Page 1

WCCM V

Fifth World Congress on

Computational Mechanics

July 7–12, 2002, Vienna, Austria

Eds.: H.A. Mang, F.G. Rammerstorfer,

J. Eberhardsteiner

Finite Element Calculation of Wave Propagation and

Excitation in Periodic Piezoelectric Systems

Manfred Hofer

?, Reinhard Lerch

Department of Sensor Technology

University of Erlangen-Nuremberg,Paul-Gordan-Str. 5, D-91052 Erlangen, Germany

e-mail: manfred.hofer@lse.e-technik.uni-erlangen.de

Norman Finger, G¨ unter Kovacs

EPCOS AG

Postfach 801709, D-81617 Munich, Germany

Joachim Sch¨ oberl, Ulrich Langer

Institute of Computational Mathematics

Johannes Kepler University Linz, Altenberger Str. 69, A-4040 Linz, Austria

Key words: Finite Element Calculation, Wave Propagation, Piezoelectricity, Periodicity

Abstract

Many sensors and actuators intechnical systems consist of quasiperiodic structures which areconstructed

by successive repetition of a base cell. Typical examples are piezoelectric composites used as ultrasonic

transducers or surface acoustic wave (SAW) devices utilized in telecommunication systems. The precise

numerical simulation of such devices including all physical effects is currently beyond the capacity of

high end computation. Therewith, we have to restrict the numerical analysis to the periodic substructure

and have to introduce special boundaries to account for the periodicity.

Due to the fact, that wave propagation phenomena have to be considered for SAW applications the peri-

odic boundary condition (PBC) has to be able to model each possible mode within the periodic structure.

That means this condition must hold for each phase difference existing at the periodic boundaries. To

fulfill this complex criteria we have introduced the Floquet theorem to the PBCs offering two different

solution strategies: the first method leads to a quadratic eigenvalue problem. Therein, a huge matrix in-

cluding all nodes not belonging to the periodic boundaries has to be inverted due to a Schur-Complement

formulation. In the other method a general non symmetric eigenvalue problem including the inner and

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Hofer et al.

the nodes at one periodic boundary has to be solved. These techniques consider every kind of propagating

mode automatically. The advantage of the latter scheme is that the matrices keep sparse. With the use of

an efficient eigensolver utilizing the Arnoldi method, which calculates only the needed eigenvalues and

takes advantage of the sparsity of the matrices, the second approach is much faster as the scheme with

the Schur-Complement which has to deal with dense matrices .

In the first part of this paper we describe the basic theory of our new PBC methods. In the second part

we show simulation results for the eigensolution of a periodic SAW structure which can be expressed

efficiently in a dispersion diagram. Such diagrams give valuable information to SAW designers such as

wave velocity and reflectivity of electrodes and help them to better understand the interaction of surface

acoustic waves (SAW) or leaky surface acoustic waves (LSAW) with radiating bulk waves. Finally, we

calculate the charge distribution of SAW structures by solving the inhomogeneous piezoelectric partial

differential equations with incorporated PBCs. Therewith, we determine the electrical admittance which

characterizes the electrical behavior of a SAW device. Further on, the computation of a voltage excitation

on arbitrary electrodes will be demonstrated using our newly developed PBCs.

1Introduction

The aid of modern sensors and actuators made our life more comfortable in various ways: The heating

controls the temperature itself with the aid of internal and external temperature sensors, irrigation is done

automatically when the humidity sensor tells the control equipment that the soil is dry, movement sensors

are used to switch on the front door light if somebody is appearing. There is a huge list of sensors and

actuators which are used in modern “smart” homes [1]. On the other hand, modern sensors and actuators

are used to increase our security. One of the most important example is the application of sensors in

modern automobiles, e.g. slipping sensors used in tires which deliver the electronic stability program

with input data or acceleration sensors which detect a collision with another car or another subject and

enforces the control circuit to inflate the airbags.

The development of modern sensors is mostly done with simulation tools, because this way is much

faster and cheaper compared with experimental measurements on prototypes. A very accurate method

for this purpose is the finite element method (FEM) [2, 3, 4]. Therein, the real behavior of the transducers

is approximated by partial differential equations (PDEs) which have to be discretized in space and time

or frequency.

In many cases, a finite element simulation of a complete sensor is not possible, because the simula-

tion time would be too large or the necessary computer resources would exceed the available amount.

