Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications
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Handbook of dielectric, piezoelectric and ferroelectric materials240
wall orientations in Fig. 9.2 are in [110] and [1
1
0] instead of [100] and
[010] shown in Fig. 9.1. Second, the interwoven structure does not have
translational symmetry in any given (001) plane like the structure showing
in Fig. 9.1. In other words, the effective 4mm symmetry of the interwoven
structure is only in a statistical sense; there is no domain superlattice in this
structure. Both types of domain structures, however, are tetragonal 4mm in
the macroscopic sense, so the [001] poled PMN–PT and PZN–PT crystals
can be treated as having tetragonal 4mm symmetry for the purpose of
macroscopic property characterization.
In more rigorous analysis, the volume ratios and the inclusion of different
types of permissible domain walls in the domain pattern will also change the
effective symmetry of the multi-domain structure.
13,14
Based on group
theoretical analysis, one can derive the average symmetry of a multi-domain
system when given the number of variants involved in the domain
pattern.
6–8
For the [001] poled rhombohedral phase crystals, there are four
possible domain states remaining after poling. There are also different
permissible domain walls, both charged and non-charged types, which divide
these domains. If we consider the walls together with the four domain states
while forming the domain patterns, more domain patterns can be generated.
The effective domain pattern symmetries can be as high as tetragonal 4mm
and as low as triclinic of 1.
14
This situation is illustrated in Fig. 9.3. The four
remaining domain states after poling along [001] are represented by the four
dots on the unit cell at the upper left corner. The average domain pattern
symmetry without the involvement of domain walls is tetragonal 4mm. However,
if we are putting in the permissible domain walls, many interesting domain
patterns can be formed. As shown in the figure, there are two patterns having
tetragonal 4mm symmetry, one pattern having orthorhombic mm2 symmetry,
three patterns having monoclinic m symmetry and one pattern having triclinic
1 symmetry.
14
The complication of the domain pattern symmetry directly affects the
complete set of material property characterization because the geometry of
the samples affects the boundary conditions, while the boundary conditions
directly regulate the domain pattern formation. Hence, the resulting anisotropy
of the domain pattern can be different for different geometry samples, which
will lead to large property variation from sample to sample. Because the full-
set material properties must be obtained from measurements on several samples
owing to the large number of independent constants to be determined, such
sample to sample variation will cause inconsistencies among data collected
from different geometry samples. Therefore, with this symmetry consideration
in mind, we must use fewer samples with similar geometries. In addition, it
is important to maintain the same boundary conditions when measuring the
full-set material constants if possible.
WPNL2204
Full-set material properties and domain engineering principles 241
4
mm
[001]
[100]
[010]
[111]
[111]
[111]
[111]
[001]
[100]
[010]
[111]
[111]
[111]
[111]
mml
4
mm
4
mm mm
2
m
(100)
(110)
(110)
(010)
(100)
(010)
(100)
(110)
(100)
(110)
(010)
(010)
(011)
(001)
(011)
(010)
(001)
(001)
(011)
(011)
9.3
Possible symmetries of domain patterns made of the four remaining domain states plus permissible domain walls
for a rhombohedral symmetry crystal. The patterns are illustrated looking down from [001] and the solid arrows are
tilting up while the dashed arrows are tilting down.
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Handbook of dielectric, piezoelectric and ferroelectric materials242
9.2.2 Factors affecting self-consistency of full-set data
Different methods are used to obtain the complete set of material properties.
The most commonly used characterization method for piezoelectric materials
is the resonance method. The principle of this technique is described in
detail by Berlincourt et al. and in the IEEE standard for piezoelectricity.
15,16
However, practical utilization of the resonance method for these multi-domain
systems produced several challenges. First of all, the multi-domain structures
produced in these crystals depend on sample geometry as described in Section
9.2.1, while the resonance method requires different geometry samples in
order to isolate different modes. Therefore, the change of domain patterns in
different geometry samples makes it impossible to get self-consistent property
data.
Second, the derived constants could have very large errors because some
of the constitutive relations contain large error amplification factors. For
example, the most obvious error amplification factor is
1/(1 – )
33
2
k
, which
can be very large since k
33
ranges from 0.90 to 0.96 in these crystals. Detailed
analysis revealed that the error amplification factor in some derived quantities
could be more than 100!
