Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications
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Handbook of dielectric, piezoelectric and ferroelectric materials250
Table 9.4
Measured and derived full-set material properties of PZN–7.0% PT single crystal poled along [001]
Density: ρ = 8350 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
11.30 10.30 10.50 10.91 6.30 7.10 11.37 10.37 10.03 14.03 6.80 7.10
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
85.87 – 14.13 – 69.02 142.0 15.87 14.08 56.74 –43.25 –9.64 20.91 14.71 14.08
Piezoelectric constants:
e
kλ
(C/m
2
),
d
kλ
(10
–12
C/N),
g
kλ
(10
–3
V m/N)
h
kλ
(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
11.09 –2.29 15.06 176 –1204 2455 6.63 –24.19 49.32 4.51 –3.14 20.69
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
T
k
15
k
31
*
k
33
*
k
t
*
2779 822.5 3000 5622 3.598 12.16 3.33 1.78 0.271 0.582 0.923 0.471
* Directly measured properties.
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Full-set material properties and domain engineering principles 251
Table 9.5
Measured and derived full-set material properties of PZN–8.0% PT single crystal poled along [001]
Density: ρ = 8315 kg/m
3
Elastic stiffness constants:
c
αβ
(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
11.50 10.50 10.90 11.51 6.34 6.50 11.80 10.80 10.00 14.25 6.76 6.50
Elastic compliance constants:
s
αβ
(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
86.89 –13.11 –69.87 141.0 15.77 15.38 55.84 –44.16 –8.19 18.52 14.79 15.38
Piezoelectric constants:
e
kλ
(C/m
2
),
d
kλ
(10
–12
C/N),
g
kλ
(10
–3
V m/N),
h
kλ
(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
10.06 –5.09 15.45 158.6 –1455 2890 6.18 –21.34 42.39 4.18 –5.84 17.72
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
T
k
15
k
31
*
k
33
*
k
t
*
2720 984 2900 7700 3.68 10.16 3.45 1.30 0.249 0.598 0.932 0.438
* Directly measured properties.
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Handbook of dielectric, piezoelectric and ferroelectric materials252
Table 9.6
Measured and derived full-set material properties of PMN–30%PT single crystal poled along [001]
Density: ρ = 8038 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
11.71 10.30 10.10 10.74 7.14 6.60 11.76 10.35 9.51 17.41 7.77 6.60
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
51.98 –18.94 –31.05 67.66 14.01 15.15 39.72 –31.20 –4.66 10.83 12.87 15.15
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
13.58 –2.41 27.07 190.3 –920 1981 5.97 –13.32 28.68 4.64 –2.19 24.60
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
T
k
15
k
31
*
k
33
*
k
t
*
3308 1243 3600 7800 3.02 8.05 2.78 1.28 0.285 0.486 0.916 0.618
* Directly measured properties.
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Full-set material properties and domain engineering principles 253
Table 9.7
Measured and derived full-set material properties of PMN–33%PT single crystal poled along [001]
Density: ρ = 8060 g/m
3
Elastic stiffness constants:
c
αβ
(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
11.50 10.30 10.20 10.38 6.90 6.60 11.69 10.49 9.05 17.31 7.70 6.60
Elastic compliance constants:
s
αβ
(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
70.15 –13.19 –55.96 119.6 14.49 15.15 45.60 –37.74 –4.11 10.08 12.99 15.15
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
10.08 –3.39 20.40 146.1 –1335 2820 10.31 –18.39 38.84 7.94 –5.64 33.94
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
T
k
15
k
31
*
k
33
*
k
t
*
1434 679 1600 8200 6.97 14.73 6.25 1.22 0.322 0.592 0.95 0.633
* Directly measured properties.
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Handbook of dielectric, piezoelectric and ferroelectric materials254
applications. It increases as the composition gets closer to the MPB and
reaches 2890 pC/N for PZN–8%PT. Beyond 8%PT, the crystals will be in the
tetragonal phase and the d
33
value drastically decreases. The d
31
coefficient
follows the same trend as d
33
and reaches the highest value of –1455 pC/N
for PZN–8%PT. On the other hand, the shear piezoelectric coefficient d
15
(~ 160 pC/N) does not change much with composition and is also rather
small compared with that of other known piezoelectric materials. The dielectric
constant
ε
33
increases with PT content from the rhombohedral phase side of
the phase diagram toward the MPB composition and reaches a peak value of
7700 for PZN–8%PT at room temperature, but the
ε
11
(~3000) value seems
to be insensitive to the compositional change. One may also notice that the
electromechanical coupling coefficients k
33
, k
31
, k
15
and k
t
are not as sensitive
to compositional change in the PZN–PT system. Although the k
33
value is
very high compared with other known piezoelectric materials, the k
t
value is
slightly less than that of soft Pb(Ti, Zr)O
3
(PZT5H) ceramics and the k
15
value is also rather small compared with that of PZT. In other words, the
[001] poled crystals are only advantageous for piezoelectric devices based
on k
33
and k
31
modes.