Therefore, the models have to be reduced in complexity or size by incorporating special properties or by

neglecting features which contribute only in a diminutive way to the final output of the simulated device.

Such complexity reduction can be done e.g. by regarding symmetry planes or by neglecting hysteresis

effects or nonlinearities.

Many sensors, e.g. composite transducers [5] or SAW devices [6, 7] consist of periodic structures. This

periodicity can be used to reduce the size of the FE model tremendously. In this paper, two different

methods regarding the wave propagation in periodic structures are described in detail. A special focus

will be given on periodic SAW structures.

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WCCM V, July 7–12, 2002, Vienna, Austria

2 Surface Acoustic Wave (SAW) Devices

A surface acoustic wave device consists of a piezoelectric substrate carrying metallic structures such as

interdigital transducers (IDTs) and reflecting gratings or electrodes (see Fig. 1) [7]. Such SAW devices

{{

Interdigital

transducer

(IDT)

{{

{

Reflector?2

Reflector?1

Piezoelectric

substrate

Figure 1: Sketch of a SAW resonator filter with an interdigital transducer (IDT) and two reflectors

are mainly used in telecommunications, e.g. as delay lines in television sets, intermediate frequency

filters for mobile phones or convolvers for spread-spectrum communications [8]. They are also used in

different sensor applications, e.g. for measuring temperature and pressure [9], flow [10] or humidity [11].

The working principle of a SAW device is illustrated in Fig. 2. A time harmonic voltage excitation of

EE

U~

SAWSAW

BAW

BAW

Figure 2: Working principle of a SAW device, generation of surface- and bulk acoustic waves (BAW)

the electrodes of an IDT leads to a deformation of the surface due to the inverse piezoelectric effect [12].

Depending on the excitation frequency, various wave modes are excited, e.g. surface acoustic waves or

bulk acoustic waves (BAWs). On the other hand, due to the direct piezoelectric effect, surface displace-

ments lead to a voltage in the electrodes and therewith, the surface strain can be transformed back into

an electrical signal.

Typical SAW devices like SAW reflector filters may consist of up to thousands of electrodes, especially if

they are built on piezoelectric substrates with low electromechanical coupling coefficient

[7]. Therefore, a FEM simulation of a complete device is not possible due to the size of the evolving

finite element grid. By regarding the periodic structure, the FEM model can be reduced to a manageable

size.

?

?like quartz

3

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Hofer et al.

3Wave Propagation in Periodic Structures

A general periodic structure can be imagined as successive repetition of a base cell

?

?(see Fig. 3). In the

? ?

?

?4

?p

x1

x2

p

h

? ?

r

?2

?1

?3

……

02p-pp

Periodic?disturbance

Figure 3: Base cell

?

?with periodicity p

following, the boundary

?

?of Fig. 3 will be referred to as “left periodic boundary (

?

?)” and boundary

two ( ?

?) will be termed as “right periodic boundary (

?

?)”.

The main goal of our examination is to simulate arbitrary waves propagating in periodic structures.

Therefore, it is not sufficient to restrict the results to waves with wavelengths

width

degrees of freedom of the right periodic boundary

the negative of those from

if arbitrary waves have to be considered. Therefore, a more fundamental approach has to be pursued.

? of a fraction of twice the

? of the base cell ( ?????? with

???

?). These modes could be easily achieved by setting the

?

?equal to those on

?

?( ???? with

???

?), or to

?

?( ?????? with

???

?) [13, 14]. This kind of periodicity cannot be used

Due to the periodicity of the geometry, the resulting field distribution must be also quasi-periodic, leading

to

???????

???

??????

???

?

???

?

(1)

with

describes the decay behavior and

time harmonic excitation is assumed. Therefore, the term

With a periodic function

? denoting the field distribution and

?????? the complex propagation constant. The variable

?

? stands for the phase propagation constant of the wave. In general, a

?

???will be omitted further on.

?

?

???????

?

???, eq. (1) can be written as

??????

?

????

???

?

(2)

Expanding this periodic function in a Fourier series

?

?

????

?

?

????

?

?

?

??

???

?

?

(3)

with

plane waves

?

?as Fourier coefficient, the complete field distribution can be written as superposition of damped,

?????

?

?

????

?

?

?

?

?

???

???

?

?

?

?

