1. Plate acoustic waves in ferroelectric wafers V. A. Klymko Department of Physics and Astronomy University of Mississippi

2. 2 Why study plate waves in ferroelectrics? Current applications for lithium niobate plates  Transducers  Actuators  Delay lines  Acousto-optical waveguides  Optical detectors Possible future applications  Ferroelectric memory for hard drives  New acoustical and RF filters  Phononic materials featuring stop bands

3. 3 Outline Plate waves in single crystal LiNbO 3  Method of partial waves  Experiment  Piezoelectric coupling coefficient Plate waves in periodically poled LiNbO 3  Finite Element method  Numerical results  Experimental data  Group velocity dispersion curves Conclusions

4. 4 Numerical solution: equations Equation of motion Piezoelectric relations General solution Z X b/2 - b/2 βββ

5. 5 Numerical solution: boundary conditions Zero normal component of the stress Continuous electric displacement. X3X3 X1X1 b/2 - b/2 βββ

6. 6 Dispersion curves: single crystal LINbO 3 4 8 7 5 3 2 1 4 8 7 6 5 3 2 1 Accepted to IEEE Trans. on UFFC Numerical solution and experiment 1- A 0, 2 – SS 0, 3 – S 0, 4- SA 1, 5 – A 1, 6 – S 1, 7 – SS 1, 8 – S 2

7. 7 Mode identification The modes are identified by the dominant component of acoustical displacement S2S2 10.0130.5410.17.448 SS 1 0.02510.0580.17.097 S1S1 0.6970.0310.16.926 A1A1 0.1010.0110.14.235 SA 1 0.00310.0230.13.604 S0S0 0.0490.0210.10.643 SS 0 0.00110.0250.10.432 A0A0 10.0010.1460.10.081 Mode type uzuz uyuy uxux  /2  (mm -1 ) f (MHz) Mode number IEEE UFFC, N12, 2008, accepted.

8. 8 Plate acoustic modes X3X3 X1X1 β S 0 (3) X3X3 X1X1 β S 1 (6) X3X3 X1X1 β S 2 (8) X3X3 X1X1 β A 0 (1) X3X3 X1X1 β A 1 (5) X3X3 X1X1 X2X2 β SS 0 (2) β SA 1 (4) X3X3 X1X1 X2X2 SS 1 (7) X3X3 X1X1 X2X2

9. 9 Piezoelectric coupling coefficient (K 2 ) K 2 = 2(V 0 -V m ) / V 0 (Kempbell, Jones, Ingebrigsten) V 0 - phase velocity with free surfaces V m - phase velocity with one surface metallized Note: For surface waves K 2 ~0.03 1 – A 0, 2 – SS 0, 3 – S 0, 4 – SA 1, 5 – A 1, 6 – S 1, 7 – SS 1, 8 – S 2 IEEE UFFC, N12, 2008, accepted.

10. 10 Delay line Calculated and measured transmission coefficient paw RF in out (A 1 ) 6 (S 1 ) (S 2 ) (A 1 ) 6 (S 1 ) (S 2 ) IEEE UFFC, N12, 2008, accepted.

11. 11 FEM model for periodically poled LiNbO 3 The functional of the total energy is minimized LiNbO 3 air Input transducer X3X3 X1X1 Absorbing load Absorbing load - kinetic -energy of electric field - elastic - energy of excitation i = 1..6, n = 1..N

12. 12 FEM dispersion curves for sample #1 Plate with free surfaces, N = 150 domains, D = 0.6 mm. 45mm 75mm b D=0.6 mm 4 8 7 6 5 3 2 1 λ=D 1- A 0, 2 – SS 0, 3 – S 0, 4- SA 1, 5 – A 1, 6 – S 1, 7 – SS 1, 8 – S 2

13. 13 Periodically poled LiNbO 3 (sample #1) Periodic domains in polarized light Domain with inverted piezoelectric field Original crystal D=0.6 mm X -Y

14. 14 Experiment: sample #1 Plate with free surfaces, N = 150 domains, D = 0.6 mm. 4 8 5 3 2 6 λ=D 5 4 1 45mm 75mm b 0.6 mm λ = D 1- A 0, 2 – SS 0, 3 – S 0, 4- SA 1, 5 – A 1, 6 – S 1, 7 – SS 1, 8 – S 2

15. 15 Experiment: sample #2 Plate with free surfaces, N = 84 domains, D = 0.9 mm. 40mm 50mm b 0.9 mm 3 1 5 λ=D 1 1- A 0, 2 – SS 0, 3 – S 0, 4- SA 1, 5 – A 1, 6 – S 1, 7 – SS 1, 8 – S 2

16. 16 Experimental group velocity Group velocity of modes A 0 and SA 1 is zero at stop-bands V g =d  /dβ (1) (4)

17. 17 Conclusions Dispersion curves are computed for PAW in ZX-cut LiNbO 3.The modes can be identified by their dominant components near cutoff frequencies. In ZX-cut LiNbO 3, modes A 1 and S 2 have high piezoelectric coupling: 23% (A 1 ) and 13% (S 2 ), which is promising for applications in telecommunication. Dispersion curves in periodically poled LiNbO 3 (PPLN) are computed and experimentally verified for the first time. Stop-bands are revealed for the first time in the dispersion curves of plate waves propagating in PPLN. The group velocity of plate waves decreases to zero at stop-band. The developed FEM model can be applied for design of ultrasonic transducers and delay lines.

18. 18 Acknowledgements I would like to thank our faculty, staff, and students for their interest in my work I am grateful to Drs. Lucien Cremaldi, Mack Breazeale, Josh Gladden, James Chambers for many useful comments and suggestions I would like to thank my advisor Dr. Igor Ostrovskii for interesting research topic and guidance. I appreciate the help of my colleague Dr. Andrew Nadtochiy with development of FEM codes. The support of the Department of Physics and Astronomy and the Graduate School was essential for the completion of this work

19. 19 Numerical solution: method of partial waves Equation of motion and equations of state with the general solution yield Christoffel equation

20. 20 Determinant of the Christoffel equation is solved for the propagation constants of partial waves General solution is the sum of partial waves Method of partial waves (2)

21. 21 Numerical solution: boundary conditions Stress-free surfaces in the air Stress-free surfaces, plate is on a metal substrate. Z X b/2 - b/2 βββ

22. 22 Numerical dispersion curves The dispersion curves for three boundary conditions 4 8 7 6 5 3 2 1 Asymmetric: 1 – A 0 5 – A 1 Symmetric: 3 – S 0 6 – S 1 8 - S 2 Shear: 2 – SS 0 4 – SA 1 7 – SS 1

23. 23 Experimental setup Electric potential is measured using metal electrodemeasured LiNbO 3 Input transducer Output transducer Shield Metal substrate Stage Amplifier X Electric potential is measured using metal electrodemeasured

24. 24 Fabrication of a sample with periodic domains (Poling) 22 kV/mm electric field is applied to the wafer surface Microscope Polarizer LiNbO 3 Plastic basin with water Needle Electrode (+11 kV) Grounded electrode Moving stage Greese