SIMULATION METHODOLOGY OF MICRO-ELECTRO-MECHANICAL … · MICRO-ELECTRO-MECHANICAL SYSTEMS 351 4.1...

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 13, Number 4, Winter 2005

SIMULATION METHODOLOGY OF

MICRO-ELECTRO-MECHANICAL SYSTEMS.

PART 1: FULL 3D-MODELS

ALEXANDRE A. SEROV

ABSTRACT. The paper is dedicated to a state-of-the-artreview of the numerical methods used in the simulation ofMicro-Electro-Mechanical Systems (MEMS). We treat all thebasic approaches that are used for the realization of modernmethods of MEMS simulation. We also classify the numericalmodels in current use. We describe the basic methods used forfull three-dimensional simulation of MEMS, and present exam-ples of numerical models of MEMS. We present some prospectsfor development of systems components suggested by the sim-ulation methods.

1 Introduction The origin of the field of Micro-Electro-MechanicalSystems can be traced to the beginning of the 1980’s [38], when thefirst papers appeared, opening a path to the study of nano-electronicdevices. At present it is possible to find examples of MEMS applica-tion in practically every area of instrument making: from the automo-bile industry [57] and telecommunications industry [7] to microbiologyand the development of medical equipment [22, 23, 55]. During thelast two decades the study of Micro-Electro-Mechanical Systems hasevolved from a field of only academic researches into an integral partof micro- and nano-technological bases for modern instrument making[13, 14, 22, 45]. MEMS-based devices have now numerous applica-tions, such as microsensors, microactuators, microaccelerometers, micro-phones, cellular phones and microelectromechanical filters. MEMS ele-ments that have appeared to date include rotary motors, linear motors,resonators, springs, gears, grippers, diaphragms and arrays of mirrorsfor display technology. Now the techniques for creating Micro-Electro-Mechanical Systems are capable of producing composite systems inte-grally, including micro-mechanical parts, analog circuits and digital logic[11]. The manual development of these types of device is extremely dif-

Copyright c©Applied Mathematics Institute, University of Alberta.

345

346 ALEXANDRE A. SEROV

ficult owing to the interdisciplinary nature of a developed system andthe great number of intra-system interactions required of the model.MEMS is now an essential interdisciplinary field of electronics, bring-ing together studies in mechanical engineering, electrical engineering,electronics, fluid mechanics, optics, chemistry and chemical engineering.The presence of interdisciplinary physical processes in a system underdesign generally requires the use of hybrid systems simulation tools. Thedevelopment and use of these tools is quite difficult. The rapid increasein the complexity of systems being designed, and the cutting edge level ofthis complexity, leads today to the necessity of creating general-systemdesign methods that integrate development tools from each of the rele-vant fields of physics and engineering [28, 29, 31, 40, 42, 46]. A basicapproach, one that is widely used at present for MEMS design, is thehierarchically structured design [31, 42], [12, 27, 30, 39, 41, 46]. Itsflowchart is presented in Figure 1.

According to this methodology, the design of the device begins withthe stage of design topology. The behavioral simulation of MEMS mustbe done after this first stage. The multicomponent physical system issplit into subsystems and components. MEMS simulation runs in twosteps, in the first of which the behavioral simulation of components ismodeled. The second step is associated with simulation of MEMS asa whole. After the system simulation stage, the MEMS description iscreated, and finally the extraction of parasitic capacitances and finalverification of the system must be carried out. These stages of MEMSdesign correspond to the set of tools used in the automatic design ofdevices. The set of tools for geometric design is one of most importantComputer Aided Design (CAD) components [33, 37, 56]. It is necessaryto take the production technology into consideration during this stageof MEMS design. This leads in turn to the creation of tools for techno-logical process simulation inside of MEMS CAD programs [18, 19, 24].Use of these tools, for example with the CAD program CoventorWare(Coventor Inc.), makes possible not only the calculation of device geo-metric parameters on the basis of technology process characteristics anddescription of designed system; it also makes it possible to visualize thedesigned MEMS that will be produced by some definite technology. Theparadigm of structured design comes into the MEMS design field fromthe area of Very Large Scale Integration (VLSI) Systems. ComputerAided Design for VLSI spans many levels of abstraction from materials,devices, circuits, logic, register to the level of the system. At each ofthese levels a design can be viewed in physical, structural (schematic)or behavioral form. One of the main tasks in development of structured

MICRO-ELECTRO-MECHANICAL SYSTEMS 347

FIGURE 1: Flowchart of structured design of MEMS [28]

MEMS design tools is the formation of standard data representationsand standard cell libraries.

