Smooth scaling of valence electronic properties in fullerenes: From one carbon atom, to C60, to graphene

Abstract

Scaling of quantum capacitances and valence electron detachment energies is studied for icosahedral and nonicosahedral fullerenes. Scaling trends are considered from zero to infinite average radius, where a fullerene's local surface properties are similar to those of graphene. Detailed density-functional-theory calculations are performed to determine the geometries and detachment energies of icosahedral fullerenes, while values of these quantities are obtained for nonicosahedral species from previously published experimental results. Strongly linear, quasiclassical scaling versus average radii ${\overline{r}}_{n}$ is seen for the quantum capacitances, but on two different scaling lines for icosahedral and nonicosahedral species, respectively. By contrast, nonclassical, nonlinear scaling versus $1/{\overline{r}}_{n}$ is seen for the electron detachment energies, i.e., the valence ionization potentials and electron affinities. This nonlinearity is not accounted for by classical theories that are used to explain trends in electronic properties of fullerenes and usually give accurate quantitative estimates. Instead, simple quantum equations are derived to account for nonlinearities in the metal-particle-like electron detachment energy scaling and to show that these are responsible for nonclassical, nonzero intercepts in the capacitance scaling lines of the fullerenes. Last, it is found that points representing the carbon atom and the graphene limit lie on scaling lines for icosahedral fullerenes, so their quantum capacitances and their detachment energies scale smoothly from one C atom, to C${}_{60}$, to graphene.