17
To illustrate the errors produced by unstable
mathematical formulas, let us look at the relative errors in the derived
piezoelectric strain constants e
33
and e
31
caused by the relative errors of
c
33
E
and
c
13
E
. The error amplification factor ranges from 7.6 to 125 as shown in
Table 9.1 for five different compositions of PMN–PT and PZN–PT crystals.
The amplitude of error amplification for single crystal BaTiO
3
is also listed
in the table for comparison. One can see that the formulas for BaTiO
3
are
fairly stable.
The third challenge is the presence of chemical inhomogeneity in these
crystals. The composition variation from one end to the other of a 2″ long
(50 mm) crystal boule could be as large as 5–10%. Because the material
properties are extremely sensitive to composition variation in these crystals,
particularly near the MPB composition, slight compositional variation from
sample to sample would totally destroy the self-consistency of the complete
data set.
Therefore, in order to get self-consistent full-set material constants, it is
important to reduce the number of samples, ensure composition uniformity
in the samples used, avoid too much geometry difference among different
samples and try to avoid using unstable formulas for derived quantities. To
meet these challenges, a systematic procedure has been developed over the
past few years, which utilizes both ultrasonic as well as resonance methods
in an effort to obtain the maximum number of measurements from the least
number of samples.
18
The property variation due to compositional fluctuation becomes serious
for compositions near the MPB because the desired piezoelectric properties
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Full-set material properties and domain engineering principles 243
Table 9.1
The ratios of relative errors of
e
33
and
e
31
to the relative errors of
c
12
,
c
13
and
c
33
for PZN–4.5%PT, PZN–7%PT, PZN–8%PT, PMN–
30%PT, PMN–33%PT and BTiO
3
PZN–4.5%PT PZN–7.0%PT PZN–8.0%PT PMN–30%PT PMN–33%PT BaTiO
3
∆
∆
e
e
c
c
31
31
12
E
12
E
21.5 54.2 52.5 39.2 59.6 2.3
∆
∆
e
e
c
c
31
31
13
E
13
E
43.8 112 109 82.6 125.1 4.9
∆
∆
e
e
c
c
33
33
33
E
33
E
14.9 17.9 19.2 7.6 15.2 3.9
∆
∆
e
e
c
c
33
33
13
E
13
E
28.7 34.4 36.3 14.3 30.1 7.2
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Handbook of dielectric, piezoelectric and ferroelectric materials244
exist only in the rhombohedral phase, not in the tetragonal phase. The
piezoelectric properties reach their peaks at the MPB composition, then
drastically decrease when the PT content is increased beyond the MPB. On
the rhombohedral side of the phase diagram, although the change in the
electromechanical coupling coefficient k
33
among different compositions is
not so drastic, other properties, such as the piezoelectric d
33
coefficient,
show a severe decrease. The property changes with composition on the
tetragonal side are much more pronounced.
9.2.3 Experimental procedure and error analysis
The ultrasonic pulse-echo method is another classic technique for material
property characterization, which has been widely used for the measurements
of elastic properties in solids and for the study of phase transitions involving
the softening of Brillouin zone center acoustic phonons. However, for
piezoelectric materials, people are more likely to use the resonance technique
since it is easier and less involved with instrumentation. On the other hand,
the passive nature of the ultrasonic technique has a unique advantage: it does
not suffer from mode coupling when two or more dimensions of the sample
are similar, while mode coupling makes the resonance technique unusable
for such cases because the shifting of the resonance peaks. Therefore, there
is no need to use high aspect ratio samples in ultrasonic measurements for
mode isolation. In addition, one can get several measurements from one
crystal since ultrasonic wave velocities along different crystal orientations
involve different combinations of elastic and piezoelectric coefficients, and
the shear and longitudinal waves also give independent measurements in a
given propagation direction.
For [001] poled PMN–PT and PZN–PT crystals, the effective macroscopic
symmetry is tetragonal 4mm as discussed in Section 9.2.1. There are a total
of 11 independent material constants needed to fully describe the system: 6
elastic, 3 piezoelectric and 2 dielectric constants. Using the ultrasonic pulse-
echo method, the phase velocities of longitudinal and shear waves can be
measured along the three pure mode directions: [100], [001] and [110].
