Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications
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Handbook of dielectric, piezoelectric and ferroelectric materials20
T
C
increased from 121 to 142°C (i.e. 26.5% PT to 32% PT). There was a
strong linear dependence of
s
33
E
on the PT content near the MPB (Fig. 1.10 b).
d
33
and g
33
Both (a) the measured d
33
values obtained on the plates and (b) the calculated
d
33,Calc
values for bars and plates (Section 1.3.3) show a strong dependence
on PT content (Fig. 1.11b). The values for the directly measured d
33
increased
from 1660 to 3140pC/N (by ~90%). The calculated d
33,Calc
values increased
from 950 to 1820pC/N, a similar increase of ~90%. There was a strong
linear dependence of d
33
with T
C
, and thus PT content. Recall that the k
33
values were relatively constant in this MPB region.
Figure 11(b) illustrates slight changes in g
33
from 42 to 50 × 10
–3
Vm/N.
By comparison, the calculated g
33,Calc
values increased initially from 24 to
31 × 10
–3
Vm/N, but then decreased to 29 × 10
–3
Vm/N at the T
C
of 142°C
(to be discussed later). The measured values for g
33
approached a steady
value of ~ 50 × 10
–3
Vm/N. By contrast, g
33,Calc
reached a maximum value
of 31 × 10
–3
Vm/N at a T
C
of 137°C.
Elastic properties and electromechanical coupling
Figure 1.12 illustrates the relationship between
s
33
E
and k
33
. Data are presented
with y-axis variable of
k
33
2
(i.e., energy conversion efficiency) and x-axis
Table 1.1
Coupling factors for bars and plates (as determined by resonance and
dielectric measurements, respectively)
Specimen no.
T
C
(°C)
k
33
(bar)
k
33
(plate)
0 121.0 0.836 0.887
1 125.6 0.869 0.904
2 126.0 0.870 0.907
3 126.5 0.873 0.908
4 127.7 0.875 0.911
5 128.8 0.882 0.914
6 130.2 0.891 0.915
7 131.1 0.892 0.916
8 132.1 0.896 0.916
9 132.7 0.899 0.921
10 134.0 0.905 0.925
11 135.8 0.910 0.927
12 136.9 0.912 0.929
13 137.6 0.913 0.932
14 138.6 0.915 0.935
15 140.8 0.917 0.939
16 141.8 0.919 0.944
Average n/a 0.896 0.921
WPNL2204
Bridgman growth and properties 21
variable of
1/
33
E
s
(related to stiffness). The values of
s
33
E
and k
33
increase
with PT content along the growth direction, as indicated by the arrow.
1.4.5 Interrelationship of properties
Measured and calculated d
33
and g
33
The d
33
values obtained from the PIEZO d
33
meter for the plate specimens
are higher than the corresponding calculated d
33,Calc
values by an average of
Growth direction
16
R
2
= 0.993
120 125 130 135 140 145
T
C
(°C)
(a)
3000
2000
1000
d
33
(pC/N)
0
d
33,Calc
(pC/N)
2000
1500
1000
35
30
25
Growth direction
16
0
50
45
40
g
33
(× 10
–3
V m/N)
g
33,Calc
(× 10
–3
V m/N)
120 125 130 135 140 145
T
C
(°C)
(b)
1.11
Comparison of measured and calculated piezoelectric properties
for selected plates.
WPNL2204
Handbook of dielectric, piezoelectric and ferroelectric materials22
66% (Fig. 1.11a). Since the calculations of d
33,Calc
use parameters obtained
from standard geometries (i.e. bars for k
33
and
s
33
E
, plates for
K
33
T
), d
33,Calc
is considered closer to the true d
33
value. The measured values for bars on
the PIEZO meter were not reproducible and varied with the applied pre-
force, especially for compositions with higher T
C
(closer to the MPB). This
was attributed to the larger stresses resulting from the smaller cross-sectional
area of the bar (0.49mm
2
). A larger applied stress (e.g. >2MPa for a 1N pre-
force) may lead to non-linear contributions to d
33
[38]
. Such effects would be
more evident for more compliant specimens closer to the MPB (i.e., higher T
C
).
g
33,Calc
was calculated from d
33
and
K
33
T
Eq.(2), where the d
33
values were
obtained from two different methods and
K
33
T
was measured only for crystal
plates. Consequently, g
33,Calc
had two corresponding sets of values (Fig.
1.11b), with g
33,Calc
closer to the true values for the reasons discussed above.
