limit and compare it with the results of direct numerical computation of the Chern numbers of the Floquet bands, and show that at sufficiently large frequencies, the B-W theory high frequency expansion works well even in the presence of spin-orbit coupling terms.

We investigate charge conductance and spin and valley polarization along with the tunnelling magneto-resistance (TMR) in silicene junctions composed of normal silicene and ferromagnetic silicene. We show distinct features of the conductances for parallel and anti-parallel spin configurations and the TMR, as the ferromagnetic−normal−ferromagnetic (FNF) junction is tuned by an external electric field. We analyse the behavior of the charge conductance and valley and spin polarizations in terms of the independent conductances of the different spins at the two valleys and the band structure of ferromagnetic silicene and show how the conductances are affected by the vanishing of the propagating states at one or the other valley. In particular, unlike in graphene, the band structure at the two valleys are independently affected by the spin in the ferromagnetic regions and lead to non-zero, and in certain parameter regimes, pure valley and spin polarizations, which can be tuned by the external electric field. We also investigate the oscillatory behavior of the TMR with respect to the strength of the barrier potential (both spin-independent and spin-dependent barriers) in the normal silicene region and note that in some parameter regimes, the TMR can even go from positive to negative values, as a function of the external electric field.

We analyze the confinement of electronic surface states in a model of a topological insulator nanowire. Spin-momentum locking in the surface states reduces unwanted backscattering in the presence of non-magnetic disorder and is known to counteract localization for certain values of magnetic flux threading the wire. We show that intentional backscattering can be induced for a range of conditions in the presence of a nanowire constriction. We propose a geometry for a nanowire that involves two constrictions and show that these regions form effective barriers that allow for the formation of a quantum dot. We analyze the zero-temperature non-interacting electronic transport through the device using the Landauer-Büttiker approach and show how externally applied magnetic flux parallel to the nanowire and electrostatic gates can be used to control the spectrum of the quantum dot and the electronic transport through the surface states of the model device.

We study the topological phase transitions induced in spin-orbit coupled materials with buckling like silicene, germanene, stanene, etc, by circularly polarised light, beyond the high-frequency regime, and unearth many additional topological phases. We also study the robustness of these phases in the presence of uniform disorder. These phases are characterised by the spin-resolved topological invariants, C 0 ↑ , C 0 ↓ , C π ↑ and C π ↓ , which specify the spin-resolved edge states traversing the gaps at zero quasi-energy and the Floquet zone boundaries respectively. We show that for each phase boundary, and independently for each spin sector, the gap closure in the Brillouin zone occurs at a high symmetry point.