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

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.

Figure 1

A schematic form of the potentials $V_{NN}\left(y\right)$ provided in Eqs. (2), (11), and (14) at $x_0=0$, quasidiscrete $n$ (23), and continuous $k_N$ (24) spectra adjacent to the ground $N=0$ (discrete states) and first $N=1$ and second $N=-1$ size-quantized levels ${\varepsilon }_N$ (7) in the GNR of width $d$.

Figure 2

The binding energy $E_{0n}^{\left(b\right)}$ (44) of the ground state $n=0$ calculated for ($q=0.13,0.24$) as a function of the reciprocal width $\frac{1}{d}$ of the GNR. Impurity is placed symmetrically to the boundaries $(x_0=0)$. The parameter $\tilde{\sigma }=0.3$.

Figure 3

The dimensionless binding energy ${\tilde{E}}_{N0}^{\left(b\right)}=E_{N0}^{\left(b\right)}/\pi p{d}^{-1}$ calculated from (44), (21), and (7) with $q=0.20$ for the ground state $(n=0)$ plotted as a function of the effective index of the corresponding subband $v=|N-\tilde{\sigma }|$ and the relative impurity position $s=\frac{2x_0}{d}$ in the GNR of width $d$.

Figure 4

The resonant width ${\Gamma }_{10}$ (34) of the ground impurity state $(n=0)$ relative to the corresponding threshold ${\varepsilon }_1$ (7) $({\tilde{\Gamma }}_{10}={\Gamma }_{10}/{\varepsilon }_1)$ versus the relative impurity position $s=\frac{2x_0}{d}$ in the GNR of width $d$ providing the parameter $\tilde{\sigma }=0.3$. The parameter $q$ is taken to be $q=0.13,0.24$.

Figure 5

The resonant width ${\Gamma }_{-10}$ (42) of the ground impurity state $(n=0)$ relative to the corresponding threshold ${\varepsilon }_{-1}$ (7) $({\tilde{\Gamma }}_{-10}={\Gamma }_{-10}/{\varepsilon }_{-1})$ versus the relative impurity position $\frac{2x_0}{d}$ in the GNR of width $d$ using the parameter value $\tilde{\sigma }=0.3$. The parameter $q$ is taken to be $q=0.13,0.24$.

Figure 6

The dimensionless binding energy ${\tilde{E}}_{00}^{\left(b\right)}=E_{00}^{\left(b\right)}/{\varepsilon }_0$ of the quasiclassical ground state $(N=n=0)$ found from (47) for $E_{00}$ and from (7) with $\tilde{\sigma }=0.3$ for ${\varepsilon }_0$ versus the parameter $q$.