Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications

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Handbook of dielectric, piezoelectric and ferroelectric materials630

for precise measurement of the period and duty cycle of the engineered

periodic domain patterns in LN and LT using (HF) acid-based etchants (Ohnishi

and Iizuki, 1975; Alekseeva et al., 1986; Mizuuchi and Yamamoto, 1992;

Sones et al., 2002). The observation of revealed domain pattern can be

carried out by optical or scanning electron microscopy (SEM). Moreover,

the etch relief can be measured with high spatial resolution by the topographic

mode of SPM. Nevertheless, the interpretation of obtained images must be

done with care, because the etching relief for nanoscale domains can be

essentially different from the domain pattern due to strong anisotropy of

etching rate in the Y direction.

It is generally accepted that the etching procedure does not significantly

affect the domain structure. Thus, the etched relief structure must correlate

one-to-one with the existing stable domain pattern. It has been demonstrated

recently that this statement is not always true (Shur et al., 2005c). It has been

shown experimentally using SPM measurements that in MgO-doped

stoichiometric LT (MgO:SLT) crystals, contrary to current opinion, the domain

walls can be shifted considerably (about tenths of micrometers) from their

initial positions during etching. It has been found that the wall motion proceeded

even after the termination of the etching procedure. It has been revealed that

the whole consequence of the domain wall positions during etching is recorded

in the etching relief height and can be extracted with high spatial and temporal

resolution (Shur et al., 2005c). The main drawback of the etching technique

is that it is a destructive method, which is useless for in situ study of the

domain kinetics, and gives information only about the surface domain structure.

The SPM methods can be used for visualization of the static domain

patterns with nanoscale resolution even without preliminary etching. PFM

analyzes the piezoelectric response of the volume fraction mechanically

deforming under the locally applied ac electric field. Since any ferroelectric

is piezoelectric, it is possible to obtain a two-dimensional (2D) map of the

complete 3D piezoelectric tensor, which gives insight to the different

ferroelectric domain structures. In the case of investigated LN and LT possessing

the simple domain structure with 180° walls only the scanning of the polar

face allows one to distinguish the domains with opposite polarization sign

by recording the phase of the signal. The domain walls can be visualized by

recording the signal amplitude.

The widely accessible optical methods which present the unique possibility

of in situ observation of domain evolution using polarizing microscope are

the most informative and suitable (Burfoot and Taylor, 1979). These methods

are non-invasive, if the properly chosen intensity and wavelength of illumination

do not affect the domain kinetics. Even super-fast switching can be investigated

by using pulse lighting and high-speed photography (Malozemoff and

Slonczewski, 1979; O’Dell, 1981). The time resolution is determined by the

light pulse duration and can be above 10ns. The domains separated by 90°

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Nano- and micro-domain engineering 631

walls can be easily distinguished by polarizing microscopy due to the different

values of spontaneous birefringence (Blattner et al., 1948; Matthias and

Hippel, 1948). This effect has been used to study many ferroelectrics. But it

is unusable for antiparallel domains with 180° walls because of the identity

of optical indicatrix in neighboring domains. Some special techniques have

been developed for optical visualization of such domains.

In BaTiO

antiparallel domains become optically distinguished when the

electric field (Little, 1955; Merz, 1956) or corresponding mechanical forces

(Hooton and Meiz, 1955) are applied perpendicularly to the polar axis. The

rotation of the optical axis in opposite directions in different domains leads

to a difference between extinction positions of antiparallel domains, allowing

their visualization.

Miller and Savage (1959) and Savage and Miller (1960) have shown that

the antiparallel domains in BaTiO

can be observed in transmitted polarized

light during polarization reversal in a longitudinal field. Kobayashi et al. (1963)

showed that in this case domains are distinguishable only under the illumination

directed at small angles to the polar axis and explained this effect by the

existence of strain-induced birefringence in the vicinity of domain walls.

The linear electro-optic effect is the versatile property making possible

optical visualization of domains and/or domain boundaries. In the case of

anti-parallel 180° domains the electric field applied along the optical axis

induces the difference in refractive indices in Z+ and Z–domains. In LN and

LT the perturbation of the refractive index in the area near the domain walls

is induced by extremely high internal bias field. This effect allows the domains

to be visualized using polarized and phase contrast modes of the optical

microscopy. This also presents the unique opportunity of visualizing the

domains with charged domain walls embedded in the bulk. The drawback of

the optical methods imposed by the diffraction limiting the spatial resolution

to 0.5µm is outweighed by the ability to realize a high temporal resolution.

