parametric simulation of pzt diameter to hole …...parametric simulation of pzt diameter to hole...

Parametric Simulation of PZT Diameter to Hole Ratio forOptimized Membrane Displacement

Arpys Arevalo*1, David Conchouso1, David Castro1, and Ian G. Foulds1,21Computer, Electrical and Mathematical Sciences and Engineering (CEMSE), King Abdullah University ofScience and Technology (KAUST), 2The University of British Columbia (UBC), School of Engineering,Okanagan Campus.*Corresponding author: 4700 KAUST, Kingdom of Saudi Arabia, [email protected]

Abstract: In this paper, the parametric simulationof an acoustic transducer is presented. Themodeling and analysis in COMSOL Multiphysics5.0, shows the optimal configuration for thelargest displacement of the membrane with theproposed material layers used. The Piezoelectricmodule was used to simulate the deflection of themembrane with an applied voltage. For this work,we varied the size of the piezoelectric materialtri-layer (Pt/PZT/Pt) diameter and the hole wherethe membrane is clamped. The results show theoptimal parameter to be used as the ratio betweenthe PZT diameter and the total diameter of themembrane.

Keywords: acoustic transducer, PZT, MEMS,polyimide, ultrasonic transducer.

1. Introduction

Electronic systems have been continuouslygrowing to a point that they play an importantrole in an average user. The demand of qualityin the electronics industry products, requires thedevelopment of new components with better char-acteristics [1, 2]. Devices such as micro-speakers,microphones, accelerometers, gyroscopes, humid-ity sensors, lenses, cameras, amongst others, needbetter characteristics to keep up with the demandfor quality.

An interesting gap to improve audio technologyexists in one of the oldest components, the acous-tic transducer. To improve the performance of thesound reproduction, we could potentially elimi-nate components that introduce noise when digitalsignals are converted to analog signals. The latterprocess usually occurs when reproducing soundin commercial loudspeaker drivers. One of the

approaches we could take in order to tackle thisproblem, is the development of a system that candirectly communicate from the digital audio signalsource to a digital acoustic transducer, without theneed to use a Digital-to-Analog converter (DAC).Such a system can be achieved with the concept of”Digital Sound Reconstruction” (DSR). The DSRconcept can be implemented with a Digital Trans-ducer Array Loudspeaker (DTAL) that was firstproposed by Huang et al. in [3]. The device can beused to reproduce binary pulses that can be addedtogether to reconstruct the analog audio signal.Actual problems or deficiencies associated withthe current acoustic transducer, such as: frequencyresponse and linearity, can be attenuated with thisapproach [4–6]. The DTAL can be organized bysets of transducers that are grouped in associationwith a bit-number. This configuration is refereedas ”binary weighted group” [7]. For example, a 3-bit micro-speaker would have three sets of acous-tic transducers, each group would have 2n trans-ducers (where n is the bit number). The most sig-nificant bit (MSB), would have 4 transducers (22

transducers). The second set would have 2 trans-ducers (21 transducers), and the least significantbit (LSB) would have 1 transducer (20 transduc-ers).

The DTAL device operates as follows: when alower pressure is needed, fewer transducers areactivated and when higher pressure is needed,more transducers are used. Each transducer con-tributes to a small pressure change in the system,which is a contribution of the total sound pressurechange generated by the entire device.

The response time of an individual elementonly depends on the digital clock that synchro-nizes the audio reconstruction process. Therefore,each individual device is independent of the re-

Excerpt from the Proceedings of the 2015 COMSOL Conference in Grenoble

constructed frequency and this enables the recon-struction of a wide range of frequencies. As a re-sult, each membrane is not required to operate in aspecific frequency range, in contrast to the currentdesign rules of loudspeakers.

In this work, we use a piezoelectric materialas the driving mechanism of our membrane, andpolyimide as the structural material. Polyimide isa very attractive polymer for MEMS fabricationdue to its low coefficient of thermal expansion,low film stress, lower cost than metals and semi-conductors and high temperature stability com-pared to other polymers [8–10]. Polyimide hasbeen previously used in the microelectronics in-dustry for module packaging, flexible circuits andas a dielectric for multi-level interconnection tech-nology [11, 12]. Recently, the polymer has beenwidely used as an elastic flexible substrate forpolymerMEMS [13–15] and also as structural ma-terial for several devices [8, 13–27], which showsthe extent of applications of the material.

Previously, the feasibility of such structure wasreported [28, 29], but the performance was not asexpected. Due to the low performance, we at-tempted to optimize the design of the membrane.We used Lead Zirconate Titanate (PZT) as thepiezoelectric material with a bottom and top elec-trode using platinum (Pt), and a layer of polyimideas part of the structural material of the bimorphactuator.

2. Computational Methods

The interaction of the mechanics and the electricalfields of the studied structure is called piezoelec-tricity. The interactions are modeled as a couplingof the linear elasticity equations and charge re-laxation time equations, using electric constants.Piezoelectricity can be described mathematicallyusing the material’s constitutive equations [?, 30,31]. Piezoelectric materials become electricallypolarized when they are subject to a strain. In amicroscopic perspective, the atoms displacementwhen the solid is deformed causes electric dipoleswithin the material. In some cases, the crystalstructures can give an average macroscopic dipolemoment or electric polarization. This effect isknown as the direct piezoelectric effect. Also itsreciprocal exist, the converse piezoelectric effect,in which the solid contracts or expands when anelectric field is applied.

