MATHEMATICAL MODELING OF ELECTROSTATIC MEMS WITHTAILORED DIELECTRIC PROPERTIES∗

JOHN A. PELESKO†

SIAM J. APPL. MATH. c© 2002 Society for Industrial and Applied MathematicsVol. 62, No. 3, pp. 888–908

Abstract. The “pull-in” or “snap-down” instability in electrostatically actuated microelec-tromechanical systems (MEMS) presents a ubiquitous challenge in MEMS technology of great im-portance. In this instability, when applied voltages are increased beyond a critical value, there isno longer a steady-state configuration of the device where mechanical members remain separate.This severely restricts the range of stable operation of many devices. In an attempt to reduce theeffects of this instability, researchers have suggested spatially tailoring the dielectric properties ofMEMS devices. Here, a mathematical model of an idealized electrostatically actuated MEMS deviceis constructed and analyzed for the purpose of investigating this possibility. The pull-in instabilityis characterized in terms of the bifurcation diagram for the mathematical model. Variations in thisbifurcation diagram for various dielectric profiles are studied, yielding insight into how this techniquemay be used to increase the stable range of operation of electrostatically actuated MEMS devices.

Key words. MEMS, microelectromechanical system, exact shooting, semilinear elliptic problem

AMS subject classifications. 34, 35, 74, 78

PII. S0036139900381079

1. Introduction. The advent of microelectromechanical systems (MEMS) hasrevolutionized numerous branches of science and industry. Already firmly establishedas an essential component of modern sensors, such as those used for automobile air-bag deployment [1], MEMS are making inroads into areas as diverse as the biomedicalindustry [2], space exploration [3], and telecommunications [4].

Spurred by rapid advances in integrated circuit manufacturing, microsystem pro-cess technology is already well developed [5]. As a result, researchers are increasinglyfocusing their attention on device engineering questions. Foremost among these is thequestion of how to provide accurate, controlled, stable locomotion for MEMS devices.As has been recognized for some time [6], it is neither feasible nor desirable to attemptto reproduce modes of locomotion used in the macro world. In fact, the unfavorablescaling of force with device size prohibits this approach in many cases. For example,magnetic forces, which are often used for actuation in the macro world, scale poorlyinto the micro domain, decreasing in strength by a factor of ten thousand when lin-ear dimensions are reduced by a factor of ten [6]. This unfavorable scaling rendersmagnetic forces essentially useless. At the micro level, researchers have proposed avariety of new modes of locomotion based upon thermal, biological, and electrostaticforces. Each of these forces scales favorably as the linear dimensions of the systemare decreased.

Rapidly becoming the method of choice, the use of electrostatic forces to providelocomotion for MEMS devices is already employed in devices such as accelerometers[1], optical switches [7], microgrippers [8], micro force gauges [9], transducers [10], andmicro pumps [11]. In this approach voltage differences are applied between mechanicalcomponents of the system. This induces a Coulomb force between components which

∗Received by the editors November 13, 2000; accepted for publication (in revised form) July 24,2001; published electronically January 18, 2002. This work was supported by the National ScienceFoundation under NSF 0071474.

http://www.siam.org/journals/siap/62-3/38107.html†School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160 (pelesko@

math.gatech.edu).

888

MODELING OF ELECTROSTATIC MEMS 889

is varied in strength by varying the applied voltage. That electrostatic actuationsuffers from an instability, limiting its effectiveness, was known to even the earliestMEMS researchers such as Nathanson et al. [12], who in 1967 first identified anddescribed the ubiquitous “pull-in” voltage instability. In this instability, when appliedvoltages are increased beyond a certain critical voltage, there is no longer a steady-state configuration of the device where mechanical members remain separate. Thisinstability severely restricts the range of stable operation of many devices [1, 7, 8,9, 10, 11]. Consequently, the understanding and control of this instability presents achallenge of great technological importance.

Several approaches to overcoming the pull-in instability have been discussed inthe literature. Chu and Pister [8] proposed a voltage control algorithm, while Seegerand Crary [13, 14], Chan and Dutton [15], Pelesko and Triolo [16, 17], and Peleskoand Chen [18] studied capacitive control schemes. An alternative to the voltage orcapacitive control schemes, which both impose some form of external control, is todesign the device itself for greater stability. This may involve structuring the shape ofthe device [19], embedding fixed charges within the device, or tailoring the dielectricproperties of device components.

In this paper, we explore the idea of using tailored dielectric properties of thedevice components to ameliorate the effects of the pull-in voltage instability. In par-ticular, we model the device shown in Figure 2.1, which consists of an elastic stripsuspended above a rigid plate. The strip is held fixed at two ends and remainsunsupported along the other edges. This structure is a common MEMS buildingblock. It is easy to fabricate using a variety of materials and is particularly ver-satile. Examples of the fabrication and use of this structure appear in references[20, 21, 22, 23, 24, 25, 26, 27]. A voltage difference is applied across the device in or-der to cause mechanical deflection. In particular, the upper surface of the membraneis held at potential V , while the ground plate is held at zero potential. In our model,we treat the deformable elastic membrane as a dielectric of finite thickness. We notethat this is in contrast to almost all analytical studies of electrostatically actuatedMEMS devices [16, 28, 15, 13, 12, 19]; the standard approach is to model MEMScomponents as perfect conductors. However, as was pointed out by Price, Wood, andJacobsen [29], few MEMS components are perfectly conducting. Hence, even in thecase of constant dielectric properties, our model yields an improvement over otheranalytical studies. Treating the membrane as a deformable dielectric does, however,introduce an analytical complication. In order to solve the electrostatic problem ex-actly, it is now necessary to solve coupled Laplace equations for a rather arbitrarycurved pair of domains. Closed form solutions are, of course, unavailable. Here weavoid this complication by exploiting the fact that most MEMS devices are of smallaspect ratio, l/L 1 in Figure 2.1, and use thin components, d/l 1 in Figure 2.1,to derive an approximate solution.

