## System-Level MEMS Design: An Evolutionary Path to Standardization

*By: Qi Jing, Technical Marketing Engineer, Mentor Graphics Corporation*

**Introduction**

Successful design of highly-integrated IoT systems (Figure 1) requires simulating MEMS components together with the peripheral circuitry. However, MEMS devices are traditionally designed using CAD tools that are completely different from IC design tools. In the past two decades, both academia and industry have been seeking new methodologies and have chosen to implement multi-disciplinary MEMS design within the IC design environment. Performing MEMS-IC co-simulation in IC design environment allows designers to take advantage of advanced analog circuit solvers and the system verification capabilities that IC tools offer.

A good system-level design methodology should facilitate MEMS device models and structure representations that are compatible with the IC design flow, and provide simulation accuracy and speed that are comparable or superior to typical analysis tools in the appropriate physical domains. It should also provide broad coverage of physical effects, and be able to support large systems. The three methodologies in use today for system-level MEMS modeling and simulation are:

- Lumped-element modeling with equivalent circuits
- Hierarchical abstraction of MEMS and analytical behavioral modeling
- MEMS behavioral modeling based on Finite Element Analysis (FEA) and Boundary Element Analysis (BEA)

**Lumped-Element Modeling with Equivalent Circuits**

To implement SPICE-compatible modeling and simulation for MEMS, the most straightforward method is to create equivalent circuits for MEMS based on lumped-element modeling. For example, Figure 2(a) shows a spring-mass-damper system. A formal analogy can be derived between the mechanical and electrical elements, leading to an equivalent circuit “in series” topology as Figure 2(b) shows. Similarly, an “in parallel” circuit analogy can be derived as Figure 2(c) shows.

Although the equivalent-circuit methods appear straightforward, designers must be aware of their viability and limitations. First, analogies shown in Figure 2 are based on the assumption that MEMS structure can be significantly simplified into a spring-mass-damper system and that the effective mass, stiffness, and damping factor can be derived. This is only suitable for simple MEMS devices. For complex devices, the derivation could be too complicated and thus impractical to perform.

Secondly, the equivalent circuits are not easy to extend. Designers have to re-derive new models in order to account for additional physical effects or to adapt to changes in geometry, topology, or boundary conditions of the design.

Therefore, it is not uncommon for designers to determine that equivalent-circuit methods are too difficult or impossible to implement. More advanced methodologies are needed.

**Hierarchical Abstraction of MEMS and Analytical Behavioral Modeling**

In IC design, complex systems are built up hierarchically using building blocks at different abstraction levels. Hierarchical schematics are created to represent systems as structural networks comprising instances of these building blocks, connected together based on design topologies. Similar ideas have been explored and applied to MEMS design.

Figure 3 provides an example of the hierarchical abstraction of a folded-flexure resonator that contains a MEMS transducer and an electrical interface circuit. The MEMS transducer is an electrostatic device that is hierarchically built using a set of functional-level elements, each of which are further decomposed into atomic-level elements.

Behavioral models for MEMS elements can be written in analog hardware description languages such as Verilog-A, Verilog-AMS, and VHDL-AMS. Resulting models are compatible with SPICE simulators, thus serve well for co-simulation purposes. Analytical behavioral models for MEMS contain the following:

- Definition of terminals, with the associated physical disciplines specified.
- Definition of model parameters, including material and process properties as well as geometric sizing and layout orientation parameters.
- Description of model behavior using a series of Differential Algebraic Equations (DAEs) that govern the relationship between, across and through variables of the terminals, with coefficients formed by parameters and internal variables.

It’s crucial to obtain precise values of the material and process parameters in order for the models to match silicon. For standardized MEMS designs, foundries have started to develop and offer MEMS PDKs. For novel MEMS designs, designers have to fabricate test structures first then extract the parameters from lab measurement results.

