Macromodel-based simulation and measurement of the dynamic pull-in of viscously damped RF-MEMS switches

Sensors and Actuators A 172 (2011) 269–279

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Macromodel-based simulation and measurement of the dynamic pull-in of viscously damped RF-MEMS switches Martin Niessner a,∗ , Gabriele Schrag a , Jacopo Iannacci b , Gerhard Wachutka a a b

Institute for Physics of Electrotechnology, Munich University of Technology, Munich, Germany MEMS Research Unit, Fondazione Bruno Kessler (FBK), Povo di Trento, Italy

a r t i c l e

i n f o

Article history: Available online 14 May 2011 Keywords: RF-MEMS switch Dynamic pull-in Squeeze-film damping Rarefaction Macromodel

a b s t r a c t We present a physics-based multi-energy domain coupled macromodel that allows for the efficient simulation of the dynamic response of electrostatically controlled and viscously damped ohmic contact RF-MEMS switches on the system-level. The predictive power of the macromodel is evaluated w.r.t. white light interferometer and laser vibrometer measurements. Furthermore, the macromodel is, concerning accuracy and performance, benchmarked versus two alternative state-of-the-art system-level models. The results obtained with the presented macromodel are in very good agreement with the measured quasi-static pull-in characteristics as well as the pull-in and pull-out transients. Due to its capability to account for multiple structural modes, the presented macromodel produces, among the evaluated models, the result that is closest to the measured phase of initial contact during dynamic pull-in. Moreover, a detailed experimental evaluation of the damping model shows a very good agreement (maximum relative error does not exceed 10%) for ambient pressures ranging from 960 hPa down to approximately 200 hPa. Compared to other damping models, this constitutes a very good result, especially because the models contain only geometric parameters and no problem-specific fit factors are needed to obtain this accuracy. The resulting macromodel is physics-based and, hence, scalable and predictive. Due to its generic nature it can be – in general – adapted for any electrostatically actuated device working in contact mode. © 2011 Elsevier B.V. All rights reserved.

1. Introduction and motivation The simulation of the pull-in and pull-out transients is essential for predicting the switching time of new radio frequency micromechanical (RF-MEMS) switch designs. Moreover, simulations of the pull-in transients can help to assess and reduce the impact energy and, thus, increase the reliability and lifetime of the movable micromechanical components [1]. However, simulations of this kind make high demands on the utilized model, since, in addition to the electrostatic, mechanical and fluidic domains as well as their nonlinear interactions, also the mechanical contact has to be considered. Furthermore, when multiple coupled energy domains with large mesh deformations including contact situations have to be analyzed, finite element (FE) models become computationally expensive and difficult to handle. Consequently, macromodels with a reduced number of degrees of freedom and, thus, reduced computational expense are necessary to enable a fast simulation of pull-in/-out transients. By nature, the resulting short computing times in the magnitude of only a few minutes do not come for free: a macromodel is a reduced description of the original problem

∗ Corresponding author. 0924-4247/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2011.04.046

and the validity of the simulated results strongly depends on the respective approach chosen for its derivation. In this work, a physics-based and, hence, scalable and predictive macromodel of the electrostatically actuated and viscously damped ohmic contact RF-MEMS switch shown in Fig. 1 is systematically derived, evaluated w.r.t. measurements performed with a white light interferometer (WLI) and a laser vibrometer (LV) and benchmarked versus two other state-of-the-art macromodels. 2. Investigated RF-MEMS switch The RF-MEMS device under consideration is fabricated at Fondazione Bruno Kessler (FBK) [2–4]. The device consists of a movable perforated gold membrane suspended above a fixed polysilicon electrode by four straight gold beams. The measured frequency of the fundamental resonance mode of the switch is 14.7 kHz. The polysilicon electrode acts as biasing electrode of the switch and is designed as an interdigitated structure consisting of seven lateral fingers connected together. By applying a voltage higher than the so-called pull-in voltage, the membrane collapses onto 12 elevated contact pads (cp. Fig. 2) and closes the ohmic contact between the RF ports. Due to the different heights of the electrode and the contact pads, the air gap underneath the suspended membrane is position-dependent. This needs to be taken into account in order

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3. Modeling

Fig. 1. SEM image of the investigated RF-MEMS switch design.

Fig. 2. White light interferometer image of the ground electrode and the contact pads. The membrane was removed from one sample to perform this measurement.

In order to derive a system-level model of the switch with the highest achievable level of transparency and accuracy, we followed the hierarchical modeling approach reported in [5]. In this approach, the RF-MEMS switch is decomposed into subsystems that constitute single functional blocks represented by physicsbased macromodels, thus allowing for predictive simulation. These submodels are derived solely on the basis of flux-conserving reduced-order and compact modeling techniques and are hence rigorously formulated in terms of conjugated variables (“across” and “through”-variables). This allows to use the generalized Kirchhoffian network theory [6] as theoretical framework for the formulation of the entire system model. The exchange of energy and other quantities between the interlinked submodels within this network is then intrinsically governed by Kirchhoff’s laws. Another advantage of a GKN-based macromodeling approach is that both compact and distributed models can be combined within the same simulation environment [5], which enables dedicated tailoring of the model complexity according to the needs and required accuracy. For the simulation of the dynamic pull-in and pull-out of the switch, the formulation of the four following submodels is required: a mechanical submodel comprising the membrane and its suspensions, a model of the electrostatic forces attracting the membrane, a model accounting for the damping forces exerted by the surrounding air and a contact model that correctly reproduces the landing of the membrane. The complete macromodel of the RFMEMS switch is then obtained by interlinking all submodels. Using coupling through Kirchhoff’s current law, it reads in its most general formulation: Mmech = Mfluidic + Melectrostatic +



Mx

(1)

x

Fig. 3. Magnification of the view on the switch in its closed state from a 30◦ angle (FIB image). The electrode with the dielectric layer on top is marked with (1). The substrate with its dielectric layer on top is marked as (2). The contact pad is marked as (3). The membrane is marked as (4). As can be seen in the picture, the surface roughness of the membrane is considerable. In the closed state, residual air gaps between the contact pads and the membrane up to 50 nm are locally available.

