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Note: The notation with the principal quantum number is shown in parentheses. Effective atomic units are used.

Note: The notation with the principal quantum number is shown in parentheses. Effective atomic units are used.

the quantum levels are dependent on both n and l and only degenerate with respect to the m. The total degree of degeneracy of a quantum level with n and l is equal to 2l + 1 (excluding spin degeneracy). It is worthwhile to point out that the degeneracy can be lifted in the other kinds of quantum dots. In quantum boxes with circle cross-sections, for example, the degeneracy is lifted partly. Now, we can conclude that the quantum level sequence and degeneracy for an electron in a spherical quantum dot are quite different from those in a Coulomb field, and that this distinguishing feature of levels might cause new phenomena in this type of GaAs-Ga1—xAlxAs structure.

In Figure 24, the ground and first excited energy levels of an electron in a spherical quantum dot as a function of r0 for an infinite barrier height and two finite barrier heights V0 = 40 and 80R* were, respectively, plotted. It is shown that the differences of energy levels between different barrier heights increase as r0 is decreased and that the difference of the first excited state energy is larger than that of the ground state energy for a fixed value of r0. It is also shown that there are no bound states for a spherical quantum dot with a finite V0 if r0 < Rc, as mentioned earlier. In Figure 25, the binding energies of the ground and first excites states of a donor in a spherical quantum dot as a function of the r0 for three barrier heights V0 = 80, 60, and 40R*, respectively, were shown. It is readily seen that as r0 decreases both the binding energies increase continuously until their maxima and then decrease fast. The values of the binding energies can be much larger than those of quantum well wire and a two-dimensional quantum well as r0 is smaller. It is interesting to point out the ratio e1b(0)/e1b(1) increases as r0 increases from some small value. 61b(0) and s1b(1) are almost independent of V0 and respectively equal to 1.192 and 0.576R* at r0 = 7.0aB. However, the ratio 1.192/0.576 is still much less than 4, which is the limit value of a three-dimensional hydrogenic donor as r0 (approaches infinity).

In Figures 24 and 25, it is easily seen that as the r0 decreases the binding energies with respect to different states of a donor in a spherical quantum dot increase until

Figure 24. Ground state energy (e10) and first excited state (e11) energy levels of an electron in a spherical quantum dot versus the dot radius r0. The top and middle dashed curves represent the levels sn and e10 of the dot of V0 = x, respectively. The solid curves a, b, c, and d represent the levels e11 and e10 of the wells of V0 = 80 and 40Ry, respectively. 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 24. Ground state energy (e10) and first excited state (e11) energy levels of an electron in a spherical quantum dot versus the dot radius r0. The top and middle dashed curves represent the levels sn and e10 of the dot of V0 = x, respectively. The solid curves a, b, c, and d represent the levels e11 and e10 of the wells of V0 = 80 and 40Ry, respectively. 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.

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