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We present a theoretical treatment of four two-dimensional electrons in a harmonic confinement potential in the presence of an external magnetic field using the exact diagonalization approach. The ground state properties and the spin and angular momentum transitions for different electron interaction strengths and magnetic fields are obtained. A magnetic field-confinement strength phase diagram is presented indicating a rich variety of ground states. An interesting feature of this system is the depolarization of spins by application of a magnetic field. The results are compared to several approximate theories.

Electrostatic ͑gated͒ quantum dots are studied by computational methods. Electronic properties of the electrostatic quantum dots are determined by the confinement potential, which is created by external voltages, applied to the electrodes, and band offsets. We have solved the Poisson equation for the two-terminal quantum dot nanodevice made of several GaAs and AlGaAs layers and obtained the confinement potential profile in the entire nanodevice. We show how the confinement potential profile can be modeled, which allows us to design-to some extent-the required electronic properties of the nanodevice. The results have been confirmed by a good agreement with experimental data. We have discussed the similarities and differences between the two-and three-terminal quantum dot nanodevices studied experimentally by Ashoori et al. ͓Phys. Rev Lett. 71, 613 ͑1993͔͒ and Tarucha et al. ͓Phys. Rev. Lett. 77, 3613 ͑1996͔͒, respectively.

A problem of interacting charge carriers confined in quasi-one-dimensional ͑1D͒ semiconductor nanostructures has been studied. We have derived an analytical 1D formula for the effective interaction potential between the confined charge carriers. We have applied both the 1D model with the effective potential and the full three-dimensional ͑3D͒ approach to an electron pair confined in a single and double quantum dot as well as to an exciton confined in a quantum wire. Comparing the results of the 1D and 3D approaches we have discussed the applicability of the effective 1D interaction potential to the real 3D nanostructures. We have shown that the present effective interaction leads to accurate results for weakly coupled multiple quantum dots and wire-like nanostructures, i.e., the quantum wires and dots with the lateral confinement much stronger than the longitudinal one.

Due to an oversight, the 2-7 phase for the 9-electron system was omitted in Fig. 5. Below we include the corrected figure. This correction does not affect our results and conclusions. FIG. 5. Phase diagram for N-electron Wigner molecules. The lowest-energy shell structures are shown as a function of magnetic field B. The boundary of the MDD-stability region is depicted on the left part of the plot. (Ã) For Nϭ19 the 1-6-12 phase appears at Bϭ15.8 T. PHYSICAL REVIEW B 67, 159902 ͑2003͒