Electronic energy spectrum and the concept of capacitance in quantum dots

The chemical potential and the capacitance of a model quantum dot have been computed, including contributions of exchange and correlation in the limit of 0 K temperature. The Schro¨dinger equation has been solved self-consistently, taking into account the electron-electron Coulomb interaction and many-body effects within the framework of density-functional theory. We have also studied the effect of conducting backgates and of nearby electrodes using the method of images. Depending on the size of the dot, we derive a prevalance of either the quantization energy or the electrostatic energy: there is a smooth transition from predominant quantum effects for small dots to classical capacitance behavior for large dots. Our simulation reproduces characteristic effects that have been experimentally observed, such as the capacitance increase for increasing electron numbers and irregularities in the chemical potential values when randomly distributed charged impurities are present.