  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.
