multi field modeling of a microelectromechanical speaker system with electrostatic driving principle

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Microsyst Technol (2014) 20:995–1006DOI 10.1007/s00542-014-2102-2

TechnIcal PaPer

Multi field modeling of a microelectromechanical speaker system with electrostatic driving principle

David Tumpold · Manfred Kaltenbacher · Christoph Glacer · Mohsin Nawaz · Alfons Dehé

received: 29 July 2013 / accepted: 4 January 2014 / Published online: 11 February 2014 © Springer-Verlag Berlin heidelberg 2014

acoustic propagation domain with open domain character-istics. Finally, we present an optimization method taking advantage of stress induced self-raising realized with vari-ous merged layers with different intrinsic pre-stress. The buckling back plate concept can be compared to bimetal characteristics.

1 Introduction

nowadays technology trend of battery powered devices like laptops, tablets or smart phones goes towards smaller and thinner cases. For this reason, energy consuming parts are mostly replaced with energy efficient technology. Micro-electro-Mechanical-Systems (MeMS) loudspeakers, fabri-cated in complementary metal oxide semiconductor (cMOS) technology merge energy efficient driving technology with cost economical fabrication process. nowadays conventional MeMS speakers used in smart phones or tablets are based on the electro-dynamic driving principle. complexity in fab-rication, size in dimension and coefficient of efficiency are the disadvantages of electro-dynamic systems, but outline the advantages of electrostatically actuated MeMS speak-ers. The design and evaluation process of such electrostatic cMOS MeMS speakers is time consuming and expensive, but can be supported and optimized with computer aided engineering methods. We present a ternary coupled model, describing the interaction of the electrostatic-mechanical and acoustic field.

There are several systems available for micro speakers, but the most common working principle is the electro-dynamic driving principle. Some examples for electro-dynamic MeMS speakers can be found in (lee and hwang 2011a; Shahosseini et al. 2010, 2013; Bai et al. 2008; cohen et al. 2010a, b, c) for circular shaped membranes

Abstract The need for computer modeling tools capable of precisely simulating multi-field interactions is increas-ing. The accurate modeling of an electrostatically actu-ated Micro-electro-Mechanical-Systems speaker results in a system of coupled partial differential equations (PDes), describing the interactions between electrostatic, mechani-cal and acoustic fields. a finite element (Fe) method is applied to solve the PDes efficiently and accurately. In the first part of this paper, we present the driving technology of an electrostatic actuated Micro-electro-Mechanical-Systems speaker, where the electrostatic mechanical cou-pling is realized with reduced order electro mechanical transducer elements. The electrostatic attracting force is derived from the capacity to gap relation of our device. In a second investigation, we focus on generation of the gen-erated sound including open domain characteristics and propagation region optimization. The sound pressure level is computed with Kirchhoff helmholtz integral as well as with FeM by using cFS++. We use the Kirchhoff helm-holtz model to characterize the interactions of multiple speaker cells in arrays and the Fe tool for single speaker cell investigations. at the acoustic Fe model, the focus is on mesh generation and optimization of the propagation region using non-conforming grids (Mortar FeM) and in addition at the boundary region to model open domain characteristics. We apply a recently developed perfectly matched layer technique, which allows us to truncate the

D. Tumpold (*) · M. Kaltenbacher Institute of Mechanics and Mechatronics, University of Technology, Vienna, austriae-mail: [email protected]: http://mec.tuwien.ac.at

c. Glacer · M. nawaz · a. Dehé Infineon Technologies aG, Munich, Germany

996 Microsyst Technol (2014) 20:995–1006

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and in (lee and hwang 2011b) for rectangular mem-brane design. Integrated speakers open up the opportunity to design small transducer arrays as described by (cohen et al. 2010a, b, c, d). as mentioned before, the electro-static MeMS speaker is an increasing driving technology, therefore papers and patents about various fabrication and processing steps can be found in (Zhu and ciferno 2005; Gabriel and Zhu 2006; Diamond et al. 2010; Diamond and Matthew 2011; Dehè 2013; Kim et al. 2005; cohen et al. 2011), but no marked-ready products are available yet. Piezoelectric speakers form the third MeMS speaker driv-ing technology, for single frequency buzzers, up to wide frequency speaker systems. Wide frequency in this context is referred to human auditory. Most actuators show square shaped membranes (Kim et al. 2009; Ko et al. 2003; Yi et al. 2009), oscillating cantilevers (lee and White 1998) or octagonal shaped structures (cho et al. 2008), but all of them are single speaker investigations. a well documented overview is provided in (Glacer et al. 2013) and (Glacer 2011).

