Due to the nature of domains, ferroics, including ferromagnetic, ferroelectric, and ferroelastic materials, exhibit hysteresis phenomena with respect to external driving fields (magnetic field, electric field, or stress). In principle, every ferroic material has its own hysteresis loop, like a fingerprint, which contains information related to its properties and structures. For ferroelectrics, many characteristic parameters, such as coercive field, spontaneous, and remnant polarizations can be directly extracted from the hysteresis loops. Furthermore, many impact factors, including the effect of materials (grain size and grain boundary, phase and phase boundary, doping, anisotropy, thickness), aging (with and without poling), and measurement conditions (applied field amplitude, fatigue, frequency, temperature, stress), can affect the hysteretic behaviors of the ferroelectrics. In this feature article, we will first give the background of the ferroic materials and multiferroics, with an emphasis on ferroelectrics. Then it is followed by an introduction of the characterizing techniques for the loops, including the polarization–electric field loops and strain–electric field curves. A caution is made to avoid misinterpretation of the loops due to the existence of conductivity. Based on their morphologic features, the hysteresis loops are categorized to four groups and the corresponding material usages are introduced. The impact factors on the hysteresis loops are discussed based on recent developments in ferroelectric and related materials. It is suggested that decoding the fingerprint of loops in ferroelectrics is feasible and the comprehension of the material properties and structures through the hysteresis loops is established.
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Ferroic materials, including ferromagnetic materials, ferroelectric materials, and ferroelastic materials etc., have been extensively studied since their discoveries, due to the fact that the subject of ferroic materials covers a large number of topics in material science, physics, and engineering aspects.[1, 2] A ferroic is a material that adopts a spontaneous and switchable internal alignment, where the alignment of spontaneous magnetization can be switched by a magnetic field in ferromagnetics, whereas the spontaneous polarization alignment can be switched by an electric field in ferroelectrics and the spontaneous strain alignment can be switched by a stress field in ferroelastics; thus, ferroic materials are those which involve at least one phase transition changing the directional symmetry of their respective prototype symmetry.[3-6] For ferroic materials, the magnetization, polarization, and spontaneous strain can be readily controlled by their conjugate magnetic-, electric-, and stress-fields, respectively, thus promising for various functional devices, such as magnetic memories, nonvolatile FRAM (ferroelectric random access memory), electromechanical devices, and shape memory alloy, to name a few.[7-10] In addition, the ferroic orderings can also be tuned by fields other than their conjugates, giving rise to “multiferroic,” which combines any two or more of the primary ferroic ordering in the same phase.[11, 12] One of the most appealing aspects of multiferroics is the magnetoelectric coupling, for example, not only electric field may control magnetization but also the polarization may be tuned by magnetic field. Figure 1 shows the phase control in ferroics and multiferroics. In one multiferroic material, four physical quantities affect others mutually, significantly expands the scope of functional material research and sheds light on the exploration of new functional devices design.
There are many common features among ferroic materials, such as the existence of domains, which can be switched with respect to external driving fields; the existence of phase transition from parent phase to ferroic phase, which can be regarded as derived by a small distortion of the prototype symmetry; anomalous responses near the ferroic phase transition [polymorphic phase transition (PPT)] showing a strong nonlinear behavior; and strong temperature-dependent properties in the proximity of the PPT, etc.[1, 2] In this feature article, ferroelectric materials will be surveyed, with emphasis on the hysteresis loops. According to the principle of crystallographic symmetry, ferroelectricity can only be found in crystals with unipolar axis (10 point groups, including 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, and 6mm). Among ferroelectric family, materials with an ABO3 perovskite structure have been extensively studied. As proposed by Slater, the ferroelectric distortion is due to the B cation “rattling” in rigid ion (oxygen) cage. Later Cochran suggested that a lattice mode involving all ions could soften and lead to the displacive instability.[14, 15] Based on ab initio calculation including the effects of charge distortion and covalency, it was demonstrated that the hybridization between the titanium 3d states and the oxygen 2p states in Ti–O octahedra is essential for ferroelectricity. Although the origin of the ferroelectricity is still controversial, these models help us understand the ferroelectricity much deeper than before.
1.2 Ferroelectric Materials
It is almost one century since the discovery of ferroelectricity in Rochelle salt,[17-19] after which, there are many developmental milestones along the way, which have been well reviewed by Kanzig, Cross and Newnham, Fousek, and Haertling. Here, we only survey some important events in the development of ferroelectric materials. In 1921, it was Valasek who first discovered the ferroelectricity in Rochelle salt, later on, ferroelectricity was reported in a new system KH2PO4 (KDP) in 1935. It was generally accepted that ferroelectricity was highly correlated with the hydrogen bonds in the early Rochelle salt and KDP periods. However, with the discovery of the ferroelectricity in simple perovskite BaTiO3 (BT) in 1944, such a hypothesis became invalid.[21, 23] The discovery of BT is very important to the development of ferroelectrics, as this finding opened a door to explore ferroelectric systems with similar crystalline structure, leading to the “perovskite” era. Meanwhile, it has challenged the scientists to model the ferroelectricity without the interference by the complex crystalline structure as encountered in Rochelle salt and KDP. In 1951, the concept of antiferroelectricity was proposed by Kittel, where the chains of ions in the crystal are spontaneously polarized, but with neighboring chains polarized in antiparallel directions. In the same year, this idea was firstly identified in PbZrO3 (PZ) material by Sawaguchi et al. and Shirane et al. As triggered by Shirane et al., the solid solution between PZ and PT [Pb(Zr1−xTix)O3, or PZT] has been extensively studied.[27-29] In 1954, Jaffe et al. reported a morphotropic phase boundary (MPB) in PZT solid solution. The boundary between tetragonal and rhombohedral phases is nearly vertical, demonstrating temperature-independent properties. Of particular importance is that PZTs with composition at MPB show maximum dielectric and piezoelectric properties, being much higher than those of BT. In addition, dopant strategies have been adopted in PZT ceramics, leading to a series of “hard” and “soft” materials.[31, 32] The technologically importance of PZT was established in 1960s and dominated the piezoelectric material market up to now. From that time on, searching high-performance piezoelectric materials in solid solution with an MPB became a standard approach for material scientists. It should be noted that during this period, most of today's practical ferroelectric materials were discovered, including LiNbO3 with corundum structure, KNbO3 /(K,Na)NbO3,[34, 35] and (Na0.5Bi05)TiO3[36, 37] with perovskite structure and PbNb2O6 with tungsten bronze structure, to name a few. In 1961, a new type perovskite ferroelectric Pb(Mg1/3Nb2/3)O3 (PMN) was firstly synthesized by Smolenskii, which was categorized to relaxor ferroelectric, manifesting itself by the diffused phase transition and strong dielectric dispersion as a function of frequency.[39, 40] Meanwhile, a series of relaxor ferroelectrics with complex perovskite structure, such as Pb(Zn1/3Nb2/3)O3 (PZN), Pb(Yb0.5Nb0.5)O3 (PYN) and Pb(Sc0.5Nb0.5)O3 (PSN), etc., have been widely studied in 1970s. Analogous to PZT, relaxor end-members were reported to form solid solutions with classical ferroelectric PbTiO3 (PT), with MPB compositions being located at PT ~ 8%–35%, exhibiting high dielectric and piezoelectric properties. Of particular significance is that some relaxor-PT ferroelectrics can be grown into single crystal forms, demonstrating superior piezoelectric and dielectric properties when compared to their polycrystalline ceramic counterparts, which have been studied in 1980s.[42, 43] However, the research fell into troughs at the beginning of 1990s due to the lack of large-size crystals. It was until 1997, Park and Shrout reported large-size relaxor-PT single crystals and their ultrahigh-field–induced strains, being on the order of 1.7%, attracted extensive attentions from both the material scientists and physicist.[44, 45] It was believed as a breakthrough in the past 50 yr for the ferroelectric materials.[46, 47] Stimulated by the relaxor-PT single crystals, together with the interest from the lead-free piezoelectric materials and multiferroics, ferroelectrics receives its resurgence at the beginning of this century.[10, 37]
1.3 Hysteresis Loop
The most common feature of the ferroic materials is the occurrence of domain structure through the spontaneous breaking of the prototype symmetry, manifesting themselves as the hysteresis loops with the respective conjugate field. The appearance of the domains is to minimize the free energy when ferroic materials undergo a phase transition from high-temperature symmetric phase to low-temperature phase with a low symmetry. The symmetry of any ferroic phase of a material can be regarded as derived by a small distortion of the prototype symmetry. For example, a BT crystal transfers from a cubic paraelectric phase into tetragonal ferroelectric phase at 120°C during cooling. A boundary separating two adjacent domains is named as domain wall, which is determined by the crystalline symmetry. Due to the large internal stress generated during further cooling, a large domain usually splits into many small domains. The mainstream of ferroelectric research is to study the domains and their response with respect to external impacting factors, such as temperature, electric field, stress, chemical forces, and others. Hysteresis loop, as a simple and effective tool, is the most generally accepted method to understand ferroelectric materials.
