Fluidmechanical Damping Analysis of Resonant Micromirrors with Out-of-plane Comb Drive

Transcription

1 Excerpt from the Proceedings of the COMSOL Conference 2008 Hannover Flidmechanical Damping Analsis of Resonant Micromirrors with Ot-of-plane Comb Drive Thomas Klose 1, Holger Conrad 2, Thilo Sandner,1, and Harald Schenk 1 1 Franhofer Institte Photonic Microsstems (FhG-IPMS), 2 TU Dresden, Semicondctor and Microsstems Technolog Laborator Corresponding athor: Maria-Reiche-Str. 2, Dresden, German, thilo.sandner@ipms.franhofer.de Abstract: Damping is the limiting factor for the reachable maximm deflection. Ths, it is a ver important isse for resonant microsstems. In this paper, we present a damping model for ot-of-plane comb driven resonant micromirrors. The basic concept of this model is to attribte viscos damping in the comb gaps as the dominant contribtor of damping moments. The model is extended b findings from a flidmechanical FEM model of an electrode finger and the moving mirror plate with a cavit nderneath. It also considers effects from pressre and temperatre changes. The reslts are verified and discssed in the context of experimental data. The primar goal of damping analsis and optimiation is to minimie power consmption and to redce driving voltage. The presented methods and models create the prereqisites for this task. Kewords: MEMS, Micromirror, comb drive, Damping, Navier-Stokes, FEM. 1 Introdction The resonant micromirrors developed at Franhofer IPMS in Dresden are electrostaticall driven MEMS devices. The driving moment is spplied b an ot-of-plane comb drive which is realied b strctring the mirror plate along its edge (Fig. 1). The micromirrors can be actated sing a plseshaped voltage with a plse freqenc approximatel twice the resonance freqenc of the mechanical sstem. The achievable maximm deflection angle for a certain device is determined onl b the ratio of the energ feed-in and the energ feed-ot. The energ feed-in is determined b the driving voltage, the driving scheme (e.g. freqenc, dt-ccle), and the change of the capacitance of the comb drive dring the oscillation. The energ feed-ot is determined b the damping of the sstem. Since the devices are processed in monocrstalline silicon and are operated in ambient pressre, flidmechanical damping is considered to be the onl relevant. Inslation Comb electrode Mirror plate Torsional flexre Bondpad b) 10µm a) 100µm Figre 1: Micromachined scanning micromirror with ot-of-plane comb drive actation. a) Microscopic photograph of a die, b) Detail view of the comb drive, c) REM image of the finger electrodes. c)

2 Figre 2: Schematic configration of the ot-of-plane comb drive and its orientation in the model s coordinate sstem For description of the oscillation properties damping effects have to be thoroghl nderstood. In this paper we present a damping analsis of ot-of-plane comb driven resonant micromirrors. It is based on parameteried flidmechanical 3D FEM models which are realied in COM- SOL Mltiphsics TM tiliing the application mode Incompressible Navier-Stokes. The analsis incldes slide-film damping effects within the electrostatic comb drive as well as sqeee and drag damping affecting the mirror plate. Within the validit range of theories of continos flow, pressre and temperatre effects are considered b introdcing an effective dnamic viscosit. A dedicated damping strctre is available for the verification of simlation reslts and the extraction of empiric data. It is an one-dimensional micromirror with a circlar mirror plate, with a diameter of 1.5 mm, and an oscillation freqenc of 1 kh (Fig. 1). To compare the properties and efficiencies of different comb geometries, six different comb designs, varing the dimensions of the electrode fingers and the widths of the trenches between the electrodes were realied [4]. The reference damping strctre shown in Fig. 1 has a finger length of L = 58.5 µm, a finger width of b = 1.5 µm and a trench width of g = 5 µm. The height of the mirror plate and the comb strctre is h = 30 µm. Figre 2 shows the schematic configration of the ot-of-plane comb drive. Frthermore it implies the description of dimensions sed in this paper. The transient behavior of resonant micromirrors can be described b an eqilibrim of torqes which reslts in a second order ordinar differential eqation: J θ + M d ( θ, θ) + k θ = M el (θ, U) (1) with J as moment of inertia of the oscillating strctre, θ = f(t) as deflection angle, M d ( θ, θ) as damping torqe, k as torsional spring constant, and M el (θ, U) as electrostatic torqe which is generated b the otof-plane comb drive. This driving torqe depends on the deflection angle and the driving voltage U. Since the terms M d ( θ, θ) and M el (θ, U) are strongl nonlinear, the exact soltion for θ can onl be compted with nmerical methods. For oscillation amplitdes which are in the range of several degrees applies J θ + k θ Md ( θ, θ) + Mel (θ, U) Ths, the eqilibrim soltion of θ can be assmed to be sinsoidal: θ ˆθ sin(2πf t + ϕ) (2) with f as oscillation freqenc, ϕ as phase shift and ˆθ as oscillation amplitde, which still depends on M d ( θ, θ) and M el (θ, U). Afterwards the phase shift ϕ is assmed to be ero. To prove the damping model and its range of validit it is necessar to compare simlation reslts with experimental data. Since the transient characteristics of the damping moment can not be measred

