FreeFEM++ WorkShop Meeting, December the 7th 2011 1

Optimized Surface Acoustic Waves Devices With FreeFem++ Using an Original FEM/BEM Numerical Model

P. Ventura*, F. Hecht**, Pierre Dufilié***

*PV R&D Consulting, Nice, France Laboratoire LEM3, Université Paul Verlaine, Metz,

France

**Laboratoire Jacques Louis Lions, Université Pierre et Marie Curie, Paris, France

***Phonon Corporation

Simsbury, Connecticut, USA

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Outlines

  Introduction

  Physical model   Electro Acoustical Computation : FreeFem++ Model

  Include temperature variations

  Numerical and experimental results   Conclusions

  Future works

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SAW IDT components How is built a SAW device

  Piezoelectric substrate

  SAW IDT transducer

  The Surface Acoustic Wave is   launched and detected   propagating

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Physical model   Geometry

x

x

y

pΩ

dΩ

d∞Ω

p∞ΩdownB

upBulB

urB

electrode

Piezoelectric medium

Dielectric medium

dlB d

rB

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  Assumptions :

  2D analysis (very long electrode) : plain strain approximation

  p periodic along the x axis

  harmonic electrical excitation of the electrodes:

  electrical assumption: no dielectric losses in the electrode

 mechanical assumption : the metallic electrode are homogeneous isotropic, elastic materials

Physical model

( ) 20

j nnV V e π γγ −=

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  The Piezoelectric domain and the Elastic domain obeys

Newton’s second law:

  The Piezoelectric domain , the Elastic domain , and the Dielectric domain obeys the quasistatic Maxwell’s equation:

Physical model

pΩ EΩ

2

2tρ∂

∇⋅ =∂

uT

pΩ EΩDΩ

0∇⋅ =D

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 The constitutives equations:

Physical model

PΩ EΩFor For

ij ijkl kl ijk k

i kli kl ik kε

⎧ = −⎪⎨

= +⎪⎩

E

S

T C S e E

D e S E ( ) 2ij ij kk ijλ µ δ µ= + −T S S

For DΩ

i ik kε=D E

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  periodic boundary conditions:

  For the interfaces and :

 For the interfaces and :

Physical model

γluB

ruB ( ) ( )22, 2,jp y e p yπγ−Φ + = Φ −

ldB r

dB ( ) ( )( ) ( )

2

2

2, 2,

2, 2,

j

j

p y e p y

p y e p y

πγ

πγ

−

−

⎧ + = −⎪⎨Φ + = Φ −⎪⎩

u u

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Finds in (satisfying in the electrode)

such that for all in (satisfying in the electrode)

is the mathematical space of satisfying γ-harmonic periodic

boundary conditions

Physical model – Weak formulation

( ) ( )

( ) ( ) ( )( )

( ) ( )( )

2

p E p E

p E D

d u dB B B

d d

u d

d d

ρω

ψ ε φ

ψ φ

Ω ∪Ω Ω ∪Ω

Ω ∪Ω ∪Ω

∪

⋅ Ω − ⋅ Ω

− ⋅ + Ω

= ⋅ ⋅ Γ + ⋅ Γ

∫ ∫

∫

∫ ∫

S v T u v u

E eS E

v T n D n

( ),φu ( ) ( )3 3p eV Vγ γΩ ∪Ω × Ω

( ),ψv ( ) ( )3 3p eV Vγ γΩ ∪Ω × Ω

1φ =0φ =

( )Vγ Ω ( )2L Ω

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  Incorporates periodic boundary conditions in the variational

formulation:

  The idea is to transform a periodic problem into a periodic problem using the relationship:

  Modify the variational formulation which allows to use only the periodic option in FreeFem++

Numerical model γ

γ

( ) ( ) ( ), ,u x y x u x yγϕ= % ( )2

exjpx

πγ

γϕ−

=

Periodic function

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First Results

  Harmonic admittance computation of buried IDT (comparison with published analytical models)

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First Results

  Harmonic admittance computation of buried IDT (comparison with published analytical models)

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Include temperature dependence

  Taylor expansion of the piezoelectric elastic tensor and mass density around nominal temperature:

  Piezoelectric substrate expansion:

  Temperature effect on the Young’s modulus and Poisson’s ratio (first order

  Electrode width follows the substrate expansion   Electrode thickness follows the metal expansion

( ) ( ) ( )( )2 30 1 0 2 0 3 01p T T T T T Tα α α= + − + − + −E C C CC C

( ) ( ) ( )( )2 30 1 0 2 0 3 01 T T T T T Tρ ρ ρρ ρ α α α= + − + − + −

( ) ( ) ( )2 31 0 2 0 3 01 S S SL L L

SL T T T T T Tα α α= + − + − + −

9

9

0.0375 10 Pa/°C

0.0149 10 Pa/°C

EdEdTddT µ

α

µα

⎧ = = − ⋅⎪⎪⎨⎪ = = − ⋅⎪⎩

( )1 01 mLmL T Tα= + −

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Numerical results

  Quartz cut turnover temperature computations

-0.00025

-0.0002

-0.00015

-0.0001

-5e-005

0

-40 -20 0 20 40 60 80

Re

so

na

nt

fre

qu

en

cy

(re

lati

ve

va

ria

tio

n)

Temperature (C)

Sensitivity of the resonant frequency on ST Quartz to the temperature

PerIDTAnalytical

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Experimental results

  Schematic of a one port STW Single port resonator

 pT  pG

 NT  strips  NG  strips

 L

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Experimental results

  Actual packaged STW resonator

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Numerical results

  Conductance of a non buried STW resonator (Q=6300)

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Numerical results

  Conductance of a buried STW resonator (Q = 10700)

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Conclusions

  A periodic Analysis of Surface Acoustic Waves Transducer has been developed using FreeFem++

  The variational formulation incorporates:

o  The Green’s function for the Piezoelectric semi-space o  The periodic boundary conditions

  The temperature dependence has been included using simplified

assumptions   Partially buried electrode STW resonators with improved Q have

been developed with the aid of the mixed FEM/BEM numerical model PerIDT

γ

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Future works

  Develop a 3D periodic model using FreeFem++ 3D to take into

account transverse wave guiding effects

  code parallelization