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1.5o25

1.5ooo

o.4o75

2.5o12

2.5ooo

Note: Exact and variational calculations are compared. Energies in units of ha. Radii in units of (S/mw)1/2. a Reference [168].

Note: Exact and variational calculations are compared. Energies in units of ha. Radii in units of (S/mw)1/2. a Reference [168].

shifted 1 /N expansion method is used to solve the effective mass Hamiltonian [172]. The influence of the electron-electron interaction on the ground state energy and its significant effect on the energy level crossings in states with different angular momentum are shown. The dependence of the ground state energy on the magnetic field strength for various confinement energies is presented.

The magnetic field dependence plays a useful role in identifying the absorption features. The effects of the magnetic field on the state of the impurity [173] and excitons [95, 159, 161, 174-176] confined in quantum dots have been extensively studied. Kumar et al. [177] have self-consistently solved the Poisson and Schrodinger equations and obtained the electron states in GaAs-GaAlAs for both cases: in zero and for magnetic fields applied perpendicular to the het-erojunctions. The results of their work [177] indicated that the confinement potential can be approximated by a simple one-parameter adjustable parabolic potential. Merkt et al.

[178] have presented a study of quantum dots in which both the magnetic field and the electron-electron interaction terms were taken into account. Pfannkuche and Gerhaxdts

[179] have devoted a theoretical study to the magneto-optical response to far-infrared radiation (FIR) of quantum dot helium, accounting for deviations from the parabolic confinement. More recently, De Groote et al. [180] have investigated the thermodynamic properties of quantum dots taking into consideration the spin effect, in addition to the electron-electron interaction and magnetic field terms. The purpose of this section is to show the effect of the electron-electron interaction on the spectra of the quantum dot states with nonvanishing azimuthal quantum numbers and the transitions in the ground state of the system as the magnetic field strength increases.

Here we shall use the shifted 1/N expansion method to obtain an energy expression for the spectra of two confined electrons in a quantum dot by solving the effective mass Hamiltonian including the following terms: the electron-electron interaction, the applied field, and the parabolic confinement potential.

4.4.1. Theory and Model

Within the effective mass approximation, the Hamiltonian for an interacting pair of electrons confined in a quantum dot by parabolic potential of the form m*ew0r2/2 in a magnetic field applied parallel to the z-axis (and perpendicular to the plane where the electrons are restricted to move) in the symmetric gauge is written as

h2 vf

where the two-dimensional vectors r1 and r2 describe the positions of the first and the second electron in the (x, y) plane, respectively. L\ stands for the z-component of the orbital angular momentum for each electron and wc = eB/m*ec, m*e, and k are the cyclotron frequency, effective mass, and dielectric constant of the medium, respectively.

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