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We work with the reduced  silicene/germanene single-particle Hamiltonian in buckled 2D hexagonal lattice expressed in terms of Pauli matrices associated with the pseudo-spin. The Hamiltonian of these systems comprises of the Dirac kinetic energy, a mass term and the spin–orbit coupling. The mass term breaks the sub-lattice symmetry of the system’s honey-comb structure and generates spin-split band gap with the help of the spin–orbit coupling. The buckled lattice generates a staggered sub-lattice potential between silicon atoms at A sites and B sites for an applied electric field E  z  perpendicular to its plane. The physics of the system is determined by two Dirac-like cones at K and K′ points. By tuning E  z , one attains a critical value at which the system at the topological insulating phase goes to a semi-metal state, where the ‘spin-down’ upon the ‘spin-up’ band gap ratio (r) ≪ 1 for the valley K and r ≫ 1 for the valley K′. This state is termed as the ‘valley-spin-polarized-metal’ due to the opposite spin-polarization of the K and K′ valleys. Upon increasing E  z  further, the system turns into a trivial insulator. This event is associated with the valley magnetic moment reversal. Our preliminary investigation have shown that, as long as the (non-magnetic) impurity scattering strength V 0 is moderate, i.e. V 0 is of the same order as the intrinsic spin–orbit coupling t  so (~4 meV), the ‘valley-spin-polarized-metal’ phase is protected. The substantial enhancement in V 0, however, leads to the disappearance of this 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.

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