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

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Handbook of dielectric, piezoelectric and ferroelectric materials760

correspond to polarization in-plane. Their origin is thus not polar but elastic.

This figure also illustrates how the domain width grows with the thickness

of the crystal.

The proportionality constant between domain size and film thickness is

(a)

Thickness

500 nm

(b)

400 nm

Thickness gradient

(increasing thickness

25.3

TEM images of 90° domains in single crystal films of BaTiO

The orientation of the polarization is in-plane, and hence the

domains are not due to depolarization but to stress (ferroelastic). The

films were not parallel-faced but wedge-shaped, so that there is a

gradient of thicknesses within the sample: the domains are wider in

the thicker side of the wedge, according to Kittel’s law. The figure

also illustrates the mechanisms for domain width change: continuous

change when the domain walls are perpendicular to the thickness

gradient(a), discrete bifurcation when they are parallel to it (b)[14].

Physical Society.

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Symmetry engineering and size effects in ferroelectric thin films 761

different for ferroelectrics and for ferromagnets, being usually bigger for the

latter [14]. In fact, the scaling factor is proportional to the width of the

domain wall [15–17]. It is thus possible to rewrite Kittel’s law by renormalizing

the square of the domain size by the domain wall thickness. If this is done,

all ferroics (whether ferromagnetic or ferroelectric) fall in the same parent

curve (see Fig. 25.3), meaning that the renormalized (adimensional) scaling

factor for 180 domains is the same for ferroelectrics and ferromagnets. This

scaling factor can be calculated analytically, and is [17]:

7 (3)

2.455

≡≅

25.3

(where χ

, χ

are the dielectric permittivities perpendicular and parallel to

the direction of the polarization and δ is the domain wall halfwidth ζ is

Reimains zeta function, ζ(3) ≅ 1.202). The interest of this equation is that it

provides a simple (though indirect) way of measuring domain wall thickness:

one needs only measure the domain size and the dielectric anisotropy instead.

For ferroelectrics, domain wall thickness turns out to be of the order of only

a few angstrom.

For ferroelectrics, Kittel’s law is valid down to a thickness of only a few

nanometres [8, 14]. Nevertheless, the energy cost of introducing domain

walls still causes the critical temperature (T

) of the ferroelectric material to

decrease [18]:

= 1 – π













25.4

Therefore, ferroelectricity will still disappear below a critical thickness (d

= π

25.5

This critical thickness is proportional to the domain wall half-width, which

in ferroelectrics is thin: about 2.5 Å for PbTiO

[17], for which d

turns out

to be of the order of only ~6 Å.

The above equations consider a ferroelectric where d>>w, so that the two

surfaces of the ferroelectric can be treated as independent. For very small

thicknesses this is not true, and the interaction between the two surfaces can

actually lower the depolarization field and thus prevent the formation of

domains [19].

Screening

The other mechanism for preventing depolarization is charge screening. This

can happen naturally when ionic impurities aggregate at the exposed surfaces

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Handbook of dielectric, piezoelectric and ferroelectric materials762

of a ferroelectric [9]. Likewise, if the ferroelectric has electrodes and these

are connected in short-circuit, the charges at the electrodes neutralize the

depolarization field. This, however, is true only for films with perfect electrodes.

In reality, electrodes have a finite Thomas–Fermi screening length, which

the depolarization field can penetrate [20]. Consequently, the depolarization

field is not completely screened in real systems. Calculations for monodomain

BaTiO

ferroelectric films with realistic SrRuO

electrodes predict a

depolarization-induced critical thickness of six unit cells [21]. These

calculations, however, do not incorporate the extra screening that comes

from the displacement of the ionic charges in the electrodes. Calculations

that incorporate these yield a considerably lower critical thickness of only

three unit cells [22].

