Plate acoustic waves in ferroelectric wafers

V. A. Klymko

Department of Physics and Astronomy

University of Mississippi

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

Outline

Plate waves in single crystal LiNbO3 Method of partial waves Experiment Piezoelectric coupling coefficient

Plate waves in periodically poled LiNbO3 Finite Element method Numerical results Experimental data Group velocity dispersion curves

Conclusions

4

Numerical solution: equations

Equation

of motion

Piezoelectric

relations

General solution

elastic constants,

piezoelectric constants,

el. field, el. displacement

ijkl

mij

c

e

E D

2

2,iji

j

Tu

t x

density,

displacement,

stress

u

T

,E kij ijkl m ij m

l

uT c e E

x

, , , , 1..3i j k l m

,Ski ikl ij j

l

uD e E

x

Z

X

b/2

- b/2

βββ

0 expi iu u i Vt x z

5

Numerical solution: boundary conditions

Zero normal component of the stress

Continuous electric displacement

33

2

0,bi xT

.

33

2

0,bi xT

3 33 30 0

2 2

b bx x

D D

1,2,3i X3

X1

b/2

- b/2

βββ

3 33 30 0

2 2

b bx x

D D

6

Dispersion curves: single crystal LINbO3

4

8 75

3

21

4

8 76

5

3 21

Accepted to IEEE Trans. on UFFC

Numerical solution and experiment

1- A0, 2 – SS0, 3 – S0, 4- SA1, 5 – A1, 6 – S1, 7 – SS1, 8 – S2

7

Mode identification The modes are identified by the dominant component of

acoustical displacement

S210.0130.5410.17.448

SS10.02510.0580.17.097

S10.6970.0310.16.926

A10.1010.0110.14.235

SA10.00310.0230.13.604

S00.0490.0210.10.643

SS00.00110.0250.10.432

A010.0010.1460.10.081

Mode typeuzuyux

/2 (mm-1)

f (MHz)

Mode number

IEEE UFFC, N12, 2008, accepted.

8

Plate acoustic modes

X3

X1

β

S0(3)

X3

X1

β

S1(6)

X3

X1

β

S2(8)

X3

X1

β

A0(1)

X3

X1

β

A1(5)

X3

X1X2 β

SS0(2)

β

SA1(4)

X3

X1X2

SS1(7)

X3

X1X2

9

Piezoelectric coupling coefficient (K2) K2 = 2(V0-Vm) / V0 (Kempbell, Jones, Ingebrigsten)

V0 - phase velocity with free surfaces

Vm- phase velocity with one surface metallized

Note:For surface waves

K2~0.03

1 – A0,

2 – SS0,

3 – S0,

4 – SA1,

5 – A1,

6 – S1,

7 – SS1,

8 – S2

IEEE UFFC, N12, 2008, accepted.

10

Delay line Calculated and measured

transmission coefficientpaw

RF in out

(A1)

6 (S1)

(S2)(A1)

6 (S1)(S2)

IEEE UFFC, N12, 2008, accepted.

11

FEM model for periodically poled LiNbO3

The functional of the total energy is minimized

LiNbO3

air

air

Input transducer

X3

X1

Absorbing load

Absorbing load

- kinetic

-energy of electric field

- elastic

21

2kinE dx dy

u

1

2t

dE dx dy D E

1:

2t

stE dx dy S T

- energy of excitation

0 dtWEEE stdkin

dxqW s 3 31

n

i ie e

i = 1..6,n = 1..N

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

65

32

1λ=D λ = D

1- A0, 2 – SS0, 3 – S0, 4- SA1, 5 – A1, 6 – S1, 7 – SS1, 8 – S2

13

Periodically poled LiNbO3 (sample #1) Periodic domains in polarized light

Domain with inverted piezoelectric field

Original crystal

D=0.6 mm

X

-Y

14

Experiment: sample #1 Plate with free surfaces, N = 150 domains, D = 0.6 mm.

4

8

5

32

6

λ=D

5 4

1

45mm

75mm

b

0.6 mm

λ = D

1- A0, 2 – SS0, 3 – S0, 4- SA1, 5 – A1, 6 – S1, 7 – SS1, 8 – S2

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

λ = D

1- A0, 2 – SS0, 3 – S0, 4- SA1, 5 – A1, 6 – S1, 7 – SS1, 8 – S2

16

Experimental group velocity Group velocity of modes A0 and SA1 is zero at stop-bands

Vg=d/dβ

5 6 7 8 9

0

1

2

3

4

5

SA1

A0

Vg

(km

/s)

f (MHz)

A0 SA

1

(1) (1) (4) (4)

17

Conclusions• Dispersion curves are computed for PAW in ZX-cut LiNbO3.The

modes can be identified by their dominant components near cutoff frequencies.

• In ZX-cut LiNbO3, modes A1 and S2 have high piezoelectric

coupling: 23% (A1) and 13% (S2), which is promising for

applications in telecommunication.

• Dispersion curves in periodically poled LiNbO3 (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

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

Numerical solution: method of partial waves

Equation of motion

and equations of state

with the general solution

yield Christoffel equation

2

2,iji

j

Tu

t x

0 expi iu u i Vt x z

density,

displacement,

stress

u

T

,E kij ijkl m ij m

l

uT c e E

x

, , , , 1..3i j k l m

elastic constants,

piezoelectric constants,

el. field, el. displacement

ijkl

mij

c

e

E D

,Ski ikl ij j

l

uD e E

x

2 0,pq pq qV u , 1..4p q

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)

2det 0pq pq V m V

8

01

expi i mm

mu u A i Vt x z

1..8m

2( ( ) )ij i j i j j i i jR x x x x

44 ,ij ijR ,pl iplj ijc R 4 4 ,p p ijp ije R

21

Numerical solution: boundary conditions

Stress-free surfaces in the air

Stress-free surfaces, plate is on a metal substrate.

33

2

0,bi xT

3 33 30 0

2 2

b bx x

D D

1,2,3i

33

2

0,bi xT

3 33 30 0

2 2

,b bx x

D D

3

1 02

0,bx

E

3

2 02

0bx

E

Z

X

b/2

- b/2

βββ

22

Numerical dispersion curves

The dispersion curves for three boundary conditions

4

8 7

65

32

1

Asymmetric:

1 – A0

5 – A1

Symmetric:

3 – S0

6 – S1

8 - S2

Shear:

2 – SS0

4 – SA1

7 – SS1

23

Experimental setup Electric potential is measured using metal electrode

LiNbO3

Input transducer

Output transducer

Shield

Metal substrate

StageAmplifier

X

Electric potential is measured using metal electrode

24

Fabrication of a sample with periodic domains (Poling) 22 kV/mm electric field is applied to the wafer surface

Microscope

Polarizer

LiNbO3

Plastic basin with water

Needle

Electrode (+11 kV)

Grounded electrode

Moving stage

Greese