Fluidmechanical Damping Analysis of Resonant

Micromirrors with Out-of-plane Comb Drive

Thomas Klose1, Holger Conrad2, Thilo Sandner∗,1, and Harald Schenk1

1Fraunhofer Institute Photonic Microsystems (FhG-IPMS),2TU Dresden, Semiconductor and Microsystems Technology Laboratory∗Corresponding author: Maria-Reiche-Str. 2, 01109 Dresden, Germany, thilo.sandner@ipms.fraunhofer.de

Abstract: Damping is the limiting fac-tor for the reachable maximum deflection.Thus, it is a very important issue for res-onant microsystems. In this paper, wepresent a damping model for out-of-planecomb driven resonant micromirrors. The ba-sic concept of this model is to attribute vis-cous damping in the comb gaps as the domi-nant contributor of damping moments. Themodel is extended by findings from a fluid-mechanical FEM model of an electrode fin-ger and the moving mirror plate with a cav-ity underneath. It also considers effects frompressure and temperature changes. The re-sults are verified and discussed in the con-text of experimental data. The primary goalof damping analysis and optimization is tominimize power consumption and to reducedriving voltage. The presented methods andmodels create the prerequisites for this task.

Keywords: MEMS, Micromirror, combdrive, Damping, Navier-Stokes, FEM.

1 Introduction

The resonant micromirrors developed atFraunhofer IPMS in Dresden are electrostat-ically driven MEMS devices. The drivingmoment is supplied by an out-of-plane combdrive which is realized by structuring themirror plate along its edge (Fig. 1). Themicromirrors can be actuated using a pulse-shaped voltage with a pulse frequency ap-proximately twice the resonance frequencyof the mechanical system. The achievablemaximum deflection angle for a certain de-vice is determined only by the ratio of theenergy feed-in and the energy feed-out. Theenergy feed-in is determined by the drivingvoltage, the driving scheme (e.g. frequency,duty-cycle), and the change of the capaci-tance of the comb drive during the oscilla-tion. The energy feed-out is determined bythe damping of the system. Since the de-vices are processed in monocrystalline sili-con and are operated in ambient pressure,fluidmechanical damping is considered to bethe only relevant.

Insulation Mirror plateComb electrode Torsional flexure Bondpad

100µm

10µm

a)

b)

c)

Figure 1: Micromachined scanning micromirror with out-of-plane comb drive actuation. a) Microscopicphotograph of a die, b) Detail view of the comb drive, c) REM image of the finger electrodes.

Excerpt from the Proceedings of the COMSOL Conference 2008 Hannover

Figure 2: Schematic configuration of the out-of-plane comb drive and its orientation in the model’scoordinate system

For description of the oscillation prop-erties damping effects have to be thor-oughly understood. In this paper wepresent a damping analysis of out-of-planecomb driven resonant micromirrors. It isbased on parameterized fluidmechanical 3DFEM models which are realized in COM-SOL MultiphysicsTM utilizing the applica-tion mode Incompressible Navier-Stokes.

The analysis includes slide-film dampingeffects within the electrostatic comb drive aswell as squeeze and drag damping affectingthe mirror plate. Within the validity rangeof theories of continuous flow, pressure andtemperature effects are considered by intro-ducing an effective dynamic viscosity.

A dedicated damping structure is avail-able for the verification of simulation resultsand the extraction of empiric data. It is anone-dimensional micromirror with a circularmirror plate, with a diameter of 1.5 mm, andan oscillation frequency of 1 kHz (Fig. 1).To compare the properties and efficiencies ofdifferent comb geometries, six different combdesigns, varying the dimensions of the elec-trode fingers and the widths of the trenchesbetween the electrodes were realized [4]. Thereference damping structure shown in Fig. 1has a finger length of L = 58.5µm, a fingerwidth of b = 1.5µm and a trench width ofg = 5µm. The height of the mirror plateand the comb structure is h = 30µm. Fig-ure 2 shows the schematic configuration ofthe out-of-plane comb drive. Furthermore itimplies the description of dimensions used inthis paper.

