Chirality dependent empirical modeling of optical transitions and bandgapsin carbon nanotubes and graphene nanoribbons

Abstract

Single wall carbon nanotubes (SWCNTs), double wall carbon nanotubes (DWCNTs) and graphene nanoribbons (GNRs) have unique electronic and optical properties. SWCNT is uniquely characterized by two positive integers (n, m), termed as chirality and DWCNT is identified by constituent SWCNTs whereas GNR is identified by type of its cross-section at the edges and number of dimer lines N. Selection of appropriate CNTs or GNRs for various applications requires prior information of their chirality and interband optical transition energies. Each SWCNT has a unique set of interband optical transition energies which depends on their chirality and other factors. Optical transitions of DWCNTs vary according to constituent SWCNTs whereas bandgaps of GNRs vary according to type and width of GNRs. Calculation of optical transitions of CNTs and GNRs from existing models was found to be deviated significantly from experimental results. It also ignored excitonic effects in optical transitions. A set of empirical models is proposed to predict different optical transitions in CNTs and bandgaps of GNRs. Experimental values of optical transitions of a large number of SWCNTs species (4, 2) to (35, 34) having diameter range 0.42 nm to 4.75 nm are considered here. There are total 654 SWCNTs in between these two chiral indices where 426 are semiconducting tubes and 228 are metallic tubes. Besides, DWCNTs and armchair GNRs of different chiralities synthesized and reported so far are studied. Based on the observations and findings, the empirical model is developed that gives a set of effective empirical equations to predict optical transitions in semiconducting and metallic SWCNTs and DWCNTs as well as bandgaps in GNRs with high accuracy. Calculated values from the empirical relations showed excellent agreement with experimental values. Such relations also lead to new method for characterizing CNTs or GNRs after synthesis. Empirical relations and family behavior of SWCNTs are also exploited to find a new technique for chirality assignment of individual SWCNT. Implications of the proposed empirical model in CNT-based devices are also demonstrated.