Bound and resonant impurity states in a narrow gapped armchair graphene nanoribbon

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An analytical study of discrete and resonant impurity quasi-Coulomb states in a narrow gapped armchair graphene nanoribbon (GNR) is performed. We employ the adiabatic approximation assuming that the motions parallel (``slow'') and perpendicular (``fast'') to the boundaries of the ribbon are separated adiabatically. The energy spectrum comprises a sequence of series of quasi-Rydberg levels relevant to the slow motion adjacent from the low energies to the size-quantized levels associated with the fast motion. Only the series attributed to the ground size-quantized subband is really discrete, while others corresponding to the excited subbands consist of quasidiscrete (Fano resonant) levels of nonzero energetic widths, caused by the coupling with the states of the continuous spectrum branching from the low lying subbands. In the two- and three-subband approximation the spectrum of the complex energies of the impurity electron is derived in an explicit form. Narrowing the GNR leads to an increase of the binding energy and the resonant width both induced by the finite width of the ribbon. Displacing the impurity center from the midpoint of the GNR causes the binding energy to decrease, while the resonant width of the first excited Rydberg series increases. As for the second excited series, their widths become narrower with the shift of the impurity. A successful comparison of our analytical results with those obtained by other theoretical and experimental methods is presented. Estimates of the binding energies and the resonant widths taken for the parameters of typical GNRs show that not only the strictly discrete but also some resonant states are quite stable and could be studied experimentally in doped GNRs.