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Figure 17. (a) Binding energy of an off-axis hydrogenic impurity enclosed within a penetrable quantum well wire of radius p0 = 1.0 Bohr as a function of the relative nucleus position and potential barrier height. (b) Binding energy of an off-axis hydrogenic impurity enclosed within a penetrable GaAs-Ga1 -xAlxAs quantum well wire of radius p0 = 1.0 Bohr as a function of the relative nucleus position and potential barrier height. 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 17. (a) Binding energy of an off-axis hydrogenic impurity enclosed within a penetrable quantum well wire of radius p0 = 1.0 Bohr as a function of the relative nucleus position and potential barrier height. (b) Binding energy of an off-axis hydrogenic impurity enclosed within a penetrable GaAs-Ga1 -xAlxAs quantum well wire of radius p0 = 1.0 Bohr as a function of the relative nucleus position and potential barrier height. 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.

for p0 = 2.0, 3.0, and 4.0 Bohrs and several potential barrier heights for the off-axis case.

In Figure 14, we show the ground and first excited states energies of the hydrogenic impurity confined within cylindrical quantum dot with the nucleus on the symmetry axis, as a function of p0 for several V0 values. The ground state energy has a similar behavior as in [65], as well as for the hydrogen atom and for electron systems (like H-, He, Li+, and Be2+) within a penetrable spherical box [78]; that is, the energy diminishes as the confinement box size increases (for a given value of V0), and for a given size of the box the energy increases as V0 increases. Similar behavior is found by Nag and Gangopadhayay [79] who show graphically qualitative results for heavy holes and electrons within a quantum well wire with cylindrical and elliptic cross-section. They have used the physical constants of the Ga047In0 53As/InP system.

Figure 18. 1s-2pz oscillator strength of the hydrogenic impurity confined in a penetrable quantum well wire with the nucleus on the axis as a function of the wire radius and potential height. 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 18. 1s-2pz oscillator strength of the hydrogenic impurity confined in a penetrable quantum well wire with the nucleus on the axis as a function of the wire radius and potential height. 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.

This energy behavior can be understood on the basis of the uncertainty principle because when confinement dimensions are reduced the energy increases. If dimensions continue decreasing there will be a point at which the kinetic energy will be greater than the potential energy associated with internal interactions of the system. In an extreme situation of confinement the potential energy is only a perturbation of total energy for a free particle system. For attractive potentials, like Coulomb's potential, there is a very clear competition between kinetic energy and potential energy. If p0 decreases, then potential energy decreases, but kinetic energy always increases because of the localization of wavefunction.

Figure 19. 1s-2pz oscillator strength of an off-axis hydrogenic impurity enclosed within a penetrable quantum well wire of radii p0 = 2.0, p0 = 3.0, and p0 = 4.0 Bohr as a function of the relative nucleus position and potential height. 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 19. 1s-2pz oscillator strength of an off-axis hydrogenic impurity enclosed within a penetrable quantum well wire of radii p0 = 2.0, p0 = 3.0, and p0 = 4.0 Bohr as a function of the relative nucleus position and potential height. 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 15a, we display the binding energy of a hydrogenic impurity in a quantum well wire with finite potential barrier, as a function of the width of the wire. The results show that, due to the finite confining capacity of the quantum well wire, there is a critical radius for which the electron is no longer confined. Indeed, as the confining barrier increases this critical radius become smaller, as expected from simple physical considerations. For a given value of the finite barrier potential the binding energy increases from its bulk value as the wire radius is reduced, reaches a maximum value, and then drops to the bulk value characteristic of the barrier material as the wire radius goes to zero. This is due to the fact that as the wire radius is decreased the electron wavefunction is compressed thus leading to an enhancement of the binding energy. However, below a certain value of p0 the leakage of the wavefunction into the barrier region becomes more important, and the binding energy begins to decrease until it reaches a value that is characteristic of the barrier material as p0 ^ 0. This effect has been studied for heavy and light excitons confined within quantum wires [80], for hydrogen impurities within quantum well wire of GaAs1-xAlxAs [81, 82], and excitons within quantum well wire in the presence of a magnetic field [83]. In Figure 15b, we show the same results for the quantum well wire of GaAs1-xAlxAs for the barrier potentials V0 = 265.0, 53.0, 10.6, 5.3 meV that correspond to the x = 0.36, 0.08, 0.016, 0.008, Al concentrations, respectively. The quantitative comparison of the curves in Figure 15b with the results of [80, 82, 83] show that our variational calculations lead to the same results.

The peak in the binding energy occurs for the smallest value of p0 for which the probability of electrons to be found outside the well is not significant; that is, the enhancement in binding energy occurs because the confining potential is forcing the electrons to move only in a smaller space and to spend most of their time closer the nucleus. This strong enhancement of the binding energy has important consequences for optical and transport properties of quantum well wires.

