Gate-induced localized states in graphene: Topological nature in their formation

In this work, we show that an NC-chain gate potential along the armchair chain (direction ŷ) of a graphene sheet gives rise to topological localized states (LSs): one branch for NC=1 and two branches for NC≥2. These LSs are shown to form whenever the gate-induced potential V0 is nonzero. The topological nature behind the formation of these LSs is revealed (for NC=1,2) by showing, for V0≠0, that the LS-secular equation can be cast into a pseudospin-rotation form on which rotation upon a valley-associated pseudospin is to equate with another valley-associated pseudospin. Both pseudospins are on the same side of the gate potential. That the rotation angle of the pseudospin-rotation operator falls within the range of variation of the relative angle Δθp between the two pseudospins, as the energy E varies across the entire energy gap for a given ky, demonstrates the topological nature and the inevitability of the LS branch formation. These topological LS branches exhibit Dirac-point characteristics, with dispersion relations leading out from the Dirac point (at ky=0). For general multiple (NC>1) carbon chain gate-potential cases, the number NLS of LS branches are found to increase with V0, up to a maximum of NLS,max=NC. Yet LS branches carrying the Dirac-point characteristics are found to be fixed at two.