The above works analyse either domain formation in unscreened

ferroelectrics, or screening in mono-domain ferroelectrics. The work of

Bellaiche and coworkers [23] looks at both effects simultaneously and shows

that, if domain formation is incorporated to realistic models of ultra-thin

films, the local structure of the dipoles within the domains shows new types

of arrangements, with rotational gradients of polarization as a function of

depth within the film. Such structures depend on both the screening efficiency

of the electrode and the strain state of the film.

25.2.2 The dielectric size effect

From the above discussion one might conclude that the size effect is irrelevant

for real ferroelectric film devices, which operate at bigger thickness (typically

tens or hundreds of nanometres), and have electrodes. Yet, in reality,

ferroelectric thin films are generally seen to behave differently from their

bulk counterparts even at such relatively large thicknesses. One of the

most ubiquitous effects is the ‘dielectric size effect’, i.e. the progressive

lowering of the dielectric constant in thin films with respect to their bulk

counterparts.

Bulk ferroelectric materials normally show a sharp dielectric anomaly at

the critical temperature of the ferroelectric transition. But, for thin films, the

peak essentially disappears and the dielectric constant can decrease by one

or two orders of magnitude, even for thicknesses of the order of hundreds of

nanometres. Quite often, too, the decrease in permittivity with decreasing

thickness is such that it can be quantitatively described as if there were a

low-permittivity ‘dead’ layer at the interface between the dielectric and the

electrode. The ‘dead layer’ model seems to hold down to thickness of only

a few nanometres [24] even when there is no evidence of the actual

existence of an actual dead layer chemically or morphologically distinct

from the rest of the film. There have been several proposed explanations for

the dielectric size effect: dead layers in the ferroelectric [25], finite screening

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Symmetry engineering and size effects in ferroelectric thin films 763

length in the electrode [26], Schotky barriers [27], phonon hardening [28],

dissimilar electrodes [29] and strain gradients [30] are but a few.

One challenge is to assess how much of the decrease in dielectric constant

is intrinsic, and thus impossible to get rid of, and how much is extrinsic. This

was addressed by the group of Marty Gregg in Belfast, who analysed the

dielectric behaviour of free-standing ferroelectric single crystal lamellae

only 70 nm thick [31]. The dielectric behaviour was found to be essentially

identical to that of the bulk crystal (see Fig. 25.4), thus proving that the

dielectric smearing previously seen in other thin films was not due to intrinsic

properties of the films or the film–electrode interface, but rather to extrinsic

effects. Two strong contributors to the extrinsically generated dielectric size

effect are dissimilar electrodes [29] and strain gradients [30] which may

arise from epitaxial strain relaxation [30, 32] or from local strain gradients

caused by point defects [33].

Intrinsic effects can nevertheless still play a role, as recently shown by the

ab-initio calculations of the dielectric constant of SrTiO

with symmetric

SrRuO

electrodes [34], which show that the screening length in the electrodes

acts as a dielectric ‘dead layer’ of low dielectric constant in series with the

‘proper’ dielectric SrTiO

. Although the effect of this dead layer is big for

films with a thickness of a few unit cells, a film of 70nm (such as the one

experimentally analysed by Saad et al. [31], would experience a reduction in

(a)

(b)

250 300 350 400 450 500 550

Temperature (K)

Relative permittivity

Loss tangent (tan δ)

30000

20000

10000

0.015

0.010

0.005

25.4

Dielectric constant of a free-standing thin single crystal of

BaTiO

with a thickness of 70nm, by M. Saad

et al.

[31] The dielectric

response is identical to that of bulk. Figure from [31]. Reproduced

with permission from IOP Publishing Limited.

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Handbook of dielectric, piezoelectric and ferroelectric materials764

dielectric constant of only 15%, which is negligible in comparison with the

effect of the other extrinsic ingredients.