9.3.2 PMN–PT multidomain crystals poled along [001]
Tables 9.6 and 9.7 are properties of two compositions of [001] poled PMN–
PT domain engineered crystals. One can clearly see much more pronounced
property degradation compared with that of the PZN–PT system when the
composition is away from the MPB. From PMN–33%PT to PMN–30%PT,
the piezoelectric constant d
33
is reduced from 2820 to 1981 pC/N and the d
31
is reduced from –1335 to –920 pC/N, respectively.
One may notice that the elastic stiffness constants c
λµ
change very little
among different compositions. But the elastic compliance constants s
λµ
change
noticeably with composition. Moreover, it seems that the superior piezoelectric
properties of these systems are directly related to the large
s
33
E
value, which
is much larger than that of PZT ceramics and also becomes larger when the
composition gets closer to the MPB.
When the PT composition is more than 39% in the PMN–PT system, the
crystal will be in the tetragonal phase. In order to get a complete picture of
the composition influence to the material properties, the material
properties of 0.58Pb(Mg
1/3
Nb
2/3
)O
3
–0.42PbTiO
3
[PMN–42%PT] are listed
in Table 9.8.
24
It is clearly demonstrated that the piezoelectric properties
become much worse in this single domain single crystal with tetragonal
crystallographic symmetry. The piezoelectric properties of tetragonal PMN–
PT crystal are even worse than regular PZT ceramics. There is a very strong
dielectric anisotropy with the dielectric constant
ε
11
T
> 8600 but
ε
33
T
only
~660.
WPNL2204
Full-set material properties and domain engineering principles 255
Table 9.8
Measured and derived full-set material properties of PMN–42%PT single domain single crystal poled along [001]
Density: ρ = 8100 kg/m
3
Elastic stiffness constants:
c
αβ
(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
17.51 8.51 8.3 10.5 2.85 8.0 17.7 8.7 7.18 16.99 8.05 8.0
Elastic compliance constants:
s
αβ
(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
9.43 –1.68 –6.13 19.21 35.09 12.5 8.02 –3.10 –2.08 7.64 12.42 12.5
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
37.5 –2.1 12.2 131 –91 260 17.23 –15.57 44.5 13.87 –9.16 53.22
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
T
k
15
k
31
*
k
33
*
k
t
*
3054 259 8627 660 3.27 38.61 1.16 15.15 0.8 0.39 0.78 0.62
* Directly measured properties.
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Handbook of dielectric, piezoelectric and ferroelectric materials256
9.3.3 PZN–PT multi-domain crystals poled along [011]
Domain patterns in these domain engineered single crystals can be manipulated
by off-polar direction poling, and the poling field does not have to be applied
along [001]. In fact, if the optimized properties are d
32
or d
15
, [011] direction
poled crystals can be better than [001] direction poled crystals. This can be
seen from the complete set of properties characterized on a [011] poled
PZN–7%PT crystal system.
25
For the [011] direction poled crystals, the
effective domain pattern symmetry is orthorhombic mm2, so that 17
independent coefficients must be measured to form the complete data set.
26
From Table 9.9 we can see that the [011] direction poled crystal has much
superior d
32
(–1460 pC/N) with the electromechanical coupling coefficients
k
32
reaching 0.87. Although the d
31
value of [001] poled crystals can also
reach –1400 pC/N, their k
31
values are less than 0.58. Therefore, the [011]
poled crystals are superior in transverse mode piezoelectric device applications.
In addition, the [011] direction poled crystals have a very large shear
piezoelectric coefficient d
15
(> 1800 pC/N), while the shear piezoelectric
coefficient of [001] poled crystals is much smaller (d
15
< 180 pC/N).
