**1.3. Results and discussion**

2 L S

(3)

è ø (4)

Poisson's ratio (-) Ratio of

2 =ρVG <sup>S</sup>

 

 

−=σ

2 1 (2)

σ

Y

Young's modulus (Pa)

( ) <sup>+</sup><sup>σ</sup> <sup>=</sup> <sup>12</sup> <sup>Y</sup> <sup>G</sup>

K

Bulk modulus (Pa)

G

 

1

Elastic constants

− 

2 S L

V V <sup>1</sup> <sup>1</sup>  

Rigidity (Pa)

2 <sup>L</sup> <sup>2</sup> S VV V 3 <sup>4</sup> <sup>V</sup>

−

V3Y <sup>−</sup>

2 <sup>L</sup> V 3 4

= ρ

 −= <sup>2</sup>

ρ VK

2 S 2 L

> 

S

1600 2000 2400 2800 3200

PZT

Alka li Nioba te

Alkali Bismuth Tita nate

> r=0.49\* r=0.74\*\*

r=0.89\* r=0.97\*\*

PLT, PT

V<sup>S</sup> (m/s)

0.2 0.25 0.3 0.35 0.4 0.45

σ (-)

6 8 10 12 14

K (10<sup>10</sup> N/m<sup>2</sup>)

2 S

2

<sup>2</sup> <sup>V</sup> <sup>1</sup> V

> *V* <sup>2</sup> G <sup>S</sup> = r

<sup>4</sup> K V

r

constants" into "Elastic constants".

Coupling factor (%)

kP

Piezoelectric strain constant (C/N)

36 Ferroelectric Materials – Synthesis and Characterization

V<sup>L</sup>

Longitudinal wave velocity (m/s)

ε<sup>r</sup> d<sup>33</sup>

stants.

V<sup>S</sup>

constants.

Transverse wave velocity (m/s)

fc<sup>P</sup>

Frequency constant

Relative dielectric constant (-)

4 Fig.2 Figure 2 Replace with the figure below because of lack of

5 28 open marks and dash lines in Figure 3) red marks and dash lines in Figure 3)

6 Fig. 3 Figure 3 Replace with the figure below because of lack of

14 35 with FT vs. 2fcp[fcp is shown in Figure 13(c) with FT vs. 2fcp(fcp is shown in Figure 13(c))

k<sup>P</sup> (%)

PZT

PZT

k<sup>P</sup> (%)

> k<sup>P</sup> (%)

17 10 (Fig. 14(c) (Fig. 14(c))

17 13 (Fig. 14(d) (Fig. 14(d))

17 15 (Fig. 14(d) (Fig. 14(d))

17 16 (Fig. 13(a) (Fig. 13(a))

18 21 (Figure 16(b) (Figure 16(b))

18 33 (Figure 13(b) (Figure 13(b))

20 3 at 1,300-1, at 1,300-

20 4 360℃ 1,360℃

18 33 (Figures 18(a)-(b) (Figures 18(a)-(b))

5 10 the case of after poling (closed marks and solid

lines in Figure 3)

*V* 2 2 L S

æ ö = - ç ÷

figure, one of the titles is changed "Material

Dielectric and piezoelectric constants

3

letters (ρ) in the equations in G, Y and K. In this

where ρ is the bulk density of the ceramic disks. Figure 2 shows the relationships between the ratio of sound velocities (VS/VL), the elastic constants, and dielectric and piezoelectric con‐

the case of after poling (black marks and solid

**Figure 2.** Relationships between the ratio of sound velocities (VS/VL), elastic constant and dielectric and piezoelectric

k<sup>P</sup> (%)

soft PZT hard PZT SZ KBT BT PLT PT

k<sup>P</sup> (%)

r=-0.91\* r=-0.93\*\*

r=-0.91\* r=-0.92\*\*

Alka li Nioba te

k<sup>P</sup> (%)

ρ

ρ <sup>G</sup> <sup>V</sup><sup>S</sup> <sup>=</sup>

sound velocities

VS/V<sup>L</sup>

ρ K 3 <sup>4</sup> <sup>G</sup>

+

(Hz・m)

VL

=

Bulk density (kg/m<sup>3</sup>)

lines in Figure 3)

letter (σ) in the figure of σvs. kp.

