CONSTITUTIVE COMPUTATIONAL MODELLING FOUNDATION OF PIEZOELECTRONIC MICROSTRUCTURES AND APPLICATION TO HIGH-FREQUENCY MICROCHIP DSAW RESONATORS*

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1 ACTA MECHANICA SINICA (English Series), Vol.18, No.2, April 2002 The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press, Beijing, China Allerton Press, INC., New York, U.S.A. ISSN CONSTITUTIVE COMPUTATIONAL MODELLING FOUNDATION OF PIEZOELECTRONIC MICROSTRUCTURES AND APPLICATION TO HIGH-FREQUENCY MICROCHIP DSAW RESONATORS* Zhang Wu (~ ~) Tang Jinchun ( ~ ) (Department of Civil Engineering, Zhejiang University, Hangzhou , China) ABSTRACT: This paper establishes a piezoelectric constitutive computational approach based on generalized eigenvalue and multivariable finite element solutions with potential applications to accurate and effective analysis of layered piezoelectric microstructures of arbitrary geometries and different anisotropic materials, to ease the limitation of current computer capacity in analyzing large-scale high-frequency disturbed surface acoustic waves (DSAW) by mounted electrodes in piezoelectric devices such as microchip SAW resonators. A new incompatible generalized hybrid/mixed element GQM5 is also proposed for improving predictions of the piezoelectric surface mount thermal stresses that are shear-dominated. The (generalized) plane strain constitutive model is numerically verified for piezoelectric finite element computation. With the help of computational piezoelectricity (electro-mechanics) for general layered structures with metal electrodes and anisotropic piezoelectric substrates, some new interesting, reliable and fundamental constitutive finite element results are obtained for high-frequency piezoelectric and mechanical SAW propagations and can be used for further applications. The ST-cut FEA results agree quite well with available exact and lab solutions for free surface case. KEY WORDS: constitutive finite element, layered anisotropic solid, computational piezoelectricity/electro-mechanics, disturbed surface acoustic waves (DSAW), surface mount stress 1 INTRODUCTION Nowadays, surface acoustic wave (SAW) devices axe applied in a continuously growing number (Approx. 5 Million parts per day) in a lot of microelectronics systems and for communication engineering, and how to accurately design and analyze the high frequency and high quality SAW devices has always been an expensive and challenging issue. In the piezoelectric SAW technology, the objective is to generate and control higher frequency surface acoustic waves in piezoelectric substrates with arbitrarily shaped metal electrodes where the acoustic fields generate electric fields and vice versus [1]. The reliable structural finite element method is selected and extended to piezoelectric constitutive finite element method Received 7 November 2000, revised 1 November 2001 * The project supported by SRF for ROCS, SEM of China, the past Rutgers Univer-Seiko Epson Joint Fund and Zhejiang Provincial NSF

2 Vol.18, No.2 Zhang & Tang: Constitutive Modelling of Piezoelectric Microstructures 171 for electro-mechanically coupled structures with mechanical displacements and electric potentials in microchip SAW resonators in this paper. To improve the computational accuracy of 3-node or 6-node triangular elements [2] for acoustic waves and the higher order plate elements for piezoelectric plate vibrations[3,4], this paper reports the adaptive Lagrange 4-16-node quadrilateral piezoelectric constitutive finite element method to determine the aforementioned piezoelectric interaction and new interesting high accuracy high frequency SAW results are obtained by combining computational and constitutive(electro-mechanics) approaches for the piezoelectric SAW resonators made of anisotropic quartz crystal substrates [5]. The piezoelectric and mechanical results by the FEA programs are verified to be correct by analytical and experimental approaches, respectively, for the case of no metal electrodes. 2 PIEZOELECTRIC DYNAMICS AND CONSTITUTIVE FINITE ELEMENT METHOD The unified constitutive finite element method for computational piezoelectricity is based on the piezoelectric variational principle and the corresponding piezoelectric dynamics equations IKo [Mo~ ;]{r Kr162 (1) where M~, K~, K~r and Kr162 are the mechanical mass, mechanical stiffness, piezoelectric coupling and dielectric matrices, respectively, where U, r F and Q are the mechanical displacement, electric potential, mechanical force and electric charge vectors, respectively. By setting Mu~ = 0, the piezoelectric dynamics relations in Eq.(1) degenerate to the static piezoelectric equations for stress analysis. The above standard piezoelectric dynamics equations are electro-mechanically coupled and perhaps well literatured, but there remain many challenging problems related to these piezoelectric equations which need to be solved analytically, numerically or experimentally, when the piezoelectric structures are composed of mixed materials and subjected to various boundary conditions. To effectively and accurately analyze and design the high frequency piezoelectric SAW resonators with layered materials and hundreds of electrodes is one of them. In order to determine the piezoelectric SAW propagations, we need to solve the following piezoelectric generalized eigenvalue equations in matrix form as described in Ref.[6] LK~r T Kr = 0 As Kr162 is singular and the dielectric masses are zeros, Eq.(2) can be solved globally or by static condensation together with perturbation to remove the singularity. Because the dielectric matrix Kr162 is not on the same scale in value as K~, and K is expanded with the dielectric part and requires a huge amount of computer storage and memory in an eigenvalue solution, a special efficient eigen solver has to be developed. The condensed equations corresponding to Eq.(2) are (K* - ~M~) V = 0 (3a)

