Page 373 - Book Hosokawa Nanoparticle Technology Handbook

6.4 ELECTRIC PROPERTIES FUNDAMENTALS
An increase in I leads to a higher tan . The dielectrics to a share strain S and S (S S , the proportionality
5
4
5
4
R
with a relatively high resistance exhibit a tan less than factor is d ).
15
0.05. A tan higher than 0.5 interferes with precise
measurements of I , i.e., . A tan higher than 1 which 6.4.1.5 Domain structure and domain probing
C
has often observed for high-temperature dielectric In general, two kinds of domain walls are observed
properties is due to a higher electrical conductivity for tetragonal ferroelectrics: one is 180 domain wall
(higher I ). Except for ferroelectric materials, the main across which the direction of P is different by 180
s
R
factor of R is leakage current. The dielectric loss of fer- (antiparallel), and the other is a 90 domain wall
roelectrics is influenced by the movement of domain across which the P vector rotates by 90 . The mini-
s
walls as well as by leakage current, as described later. mization of strain energy results in the 90 domain
A peak in the temperature dependence of tan near T C structure, while the minimization of electrostatic
is ascribed to domain wall motion. energy requires the 180 domain structure. The
domain walls in the 90 and 180 domain structures
6.4.1.4 Piezoelectric constant have usually charge-free configurations. The domain
–2
In elastic bodies, an application of stress T (N m ) gen- in which the P vector is parallel to the direction of
s
erates a strain S (dimensionless). In the case of piezo- observation is called “c domain”, while the domain
electrics with a non-centrosymmetric crystal structure, with a P vector normal to it is “a domain”. For PFM
s
T induces not only S but also polarization (P). Using observations, the c domains are visualized by the out-
–1
piezoelectric constant d (C N ) and elastic compliance of-plane mode, while the a domains are revealed by
2
–1
s (m N ), we express the piezoelectricity (d type) as the in-plane mode.
Here, let us consider the piezoelectric vibration
D T E dT (6.4.6) induced by applying an E. For longitudinal vibration,
the domains with a P parallel to E elongate in pro-
s
portion to d , and the domains with a P antiparallel
E
S dE s T 33 s
to E shrink. When the direction of E is away from the
T
where denotes dielectric constant ( ) with a con- P vector, E induces thickness-share vibration in pro-
s
0 ij
E
stant T. Similarly, s shows s with a constant E. Since portion to d 15 as well as longitudinal one. For the
S and T are second-rank tensors, the coefficient s thickness-share vibration, the slip direction of the
between them becomes fourth-rank tensor. The coef- crystal surface depends on the direction of P .
s
ficient d between T and first-rank tensor D is a third- PFM can visualize the P vector from the piezo-
s
rank tensor. Here, only d is focused because the signal response signal detected by the tip of the cantilever
of PFM is directly related to d. of SPM. Fig. 6.4.4 shows the schematic representa-
Considering the crystal symmetry (P4mm) of tetrag- tion of the PFM measurement system used in this
onal BaTiO and PbTiO , we can simplify d as follows: study. An ac voltage is applied between the conduc-
3 3 tive cantilever and bottom electrode and induces
piezoelectric vibration just beneath the tip of the
0 ⎡ 0 0 0 d 0⎤ cantilever. The cantilever acts as a detector of the
⎢ 15 ⎥ piezoelectric vibration as a result of the deflection
d 0 0 0 d 15 0 0 ⎥ (6.4.7) or torsion of the cantilever as well as the top elec-
⎢
⎣ d ⎢ 31 d 31 d 33 0 0 0 ⎥ ⎦ trode. In theory, local dielectric constant such as d 33
and d 15 can be determined by PFM investigations.
where d , d and d are the individual parameters rep- When E is parallel to P , the cantilever deflects ver-
s
31
15
33
resenting piezoelectric response. For piezoelectric tically, i.e., parallel to E, due to piezoelectric longi-
ceramics, the poling direction is set to be the “3” direc- tudinal vibration (the out-of-plane mode in
tion and the properties along the “1” and “2” axes are Fig. 6.4.4a). For the domains in which the direction
the same. Thus, the dielectric permittivity and piezo- of P is parallel to the crystal surface, i.e., E normal
s
electric d constant are described by equations (6.4.4) to P , thickness-share vibration induces the torsion
s
and (6.4.7), respectively. If T 0, equation (6.4.6) is of the cantilever (the in-plane mode in Fig. 6.4.4b).
simplified as S d E , S d E , S d E , S The combined observations of the out-of-plane and
31 3
4
33 3
1
31 3
2
3
d E , S d E . For tetragonal BaTiO crystals the in-plane PFM images enable us to obtain a three-
15 1
5
15 2
3
–1
values of d 35 pC N , d 86 pCN –1 and dimensional domain structure near the crystal
31 33
–1
d 590 pC N have been reported [13]. When an E surface [14].
15
is applied along the direction of P (the “3” axis), the
s
crystals elongate along the “3” direction by S (the 6.4.1.6 Domain structure of PbTiO crystals observed
3
3
proportionality factor is d ) while the crystals shrink by PFM
33
along the “1” and “2” axes by S ( S , the propor- Fig. 6.4.5 represents the example of the domain
2
1
tionality factor is d , and d is a negative constant). structure of PbTiO crystals observed by PFM [5]. In
3
31
31
The application of E normal to the P direction the measurements, an E is applied along the a or c
s
induces thickness-share piezoelectric vibration, leading axis. Comparative investigations of the amplitude (A)
347