The list of modern subfields of activity from the area of structuredMEMS design includes:

• standard MEMS data representations and interchange formats• standard MEMS cell libraries supporting behavioral, schematic and

physical views at all levels of abstraction. These libraries must includematerials database, layout cells, schematic element library, systemmacro-model library

• standard MEMS process-module libraries and standard process flows• process simulation and visualization


FIGURE 2: Modelling and simulation in MEMS design: different phys-ical domains and levels of abstraction [42]

• process synthesis and technology file extraction• Three-dimensional (3D) rendering and animation• 3D generation from layout and technology files• layout of arbitrarily shaped objects with design rule checking• layout synthesis and verification• fast modeling and verification tools: coupled multi-domain numerical

analysis tools• parasitic extraction to schematic and behavioral views• macro-model parameter extraction from physical and schematic views• multi-domain schematic capture: schematic view of connectivity be-

tween mechanical, electromechanical, thermal and circuit lumped-parameter elements

• mixed-signal multi-level multi-domain simulation.

A vital problem for modern CAD systems used for MEMS design [6] to-day is the poor level of development of general-system simulation meth-ods, the basic features of which are: the low level of general-system ver-ification methods, the limitations placed on the amount of complicationallowed in developed devices and the significant increase in expendituresfor design involving increases of complexity in MEMS. Methods for thedevelopment of mathematical modeling for technical systems, includ-ing various physical components, is now the main problem facing CADin this field of microelectronics. In particular, one of central problemsof simulation methods development is the creation of macromodels for


possible MEMS components which precisely describe the dynamics ofdevice functioning.

2 The basic approaches used for development of MEMS sim-

ulation methods There are presently two main directions for thedevelopment philosophy of MEMS simulation. The first is the systemsapproach, in which the architecture of the system is built by a principleof unification, with the integrated use of specialized simulators. Eachof these simulators is associated with a definite branch of physics or adefinite level of MEMS description for the system being considered. Ac-cording to the second approach, the heterogeneous physical componentsof MEMS are considered within the framework of a unified simulationmedium on the basis of methods designed for their homogeneous descrip-tion. Both of these directions are being actively developed at present.The multicomponent physical system as the core of a MEMS mathe-matical model can be described by a set of differential equations. Forthe study of processes which occur in such systems, the methods of fullthree-dimensional simulation are applied. Three-dimensional models areused both for the in-depth study of the dynamics of a system, and forproblem solving directly tied to the development of reduced-order mod-els. The application of 3D-models in numerical analysis is usually asso-ciated with considerable expenditure of computer time. The demandsfor resources indispensable for the numerical analysis of a design canbe essentially reduced through the application of reduced-order mod-els (macromodels). Development of methods for full three-dimensionalMEMS simulation together with the construction of macromodels, con-stitute two basic branches of modern tools development for the numericalmodeling of Micro-Electro-Mechanical Systems. It is necessary to notethat the development of macromodeling methods and the developmentof 3D-models do advance by independent paths. This is associated withthe fact that three-dimensional models in some cases are required at levelof making macromodels [15]. Secondly, an ideological link between ourtwo approaces also exists at the level of using the same conceptual ap-proaches, as it was, for example, with the use of the method of Multiple-Energy Domain Representation. This approach was presented in [15] asa basis of a method of MEMS macromodel development. Later it wasused in the well-known set of programs SUGAR for the development ofthree-dimensional models [10].