3 172 ACTA MECHANICA SINICA (English Series) 2002 where w is the circular natural frequency and K* = K~ - K~r (3b) K* is still symmetric but no longer sparse. The efficiency is relatively improved by use of static condensation, but unlike other piezoelectric materials such as lithium niobate and aluminum nitride, piezoelectric quartz crystals have relatively strong coupling coefficients, so the computational accuracy becomes very low due to the static condensation and perturbation. Therefore, Eq.(2) will be directly solved globally without static condensation or perturbation in this paper. From Eq.(3b), we see the similarities between piezoelectric element stiffness matrix and incompatible hybrid/mixed element stiffness matrix [7]. The differences are that the piezoelectric formulation is electro-mechanically coupled although the electric and mechanical fields are completely different, and consequently, the dielectric matrix Kr162 becomes singular, while the incompatible-type element formulation should decouple the conforming and nonconforming displacements to ensure the element stability for various element geometric configurations. One objective of solving Eq.(2) is to seek the right angular frequency corresponding to surface acoustic waves in piezoelectric crystal substrates. From w and wavelength, the SAW velocity is then calculated for piezoelectric device design. To formulate and select a finite element with good numerical performance is another key aspect of solving a general piezoelectric problem with specific boundary conditions. For piezoelectric element methods, the element mass and stiffness matrices are determined by the following element field interpolation for displacements and potential ~D i=1 where Ni(i = 1, 2,-.. nen) are the shape functions of an element with nen nodes. In the finite element formulation, the element mass and stiffness matrices corresponding to Eq.(2) can be generated accordingly and are given as follows, respectively M~u = f~ pnt Ndv = [fvbtcebe dv ~BTePeBd dv] K= fvbtcsdv [~BTpe~edv f B[DeBedvJ (5) (6) where N3x. is the element shape function matrix for mechanical part, the expanded element shape function matrix N~x. for piezoelectric finite element method is implied here, and p is the mass density of piezoelectric materials like quartz crystals. For quartz crystal, p -= 1650kg/cm 211'5]. In the element stiffness matrix, Be and Bd reflect azfisotropic elastic and dielectric geometric matrices for a piezoelectric finite element under certain axis of cut angles, corresponding to conforming and incompatible parts of a nonconformingdisplacement-enhanced generalized hybrid/mixed element formulation, while Ces Pe2 and De2 are the anisotropic mechanical stiffness constants, piezoelectric constant and dielectric permittivities, respectively, for the generalized plane strain model, and the material

