Read an Excerpt
Chapter 2: An Approach to Mems Design
The physical level addresses the behavior of real devices in the threedimensional continuum. The governing equations are typically partial differential equations (PDE's). Various analytical methods can be used to find closed-form solutions in ideal geometries, but the modeling of realistic devices usually requires either approximate analytical solutions to the PDE's or highly meshed numerical solutions. A variety of numerical modeling tools using either finite-element, boundary-element, or finite-difference methods are available for simulation at the physical level.
While meshed representations of the PDE's of continuum physics are useful in physical simulation, such representations are typically too cumbersome when dealing with entire devices and their associated circuitry. Instead, we go to the device level and create what are called macro-models or reduced-order models in a form that captures the essential physical behavior of a component of the system, and simultaneously is directly compatible with a system-level description.
An ideal macro-model is analytic, rather than numerical. Designers can think more readily with analytic expressions than with tables of numeric data. The macro-model should capture all the essential device behavior in a form that permits rapid calculation and direct insertion into a system-level simulator. The macro-model should be energetically correct, conserving energy when it should, and dissipating energy when it should. It should have the correct dependence on material properties and device geometry. It should represent both static and dynamic device behavior, both for small-amplitude(linear) excitation, and large-amplitude (presumably nonlinear) excitation. Finally, the macro-model should agree with the results of 3D simulation at the physical level, and with the results of experiments on suitable test structures. This is a tall order! Not every model we shall encounter will reach this ideal. But the requirements are clear.
An important feature of Fig. 2.2 is the presence of the various Designer Inputs. The designer can create models directly at the system level, or directly at any of the lower levels. For example, one can specify a physical device description along with all its dimensions and material properties, then use physical simulation tools to calculate device behavior, capture this behavior in a reduced-order model, and insert it into a system-level block diagram. Or, if one is early in the design process, one can simply use a parameterized reduced-order model to represent a particular device and defer until later the specification of device dimensions to achieve the desired performance.
Generally, when one moves to a lower level to get information needed at a higher level, we use the term simulation, although the term modeling would do as well. And when one starts at a lower level and performs analysis to move to the next higher level, we use the term verification, although, again, the term modeling would serve equally well. The word "verification" is puzzling to some. A way to think about it is as the testing of a hypothesis. Suppose at the device level a designer thinks that a particular shape of device is capable of producing a particular capacitance change when a force is applied to it. The designer can either test this hypothesis by building the device and performing experiments, or he3 can move down one level, to the physical level, and use the laws of elasticity and electricity to predict the change in capacitance in response to the applied force. The results of that modeling activity, whether done analytically or numerically, are returned to the device level and can be used to test, i.e. verify, the designer's hypothesis.
Experiment, of course, is the ultimate verification, but when designing devices and systems, one hopes to learn a great deal about the system through modeling before the relatively costly step of building experimental prototypes. 4 Just prior to committing to prototypes, one would like to be able to take the proposed prototype design, described at the process level with masks and a specific process sequence, and go up each step of the verification ladder using the actual detailed design.
2.2.1 Analytical or Numerical?
An important strategic issue is what mix of analytical and numerical tools to use in design. In this book, we shall make substantial use Of SIMULINK for system-level modeling using block diagrams, and MATLAB for selected numerical device-level and physical-level simulation. However, intelligent use of such numerical tools ultimately depends on a solid understanding of the underlying principles of the devices and their operation, and this understanding is best achieved by studying analytical (algebraic) models in detail. Furthermore, the design insights provided by analytic models are invaluable, especially the insights into the effects of varying either device dimensions or material properties. Thus, for the purposes of this book, the answer is: "analytical first, numerical second."
In the commercial MEMS world, especially as devices enter high-volume manufacturing, there is increased emphasis on numerical tools at all levels. Since the mid-1990's, a merchant CAD industry targeted at the MEMS and Microsystems fields has developed. The advantage of using numerical CAD tools, especially at the physical level, is that subtle second-order effects can be accurately captured without requiring detailed analytical model development. For some of the examples in this book, comparisons will be made between the approximate analytical models developed here and the corresponding, more accurate results of three-dimensional physical simulation, in order to give a feeling for the accuracy and limitations of the analytical models.
Before exploring an example that illustrates the various modeling levels, we will further expand this picture to illustrate both the richness and the complexity of the modeling challenge.
2.2.2 A Closer Look
Figure 2.3 shows a much expanded view of the "Modeling and Analysis" block of Fig. 2.1. The unshaded blocks capture the basic modeling levels illustrated in Fig. 2.2. The shaded blocks are new, and reflect some of the added complexity of designing real microsystems.
At the top of Fig. 2.3, a product idea is converted into a system architecture. This high-level architectural step requires immediate attention to a very detailed design question: whether or not to merge the system electronics with the non-electronic devices as a single monolithic device. This decision about physical system partitioning affects everything in the design, especially the device packaging.
While numerical prototyping may assist in making this very important decision, business judgment may play an even bigger role. As an example, one manufacturer of automotive accelerometers committed from the start to a fully integrated monolithic design, while several others adopted the so-called "twochip" design, in which a custom integrated circuit and a mechanical MEMS device are fabricated separately, then packaged together. Obviously, the design details are very different for these two approaches. The relative success in the marketplace depends on device performance and manufacturing cost. Because the full manufacturing cost (including device yield) cannot be predicted accurately prior to the implementation of the complete product technology, there is significant business risk associated with this kind of high-level architectural decision.
In addition to the physical partitioning of the system into components, the system is usually also partitioned into subsystems for analysis purposes. What is shown in the unshaded blocks of Fig. 2.3 is a representation of the various modeling levels and their interrelationships for one of the subsystems. Working down, one finds lumped-element subsystem modeling, and below that, the construction of individual device models, both for the electronic and nonelectronic components of the subsystem. Interactions with the package must be included at this level. For both kinds of components, individual device models are needed, and these depend on detailed device dimensions and on the underlying physical laws governing device behavior. The device dimensions and material properties are directly related to the the photomasks and process steps...
