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Self-consistent numerical solutions of the Poisson and Schr\"odinger equations have been obtained for electron states in a GaAs/AlₗGa₁-ₗAs heterostructure with confinement in all three spatial dimensions. The equations are solved in the Hartree approximation, omitting exchange and correlation effects. Potential profiles, energy levels, and the charge in the quantum dot are obtained as functions of the applied gate voltage and magnetic field. First, the zero-magnetic-field case is considered, and the quantum-dot charge is allowed to vary continuously as the gate voltage is swept. Then, in connection with the phenomenon of Coulomb blockade, the number of electrons in the quantum dot is constrained to integer values. Finally, the calculation is extended to examine the evolution of levels in a magnetic field applied perpendicular to the heterojunction. Our results indicate that the confining potential has nearly circular symmetry despite the square geometry of the gate, that the energy levels are quite insensitive to the charge in the quantum dot at a fixed gate voltage, and that the evolution of levels with increasing magnetic field is similar to that found for a parabolic potential.
Kumar et al. (Sat,) studied this question.