The size dependence of the one-particle band gap and the Coulomb and exchange excitonic corrections of spherical quantum dots are calculated using the effective-mass approximation with finite confining potentials. Full analytical expressions are found for the three magnitudes, and it is shown that the Coulomb and exchange excitonic corrections are in good qualitative and quantitative agreement with available state-of-the-art calculations ͑for Si, GaAs, and CdSe͒ and experiments ͑for InP͒. ͓S0163-1829͑99͒16435-X͔The experimental and theoretical study of quantum size effects in quantum dot ͑QD͒ semiconductor heterostructures has become a very active research area, both because of their unique physical properties and prospect for applications. 1 As the size of the QD is reduced, both the single-particle band gap increases ͑blueshift͒ and the electron-hole excitonic correction becomes more pronounced ͑redshift͒. However, as the size dependence of the former is usually stronger than the exciton size dependence, this results in an overall blueshift of the optical-absorption spectrum ͑as compared with the bulk͒. Additional impulse to these studies was provided by the discovery of visible luminescence from porous Si. 2 Although the microscopic mechanism, which is behind the photoluminescence, is still under debate, there exists a growing consensus that quantum confinement is involved in producing this phenomenon. 3 From the theoretical point of view, the electronic structure of small quantum dots has been studied by a variety of methods: single-band effective-mass approximation ͑EMA͒, 4 multiband effective-mass approximation with infinite confining barriers, 5 empirical tight-binding ͑ETB͒, 6 empirical pseudopotential method ͑EPM͒, 7 and ab initio pseudopotential calculations. 8 There is a tendency to disregard the EMA as a quantitative and even qualitative method for the study of these nanocrystallites, mainly because the comparison of the EMA with the latter more sophisticated and reliable techniques shows large discrepancies, as, for instance, a gross EMA overestimation of the one-particle band gap. This is an important issue, as the great advantage of the EMA is its flexibility and versatility, in addition to allowing a quite natural extension to situations with electric and magnetic external fields, the presence of impurities, etc. 9 A point worth noting is that most often ͓i.e., Refs. 6͑c͒, 6͑d͒, 7, and 8͔ EMA is associated with the infinite barrier approximation for the quantum dot confining barrier ͑IEMA͒; this is clearly the simplest version of the EMA, but obviously the less accurate. It is the aim of this work to demonstrate that just by relaxing this hard-wall boundary condition, the finite barrier version of the EMA ͑FEMA͒ gives quantum size effects for Coulomb and exchange exciton energies in quite good agreement with the more accurate calculations available to date.Using the envelope function approach to the effectivemass approximation, the Hamiltonian of the electron-hole system in a spherical dot 10 of r...
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