to enable a reliable estimation of viscous damping. A focused ion beam (FIB) setup was used to determine the exact widths of the gap and the surface roughness of the membrane (cp. Fig. 3). The technical data of the switch is summarized in Table 1. Table 1 Technical data of the investigated RF-MEMS switch. Membrane Length × width × thickness Side length of holes Spacing between holes

260 ␮m × 140 ␮m × 5.2 ␮m 20 ␮m 20 ␮m

Suspension beams Length × width × thickness

165 ␮m × 10 ␮m × 2 ␮m

Gap widths Membrane to contact pad (gc ) Membrane to electrode (ge ) Membrane to contact pad (gsub ) Effective residual air gap (gmin )

1.7 ␮m 2.7 ␮m 3.4 ␮m ≈20 − 50 nm (see Fig. 3)

For the electrode and the substrate, the gap width is given between the membrane and the dielectric layers situated on top of the electrode and the substrate, respectively.

Here, Mmech denotes the mechanical moment, Mfluidic the moments acting on the membrane due to viscous damping forces, Melectrostatic the moment due to the electrostatic actuation and Mx any other moments that need eventually to be taken into account. The submodels that are implemented to calculate the different contributions to this equation and their underlying theoretical fundamentals are outlined in the following subsections. 3.1. Mechanical submodel A well-established and widely approved approach to derive linear reduced-order models for a given problem is the Galerkin method, where the dynamics of the system is described by a superposition of – ideally only a few – eigenvectors of the underlying system of equations. In order to check whether this approach can be applied with sufficiently high accuracy to the mechanics of the investigated RF-MEMS switch we performed an initial FE analysis in ANSYS, where we applied a uniform pressure to the membrane and determined the vertical displacements zd of the membrane center employing linear as well as nonlinear finite element formulations. Intrinsic stresses have been taken into account by adjusting the fundamental resonance frequency of the model to the measured value. The results are compared in Fig. 4. The error amounts to 1.5% or less for a membrane displacement up to one-third of the gap ge , and up to 4.6% in the collapsed state (zd = gc ), i.e. the mechanical restoring force is slightly underestimated in this region by the linear model (cp. Fig. 4) and a small error in the calculation of the release point is to be expected. Since for the considered RF-MEMS switch this is acceptable for further investigations we derived a linear mechanical model of the membrane and its suspensions where the mechanical eigenmodes of the structure serve as a set of basis functions. Thus, the vector of displacements z(t) of the discretized

M. Niessner et al. / Sensors and Actuators A 172 (2011) 269–279

E1V,j (q). The corresponding continuous capacitance function is then obtained as:

Displacement [µm]

0 Nonlinear FEM model Linear mechanical model

−0.5

271

Cj (q) = 2

−1

E1V,j (q)

(4)

(1 V )2

−1.5

In a third step, Lagrangian functionals Mel,i are derived for each respective eigenmode. From the physical point of view they act as modal moments and have to be incorporated as mechanical “through”-variables in Eq. (1), which realizes straightforwardly the coupling between the mechanical and electrostatic energy domain. In terms of the capacitance functions of the seven electrode fingers and the voltage V, the Lagrangian functionals read:

−2 −2.5 −3 0

200

400

600

800

1000

1200

1400

1600

Pressure load [Pa] Fig. 4. Vertical displacement zd of the membrane center versus a pressure load that was uniformly applied across the membrane. The results were obtained from the FE analysis of a calibrated RF switch model in ANSYS.

Mel,i (q) =

7  ∂Ej (q) j=1

suspended membrane can be approximated by superposing a number m of linearly independent, weighted and discretized eigenmode shape functions i and adding an offset z 0 accounting for the displacement in the equilibrium state [7,8]: z(t) ≈ z 0 +

m 

qi (t)i

(2)

q¨ i + ω2 qi = Mexternal,i

I=

7 

Q˙ el,j =

j=1

(3)

Here, ωi denotes the angular eigenfrequency of the i th eigenmode and Mexternal,i an external modal moment, e.g. due to the electrostatic actuation, that couples with the i th equation of motion. For the considered RF-MEMS switch it proved sufficient to take into account only two structural modes: the fundamental mode at 14.7 kHz and the next higher, completely symmetric mode at 130 kHz. Equation system (3) constitutes the mechanical reduced order model with the modal amplitudes qi and the modal moments Mi being the “across” and “through”-variables, respectively. 3.2. Electrostatic submodel Since the deflection of the suspended membrane is relatively small compared to its lateral dimensions, the geometry of the electrostatic problem with the membrane and the electrode fingers is very close to a parallel plate capacitor configuration. Thus, the following, straightforward three-step procedure is applied for generating the electrostatic compact model. First, the electrostatic energy stored between the suspended plate and the electrode fingers is calculated. This is accomplished by segmenting the area of each electrode finger, introducing an elementary parallel plate capacitor for each segment and, eventually, by adding up the electrostatic energy stored in the single parallel plate capacitors. The 700 nm thick dielectric layer on top of the electrode fingers is also included in the model. By varying the modal amplitudes qi , the electrostatic energy E1V,j,discrete of the j th finger for a voltage of 1 V is calculated as a discretized function in terms of the considered eigenmodes q = [q1 ..qi ..qm ]T . Second, a polynomial in q is fitted to the specific energy function in order to get the continuous electrostatic energy function

(5)

j=1

To complete the electrostatic model we have to introduce additionally the relation between the total electric current I through the capacitor formed by the electrode fingers and the voltage V:

i=1

The physical interpretation of the scalar qi (t) that scales a corresponding eigenmode shape i is then that of a modal amplitude. Ideally, the number m of eigenmodes that are necessary to achieve an adequate accuracy is relatively small (in the order of two to ten) so that, after applying a modal transformation [9] scheme, the undamped mechanical behavior of the suspended membrane can be described through a small number of m uncoupled equations of motion:

∂qi

V 2  ∂Cj (q) 2 ∂qi 7

=

7  d

dt

(Cj V ) =

j=1

 m  7   ∂Cj j=1

i=1

∂qi



 q˙ i V

+ Cj V˙

(6)