Next we use our approximate solution for the electrostatic fields in the equationgoverning the mechanical motion of our deformable membrane. We treat the finitethickness deformable dielectric as an infinitely thin elastic membrane. While thisapproximation will not be valid for all MEMS devices, it will be valid for many, such asmicropumps made of thin glassy polymers [11, 19] or devices such as the grating lightvalve (GLV), composed of thin ribbons held under tension [31]. Also, it will be a goodapproximation for devices whose thickness is many times less than their deflection[30], such as those discussed in [8, 27, 32, 33, 34, 35]. Further, even for devices forwhich bending stresses cannot be ignored, we expect our model to yield qualitative

890 JOHN A. PELESKO

z'

x'

y'

GroundPlate

Thickness d

Membrane

z'=u'(x')FixedEdge

FixedEdge

Free Edge

Free Edge

L

w

l

Fig. 2.1. Geometry of our idealized MEMS device.

if not quantitative understanding. When the approximate electrostatic potential andthe elastic membrane theory are combined, we arrive at a single semilinear ellipticequation governing the deflection of the elastic membrane. In section 3, we presentsome general results about this equation. We prove, for physically realistic dielectricprofiles, that the pull-in voltage instability cannot be avoided. We develop an existencetheory and use this to obtain rigorous bounds for the pull-in voltage.

Next we ask what can be achieved by tailoring the dielectric properties. In section4, we present a detailed analysis of the strip model for power law dielectric propertyprofiles. We show that the choice of profile can have a profound effect on the numberof solutions, the pull-in voltage, and the stable range of operation of the device. Thisanalysis is repeated for a disk shaped device. Finally, we show how our analysis maybe used to produce design curves where the engineer may choose a dielectric profileto balance stable operating range against power requirements.

2. Formulation and simplification of the model. In this section we presentand simplify the governing equations for the behavior of our idealized electrostaticallyactuated MEMS device. We begin by formulating the equations governing the elec-trostatic field surrounding and within device components. Our device consists of athin elastic membrane suspended above a rigid plate. A potential difference is appliedbetween the top surface of the membrane and the plate. Both membrane and platehave width w, length L, and in the undeformed state are separated by a gap of lengthl. The geometry is sketched in Figure 2.1. The elastic membrane is assumed to bea dielectric material of uniform thickness 2d. The ground plate, located at z′ = 0, isassumed to be a perfect conductor. The deflection of the membrane, which is assumedto be a function of x′ alone, is specified by the deflection of its center plane, locatedat z′ = u′(x′). Hence the top surface is located at z′ = u′(x′)+d, while the bottom ofthe membrane is located at z′ = u′(x′)− d. With these assumptions in mind, we first

MODELING OF ELECTROSTATIC MEMS 891

formulate the equations governing the electrostatic potential surrounding the deviceand within the membrane. The potential between the membrane and ground plate,Ψ1, satisfies

′Ψ1 = 0,(2.1)

Ψ1(x′, y′, 0) = 0, x′ ∈ [−L/2, L/2], y′ ∈ [−w/2, w/2],(2.2)

where the prime on the Laplace operator denotes derivatives with respect to primedvariables. The potential inside the membrane, Ψ2, satisfies

∇′ · (ε2(x′)∇′Ψ2) = 0,(2.3)

Ψ2(x′, y′, u′(x′) + d) = V, x′ ∈ [−L/2, L/2], y′ ∈ [−w/2, w/2],(2.4)

where here ε2(x′) is the permittivity of the material comprising the membrane. It

is assumed to be a function of x′ and is related to the dielectric constant of themembrane, K2, through ε2(x

′) = ε0K2, where ε0 is the permittivity of free space. Atthe interface between the gap and the membrane, the tangential electric fields andnormal displacement fields must be continuous. Hence we impose

Ψ1(x′, y′, u′(x′)− d) = Ψ2(x

′, y′, u′(x′)− d),(2.5)

ε2(x′)∇′Ψ2 · n = ε0∇′Ψ1 · n at z′ = u′(x′)− d.(2.6)

Here n is an outward pointing unit normal.Next we introduce dimensionless variables and rewrite the equations governing

the electrostatic field in dimensionless form. We define

ψ1 = Ψ1/V, ψ2 = Ψ2/V, u = u′/l, x = x′/L, y = y′/w, z = z′/l(2.7)

and substitute these into (2.1)–(2.6). This yields

l2

L2

(∂2ψ1

∂x2+

L2

w2

∂2ψ1

∂y2

)+

∂2ψ1

∂z2= 0,(2.8)

ψ1(x, y, 0) = 0, x ∈ [−1/2, 1/2], y ∈ [−1/2, 1/2],(2.9)

l2

L2

(∂

∂xε2(x)

∂ψ2

∂x+ ε2(x)

L2

w2

∂2ψ2

∂y2

)+ ε2(x)

