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
FreeFEM++ WorkShop Meeting, December the 7th 2011 2
Outlines
Introduction
Physical model Electro Acoustical Computation : FreeFem++ Model
Include temperature variations
Numerical and experimental results Conclusions
Future works
FreeFEM++ WorkShop Meeting, December the 7th 2011 3
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