Enabling Technologies for System-Level Simulation of MEMS Tamara Bechtold1, Gabriele Schrag2, Lihong Feng3 Institute for Microsystems Engineering - IMTEK, Freiburg University, Germany 2 Institute for Physics of Electrotechnology, Technische Universität München, Germany 3 Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany corresponding author:
[email protected] 1
Abstract The rapid progress in microelectromechanical systems (MEMS) and the evolution from a limited set of well-established applications, examples, in automotive industry, print heads, and digital light projection, to other fields mainly driven by consumer applications (such as image stabilization, smart phones, game consoles, etc.) opens a new market for these devices. It simultaneously creates the need for fast, efficient, and adequate design and optimization tools not only for stand-alone devices, but also for entire multiphysics microsystems comprising the MEMS component, the attached control and read-out circuitry and the package, while additionally considering environmental impacts potentially affecting the system functionality. This paper provides an overview of system-level modelling methodologies and tools. 1. Introduction Owing to their nature as transducing elements, an inherent feature of many MEMS devices is that multiple energy domains and their couplings determine their operation. A sampling of coupling between a subset of physical energy domains of major importance in microsystems is illustrated in Figure 1.
Figure 1: A sampling of multiple couplings between different energy domains in microsystems [2] . Furthermore, MEMS often exhibit complex geometrical structures. Modeling these features on continuum level (by partial differential equations) and numerically simulating them by e. g. the finite element method (FEM) may be feasible for a single device. However, it becomes prohibitive when its interplay with the package, the surrounding electronics or other
transducers has to be taken into account. Hence, compact modeling methods, which are able to reduce the number of degrees of freedom (DOF) and hence, to make these device models tractable, are sought after. In the field of semiconductor devices and modules, comprehensive simulation environments and methodologies for the top-down and bottom-up design are well established since many years and environments for the virtual prototyping of devices and systems including their compact modeling are in a mature state [23]-[26]. In contrast, this evolution for MEMS has just started within the last decades and is still in progress. So far, several approaches for systemlevel modeling of MEMS have been established within the world-wide community and found their way into practical application and – to a certain extent – into commercially available software implementations. A good overview of the state-of-the-art in this field is given in [1]. When speaking about the compact models of MEMS components, one differs between two main technologies: lumped-element modelling and reducedorder modelling. A lumped-element model specifies a compact- modelling approach where spatially distributed physical behavior is ‘‘lumped’’ into a finite set of ‘‘elements’’ that approximate behaviour at discrete points in space. A common and very simple mechanical example is a 1- DOF mass-spring-damper model. A common electrical example is an inductor– capacitor network approximation of a transmission line. A reduced-order model is a compact model formed by mathematically reducing the order of a highDOF model, which typically emerges from numerical continuum simulations. Mathematical methods involved are known under the common name of model order reduction (MOR). 2. Compact Modeling of MEMS by Lumped Elements Lumped element models describe the operation of devices or systems through a small set of concentrated (lumped) variables. They can be as simple as an equivalent electrical network or more complex and physics-based. The former employ standard electrical elements such as resistors, inductors, and capacitors to emulate the nonelectrical behavior. The latter incorporate the governing physical equations and dependencies from design and external parameters, usually implemented directly in a hardware description language (HDL).
The strengths of physics-based lumped-element models are their efficiency with respect to computational expense and speed, and their adaptability to different design variations, e. g. change of geometry. Therefore, they are very well suited for design and parameter studies as well as for optimization processes. However, an automated model derivation is not possible for this kind of models. An experienced engineer or expert user with profound knowledge on the given problem has always to be involved in the model derivation process in order to properly incorporate all important physical effects adequately. An illustrative example for physics-based, lumped element system-level modeling is the electrostatically driven micropump, shown in Figure 2 and described in more details in [3] and [4]. This complex device, whose operation is governed by the coupling of mechanical, electrical, and fluidic domain is divided into functional blocks according to Figure 3, each described by a physics-based analytical expression and implemented in HDL. All submodels can be connected within a circuit- or system-level simulator (e. g. Spice, Spectre, Saber, Simplorer) according to the generalized Kirchhoffian network methodology, described in section 4. The comparison of the simulated frequencydependent pump rate to measurements is shown in Figure 4. It is worth mentioning that, the once calibrated compact model can be used for further designs with different geometrical dimensions without additional recalibration as demonstrated in [3].
