Quantum Size Effects A Theoretical Overview

A theoretical description of the above-noted size-dependent behavior has been proposed by L. E. Brus [113,114] and Y. Kayanuma [124]. 

As the size of a semiconductor crystal becomes small a regime is entered in which the electronic properties, e.g. ionization potential and electron affinity, are determined by size and shape of the crystals [113], When a quantum of light (hv) with energy exceeding the band gap falls on the surface of a semiconductor crystal there appears a bounded electron-hole pair known as an exciton 

As the radius of a semiconductor crystallite approaches the exciton-Bohr-radius its electronic properties begin to change, whereupon quantum size effects can be expected. The Bohr radius ub of an exciton is given by 

10 eV s), rUe and W are the effective masses of the electron and the hole, s is the dielectric constant of the semiconductor, and e is the electron charge. The exciton radius (Rexc) can be calculated [127]  

The quantitative size-dependence of the minimum electronic excitation energy of a nanoparticle can be approximated taking into account the uncertainty in position (Ax) and momentum (Ap) of the exciton  

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