Electronic Structure and Scaling of Coulomb Defects in Carbon Nanotubes from Modified H\"uckel Calculations

Controlled doping and understanding its underlying microscopic mechanisms is crucial for advancement of nanoscale electronic technologies, especially in semiconducting single-wall carbon nanotubes (s-SWNTs), where adsorbed counterions are known to govern redox-doping levels. However, modeling the associated 'Coulomb defects' is challenging due to the need for large-scale simulations at low doping levels. Using modified H\"uckel calculations on 120 nm long s-SWNTs with adsorbed $\rm Cl^-$ ions, we study the scaling properties of shallow Coulomb defect states at the valence band edge and quantum well (QW) states in the conduction band. Interestingly, the QW states may underlie observed exciton band shifts of inhomogeneously doped semiconductors. Binding energies of Coulomb defects are found to scale with counterion distance, effective band mass, relative permittivity and counterion charge according to $d^{\alpha-2}m^{\alpha-1}\epsilon_r^{-\alpha}|z_j|^{\alpha}$, with $\alpha$ as an empirical parameter, deepening our understanding of s-SWNT doping.