Rf Mems: Development of Design Rules

The modeling of MEMS passive electrical components is presented. The devices are modeled using the MRTD (multiresolution time-domain) and FDTD (finite-difference time-domain) electromagnetic modeling techniques. Methods are presented that allow these time-domain electromagnetic models to be combined with time-domain motion models of MEMS devices. INTRODUCTION MEMS devices have several characteristics that make them attractive to use as circuit components. Their low loss characteristics, as well as variability, make them unique among currently available technology. However, regardless of their gains, the effects of design choices on the performance of the device are largely unknown. One reason for this is that the devices are difficult to model. MEMS electrical components have both electromagnetic and mechanical interaction, meaning that a simulator for these devices must be able to model both phenomena. The multi-resolution time-domain (MRTD) and finitedifference time-domain (FDTD) techniques have been successfully used to model the electromagnetic characteristics of many devices. MRTD has several advantages over FDTD that make it the method of choice for modeling complex structures such as MEMS, the most important being the adaptive grid provided by wavelet analysis. This paper outlines a technique in which these time-domain methods can be combined with a time-domain motion model of MEMS devices. This is done using the example of a MEMS variable capacitor. This MEMS simulator can be used to determine the performance of MEMS devices prior to fabrication, and thus aid in the development of MEMS design rules. FDTD AND MRTD BACKGROUND The FDTD electromagnetic modeling technique [1] utilizes a finite difference discretization of Maxwell’s electromagnetic equations to numerically model the electromagnetic interaction of structures. It has been used to simulate a variety of geometries and determine the characteristics of a variety of structures [2]. There are, however, several well-known limitations of the FDTD technique. The FDTD scheme models a structure by creating a discrete grid that represents the fields on the structure. The most constraining requirement of the FDTD scheme is that all cell dimensions must be at most one tenth of the maximum wavelength that will be used in the structure [2]. This limitation makes the modeling of highfrequency structures very difficult. The MRTD scheme utilizes a scaling and wavelet function discretization of the electromagnetic fields, as opposed to the pulse scaling functions employed by the Yee FDTD technique. The wavelet functions allow the cell sizes used in the MRTD technique to approach the Nyquist limit (λ/2) [3]. The wavelet functions act as band-pass filters, complementing the low-pass filter action of the scaling function. Thus, the wavelet functions increase the frequency content modeled in the simulation. This property of wavelet analysis lays the foundation for the adaptive gridding technique. MRTD algorithms have demonstrated unparalleled properties when applied to the analysis of structures with medium to large sized computational domains. Through a twofold expansion of the fields in scaling and wavelet functions with respect to time/space, memory and execution time requirements are minimized while a high resolution in areas of strong field variations or field singularities is achieved through the use of sufficiently large number of wavelet resolutions. The major advantage of the MRTD algorithms is their capability to develop real-time time and space adaptive grids through the efficient thresholding of the wavelet coefficients. Various expansion basis have been utilized for the implementation of the MRTD algorithms. The Battle-Lemarie basis offers a reduction in memory by 2-3 orders of magnitude for 3D structures. Nevertheless, the entire-domain character of these functions adds a significant computational overhead in the approximation of the field derivatives in Curl Maxwell equations. In addition, Hard Boundaries (e.g. PEC’s) cannot be