3 Physical and mathematical models of micro-electro-mech-

anical systems It is possible to sub-divide into two groups the col-lection of mathematical interpretations of physical analogs used withinthe framework of MEMS simulation methods. The first type of mathe-matical model has in the literature the title of non-lumped parametersmodels. Within the framework of this type of description, the modeledsystem is considered as a continuum. The physical properties of a ho-mogeneous system are described by continuous functions. The presencein a simulated system of several different technological materials (in theelementary case) is accounted for through discontinuities of functionsat the applicable boundaries. By virtue of its specificity, this approachis used extensively for the development of full three-dimensional modelsof Micro-Electro-Mechanical Systems. The second type of mathematicalmodel has the title of lumped parameters models. Within the frameworkof this description the modeled system is considered as combination ofdiscrete elements. Definite physical characteristics are assigned to eachof these elements . In particular, each of the systems components canbe considered as an absolutely rigid body, or as a deformable body, de-pending on the properties of the modeled system and demands to be metby the mathematical model. This approach is successfully used both atthe development stage for three-dimensional models, and in problems ofmacromodelling. Use of these two approaches for development of MEMSmathematical models is completely determined by the relevant physicson which is based the functioning principles of the particular device. Ac-counting for electro-elastostatic or electro-elastodynamic phenomena inthe model gives the advantage of using models with lumped parameters.The necessity of analysis of phenomena from the field of air- or hydro-dynamics or adjoining these areas to other areas of physics results inthe necessity of using non-lumped parameter models. The convenienceand possibility for use of each type of mathematical model is dictatedby demands for the realization of an effective mathematical method andthe correct formulation of the physical problem.

4 Methods of development of full three-dimensional MEMS

models For the development of full three-dimensional simulations forMicro-Electro-Mechanical Systems both lumped- and non-lumped pa-rameters models can be used. The type of mathematical descriptionselected for a physical system completely determines which mathemat-ical apparatus can be used for development of a method of numericalmodeling.


4.1 Non-lumped parameters models of a system Usage of thistype of model for interpreting a system as a continuous medium impliesapplication of the mathematical apparatus of mechanics and electrody-namics of continua, hydrodynamics, and perhaps other fields of theoret-ical physics. The dynamics of a multicomponent physical system is thusdescribed by a set of partial differential equations. The discretizationof a continuum model leads to the methods of finite differences, finiteelements, boundary elements and other methods usually used in theapplicable areas of science. In general, a singular characteristic of three-dimensional MEMS models development is the necessity of accountingfor not only the composite geometry of a considered system, but alsoits non-steady state. The modifications of MEMS topology, character-istic, for example for electrostatic MEMS, are associated with physicalprocesses created in the given topological structures by variations in theoutside influences. The mathematical description of this physical prob-lem results in one of the most difficult problems in the field of numeri-cal methods: problems with mobile (free) boundaries [17], inherent notonly for Micro-Electro-Mechanical Systems [5, 34–36],but also for otherfields of application of simulation methods [43]. The method of MEMSsimulation, realized in the simulator AutoMEMS (Coyote Systems Inc.)is described in the paper [25] (see Figure 3). The computational schemebeing more or less common to 3D models, well characterizes this classof models. For MEMS simulation in an autonomous mode the followingfour operations implement the process: the generation of a 3D modelon the basis of data about photolithographic masks; the introductionof information about boundary conditions and properties of materialsused for construction; generation of a computational grid; the solutionof the system of partial differential equations describing the model ofthe MEMS device. The numerical model introduced in paper [25] isbased on the use of the boundary elements method (BEM). Realiza-tions of other methods of MEMS simulation [16, 21, 20] are based onsome combination of a finite element method (FEM) and BEM. Thesemethods differ by the various degrees of automation for the describedsimulation stages.

The paper [5] presents a numerical model of an idealized electrostati-cally actuated MEMS device. The device consists of an elastic membranesuspended above a rigid plate (Figure 4). The membrane has width L

and is separated from the ground plate by a distance l in its undeformedstate. The model assumes that both membrane and ground plate areinfinite in the z direction. It is also assumed that both membrane andplate are perfect conductors and a potential difference is applied between


FIGURE 3: Dataflow at MEMS simulation in AutoMEMS [25]

them. The equations of the mathematical model associate the electro-static potential in the region surrounding the device and the elastic dis-placement of the membrane. The numerical problem of MEMS devicesimulation is formulated in the paper as a steady-state boundary-valueproblem.

