Introduction and theoretical considerations [609954]

13

CAPTER 1

Introduction and theoretical considerations

1.1 Introduction

Ferroelectricity is the key phenomen on for many applications of modern technologies,
ranging from sensors and actuators to optical or memory devices [ 1-2]. Nevertheless, in spite of
considerable experimental and theoretical work, this field remains full of new challenges and open
questions . The antiferroelectrics (AFEs) are more complex when compared to ferroelectrics (FEs) .
In spite t he AFE properties were often observed and reported, they only rarely were explained by
apropriate models. Only recently, the progress in the understanding the physics associated with
antiferroelectric ity started to emerge , as result of emergent applications rela ted to storage energy
[3]. New publications provide some deeper insights in the field of antiferroelectricity as origin of
their functional properties [ 3-5]. In this chapter, a short introduction and a comparative
presentation of the available studies on phenomenology of ferroelectricity and antiferroelectricity
is given, with particular emphasis on their properties and applications.

1.2 Brief history

FEs and AFE s have been initially been classified according to experimental observations .
Even it was believed that the discovery of Rochelle salt (sodium potassium tartrate tetrahydrate,
NaKC 4H4O6·4H 2O) in 1665 was marked as the starting point of study on FEs, the ferroelectricity
was described only in 1921 by Valasek, who obtained electric field dependen t of polarizations P –
E hysteresis curves for Rochelle salt analogous to the B –H curves in ferromagnetism. He studied
the electric hysteresis and piezoelectric response of the crystal in more detail [ 6-7]. During the
Second World War this phenomen on was dis covered in other materials like BaTiO 3 [8]. Since
then, the science of ferroelectricy had expanded very fast . More materials have been discovered in
the last decades. Various FE formulations, their form (bulk, films , particles, dots ), their fabrication,
function (properties), were described in relation to their FE behaviour and prospective specific
areas of application [ 9-10]. An intensive understanding of properties of FE materials as a function

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of their micro – and nanostructures have been gained and numerous breakthroughs occurred thanks
to both theoretical calculations and experimental efforts [ 11-13]. Today, the FEs are successfully
used at industrial scale. They have been the heart of a lot of application in modern electronics, like
high-dielectric -constant capacitors to later developments in piezoelectric transducers, positive
temperature coefficient devices, and electrooptic light valves, medical ultrasonic composites, high –
displacement piezoelectric actuators (Moonies, RAINBOWS), photostrictors, and thin and thick
films for piezoelectric and integrated -circuit applications [ 9].
The phenomena assigned to antiferroelectricity w ere mentioned only 30 years after
discovery of ferroelectricity. Kittel may be considered the father of antiferroelectricity since he
was the first to develop a phenomenological model of antiferroelectricity in 1951 , even before
experimental studies had revealed the existence of the AFE structure. He defined the
antiferroelectricity in analogy with ferroelectricity and classifi ed the AFE s as a subclass of FE
materials [ 14]. In parallel, in June 1950, E. Sawaguchi et al. established that led zirconate (PbZrO 3)
has a n AFE structure [15 -18]. Despite of a study which show evidence of ferroelectricity in PbZrO 3
[19], the PbZrO 3 remai ns the prototype system for AFE materials. Among various examples of
AFE perovskites such as PbHfO 3 [20], Pb(B' 1/2B"1/2)O3 [21], (Bi 1/2Na1/2)TiO 3 [22], NaNbO 3[23-
25], AgNbO 3 [26], and order -disorder compounds NH 4H2PO 414 [27-28] and
Cu(HCOO) 2•4H 2O15[29], (Pb,La)(Zr,Sn,Ti)O 3 (PLZST) [ 30] the Pb(Zr,Sn,Ti)NbO 3 (PNZST) [ 31]
appeared to be the most promising AFE’s. AFE s are useful in many applications like large digital
displacement transducers [ 32], bistable optical devices [33] and energy -storage capacitors [ 34].
Despite their technological importance, the reason why materials become AFE remained
hypothetical since four years ago [ 5].

1.3 Definition of Ferroelectricity and Antiferroelectricity

The term ferroelectricity is used in analogy to ferromagnetism referring to a material that
exhibits a permanent magnetic moment. The FE phenomenon is defined as [ 35-36]:
Ferroelectricity refers to a property of certain crystals that have a spontaneous electric
polarization that can be reoriented by the applied external electric field.
The FE materials show some typical properties like: FE domains, structural phase
transit ions, temperature dependence of electrical, mechanical an optical properties. The adjacent
dipoles with the same polarization orientation, form domains characterized by a spontaneous

15
polarization. The spontaneous polarization depends on temperature (it decreases to zero when
temperature is reduced down to a acritical temperature denominated as “Curie temperature”) and
can be aligned or reoriented by an external electric field . The field induced polarization re –
orientation ( FE switching) is accompanie d by hysteresis. The structural requirement for a material
to be FE is to have a noncentrosymmetric polar structure. The structure is dependent on
temperature as it show a transition between a polar state ( FE phase) to a non -polar state
(paraelectric phase ) at the Curie temperature. FEs have other important properties like
piezoelectricity , pyroelectricity and electrooptical properties [35-36].
Antiferroelectricity was first defined by Kittel [ 14] as “a state in which lines of ions in the
crystal are spontaneously polarized, but with neighbours’ lines polarized in antiparallel
directions, so that the spontaneous macroscopic polarization of the crystal as a whole is zero. ”
Therefore, AFE s have nonpolar structure with the dipoles oriented antiparallel to
adjacent ones, resulting in a net zero microscopic polarization . In such structures, a FE state
can be in duced through a dipole reorientation . After the FE state is induced, in a similar way
as in FEs, the spontaneous polarization can be further oriented by the appli cation of external
fields.
It is worth to note that AFEs are non -polar materials , but not all the nonpolar phases are
AFE .
A FE material has a polar distortion which allows at least two orientation states of its
spontaneous polariz ation in the absence of an electric field . The existence of this spontaneous
polarization is the key reason of the high interest and the wide application of FEs [35-36]. By
applying an external electric field , the polarisation can be shifted from one to another of these
states. AFE systems show some characteristics which are quite different from an FE system. When
an AFE material is exposed to a sufficiently high electric -field, E AF , the dipoles tend to shift and
align along the field direction, so that a net p olarization parallel to the field is induced, and
therefore, an AFE-to-FE field transition was realised . The phase switching behavior takes place
when the difference in the free energ ies between AFE and FE phase s is small [ 14, 31]. Together
with the AFE -to-FE phase switching, a field induced structural transition takes place. There must
be an alternative FE structure with close -free energy obtained from a polar distortion of the same
high-symmetry reference structure, and an applied field must induce a first -order transition from
the AFE to this FE structure. This field -forced phase transition is accompanied by a large volume
change which is the key for applications like transducers and actuators [ 37]. Contrary to FEs,