A promising new integral optical technique is based on the physical

phenomenon of light deflection at the domain boundaries. In these experiments,

a partially deflected laser beam produces, in the far-field, the light pattern

which carries information about the orientation of the domain walls. In LN

the light pattern is a six-point star, because in this material the individual

domains are mainly hexagons. The same experiments carried out in LT

demonstrate a three-point star pattern, because the observed domain shapes

are usually triangles. This method can be used for the statistical characterization

of the wall orientation in complicated multidomain patterns.

21.4 General consideration

In this chapter all stages of the domain structure evolution will be considered

from a unified point of view as a manifestation of the various nucleation

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Handbook of dielectric, piezoelectric and ferroelectric materials632

processes. This commonly accepted approach was proposed and developed

by Fatuzzo and Merz (1967) and Miller and Weinreich (1960). The nucleation

and growth approach is widely used for the description of the kinetics of

aggregative state changes during first-order phase transformation, e.g.

crystallization and melting (Shur, 2005). In our case the coexisting domains

with different orientation of the spontaneous polarization are considered as

an analogue of volumes occupied by different phases, while the domain

walls represent the phase boundaries. It has been shown by direct experiments

that the width of ferroelectric walls is a few unit cells (Lin and Bursill, 1982;

Bursill and Lin, 1986). This result supports the consideration of the domain

structure evolution as a discrete nucleation process. Following this approach

the domain kinetics during all the above-mentioned stages has to be considered

as a result of numerous elementary processes of thermally activated generation

of nuclei with a preferred orientation of the spontaneous polarization.

21.4.1 The elementary nucleation processes

The domain kinetics, similar to any first-order phase transformation kinetics,

is achieved through competitive formation of nuclei of three different

dimensionalities (Fig. 21.3). Each nucleus represents the nanoscale domain

with changed orientation of the spontaneous polarization.

The appearance of the optically observed new domains (‘nucleation of

new domains’) is a result of the expansion of either newly formed or residual

3D nanoscale nuclei. Such ‘invisible expansion’ occurs by 1D- and 2D-

nucleation (Fig. 21.3). In the real systems the heterogeneous nucleation

takes place during first-order phase transformation, and the position of the

nucleation sites is determined by imperfections and point defects. The

predominance of the nucleation at the sample surface is the special feature of

ferroelectrics caused by the existence of the intrinsic surface dielectric gap

playing the role of the ‘defect plane’.

(a) (b)

dep

scr

21.3

Schemes: (a) of the nucleation processes of various

dimensionalities, (b) of the ferroelectric with intrinsic dielectric

surface layer.

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The growth of visible domains by motion of the walls (‘sideways domain

growth’) is achieved through 1D- and 2D-nucleation, as shown in Fig. 21.3.

The elementary steps with the thickness about one unit cell are generated by

2D-nucleation at the wall. The subsequent motion of the steps along the wall

is a result of 1D-nucleation. The crystal symmetry imposed the restrictions

on the orientation of the elementary steps thus leads to experimentally observed

anisotropy of the domain growth. The shape of isolated domains usually

observed during ‘equilibrium’ switching conditions is defined by the crystal

symmetry: e.g. square domains are formed in tetragonal phase of BaTiO

(Iona and Shirane, 1962) and hexagonal domains – in LN (C

or R

symmetry)

(Prokhorov and Kuzminov, 1990).

The nucleation probability determines the rates of all switching processes.

For any first-order phase transformation it is determined by the driving

force value. The electric field averaged over the volume of the order of the

nucleus size (named in the following as ‘local field’, E

loc

) plays the role of

the driving force of nucleation during polarization reversal in ferroelectrics

(Shur, 1996). E

loc

is macroscopic electric field obeying macroscopic

electrodynamics equations with proper boundary conditions (Lifshitz et al.,

1985). E

loc

is essentially spatially inhomogeneous and changes during

polarization reversal. It is necessary to point out that E

loc

has nothing in

common with the local field which is analyzed while discussing the interaction

of the crystal lattice ions.

loc

in a ferroelectric capacitor is the sum of: (1) the external field applied

by electrode pattern, (2) the depolarization field produced by bound charges

developing as a result of spatial inhomogeneity of the spontaneous polarization

in multidomain state, (3) the external screening field originating from the

redistribution of the charges at the electrodes, and (4) the bulk screening

field governed by various bulk screening processes.