Since shear waves can have their displacements parallel or perpendicular to
the poling direction, one can obtain eight independent velocities in these
three pure mode directions. The relationships between the velocities and the
elastic stiffness constants for a tetragonal symmetry single crystal are given
in Table 9.2. It can be seen that there is more than one way to determine
c
44
D
and
c
12
E
, which can provide some consistency checks.
In addition to the ultrasonic measurements, two length-extensional and
one thickness resonance measurement can be used to obtain the corresponding
electromechanical coupling coefficients k
33
, k
31
and k
t
, as well as elastic
compliances
ss
11
E
33
D
,
and the elastic stiffness
c
33
D
. The dielectric measurements
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Full-set material properties and domain engineering principles 245
at low frequencies, generally at 1 kHz, can give the dielectric constants
ε
11
T
and
ε
33
T
. The piezoelectric strain constant d
33
may also be checked by the
quasi-static method using a d
33
meter. Combining all the above-mentioned
measurements, the following material constants can be directly obtained:
ccccc
11
E
12
E
44
E
66
E
33
D
, , , ,
and
cs
44
D
11
E
;
and
s
33
D
11
T
; ε
and
ε
33
T
33
; d
as well as the
coupling coefficients k
33
, k
31
and k
t
. Altogether 14 constants can be obtained
directly from 17 independent measurements. This provides consistency checks
and, more importantly, the flexibility to throw out unreliable measurements
since there are only 11 independent coefficients to be determined for the
4mm symmetry.
In order to illustrate the concept of error propagation, let us look at a few
examples using the data of PMN–30%PT as numerical inputs. For tetragonal
symmetry, there are six independent elastic stiffness constants
c
αβ
E
. However,
c
33
E
and
c
13
E
cannot be directly measured. The quantity
c
33
E
may be obtained
from the measured
c
33
D
and k
t
by using the relation
cc k
33
E
33
D
t
2
= (1 – )
9.1
The errors in the measurements of
c
33
D
and k
t
will propagate to the calculated
c
33
E
,
∆∆
∆
c
c
c
c
k
k
k
k
33
E
33
E
33
D
33
D
t
t
t
2
t
2
= + 2
1 –
9.2
Equation (9.2) says that the error of calculated
c
33
E
comes from two sources,
the relative error of k
t
and
c
33
D
. Because the relative error of k
t
is only 1.1%,
and the relative error of
c
33
D
from the ultrasonic measurement is about 0.9%
for a 5 mm thick sample, the relative error of
c
33
D
from Eq. (9.1) is about
2.0% at the most. Therefore, such formulas are stable.
In order to determine
c
13
E
, we can use the following formula:
c
sc c
s
13
E
13
E
11
E
12
E
33
E
= –
( + )
9.3
which requires
s
13
E
and
s
33
E
to be determined first. The IEEE standard on
piezoelectricity suggests calculating
s
33
E
from the measured
s
33
D
and k
33
by
the following relation:
Table 9.2
Relationships between phase velocities and elastic constants for a
system with tetragonal
4mm
symmetry. They apply to the multi-domain PMN–PT
and PZN–PT single crystals poled in [001]
v
v
l
[001]
v
s
[001]
v
l
[100]
v
s
[100]
⊥
v
s||
[100]
v
l
[110]
v
s
[110]
⊥
v
s||
[110]
ρ
v
2
c
33
D
c
44
E
c
11
E
c
66
E
c
44
D
1
2
( + + 2 )
11
E
12
E
66
E
cc c
1
2
( – )
11
E
12
E
cc
c
44
D
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Handbook of dielectric, piezoelectric and ferroelectric materials246
s
s
k
33
E
33
D
33
2
=
1 –
9.4
In this case, the relative error of
s
33
E
is given by
∆∆
∆
s
s
s
s
k
k
k
k
33
E
33
E
33
D
33
D
33
33
33
2
33
2
= + 2
1 –
9.5
For a [001] poled PMN–30%PT the measured electromechanical coupling
coefficient k
33
= 0.92, then
2 /(1 – )
33
2
33
2
kk
= 11.0. Hence, if there was 0.5%
error in the measured k
33
, the calculated
s
33
E
would have more than 5.5%
error. Considering the relative error of
s
33
D
could reach 0.9%, the relative
error of
s
33
E
could be larger than 6.4%. Although this error level is still
acceptable, we need to remember that the reliability of
s
33
E
is not as high as
the directly measured quantities.