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which show a remanent polarization after application of electric field [ 36], in the AFE materials,
upon the release of the external electric field, the dipoles will recover back to their AFE state and
thus, the net polarization of the crystal will return back to zero [ 31]. Howe ver, similar to FEs, the
net macroscopic polarization in the virgin state AFEs is zero in the absence of electric field and
those materials display single P -E curves or double hysteresis loops at a sufficiently high electric
field. In both, FE’s and AFE’s, the domain structure, the polarization switching and phase
transitions depend strongly on temperature, electric field , strain -stress (electrical and mechanical
boundary conditions) and on the nano/ microstructure s [37-38].

1.4 Basic properties of ferroelectric and antiferroelectric materials

1.4.1 Perovskite Structure

Oxide m aterials with perovskite structure with general formula ABO 3 (Fig. 1.1) are very
important in material science and technology. This structure seems to be a particularly favourable
configuration because exhibits a high flexibility of composition and structural distortion, leading
to a wide range of properties such as the piezoe lectric, pyroelectric, FE, and magnetic properties
[39-42].

Figure 1.1 An ideal cubic ABO 3 perovskite unit cell

The fundamental structure of perovskite is cubic , with oxygen ions situated on A site s at
each of the six face centres of the cube, forming a corner -linked array of octahedra. The larger A –
cation occupies the space between the oxygen octahedral, while the smaller B -cation sits at the
center of the octahedral (Fig. 1.2) . Valence of A cations takes value from +1 to +3 and of B cations
from +3 to +6 , and t he coordination numbers of A – and B – cations are 12 and 6 , respectively
[42,43].

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Doping at A – and/or B -site with different elements having dissimilar electronic
configuration can lead to dram atic effects associated with the change of crystalline structure and
of the electro nic configuration. In this way, the functional properties may be changed in order to
obtain a desired material with specific properties requested for a well -defined application. The
BO 6 octahedra play a critical role for inducing FE properties [ 42, 44 ]. The stability, rotation or
distortion of crystal structures vary with different types and sizes of A – and B – cations. The
tolerance factor (t) is an indicator for predicting the stability of perovskite structure, [ 42] and it is
defined by the following equation:

𝑡=𝑅𝐴+𝑅𝑂
√2(𝑅𝐵+𝑅𝑂) (1.4.1)

where R A and R B are the average A -site and B -site ionic radii and R O is the oxygen ionic
radius in the appropriate coordination [ 42]. It has been shown that FE phase is stabilized for 𝑡>
1 and AFE phase is stabilized for <1 .

Figure 1.2 Network of corner -linked octahedral where the A sites (yellow ball) is inside
an octahedral cage of Oxygens

The large m ajority of FE materials like Barium Titanate (BaTiO 3), Lead Titanate (PbTiO 3),
Lead Zirconate Titanate (PZT), Lead Lanthanum Zirconate Titanate (PLZT) have the perovskite
type structure. Barium titanate BaTiO 3 is the first laboratory prepared FE and it is the prototype
perovskite FE structure. Barium ions (A ions), occupy the corner sites, titanate ions (B ions) are
located in the centers of the c ube (the oxygen octahedral) and oxygen anions are on the face –

18
centers. The BO 6 building block led to the finding of a series of FE crystals with similar structure,
such as KNbO 3, KTaO 3, LiNbO 3, LiTaO 3, PbZrO 3 and PbTiO 3. [43]. Pb(Zr,Ti)O 3 is the solid
solution between AFE Lead Zirconate (PbZrO 3) and FE Lead Titanate (PbTiO 3). Lead zirconate –
titanates and have the formula Pb(Zr 1-xTix)O3 (PZT). Crystalline structure and physical properties
of PZT strongly depend on (x) lead titanate content. Titanium -rich compositions transform into
tetragonal perovskite structures whereas the phase transformation in Zr -rich compositions is more
complex. At low Ti content (x > 95%) and at room temperature the structure is orthorhombic and
for x < 90% the structure is rhombohedral [ 45]. However, scientists started to pay more attention
to the MPB in simple -structured pure FE compounds such as FE oxides. The morphotropic phase
boundaries are present in many solid solutions and of particular interest are compositions PZT,
relaxor FEs such as Pb(Zn 1/3Nb2/3)O3 – PT and Lead Magnesium niobate -lead titanate (1 –
x)PbMg 1/3Nb2/3O3-xPbTiO 3). For example, PZT is a perov skite FE with a MPB between the
tetragonal and rhombohedral FE phases in the temperature -composition phase diagram. High
resolution x -ray powder diffraction measurements on homogeneous PZT sample s have shown that
there is a monoclinic phase exists between the well -known tetragonal and rhombohedral phases
[46-48].
AFEs have centrosymmetric structures which can be described by unit cells with oppositely
directed dipoles generated by ionic displacements from a higher -symmetry reference structure.
AFE class of materials include certain niobate perovskites, vanadates and complex perovskite
oxides which satisfies the structural and energetic criteria for antiferroelectricity. Among them,
PbZrO 3 is the most studied AFE oxide. It has Pbam orthorhombic distorted pero vskite structure
[49]. Isostructural with PbZrO 3, AFE PbHfO 3 was discovered soon after the identification of
antiferroelectricity in PbZrO 3 [20]. NaNbO 3 and AgNbO 3 are both perovskite AFEs with
orthorhombic space group Pbcm with eight formula units per uni t cell [50-51]. Some d ouble
perovskites may also show antiferroelectricity [ 20].

