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Figure 26. Maximum binding energies ebmax for the hydrogenic donor ground (l = 0) and the first excited (l = 1) states in a spherical quantum dot versus the barrier height V0. Arrows indicate the relevant scales. 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 26. Maximum binding energies ebmax for the hydrogenic donor ground (l = 0) and the first excited (l = 1) states in a spherical quantum dot versus the barrier height V0. Arrows indicate the relevant scales. 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.

the maxima of the two-dimensional quantum well [89] and spherical quantum dot are very close to (slightly larger than) the maxima for the quantum well wire [82]. We should point out that the maximum binding energies of higher excited states can also be used to present the dimension features.

The radial Eq. (147) was solved and the exact solutions of confined electron and donor states in a spherical quantum dot were obtained. The quantum levels and binding

Figure 27. Maximum binding energies ebmax of a donor ground state in a quantum well versus the well dimensionality and the barrier height V0. For the quasi-one-dimensional case, the dashed lines represent the maximum binding energies of quantum well wires and the solid lines represent the mean values of the maxima of the two-dimensional quantum wells and the spherical quantum dots. 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 27. Maximum binding energies ebmax of a donor ground state in a quantum well versus the well dimensionality and the barrier height V0. For the quasi-one-dimensional case, the dashed lines represent the maximum binding energies of quantum well wires and the solid lines represent the mean values of the maxima of the two-dimensional quantum wells and the spherical quantum dots. 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.

energies of a donor in the spherical quantum dot are calculated numerically. The numerical results reveal that the values of the quantum levels of a confined electron in a spherical quantum dot with a finite barrier height are different from those with an infinite barrier height. The differences increase as r0 decreases. However, the quantum level order is the same for both infinite and finite harrier heights. It is also shown that the quantum level sequence and degeneracy for an electron in a spherical quantum dot are similar to those of a superatom and different from those in a Coulomb field. The quantum level structure of a donor in a spherical quantum dot is similar to that of an electron only confined by the spherical quantum dot as the quantum confinement due to the well potential is stronger than that due to the donor potential. It is useful for understanding the shell model [97] in microclusters.

On the basis of the calculated results, the crossover from three-dimensional to zero-dimensional behavior of the donor states in a spherical quantum dot is shown when the radius becomes small. The binding energy of a hydrogenic donor state in the well of GaAs-GaxAl1-xAs and its maximum are strongly dependent on the well dimensionality and the barrier height and there is a larger confinement and binding energy of a donor state in a spherical quantum dot than in a quantum well wire and quantum well. Using calculated results of quantum well and quantum well wire, it has been shown that the maxima of the binding energies of hydrogenic donors in quantum wells, quantum well wire, and spherical quantum dots of GaAs-GaxAl1-xAs can be used to present, respectively, quasi-two-, one-, and zero-dimensional features of the hydrogenic donor states. Further, it has been found that the maximum of the binding energy of donor ground state in a GaAs-GaxAl1-xAs quantum well wire is about half of the summation of the maximum binding energies in the corresponding two-dimensional quantum well and spherical quantum dot.

It should be pointed out that impurities could be located anywhere in a spherical quantum dot and that the binding energies will decrease and the level ordering will change as the impurity location shifts to the edge or out of the spherical quantum dot. Based on the exact solutions obtained, the quantum levels and binding energies of a donor located out of the center of a spherical quantum dot can be obtained by use of a variation method. The exact solutions are also useful for the calculation of excited states in a spherical quantum dot, which is a kind of quantum dot. It will be interesting to compare the calculated results about quantum levels and binding energies of impurity and exciton states in a spherical quantum dot with those of other kinds of quantum dots.

3.5. Shallow Donors in a Quantum Well Wire: Electric Field and Geometrical Effects

In this section the effects of an external electric field on donor binding energies in quantum well wires with cylindrical and square cross-sections are investigated. A system with a GaAs quantum well surrounded by Alx Ga1-xAs potential barriers in the x, y plane has been chosen. The electron is thus free to move in the z direction, in the absence of a Coulomb center binding the electron. A realistic finite potential well model is considered [98].

The behavior of eb under an electric field is different for quantum well wires of rectangular and cylindrical cross-section. While the direction of the electric field is immaterial for cylindrical wires, it is very important for wires with rectangular cross-section.

It is found that the binding energy of the hydrogenic impurities is a rather sensitive function of the geometry of the wire especially under the influence of an electric field.

It is also found that the electric field effects on eb are extremely sensitive to the impurity position in or outside the wire.

### 3.5.1. Theory and Calculation

It is convenient to use the Cartesian coordinates for wires of rectangular cross-section and cylindrical coordinates for wires of circular cross-section. The Hamiltonian for the wire of rectangular cross-section, lying along the z direction, is h2

Lx 2

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