∂ψ2

∂z2= 0,(2.10)

ψ2(x, y, u+ d/l) = 1, x ∈ [−1/2, 1/2], y ∈ [−1/2, 1/2],(2.11)

ψ1(x, y, u− d/l) = ψ2(x, y, u+ d/l), x ∈ [−1/2, 1/2], y ∈ [−1/2, 1/2],(2.12)

l2

L2

(ε2(x)

∂ψ2

∂x− ε0

∂ψ1

∂x

)−(ε2(x)

∂ψ2

∂z− ε0

∂ψ1

∂z

)= 0 at z = u− d/l.(2.13)

892 JOHN A. PELESKO

Next we note that the square of the aspect ratio, l/L, appears as a nondimensionalparameter in our scaled equations, (2.8)–(2.13). For typical devices, this is a smallparameter [8, 31], and hence we simplify our system by ignoring terms of order l2/L2.In engineering parlance, we are ignoring fringing fields. With this simplification,(2.8)–(2.13) reduce to a pair of linear coupled ordinary differential equations that arereadily solved to yield

ψ1 =K2z

(u(x) + δ) + (K2 − 1)(u(x)− δ),(2.14)

ψ2 =z + (u(x)− δ)(K2 − 1)

(u(x) + δ) + (K2 − 1)(u(x)− δ),(2.15)

where δ = d/l and K2 = ε2/ε0 is the dielectric constant of the deformable membrane.If we make the additional assumption that the membrane is thin compared to the sizeof the gap, i.e., δ 1, our approximate solutions further simplify to

ψ1 =z

u(x),(2.16)

ψ2 =z + (K2 − 1)u(x)

K2u(x).(2.17)

Next we turn our attention to the elastic part of the problem. As discussed in theintroduction, we model our deformable dielectric slab as an elastic membrane whosedeflection is a function of x′ alone and is governed by

Td2u′

dx′2 =ε22|∇′Ψ2|2 ,(2.18)

where T is the tension in the membrane and the gradient of Ψ2 is to be evaluatedat z′ = u′(x′). Using the same nondimensionalization introduced in (2.7) and using(2.17) to evaluate the gradient, we simplify this to

d2u

dx2=

λf(x)

u(x)2,(2.19)

where

λ =V 2L2ε02T l2

(2.20)

and

f(x) =1

K2(x).(2.21)

The parameter λ is a dimensionless number which characterizes the relative strengthsof electrostatic and mechanical forces in the system. As λ is proportional to theapplied voltage, it serves as a convenient bifurcation parameter. We can alreadycharacterize the pull-in instability in terms of λ. If solutions to our model equationsfail to exist, then the device must be in a collapsed or pulled-in state. Hence char-acterizing existence of solutions in terms of λ and studying the pull-in instability are

MODELING OF ELECTROSTATIC MEMS 893

equivalent. We note that λ ≥ 0 is the physically relevant regime. Finally, as per ourassumption that the left and right edges of the membrane are fixed, we impose theboundary conditions

u(−1/2) = u(1/2) = 1.(2.22)

Equations (2.19)–(2.22) govern the behavior of our idealized MEMS device with tai-lored dielectric properties. Throughout the remainder of this paper, we shall studythe model equations (2.19)–(2.22) and generalizations thereof.

3. General properties. In deriving our model, (2.19), (2.22), we carried outour analysis with respect to the strip geometry shown in Figure 2.1. In section 4,we will present a detailed analysis of this one-dimensional (1D) model for the stripgeometry. However, it is easy to see that the derivation of the reduced model, (2.19),(2.22), was not strongly dependent on the choice of geometry. In fact, if we keptthe small aspect ratio assumption, l/L 1, and the thin membrane assumption,d/l 1, but relaxed the requirement that the deflection be a function of x alone andthat the geometry be that of a rectangular strip, the analysis of the previous sectiongoes through with little alteration. Omitting the details, we present

u =λf(x, y)

u2in Ω ⊂ R2,(3.1)

u = 1 on ∂Ω,(3.2)

as the obvious generalization of our model to planar arbitrarily shaped electrostaticallyactuated MEMS devices with spatially tailored dielectric profiles. For convenience,we change variables u → 1+u and restate with the homogeneous boundary conditionas

u =λf(x, y)

(1 + u)2in Ω ⊂ R2,(3.3)

u = 0 on ∂Ω.(3.4)

We assume that Ω is a bounded domain with smooth boundary. The boundarycondition, (3.4), has been stated in its simplest form; extensions to allow free orelastically supported portions of the boundary are straightforward and will not bediscussed.

We offer the generalization above because the results of this section are as easilystated and proved for (3.3)–(3.4) as they are for (2.19), (2.22). Before stating theseresults, we recall that f(x, y) = 1/K2. That is, f is the reciprocal of the dielectricconstant of our deformable membrane. If the membrane is assumed to be a dielectricmaterial everywhere, then f must satisfy

1 ≥ f(x, y) ≥ c > 0 ∀(x, y) ∈ Ω.(3.5)

If parts of the membrane are allowed to be perfectly conducting, then this conditionmay be relaxed by taking c = 0. Throughout the remainder of this section, we assumethat f satisfies the inequality (3.5). With this assumption in mind, we may prove thefollowing theorem.

894 JOHN A. PELESKO

Theorem 3.1. There exists a λ∗ such that no solution to (3.3)–(3.4) exists forλ > λ∗.