The transient behavior of resonant mi-cromirrors can be described by an equilib-rium of torques which results in a second or-der ordinary differential equation:

J θ + Md(θ, θ) + k θ = Mel(θ, U) (1)

with J as moment of inertia of the oscillat-ing structure, θ = f(t) as deflection angle,Md(θ, θ) as damping torque, k as torsionalspring constant, and Mel(θ, U) as electro-static torque which is generated by the out-of-plane comb drive. This driving torque de-pends on the deflection angle and the driv-ing voltage U . Since the terms Md(θ, θ) andMel(θ, U) are strongly nonlinear, the exactsolution for θ can only be computed with nu-merical methods. For oscillation amplitudeswhich are in the range of several degrees ap-plies∣∣J θ∣∣+

∣∣k θ∣∣� ∣∣Md(θ, θ)∣∣+∣∣Mel(θ, U)

∣∣Thus, the equilibrium solution of θ can beassumed to be sinusoidal:

θ ≈ θ sin(2πf t+ ϕ) (2)

with f as oscillation frequency, ϕ as phaseshift and θ as oscillation amplitude, whichstill depends on Md(θ, θ) and Mel(θ, U). Af-terwards the phase shift ϕ is assumed to bezero.

To prove the damping model and itsrange of validity it is necessary to com-pare simulation results with experimentaldata. Since the transient characteristics ofthe damping moment can not be measured

directly, a mean damping coefficient is intro-duced:

D =1T

∫ T

0

Md(θ, θ)θ

dt , T =1f

(3)

This mean damping coefficient can bemeasured[4]. Thus, it is possible to verifysimulation results with experimental data.

2 Governing equations

2.1 Flow properties

For determination of valid approaches andmodels which can be applied to a given flu-idmechanical problem there are several phys-ical measures available.

The Knudsen-number measures the ra-tio between mean free path length within thefluid and a characteristic geometrical lengthof the structure. It is used to determinewhether the continuum mechanics formula-tion of fluid dynamics can be used. It isdefined to

Kn =Ld

(4)

with L as mean free path length and d ascharacteristic structure length.

The continuum condition of Navier-Stokes equations is considered to be ful-filled if Kn < 0.1. With the definition ofmean free path length [1] the following con-dition for temperature and pressure of thefluid can be deduced:

p

T>

k√2πKn d σ2

∣∣∣∣Kn<0.1

(5)

With T as absolute temperature, p as am-bient pressure, k as Boltzmann constant,and σ as molecule diameter of the fluid.

Assuming a characteristic length of d =g = 5µm, a maximum absolute temperatureof 333 K (≈ 60 ◦C), and a molecule diameterof 0.35 nm (nitrogen), the continuum condi-tion is fulfilled for ambient pressures withp > 1.7 · 104 Pa.

Navier-Stokes equations with no-slipboundary conditions are only valid if the ad-ditional condition Kn < 10−3 is fulfilled. Forlarger Knudsen numbers a slip occurs at theinterface between structure and fluid whichhas to be considered by the model. This can

be done by introducing a modified (effective)dynamic viscosity [6]:

ηeff :=η

1 + f(Kn)(6)

It should be noted that differently from thedynamic viscosity η the effective dynamicviscosity depends on the temperature as wellas the pressure of the fluid. Since the Knud-sen number depends on the local geometri-cal properties of the structure (characteristiclength) the effective dynamic viscosity is alsoa function of location.

It was shown in earlier publications [4, 2]that for micromechanical devices the empir-ical approximation of Knudsen is a goodchoice for the function f(Kn):

f(Kn) :=Z Kn0.1474

, Z =Kn + 2.507Kn + 3.095

(7)

The Reynolds number of a flow is usedto determine whether it is laminar or turbu-lent. It is defined to

Re =ρm

ηu d (8)

With ρm as density and u as characteris-tic velocity of the fluid [1]. The maximumReynolds number is determining the prop-erties of the flow. Is a critical value Recr ≈2000 exceeded, turbulences appear. Other-wise the flow is laminar.

Considering the geometrical properties itcan be shown that the maximum Reynoldsnumber of a micromirror with out-of-planecomb drive can be denoted as [2]:

max Re =π ρm

ηθ f D d (9)

With D as diameter of the mirror plate.Even in worst case scenarios (large, fastoscillating mirror) the Reynolds numberdoes not exceed a value of 100. Thus, theo-ries of laminar flow can be applied withoutlimitations.