In principle, every ferroelectric material has its own unique hysteresis loop, as a fingerprint. Through the hysteresis loops, the ferroelectricity could be identified directly. Figure 2 is a typical ferroelectric hysteresis loop, through which the characteristic parameters, such as spontaneous polarization (Ps), remnant polarization (Pr), and coercive field (Ec), can be determined. Owing to the requirement of the energy minima, the grains in polycrystalline materials are always splitting into many domains. The directions of the domains are randomly distributed in such a way to lead to zero net macroscopic polarization. When the external field exceeds the Ec, the polycrystalline ferroelectric ceramic may be brought into a polar state. As shown in Fig. 2, a macroscopic polarization is induced gradually by increasing the electric field strength. The drastic variation in the polarization in the vicinity of Ec is mainly attributed to the polarization reversal (domain switching), while at high field end, the polarization is saturated and the material behaves as a linear dielectric. When the electric field strength starts to decrease, some domains would back-switch, but at zero field the net polarization is nonzero, leading to the remnant polarization Pr. To obtain a zero polarization, an electric field with opposite direction is needed. Such field strength is called the coercive field (or coercivity). With increasing the opposite field strength, a similar rearrangement of the polarization is observed in the negative field part. For ferroelectric materials, the spontaneous polarization Ps may be estimated by intercepting the polarization axis with the extrapolated linear segment, as shown in Fig. 2. Since ferroelectrics usually possess ferroelastic domains (with the exception of LiNbO3, which only has 180° ferroelectric domains), spontaneous strain is also induced with the external electric field simultaneously. Therefore, if the strain is monitored as well as the polarization, a strain–electric field curve, like “butterfly,” can be observed.
For ideal ferroelectric system, the observed hysteresis loops should be symmetric. The positive and negative Ec and Pr are equal. In reality, the shape of the ferroelectric hysteresis loops may be affected by many factors, such as thickness of the samples, material composition, thermal treatment, presence of the charged defects, mechanical stresses, measurement conditions, and so on. Their effects on material properties could be well reflected through the loops. Therefore, by decoding the hysteresis loops, we could comprehend the material properties and structures.
There are many classical publications reviewing the hysteresis phenomena in ferroic materials.[50-53] Most of these articles emphasize the physical significance behind the experimental results. However, to researchers working on ferroelectric materials, understanding the relationship between microscopic structures and macroscopic properties through the hysteresis loops is very important, which is the purpose of this feature article. The structure of this feature article is arranged as follows. Starting from the measurement, measuring principles and techniques are introduced. The factors, especially the conductivity, which will give rise to misunderstanding of the loops, are discussed. Based on the features of the loops, four kinds of ferroelectric loops are discussed in terms of the structure and property relationship, the information which can be achieved from the loops is proposed. Following the classification, the factors affecting the hysteresis loops are introduced and discussed with an emphasis on the connection of the microscopic structures and macroscopic loop characteristics. Finally, summary and future perspectives are given.
2 Measurement of Hysteresis Loops
2.1 Polarization Versus Electric Field Hysteresis Loops
Before decoding the information from the ferroelectric loop, we should make sure that the obtained loops represent the real properties of materials, instead of the potential artifacts.[54, 55] In this section, we will introduce the methods to characterize the loop in ferroelectrics. Cautions are made to avoid some obvious misinterpretation, normally due to the existence of the conductivity.
Although the first ferroelectric loop was reported in Rochelle salt by Valasek, the development of the ferroelectric study is rather slow. One of the reasons may be due to the difficulty in determination of the ferroelectric hysteresis loop. It was not until 1930, Sawyer and Tower developed the first electronic circuit to characterize the ferroelectric properties of Rochelle salt. Figure 3 shows the schematic connection of the so-called Sawyer–Tower circuit. Through it, an ac voltage is imposed on the surface of an electroded ferroelectric sample, placed on the horizontal plates of an oscilloscope; thus, the quantity plotted on the horizontal axis is proportional to the field across the crystal. A linear capacitor C0 is connected in series with the ferroelectric sample. The voltage across C0 is proportional to the polarization (P) of the ferroelectric sample. In fact, dielectric displacement (D) and polarization are connected by Eq. (1):
where D is the charge density collected by C0. Compared to the larger value of P, the contribution by ɛ0E can be omitted. Therefore, the obtained D is considered as P in practice. With the well-developed electronic techniques, the Sawyer–Tower circuit is no longer used as its original form. Most testing systems for P–E relationship can be achieved by commercial apparatus mainly from two companies, that is, aixACCT and Radiant. In these equipment, P is collected through charge or current integration technique following with compensation by a variable resistor R (see schematic circuit in Fig. 3).
Although the measuring technique is not the main obstacle to study the P–E hysteresis loops, interpreting the loops is much more challenging to the neophyte researchers working on ferroelectrics. Misinterpretations of the P–E hysteresis loops are frequently made due to the strong interference by conductivity. To understand the information from the P–E loops, let us consider three types samples with different electric features. Figure 4 shows the relationships among electric field (E), current (I), and polarization (P) with respect to ac electric field for a linear resistor, a linear capacitor, and a ferroelectric crystal, respectively. Here, the term “linear” means that the resistance or the capacitance does change with respect to electric field (both triangular and sinusoidal wave forms can be used for the measurements). For linear resistor, I keeps the same phase with E, resulting in a linear relationship between them. The corresponding P–E characteristic is shown in Fig. 4(a). Clearly this resistor exhibits a symmetric loop with respect to both horizontal and vertical axis. The case for linear capacitor is shown in Fig. 4(b). Since I of the capacitor is the differential of E with time, a constant I is obtained with respect to the constant E, but changes the sign for the reversal of E direction. As observed in the I–E curve, a square loop is illustrated with a clockwise flowing direction. Note that the corresponding P–E characteristic of such a linear capacitor shows a linear feature. In contrast, the case for a real ferroelectric sample is totally different from the above two cases. As shown in Fig. 4(c), with increasing the electric field amplitude, there are obvious current peaks through the sample. The electric field correlating to the maximum current is identical to the Ec determined by the P–E loop. It should be noted that the current maxima do not occur at the maxima of electric field, due to the fact that the current is related to the domain reversal, other than the conductivity. Only the current maxima resulting from the domain reversal or switching is the sign of ferroelectricity. In this case, the P–E relationship exhibited real hysteresis loop, which always possessing an anticlockwise feature between P and E.
Note that we only determine the charge owing to the switching of the polarization using the Sawyer–Tower circuit. In this case, we assume an ideal ferroelectric insulator:
However, the real case of ferroelectric determination is more complex, as conductivity always coexisting with the capacitive ferroelectric samples. Thus:
where σ is the electrical conductivity, t is the measuring time. It will deteriorate the identification of the ferroelectricity if the conductivity is large. According to the criterions given by Dawber et al., “large” in this case is σ > 10−6 S/cm, whereas “small” is σ < 10−7 S/cm. As shown in Fig. 5(a), for a real ferroelectric sample, a transition from typical ferroelectric hysteresis loop into round loop was observed with increasing the conductivity gradually. With large enough conductivity, the polarization contributed by domain switching is totally submerged. It can be considered as a combination of the P–E characteristics shown in Figs. 4(a) and (c). In this case, there is evidence for ferroelectricity; however, it is difficult to be extracted from the conductivity. Figure 5(b) gives another example, in which a capacitor and a resistor are combined. Though hysteresis features appear with increasing conductivity, neither saturated polarization nor switching current can be observed in this combined system, thus not relate to the ferroelectricity. In undoped BiFeO3 thin film, similar P–E loops as shown in Fig. 5(a) were reported. It was also found that significant improvement of the ferroelectric properties of BiFeO3 thin film was achieved through control of electrical leakage by Nb doping.