15 30 45 60

5 10 15 20

Alkali Bismuth Titana te

PLT, PT

Alka li Nioba te

4000 4500 5000 5500 6000

Alkali Bismuth Titanate

PLT, PT

V<sup>L</sup> (m/s)

Y (10<sup>10</sup> N/m<sup>2</sup>)

G (10<sup>9</sup> N/m<sup>2</sup>)

ì ü ï ï ï ï = - í ý ï ï æ ö ï ï ç ÷ - ç ÷ î þ è ø

1 1 σ 1

## *1.3.1. Dependence of planar coupling factor on elastic constant*

Figure 3 shows the relationships between longitudinal (VL) and transverse (VS) wave velocities, Young's modulus (Y), Poisson's ratio (σ), modulus of rigidity (G), and bulk modulus (K) vs. planar coupling factors (kp) of disk in (1- *x*)(Na, K, Li, Ba)(Nb0.9Ta0.1)O3-*x*SZ (abbreviated to "SZ"), (1-*x*)NBT-*x*KBT ("KBT"), 0.79NBT-0.20KBT-0.01BFT ("KBT"), and (1-*x*)NBT-*x*BT ("BT") lead-free ceramics compared with 0.05Pb(Sn0.5Sb0.5)O3-(0.95-*x*)PbTiO3-*x*PbZrO3 ceramics with ("hard PZT") and without 0.4 wt% MnO2 ("soft PZT"), and with 0.90PbTiO3-0.10La2/3TiO3 ("PLT") and 0.975PbTiO3-0.025La2/3TiO3 ("PT") ceramics before and after fully DC poling. In the case of after poling (marks and solid lines in Figure 3), although the VL values of the PZT ceramics were almost constant at approximately 4, 600-4, 800 m/s independently of the composition *x*, their VS values linearly decreased from 2, 500 to 1, 600 m/s with increasing kp from 20% to 65% (solid line). In addition, the VL and VS values of the PZT ceramics were smaller than those of the lead-free ceramics (VL = 5, 000-5, 800 m/s and VS = 2, 600-3, 000 m/s; solid lines). Although the VL values of the PT ceramics were almost the same (4, 800 m/s) as those of the PZT ceramics, the VS values of the PT ceramics were approximately 2, 700 m/s. On the other hand, the VL values of the SZ ceramics were relatively high (5, 500-5, 800 m/s); further‐ more, the VS values of the SZ ceramics also increased (2, 600-2, 700 m/s) and linearly decreased with increasing kp from 25% to 50% (solid line), the behavior of which was almost the same as that of the VS values of the PZT ceramics. The VL values of the KBT, BT, and PLT ceramics (5, 000-5, 400 m/s) were between those of the PZT, PT, and SZ ceramics. However, the VS values of the KBT, BT, and PLT ceramics (2, 800-3, 000 m/s) were the highest. Therefore, it was possible to divide VL and VS into three material groups, namely, PZT and PT/ KBT, BT (alkali bismuth titanate), and PLT/ SZ (alkali niobate). In addition, kp increased from 4% to 65% with decreasing Y from 15 × 1010 to 6 × 1010 N/m2 and increased with increasing σ from 0.25 to 0.43. It was clarified that higher kp values can be realized at lower Y and G, and higher σ and K.

In comparison with the values of before poling (the kp was made use of the values after poling; marks and dash lines in Figure 3), VL, σ, and K increase and VS, Y, and G decrease after poling because of ferroelectric domain alignment. In addition, while the correlation coefficients in the kp vs. Y, σ, and G were almost independent of poling treatment, the coefficients in the kp vs. K after poling increases from 0.49 to 0.74. It is thought that the increase in K is significant to realize piezoelectricity as mentioned below.