The Mach number indicates the ratio ofinertia forces and compression forces. Thiscorresponds to the quotient of flow velocityand the speed of sound c within the medium:

Ma =u

c(10)

Is the condition Ma < 0.3 fulfilled, compres-sion effects can be neglected. Since the max-imum Mach number which occurs within

a flow at typical micromirrors with out-of-plane comb drive is in the range of < 0.1 [2]the fluidmechanical problem can be solvedusing simplified (incompressible) Navier-Stokes equations.

2.2 General flow model

According to the findings in the prior sectionthe fluidmechanical damping of a micromir-ror with out-of-plane comb drive can be de-scribed by incompressible Navier-Stokesequations with slip correction:

ρm

(∂~u

∂t+ (~u · ∇)~u

)= −∇p+ ηeff ∆~u

∇ · ~u = 0 (11)

~ubc = ~vbc

Thereby p = f(x, y, z) is the pressure and~u = ~f(x, y, z) is the velocity of the fluid; ~ubc

and ~vbc are the flow velocity at the interfacebetween fluid and structure and the velocityof the structure itself.

The obstruction of a moving structure ina fluid can be determined from the resultsof Eq. (11). The effective damping force re-sults from integration of pressure p over theinterfaces (normal component) and the ve-locity gradient in direction of the interface’snormal ∂~u

∂~n0(tangential component):

~Fd =∫

(A)

∫p~n0 dA+

∫(A)

∫ηeff

∂~u

∂~n0dA (12)

Now the damping torque, effecting a tiltingmirror plate can be expressed by:

~Md =∫

(A)

∫~r × d~Fd (13)

With ~r as position vector which is alwaysorthogonal to the rotation axis of the tiltingmirror.

3 Theory

3.1 Damping of comb drive

It has been shown that damping of mi-cromirrors with out-of-plane comb drive isdominated by fluidmechanical interactionwithin the comb structure [4]. To describethese damping mechanisms it is useful todistinguish three different states of the elec-trodes [2]:

v0

z

y

v=0

...... ...

v0

z

y

a) b)

uu0

u

...

Figure 3: Flow profiles between drivingelectrodes. a) engaged, b) disengaged state

θ ≈ 0 The comb structure is fully engaged.Since the distance of the electrodes issmall in comparison to the length andheight, the flow between them can beconsidered as Couette flow, indicatedby a linear velocity profile (Fig. 3a, [3,1]). According to Eq. (12), the dampingforce can be expressed by:

~Fd ≈ ~FCouette = −ηeffAs

g~v (14)

With As ≈ 2Lh as interacting surfacearea and ~v as velocity of the movable elec-trodes.

|θ| > θc The comb structure is fully disen-gaged. Since the distance between theelectrodes is still small in comparisonto the length and height, the resultingflow between them can be considered asHagen-Poiseuille flow (Fig. 3b, [2]),indicated by a parabolic velocity pro-file. According to Eq. (12), the resultingdamping force can be expressed by:

~Fd ≈ ~FPoiseuille = −4 ηeffAs

dh~v ξ (15)

with

ξ =(

1− |~u0||~v|

)and dh = 4

A

U

A ≈ L(2g + b) is the hydraulic diameter,U is the perimeter of the flow channel.

0 < |θ| < θc The comb structure is in tran-sition between engaged and disengagedstate. Couette flows as well as Hagen-Poiseuille flows are appearing betweenthe electrodes.

3.2 Damping of mirror plate

In terms of damping mechanisms at tiltingplates, two different effects have to be dis-tinguished:

Drag damping is affecting every struc-ture moving within a fluid. The dampingforces result according to Eq. (12) from inte-gration over pressure which can be describedby Navier-Stokes equations (11):

~Fdrag = − ~v

|~v|cd ρm |~v|2

Ap

2(16)

With Ap as base area of the mirror plateand cd as empirical factor which can be de-termined by experiments or numerical anal-yses. From Eq. (13) we can derive the damp-ing torque of a tilting plate caused by drageffects:

~Mdrag ≈ −cd ρm

2~ω |~ω|

∫(Ap)

∫|~r|3 dA (17)

With ~ω = ~θ as angular velocity of the mir-ror plate (Fig. 4). It should be noted thatthe daming torque depends quadratically onthe angular velocity ~ω. Since |~ω| ∝ fθ, agrowing influence of drag damping can beexpected for micromirrors with high frequen-cies or deflection angles.