1.4.2 Domain switching: Ferroelectric versus antiferroelectric behaviour

a) Domain switching in ferroelectric materials

The regions of the crystal with uniformly oriented spontaneous polarization are called FE
domains [ 53-54]. In an as -processed ceramic, each grain consist of distinct types of uniform
domains oriented in a preferential direction but polarizations of different grains could have
different orientations governed by the local crystallographic symmetry. In the absen ce of electric

19
field, domains are oriented with equal probability along one of the crystallographic equivalent
directions , resulting in the lack of piezoelectric effect. Thus the grains in a virgin polycrystalline
sample are not uniformly polarized and the initial macroscopic polarization is zero ( point A from
Fig.1.3 ). When an electric field is applied, domains start to orient along the applied field. This
gives rise to an increase in polarization and a nonlinear behaviour describing the hysteresis curve.
The study of the hysteresis loops, namely current -electric field (I -E), polarization -electric field (P –
E) and strain -electric field (S -E), is one of the most important tools to investigate the behaviour
and to assess the properties of FE/ferroelastic/ AFE materials [ 38]. The e lectrical parameters of
interest for FEs include : coercive field (E c), spontaneous polarization (P s) and remanent
polarization (P r). Spontaneous polarization (P s) is the polarization at maximum saturation field
minus the induced contribution, remanent polarization (P r) is the polarization that persists at zero
field, and coercive field (E c) is the field required to reverse the remanent polarization back to zero.
A typical polarization reversal for FEs is sown in Fig. 1.3 When elec tric field is applie d on a virgin
FE ceramic, at small values of the alternating electric field, the polarization increases linearly with
the field amplitude. In this region, still no domain switching takes place. As the field is increased,
the polarizatio n of domains with unfavourable direction of polarization will start to orient along
the directions of the field. The polarization response in this region is high and strongly nonlinear.
Further increasing of the fie ld will bring the system into the saturat ion of the polarization (point
B). Subsequently, by decreasing the field, a remnant polarization P r remains when the E field is
reduced down to zero (Point C). However when the field is applied in the opposite direction the
polarization decreases down to z ero at a certain value of electric field named the coercive field Ec
(point D). A new alignment of dipoles and saturation (point E) takes place if the field in the
negative direction is further applied. When the field strength is then reduced to zero the
polarization reverses to complete the loop [36, 38 ].
During the domain switching a macroscopic deformation and unit cell distortion take s
place. For example the spontaneous polarization in PbTiO 3 lies along the c -axis of the tetragonal
unit cell and the crystal distortion is usually described in terms of the shifts of O and Ti ions re lative
to Pb. Under the application of an electric field along the spontaneous polarization direction, the
Ti displacement is shifted along the field direction while c -axis of the lattice elongates and the a –
axis of the lattice shrinks. This is the reason for the high performance piezoelectric ity in such
ceramics [ 55]. It is worth mentioning that the domain switching takes place only on polar materials
in the direction of polar axis (easy polarization axis). The polarization (order parameter) is aligned
along definite

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Figure 1.3 Schematic illustrations of the polarization switching: (A -C) the initial poling,
(C-E) the electrical reversal, and (F -A) the electrical cycling. Under the application of an electric
field, the B cations displacement is shifted along the electric field dir ection, giving rise to the
lattice distortion. (The rectangles with blue arrows represent schematically the repartition of the
two polarization states in the material ( e.g. in the cermic grains) at different fields .

crystallographic directions ( e.g. the [111] directions for the rhombohedral phase). When the
number of allowed directions is large, domain switching takes place quasi -continuous and the
polarization is enhanced. The tetragonal structure show s a sizeable elongation along [001] and a
large spont aneous polarisation in the same direction. There are six equivalent polar axes in the
+ECPs
-PrPrP
E -EC
Before poling
𝑬
Under electric field
𝑬
Under electric field
After poling
A
B
C
D
E
F

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tetragonal p hase corresponding to [100], [100], [010], [010], [001], and [001 ] directions of the
cubic paraelectric state. A rhombohedral FE structure shows distortion an d polarisation are along
[111] directions, giving rise to eight pos sible domain states: [111], [ -1-11], [1 -1-1], [11 -1], [1 -1-
1 ], [ 1 -1-1], [-1-1-1 ], and [ -1-1 1 ]. There are fourteen possible poling directions on monoclinic
structure over a very wide t emperature range, which may in part explain the ceramic pi ezoelectric
behaviour near the MPB boundary [ 56].