It should be emphasized here that the conductivity in ferroelectrics also presents itself in hysteresis loops through another way, that is, a large gap between the starting and ending applied fields. Figure 6 gives the P–E hysteresis loops measured on BiFeO3 ceramics, exhibiting a large discrepancy between the starting and ending fields, that is, so-called “gap”, inherently associated with the high conductivity due to the existence of Fe2+–Fe3+.
2.2 Strain Versus Electric Field Curves
To measure the induced strain as a function of electric field, the key point is the determination of the displacement, which usually ranges from 1 nm to 100 μm, depending on the studied material and its dimension. There are two main methods (A) and (B) have been generally used.
2.2.1 Magnetic Induction Method (LVDT)
The critical part for realization of this method is a linear variable differential transformer (LVDT), which is an electromechanical transducer that produces an electrical output proportional to the displacement of a separate, moveable, high-permittivity core. Normally the transformer consists of three windings or coils, one primary and two secondary. The two secondary windings are connected in series. Application of an ac signal to the primary winding Vp induces a magnetic field inside the transformer. The magnetic flux couples to the secondary windings via the moveable core, which leads to an output signal Vs. The two secondary windings are connected in such a way that the output voltage is the difference between the induced signals in each winding. Practically, the displacement of the core could be coupled to the displacement of a sample under applied field via a stiff and nonmagnetic metal rod. For more information, readers could refer to Ref. .
2.2.2 Optical Method
This method is based on the reflection of monochromatic light. To this method, the photonic sensors by MTI Instruments Inc. (Albany, NY) are frequently used. The photonic sensor is a fiber-optic measurement system designed for both vibration and displacement measurement. In principle, it can measure displacement in the range from 10 nm to 5 mm statically or at frequency up to 150 kHz. The probe consists of a bundle of optical fibers, half of which emit light, while the other half receives the light reflected back from an optically reflective surface. The intensity of reflected light detected by the sensor is a function of the distance between the probe and the sample surface. When the probe is in contact with the reflective surface (x = 0), no light is reflected back to the senor (I = 0). When the probe is place a long way from the reflective surface (x = ∞), no light is reflected back and again the intensity is zero. Between these two points, there is a position of maximum intensity (Imax). To practice, there is a region of linearity between the maximum in intensity (I = Imax) and the zero point (I = 0). By calibrating the position of maximum intensity for surface of varying reflectivity, and therefore different reflected intensities of light, the sensor can give an accurate measure of the distance between the surface and probe and thus the distortion of the sample.
Recently, interferometric methods including homodyne, heterodyne, and Fabry–Perot techniques are also frequently used to measure small displacements. Two main optical schemes have been proposed for homodyne interferometer: the single-beam Michelson interferometer, in which the displacement of only one of the two major surfaces of the sample is monitored, and the double-beam Mach-Zender interferometer in which the difference of the displacements of both major surfaces of the sample is taken into account.
2.2.3 Piezoresponse Force Microscopy
In addition, piezoresponse force microscopy (PFM) has been used to determine the local movement of the domain walls, with a resolution of the order of nanometer scale.[69, 70] By applying the electric field to the sample through the needle of the PFM, not only domain wall evolution with respect to the driving field can be readily recorded by this technique but also the electric field-dependent piezoelectric properties can be obtained. This method offer more information of the local piezoelectric properties than traditional methods.
2.2.4 Strain Gauge
Strain gauge method has also been used for measuring the strain versus electric field curves of material. In this method, the strain of materials is monitored by strain gauge. This method is preferable to measure the coupling effect of strain versus electric field under mechanical stress loading.[71-73]
3 Classification of Hysteresis Loops
There are many kinds of ferroelectric systems reported since the discovery of the ferroelectricity in Rochelle salt, and each system possesses unique characteristic hysteresis. In this section, the ferroelectric loops are categorized into four groups according to their morphological features, then the information obtained from the hysteresis loops is discussed and the potential applications related to these loops are introduced. Finally, the accompanying strain–electric field (S–E) curves are discussed based on the classification of P–E hysteresis loops.
3.1 Classification of Polarization–Electric Field Hysteresis Loops
3.1.1 Classic Ferroelectric Loops
The most common ferroelectric hysteresis loop possesses the feature as shown in Fig. 2, which can be categorized into classic hysteresis loop, in which the polarization shows a drastic variation when the external electric field is in the vicinity of the coercive field, demonstrating that most domains are reversed or switched by the applied external field with a strength of coercive field. Strictly speaking, the P–E loop near the Ec is not absolutely perpendicular to the horizontal axis, suggesting that the domain wall motion or switching yet exists above the coercive field. This phenomenon is attributed to the clamping of domain walls, which can be electrical, mechanical, or chemical driven. For different materials, the slope of this part varies significantly. At high field region, most of the domains align themselves along the electric field direction and the polarization approaches its saturated value, with a linear response between the polarization and electric field.
From the P–E loops, the spontaneous polarization Ps, remnant polarization Pr, and coercive field Ec can be easily achieved. The polarization is very important for the evaluation of piezoelectric characteristics of ferroelectric materials, following Eq. (4)[74, 75]:
where ε is the dielectric permittivity and Q is the electrostrictive coefficient. Generally, the Pr values for ferroelectric materials fall in a wide range, being on the order of 0.2–0.5 C/m2 for perovskite bulk materials, whereas the highest value of Pr was reported to be ~0.7 C/m2 for LiNbO3 crystals. Meanwhile, the coercive field, which is the indicator of the “hardness” of the domain reversal, was found to be in the range of 0.5–150 kV/cm, where the BiInO3–PbTiO3 perovskite ceramic with tetragonal phase was reported to possess a very high value of 125 kV/cm.
It should be noted even for the same material system, ferroelectric hysteresis loops of single crystalline and polycrystalline ceramic samples show a large difference. This is mainly attributed to the clamping effect of domains with respect to grain boundaries. It can be seen in Fig. 7, the shape of the P–E loop for BT crystal is rather square, whereas for BT ceramics the loop is slanted at a certain degree. By applying the same electric field, saturated polarization can be induced in BT crystal instead of ceramic, suggesting that the polarization switching in crystal is much easier than that in ceramic, owing to the absence of grain boundary. In addition, due to the crystal symmetry, domains in tetragonal BT single crystal could be switched completely with respect to the external field applied along  direction. In contrast, because of the random distribution of the grains, maximum 83% polarization can be switched in ceramics without considering the clamping effect by adjacent grains. This partially explained the lower spontaneous polarization observed in BT ceramic. Thus, BT single crystal was found to possess a lower Ec with a higher Pr, when compared to those of BT ceramics. Except the strong anisotropic characteristic observed in single crystals, other effects, such as phase, grain size, and density, will also contribute to the magnitude of Pr, Ec, and squareness of the P–E loop, which will be discussed in detail in Section IV.
Figure 7 shows ferroelectric hysteresis loops for BT crystal, coarse grain, and fine grain ceramics, exhibiting a transition from square loop to slim loop with decreasing the grain size. Here, crystal sample can be considered as a very large grain without grain boundary. Therefore, the clamping effect due to the neighboring grains is absent in crystals. It should be noted that the P–E loop of crystal is measured along its spontaneous polarization direction, whereas the P–E along other directions behave very differently, due to the strong anisotropy of the crystals, which will be discussed in following section.