In the first part of the paper the electrostatic driving technology is presented for the single ended principle and the optimized push pull or pull-pull principle. next, the full MeMS speaker model is treated, where firstly the focus is on the electrostatic-mechanical interactions. The non-linear voltage to force relation, taking into account insu-lation layers with various permittivity layers and snap-in characteristics are discussed. native anSYS is used to model the electro-mechanical interaction, where reduced order elements are applied. The accurate implementation of these reduced order elements plays a major role in the electrostatic-mechanical interaction, as previous work has shown (Tumpold et al. 2013). The second main topic is about the mechanical-acoustic model, where the mechani-cal displacement results in an acoustic wave propagation. This part deals with modeling a propagation region includ-ing open domain characteristics. Due to sub micrometer mesh at the membrane level and well-adjusted mesh size concerning to the wavelength of our acoustic fields, mortar FeM with non conforming grid method was applied. This non conforming grid method allows us to properly adapt the mesh size between the MeMS structure and the propa-gation region. Perfectly matched layers (PMl) are used to model open domain characteristics by coincidentally trun-cating the propagation regions. Single transducer investiga-tions were performed via FeM using cFS++, where the full eight-bit speaker array (matrix with 16 by 16 speaker cells) was computed with Kirchhoff helmholtz inte-gral. Finally, an optimization method to increase the gap size between membrane and back plate is presented. This method allows us to fabricate the speaker system within lit-tle fabrication time as a flat system and receive a strongly buckling back plate system as a result. The buckling back

plate method uses various layers with different intrinsic pre-stress to force the flat back plate into a buckling shape. Due to the increased gap size of the buckling shape, the possible membrane stroke level and therefore the maxi-mum sound pressure level (SPl) increases too.

2 Electrostatic driving principle

2.1 Single ended system

Single ended electrostatic speaker systems work very sta-ble in comparison to push-pull or pull-pull systems. This is a direct result of the simple mechanical setup, where two electrodes are used in parallel. The top electrode (sta-tor or back plate) is mechanically fixed while the second electrode representing the membrane is flexible. By apply-ing a voltage or potential difference between both elec-trodes the electrostatic attracting force pulls the flexible membrane towards the stator. Both forces, first the elec-trostatic attracting force caused by the voltage between the electrodes and second the mechanical restoring force of the membrane are in equilibrium state. The restoring force can be adjusted by tensile pre-stress in the membrane due to fabrication process and must be computed nonlin-early due to large deformation and tensile pre-stress. The operation point of the membrane can be adjusted by a bias voltage. By applying a modulated voltage, the membrane starts to move forth and back, depending on which force is stronger (mechanical or electrical). Both forces are in equilibrium state for this operation mode. The quad-ratic relation between electrical force and voltage applied results in a low-distorted system for small membrane dis-placements. The voltage to displacement relation increases

Fig. 1 Working principle of single ended electrostatic speaker sys-tem with three states; a operating point with bias voltage, b force state and c release state

997Microsyst Technol (2014) 20:995–1006

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nonlinearly up to the snap in point, where the electrostatic force and the mechanical pull back force are out of bal-ance and the system jumps onto an instable mode (snap through characteristic). at the snap in point the membrane is abruptly pulled towards the back plate until mechani-cal contact occurs. In Fig. 1 the basic working principle is displayed. Two electrodes, first the fixed stator and second the flexible membrane with a voltage source. By apply-ing a voltage, the membrane starts to displace towards the stator which is called forcing state. If the membrane is attracted or loaded and the supply is switched off, the stored potential energy in the membrane is transferred into kinetic energy and flips away in opposite direction; hence this state is called release state.

