Aromatic trends in single-walled carbon nanotubes: diamagnetic anisotropy for arbitrary chiralities

The chirality dependence of single-walled carbon nanotube (SWNT) properties often leads to ``fan-out'' diagrams whose departure from the large diameter scaling limit is of fundamental interest. ~Here we present the first experimental indication of fan-out behavior for orbital magnetic anisotropy ($\Delta $\textit{$\chi $}), which has long been an important probe of electronic structure in aromatic molecules. ~We will discuss the experimental approach (polarized resonant photoluminescence) that made this background-free measurement possible, and explain how these results can be used to predict $\Delta $\textit{$\chi $} for \textit{arbitrary} SWNT chiralities. ~Taking into account general symmetry considerations, \textit{ab initio} calculations, large-diameter tight-binding theory, and our experimental data, we obtain a chiral expansion for $\Delta $\textit{$\chi $} using a single fitting parameter. ~The results show (2n+m) family trends whose asymmetry between ``mod 1'' and ``mod 2'' semiconducting families is reminiscent of those seen in other SWNT optical, phonon, and exciton properties. ~Finally, we discuss the (n,m) dependence of zone-folding tight binding calculations when applied to realistic tube sizes.