3.1.2 Double Hysteresis Loops
It is well-known that in antiferroelectrics, two opposite polarizations arrange at two nearby crystalline lattice, as shown in Fig. 8. Due to such arranging characteristic, the net polarization of antiferroelectrics is zero in their virgin state (or without exposing to external electric field). With applying high enough electric field, at which macroscopic polarization can be induced, an antiferroelectric-to-ferroelectric phase transition occurs. The induced polarizations switch back to zero through another ferroelectric-to-antiferroelectric phase transition after removing the external electric field. These features of antiferroelectrics determine their unique hysteresis loops, that is, so-called double hysteresis loop. Figure 9(a) shows a typical antiferroelectric double hysteresis loop. It can be seen that before applying electric field, macroscopic net polarization is zero. With increasing field to a critical value (Ef), an antiferroelectric-to-ferroelectric phase transition is induced following with a rapid increase in the polarization, corresponding to the polarization of the ferroelectric phase at field above Ef. As the induced ferroelectric phase is metastable, a ferroelectric-to-antiferroelectric phase transition occurs with decreasing electric field, approaching the value of Ea. In the negative electric field part, a similar loop also exists. It should be noted that from a thermodynamic point of view, the observed double hysteresis loops occur only when the antiferroelectric phase' free energy is slightly lower than that of the ferroelectric phase. There is another kind of antiferroelectric material, in which its virgin antiferroelectric phase is a metastable state. Once a high enough electric field is applied, a stable ferroelectric phase is induced at the very first field cycle [Fig. 9(b)], which will maintain and exhibit a typical ferroelectric loop in the following cyclic field. To such a kind of antiferroelectric material, heating through an inversion to a nonferroelectric form and cooling would restore the metastable antiferroelectric phase again.
The above-mentioned critical fields Ef and Ea are related to the antiferroelectric-to-ferroelectric and ferroelectric-to-antiferroelectric phase transitions, respectively, which can be determined by two approaches. One is that the values can be determined from the measured loops. Although in schematic plot, the polarizations at Ef and Ea are almost perpendicular to the field axis, the real case is that the rapid variation in the polarizations occurs in a region instead of at a point. Generally, Ef and Ea are determined at the electric fields corresponding to the half value of the maximum induced polarization. The other method is that the values can be extracted from the current–electric field characteristic curve. There is only one current peak with either positive or negative field for classic ferroelectric materials, whereas for antiferroelectrics there are two peaks in the I–E curve. Each peak corresponds to a critical field, as shown in Fig. 10. In the positive applied field region, the positive current peak correlates to Ef, whereas the negative current peak correlates with Ea.
Except the critical fields Ef and Ea, other information can also be achieved from the P–E loop. The dielectric constant of the antiferroelectric materials can be calculated from the slope of the linear part of P–E loop at low field. In addition, the stored energy can be calculated by the integration of the shaded areas, as marked in Fig. 11. For energy storage application, the dielectric constants of the antiferroelectric and the induced ferroelectric, the critical fields Ea and Ef, hysteresis are important factors to evaluate the materials, which are closely associated with the material compositions, which will be discussed in Section IV.
It was generally accepted that the double hysteresis loop is the fingerprint of antiferreoelctricity in early stage of ferroelectric study. With the development of ferroelectric materials, however, this viewpoint was challenged by the hysteresis loops observed in unpoled acceptor-doped hard PZT and ferroelectrics with first-order ferroelectric–paraelectric phase transition at T0 < T < TC.[79, 80] Compared to classical loops observed in undoped and donor-doped soft PZTs, the P–E loop of hard PZT shows a constraining effect on Pr by removing the bias field from both positive and negative directions, leading to a pinched loop, which is similar to the double P–E loop as observed in antiferroelectrics. The difference between the pinched and double loops from the morphology is that for pinched loop, the polarization at zero ac field not necessarily be zero. In these materials, domains/domain walls are mainly pinned by the charged defects dipoles. Even the applied electric field can induce a macroscopic net polarization, the polarization will switch back to its unpoled (virgin) state by the restoring force, which generated by the coupling of defects dipoles and domains/local polarizations. This is totally different from the double hysteresis loop achieved in antiferroelectrics, where the zero net polarization at its virgin state is due to the opposite arrangement of the unit cell distortions. Double hysteresis loops with an open gap at original point are frequently reported in antiferroelectric materials, see Fig. 12. Strictly speaking, however, such an open loop is mainly due to the coexistence of the ferroelectric and antiferroelectric phases.
It should be noted that the debates on recently reported (Bi0.5Na0.5)TiO3 (BNT)-based lead-free system, where the double hysteresis loop was attributed to the antiferroelectric phase,[82-84] whereas others associated these phenomena to the nonpolar nanoregion. Other factors giving rise to double hysteresis loop include, but not limited to, fatigued ferroelectric samples due to the space charge, which will be discussed in Section IV.
3.1.3 Asymmetric Hysteresis Loops
In above two kinds of hysteresis loops, the common feature is that both of them exhibit a symmetric shape with respect to the origin point. However, for poled hard ferroelectrics, the hysteresis loops always showing an asymmetric feature as illustrated in Fig. 13. It can be seen that both the negative and positive shift to right or left with respect to the horizontal axis at a certain degree. Normally this shift is attributed to the existence of an internal bias field, which is defined as . In “hard” ferroelectrics, due to the pining effect by defect dipoles or defect clusters on domain walls, the poling and depoling processes are much difficult than those of ferroelectrics without pining effect. A macroscopic polarization is built up during the poling process, which is stabilized through the internal bias field during aging. On the contrary, even a net polarization is induced by poling in soft ferroelectrics, the measured hysteresis loops will transfer from asymmetric to symmetric shape with electric field strength exceeding Ec, due to the lacking of pining defect dipoles. The evolution of this transition is illustrated in Fig. 14.
From the asymmetric loop, the Pr, Ps, Ec, and Ei can be obtained. The internal bias is closely related to the acceptor–oxygen vacancy defect dipoles, other factors, such as doped level, grain boundary, sintering condition, poling status, aging time, measured temperature, will affect the value of the internal bias, which will be discussed in Section IV. The internal bias is important parameter to evaluate the hard ferroelectric materials, which is associated with the clamping effect of the domain wall motion and polarization rotation, account for the decreased power dissipation (such as dielectric loss and mechanical loss).[87, 88] Therefore, ferroelectric materials with higher internal bias are desirable for high power electromechanical applications, due to the fact that the internal bias is proportionally related to the mechanical quality factor. Although the coercive field is not necessarily related to the power dissipation, higher coercive field is desired for high power applications, by offering higher field stability.[90, 91]
As proposed by Carl and Härdtl, there are two methods to determine Ei with respect to the poled and unpoled states. The determination of Ei can be accomplished through either P–E or I–E characteristic curves. Note that the Ei determined from unpoled and poled states shows a large difference. This may be due to the different domain structures in unpoled and poled states.
3.1.4 Slim Hysteresis Loops
It is generally accepted that the domain wall motion accounts for the hysteresis phenomena.[50, 52] Although X-ray diffraction results suggest a pseudocubic phase structure of relaxor ferroelectrics, microdomains or polar nanoregions (PNRs) instead of the macroscopic domains do exist in a wide temperature range in the vicinity of the phase transition.[92, 93] Due to their much smaller characteristic size, microdomains response to the external field much faster than macroscopic domains and frequently result in a slim loop, as shown in Fig. 15. Apart from the typical ferroelectric loop, the slim loop does not possess obvious hysteresis but with nonlinearity, revealing the existence of the microdomains, due to the fact that the macroscopic domain switching gives rise to hysteresis, while microdomains or PNRs lead to nonlinear response with respect to external field.[95, 96] The slope of the curve, which relates to the dielectric constant, decreases as electric field increases. At low field, the microdomains are switched by the external field and result in a high dielectric response, which is reflected by the steep slope of the loop, whereas at high field ends, macroscopic polarizations are induced and aligned along the direction of the field. However, without contribution by the domain wall motion, the slope of the loop approaches to its saturated value. Due to the high dielectric constant of relaxor ferroelectrics, being in the range of 5000–10000, low hysteresis, and the diffused dielectric maxima, relaxor ferroelectrics can be used for energy storage media. From the P–E loop, the integration area between the polarization axis and the discharge curve can be calculated, which is related to the stored energy, being much higher than that of normal ferroelectrics.