The advantage of this method is the ease of fabrication in cMOS technology. however, the disadvantages are the strong nonlinear force versus displacement relation, as well as the only attracting force possible in this setup which results in a small stroke level. Furthermore, gap deviations in etching processes show quadratic impact on the move-ment characteristics due to quadratic linkage between gap and force.

as already explained, there are two driving states; first, the non-snap in mode, where the membrane follows the applied acoustical voltage signal. In this driving state the system is in equilibrium of forces (restoring force and electrostatic attracting force), the membrane stroke level is low and the distortion is quadratic. The second mode is the snap-in mode, where the restoring force is lower than the electrostatic attracting force and the membrane is pulled abruptly towards the stator, where mechanical contact occurs. In this state the stroke level is higher compared to the non-snap in mode. Below the snap in point, the mem-brane can be controlled directly via supply voltage, while above the snap in point the membrane is forced rapidly towards the back plate.

2.2 Push pull system

a simple improvement of the single ended system is the push-pull or pull-pull system, where the major disadvan-tages (gap size and non-linear voltage to force relation) can be almost avoided by the mechanical setup. The basic principle is depicted in Fig. 2. In this setup the membrane (electrode two) is located between two stators (electrode one and three). Both stators are perforated to be acousti-cally transparent and reduce squeeze film damping effects. One key benefit of this method is the doubling of the gap size and the linearized electrostatic force to voltage relation near the operation point.

at the idle mode, all electrodes have the same charge level and the membrane is ideally located exactly in the middle. By increasing the charge level on one stator, the

membrane is attracted towards the direction with the larger potential difference. Based on the mechanical setup the membrane stroke level is doubled. all simulation data and measurements were based on the single ended system, nev-ertheless the real device and the model can be adapted eas-ily to the pull-pull system at little costs.

3 Modeling methods

In this section an overview about the multi-domain model is given, starting with a total overview, followed by the electrostatic-mechanical model and the mechanical-acous-tical model.

Fig. 2 Working principle of push-pull or pull-pull electrostatic speaker with three electrodes in a idle state and operation point, b attracting force with respect to bias and modulated voltage towards stator a and c towards stator B

Fig. 3 Major parts for modeling the MeMS speaker with two sub models for electrostatic-mechanical coupling and mechanical acous-tic wave propagation, starting from a voltage resulting in a sound pressure level


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3.1 Modeling overview

The electrostatically actuated MeMS speaker modeling chain can be separated at the membrane displacement into two sub models. First the electrostatic-mechanical model, where an input voltage results in a mechanical movement of the membrane including all necessary nonlinearities for electrostatic field coupling and geometric nonlineari-ties at the mechanical field. The second part of our model projects the membrane movement into an acoustical field with open domain characteristics to compute the SPl at the point of interest (POI). Figure 3 describes our modeling approach for designing a physical abstraction of the MeMS micro speaker. The mechanical part is surrounded with a solid line; this part is modeled totally with native anSYS. The acoustic block is highlighted with a dashed line and modeled on the one hand with FeM (coupled Field Simu-lator cFS++) and on the other hand in MaTlaB using the Kirchhoff helmholtz ansatz, which is highlighted by a dash-dotted line in the acoustic model.

The model split up at the membrane displacement is valid, because the counteracting force of air surrounding the membrane can be neglected at the mechanical-acous-tic model (Kaltenbacher 2007). In addition, this opens up the opportunity to verify the electrostatic-mechanical model with measurements and feed the acoustic model with these measurements. To minimize computational time and amount of memory, the complex three dimen-sional structural model was reduced into a two dimen-sional axis symmetric model. hence, parameter varia-tions and optimization routines can be applied. Material parameters of the axis rotational back plate structure,

including perforation hole characteristics were deter-mined by (Füldner 2004) in a prior work on a silicon MeMS microphone with the same dimensions and mate-rials used.

