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r n-. m r r by requiring an agreement between 1/k expansion and the exact analytic results for the harmonic and Coulomb potentials. From Eq. (329) we obtain a = 2 - (2nr + 1)(b (330)

where nr is the radial quantum number related to the principal n and magnetic m quantum numbers by the relation nr = n - \m\ - 1. Energies and lengths in Eqs. (321)-(330) are expressed in units of R*y and a*B, respectively.

For the two-dimensional case, N = 2, Eq. (325) takes the following form:

Once r0 (for a particular quantum state and confining frequency) is determined, the task of computing the energy is relatively simple.

The results are presented in Figures 64-68 and Tables 6 and 7. The relative ground state energy \00) of the relative motion, for the zero magnetic field case, against the confinement length is displayed in Figure 64. The present results (black dots) clearly show an excellent agreement with the numerical results of [178] (dashed line). In Figure 65, the first low energy levels \00), \10), and \20) of the relative Hamiltonian are presented as a function of the effective confinement frequency c, using parameters appropriate to InSb, where the dielectric constant k = 17.88, electron effective mass m* = 0.014m0, and confinement energy hc0 = 7.5 meV [180]. The energy levels obviously show a linear dependence on the effective frequency. As the effective frequency c increases the confining energy term dominates the interaction energy term and thus the linear relationship between the energy and the frequency is maintained. This result is consistent with [180].

To investigate the effect of the electron-electron interaction on the energy spectra of the quantum dot, we plotted

Figure 65. The low-lying relative states \00), \10), and \20) for two electrons in a quantum dot made of InSb as a function of confinement frequency c. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Figure 65. The low-lying relative states \00), \10), and \20) for two electrons in a quantum dot made of InSb as a function of confinement frequency c. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

in Figure 66 the total ground state energy \00; 00) of the full Hamiltonian for independent (solid line) and interacting (dashed line) electrons as a function of the ratio cc/c0. The figure shows, as we expect, a significant energy enhancement when the electron-electron Coulombic interaction term is turned on. Furthermore, as the magnetic field increases, the electrons are further squeezed in the quantum dot, resulting in an increase of the repulsive electron-electron Coulombic energy and in effect the energy levels.

V aB

Figure 64. The relative ground state energy \00) for the electrons in a quantum dot as a function of confinement length i0 = (h/m*e c0)1/2 for the zero magnetic field. This section's calculations: closed circles; Ref. [178]; solid line. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Figure 66. The total ground state energy |00; 00) for two electrons in a quantum dot as a function of the ratio wc/w0. For independent (solid line) and interacting (dashed line) electrons. Reprinted with permission from [205], J. L. Marín et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Figure 66. The total ground state energy |00; 00) for two electrons in a quantum dot as a function of the ratio wc/w0. For independent (solid line) and interacting (dashed line) electrons. Reprinted with permission from [205], J. L. Marín et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

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