Proof. Let α1 be the lowest eigenvalue of

−u = αu on Ω,(3.6)

u = 0 on ∂Ω,(3.7)

with w1 the associated eigenfunction. It is well known that α1 is simple and that w1

may be chosen strictly positive in Ω. Now we rewrite (3.3) as

−u− α1u = −λf(x, y)

(1 + u)2− α1u.(3.8)

The solvability condition for this problem is then

∫Ω

(λf(x, y)

(1 + u)2+ α1u

)w1 = 0.(3.9)

Since w1 is strictly positive, the term in parentheses either must be identically zero,or it must change sign. That it is not zero is clear. Hence we are led to considerλf(x,y)(1+u)2 +α1u. If this expression is to change sign, at some u, we must have λf(x,y)

(1+u)2 =

−α1u. If this is not possible when f achieves its minimum, it is not possible at all.Hence we plot each side of the expression λc

(1+u)2 = −α1u as a function of u. An

easy calculation then reveals that as λ is increased beyond 4α1

27c the two curves do not

intersect, and hence no solution exists for λ > 4α1

27c .We note that the proof is standard and the argument is similar to that used to

prove the existence of “blow-up” in reaction diffusion systems, nonexistence in theBratu problem, etc. [36]. To stress the importance of this result here, we state as acorollary its physical counterpart.

Corollary 3.2. No choice of profile for the dielectric constant f is possiblewhich completely removes the pull-in instability.

That is, the nonexistence of solutions to (3.3)–(3.4) corresponds to the devicebeing in a collapsed or “pulled-in” state. What we have shown is that regardless ofspatial tailoring of the dielectric constant, f(x, y), there is a voltage, λ∗, called thepull-in voltage, beyond which no uncollapsed states exist. Hence the pull-in instabilitypersists.

Now it is also useful to have an existence result and a bound on the pull-in voltage.We use the well-known method of upper and lower solutions [37, 38, 39], where wedefine the following.

Definition 3.3. The function u ∈ C2(Ω) is called an upper solution if

−u ≥ −λf(x, y)

(1 + u)2in Ω,(3.10)

u ≥ 0 on ∂Ω.(3.11)

The function u ∈ C2(Ω) is called a lower solution if the opposite inequalities aresatisfied.

An upper solution is easy to find as is shown in the following lemma.

MODELING OF ELECTROSTATIC MEMS 895

Lemma 3.4. Any positive constant, α > 0, is an upper solution for all λ ≥ 0.Proof. The proof is obvious.Now, to construct a lower solution, we first define the domain Ω′ as a bounded

domain with smooth boundary which contains Ω as a proper subset. Next considerthe eigenvalue problem

−u = µu on Ω′,(3.12)

u = 0 on ∂Ω′(3.13)

on this enlarged domain, and let µ1 be the lowest eigenvalue with w1 as the associatedeigenfunction. Notice that since Ω is a proper subset of Ω′, w1 may be chosen strictlypositive on Ω. Now we attempt to construct a lower solution of the form Aw1, whereA is a scalar. We need Aw1 ≤ 0 on ∂Ω, which means we must require A < 0. Nextwe need

−(Aw1) ≤ − λf(x, y)

(1 +Aw1)2(3.14)

to hold in Ω. However, since the Laplacian is a linear operator and w1 is an eigenvec-tor, we may rewrite this requirement as

−µ1Aw1 − λf(x, y)

(1 +Aw1)2≥ 0.(3.15)

There are two difficulties with satisfying the inequality (3.15). First, we must ensurethat 1 +Aw1 > 0. Suppose w1 is normalized so that its maximum over Ω is one, andlet m be its minimum over Ω. We can satisfy 1+Aw1 > 0 by satisfying 1+A > 0 or,equivalently, A > −1. So thus far we must choose A such that 0 > A > −1. Now westill must choose A so that (3.15) is satisfied. It is sufficient to satisfy

−µ1Am− λ

(1 +Am)2≥ 0.(3.16)

However, it is easy to see by graphical analysis that this inequality may be satisfiedfor a range of λ by choosing A such that −1 < A < −1/(3m) provided m > 1/3.Now it is clear that the domain Ω′ must be chosen so that the minimum of the firsteigenfunction, w1, is greater than 1/3 over Ω. Finally, an easy calculation showsthat this choice of Ω′ and A leaves the inequality (3.16) satisfied for λ ≤ 4

27µ1. Tosummarize, we have shown the following.

Lemma 3.5. There exists a constant A such that the function Aw1 is a lowersolution for all λ ≤ 4

27µ1.Now, using Lemmas 3.4 and 3.5 and the well-known theorem that a solution exists

between an ordered pair of upper and lower solutions, we obtain the existence result.Theorem 3.6. There exists a solution to (3.3)–(3.4) for all λ ≤ 4

27µ1.As an important corollary, by combining Theorem 3.6 and the proof of Theorem

3.1, we obtain bounds on the pull-in voltage.Corollary 3.7. The pull-in voltage for (3.3)–(3.4), λ∗, satisfies 4

27µ1 ≤ λ∗ ≤4

27cα1.

