Finite Element Calculation of the Dispersion

Finite Element Calculation of the Dispersion Relations of Infinitely Extended SAW Structures Including Bulk Wave Radiation Manfred Hofera , Norman Fingerc , Sabine Zaglmayrb , Joachim Sch¨oberlb , G¨ unter Kovacsc , Ulrich Langerb and Reinhard Lercha a

Department of Sensor Technology, Paul-Gordan-Str. 5, Erlangen, Germany b Institute of Computational Mathematics, Altenberger Str. 69, Austria c EPCOS AG, Postfach 801709, Munich, Germany

ABSTRACT In the design procedure of surface acoustic wave (SAW) devices simple models like Equivalent Circuit Models or the Coupling of Modes (COM) Model are used to achieve short calculation times. Therefore, these models can be used for iterative component optimization. However, they are subject to many simplifications and restrictions. In order to improve the parameters required for the simpler models and to achieve better insight to the physics of SAW devices analysis tools solving the constitutional partial differential equations are needed. We have developed an efficient calculation scheme based on the finite element method. It makes use of newly established periodic boundary conditions (PBCs) allowing the simulation of an infinitely extended SAW device. This is a good approximation of many SAW devices which show a large number of periodically arranged electrodes. We have developed two different formulations for the PBCs: One leads to a small quadratic eigenvalue problem in which an additional matrix inversion has to be performed and the other to a linear eigenvalue problem operating on a larger matrix. These formulations allow the calculation of the complete dispersion relation. Bulk waves which are generated due to mode conversion at electrode edges are allowed to leave the calculation area nearly without reflection. Therefore, the calculation scheme also considers damping coefficients caused by the conversion of surface waves into bulk waves. This behavior coincides well with real SAW devices, in which bulk waves are scattered in all directions at the substrate bottom. In the paper, a short introduction to the basic theory of the numerical calculation scheme will be given first. The applicability of the calculation scheme will be demonstrated by comparing analytical, measured and simulated results. Keywords: Surface Acoustic Waves, Finite Element Method, Periodicity, Dispersion

1. INTRODUCTION A surface acoustic wave device consists of a piezoelectric substrate carrying metallic structures such as interdigital transducers (IDTs) and reflecting gratings or electrodes1 (see Fig. 1). Such devices are used in a wide range of applications, mainly as delay lines and filters in the UHF range2 but also as sensors for different applications such as temperature and pressure,3 chemical,4 flow5 or humidity6 measurements and many other types of physical quantities. It is an advantage of these types of sensors that Further author information: (Send correspondence to Manfred Hofer) E-mail: [email protected]

{

Interdigital transducer (IDT)

Piezoelectric substrate

Acoustic absorber

Figure 1: Sketch of a SAW filter with two Interdigital Transducers (IDTs)

they can be used in a wireless passive way and therefore don’t need a separate power supply.7–9 They can be placed on particular difficult accessible locations and are even usable in harsh environments.10 Despite this wide range of applications in sensor technology, the main area of application is, as mentioned already before, in telecommunications.11 SAW filters achieve superior characteristics not attainable with discrete electric components and are therefore used in different types of telecommunication devices such as receivers, e.g. in mobile phones or television sets. The improvement of this sophisticated low-loss devices needs highly developed design and simulation tools. The design of SAW devices is based on signal, network and field theory, whereas every type of theory is based on a different level of abstraction. To gain a deeper insight to the physics of SAW devices field theory is the most appropriate one. Therein, the complete partial differential equations are considered. Therefore, all second order effects are regarded automatically. The models used can be kept abstract and, only few additional assumptions are needed. A widely used method which relies on a numerical treatment of field theory is the Finite Element Method (FEM).12, 13 Therein, the full partial differential equations describing the behavior of piezoelectric devices are discretized and numerically solved. The calculation of a complete SAW filter with the FEM is at present impossible: A conventional two-port SAW filter consisting of two IDTs and two resonators (see Fig. 2) may have a length of up to thousands of wavelength and an aperture as far as a hundred wavelengths. The substrate which carries the electrodes has - depending on the working frequency - also a depth of up to hundred wavelengths. Taking into account, that in the FEM every wavelength should be discretized with at least ten first order finite elements and that an arbitrary piezoelectric material has four degrees of freedom this leads to 4 × 108 unknowns in the three dimensional case. Nowadays computer resources and algorithms can’t handle this amount of unknowns. Therefore, additional assumptions must be made. SAW reflectors consist of up to thousand of electrodes. Thus, it’s allowed to model them as an infinitely extended periodic structure with a single electrode or grating as base cell. Furtheron, the aperture of the electrodes is, compared to the electrode gap, up to two hundred times larger. Lateral effects have therefore only a neglible impact. These two assumptions lead to a simple two dimensional model with periodic boundary conditions (PBCs) which are described in section 3. A restriction of the FEM is the fact that it must operate on a self-contained area. This is a problem especially when wave propagation phenomena have to be considered. In this case special conditions at the boundaries must be fulfilled. For this purpose, so called absorbing boundary conditions (ABCs) can be used which are explained in section 4.

