Section 5. Optical properties Modeling the dielectric functions of silicon-based ®lms in the amorphous, nanocrystalline and microcrystalline regimes A.S. Ferlauto a , Joohyun Koh a , P.I. Rovira a , C.R. Wronski a , R.W. Collins a, * , Gautam Ganguly b a Department of Physics, Materials Research Laboratory, and Center for Thin Film Devices, The Pennsylvania State University, University Park, PA 16802, USA b Solarex, 3601 LaGrange Parkway, Toano, VA 23168, USA Abstract We describe simple expressions that use a minimum number of free parameters to ®t the dielectric function spectra of a variety of Si-based ®lm materials ranging from amorphous silicon (a-Si:H) and its alloys with Ge and C to nano- crystalline silicon (nc-Si:H) and microcrystalline silicon (lc-Si:H). Three applications of these formulas are presented. First, we demonstrate how the expressions can be used in optical modeling of multijunction solar cells. Second, we analyze a-Si:H materials prepared versus the H 2 -dilution ¯ow ratio, R H 2 =SiH 4 , and observe that improved or- dering is obtained at larger R. Finally, we analyze Si ®lms as a function of thickness across the a ! lc phase boundary and quantify eects of electronic con®nement in the nc-Si:H regime and grain development in the lc-Si:H re- gime. Ó 2000 Elsevier Science B.V. All rights reserved. 1. Introduction Analytical expressions that describe the optical functions, i.e., the real and imaginary parts of the dielectric function versus wavelength or photon energy, have important applications in amorphous and microcrystalline semiconductor science and technology [1]. First, in materials studies such ex- pressions can be used to analyze optical data by least-squares regression and in this way, extract useful parameters. The optical data of interest in- clude transmittance, re¯ectance and ellipsometric spectra, collected ex situ or in real time, and the deduced parameters include the optical gap, criti- cal point and broadening energies that provide information on composition, ordering, phase and grain size. Second, such expressions constitute a database for the optical functions of the individual layer components of multilayer stacks, allowing one to simulate the optical response of complex multilayer devices. 2. Experimental In our modeling of the dielectric function spectra, eE e 1 E ie 2 E, we apply simple expressions of the form e 2 E HE  E G  GE f P N i1 L i Eg, where E G and H designate the op- tical gap and the Heaviside unit step function, re- spectively. In addition, GE describes e 2 E for Journal of Non-Crystalline Solids 266±269 (2000) 269±273 www.elsevier.com/locate/jnoncrysol * Corresponding author. Tel.: +1-814 865 3059; fax: +1-814 865 2326. E-mail address: rwc6@psu.edu (R.W. Collins). 0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 8 3 4 - 0