896 JOHN A. PELESKO

4. Analysis for power law profiles. The analysis of the previous section es-tablished that, regardless of how the dielectric constant profile is chosen, the pull-ininstability cannot be avoided. However, as a practical question it is reasonable toask: What can be accomplished by tailoring the profile? That is, we would like tounderstand how changes in the dielectric profile affect the structure and multiplicityof solutions, the pull-in voltage, and the stable range of operation of the device. To-ward this end, we investigate solutions to our model equations, (2.19)–(2.22), for thespecial case when f is chosen to satisfy a symmetric power law. The motivation forinvestigating this choice for f is physical. Heuristically speaking, the “most unstable”portion of our device is its center, where the influence of the electrostatic field is strongand the influence of the supporting boundaries are weak. Hence it is reasonable toattempt to minimize the electric field in this region. Conversely, it seems reasonableto allow the electric field to be strong near the “most stable” portion of the device,i.e., near the boundaries. With this motivation for choosing a power law profile for f ,we carry out the analysis for both our original strip geometry as well as a disk shapedgeometry.

4.1. Analysis of the strip for f(x) = |x|α. We begin by setting f(x) = |x|αin (2.19)–(2.22). We assume α ≥ 0. For the convenience of the reader, we restate theproblem

d2u

dx2=

λ|x|αu2

,(4.1)

u(−1/2) = u(1/2) = 1.(4.2)

A few elementary properties of solutions to (4.1)–(4.2) follow almost by inspection.First, we see that the equation is invariant under the transformation x → −x, andhence the solutions are symmetric; i.e., u(x) = u(−x). Second, since the secondderivative is everywhere positive for positive λ, the solution must be concave upeverywhere or, more succinctly, convex. Combining the convexity result with thesymmetry result implies that u(x) ≥ u(0) for all x ∈ [−1/2, 1/2]. Finally, since thesolution must be one on the boundaries and remain convex, we have that u(x) ≤ 1for all x ∈ [−1/2, 1/2] as well. We summarize these results in the following theorem.

Theorem 4.1. Any smooth solution u ∈ C2[−1/2, 1/2] to (4.1)–(4.2) satisfies(i) u(x) = u(−x),(ii) u is convex,(iii) u(x) ≤ 1 for all x ∈ [−1/2, 1/2].Now, reflectional symmetry allows us to replace the problem (4.1)–(4.2) with the

problem

d2u

dx2=

λxα

u2,(4.3)

du

dx(0) = 0,(4.4)

u(1/2) = 1.(4.5)

This reduces the interval to [0, 1/2] and provides a homogenous boundary conditionat x = 0. Next we observe that (4.3) contains an additional symmetry. In particular,

MODELING OF ELECTROSTATIC MEMS 897

this equation is invariant under the one parameter stretching group of transformations

u∗ = eεu,(4.6)

x∗ = exp

(3ε

2 + α

)x.

Such an invariance implies that our boundary value problem may be converted toan initial value problem. This technique, which is well known in fluid dynamics, issometimes called exact shooting and was first introduced by Toepfer [40] in 1912 inconnection with the Blasius problem. In the 1960’s and 1970’s it was extended to awide class of problems primarily through the work of Klamkin and Na. The interestedreader is referred to Chapter 7 of [41] for a discussion and full list of references. Herewe proceed by noting that u(x) is a solution of (4.3)–(4.5) if and only if

u(x) = aw(η),(4.7)

where

η = bx,(4.8)

a =1

w(b/2),(4.9)

λ =b2+α

w(b/2)3,(4.10)

and w satisfies

d2w

dη2=

ηα

w2,(4.11)

w(0) = 1,(4.12)

dw

dη(0) = 0.(4.13)

With these identifications, the bifurcation diagram for our boundary value problem,(4.3)–(4.5), is parameterized in terms of b and w. That is, to understand solutions, wewish to plot λ versus ||u|| for some suitable norm. From Theorem 4.1, it is easy to seethat the maximum deflection will occur at x = 0, and hence the maximum or infinitynorm is particularly convenient. That is, we may write ||u||∞ = 1 − u(0). Then,we see that both λ and ||u||∞ are parameterized in terms of b and w. In particular,λ is parameterized through (4.10), and ||u||∞ is parameterized by observing that||u||∞ = 1− u(0) = 1− 1

w(b/2) .

It is easy to numerically integrate the initial value problem (4.11)–(4.13) and usethe result to compute the complete bifurcation diagram for (4.3)–(4.5). The result ofsuch a computation is shown in Figure 4.1. We note that as η → ∞, w and hence λand u change very little. This makes it difficult to compute the bifurcation diagramas it approaches the barrier ||u||∞ = 1. Also, as α is varied, we notice that the

898 JOHN A. PELESKO

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

||u|| ∞

α=0.5

α=1

α=2

Fig. 4.1. Bifurcation diagrams for the strip model for various values of α in the dielectricconstant profile, f(x) = xα.

multiplicity of solutions changes. Hence it is useful to further study the equations forw analytically. We follow a procedure similar to that used by Joseph and Lundgren[42] or Keller and Fier [43]. In fact, our analysis in this subsection and the nextrepresents a partial generalization of Joseph and Lundgren [42], who studied radialsolutions of (3.1)–(3.2) in n dimensions, n > 2, for the special case when f(x, y) = 1.