Figure 4 shows the relationships between kp vs. changes (Δ) in longitudinal (VL) and transverse wave velocities (VS) [ΔVL/ΔVS], and changes in Young's modulus (Y), Poisson's ratio (σ), bulk modulus (K), and rigidity (G) [ΔY/Δσ/ΔK/ΔG] before and after DC poling in soft and hard PZT, PbTiO3 (PT/PLT), alkali niobate (SZ), and alkali bismuth titanate (KBT/ BT). Higher kp was realized in the regions of large +ΔVL and +ΔK, and larger -ΔVS, -ΔY, and -ΔG. There were thresholds regarding kp vs. ΔVS, ΔY, and ΔG around -5%, -7%, and -10%, respectively. On the other hand, there were no thresholds in the cases of kp vs. ΔV<sup>L</sup> and ΔK, especially Δσ. As there were kp peaks regarding Δσ in hard and soft PZT ceramics and kp maximum in alkali niobate (SZ) ceramics, the compositions in kp peaks and kp maximum correspond to a morphotropic phase boundary (MPB) in PZT [25] and to take lowest value of VS/VL in SZ (see the following lines in Figure 3)

4 Fig.2 Figure 2 Replace with the figure below because of lack of

5 28 open marks and dash lines in Figure 3) red marks and dash lines in Figure 3)

6 Fig. 3 Figure 3 Replace with the figure below because of lack of

5 10 the case of after poling (closed marks and solid

18 21 (Figure 16(b) (Figure 16(b))

18 33 (Figure 13(b) (Figure 13(b))

20 3 at 1,300-1, at 1,300-

20 4 360℃ 1,360℃

18 33 (Figures 18(a)-(b) (Figures 18(a)-(b))

lines in Figure 3)

the case of after poling (black marks and solid

ρ

ρ <sup>G</sup> <sup>V</sup><sup>S</sup> <sup>=</sup>

sound velocities

VS/V<sup>L</sup>

ρ K 3 <sup>4</sup> <sup>G</sup>

+ =

(Hz・m)

VL

Bulk density (kg/m<sup>3</sup>)

letters (ρ) in the equations in G, Y and K. In this

σ

Y

Young's modulus (Pa)

( ) <sup>+</sup><sup>σ</sup> <sup>=</sup> <sup>12</sup> <sup>Y</sup> <sup>G</sup>

K

Bulk modulus (Pa)

G

 

− 

<sup>1</sup> <sup>V</sup> V <sup>1</sup> <sup>1</sup> <sup>2</sup> 1

2 S L

 

Elastic constants

Poisson's ratio (-) Ratio of

2 =ρVG <sup>S</sup>

V3Y <sup>−</sup>

2 <sup>L</sup> V 3

= ρ

 −= <sup>2</sup>

<sup>4</sup> <sup>ρ</sup> VK

 

 

−=σ

Rigidity (Pa)

−

2 S 2 L

> 

S

2 S 2 <sup>L</sup> <sup>2</sup> S VV V3 <sup>4</sup> <sup>V</sup>

figure, one of the titles is changed "Material

Dielectric and piezoelectric constants

constants" into "Elastic constants".

Coupling factor (%)

kP

Piezoelectric strain constant (C/N)

V<sup>L</sup>

Longitudinal wave velocity (m/s)

ε<sup>r</sup> d<sup>33</sup>

V<sup>S</sup>

Transverse wave velocity (m/s)

fc<sup>P</sup>

Frequency constant

Relative dielectric constant (-)

14 35 with FT vs. 2fcp[fcp is shown in Figure 13(c) with FT vs. 2fcp(fcp is shown in Figure 13(c)) **Figure 3.** VL, VS, Y, σ, G, and K vs. kp before\* (red marks, dash lines) and after poling\*\* (black marks, solid lines) in lead-containing and lead-free ceramics (r: correlation coefficient).

