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Effect of disorder in the transition from topological insulator to valley-spin polarized state in silicene and Germanene

We start with the silicene or germanene single-particle Hamiltonian in buckled 2D hexagonal lattices expressed in terms of Dirac matrices in the Weyl basis. The Hamiltonian of these systems comprises of the Dirac kinetic energy, a mass gap term, and the spin-orbit coupling. The second term breaks the sub-lattice symmetry of the honey-comb lattice structure and generates a gap. The buckled structure generates a staggered sub-lattice potential between silicon atoms at A sites and B sites for an applied electric field perpendicular to its plane. Tuning of the electric field, allows for rich behavior varying from a topological insulator to a band insulator with a valley spin-polarized metal at a critical value in between. Thus, the mobile electrons in silicene or germanene are coupled differently compared to graphene to an external electric field. Our preliminary investigation have shown that, as long as the non-magnetic impurity scattering strength is moderate, i.e. the strength is of the same order as the intrinsic spin-orbit coupling of 4 meV, valley-spin polarized metal phase is protected. The effective "two-component Dirac physics" remains valid in this phase. The increase in the electric field beyond the critical value leads to the valley magnetic moment reversal. The enhancement in the impurity scattering strength, however, leads to the disappearance of the polarized metal phase due to accentuated intra- and inter-valley scattering processes. This disappearance does not occur due to the increase in Rashba spin-orbit coupling effect.