(4)

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WCCM V, July 7–12, 2002, Vienna, Austria

3.1Dispersion diagram

It is important to know the relation between the complex propagation constant

be able to describe wave propagation in periodic structures. This dependency can be illustrated in a so

called dispersion diagram (Fig. 4).

? and the frequency

? to

Surface-

wave

?

?

????

?

p

??

p

?c

?2

?1

????

??

p

???

p

0

Bulk-

acoustic

wave

Figure 4: Dispersion diagram of a periodic structure

In general, two different regions can be distinguished:

? Wave propagation: At frequencies below the lower stopband edge

band edge

to backscattered bulk waves occurs. This results in loss of energy of the propagating wave mani-

festing in a nonzero, positive damping coefficient

?

?and above the upper stop-

?

?the considered modes are propagating ones. At the onset-frequency

?

?a conversion

?.

? Wave reflection: At frequencies in the stopband ( ?

riodic disturbances (like electrodes) existing in the simulation area. These reflections add coher-

?

????

?) the waves are reflected at pe-

ently, therefore, no propagating modes exist and the waves are damped exponentially ( ??? at

?

disturbance [15]. If no disturbance is present, the stopband vanishes (dotted line in Fig. 4).

?????????? with

???). The width of the stopband is proportional to the reflection per

4FE Formulation of the Periodic Boundary Conditions (PBCs)

The fundamental equations for the FE simulation of piezoelectric media has been published many times.

Therefore, our derivation directly starts at the FE matrix form of a harmonic piezoelectric problem [16]

?

?

??

????

??

??

?

?

??

?

??

?

??

??

??

?

?

???

?

?

?

?

?

?

?

?

?

?

?

?

(5)

where the notation has been chosen to

5

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Hofer et al.

?

??

mechanical mass matrix

mechanical damping matrix

mechanical stiffness matrix

dielectric stiffness matrix

piezoelectric coupling matrix

nodal vector of external mechanical forces

nodal vector of electric charges

vector of mechanical displacements

vector of electric potentials

For convenience, the matrix

tions, the vector of unknowns will be denoted as

side as

?

??

?

??

?

??

?

??

?

?

?

?

?

?has been introduced as combination of all FE matrices. In further deduc-

? ( ??????℄

?) and the source term on the right-hand-

? ( ??????℄

?). These definitions allow us to write (5) as

?

?

????

(6)

4.1Schur-Complement Method

To be able to incorporate the periodicity condition (1) we split the unknowns

boundary nodes

vector

? into inner nodes

?

?and

?

?( ?

?

???

?

??

?

℄). The same has to be done with the matrix

?

?and the right-hand-side

?

?

?

?

??

?

?

??

?

?

??

?

?

??

??

?

?

?

?

?

?

?

?

?

?

?

?

(7)

The forces in the interior of the simulation area stay in equilibrium. Therefore, the right hand vector

contributing to the inner nodes is a zero vector. This fact can be used to take the Schur-Complement of

(7) resulting in

???

?

??

?

???

??

?

?

??

??

?

??

?

?

?? ?

?

?

?

??

?

?

(8)

which can also be written as

?

?

??

?

??

?

??

?

??

??

?

?

?

?

?

?

?

?

?

?

?

?

?

(9)

Incorporation of the periodicity condition (1) by replacing

sign is required for the equilibrium of the appearing forces on the periodic boundaries) results in a

quadratic eigenvalue problem in

?

?with

??

?and

?

?with

???

?(the minus

???

???

?

?

?

??

?

?

????

??

??

??

??

?

??

??

?

?

???

(10)

This equation can be solved for example by inverse iteration [17], with a two-sided Lanczos method [18],

or by linearizing the system [19] which doubles the matrix size. The latter solution method is straight

forward to implement and therein the standard non-symmetric eigenvalue solver from LAPACK [20] can

be used.

The matrices in (10) are dense but very small: the size is proportional to the number of unknowns of

one periodic boundary. Due to the Schur-Complement, an inversion of the matrix

the unknowns of all inner nodes and is therefore of notable size, has to be performed.

frequency weighted mass and damping matrices and thus, the inversion has to be performed for every

frequency step separately.

?

?

??, which contains

?

?

??includes the

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WCCM V, July 7–12, 2002, Vienna, Austria

4.2Transformation to General Linear Eigenvalue Problem

A different scheme can be developed by starting at an equivalent formula to (7)

?

?

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

?

???

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

(11)

and incorporating the periodicity condition (1) in a similar way as before yields

?