The external field is determined by the shape, sizes, and period of the

electrode pattern. The field singularities in the surface layer at the electrode

edges due to the fringe effect play the crucial role in the nucleation of the

new domains. The switching process usually starts through formation of the

nuclei at the electrode edge or at the tips of the stripe electrodes. Due to the

fringe effect the real value of E

loc

initiating the switching process in single-

domain state (threshold field) essentially exceeds the field value calculated

as potential difference divided by sample thickness. The pronounced field

concentration in the vicinity of the ends and corners of the stripe electrodes

must be accounted for while creating the tailored domain structures (Shur,

2005).

The depolarization field E

dep

is produced by bound charges existing at the

polar surfaces of the sample and at the charged walls of the encountering

domains (‘head to head’ or ‘tail to tail’). The bound charges density (ρ

) is

determined by spontaneous polarization (P

) by the following equation:

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Handbook of dielectric, piezoelectric and ferroelectric materials634

= – div P

21.1

For single-domain plate of typical ferroelectrics the depolarization field strength

can reach 10

V/m. It is clear that E

dep

far exceeding the experimentally

observed threshold or coercive fields leads to break-up of the single domain

state and fragmentation of the domain structure into narrow domains of

opposite sign.

In the volume of the isolated domain of ellipsoid shape:

dep

= L P

/εε

21.2

where L is the depolarization factor, which is essentially reduced for the

elongated domains. Thus, the wedge and needle-like shapes are the most

favorable for new domains due to minimization of the depolarization energy.

The domain growth by nucleation at the existing domain wall is determined

by the value of E

dep

at the wall front in the interface between the dielectric

gap and the ‘ferroelectric bulk’. This field is essentially spatially inhomogeneous

and strongly depends on the distance from the wall (Drougard and Landauer,

1959). For the sideways domain growth stage E

dep

increases with domain

enlargement, thus decreasing the nucleation probability at the moving wall.

This effect leads to experimentally observed deceleration and termination of

the wall motion (Drougard and Landauer, 1959; Shur, 1996, 2005a).

The effect of E

dep

is reduced by screening processes. For effective screening

of E

dep

the complete switching can be achieved even in a comparatively low

electric field. Only a small shift of the existing domain walls is observed for

ineffective screening due to the screening retardation. The incomplete screening

of E

dep

is the main source of the spontaneous backswitching (‘flip back’)

representing the partial recovering of the initial domain state after the external

field switch off. Two types of screening processes must be distinguished:

external and bulk.

External screening in ferroelectric capacitors is due to redistribution of

charges in the electrodes (the current in external circuit). The screening rate

is determined by the characteristic time of the external circuit τ

(product of

effective resistance and capacity). Usually τ

ranges from a few nanoseconds

to microseconds. The external screening field E

scr

(r,t) effectively compensates

dep

(r,t) thus allowing the long-distance-shift of the domain wall and drastically

diminishes the value of the applied external field E

(r), which is necessary

for the complete switching. The experimentally observed switching time t

always exceeds τ

as the complete switching can be obtained only if the

depolarization field is sufficiently screened.

The fast external screening never completely compensates E

dep

due to the

existence of the intrinsic dielectric surface layer (‘dielectric gap’ or ‘dead

layer’), in which spontaneous polarization is absent (Fig. 21.3b) (Fridkin,

1980; Shur, 1996).

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Nano- and micro-domain engineering 635

For a ferroelectric capacitor of thickness d the bulk residual depolarization

field E

remains in the area freshly switched from one single-domain state

to another one even after complete external screening due to existence of the

intrinsic surface dielectric layer of the thickness L (Fridkin, 1980; Shur,

1996):

= E

dep

– E

scr

= 2L/d [P

/(ε

)] 21.3

where ε

is the dielectric permittivity of the surface dielectric gap.

is several orders of magnitude less than E

dep

, but it is of the order of

the experimentally observed threshold/coercive fields. Thus, the effective

compensation of E

by comparatively slow bulk screening processes is

necessary for complete switching. That is why bulk screening plays an essential

role in the evolution of the domain structure.