3 directl, a mean damping coefficient is introdced: D = 1 T T 0 M d ( θ, θ) θ dt, T = 1 f (3) This mean damping coefficient can be measred[4]. Ths, it is possible to verif simlation reslts with experimental data. 2 Governing eqations 2.1 Flow properties For determination of valid approaches and models which can be applied to a given flidmechanical problem there are several phsical measres available. The Kndsen-nmber measres the ratio between mean free path length within the flid and a characteristic geometrical length of the strctre. It is sed to determine whether the continm mechanics formlation of flid dnamics can be sed. It is defined to Kn = L d (4) with L as mean free path length and d as characteristic strctre length. The continm condition of Navier- Stokes eqations is considered to be flfilled if Kn < 0.1. With the definition of mean free path length [1] the following condition for temperatre and pressre of the flid can be dedced: p T > k 2 π Kn d σ 2 (5) Kn<0.1 With T as absolte temperatre, p as ambient pressre, k as Boltmann constant, and σ as molecle diameter of the flid. Assming a characteristic length of d = g = 5 µm, a maximm absolte temperatre of 333 K ( 60 C), and a molecle diameter of 0.35 nm (nitrogen), the continm condition is flfilled for ambient pressres with p > Pa. Navier-Stokes eqations with no-slip bondar conditions are onl valid if the additional condition Kn < 10 3 is flfilled. For larger Kndsen nmbers a slip occrs at the interface between strctre and flid which has to be considered b the model. This can be done b introdcing a modified (effective) dnamic viscosit [6]: η eff := η 1 + f(kn) (6) It shold be noted that differentl from the dnamic viscosit η the effective dnamic viscosit depends on the temperatre as well as the pressre of the flid. Since the Kndsen nmber depends on the local geometrical properties of the strctre (characteristic length) the effective dnamic viscosit is also a fnction of location. It was shown in earlier pblications [4, 2] that for micromechanical devices the empirical approximation of Kndsen is a good choice for the fnction f(kn): f(kn) := Z Kn Kn , Z = Kn (7) The Renolds nmber of a flow is sed to determine whether it is laminar or trblent. It is defined to Re = ρ m η d (8) With ρ m as densit and as characteristic velocit of the flid [1]. The maximm Renolds nmber is determining the properties of the flow. Is a critical vale Re cr 2000 exceeded, trblences appear. Otherwise the flow is laminar. Considering the geometrical properties it can be shown that the maximm Renolds nmber of a micromirror with ot-of-plane comb drive can be denoted as [2]: max Re = π ρ m η θ f D d (9) With D as diameter of the mirror plate. Even in worst case scenarios (large, fast oscillating mirror) the Renolds nmber does not exceed a vale of 100. Ths, theories of laminar flow can be applied withot limitations. The Mach nmber indicates the ratio of inertia forces and compression forces. This corresponds to the qotient of flow velocit and the speed of sond c within the medim: Ma = c (10) Is the condition Ma < 0.3 flfilled, compression effects can be neglected. Since the maximm Mach nmber which occrs within

