Modeling of MEMS Resonator Piezoelectric Disc Partially Covered with Electrodes

The Legendre polynomial method has been extended to the modeling of MEMS resonator disc partially covered with electrodes. The disc has been divided into two areas: one with electrodes and the other without electrodes. For each area, The Maxwell equations and the piezoelectric constitutive equations of motion are studied and solved to yield a frequency response and electrical behavior of the MEMS resonator applying a semi analytical method based on a Legendre polynomials series and trigonometric functions. However, the method allows incorporating the boundary conditions directly into the governing equations by assuming position-dependent of elastic constants, mass density and delta functions. The alternating electrical source is described by specific terms which are also introduced into the equation of motion. The formalism has been developed which allows for both harmonic and modal analyses. In order to validate our polynomial approach, numerical results are presented such as resonant and anti-resonant frequencies, electric input admittance, electromechanical coupling coefficient and field profiles of fully and partially metallized PZT5A resonator discs. The results obtained were compared with those obtained by an approximated analytical method. The developed software proves to be very efficient to retrieve the contour modes of all orders.


Introduction
When a piezoelectric material is subjected to a mechanical strain, electrical charges were generated and conversely. This phenomenon was widely exploited in various engineering applications such as Micro-Electro-Mechanical systems (MEMS) technology. MEMS technology has been obtained significant growth in its field of application such as in electro-optic modulators, ultrasonic detectors, accelerometers, transducers, oscillators, electromechanical sensors and actuators [1][2][3][4][5][6].
Although, the vibration characteristics of piezoelectric materials are extracted from the piezoelectric constitutive equations, linear piezoelectricity and the Maxwell equations [7][8].
Several methods allow calculating vibration characteristics of piezoelectric devices. In 1967, Eer Nisse [9] presented a vibrational method to analyze the vibrational behavior of piezoelectric disks. C.H.Huang and C.C Ma used Electronic speckle pattern interferometry method to study vibration characteristics for piezoelectric cylinders [10]. Kharouf et al [11] studied the vibrational axisymmetric characteristics of hollow cylinders and disks made from piezoelectric materials by using the Rayleigh-Ritz method. These methods give accurate results especially for simple geometries. In the case of complicated geometries, alternative methods are needed. The finite elements method was used to analyze vibrational modes of ceramic disk composed of PZT5A with the diameter to thicknesses ratio range from 0.2 to 10 [12]. By the same method, Guo et al [13] have been calculated the vibration characteristics of disks composed of PZT5A with diameter to thicknesses ratio ranging from 10 to 20.
Moreover, Legendre polynomial method is one of the most methods used recently to check the electrical response of piezoelectric resonators and gives excellent precision for various structures [14][15][16][17][18][19][20][21]. This method is based on expressing the components of electrical and mechanical displacement by double series of Legendre polynomials.
In this article, polynomial method was extended for studying the frequency spectrum of a piezoelectric PZT5A disc resonator divided into two areas. Centralized area is electroded whereas the annular area is free of electrodes.

Formulation of the Problem
As shown in figure 1, the studied geometrical configuration is a homogeneous cylinder of finite length, divided into two areas named N and M. M area contains electrodes from the top and the bottom. H and R are the thickness and the radius of the cylinder respectively. Studying the structure in cylindrical coordinates Or z ϕ .
The z-axis coincides with the axis of anisotropy and the origin is placed at the center of the disc.
The vibration characteristics of a piezoelectric materials are studied by following equations [22][23]: are the components of electrical displacements and σ denotes the stress tensor.
( , ) ( , , ) = U M t u v w are the components of mechanical displacement in the radial, circumferential and axial direction.
Under the assumption of small deformations, the strain mechanical displacements relations in terms of cylindrical coordinates system are expressed by [24].
Along this paper, we adopt the following change of variables:   (1) can be written as: Taking in account the axisymmetric character of electrical source and studied structure, the variation of wave field does not depend on ϕ direction. Moreover, the mechanical displacement in circumferential direction is uncoupled from the electric field. We can only tike in consideration the equations (3a), (3c) and (3d).
Two rectangular window functions 3 ( ) q π and ( ) 1 q θ express the boundary conditions related to our structure.
The equations expressed in (4) are directly incorporated in the constitutive equations of our problem.
The normalized stress components are expressed by )  10  11  12  13  31  1  1  3  3   1  1  2  1 The electric displacement components: The index j=1, 2 refers to area M and area N respectively. Both for the metalized and non-metalized area, we express the field of governing equations (4) and their corresponding boundary, symmetry and continuity conditions.
For the metallized area, we have: R R In this area, the boundary, symmetry and continuity conditions are defined by following relations: The field of equations (3) becomes: For the non-metalized area, we have: 0 1 1 ≤ ≤ a q and We express also the boundary, symmetry and continuity conditions defined by following relations: The field of equations (3) becomes: ( ) (2) (2) (2) (1) Mechanical and electrical displacement components for area M and N are defined respectively as follow:  Substituting the mechanical and electrical displacements (8) and (9) in the field of equations (6) and (7) where: Based on foregoing formulation, we present simultaneously modal and harmonic analyses.

Harmonic Analyses
The harmonic analysis allows determining the normalized electric impedance which is expressed by using the expression of displacement current density in our piezoelectric structure = z J iwD . From the average electrical current that flows through the electrode of surface S, we determine the normalized electric input admittance by following relation:

Modal Analyses
Through modal analyses, resonant ( Ω r ) and anti-resonant frequencies ( Ω a ) are calculated by following relations, respectively.

Numerical Results
We start our calculations with properties in table (1), the computer program is established using Matlab software. The radius and the thickness of the studies resonator disc are H=1mm, R=15mm. The truncation number for our calculation is limited on M=N=12.

Resonant and Antiresonant Frequencies
The obtained results using polynomial approach are compared with results calculated analytically [25]. Analytical method is presented in appendix A. Tables 2 and 3     As shown in those tables, the higher error is less than 0.8% which means good agreement was found.

Normalized Electric Admittance
Our model allows also determining normalized electric input admittance of piezoelectric disc resonator for different rates of metallization. Figure.2 and Figure 3 displays normalized electric input admittance of a MEMS PZT5A resonator as a function of normalized frequency of contour modes for metallization rate 50% and 60% respectively.   , electrical and mechanical displacement profiles are calculated and presented in a 3D configuration for 80% rate of metallization. Figure 4 and Figure 5 shows the displacements profiles in radial and axial direction respectively. In Figure 6, electrical displacement profiles were presented. We can denote that the rate of metallization is an important factor in controlling the frequency variations of the resonator. Therefore, resonant and anti-resonant frequencies increase with the decrease of the length on the electrode.

Conclusion
Along this paper, Legendre polynomial method was successfully extended for studying the vibrational characteristics of a piezoelectric MEMS resonator partially metallized with centralized metallization. As an optimization tool, Legendre polynomial approach provides satisfied results such as resonant and anti-resonant frequencies, normalized electric impedance, electromechanical coupling coefficient, mechanical and electrical displacement profiles in three dimensional configurations. The formulation is easy to implement in which boundary conditions and electric excitation source are directly incorporated in the equation of motion. . ( ) 1