Covariation of
K
33
T
and d
33
The results show that both d
33
(Fig. 1.11a) and
K
33
T
(Fig. 1.9) increase
significantly (>80% and >50%, respectively) with a relatively small change
in composition (i.e. 4% change in PT content). The increasing trend in both
d
33
and
K
33
T
can be attributed to changes in stress-induced and field-induced
polarization P
3
30
, where:
P
i
= d
ijk
T
jk
1.7
and
1.5 2.0 2.5 3.0
1/
s
33
(× 10
10
N/m
2
)
Growth direction
16
0
85
k
33
(%)
R
2
= 0.999
80
70
75
E
2
1.12
Relationship between longitudinal energy conversion efficiency
and reciprocal compliance.
WPNL2204
Bridgman growth and properties 23
P
i
= ε
0
χ
ij
E
j
1.8
Thus, the dielectric susceptibility
χ
33
T
is essentially equivalent to
KK
33
T
33
T
33
T
( = 1 + )χ
for high-K materials (
K
33
T
> 4000 for PMN–PT);
therefore,
P
i
= d
ijk
T
jk
; ε
0
K
ij
E
j
1.9
and it is clear from Eq. (1.9) that both d
33
and
K
33
T
would change in the same
sense with displaced polarization P
3
.
Equation (1.6) further explains why d
33
exhibits a greater variation (>80%)
than
K
33
T
(>50%). Both k
33
and
s
33
E
increase with T
C
(Fig. 1.10). In addition,
s
33
E
shows a larger increase than
K
33
T
(Fig. 1.9) for the same composition and
T
C
range (121–142°C). As a result, the product of
s
33
E
and
K
33
T
gives a
larger relative increase with T
C
, compared with
KKK
33
T
33
T
33
T
( )≡
. Thus,
the triple product of k
33
(which increases with T
C
) with
s
33
E
and
K
33
T
, i.e,
dksK
33,Calc
33
33
E
33
T
=
, should have a larger relative increase than
K
33
T
alone. This is indeed observed in the T
C
range of 121–142°C (i.e. near the
MPB).
Bokov and Ye
39
have suggested that the superior d
33
values for PMN–PT
crystals before the MPB were due to the large dielectric response (
K
33
T
). Our
results confirm that an increase in
K
33
T
is important for an increase in d
33
, but
k
33
and
s
33
E
are also important. For example, softening in the vicinity of the
MPB increases
s
33
E
and gives rise to enhanced dielectric and piezoelectric
displacements, as discussed before.
Small increase in the piezoelectric voltage coefficient
In contrast to the relatively large increases in d
33
and
K
33
T
with T
C
(and PT
content), g
33
shows a weaker dependence on T
C
(Fig. 1.11b). This is expected
from Eq. (1.2), where d
33
and
K
33
T
share similar offsetting increasing trends
with T
C
. Specifically, g
33
can be derived from Eqs. (1.2) and (1.6) as:
g
ks
K
33
33
33
E
33
T
=
1.10
Similar to the discussion on d
33
(above), the relative increases in
s
33
E
and
K
33
T
are of the same magnitude. Therefore, the resulting g
33
exhibits a weak
dependence on T
C
(and PT content).
The decrease in g
33,Calc
for the two specimens with the highest T
C
(Fig.
1.11b) can now be considered in terms of the changes in k
33
,
s
33
E
and
K
33
T
in
Eq. (1.10): k
33
approaches a constant value as T
C
increases,
s
33
E
increases
with a linear dependence, and
K
33
T
exhibits an exponential increase with T
C
.
Therefore, as the composition changes towards the MPB, the exponential
WPNL2204
Handbook of dielectric, piezoelectric and ferroelectric materials24
increase in
K
33
T
out-competes the combined increase in k
33
and
s
33
E
, thus
resulting in a net decrease in g
33,Calc
, as observed.
Mechanical softening and property enhancement
Property variations occur along the growth direction in PMN–PT crystals
with composition towards the MPB. Both k
33
and
s
33
E
increase with T
C
(PT
content) (Fig. 1.12). These trends are attributed to an increasing softening
that occurs as the composition approaches the MPB (35% PT), where a
phase transformation occurs to the tetragonal state. Softening at phase
transformations is considered an important factor for induced displacements
and the superior properties for
K
33
T
, d
33
and k
33
.
The reported anisotropies in property coefficients for PMN–PT single
crystals corroborate our argument that mechanical softening is important for
piezoelectric enhancement
40
. For stress-induced displacements, superior
piezoelectric properties occur along the softest direction. For the compositions
reported above, the lowest value of the Young’s modulus (largest elastic
compliance) was along <001>.
