Exact solution of the Thomas-Fermi two-dimensional N-electron parabolic quantum dot

The Thomas-Fermi approach is applied to the problem of the two-dimensional parabolic quantum dot. The equation is solved exactly in conjunction with Poisson’s equation for a circularly symmetric parabolic confinement. The solutions depend only on the ratio between the square of the product of the confinement constant and the dielectric constant of the host material and on the number of electrons. Asymptotic solutions for weak and strong confinement were also obtained for the chemical potential, the total energy, and the differential capacitance, reproducing the correct trends. For bounded parabolic potentials, an estimate of the maximal number of electrons that a dot can support is given. Appropiate Gaussian asymptotic behavior for the density is obtained by including a Weizsäcker-type kinetic energy term.