E I

Figure 1.4 Current vs. field during domain switching in FEs (“current hysteresis”)

Domain switching in FE ceramics is a complex process that is not very well understood.
In order to induce piezoelectricity in the FE ceramic, those may be brought into a polar state by
applying a strong electric field (10 -100 kV/cm) for a long time, usu ally at elevated temperatures ,
then cooled down to the room temperature under field . During this process (called poling ), the
domains are realigned as close as possible to the field direction , thus making the ceramic become
uniaxial piezoelectric. The poli ng process is essential for the technological applications of FE
ceramics in sensors, actuators, and transducers that exploit the piezoelectric effect [ 57].
As clearly pointed out by J.F Scott in “Ferroelectric go bananas” [ 58], it is possible to
observe a hysteresis even without any ferroelectricity. In order to assign the FE behaviour to a
material the current curve versus electric field must be considered in order distinguish FE
switching from artefacts. A schematic representation of a typical I(E) curve , representative for a Electric Field (E)
Current (I)

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FE, is shown in Fig.1.4 . The linear field ramp is positive through the ascent and the same, but
inverted value through descent of the ac signal. During the FE switching a peak of this current is
observed at a certain field , both in the rising part of the positive fields and in the falling part of the
negative fields and corresponds to the coercive field. In non -FE materials which show FE-like
hysteresis, the leakage current simply increases with the electric field and no FE switching peaks
would appear in the transient current response.

b) Domain switching in antiferroelectric materials

Application of an electric field or pressure to the AFE can stabilize the parallel dipole
arrangement and thus , lower its free energy. The dipole switching in AFE s under the application
of electric field is a combination of strain behaviour and preferred orientation of AFE domains. A
step-by step sequence is shown in Fig. 1.5 . Initially, in the virgin state of the AFE ceramics, the
dipoles are rando mly oriented. After exposure to a low electric field value 𝐸<𝐸𝐴𝐹, the AFE
domains are preferentially oriented but with its c axes still perpendicular to the field direction. In
this stage the unit cell is deformed along the field direction: the size of the c axis is increased along
longitudinal direction while unit cell size is decreased along the transverse direction. Once an E
field applied is large enough 𝐸=𝐸𝐴𝐹 to induce the AFE -FE transition, the switching from non –
polar state to polar state tak es place with a sudden increas e of polarization. At this step the oriented
AFE dipoles change to oriented FE dipoles along the electric field direction and the structure of
the unit cell is changed to a new FE structure. In comparison to sample exposed to an electric field,
the primitive unit cell shows a slight decrease in the longitudinal dimension and a large increa se
in the transverse one. If higher field is applied , the domains will continue to switch as close as
possible to the field direction and the refore , the oriented FE will become poled , with piezoelectric
character . After the field removal , there are two possible situation s, depending how far the system
is from the AFE -FE phase boundary: 1) the FE phase return s to the oriented AFE state ( reversib le
AFE to FE field induced transition ) or 2) the FE phase remain s in the poled state (irreversible
field assisted AFE -to-FE transition ). However in both situations, the ceramic does not return to
its virgin state, unless it is heated above its Curie temperature [3,31,59 ]. In conclusion, AFE
materials can be divided into two subcategories, according to the stability of the ir induced FE state.
If the induced FE state transforms back to the AFE phase after the field removal, it is named
reversible AFE. I f the induced FE state persists after the applied field is removed, it is called

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irreversible AFE. However, the induced FE state is not stable, it can return to AFE state at high
temperatures [ 60], by applying a sufficiently high pressure [ 61] or by applyi ng an electric field of
reverse polarity [ 62]. Usually the reversibility is recognized from the hysteresis loop shape: a
double hysteresis loop is displayed for a reversible AFE -to-FE field induced transition or a single
hysteresis loop for the AFE-to-FE field assisted irreve rsible transition. Figure 1.6 shows the typical
P-E response for an AFE when the applied electric field strength is high enough to induce the FE
state. The polarization shows a linear response (similar to a linear dielectric ) at low ele ctric field s,
then the AFE -to-FE transit ion is induced at a critical field value E AF accompanied by a sudden
increase in polarization. The as -induced FE phase turn s to the AFE state in the first quadrant during
the field reversal at a lower field value, E FA, describing a hysteresis loop. Similarly, AFE -to-FE
and

Figure 1.5 Schematic diagram of the e volution of AFE -FE phase switching during
application and rel easing of adequate electr ic field. Figure adapted from [ 59]

FE–to-AFE transitions take place during the third quadrant on reversing electric field and
subsequently increasing electric field [63 -67].
Random AFE,
virgin stateOriented AFE,
EOriented FE,
E
1 2
Poled FE,
E3
Piezoelectric effect
E
Poled FEOriented AFE
OrReversible AFE to FE
field induced transition
Irreversible AFE to FE
field assisted transition456

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Figure 1.6 Representative electric -field-induced polarization hysteresis loop of AFE
materials.

FE-AFEFE-AFE
AFE-FEE I
AFE-FE

Figure 1.7 Current -field dependence of AFEs

A representative current –field (I –E) hysteretic curve of AFE ceramics is illustrated in Fig.
1.7. The high increase in current is caused by the switching of the dip lles. As can be seen in Fig.
1.7 four obvious peaks are observed in the I(E) curve during electrical loading. There is a sharp
increase in current when the applied field just reaches the threshold values for the phase transition
between AFE and FE state and a broad peak durin g decreasing of electric field just below the
critical one that induced dipole re-orient ation back to the original anti -parallel state. Similarly,
EAF
EFAEFA
Electric Field (E) Polarisation (P) EAFElectric Field (E)
Current (I)

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two peaks are present in the fourth quadrant during the field reversal with negative amplitude [ 3,
31].

1.4.3 Tunability

a) Tunability in ferroelectrics

FEs show electric field tunable dielectric properties which have attracted extensive
attention in recent years due to their potential applications for microwave devices such as tunable
filters, phased array antennas, delay lines and phase shifters [ 68-69].Tunability, n, is defined as
the ratio of its permittivity at zero applied electric field to its permittivity at a specific non -zero
field [ 69]:
𝑛(𝐸)=𝜀(0)
𝜖(𝐸), (1.4.2 )
Relative tunability can be defined as
𝑛𝑟(𝐸)=𝜀(0)−𝜀(𝐸)
𝜖(0), (1.4.3 )
A typical “butterfly” permittivity -field dependence specific to FE materials is shown in
Fig. 14, where the maxim a in permittivity appear at the coercive field ±E c.

