I.
INTRODUCTION
Ferroelectric materials (crystals) display a spontaneous electrical polarization, which classifies them into
the pyroelectric family. A crystal is considered to be ferroelectric when its polarization can be shifted
from one (or more) orientational state(s) to another by an electric field. 1 The crystal structure of any
two orientational states is identical, and the only difference between them lies with the electric
polarization vector in the absence of an electric field. Therefore, ferroelectricity is a concept that is
based not only upon the structure a crystal, but also upon the dielectric behavior of the crystal. The
centers of the positive and negative charges of ferroelectric materials may not match even without the
exposition to an external electric field. 2
Ferroelectric materials are used to manufacture various devices such as pyroelectric sensors, electrooptic devices, high-permittivity dialectrics, piezoelectric devices, and PTC (Positive Temperature
Coefficient of resistivity) components. However, ferroelectric devices are not used commercially in some
areas where competitive materials hold an advantage. For instance, light sensors are usually made from
semi-conductive materials which are more sensitive and respond faster than ferroelectrics.
Furthermore, magnetic devices are more popular than ferroelectric materials for memory applications,
and liquid crystals are typically used for optical displays, even though ferroelectrics could be used as
well. One reason for this is the lack of well-documented knowledge on ferroelectric materials. 4 Perhaps
the biggest impact of ferroelectric crystals today is its utilization in dielectric ceramics for capacitor
applications, ferroelectric thin films for non-volatile memories, piezoelectric materials for medical
ultrasound imaging and actuators, and electro-optic materials for data storage and displays. 7
Polarization:
The main feature that differentiates ferroelectrics and other pyroelectrics is that the spontaneous
polarization for ferroelectrics can be inverted, at least partially, by applying an electric field.
Ferroelectric crystals have areas with homogeneous polarization called ferroelectric domains, which
means that all the electric dipoles are aligned in the same direction. There may be many domains in a
crystal, separated by interfaces called domain walls. Originally (before an electric field is applied), these
domains are oriented at random, without a net microscopic polarization. 3 Applying an electric field
induces movement of the domain walls, which results in the orientation of the domains along the
applied electric field in a net microscopic polarization. A very strong field could even lead to the reversal
of the polarization in the domain, known as domain switching. 7
When the electric field is reduced, the friction in the domain wall inhibits a complete reduction of the
polarization to zero, which means the material will preserve a remanent polarization, which can be
utilized in a variety of applications such as non-volatile semiconductors, memories and ferroelectric
cathodes. 4 In order to depolarize the material, a negative field is applied, known as the coercive field. 3
Crystal perfection, electrical conductivity, temperature and pressure are all factors which affect the
reversibility of polarization. However, the spontaneous polarization of a material is mostly temperaturedependent and its existence can be perceived by observing the flow of charge to and from the surfaces
as the temperature changes. 2 Changes in the macroscopic polarization can be effective by heating the
material above the Curie temperature, which causes the ferroelectric domains to randomize. 3
Curie Temperature:
Above Curie temperature (Tc), the material is not ferroelectric, but still possesses interesting dielectric
properties. In the high temperature phase, the crystal is called a paraelectric material, and it changes
from the high-temperature paraelectric phase to the low-temperature ferroelectric phase at Tc. In the
ferroelectric phase, the crystal structure is different to that of the paraelectric phase, due to small
displacements of the atoms in the unit cell. The values of Tc are different for each material. Examples
for some perovskites are shown in Table 1. Above the highest value listed, the material is in its cubic
phase. At the first (highest) value of Tc, the crystal distorts into a lower symmetry phase. 6
Table 1. Ferooelectric transition temperatures in kelvins in some perovskites(6)
Crystals
SrTiO3
BaTiO3
PbTiO3
NaTaO3
NaNbO3
KNbO3
Tc(K)
105
403, 278, 183
763
903, 823, 753
916, 845, 793, 753, 638, 73
691, 498, 263
Hysteresis Loops:
As previously mentioned, the most important feature of ferroelectric materials is the polarization
reversal that occurs when an electric field is applied. They also feature multiple domains. Furthermore,
when an electric field is applied, domain walls are either reduced (in ceramics) or completely removed
(in crystals). 3 Perhaps one of the most recognized results of the domain-wall switching in ferroelectric
materials is the occurrence of the ferroelectric hysteresis loop (Figure 1). At small values of the
alternating electric field, the polarization increases with the field amplitude in a linear fashion (segment
AB in Figure 1). In this area, the electric field is too weak to switch domains with unfavorable direction of
polarization. When the field is increased, the polarization of domains with unfavorable direction of
polarization starts to switch along directions that are as close as possible to the direction of the field,
increasing the measured charge density (segment BC) quickly in a strong nonlinear fashion. 8
After all the domains are aligned (point C) the ferroelectric once again acts as a linear dielectric
(segment CD), as seen in Figure 2. As the electric field begins to weaken, some domains will switch back.