The governing equations for the electrostatic potential satisfy La-place’s equation:

(1) ∆ψ = 0


FIGURE 4: Geometry of electrostatically actuated device [5]

with boundary conditions:

ψ = 0, at y = −l, −L

2< x <

L

2(2)

ψ = V, at y = u, −L

2< x <

L

2(3)

where V is the applied voltage and u(x, t) is the displacement of themembrane from the y = 0 position, x and y are coordinates, t is time.In the mathematical model the displacement of the membrane satisfiesthe following equation:

(4) ρ∂2u

∂t2+ ν

∂u

∂t− T

∂2u

∂x2=

| E |2

8π.

Here ρ is the linear mass density of the membrane, ν is a dampingcoefficient, T is tension and E is the electric field evaluated along themembrane surface. The first term on the left side of equation (4) isassociated with the inertia of the moving membrane. The second termdescribes the force of mechanical resistance to motion of the membrane.The third term approximates the force arising in the membrane due itselastic strain. The term on the right side of this equation is associated


with the force of electric attraction acting on the membrane. Equation(4) must be solved with the following boundary conditions:

(5) u

(

−L

2, t

)

= u

(

L

2, t

)

= 0.

This equation describes the type of fastening of the membrane edges. Itis necessary to note that equation (4) has both steady-state and non-steady-state solutions. For a numerical solution one can set time deriva-tives equal to zero on the left side of (4). In this case it will be possibleto investigate only steady-state solutions of this MEMS simulation prob-lem. Another method of solution is to use iteration methods. In thiscase it is possible to solve both steady-state and transient problems.This works because the characteristic time for electric field stabilizaa-tion is much less than for membrane motion. A method for quasi-three-dimensional numerical simulation of Micro-Electro-Mechanical Systemswith fixed-fixed beam geometry is developed in the paper [44]. Themathematical problem formulation is based on geometric domain decom-position into two parts: elastic beam and air gap domains (Figure 5).

The state of this system is described by the Laplace equation for thisMEMS model:

∆ψ1 = 0(6)

∇(εp∇ψ2) = 0(7)

where εp is dielectric permittivity of plate material, ψ2(x, z, t) is theelectric potential in the elastic beam, ψ1(x, z, t) is the electric potentialin the air gap. A method of mathematical description for fixed-fixedbeam movement depending on the voltage applied between the beamand substrate is proposed in a paper in preparation. The equation ofplate motion is introduced in this paper under the assumption of sheardeformation of the elastic beam:

(8) m~a = ~Fd + ~Fs + ~Fe

where m is the mass of the beam element, a is the acceleration of thebeam element, Fd is the force of viscous damping, Fs is the elastic forceacting on the element, Fe is the force of the electric field on the element.The resulting plate shape dynamics has the following mathematical for-multion:


FIGURE 5: Geometric domain of numerical problem solution at elec-trostatic MEMS simulation. Here l is the depth of air gap in the absenceof a potential difference between the beam and substrate, d is the half-thickness of the elastic beam, u(x, t) is the mid-line of the beam, so thatu(0, t) = l + d

(9) ∆ttu+De

dd·

1

1 + (∇xu)2

×

(

χd1

(1 + (∇xu)2)1/2(∇tu)

2 + χd2(∇xu)2∇tu

)

= Ga ·1

1 + (∇xu)2

(

χe1

(1 + (∇xu)2)1/2(∆xxu)

+χe1

dd(∇xu)

2 +χs

2(∇xu)

2∇zψ

ψ

)

+Em ·χe2

dd((∇xψ)2 + (∇zψ)2)

where χd1 and χd2 are parameters depending on both the properties ofthe substance between the moving plate and substrate and the positionof the element; χe1, χs are parameters depending on the elastic proper-


ties of the beam; χe2 is a parameter that depends on electric propertiesof the substance between the beam and substrate; De, Ga and Em arethe similarity parameters for the problem. Our method for the numeri-cal solution of equations (6), (7) and (9) developed in [44], is based onthe Boundary Elements Method. The numerical solution of the electricpotential equations and the equation for the function of air gap size iscarried out by minimizing of a functional derived on the basis of theLaplace equation:

I =

∫

V

(

α

2

(

∂ψ

∂x

)2

+β

2

(

∂ψ

∂z

)2)

dV +

∫

S

ϕ(x, z)ψ dS

where α and β are certain constants; the function ϕ describes the bound-ary conditions for the numerical problem; V and S are the volume andthe boundary, respectively. The method proposed in [44] is based onthe non-lumped parameters model paradigm and may be used both forthe investigation of time-dependent processes arising from changes in thegoverning voltages of the MEMS and for the study of static MEMS char-acteristics. The use of this method makes it possible to solve the prob-lem of optimizing constructional parameters for electrostatic MEMS.The obvious advantage of simulation methods based on the non-lumpedparameter paradigm is a maximum level of correspondence between aMEMS device’s mathematical model and the physical behavior of thedescribed system. The use of the machinery of deformed body mechanicstogether with the equations of electro- and thermal physics and hydro-and aerodynamics allows the development of numerical methods corre-sponding to pre-given properties of conservation for the considered sys-tem. That is presently not possible for the majority of methods usinglumped-parameter models. Another important feature is the continu-ity of development, appropriate for this branch of MEMS simulationmethods. The feasibility of using developments accumulated in the fieldof physical systems simulation during more than a half-century, allowsavoidance of the solution of a series of very hard scientific problems,inherent for numerical problems in various branches of physics. Amongdeficiencies for this group of methods it is possible to refer, first of all, tothe lack of uniformity and universality of the description of a consideredphysical system. This results in the necessity of using the descriptiveapparatus from separate areas of science, which requires a careful studyof a physical analog for the considered system. Another current prob-lem is the lack of a capability of macromodels development, genericallyassociated with a method of full three-dimensional simulation.


4.2 Lumped-parameters models of a system The use of lumped-parameters models implies the application of a mathematical apparatusto describe systems which consist of discrete elements. The dynamicsof MEMS thus is described by a set of ordinary differential equations.For the discretization of a continuum the same set of methods is uti-lized as with non-lumped parameters models. The philosophy underlyingpresent methods of MEMS simulation is substantially determined by thespecific nature of the problem, for the solution of which an appropriatemethod will be used. It is common for methods of simulation developedfor independent problem solving in some definite area of physics to sig-nificantly differ from the methods used for the solution of the same classof problems, originating from research on Micro-Electro-Mechanical Sys-tems. Another reason for these differences is the desire of investigatorsto find a general-purpose method for the description of all constituents ofa multicomponent physical system. Both of these reasons exert a signifi-cant influence on the development of methods for 3D-models. One of themethods used for MEMS description within the framework of lumped-parameters models is a method universally describing a multicompo-nent system, based on a formalism of Lagrange [11]. This approach,in the fundamentals of which lies the decomposition of a system intogeometrical primitives, uses the apparatus of deformed body dynamicsfor a description of these elements. This approach is common for fieldsof construction both of full three-dimensional models and macromodels.The simplest way to integrate simulation tools designed for systems withcomponents of different physical natures can be the selection of one ofthe components as dominating and the subservience of all the remainingcomponents to laws adopted for the description of this component. Thismethod has naturally appeared in the development of MEMS simulationtools. Another factor moving the development of simulation methods forMicro-Electro-Mechanical Systems in this direction was the high level ofefficiency of the tools and methods of electrical circuits simulation. Fromthe idea to join the tools of simulation with the electrical simulator camethe method known as Modified Nodal Analysis [9]. Nodal analysis [32]is presently applied to the formulation of the systems of equations forelectrical circuits. This system is used in the simulators that are widelyknown as SPICE. The method of Modified Nodal Analysis extends andsupplements a set of arguments used for the characterization of a mod-eled system. Many MEMS can be described within the framework of


lumped-parameters models by a system of ODEs of the following form:

(10) Mq̈ +Dq̇ +Kq = F (t, q, q̇)

where M , D and K are matrices of masses, friction coefficients and elas-ticity, respectively. The discretization of the system’s geometry on N