3.2 electrostatic-mechanical modeling

Since we used native anSYS to model the electrostatic-mechanical coupling, there are basically two methods. First the multi-field ansatz, where the model is separated into its two field domains, the electrostatic domain and the mechanical domain. This ansatz was presented in (Tumpold et al. 2013) in detail and verified with an ana-lytical model and will not be treated in this paper. The second modeling method uses reduced order electro-mechanical-transducer (eMT) elements (TranS126). They feature a single degree of freedom in transla-tion and for electric voltage applied and are very effec-tive from computational point of view. as displayed in Fig. 4a, the eMT element is located between both elec-trodes. The top electrode represents the back plate and is coated with an insulation layer with a relative permittiv-ity of εr = 7.5 and a high tensile pre-stress of 1,000 MPa compared to the tensile pre-stress of the poly silicon with 165 MPa for the back plate and 43 MPa for the mem-brane. Pre-stress values were determined due to measure-ments by (Brueckner et al. 2009) as well as provided by manufacturer data and adapted on previous models (Fül-dner 2004; Tumpold 2013).

The eMT is a nonlinear spring element, where the nonlinear force displacement relation is derived from the capacitance to gap relation [see (8)], hence the input data is a polynomial function fifth order as described in (anSYS 2009a). as a result, the gap of both electrodes is modeled with capacitances in series as can be seen in Fig. 4b. The mechanical properties of the insulation layer lead to a fixed capacitance to gap ratio, whereas the capacity of the air gap changes with the membranes displacement. The insulation layer capacitance specifies the maximum capacitance possible, if the membrane is mechanically in contact with the back plate and the gap is closed. The gap versus capacitance relation is depicted in Fig. 4c, where Gapmax is related to the minimum capaci-tance (or initial capacitance) and Gapmin is related to the maximum capacitance. C(y) according to Fig. 4, is com-puted with

where the shift is computed as

(1)C(y) =�

Gmin − G0

,

(2)G0 =CmaxGmin − CminGmax

Cmax − Cmin

(a)

(b) (c)

Fig. 4 a eMT nonlinear TranS126 implementation between nodes I and J results in b analogue circuit diagram and corresponding gap versus capacitance relation in c


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and the scaling by

The two characteristic capacitance Cin and CGap are com-puted by

and CGap

where A is the axis rotational surface of the layer of one eMT element due to the ring shaped area, d1 the thick-ness of the insulation layer, d2 the thickness of the initial gap and εr the relative permittivity for the insulation layer. as displayed in Fig. 4 b, the minimum capacitance can be computed by

The implemented function in anSYS has to be done with a polynomial function of fifth order (anSYS 2009b). hence, we are using a matrix A with the polynomial func-tion values, the known capacity values as b computed with (1) and our unknowns x

each electro-mechanical transducer element along the radius owns an individual capacitance to gap relation, which must be computed and set separately. The elec-trostatic force is computed by the virtual work principle resulting for our case in (anSYS 2009a)

with V the voltage applied. We found out that the capaci-tance to gap relation can not be fitted with the polynomial function for every setup. Oscillations in the force charac-teristics can occur and lead to alternating signs of the elec-trostatic attracting force, caused by the the strong gradient of the capacity gap relation (8). a details description of this problem can be found in (Tumpold and Kaltenbacher 2013).

We fully take geometric nonlinearities into account, hence large deformation and stress stiffening are included in our model. Material properties were taken from prior models of the silicon microphone (Kaltenbacher 2007). additional computational background information about structural nonlinear effects are listed in (Füldner 2004; anSYS 2009a) for FeM and in (hibbeler 2005) for struc-tural mechanics.

(3)� = Cmax (Gmin − C0).

(4)Cin = Cmax =ε1A

Gmin

=εrε0A

d1

(5)CGap =ε2A

G(y)=

ε0A

d1 − y,

(6)Cmin =CInCGap

CIn + CGap

=ε1ε2A

ε1d2 + ε2d1 − ε1y

(7)A · x = b.

