JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 11, NO. 5, OCTOBER 2002 505 MEMS Resonators That Are Robust to Process-Induced Feature Width Variations Rong Liu, Brad Paden, Senior Member, IEEE, and Kimberly Turner, Member, IEEE Abstract—A stability analysis and design method for MEMS resonators is presented. The frequency characteristics of a later- ally vibrating resonator are analyzed. With the fabrication error on the sidewall of the structure being considered, the first and second order frequency sensitivities to the fabrication error are derived. A simple relationship between the proof mass area and perimeter, and the beam width, is developed for single material structures, which expresses that the proof mass perimeter times the beam width should equal six times the area of the proof mass. Design examples are given for the single material and multi-layer structures. The results and principles presented in the paper can be used to analyze and design other MEMS resonators. [705] Index Terms—Frequency stability, MEMS resonator, process variation. I. INTRODUCTION R ESONATORS have been widely used as a key component in MEMS devices, such as in microgyroscopes [1]–[3], microvibromotors [4], microengines [5], and RF systems [6]. Resonators are actuated, usually electrostatically, to oscillate at their natural resonant frequency, so that the robustness of the de- sign frequency to process variations is one of the most impor- tant functional properties for the resonator design. Frequency stability of a resonator can directly affect the quality of the system in which it serves as a component. For the lateral vi- brating rate gyroscopes, the frequency matching for their two vibrating modes is important for the output sensitivity. If fre- quency of any one of the modes shifts, the output signal’s accu- racy will be decreased. Although symmetry in these gyroscopes helps the two modes to track to first order, it is useful to enhance the frequency matching by designing the resonant frequency to be insensitive to process variations. In microvibromotors, several resonators impact a bar to make it move in the plane of the chip. If the impacting frequencies of the resonators are not harmonic, the motion of the bar will be unpredictable. Similarly, two orthogonal resonators actuate the previously mentioned microengine, and the rotational sta- bility of the engine is affected by the synchrony of these two res- onators. Finally, in RF systems, the resonator is used in oscilla- tors and filters. Therefore the frequency stability of the resonator in these applications is particularly important as its frequency determines the system performance in a fundamental way. Manuscript received June 4, 2001; revised April 5, 2002. Subject Editor R. T. Howe. The authors are with the University of California at Santa Barbara, Santa Barbara, CA 93106 USA. Digital Object Identifier 10.1109/JMEMS.2002.803279. Fabrication processes induce wafer to wafer as well as on-wafer frequency variations. To identify sources of vari- ability, one looks to the associated MEMS processes such as plasma-enhanced chemical vapor deposition (PECVD), lithog- raphy, reactive ion etch (LRIE), isotropic etch, deep reactive ion etch (DRIE), and sputter metallization. Process temperature variations have a critical influence in most of these processes. Wafer-to-wafer variations depend on the temperature control of process equipment and thermal stabilization of equipment prior to use. Thermally induced on-wafer variations are caused by wafer edge effects, process chamber asymmetry, and gas flow effects. Once processing begins, surface stresses disturb wafer planarity, which, in turn, affects optical lithographic steps and on-wafer device uniformity. Transport processes are difficult to control for uniformity both on-wafer and for matching devices. Application of photo resist, the distribution of process gasses, and the diffusion of reactant in wet etching are nearly impossible to control across a wafer, and wafer to wafer variations also occur. Pressure variation is gas-phase reactions is yet another source of process variability and the list goes on. Thus, with present micromachining techniques, the fabrica- tion process variation in MEMS is inevitable and it will con- tinue to be the case when devices are miniaturized to the point of process limitations. For example, the fabrication tolerance for the width of a typical suspension beam is reported to be about 10% in [10]. Although it is known that the fabrication errors af- fect the frequency stability [9], [11], it may be that design for process variation was largely overlooked because of the struc- ture complexity and differences in micromachining methods. The fabrication errors not only change the stiffness of the beam of the resonator, but also change the mass of the proof-mass. Even the same fabrication errors will cause different frequency variations for different resonator structures. In this paper, the frequency robustness for a folded-beam lateral vibrating res- onator is analyzed. Based on the analysis, an optimum design method is presented for the resonator to obtain minimum fre- quency sensitivity. A simple relationship between the area and perimeter of proof mass and beam width is derived for single material structures. II. ANALYSIS OF A FOLDED-BEAM SUSPENDED RESONATOR WITH SINGLE STRUCTURAL MATERIAL A folded-beam suspended resonator is shown in Fig. 1, where the cross section of the beams is assumed to be rectangular. The 1057-7157/02$17.00 © 2002 IEEE