Squeeze film damping occurs when aplate is moving nearby a immovable struc-ture (e.g. according to Fig. 4). A chang-ing distance results in a progression of pres-sure and leads to a flow between the plateand the structure. Since IPMS micromirrorsusually have a small cavity underneath themirror plate this effect can also be relevantfor a damping analysis. For small distancesbetween plate and immovable structure thesqueeze film effect can be described by ananalytical approximation [3, 5]:

∂2p′

∂x2+∂2p′

∂y2= 12

ηeff

d3vz , p′bc = 0 (18)

In this Poisson’s differential equation vz =f(x, y) is the velocity of the plate in direc-tion of its normal, p′ = p − p0 is the differ-ence between pressure and ambient pressure.Analogical to Eq. (13) the damping torqueresulting from the squeeze film effect can byexpressed by:

Msq =∫

(Ap)

∫p′ y dA (19)

Solutions of Eq. (18) for simple geometriesand examples of resulting pressure regimescan be found in [3].

p

v

z

yv v

u

v=0

p'=0

rdF

p'

Drag damping

Squeeze damping

Figure 4: Damping mechanisms of tilting plates

4 Numerical model

To verify analytical approaches and for de-termination of flow parameters ξ in Eq. (15)as well as cd in Eq. (17) numerical mod-els are required. Two numerical mod-els are created using the FEM tool COM-SOL MultiphysicsTM, utilizing the applica-tion mode Incompressible Navier-Stokes.

4.1 Comb drive

In order to perform damping analyses of ar-bitrary IPMS microscanners, a parametrizedfinite element model of the out-of-planecomb is created. Comparisons of severaltwo- and three-dimensional approaches us-ing a complex reference model with threeelectrode fingers and their respective counterelectrodes (left-hand side of Fig. 5) showthat the model has to be three-dimensionalto include all relevant effects. The final FEMmodel consists of a half finger and a halfcounter electrode (right-hand side of Fig. 5).Using symmetrical boundary conditions, thefluid state at an electrode which is containedin a electrode comb with infinite number offingers, can be simulated. Since the countof electrodes is usually very high, the errorcaused by electrodes near the ends of thecomb is assumed to be small.

y

x

x

z

Model Boundary

Electrode

Fluid

Symmetry

Figure 5: Fluidmechanical FEM model withsymmetric boundary conditions

Figure 6: Velocity slice plot along the middle ofa disengaged moving electrode

Figure 6 shows a typical slice plot of the ve-locity field of the fluid. Barring changes ofthe finger geometry and location, the FEMmodel has only two degrees of freedom: thefinger velocity ~v = ~f(~ω) and the deflection ofthe finger s = f(θ). Therefore the dampingcharacteristic of a certain comb geometry atvariable mirror dimensions, frequencies, anddeflections can be completely investigated byexclusively considering variable velocity anddeflection of the electrodes.

To prove the assumptions made in theanalytic model, the velocity profiles betweenthe electrodes are extracted. Thereby an in-teresting effect can be observed. The Cou-ette flow as well as the Hagen-Poiseuilleflow, both appear at the moving elec-trodes as expected, but additionally a sec-ond Hagen-Poiseuille flow develops be-tween the counter electrodes (Fig. 7). Thisis caused by the moving electrode which actslike a piston, dragging the fluid. Accordingto the law of actio and reactio the appear-ing shear forces at the counter electrode areapplied by the moving electrode. This re-sults in an additional damping force whilethe transition between engaged and disen-gaged state. Since two different Poiseuilleflows have to be considered, two differentfactors ξ1 and ξ2 are required. Thereby, ξ1characterizes the flow between the counterelectrodes and ξ2 characterizes the flow be-tween the moving electrodes. From simula-tions with variable deflections, the charac-teristics of ξ1 and ξ2 depending on s can bederived. Using this, it is possible to utilizeEq.(14) and Eq. (15) to realize a good ap-proximation of the damping force of a mov-ing electrode:

~Fd ≈ ~FCouette +∑

~FPoiseuille (20)

u

Electrode finger

Counter electrode

Poisseulle flow

Couette flow

Poisseulle flow

x=x2

x=x1

Figure 7: Simulated velocity profiles betweenmoving electrodes and its counter electrodes

4.2 Mirror plate with cavity

Damping properties of the mirror plate canbe investigated with a simplified 3D FEMmodel (Fig. 8). To reduce complexity thecomb is not included. The tilting of theplate is considered by a location-dependentboundary condition: uz,bc = vz,bc = y |~ω|.

The damping torque for a given angu-lar velocity ~ω = f(θ, f) can be derived us-ing the integration capabilities of COMSOLMultiphysicsTM according to Eq. (12) andEq. (13) respectively.

b)

a)

x y

y

z

Substrate Mirror plate Cavity Fluid

A A

A­A:

0.3

­0.4p' / Pa

0

Figure 8: Simulation results for a 3D FEMmodel of a micromirror with standard cavity

cavity

cavity

(Mdr

ag +

 Msq

) / N

m •1

0­9

Figure 9: Damping torques for different cavities

Fig. 9 shows damping torques of a circularmirror plate with a diameter of 1.5 mm andtwo cavity heights. It is noticeable that theresulting curves are superpositions of linearand quadratic functions. By using nonlinearcurve fitting, the empirical factor cd can bederived as well as the linear damping coeffi-cient caused by the squeeze film effect.

5 Experimental results

Utilizing Eq. (3), Eq. (13) and the values forthe flow parameters ξ1, ξ2 and cd accordingto section 4, it is possible to calculate themean damping coefficient for a given deflec-tion amplitude. Fig. 10 shows a compari-son of simulated mean damping coefficientsand experimental data. Although the sim-ulated damping coefficients are consistentlysmaller, the characteristic is very similar.The difference can be explained by the un-avoidably non-ideal driving regime used inthe experimental setup [4]. Since the the-oretical maximum amplitude can never bereached in the real world the damping coef-ficient is overestimated.

0 5 10 150

0.5

1

1.5

2x 10

−12

θ^ / °

D /

(Nm

⋅ s)

Experimental dataSimulation

Figure 10: Mean damping coefficients

6 Conclusion

In this paper we presented a damping modelfor micromirrors with out-of-plane combdrive. The model bases on Navier-Stokestheory and includes the viscous gas dampingwithin the comb drive as well as the effectscaused by drag damping and squeeze filmdamping affecting the mirror plate. The va-lidity limits of this model were discussed interms of pressure and temperature changesand were proven by experimental data. Thesimulation results fit experimental resultsvery well.

With the presented models and methodsit is possible to predict the damping proper-ties of arbitrary comb and mirror geometries.This is very important for understanding thebehavior of available micromirrors. It is alsoa requirement for the optimization of thisdevices. Thereby the geometry of the combdrive, the geometry of the mirror plate andthe cavity of the device can be varied in or-der to find an improved design.

References

[1] Mohamed Gad el Hak, The mems hand-book, CRC Press, 2002.

[2] T. Klose, T. Sandner, H. Schenk, andH. Lakner, Extended damping model forout-of-plane comb driven micromirrors,SPIE Proceedings Series 6114 (2006),184–195.

[3] J. Mehner, Entwurf in der mikrosys-temtechnik, Dresden Univ. Press, 1999.

[4] T. Sandner, T. Klose, A. Wolter,H. Schenk, and H. Lakner, Dampinganalysis and measurement for a comb-drive scanning mirror, SPIE ProceedingsSeries 5455 (2004), 147–158.

[5] J. Starr, Squeeze-film damping in solid-state accelerometers, IEEE Solid-StateSensor and Actuator Workshop (1990).

[6] Timo Veijola, Heikki Kuisma, JuhaLahdenpera, and Tapany Ryhanen,Equivalent-circuit model of the squeezedgas film in a silicon accelerometer, Sen-sors and Actuators A 48 (1995), 239–248.