25.3 Heterogeneous thin films: superlattices and

gradients

25.3.1 Superlattices

Heterogeneity can be artificially introduced in ferroelectrics by, for example,

fabricating superlattices. These have been an object of very active research

over the last dozen years or so [35–45]. The initial impetus for fabricating

these structures was to try to replicate some of the advantages of relaxors

such as high dielectric constant with smooth temperature dependence. Initial

results were promising, as indeed the large, frequency-dependent dielectric

characteristics of relaxors were reproduced, roughly at the same stacking

periodicity as the naturally occurring heterogeneity of relaxors (a few

nanometres). However, most of these early results were later shown to be

probably influenced by conductive artefacts: oxygen vacancies may accumulate

at the interfaces between each slab [40–43], and thus the space charge dominates

the dielectric behaviour through the Maxwell–Wagner effect (see Fig. 25.5)

[39, 40].

More recent works are being more careful in trying to eliminate conductive

artefacts, and also are using alternative means of exploration which are not

affected by these, such as structural, rather than electric, characterization

techniques. Among the more sound results is the observation that when the

superlattice period is decreased, the material may have a transition in which

it goes from behaving as two separate components to behaving as a single

compound [38, 41]. The mechanisms for this inter-layer coupling are the

depolarization field [46] and strain [35, 42], which depend on material

parameters such as spontaneous polarization and lattice mismatch, as well as

on the stacking periodicity. The interplay between these parameters has also

yielded several new and unexpected phenomena, such as an orthorhombic

phase in normally cubic STO when it is superlatticed with BTO [42], and an

anomalous enhancement of polarization for small-period superlattices [45].

25.3.2 Gradients

Another type of heterogeneous structure is achieved by introducing a

compositional gradient across the thickness of a planar capacitor structure.

By growing films in which the composition is gradually changed from, say,

pure BaTiO

to pure SrTiO

, or from PbTiO

to PbZrO

, one introduces a

built-in bias field that self-poles the sample. This leads to rather unusual

behaviour of the polarization vs. field loops [47], although it is as yet unclear

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Symmetry engineering and size effects in ferroelectric thin films 765

how much of this is intrinsic and how much is due to conductivity artefacts

[48]. It seems that such graded ferroelectrics may present very large pyroelectric

[49] and piezoelectric coefficients [50] without the need to externally pole

the material.

Sometimes, however, the gradients are not deliberate but accidental. Thin

films grown on substrates with a different lattice parameter will try to

accommodate the strain. There are several ways to do this: twin domains

[51, 52], dislocations [52–54] and compositional segregation [55] are but a

few. It has been observed that strain relaxation may not always be homogeneous

throughout the thickness of the film, but rather it may take place in a gradual

100Hz

200Hz

500Hz

1kHz

2kHz

5kHz

10kHz

100Hz

200Hz

500Hz

1kHz

2kHz

5kHz

ε′

5000

4000

3000

2000

1000

ε′

5000

4000

3000

2000

1000

100 150 200 250 300 350 400

0.8

0.6

0.4

0.2

tan δ

2.0

1.5

1.0

0.5

50 100 150 200 250 300 350 400

Temperature (K)

50 100 150 200 250 300 350 400

Temperature (K)

25.5

Comparison between measured relaxor-like dielectric properties

in ferroelectric superlattices (above) and calculated properties

assuming Maxwell–Wagner behaviour due to interfacial space-charge

(below), showing that the relaxor-like properties are due to

conductive artefacts. Figures adapted from ref[40]. Reused with

permission from G. Catalan, D. O’Neill, R. M. Bowman, and J. M.

Gregg,

Applied Physics Letters

American Institute of Physics.

