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

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Handbook of dielectric, piezoelectric and ferroelectric materials770

piezoelectric ceramics versus single crystals is, probably, the reason why it

was not widely known that the main contribution to the

eff

in well-known

piezoelectrics such as PZT, comes from the d

piezoelectric modulus [75],

which relates the induced transversal polarization to the shear stress. Had we

carefully considered the implications of this result, already reported in the

1970s [60], the behaviour of the MPB piezoelectrics would have been

understood earlier.

25.4.2 Polarization rotation

The first step into the new understanding was offered by the seminal work

of Park and Shrout [67] on rhombohedral PZN–PT and PMN–PT single

crystals showing unprecedented high-strain values of 1.5% and high

electromechanical performance (d

= 2500 pm/V) with virtually no hysteresis.

Lack of hysteresis implies no domain wall motion, and this was not possible

to understand in the context of phase coexistence and multiple accessible

polar domains. The concept of engineered domain configuration was then

developed. According to this model, poling a rhombohedral crystal with a

field along the [001]

(pseudo-cubic) direction will favour four of the eight

possible rhombohedral domains. Upon increasing the electrical field, domain

wall motion will not take place because the four domains are symmetrically

oriented with respect to the field. Instead, the unit cell will elongate along

[001]

, forcing the polarization to rotate away from the rhombohedral

polarization direction [111]

The possibility of polarization rotation under electric field was investigated

by Fu and Cohen using calculations based on first principles [76]. They

showed that, in a monodomain rhombohedral BaTiO

crystal (easier to deal

with than the chemically complex MPB solid solutions), the minimum energy

path for the polarization to change between the rhombohedral and the tetragonal

orientations under an [001]-oriented electric field, corresponds to the continuous

rotation of the polarizarion from the [111]

to the [001]

direction. This

mechanism produces a hysteresis-free strain vs. field dependence (very similar

to those observed for the high-strain piezoelectrics [68]) and generates very

large strain values and piezoelectric coefficients. However, the electric fields

at which polarization rotation would take place are huge and less accessible

than those experimentally found for the lead MPB compounds. That was

probably the reason why this predicted behaviour was not observed in BaTiO

Thus, polarization rotation at the unit cell level was presented as a new

mechanism to enhance the electromechanical responses of ferroelectric

materials. It is very likely, for the compositions close to MPBs, that the

threshold electric fields for polarization rotation were unusually low, due to

lattice softening at the phase boundary [77–79], allowing this effect to be

experimentally observed at relatively low electric fields. These ideas were in

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Symmetry engineering and size effects in ferroelectric thin films 771

tune with an earlier report by Du and colleagues in which, by using a

phenomenological Landau–Devonshire approach, they showed that

rhombohedral PZT single crystals poled along the non-polar [001]

direction

would show enhanced dielectric and electromechanical properties [80].

Unfortunately no single crystals of PZT were available to test this result but,

two years later, Taylor and Damjanovic were able to observed this behaviour

on differently oriented PZT thin films [81].

25.4.3 Low-symmetry ferroelectric phases

Around this time, a third phase was unexpectedly observed in PZT by using

high-resolution synchrotron X-ray diffraction [82–84]. The new phase extends

over a narrow region of compositions next to the MPB, just in between the

R and T phases [83] (Fig. 25.8). This phase, with monoclinic symmetry

(space group Cm), has a mirror plane as the only symmetry element and it is,

thus, fundamentally different from other known ferroelectric phases with

symmetry axes. A symmetry axis determines the direction of the polarization

25.8

Phase diagram of PZT around the MPB, down to low

temperatures, as determined from synchrotron X-ray powder

diffraction (solid symbols). The phase boundaries are compared with

those reported by Jaffe

et al.

[60] (open symbols), also shown in Fig.

25.1(b). The MPB appears as a straight, slightly slanted line

separating the tetragonal (T) and the monoclinic (M) phases. Figure

Physical Society.

Jaffe

et al.