The other way to get
s
33
E
is from quasi-statically measured d
33
using the
formula
s
d
k
33
E
33
2
33
T
33
2
=
ε
9.6
In this case, the relative error of the calculated
s
33
E
is given by
∆
∆
∆
∆
s
s
d
d
k
k
33
E
33
E
33
33
33
T
33
T
33
33
= 2 + + 2
ε
ε
9.7
which depends only on the relative errors of k
33
and d
33
. Considering the
worst scenario, the relative error of calculated
s
33
E
from Eq. (9.6) is only
about 2.7%. Obviously, for piezoelectric materials with large k
33
it is better
to use Eq. (9.6) instead of Eq. (9.4) to determine
s
33
E
.
After obtaining
c
33
E
and
ss
33
E
13
E
,
can be calculated by
s
ssc
cc
13
E
33
E
33
E
33
E
11
E
12
E
= –
( – 1)
2( + )
9.8
Finally, the constant
c
13
E
can be calculated from Eq. (9.3). Up to this point,
all six independent elastic stiffness constants under constant electric field
have been determined. Since the elastic compliance matrix is the inverse of
the elastic stiffness matrix, all six independent elastic compliance constants
s
λµ
E
can be easily determined. The measured
s
11
E
from the resonance bar can
be used as a consistency check.
The piezoelectric stress constant d
31
is readily obtained from the measured
quantities
dsk
31
33
T
11
E
31
2
= – ε
9.9
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Full-set material properties and domain engineering principles 247
From the measured
c
44
E
and
c
44
D
, the electromechanical coupling coefficient
k
15
can be determined
k
c
c
15
2
44
E
44
D
= 1 –
9.10
The possible error of k
15
determined depends on the ratio of
cc
44
E
44
D
/
:
∆
∆∆
k
k
c
c
c
c
c
cc
15
15
44
D
44
D
44
E
44
E
44
E
44
E
44
D
=
1
2
+
–
9.11
If the ratio
cc
44
E
44
D
2
3
/ > ,
the error term
[( / ) + ( / )]
44
D
44
D
44
E
44
E
∆∆cc cc
will be
magnified. For a [001] poled PMN–30%PT single crystal
cc c
44
E
44
D
44
E
/( – )
=
10.1, therefore, the term
[( / ) + ( / )]
44
D
44
D
44
E
44
E
∆∆cc cc
will be magnified by a
factor of 5. The shear piezoelectric coefficient d
15
can be calculated by
d
k
ccc
15
2
11
T
15
2
44
E
11
T
44
E
44
D
= =
1
–
1
ε
ε
9.12
The piezoelectric stress constant e
ij
can be derived from the corresponding
formulas:
edcc dc
31 31
11
E
12
E
33
13
E
= ( + ) +
9.13
edcdc
33 31
13
E
33
33
E
= 2 +
9.14
edc
15 15
44
E
=
9.15
For a [001] poled PMN–30%PT single crystal, the estimated error of e
31
caused by the uncertainties of
c
12
E
and
c
13
E
alone is given by
∆
∆
∆
e
e
c
c
c
c
31
31
12
E
12
E
13
E
13
E
= 39.2 + 82.6
9.16
This means that a very small change in
c
12
E
and
c
13
E
will lead to very large
uncertainties in the derived value of e
31
. Also, the error of e
33
caused by
c
33
E
and
c
13
E
alone can be estimated by
∆
∆∆
e
e
c
c
c
c
33
33
33
E
33
E
13
E
13
E
= 7.6 + 14.3
9.17
From Eqs. (9.16) and (9.17), the calculated values of e
31
and e
33
are very
sensitive to the changes of
c
12
E
and
c
13
E
when d
33
and d
31
are very large,
hence they are not very stable formulas. Interestingly, the reverse is not true,
i.e., the change in the value of e
31
has very little influence on the values of
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Handbook of dielectric, piezoelectric and ferroelectric materials248
c
12
E
or
c
13
E
. Overall, the derived piezoelectric strain constants e
ij
are less
reliable than the piezoelectric stress constants d
ij
; therefore, one must assign
less weight for e
ij
coefficients when performing data analysis to get the self-
consistent set of material constants. General guidelines in obtaining reliable
data are as follows:
• Energy conservation demands the electromechanical coupling coefficients
to be always less than 1. Thus, large piezoelectric constants d
iλ
demand
large dielectric permittivity
ε
ij
. After a full set of constants has been
determined, the electromechanical coupling coefficients can be calculated
using these constants and none of them should be larger than 1.