Bulk screening is the only way to compensate E

existing after completion

of the external screening. Three groups of the bulk screening mechanisms

are usually considered: (1) redistribution of the bulk charges (Fridkin, 1980;

Shur, 1996), (2) reorientation of the defect dipoles (Lambeck and Jonker,

1986), (3) injection of carriers from the electrode through the dielectric gap

(Tagantsev et al., 2001). All bulk screening mechanisms are considerably

slower than the external ones, with typical time constants τ

ranging from

milliseconds up to days and even months.

According to the above-discussed considerations the local electric field

being the driving force of all nucleation processes is spatially inhomogeneous

and time dependent. After completion of the external screening the local

electric field at the given point and at the given moment E

loc

(r,t) is the sum

of the external field E

(r), the residual depolarization field E

(r,t), and bulk

screening field E

(r,t) (Shur, 1996, 1998; Shur and Rumyantsev, 1997 Shur):

loc

(r,t) = E

(r) + E

(r,t) + E

(r,t) 21.4

That is why the application of the short field pulse, for which the bulk

screening of the new state lags behind, is absolutely ineffective for the

creation of a modified domain pattern. The cooperative action of E

and E

remaining after the external field is switched off leads to the spontaneous

backswitching process. This effect could be observed in the areas where the

value of E

loc

(r,t) exceeds the threshold field for the switching process E

i.e.:

loc

(r,t) = –[E

dep

(r,t) – E

scr

(r,t) + E

(r,t)] = – [E

(r,t) + E

(r,t)] > E

21.5

The initial domain state can even be completely reconstructed for ineffective

screening of the modified E

dep

produced by the domain pattern formed at the

end of the electric field pulse. The conservation of the new domain configuration

can be realized only when the duration of the external field pulse is long

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Handbook of dielectric, piezoelectric and ferroelectric materials636

enough for effective bulk screening. Thus for an appropriate duration of the

field application it is possible to stabilize almost any domain pattern.

The systematic study of the polarization reversal in various ferroelectric

single crystals reveals a great variety of the domain kinetics scenarios. The

main problem in their classification lies in the fact that it is impossible to

attribute the given scenario to a definite range of applied fields and switching

rates. The critical values of the fields and rates corresponding to a replacement

of one domain kinetics scenario by another differ drastically for different

materials and experimental conditions.

Basing on the above-discussed approach we have chosen the ratio between

switching rate (1/t

) and bulk screening rate (1/τ

scr

) for quantitative

characterization of the screening efficiency. This parameter is named screening

effectiveness R (Shur, 2005, 2006(a)):

R = τ

scr

21.6

Three ranges for screening effectiveness must be distinguished: (1) R << 1

– ‘complete screening’, (2) R ~ 1 – ‘incomplete screening’ and (3) R >> 1 –

‘ineffective screening’. It was confirmed both experimentally and by computer

simulation that the proposed ranges correspond to qualitatively different

switching regimes (scenarios of domain kinetics).

The complete screening corresponds to quasi-equilibrium switching

conditions when the reconstruction of the subsystem of the bulk screening

charges is fast enough to keep pace with the changes of the domain structure.

The retardation of the bulk screening makes the switching conditions non-

equilibrium. Thus the equilibrium switching conditions corresponds to R

<< 1 and the highly non-equilibrium ones to R >> 1.

Sideways domain wall motion is a result of step generation (2D nucleation)

and subsequent step growth (1D nucleation) along the wall (Fig. 21.3a). The

shape and orientation of the moving wall strongly depend on the spatial

distribution of the nucleation sites. Two limiting variants can be considered:

stochastic and predetermined.

Stochastic nucleation with random position of the nucleation sites at the

wall is the classical approach to the problem of the wall motion (Miller and

Weinreich, 1960b; Hayashi, 1972). This approach assumes that the switching

field is homogeneous and the step generation probability is independent of

the position at the wall. As a result, the shape of the moving wall becomes

irregular. In this case the deceleration of the wall by local pinning centers is

ineffective. The wall easily flows around the pinning center, because the

decrease of the nucleation probability occurs only in the vicinity of the

center, while the rest of the wall continues to move. The shape of individual

domains under stochastic nucleation is usually irregular and does not follow

the crystal symmetry.

However, in many ferroelectrics switched under usual experimental

conditions the domains demonstrate regular shape with walls strictly oriented

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Nano- and micro-domain engineering 637

along definite crystallographic directions. In LN the domains demonstrate

hexagon shape with walls strictly oriented along Y crystallographic directions.

This behavior can be understood, if we assume that the predetermined

nucleation prevails.