?

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

??

?

?

?

?

?

?

?

??

?

???

?

?

?

?

???

?

?

?

?

?

?

?

?

?

?

?

?

?

??

?

?

?

?

(12)

with

the right hand side. The term with the squared propagation constant

matrix

problem

?denoting the identity matrix. Multiplication with anappropriate matrix from theleft side eliminates

?

?cancels out because the according

?

?

??is per definition a zero matrix. Finally, the problem can be written as a general eigenvalue

?

?

?

?

?

?

?

???

?

?

?

?

?

?

(13)

with matrices

??

?

?

?

??

?

?

??

?

?

??

?

?

(14)

??

?

???

?

??

??

?

??

???

?

??

??

?

??

?

?

?

(15)

These matrices are sparse but much larger compared with those from the Schur-Complement method.

Here, all nodes of the interior of the simulation area and additionally those on one periodic boundary

contribute to the matrices

of an eigenvalue solver for sparse, non-symmetric, complex matrices. In the future, Arnoldi methods will

be used to solve our problems.

With our LAPACK solver the Schur-Complement method is much faster as the other method. Beside the

calculation velocity both methods are equal and deliver same results.

? and

?. At present, the sparsity of the matrices cannot be used due the lack

4.3Solution with Defined Excitation

With the above methods, all possible freely propagating modes can be calculated as eigenvalues of the

described eigensystems. These give the SAW device designer valuable insight in general propagation

properties of the special chosen configuration and materials.

In addition to freely propagating modes (= homogeneous solutions of the system of PDEs) one is also

interested in particular solutions describing the excitation properties of the system: Here, the magnitude

potential on the electrodes is fixed to a predefined value. Assuming a phase difference of

to unit cell, the propagation constant takes the form

given, only a quite simple system of linear equations has to be solved. In principle, the matrices of both

? from unit cell

?????????. Since, in this case, both

? and

? are

7

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Hofer et al.

methods (the Schur-Complement method and the transformation to general linear eigenvalue problem)

can be applied, but the Schur-Complement needs an additional matrix inversion and therefore the faster

transformation method will be utilized.

For the SAW designer it’s important to know the charge distribution at an electrode. With the aid of the

SAW part of the Green’s function, one is able to calculate the excited waves of a SAW structure [6].

5Results

A typical example for wave propagation in SAW structures has been chosen to show the functionality

of our methods (see. Fig. 5). The simulations and measurements have been performed on a substrate

Pitch?(p)

Electrode

Substrate

?L

?R

wEl

hEl

z

x

Figure 5: Model for the simulation of wave propagation in periodic SAW structures

of 37.5

The height of the electrode

Ærotated quartz with aluminum electrodes. The material constants have been taken from [21].

?

??was fixed at 250nm, the pitch

? at

??? and the metalization ration

?

( ?

on the propagation properties, like stopband edge and stopband width. The substrate hight was chosen to

be eight pitches.

As an example, the propagation diagram of a SAW structure with a metalization ration of

be seen in Fig. 6(a). The measured data are slightly beneath the simulated ones. This can be explained

??

??

?? with

?

??the electrode width) has been varied to show the influence of the electrode width

????? can

0.9850.990.99511.005 1.011.015

384

386

388

390

392

394

396

β p / π

Frequency (MHz)

(a) Exemplary dispersion diagram at

??

???

0.2 0.30.4 0.5

η

0.6 0.7 0.8

386

387

388

389

390

391

392

393

394

395

396

Frequency (MHz)

Simulation

Measurement

(b) Stopband edges as a function of metal-

ization ratio

?

Figure 6: Comparison of simulation and measurement of SAW structures on 37.5

Ærotated quartz

with the finite simulation area. With increasing depth of the simulated substrate, the solution converges

8

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WCCM V, July 7–12, 2002, Vienna, Austria

against the measured results, because the i-th approximated eigenvalue is bounded from below by the

i-th exact eigenvalue while Galerkin rules are not violated (e.g. when reduced integration is employed)

[3]. This fact has experimentally proofed by simulations with a substrate height of only four pitches. In

this case, the stopband edges raise approximately 2MHz at a total stopband width of about 5Mhz.

Various metalization ratios have been examined with our method (Fig. 6(b)). Measurements are possible

only outside the stopband, therefore, the measured data have been extrapolated to the value

to obtain the measured stopband width for this comparison. The influence of the electrode width on the

stopband size mainly due to the additional mass loading [22] can be recognized clearly.

