Koren B., Vuik K. (Editors) Advanced Computational Methods in Science and Engineering

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Numerical Modeling of the Electromechanical

Interaction in MEMS

S.D.A. Hannot and D.J. Rixen

1 Introduction

Microsystems or Micro–Electro–Mechanical Systems (MEMS) are small (microm-

eter size) machines usually built by lithographic technologies originally developed

for microchips. MEMS are designed to integrate sensing and actuation (and even

data processing) on a single chip, therefore they often include moving and deform-

ing parts. Currently microsystem technologies are used for a wide variety of pur-

poses such as: read/write heads in hard-disk drives, ink-jet printheads, Digital Light

Processing (DLP) chips in video projection systems and several types of sensors for

pressure, ﬂow, acceleration or bio-elements.

Due to their very small size, the dominant driving forces differ from the ones in

the macro–world. For instance, the gravitational forces are negligible with respect to

the elastic, adhesive and electrostatic forces [43]. These scaling effects may cause

a strong coupling between different physical domains. For instance small silicon

devices can be heated by an electric current relatively fast and they will cool down

rapidly by conduction to the surrounding parts. Devices can be heated and cooled

several times per second. Due to the thermal expansion of the material this effect

can be used to actuate a device at frequencies that are high compared to macroscopic

systems [22, 23, 27].

Another coupling that becomes useful at small scales is the coupling between

structural displacement through electrostatic forces. At these small scales electro-

static forces are big enough to actuate devices, which is the type of coupling dis-

cussed in this document. Two types of MEMS that utilize the electromechanical

S.D.A. Hannot

Delft University of Technology, Faculty of Mechanical Maritime and Materials Engineering,

Mekelweg 2, 2628 CD Delft, The Netherlands, e-mail: s.d.a.hannot@tudelft.nl

D.J. Rixen

Delft University of Technology, Faculty of Mechanical Maritime and Materials Engineering,

Mekelweg 2, 2628 CD Delft, The Netherlands, e-mail: d.j.rixen@tudelft.nl

315

Lecture Notes in Computational Science and Engineering 71,

DOI 10.1007/978-3-642-03344-5_11,

B. Koren and C. Vuik (eds.), Advanced Computational Methods in Science and Engineering,

S.D.A. Hannot and D.J. Rixen

Fig. 1 Micro bridge Fig. 2 Comb drive

+ +

- -

- - - - -

V = 0 V > 0 V ≫ 0

Fig. 3 Electromechanical coupling

coupling are micro switches and comb drives. Micro switches are the oldest type

of microsystems. They are similar to magnetic relay switches, but the magnetic ac-

tuation is replaced by electrostatic actuation thanks to the very small dimensions

[34, 44, 31]. Closely related to these devices is the micro mirror used in optics,

which is basically a large switch with a reﬂecting surface. A picture of a small mi-

cro bridge that can be used as micro switch is given in ﬁg. 1. The comb drive is an

array of small ﬁngers which can be used to maximize the electrostatic force between

a moving and a ﬁxed set of ﬁngers while still allowing a large displacement of the

moving set. A picture is shown in ﬁg. 2. Modeling this electro–mechanical coupling

is particularly challenging because the resulting equations are non-linear and the

physics are strongly coupled.

1.1 Electromechanical coupling

The basic physical principle of electromechanically actuated microsystems is il-

lustrated in ﬁg. 3. Once a potential difference between a moving (or deformable)

electrode and a ﬁxed electrode is applied a charge difference is created. Coulomb

forces between the charges will make the ﬂexible electrode deform. This effect de-

pends non-linearly on the shape of the electrode and the deformation itself, because

the charge will concentrate near the point where the gap between the electrodes is

smallest, which causes even higher Coulomb forces at this point [2].

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Numerical Modeling of the Electromechanical Interaction in MEMS

A coupled electromechanical model normally consists of two domains: a struc-

tural domain and an electric domain. The structural domain contains all moving

electrodes. The electric domain in principle contains the whole model, but if it is

assumed that the electrodes are ideal conductors the electrostatic equation has to be

solved only in the domain between the conductors. The applied potentials on the

electrodes can then be modeled as ﬁxed potential boundary conditions (Dirichlet

boundary conditions).

Once the ideal conductor assumption has been made and when it is furthermore

assumed that there are no free charges in the electrical domain, the equation that

describes the electrostatics in the air gap is the electrostatic equation [18, 25]:

∂



∂φ

∂



= 0 , (1)

where

is the electric permittivity in the gap and

(x,y,z) is the electric poten-

tial. Einstein’s summation convention will be used in this document as it was used

in this equation for i = x,y,z.