(8)Fe =∂We

∂y=

1

2

∂C

∂yV2,

3.3 acoustic modeling

Weak couplings between mechanical and acoustic fields are valid, if the pressure of the fluid on the structure is negligible as described by (Füldner 2004). Special bound-ary conditions are required to model open domain charac-teristics with FeM, because setting the acoustic pressure to zero results in a acoustic soft wall (dirichlet boundary condition; pressure is zero) and reflections occur. To pre-vent reflections of waves leaving the propagation region, PMl were added surrounding our model. another method to model open domain characteristics at the propagation region can be achieved by using absorbing boundary con-ditions (aBc). aBc of first order absorbs just the normal component of impinging wave front but other angles of incident are reflected. There are investigations on higher order aBcs as (Grote and Sim 2011) with improvements on arbitrary angles of incident. Previous tests (Tumpold 2013) have shown that PMl works better for our com-putational domains. another challenge in the acoustical domain occurred at the mesh generation, starting at the sub-micrometer structure of the MeMS membrane and the dimensions of the propagation region. These dimen-sion variations results in a high amount of finite elements or strongly distorted elements. To get rid of distorted ele-ments as well as large amounts of elements, we modeled our propagation region with Mortar FeM (non-conforming grids) and truncated and surrounded it with a PMl region. The result is depicted in Fig. 5, where the smallest part rep-resents the membrane of the speaker cell and is surrounded by four shells of propagation region with air characteristics.

In the ambient region the discretization size increases from the sub micrometer (approximately 100 nm) mesh at the speaker cell to λ/10 in four steps linearly, to reduce the total amount of elements concurrently without mesh distor-tions and mapped mesh conditions. λ represents the wave-length between 3.43 cm at 1 khz and 1.71 mm at 20 khz

Fig. 5 Mechanical-acoustic model with single speaker cell and fluid structure interface (FSI), propagation region modules assembled with Mortar FeM and PMl to model open domain characteristics


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computed by λ = c/f, where c is speed of sound with 343 m/s and f the frequency. Further information on PMl can be found in (Kaltenbacher 2007; hüppe and Kalten-bacher 2012; Berenger 1994; Kaltenbacher et al. 2013) and for Mortar FeM with nonconforming grids in (Trieben-bacher 2012; Triebenbacher et al. 2010).

large arrays of speaker cells as well as large propaga-tion regions result in expensive computations. Therefore, Kirchhoff helmholtz integral solutions are applied to com-pute the acoustic SPl at any POI. The single point solution of the POI in Kirchhoff helmholtz results in fast computa-tions compared to the FeM but shows restrictions in accu-racy due to membrane movements. This method is using only the normal displacement component of the membrane, hence the movement characteristic is assumed a piston moving solid. The implemented form uses the pressure for-mulation in time domain of the second rayleigh integral of Kirchhoff helmholtz

where, ρ0 is the density of air, rk the distance between the kth speaker cell center and the POI, un the normal displace-ment component of the stroke level of the membrane and c the speed of sound in air. Ŵk represents the active mov-ing surface area of each individual cell and N the number of active speaker cells for time t. as mentioned before, the membrane movement is assumed as piston moving solid only taking the normal component of displacement into consideration. This simplification is justified, because of the large ratio between diameter (1,000 μm) and maxi-mum stroke level (2 μm). From (9) can be seen, that the displacement is linked with pressure by second time deriva-tive, which is the acceleration. This fact plays a major role in digital sound reconstruction (DSr) at the speaker array and total harmonic distortion (ThD) of the speaker system. Further details on DSr and the Kirchhoff helmholtz model is given in (Tumpold 2013).

4 Optimization

We have basically three options to increase the SPl at our MeMS speaker. First the frequency, where we are limited by human auditory between 20 hz and 20 khz. Second the surface area, where the limitation is given by dimen-sions and size as well as cost economical effects from the manufacturer side. Third the stroke level of the membrane, where the limitation is also given by manufacturer side due to technology restrictions. The challenge is to increase the gap size or stroke level with the same production steps to

(9)p(xPOI,t) =ρ0

4π

N∑

k=1

∫

Ŵk

1

rk

d2un

dt2

(

x, t −rk

c

)

dŴ,

keep the device cheap. Our idea is based on stress induced self-raising and called buckling back plate (BBP). The manufacturing process of the BBP is treated in (Glacer et al. 2013) in detail.