The essence of such kind of slim hysteresis loop is the existence of microdomains, which is a transition state between the macroscopic polar state and nonpolar state. It should be noted that in normal ferroelectric and antiferroelectric systems, “2S” slim loops are frequently observed but in a relatively narrow temperature region close to TC, in contrast to the “1S” slim loop observed in relaxor ferroelectrics over a wide temperature range. Figure 16 shows a hysteresis loop of lead lanthanum zirconate titanate (PLZT)-based antiferroelectrics, exhibiting a typical “2S” shape near its Curie temperature.
In antiferroelectrics, both Ef and Ea shift to higher values with increasing temperature, as more energy is required to keep the induced metastable ferroelectric phase. However, macroscopic ferroelectric phase cannot be induced even at higher field when temperature approaches TC, which can be confirmed by the vanish of double hysteresis and appearance of the “2S” slim loop near TC. Due to the existence of nonlinearity, metastable polar phase is highly expected in this temperature region.
3.2 Classification of Strain–Electric field Curves
For ferroelectric materials, application of electric field to the sample not only induces the polarization but also the strain. Generally, they are correlated through Eq. (5):
where Q is the electrostrictive coefficient, which is insensitive to temperature. It should be noted that, Eq. (5) is not an accurate equation to describe the relation between strain and electric field, due to the fact that the microstructure of materials, anisotropy, and nonlinear properties are not considered. According to the classification of the hysteresis loops, the S–E curves can be categorized into four groups with bipolar and unipolar strain curves, as given in Fig. 17. (1) To ferroelectrics with classic hysteresis loops, the bipolar S–E curves show symmetric “Butterfly” curves. When the dc field is relatively small, the S–E curves obey linear relationship, corresponding to the piezoelectric effect. The large hysteresis in S–E curves and the large negative strain are mainly attributed to the ferroelastic domain wall switching. In unipolar mode, the first cycle shows a very large hysteresis, mainly due to the extrinsic contribution by domain switching. After the first cycle, a large remnant strain is maintained. Relatively small hysteresis can be achieved with further unipolar electric field cycles, as shown in Fig. 17(a). Generally, typical soft PZT exhibit such an S–E characteristic, from which, the high field can be calculated by the slope of the S–E curve (usually at 20 kV/cm), while the strain hysteresis (related to the piezoelectric loss) can be evaluated by the fraction of the strain at half maximum field.[103, 104] (2) To antiferroelectrics, their bipolar S–E curves show a double loop feature. Once the net polarization is induced, much higher strain with large hysteresis is expected, as shown in Fig. 17(b). In unipolar mode, antiferroelectric has large strain, induced by the antiferroelectric-to-ferroelectric phase transition, accounts for the large displacement, which will benefit large displacement transducer or actuator applications. However, the large hysteresis, which inherently associates with the phase transition, will greatly impair precision actuation application. (3) To hard ferroelectrics, the bipolar S–E curves of unpoled aged state possess similar double S–E curves as that of antiferroelectrics, whereas strong asymmetric “butterfly” S–E curves are observed in poled aged state, as shown in Fig. 17(c). For hard materials, the unipolar strain is linear and hysteresis free, due to the fact that the internal bias stabilizes the domain wall motion, greatly reduces the extrinsic contribution. This is important for electromechanical applications where high strain precision or no dc bias is required. (4) Finally, to relaxor ferroelectrics, the bipolar S–E curves are shown in Fig. 17(d). Due to the absence of macroscopic domains, there is no or very low hysteresis being observed in S–E curves. Of particular interest is that the dc field-induced strain in relaxor ferroelectrics is hysteresis free and temperature independent, thus, finding their potential applications for high precision actuation in a wide temperature range. Analogous to antiferroelectrics, the bipolar strain curves of relaxor ferroelectrics were found to possess only positive strains, whereas ferroelectric materials were found to show both positive and negative strains as a function of ac field, due to the lack of ferroelastic domains in antiferroelectrics and relaxor ferroelectrics. In addition, it should be noted that hysteresis-free unipolar strain behavior with large strain level of 1.7% has been reported for the relaxor-PT ferroelectric single crystal. This is due to the lack of domain wall motion, which is inherently associated with the engineered domain configuration.
Besides the P–E and S–E hysteresis loops, there is another kind of hysteresis loops, that is, piezoelectric response-electric field loops, which are frequently reported in ferroelectric materials, especially in thin films. In brief, we limit the discussion on longitudinal piezoelectric coefficient d33 as a function of dc bias field Edc. Based on Landau–Ginzburg–Devonshire (LGD) theory calculations, the intrinsic piezoelectric coefficients would initially increase with an applied negative bias field until depoling and repoling set-in. Then, the d33 tunes the sign and becomes smaller gradually with increasing the bias field. In some systems, the predicted response has been observed in both bulk material and thin films.[106, 107] In fact, not only the intrinsic effects control the response during this processing, extrinsic domain wall motion also contributes to a similar morphological response as intrinsic calculation by means of the back-switching of domains. In some hard PZT systems, where domain wall motion is pinned by means of defect dipoles or charge injection (fatigue), d33−Edc loop exhibits the features as that observed in P–E hysteresis loops. A maximum d33 was observed at highest field, due to the lack of back-switching of non-180° ferroelastic domain walls.
From viewpoint of applications, prediction of the response of ferroelectric materials subjected to external field is necessary for subsequent transducer design. Hwang et al. predicted the polarization and strain for an individual grain from the imposed electric field and stress through a Preisach hysteresis model. By averaging the response of the individual grains, which were considered to be statistically random in orientation, they successfully predicted the response of the bulk ceramics with respect to the external loads. Recently, a two-dimensional model that consists of four energy wells, four saddle points, and one energy maximum was reported. Using the parameters determined from experimental data of BaTiO3 single crystals, the responses of the polarization subjected to multiaxial in-plane electric field loading at various frequencies were calculated. Then, by employing stochastic homogenization techniques, a macroscopic model suitable for nonhomogeneous, polycrystalline compounds was developed. To predict the response of ferroelectric materials under different excitations without having to perform too much experimental work, several behavioral laws linking to the electrical field, temperature, and mechanical stress were proposed by Guyomar et al.[111, 112] The scaling law could also predict the piezoelectric coefficient under stress using only pure electrical measurements, and the dielectric constant under an electrical field using pure mechanical measurements. Due to the limitation of this article, it is impossible to include all the important literatures on modeling within such a short paragraph. However, we believe that with the clues supplied from these references and references within them, readers could find lot of information readily.
4 Impact Factors on Ferroelectric Hysteresis Loops
There are many factors affecting the characteristics of ferroelectric hysteresis loop. Some factors are associated with the material itself, whereas others come from the measurement conditions. In this section, we will discuss the ferroelectric hysteresis loop with respect to the grain size, phase transition, dopants, frequency, amplitude of electric field, temperature, frequency, stress, etc.
