Localized states of a semi-infinite zigzag graphene sheet with impurity lines

The localized states of a semi-infinite zigzag graphene sheet are studied using a tight-binding model that allows for the inclusion of either one or two lines of impurities. These impurity lines of atoms are placed in rows labeled as n (n = 1, 2, 3, …), where n = 1 is the free edge. The localized defect modes associated with these impurities are studied analytically and numerically within a tridiagonal matrix formalism. For one impurity line, the modes are analyzed according to the position of that line on the sheet, whereas the modes for two impurities are studied also according to their separation and their positions relative to the edge. When an impurity line is located at the edge (n = 1), it is found that the edge states are modified. When the impurities are positioned away from an edge (n > 1), additional localized modes are found to occur that may be relatively flat in their dispersion.