4 Vol.18, No.2 Zhang & Tang: Constitutive Modelling of Piezoelectric Microstructures 173 ones for the plane strain case are also implied here. Although the piezoelectric effect can be very small, e.g. Pe(1,1)= C/m 2 for ST-cut quartz, the piezoelectric SAW velocity may have 0.35% of increase which can not be neglected in a high precision analysis and design of the high frequency micro SAW resonator. B = DN* denotes geometric matrix for the finite element method, where D is the expanded constitutive differential operator matrix for piezoelectric generalized plane strain model. The piezoelectric material parameters have to be very accurately measured, and the finite element methods should be carefully chosen and formulated; otherwise, the highprecision piezoelectric SAW modes will not be captured and predicted using finite element methods. And due to the characteristics of piezoelectric quartz crystals, the problem can be modeled as a generalized plane strain analysis, which can be viewed as a special 3-D problem with piezoelectric differential constraint O(*)/Oz = 0, which is slightly different from that of the plane strain problem in Eq.(6). 3 SPECIAL PIEZOELECTRIC EIGENVALUE SOLVERS In the piezoelectric finite element stiffness matrices, the dielectric matrix Kr162 is singular and its entries are much smaller than those of mechanical part K~, a special eigenvalue solver [e] should be developed for the piezoelectric wave numerical analysis. For ST-cut quartz, the dielectric constant De(l,1)= 3.921x F/m, which is very small. The piezoelectric eigensolver has been specialized to do the job of searching the SAW frequencies in any range of high frequencies. This is different from the common practice and program of computing the lowest frequencies relevant to structural damage and failure. Finally, the super convergent finite element method together with an efficient eigenvalue solver is developed in such a manner that only nonzero terms in stiffness and mass matrices are stored to save the memory to the maximum extent in this paper, and that dynamic memory allocation is added, the multi-mesh, Rayleigh Quotient Interaction Scheme (MRQI), is adopted to further reduce the storage requirements for piezoelectric analysis. 4 FINITE ELEMENT SELECTION FOR SURFACE MOUNT STRESSES AND SURFACE WAVES The right choice of finite elements for high-frequency SAWs is essential for high accuracy of the SAW analysis. The new types of finite element methods for improved stress computation [7] are numerically found not to produce better results when used in SAW analysis. It was also found that piezoelectric plate element methods did not provide good converged results in resonator analysis either [3]. That can be explained and attributed to the assumptions in Mindlin's plate constitutive modeling equations [4]. The Lagrange-type finite elements possess inside nodes over the elements, so it can, in principle, better represent both eigenvectors and eigenvalues, in comparison with Serenditytype finite elements such as 4-node and 8-node elements or 3-node and 6-node triangle elements with no inside nodes for stress analysis. Isoparametric element methods can be easily implemented in the program by using the same shape functions for field variables and coordinates. One major reason why quadrilateral (rectangular) finite elements, instead of triangular elements, are selected in this SAW propagation analysis is that the element alignment can

5 174 ACTA MECHANICA SINICA (English Series) 2002 be made desirable in the direction of SAW propagation from the viewpoint of optimal finite element representation of the actual field distribution. The above adopted elements for acoustic waves axe compatible and the displacements are major factors involved for SAW prediction. In this paper, 4- and 16-node isoparametric compatible Lagrange finite elements are selected for generalized plane strain piezoelectric wave analysis, in the piezoelectric coding and comparison. Although 16-node Lagrange element is rarely used for stress computation [7,s], it is found that superconvergent adaptive 16-node compatible Lagrange element method with consistent mass generation possesses excellent overall performance for high frequency (SAW) wave analysis, while we have found that the convergence accuracy of 4-node based elements is not high enough even using more than elements for half-period high frequency piezoelectric SAW analysis. This is a little different from piezoelectric mounting stress prediction using a compatible displacement finite element [a]. A 4-node based incompatible finite element with relatively refined mesh can be formulated to achieve expected stress accuracy except for shear-dominated thermo-stresses. For example, a five-node QM5 Serendity incompatible element can be formulated based on the well-known Q4 and Taylor/Wislon nonconforming element QM6. In QM5, the total element displacement vector is composed of compatible and incompatible parts (U = Uq U~) and the incompatible displacement shape functions are adopted as follows U~ = v~ = 0 2- ~2 _ ~2 A: (7) In the corresponding element strains and stiffness matrix in Eq.(6), the isoparametric Jocobian adjoint matrix J(~, y)* is replaced by g(0, 0)* at the element centroid, so that QM5 can pass the constant stress/strain patch test like QM6. To pass the axisymmetric patch test, Axi-QM5 should be further modified following the axisymmetric nonconforming element formulation of AQM6 [1~ passing the constant stress/strain patch test. For relatively refined meshes, incompatible element QM5, and new corresponding piezoelectric element GQM5 using the generalized hybrid/mixed element formulation[ 7] should be, in principle, more accurate than Q4 and more efficient than GQM6/QM6 [7] and other used compatible elements [9] for expected high electro-mechanics accuracies. 5 ADAPTIVE STRATEGIES FOR SURFACE MOUNTING STRESS AND SAW WAVE ANALYSIS A piezoelectric SAW device like a micro resonator in Fig.1 is actually a smart electroded structure made of smart piezoelectric materials such as anisotropic quartz crystals and ceramics. The quartz substrate of a SAW resonator is mounted into a ceramic package by using an adhesive. In these surface mounted resonators, the applied electric fields will generate strains and mechanical deformations like surface acoustic wave motions and vice versus. The key issue here is the actual coupled relationship between the two fields and how to accurately determine it using high-precision and reliable constitutive finite element methods. For linear and highly nonlinear situations, the computational-constitutive adaptive indicator was proposed in the adaptive finite element computation [s]. Unlike the stress

