Characterization and Modeling of Piezoelectric Integrated Micro Speakers for Audio Acoustic Actuation

An array of piezoelectric micro actuators can be used for radiation of an ultrasonic carrier signal modulated in amplitude with an acoustic signal, which yields audio frequency applications as the air acts as a self-demodulating medium. This application is known as the parametric array. We propose a parametric array with array elements based on existing piezoelectric micro ultrasonic transducer (pMUT) design techniques. In order to reach enough acoustic output power at a desired operating frequency, a proper ratio between number of array elements and array size needs to be used, with an array total area of the order of one cm square. The transducers presented are characterized via impedance, admittance, noise figure, transducer gain and frequency responses. Keywords—Pizeoelectric, Microspeaker, MEMS, pMUT, Parametric Array


Introduction
Multimedia systems and components such as micro speakers are elements of great interest for the research community, driven by the needs of mobile phone, entertainment, laptop, tablets, and computer industries.It has been long time since "full range" multimedia micro speakers are trying to be developed, often finding design conflicts limited by fundamental physics.For example, the concept of miniaturization is in conflict with the idea of full-range: the smaller the radiating source, the higher the operating frequency range.Low frequencies and therefore large wavelengths require big source sizes which mean large loudspeaker diaphragms and piston excursion.
There are usually many more conflicting requirements and tradeoffs found in the design of high sound pressure level output, low noise and distortion, full audio range, omnidirectional, and minimum size transducers.
Some studies concerning micro speakers of the moving coil type have been reported in [1][2][3][4].Within this type of speakers, performance was improved after modifying magnetic characteristics [2], optimizing diaphragm pattern design via a finite element approach [5] or introducing a combined permanent magnet [4].Although moving-coil is a well-matured technology, and probably best for mid-or low-frequency range loudspeaker units, the voice coil and the magnet do produce coupling interactions and nonlinearities difficult to deal with [6,7], introducing complexity in the design.
A different approach to micro speaker development is the piezoelectric type, as reported amongst others in [8][9][10][11][12].Driven not only by applications in the audio field but most importantly by biomedical and underwater ultrasound applications and requirements, the piezoelectric micro ultrasonic transducer (pMUT) has been widely described in [13][14][15][16][17].The fabrication details are by now well established together with their performance characteristics.Advantages of piezoelectric type transducers include smaller size, lower costs, batch fabrication, and integration with CMOS, which may allow for fabrication with required electronics such as amplifier or other signal conditioning circuits.Their main disadvantage is the difficulty of finding foundry standard processes which include piezoelectric layers for research and inexpensive prototyping, in contrast with the other main types of micro ultrasonic transducers: the capacitive type (cMUT), which can be fabricated with multiuser MEMS processes [18] or simply with standard CMOS foundry processes [19].Impedance matching with radiating media properties, in our case air, is also an important issue: the acoustic impedance of air (400 kg/m 2 s) is considerably smaller than that of piezoelectric materials commonly used in ultrasound imaging (30 × 10 6 kg/m 2 s), for which matching techniques have to be employed.
In this paper, the techniques used for pMUT design and modelling which comprise some of radio-frequency (RF) system analysis are extended to the audio acoustic frequency range for the numerical analysis and characterization of a piezoelectric unimorph micro speaker.A design is presented by which the actuating membrane is made only of isolating SiO 2 and a layer of nearly 100 micron of ZnO film, electroded on both sides.The resulting transducer, operating in the thickness mode, may be used for a variety of sound and vibration applications including active noise control, parametric arrays, beamforming arrays, and vibration sensors.Results are presented showing the most important design tradeoffs and their consequences on performance.

Transducer Design
The transducer presented is a piezoelectric unimorph with a Si-SiO 2 -Al-ZnO-Al layer stack, as shown in Figure 1, in thickness mode excitation.In order to operate at audio frequencies, transducer dimensions have to be increased by several orders of magnitude from those corresponding to micro transducers intended for ultrasonic radiation.Transducer radii range from 0.75 mm to 1.25 cm, and layer thicknesses range between 10 μm and 0.5 mm.
The described devices are intended to be fabricated in a post-CMOS technology by stacking layers of piezoelectric material on a mechanical silicon substrate.A schematic view of the layer stack and geometric variables is shown in Figure 1.The design parameter ranges are specified in Table 1.