Predetermined nucleation means that the generation of the steps (2D-

nucleation) along the wall is spatially inhomogeneous. The preferred nucleation

at the given sites determined by field singularities can be produced both

artificially and during growth of isolated domain.

For growth of isolated domain the switching field at the wall is decreased

due to incomplete screening of the depolarization field. For a polygon domain

shape this screening retardation leads to formation of the field singularities

at the vertices leading to local increase of the step generation probability. In

this case the predetermined nucleation at polygon vertices must be realized.

The position of the nucleation sites for hexagon domains in LN is shown in

the Fig. 21.4. The crystal symmetry imposes restrictions on the directions of

the growth of the elementary steps, thus leading to anisotropy of the domain

growth, which is observed experimentally (Shur, 2005). In LN crystal with

) symmetry the step propagation along three allowed directions

must prevail. The direct experimental observation of the domain kinetics

allows us to state that the steps propagate along the Y direction only (Shur,

2005, 2006). The steps generated at three Y non-adjacent vertices are growing

while the growth of the steps generated at three other vertices is completely

suppressed. The experimentally observed hexagon shape is obtained when

the step growth rate exceeds essentially the step generation rate.

The growth of the hexagon domain governed by predetermined nucleation

is demonstrated on the schematic drawing presented in Fig. 21.4. It is clearly

seen that the growing conditions for steps generated at the vertices A and B

are essentially different. The growth probability for steps generated at vertex

A must far exceed the probability at B, because the growth of the latter must

occur in the directions opposite to the preferable ones.

(a) (b)

21.4

Predetermined nucleation: (a) step generation at vertices and (b)

step growth in three

directions.

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Within this consideration we have shown by computer simulation that the

growing domain preserves its polygon shape, if the step generation probability

at the faces is neglected (Lobov et al., 2006). The additional experimental

evidence of the proposed approach was obtained while observing the

propagation of macro-steps along the wall in three Y directions only. These

micrometer-scale steps formed as a result of interaction of the moving wall

with the crystal defects can be easily visualized in situ by optical microscopy

(Shur, 2005, 2006).

21.4.2 Artificially produced nucleation sites

It is interesting to study the role of the predetermined nucleation on the

domain shape when the nucleation sites are produced artificially. The scratching

of the crystal polar surface is one of the possible ways to initiate the nucleation

in the given place. In this case the predetermined position of the nucleation

sites is caused by an artificial array of field concentrators produced by the

scratch. The slow switching using the solid liquid electrode of the sample

with a single scratch oriented along X axis produces the regular tetragons

(Fig. 21.5). It is clearly seen that the walls are strictly oriented along the Y

directions. Such behavior is an unequivocal experimental confirmation of

the prevailing role of the predetermined nucleation in LN.

21.5 Domain growth: from quasi-equilibrium to

highly non-equilibrium

21.5.1 Growth of isolated domains

It has been shown that the shape of individual domains in LN essentially

depends on experimental conditions and can differ from the above-discussed

30 µm

21.5

Tetragon domains in CLN. Predetermined nucleation initiated by

scratch oriented in

direction. Optical images of domains revealed

by etching.

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Nano- and micro-domain engineering 639

regular hexagon (Fig. 21.6). According to the mechanism of predetermined

nucleation it is possible to explain all shapes observed in LN and LT in wide

range of experimental conditions as a result of variation of the ratio between

rates of step generation and step growth. It is convenient to characterize the

step generation rate by experimentally measured vertex motion velocity V

gen

and the step growth – by motion velocity of the step edge V

. The hexagon

shape is obtained, if condition V

gen

<< V

is fulfilled.

It has been shown that the ratio between V

gen

and V

is the main parameter

which governs the shape of isolated domain. The simulation of the domain

growth for wide range of V

gen

values allows us to obtain the smooth

variation of the domain shapes from equilateral hexagon to stars with three

rays (Fig. 21.6f).

5 µm

(a)

(b)

5 µm

(c)

5 µm

(d) (e)

(f)

5 µm

10 µm

5 µm

(g)

gen

0 0.5 1 10 ∞

21.6

The domain shapes obtained as a result of switching in various

conditions in CLN: (a) hexagons, (b) irregular hexagon, (c), (d)

triangles, (d), (e) Mercedes stars, (f) star with three rays. Optical

images of domains revealed by etching. (a), (e) – dark field mode. (g)

Simulated dependence of domain shapes in LN on the ratio of step

generation to step growth rates.

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