Implications for applications
The property variations along the growth direction for PMN–PT crystals
have important implications for device applications. The superior and variable
d
33
and
K
33
T
values enable a wide selection of piezoelectric and impedance
parameters for design considerations. A most important factor is that k
33
remains relatively high and stable (k = 0.87–0.92) over a wide range of d
33
and
K
33
T
values. This allows for the design of devices that do not significantly
affect the energy conversion efficiency or signal transmission bandwidth.
Experimental results also indicate the voltage coefficient is not superior to
PZT ceramics. For example, the g
33
of PMN–PT crystals (24 ~ 31×10
–3
Vm/
N) is similar to that of PZT (20 ~ 27 × 10
–3
Vm/N). Thus, it would appear not
to be particularly beneficial to use PMN–PT for voltage generation. Instead,
the design should take advantage of the superior d
33
and k
33
coefficients of
PMN–PT crystals.
Since the property variations occur along the growth direction, it is
advantageous to choose crystals grown along preferred crystallographic
directions for specific excitation modes. This is particularly true when in-
plate homogeneity is critical. Specifically, a <001>-seeded crystal offers the
best homogeneity for a (001) plate, where the longitudinal extension mode
is used. Similarly, a <011>-grown crystal is best for (011) plates where the
transverse extension mode is desired. The direction of crystal growth becomes
increasingly important for larger crystal plates. For example, consider a
(001) plate 50mm in length, sectioned from a <011> crystal, where the plate
WPNL2204
Bridgman growth and properties 25
length direction is 45° to the growth direction. The two ends of the plate are
>35 mm apart along the growth direction. For the current melt-growth method,
such a distance would allow for significant chemical segregation to occur,
with associated property variations along the length of the plate. Therefore,
it is preferable to use PMN–PT crystals grown along certain crystallographic
directions for specific applications.
1.5 Optimization of cut directions
The piezoelectric properties of PMN–PT crystals are anisotropic, i.e. direction
dependent. To optimize piezoelectric performance for each vibration modes,
we topologically surveyed the maximum piezoelectric d coefficient through
theoretical calculation and experimental verification
26
. The survey has led to
discoveries of new cut directions that maximize piezoelectric d coefficients
for all of the possible symmetric domain configurations and all of the vibration
modes of PMN–PT crystals.
1.5.1 Method
To calculate the piezoelectric coefficients using the matrices for equilibrium
properties of the 32 crystal classes
41
we assume that the elasto-piezo-dielectric
matrices are applicable not only to the original 3m single-domain PMN–PT
crystals but also to the domain-engineered crystals with 4mm (<001> poling)
or mm2 (<011> poling) multi-domain symmetric configurations (see Fig.
1.17 on page 35). Coordinate rotations were applied on all of these three
domain symmetries. The maximum piezoelectric coefficients were calculated
based on the measured independent matrix values in the original coordinates
and verified experimentally in the rotated coordinates.
The fundamental of the transformation law is that the components of a
tensor are the components of a physical quantity, which should retain its
identity however the axes may be changed. The transformation of piezoelectric
coefficients by changing coordinate system is represented by the following
equation
42
:
′
d
ijk
= Σ a
il
a
jm
a
kn
d
lmn
1.11
where d
lmn
is the piezoelectric coefficient in the original coordinate system,
′
d
ijk
is the piezoelectric coefficient in the new rotated coordinate system, and
a
il
, a
jm
and a
kn
are the components of the transformation matrix.
The coordinate rotation was defined in the following way: rotation was
first made by angle ϕ around the z-axis, then around the new x-axis by angle
θ, and finally around the new z-axis by angle Ψ
[43]
. All of the rotations were
counterclockwise. The piezoelectric coefficients after the rotation in the three-
dimensional space were derived as functions of the independent piezoelectric
WPNL2204
Handbook of dielectric, piezoelectric and ferroelectric materials26
coefficients in the original coordinate system and the rotated Euler angles
(ϕ, θ, Ψ) using tensor calculations.
The coordinates were selected as follows: for the 3m symmetry, [111] as
z-axis, [1–10] as x-axis and [11–2] as y-axis; for mm2 symmetry, [011] as z-
axis, [100] as x-axis, and [01–1] as y-axis; for 4mm symmetry, [001] as x-
axis, [100] as x-axis, and [010] as y-axis. To obtain the independent piezoelectric
coefficients, three sets of specimens of PMN–31%PT crystal (3m, mm2 and
4mm) were prepared to cope with the scattering of the measurements due to
PT content variations. Independent piezoelectric coefficients of the three
engineered domain systems were directly measured using a modified Berlincout
meter with homemade adaptors, with the measurement error within ±5%.