(60)

Noheda

et al

. (84)

PbZr

1–

M+T

40 42 44 46 48 50 52 54

% Ti content

(K)

800

600

400

200

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Handbook of dielectric, piezoelectric and ferroelectric materials772

(any component perpendicular to that axis would cancel out by symmetry).

In an un-twinned ferroelectric crystal with such a well-defined polar axis,

two equivalent polarization directions exist, 180° apart from each other.

During switching, relatively high electric fields are needed to invert the

polarization direction. This well-known scenario of a symmetric double-

well potential with a relatively large potential barrier, becomes elusive in the

monoclinic Cm case, in which the symmetry imposes minimal constraints to

the polarization direction (namely to lie within the mirror plane, m). Switching

fields can be very small and under the application of a non-collinear electric

field contained in m, the polarization is able to rotate to align parallel to the

field.

Interestingly, the m plane in monoclinic PZT is the (1–10)

[83], which

contains both the [111]

and the [001]

directions, that is the polar axes of

the R and T phases (see Fig. 25.9). Therefore, the rotation of the polarization

between [111]

and [001]

is possible in the presence of this bridging phase

and domain wall motion is not required in order to change the polarization

direction. In this way, the link between the hysteresis-free strain curves, the

high electromechanical responses and the MPB compositions was made

through the polarization rotation mechanism in low symmetry phases.

From symmetry arguments [79, 85], the T-to-M phase transition can be of

second order (with continuous change in the order parameters). However,

the transformation between the M and R phases has to be, also by symmetry

arguments, discontinuous. Thus, although the polarization can rotate

continuously from [001] towards [111] by changing composition across the

[001]

[111]

[110]

(010)

[11 0]

[011]

25.9

Sketch of the orientations of the polarization (P) for the different

monoclinic phases of a ferroelectric perovskite: in M

and M

, P is

contained in the (1–10)

plane (light grey), while in M

, P is

contained in the (010)

plane (dark grey). M

, M

and M

allow

rotations of P between the polar axes of the tetragonal and

rhombohedral, rhombohedral and orthorhombic and orthorhombic

and tetragonal phases, respectively.

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Symmetry engineering and size effects in ferroelectric thin films 773

MPB, it cannot reach a truly rhombohedral phase in a continuous way.

Indeed, there is sufficient evidence to propose that the so-called rhombohedral

phase in PZT and related systems may be only rhombohedral in average but

monoclinic at the unit cell level [83, 86–91]. This would explain the fact that

no second phase boundary limiting the monoclinic and the rhombohedral

phase has been directly observed [90] (see dashed line in Fig. 25.8).

The intermediate monoclinic phase in PZT (at zero electric field) has

been observed by other experimental techniques [92, 93] and has been

reproduced by first-principles calculations [75, 94–96]. Ginzburg–Landau–

Devonshire phenomenological calculations are also able to reproduce

monoclinic, and even triclinic, phases, if higher order terms (eighth order for

monoclinic and twelfth order for triclinic) are taken into account [79]. This

reflects the fact that low symmetry phases are expected in the presence of

highly anharmonic energy potentials or flat energy landscapes, as are known

to exist in the proximity of a phase transition [77, 78], or in lead oxides, due

to disorder [97, 98].

The tendency of Pb to form four, instead of three, short Pb–O bonds

favours the monoclinic phase at the local level, even for Zr-rich compositions,

which show an average rhombohedral phase. It has been repeatedly shown

by synchrotron and neutron diffraction that the rhombohedral phase in PZT,

although rhombohedral on average, contains a large degree of disorder. This

disorder is very anisotropic [86]. The diffraction data could well be explained

by a model in which the lead shifts are displaced away from the [111]

direction. This deviation can be described by three equal components along

the main crystallographic axes (see Fig. 25.10). In the ‘rhombohedral’ phase

the three components are randomly distributed in such a way that the average

polarization still points along [111]

[88]

With increasing Ti content, the

[001] component is gradually preferred against the other two, favouring the

monoclinic phase, and providing the structural bridge to the tetragonal phase

[83]. The disorder existing at higher Zr contents diminishes towards the

MPB and in the monoclinic phase there is no discrepancy between the local

and the average structure. It can be said then that, by increasing Ti content,

the solid solution transforms gradually from short-range to long-range

monoclinic ordering without crossing any phase boundaries [90].