Figure 1.8 Typical field dependence of the dielectric permittivity of a tunable
ferroelectric ceramic

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b) Tunability in antiferroelectrics

Similar with FEs, the AFE materials show field -dependen t dielec tric permittivity. However
the (E) loop is quite different. As depicted in Fig. 1.9, permittivity changed in a similar way as
for FEs during the half quarter of the electric cycle: as the DC electric -field increasing from 0 to
maximum values, dielectric constant of AFEs increases sharply, and rea ches the first peak value
at A and then decreases gradually. When electric field decrease s from its maximum value down
to 0, a new dielectric maximum occurs at a lower field value. The f irst and second peak are related
to the AFE -to-FE phase switching and FE -to-AFE respectively. A similar behaviour is noticed
during the application of negative electric field. Therefore, four peaks are observed in the field
dependence of permittivity during the field cycling [70] (two “butterfl y” loops ).
There is another situation which is worth to be mentioned in this section. Fig. 1.10 show
the DC field dependence of permittivity and its corresponding P(E) loop in of an AFE materials
with irreversible AFE to FE field induced transition field-assisted irreversible AFE -to-FE phase
transition ( during reversal of electric field the FE phase remain polarized and it does not return to
AFE state) . As shown in Fig. 1.10 during i ncreasin g the bias electric field from 0, the dielect ric
constant and the dielectr ic loss increase. Wh en the electric field reached the value of the AFE -to-
FE switching field (about 30 kV/cm ), the dielectric constant drops while the FE polarization
increases. When the same is further cycled the abrupt changes in dielectric cons tant and dielectric
loss cannot be observed again and the materials respond to the electric field like a FE [71].

27
Figure 1.9 DC field dependence of pemittivity in AFEs with reverssibile AFE to FE
field induced transition [3]

Figure 1.10 a) DC field dependence of permittivity and b) its corresponding P(E) loop in
AFEs with irreversible AFE to FE field induced transition [71]

1.4.4 Energy storage properties

Parallel plate AFE capacitors are the key components for high energy storage density ,
which recently becom e increasingly important especially in pulsed power circuit applications, such
as hybrid electric vehicles (HEVs), medical devices, spacecraft, and electrical weap on systems . In
b) a)

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particular, AFEs offer enhanced energy density, lower hysteresis losses, good b reakdown strength
and high dielectric permittivity [ 72].

Losed Energy (Jloss) Stored Energy(J)P (C/cm2)
E (kV/cm)a)
b)
Lost energy (Jloss) Stored Energy (J)
E (kV/cm)P (C/cm 2)

Figure. 1.1 1 P(E) hysteresis loop and energy storage characteristics for a) FE and b) AFE
ceramic s

Fig. 1.1 1 shows the hysteresis loop and effective energy density of FEs and AFEs,
respectively . The energy storage density, energy storage loss and energy storage efficiency can be
determined from P(E) characteristics. The energy values can be calculated according to the
definition of recoverable energy density of the FE capacitor, ( i.e. at the withdrawal of applied
field) [ 71].:
𝐽=∫𝐸𝑚𝑎𝑥𝑑𝑃𝑃𝑚𝑎𝑥
𝑃𝑟 (Emax≡ applied electric field and P≡ polariza tion), (1.4.5)
And the energy loss density:
𝐽𝑙𝑜𝑠𝑠=∫𝐸𝑚𝑎𝑥𝑑𝑃𝑃𝑚𝑎𝑥
0 and energy efficiency η=J/(J+J loss) (1.4.6)

1.5 Landau -Ginzburg -Devonshire Theory o f Ferroelectric ity

Ginzburg is the first who developed a phenomenological theory for ferroelectricity [ 73].
He used the Landau theory of second -order phase transitions [ 74-75] for his formulations and
applied a similar treatment as Devonshire [ 76-78]. The main variable of FE state at equilibrium
are: the temperature (T), the spontaneous el ectric polarization (P), the electric field (E), the strain
and the stress. The main FE characteristics which are going to be discussed in the following are
polarization reversal (switching), and disappearance of spontaneous polarization above the FE
phase transition temperature T c.
The free energy F of a n homogeneous FE crystal can be generally expressed as a function
of ten variables (three components of polarization, six components of the stress tensor, and

29
temperature). If it is assumed that the easy p olarization axis has the same direction as the applied
electric field and the order parameter in the Landau theory has the same transformation properties
as the polarization vector P, the Gibbs free energy density G in the Landau -Ginzburg polynomial
expans ion in the uniaxial case can be expressed as:
𝐺=𝐹−𝐸𝑃=𝐹0+𝛼
2𝑃2+𝛽
4𝑃4+𝛾
6𝑃6−𝐸𝑃 , (1.5.1)
where 𝐹0 is the free energy density of the paraelectric phase when 𝐸=0 ,𝛼 coefficient
is pressure and temperature dependent while β, and γ are temperature -independent or less
temperature , but pressure dependent.
The equilibrium configuration is determined by finding the minima of F :
𝜕𝐹
𝜕𝑃=0, (1.5.2)
and
𝜕2𝐹
𝜕𝑃2>0. (1.5.3)
Thus the expression of electric field as a function of polarisation can be expressed as:
𝐸=𝛼𝑃+𝛽𝑃3+𝛾𝑃4, (1.5.4)
The linear dielectric susceptibility above the transition temperature can be obtained by
differentiating this equation with respect to P and then setting P = 0:
𝜕2𝐹
𝜕𝑃2=1
𝜒=𝑇−𝑇0
𝜀0𝐶, (1.5.5)
𝜒=𝑃
𝜀0𝐸=1
𝛼. (1.5.6)
Where 𝛼 expression around the Curie point is
𝛼=1
𝜒=1
𝜖0𝐶(𝑇−𝑇0), 𝛽<0,𝛾=0, (1.5.7)
The equation (1. 5.7) represent s the Landau -Ginzburg Devonsire of second -order ferro –
electric phase transitions. The transition from paraelectric to FE phase takes place at temperature
T0.
The spontaneous polarization in the FE phase is given by:
𝑃𝑠=𝑃(𝐸=0)=√𝑇0−𝑇
𝛾𝐶 𝑇<𝑇0., (1.5.8)

