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a/rn

Figure 37. The binding state energy of a hydrogenic impurity as a function of the relative position of the impurity for large dot radii. Note that there is a point from which the edge states start to appear. 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 35. The binding state energy of an off-center hydrogenic impurity as a function of the relative position of the impurity for different dot radii. 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 37. The binding state energy of a hydrogenic impurity as a function of the relative position of the impurity for large dot radii. Note that there is a point from which the edge states start to appear. 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 36, the ground state energy for different radii of quantum dot versus the relative position of the off-center impurity (a/r0) is shown. For very large quantum dot radii Levine's ground state energy of an atom located precisely on the surface of a semi-infinite medium is obtained; that is, the ground state energy changes gradually from its bulk value (—1 Rydberg) into Levine's state (—1/4 Rydberg) as the impurity reaches the surface.

In Figure 37, the ground state binding energy for different radii of quantum dot versus the relative position of the off-center impurity (a/r0) was plotted. For the too large

Figure 36. The ground state energy of a hydrogenic impurity in a spherical quantum dot of very large radius. Note that the ground state changes gradually from its bulk value (-1R*) to the Levine state (-4R*). 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 36. The ground state energy of a hydrogenic impurity in a spherical quantum dot of very large radius. Note that the ground state changes gradually from its bulk value (-1R*) to the Levine state (-4R*). 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.

quantum dot radii considered here, binding energy curves cross one another and define a point from which the edge states begin to appear. This critical point corresponds to a hydrogenic impurity placed at the relative distance a/r0 = 0.68 from the center of the quantum dot. The associated binding energy with the quantum dot of largest radius (r0 = 100 effective Bohr) decreases from 1R*y (effective Rydberg) for an impurity located far from a quantum dot surface to 1/4R* for an impurity located exactly at the surface of a quantum dot. The binding energy behavior is in reasonable agreement with that of Chen [103], who studied the problem of hydrogenic donors near semiconductor surfaces. Furthermore, starting from the critical point (near the quantum dot surface), there is a inversion of binding energy for quantum dots of different radii. This occurs because the boundary has a greater effect on binding energy when the quantum dot size increases, as a result of the strong spatial limitation due to the closeness of the impurity to the surface. When a magnetic field is applied, the critical point is removed and the binding energy curves cross one another at only one point. Moreover, crossing points are shifted away from the boundary as the magnetic field is increased, and finally a moderate field strength causes them to vanish.

3.7. Two-Electron Atomic Systems Confined within Spheroidal Boxes

The direct variational method is used to estimate some interesting physical properties of the He atom and Li+ ion confined within impenetrable spheroidal boxes. A comparative investigation of the ground state energy, pressure, polar-izability, dipole moments, and quadrupole moments with those of He atom inside boxes with paraboloidal walls is made [104, 105]. The overall results show a similar qualitative behavior. However, for Li+ there are quantitative differences on such properties due to its major nuclear charge, as expected. The trial wavefunction is constructed as a product of two hydrogenic wavefunctions adapted to the geometry of the confining boxes.

Nowadays there is special interest in investigation of atoms and molecules confined in nanostructures because their physical properties become highly dependent of the size and shape of the confinement volume [65, 77, 78, 105111]. The reduction of space where the atom is located is equivalent to subjecting it to high pressures and the results of this new spatial condition are, besides other effects, the increase in total energy of the atom and the decrease of its polarizability.

The "atom in a box" model consists of replacing its interaction with neighboring particles by a potential wall. The simplest situation is to assume an infinite potential wall since for this case the wavefunction must vanish at the boundary of the box. The latter condition simplifies the calculations extremely. When a quantum system is localized in a box, the result is that the wavefunction of the system is constrained into that region.

The study of hydrogenic impurities trapped in solids has signaled the limitations of the model of confinement inside spherical boxes. For example, the observed hyperfine splitting of atomic hydrogen in a-quartz shows the presence of nonvanishing anisotropic components [112], and moreover, the electron microscopy studies of semiconductor clusters show microparticles whose shapes vary from quasi-spherical to pyramidal [113]. This has motivated the use of other models involving nonspherical boxes.

The helium atom inside spherical boxes has been investigated for modeling the effect of pressure on atoms with more than one electron [106-108]. Recently the helium atom inside boxes with paraboloidal walls [105] and the helium atom in a semi-infinite space limited by a paraboloidal boundary [114] have been studied.

3.7.1. Variational Calculation of Ground State Energy

The prolate spheroidal coordinates (£, y, p) are defined by

where r1 and r2 are the distances from any point to two fixed points (foci) separated by a distance 2R. In these coordinates

{£ = const. - 1 < y < 1 0 < p < 2wj (195)

defines a family of ellipsoids of revolution around the z-axis, while

defines a family of hyperboloids of revolution around the z-axis. In both cases 2R is the focal distance and p is the usual azimuth angle [115].

If the nucleus of the helium atom is located on one of the foci (z = -Rez), r1 and r2 denote the location of the two electrons relative to the fixed nucleus, and r12 is the interelectronic distance, then the Hamiltonian for this system formed by two electrons and one nucleus of charge Z (= 2) is given by

0 0

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