The model in Fig. 5 has also been used to calculate the charge distribution on the electrode (see Fig. 7).

? ?????

1

2

0

0.2

0.4

0

50

100

150

x?-?position?(?m)

3

z?-?position?(?m)

Charge?per?area?(µC/m2)

Substrate

Electrode

Figure 7: Charge distribution at an aluminum electrode

For the calculation of the charge distribution at all electrode edges, the air around the electrode has to

be taken into account. In air, the electric potential has to be calculated but the mechanical field may be

neglected due to the extremely low mechanical stiffness of the surrounding air. The only thing to be

regarded is the vanishing stress contribution on the free substrate surface. Especially for materials with

low dielectric constants (relative dielectric constants of quartz:

charge of the electrode-air interfaces has to be taken into account. The charge on these interfaces can be

estimated by

electrodes on a quartz substrate one would make an error of at least 16% by neglecting the charge on the

electrode-air interfaces.

The calculated total charge at the electrode with varying phase difference can be regarded as a kind

?

?

??

???? and

?

?

??

????) the surface

???

?times the charge of the electrode-substrate interface. Calculating the total charge at

of frequency spectrum. From this dependency, the so called “harmonic admittance” Y ( ?

with

little “disturbances” in the harmonic admittance coincide exactly with propagating modes calculated as

eigenvalues of the system. In addition to the modes occurring in real SAW devices, spurious plate modes

can be seen due to the Dirichlet boundary condition on the bottom of the FE model. With the aid of

absorbing boundary conditions for piezoelectric materials, these spurious modes can be eliminated.

Atangles afew degrees beside 180

real part of the harmonic admittance. To be able to see this effect in the harmonic admittance, absorbing

boundary conditions are needed again. Another possibility to simulate bulk acoustic waves correctly is

the use of analytical solutions at the bottom boundary instead of Dirichlet conditions [24].

??????

? the total charge and

? the electrical excitation potential) can be calculated (Fig. 8(a)) [23]. The

Æbulk acoustic waves areexcited. Theycan berecognized as anonzero

9

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Hofer et al.

With the inverse discrete Fourier transformation it is possible to calculate the field distribution for an

excitation at arbitrary electrodes, e.g. at only one “hot” electrode finger of a SAW one port resonator

from the harmonic admittance.

The charge distribution on the electrode-substrate interface as a function of the phase angle

in Fig. 8(b). For angles near to

sign. Therefore, the charges on different electrodes attract each other. They concentrate mainly at the

electrode edges. On the other hand, at angles near

same potential. Thus, the attraction is not as strong and the charge is distributed more uniformly over the

electrode width.

? can be seen

???

Æthe electrode potential of neighboring electrodes have a different

?

Æor

???

Æneighbored electrodes have nearly the

050 100150 200250300350

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

θ (°)

ℑ(Y) (S)

(a) Imaginary part of the harmonic admittance

????

1

1.5

2

2.5

3

0

100

200

300

400

0

20

40

60

80

100

x−position (µ m)

θ (°)

Charge per area (µ C/m2)

(b) Charge distribution on the interface of electrode and

substrate

Figure 8: Imaginary part of the harmonic admittance Yrespectively thecharge distribution in anelectrode

as a function of the phase

? between left and right boundary (

?????,

?????MHz)

6Conclusion

Two different finite element methods for the calculation of wave modes in periodic piezoelectric config-

urations like surface acoustic wave (SAW) structures have been introduced. The first method results in a

quadratic eigenvalue problem with small, dense matrices. The second approach leads to a general linear

eigenvalue problem with large, sparse matrices. The resulting complex eigenvalues can be interpreted as

propagation and damping constants of the according mode.

With given phase angle between left and right boundary of a base cell of the periodic structure and the

electrical potential on the SAW electrodes, the complete charge distribution at the electrode can be cal-

culated. With the aid of the SAW part of the Green’s function, this charge distribution can be used to

calculate the complete propagation modes. Due to the finite simulation area and the Dirichlet boundary

condition at the bottom of the simulation area, spurious plate modes appear in the charge distribution.

They can be eliminated either with the introduction of absorbing boundary conditions or the use of ana-

lytical solutions at the bottom boundary. These methods will be examined in the next future.

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WCCM V, July 7–12, 2002, Vienna, Austria

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