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Handbook of dielectric, piezoelectric and ferroelectric materials766

manner, leading to vertical strain gradients across the dielectric layer [32,

55]. Such strain gradients break the inversion symmetry, and can couple to

the ferroelectric properties through the so-called flexoelectric effect [56–

58]. Because strain gradients break the inversion symmetry, flexoelectric

polarization can actually be induced in any material, even those that are not

ferroelectric. The introduction of flexoelectricity in the thermodynamic potential

leads to an equation of state of the form [30]:

aP bP +

= +

25.6

where E is the electric field, P is the polarization, µ is the flexoelectric

coefficient, χ the permittivity and dε/dz is the strain gradient; a and b are the

first two coefficients of the Landau thermodynamic potential. From this

equation it can immediately be seen that the effect of strain gradient is

analogous to that of an external DC electric field. Accordingly, two effects

can be predicted:

1 There will be a preferential direction of the polarization in the film (self-

poling or imprint) [59].

2 In much the same way that a dc voltage would lowers the capacitance in

a C(V) measurement, the strain gradient lowers the capacitance and

smears the dielectric peak. Calculations based on this idea reproduce

experimental results very well, as shown in Fig. 25.6 [32].

2000

1000

2000

1000

Calculated

Measured

200 300 400

(K)

400

350

300

250

Calculated

Measured

250 500 750

(nm)

25.6

Comparison between experimental dielectric properties of (Ba,

Sr) TiO

thin films and flexoelectric calculations using the strain

gradients measured by X-ray diffraction in the same samples.

American Physical Society

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Symmetry engineering and size effects in ferroelectric thin films 767

In all the above discussions, we have deliberately left out what is possibly

the most important factor in the rationalization of ferroelectric thin film

properties: the biaxial strain induced by the substrate. This will be the subject

of Section 25.5.

25.4 Symmetry and ferroelectric properties:

polarization rotation and lattice softening

25.4.1 Piezoelectricity and phase coexistence

Ferroelectric materials are not only interesting because of the properties

directly derived from their remnant polarization. The best piezoelectrics are

also to be found among ferroelectric perovskites due to the strong coupling

between the polarization and the lattice existing in most of these compounds.

Moreover, a few of these ferroelectrics show astonishingly high dielectric

and piezoelectric responses, whose mechanisms are not totally understood.

In particular, the ceramic lead titanate–lead zirconate solid solution,

PbZr

1–x

(PZT), has been for decades the most popular piezoelectric in

the actuator market. This is because, next to the low price and ease of

fabrication and handling of the ceramics, the effective longitudinal piezoelectric

coefficient

()

eff

of the x = 0.48 composition (PZT48) is as high as ~600

pm/V [60], two orders of magnitude larger than that of quartz. The highly

responsive PZT48 lies just at the boundary between the tetragonal (T) and

the rhombohedral (R) phase fields, the so-called morphotropic phase boundary

(MPB), in the phase diagram depicted in Fig. 25.7(b) [60–62]. But the relevance

of this fact to explain the unusually large electromechanical response in this

compound is, only recently, beginning to be understood.

As shown in Fig. 25.7(b), the MPB has a very weak temperature dependence

and thus only an exquisite compositional homogeneity of the ceramics and

the extension of the temperature range (by measuring at lower temperatures)

allows us to investigate the R-to-T phase transition in detail by means of

temperature variations. Changes in composition away from x = 0.48 lead to

a decrease in the piezoelectric coefficients and the electromechanical coupling

factor, as well as in the dielectric permittivity. The decrease is dramatic for

increasing Ti content (towards the tetragonal side), while relatively high

dielectric and piezoelectric responses are found in the rhombohedral part of

the phase diagram (see Fig. 25.7a).

Because of the lack of group–subgroup relationship between the

rhombohedral and tetragonal space groups (R3m and P4mm, respectively), a

T-to-R phase transition must be of first order and therefore a phase coexistence

region is expected. A coexistence region is indeed often observed around the

MPB but it is not straightforward to interpret this as a signature of a first-

order phase transition in the presence of even slight compositional fluctuations

(according to Fig. 25.7b, fluctuations in composition of ∆x = 0.01 at the

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Handbook of dielectric, piezoelectric and ferroelectric materials768

MPB can give place to a change of about 100K in the transition temperature).

