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—0.1183

—0.104 528

Note: Energy units: Hartrees, distance units: Bohrs. a Results of this subsection.

b Results of Tsonchev and Goodfriend [87] obtained by expanding the wave-functions in a basis of functions which depend on the polar angle 6.

Note: Energy units: Hartrees, distance units: Bohrs. a Results of this subsection.

b Results of Tsonchev and Goodfriend [87] obtained by expanding the wave-functions in a basis of functions which depend on the polar angle 6.

Figure 23. Ground state energy of an off-axis hydrogen atom enclosed within an impenetrable cylindrical box as a function of the position of the nucleus (relative to axis of the cylinder), for various radii of the confining box. Note that when b/p0 ^ 1 and p0 ^ 1, the energy approaches -1/8 Hartree which corresponds to the ground state of the hydrogen atom close to an infinite planar surface. 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 23. Ground state energy of an off-axis hydrogen atom enclosed within an impenetrable cylindrical box as a function of the position of the nucleus (relative to axis of the cylinder), for various radii of the confining box. Note that when b/p0 ^ 1 and p0 ^ 1, the energy approaches -1/8 Hartree which corresponds to the ground state of the hydrogen atom close to an infinite planar surface. 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.

3.4. Confined Electron and Hydrogenic Donor States in a Spherical Quantum Dot

According to hydrogenic effective mass theory, exact solutions and quantum level structures are presented for confined electron and hydrogenic donor states in a spherical quantum dot of GaAs-Ga1-xAlxAs [88]. Calculated results reveal that the values of the quantum levels of a confined electron in a quantum dot can be quite different for cases with finite and infinite barrier heights. The quantum level sequence and degeneracy for an electron in a quantum dot are similar to those of a superatom of GaAs-Ga1-xAlxAs but different from those in a Coulombic field. There is stronger confinement and larger binding energy for a hydrogenic donor in a spherical quantum dot of GaAs-Ga1-xAlxAs than in the corresponding quantum well wires and two-dimensional quantum well structures. The binding energy and its maximum of the ground state of a donor at the center of a quantum well are found to be strongly dependent on the well dimensionality and barrier height.

Because the transverse and longitudinal variables do not separate, the impurity states in two-dimensional quantum wells and quantum well wires cannot be solved exactly. Therefore approximation methods should be used. A reasonable trial function is needed to obtain a correct variational state of an impurity in two-dimensional and one-dimensional confined systems, and calculated results are more accurate if the coupling effect between the impurity and well potentials is considered using a trial function which has (or a part of which has) correctly both donor and well potential effects [89]. However, for a hydrogenic donor at the center of spherical quantum dots the exact solutions [90] can be obtained. It is interesting not only from a physical point of view but also from a mathematical point of view to compare the solutions and binding energies with those of two-dimensional and one-dimensional systems. In this section the exact solutions and quantum level structures for confined electron and hydrogenic donor states in spherical quantum dots are reported. The dependence of the quantum levels and the binding energies on the dimensionality of quantum wells is also presented.

The calculation is based on the effective mass approximation. It has been known to give excellent results for the electronic structure of GaAs-Ga1-xAlxAs two-dimensional quantum wells and (AlAs)n/(GaAs)n superlattices if the well width or n is sufficiently large. The limit is estimated to be about 30 A (n ^ 10) [91]. Therefore it should also be valid for the GaAs-Ga1-xAlxAs quantum dots, as the size (diameter for a ball) is sufficiently large. Based on the facts mentioned earlier, the limit for a ball diameter is also estimated to be the same value and equal to 30 A. Here we treat the cases where the diameter is larger than the critical size. It is interesting to point out that the maximum quantum confinement of an electron in the GaAs-Ga1-xAlxAs quantum ball is already obtained before the diameter approaches the critical value. In addition, polarization and image charge effects can be significant if there is a large dielectric discontinuity between the quantum ball and the surrounding medium [92]. However, this is not the case for the GaAs-Ga1-xAlxAs quantum system; therefore we ignore such effects.

According to hydrogenic effective mass theory, the electron bound states and their binding energies have been found in two-dimensional quantum wells and quantum well wires. Normally, the effective mass equation is reliable for weakly bound states, and one might worry that the effective mass equation is inappropriate when the binding energy is greatly enhanced in spherical quantum dots of GaAs-Ga1-XAlxAs. However, the bandgap of GaAs is 1.4 eV, while R* = 5.3 meV. Thus roughly a 100-fold enhancement of the binding energy is necessary before the effective mass equation becomes inapplicable. This difference is much greater than the enhancement seen in the cases considered here, so that the theory is still reliable for the bound states in spherical quantum dots of GaAs-Ga1-xAlxAs.

Let us for definiteness consider a hydrogenic donor at the center of the quantum dot of radius r0. The potential due to the discontinuity of the band edges at the GaAs-Ga1-xAlxAs where A is a constant, interface r = r0 is

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