Piezoelectric Actuation. Following the IEEE Standard on
Piezoelectricity [20], the linear piezoelectric constitutive equations relating the electrical field E i=1−3 in the case of actuators or the spatial polarization vector P i=1−3 in the case of sensors to the corresponding longitudinal and shear components of the mechanical stress T j=1−6 or strain S j=1−6 tensors are where d i j [m/V] and e i j [C/m 2 ] are the piezoelectric 3 × 6 constants of the piezoelectric material, and the indexes i and j denote polarization and actuation direction, respectively.C jh represents the 6 × 6 elastic stiffness of the piezoelectric material under constant electric field, and it relates the two piezoelectric constant types following the expression: The dielectric polarization P i=1−3 of the piezoelectric material is expressed as with ε i j (units of F/m) and D i (units of C/m 2 ) representing the 3 × 3 permittivity matrix and electric displacement vector, respectively, and ε 0 is the permittivity of free space.Equations (1) to (4) are the existing fundamental piezoelectric constitutive equations which describe the membrane behaviour.

Transducer as a Two-Port Network.
For the electromechanical analysis of our interest, a piezoelectric actuator can be described as a linear two-port network (Figure 2), for which the following relations hold between electrical and mechanical inputs and outputs for harmonic excitation under standard electromechanical analogy: The electromechanical transducer represented schematically as a two-port network.
where V and I denote complex voltage and current and the subscripts i and o denote input and output, respectively.F and u denote complex force and particle velocity on the front face of the transducer, and the matrix Z containing elements Z 11 , Z 12 , Z 21 , and Z 22 is the complex impedance matrix for the transducer viewed as a two-port network.By modelling the transducer with a numerical simulation environment such as Simulation Program with Integrated Circuits Emphasis (SPICE), data for the impedance matrix or particle velocity amongst other quantities can be obtained.For a more detailed analytical two-port, lumped-element modelling of a piezoelectric unimorph composite circular plate, see [21].In this work, the matrix Z was computed with Spectre after modelling the actuator with the commercial MEMS software MEMS+ by Coventor Inc.

Vibration Problem.
In order to relate the acoustics of the device to the intrinsic material properties of the membrane, the problem has to be tackled from an electromechanoacoustical perspective.In the mechanical domain considering vibrating plates, Euler's flexural equation of motion for a forced circular plate with clamped edges would apply.According to our results and following previous work on vibrating piezoelectric circular diaphragms [22], piezoelectric micromachined devices behave more like membranes (resonance frequencies proportional to R −1 ) than like thin plates (resonance frequencies proportional to R −2 ).Therefore the problem can be formulated as a forced circular membrane with clamped edges, for which the equation of motion is where w(r, t) is the membrane's displacement at a point pointed by vector r and instant t, f (r, t) is the forcing function at any point pointed by r and instant t, T is the tension on the membrane, ∇ 2 is the Laplacian operator, and ρ 0 is the material density.Solving (6) for w(r, t) with the appropriate boundary conditions corresponding to clamped edges gives an expression for the displacement at any point of the membrane at any instant t, given a known forcing function.The solution to (6) is usually not easy to compute and analytical solutions might not exist depending on the geometry and boundary conditions.Finite integral transforms might be obtained [23] or alternatively numerical solutions can be computed by the finite element and boundary element methods [24].
In our case, the vibration of the membrane was computed numerically by means of the commercial design and visualization software MEMS+.

Acoustic Radiation.
The acoustic pressure field p(r, t) at any point r and time instant t generated in free field by a radiating surface S with centre at the coordinates origin might be obtained from the Rayleigh integral [25] as where ω is the angular frequency, ρ 0 is the fluid density, k is the wavenumber, r = |r| is the scalar distance from the origin to the listening point, u is the complex particle velocity, and j is the complex unit number.In our case we can use the particle velocity u calculated numerically from (5), to obtain a good approximation of the radiated sound pressure field.
When integrated over a circumference of radius R accounting for the transducer's diaphragm surface, Rayleigh integral yields where I 0 is the complex output current numerically calculated after (5) and D(θ) represents the directivity function for a flat circular piston of radius R on an infinite baffle at an off-axis angle θ, given by with J 1 being the Bessel function of the first kind, k = 2π/λ = ω/c the wavenumber, λ the wavelength, and ω the angular frequency.
Therefore the acoustic output of the transducer can be modelled by a directivity function corresponding to a flat circular piston multiplying the current frequency response characteristic of the transducer, together with a complex exponential in time and distance and a 1/r decay factor [26].