1.5.2 Results and discussion
<001> poling
A symmetrical multi-domain configuration is formed with macroscopic
tetragonal symmetry (4mm) after poling along <001>. This 4mm symmetrical
domain structure only has three independent piezoelectric coefficients, d
15
=
d
24
, d
31
= d
32
and d
33
.
The calculation on the 4mm multi-domain was not presented in this section
because it has already been well described in the literature
44
. The maximum
of d
33
is in the <001> direction (see Fig. 1.16 on page 33).
<111> poling
A single domain PMN–PT crystal (3m) can be obtained by completely poling
along the <111> direction. The single domain crystal has four independent
piezoelectric coefficients: d
15
(= d
14
), d
16
(=d
21
/2 = –d
22
/2), d
31
(= d
32
) and
d
33
. The representation surface of the shear piezoelectric coefficient d
15
was
then calculated (Fig. 1.13). The amplitude of the surfaces represents the
absolute value of the d
15
in that orientation.
The maximum value of d
15
of 5190pC/N is in the direction of θ of 337.5°
and ϕ of 0° (zxt–22.5°). The maximum amplitude of d
15
(–5190pC/N) was
found at θ of 157.5° and ϕ of 0° (xzt 157.5°). The maximum d
15
value in the
rotated coordinate is approximately 1.1 times the original value. In particular,
the cross-talk from d
16
is eliminated. In contrast, strong crosstalk between
d
15
(4800 pC/N) and d
16
(1975 pC/N) exists before the rotation of 22.5°.
<011> poling
When poling along the <011>, a symmetric multi-domain configuration is
formed with macroscopic orthorhombic symmetry (mm2). The piezoelectric
WPNL2204
Bridgman growth and properties 27
matrices of ferroelectric crystals with orthorhombic symmetry have five
independent piezoelectric coefficients: d
15
, d
24
, d
31
, d
32
and d
33
. The
representation surface of the transverse piezoelectric coefficient d
31
was
calculated (Fig. 1.14). The direction of the largest d
31
(–1750pC/N) is in θ
= 0° or 180° and ϕ = 0° , whereas d
32
has a positive value of approximately
half or the d
31
.
330°
300°
270°
240°
210°
180°
150°
120°
90°
60°
30°
0°
2000
4000
60000
[112]
–
[011]
–
(b)
1.13
(a) Three-dimensional plot of the piezoelectric representation
surface of
d
15
for a rhombohedral single-domain PMN–PT crystal. (b)
Cross-section of the piezoelectric representation surface in the
(110)
plane.
4000
2000
[111] 0
–2000
–4000
5000
–5000
–2500
0
[112]
2500
5000
5000
–2500
–5000
0 [11 0]
(a)
WPNL2204
Handbook of dielectric, piezoelectric and ferroelectric materials28
1.14
(a) Three-dimensional plot of the piezoelectric representation
surface of
d
31
for a PMN–PT crystal with
mm2
symmetric multi-
domain. (b) Cross-section of the piezoelectric representation surface
in the (011) plane.
330°
300°
270°
240°
210°
180°
150°
120°
90°
60°
30°
0°
[011]
–
100
(b)
1000
1500
500
0
900
450
[011] 0
–450
–900
1800
–900
–450
0
[011 ]
450
900
900
–900
–1800
0 [100]
(a)
WPNL2204
Bridgman growth and properties 29
The shear piezoelectric coefficient d
36
is a dependent tensor and is zero in
the original coordinate. To explore the maximum value of d
36
in a rotated
coordinate system, the representation surface of the d
36
was calculated (Fig.
1.15). The maximum d
36
(±2600 pC/N) was obtained in the direction of θ of
0° and ϕ of ± 45° (zxt ±45°) or ±225°. The d
36
rotated about z-axis can be
expressed as:
d
36
= –2d
31
sin(ϕ)cos(ϕ) +2d
32
sin(ϕ)cos(ϕ) 1.12
330°
300°
270°
240°
210°
180°
150°
120°
90°
60°
30°
0°
[011]
–
[100]
(b)
2000
3000
1000
0
1.15
(a) Three-dimensional plot of the piezoelectric representation
surface of the
d
36
for a PMN–PT crystal with
mm2
symmetric multi-
domain. (b) Cross-section of the piezoelectric representation surface
in the (011) plane.
(a)
1000
500
[011] 0
–500
–1000
2000
–2000
–1000
0
[011 ]
1000
2000
1000
–1000
–2000
0 [100]
WPNL2204