Exact and Closed Form Solutions for the Quantum Yield, Exciton Diffusion Length, and Lifetime To Reveal the Universal Behaviors of the Photoluminescence of Defective Single-Walled Carbon Nanotubes

The exact and closed form solutions for the photoluminescence quantum yield (PL QY), effective exciton lifetime, and effective exciton diffusion length were derived for defective single-walled carbon nanotubes (SWCNTs) with randomly and uniformly distributed defects or quenching sites. Asymptotic analysis of the exact and closed form solution not only recovered but also corrected the analytical QY for defective SWCNTs recently derived by Hertel et al. (ACS Nano 2010, 4, 7161-7168). Using the exaction solutions, we reanalyzed the length-dependent QYs and exciton lifetimes of fractionated (6, 5) tubes and determined the intrinsic QY eta(0)(infinity) = 0.106, exciton diffusion length L(infinity)(0) = 272 nm, internal quenching site density c = 0.0039 nm(-1), exciton diffusion coefficient D = 2.2 cm(2)/s, radiative lifetime tau(r) = 1.6 ns, and intrinsic exciton lifetime tau = 168 ps. The effects of F4TCNQdoping on the SWCNT PL intensity were quantitatively studied as well. Through numerical studies of the PL QYs of defective SWCNTs, we elucidated the critical role of the quenching site distance distribution in determining the absolute QYs of defective SWCNTs.