Losses in Ferroelectric Materials
Gang Liu
1,2,3
, Shujun Zhang
2
, Wenhua Jiang
2
, and Wenwu Cao
1,2,*
1
Condensed Matter Science and Technology Institute, Harbin Institute of Technology, Harbin
150080, China
2
Materials Research Institute, The Pennsylvania State University, University Park, PA 16802,
USA
3
Center for High Pressure Science and Technology Advanced Research, Shanghai 201203,
China
Abstract
Ferroelectric materials are the best dielectric and piezoelectric materials known today. Since the
discovery of barium titanate in the 1940s, lead zirconate titanate ceramics in the 1950s and
relaxor-PT single crystals (such as lead magnesium niobate-lead titanate and lead zinc niobate-
lead titanate) in the 1980s and 1990s, perovskite ferroelectric materials have been the dominating
piezoelectric materials for electromechanical devices, and are widely used in sensors, actuators
and ultrasonic transducers. Energy losses (or energy dissipation) in ferroelectrics are one of the
most critical issues for high power devices, such as therapeutic ultrasonic transducers, large
displacement actuators, SONAR projectors, and high frequency medical imaging transducers. The
losses of ferroelectric materials have three distinct types, i.e., elastic, piezoelectric and dielectric
losses. People have been investigating the mechanisms of these losses and are trying hard to
control and minimize them so as to reduce performance degradation in electromechanical devices.
There are impressive progresses made in the past several decades on this topic, but some
confusions still exist. Therefore, a systematic review to define related concepts and clear up
confusions is urgently in need. With this objective in mind, we provide here a comprehensive
review on the energy losses in ferroelectrics, including related mechanisms, characterization
techniques and collections of published data on many ferroelectric materials to provide a useful
resource for interested scientists and engineers to design electromechanical devices and to gain a
global perspective on the complex physical phenomena involved. More importantly, based on the
analysis of available information, we proposed a general theoretical model to describe the inherent
relationships among elastic, dielectric, piezoelectric and mechanical losses.
For multi-domain ferroelectric single crystals and ceramics, intrinsic and extrinsic energy loss
mechanisms are discussed in terms of compositions, crystal structures, temperature, domain
configurations, domain sizes and grain boundaries. The intrinsic and extrinsic contributions to the
© 2015 Elsevier B.V. All rights reserved.
*
Corresponding author: dzk@psu.edu.
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Mater Sci Eng R Rep. Author manuscript; available in PMC 2016 March 01.
Published in final edited form as:
Mater Sci Eng R Rep. 2015 March 1; 89: 1–48. doi:10.1016/j.mser.2015.01.002.
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total energy dissipation are quantified. In domain engineered ferroelectric single crystals and
ceramics, polarization rotations, domain wall motions and mechanical wave scatterings at grain
boundaries are believed to control the mechanical quality factors of piezoelectric resonators. We
show that a thorough understanding on the kinetic processes is critical in analyzing energy loss
behavior and other time-dependent properties in ferroelectric materials. At the end of the review,
existing challenges in the study and control of losses in ferroelectric materials are analyzed, and
future perspective in resolving these issues is discussed.
Keywords
Energy dissipation; energy loss; quality factor; ferroelectrics; piezoelectric; dielectric
1. Introduction
1.1 Ferroelectricity
Ferroelectricity is the property of certain materials with a spontaneous electric polarization
(P
s
). There are more than one energetically degenerate states in the ferroelectric phase and it
is possible to switch polarization between these energetically degenerate states when a
suitably strong electric field (E) is applied [1–3]. The energy degeneracy of these
polarization states could be broken under a bias (electric or elastic) field, but polarization
switching can still be induced by electric field between these ferroelastic states. In general,
materials that have switchable spontaneous electric polarization are called ferroelectric
materials. For ferroelectric materials, except those melt before transforming into non-polar
crystalline phase, a permanent non-zero polarization is usually formed at a particular
temperature, namely the Curie temperature (or Curie point) T
C
. This is associated with a
paraelectric-ferroelectric phase transition and reflected in terms of a softening of an optical
mode at the Brillion zone center, which produces the separation of positive and negative
charge centers below the Curie temperature. Such non-zero spontaneous polarization is the
common characteristic of ferroelectrics, resulting in the nonlinear polarization-electric field
(P-E) hysteresis loops showing the variation of electric polarization with electric field and
the polarization switching behavior. The hysteresis nature of the P-E relationship in
ferroelectrics demonstrates that the polarization is dependent not only on the applied field,
but also on the history. In principle, from P-E loops ferroelectricity can be identified directly
and the ferroelectric parameters, including spontaneous polarization (P
s
), remnant
polarization (P
r
) and coercive field (E
c
), can be determined. The general features of the P-E
curves in non-ferroelectrics and ferroelectrics are illustrated in Fig. 1. One can see that
among various ferroelectrics, the maximum achievable polarization is the largest for single
crystals and smallest for thin films. Such obvious difference can be understood from the fact
that for polycrystalline ceramics the intergranular interactions leads to a strain restriction on
polarization switching, while for thin films the substrate interface further limits the poling
ability of the ferroelectric materials [2].