elements results means that the vector of coordinates of elements-nodesq has dimension 6N , as well as vectors of forces and moments of forcesF , acting on elements. The mechanical component of a system has 6Ndegrees of freedom [3] and can be described by a system of 6N equations.Accounting for the electrical degrees of freedom results in an increase inthe order of the system of equations. If the number of variables describ-ing the electrical MEMS constituents is equal to m, then the generalorder of the system of equations describing the MEMS dynamics be-comes equal to 6 ·NMechanical +m ·NElectrical. A similar approach canbe used for taking into account components of a physical nature distinctfrom electrical. The resulting set of equations, based on Newton’s laws,can be considered as a mechanical analog of the Kirchoff equations, us-ing the analogy of current to force, and electric potential of a unit tomovement of the MEMS element. The solution of full three-dimensionalproblems of static analysis and transient analysis in MEMS devices isimplemented on the basis of the given information on properties of thematerials used (elastic modulus, Poisson’s constant, coefficient of ther-mal expansion, etc.), geometry of the system, electric potential differenceand forces applied to different parts of the system. The method of Mod-ified Nodal Analysis is used by the simulator SUGAR [3, 8], designedin the Sensor and Actuator Center, University of California, Berkeley.Algorithms of a method of solution are given also in the well-known pro-gram MATLAB [26]. The systems of equations (10) may be used forthe simulation of an RF-switch, shown in Figure 6. For the linear case,the matrices M , D and K can be determined as follows:

M =ρAL

420·

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

140 0 0 70 0 00 156 22L 0 54 −13L0 22L 4L2 0 13L −3L2

70 0 0 140 0 00 54 13L 0 156 −22L0 −13L −3L2 0 −22L 4L2

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣


FIGURE 6: Fixed-free beam used as switch for an RF network

D =µLw

420δ·

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

140 0 0 70 0 00 156 22L 0 54 −13L0 22L 4L2 0 13L −3L2

70 0 0 140 0 00 54 13L 0 156 −22L0 −13L −3L2 0 −22L 4L2

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

K =E

L3·

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

AL2 0 0 −AL2 0 00 12I 12EI 0 −12I 6IL0 6IL 6EIL 0 −6IL 2IL

−EAL2 0 0 AL2 0 00 −12I −12EI 0 12I −6IL0 6IL 6EIL 0 −6IL 4IL

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

where E is Young modulus; L is the length of the beam; A is the crosssectional area of the beam; I is the moment of inertia ; ρ is the den-sity of beam material; µ is the viscosity of the substance between thebeam and substrate; w is the width of the beam; δ is the distance fromthe beam to substrate. Modified Nodal Analysis underlies the simulatorNODAS [47], designed at Carnegie Mellon University. There are charac-teristic problems with approximation of continuous physical phenomenaby discrete relations. In particular, the problems of approximation offorces damping mechanical motion create some of the major difficulties.Another idea of adjoining the tools of MEMS simulation to electricalsimulators was the direction based on development of Equivalent Cir-


cuit Models. The method of electrical analogies in its fundamentals iswidely applied in different areas of science and engineering, among themmechanics, thermal physics, acoustics and the field of Micro-Electro-Mechanical Systems [48, 49]. It is based on the mathematical analogybetween dynamical equations of a considered system and some electricalcircuit (see Table 1).

Mechanical parameter Electrical parameter

Variable Velocity, angular velocity VoltageForce, torque Current

Lumped Damping Conductancenetwork Compliance Inductanceelements Mass, mass moment of inertia Capacitance

Transmission Compliance per unit length Inductance per unit lengthlines Mass per unit length Capacitance per unit length

Characteristic mobility Characteristic impedanceImmitances Mobility Impedance

Impedance AdmittanceClamped point Short circuit

Free point Open circuitSource Force Current

immitances Velocity Voltage

TABLE 1: Electromechanical mobility analogies

The similarity of the equations implies the mathematical equivalenceof physical quantities from different areas of physics. A fundamentalproblem in this approach is the search for the electrical circuit whichwould be described by equations similar to the dynamical equations ofan investigated Micro-Electro-Mechanical System. One of most typicalapplications of the electrical analogies method is the simulator APLACof the company Aplac Solutions,which produces a realization of a simula-tion capability for MEMS, significantly different at the level of a physicalanalog. The papers [50, 54] were the ideological basis for the use of theEquivalent Circuit Model for the simulation of multicomponent systems.There it was shown that electrical, fluid, mechanical and thermal sub-systems can be jointly described on the basis of the Kirchoff formalismapplied to electric networks. It was a natural development, realizedin the simulator APLAC, to construct a library of equivalent networksintended for the simulation of electromechanical and fluid MEMS com-ponents. The most attractive property of this method is the fact thatas the MEMS models are described by electrical equivalent networks,


the description of MEMS becomes homogeneous. An equivalent net-work for an electro-mechanical system of capacitive type is shown inFigure 7. The model of mass suspended on a spring [50, 51] describesthis type of devices quite well. The spring is modeled as gyrator andcapacitance Cz . The capacitance C is the equivalent of mass. Dampingin the Micro-Electro-Mechanical System is modeled by conductance G.