6 Vol.18, No.2 Zhang & Tang: Constitutive Modelling of Piezoelectric Microstructures 175 analysis in SAW devices like a micro SAW resonator, there is usually no need to compute stresses in the SAW eigenvalue analysis. Therefore, the magnitudes of the displacement distributions are adopted to be the adaptivity indicator, considering the features of the surface acoustic waves that the displacements and potentials decay (to zero) away from the wave propagating surfaces and do not vary much along the wave propagating x-surface (Fig.l). aluminum oxide wear pure aluminum electrode Y t H ~ a r t z substrate potential z j ~- x surface mounted (a) (b) Fig.1 (a)part of the amplified cross view of a piezoelectric resonator with electrodes; (b)variation of SAW particle displacement and potential with depth y In order to systematically formulate a simple y-directional remeshing scheme, we give further consideration to the following aspects. The distribution trends of SAW displacements and potentials are known and the typical computational structures for piezoelectric resonators are trapezoids and rectangles (Fig.2). Furthermore, the SAW velocities in the piezoelectric substrates without electrodes are also known, for example, the exact SAW velocity in the ST-cut-quartz substrate is always 3 158m/s. Therefore, practical strategies for adaptive piezoelectric remeshing can be formulated for simplicity without loss of accuracy. Under the precondition that the 4- and 16-node Lagrange finite elements are correctly formulated and used, the y-directional mesh enrichment or remeshing formula along the depth(y) is given by using the simple geometric progression. rn~ -- r) h.-y t~ = ~r--nod~ (Ynodey-- 1) < hx (to ensure the nearly optimal element configuration) 1 -- r rn-i dym ~,nodey-1 (Ynodey -- 1) m = 1, 2,''', nodey(= kny -Jr- 1) dym = Yl -4- dym-1 > IOPT (to avoid any interference from the back surface) (8) (9) (lo) where hx and h t~ are element length in x direction and the top element length in y direction,,-y respectively, dy,~ -- y,~ - ym-1.