To proceed, we change variables in (4.11) by setting ζ = log(η) and w(η) = η2+α

3 v(ζ).This yields

d2v

dζ2+

(2α+ 1)

3

dv

dζ+

(2 + α)(α− 1)

9v =

1

v2.(4.14)

We reduce this to a first order system by defining p = 1/v and q = v′/v. This yields

dp

dζ= −pq,(4.15)

dq

dζ= −q2 − (2α+ 1)

3q − (2 + α)(α− 1)

9p3.(4.16)

The initial conditions on w, (4.12)–(4.13), become

limζ→−∞

p(ζ) = 0(4.17)

and

limζ→−∞

q(ζ) = − (2 + α)

3.(4.18)

Now our first order system for p and q, (4.15)–(4.16), possesses three critical points,

MODELING OF ELECTROSTATIC MEMS 899

q

p

q q

p p

A

B

C

A

B,C

A

B

C

α < 1 α = 1 α > 1

Fig. 4.2. Sketch showing motion of critical points in the p − q plane as α is varied. Note theexchange of stability as α passes through one.

which we label A, B, and C:

A : p = 0, q = − (2 + α)

3,

B : p = 0, q =1− α

3,

C : p =

((2 + α)(α− 1)

9

)1/3

, q = 0.

The critical point A coincides with the conditions at negative infinity, (4.17)–(4.18).Linearizing about this point, we find that the linear system has eigenvalues

µ1 = 1,(4.19)

µ2 =(2 + α)

3

and hence is an unstable node. Our goal for this analysis is to determine the asymp-totic behavior of w(η) as η → ∞. This will allow us to investigate the bifurcationdiagram for our original problem (4.3)–(4.5) through (4.10) and our expression for||u||∞. In our current variables, this implies that we wish to determine the asymp-totic behavior of p(ζ) as ζ → ∞ for the curve in the p− q phase plane satisfying theconditions (4.17)–(4.18). The solution which does this emanates from the unstablenode located at the point A and lives in the right half of the p− q phase plane. Therestriction to the right half plane is easy to see as it is clear from (4.11)–(4.13) that wand hence p must be positive. In Figure 4.2 we have sketched our three critical pointsand their motion as α is varied in the p − q phase plane. For α ≤ 1, the only pointin the phase plane which can determine the large ζ asymptotics for p is the point B.Linearizing about B, we find that the linear system has eigenvalues

µ1 = −1,(4.20)

µ2 =(α− 1)

3

and hence for α < 1 is a stable node. So, for α < 1, the trajectory we want is onewhich leaves point A as ζ → −∞, wanders in the right half plane, and approachespoint B as ζ → ∞. The large ζ asymptotics for p are now easy to obtain. We find

p(ζ) ∼ c0 exp

((α− 1)

3ζ

)+ c1 exp(−ζ) + o(exp(−ζ)) as ζ → ∞.(4.21)

900 JOHN A. PELESKO

This allows us to deduce the large η asymptotics for w(η), and we find

w(η) ∼ 1

c0η + o(1) as η → ∞.(4.22)

So, for α < 1, we may deduce the behavior of the bifurcation diagram for (4.3)–(4.5).First, from

||u||∞ = 1− 1

w(b/2)(4.23)

and (4.22), we see that ||u||∞ → 1 as we monotonically let b → ∞, while from (4.10)we see that λ → 0 as b → ∞. Finally, while α < 1, it is easy to see that, for trajectoriesemanating from point A, p is monotonically increasing while q < 0 and monotonicallydecreasing while q > 0. Hence we may also conclude that for α < 1 our boundaryvalue problem, (4.3)–(4.5), has two, one, or no solutions.

Next we turn our attention to the case when α > 1. In Figure 4.2, we see thatat α = 1 the points B and C have coalesced. As α is increased beyond one, C movesinto the right half plane. Further, the point B, which for α < 1 was a stable node,changes to a saddle point by (4.21). Since it must be stable along the line p = 0 from(4.15)–(4.16), its local behavior is as shown in Figure 4.2. This implies that the pointB is no longer the attractor for trajectories leaving the point A at ζ = −∞. Hencethe large ζ asymptotics for p must now be determined by the point C. Linearizingabout C, we find that the linear system has eigenvalues

µ1,2 = − (2α+ 1)

6± 1

6

√−8α2 − 8α+ 25.(4.24)

Hence for 1 < α ≤ −12 +

12

√272 the point C is a stable node, while for α > − 1

2 +12

√272

the point C is a stable spiral. We may now obtain the large ζ behavior of p(ζ) forα > 1. We find

p(ζ) ∼((2 + α)(α− 1)

9

)1/3

+ c0eµ1ζ + c1e

µ2ζ + o(e−(2α+1)

6 ζ) as ζ → ∞(4.25)

and hence

w(η) ∼(

9

(2 + α)(α− 1)

)1/3

η(2+α)/3 + o(1) as η → ∞.(4.26)

So, for α > 1, we may deduce the behavior of the bifurcation diagram for (4.3)–(4.5).First, from

||u||∞ = 1− 1

w(b/2)(4.27)

and (4.22), we again see that ||u||∞ → 1 as we monotonically let b → ∞, while from

(4.10) we see that λ → 22+α(2+α)(α−1)9 as b → ∞. Further, when α > − 1

2 + 12

√272 ,

the oscillatory behavior of p and hence w implies that the curve of solutions must

oscillate infinitely many times as it heads to the point λ → 22+α(2+α)(α−1)9 , ||u||∞ = 1.

In particular, for λ = 22+α(2+α)(α−1)9 , α > − 1

2 + 12

√272 , our boundary value problem,

(4.3)–(4.5), has infinitely many solutions. A close-up of the bifurcation diagram near||u||∞ = 1 for α = 2 is shown in Figure 4.3.