Deeper insight into the local order of these materials has been obtained

using density functional calculations by Grinberg et al. [94, 95]. They showed

that the local structure in PZT is indeed very different from the average one

offered by conventional diffraction refinements: the shifts of Zr, Ti and Pb

cations are not collinear. The most striking behaviour is displayed by the Pb

cations: shifts along [100]

, [211]

or [110]

pseudo-cubic directions are

preferred over the [111]

shifts, even in the rhombohedral phase, where the

average polarization is along [111]. Moreover, the Pb cations are highly

sensitive to their local environment. In particular, Pb displacements tend to

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Handbook of dielectric, piezoelectric and ferroelectric materials774

point towards the Ti neighbours and away from their larger Zr neighbours,

governed by simple electrostatics reasons.

At relatively low Zr contents, in the tetragonal phase, local Pb shifts will

tend to follow the average [001] polarization. However, in the rhombohedral

phase, shifts of the Pb along the overall macroscopic polarization [111]

bring the Pb and Zr neighbours closer together and are energetically

unfavourable, forcing the Pb atoms to deviate from the average polarization

direction. Clearly, this balance becomes more difficult the larger the Zr

content. These authors also show that at around the x = 0.50 composition, a

monoclinic phase made of [001]

(for Pb cations undisturbed by a Zr

neighbour) and [211]

Pb distortions (of those trying to avoid a Zr) is

expected, in agreement with the experimental finding.

In other solid solutions, intermediate low symmetry phases have also

been observed, some of them different from that in PZT (the so-called M

phase [79]), which gives a polarization P

= P

< P

, where P

, P

and P

are

the Cartesian components of the polarization along the pseudo-cubic perovskite

directions (see Fig. 25.9). The intermediate monoclinic phase that can bridge

the tetragonal and orthorhombic phases is called M

and supports polarization

with P

< P

; P

= 0. The symmetry plane in this case is the (010)

and the

space group is Pm. The intermediate monoclinic phase between the

rhombohedral and the orthorhombic phases (M

) has the same symmetry as

phase but in this case P

= P

> P

. These three phases have been

predicted by a Ginzburg–Landau–Devonshire approach [79], as well as by

25.10

In order to explain the disorder typically observed in the

rhombohedral phase (R) of PZT, Corker

et al.

[88] proposed a model

in which Pb can randomly occupy three different sites (grey dots),

while keeping the macroscopic polarization along the [111] direction.

These random positions are defined by a common component along

[111] and three shifts of the same magnitude but directed along

[100], [010] and [001], respectively. One of these sites, which can be

described as (

) + (00

) coincides with the position of Pb in the

monoclinic phase, M (right). The transition from R to M corresponds

to the preference of the [001] Pb displacement against the [100] and

[100] ones, upon decreasing Zr content. Figure adapted and reprinted

Society.

[001]

[111]

[110]

[111]

[110]

–

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Symmetry engineering and size effects in ferroelectric thin films 775

first-principles calculations [75, 96] and all of them have been also found

experimentally in different systems. The richest solid solutions seem to be

PMN–xPT [99] and PSN–PT [100, 101], in which both M

and M

phases

are observed at the MPB.

The availability of single crystals of PMN–PT and PZN–PT [102] made

it possible to observe directly the polarization rotation by detecting monoclinic

phases by X-ray and neutron diffraction. This was done in rhombohedral

compositions by applying a [001]-electric field in situ and observing the

lattice distortion. Low symmetry M phases were observed under electric

fields [103, 104] and at large enough fields a tetragonal phase was induced.

The R–M–T transformation path is irreversible [105] and the crystals remain

monoclinic upon removal of the electric field (at least during a time scale of

months). This seems to argue in favour of the stability of the intermediate

monoclinic phase, in agreement with first-principles calculations in PZT

[106]. There is, however, still some disagreement in the community and it

has been argued that these are not proper monoclinic phases but rather

monoclinic metastable states induced by strain and/or electric field [107,

108].