4 a flow at tpical micromirrors with ot-ofplane comb drive is in the range of < 0.1 [2] the flidmechanical problem can be solved sing simplified (incompressible) Navier- Stokes eqations. 2.2 General flow model According to the findings in the prior section the flidmechanical damping of a micromirror with ot-of-plane comb drive can be described b incompressible Navier-Stokes eqations with slip correction: ) ( ρ m + ( ) = p + η eff t = 0 (11) bc = v bc Thereb p = f(x,, ) is the pressre and = f(x,, ) is the velocit of the flid; bc and v bc are the flow velocit at the interface between flid and strctre and the velocit of the strctre itself. The obstrction of a moving strctre in a flid can be determined from the reslts of Eq. (11). The effective damping force reslts from integration of pressre p over the interfaces (normal component) and the velocit gradient in direction of the interface s normal (tangential component): F d = n 0 p n 0 da + (A) (A) η eff n 0 da (12) Now the damping torqe, effecting a tilting mirror plate can be expressed b: M d = r df d (13) (A) With r as position vector which is alwas orthogonal to the rotation axis of the tilting mirror. 3 Theor 3.1 Damping of comb drive It has been shown that damping of micromirrors with ot-of-plane comb drive is dominated b flidmechanical interaction within the comb strctre [4]. To describe these damping mechanisms it is sefl to distingish three different states of the electrodes [2]: v 0 v= v 0 a) b) Figre 3: Flow profiles between driving electrodes. a) engaged, b) disengaged state θ 0 The comb strctre is fll engaged. Since the distance of the electrodes is small in comparison to the length and height, the flow between them can be considered as Coette flow, indicated b a linear velocit profile (Fig. 3a, [3, 1]). According to Eq. (12), the damping force can be expressed b: F d F Coette = η eff A s g 0... v (14) With A s 2Lh as interacting srface area and v as velocit of the movable electrodes. θ > θ c The comb strctre is fll disengaged. Since the distance between the electrodes is still small in comparison to the length and height, the reslting flow between them can be considered as Hagen-Poiseille flow (Fig. 3b, [2]), indicated b a parabolic velocit profile. According to Eq. (12), the reslting damping force can be expressed b: F d F Poiseille = 4 η eff A s d h v ξ (15) with ξ = ( 1 ) 0 v and d h = 4 A U A L(2g + b) is the hdralic diameter, U is the perimeter of the flow channel. 0 < θ < θ c The comb strctre is in transition between engaged and disengaged state. Coette flows as well as Hagen- Poiseille flows are appearing between the electrodes. 3.2 Damping of mirror plate In terms of damping mechanisms at tilting plates, two different effects have to be distingished:

5 Drag damping is affecting ever strctre moving within a flid. The damping forces reslt according to Eq. (12) from integration over pressre which can be described b Navier-Stokes eqations (11): F drag = v v c d ρ m v 2 A p 2 (16) With A p as base area of the mirror plate and c d as empirical factor which can be determined b experiments or nmerical analses. From Eq. (13) we can derive the damping torqe of a tilting plate cased b drag effects: M drag c d ρ m ω ω r 3 da (17) 2 (A p) With ω = θ as anglar velocit of the mirror plate (Fig. 4). It shold be noted that the daming torqe depends qadraticall on the anglar velocit ω. Since ω f ˆθ, a growing inflence of drag damping can be expected for micromirrors with high freqencies or deflection angles. Sqeee film damping occrs when a plate is moving nearb a immovable strctre (e.g. according to Fig. 4). A changing distance reslts in a progression of pressre and leads to a flow between the plate and the strctre. Since IPMS micromirrors sall have a small cavit nderneath the mirror plate this effect can also be relevant for a damping analsis. For small distances between plate and immovable strctre the sqeee film effect can be described b an analtical approximation [3, 5]: 2 p x p 2 = 12 η eff d 3 v, p bc = 0 (18) In this Poisson s differential eqation v = f(x, ) is the velocit of the plate in direction of its normal, p = p p 0 is the difference between pressre and ambient pressre. Analogical to Eq. (13) the damping torqe reslting from the sqeee film effect can b expressed b: M sq = p da (19) (A p) Soltions of Eq. (18) for simple geometries and examples of reslting pressre regimes can be fond in [3]. v p p'=0 Drag damping v p' df r Sqeee damping v=0 Figre 4: Damping mechanisms of tilting plates 4 Nmerical model To verif analtical approaches and for determination of flow parameters ξ in Eq. (15) as well as c d in Eq. (17) nmerical models are reqired. Two nmerical models are created sing the FEM tool COM- SOL Mltiphsics TM, tiliing the application mode Incompressible Navier-Stokes. 4.1 Comb drive In order to perform damping analses of arbitrar IPMS microscanners, a parametried finite element model of the ot-of-plane comb is created. Comparisons of several two- and three-dimensional approaches sing a complex reference model with three electrode fingers and their respective conter electrodes (left-hand side of Fig. 5) show that the model has to be three-dimensional to inclde all relevant effects. The final FEM model consists of a half finger and a half conter electrode (right-hand side of Fig. 5). Using smmetrical bondar conditions, the flid state at an electrode which is contained in a electrode comb with infinite nmber of fingers, can be simlated. Since the cont of electrodes is sall ver high, the error cased b electrodes near the ends of the comb is assmed to be small. x Smmetr Electrode Flid Model Bondar Figre 5: Flidmechanical FEM model with smmetric bondar conditions x v