MODELING OF ELECTROSTATIC MEMS 901

32 33 34 35 36 37 38 39 40

0.94

0.95

0.96

0.97

0.98

0.99

1

λ

||u|| ∞

Fig. 4.3. Close-up of bifurcation diagram for the strip model showing multiple folds. Hereα = 2 in the dielectric constant profile, f(x) = xα.

4.2. Analysis of the disk for f(r) = rα. The analysis carried out in theprevious subsection for the strip geometry may be easily repeated for a disk shapeddevice. Since this geometry is also canonical, we briefly carry out such an analysishere. In particular, we study

d2u

dr2+

1

r

du

dr=

λrα

u2,(4.28)

du

dr(0) = 0,(4.29)

u(1) = 1,(4.30)

which is our generalized model, (3.1)–(3.2), taken on a circular disk of unit radius.Further, cylindrical symmetry and the power law profile, f(r) = rα, are assumed. Weimmediately reduce to an initial value problem by noting that u(r) is a solution to(4.28)–(4.30) if and only if

u(r) = aw(η),(4.31)

where

η = br,(4.32)

a =1

w(b),(4.33)

902 JOHN A. PELESKO

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

||u|| ∞

α=1 α=2

α=3

Fig. 4.4. Bifurcation diagrams for the disk model for various values of α in the dielectricconstant profile, f(r) = rα.

λ =b2+α

w(b)3,(4.34)

and w satisfies the initial value problem

d2w

dη2+

1

η

dw

dη=

ηα

w2,(4.35)

w(0) = 1,(4.36)

dw

dη(0) = 0.(4.37)

With these identifications the bifurcation diagram for (4.28)–(4.30) is parameterizedin terms of b and w. In particular, λ is parameterized through (4.34) and ||u||∞ isparameterized through

||u||∞ = 1− u(0) = 1− 1

w(b).(4.38)

Again, it is easy to sketch the complete bifurcation diagram for (4.28)–(4.30) bynumerically solving the initial value problem (4.35)–(4.37). This is done in Figure4.4. This time, we observe that for all values of α ≥ 0 many solutions may exist. Amore detailed analysis is in order. As in the previous subsection, we change variablesby setting ζ = log(η) and w(η) = η(2+α)/3v(ζ) to obtain

d2v

dζ2+

2(2 + α)

3

dv

dζ+

(2 + α)2

9v =

1

v2.(4.39)

MODELING OF ELECTROSTATIC MEMS 903

We then reduce to a first order system by setting p = 1/v and q = v′/v, which yields

dp

dζ= −pq,(4.40)

dq

dζ= −q2 − 2(α+ 2)

3q − (2 + α)2

9+ p3.(4.41)

The initial conditions on w, (4.36)–(4.37), become

limζ→−∞

p(ζ) = 0(4.42)

and

limζ→−∞

q(ζ) = − (2 + α)

3.(4.43)

Now this time our first order system for p and q, (4.40)–(4.41), possesses only twocritical points, which we label A and B:

A : p = 0, q = − (2 + α)

3,

B : p =(2 + α)2/3

91/3, q = 0.

Once again, the critical point A coincides with our conditions at negative infinity.Linearizing about A reveals that the linear system has eigenvalues

µ1 = 0,(4.44)

µ2 =(2 + α)

3.

Despite the zero eigenvalue, we can deduce that A is an unstable node. To do so, wesimply observe that, along the line p = 0, (4.40)–(4.41) reduces to an ODE for q alone

with trajectories heading away from q = − (2+α)3 . Hence the local behavior about A is

as pictured in Figure 4.5. Now we wish to deduce the behavior of p as ζ → ∞. FromFigure 4.5, we see that the critical point B lives in the right half plane for all α ≥ 0.Linearizing about B reveals that the linear system has eigenvalues

µ1,2 = − (2 + α)

3± 2

√2(2 + α)i

3(4.45)

and hence is always a stable spiral. So for all α the trajectory of interest is onewhich leaves the point A as ζ → −∞ and tends toward B as ζ → ∞. The large ζasymptotics for p are now easy to deduce. We find

p(ζ) ∼ (2 + α)2/3

91/3+ c0e

(µ1ζ) + c1e(µ2ζ) + o(e(−

(2+α)3 ζ)) as ζ → ∞(4.46)

and hence

w(η) ∼ 91/3

(2 + α)2/3η(2+α)/3 + o(1) as η → ∞.(4.47)

Finally, we may deduce the behavior of the bifurcation diagram for (4.28)–(4.30).From (4.38) and (4.47) we see that ||u||∞ → 1 as we monotonically let b → ∞, while

from (4.34) we see that λ → (2+α)2

9 as b → ∞. Further, since B is a stable spiral forall values of α, the curve of solutions will always oscillate. In particular, this implies

that (4.28)–(4.30) will have infinitely many solutions for λ = (2+α)2

9 and any value ofα ≥ 0.

904 JOHN A. PELESKO

q

p

A

B

q

p

A

B

α=0 α>0

Fig. 4.5. Sketch showing critical points in the p − q plane for disk analysis and their motionas α is varied.