4.1 Effect of Materials
4.1.1 Grain Size and Grain Boundary
Based on LGD theory in the case of isolated particles, the size dependence of the crystal lattice distortion and polarization in ferroelectric phase has been calculated.[113, 114] It is suggested that there is a critical size, below which, a transition from a ferroelectric phase to a cubic paraelectric phase will occur with the particle size being on the order of a few nanometers to a few tens nanometers accompanying with the disappearance of polarization.[113, 114] For example, the critical grain size is 4–20 nm for PT[115, 116] and 10–100 nm for BT.[117, 118] In ferroelectric polycrystalline ceramics with large grains, domain patterns are formed to balance the depolarization field, which mainly comes from the charges accumulate at the grain boundary. With decreasing the grain size from micrometer to nanometer, this process is not energetically favorable in fine grain ceramics, where the compensation via surface charges and/or polarization gradient is possible. The grain size decrease leads to the following results. First, the lattice distortion will become less pronounced as the grain size decreases.[119, 120] This effect is similar to the hydrostatic pressure effect on polycrystalline ceramics, which will be discussed in Section IV. (3). Both of them favor a high-symmetric phase, that is, promoting to a cubic paraelectric phase. As reported in BT ceramics, the tetragonal distortion (c/a−1, where c and a are the unit cell parameters) was found to decrease linearly with decreasing the grain size from 1200 to 50 nm, accompanying with a TC shift from 127°C to 105°C. Second, multidomain structure is prone to transfer to single domain state with respect to grain size decreasing. It was suggested that domain size is proportional to square root of grain size.[121, 122] However, this law can only be hold for the grain size larger than hundreds nanometer, below which it is invalid. As reported by Arlt, there is a transition from multidomain state into single domain state with decreasing the grain size in BT ceramics. In this case, both domain wall motion and domain switching become difficult due to the increased clamping effect by the neighboring grains and the absence of domain walls. Based on the above, contributions from both intrinsic lattice and extrinsic domain wall to the polarization are reduced with decreasing the grain size, resulting in a smaller value of Pr. Thus, the ferroelectric hysteresis loop is expected to transfer from square shape to slanted one. This was clearly illustrated in Fig. 7, which gives the experimental results for grain size effect in BT system. In other ferroelectric systems, such as PZT,[123-125] PMN–PT,[126, 127] BiScO3–PbTiO3, BaTi0.995Mn0.05O3, the grain size effect has also been extensively studied, similar results have been achieved as observed in BT ceramics. Figure 18 shows two representative hysteresis loops observed in PMN–PT and BiScO3–PbTiO3 systems with different grain sizes. The common feature among these data is that with decreasing grain size, the loops transfer from square shape to slanted shape and finally to slim shape, accompanying by a decrease of Pr as a function of grain size. These results could be well understood from both intrinsic and extrinsic points of views. Recently, Liu et al. proposed that the existence of low permittivity paraelectric grain boundary and its influence on the grain microstructures played a key factor to the grain size effect. A two-dimensional polycrystalline phase-field model was developed to simulate the hysteresis behaviors of the nanoscale BT ceramics, showing apparent grain size dependence of the hysteresis loops, where the noticeable vortex polarization structures were observed as the grain size reduced to tens of nanometers. According to this simulation, a similar evaluation of the hysteresis loops as observed in above-mentioned systems was confirmed.
Compared to remnant polarization, the scenario of coercive field was found to be more complicated. In some systems, it was found that Ec decreased gradually with decreasing grain size, whereas in some other systems, the Ec was reported to maintain the same values with grain size.[126, 131] Theoretically, intrinsic Ec corresponds to a field strength at which, polarization is switched completely without considering the extrinsic domain wall contribution. However, in practice, the real electric field required for polarization switching is much lower than the theoretical value, being about 10% of intrinsic Ec, due to the aid of domain nucleation and domain wall motion. Generally, Ec is determined from the hysteresis loops, being equal to the electric field at which the polarization is zero. However, for a very slanted or slim ferroelectric hysteresis loop, ferroelectric property is greatly reduced, which is demonstrated by the relatively small Pr. This hysteresis loop can be deemed as the combination of linear dielectric response and conductivity. In this case, even the P–E characteristic curve has an intercept with respect to the electric field axis. It is meaningless to consider the value as Ec, as shown in Fig. 16(a) for the 0.8PMN–0.2PT ceramics with grain size less than 140 nm.
It is important to note that the coercive fields of single crystals are much lower than those of ceramics with the same composition, for example, the coercive field was found to be on the order of ~2.5 and ~5 kV/cm for PMN–PT crystal and ceramic, respectively, due to the fact that no grain boundaries exist in single crystals, which clamp the domain reversal and domain wall motion.
4.1.2 Phase and Phase Boundary
Ferroelectric phase plays an important role on hysteresis loop, as the symmetry determines the spontaneous polarization directions, where the polarization may exist at certain external conditions, such as temperature, chemical composition, pressure, electric field, etc. For example, polarization can be developed along six <100> directions in tetragonal phase, while it can be developed along eight <111> directions in rhombohedral phase. Since the discovery of the MPB in PZT solid solution, much attention has been devoted to the ferroelectrics with an MPB. The end-members of MPB systems have different phase structures. Taking Pb(Zr1−xTix)O3 as an example, PZ has a rhombohedral symmetry while PT has a tetragonal symmetry at room temperature. With increasing the PT concentration, rhombohedral PZT will transfer to tetragonal phase, where the Zr/Ti ratio at MPB region is around 52/48. As shown in Fig. 19, this phase transition is an abrupt structural change as a function of composition, with the phase boundary being nearly independent of temperature, which ensures the unique performance of the MPB composition.
Figure 20 shows P–E loops for Nb modified PZT ceramics with different phases (Zr/Ti ratio), where the Pr was found to gradually reduce from 0.4 C/m2 for rhombohedral phase, through 0.28 C/m2 for MPB, and finally to 0.2 C/m2 for tetragonal phase. This trend is consistent with the data of pure PZT ceramics reported by Jaffe et al. Interestingly, the highest Pr was achieved in PZT composition with 56 mol% Zr. A simple explanation of this phenomenon is that rhombohedral phase has eight equivalent direction of domain state, while it is only six for tetragonal phase, therefore higher Ps and Pr can be expected in rhombohedral phase. The ferroelastic domains (109° and 71° domains) in rhombohedral phase make the domain switching easier than that in tetragonal phase (90° ferroelastic domains), where the lattice distortion is induced by the gradually increased tetragonality (c/a ratio) as the PT concentration increases. In this case, due to the very large stress induced by the lattice distortion, domain walls are highly clamped, leading to higher Ec and lower Pr, as reflected by the ferroelectric hysteresis loops of the tetragonal phase.
However, the above discussion seems to be not well held in lead-free piezoelectric materials. For x(Bi1/2Na1/2)TiO3–(1−x)(Bi1/2K1/2)TiO3 (BNKT), the Pr at rhombohedral phase is larger than that in tetragonal phase, similar to PZT, while the Ec exhibits contrary tendency to PZT, with higher Ec in rhombohedral phase, as given in Fig. 21. The c/a ratio for tetragonal BNKT (x = 0.7) is about 1.015, which is smaller than that observed in tetragonal PZT. Due to the smaller lattice distortion, domain switching may be easier when compared to the tetragonal PZT, this may explain the above-mentioned phenomena. In addition, the Pr of MPB composition exhibited the highest values in BNKT ceramics, similar results have also been reported in BNT–BKT–BT lead-free system.
The hysteresis loop of ferroelectrics around the phase transition point shows interesting characteristics, and no matter it is induced by composition or temperature. For example, in BT single crystals, a double hysteresis loop was reported at temperature approaching TC, due to the electric field-induced ferroelectric–paraelectric phase transition, as shown in Fig. 22(a). Similarly, a triple hysteresis loop was also reported in BT crystals at temperature slightly below the orthorhombic–tetragonal phase transition temperature with the electric field applied along c axis [Fig. 22(b)]. The middle part of the loop is induced by the domain switching of the orthorhombic phase, whereas the upper positive and lower negative parts are the result of orthorhombic–tetragonal phase transitions. Similar phenomena were also observed in PMN–PT single crystals, where a “2S” slim hysteresis loop was achieved at temperature of TC, while triple hysteresis loop exists for PMN–PT with MPB compositions.
As mentioned above, PZTs exhibit soft and hard characteristics through adding of donor and acceptor dopants, respectively. Normally, in soft PZT, lead vacancies are generated to keep electric charge equilibrium. It suggests that domain wall mobility is increased in soft PZT, resulting in facilitated domain switching and domain wall motion. Therefore, a square ferroelectric hysteresis loop with a larger Pr and lower Ec is frequently observed. In contrast, due to the existence of oxygen vacancies in hard PZTs, which are generated through the acceptor doping, domain wall mobility is reduced. In this case, higher field is needed to induce polarization. It is believed that small amount of dopant in ceramic materials mainly affected extrinsic domain wall mobility, while having minimal effect on the intrinsic crystalline lattices.[139, 140] The corresponding ferroelectric hysteresis loop is shown in Fig. 17. Hard PZT in unpoled state could be confused as antiferroelectrics because of the similar double hysteresis loop.