Figure 3: Division of the micropump into functional blocks, each to be described by physics-based analytical expressions which are subsequently implemented in HDL [4].
Figure 4: Frequency-dependent pump rate of the electrostatically operated micropump comparison between measured and simulated data of the lumpedelement-based compact model for the design under consideration [4]. Further examples for the application of such physics-based compact models are given in [5] – [8].
Figure 2: Schematic view of the electrostatically driven micropump, which consists of a pump chamber with an electrostatically displaceable membrane as driving element, passive inlet and outlet valves, and externally attached tubes [3].
3. Compact Modeling of MEMS by Model Order Reduction Spatial discretization of the governing partial differential equations (PDEs) via e. g. finite elements method leads to large-scale systems of ordinary differential equations (ODEs) or differential algebraic equations (DAEs). Time integration of such systems is usually computationally too expensive to allow for efficient design optimization. Developed from well-established mathematical theories and robust numerical algorithms, model order reduction has been recognized as being very efficient in reducing the simulation time for such large-scale dynamical systems. Through MOR, the smaller systems of ODEs or DAEs are derived, as schematically represented in Figure 5.
state (temperature) vector, the load and the output vector respectively and is the dimension of the system, i. e. the number of degrees of freedom in finite element model.
Figure 5: Schematical representation of model order reduction for an ODE system in the right-hand-side and representation with system matrix ∈ . input and output distribution arrays, , ∈ Dimension of the reduced system is ≪ . Figure 7: FEM mesh of the three-dimensional model with 60.000 nodes [10] The reduced model is simulated instead, and the solution of the original PDEs can be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. To date, MOR has been widely applied to simulation of MEMS and has achieved much success in enhancing traditional simulation tools [9]-[14]. Main advantage of MOR approach is that it can be automated, i. e. the designer must not necessarily be an expert in physics. On the other hand side, preservation of important properties like stability and passivity of the multiphysical models is an issue [15]-[18]. Furthermore, parametric and nonlinear methods are still under development [19]-[21]. An illustrative example for MOR-based compact modeling of MEMS is a silicon-based micro-hotplate, shown in Figure 6. Another example is a MEMS piezoelectric energy harvester described in a separate contribution of the present proceedings.
Projection-based MOR (Block–Arnoldi algorithm with deflation [9]) has been applied to system (1). This process is automatic, in the sense that it requires from the designer to solely specify outputs of interests and the size of the reduced model. It is worth mentioning that the exact mathematical proof for error bounds for this kind of reduction are still not available. The reduced model will have the same form as (1), but much smaller dimension. It can easily be implemented in HDL form. Figure 8 shows an excellent match between the full-scale model and the reduced-order model, which now can be used for system-level simulation, control, or design optimization.
