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

Abstract

In this work, we show that an N-C-chain gate potential along the armchair chain (direction (y) over cap) of a graphene sheet gives rise to topological localized states (LSs): one branch for N-C = 1 and two branches for N-C >= 2. These LSs are shown to form whenever the gate-induced potential V-0 is nonzero. The topological nature behind the formation of these LSs is revealed (for N-C = 1, 2) by showing, for V-0 not equal 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 Delta theta(p) between the two pseudospins, as the energy E varies across the entire energy gap for a given k(y), 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 k(y) = 0). For general multiple (N-C > 1) carbon chain gate-potential cases, the number N-LS of LS branches are found to increase with V-0, up to a maximum of N-LS,(max) = N-C. Yet LS branches carrying the Dirac-point characteristics are found to be fixed at two.