The static mechanical equilibrium equations, when assuming small deforma-

tions, can be written as [15]:

∂

+ F

= 0 , (2)

where

is the mechanical stress tensor and F the external force ﬁeld which incor-

porates all applied forces. For simplicity it is assumed that the only applied forces

are the Coulomb forces due to the electrostatic ﬁeld. When there are no moments

proportional to a volume, which is the case for most solid materials the stress tensor

is symmetric:

i j

. (3)

To compute the displacement caused by an applied load a relation between stress

and displacement u is required. This is done in two steps. First the strain tensor

is determined as a function of u and then the stress as a function of the strain.

These relations are in principle non linear but once it is assumed that the stresses

and strains remain relatively small the relations can be assumed to be linear

, which

yields the linear relations:

i j

= C

i jkl

, (4)

i j



∂



, (5)

where C is Hooke’s tensor which has constant coefﬁcients.

Thus the electric problem is linear and the mechanical problem is assumed to be

linear, however the coupled problem is non-linear. The non-linearity stems on one

Note that when modeling thin MEMS the non-linear mechanical equations need to be consid-

ered. For simplicity linear mechanics is assumed but all the concepts discussed in this text can be

extended to non-linear mechanics

317

S.D.A. Hannot and D.J. Rixen

hand from the fact that the electrostatic domain is deﬁned by the deformed position

of the electrodes, and on the other hand from the fact that the electrostatic forces

depend on the accumulated charges on the surfaces of the electrodes. There the

distributed force follows from Coulomb’s law [18]:

= qE

, (6)

where q is the distributed charge on the surface and E the electric ﬁeld generated by

the other electrode. In the vicinity of charges on the surface of a conductor one has

to note that half of the electric ﬁeld is generated by those charges themselves and

does therefore not contribute to the force. Hence, if E is the total electric ﬁeld in the

domain one must use F

= q(E

/2) [18].

According to Gauss’ law the surface charge q on a conductor can be written as a

function of the electric ﬁeld at the boundary and the unit normal vector n pointing

towards the electric domain:

q =

. (7)

The electric potential is deﬁned such that the electric ﬁeld is (minus) the gradient

of an electric potential

, therefore the force can be deﬁned as a function of this

potential [18]:



∂φ

∂

∂φ

∂



, (8)

where it has to be noted that the force is always pointing into the electric domain.

Again we stress that the non-linear coupling is caused by the non-linear dependency

of F on

and the fact that the electric domain is shaped by the structural displace-

ments.

1.2 A one-dimensional example

The consequences of this non-linear relation between structural displacements and

electrostatic forces can be explained best by a simple one-dimensional model. Fig-

ure 4 gives such a very simple model of a micro actuator. In an air gap, the electro-

static problem is described by Laplace’s equation, which reduces to the following

equation in 1D:

∂

∂ξ

= 0 , (9)

where

is the electrostatic potential,

is the spatial coordinate and

the electric

permittivity. The domain on which this differential equation has to be solved is

[0,x], x describing the position of the moving electrode. The boundary conditions

are:

= V at

= x and

= 0 at

= 0. The simple solution to this boundary value

problem is:

. (10)

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Numerical Modeling of the Electromechanical Interaction in MEMS

Fig. 4 1D micro resonator

The electric ﬁeld is then by deﬁnition:

E = −

∂φ

∂ξ

= −

, (11)

and the electrostatic force on the mass is:

. (12)

The mechanical force applied to the electrode arising from the spring is

mech

= k (x −x

) , (13)

with k the linear stiffness of the spring and x

the position when the spring is at its

natural length. The static equilibrium is found by summing all applied forces:

k(x −x

) +

= 0 , (14)

Clearly, to a given applied potential V corresponds an equilibrium position x and

vice-versa. It can be shown that there exists a voltage and corresponding displace-

ment for which

∂

= 0, thus for which the voltage as function of displacement

reaches a maximum. This voltage equals: V

8k x

. Using this maximum volt-

age to normalize equation (14) it can be written as:







−x



−x



−1



. (15)

From the two solutions obtained for this equation it can be shown that the solution

closest to the bottom conductor is unstable: at this position a small perturbation of

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S.D.A. Hannot and D.J. Rixen

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−x

Fig. 5 1D voltage-displacement curve

the moving electrode will make it either return to the equilibrium position close

to the undeformed position or smash against the ﬁxed electrode (the electrodes are

pulled together). The other solution is stable and can be used to perform a linearized

vibrational analysis. There exists a voltage above which there exist no more static

solutions, this voltage (equal to V

mentioned above) is called the pull-in voltage.

