121 of the influence of the
d
31
component in the film when a deformation of the structure occurs.
Figure 5-12: Diagram of a free-standing film in expansion a and contraction b compared to a clamped film in expansion c and contraction d.
Theoretical analysis [82] shows that a reduction of measured
d
33
is inevitable for a clamped sample according to
5-1
By substituting the parameters for the properties of an alumina substrate and a clamped thick-film as listed in Table 5-3 into equation 5-1, the unclamped
d
33
can be estimated, which is slightly more than 80 compared to the measured value of a clamped sample.
Another problem associated with the determination of
d
33
is the fact that the system of substrate-piezoelectric film acts as a natural bending element. Therefore, to determine
d
33
correctly, the change in thickness of a specimen between two opposite points at the upper and the lower side of the sample must be measured [84].
Free-standing films are not completely free from the clamping effect: for example, the electrode itself may cause a mechanical clamping. However the thickness of the
electrodes is much smaller than the piezoelectric films and furthermore the elasticity of AgPd electrode is greater than the piezoelectric film, therefore the clamping effect of
the electrode-PZT can be neglected [84].
2
12 11
13 31
33 33
E E
E su b str a te
su b str a te u n cla mp ed
cla mp ed
s s
s e
d d
d
a
c b
d
122
Table 5-3: Parameters for 96 alumina substrate [82].
Parameter Value
Poisson ratio dimensionless
0.25 Young’s modulus
× 10
9
Pa 331
Elastic compliance × 10
-12
m
2
N 16.4
× 10
-12
m
2
N -4.78
× 10
-12
m
2
N -8.45
5.5.2 Decay of d
33
over Time
The measured piezoelectric charge coefficient decayed as a continuously varying stress was applied to the materials. This is a common phenomenon for piezoelectric materials
and arises because of several factors, including the presence of a defective interface layer, which can give rise to the backswitching of domains [106].
A series of experiments were carried out to determine
d
33
for clamped and unclamped samples. A clamped sample, as shown in Figure 5-13 a, was printed directly on an
alumina substrate without a carbon sacrificial layer. The sample was fabricated with the process similar to sample D series, as described in Table 4-. The only difference is that
the sample was not covered with a non-active PZT layer on both sides of the electrodes. This clamped sample will be compared with an unclamped sample similar to those of
sample D series.
The measurements were taken at two different periods of time; one of which was taken just after the samples had been polarised and the other was taken after six months
following polarisation. Comparisons were also made on
d
33
with two different co-firing profiles, 850 °C and 950 °C, for unclamped samples, as shown in Figure 5-13 b.
123
Figure 5-13: Photographs of a a clamped sample printed across a score line on a substrate and b an unclamped sample held with a pair of tweezers.
Figure 5-14 shows the
d
33
measurement results for the clamped sample at a continuous alternating mechanical force of 0.25 N at 110 Hz for 15 minutes at two conditions; one
of which is taken just after polarisation and the other one is taken after six months from the first measurement. At the beginning of the measurement, a
d
33
value of 42 pCN was measured which gradually dropped to 31 pCN after 15 minutes. At the second
measurement, 6 months after polarisation, the initial value decreased to 32 pCN and after 15 minutes of continuous application of alternating force, the value decreased
further to 22 pCN, which shows a decaying rate of 29 over a period of 6 months.
Figure 5-14:
d
33
as a function of time elapsed over 15 minutes for measurements taken just after polarisation and six months after polarisation for a clamped sample.
Tweezer
Exposed Electrode Multilayer Films
Detached
b Substrate
Score Line Multilayer Films
a