Figure 10). We believe that the origin of piezoelectricity in piezoelectric ceramics was due to large change in VS (-ΔVS) while applying DC poling field parallel to the thickness of disks. Therefore, the larger changes in VS (-ΔVS) correspond to larger changes in Y (-ΔY) and G (-ΔG). These phenomena mean that the origin of high piezoelectricity was due to the mechanical softness of the materials under compress stress (large +ΔVL and +ΔK). In addition, the realization of high piezoelectricity is easy deformation by DC poling field in diameter (large - ΔG) as well in thickness (large -ΔY). 17 10 (Fig. 14(c) (Fig. 14(c)) 17 13 (Fig. 14(d) (Fig. 14(d))

Figure 5 shows schematic charts between domain alignment and the changes in Y, σ, G, and K after DC poling. Comparing domain alignment before poling to the alignment after poling, same charges (+ or -) are generated and gathered in the regions of each ends by opposite charge due to DC poling field, namely orientation polarization which occurs by domain alignment. The orientation polarization acts by reducing of Y and G by repulsion to each other because there are same charges in domain alignment. The enhancing σ and K can be explained by the same phenomena. Therefore, it can be said that higher domain alignment leads to large changes in Y, G, and K. However, large change in σ (+Δσ in Figure 4) does not lead to higher domain alignment since σ value is decided by the combinations of Y (G) and K after poling. 17 15 (Fig. 14(d) (Fig. 14(d)) 17 16 (Fig. 13(a) (Fig. 13(a))

2

Origin of Piezoelectricity in Piezoelectric Ceramics from the Viewpoints of Elastic Constants Measured by… http://dx.doi.org/10.5772/60793 39

**Figure 4.** Changes (Δ) in VL, VS, Y, σ, G, and K vs. kp before and after poling in lead-containing and lead-free ceramics.

Figure 10). We believe that the origin of piezoelectricity in piezoelectric ceramics was due to large change in VS (-ΔVS) while applying DC poling field parallel to the thickness of disks. Therefore, the larger changes in VS (-ΔVS) correspond to larger changes in Y (-ΔY) and G (-ΔG). These phenomena mean that the origin of high piezoelectricity was due to the mechanical softness of the materials under compress stress (large +ΔVL and +ΔK). In addition, the realization of high piezoelectricity is easy deformation by DC poling field in diameter (large -

**Figure 3.** VL, VS, Y, σ, G, and K vs. kp before\* (red marks, dash lines) and after poling\*\* (black marks, solid lines) in

soft PZT hard PZT SZ KBT BT PLT PT

k<sup>P</sup> (%)

r=-0.91\* r=-0.93\*\*

r=-0.91\* r=-0.92\*\*

Alka li Nioba te

k<sup>P</sup> (%)

k<sup>P</sup> (%)

letters (ρ) in the equations in G, Y and K. In this

σ

Y

Young's modulus (Pa)

( ) <sup>+</sup><sup>σ</sup> <sup>=</sup> <sup>12</sup> <sup>Y</sup> <sup>G</sup>

K

Bulk modulus (Pa)

G

 

− 

<sup>1</sup> <sup>V</sup> V <sup>1</sup> <sup>1</sup> <sup>2</sup> 1

2 S L

 

Elastic constants

Poisson's ratio (-) Ratio of

2 =ρVG <sup>S</sup>

V3Y <sup>−</sup>

2 <sup>L</sup> V 3

= ρ

 −= <sup>2</sup>

<sup>4</sup> <sup>ρ</sup> VK

 

 

−=σ

Rigidity (Pa)

−

2 S 2 L

> 

S

1600 2000 2400 2800 3200

PZT

Alka li Nioba te

Alkali Bismuth Tita nate

> r=0.49\* r=0.74\*\*

r=0.89\* r=0.97\*\*

PLT, PT

V<sup>S</sup> (m/s)

0.2 0.25 0.3 0.35 0.4 0.45

σ (-)

6 8 10 12 14

K (10<sup>10</sup> N/m<sup>2</sup>)

2 S 2 <sup>L</sup> <sup>2</sup> S VV V3 <sup>4</sup> <sup>V</sup>

figure, one of the titles is changed "Material

Dielectric and piezoelectric constants

the case of after poling (black marks and solid

ρ

ρ <sup>G</sup> <sup>V</sup><sup>S</sup> <sup>=</sup>

sound velocities

VS/V<sup>L</sup>

ρ K 3 <sup>4</sup> <sup>G</sup>

+ =

(Hz・m)

VL

Bulk density (kg/m<sup>3</sup>)

lines in Figure 3)

letter (σ) in the figure of σvs. kp.

constants" into "Elastic constants".