This is graphically indicated in ﬁg. 5. The pull-in voltage is the maximum of the

curve. The solid part of the curve gives stable solutions, the dashed part being un-

stable solutions.

The result in ﬁg. 5 also provides an indication about the real behavior of electro-

static microsystems. As mentioned the pull-in voltage is the maximum of the curve.

This means that when a voltage above this pull-in voltage is applied to the system,

the moving electrode will snap to the ﬁxed electrode and they will stick together,

possibly short-circuiting the device. Since this pull-in phenomenon is a very criti-

cal mode of operation, an important goal of MEMS modeling is the prediction of

reliable pull-in voltages and displacements [8, 12].

An important characteristic of pull-in is its similarity to limit point buckling.

Limit point buckling is deﬁned by a maximum point of the applied force in the force-

displacement relationship for a non-linear mechanical problem [6, 41]. A simple

version of limit point buckling is presented in ﬁg. 6. At this buckling point the

derivative of the applied load with the generalized coordinate is zero:

∂

∂α

= 0 , (16)

which is very similar to the deﬁnition of the pull-in point:

∂

= 0 . (17)

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Numerical Modeling of the Electromechanical Interaction in MEMS

Fig. 6 Limit point buckling of a two spring model actuated by force P [41]

There exist many numerical techniques designed to handle the speciﬁc problems of

limit point buckling. Several of those techniques can be adapted for pull-in compu-

tations as explained in the following sections.

1.3 Numerical modeling

The coupled equilibrium equations (1) and (2) are partial differential equations. To

solve these equations numerically they have to be discretized. Different discretiza-

tion techniques can be applied, but the mechanical equations (2) are almost always

handled with the Finite Element Method (FEM) for which abundant literature ex-

ists (e.g. [4, 10]). The electrostatic problem (1) is a Laplace equation which also

describes other physical ﬁelds such as thermics, potential ﬂuid ﬂow or membrane

displacements. Therefore scientists from several application ﬁelds have investigated

the Laplace equation. Structural engineers prefer to use the Finite Element Method

for solving this equation, whereas engineers from the ﬂuid mechanics ﬁeld typically

apply the Finite Volume Method (FVM) [42, 14] or Finite Difference schemes. The

Boundary Element Method (BEM) is also quite popular because it handles inﬁnitely

extending domains in a natural manner [26].

Once such a discretization procedure has been chosen the discretized equations

have the following form:

u = f

ext

(u,v) , (18)

(u)v = q(u,V) , (19)

where K

is the electric stiffness matrix, K

the mechanical stiffness matrix, v the

set of discretized potentials (nodal potentials for a FEM model), u the set of dis-

cretized displacements, q are applied charges (on the boundaries or in the electric

domain) and V the imposed potentials on Dirichlet boundaries. f

ext

are the applied

mechanical forces that include Coulomb forces obtained from eq. (8) or, alterna-

tively, by formulations based on variational calculus and total energy approaches.

The latter methods to construct the electrostatic forces are more efﬁcient when the

structure exhibits sharp corners [37, 21]. From the discretized equations (18) it can

be clearly seen that the non-linear electrostatic coupling originates on one hand from

321

S.D.A. Hannot and D.J. Rixen

a. Co-simulation b. Coupled simulation

ext

(

nodes with structural degrees of freedom

nodes with electric degrees of freedom

Fig. 7 Modeling strategies

the fact that the electrostatic forces applied on the structure are non-linear functions

of the potential ﬁeld, and on the other by the fact that the electrostatic domain (hence

the operator K

) depends on the structural response u.

Deﬁning the internal forces and internal charges as

int

= K

u , (20)

int

= K

(u)v , (21)

the coupled equilibrium equations can be written in the following monolithic form

[37]:

ext

(U,V) −F

int

(U) = 0 , (22)

where

U =





, (23)

F =





, (24)

There are two ways to obtain these sets of equilibrium equations: co-simulation

or fully coupled simulation.

