Electrical modeling of nanostructured thin films A. Santabarbara, S. Spadaro, G. Conte ⁎ Electronic Engineering Department, University Roma Tre, Via della Vasca Navale, 84-00146 Rome, Italy abstract article info Available online 1 February 2011 Keywords: Impedance spectroscopy Degenerate networks Matrices transformations Nanocrystalline diamond Electric circuit networks based on lumped components and distributed impedance elements are considered in the analysis of impedance spectroscopy data of nano-structured materials. A procedure based on transformation among degenerate networks is used as synthesis of linear, passive, one-port electric schemes and non-distinguishable distributed elements due to the noise level in observed data. Experimental results obtained on nitrogen doped ultra-nanocrystalline-diamond thin films are used to illustrate this process. © 2011 Elsevier B.V. All rights reserved. 1. Introduction There are electric circuit involving three or more lumped elements and non-linear frequency dependent components that can be rear- ranged in various ways and still yield exactly the same impedance as a function of frequency. Obviously, for the different interconnections the values of the elements will have to be different to yield the same impedance spectrum. Electric circuits with lumped linear and non- linear components are today widely used to analyze the ac response of solid state devices [1] and impedance spectra of nano-structured materials [2–5]. Data analysis by fitting a trial mathematical model of an electric scheme is commonly used due to the existence of commercial software for the analysis of impedance spectra. Frequent- ly, parameters like χ 2 are optimized by using electric schemes with poor or none correspondence to the physical system under study. A problem arises when we try to transform such optimized schemes, and related component values, with an electric scheme more adhering to the physical system under study. The possibility to transform a linear passive electric circuit in an equivalent scheme was initially stated by W. Cauer [6]. He empha- sized as electrical networks formed a group with the impedance function as an absolute invariant and that it was possible to proceed in a continuous manner from one network to its equivalent network by a linear transformation of the instantaneous mesh currents and charges of the network. The theory of the affine transformations stated by Cauer was successively analyzed by N. Howitt [7]. He point-out as in Cauer's work was already addressed the possibility to represent the impedance function of a n-mesh network by a linear combination of quadratic forms, positive and definite. The quadratic forms corre- sponding to the rate of dissipation of energy in heat, stored electro- magnetic and electrostatic energies. This linear combination can also be written as a n×n matrix or decomposed in three different n×n matrices by using the superposition principle. Successively, was demonstrated the possibility of simultaneously subject the triple of quadratic forms to a group of real affine transformations, leading to equivalent passive electric circuits. Tables of such degenerate elec- trical networks for use in the equivalent circuit analysis were reported by S. Fletcher [8]. Observed data fitting with the mathematical model of an electric circuit containing one or more non-linear frequency-dependent components (CPEs) is frequently encountered in impedance spectra analysis. CPEs have been used to describe the power-law dependence of the impedance components on frequency widely observed in many materials and well known as Jonscher Law [9], carriers hopping among defective sites [10] and distribution of charge carriers paths as a consequence of a distribution in the grains dimension [11]. As a CPE is a non-linear component, an affine linear transformation involving such element cannot be in principle used. The possibility to identify an electric circuit among different equivalent networks containing CPEs has been studied by F. Berthier et al. [12]. They conclude that in the case of noisy impedance spectral data, CPEs contained in equivalent schemes cannot be distinguished. In other words, the n-exponent in the expression of CPE impedance: Z CPE = 1/Qs n remains constant, while Q-value changes, being s the Laplace variable. 2. Material and methods UNCD thin films were grown on p-type silicon by microwave plasma CVD technique at 800 °C starting from a methane–argon mixture with 5% nitrogen (see for details [13]). The thickness of the sample was 1.5 μm. 200 nm of Silver were deposited on the UNCD film through a shadow mask to carry out planar measurements. Measure- ments were performed under vacuum into an APD 202 S helium gas exchange cryostat, equipped with an Oxford ITC4 controller, in the range 115–300 K with an accuracy of ± 1 K. Small amplitude ac signal response was analyzed as a function of temperature employing a Thin Solid Films 519 (2011) 4018–4021 ⁎ Corresponding author. E-mail address: gconte@uniroma3.it (G. Conte). 0040-6090/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2011.01.195 Contents lists available at ScienceDirect Thin Solid Films journal homepage: www.elsevier.com/locate/tsf