Different from the regularity summarized from PZT ceramics, the doping effect in antiferroelectric ceramics is much more complicated. Pb(Zr,Sn,Ti)O3, as the most studied antiferroelectric system, both the Ea and Ef are increased but with a narrowing of the loop width ΔE = Ef − Ea when the Ti component decreases, demonstrating decreased hysteresis and energy loss, with increased energy storage density, as shown in Fig. 23. In addition, various impurities have been added in this system, for example, Sr2+ and Ba2+ for Pb2+ on A site and Nb5+ for Zr4+/Ti4+ on B site, were used for reducing the antiferroelectric–paraelectric phase transition temperature and increasing the mobility of the domain walls, respectively, while the doping of La3+ may induce transition from normal antiferroelectrics into relaxor antiferroelectrics.
For ferroelectric ceramics, due to the random distribution of grains, the properties measured along any direction in principle are the same. However, for ferroelectric single crystals and textured ceramics, the properties are strongly dependent on crystallographic directions, which can be clearly demonstrated by the hysteresis loops. Figure 24(a) shows the polarization hysteresis for rhombohedral PMN–0.30PT single crystals measured along crystallographic orientations , [110,] and , revealing that both the remnant polarization Pr and coercive field Ec are orientation dependent. The values of Pr were found to be on the order of 0.24, 0.34, and 0.41 C/m2 for , , and  orientations, whereas the values of Ec were 2.5, 2.9, and 3.2 kV/cm, respectively. It is known that there are eight possible polarization orientations along the pseudocubic <111> direction for rhombohedral PMN–PT single crystals (3m symmetry). Upon applying an electric field, the dipoles reorientate as close as possible to the applied electric field direction. For  poled crystals, there are four equivalent polar vectors along the <111> direction, with an inclined angle of 54.7° from the poling field. Following “4R,” “2R,” “1R”, and others are technical nomenclatures used in the framework of domain engineering as proposed in relaxor ferroelectric single crystals.[10, 144, 145] The four <111> domains are equivalent with a domain engineered configuration “4R”, resulting in a macrosymmetry 4mm. For  poled crystals, there are two equivalent polar vectors along the <111> direction, which will rotate 35.5° toward the applied field direction of  with a designated domain engineered configuration “2R.” For this case, the macroscopic symmetry is mm2. In contrast, there is only one polar vector along the <111> direction for  poled ferroelectric crystals, thus it will form a monodomain state, designated “1R,” exhibiting macroscopic symmetry 3m. According to the domain engineered configurations, the polarization level derived from the hysteresis loops should correspond, in theory, to the intrinsic value along the polar axis of the monodomain crystal Ps, following .[44, 145] As expected, the Pr values for the different orientations obtained from Fig. 24(a) are in good agreement with the predicted values. For lead-free ferroelectric crystals, such as BNT, the ferroelectric hysteresis loops measured along different crystallographic directions also exhibited large difference, as shown in Fig. 24(b).
For ferroelectric ceramics with textured configurations, the anisotropy reflected in the hysteresis loop is obvious due to the orientated growth of the grains. Figure 25 presents three groups of ferroelectric hysteresis loops determined from randomly oriented and textured ceramics with different structures. Figure 25(a) shows the P–E hysteresis loops for randomly oriented and textured (Na1/2Bi1/2)TiO3–5.5 mol% BaTiO3 ceramics with perovskite structure. As these ceramics have a rhombohedral symmetry, the Pr determined along <001> direction must be of that determined along <111> direction, with an angle of 54.7°. In ceramics, however, grains are randomly distribution in three dimensions. The angles between the measurement direction and the spontaneous polarization direction (<111>) of various grains change from 0° to 54.7°. Therefore, the averaging of polarization in three-dimensional space, randomly oriented ceramics has a higher Pr than <001> textured materials. For Sr0.53Ba0.47Nb2O6 with tungsten bronze structure and Bi4Ti2.96Nb0.04 O12 with bismuth layer structure, the maxima Pr are observed along their c directions. Therefore, in textured ceramics, the higher Pr is observed along the c axis. In contrast, lower Pr is observed in the direction perpendicular to the c axis. For randomly oriented ceramic samples, the values of Pr are between the values of these two limitations, as illustrated in Figs. 25(b) and (c).
Analogous to the P–E characteristics, the bipolar S–E curves of ferroelectric crystals are also associated with the crystal orientation. Figure 26 shows the Strain–Electric field behaviors for - and -oriented rhombohedral PMN–0.29PT crystals, where strong orientation dependence of S–E behaviors was observed. The positive strain of -oriented rhombohedral crystal is much higher than that of -oriented crystal. This is because that  poled rhombohedral crystal is in “4R” domain engineered configuration, whereas  poled rhombohedral crystal is in “1R” single domain state. The piezoelectric coefficient of domain engineered crystal is much higher than that of single domain state, due to the “polarization rotation process,” which contributes to a high and linear strain. For the part of negative strain, however, it can be seen that the strain of  poled PMN–0.29PT crystal is about five times higher than that  poled crystal at the coercive field. As shown in Fig. 26(b), at the coercive field, the negative strain induced by domain switching was minimal for -oriented rhombohedral crystal because the spontaneous strain of eight possible domains (, [ 1 ], , , , [ 11], [1 1] and [ 1]) are equivalent with respect to the  direction. Thus, the negative strain of -oriented rhombohedral crystal is induced by linear piezoelectricity, being equal to −d33Ec approximately. On the contrary, due to the spontaneous strain of eight possible domains are not equivalent to  direction, the negative strain for  poled rhombohedral crystals was induced by the non-180° ferroelastic domain switching from the  domain to , [ 1 ], [1 ], [ 11], [1 1] or [ 1] domains, as shown in Fig. 26(c), resulting in high level of negative strain. At higher electric field, admittedly, every domains are transformed to  domain, being again in single domain state.
4.1.5 Thickness of the Sample
For bulk ferroelectrics, including the ceramics and crystals, the studied thickness is generally higher than 200 μm, which has limited impact on their ferroelectric properties. However, the Pr was reported to decrease and Ec increase when the sample thickness further down to <150 μm, as shown in Figs. 27(b) and (d), due to the fact that the domain size is on the order of the sample scale, thus the sample boundary clamps the domain reversal and restricts the polarization rotation, accounting for the decreased Pr and increased Ec. Of particular significance is that the fine grain PMN–PT ceramics and Pb(In1/2Nb1/2)O3–Pb(Mg1/3Nb2/3)O3–PbTiO3 (PIN–PMN–PT) crystals exhibit minimal scaling effect, as given in Figs. 27(a) and (c), which can be explained by the smaller domain sizes, being on the order of ~2 μm, much smaller when compared to the sample thickness.[151, 152]
On the contrary, for ferroelectric thin films, an unavoidable topic needs to be addressed is the thickness effect on their properties. By decreasing the thickness of the films from micrometer to nanometer scale, such an effect becomes more pronounced. Ma et al. studied the hysteresis loops of (Pb0.92La0.08)(Zr0.52Ti0.48)O3 films with the thickness varying from 3100 to 350 nm and found that the PE loops transformed from square shape to slanted shape, as shown in Fig. 28. The Pr decreased with decreasing the thickness, while Ec showed the opposite trend. This observation could be well interpreted in terms of the domain wall clamping by the tensile stress. Due to the different thermal expansion coefficients and the lattice mismatch between the substrate and ferroelectric materials, there exists a tensile stress at the interface between them. This tensile stress deceases gradually from the interface to the ferroelectric films. With the in-plane tensile stress, the domain wall motion and domain switching are clamped. Higher field is required to move or switching the domain wall and the switching regions are expanded compared with free standing state of the materials.[154-156] As the polarization measured by the Saywer–Tower circuit is a collective response of the dipoles, the fraction of the clamped domain volume becomes higher with decreasing film thickness. Therefore, a decrease of Pr accompanied with an increase of Ec was observed with decreasing the film thickness.
