Qs7

—13.5959

—13.6539

Note: Energies given in Hartrees. Values in the last column for Li+ and Be2+ correspond to self-consistent field calculations by Ludena and Gregori [107].

a For V0 = m the energy values are taken from [121].

Note: Energies given in Hartrees. Values in the last column for Li+ and Be2+ correspond to self-consistent field calculations by Ludena and Gregori [107].

a For V0 = m the energy values are taken from [121].

for H- becomes e = -0.513 Hartree, a value closer to that of -0.527 reported by Pekeris [129] using a 1078-parameter wavefunction. Hence, we believe that a more flexible trial wavefunction in our treatment of two-electron confined atoms will improve the quality of the predictions.

3.10. Optical Properties of Impurity States in Semiconductor Quantum Dots

The optical absorption spectra associated with transitions between the n = 1 valence level and the donor impurity band have been calculated for spherical GaAs quantum dots with infinite potential confinement, using a variational procedure within the effective mass approximation, following [130]. We show results either for one impurity or for a homogeneous distribution of impurities inside of the quantum dot. The interaction between the impurities has been neglected. The main features found in the theoretical spectra were an absorption edge associated with transitions involving impurities at the center and a peak related to impurities at the edge of the dot. For all sizes of the quantum dot the peak associated with impurities located next to the edge always governs the total absorption probability.

Oliveira and Perez-Alvarez [131], Porras-Montenegro and Oliveira [132], and Porras-Montenegro et al. [133] have studied the optical absorption spectra associated with shallow donor impurities for both finite and infinite barrier GaAs-GaAlAs quantum wells and quantum well wires. The main features were an absorption edge associated with transitions involving impurities at the center and a peak related to impurities at the edge of the wire. For quantum well wires the situation of the bulk material is reached for radii of 1000 A and the peak associated with transitions involving impurities at the center decreases as the radius of the structure is diminished.

Porras-Montenegro et al. [134, 135] have calculated the impurity binding energies as functions of the radius, impurity position as well as the density of impurity states in spherical GaAs-Ga1-xAlxAs quantum dots, finding two structures associated with impurities located at the center and at the edge of the quantum dots, which are expected to show up in absorption and photoluminescence spectra associated with shallow hydrogenic impurities in quantum dots, as was the case in detailed calculations of the impurity-related optical absorption spectra in GaAs-GaAlAs quantum wells [131] and quantum well wires [133, 136].

Helm et al. [137] performed a far-infrared absorption study in lightly doped GaAs-Ga1-xAl^As superlattices and found that, at low temperatures, the absorption spectra are dominated by donor transitions. They studied the 1s-2pz donor transitions experimentally and theoretically, obtaining excellent agreement.

Otherwise experimental progress through spectroscopic techniques made possible a detailed analysis of the effects of confinement on shallow impurities in quantum wells. Far-infrared magnetospectroscopy measurements on shallow donor impurities in GaAs-Ga1-xAlxAs multiple quantum well structures were performed by Jarosik et al. [138], who assigned structures in the transmission spectra to intraimpu-rity 1s-2p± transitions. Work on magnetic field effects on shallow impurities in GaAs-Ga1-xAlxAs multiple quantum well structures was also recently reported by Yoo et al. [139].

3.10.1. Model and Theory

The Hamiltonian of a shallow hydrogenic impurity in a spherical quantum dot of GaAs can be written in the effective mass approximation as

where Vb(r) is the confining potential which is zero for r < r0 and infinite for r > r0, r0 being the radius of the dot. The impurity position is denoted by R.

The eigenfunction of the Hamiltonian in the absence of the impurity for the ground state (n = 1 and l = 0) and for the infinite potential well is [134]

where r is (r, 0, (p). In order to satisfy the boundary conditions j(r = ro) = 0, the eigenenergies corresponding to Eqs. (245) and (246) are = h2k1o/2m*e with = w/ro.

With inclusion of the impurity potential, one should use a variational approach to determine the ground state binding energy. The trial wavefunction considered is j(r) = { N(è>X)^^^exp(—X\r — R\) r<ro

o r>ro with A being a variational parameter and N(R, A) the normalization factor. The binding energy of the impurity is given by

0 0

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