Effective Medium Approach For Calculation of Linear and Nonlinear Properties of Porous Semiconductor Composites.
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Effective Medium Approach For Calculation of Linear and Nonlinear Properties of Porous Semiconductor Composites. Vladimir Kochergin1 and Helmut Föll2 Lake Shore Cryotronics, Inc., Columbus, OH 43082, USA Tel. (614) 891 2243; Fax. (614) 818 1607; e-mail [email protected] 2 Materials Science, Faculty of Engineering, Christian-Albrechts-University of Kiel, Kaiserstr. 2, 24143 Kiel, Germany
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ABSTRACT The general methodology of calculating linear and nonlinear properties of nanoporous and nanostructured semiconductor materials and composites is presented. A Maxwell-Garnett approach is generalized for the case of porous semiconductor materials composed of a number of differently oriented pore lattices. Specifically, the cases of electrochemically etched mesoporous silicon on (110)-oriented substrate and electrochemically-etched porous InP and GaAs materials on (100) substrates are considered. The observed optical anisotropy of mesoporous Si is explained. A biaxial anisotropy of the porous InP or GaAs material with crystallographic pores is predicted. INTRODUCTION Pore formation in semiconductors by means of electrochemical etching emerged as a promising technique to tailor the optical properties of these materials (see, for example, [1] and references therein). Semiconductors are turned porous by electrochemical etching of the nonporous semiconductor substrates in suitable electrolytes. It was shown that many types of semiconductors could be made porous by this method, including Si, Ge and III-V compounds (e.g. InP, GaP, GaAs, GaN). Porous layers with many kinds of morphologies were demonstrated [1]. Considerable experimental efforts were devoted to investigations of linear and nonlinear optical properties of various porous semiconductors and composites (see, e.g. [2-5]), but a general theoretical approach is still missing. Previously [5], the generalized Bruggeman method [6] was used for calculations of the optical effects in these materials. While providing fair estimates for mesoporous (110)-oriented Si, this approach is not applicable for more complex structures, employing e.g. multiple pore lattices, or for materials containing pores with noncircular cross sections. A “Looyenga formulation” [7] was also applied to calculations of effective dielectric constants of porous silicon [ 8], however, this method is not applicable to the case of an anisotropic medium. The methodology presented here has the capability to analyze such structures over some range of parameters. EFFECTIVE PERMITTIVITY TENSOR CALCULATIONS In this approach, several assumptions are made: 1) pores are represented by a sequence of elongated cavities filled with air (or other materials) with elliptical (for the case of Si) or
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triangular (for the case of InP or GaAs) shapes; 2) pores are uniformly and randomly distributed in the semiconductor material such that the pores belonging to one of M lattice subsets have their axes essentially parallel to each other; 3) the bulk semiconductor material is isotropic and has a relative
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