Capacitance: A property of nanoscale materials based on spatial symmetry of discrete electrons

Capacitance is a measure of the ability to store electrons and is conventionally considered to be a constant dependent upon the shape of metal contacts and the dimensions of the system. In general, however, equipotentials of dielectric systems without metal contacts take the shape of very complex three-dimensional surfaces resulting from the spatial distribution of discrete electrons. The fundamental definition of capacitance, C≡Q/V, in which V is the potential within which electrons are confined, requires that the total capacitance take into account local capacitances of every electron and all cross-capacitances. To circumvent this complexity, the average total electrostatic potential experienced by each electron is utilized to obtain a capacitance expression generally appropriate to dielectric systems consisting of few excess electrons without metallic contacts. The capacitance may then be expressed as an exact function of the total electrostatic potential energy of the system. The integrity of this expression is demonstrated using a representative system of N excess electrons confined to a dielectric sphere. The capacitance expression is shown to be consistent with the conventional capacitance for a single electron dielectric sphere and with C=4πε0ε′a for metallic spheres. A relatively large sphere size is chosen such that the magnetic moment interaction energy is negligibly small. The capacitance exhibits a non-uniform relationship with respect to N coincident with shell-filling patterns of the natural atomic system. This classical electrostatic interactions approach is particularly appealing to the practical development of nanoscale materials and devices as it circumvents immediate recourse to often unintuitive and complicated quantum mechanical descriptions.