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A similar equation was found in [151] for a hydrogenic impurity confined within a quantum well wire.

The result of averaging the two-dimensional Coulombic attraction over the fast y motion is such that Eq. (281) can be interpreted as Schrodinger equation for the one-dimensional atom under the potential Veff(x).

Moreover, a careful analysis of Veff (x) in the limit y0 ^ 0 reveals that this potential has the same behavior as the one-dimensional Coulomb potential, as was also pointed out by Jan and Lee [151] in the case of a three-dimensional hydrogen atom confined within a cylinder whose radius approaches zero. The energy ex in this limiting situation becomes the same as that obtained by Loudon [148], that is, —x. It can also be noted that when y0 ^ 0, ey ^ tx and thus the total energy for the confined one-dimensional atom e = ex + ey ^ 0. The latter fact supports the results obtained in cases when the open confining curve approaches a straight line.

4.2. Geometrical Effects on the Ground State Energy of Hydrogenic Impurities in Quantum dots

The effect of nonsphericity of quantum dots on the ground state energy of hydrogenic impurities is studied in the frame of the effective mass approximation and the variational method [152]. The difference in composition of the quantum dot and the host material is modeled with a potential barrier at the boundary of the dot. To make the analysis, two symmetries are considered for the quantum dot: spherical and spheroidal. In this way, the ground state energy is calculated as a function of the volume of the quantum dot, for different barrier heights. The results show that the ground state is strongly influenced by the geometry of the dot; that is, for a given volume and barrier height, the energy is clearly different if the dot is spherical or spheroidal in shape when the volume of the dot is small (strong confinement regime). As the volume of the dot increases (weak confinement regime) the geometry becomes irrelevant, as expected.

The confinement of excitons in quantum dots and other microstructures such as quantum wells, quantum well i wires, etc. was corroborated experimentally in the past [113, 153-156]. The main feature of this effect corresponds to a frequency shift of the first excitonic peak to higher energy, compared to the mean bulk electronic peak. There have been many theoretical efforts to explain quantitatively this quantum mechanical effect [92, 157-163]. However, in all these works, the dot is assumed to be of spherical shape. This last assumption is not strictly true, since the electron microscopy studies of these systems show microparticles whose shapes vary from quasi-spherical to pyramidal [113]. The aim of this work is to analyze quantitatively the dependence of the ground state energy of a hydrogenic impurity on the geometry of the quantum dot. The latter would be of interest since it might constitute a qualitative study of the behavior of excitons confined within these microstructures. We assume that the impurity is confined within a potential barrier of finite depth, which emulates, on the average, the surrounding medium in which the dot is embedded. The calculation of the impurity ground state energy is performed using a variational approach, which previously was shown to be useful in dealing with this type of confined system in a fairly good fashion [66, 78, 111, 121]. The model Hamil-tonian for the impurity is assumed to be valid within the effective mass approximation and we have considered, for the sake of comparison, spherical and spheroidal dots of the same volume and confining barrier. In this way, the ground state energy of the impurity is calculated as a function of the volume and the confining barrier heights.

### 4.2.1. Brief Description of the Method

If {qi} denotes the orthogonal set of coordinates compatible with the symmetry of the confining boundary, the trial wavefunctions can then be written as and

where a and fi are parameters to be determined, q0 is associated with the size and symmetry of the confining boundary, A and B are normalization constants, $0 is the free system wavefunction, $ is a function with the proper asymptotic behavior as q0 ^ œ, and f is an auxiliary function that allows the condition

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