Whatever the origin, such a coexistence region allows for a large number of

available polarization orientations o domains (8 for the R phase plus 6 for

the T phase) and this was believed to cause the huge response enhancement

observed at the MPB compositions [63].

However, the width of the coexistence region seemed to be determined

not only by the compositional uncertainty [64], but also by grain size effects

PbZr

1–

10 20 30 40 50 60 70

%Ti content

(a)

Relative dielectric permittivity, ε

1600

1200

800

400

0.4

0.3

0.2

0.1

0.0

Electromechanical coupling factor,

25.7

(a) Relative dielectric permittivity (open symbols) and

electromechanical coupling factor (solid symbols) of PbZr

1–

(PZT) at room temperature, as a function of Ti content, showing a

large enhancement of both properties at the MPB composition,

0.48. (b) Phase diagram of PZT as determined by Jaffe

et al.

[60]

from measurements of the elastic moduli. Figures (a) and (b)

modified and reprinted, respectively, from ref. [60] with permission

of Elsevier.

Temperature (°C)

500

450

400

350

300

250

200

150

100

0 10203040 5060708090100

PbZrO

Mole % PbTiO

PbTiO

R(LT)

R(HT)

MPB

(b)

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Symmetry engineering and size effects in ferroelectric thin films 769

[65]. In particular, large grain ceramics were shown to display coexistence

regions of less than 0.1 mole% [64]. More interestingly, the maximum

electromechanical response was found always at a very specific composition

(x = 0.48 at RT) independently of the extent of the coexistence region and the

phase population. In other words, the MPB appeared as an amazingly robust

line. This was early realised by Jaffe et al. [60] and it was, in fact, by

measuring the maxima of the piezoelectric response how the MPB in the

phase diagram of Fig. 25.7(b) was determined. The same authors already

remarked that the composition of maximum dielectric response (x = 0.48)

was shifted towards the tetragonal side with respect to that for which the

rhombohedral and tetragonal compositions appeared at equal ratios (x =

0.465). Later it was reported that, in very homogeneous ceramics, the maximum

response was indeed located in the tetragonal side of the MPB and,

unexpectedly, it was outside the coexistence region [66]. This worked against

the argument of the maximum number of available domains being the main

origin of the large observed piezoelectricity. Moreover, a significant dependence

of the effective d

with doping and changes from sample to sample had

suggested a different extrinsic origin, such as domain wall contributions, for

this remarkable behaviour.

PZT was considered a fortunate rare case for many years, leading the

actuator market in isolation. However, since 1995, other related solid solutions,

such as Pb(Mg

1/3

2/3

)

1–x

(PMN–PT), Pb(Zn

1/3

2/3

)

1–x

(PZN–

PT) and Pb(Sc

1/2

)

1–x

(PSN–PT), with very similar phase diagrams

to that of PZT, have been shown to display even larger piezoelectric coefficients

(of about 2500pm/V) for compositions lying at the boundary between the

tetragonal and rhombohedral phases [67–69]. A detailed structural investigation

of the MPBs then seemed necessary and the study of these materials by high-

resolution X-ray and neutron diffraction in an extended temperature range

was systematically carried out [70–74].

It is well known that the piezoelectric response of materials is highly

anisotropic. The most commonly reported piezoelectric coefficients are the

longitudinal d

, for which the induced polarization and the normal stress are

collinear, and the transversal one, d

, in which the direction of the induced

polarization is perpendicular to the normal stress. Configurations exploiting

other piezoelectric components, as those that involve shear strain, require

less straightforward geometries and have been less investigated. However,

for polycrystalline materials, the ones commonly used in applications, the

measured effective coefficients are an average of the components of the

single-crystal piezoelectric tensor. In the tetragonal case, the effective

longitudinal coefficient can be written as

eff

∫

sin

θ + d

sin

+ d

cos

θ) cosθsinθdθ, where θ is the angle between the crystallites polar

axis and the poling axis of the ceramic pellet [75]. The prominence of

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