Reflector

IDT 1

IDT 2

Substrate

Figure 2: Conventional two-port SAW resonator filter

2. FE FORMULATION FOR PIEZOELECTRIC EFFECTS To keep our presentation self-consistent, we will give a short introduction to the simulation of piezoelectric devices applying the FE method. Starting points of the derivation of our FE formulation are the constitutive equations of piezoelectricity14 T = c E S − et E D = eS + εS E ,

(1) (2)

with T S E D cE e εS

tensor of mechanical stress tensor of mechanical strain vector of electric field vector of electric displacement tensor of mechanical coefficients at constant electric field tensor of piezoelectric coefficients permittivity tensor at constant mechanical strain .

Furtheron, the equation of motion ¨ − fV ∇ · T = ρd

(3)

wherein ρ denotes the density, d the vector of displacements and fV the vector of external volume forces has to be taken into account. The dot notation refers to differentiation with respect to time. Supposing a harmonic excitation d(x, t) = d(x)ejωt

(4)

∇ · T = −ρω 2 d − fV .

(5)

eq. (3) can be rewritten as Furthermore, one has to bear in mind that piezoelectric materials are insulators and therefore no free volume charges may exist. This can be expressed by ∇·D=0. (6) Applying Galerkin’s method15 to eq. (1) to (6) by multiplication with an appropriate weighting function, using Green’s formula15 and discretization in space one obtains a system of linear equations in matrix form16     uu  d F K + jωCuu − ω 2 Muu Kuφ = , (7) φ Q Kφu −Kφφ {z } | K? where the notation has been chosen to

Muu Kuu Kφφ Kuφ F Q φ

mechanical mass matrix mechanical stiffness matrix dielectric stiffness matrix piezoelectric coupling matrix nodal vector of external mechanical forces nodal vector of electric charges vector of electric potentials

The damping matrix Cuu which describes the behavior of the absorbing boundaries has already been incorporated and will be depicted in section 4. For convenience, the FE matrices have been combined to the matrix K? , the vector of unknowns will be denoted as u (u = [d, φ]T ) and the source term on the right-hand-side as R (R = [F, Q]T ). These definitions allow us to write eq. (7) in the following convenient form K? u = R .

(8)

3. PERIODIC BOUNDARY CONDITION SAW resonators may consist of up to thousand of electrodes. Therefore, only a neglible error is made by regarding the reflector as an infinitely extended periodic structure (see Fig. 3). The substrate - in the case of SAW devices a piezoelectric material - spreads over the lower half space (z ≤ 0), whereas the periodic placed electrodes with arbitrary shape range from 0 ≤ z ≤ h. In the upper half space, air is considered. An optional protection cover may be additionally introduced into the model. The period of the structure is a pitch (p). z

Cover

Period (p)

Electrodes

h x Substrate

Figure 3: Periodic SAW structure

In every base cell of Fig. 3 an impinging (surface-) wave from the left is mainly transmitted but also partly reflected due to four main effects: a) piezoelectric shorting, b) geometric discontinuities, c) electrical regeneration and d) mass loading.17 A small part of the energy is converted into heat due to ohmic losses or conversion into bulk acoustic waves.