Coupling factor (%)

kP

Piezoelectric strain constant (C/N)

V<sup>L</sup>

Longitudinal wave velocity (m/s)

ε<sup>r</sup> d<sup>33</sup>

V<sup>S</sup>

Transverse wave velocity (m/s)

fc<sup>P</sup>

Frequency constant

Relative dielectric constant (-)

Figure 5 shows schematic charts between domain alignment and the changes in Y, σ, G, and K after DC poling. Comparing domain alignment before poling to the alignment after poling, same charges (+ or -) are generated and gathered in the regions of each ends by opposite charge due to DC poling field, namely orientation polarization which occurs by domain alignment. The orientation polarization acts by reducing of Y and G by repulsion to each other because there are same charges in domain alignment. The enhancing σ and K can be explained by the same phenomena. Therefore, it can be said that higher domain alignment leads to large changes in Y, G, and K. However, large change in σ (+Δσ in Figure 4) does not lead to higher domain

alignment since σ value is decided by the combinations of Y (G) and K after poling.

2

ΔG) as well in thickness (large -ΔY).

15 30 45 60

5 10 15 20

Alkali Bismuth Titana te

PLT, PT

Alka li Nioba te

4000 4500 5000 5500 6000

Alkali Bismuth Titanate

PLT, PT

V<sup>L</sup> (m/s)

Y (10<sup>10</sup> N/m<sup>2</sup>)

G (10<sup>9</sup> N/m<sup>2</sup>)

lead-containing and lead-free ceramics (r: correlation coefficient).

k<sup>P</sup> (%)

PZT

PZT

38 Ferroelectric Materials – Synthesis and Characterization

k<sup>P</sup> (%)

> k<sup>P</sup> (%)

4 Fig.2 Figure 2 Replace with the figure below because of lack of

5 28 open marks and dash lines in Figure 3) red marks and dash lines in Figure 3)

6 Fig. 3 Figure 3 Replace with the figure below because of lack of

14 35 with FT vs. 2fcp[fcp is shown in Figure 13(c) with FT vs. 2fcp(fcp is shown in Figure 13(c))

17 10 (Fig. 14(c) (Fig. 14(c))

17 13 (Fig. 14(d) (Fig. 14(d))

17 15 (Fig. 14(d) (Fig. 14(d))

17 16 (Fig. 13(a) (Fig. 13(a))

18 21 (Figure 16(b) (Figure 16(b))

18 33 (Figure 13(b) (Figure 13(b))

20 3 at 1,300-1, at 1,300-

20 4 360℃ 1,360℃

18 33 (Figures 18(a)-(b) (Figures 18(a)-(b))

5 10 the case of after poling (closed marks and solid

lines in Figure 3)

**Figure 5.** Effect of ferroelectric domain alignment by DC poling on Y, σ, G, and K; generated charges (- → +) accompa‐ nied with domain alignments, displacement (→) and changes in Y, σ, G, and K (↘: decrease, ↗: increase) after poling are also shown in this figure.

### *1.3.2. Design for research and development on lead-free piezoelectric ceramics*

Figure 6 shows the relationship between VS/VL vs. kp. The kp linearly increased with decreasing VS/VL in lead-free ceramics as well as lead-containing ceramics such as PZT, PLT, and PT. Furthermore, it was confirmed that VS/VL was an effective figure to evaluate both the elastic constants and the piezoelectric constants.

**Figure 6.** Relationship between ratio of sound velocities (VS/VL) and planar coupling factor (kp) in lead-free and leadcontaining ceramics.

When we research and develop new piezoelectric ceramics with high piezoelectricity in leadfree ceramics, we must need a new concept different from the conventional research looking for chemical compositions such as MPB [25]. From the equations (1)-(4) and the change in VL and VS before and after poling as mentioned previously (Figure 4), we focused on the ceramic bulk density (ρ). Figure 7 shows the relationship VS/VL vs. ρ: ρ of lead-containing ceramics (PZT, PLT, and PT) was independent of VS/VL. However, in lead-free ceramics (SZ, KBT, and BT) ρ decreased with decreasing VS/VL. In the case of ρ vs. kp in Figure 8, although kp in leadcontaining ceramics (PZT, PLT, and PT) is independent of VS/VL, kp in lead-free ceramics (SZ, KBT, and BT) increased with decreasing ρ. From the above results, we came to an important concept to obtain lead-free ceramics with high piezoelectricity, namely the R&D on lead-free ceramics with lower bulk density. As a result, it confirmed the importance of measuring VL and VS to evaluate the piezoelectricity.