Results
The commercial software MEMS+ from Coventor Inc. was used for transducer design and visualization in conjunction with Cadence Virtuoso as a numerical calculation engine.An AC analysis was performed to calculate the frequency response voltages and currents.An s-parameter analysis was used for impedance, admittance, noise figure, and transducer gain calculation.Both analyses were computed between 100 Hz and 100 kHz.
As it can be seen in Figure 3, impedance magnitude goes down and impedance resonance frequency lower as radius increases.Admittance behaviour is obviously the opposite: magnitude goes up and resonance frequency also goes down in the same fashion as radius increases.To illustrate how resonance frequencies vary as a function of radius or piezoelectric layer thickness, the values of the resonance peaks can be plotted for each radius or thickness, extending the data presented in Figure 3 to that shown in Figure 4, where radii ranged from 7.5 to 12.5 mm and thicknesses ranged from 10 to 500 micron.Plotting the data of Figure 4(a) against inverse radius, a straight line is found (Figure 5), which means that the transducer first resonance frequency is inversely proportional to the radius, corroborating the findings of [22] which mean that these diaphragms would behave more as membranes (R −1 resonance frequency dependence) than as thin plates (R −2 resonance frequency dependence).
Three sample transducer frequency responses for radii of 8.3, 9.7, and 12.5 mm and piezoelectric thicknesses of 0.08, 220, and 500 micron are shown in Figure 6.
Depending on the values of transducer radii or piezoelectric thicknesses, the resonance peaks in transducer FRF magnitude might increase more or less with respect to each other.To have a complete picture of how radii and thicknesses affect Table 2: Main parameters calculated after sp and ac analyses versus each of the independent variables: frequency, piezoelectric layer thickness, and transducer radius, keeping in each case two of them constant from values f = 10 kHz, t PZE = 80 um, and R = 1 cm.Noise figure and transducer gain are characterized in a similar manner: in the case of noise figure (NF), for a fixed radius of 1 cm, the values of NF versus frequency were obtained from the Cadence design environment for each piezoelectric layer thickness and are shown as a 3D plot in Figure 9.In the case of transducer gain, its calculated values from the Cadence design environment are similarly shown against frequency for a fixed radius of 1 cm and for each piezoelectric thickness (Figure 10).

Discussion
The results presented characterise the transducer design space and yield information and details on impedance, resonance frequency, noise figure, and transducer gain as a function of frequency, piezoelectric layer thickness, and transducer radius.
The main variables considered in this study induce to treat the problem as a 4-variable problem: frequency ( f ), piezoelectric layer thickness (t PZE ), radius (R), and one other variable for the z-axis which can be either frequency response (V ), impedance (Z), admittance (Y ), noise figure (NF), or transducer gain (G).
Therefore, 3D plots can be made of any of the z-axis variables with respect to two other independent variables after fixing the remaining one.For example, a 3D plot can be obtained of |Z| versus piezoelectric thickness and transducer radius at any fixed frequency.Similarly, if transducer radius was fixed, the plot of |Z| versus piezoelectric thickness and frequency could be produced.Alternatively, two variables might be fixed and the remaining two can be plotted against each other, which yield a plot matrix.Table 2  example of a matrix plot obtained for fixed values of f = 10 kHz, R = 1 cm, and t PZE = 80 μm.This approach introduces a complete way of characterizing the piezoelectric transducer in terms of impedance, frequency response, admittance, transducer gain, and noise figure.Any two independent variable values might be set and the relation with the two others might be obtained numerically.After the design space is characterised the next logical step would be to introduce a multiobjective optimization of the transducer design variables.This approach will hopefully help improve the process steps involved in micromachining and manufacturing this kind of piezoelectric micro speakers.

Conclusion
The design and analysis of a piezoelectric micro speaker based on pMUT operation modelling and characterization is presented.The analysis is based on s-parameter analysis  for a two-port network, after which values for impedance, admittance, frequency response, noise figure, and transducer gain have been presented.Relations for transducer radius, piezoelectric thickness, and operating frequency were obtained which yield diaphragm displacement and therefore sound pressure level and noise figure values.Results represent useful tradeoffs for piezoelectric micro speaker design.

Figure 1 :
Figure 1: Schematic view of simulated actuator.Layers, from top to bottom: top electrode, piezoelectric layer(s), bottom electrode, oxide layer, and Si-structural layer.

Figure 3 :Figure 4 :
Figure 3: Transducer impedance magnitudes (a) and admittance magnitudes (b) calculated for three different radii, all of them for a piezoelectric layer thickness of 80 μm.

5 f 6 fFigure 7 :
Figure 7: Peaks of impedance versus radius and piezoelectric layer thickness at four fixed frequencies.
th ic k n e s s ( µ m )V (V) R (m m )

6 R
(m m ) f = 20 kHz P Z E th ic k n e s s ( µ m ) V (V) (d)

Figure 8 :
Figure 8: First resonance in FRF magnitude versus radius and piezoelectric layer thickness at four fixed frequencies.
t h ic k n e s s ( µ m ) NF (dB)

Figure 9 :
Figure 9: Noise figure versus frequency and piezoelectric thickness at fixed radius = 1 cm.
t h ic k n e s s ( µ m )

Figure 10 :
Figure 10: Maximum stable power gain versus frequency and piezoelectric thickness at fixed radius = 1 cm.
shows an