3.1. Wave Propagation in Periodic Structures Many authors have studied wave propagation in periodic structures.18–22 Some of them deal with a strict periodicity of a half or a full wavelength.23–25 This kind of strict periodicity can’t be used for determination of the dispersion behavior of periodic structures. Therefore, a more fundamental approach has to be pursued. Due to the periodicity of the geometry, the resulting field distribution must be also quasi-periodic, leading to u(x + p)ejωt = u(x)eγp ejωt ,

(9)

with u denoting the field distribution and γ = α + jβ the complex propagation constant. The variable α describes the decay behavior and β stands for the phase propagation constant of the wave.

In general, a harmonic excitation is assumed. Therefore, the term ejwt indicating harmonic waves will be omitted in further formulae. The aim of our simulations is to determine the dependency of the propagation constant γ from the excitation frequency. This connectivity can be sketched in a dispersion diagram (see Fig. 4). In the stopband area (from ω1 to ω2 ) the propagation of the SAW is not possible. In this region the wave has a damping constant α larger than zero. Beginning at the onset-frequency ωc the back scattered surface wave is partly converted into a non guided bulk acoustic wave. This phenomenon manifests again in a damping constant α ≥ 0.

w

Bulkacoustic wave

Rayleighwave

ax(w)

bx(w)

wc w2 w1

-2p p

-p p

2p p

p p

0

bx

Figure 4: Dispersion diagram of a periodic structure

3.2. FE Formulation for Periodic Structures Starting at eq. (8) we split the unknowns in u into those of inner nodes ui and boundary nodes ub . The same has to be done with the matrix K? and the right-hand-side vector R 

K?ii K?bi

K?ib K?bb



ui ub



=



0 Rb



.

(10)

In the inner region of the calculation area all forces must stay in equilibrium, which results in a zero vector Ri . This fact has already been accommodated in eq. (10). 3.2.1. Schur-Complement Formulation One possible formulation for the PBCs can be developed in taking the Schur-Complement of eq. (10) ? ? ? (−Kbi Kii?−1 Kib + Kbb ) ub = Rb . | {z } S

(11)

The remaining unknowns ub are those on the periodic boundary. Splitting them into left and right nodes using indices l and r (ub = [ul , ur ]T ) and incorporating the periodicity condition according to eq. (9) leads to a quadratic eigenvalue problem in η = eγp 26 η 2 Slr ul + η(Sll + Srr )ul + Srl ul = 0 .

(12)

Therein, only the nodes on the left boundary appear. The quadratic eigenvalue problem can be solved for example by inverse iteration,27 with a two-sided Lanczos method,28 or by linearization of the system which doubles the matrix size.29 The latter solving method is straight forward to implement and therein the standard non-symmetric eigenvalue solver from LAPACK30 can be used.

3.2.2. Transformation to a General Linear Eigenvalue Problem Another method to yield a formulation for the (9) directly into the formula  ? Kii Kil?  Kli? Kll? ? ? Kri Krl | {z K

PBCs would be to incorporate the periodicity condition of eq.     ? ui 0 Kir ?  ul  =  Rl  , Klr ? ur Rr Krr }

(13)

before taking the Schur-Complement. To be able to formulate an eigenvalue problem, eq. (13) must be transformed in such a way that the right-hand-side gets zero. By taking proper transformation matrices one yields a general eigenvalue problem of the form     ui ui A = ηB (14) ul ul with the matrices A and B defined as A =



Kii? ? Kri

B



0 −Kli?

=

Kil? 0



(15)

? −Kir ? ? −(Kll + Krr )



.

(16)

Compared to the Schur-Complement formulation no matrix inversion is needed. On the other hand the size of the eigenvalue problem is much larger: Here, all inner nodes and additionally the nodes on one periodic boundary contribute to the eigenvalue matrix, but, fortunately, the matrices A and B keep sparse. In the Schur-Complement formulation, the eigenvalue problem must be solved only for the nodes on one periodic boundary, but there the matrices are dense.

