Quantum and many-body effects on the capacitance of a spherical-shell quantum dot

We calculate exactly, using finite-size techniques, the quantum mechanical and many-body effects to the self-capacitance of a spherical-shell quantum dot, i.e., the motion of electrons is confined to the surface of the dot, in the regime of extreme confinement, where the radius of the sphere is much smaller than the effective Bohr radius. We find that the self-capacitance oscillates as a function of the number of electrons close to its classical value. We also find that the electrostatic energy as a function of the number of electrons extrapolates to zero when N=1, suggesting that the energy scales like ${\mathit{e}}^2$N(N-1) instead of (Ne${)}^2$. We also provide evidence that the main deviations from the semiclassical description are due to the exchange interaction between electrons. This establishes, at least for this configuration, that the semiclassical description of Coulomb charging effects in terms of capacitances holds to a good approximation even at very small scales.