9.4 Correlation between single domain and multi-
domain properties and the principle of property
enhancement in domain engineered
ferroelectric single crystals
In order to understand the principles of property enhancement through the
domain engineering process, it is necessary to find the correlation between
single domain properties and multi-domain properties. There are two main
contributions from multi-domain structures that do not exist in single domain
systems. One is the domain wall contribution, as often mentioned in ceramic
piezoelectric materials, and the other is the orientation effect.
A full set of data had been obtained for single domain single crystal
PMN–33%PT,
27
which has rhombohedral 3m symmetry. There are altogether
12 independent material constants to fully describe the crystal: 6 elastic, 4
piezoelectric and 2 dielectric constants. In order to define the property matrix
and find the correlations to the multi-domain data, we must define the coordinate
system of the rhombohedral phase with respect to the cubic phase coordinates.
As illustrated in Fig. 9.4, if we use the subscript ‘r’ to represent the
rhombohedral phase, their relationships with cubic coordinate systems are:
[100]
r
↔ [1
1
0]
, [010]
r
↔ [11
2
], [001]
r
↔ [111]
Here the conversions between the cubic and the tetragonal coordinate
systems are different from those used in reference 27 therefore, some signs
of the constants listed in Table 9.10 are different from those given
WPNL2204
Full-set material properties and domain engineering principles 257
Table 9.9
Complete set of material coefficients of PZN–7%PT single crystal poled along [011]
Density: ρ = 8350 kg/m
3
Elastic stiffness coefficients:
c
λµ
(10
10
N/m
2
)
c
11
E
c
12
E
c
13
E
c
22
E
c
23
E
c
33
E
c
44
E
c
55
E
c
66
E
14.5 15.32 12.67 18.02 15.00 14.10 6.47 0.34 7.10
c
11
D
c
12
D
c
13
D
c
22
D
c
23
D
c
33
D
c
44
D
c
55
D
c
66
D
17.40 21.16 11.43 29.82 12.50 14.63 6.54 0.41 7.10
Elastic compliance coefficients:
s
λµ
(10
–12
m
2
/N)
s
11
E
s
12
E
s
13
E
s
22
E
s
23
E
s
33
E
s
44
E
s
55
E
s
66
E
67.52 –60.16 3.36 102.00 –54.47 62.02 15.45 291.55 14.08
s
11
D
s
12
D
s
13
D
s
22
D
s
23
D
s
33
D
s
44
D
s
55
D
s
66
D
59.40 –35.38 –16.17 26.30 5.16 15.05 15.30 245.99 14.08
Piezoelectric coefficients:
e
iλ
(C/m
2
),
d
iλ
(10
–12
C/N),
g
iλ
(10
–3
V m/N) and
h
iλ
(10
8
V/m)
e
15
e
24
e
31
e
32
e
33
6.25 3.24 –8.64 –17.44 3.69
d
15
d
24
d
31
d
32
d
33
1823 50 478 –1460 1150
g
15
g
24
g
31
g
32
g
33
24.99 3.03 16.98 –51.85 40.84
h
15
h
24
h
31
h
32
h
33
1.02 1.98 –33.53 –67.65 14.33
Dielectric coefficients: ε
ij
(ε
0
) and β
ij
(10
–4
/ε
0
)
ε
11
S
ε
22
S
ε
33
S
ε
11
T
ε
22
T
ε
33
T
6953 1847 291 8240 1865 3180
β
11
S
β
22
S
β
33
S
β
11
T
β
22
T
β
33
T
1.44 5.42 34.34 1.21 5.36 3.14
Electromechanical coupling coefficients:
k
k
15
k
24
k
31
k
32
k
33
k
t
0.40 0.10 0.35 0.86 0.87 0.19
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Handbook of dielectric, piezoelectric and ferroelectric materials258
[001]
[100]
[010]
[112] or [010]
r
[111] or [001]
r
[110] or [100]
r
9.4
The rhombohedral coordinates defined for PMN–0.33PT and their
relationship to the cubic coordinates.
