Theory of the Physical Properties of Si Nanocrystals
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M. LANNOO, C. DELERUE, G. ALLAN and E. MARTIN IEMN Ddpartement ISEN, 41 Bd Vauban, 59046 LILLE Cedex, FRANCE
ABSTRACT This paper reviews calculations concerning several aspects of silicon crystallites and their relevance for porous silicon. This begins with the optical properties of perfect crystallites: gap versus size, radiative recombination time, relative importance of phonon assisted transitions. A second part is devoted to the determination of the excitonic exchange splitting and of the Stokes shift which are found to bring a similar contribution (-10 to 20 meV). The effect of surface defects like dangling bonds is then investigated with their contribution to the recombination time. The Auger non radiative recombination time is also calculated and found to be short (-1 nsec). This is confirmed by some experiments on porous silicon which show a saturation effect of the photoluminescence under intense optical excitation or under cathodic polarization in aqueous solution, Auger recombination preventing the existence of more than one electron-hole pair per crystallite. Donor and acceptor impurities are studied in detail (screening of Coulomb potential, notion of ionization energy) with the conclusion that they are ionized. A final discussion shows the present level of understanding and identifies problems remaining to be solved.
INTRODUCTION A natural explanation of the photoluminescence of porous silicon [1] is the quantum confinement in silicon crystallites [2-5] but states localized at the surface of the crystallites, e.g. due to dangling bonds should play an important role [2,6]. Our aim here is thus to develop some aspects of the theory of silicon crystallites which could be helpful for the interpretation of experimental data. The followifig points are first briefly summarized: size dependence of the energy gap, predicted radiative recombination times and the importance of phonon assisted transitions, magnitude of the excitonic exchange splitting and Stokes shift, non radiative recombination at dangling bonds. We then discuss in some detail the important question of Auger recombination. Finally we present recent results on the dielectric screening in crystallites with their consequences for hydrogenic impurities, excitons and Coulomb charging effects.
OPTICAL PROPERTIES OF CRYSTALLITES Let us start with the size dependence of the energy gap. Fig. 1 shows the results of an empirical L.C.A.O. (Linear Combination of Atomic Orbitals) calculation detailed in ref [7], giving an extremely accurate description of the bulk silicon bands. When applied to the determination of the electron states of silicon clusters it allows to treat crystallites with Td symmetry containing up to 2059 atoms (cluster size = 4.3 nm). In practice these are saturated by hydrogen atoms to simulate ideal boundary conditions with no dangling bond at the surface [2]. Results for the energy gap are plotted on fig. 1 versus the diameter, not only for O-D crystallites but also for cylindrical wires of (100), (110) and (111) axes. As expected t
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