Resonating Microelectromechanical Structures for Metrology
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Resonating Microelectromechanical Structures for Metrology A. M. Allen and G. C. Johnson Department of Mechanical Engineering University of California, Berkeley, CA 94720-1740, U.S.A. ABSTRACT A method for determining geometric process errors in MEMS devices from the resonant frequencies of simple structures is presented. The ability to accurately determine the as-built geometry of MEMS devices is important given the magnitude of the geometric uncertainty relative to the dimensions of these devices. As MEMS become smaller, the need for reliable metrology becomes even more important. The method presented here provides a way to determine the etch offset (the difference between the design width of a planar feature and the as-built width), the average angle of the side-walls and the compliance of the support of the structure. An important feature of the approach presented is that by using frequency ratios for various structural modes, neither the elastic modulus nor the mass density of the film need be known. INTRODUCTION The ability to measure the as-built geometry of micro-electro-mechanical systems (MEMS) is crucial in the analysis of the their performance. Small variations from the designed geometry can arise due to mask misalignment, over or under etching, and other uncertainties in the fabrication process. An extensive list of examples chronicling the sensitivity of commercial and research devices to geometry variation has been compiled by Gupta [1]. In this same work, he addresses the problem of determining geometry variation. However, his method requires extensive finite element analysis and assumes a priori that the mechanical properties of the materials involved are well known. It has been well documented in the literature of MEMS material characterization that the mechanical properties of most thin films used are not known with precision. Indeed, for the Young’s modulus of the most common MEMS material, poly-crystalline silicon, values ranging from 130 to 200 GPa have been reported within the last five years [2]. In this paper, we address the problem of determining variations from the designed geometry using lateral resonant structures made of materials whose mechanical properties need not be known. By comparing the resonant frequencies of structures whose nominal beam widths are different, the as-built beam width can be accurately determined. Unlike many ex situ conventional metrology measurements, such as SEM, surface profilometers, and interferometers, the in situ analytical metrology technique described here requires only the measurement of resonant frequencies of electrostatic comb drive resonators. The basis for this approach is an analytic equation for the resonant frequency of a simple structure. Effects of support compliance are included and an examination of the sensitivity to uncertain measurements of frequency is presented. OVERVIEW OF THE TECHNIQUE The structures used in this approach are built of a single film of uniform thickness t, and consist of a large end mass of area A supported by two par
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