7 176 ACTA MECHANICA SINICA (English Series) 2002 Fig.2 Part of graded Lagrange finite element mesh for piezoelectric SAW devices with multi-sloped metal electrodes (r =0.992 for SAW, and r =1/0.992~1.87 for mounting stresses) The above adaptive enrichment strategy is characterized by tactfully relocating the nodes to refine and optimize the mesh refinement and alignment for SAW analysis, while the total number of elements can be kept the same. Unlike the adaptive meshing in stress analysis, the element alignment is so important in SAW propagation analysis that simple bisections appropriately graded are often used to mesh the substrate structures. Numerical results show that this enrichment strategy for piezoelectric SAW finite element analysis is quite effective for mechanical SAW propagation too. 6 VERIFICATION AND APPLICATION PROBLEMS In the SAW analysis, the enforced finite element boundary conditions for half-wavelength (PT) analysis of SAW propagation can be given by u~eft(x = xl) = -u~ght(x = xl + PT) u, v, w, r (11) These SAW constraint conditions are different from the ordinary boundary conditions in stress/strain analysis in general finite element computations, and require special treatment to be.enforced. The first benchmark test problem is to numerically compute the SAW velocity in an ST-cut rectangular quartz crystal substrate without electrodes in Fig.2, so as to compare the computed results with the exact and experimental solutions; the exact SAW velocity for ST-cut quartz crystals should be a constant of m/s for any frequencies. The detailed results for SAW and surface mount stresses are given in Table 1 and Figs.3 and 4. Table 1 Computed piezoelectric SAW velocities vs cut angle on free quartz substrate surface, using adaptive 16-node Lagrange finite element mesh (r , 4 elements, specified relative error~ 0.04%) SAW velocity SAW velocity Exact Experimental Cut angle (m/s) (m/s) (Datta, 1986) (Seiko Epson) piezo, mech. piezo, mech ST-cut, Note: If using a uniform mesh(r=l.0), 4 400(=1 600 elements) is required to obtain the results with same accuracy

8 Vol.18, No.2 Zhang & Tang: Constitutive Modelling of Piezoelectric Microstructures 177 Table i indicates that the adaptive 16-node Lagrange FEM for piezoelectric and mechanical SAW analyses is accurate and with about 75% saving in the number of elements as compared with the ordinary uniform meshing method. Figure 3 shows coarse mesh accuracy differences of piezoelectric surface mount von Mises stresses by using the piezoelectric thermal (PT) compatible finite element Q4 and incompatible generalized hybrid/mixed element GQM5. The most dangerous points are close to the left and right corners of the surface mounted substrate. As the quartz crystals are very hard (for ST-cut quartz crystals, Ce(1,1)= 8.674x101~ Pa) and thermal mount stresses are mainly shear ones, the new types of finite element methods for high accuracy stress prediction become as significant as for high frequency acoustic waves. "~ PT-Q4 ] --O- PT-GQM5J i 0.5 I Ii 13 node location of surface mount interface Fig.3 ST-Cut thermal piezoelectric von Mises stress curves along the surface mount interface using a x-directional uniform coarse mesh, CTE(1, 1) = /C ~ The piezoelectric and mechanical SAW motions and corresponding computed SAW velocities for any cut angles are given in the second problem(fig.4). The motions at a low frequency of 39.5 MHz can also be seen on an atomic scale with nanometer resolution experimentation [11]. However, the cost is high and is not providing required accurate data like piezoelectric acoustic wave velocities and potentials in this paper cut angle for piezo.(upper) and mech.(lower) Fig.4 Unmetallized piezo, and mech. SAW velocity curves for a free surface case, PT=5 tim, wave frequency=314.7~330 MHz. (Note: prescribed relative error~0.04%) The second test problem is to analyze a practical DSAW resonator with layered materials of quartz substrate and Aluminum electrodes (Figs.l, 2 & Table 2). The quantitative relationship between them is studied by the proposed adaptive constitutive finite element method, as the exact or accurate experimental results are hard to acquire or not yet available for this case. There could be many vibrating frequencies for an electroded resonator structure, but only two of the computed displacement and potential modes are piezoelectric surface acoustic