However, in the absence of a field, the polarization is not zero (point PR), and it is known as remanent
polarization. In order to reach the null polarization state, the field must actually be reversed (point F),
and it is identified as the coercive field, EC. A further increase of a negative field causes a new alignment
of dipoles and saturation (point G). Then, to complete the cycle, the field strength is reduced to zero and
reversed. 8
Fig 1. A hysteresis loop illustrating the coercive field Ec, the spontaneous polarization Ps,
and the remanent polarization PR. The hexagons with gray and white regions represent
schematically repartition of two polarization states in the at different fields. (8)
Fig 2. Dielectric polarization (3)
An ideal hysteresis loop is symmetrical, so the positive and negative remanent polarizations, as well as
the positive and negative coercive fields should be equal. However, factors like the thickness of the
sample, presence of charged defects, mechanical stresses, preparation conditions, and thermal
treatment can affect the coercive field, the spontaneous and remanent polarization, and the shape of
the loop. 8 Experimentally, a single hysteresis loop is seen when the spontaneous polarization of a
ferroelectric is reversed, and a double hysteresis loop is observed for the induced phase transition of
antiferroelectrics, when a low frequency ac field of appropriate strength is applied. Ferroelectric
materials exhibit similar behavior, expect for the fact that the hysteresis is in electric polarization (P) as a
function of applied electric field (E). 1
Mean Field Theory:
The occurrences known as critical phenomena, which take place very close to a second-order phase
transition have been given considerable attention. This special interest focuses on the nature of the
long-range spatial correlations and thermodynamic singularities which develop at a critical point, and on
increasing the knowledge of the basic simplicity and apparent resemblance of critical behavior in very
dissimilar physical contexts. To better comprehend this concept, it is necessary to examine the
divergence of the dielectric susceptibility itself. In thermodynamic theory, identifying the origin of the
driving mechanism is not as important as identifying the primary pair of conjugate variables (electric
displacement (Di) and electric field (Ei)) and defining lowest-order non-linearities in the equation of state
involving them. 8
Generally the most convenient potential is the elastic Gibbs function (G1) expressed as a function of
temperature, stress, and displacement. The simplest theoretical situation results with the assumption
that Di=D is directed along one of the crystallographic axes only (i.e. that spontaneous polarization and
the applied electric field both occur along this direction), that all stresses are zero, and that the nonpolar phase is centrosymmetric. Even though these restrictions are not essential, with their presence the
free energy G1 can be expressed in the straightforward polynomial form
𝛼
𝛾
𝛿
𝐺1 = ( ) 𝐷 2 + ( ) 𝐷 4 + ( )𝐷 6
2
4
6
where energy is calculated from the non-polar phase and the polynomial is terminated arbitrarily at D6
for mathematical simplicity. Assuming 𝛾 𝑎𝑛𝑑 𝛿 are independent of temperature, the order of the
transition depends on the sign of 𝛾 (with 𝛿 being positive for stability reasons because 𝐷 → ∞). 2
Second-order ferroelectric phase transitions
Distinguishing the elastic Gibbs function with respect to D at constant temperature results in the
dielectric equation of state
𝐸 = 𝛼𝐷 + 𝛾𝐷 3 + 𝛿𝐷 5
It is clear from Figure 3 (a) that P goes through a continuous (second-order) phase transition as 𝛼 passes
through zero, but based on 𝐸 = 𝛼𝐷 + 𝛾𝐷 3 + 𝛿𝐷 5
it is evident that 𝛼 is in fact the reciprocal
permittivity at zero field in the non-polar phase. Conventional phenomenological theory assumes that
near the Curie point (Tc), 𝛼 depends on temperature linearly: 𝛼 = 𝛽(𝑡 − 𝑇𝑐) = 𝑘 𝑥,𝑇>𝑇𝑐 with 𝛽 a
positive constant and k the reciprocal of the dielectric constant (𝜀). Experimentally, this is the most
prevalent form of reciprocal permittivity. It is also the form that results from a simple mean-field
statistical model as 𝑇 → 𝑇𝑐 where the zero-field susceptibility x is related to the dimensionless dielectric
constant 𝜀 by 𝜀 = 1 + 𝑥. 2 The spontaneous polarization and isothermal dielectric constant below Tc are
calculated as follows:
𝑃= 𝛽
(𝑇𝑐 − 𝑇)
𝛾
𝑎𝑛𝑑 𝑘 𝑥,𝑇 = 𝛽(𝑇 − 𝑇𝑐) + 3𝛾𝑃2
After one is substituted into the other,
