Post on 16-Jan-2016
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
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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)
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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.
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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
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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