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Quantum Effects on the Dielectric Function of Porous Silicon M. Cruz1, S. F. Díaz1, C. Wang2, Y. G. Rubo3, and J. Tagüeña-Martínez3 1 Escuela Superior de Ingeniería Mecánica y Eléctrica–UC, IPN, Av. Santa Ana 1000, 04430, México, D.F., México. 2 Instituto de Investigaciones en Materiales, UNAM, A.P. 70-360, 04510, México, D.F., México. 3 Centro de Investigación en Energía, UNAM, A.P. 34, 62580, Temixco, Morelos, México.

ABSTRACT In this work, the imaginary part of the dielectric function of porous silicon is studied by means of both the tight-binding and the effective medium approaches, in the latter exact result is obtained for the case of 50% porosity. Within the tight-binding approximation, the dielectric function is calculated by using the interconnected and chessboard-like supercell models for the Si skeleton. These microscopic models give quantitatively similar results, which are by a factor of three larger than those from the effective medium theory.

INTRODUCTION The study of the dielectric properties of porous materials is one of the most simple and efficient methods to get insight into its microscopic structure. From the classical point of view, the porous material is characterized by an effective dielectric constant ( ε eff ), which depends on both the dielectric constants of the bulk material and inclusions (pores), and the geometry of the system. The geometrical features of the composite material (like porosity, morphology) come into the expression for ε eff through the so-called spectral function [1]. The spectral function is independent of the parameters other than geometry (e.g., the frequency of the light, temperature, etc.), and, in principle, can be extracted from the experimental data [2]. Clearly, this analysis is based on the assumption that the dielectric constants of the components of the composite material are not substantially different from their bulk values. When the grain sizes are small enough, so that spatial atomic discreteness and quantum mechanical effects become important, the effective dielectric medium approach (EDMA) fails. Nevertheless, due to its simplicity, the EDMA is frequently used to analyze the nano- and microcrystalline porous materials, in particular, the porous silicon (por-Si) [3,4]. In spite of that one should not expect this method to describe the most evident effects of spatial quantization, like the blue shift of the absorption edge, and the appearance of the interface states, still it is not clear whether the effective medium calculations can give a reasonable approximation for the dielectric function in the high-frequency region, far from the absorption edge. To find the limitations of the EDMA and to establish the typical microscopic size of the inclusions where the quantum effects become important, we perform here a comparison of EDMA and the microscopic tight-binding calculation using the supercell model for porous silicon [5,6]. In this model, the porous silicon is modeled as a superlattice of free-standing Si columns (or a superlattice of columnar pores i