Effective Conductivity Modelling of Polycrystalline ZnO Thin Films
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This work presents a theoretical model for the electrical conductivity of chemically deposited thin films. We are specially interested in materials produced by chemical reactions in aqueous solutions, like Chemical Bath Deposition (CBD) [4], Spray Pyrolysis (SP) [4] or Successive Ion Layer Adsorption and Reaction (SILAR) [2]. These deposition processes often present inhomogeneities in a scale that may be considered mesoscopic, where a characteristic length is small to the human eye, but big compared with the atomic size[5]. We are interested in geometrical characteristics that can be studied under a continuous medium model. The ZnO thin films obtained by SP are good examples, because the deposition by reaction of small droplets makes circular marks on the surface (figure la). In these regions the presence of impurities (substitutional Cl ) increases the effective electrical conductivity[1]. In order to analyze the influence of geometrical characteristics we propose a bidimensional model of two compounds: a homogeneous host material with inclusions. We choose the inclusions to be elliptical to cover the model's application range from circles to needles. The elliptical inclusions are randomly distributed with random orientations. Also, we can include the probability of overlapping ellipses (Figure lb). The model proposed has two main features: it is bidimensional and statistically isotropic. With these conditions we can use the "reciprocity theorem" [6] to find analytical interpolation formulas for the effective conductivity. This is developed in the next section. Then, we 447
Mat. Res. Soc. Symp. Proc. Vol. 403 01996 Materials Research Society
I(Xp pm
a)
b)
Figure 1: a)Optical microscopy of spry pyrolysis ZnO thin films[l], obtained at 460 °C , and 8.6 l/m, 9.6 l/m, 10.6 I/m, air flow rate through spry system respectively . b) Two dimensional system of elliptical inclusions to model a ZnO thin film. discuss the application of this formula to ZnO chemically deposited thin films. Finally, we end with some conclusions and remarks. INTERPOLATION FORMULA FOR RANDOM ELLIPTICAL INCLUSIONS In order to build an analytic formula for the effective electrical conductivity of a two components system the reciprocity theorem[6], [7] is very useful. This theorem is a restriction equation for systems with cyhndrical symmetry, and its main apphcation is in randomly distributed two dimensional systems of inclusions. The reciprocity theorem has two immediate implications: 1. The dimensionless effective conductivity X] given by
• (•o, •:)
•i (•°' •:),
(1)
6ro
where a0, a: are the conductivities of component 0 and 1, respectively and ae! (a0, a:) is the effective conductivity of the composite system, may be written as a single function of the parameter 7 = Ol+aO 2. When we expand X] (7) in terms of the area fraction f occupied by the ellipses, the relation is valid for G (7) even and F (7) odd functions of % In the model of randomly distributed and oriented elliptical inclusions in a continuous media, we have to introduce the
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