It was said that the direction of the R&D on lead-free piezoelectric ceramics with high piezoelectricity was looking for ceramics with lower bulk density. For example, in perovskite structure, small cations at A and B sites in perovskite structure of ABO3 were selected. In

*1.3.2. Design for research and development on lead-free piezoelectric ceramics*

constants and the piezoelectric constants.

40 Ferroelectric Materials – Synthesis and Characterization

0

10

20

30

**kP (%)**

and VS to evaluate the piezoelectricity.

containing ceramics.

40

50

60

70

Figure 6 shows the relationship between VS/VL vs. kp. The kp linearly increased with decreasing VS/VL in lead-free ceramics as well as lead-containing ceramics such as PZT, PLT, and PT. Furthermore, it was confirmed that VS/VL was an effective figure to evaluate both the elastic

0.35 0.40 0.45 0.50 0.55 0.60 0.65

**VS/VL**

soft PZT hard PZT SZ KBT BT PLT PT

**Figure 6.** Relationship between ratio of sound velocities (VS/VL) and planar coupling factor (kp) in lead-free and lead-

When we research and develop new piezoelectric ceramics with high piezoelectricity in leadfree ceramics, we must need a new concept different from the conventional research looking for chemical compositions such as MPB [25]. From the equations (1)-(4) and the change in VL and VS before and after poling as mentioned previously (Figure 4), we focused on the ceramic bulk density (ρ). Figure 7 shows the relationship VS/VL vs. ρ: ρ of lead-containing ceramics (PZT, PLT, and PT) was independent of VS/VL. However, in lead-free ceramics (SZ, KBT, and BT) ρ decreased with decreasing VS/VL. In the case of ρ vs. kp in Figure 8, although kp in leadcontaining ceramics (PZT, PLT, and PT) is independent of VS/VL, kp in lead-free ceramics (SZ, KBT, and BT) increased with decreasing ρ. From the above results, we came to an important concept to obtain lead-free ceramics with high piezoelectricity, namely the R&D on lead-free ceramics with lower bulk density. As a result, it confirmed the importance of measuring VL

It was said that the direction of the R&D on lead-free piezoelectric ceramics with high piezoelectricity was looking for ceramics with lower bulk density. For example, in perovskite structure, small cations at A and B sites in perovskite structure of ABO3 were selected. In

**Figure 7.** Relationship between ratio of sound velocities (VS/VL) and ceramic bulk density (ρ) in lead-free and lead-con‐ taining ceramics.

addition, for practical use, the Curie point (Tc) or depolarization temperature must be over 250 ℃. From the two items we will expect the candidates for new lead-free ceramics such as SrTeO3 (ρ = 4.82 g/cm<sup>3</sup> , Tc = 485 ℃) and YMnO<sup>3</sup> (Tc = 640 ℃), in addition to KNbO3 (ρ = 4.62 g/ cm3 , Tc = 418 ℃) and LiNbO3 (ρ = 4.46 g/cm<sup>3</sup> , Tc = 1, 210 ℃), respectively [26, 27]. In fact, the bulk density (ρ) of SZ, which possessed the highest kp [5, 8] in lead-free ceramics we investi‐ gated, was around 4.6 g/cm<sup>3</sup> .

**Figure 8.** Relationship between ceramic bulk density (ρ) and planar coupling factor (kp) in lead-free and lead-contain‐ ing ceramics.

