Two Electron Quantum Dot - A Variational Treatment For The Ground State

A variational treatment for a two-electron quantum dot (the artificial helium atom) is proposed which leads to exact answer for the ground state energy. Depending on the magnetic field value the singlet-triplet and triplet-triplet transitions of the ground state take place, which are captured in our solution.Using the same variational technique we find corrections to wave function and energy and the transition field srengths in a realistic dot where electron wave function has a finite extent in the direction perpendicular to the x-y plane in which usually 2-D dot electrons are confined. Using the variational wave function we show that photoemission cross-section as a function of magnetic field has sharp discontinuities, which can be used for experimental verification of the singlet-triplet transitions. Quantum dots [1] are little two-dimensional islands of electrons, which are laterally confined by an artificial potential. They can be thought of as artificial atoms with the field of nucleus replaced by an imposed external potential.The artificial hydogen atom is a single electron in a two dimensional circular geometry confined by a harmonic potential. The problem becomes interesting in the presence of magnetic field in the perpendicular direction and wave funcions for this case were worked out by Fock [2] shortly after the Schroendiger equation was established. The artificial ’helium atom’ problem was, however taken up more than forty years later. In an extensive numerical work Maksym and Chakraborty [3] and Wagner et al [4] found extremely interesting effect of the competition between magnetic field and Coulomb repulsion between electrons in two-electron quantum dots. In particular these authors found that the ground state can change character as the magnetic field changes, leading 1 Author to whom all correspondence should be addressed