Other authors have explained the intermediate structures as ‘adaptive

phases’ similar to those taking place in martensites [109, 110]. According to

this model, crystallographic nano-twins with tetragonal or rhombohedral

symmetry exist and will appear as a uniform monoclinic phase under long-

range probes such as X-ray diffraction. The macroscopic rotation of the

polarization occurs under electric field by means of de-twinning. Nano-

twinning is indeed expected close to the MPBs, and also in truly monoclinic

phases, due to the very small anisotropy, which gives place to a low domain

wall formation energy. Recently, it has also been suggested that domain wall

broadening occurs under an electric field and that the intrinsic piezoelectric

response of the domain walls are at the origin of the unusual behaviour

observed in morphotropic systems [111], in which there is a high density of

domain walls.

As it seems, first-principles calculations and local probe techniques in

PZT favour truly monoclinic phases, while structural and optical investigations

on PMN–PT and PZN–PT single crystals seem to be in good agreement with

the postulates of the adaptive theory. The diffraction patterns of PZT48 at

low temperatures show clear monoclinic peak splitting [83], while the

monoclinic distortion in the diffraction patterns of PMN–PT or PSN–PT

appears more often as peak broadening and asymmetry in the diffraction

peaks, due to a larger amount of disorder in this chemically more complex

systems.

As reminded [112], the phase boundary between two different ferroelectric

phases (with different direction of the polarizations), as in the case of the

MPBs, involves the softening of the lattice in a direction different from the

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Handbook of dielectric, piezoelectric and ferroelectric materials776

polar direction and, therefore, causes large dielectric anomalies that are

responsible for the ease of rotation of the polarization and for the enhancement

of the piezoelectric response due to shear tensor components [113–115]. The

proximity of a critical point will also have the effect of softening the lattice

and allowing anomalously high piezoelectric deformations [116]. And, as

has been recently pointed out, these descriptions are not unrelated to one

another [117]. Thus, over the past years it has become clear that very large

dielectric (and piezoelectric) responses are expected when polarization rotation

is allowed, either at the unit cell level (by the softening of the lattice at a

phase transition, and/or by inducing phases with no symmetry axis) or

macroscopically (by de-twinning of 90°-like domains).

Despite the recent understanding, it is still puzzling that so very few

materials actually display the above behaviour. Moreover, the ones that do

so are chemically complex and thus difficult to control and model. It would

be desirable to find simple materials behaving in such a responsive way. As

described above, BaTiO

is one such material but polarization rotation happens

at impractically high electric fields. Rotation can become easy close to the

transition temperatures [112] but this strong temperature dependence is also

not practical.

The question then arises whether it is possible to design and build new

materials displaying any of the polarization rotation mechanisms so far detected.

One possible approach is using strain engineering by means of thin film

deposition techniques, as described in the next section. The advantage of this

approach is that it provides us with very thin films for integration and

miniaturization of dielectric, ferroelectric and piezoelectric components, which

are still missing in the applications market. The disadvantages are related to

high production costs, clamping effects and size effects that, as shortly reviewed

in the previous section, are an interesting topic on their own.

25.5 Strain effects on ferroelectric thin films

As described in Section 25.4, substituting Ti

in the classical ferroelectric

perovskite PbTiO

with a larger size isovalent cation, such as Zr

or a

mixture of (Mg

1/3

2/3

)

, (Zn

1/3

2/3

)

, (In

1/2

)

and (Sn

1/2

)

the tetragonal ferroelectric ground state, with polarization along [001]

transforms into a rhombohedral ferroelectric phase with polarization along

[111]

. At the boundary between these two, the electromechanical and dielectric

responses are astonishingly large. It seems plausible that a similar effect

could be attained by enlarging the unit cell via external pressure or epitaxial

stress. Indeed, first-principl-es calculations on PbTiO

[118] and PZT [119]

under hydrostatic pressure, as well as structural and Raman experiments

[119, 120], confirm that polarization rotation also takes place under external

pressure.