### *1.3.3. Ferroelectricity in almost same ceramic bulk density*

In addition to the relationships between VS/VL vs. kp in Table 1, we evaluated the chemical composition dependence of VS/VL in piezoelectric ceramics. Figure 9 shows the composition *x* dependence of kp and d33 in (1-*x*)(Na, K, Li, Ba)(Nb0.9Ta0.1)O3-*x*SrZrO3 (SZ) (*x* = 0.00, 0.02, 0.04, 0.05, 0.06, 0.07). While the kp in SZ compositions of *x* = 0.00 - 0.07 was independent of ρ in Figure 8, the maximum kp and d33 were obtained at *x* = 0.04 and 0.05, respectively (Figure 9). Figure 10 shows the *x* dependence of VS/VL in SZ before and after poling. The minimum VS/VL after poling were obtained at *x* = 0.04 - 0.05. From both figures, it was concluded that the highest kp appeared in the case of the minimum VS/VL. However, there is a composition without MPB at *x* = 0.04 - 0.05 in SZ [5, 8]. Therefore, we could introduce the novel method to evaluate the piezoelectricity by measuring acoustic wave velocities of VL and VS in spite of the existence of MPB.


**Table 1.** Relationships between longitudinal wave velocity (VL), transverse wave velocity (VS), ratio of sand velocities (VS/VL), ceramic bulk density (ρ), and planar coupling factor (kp) in lead-free and lead-containing ceramics.

**Figure 9.** Composition *x* dependence of kp and d33 in (1‒*x*)(Na, K, Li, Ba)(Nb0.9Ta0.1)O3-*x*SrZrO3 (SZ) (*x* = 0.00, 0.02, 0.04, 0.05, 0.06, 0.07).

Origin of Piezoelectricity in Piezoelectric Ceramics from the Viewpoints of Elastic Constants Measured by… http://dx.doi.org/10.5772/60793 43

*1.3.3. Ferroelectricity in almost same ceramic bulk density*

42 Ferroelectric Materials – Synthesis and Characterization

Longitudinal

Transverse

Ceramic bulk density (ρ)

**kP (%)**

0.05, 0.06, 0.07).

of MPB.

In addition to the relationships between VS/VL vs. kp in Table 1, we evaluated the chemical composition dependence of VS/VL in piezoelectric ceramics. Figure 9 shows the composition *x* dependence of kp and d33 in (1-*x*)(Na, K, Li, Ba)(Nb0.9Ta0.1)O3-*x*SrZrO3 (SZ) (*x* = 0.00, 0.02, 0.04, 0.05, 0.06, 0.07). While the kp in SZ compositions of *x* = 0.00 - 0.07 was independent of ρ in Figure 8, the maximum kp and d33 were obtained at *x* = 0.04 and 0.05, respectively (Figure 9). Figure 10 shows the *x* dependence of VS/VL in SZ before and after poling. The minimum VS/VL after poling were obtained at *x* = 0.04 - 0.05. From both figures, it was concluded that the highest kp appeared in the case of the minimum VS/VL. However, there is a composition without MPB at *x* = 0.04 - 0.05 in SZ [5, 8]. Therefore, we could introduce the novel method to evaluate the piezoelectricity by measuring acoustic wave velocities of VL and VS in spite of the existence

wave velocity (VL) VL↗ kP↗ VL; independent of kP

VS/VL VS/VL↘ kP↗ VS/VL↘ kP↗

wave velocity (VS) VS↘ kP↗ VS↘ kP↗

VS/VL↘ ρ↘ ρ↘ kP↗

**Table 1.** Relationships between longitudinal wave velocity (VL), transverse wave velocity (VS), ratio of sand velocities (VS/VL), ceramic bulk density (ρ), and planar coupling factor (kp) in lead-free and lead-containing ceramics.

d33=**261**pC/N

d33=**125**pC/N

0.00 0.02 0.04 0.06 0.08

d33=**302**pC/N

d33=**331**pC/N

**x (mole fraction)**

**Figure 9.** Composition *x* dependence of kp and d33 in (1‒*x*)(Na, K, Li, Ba)(Nb0.9Ta0.1)O3-*x*SrZrO3 (SZ) (*x* = 0.00, 0.02, 0.04,

Lead-free Lead-containing

ρ; independent of VS/VL and kp

d33=1**31**pC/N

d33=**280**pC/N

**Figure 10.** Composition *x* dependence of ratio of sound velocities (VS/VL) in (1‒*x*)(Na, K, Li, Ba)(Nb0.9Ta0.1)O3-*x*SrZrO3 (SZ) (*x* = 0.00, 0.02, 0.04, 0.05, 0.06, 0.07).