• For 4mm symmetry crystals, the elastic compliances
s
12
E
and
s
13
E
must be
negative and the volume compressibility must be positive, i.e.
2 + 2 + 4 + > 0
11
E
12
E
13
E
33
E
ssss
.
• The elastic strain energy of crystals should be positively definite. For
4mm symmetry crystals this requires
cccc
44
E
66
E
11
E
12
E
> 0, > 0, > | |
and
( + ) > 2()
11
E
12
E
33
E
13
E2
ccc c
. Also, the condition
( + ) > 2()
11
E
12
E
33
E
13
E2
sss s
needs to be satisfied.
• From similar arguments as in the above point d
31
must be negative for
4mm symmetry crystals when d
33
is positive. For the same reason, e
31
,
g
31
and h
31
should all be negative. For the orthorhombic [011]
poled crystals, d
33
and d
31
can be both positive, but d
32
must be very
negative.
9.3 Complete set material properties for a few
compositions of PMN–PT and PZN–PT single
crystals
Using the experimental procedure described in Section 9.2, a few complete
sets of material properties have been obtained for domain engineered
0.955Pb(Zn
1/3
Nb
2/3
)O
3
–0.045PbTiO
3
[PZN–4.5%PT],
19
0.93Pb(Zn
1/3
Nb
2/3
)O
3
–0.07PbTiO
3
[PZN–7%PT],
20
0.92Pb(Zn
1/3
Nb
2/3
)O
3
–0.08PbTiO
3
[PZN–8%PT],
21
0.70Pb(Mg
1/3
Nb
2/3
)O
3
–0.30PbTiO
3
[PMN–30%PT]
22
and
0.67Pb(Mg
1/3
Nb
2/3
)O
3
–0.33PbTiO
3
[PMN–33%PT]
23
single crystals. The
results are listed in Tables 9.3, 9.4, 9.5, 9.6 and 9.7, respectively. All multi-
domain systems are treated as having effective tetragonal 4mm symmetry
following the arguments given in Section 9.2.1.
9.3.1 PZN–PT multidomain crystals poled along [001]
From Tables 9.3, 9.4, and 9.5, we can clearly see property changes in different
compositions of PZN–PT crystals poled along [001]. The effective piezoelectric
coefficient d
33
is the most important parameter for transducer and sensor
WPNL2204
Full-set material properties and domain engineering principles 249
Table 9.3
Measured and derived full-set material properties of PZN–4.5%PT single crystal poled along [001]
Density: ρ = 8310 kg/m
3
Elastic stiffness constants:
c
λµ
(10
10
N/m
2
)
c
11
E*
c
12
E
c
13
E
c
33
E
c
44
E*
c
66
E*
c
11
D
c
12
D
c
13
D
c
33
D*
c
44
D*
c
66
D
11.10 10.20 10.10 10.526 6.40 6.30 11.28 10.38 9.38 13.48 6.70 6.30
Elastic compliance constants:
s
λµ
(10
–12
m
2
/N)
s
11
E*
s
12
E
s
13
E
s
33
E
s
44
E
s
66
E
s
11
D
s
12
D
s
13
D
s
33
D*
s
44
D
s
66
D
81.63 –29.48 –50.04 105.5 15.63 15.87 61.36 –49.75 –8.08 18.65 14.93 15.87
Piezoelectric constants:
e
iλ
(C/m
2
),
d
iλ
(10
–12
C/N),
g
iλ
(10
–3
V m/N),
h
iλ
(10
8
V/m)
e
15
e
31
*
e
33
*
d
15
d
31
d
33
*
g
15
g
31
g
33
h
15
h
31
h
33
8.87 –3.76 15.39 139 –966 2000 5.05 –20.98 43.44 3.38 –4.70 19.22
Dielectric constants: ε
ij
(ε
0
), β
ij
(10
–4
/ε
0
) Electromechanical coupling constants:
k
ε
11
S
ε
33
S
ε
11
T*
ε
33
T*
β
11
S
β
33
S
β
11
T
β
33
S
k
15
k
31
*
k
33
*
k
t
*
2961 904 3100 5200 3.38 11.06 3.23 1.92 0.217 0.498 0.907 0.468
* Directly measured properties.
WPNL2204