Domain Wall Contributions to the Piezoelectric Properties of Ferroelectric Ceramics and Thin Films, and Their Significan
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15 Mat. Res. Soc. Symp. Proc. Vol. 459 ©1997 Materials Research Society
FIELD DEPENDENCE OF THE DIELECTRIC PIEZOELECTRIC COEFFICIENT
PERMITTIVITY AND THE
Ferroelectric ceramics The experimental procedure for measurements of the direct piezoelectric effect in bulk ceramics and preparation of the ceramic samples are described in detail in Ref. [5,7]. Figure 1 shows the piezoelectric coefficient of Bi4Ti3Ol2 doped with -1.3at% Nb (BTO12-Nb) and PZT (63/37) doped with 4at%Nb as a function of the amplitude of ac pressure. In both materials the dependence of the piezoelectric coefficient on the field may be described by the Rayleigh equation [8]: d33 = dinit + aX 0
(1)
where d33 is the direct piezoelectric coefficient, X 0 is the amplitude of the ac pressure X(t), dinit is the sum of the lattice (so called intrinsic) contribution, and the contribution from reversible displacement of the walls. In piezoelectric ceramics, dinit depends also on the poling conditions. The Rayleigh coefficient a is due to irreversible displacement of the walls. The Rayleigh law is valid in the limit of weak field signal, where domain walls do not interact and where restoring forces due to stray fields can be neglected. Therefore, displacement of the walls is controlled only by pinning of the walls on defects. One consequence of the domain wall pinning and depinning as the field is cycled is the appearance of piezoelectric hysteresis. For the direct piezoelectric effect, the charge density (Q) vs. pressure hysteresis may be written in terms of the Rayleigh parameters as: (X2 _ X2)(2
+ X)
(2)
Q(X) = (dinit + aXo)X
170
22 d =118.5 + 17.77 X
d3= 16.8 + 0.58 X pC/N
21
33
0
33
160 0
1 50Z
20
o
00
0
140
•19
130
18 - ",I(a) . . .
-
17
0
1
2
3
4
5
6
0
I.. .
0.5
. • . .. .
1
(b)
. ..... .
1.5
2
2.5
12 0
3
Amplitude of ac pressure (MPa)
Amplitude of ac pressure (MPa)
Fig. 1. The direct longitudinal piezoelectric coefficient of the BTO12-Nb (a) and PZT (63/37) -4%Nb (b) as a function of the amplitude of ac pressure. Full symbols represent experimental data and solid lines are linear fits with the Rayleigh equation (1). The frequency of the ac field was 1 Hz. (5,9]
16
600
150
-
400
E E
-100
so
200
50
0
0 E
R -50
-200
0:
(b)
.40
-100
-6
-4
-2
0
2
4
-.... -2
600 ,. -3
-150 6
-1
0
1
2
Pressure (MPa)
Pressure (MPa)
Fig. 2. Experimental loops (circles) and loops calculated from the Rayleigh law and Eq. (2) (solid lines) for BT012-Nb (a) and PZT (63/37)-Nb (b). [5,91
where "+" stands for decreasing and "-" for increasing field. Figure 2 compares calculated [Eq. (2)] and experimental loops for BTO I2-Nb and PZT (63/37)-Nb. The area of the hysteresis is related to the piezoelectric phase angle 8p (or the piezoelectric "loss"). The phase angle 8p may be calculated as a function of X 0 from the Rayleigh relationships as [7,10]: 2.p 4aX°
(3)
37d33
The calculated and experimentally determined piezoelectric phase angle are shown in Figure 3
for BTO12-Nb. The measured phase angle
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