3.3. Verification of the PBCs Verification of the presented methods was done by comparing the analytic solution of pure mechanical symmetric plate modes31 4β 2 ktl kts tan kts b/2 =− 2 , (17) tan ktl b/2 (kts − β 2 )2 with b denoting the plate thickness, ktl the wave number of the longitudinal wave, kts the wave number of the shear wave and β the propagation constant with the simulated results. The dependencies between wave numbers and propagation constant can be written as 2 ktl 2 kts

=



ω cL

2

− β2

(18)

=



ω cS

2

− β2 ,

(19)

with cL the velocity of the longitudinal wave and cS the velocity of the shear wave. The FE model for the simulation of symmetric plate modes can be seen in Fig. 5. Therein, p stands for the width of the base cell and H is the half plate thickness. The boundaries denoted with Γl and Γr are the left and right periodic boundary. On Γ1 displacements in x2 direction have been suppressed to receive only symmetric lamb modes. On Γ3 a force free boundary is assumed.

G3

Gl=G4

x2

Wp

Gr=G2

H

x1 G1 p Figure 5: FE Model for the calculation of plate modes

The calculation was performed on the basis of the two methods introduced above. Both methods delivered similar results. The Schur-Complement method was approximately three times faster, because the used LAPACK eigenvalue solver could handle only dense matrices. A comparison of analytic and simulated results can be seen in Fig. 6. The simulation coincides very well qualitatively and quantitatively with the analytic solution.

Frequency (kHz)

5 4 3 2 1

(a) Dispersion diagram of three lamb waves of lowest order (after Mindlin: see31 )

(b) Simulated plate modes

Figure 6: Qualitative comparison of analytic and simulated results of symmetric plate modes

4. ABSORBING BOUNDARY CONDITION FE calculations need a self-consistent simulation area to operate on. Especially in the case of wave propagation special boundary conditions have to be incorporated into the calculation scheme. Standard FE boundaries such as Neumann or Dirichlet conditions are not usable. They would cause strong reflection not occurring in real devices. Therefore, many authors have proposed special absorbing boundary conditions (ABCs) for the FE calculation of elastic waves. One of the first was suggested from Lysmer and Kuhlemeyer32 σ τ

= ρcL u˙ n = ρcS u˙ t .

(20) (21)

In the above equation σ and τ denote the normal- and shear stress on the boundary and u˙ n and u˙ t the time derivatives of the mechanical displacements in normal and tangential direction. A system of hierarchical ABCs has been published by Engquist and Majda.33, 34 The first order ABC is similar to the ABC of Lysmer and Kuhlemeyer. For piezoelectric materials the velocities can not be calculated globally, they must be computed for every boundary element separately, because velocities change with the direction of the boundary. For this purpose, the piezoelectric stiffened Christoffel equation was used.14 In the simulation of bulk acoustic wave radiation of SAW devices, the radiation angles of these waves are known and have been measured recently.35 Therefore, a focus has to be given to these angles. Higdon36 proposed a method which is able to absorb waves impinging from selected directions with angles αj (which are defined as angles normal to the boundary) "m  # Y ∂ ∂ (cosαj ) − c u=0. (22) ∂t ∂x i=1 This ABC is a generalization of those from Engquist and Majda. The first order ABC of Higdon has been adapted to the piezoelectric case and has been implemented into our code.

5. RESULTS The model illustrated in Fig. 7 has been used for the calculation of surface acoustic wave (SAW) effects. Electrode Substrate

GL

Pitch (p)