We applied this novel method to 0.05Pb(Sn0.5Sb0.5)O3-(0.95-*x*)PbTiO3-*x*PbZrO3 (*x* = 0.33, 0.45, 0.48, 0.66, 0.75) with (hard PZT) and without 0.4 wt% MnO2 (soft PZT) to confirm the effec‐ tiveness of our developed VS/VL evaluation. While kp in PZT compositions of *x* = 0.33 - 0.75 was independent of ρ in Figure 8, the maximum kp was obtained at MPB around *x* = 0.45 - 0.48 [11, 21]. Figure 11 shows the *x* dependence of VS/VL in PZT before and after poling. The minimum VS/VL before and after poling appeared around *x* = 0.45 - 0.48. From the relationships between *x* vs. kp and VS/VL, it was concluded that the highest kp was realized in the case of minimum VS/VL. These compositions around *x* = 0.45 - 0.48 correspond to MPB in hard and soft PZT. Therefore, it was said that we could introduce the novel method to evaluate the piezoelectricity by measuring acoustic wave velocities of VL and VS in the compositions with MPB.

### *1.3.4. Relationship between the ratio of transverse wave velocity to longitudinal wave velocity and Poisson's ratio*

Figure 12 shows the relationship between VS/VL vs. σ in solids including piezoelectric ceramics. The equation between σ, VL, and VS is shown in this figure. All of the σ in solids was plotted on a line of the equation. The VS/VL regions of Poisson's ratio in soft PZT, hard PZT, lead-free and PbTiO3 (PLT and PT) were also shown in this figure. The regions of σ regarding VS/VL increased from PbTiO3, lead-free, hard PZT to soft PZT with increasing piezoelectricity. In addition to higher Poisson's ratio, it was clarified that higher kp can be realized in lager K as shown in Figure 3. We believe the physical meaning of this behavior toward σ is as follows: increasing mechanical softness (lower Y) in piezoelectric materials, it becomes easy to deform by DC poling field. The σ becomes larger while the materials become softer, and furthermore, the K must become larger in order to transmit effectively from longitudinal deformation (the directions of poling and applying electric field are thickness direction of disk) to transverse deformation (the radial direction of disk). In this study, it was described a road map in

k<sup>P</sup> (%)

18 9 firing at 1,\_300-1,\_360℃ firing at 1,300-1,360℃

70

k<sup>P</sup> (%)

letter (σ) in the figure of σvs. kp.

5 10 15 20

Y (10<sup>10</sup> N/m<sup>2</sup> )

Figure 23 Replace with the figure below because oflack of

22 Fig.

12 Fig.

12

23

k<sup>P</sup> (%)

k<sup>P</sup> (%)

0.2 0.25 0.3 0.35 0.4 0.45

σ (-)

**Figure 11.** Composition *x* dependence of ratio of sound velocities (VS/VL) in 0.05Pb(Sn0.5Sb0.5)O3-(0.95‒*x*)PbTiO3 *x*PbZrO3 (*x* = 0.33, 0.45, 0.48, 0.66, 0.75) with (hard PZT) and without 0.4 wt% MnO2 (soft PZT) ceramics before and after poling. Figure 12 Replace with the figure as below because of lack of letter (σ) in the figure of VS/VL vs.σ.

4 2 the material constants the elastic constants **Figure 12.** Relationship between ratio of sound velocities (VS/VL) and Poisson's ratio (σ) in solids including piezoelec‐ tric ceramics.

piezoelectric ceramics regarding the relationships between longitudinal and transverse wave velocities, Young's modulus, Poisson's ratio, modulus of rigidity and bulk modulus to research and develop new piezoelectric ceramic materials, especially lead-free ceramics with high piezoelectricity. 4 5 material constant in "the caption of Figure 2" elastic constants

3