The constitutive relation between the strain andthe electric field in a piezoelectric material isshown below (strain-charge form):

S = sET+dT ED = dT+ εT E

(1)

Where, S is the strain, T is the stress, E isthe electric field, and D is the electric displace-ment field. The materials parameters sE ,d and εT ,correspond to the material compliance, the cou-pling properties and the permittivity of the mate-rial. These parameters are tensors of rank 4, 3 and2 respectively. However they can be representedas matrices within an abbreviated subscript nota-tion, as it is more convenient to handle. In COM-SOL Multiphysics, the Piezoelectric Devices in-terface uses the Voigt notation, which is standardin the literature of piezoelectricity but differs fromthe defaults used in the Solid Mechanics interface.The latter equation (1) can be expressed in thestress-charge constitutive form, which relates thematerial stresses to the electric field:

T = cES− eT ED = dS+ εSE

(2)

The stress-charge form is usually used in the fi-nite element method due to the useful match tothe PDEs of Gauss’ law (electric charge) and theNavier’s equation (mechanical stress) [?]. Usuallymost material’s properties are given in the strain-charge form. The material properties, cE ,e and εTare related to the parameters sE ,d and εT , and canbe transformed between each other by the conver-sion equations shown below:

cE = s−1E

e = ds−1E

εs = ε0εrT −ds−1E dT

(3)

2.1 Governing Equations

The Piezoelectric equations used in COMSOL,combine the momentum equation,

ρ0δ2uδt2 = ∇X (FS)+FV (4)


with the charge conservation equation of Elec-trostatics,

∇ ·D = ρV (5)

where the ρV is the electric charge concentra-tion. The electric field is computed from the elec-tric potential V as:

E =−∇V (6)

In both Equations (4) and (5), the constitutiverelations of Equation (3) are used, which makesthe resulting system of equations closed. The de-pendent variables are the structural displacementvector u and the electric potential V [?].

3. Design and Simulation Setup

The components of the piezoelectric membraneare: a 300nm platinum (Pt) bottom electrode, a250nm piezoelectric layer (PZT), a 300nm Pt topelectrode and a 3µm thick polyimide structurallayer, to complete the bimorph membrane. InFig. 1, the first version of our membrane is shown.

Figure 1: (Left) Isometric view of a membraneon silicon substrate (Right) Conceptual explodedview of the first design of the micro-membrane.

As it can be seen in the image, the piezoelectricactuator dimensions are larger than the hole area.This is the parameter that needs to be optimized,for a larger displacement of the membrane. Inour simulation, we used the Piezoelectric module,whereby the structure was setup in a 2D environ-ment. The structure was simulated as a cantilever,which is clamped from both sides, as shown inFig. 2.

For the mechanical constraints the six verticalboundaries (edges) to each side of the structurewas set to be fixed. All the other boundaries wereset to be free. For the AC/DC interface the bottom

Figure 2: Schematic of the cross-secional view ofthe piezoelecric membrane.

electrode was set to be the ground, and the topelectrode was set to be a Terminal with a potentialof 10V .

A stationary study was selected and a paramet-ric sweep was setup, to be able to change the ge-ometry for different dimensions of the actuator.The PZT diameter dimension (shown in Fig. 2)will be constrained proportionally to the ratio ”a”,as shown below in Equation 7.

PZTd = a∗Holed (7)

4. Results

The first version of the fabricated devices did notmeet the expected performance, even though thetransducers were able to reproduce sound waves.From the COMSOL simulation results, we wereable to see that the original design was out of theoptimal range parameters for a larger membranedisplacement, see Fig. 3 and 4.

As seen above, the original design was reallyout of range for large membrane displacement.The range of displacement was in the range ofhundreds of pico-meters. In Fig. 3, a 3D view pro-duced from a revolution of the results is shown.

Due to these results, we were able to modifythe device accordingly. In Fig. 5, a modified ver-sion of the membrane is shown. Where the desiredPZT/Hole ratio ”a” should be between 0.8 to 0.9,i.e. the Pt/PZT/Pt layers must have a diameter be-tween 80%−90% of the hole diameter area.

As seen in the conceptual view of Fig. 5, thepiezoelectric stack has 4 arms. This arms will bethe interconnection with the next element in thedevice array.


Figure 3: 3D representation from a partial revo-lution of the simulation results, showing the de-formation of the membrane and the internal layers(colors depict the different materials in the struc-ture).

5. Conclusions

After analyzing the simulation results, we no-ticed that the original parameter designs were notoptimal. New designs were developed to get abetter performance of the array of actuators. Thenext steps in the project include simulations of theacoustic energy generated by the acoustic trans-ducers. We have fabricated and diced the chipsfrom a four inch silicon wafer using our in-housedicing method [32]. An interesting characteristicthat we will also consider, is the directivity of thebeam forming pattern, which is an inherit charac-teristic of the final transducer array.

The device will work as a directional loud-speaker, either using the Digital Sound Recon-struction concept or by signal modulation using an

Figure 4: Displacement vs Diameter to Hole Ratioof the acoustic transducer.

Figure 5: Conceptual view of the new membranedesign for the Pt/PZT/Pt layer stack for optimizedmembrane displacement.

ultrasonic signal, which contains the audible sig-nal. This characteristic, has great potential for awide range of applications for our MEMS micro-loudspeaker, such as: separate multi-user inten-sity and signal control of the audio source, pri-vate audio, medicine, underwater communication,to name but a few.

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