Here, Qel,j denotes the electric charge stored on the j th electrode finger. Eq. (6) represents the Kirchhoffian network formulation for the electrical energy domain in terms of the “across”-variable V and the “through”-variable I. 3.3. Fluidic submodel The modeling of viscous damping effects is in many respects a challenging task, since it is, by its nature, a distributed effect which, in the most general and rigorous physical approach, is described by the complex and nonlinear Navier–Stokes equation. Under the assumption of small Reynolds numbers (laminar flow) and certain geometrical features (small air gaps underneath relatively large, laterally extended membrane structures), which applies for a variety of microdevices, the well-known and much simpler Reynolds equation can be employed instead to calculate the so-called squeeze-film damping (SQFD) exerted by the surrounding atmosphere [10]. Furthermore, for the investigated RF-MEMS switch, the lateral velocities accounting for the Couette flow can be omitted, since the membrane is moving in the vertical direction. The resulting Reynolds equation then reads:

 ∇



g 3 ∇p 12

=

∂g ∂ +g ∂t ∂t

(7)

Here,  denotes the density of air,  the viscosity of air and g the gap width. However, Eq. (7) does neither account for the loss of fluid flowing through the perforations, which are often present in microstructures, nor for the pressure drop at the outer border of the suspended membrane. In order to include these effects, the mixed-level approach as presented in [11] is adapted for the modeling of the damping forces in this work. The general idea of this approach is to evaluate the nonlinear Reynolds equation where applicable using a finite network and to employ physics-based compact models to account for holes and effects at the outer border. Since the approach comprises both distributed and lumped element models, it is further referred to as mixed-level modeling (MLM) approach. The following paragraphs briefly outline the special features of this fluidic mixed-level model.

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time-varying gap height due to the moving membrane. Here, dkl denotes the distance between two neighbouring nodes, gkl the average gap width of two adjacent nodes, wkl the width of the Poiseuille flow channel between two nodes, Ak the nodal area of the k th node, P0 the ambient pressure and Pk0 the pressure difference between the pressure at the k th node and the ambient pressure. ϒ Reynolds is a correction factor, which accounts for gas rarefaction occurring at low pressures and/or for small device dimensions (for details please refer to the Appendix A). All geometrical parameters are directly obtained from the discretization of the Reynolds equation and by an automated model generation procedure, as described in Section 3.6. For the derivation of QC,k0 , ideal gas was assumed, i.e. the change of  to be proportional to the change of the pressure Pk0 .

Fig. 5. Rectangular plate moving towards a fixed rectangular plate (upper graph) and a finite network model of this situation (lower graph). The term on the left-hand side of Eq. (7) is modeled by a resistor. The first term on the right-hand side of Eq. (7) is modeled by a source and the second term by a capacitor.

3.3.1. Evaluation of Reynolds equation by a finite network According to [11], the Reynolds equation is first spatially discretized in a flux-conserving way, so that a fluidic finite network consisting of nonlinear fluidic resistors, sources and capacitors is obtained (see Fig. 5). This Kirchhoffian network is governed by the pressure differences Pkl between nodes and the flow rates Qkl along network edges as “across”- and “through”-variables. The coupling between the fluidic and the mechanical model, and, thus, the calculation of the damping force on the structure, is automatically realized in this approach through the air gap g, which is simultaneously an input parameter of the Reynolds equation (Eq. (7)) and an output parameter of the mechanical model. The local gap width gk at the k th network node reads: gk (q, t) = g0,k +

m 

i,k · qi (t)

(8)

i=1

Here, g0,k denotes the initial gap width at the k th node and i,k denotes the value of the discretized eigenmode shape of the i th eigenmode at the k th fluidic network node. In that way, the described finite network approach enables also the modeling of non-uniform air gaps as it is the case of the RF-MEMS switch investigated here. All flow rates governing the above mentioned fluidic network can be directly derived from Eq. (7) and read [11]: Qkl =

3 wkl · gkl

12 · dkl

QC,k0 =

· Pkl · ϒReynolds

Ak · gk · P˙ k0 P0

QS,k0 = Ak · g˙ k

(9) (10) (11)

In its physical interpretation, Qkl is the flow rate between two neighbouring nodes and is responsible for the dissipative losses, QC,k0 represents the flow through a fluidic capacitor and models the compression of the air inside the varying gap, and QS,k0 is imprinted by a fluidic source, which stands for the fluid flow induced by the

3.3.2. Physics-based compact models In this section we present compact models that account for additional pressure drops and, thus for dissipative losses, which cannot be covered by the Reynolds equation. These occur at the edges of the moving structures and at perforations if present. An air flow leaving and entering the gap underneath the membrane through the outer borders can be considered similar to an air flow passing through an elliptical orifice [12]. Consequently, a certain pressure drop occurs between the nodal pressures at the outer border and the ambient pressure, whereas, originally, in Eq. (7) ambient pressure is assumed at the structure’s boundaries. The additional pressure drop is modeled by means of a fluidic resistor added at these locations. Sattler [12] derived the respective border resistance RB,k analytically by calculating the flow through an elliptical orifice: RB,k =

Pk0 3  −1 =· 2 · ϒBT Qk0 gk · bk

(12)

Here,  = 0.84 denotes a parameter, which accounts for geometrical non-idealities which cannot be covered by this analytical formula. The parameter  was fitted by 3D FE simulations and showed to be constant throughout all geometrical variations. bk is the length of the outer border assigned to the k th network node and ϒ BT is again a correction factor for gas rarefaction (see Appendix A). Similarly, the gas flowing through each r th perforation hole of the suspended membrane is modeled by introducing three fluidic resistors [12,13], each accounting for a different contribution of the fluid flow to the additional pressure drop occurring at each hole (see Fig. 6). The first resistor RT,r models the flow of air from the gap underneath the membrane into the hole (transition region). The second resistor RC,r models the resistance of the channel-like part of the perforation the fluid has to pass through. The third resistor RO,r includes the contribution, when the fluid leaves the channel and is modeled as orifice flow. The equations that account for these resistances are analytically derived by Sattler [12] and read: RT,r =  · RC,r = RO,r =

3  gr2 br

12Lr 0.42sr4 21 2sr3

−1 · ϒBT

· ϒC−1

· ϒO−1

(13)

(14)

(15)

Here, gr denotes the gap width at the r th hole, sr the side length of the square hole, br = 4sr the perimeter of the square hole and Lr the length of the fluidic channel, i.e. the thickness of the membrane. ϒ C and ϒ O are correction factors for rarefaction (see Appendix A). The employed compact models RT,r , RC,r and RO,r are only valid for square holes. Compact models for circular perforations can be found in [12].