4.1 Buckling back plate

The basic idea behind the BBP is based on an increased volume flow resulting in an increased SPl and on a flat cheap fabrication technology. By covering the softer poly-silicon layer (electrode) with the stiffer silicon-nitri-de layer (insulation), mechanical forces acting as lever occur and start to bend the material towards the stiffer coating. an analogue effect can be found in bi-metal structures with various temperature expansion coefficients—one material contracts heavily, the second layer less, which results in a mechanical bending deformation. The design process of the BBP can be seen in Fig. 6. In the first draft (a) the silicon microphone structure is displayed, where the poly silicon layer is fully covered with the nitride layer. This results in a flat structure with high stiffness but little gap size. The second design process (b) shows stages of corrugation rings. another disadvantage is the exposure of the electrode layer, which can lead to short circuits at mechanical contact. The third case (c) displays an opti-mized structure with deep corrugation rings for soften-ing the poly-silicon layer on the one hand and increasing the leaverage of the silicon-nitride layer. additionally, the short circuit problem is reduced due to the corrugation ring structure.

additional information on this BBP system can be found in (Glacer et al. 2013) for influence of process and layout parameters and (Dehè 2013) for the basic idea.

(a)

(b)

(c)

Fig. 6 Buckling back plate with full silicon-nitride coating (a), with corrugation rings (b) and an approach for an optimized corrugation ring shape (c)


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5 Results

5.1 electro-mechanical model verification

as mentioned before, the flat structure using the single ended driving technology is used for all investigations. To verify the electro-mechanical model, the membrane dis-placement was measured optically with a laser scanning vibrometer and additionally electrically for the snap in determination with the capacitive response via capacitive voltage divider as discussed in (Triebenbacher et al. 2010; Tumpold 2013) in more detail. Figure 7 displays the mem-brane displacement of the flat speaker system (see Fig. 6a as axis rotational result. On the left side the center point displacement can be seen, where the snap in point is com-puted by FeM between 10.9 and 11 V. The right side shows the axis rotational membrane bending towards the back plate. The nonlinear force or voltage to displacement ratio can be seen very well; the displacement at 5 V is about 8 % of gap level while the displacement at a load of 10 V is about 31 %.

The snap-in occurs, when the electrostatic force and the mechanical restoring force of the membrane are out of balance and the system falls into an unstable mode - also known as snap through characteristic (Zhang et al. 2007; Krylov et al. 2008; Gerson et al. 2011). From acoustical point of view the snap in results in a wide frequency stim-ulus, because of the large positive and negative accelera-tions and a bad ThD is the consequence. The advantage of the snap in mode is the increased SPl according to the

increased membrane stroke level. electrical measurements have shown a snap in voltage between 11 and 15 V for various speaker cells. We found out that these variations are caused on the on hand due to fabrication tolerances in gap size (quadratically linked in electrostatic force) and on the other hand due to mechanical imperfections on the membrane which lead to mechanical weak spots. These imperfections are not taken into consideration at our axis-symmetric simulation. In addition to this the per-forated back plate with its spider suspension fixing was assumed as solid body. This results in an effective lower surface and higher electric fringe field rates, hence a lower effective capacitance. as a result the electrostatic force is stronger in the model compared with the device. That effect was investigated for effective capacitance modeling for the microphone in (Kaltenbacher 2007) before and confirms our results. considering and understanding these facts, the snap in point is determined accurately within the simulation.

The flat back plate structure was adapted to the buck-ling back plate structure with corrugation rings and lever arms. exactly these weak spots at the leaver arms change the mechanical behavior of the entire structure from the electrostatic point of view. The single snap in point of the flat system, where the total back plate can be represented as single mechanical spring is replaced by a multi snap in point caused by multiple corrugation rings. each of these rings show different mechanical properties defined by their length and thickness, a coupled mass spring system with different mechanical snap in points is the consequence.

