4.1 Piezoelectricity
57
Pb
2+
O
2−
Ti
4+
T < TC
T > TC
P
Fig. 4.1 The configurations of PbTiO 3 crystalline structure
P 3
ΔP 3
P 3
ΔP 1
ΔP 3
P 3
E 3
P 3
E 1
E 3
P 3
P 3
σ 3
σ 3
ε 1 (ε 2 )
ε 3
ε 5 (ε 4 )
σ 5 (σ 4 )
σ 1
(σ 2 )
(σ 2 )
σ 1
T < T C
P
Direct piezoelectric effect
Converse piezoelectric effect
x3
x1
x2
x3
x3
x1(x2)
x1(x2)
Fig. 4.2 The direct and converse effects of piezoelectric material
shifted to one side of the crystalline structure when the temperature is below the
Curie point. As a consequence, the center of the positive electric charges of the unit
cell is different from that of the negative ones. The crystal is then called polarized.
The piezoelectric material has two effects, namely the direct and converse effects,
which are shown in Fig. 4.2. Applying a stress in direction x 1 , it will decrease the
distance between the ion of titanium and the geometric center of the unit cell. This
can be understood as an additionally generated polarization, which results in extra
electric charges due to the stresses. Similarly, applying a normal stress σ 33 or shear
stress σ 13 , one produces electric charges as well. Those phenomena are called direct
piezoelectric effect, which can be expressed separately as
ΔP 1 = d 15 σ 5 ,
ΔP 2 = d 24 σ 4 ,
ΔP 3 = d 31 σ 1 + d 32 σ 2 + d 33 σ 3 ,
(4.1)
where ΔP i denotes the extra polarization in x i direction.
In an analogous way, the physical meaning of the converse piezoelectric effect
can be observed. Applying an electric field along the polarization direction will move
the ion of titanium off the center in x 3 direction. This will result in stretching the cell
along direction x 3 and squeezing along direction x 1 and x 2 , which yields additional
57
Pb
2+
O
2−
Ti
4+
T < TC
T > TC
P
Fig. 4.1 The configurations of PbTiO 3 crystalline structure
P 3
ΔP 3
P 3
ΔP 1
ΔP 3
P 3
E 3
P 3
E 1
E 3
P 3
P 3
σ 3
σ 3
ε 1 (ε 2 )
ε 3
ε 5 (ε 4 )
σ 5 (σ 4 )
σ 1
(σ 2 )
(σ 2 )
σ 1
T < T C
P
Direct piezoelectric effect
Converse piezoelectric effect
x3
x1
x2
x3
x3
x1(x2)
x1(x2)
Fig. 4.2 The direct and converse effects of piezoelectric material
shifted to one side of the crystalline structure when the temperature is below the
Curie point. As a consequence, the center of the positive electric charges of the unit
cell is different from that of the negative ones. The crystal is then called polarized.
The piezoelectric material has two effects, namely the direct and converse effects,
which are shown in Fig. 4.2. Applying a stress in direction x 1 , it will decrease the
distance between the ion of titanium and the geometric center of the unit cell. This
can be understood as an additionally generated polarization, which results in extra
electric charges due to the stresses. Similarly, applying a normal stress σ 33 or shear
stress σ 13 , one produces electric charges as well. Those phenomena are called direct
piezoelectric effect, which can be expressed separately as
ΔP 1 = d 15 σ 5 ,
ΔP 2 = d 24 σ 4 ,
ΔP 3 = d 31 σ 1 + d 32 σ 2 + d 33 σ 3 ,
(4.1)
where ΔP i denotes the extra polarization in x i direction.
In an analogous way, the physical meaning of the converse piezoelectric effect
can be observed. Applying an electric field along the polarization direction will move
the ion of titanium off the center in x 3 direction. This will result in stretching the cell
along direction x 3 and squeezing along direction x 1 and x 2 , which yields additional