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1.2 Ferroelectric materials and their applications
In 1920, the ferroelectric phenomenon was discovered in Rochelle salt by J. Valasek, who
demonstrated for the first time that ‘permanent polarization is the natural state’ of Rochelle
salt and published the first hysteresis loop of a ferroelectric material [4–5]. After that, it was
not until the 1940s, the discovery of ferroelectricity in barium titanate (BaTiO
3
) forwarded
the study of ferroelectricity to practical application research from pure scientific curiosity
[6–8]. In 1950s, the discovery of lead zirconate titanate ceramics (Pb(Zr
1−x
Ti
x
)O
3
, PZT for
short) is believed to be a milestone in ferroelectric research [9–10]. The investigations on
this binary system clearly demonstrated the importance of composition-induced
ferroelectric-ferroelectric phase transitions, emphasizing on the concept of morphotropic
phase boundary (MPB) that benefits piezoceramic research [9–12]. In 1980s and 1990s,
ultrahigh piezoelectricity and superior electromechanical coupling properties were reported
in relaxor-PbTiO
3
(relaxor-PT) ferroelectric single crystal systems [13–14]. These materials
exhibit a piezoelectric effect that is three to ten times larger than that of conventional
piezoelectric ceramics, and the research on this new generation of single-crystal
ferroelectrics has progressed tremendously in the past two decades [14–22]. The giant
piezoelectric coefficient and high electromechanical coupling factors of relaxor-PT single
crystals have been extensively studied, which are closely associated to engineered domain
configurations [14, 23–32]. The relaxor-PT crystals have cubic perovskite structure in the
paraelectric phase, while ferroelectric state can be either in rhombohedral (R), monoclinic
(M), orthorhombic (O), or tetragonal (T) phases, depending on the PT content and poling
process [25–27]. Various artificial domain configurations can be formed in the ferroelectric
state by poling the crystals along different crystallographic directions, which produces very
different piezoelectric, electromechanical and mechanical properties, as listed in Table 1
[28–30]. For example, the highest longitudinal piezoelectric coefficient d
33
can be achieved
in the engineered ‘4R’ (where ‘4’ means four equivalent domain variants and ‘R’ denotes
the rhombohedral phase crystals, as listed in Table 1) domain configuration, while the
highest shear piezoelectric coefficient d
15
was observed in the ‘1R’ single domain
configuration. In addition, the mechanical quality factor (the parameter evaluating energy
loss, detailed in later) was reported to be on the order of ~80 for the longitudinal mode in the
‘4R’ domain state, increased to about 600 in the ‘2R’ domain state and further enhanced to
over 1000 in the single domain ‘1R’ state. Fig. 2 shows the piezoelectric activity of typical
ferroelectrics that included both lead-containing and lead-free materials. It is clear that the
lead-based ferroelectrics with ABO
3
perovskite structure are still the best piezoelectric
materials known today.
Ferroelectric materials have been extensively used for a wide range of applications,
including resonators, actuator, transducers, transformers, sensors, non-volatile FeRAM,
capacitors, etc. [37–40] The main feature of ferroelectric materials is the presence of
spontaneous polarization, which can be reoriented by applying an external field, including
electrical and stress fields. The regions with uniform polarization are called ferroelectric
domains, and the interface boundary between two adjacent coherent domains is called a
domain wall, which is often a crystal plan with discrete orientations allowed by the strain
compatibility relation in a given symmetry. Both polarization rotation/elongation and
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domain wall motions were thought to be the dominant factors controlling the macroscopic
properties of ferroelectrics [15, 41–43].