30
Where , 𝑇>0 is the Curie temperature, and 𝐶>0 is the Curie Weiss constant. The Curie
Weiss temperature T 0, where α changes sign , is close but not exactly coincident with the Curie
temperature T C. The equation (1.3. 7) captures the Curie -Weiss behaviour observed in t he majority
of the dielectrics in the paraelectric state, for 𝑇>𝑇0. The first and second derivatives of equation
1.5.4 are continuous and discontinuous, respectively , with temperature. In both situations, the
difference of entropy (the first derivative o f free energy) is zero at T 0. These last two expressions
suggest that Ps vanishes at T = T 0 and consequently , the dielectric susceptibility 𝜒 diverges. In the
case of P S ≠ 0, β>0 corresponds to a second -order phase transition, while β< 0 corresponds to a
first-order phase transition. For a second -order transition which occurs at T = T 0, the free energy
will evolve continuously when decreasing temperature from the first curve with a single minimum
corresponding to P = 0 (paraelectric sta te) in Fig 1 .12 to the F(P) dependence with two symmetric
minima at finite polarizations P = ±P 0, corresponding to the FE state. The signs ± indicate that the
polarization can be displayed in both direction s along the symmetry axis and corresponds to the
two energetically equivalent polar states at zero field. In the paraelectric state, P s = 0 which means
that α is a positive value when the system is in a stable paraelectric state.

Figure 1.12: Second order phase transition. (a) Free energy (F) as a function of the
polarization (P) at T > T 0, T = T 0, and T < T 0; (b) Spontaneous polarization P 0 (T) as a function
of temperature (c) The susceptibility χ and its inverse, at the equilibrium condition P0(T)

The variation of the F, P and 𝜒 associated with this second order phase transition as
described by Landau -Devonshire theory are displayed in Fig. 1 .12. The free energy as a function
of P for three typical temperatures is plotted in Fig. 1 .12 (a). The double wells in the energy
P
T
TP0
T0
T0a) b)
c)T0
T0
T0
PF

31
diagram at temperatures lower than T 0 (=T C) correspond to two stable FE states. As temperature
goes below the critical temperature, the extreme at E = 0 becomes a local unstable maximum,
while two minima emerge at ±E 0. The order parameter grows continuously from zero as the
temperature is decreased from the critical point. It can be observed in Fig. 1 .12 (c) that as the
temperature is raised, the bulk polarisation decreases continuously and vanishes at a temperature
T0.
If the Landau parameters meet the condition: β<0 and γ>0, the phase transition is of first
order.
𝛼=1
𝜀0𝐶(𝑇−𝑇0),𝛽<0, 𝛾>0 . (1.5.9)
The procedure for finding the spontaneous polarization and the linear dielectric
susceptibility is conceptually the same as before, but now the quartic and sixth order terms cannot
be neglect ed.

Figure 1.13: First order phase transition. a) Free energy as a function of the polarization
at T > TC, T = TC, and T = T 0 < TC; (b) Spontaneous polarization P as a function
of temperature (c) The susceptibility χ and its inverse

PP
T
TF
TCT0
TC T0P0
a)b)
c)T0
T0
TC

32
The temperature at which the transition takes place is, by definition, the Curie temperature
TC which exceeds T0 .
For temperature lower than this, the polarization at equilibrium is:
𝑃𝑠=(𝑃(𝐸=0))=±√1
2𝛾(⌈𝛽⌉+√𝛽2−4
𝐶(𝑇−𝑇0)𝛾) , 𝑇<𝑇0,
(1.5.1 0)
The contribution of the polarization to the dielectric polarizability in the paraelectric and
FE phases is calculated from th e electric field expression (1.5 .4) as follows:
1
𝜒=[𝜕2𝐹
𝜕𝑃2]=𝑇−𝑇0
𝐶, 𝑇>𝑇𝐶,
(1.5.1 1)
1
𝜒=[𝜕2𝐹
𝜕𝑃2]=8(𝑇−𝑇𝐶)
𝐶+3𝛽2
4𝛾, 𝑇<𝑇𝐶,
(1.5.1 2)
In this case the dielectric stiffness (inverse of the linear susceptibility) that does not vanish
at T0, shows a finite jump in both the susceptibility and the spontaneous polarization at the
transition. The spontaneous polarization changes suddenly by the magnitude (for E= 0 ). At any
temperature between TC and T0 the unpolarized phase exists as a local min imum of the free energy.
At T = Tc the three minima are energetically degenerate. As a consequence, the system’s behaviour
at T = Tc will depend on whether is approaching Tc from lower or higher temperatures. More
clearly, the system will be in one of the two finite polarization minima if it is heated from an initial
low temperature Ti < TC, whereas it will be in a paraelectric state ( P = 0) if the initial temperature
is high (Ti > TC). Fig 1.13. shows the dependences of free energy F on the parameters P and χ and
χ-1 on temperature for the first order phase transition. The FE state is stable for T < T C,
corresponding to the double wells. When temperature increases to T 1 (T1 > T C), the paraelectric
state becomes mo re stable and the phase transition occurs at T = T C. At this temperature, the
polarized state shows the same energy as the paraelectric state (the middle well). However, the
reverse phase transition from paraelectric to FE phase takes place at a different te mperature T2 (T2
< T C). Therefore, thermal hysteresis occurs between heating and cooling for the first order phase
transitions. This is a the key characteristics of this type of phase transitions. During heating,
polarization decreases abruptly and discontinuously to zero at TC (Figure 1.13(b)) and the
reciprocal of susceptibility χ-1 changes abruptly at TC.