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273

3.4. Contact submodel

ambient pressure P0

The elastomechanical contact between the membrane and the contact pads is modeled by formulating elastic spring forces FCF,s for each s th contact pad:

RO

FCF,s = kc · gs (q)

moving plate

Here, kc denotes the contact stiffness that models the additional stiffness added to the system when the membrane is penetrating the pads and gs denotes the gap width between the membrane and the pads. The contact forces are only present if penetration occurs, i.e. if:

moving plate

RC

(20)

gs (q) ≤ 0

(21)

In order to enhance the numerical stability of the simulation, a tanh-function s (q) similar to [14] is used to implement this condition:

RT



s (q) = 0.5 ·

finite network

1 − tanh

finite network fixed plate

Fig. 6. Illustration of the MLM approach. A hole is modeled and embedded within the finite network by connecting all nodes of the finite network that are located next to it to the resistor RT modeling the transition region when the fluid enters the channel. The resistor RC models the channel resistance. The resistor RO models the orifice flow.

    gs q

The s (q)-function ranges between the values 0 and 1. The parameter ˇ determines the steepness of the transition between these values. For this study, ˇ is chosen to ˇ = gc × 10−9 , i.e. a very steep transition. The modal contact model is then obtained by multiplication with an averaged eigenmode shape function value i,s for the s th contact pad: Mcontact,i =

12 

i,s · s (q) · FCF,s =

s=1

The total gas flow through a perforation hole can then be written in terms of the pressure drop Pr0 and the flow rate Qr0 : Pr0 = (RC,r + RO,r + RT,r ) · Qr0

(22)

ˇ

12 

i,s · s (q) · (kc · gs (q)) (23)

s=1

3.5. Fully coupled model for viscously damped MEMS devices

In addition to Qr0 , also the relative motion of the air in the channel of the moving perforated membrane has to be accounted for. Eq. (16) extended with this “moving channel correction” reads:

According to the general Eq. (1) in Section 3 the subdomain models derived in Sections 3.1–3.4 can now be used to assemble the fully coupled macromodel. To this end, Mexternal,i in Eq. (1) has to be replaced with the sum of the modal moments from the fluidic and contact submodels:

Pr0 = (RC,r + RO,r ) · (Qr0 + Qr0,rel ) + RT,r · Qr0

˙ P0 ) + Mshear,i (q, q, ˙ P0 ) q¨ i + ω2 qi = Mel,i (q, V ) + Mreynolds,i (q, q,

(16)

(17)

Here, Qr0,rel = sr2 g˙r is the relative air flow in the channel with respect to the moving perforated membrane, where g˙r is the velocity of the membrane. 3.3.3. Calculation of the total damping force The total damping force on the structure can be calculated from the nodal pressures Pk and the nodal areas Ak according to Fk = Pk · Ak . In order to obtain the modal moment, which has to be included as “through”-variable in the total system model (Eq. (1)), the nodal forces are multiplied with the discretized eigenmode shape function [9]: Mreynolds,i =

n 

i,k · Fk =



k=1

i,k · Pk · Ak

(18)

k

Here, n denotes the number of nodes of the fluidic finite network. In addition to the nodal damping forces exerted by squeezing the air out of the gap underneath the membrane (Fk ), also the shear forces Fr along the channel walls of the perforations have to be accounted for. Using the analytic expression derived by Sattler [12], the moment due to the shear forces in the holes reads accordingly: Mshear,i =

18  r=1

i,r · Fr =

 r



r,k · sr2 · RC,r · Qr0 + Qr0,rel



(19)

+ Mcontact,i (q)

(24)

In this formulation, the modal mechanical eigenmode equations are now no longer independent from each other. The interaction with each other, which means the coupling between the single energy domains, enter Eq. (24) through the right-hand side, where the electrostatic, fluidic and contact models are included. Additionally, we would like to point out that the derived submodels are generic and, up to this point, no device specific assumptions or simplifications are made, so that they can be applied to any viscously damped and electrostatically actuated microdevice. Since all models are formulated in terms of Kirchhoffian network variables and rules, this macromodel can be implemented in any standard circuit simulator that offers the feature to implement models via a hardware description language. Some technical aspects of the macromodel derivation and its software implementation are addressed in the following section. 3.6. Automated model generation Once generic models for microdevices or parts of them have been derived and evaluated successfully, the problem- and geometry-specific modeling should be facilitated and automatized as far as possible. To this end, a MATLAB-based toolbox has been developed at our institute [15–17]. Starting from a given FE model,

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Fig. 8. Block diagram of the setup for pressure-dependent measurements. The DUT is enclosed in an on-purpose built pressure chamber with optical window. The optical unit directs the laser beam towards the DUT, so that the out-of-plane velocity can be measured. A combined data acquisition and arbitrary waveform generator unit synchronizes the actuation and measurement signals. Two pressure sensors serve as readout. The sensors are connected to the PCU (lines are omitted for the sake of clarity). The PCU steers two valves. One valve connects to the ambient atmosphere and the other one to an ADIXEN hybrid membrane and turbo molecular vacuum pump.

