The spectrum of an exciton in a nano crystal

The spectrum of an exciton in a small semiconductor nano-crystal has been inves-tigated in the case where its size was restricted by the condition  

The last term in the exciton spectrum given elsewhere represents the spectrum of a hole undergoing oscillations of frequency.  

When the radius a of the SNc is greater than or equal to the critical value Uq, a bulk exciton can be formed and become localized as a complete entity, in the SNc. This type of threshold for the appearance of the bulk exciton in the small SNc of radius a Oc can be predicted by simple qualitative estimates. The necessary conditions for the formation of a bulk exciton in the SNc is that its binding energy Ei, must be greater than the energy of the polarization interaction ( e /e2a) between electron and hole on the one hand, and the SNc surface, on the other  

It follows from this inequality that the bulk exciton will appear in the SNc only if its radius a is greater than the critical size a of the SNc  

We must now explain, at least qualitatively, the existence of the critical radius (equation given in Ref. 11). In an infinite semiconductor medium a large-radius exciton appears as a result of the Coulombic attraction Feh( e, between an electron and a hole. The Hamiltonian of an exciton moving within a small SNc contains not only the Coulombic attraction but also the terms Tee (c, 

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