Fig. 7 Static membrane dis-placement result under applied voltage (right side) and center displacement versus applied voltage with snap in (left side)

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

radius (normed)

Radius vs. membrane displacement under voltage

0 5 10 150

0.2

0.4

0.6

0.8

1

voltage (V)

cent

er d

ispl

acem

ent

(nor

med

)

Voltage vs. center displacement

30 V

20 V

11 V

10.9 V10.5 V

9 V

5 V

0.5 V


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Figure 8 illustrates two different types of buckling back plate systems. The solid line represents our standard ini-tial or reference system while the dashed line a system with doubled poly silicon thickness. Both systems were electrostatically coupled with eMT elements and a linear increasing voltage up to 40 V was applied. The blue lines correspond with the back plate center point displacement (top to bottom) and the red lines the membrane center point displacement (bottom to top). The first and at once largest snap in occurs at the initial system at about 17 V applied. at this point the membrane is mechanically in contact with

the back plate near the edge area. By continually increas-ing the voltage over the first snap in level, the membrane and back plate starts to stick in center direction with addi-tional snap in characteristics. The following snap in points occur with repetitive pattern and decreasing displacement level, since the system is axis rotational and no imperfec-tions are included. By increasing the stiffness or poly thick-ness the maximum buckling shape is reduced. a direct result is a higher electrostatic attracting force and an ear-lier first snap in point at 16 V. Figure 8 also displays, that the back plate movement decreases with approximately the

Fig. 8 Multi-snap in character-istic of two systems, first with 330 nm poly thickness (solid) and second 660 nm poly silicon thickness (dashed). electro-statically loaded with a linear increasing voltage from zero to 40 V

0 5 10 15 20 25 30 35 40-2

-1

0

1

2

3

4

Voltage (V)

disp

lace

men

t (µ

m)

Membrane 330nm(330nm poly)Buckling Back Plate (330nm poly)Membrane (660nm poly)Buckling Back Plate (660nm poly)

Fig. 9 Multi snap in charac-teristic displayed for capacity versus voltage between FeM result and measurement of fabricated device

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 302

4

6

8

10

12

14

16

voltage (V)

capa

city

(pF

)

MeasurementFEM Result


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same factor as the thickness increases, which is a result of the flexibility and stiffness of the BBP. The multi-snap in verification can be done on the one hand directly with an optical measurement or on the other hand with an indirect method via capacity. The latter method can be used simul-taneously to check the electrostatic attracting force level, hence this method was applied. Figure 9 compares the measured results with the Fe computation. The capacitive response is modeled accurate during the pre-snap in phase and the snapped device while the snap in point is not com-puted correctly. as mentioned before, this is a result of the axis rotational model we are using and imperfections in the fabricated device.

In addition to this, the measured system shows a creep-ing release state. This effect was already discovered in (O’Mahony et al. 2005). The effect is known as latch-ing and is caused by charges in the insulation layer which create an additional electric field, hence an additional electrostatic force. We focused on the multi-snap in com-putation, therefore the mutli-release was not computed in Fig. 9. additional snap in and release results for the basic flat MeMS speaker design can be found in a prior work in (Tumpold 2013).

5.2 acoustic model verification

To verify the acoustic model, we measured the SPl of a small speaker array of 16 speaker cells simultaneously driven. The SPl results are displayed in Fig. 10. The flat back plate speaker design was used for these investiga-tions. This small array increases the SPl theoretically by +24dBSPL (+6dBSPL each time doubling the active area)

and lead to a better signal to noise ratio. For the measure-ment the speaker was driven in non snap in mode, with a bias voltage of 10 V and an audio signal level of 1–4 V peak to peak with sinusoidal characteristics. The mem-brane stroke level was determined with the static voltage displacement computation of the electrostatic mechanical model (with 4 V audio excitation) as depicted in Fig. 7. The deviation between measurement and simulation is on the one hand a result of the nonlinear force displacement rela-tion and on the other hand the achieved membrane stroke level was not determined accurately enough. This results in an SPl offset.

