Effect of Geometrical Irregularities on the Band Gap of Porous Silicon
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found in the study of the fractal drum described in references [2,4]. The present work is devoted to describe what can be the possible links between the properties and structure of porous silicon and the known properties of irregular or fractal resonators with eigenmodes named "fractinos". We do not consider here that porous silicon is indeed a mass fractal as in reference [5]. We restrict the discussion of the possible effects of a simple geometrical irregularity of the prefractal type. We will show that there exists a strong increase in the quantum confinement effect due to the irregularity. This indicates that irregular silicon crystallites or wires might display a strongly increased blue shift in the luminescence. Qualitatively speaking this tells us that from the point of view of electrons or holes, irregular solids appear to be smaller than their actual size. More generally the main result of this study is that, at small energy, the density of states in a solid depends strongly on the shape of the solid. This differs from the usual statement in solid state physics that the density of states depends on the volume only, a statement which is only true in the asymptotic (small wavelength) limit. This is not the case for those states of small energies which participate to the transport and electro-optic properties of small semiconductor clusters. Up to now, little has been done to take care of the shape dependence of the properties of small quantum structures. In this work we aim at describing the overall qualitative effects of the irregularity on a specific example. We consider what we call a Sommerfeld regular or irregular wire or crystallite as pictured in Fig. 1. It is a cylinder of height Lz, the base of the cylinder being an irregular shape as shown in the figure. It is considered here as an empty box with infinite potential walls. Because we wish to discuss the effect of the irregularity alone we will compare irregular wires (or crystallites) with regular wires (or crystallites) having the same volume [4]. 37 Mat. Res. Soc. Symp. Proc. Vol. 358 0 1995 Materials Research Society
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Fig. 1: The two geometries that we compare: On the left a square wire and on the right the irregular structure that we discuss. It is a cylinder of height Lz with an irregular base. If L >> L L it pictures an irregular wire. We will also discuss the case of L = L = L = L which correspxondys to an irregular crystallite or quantum dot. The geometry of t&e base, an irregular "prefractal", is shown at the bottom. The area of this irregular base is the same as that of the square from which it is issued. Hence classically, the density of states for these two systems should be the same independently of their shape. Figs. 2 and 3 show that this is far from being true. ELECTRONIC STATES IN IRREGULAR GEOMETRY: In this first approach we forget the crystalline structure of silicon and look only for the envelope wave-functions in the framework of the effective mass theory. First principle pseudopotential calculations [6,7] for regular wires
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