Fig. 7. Workflow for the generation of multi-energy domain coupled macromodels using an especially developed MATLAB-based toolbox.

it enables the automated generation of hardware description language (HDL)-based macromodels, which can be evaluated using a standard circuit simulator. The basic model generation flow is shown in Fig. 7. In this work, the simulator SPECTRE from CADENCE was used. 4. Results and discussion For the characterization, a white light interferometer (WLI) and a laser vibrometer (LV) embedded within a dedicated experimental setup as described in Section 4.1 were used to measure several samples of the considered RF MEMS switch design. The derived macromodel was evaluated w.r.t. the performed measurements (see Section 4.2) as well as w.r.t. alternative macromodeling approaches that also use GKN as a theoretical framework (see Section 4.3). 4.1. Experimental setups The geometry of the switches (Section 2) was mainly characterized with a WYKO NT1100 DMEMS white light interferometer (WLI) from VEECO, which is able to provide a magnified, highresolution static 3D profile of a MEMS structure, and, thus, to measure both the planar dimensions and the height of a given device. Transient measurements were performed with a second experimental setup based on an OFV-5000 single spot laser vibrometer (LV) from POLYTEC. As, here, the measurement principle is

based on the Doppler effect, the primary measured quantity constitutes the velocity of the structure and transients are recorded in real-time. This offers the possibility to carry out a large number of measurements and thereby to broaden and consolidate the experimental basis for model evaluation, and, besides of that, also enables the investigation of non-periodic transients (like switching) of a mechanical device. An especially designed pressure chamber has been added in order to enable pressure-dependent transient measurements for a thorough evaluation of the fluidic model with its correction factors for the low-pressure regime. Fig. 8 shows a block diagram of the complete setup. An essential component here is the on-purpose built pressure control unit (PCU) that determines the pressure inside the chamber via two high-precision capacitive pressure sensors and controls it by steering two electric valves.

4.2. Experimental verification of the derived macromodel The experimental verification of the presented macromodel was carried out in three steps. First, the quasi-static pull-in and pull-out characteristics of the devices were measured in order to evaluate the submodels that describe the electromechanical coupling and the quasi-static contact situation of the switch without dynamic impact and viscous damping. Second, the fluidic submodel, which accounts for the viscous damping, was verified by transient pressure-dependent measurements. Finally, pull-in and pull-out transients of the devices were measured in order to evaluate the coupling of all submodels and, thus, the full macromodel of the device, which will be abbreviated by the acronym “MLM model” (mixed-level macromodel) in the following sections.

M. Niessner et al. / Sensors and Actuators A 172 (2011) 269–279

400

275

Damped oscillations after release

200

Displacement [nm]

0 −200 −400 −600 −800 −1000 −1200 −1400 −1600 −1800 0

LV measurement MLM model

5

10

15

20

25

30

Bias voltage [V] Fig. 9. Measured (laser vibrometer) and simulated quasi-static pull-in/-out characteristics of the membrane as a response to a triangular waveform. Only the first half of the cycle with positive voltages rising from 0 to 35 V and decreasing from 35 V to 0 V is shown.

4.2.1. Quasi-static pull-in and pull-out characteristics The measurement of the quasi-static pull-in/-out characteristics was performed using the LV. In order to minimize the effect of dielectric charging [18,19], the switch was actuated by a voltage of 70 V pp (peak-to-peak) with triangular waveform at a frequency of 1 Hz and a mean value of 0 V. The pull-in of the inspected samples occured between 29 V and 31 V. The release occurred between 22 V and 26 V. Fig. 9 shows a comparison of the measured and the simulated data. For the latter, the same actuation signal as in the measurement was applied to the MLM model. Concerning the pull-in characteristic and the pull-in voltage, the simulation and measurement are in very good agreement, which proves that the modeling approaches employed for the mechanical and electrostatic subdomain and the resulting macromodels cover the physics underlying the device operation accurately in this regime. On the other hand, the MLM model calculates a too early release. Dielectric charging is, in the time scale of this measurement, not likely to be responsible for the additional forces leading to the lower release voltage, as the measurement proved to be reproducible for different frequencies and a larger number of cycles without notable shift in the pull-in or release voltages. Adhesion forces may be relevant, but for the investigated switch design, state-of-the-art physics-based gold-to-gold adhesion models as presented in [20,21] predict opening times, i.e. the time that elapses between turning off the hold voltage and the initial release of the membrane, of 1 ms and higher, whereas the opening times of the inspected switches were constantly around 13 ␮s for different actuation waveforms and frequencies. As the release voltages measured for the set of specimens vary significantly, and, particularly, show also release voltages in the magnitude of the simulated release voltage, the authors decided to accept this uncertainity in the quasistatic release voltage for this work. 4.2.2. Pressure-dependent characterization A crucial point in modeling the dynamics of micromechanical actuators is the correct inclusion of viscous damping effects into the model, since damping is, by its nature, a distributed effect and depends strongly on the device topology. Recent results presented by Veijola et al. [22] and De Pasquale et al. [23] showed that fluidic macromodels may deviate tremendously from measurements and should therefore be thoroughly validated by adequate experiments. The quality factor Q is chosen as measurand for the quantification of the damping forces. Lerman and Elata [24] showed that the quality factor may also depend on the respective primary source of the measured data, i.e. the electric current, the displacement

Fig. 10. Measured (laser vibrometer) and simulated quality factors. In the experiment, the device was biased with 5 V DC and actuated with white noise of 1 V amplitude. In the simulation, the device was actuated with a voltage step and the quality factor was extracted from the envelope of the damped oscillation. The RFMEMS switch sample investigated in this experiment had a different gap width than given in Table 1: gc = 2 ␮m, ge = 3 ␮m, gsub = 3.7 ␮m, respectively. Table 2 Relative error Edamp,rel of the measured and simulated quality factors at different pressures P0 . P0 [hPa]