Because of the small array dimensions and the small stroke level, sound perception starts at 1 khz in addition to this all measurements were taken in laboratory envi-ronment, hence the ambient noise is about 50 dB. The simulated results of Kirchhoff helmholtz and FeM can be seen on the right side of Fig. 10, where the devia-tions are negligible over the total frequency band. as a result, the simplifications due to piston moving solid are valid for the Kirchhoff helmholtz ansatz in our case, due to the large diameter to stroke level ratio. There-fore, Kirchhoff helmholtz is a powerful method for SPl computation.

5.3 Buckling back plate verification

a white light interferometer measurement setup was used to determine the three dimensional shape of the buckling back plate. To verify simulated axis symmetric results with measurements a X-Z plane was extracted out of the three dimensional model. The three dimensional raw data

Fig. 10 SPl measurement of non snap in moving membrane with sinusoidal characteristics with operation point load due to bias voltage and acoustic audio signal versus FeM and Kirchhoff helmholtz (left) and difference in dBSPl between FeM and Kirchhoff helmholtz (right)

103

104

40

50

60

70

80

90

100

frequency (Hz)

SP

L (d

B)

Measurement: 10V ± 1VMeasurement: 10V ± 2VMeasurement: 10V ± 3VMeasurement: 10V ± 4VFEMKirchhoff Helmholtz

103

104

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

frequency (Hz)

SP

L (D

B)


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with the buckling shape and the X-Z plane can be seen in Fig. 11.

The measurement shows, that the basic idea of vari-ous tensile pre-stress layers is working for MeMS speaker buckling back plate systems. Due to tolerances in fabri-cation process, in-plane imperfections can occur. These imperfections can lead to inhomogeneous buckling shapes. Since we are using an axis-rotational model, in-plane imperfections are not taken into account, hence we have to select a homogenous buckling device to extract the

X-Z plane for comparable information. The comparison between measured displacement and Fe model result can be seen in Fig. 12.

as mentioned before, the back plate is designed acoustically transparent; hence the surface is perforated with holes. The white light interferometer beam pene-trates the perforation holes directly or diffraction occurs at the whole edges. This results in alternatively disap-pearing measurement signals with high signal to noise ratio.

Fig. 11 White light interferom-eter measurement of buck-ling back plate structure and extracted plane for Fe result verification

Fig. 12 FeM result versus measurement of buckling back plate static deformation past stress induced self-raising

0 100 200 300 400 500 6000

0.1

0.2

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6 Summary and conclusion

a finite element model has been developed to study elec-trostatic driven MeMS loudspeaker. The electrostatic, mechanical and acoustic fields as well as their interactions have been modeled by partial differential equations, where native anSYS was used to model the electrostatic mechani-cal behavior and cFS++ was used as tool to solve for the acoustic field. In addition, the SPl has been computed with the help of Kirchhoff helmholtz integral for large array dimension (up to 255 elements). The nonlinear electrostatic mechanical coupling was implemented with electro mechan-ical transducer elements (TranS126) representing a non-linear force to gap relation derived from the capacity to gap function of a lumped element model. Geometric nonlineari-ties as large deformation and stress stiffening effects were taken into account. The latter opened up the opportunity to optimize the flat structure of the first speaker prototype with an self induced stress-raising technique, which can be com-pared to bi-metal characteristics. all finite element results were verified with measurements and establish a solid base for further investigations. From the acoustic point of view, the SPl is to weak, this is a result of the membrane stroke level limitation caused by fabrication technologies. The non snap in mode results in low distorted sound but with little amplitude while the snap in driving mode increases the SPl possible utilizing the full gap between membrane and sta-tor. Two optimization approaches were presented in order to increase the SPl; first the buckling back plate with the flat fabrication technology and the stress induced self-rising and second the push-pull or pull-pull system, where the gap size can be doubled easily.

Acknowledgments This project has been supported within the cOMeT—competence centers for excellent Technologies Pro-grams by BMVIT, MBWFJ and the federal provinces of carinthia and Styria.

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