9 178 ACTA MECHANICA SINICA (English Series) 2002 wave modes if captured and should be displayed as in Figs.1 and 5. ~ 2 3x x 10-4 i,~ 0 Jl Q9 --I X 10-4 :. ot "~ X 10-4 Fig.5 The variations of piezoelectric (electro-meeh coupled and metallized) DSAW particle displacement in z and potential along the depth(y) An interesting phenomenon is shown in Table 2 that when there axe aluminum electrodes interfering with the SAW propagation, the wave velocities will decrease and there axe two DSAW velocities of a stopband, rather than one standing-wave velocity in the case of no electrodes. The geometric and material features of electrodes will rigorously influence the SAW frequencies and velocities. Both ST-cut lower and upper limit velocities tend to be m/s as the electrode decreases in size and weight. By use of the lower and upper limit frequencies fl and fu computed in Table 2, the stopband width is obtained as 6f = fu - fb and center frequency can be calculated as fr = 0.5(fl + fu). The above accurate quantitative and qualitative relationship discovered by adaptive finite element computation based on generalized plane strain constitutive model is valuable and essential to piezoelectric SAW micro resonator design and analysis. Table 2 FEA results of the lower and upper limit frequencies g~ velocities of ST-cut quartz substrate with aluminum electrodes metallized (H pro, LT=2.5 pm, expected relative error<0.04%) Wave length No. of found Substrate Electrode SAW mode (~m) (eigenvalues) elements elements displacements x SAW Normlized Actual SAW u v w mode frequency(hz) frequency(mhz) velocity (m/s) 1 0, x10 0,311616x x x x x , CONCLUSION AND DISCUSSION In computational piezoelectricity(electro-mechanics) for this microchip DSAW resonator analysis and design, only the unified constitutive finite element method will lead to general acceptable results, and this needs profound understanding of the theories of highprecision piezoelectric finite element methods and electro-mechanics models like the (generalized) plane strain model, which develops into a new concept of generalized computational mechanics in this paper.

10 Vo1.18, No.2 Zhang & Tang: Constitutive Modelling of Piezoelectric Microstructures 179 From the computed results in this paper, it is evident that the suggested constitutive computational approach based on finite element and generalized eigenvalue solutions is very powerful for evaluating how the metal electrodes affect the SAW propagating frequencies and velocities in the piezoelectric substrates on the micrometer scale, and also very accurate for discovering the actual relationship between the acoustic mechanical fields and electric fields. We can further conclude that: (1) The adaptive 16-node piezoelectric constitutive finite element method and results are valuable and accurate for piezoelectric DSAW resonator analysis, while the 4-node based piezoelectric generalized plane strain finite elements for acoustic waves can be used for comparison but is not accurate enough for low convergence rate. In this respect, the study has attracted our attention in further numerical vibration computations in piezoelectric crystal plate and piezoelectric ceramic shell analyses [3,12]. (2) The piezoelectric generalized plane strain constitutive theory is numerically proved to be correct by the suggested adaptive piezoelectric constitutive finite element computation for the free surface SAW propagation. The assumed stress element methods[ y] are not for acoustic waves but for improved computation of mounting stresses in piezoelectric devices, and the plane strain finite element method(w - 0) will not converge to the exact solutions of the free-surface ST-cut piezoelectric SAW velocities, even though the monotonously convergent finite element methods are adopted. (3) The arbitrarily shaped metal electrodes will rigorously affect the high SAW frequencies and velocities in anisotropic piezoelectric quartz crystal substrates, and the generalized plane strain FEA procedure and codes for one-period analysis are very useful for accurately analyzing the relationship between the DSAW velocities and the geometric and material characteristics of the electrodes, which is analytically and experimentally verified for the case of no electrodes, and also an advantageous alternative for the case of multi-layered piezoelectric structures in microchip DSAW resonator design and analysis. (4) Piezoelectric SAW analysis greatly increases computational time as compared with mechanical ones, so only nonzero entries in the unified constitutive finite element stiffness and mass matrices are stored for the specialized eigensolver in this paper, to save the computer storage to the full extent. However, due to the capacity limitation and generality of modern available computing facilities and methods, we still can not capture the piezoelectric and mechanical SAW motions in the full-scale resonator structures with hundreds of electrodes using sufficiently refined meshes, even though a 3-D DSAW codes for the full-scale piezoelectric SAW analysis have been finished[ 13]. This has been and will be a challenging issue in further research on more accurate assumed (nonconforming) displacement finite element methods and more general electro-thermo-mechanics theories in computational piezoelectricity. There are great potential applications of the piezoelectric constitutive finite element method(cfem) for high-frequency acoustic waves and incompatible constitutive finite element method(ic-fem) for surface mounting stresses, respectively. After a period of intense research and application, fundamental CFEM and IC-FEM discussed above and some unique reliable results essential to the analysis and design of new high-quality piezoelectric DSAW resonators have been obtained. Their further applications to various DSAW devices including SAW sensors [14] should be investigated using the unified element methods as before if possible, and looking back on the past achievements that bear testimony to the potential of

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