960

400

200

100

10

1

Edamp,rel

3.7%

8.5%

9.4%

14.4%

28%

126.4%

or the velocity. Especially in the presence of multiple resonance frequencies, the displacement gain often decreases for successive peaks while the velocity gain remains constant. Consequently laser-Doppler vibrometry, which determines the velocity of the membrane directly, is an advantageous characterization method for viscous damping effects and will be used as primary source of data in this work. Several samples were bonded and mounted in the pressure chamber of the LV-based setup for carrying out a systematic transient and pressure-dependent characterization. Fig. 10 shows a comparison between the measured and the simulated quality factors of the RF-MEMS switch design. In the experiment, the sample was biased with a DC offset of 5 V and actuated by a white noise signal with an amplitude of 1 V. This way, the DC-induced deflection of the membrane could be neglected, the ACinduced amplitude was just in the range of several nanometers. The experimental quality factors were extracted from the measured frequency spectrum where the step size has been chosen to 3.125 Hz. In the simulation, we did not use the SPECTRE small-signal analysis feature as it uses a linearized model only. Instead, we used the fully nonlinear coupled macromodel, applied a voltage step in the time domain in order to generate a damped oscillation and extracted the quality factor from the envelope of this oscillation. As shown in Fig. 10, the simulated and measured quality factors match very well at high ambient pressures, but the MLM model starts to first overestimate and then to underestimate the damping, increasingly with falling pressure. The underestimation at low pressures is due to other damping mechanisms such as thermoelastic damping, anchor loss or electrical damping [25] that are not included in the MLM model and limit the quality factor of the device in this pressure regime. In Table 2 the relative error Edamp,rel for different pressure values is listed. It exceeds a threshold of 10% for pressures below approximately 200 hPa and increases with lower pressure. Nevertheless, our experiments showed that the fluidic submodel, described in Section 3.3, provides good accuracy for varying pressure. As the correction factors are functions of both the pressure and the gap width (see Appendix A), this finding implies

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1.5

0 LV measurement MLM model

Displacement [nm]

Displacement [μm]

0.5 0 −0.5 −1

−400 −600 −800 −1000 −1200 −1400 −1600

−1.5

−1800 0

MLM model Architect3D model Iannacci model LV measurement

−200

1

1

2

3

4

5

0

10

20

30

Fig. 11. Measured (laser vibrometer) and simulated pull-in/-out transient of the membrane of an RF-MEMS switch sample. The sample was actuated with a rectangular voltage pulse at 200 Hz with an amplitude of 35 V (on) and 0 V (off), respectively. Duty-cycle 50%; ambient pressure 960 hPa. The measured velocity was integrated to obtain the membrane displacement.

40

50

60

70

80

Time [µs]

Time [ms]

Fig. 12. Measured (laser vibrometer) and simulated pull-in transient of the membrane of one RF-MEMS switch sample. The sample was actuated with a rectangular voltage pulse at 200 Hz having amplitudes of 35 V (on) and 0 V (off), respectively. Duty cycle: 50%; ambient pressure: 960 hPa. Only the first 80 ␮s are shown. The measured velocity was integrated to obtain the membrane displacement.

0

• During the landing phase (pull-in transient), a mode at 218 kHz dominates the behavior of all measured structures. For some samples, also a second, but by magnitudes weaker mode at 87 kHz is present. By scanning different spots of the membrane with the LV, we assured that there are no others than these two modes present during the switching operation. • During the release phase after pull-in/contact occurred, two modes at 14.7 kHz and 136 kHz are superimposed. • During the release phase, after an actuation with a voltage lower than the dynamic pull-in voltage, only one eigenmode at

Peak−normalized amplidute (dB)

4.2.3. Pull-in and pull-out transients Finally, the fully coupled macromodel is now evaluated on the basis of transient pull-in and pull-out characteristics. To this end, the complete pull-in and pull-out transients of several switches were recorded by the LV at a sampling rate of 2.56 MHz. The samples were actuated by a voltage of 35 V with a rectangular waveform at 200 Hz and a duty-cycle of 50%, so that the membrane first collapses onto the contact pads and then, after being released, returns after a damped oscillation to its initial (rest) position. The equivalent actuation was applied in the simulation, which is compared to the measured transients in Fig. 11. The MLM model reproduces the pullout transient, i.e. the damped oscillation after release, with very good agreement. This was to be expected because the mechanical, as well as the fluidic submodel, already proved to deliver accurate results. The pull-in of the membrane was simulated without any convergence problems. Whilst in contact, the gap width between the membrane and contact pad is fixed at gmin as a lower limit (see Table 1). Fig. 12 shows the simulated and measured pullin transients of the membrane during the first 80 ␮s. Obviously, the measured pull-in transient resembles a vibration with several mechanical eigenmodes involved. A FFT analysis of the landing phase (actuation with 35 V, cp. Fig. 13), of the release phase after actuation with 35 V (and after pull-in/contact occurred, cp. Fig. 14) and of the release phase after actuation with 25 V (so that no pullin/contact occurred, also cp. Fig. 14) of different samples reveals the following findings:

−10 218 kHz

87 kHz −20 −30 −40 −50 −60 0

50

100

150

200

250

300

Frequency (kHz) Fig. 13. Frequency spectrum of the landing phase of the membrane of an RF-MEMS switch sample measured by laser vibrometry (actuation voltage 35 V, frequency 250 Hz). A Hanning window was used for the FFT in MATLAB.

0 136 kHz

35V (after pull−in) 25V (no pull−in)

−10

Peak−normalized amplitude (db)

that our fluidic submodel will also perform well in calculating the damping for varying, and especially decreasing, gap widths whilst actuation at normal pressure, what is essential for the modeling of RF-MEMS switches. One of our recent works further confirms this statement [26].

14.7 kHz −20 −30 −40 −50 −60 0

50

100

150

200

250

300

Frequency (kHz) Fig. 14. Frequency spectrum of the release phase of the membrane of an RF-MEMS switch sample measured by laser vibrometry (actuation voltages 35 V and 25 V, frequency 250 Hz). A rectangular window was used for the FFT in MATLAB.

