We study the electronic states of graphene nanoribbons with modified edge structures by attaching Klein's bearded bonds as a minimal model of edge modification. The partial attachment of Klein's bearded bonds to graphene nanoribbons gives rise to the partial flat bands at zero-energy even under the condition of jN A À N B j ¼ 0, where N A ðN B Þ is the number of A ðBÞ-sublattice sites. Using transfer matrix method, we successfully derive the analytic representation of edge states for modified zigzag edge. The modification of armchair edges causes the complete flat bands, where the wavefunction has the character of valley polarization. We also applied the density functional theory to optimize the lattice structure and estimate the spin density. Our results indicate that the chemical and structural modification of graphene edge will serve to design and stabilize the spin polarized edge states.
We study the electronic states of graphene nanoribbons whose edge structures are modified in the manner that Klein's bearded bonds are periodically distributed. In the triply periodic system, complete flat bands are obtained in the case that two Klein's bonds are attached to edge per unit cell. However, there is no flat band in one Klein's bond case. We discuss the existence of flat bands in the triple periodicity system in relation to the linkage between wavefunctions; the analytic expression of the wavefunction is consecutively derived by the transfer matrix method.
Nanoelectronics based on graphene has become a fast growing field with a number of technical applications. In these circumstances, we perform numerical study of the existence of flat bands (FB) in graphene ribbons, in consideration of zigzag and Klein's bonds with periodic distribution for all cases in which N ≦ 10, where N denotes the period of the edge pattern. As a result, the electronic state at Fermi energy of graphene ribbons can be categorized into four types, depending on the period and the density of the Klein's bonds (NK), R ≡ NK/2N; These are type (i) where FB disappears, type (ii) ribbons that have only FB, type (iii) ribbons possessing only partially flat band (PFB), and type (iv) containing both FB and PFB. When N increases in type (iv), double degeneracy of PFB is maintained, while degeneracy of FB increases. Systems of N = 3n are classified into categories types (i) and (ii), while systems of N ≠ 3n belong to types (iii) and (iv). We would like to emphasize that above properties for appearances and disappearances of PFB and FB are dominated only by numbers of Klein's bonds in corresponding unit cells for N periods. Namely, those are independent of positions of Klein's bonds. The relationship between those properties in ribbons and the role of Dirac K-points originating from graphene is discussed.
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