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Symmetry engineering and size effects in ferroelectric thin films 777

In thin film crystals, biaxial strain provided by a crystalline substrate can

be applied permanently in the plane of the film. The perpendicular lattice

parameter will be strained in the opposite direction according to the Poisson

ratio:

sss s

zxy

= ( + )

1 –

25.7

where s

is the out-of-plane strain (with respect to the unclamped structure),

and s

, s

are the lattice mismatches between the in-plane lattice parameters

of the film and those of the substrate, which reduce to a single mismatch

value s

in the particular case of square in-plane lattices for both substrate

and film. Parenthetically, it is worth noting here that reliable values of the

Poisson’s ratio of perovskites are hard to come by. The convention is to cite

values in the region of 0.3–0.4, but these are often based on hearsay. It is

easier to find reliable values for the elastic compliances. The Poisson’s ratio

can then be related to the elastic compliances using ν = S

. For SrTiO

(STO) single crystals, the Poisson’s ratio is actually about 0.22–0.24,

considerably lower than often assumed.

Thin films are normally grown (or crystallized) at high temperatures on

rigid substrates. Since the film and substrate rarely have the same thermal

expansion coefficient, strains will generally develop upon cooling from the

deposition temperature. In the case of epitaxial films, this process can be

controlled by selecting substrates with a structure and lattice parameters

closely matched to those of the film. In this way one can achieve coherent

growth and hence, provided there is no relaxation, gain complete control

over the in-plane strain state of the film. Thus, in any phase transition where

there is a degree of spontaneous strain associated, one can use epitaxial

strain to control, to some extent, the film properties. This trick has been used

in the past to change critical temperatures in superconductors [121, 122], metal-

insulator transitions [123, 124] and, of course, ferroelectrics [125–127].

Polarization requires a relative displacement of the ions in the unit cell,

and hence a structural deformation. Accordingly, the onset of spontaneous

polarization in ferroelectrics causes a spontaneous strain with respect to the

paraelectric, more symmetric phase. Conversely, externally induced

deformations of the crystal can change the polarization state and/or the polar

symmetry. In the case of perovskite ferroelectrics (by far the most studied

family), the onset of ferroelectricity is generally associated with a stretching

of the unit cell along the polar direction, usually in a cubic–tetragonal transition.

Trivially, then, if one forces the unit cell to be tetragonal by the use of

compressive in-plane strains, the Curie temperature will rise and the room

temperature polarization will also increase. Using standard Landau theory of

phase transitions, it is possible to quantify the shift as a function of strain,

which is given by the formula [30, 125, 126]:

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Handbook of dielectric, piezoelectric and ferroelectric materials778

∆TCQ

sCQ

C012

11 12

012

= 2

= 4

1 –

εε

25.8

where C is the Curie constant, s

is the lattice mismatch between the in-

plane lattice parameters of the film and the substrate, Q

and S

are the

transverse electrostrictive coefficient and elastic compliances, and Y,

are

Young’s modulus and the Poisson ratio, respectively. Since both Q

and s

are negative, compressive in-plane strains will increase the critical temperature

of the ferroelectric. This effect has been known for a while [125–129], but it

has recently been taken to new extremes, with shifts in the critical temperature

of BaTiO

from ~125 to ~540 °C [126]. A tensile in-plane strain, on the other

hand, has been used to achieve room temperature ferroelectricity in SrTiO

(STO) [130]. Since this material is normally paraelectric at all temperatures,

this implies a shift in T

from 0 to around 300K. The shifts in T

for strained

BaTiO

(BTO) and STO films are the biggest ever reported for any material.

The recent research on SrTiO

also illustrates another of the potential uses

of strain, which is to achieve crystallographic symmetries in the film that do

not exist in bulk form [131]. This possibility was first theoretically predicted

by Pertsev et al. [126], and has so far been experimentally achieved in STO

[131], BiFeO

(BFO) [132, 133] and PbTiO

(PTO) [134].