14.7 kHz, namely the fundamental eigenfrequency of the device, occurs. • Since the mode at 136 kHz is only present after contact, these findings substantiate the assumption that multiple structural modes, which are also different than the fundamental one, are activated whilst impact. The modes at 14.7 kHz and 136 kHz occurring during the release phase correspond to the fundamental and the next higher,

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4.3. Comparison with alternative state-of-the-art macromodels Having validated the MLM model experimentally (Section 4.2), it is now benchmarked versus two other state-of-the-art macromodels using also GKN as a theoretical framework. The first model presented by Iannacci [27] (abbr. “Iannacci model”) uses solely lumped element models: four lumped linear beam models for the suspensions, a rigid plate model with both translational and rotational degrees of freedom for the membrane, and a lumped capacitor for the electrostatic domain. The viscous damping is calculated through a modified Reynolds equation presented by Bao [28] that accounts also for perforations. The contact is modeled by a linear elastic force with a contact stiffness similar to Eq. (20). The second model also consists of lumped elements and is compiled in the commercial software environment Architect3D from COVENTOR [29] (abbr. “Architect3D model”). Here, a high-order nonlinear beam model is used for the suspensions. The membrane is modeled by the mixed interpolation of the tensorial components approach (MITC), the electrostatic domain by a lumped capacitor, and the viscous damping by a generic lumped submodel that is based on the Reynolds equation without any modifications for geometrical specifics. For complex geometries, the user manual recommends to perform a FE analysis of the fluidic problem and to extract a mathematically reduced model for the simulation in Architect3D. In this work, the lumped Reynolds-based model is used. For the mechanical contact, an available standard model is included in the simulation which is, however, not described in detail in the user manual. Both models performed equivalently to the MLM model in case of the quasi-static pull-in, but the models also predict a too low release voltage. As the Iannacci model performed very well for other RF-MEMS devices in this point [27], this finding substantiates the assumption that contact-related phenomena, not yet considered in the models, are responsible for the mismatch between simulation and measurement in the case of the investigated samples, and not the quality of the electromechanical models themselves. Both models were also able to simulate the dynamic pull-in transient, but a lot of adjustments were necessary concerning the solver parameters in order to achieve convergence of the simulation within Architect3D. Here, another challenge was also to calibrate the very generic damping model so that reasonable results were obtained. Even with the best parametrization, the Architect3D model still overestimates the damping whilst the oscillation after the release of the membrane (cp. Fig. 15) whereas the Iannacci model performed equivalently to the MLM model. In case of the initial contact phase between the membrane and the pads, both models also failed to exactly reproduce the real behavior of the membrane (cp. Fig. 12) and perform worse than the MLM model. Architect3D overestimates the closing time, which

1500 Architect3D model LV measurement

1000

Displacement [nm]

completely symmetric mode of the structure, which are also included in the mechanical submodel of the MLM model (see Section 3.1). Differently, the mode at 218 kHz could not be identified clearly. A modal analysis performed in ANSYS predicts several modes around this frequency, but it might also be possible that the high contact stiffness causes a shift of a mode at a lower frequency to 218 kHz. Up to now, locally resolved LV measurements did not give enough evidence for a clear identification of the shape of this vibration. Nevertheless, the MLM model reproduces the landing phase already with a remarkably good agreement (see Fig. 12), which is certainly due to the fact that the underlying approach includes, by its nature, different (and also higher) structural modes in the mechanical model as well as in the contact model (see Sections 3.1 and 3.4).

277

500 0 −500 −1000 −1500 2.5

3

3.5

Time [ms] Fig. 15. Measured (laser vibrometer) and simulated pull-in/-out transient of an RFMEMS switch sample. The sample was actuated with a rectangular voltage pulse at 200 Hz having amplitudes of 35 V (on) and 0 V (off), respectively. Duty cycle: 50% duty-cycle; ambient pressure: 960 hPa. For the sake of clarity, only the release phase t = 2.5–3.5 ms is shown and the results from the Iannacci model are omitted.

might originate from still suboptimal solver settings, and the overestimation of damping calculated by the very generic fluidic model. Concerning simulation time, the Iannacci model is, due to its lumped descriptions, the fasted model. The simulation of a complete pull-in and pull-out transient is performed within only 10 s (using a single CPU at 2.8 GHz). As the MLM model explicitly includes the eigenmode at 130 kHz and employs a full mixed-level model for the evaluation of damping, it is the slowest model, taking 300 s to simulate. However, the simulation time of MLM model can potentially be reduced drastically [16,17]. The simulation time of the Architect3D model is with 100 s between the Iannacci and MLM models. 5. Conclusions We derived a multi-energy domain coupled macromodel of an RF-MEMS switch applying a systematic hierarchical modeling approach. The presented macromodel consists of lumped and distributed reduced-order models formulated in a generalized Kirchhoffian network description (mixed-level macromodel—MLM model). It is physics-based and, thus, shows very good agreement with the measured quasi-static pull-in characteristic, the noncontact transient measurements for ambient pressures down to about 200 hPa, and the pull-out transient after contact. Due to its capability to explicitly account for multiple structural modes, the MLM model produces a pull-in transient that is close to the real situation: the two modes of the mechanical submodel are coupled whilst impact through the mechanical contact force, and result in a superimposed vibration. However, the MLM model fails in predicting the correct release voltage in the quasi-static pull-out scenario. Obviously, this is due to contact-related phenomena, other than dielectric charging and gold-to-gold adhesion force, that still have to be identified and implemented in the model. When benchmarked against the Iannacci and Architect3D models, the MLM model performs equivalently, if not better in various aspects. Having a closer view to the initial contact of the membrane to the counter electrode, the MLM model reproduces the measured curve with better agreement than the Iannacci and Architect3D models. This is certainly due to the fact, that different mechanical eigenmodes are included in this model. The predictive simulation of viscous damping seems to remain a challenging task, although a lot of work has been done in this field by Schrag [11], Sattler [12] and Veijola [22]. As can be seen in the case of the Architect3D model, the strong influence of the topography of the moving structure on the fluidic damping prohibits the use of generic models,