GR

Figure 7: Model for the simulation of SAW effects

First, a simulation with aluminum electrodes and YZ-LiNbO3 as substrate material was performed. The material data were taken from 37 . The pitch was chosen to be 2 µm and the electrode height 0.35 µm. The substrate thickness of the FE model was determined with five pitches. The results of the simulations with these presumptions can be seen in Fig. 8. The ABCs absorb waves impinging from a certain, predefined direction. Since bulk acoustic waves are radiated very flat at the onset-frequency and this radiation angle increases with frequency, the absorption frequency drops with increasing absorption angle. This fact can be clearly recognized by comparing the left and right pair of pictures in Fig. 8. For a different periodic structure, simulation results have been compared to measured data. A periodic structure with 1.33 µm pitch, aluminum electrodes of 0.2 µm height and a substrate material of 36◦ YX-LiTaO3 has been considered. At this cut of LiTaO3 , the displacements of the surface wave do not keep in the sagittal plane. Therefore, the displacements in all three space directions and the full anisotropy of the material have to be taken into account, although the model itself may kept two dimensional. The comparison of the simulated and measured results are shown in Fig. 9. It was not possible to obtain measured results for stopband frequencies. Therefore, just the propagation branches of the surface wave can be compared to the simulated results. In Fig. 9, many types of propagation modes can be identified. The one

0.92

0.92

0.92

0.92

0.9

0.9

0.9

0.9

0.88

0.88

0.88

0.88

0.86

0.86

0.86

0.86

0.84

0.84

0.84

0.84

0.82

0.82

0.82

0.82

0.8 0.9

1 βp/π

1.1

0.8

Frequency (GHz)

Frequency (GHz)

Bulk wave radiation

−0.05

0 αp

0.8

0.05

0.95

a) Absorption angle: 0◦

1 βp/π

1.05

0.8

−0.05

0 αp

0.05

b) Absorption angle: 88◦

Figure 8. Simulation results for a periodic structure on YZ-LiNbO3 at two different absorption angles (The fragments of the damping coefficients α belong to the phase constant β entering or leaving the appropriate picture on the left.)

1.58 1.56

Frequency (GHz)

1.54 1.52 1.5 1.48 1.46 1.44 1.42 1.4 0.92

0.94

0.96

0.98

1 βp/π

1.02

1.04

1.06

1.08

Figure 9: Comparison of simulated (dots) and measured (crosses) dispersion data

starting at low frequency in the middle of the picture is a Rayleigh mode. Bulk acoustic wave modes can be determined also which interact with the surface wave. The interesting mode, which is compared to measured data, is a leaky surface wave (LSAW). The simulation leads to slightly higher stopband edges as they should be compared to the measured data. This is a result of the chosen FE grid. Finer meshes lead to larger eigenvalue problems which can’t be handled efficiently at the moment. For better performance eigenvalue solvers on the basis of an Arnoldi Algorithm38 will be used in the future. These are able to handle sparse matrices and needn’t calculate all eigenvalues. Restriction to the needed eigenvalues is possible and therefore, much solving time can be saved.

6. CONCLUSION Two new schemes for the calculation of wave propagation in piezoelectric periodic structures have been introduced. The first method leads to a quadratic eigenvalue problem which has a size proportional to the number of nodes on a periodic boundary. Therein, the matrix consisting of the inner nodes has to be inverted due to a

Schur-Complement formulation. The eigenvalue matrices are dense. The second method leads to a general eigenvalue problem of size proportional to the sum of inner nodes and the nodes of a periodic boundary. The matrices keep sparse and are well conditioned. Due to the use of the non-symmetric eigenvalue solver of LAPACK which is not able to handle sparse matrices, the method with the Schur-Complement is at present three times faster compared to the second method. In the near future, Arnoldi Solvers which take the sparsity pattern into account and do not need to calculate all eigenvalues, will be used. Therewith, the performance should be improved tremendously. The two methods have been verified numerically by comparing the simulation results of symmetric plate modes to the analytic solution. In order to simulate bulk acoustic wave radiation, absorbing boundaries for piezoelectric materials have been developed. The full anisotropy of theses materials is considered. Simulations with the aid of the new periodic boundary conditions (PBCs) and the new absorbing boundary conditions (ABCs) have been performed and compared to measured data. It could be shown that bulk acoustic waves impinging from certain directions at the absorbing boundary are absorbed very well. To consider a wider range of wave angles, higher order ABCs for piezoelectric elastic waves have to be implemented in the future.

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