5. Discussion. We set out with the goal of exploring the idea of using tailoreddielectric properties of MEMS components to control or ameliorate the effects ofthe pull-in voltage instability. Our first step along this path was to construct amathematical model of an idealized electrostatically actuated MEMS device. Themodel consisted of two parts. First, the electrostatic potential on and within the devicewas assumed to satisfy Laplace’s equation with the appropriate boundary conditions.In contrast to previous analytical studies of electrostatic MEMS devices, our movablecomponent was treated as a dielectric material rather than as a perfect conductor.Next, we noted that the potential equations were coupled to the shape of the movablecomponent. This precluded an exact solution to the coupled Laplace equations. Sowe relied upon the fact that most MEMS devices are of small aspect ratio and arecomprised of thin parts in order to obtain an approximate solution. In particular,we ignored fringing fields. We note that the accuracy of this approximation hasbeen explored in [16, 17] by direct comparison with a full numerical solution of theLaplace equation. It was found that, for even moderately small aspect ratios, fringingfields may be neglected. The second part of our model consisted of the equationsgoverning the elastic behavior of the deformable membrane. Here we assumed thatthe rectangular plate was thin, that bending stresses could be ignored, and thatapproximating deflections by those of a membrane under tension was sufficient. Onceagain, we note that this will not be valid for every electrostatically actuated MEMSdevice; variety precludes this. However, it will be valid for many key devices such asmicropumps [11, 19] and should serve as a reasonable first approximation for deviceswhose displacement is large compared to the membrane thickness [30].

Our next step was to consider a generalized form of our model and to establishboth nonexistence and existence results. Here, establishing that solutions fail to existbeyond some critical value of the dimensionless applied voltage λ was tantamountto proving that no spatial tailoring of the dielectric profile could completely removethe pull-in instability. Further, using the method of upper and lower solutions, weestablished existence for a range of λ. This allowed us to obtain rigorous bounds onthe pull-in voltage.

MODELING OF ELECTROSTATIC MEMS 905

0 1 2 3 4 5 6 7 8 9 100.35

0.36

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

α

Pul

l-in

Dis

tanc

e

Fig. 5.1. A sample design curve showing the nondimensional pull-in distance as a function ofthe exponent, α, in the dielectric constant profile.

Then, recognizing that pull-in could not be completely eliminated, we set out toexplore what could be accomplished via spatial tailoring of the dielectric constant. To-ward this end, we investigated solutions for our 1D strip geometry when the dielectricconstant spatially varied according to a power law. The motivation for consideringthis profile was that it reduced the effect of the Coulomb force in regions where theelastic force was small. Hence we expected a stabilizing influence. We found thatthe bifurcation diagram changed dramatically as the exponent in the power law wasvaried. In particular, for values of the exponent α less than one, our model possessedexactly two solutions for all λ up to the pull-in. Of course, at the pull-in voltage or,equivalently, at the “nose” of the fold in the bifurcation diagram, only one solution

was found to exist. For α between one and the critical value − 12 + 1

2

√272 , our bi-

furcation diagram still consisted of a single fold, but the endpoint had moved. As aconsequence, for a range of λ the solution to our model was unique. Note that thisalready implies enhanced stability of the device. That is, for the range of λ for whichthe solution is unique, the solution is stable to any size perturbation. For α greater

than the critical value − 12 + 1

2

√272 , an even more dramatic change in the bifurcation

diagram was found to occur. Recall from Figure 4.2 that as α passed through one,the control of the behavior of the bifurcation diagram was passed from the critical

point B to the critical point C. Further, as α passed through − 12 +

12

√272 , C changed

from a stable node to a stable spiral. This implied that the bifurcation diagram de-veloped multiple folds—infinitely many, in fact. These additional solutions may haveimplications for MEMS design. The fact that these additional states were created byour spatial tailoring of the dielectric constant suggests that much could be achieved inthis direction. That is, we have shown that tailored dielectric properties comprise a

906 JOHN A. PELESKO

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

40

α

λ

Fig. 5.2. A sample design curve showing the nondimensional pull-in voltage as a function ofthe exponent, α, in the dielectric constant profile.

valuable new design parameter for electrostatic MEMS. Of course, our model requiresexperimental verification. While the pull-in instability has been experimentally ob-served, verifying that aspect of our model, little experimental work has been done ondevices with tailored dielectric properties. What about implications for our originalgoal of reducing the effects of the pull-in instability? In Figure 4.1, we have plottedthe bifurcation diagram for our strip model for various values of α. Observe that thefirst “nose” of the curve moves up and to the right as α is increased. If we focus onutilizing the lowest branch of the curve, this implies that the pull-in distance, i.e., themaximum achievable stable displacement, increases with increasing α. In Figure 5.1,we plot the pull-in distance as a function of α. If our model was experimentally vali-dated, then this plot would constitute a design curve allowing an engineer to choosea displacement and then match the dielectric profile to achieve it. The price we payis in increased power requirements. This is seen most clearly in Figure 5.2, where weplot the pull-in voltage versus α.

Next, we repeated our strip analysis for a disk shaped device. Here the bifur-cation diagram was always found to have infinitely many folds. This should not besurprising as a similar result was found for the case α = 0 in [18]. Here we again notethat tailoring of the dielectric profile allows us to move the “nose” of the bifurcationdiagram up and to the right, enhancing stability.

Finally, we again emphasize the fact that while spatial tailoring of the dielectricconstant cannot completely remove the pull-in instability, it is a useful design pa-rameter in electrostatically actuated MEMS. The contrasts between the behavior ofour strip and disk shaped devices in the case of constant dielectric properties furthersuggests that shape and dielectric properties should form a useful complementary pairof design parameters.

MODELING OF ELECTROSTATIC MEMS 907

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