𝑘 𝑥,𝑇 = 2𝛽(𝑇𝑐 − 𝑇)
𝑇 < 𝑇𝑐
𝑃→0
These temperature dependencies are qualitatively sketched in Figure 3 (b) and (c). Particularly, the slope
of the reciprocal permittivity 𝑘 𝑥,𝑇 vs. temperature curve below Tc is negative and twice as large as
above Tc. 2
Fig 3. Qualitative temperature dependence of the free energy vs. spontaneous polarization (Ps) and reciprocal isothermal
permittivity (𝑘 𝑥,𝑇 ) near a second-order ferroelectric transition. (2)
On the other hand, most ferroelectric phase transitions are of first-order (not second), with a gap in the
first derivatives of the thermodynamic potential. Granted, a first-order transition point is not a
singularity but simply a point at which the thermodynamic potentials of the two phases are ponds to an
ordinary equilibrium state, and the stable state jumps from one to other at the transition. 2
Considering the free energy for 𝛾 negative and 𝛿 positive, it is possible for the potential G1 to develop
equal minima at D=0 and at non-zero values D = ±𝐷𝑐 (P =±𝑃𝑐 ) and to define a first-order transition. The
parameter 𝛼 is still the reciprocal isothermal permittivity at constant stress in the non-polar. 𝛼 =
𝛽(𝑡 − 𝑇𝑜) but now, as seen below, To is called Curie-Weiss temperature and is not equal to the
transition temperature Tc. Writing 𝛾 = −𝛾′
𝛽
𝛾′
𝛿
𝐺1 = ( 2 ) (𝑡 − 𝑇𝑜)𝐷 2 − ( 4 ) 𝐷 4 + (6 ) 𝐷 6
If 𝛽, 𝛾 ′ , 𝑎𝑛𝑑 𝛿 are assumed to be positive constants, the dielectric equation of state becomes
3
𝐸 = 𝛽(𝑇 − 𝑇𝑜)𝐷 − 𝛾 ′𝐷 + 𝛿𝐷 5
As a function of temperature, the curves G1 take the form shown in Figure 4 (a). As shown, a zero-field
first-order transition takes place when G1 and its first derivative with respect to D are both zero for a
non-zero value of D; that is when are simultaneously satisfied. 2 The solution is
3
3𝛾′
𝑇 = 𝑇𝑐 = 𝑇𝑜 + ( ) (𝛾 ′ )2 (𝛽𝛿)−1 𝑎𝑛𝑑 𝑃2 = 𝑃𝑐 2 =
16
4𝛿
𝑇 = 𝑇𝑐
for the value of spontaneous polarization at Tc. Zero-field permittivity can be deduced as
2
𝑘 𝑋,𝑇 = 𝛽(𝑇 − 𝑇𝑜) − 3𝛾 ′𝑃 + 5𝛿𝑃4
Above Tc, 𝑘 𝑋,𝑇 = 𝛽(𝑇 − 𝑇𝑜). Below Tc, the spontaneous polarization takes effect, and solving the
equation of the Gibbs energy for spontaneous polarization (D=P when E=0) and substituting in the
previous equation gives
2
𝑘 𝑋,𝑇 =
3𝛾 ′
+ 8𝛽(𝑇𝑐 − 𝑇)
4𝛿
𝑇 → 𝑇𝑐
in the limit of T approaching the transition point Tc from below. Since the paraelectric reciprocal
permittivity can be written as
2
𝑘 𝑋,𝑇 =
3𝛾 ′
+ 𝛽(𝑇 − 𝑇𝑐)
16𝛿
𝑇 → 𝑇𝑐 +
it is deduced that the permittivity is finite but discontinuos at Tc and that the ratio of the slope dk/dT
immediately below Tc to that immediately above Tc is -8. The diagrams for spontaneous polarization
and reciprocal permittivity are shown in the Figure 4 (b) and (c). 2
Fig 4. Qualitive temperature dependence vs. spontaneous polarization (Ps) and
reciprocal isothermal permittivity (𝑘 𝑋,𝑇 ) for a first-order ferroelectric transition. (2)
Material Defects:
Inhomogeneities, such as impurities and radiation damage, have an impact on the dielectric properties
and switching behavior of ferroelectrics. Defects in any crystalline pattern usually cause deformation of
the surrounding volume and alteration of the local fields. These adverse impacts are more difficult to
quantify than the effects of domain walls because there are no mechanical compatibility conditions and
the degree of the crystal deformation depends on the nature of the defect, its site in the crystal, and the
interaction between the host and the defect. 2
The macroscopic average value of the polarization ∆𝑃 due to the defects may or may not be inverted. If
it is not, then the coercive field will depend on both the field required to switch the defects and the sign
and magnitude of ∆𝑃. The presence of defects tends to increase the coercive field. Based on the
distribution of ∆𝑃 throughout the crystal volume, the defects can have a significant impact on the
switching properties if ∆𝑃 does not reverse in an external field. If all the dipoles have the same
orientational state, the hysteresis loops will appear biased as shown in Figure 5 (a). If the dipole
orientations are ordered over large regions, but different regions are antiparallel in the same way as
ferroelectric domains, then the hysteresis loops could be seen as in Figure 5 (b). If the dipoles are
completely random, which is possible if they were introduced into the crystal in a nonpolar phase, then
the loop would look normal with an increased coercive field. 2
Fig 5. Biased hysteresis loops which may arise owing to the presence of defects in a
ferroelectric crystal.