Semiclassical analysis of a two-electron quantum dot in a magnetic field: Dimensional phenomena

It is shown that with the inclusion of the vertical extension of a quantum dot the experimental findings of Ashoori [Phys. Rev. Lett. 71, 613 (1993)] can be modeled consistently with a parabolic confinement. Furthermore, the magnetic properties such as the magnetic moment and the susceptibility are sensitive to the presence and strength of a vertical confinement. Using a semiclassical approach the calculation of the eigenvalues reduces to simple quadratures providing a transparent and almost analytical quantization of the three- dimensional quantum dot energy levels that differ from the exact energies only by a few percent. While the dynamics for three-dimensional axially symmetric two-electron quantum dot with parabolic confinement potentials is in general nonseparable due to the Coulomb interaction we have found an exact separability for specific values of the magnetic field