812 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 3, JUNE 2004 Application of Conformal Mapping Approximation Techniques: Parallel Conductors of Finite Dimensions Nadine Pesonen, Walter K. Kahn, Life Fellow, IEEE, Richard A. Allen, Senior Member, IEEE, Michael W. Cresswell, Fellow, IEEE, and Mona E. Zaghloul, Fellow, IEEE Abstract—This paper describes a novel approach to the application of conformal mapping to capacitance evaluation. A particular structure composed of an array of identical lines and located below a conductive plate is studied. Results show the application of conformal mapping can reduce computing time when using three-dimensional electrostatic modeling and it can be the basis of algorithms of practical applications, especially in the semiconductor industry. Index Terms—Capacitance evaluation, conformal mapping, fringing effect. I. INTRODUCTION I N THE semiconductor industry, integrated circuit fabrica- tion relies on optical lithography to transfer patterns printed on a photomask to a silicon wafer. Optical lithography uses light and, in general, reduction optics to print features on the wafer four or five times smaller than on the corresponding features on the photomask. Since a duplicate of the image of the photomask is trans- ferred, any defect in the photomask patterns conveys directly to the silicon wafer. In recent years, it has become increasingly challenging to certify that a photomask is free of any defect big enough to cause a failure in the resulting integrated circuits, es- pecially since integrated circuit dimensions have now reached the nanometer level. A novel technique first introduced in a previous article [1] uses a noncontact capacitance sensor to examine the patterns of the mask, measure key dimensions, and give indications on whether certain types of defects are present on the mask. The capacitance sensor uses a sensing plate positioned above the mask, on which patterns of lines of identical width are printed. In this paper, several methods, described in the following Manuscript received October 15, 2002; revised February 17, 2004. This work was supported in part by VTT Electronics, Espoo, Finland, and in part by the NIST Office of Microelectronics Programs. N. Pesonen is with the MEMS Sensors Group, National Research Center of Finland, FIN-02044 VTT, Finland (e-mail: Nadine.Pesonen@vtt.fi). R. A. Allen and M. W. Cresswell are with the Semiconductor Electronics Division, National Institute of Standards and Technology, Gaithers- burg, MD 20899-8123 USA (e-mail: Richard.Allen@nist.gov; michael. cresswell@nist.gov). W. K. Kahn and M. E. Zaghloul are with the Department of Electrical and Computer Engineering, George Washington University, Washington, DC 20052 USA (e-mail: wkkahn@gwu.edu; Zaghloul@gwu.edu). Digital Object Identifier 10.1109/TIM.2004.827065 section, are first considered for an algorithm to link the capac- itance to the average width of the patterned lines. From these methods, a final algorithm based on conformal mapping is chosen as the optimal method of linewidth determination. The results obtained from conformal mapping techniques are then compared to results given by existing capacitance evaluation methods, such as the use of an electromagnetic field simulators and the identification to a microstrip. In a companion paper, detailed experimental results are described, showing how this theory can be applied in the determination of the dimensions of features on a photomask [2]. II. CAPACITANCE CALCULATIONS REVIEW Most capacitance calculations for parallel-plate condensers assume that the plates are infinite planes and neglect the fringing effect [3]. This assumption is equivalent to saying that only a uniform homogeneous field reigns between the two plates. However, when length, width, and thickness of the plates are in the same order as the distance separating the plates, the resulting fringing field has a strong effect on the value of capacitance and should, therefore, be taken into consideration when one wants to obtain a precise value of the capacitance. Several methods exist to quantify the fringing effect [4]. The method of images, Freehand flux plotting, the use of Green’s functions, the development of the electric field into a series or finite elements, and the method of transmission lines networks are powerful methods for determining the distribution of the electrical field in a specified area, but they are cumbersome and time consuming for calculations of capacitance. For problems involving two-dimensional (2-D) electric and magnetic fields, the principles of conformal mapping can be applied. Conformal mapping is a well-known transformation that maps a problem with specific boundaries and having a complex geometry onto another equivalent problem of simpler geometry [5]. This paper takes a different approach to the traditional use of conformal mapping to evaluate the capacitance comprised between an array of identical conductive lines of finite dimen- sions and a conductive plate located above at a short distance. The three-dimensional (3-D) capacitance problem is trans- formed into a succession of 2-D well-known configurations exhibiting uniform homogeneous fields for which capacitance calculations are familiar. 0018-9456/04$20.00 © 2004 IEEE