33
The electric field can be calculated from the free energy with the relation :
𝐸=𝜕𝐺
𝜕𝑃=𝛼𝑃+𝛽𝑃3+𝛾𝑃5,
(1.5.1 3)

Figure 1.14. Schematic hysteresis in an idealized ferroelectric

The equation (1.5.16 ) indicates a nonlinear dependence of the polarization P on electric
field E. On the graphical representation it leads to a FE hysteresis loop. An ideal hysteresis loop
is shown in Fig. 1.14. In a FE state (correspond to the condition 𝑇<𝑇𝐶) there are (at least) two
minima of the free energy, corresponding to spontaneous polarizations of different spatial
orientations. There is a barrier between these minima and a certain value of electric field is
requested to switch the polarization. The Landau -Devonshire theory described previously predicts
a FE hysteresis as shown schematically in Figure 1.14, for an ideal situation where all the dipoles
have to be overturned together to switch from one polarization orientation to the other.

1.6 Landau Theory of Antiferroelectrics
P
E

34

Similarly to FEs, an AFE material transforms to a paraelectric phase at the Curie
temperature TC. This critical temperature is related to a structural phase transition between two
non-polar phases , with a dielectric anomaly at the high temperature side. In 1951, the macroscopic
phenomenological theory of AFEs was firstly proposed by C. Kittel [ 14], together with the
description of some intrinsic characteristics. Appling a similar formalism, as Landau for FEs, it
was found that the susceptibility of AFEs will be continuous and nearly constant vs. temperature
until the Curie point is reached , where a small discontinuity in the temperature coefficient may be
present. The magnitude of the anomaly depends of the nature of the transition , but is much lower
than for FEs . The AFE may obey the Curie Weiss law , but this is not an indicative of AFE cha racter
[14]. This theory is limited and quite intriguing, because it does not explain some important
phenomena observed in AFE materials , as for example the mechanism s to drive the relative spatial
positions of the two sub lattices and the cell doubling du ring AFE phase transitions. Hatt et al.
suggested a Landau -Ginzburg model for AFE phase transitions based on microscopic symmetry
[79] Recently, Pierre Tolédano et al . extended Landau theoretical model to AFEs and
demonstrated that there are symmetry criteria defining AFE transitions [ 81].
However the Kittel theory remain s the basis for describing antiferroelectricity. According
with his model the AFEs have two equivalent lattices, Pa and Pb, which could be polarized
independently and have an interaction between them. The sublattices polarizations Pa and Pb in
the Kittel model can be regarded in terms of the molecular dipole moments oriented antiparallel
to their adjacent dipoles, which results in a zero net polarization. In this case, in t he AFE state the
formula units containing a positive dipole -moment component in the a direction form one
sublattice with polarization Pa, and the formula units containing a negative dipole -moment
component in the a direction form the other sublattice with polarization Pb.
The free energy of an AFE system can be written as:
∆𝐺=𝑓(𝑃𝑎2+𝑃𝑏2)+𝑔𝑃𝑎𝑃𝑏+ℎ(𝑃𝑎4+𝑃𝑏4), (1.6.1)
where f, g, h are phenomenological coefficients. He suggested a second -order transition
from the paraelectric state to the AFE state, by truncating the free energy at the fourth order. The
difference ∆𝐺 between AFE and FE state is small and therefore , an external electric field can
induce a phase transition from AFE -to-FE state. When the field strength becomes sufficiently
large, the polarization in the direction opposite to the field abruptly switches its orientation to
become parallel to the field, resulting in a FE state. If 𝑔>0, the transition will favour the
antiparallel orientation of 𝑃𝑎 and 𝑃𝑏, making the low -temperatu re phase to be AFE. On the other

35
hand, if 𝑔<0, the transition will favour the parallel orientation of 𝑃𝑎 and 𝑃𝑏, and the transition
will lead to a FE state.
However, Kittel model [ 14] is not realistic because 𝑃𝑎 and 𝑃𝑏 are assumed at the same
location in space (or anywhere in the space) and it does not consider the symmetry change during
the AFE-to-FE field induced transition. Tolédano et al. [81] developed a Landau model able to
account more aspects of the AFE states, including local dipole orientation and crystalline structure
changing , without the need of assum ing sublattices. In the following , the main characteristics of
AFEs described by th is approach are summarised .
The emergence of polar sites constitute s a necessary condition but it is not sufficient to
explain the existence of a PE–AFE transition. In order to preserve the site symmetries at the
macroscopic level and to allow a subsequent stabilization of a FE phase under applied electric field
two condi tions are required:
Condition 1 : At the PE –AFE transition, a set of crystallographic sites undergo a symmetry
lowering that results in the emergence of polar sites and give rise to a local polarization .
Condition 2 : The AFE space -group has a symmorphic polar subgroup coinciding with the
local symmetry of emerging polar sites .
Tolédano et al. suggested a definition of PA –AFE transitions, which does not imply the
stabilization of a FE phase under applied field or a double hysteresis loop, and extends the current
characteristics of AFEs to the larger class of materials in which a polar field -induced phase
emerges from a non -polar phase:
PE–AFE transitions are structural transitions between non -polar phases where the
symmetry of crystallographic polar sites emerging at the local scale coincides with the symmetry
of a polar symmorphic subgroup of the AFE space -group, allowing the emergence of an electric
field induced polar phase at the macroscopic scale.
The dielectric properties of AFE transitions are derived from the Landau potential:
𝜙(𝜂,𝑃,𝑇)=𝜙0(𝑇)+𝛼
2𝜂2+𝛽
4𝜂4+𝛾
6𝜂6+𝑃2
2𝜒0+𝛿
2𝜂2𝑃2−𝐸𝑃 (1.6.2)
Where
𝛼=𝑎(𝑇−𝑇𝐶) (1.6.3)

