Scaling law of quantum confinement in single-walled carbon nanotubes

Quantum confinement significantly influences the excited states of sub-10 nm single-walled carbon nanotubes (SWCNTs), crucial for advancements in transistor technology and the development of novel optoelectronic materials, such as fluorescent ultrashort nanotubes (FUNs). However, the length dependence of this effect in ultrashort SWCNTs is not yet fully understood in the context of the SWCNT exciton states. Here, we conduct excited state calculations using time-dependent density functional theory on geometry-optimized models of ultrashort SWCNTs and FUNs, which consist of ultrashort SWCNTs with sp3 defects. Our results reveal a length-dependent scaling law of the E11 exciton energy that can be understood through a geometric, dimensional argument, which departs from the length scaling of a 1D particle-in-a-box. We find that this scaling law applies to ultrashort (6,5) and (6,6) SWCNTs, as well as models of (6,5) FUNs. In contrast, the defect-induced Esp3 transition, which is redshifted from the E11 optical gap transition, shows little dependence on the nanotube length, even in the shortest possible SWCNTs. We attribute this relative lack of length dependence to orbital localization around the quantum defect that is installed near the SWCNT edge. Our results illustrate the complex interplay of defects and quantum confinement effects in ultrashort SWCNTs and provide a foundation for further explorations of these nanoscale phenomena.