silicon nitride membrane tunable cavity
thin-film heater
air silicon substrate
Figure 6: A silicon nitride membrane with integrated heater and sensing element [10]. A three-dimensional thermal-domain finite element model of the microhotplate, displayed in Figure 7, incorporates 60.000 DOF, which correspond to 60.000 ODEs of the form: U 2 (t ) C T K T F R (T ) y E T
where C , K R nn
(1)
are the global heat capacity and
heat conductivity matrices, T (t ), B, E R n are the
Figure 8: Temperature response over time at membrane’s center for the full-scale numerical model with 60.000 DOF and for the reduced order model with only 30 DOF [10]. Speed up in computational time is by several orders of magnitude. 4. System-Level Modeling of MEMS Using Generalized Kirchhoffian Networks System-level modeling and simulation of MEMS goes beyond a compact modeling of a single transducer. Its goal is rather, to describe the interplay of each single transducer with the package, the surrounding electronics and other transducers. The most appropriate approach for complex systems is a “divide and conquer” one, that is to define simpler subsystems, in a similar manner as suggested in the micropump example. Given that the entire system is
properly decomposed into suitable subsystems– the compact modeling approaches presented in the previous sections (based on either lumped elements or on MOR) can be applied to each subsystem. A proper framework for the final assembly of the subsystem compact models to a system model and, thus, for the co-simulation of all electrical and nonelectrical parts is provided by the Kirchhoffian network theory, which is a well-established methodology for the mathematical representation of electrical networks, and can be generalized to other energy domains [4]. The models that are derived for the subsystems are linked via properly chosen, consistent flux-conserving interface conditions (e.g., balance of forces, energy, and electrical flux,) according to the physical quantities exchanged between the subsystems under consideration, as schematically shown in Figure 9.
voltage, causing a volume change and, thereby, driving the fluid flow w.
Figure 10: Schematic view of an electrofluidic Kirchhoffian network. The conservation laws for volume flow w and hydrostatic pressure p are reflected in sum rules at the nodes and along closed loops [4]. Advantages of a generalized Kirchhoffian network formulation for microsystems are that it constitutes a physics-based description of the system, that the couplings between different energy domains are realized in a natural, thermodynamics-based way and that the Kirchhoffian network models can be incorporated in any standard circuit- or system-level simulator that allows for model implementation in terms of a HDL.
Figure 9: Partition of a system into two subsystems A and B. Along the interfaces, each of the quantities (fluxes) is exchanged in a physically consistent manner (subject to balance equations) [4] The biggest challenge in this approach consists in setting-up adequate compact models for the chosen subsystems, so that the distributed field variables are lumped and mapped onto the so-called terminals (or nodes) along the interfaces of the subsystems. Each device operation is then described by exchanging the values of these lumped variables via these terminals. Note that in case of using projection-based MOR for compact modeling of subsystems, a potentially large sacrifice of precision at the model’s surfaces would be required for such lumping [22]. Figure 10 illustrates the application of the generalized Kirchhoffian network theory schematically for an electrofluidic transducer (e.g., electrostatically actuated micropump, described in section 2). The fluidic domain is described by the hydrostatic pressure p and the volume flow w, and the electrical domain is described by the electrical voltage U and current I. The coupling between electric and fluidic domain is realized via the transducer element B1, which is a fourterminal element with the respective terminals in each domain. The transduction mechanism between electric and fluidic domain can be realized by a mechanical transducer, for example, an electrostatically actuated membrane. The membrane is deflected by an electrical
4. Conclusions The real challenge in system-level modeling of MEMS consists in constructing adequate compact models for the chosen subsystems, that is how to optimally fill the boxes in Figure 10. Here, a universal approach that could be applicable for any kind of MEMS structure and subsystem cannot be identified, as the choice of the available methodologies depends strongly on specified function and application, the context, in which the model is supposed to be used, the system architecture, the required accuracy, and the desired reusability of the model for different design variants. In general, lumped element models preserve physical transparency, require however, a modeling expert. Model order reduction, on the other hand side, can be automatized, but remains rather mathematical, and still to be developed in the sense of connecting parameterized and nonlinear reduced order models. Acknowledgments We would like to thank to all authors of [1] for their valuable contributions. References 1. T. Bechtold, G. Schrag, L. Feng (eds), “System-Level Modeling of MEMS”, (WileyVCH Verlag GmbH & Co. KGaA, ISBN: 978-3527319039, (2013). 2. G. K. Fedder, T. Mukherjee, " Introduction: Issues in Microsystems Modeling" in "System-Level Modeling of MEMS”, Wiley-
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