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

TL;DR

This paper analytically investigates impurity states in narrow armchair graphene nanoribbons, revealing how confinement and impurity position affect energy levels and widths, with results aligning well with existing theories and experiments.

Contribution

It provides explicit analytical expressions for impurity energy spectra in narrow GNRs, including effects of ribbon width and impurity displacement, advancing understanding of impurity states in these systems.

Findings

Narrower GNRs increase binding energies and resonant widths.

Impurity displacement affects binding energies and widths differently for various series.

Analytical results agree with other theoretical and experimental data.

Abstract

An analytical study of discrete and resonant impurity quasi-Coulomb states in a narrow gaped 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 sub-band is really discrete, while others corresponding to the excited sub-bands consist of quasi-discrete (Fano resonant) levels of non-zero energetic widths, caused by the coupling with the states of the continuous spectrum branching from the low lying sub-bands. 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 centre from the mid-point 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 the some resonant states are quite stable and could be studied experimentally in doped GNRs.