FIGURE 7: The equivalent network of MEMS that is described as amass-spring system [48]

Figure 8 presents the structure and operating principles of a capacitiveMEMS switch [53]. The bridge is perforated to reduce the damping ofair flow.

The differential equation describing the motion of the bridge has thefollowing form:

(11) m∂2z

∂t2+ γ

∂z

∂t+EIy

∂4z

∂x4+ hwS

∂2z

∂x2= f(x, z)

where m is the mass of the bridge, z is the displacement, l is the co-ordinate along the beam, γ is the damping term due to the gas flow,E is the effective Youngs modulus, I is the moment of inertia, S is thestress of the beam, h is the height of the beam, w is the width of thebridge, f is the distributed load due to electrostatic actuating force andthe mechanical contact force. This model of an RF MEM switch wasconstructed from elementary finite difference sections. These sectionsconsist of elements that model bridge deflection, electrostatic actuationforce, gap capacitance and mechanical contact force. The construction


FIGURE 8: Structure of RF MEM switch [53]

of the switch model is shown in Figure 9. The two first and two last sec-tions model the clamped-clamped boundary conditions. All other modelsections are identical.

Each section contains a model of beam deflection (BE), mechani-cal contact (NC), gas damper (PGD) and electromechanical transducer(NTR) with electrostatic actuation force and gap capacitance. The ca-pacitances of sections are connected in parallel. The paper [53] presentsin addition a calculation method for the air flow through the perforationholes. Numerical modeling of the micro-electro-mechanical componentfunctioning of a system can be performed together with the model fornetworks of electronic control both in frequency and in time domains[48, 50]. The method of electrical equivalent networks has obvioust ad-vantages, but also has deficiencies. One is associated with the precision


FIGURE 9: Block diagram of switch model [53]

of approximation in the case of essentially nonlinear systems [52]. Inparticular, the models used in microfluidic MEM subsystems approxi-mate the process of gas flow well enough only in the case when the gaspressure and change in position of MEMS plate are small enough incomparison with static values.

5 Conclusion Modern methods of MEMS simulation have passedthrough a long period of development. They have been realized throughsuch tools as MEMCAD (M.I.T.), CAEMEMS (University of Michigan),CAPSIM (Catholic University of Leuven), SENSOR (Fraunhofer Insti-tute), SESES (ETH), IntelliCAD (IntelliSense Corp.), CoventorWare(Coventor Inc.), APLAC (Aplac Solutions), NODAS (Carnegie MellonUniversity), SUGAR (University of California at Berkeley) and manyothers. And the problems of MEMS simulation methods design havenot lost importance since the beginning of the 90s because of the ar-


rival of new material technologies and new devices types. The solutionof problems of miniaturization for the field of Micro-Electro-MechanicalSystems results in an increase in financial support for the developmentof nano-size devices [1, 2, 4]. The result is the current appearance ofdevices, the principle of operation of which is based on effects from quan-tum physics. Independent progress in different areas of nanoscopic sys-tems, for example in nano-electro-mechanical systems (NEMS), single-electron devices (SED), nano-fluidic systems (NFC), will in several yearslead to the creation of new areas of science and engineering. Not only thedevelopment of the design tools for these devices, but also the methodsof their simulation, will differ considerably from ones existing now. Oneof the features of models of nanoscopic devices is that the description ofa system must be done at the molecular and atomic level. The numer-ical effectiveness of methods of simulation for these systems is in manyrespects determined by the effectiveness of their description by meansof stochastic processes. Different levels of physical models used for thestudy of dynamic phenomena in nanoscopic system will already in thenear future create the necessity of the development of approaches usingmethods of molecular dynamics as a base for the numerical methods,and the theory of control of stochastic systems as a fundamental for theconstruction of automated design systems of a new generation—CAD ofnano-electro-mechanical systems.

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Senior Researcher, PhD, Research Center Scan-plus, Moscow, Russia

E-mail address: [email protected]

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