After models are ready, they form model libraries that can be used for many designs in the appropriate design space. For example, atomic-level elements shown in Figure 3 not only serve as the foundation for folded-flexure resonators, but also work for many other typical suspended MEMS designs, such as accelerometers, gyroscopes, resonator filters, micro mirrors, and RF switches. Model libraries make it possible for people unfamiliar with MEMS to use the models for system integration, and help protect MEMS IP.

Due to the large variety of MEMS designs in underlying physics, fabrication processes and design styles, no model library can be a universal solution that fits all. If the device employs unique, irregular geometries, or if the device involves physics mechanisms that are not well-understood, a new model has to be developed from scratch.

**MEMS Behavioral Modeling Based on FEA/BEA**

Because geometry shapes supported by analytical models are discrete and limited, MEMS designers sometimes resort to Finite Element Analysis (FEA) and Boundary Element Analysis (BEA) tools. FEA/BEA tools use conventional numerical analysis methods for simulations in mechanical, electrostatic, magnetic, and thermal domains. They often rely on auto-meshers to partition a continuum structure into a mesh comprised of low-order finite elements. The tools then construct system matrices based on the meshing and solve the matrices within boundary conditions.

Efficient simulation of coupled physical domains is often a challenge to FEA/BEA-based tools. For example, to model the interaction between mechanical and electrostatic domains, some FEA/BEA tools must perform analyses for each domain separately and iteratively until a converged solution is found. Superior tools can simulate coupled domains all-together, but the simulation is computationally expensive and may result in unacceptable run times.

To alleviate limitations of FEA/BEA-based methods, while still utilizing their strength, Reduced Order Modeling (ROM) has been deployed, effectively bridging the gap between traditional FEA/BEA tools and electrical circuit simulators. ROM is a numerical methodology that attempts to reduce the degrees of freedom within system matrices to create macro models for MEMS devices. The resulting models can be constructed in languages like Verilog-A, then exported into SPICE simulators for co-simulation.

Up-to-date ROMs can be built not only from FEA/BEA results, but also from user-defined analytical equations and experimental data. Parameters in the reduced models can be preserved, so that design variations can be evaluated without going through the FEA and model order reduction process again. This enhances the coverage and efficiency of model libraries based on FEA/BEA and ROM.

Like all modeling methodologies, FEA/BEA-based methods cannot fully cover the entire MEMS design space either. Physical effects, as well as design and process imperfections, must be pre-defined in the original FEM/BEM model in order to be captured. In addition, creation of accurate models not only requires solid understanding of the underlying physics of MEMS devices, but also knowledge in both FEA/BEA tools and the model order reduction process.

**Conclusion**

To meet the need for MEMS-IC co-simulation, multiple modeling and simulation methodologies have been proposed, explored, and developed over the past two decades. Equivalent-circuit methods, structural analytical behavioral modeling, and reduced-order modeling based on FEA/BEA, are all effective methods and each has its own advantages and limitations. Knowing when to use which type of modeling method is important:

- When the design is small and simple, equivalent-circuit methods are the most straightforward.
- When the design is decomposable and the geometry, process, and dominant physical effects are close to what was used in the creation of primitive model libraries, hierarchical analytical modeling and structural system composition are the best choice.
- For unique designs using complex geometries, ROM methods based on FEA/BEA are more flexible and powerful.

For IC design, it took decades of academia and industrial endeavors for models, SPICE simulators, and foundry PDKs to emerge, mature, and converge into well-adopted industry standards. The MEMS modeling and simulation counterparts need to go through the same evolutionary path. This path has even more challenges than IC design, due to the much broader multi-physics coverage of MEMS and the diversity of MEMS manufacturing processes, applications, and design styles. Joint effort from design companies, foundries, and EDA tool vendors is required to enable this evolution. For more information about system-level MEMS modeling and simulation, download the whitepaper “System-Level MEMS Design – Exploring Modeling and Simulation Methodologies”.