36
The model involves one symmetry breaking parameter 𝜂 and the polarization P is a field –
induced order -parameter. For "proper" AFE transitions η can be correlated with local dipole
distribution either in a continuous formalism, as a polarization wave amplitude, or a combination
of local dipoles belonging to antiparallel arrays of emerging polar sites. For "improper" AFE
transitions η represents a structural (displacive or ordering) mechanism which typifies the lowering
of symmetry at the transition, the emergence of an antiparallel polarization wave amplitude being
an induced secondary effect of the preceding primary mechanism.
The equations of st ate can be obtained if φ is minimized with respect to η and P:
𝜂(𝛼+𝛽𝜂2+𝛾𝜂4+𝛿𝑃2)=0 (1.6.4)
𝑃=(1+𝛿𝜒0𝜂2)=𝜀0𝜒0𝐸 (1.6.5)
When 𝐸=0, the last two equations yield two possible stable phases: the P E phase (𝜂=
0,𝑃=0) and the AFE phase ( 𝜂≠0,𝑃=0). If 𝐸≠0 two phases are stabilized: a FE phase ( 𝜂=
0,𝑃≠0) and a phase in which the AFE has a non -zero total polarization ( 𝑃≠0) (weak FE of
ferrielectric order may be present).
Temperature -field T -E phase diagram is shown in Fig. 1.15. For 𝑇≥𝑇𝐶 the paraelectric
phase is stable at 𝐸=0, and transform into a FE phase for 𝐸≠0. For 𝑇𝐶>𝑇>𝑇0 , the AFE
phase [𝜂=±(−𝛼
𝛽)1/2
,𝑃=0] which is stable at 𝐸=0, transform in a ferrielectric ( FI) or FE
phase when 𝐸≠0. The equilibrium value of 𝜂 and 𝑃 of the induced FI or are given by
𝜂=±[(−𝛼−𝛿𝑃2)/𝛽]1/2 (1.6.6)
where 𝑃 is a real root of the Cardan equation:
𝛿2
𝛽𝑃3−(1
𝜒0−𝛼𝛿
𝛽)𝑃+𝐸=0 (1.6.7)
With the increasing of electric field the FI phase transforms across the second -order transition
curve into a FE phase:
𝐸=±1
𝜒0(−𝛼
𝛿)1/2
(1.6.8)
For 𝑇<𝑇0, the transition of the FI into the FE phase is of first order and it cross the region of the
coexistence of the FI and FE phase. As the electric field increases, the FI phase tr ansforms
discontinuously into the FE phase and the limit of stability of the FI (in the figure indicated with
𝐸𝑐1) correspond s to:

37
3𝛿3
𝛽𝑃4+𝛿𝑃2(4𝛿𝛼
𝛽−1
𝜒0)+𝛼(𝛿𝛼
𝛽−1
𝜒0)=0 (1.6.9)
When the electric field decreases, the FE phase show s a limit of stability on the 𝐸𝑐2 curve.
The meeting point of 𝐸𝑐1 and 𝐸𝑐2 provides the values of 𝑇0 and 𝐸0.

Figure 1.15. Theoretical temperature –electric field (T –E) phase diagram associated with
the free -energy given by Eq. (1. 6.2) for 𝛽 > 0 and 𝛼 >0. Hatched and hatched -dotted curves
represent, respectively, second -order transition and limit of stability curves. The thermodynamic
paths for T > T C, TC > T > T 0 and T < T 0 are described in the text.

The dielectric susceptibility at the P E-AFE transition can be deduced from following
equation . For a second order transition (𝛽>0) and below 𝑇𝐶 :
𝜒(𝑇)=𝜒0
1+𝛿𝛼𝜒
0𝑇𝑐−𝑇
𝛽 (1.6.10)
The dependence of susceptibility on temperature depends on the sign of the 𝛿 parameter (Fig. 1.16
a)). Fig. 1.16 b)) shows the temperature dependence of a first order transition ( 𝛽<0) which occur
when 𝑇1>𝑇𝐶. The dielectric permittivity upward (𝛿<0) or downward (𝛿>0) showing

38
discontinuity which reflect the attractive coupling between η and P requir ed for compensating the
repulsive interactions between parallel dipoles.

Figure 1.16. Temperature dependence of the dielectric sus ceptibility as given by Eq.
(1.6.10 ) across a second -order (a) and first -order (b) transition

The macroscopic hysteresis describes the dipole moments associated the AFE state. Antiparallel
dipole ordering is energetically more stable than the parallel dipole ordering.
However, if an external electric field is applied to domains, parallel to the Pa direction, this
field will interact with the dipoles, causing Pa to increase and Pb to decrease in magnitude. In this
way the parallel dipole arrangement is stabilized and thus , its free energy is lowered. At sufficiently
large field strength EAF, the dipoles with Pa antiparallel to the external field will switch , so that Pa
becomes parallel a nd the phase transition from AFE to FE phase is induced. This results in a state
in which the dipole components are parallel along the a direction and antipa rallel along the b
direction. It is a high field FE state with the polarization along a and with a cell size doubled that
of the Kittel model [ 80]. When the field decreases to a critical value, E AF, the minimum energy of
FE ordering restores back to the in itial magnitude and as a consequence the material recover its
AFE phase. As a result, at high positive or negative fields a field -induced FE hysteresis loop
appears, giving rise to the double hysteresis loop, as shown in Fig. 1.17 , which is the main feature
of AFEs.

39

Figure 1.17 Typical electric field dependence P(E) of the polarization of an AFE material
together with the energy diagrams explaining the double hysteresis loops

P
E
EFAEAF

40

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