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and requires a tailored modeling strategy, as used in the MLM and Iannacci models. The presented study identifies three issues for future research. First, as none of the three evaluated models was able to capture the correct release voltage, efforts need to be directed towards the identification of the effects leading to this mismatch. Adhesion models as presented in [30] should be considered. Second, the structural mode at 218 kHz that dominates the initial contact phase should be identified, e.g. using a laser scanning vibrometer, in order to get a deeper understanding of the contact physics of this phase. The third very important issue is the experimental evaluation of the fluidic mixed-level submodel. Our investigations showed that a very good agreement of our MLM model with the measurements was obtained down to pressures of approximately 200 hPa; an error threshold of 10% is exceeded for lower pressures only. This is contradictory to the results of De Pasquale [23], who found high relative errors of 38% and more at normal pressure. Using the setups described in Section 4.1, we carried out a systematic and pressuredependent study for the evaluation of the mixed-level approach used in Section 3.3. First results [31] show that the approach delivers equivalently low relative errors also for other structures and, indeed, performs better than the Veijola models used by De Pasquale, making this modeling technique a very promising option for the calculation of squeeze-film damping in MEMS. This is true especially because only geometry- and design-based parameters are included in the mixed-level model, and it can therefore be automatically generated from a given discretization, as described in Section 3.6, and applied to arbitrary device topographies. Acknowledgments The authors would like to thank H. Mulatz and M. Becherer from the Institute for Technical Electronics of the Munich University of Technology for their efforts to bond samples of the devices and for taking SEM and FIB images with the equipment available at their institute. The authors furthermore acknowledge the Dr. Johannes Heidenhain Stiftung for their financial funding of this work. Appendix A. This appendix contains correction factors that take into account the effect of gas rarefaction, and reduce the fluidic resistances given in Section 3.3, accordingly with decreasing ambient pressure and/or air gap dimensions. The factors ϒ BT , ϒ C and ϒ O are taken from Sattler [12], who did a thorough review of publications by Beskok [32,33], Sharipov [34–36] and Veijola [37–40]. The factors ϒ Reynolds , ϒ BT and ϒ O are directly taken from (or based on) publications by Veijola [38–40] and the factor ϒ C is taken from Beskok [33]. The expressions read: ϒReynolds = 1 + 9.638Kn1.159

(25)

1 + 0.5D−0.5 · 30−0.238 1 + 2.471D−0.659

  √ 6Kn ϒC = 1 + 1.085Kn · arctan (8 Kn) 1 + 1 + Kn ϒBT ≈ ϒReynolds ·

ϒO =



1+

6.703Kn(1.577 + Kn) 2.326 + Kn

(26) (27)

 1 + 0.688D−0.858 −0.125  0 ·

1 + 1.7D−0.858 (28)

Here, Kn = f /df denotes the Knudsen number, where f is the mean free path, which changes with the ambient pressure P0 , and df denotes the characteristic length of the flow. The characteristic length is different for each correction factor (cp. Table 3). Further√ more, D = /(2Kn) represents the parameter of rarefaction and

Table 3 Correspondence of correction factors and characteristic lengths. Correction factor

df

ϒ Reynolds (Eq. (25)) ϒ BT (Eq. (26)) ϒ C (Eq. (27)) ϒ O (Eq. (28))

gk gk sr sr /2

√ 0 = Lr /(sr / ) the ratio of the channel length Lr and the side length of the hole sr . The presented correction factors are valid for square holes only.

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Biographies Martin Niessner received the B.Sc. (with honors) and diploma degrees (with honors) in electrical engineering from the Munich University of Technology, Germany, in 2004 and 2005, respectively. He was a visiting graduate student at the University of Illinois, Urbana-Champaign, from 2004 to 2005. In 2006 he joined the MEMS modeling group at the Institute for Physics of Electrotechnology at the Munich University of Technology and is currently working towards the doctorate degree in electrical engineering. His main research interests are the modeling and simulation of coupled effects in microsystems and model verification. Gabriele Schrag received her diploma in physics from the University of Stuttgart in 1993, where she continued her work until 1994. In 1995 she joined the Institute for Physics of Electrotechnology at the Munich University of Technology, working on modeling methods for microdevices and microsystems, with a special focus on fluid–structure interaction and viscous damping effects. In 2002 she received her doctorate degree (with honors) from the Munich University of Technology, her thesis covering the “Modeling of Coupled Effects in Microsystems on Device and System Level”. Since 2003 she has been heading the MEMS modeling group at the Institute for Physics of Electrotechnology. Her research activities are focused on methodologies for the predictive simulation and optimization of microdevices and microsystems, parameter extraction and model verification. Jacopo Iannacci received the M.Sc. degree in electronic engineering from the University of Bologna (Italy) in 2003 and the Ph.D. in information technology in 2007 from the ARCES Research Center (University of Bologna). In 2005 and 2006 he worked at the HiTeC DIMES Technology Center (Technical University of Delft, the Netherlands) in developing packaging solutions for RF MEMS and from October 2007 he joined Fondazione Bruno Kessler (FBK) in Trento (Italy) as Researcher on MEMS technology. His scientific interest is focused on compact modeling, design, optimization, integration, packaging and reliability, of MEMS/RF-MEMS devices and networks for sensors and telecommunication systems. He authored and co-authored about 80 scientific contributions in international conferences proceedings and journal papers. Moreover, he also authored a few book chapters and a couple of books in the field of RF-MEMS technology. Gerhard Wachutka received the doctorate degree from the Ludwig-MaximiliansUniversität, Munich, Germany, in 1985. From 1985 to 1988, he was with Siemens Corporate Research and Development, Munich, where he headed a modeling group active in the development of modern high-power semiconductor devices. In 1989, he joined the Fritz-Haber-Institute of the Max- Planck-Society, Berlin, Germany, where he worked in the field of theoretical solid-state physics. From 1990 to 1994, he was head of the microtransducers modeling and characterization group of the Physical Electronics Laboratory at the Swiss Federal Institute of Technology (ETH), Zurich. There, he also directed the micro-transducers modeling module of the Swiss Federal Priority Program M2S2 (Micromechanics on Silicon in Switzerland). Since Spring 1994, he has been heading the Institute for Physics of Electrotechnology at the Munich University of Technology, where his research activities are focused on the design, modeling, characterization, and diagnosis of the fabrication and operation of semiconductor microdevices and microsystems. Professor Wachutka is member of the IEEE, the American Electro-chemical Society, the American Materials Research Society, the ESD Association, the VDE Association for Electrical, Electronic and Information Technologies, the VDI Association of German Engineers, the German Physical Society, the American Physical Society, and the AMA Society for Sensorics.