In Figure 16, we show the ground and first excited state energies for the asymmetric case. The ground state energy approaches the value calculated in [65] for the infinite potential barrier case, as the potential barrier is increased, and when the size of the quantum well wire becomes infinite and the nuclei is close to the boundary. For a given value of the finite barrier potential the energy of the 2pz excited state is almost independent of the relative position of the nuclei within the quantum well wire, as compared with the variation of the ground state energy. This is due to the orientation of the orbital in which the electron moves; a similar effect has been studied for a hydrogen atom enclosed between two impenetrable parallel planes and for a heavy exciton in a CdS film in [84].

Also, we can note that for p0 = 1 Bohr, the ground state energies calculated for V0 = 1,2,10, 50 Hartrees are in exact agreement with the ground state energies when the atom nucleus is on the symmetry axis as we can see in Figure 14. The same is true for the first excited state energies (compare Fig. 16 with Fig. 14).

In Figure 17a, we can note that the value of the binding energy for the asymmetric case decreases as the nucleus approaching the quantum well wire is increased. A similar behavior occurs for the binding and the ground state energy of a hydrogenic impurity placed in a rectangular cross-section quantum well wire of GaAs-GaxAl1—xAs [81].

In Figure 17b, we show the same results as in Figure 17a, for the case of a hydrogenic impurity within a cylindrical quantum well wire of GaAs-GaxAl1—xAs with p0 = 103.4 A. The calculations were carried out for barrier potentials V0 = 265.0, 53.0, 10.6, 5.3 meV that correspond to aluminum concentrations x = 0.36, 0.08, 0.016, 0.008. The quantitative comparison of Figure 16b with the results of [81] shows the effect of the geometry on the binding energy.

The binding energy for p0 = 1 Bohr is in agreement with the binding energy when the nucleus is on the axis of the wire (compare Fig. 17a with Fig. 15a).

In Figure 18, the 1s-2pz transitions of the studied states in the symmetric case are shown. We note a similar behavior to that found in other systems which are confined in regions with different symmetries, for example, the heavy exciton case confined in a KCl ionic sphere for several radii and penetrabilities, in which the excitonic transitions vanish for given sphere sizes, as a consequence of the finite confining potential value, and moreover in that case the absorption peak is shifted to high energies as the sphere radius decreases. This effect has been experimentally observed in SiO2 spheres [84]. In addition, for V0 = x, there is a critical radius of quantum well wire for which the absorption has a maximum and then returns to bulk value as the radius continues decreasing.

In Figure 19, we show the oscillator strength f0 (for el || z) as a function of relative nucleus position (b/p0) for p0 = 2.0, 3.0, and 4.0 Bohrs, with the following potential barrier values: V0 = 0.1 and 0.2 Hartrees for the off-axis case. The maximum in the absorption peak allows us to predict the optimum site for the location of the hydrogenic impurity within the quantum well wire. We can note that, for the same parameter values as those mentioned earlier, the transition intensities are in agreement with the transition intensities of the studied states in the symmetric case (compare Fig. 19 with Fig. 18).

3.3. Asymmetric Confinement of Hydrogen by Hard Spherical and Cylindrical Surfaces

The variational method is used to calculate the ground state energy of the hydrogen atom confined within hard spherical and cylindrical surfaces, for an atomic nucleus, which is off the center of symmetry of the confining boundary. It is shown that the wavefunction for the free hydrogen atom in spherical coordinates (referred to the center of symmetry of the confining surface) can be used, without further assumptions, to construct the trial wavefunction systematically. The latter is assumed to be the product of the free wavefunction and a simple cutoff function that satisfies the appropriate boundary conditions.

In order to show the advantage of the method we present in detail two cases: the hydrogen atom confined within hard spherical and within cylindrical surfaces. The asymmetry of these systems is due to the nucleus of the atom being shifted off the center of symmetry of the confining surface (i.e., either off the center of the sphere or off the axis of the cylinder). Of course, in such cases the corresponding Schrodinger equation is no longer separable.

### 3.3.1. Applications of the Method

In this section, two explicit examples using the variational method application will be described. Both involve the hydrogen atom confined within a domain with impenetrable boundaries. In the first example, we shall consider the atom within a spherical surface where the center of surface and the nucleus of the atom differ by a constant distance a. Referring to Figure 20, the coordinates of the electron of the hydrogen atom with respect to the nucleus (r') and with respect to the center of the confining sphere (r) are related by r = r — a

The corresponding Hamiltonian can now be written as

2 r — a where Vb is a confining potential defined as

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