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

Transcription

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 7 th

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 7 th

3 SAW IDT components How is built a SAW device Piezoelectric substrate SAW IDT transducer The Surface Acoustic Wave is launched and detected propagating FreeFEM++ WorkShop Meeting, December the 7 th

4 x Physical model Geometry y Ωd B up u B l d Bl Dielectric medium electrode Piezoelectric medium Ω d Ω p u B r x d B r B down Ω p FreeFEM++ WorkShop Meeting, December the 7 th

5 Physical model Assumptions : 2D analysis (very long electrode) : plain strain approximation p periodic along the x axis harmonic electrical excitation of the electrodes: Vn j2 ( γ ) = Ve 0 πnγ electrical assumption: no dielectric losses in the electrode mechanical assumption : the metallic electrode are homogeneous isotropic, elastic materials FreeFEM++ WorkShop Meeting, December the 7 th

6 Physical model Ω The Piezoelectric domain p and the Elastic domain obeys E Newton s second law: 2 = ρ u T 2 t The Piezoelectric domain p, the Elastic domain E, and the Dielectric domain obeys the quasistatic Maxwell s equation: Ω D Ω Ω Ω D =0 FreeFEM++ WorkShop Meeting, December the 7 th

7 The constitutives equations: Physical model For Ω P T = C S e E S Di = ekliskl + εikek E ij ijkl kl ijk k For Ω E ( ) 2 T = λ + µ δ S µ S ij ij kk ij For Ω D D = ε E i ik k FreeFEM++ WorkShop Meeting, December the 7 th

8 γ Physical model periodic boundary conditions: l B u For the interfaces and : l For the interfaces B r and : d j2πγ B Φ+ ( p 2, y) = e Φ ( p 2, y) r u j2πγ B ( ) ( ) d u + p 2, y = e u p 2, y Φ j2πγ ( + p 2, y) = e Φ ( p 2, y) FreeFEM++ WorkShop Meeting, December the 7 th

9 Physical model Weak formulation (,φ ) 3 u ( ) 3 V Ω Ω V ( Ω) Finds in γ p e γ (satisfying in the electrode) 3 such that for all ( v,ψ ) in ( ) 3 V ( ) (satisfying in the γ Ωp Ω e Vγ Ω φ = 0 electrode) Ω Ω ( ) ( ) is the mathematical space of boundary conditions 2 dω ρω dω Ω Ω p E p E p E D ( ψ) ( u) ε ( φ) ( ) E es + E dω Ω Ω Ω ( ) d ψ ( φ) B B B d u d ( ) = v T n Γ+ D n dγ S v T u v u V γ ( Ω 2 ) L ( Ω) φ =1 satisfying γ-harmonic periodic FreeFEM++ WorkShop Meeting, December the 7 th

10 Incorporates formulation: γ Numerical model periodic boundary conditions in the variational γ The idea is to transform a periodic problem into a periodic problem using the relationship: u x, y x u x, y = % ( ) ϕ γ ( ) ( ) ( ) ϕ γ x = e x j2πγ p Periodic function Modify the variational formulation which allows to use only the periodic option in FreeFem++ FreeFEM++ WorkShop Meeting, December the 7 th

11 First Results Harmonic admittance computation of buried IDT (comparison with published analytical models) FreeFEM++ WorkShop Meeting, December the 7 th

12 First Results Harmonic admittance computation of buried IDT (comparison with published analytical models) FreeFEM++ WorkShop Meeting, December the 7 th

13 Include temperature dependence Taylor expansion of the piezoelectric elastic tensor and mass density around nominal temperature: ( 1 α ( ) α ( ) 2 α ( ) 3 ) ( ) ( ) 2 ( ) 3 Cp = C + T T + T T + T T E C C C ( ) ρ ρ ρ ρ = ρ 1+ α T T + α T T + α T T Piezoelectric substrate expansion: ( ) ( ) ( ) L L 2 L S S S L = 1+ α T T + α T T + α T T S Temperature effect on the Young s modulus and Poisson s ratio (first order de 9 = α = Pa/ C E dt dµ = α µ = dt Pa/ C Electrode width follows the substrate expansion Electrode thickness follows the metal expansion ( ) m L = 1+ α T T m L 1 0 FreeFEM++ WorkShop Meeting, December the 7 th

14 Numerical results Quartz cut turnover temperature computations Sensitivity of the resonant frequency on ST Quartz to the temperature 0 PerIDT Analytical -5e Temperature (C) FreeFEM++ WorkShop Meeting, December the 7 th

15 Experimental results Schematic of a one port STW Single port resonator p T p G L N T strips N G strips FreeFEM++ WorkShop Meeting, December the 7 th

16 Actual packaged STW resonator Experimental results FreeFEM++ WorkShop Meeting, December the 7 th

17 Numerical results Conductance of a non buried STW resonator (Q=6300) FreeFEM++ WorkShop Meeting, December the 7 th

18 Numerical results Conductance of a buried STW resonator (Q = 10700) FreeFEM++ WorkShop Meeting, December the 7 th

19 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 γ 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 FreeFEM++ WorkShop Meeting, December the 7 th

20 Future works Develop a 3D periodic model using FreeFem++ 3D to take into account transverse wave guiding effects code parallelization FreeFEM++ WorkShop Meeting, December the 7 th