The case of BiFeO

, a ferroelectric antiferromagnet, has attracted much

attention in the past few years for being one of the few simple materials

displaying ferroelectric and magnetic order at room temperature. Although

BiFeO

is rhombohedral (space group R3c) in bulk and, therefore, has the

spontaneous polarization along the [111]

direction, the epitaxial growth of

BiFeO

films on [001]-oriented SrTiO

substrates, which infer a misfit

compressive strain of 1.1%, induces the tetragonal symmetry in very thin

films, in which the polarization aligns normal to the film plane. Partial

relaxation of the strain as the thickness is increased tilts the polarization and

reduces the symmetry down to monoclinic [132, 133, 135], with a unit cell

of the same type than that of monoclinic PZT (space group Cm, see Section

25.4). Further increment of the thickness to about 100 nm can rotate the

polarization completely to lie along the [111]

direction and relax the structure

to the bulk rhombohedral symmetry. Therefore, compressive biaxial strain in

rhombohedral BiFeO

has the same effect as chemical pressure in rhombohedral

PbZr

1–x

(increasing Ti content). Surprisingly, the strain and the associated

change of symmetry barely affect the functional properties – or at least the

remnant polarization – compared to bulk. While early works suggested

a very low bulk polarization of only 5 µC/cm

in bulk [136], recent

investigations in single crystals have shown polarizations of 60 µC/cm

along the [001]

pseudocubic

direction, which correspond to ~ 100 µC/cm

along

the polar [111] direction [137] and which are almost identical to those

measured in thin films. In spite of the large polarization, no huge piezoelectric

responses have been detected in BFO, which may be due to the small

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Symmetry engineering and size effects in ferroelectric thin films 779

polarization–strain coupling in BiFeO

compared with that of PbTiO

BaTiO

[138]. There are still many unknowns about this interesting ferroelectric,

but the large number of groups currently dedicated to investigating it may

change this picture soon.

Because of the clamping to the substrate, strain may change the polar

symmetry of the films without changing the symmetry of the unit cell. The

decoupling between the unit cell parameters (micro-strain) and the polarization

due to substrate clamping has been observed in SrTiO

[131] and in PbTiO

thin films [134]. This means that (c/a–1), a and c being the tetragonal lattice

parameters, is no longer an order parameter of the ferroelectric transition,

which complicates characterization of ferroelectricity in very thin films.

25.5.1 Fully strained films: novel crystal symmetries and

depolarizing fields

The discussion in the introduction focuses on a rather simplified picture,

where every unit cell of the film experiences a strain equal to the mismatch

with the substrate or misfit strain (s

= a

subs

bulk

–1). This is true for non-

relaxed, un-twinned films. For relaxation not to take place, the mismatch

with the substrate ought to be small enough. Even for low misfits, because

the strain energy scales with thickness, there will be a certain critical thickness

above which formation of dislocations or formation of twin walls will

take place to relax the strain. The critical thickness for the appearance of

dislocations (t

) is smaller for larger misfit strain values, according to the

equation [53, 54]:

bb b

xy z

22 2

+ + (1 – )

8 (1 + )

ν

πν









25.9

where b

are the coordinates of the Burger’s vector of the dislocation, r

the lattice parameter, and β is an adimensional factor that is approximately

4 for perovskites [54]. Since there is not much choice of substrates, having

fully strained and dislocation-free films is not easy and, when possible, it

usually means dealing with very thin films, whose properties and structure

are difficult to characterize.

Predictions about untwined, defect-free films are somewhat easier (or so

we believe) for theorists. Phenomenological Landau–Devonshire calculations,

full first principles, as well as first-principles-based methods combined with

Monte Carlo simulations have been used to predict phase stability under

strain and/or the temperature vs. strain phase diagrams of single domain

slabs of ferroelectric perovskites such as BaTiO

, PbTiO

and PbZr

1–x

[126, 136–142]. For BaTiO

single-domain films, most of the reported methods

agree qualitatively on the strain phase diagram [126, 136, 142]. For relatively

large compressive misfit strain values, the films have tetragonal symmetry

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