6 Poisselle flow 2 Coette flow Figre 6: Velocit slice plot along the middle of a disengaged moving electrode Figre 6 shows a tpical slice plot of the velocit field of the flid. Barring changes of the finger geometr and location, the FEM model has onl two degrees of freedom: the finger velocit v = f( ω) and the deflection of the finger s = f(θ). Therefore the damping characteristic of a certain comb geometr at variable mirror dimensions, freqencies, and deflections can be completel investigated b exclsivel considering variable velocit and deflection of the electrodes. To prove the assmptions made in the analtic model, the velocit profiles between the electrodes are extracted. Thereb an interesting effect can be observed. The Coette flow as well as the Hagen-Poiseille flow, both appear at the moving electrodes as expected, bt additionall a second Hagen-Poiseille flow develops between the conter electrodes (Fig. 7). This is cased b the moving electrode which acts like a piston, dragging the flid. According to the law of actio and reactio the appearing shear forces at the conter electrode are applied b the moving electrode. This reslts in an additional damping force while the transition between engaged and disengaged state. Since two different Poiseille flows have to be considered, two different factors ξ 1 and ξ 2 are reqired. Thereb, ξ 1 characteries the flow between the conter electrodes and ξ 2 characteries the flow between the moving electrodes. From simlations with variable deflections, the characteristics of ξ 1 and ξ 2 depending on s can be derived. Using this, it is possible to tilie Eq.(14) and Eq. (15) to realie a good approximation of the damping force of a moving electrode: F d F Coette + FPoiseille (20) Electrode finger Poisselle flow 1 Conter electrode Figre 7: Simlated velocit profiles between moving electrodes and its conter electrodes 4.2 Mirror plate with cavit Damping properties of the mirror plate can be investigated with a simplified 3D FEM model (Fig. 8). To redce complexit the comb is not inclded. The tilting of the plate is considered b a location-dependent bondar condition:,bc = v,bc = ω. The damping torqe for a given anglar velocit ω = f(θ, f) can be derived sing the integration capabilities of COMSOL Mltiphsics TM according to Eq. (12) and Eq. (13) respectivel. A a) b) A A: Sbstrate Mirror plate Cavit Flid x A p' / Pa Figre 8: Simlation reslts for a 3D FEM model of a micromirror with standard cavit

7 6 Conclsion (Mdrag + Msq) / Nm 10 9 cavit cavit Figre 9: Damping torqes for different cavities Fig. 9 shows damping torqes of a circlar mirror plate with a diameter of 1.5 mm and two cavit heights. It is noticeable that the reslting crves are sperpositions of linear and qadratic fnctions. B sing nonlinear crve fitting, the empirical factor c d can be derived as well as the linear damping coefficient cased b the sqeee film effect. 5 Experimental reslts Utiliing Eq. (3), Eq. (13) and the vales for the flow parameters ξ 1, ξ 2 and c d according to section 4, it is possible to calclate the mean damping coefficient for a given deflection amplitde. Fig. 10 shows a comparison of simlated mean damping coefficients and experimental data. Althogh the simlated damping coefficients are consistentl smaller, the characteristic is ver similar. The difference can be explained b the navoidabl non-ideal driving regime sed in the experimental setp [4]. Since the theoretical maximm amplitde can never be reached in the real world the damping coefficient is overestimated. D / (Nm s) x Experimental data Simlation θ^ / Figre 10: Mean damping coefficients In this paper we presented a damping model for micromirrors with ot-of-plane comb drive. The model bases on Navier-Stokes theor and incldes the viscos gas damping within the comb drive as well as the effects cased b drag damping and sqeee film damping affecting the mirror plate. The validit limits of this model were discssed in terms of pressre and temperatre changes and were proven b experimental data. The simlation reslts fit experimental reslts ver well. With the presented models and methods it is possible to predict the damping properties of arbitrar comb and mirror geometries. This is ver important for nderstanding the behavior of available micromirrors. It is also a reqirement for the optimiation of this devices. Thereb the geometr of the comb drive, the geometr of the mirror plate and the cavit of the device can be varied in order to find an improved design. References [1] Mohamed Gad el Hak, The mems handbook, CRC Press, [2] T. Klose, T. Sandner, H. Schenk, and H. Lakner, Extended damping model for ot-of-plane comb driven micromirrors, SPIE Proceedings Series 6114 (2006), [3] J. Mehner, Entwrf in der mikrosstemtechnik, Dresden Univ. Press, [4] T. Sandner, T. Klose, A. Wolter, H. Schenk, and H. Lakner, Damping analsis and measrement for a combdrive scanning mirror, SPIE Proceedings Series 5455 (2004), [5] J. Starr, Sqeee-film damping in solidstate accelerometers, IEEE Solid-State Sensor and Actator Workshop (1990). [6] Timo Veijola, Heikki Kisma, Jha Lahdenperä, and Tapan Rhänen, Eqivalent-circit model of the sqeeed gas film in a silicon accelerometer, Sensors and Actators A 48 (1995),