In co-simulation separate structural and electric models are handled by separate

codes to solve the different physical domains, and the results are communicated

between the codes at some synchronization moments in the solution procedure: me-

chanical displacements are sent to the electric model to update its geometry and

electric potentials or ﬁelds are sent to the structural model to generate electrostatic

forces. This is schematically depicted ﬁgure 7a. An advantage of co-simulation is

that different discretization procedures can be used for the different domains and es-

tablished specialized software can be used for each physical domain. For instance a

322

Numerical Modeling of the Electromechanical Interaction in MEMS

structural FEM model and a separate electrostatic FEM model can be co-simulated

[24, 45], but it is also common in the literature to combine a structural FEM model

with electrostatic BEM approach [2, 32, 17]. Structural FEM models coupled to

electrostatic FVM models are rare, which is surprising because structural FEM mod-

els coupled with ﬂuid FVM models are very common [13], even structural FVM

with ﬂuid FVM does exist [39]. Since FVM handles singularities at corners bet-

ter than FEM, coupling FEM and FVM in electrostatics is probably an interesting

aspect for future research.

In fully coupled simulation the entire problem is handled in a single code: only

one mesh is used to model both domains as schematically explained in ﬁgure 7b. In

that case all the properties of the multiphysical problem are known in a single code,

which greatly simpliﬁes analyzing global properties such as eigenfrequencies for

instance (see section 4). An advantage of the coupled simulation is that it is much

easier to obtain a linearization of the coupled equations, which enables the creation

of very efﬁcient coupled solvers [37].

2 Solution techniques

Traditionally non-linear solvers rely on successive linear iterations to ﬁnd a solution

to (22). In this chapter the distinction will be made between solvers that solve the

coupled problem in a staggered manner and solvers that solve physics in monolithic

way.

A staggered solver basically solves one physical domain at a time. Figure 8

presents a simple diagram for a staggered solver: at iteration k ﬁrst the electrostatic

force is computed with the electric solution from step k −1, next the new displace-

ments are computed by the mechanical solver and sent to the electric solver, which

then can compute the new electrostatic solution.

ext

k−1

ext

k −1 k k + 1

Electric solver

Mechanic solver

Fig. 8 Staggered solution procedure

This scheme can be interpreted from a mathematical point of view as a block

Gauss-Seidel iteration and can be written as [3, 7]:

= K

−1

k−1

)q(u

k−1

,V) , (25)

323

S.D.A. Hannot and D.J. Rixen

= (K

)

−1

ext

k−1

) . (26)

This can be repeated until convergence. The algorithm is converged when the update

for the displacements is smaller than a speciﬁed tolerance (||u

−u

k−1

|| < tol).

It is clear that this algorithm can be implemented when the problem is handled

by co-simulation, that is when the structural and electrostatic equations are solved

in separate codes. But obviously the staggered approach can also be used when the

discretization is done in a fully coupled simulation. An advantage of a staggered

solver is that the structural and electrostatic problems taken separately are, for many

applications, linear. Also, for each subproblem, efﬁcient specialized solvers can be

used.

Monolithic solvers attempt to solve the fully coupled equations as a whole and

lead to speciﬁc solution procedures [7] (see ﬁgure 9). Clearly, monolithic solvers

need to have access to the fully coupled model and are not appropriate for co-

simulation strategies. So far mainly FEM-FEM coupling can be found in the lit-

erature [19, 37]. The advantage is that the convergence rate tends to be faster but,

the full equilibrium equation being non-linear, the monolithic approaches are typ-

ically using successive linearized problems and thus often require computing the

global tangent stiffness matrix K =

∂

k−1

k −1 k k + 1

Coupled solver

Fig. 9 Monolithic solution procedure

Using equations (18) and (19) the monolithic linearized problem can be written



(u,v) K

(u,v)

(u,v) K

(u)



∆u

∆v





∆f(u,v)

∆q(u,V )



, (27)

where

= K

∂

ext

(u,v)

∂

, (28)

∂

ext

(u,v)

∂

. (29)

It can be shown that the tangent matrix is symmetrical, namely that K

= K

[37].

Obtaining this tangent stiffness matrix from for instance the FEM discretization can

be intricate [19, 36, 37] and solving the linearized problem (27) is time consuming.

Without going in further speciﬁc details it can be seen that all four sub-matrices of

K depend on the state vector U and thus need to be updated regularly in the solution

process. Hence generally speaking one can state that monolithic approaches exhibit

good convergence rates and are robust for badly conditioned problems, but every

iteration incurs signiﬁcant computational effort.

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