Innovations in electromechanical devices continue to be the driving force for the
development of new ferroelectric materials, and the understanding of external field induced
behavior, including piezoelectric response and energy loss (or dissipation) behavior. For a
piezoelectric material, the total loss can be separated into three parts: elastic, dielectric and
piezoelectric losses [23, 44–47]. These losses are associated with non-equilibrium
thermodynamics. Investigations on the energy loss of ferroelectrics are critically important,
which can lead to a better understanding on anelasticity, dielectric relaxation and
piezoelectric hysteresis [48–51]. From a practical application viewpoint, energy loss may
cause significant heat generation in electromechanical devices under strong field driving
condition, leading to device failure, especially in high-power transducers and ultrasonic
motors [23, 44, 47, 52–54]. Although high mechanical loss is associated with broader
bandwidth in sensors and transducers, but it will reduce device sensitivity [47]. For very
high frequency ultrasonic transducers, the dimensions of each element may be down to
micrometers so that the sensitivity is very important. High attenuation at high frequencies
will significantly affect the performance of high frequency transducers [55–56].
1.3 Motivation to study losses
Up to date, the complex energy loss behavior in ferroelectrics is far from being fully
understood, especially in ferroelectric perovskite materials with high electromechanical
coupling factors, such as soft Pb(Zr,Ti)O
3
(PZT) ceramics (k
33
~ 0.75) and relaxor-PbTiO
3
single crystals (k
33
~ 0.90) [23–24, 47]. In practice, the Butterworth Van Dyke equivalent
circuit and the ‘half power point’ measurements (3 dB method, see Fig. 3) have been
extensively used to measure the mechanical quality factor Q
m
of a piezoelectric resonator
[57]. Nevertheless, such approaches produce some confusion due to the difference between
values measured at resonance and antiresonance frequencies, and could not bridge those
engineering parameters to the origin of energy loss in ferroelectrics, which are strongly
associated with various material characteristics, such as piezoelectric response, polarization
switching, and ferroelectric/ferroelastic domain microstructures [24, 58]. In addition, the
measured mechanical quality factor Q
m
is easily affected by measurement conditions,
including but not limited to, frequency, vibration mode, sample surface condition and
adhesion of the electrode. [23] Thus, it is necessary to systematically study the energy loss
behavior in ferroelectric materials, explore the origin of these energy dissipations/losses and
understand their correlation with material composition, crystal phase, domain state and
external conditions, such as temperature, field, frequency, etc.
The intension of this review is to survey the current status of research on different losses in
ferroelectric materials, analyze available data to clear some confusions and more
importantly, to clarify the inherent energy loss mechanisms and propose reliable methods for
the characterization of dielectric, elastic and piezoelectric losses to provide a resource for
researchers and engineers interested in high power electromechanical devices.
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1.4 Scope of this article
The article will be divided into 7 sections. In Section 1, the background of ferroelectricity is
briefed, and the motivation in the study of energy losses in ferroelectric materials is given.
In Section 2, theories on energy losses in ferroelectrics will be surveyed, with the emphasis
on the relationship between mechanical Q
m
and experimentally measured quality factors Q
r
or Q
a
. Then, a universal method describing various losses in ferroelectrics is proposed. In
Section 3, various methods for ferroelectric materials characterization, especially those for
evaluating the loss behavior, will be surveyed and discussed. In Sections 4 and 5, the
mechanisms of energy loss will be investigated in terms of intrinsic and extrinsic
contributions, which correspond to polarization elongation/rotation (lattice deformation),
and ferroelectric domain and/or grain effects. In Section 6, loss behavior of selected
ferroelectric materials, including perovskite ferroelectric ceramics, relaxor-PbTiO
3
single
crystals, as well as lead-free piezoelectric single crystals, are analyzed. Finally, in Section 7,
a summary and future perspective will be given and emerging future topics are suggested.
2. Theories on energy losses
2.1 Basic concepts
2.1.1 Elastic, dielectric and piezoelectric losses—The material constants reported in
the literature usually do not include energy losses, while in reality, materials constants of the
ferroelectrics are complex quantities, which could be expressed in the following form [47,
59–62]:
(1a)
(1b)
(1c)
where tan γ, tan δ and tan θ are the elastic, dielectric and piezoelectric losses, respectively.
These imaginary parts reflect delayed responses under corresponding external stimuli.
Please note that the power dissipation of a passive material must always be positive,
however, the piezoelectric loss tan θ might either positive or negative and there are many
constraints on the permitted values of the elastic, dielectric, and piezoelectric losses [51, 59].
Some of these “losses” only represent the time shifted response, may not be real “energy
loss”
2.1.2 Mechanical quality factor—The mechanical loss of the material is generally
described by its reciprocal quantity, i.e., the mechanical quality factor Q
m
, which is
proportional to the ratio of the total stored mechanical energy over the energy loss within
one complete vibration cycle,
(2)
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