Table 9.10
Measured and derived material properties of single domain PMN–
33%PT single crystal poled along [111]
Elastic stiffness constants:
c
λm
(10
10
N/m
2
)
c
11
E*
c
12
E*
c
13
E
c
14
E*
c
33
E
c
44
E*
c
66
E*
20.12 7.36 11.50 –4.15 17.12 2.90 6.38
c
11
D
c
12
D
c
13
D
c
14
D*
c
33
D*
c
44
D*
c
66
D*
20.43 7.62 10.32 –3.87 21.94 5.63 6.41
Elastic compliance constants:
s
λµ
(10
–12
m
2
/N)
s
11
E*
s
12
E
s
13
E
s
14
E
s
33
E
s
44
E
s
66
E
62.16 –53.85 –5.58 166.24 13.34 510.98 232.02
s
11
D
s
12
D
s
13
D
s
14
D
s
33
D*
s
44
D
s
66
D
9.39 –3.94 –2.57 9.15 6.97 30.33 26.65
Piezoelectric coefficients:
e
iλ
(C/m
2
),
d
iλ
(10
–12
C/N),
g
iλ
(10
–3
V m/N) and
h
iλ
(10
8
V/m)
e
15
*
e
22
*
e
31
e
33
d
15
d
22
d
31
*
d
33
*
7.52 –0.78 –2.88 11.83 4100 –1340 –90 190
g
15
g
22
g
31
g
33
h
15
h
22
h
31
h
33
11.72 –3.83 –1.59 3.35 36.38 –3.78 –9.94 40.78
Dielectric constants: ε
ij
(ε
0
) and β
ij
(10
–4
/ε
0
)
ε
11
S*
ε
33
S
ε
11
T
ε
33
T*
β
11
S
β
33
S
β
11
T
β
33
T
233 328 3950 640 42.84 30.53 2.53 15.63
Electromechanical coupling constants
k
and density ρ
k
15
k
31
*
k
33
*
k
t
*
ρ(kg/m
3
)
0.70 0.15 0.69 0.47 8060
* Directly measured properties.
in reference 27. Detailed explanation of the coordinate transformation has
been described by Zhang and Cao.
28
The full-set single domain material
properties were measured under a bias field along the polar direction [111]
WPNL2204
Full-set material properties and domain engineering principles 259
to ensure that the crystals are always in the single domain state because the
single domain state seems less stable than the multi-domain state. One must
note that the application of a bias electric field will cause some property
change since most properties are field dependent.
The measured complete set of material properties under bias for the PMN–
33%PT single domain single crystal is given in Table 9.10.
27,28
One can find
many interesting facts from the single domain data set in comparison with
the multi-domain data set of the same composition given in Table 9.7. The
electromechanical coupling coefficient k
33
and the piezoelectric constant d
33
for single-domain PMN–33%PT single crystals are only 69% and 190 pC/N,
respectively. These values are much smaller than those of domain engineered
PMN–33%PT single crystals. However, the shear piezoelectric constant d
15
of single domain PMN–33%PT single crystals reaches 4100 pC/N even under
bias, which is much larger than that of multi-domain PMN–33%PT single
crystals. There is also a report of d
15
value for the single domain PMN–PT
systems as high as 7000 pC/N without bias.
29
This means that part of the d
15
value in the single domain state has been converted to effective d
33
in the
domain engineered crystals.
To see this conversion from the orientation effect, we can perform matrix
rotation using the single domain data given in Table 9.10. The transformations
of the dielectric constants piezoelectric constants and elastic compliance
from one orientation to another can be performed based on their matrix
definitions. Let {b
i
} and
{}
*
b
i
(i = 1, 2, 3) be orthonormal bases representing
the original and rotated coordinates, respectively. The component a
ij
of the
transformation matrix A is given by
a
ij i j
=
*
bb•
9.18
In the Voigt notation, the material constants in the rotated coordinate
system can be calculated by
30
ε* = AεA
t
9.19
d* = AdN
t
9.20
s* = NsN
t
9.21
where A
t
and N
t
represent the transpose of matrices A and N, respectively,
with the matrix N given by
N
aaa aa aa aa
aaa aa aa aa
aaa aa aa aa
aa aa aa aa aa aa aa aa aa
aa
=
2 2 2 + + +
2
11
2
12
2
13
2
12 13 11 13 11 12
21
2
22
2
23
2
22 23 21 23 21 22
31
2
32
2
33
2
32 33 31 33 31 32
21 31 22 32 23 33 22 33 23 32 21 33 23 31 21 32 22 31
11 3131 12 32 13 33 12 33 13 32 11 33 13 31 11 32 12 31
11 21 12 22 13 23 12 23 13 22 11 23 13 21 11 22 12 21
2 2 + + +
2 2 2 + + +
aa aa aa aa aa aa aa aa
aa aa aa aa aa aa aa aa aa
9.22
WPNL2204