For antiferroelectric PbZrO3 thin films, smaller thickness was reported to favor the ferroelectric phase due to the large compressive stress as the thickness decreases. A transformation from antiferroelectric double loop to ferroelectric single loop was found to occur in PbZrO3/SrRuO3/SrTiO3 epitaxial heterostructures when the PbZrO3 layer thickness is below 22 nm, as shown in Fig. 29(a). There is a compressive stress between the PbZrO3 and SrRuO3 interface, due to their different lattice parameters. Figure 29(b) gives the compressive stress as a function of the layer thickness, where it was found that the compressive stress reached 2 GPa when the layer thickness decrease to 22 nm. By such a large stress, the mechanical stress presenting in the PbZrO3/SrRuO3 interface overcomes the small free energy difference between the orthorhombic antiferroelectric phase and rhombohedral ferroelectric phase, stabilized the ferroelectric phase and prevented any further phase transition.[25, 158, 159] It should be noted that the Pr obtained from the ferroelectric loop is still low, due to the fact that even the ferroelectric phase was stabilized by the mechanical stress, the domain wall motion is yet hard due to the clamping effect at the interface.
4.2.1 Aging Without Poling
Aging process includes aging without poling and after poling, to release the stress induced by the high temperature or electric field. During the cooling through the TC, domain realignment does occur slowly in small steps as time elapses. The mechanism of this aging process is mainly attributed to the presence of mobile charge species, such as defect or defect dipoles, which usually stabilize the domain pattern and decrease the domain wall contribution to the polarization response.[160, 161] This phenomenon is more obvious in accepted-doped ferroelectric materials, such as Fe3+-doped PZT ceramics,[79, 162] where the oxygen vacancies are present to keep the electronic equilibrium due to the lower valence replacement of the Fe3+ ions for (Zr4+,Ti4+) ions, forming the defect dipoles , which clamp the domain wall motion. However, the exact mechanism dominating the stabilization of the domain walls is still under debate.[87, 163-169] Three models have been proposed to answer this question. (1) the bulk effect (charged defects align along the polarization within ferroelectric domains),[163-165] (2) the domain wall effect (charged defects diffuse to domain walls creating pinning centers),[166, 167] and (3) the interface effect (charged defects drift and accumulate at the grain boundaries and other interfaces).[168, 169] Although the bulk and domain wall models are based on experimental observations, the contribution by the surface interface cannot be excluded.
It should be noted that classical ferroelectric hysteresis loop can be observed in samples with pinning centers, if the aging state is disturbed. For example, when the sample is quenched from high-temperature paraelectric state, normally square hysteresis loops are observed, due to the fact that the pinning centers distributes randomly with respect to the polarization and cannot form the restoring force immediately. However, as shown in Fig. 30, when time elapses, domain patterns are stabilized by the charge defects gradually and the macroscopic net polarization becomes zero. A polarization is only induced when the sample subjects to an external electric field which is larger than the pinning force generated by the charged defects. However, the distribution of the charged defects cannot be disturbed immediately. Upon removing the field, there is a restoring force switching the domain pattern back to the original state as much as possible due to the existence of the well-distributed charged defects, resulting in a constriction effect on the loop (pinched loop).[79, 165]
Of particular interest is that similar double hysteresis loops are also observed in antiferroelectrics, the mechanisms of the double hysteresis between the aged ferroelectrics and antiferroelectrics, however, are completely different. In antiferroelectrics, there are two phases, including stable antiferroelectric phase and field-induced metastable ferroelectric phase during the dynamic hysteresis measurement, while there is only one ferroelectric phase in aged ferroelectrics. In addition, in antiferroelectrics, ac field loading cannot affect the morphology of the loop, while the ac loading can alleviate the pinning effect and open the pinched loop gradually in aged ferroelectrics, as shown in Fig. 30. Thus, this feature can be a criterion to distinguish the aged ferroelectric from real antiferroelectric.
4.2.2 Aging After Poling
The above discussion was focused on the double hysteresis loops observed in aged ferroelectrics after sintering or electroding in an unpoled state. Here, the aging behavior for samples after poling and the corresponding P–E loop characteristics will be discussed.
Poling is a reorientation process of domains. As discussed in earlier section, B-site acceptor dopants in ABO3 perovskite ferroelectrics are believed to substitute high valence B cations by low valence of cations, such as Mn2+, Mn3+, and Fe3+, resulting in oxygen vacancies, leading to the development of acceptor–oxygen vacancy defect dipoles. The dipoles align themselves along a preferential direction for the spontaneous polarization, and/or move to the high-stressed areas of domain walls or grain boundaries by diffusion, pin the walls, and stabilize the domains. The build-up of these parallel defect dipoles to the local polarization vector leads to an offset of P–E behavior or internal bias (see Fig. 31 of Mn-doped PMN–PZT crystal). Compared to the obvious internal bias observed in PMN–PZN–Mn crystal, PMN–PZT crystal exhibits a symmetric hysteresis loop with respect to the ac electric field, although it has already been poled before the P–E characterization.
Besides the pining effect due to the defect dipoles, the space charge accumulated during aging process should also be considered to explain the evolution of the hysteresis loops during aging in poled ceramic samples. Okazaki and Nagata have proposed a model to include the space charge effect for aged piezoelectric ceramics, as shown in Fig. 32, in which the smaller rectangles represent the grain boundary, whereas the larger one represents grains. Based on this model, the aging process is divided into four steps: (1) In initial state, the total polarization Pr is a sum over all domains and spontaneous polarization is P1. (2) After poling, domains arrange with the poling field direction, and the charges are induced at the boundaries (P3). (3) In the aging process, Pr gradually reduces, due to the fact that the space charges accumulate and lead to P2 in the grain. Meanwhile, the induced charges at the boundaries are reduced too. (4) In the final state, the polarization of the domain is screened completely by space charges, which accumulate at the ends of the domain. In every state, the corresponding P–E hysteresis loop is shown in Fig. 32, showing an internal bias field. It should be noted that the internal bias field will increase as a function of aging time after the poling process, following a time law of the form[172, 173]:
where A depends on the temperature: with rising temperature, Ei builds up more quickly. Furthermore, A depends somewhat on the previous history of the samples. This will lead to a more asymmetric behavior of the P–E loops.
4.3 Effect of Measurement Conditions
4.3.1 Applied Electric Field Amplitude
Ferroelectric materials exhibit strong nonlinear dielectric and piezoelectric responses with respect to the electric field strength. According to the field strength, the P–E behavior can be divided into three regions as shown in Fig. 33. At field region below the threshold field Et, the dielectric permittivity ε′ is almost independent of the field strength. Between the Et and Ec is the intermediate Rayleigh region, where ε′ shows a linear increase with the electric field strength. It is believed that in this region some large-scale domain wall translation occurs. At high field region above the Ec, normally there is very large hysteresis in P–E behavior and an obvious increase in the dielectric permittivity apart from the linear relationship between the ε′ and E0, due to the high degree of irreversible process, that is, domain nucleation and growth. It should be noted that the boundary between the low-field region and Rayleigh region is blurry in some systems without random pinning defect dipoles. For soft PZT, the threshold field Et is too low to be distinguished from the Rayleigh region, whereas for BS–PT ceramics, there is no evidence to support the existence of Et. Furthermore, the transition from Rayleigh region to high-field region is not accurately separated by Ec.
In fact, Rayleigh region is of special interest due to the fact that the Rayleigh law proposed in 1887 has been successfully used to model the nonlinear hysteresis loop for ferroelectric materials at the intermediate field level. The Rayleigh relation for dielectric response can be expressed as
where α is the dielectric Rayleigh coefficient, and ε′(0) is the initial dielectric permittivity, which is a field-independent term and represents the contribution by intrinsic lattice and contribution from reversible domain wall vibration, ε0 is the vacuum dielectric permittivity, E0 is the amplitude of the ac electric field. Normally the dielectric response at Rayleigh region can be well depicted by Eq. (7). The term αE0 represents the contribution by the irreversible domain wall motion. In Eq. (8), the “+” sign corresponds to the decreasing field, whereas the “−” sign corresponds to the increasing field. In practice, the maximum E0 used for Rayleigh modeling is limited below 0.5 Ec, because above 0.5 Ec there is an obvious deviation of P–E loops from symmetric shape into asymmetric shape. Figure 34 gives an example of the Rayleigh fitting to the measured loop of a PLZT film. Nowadays, the Rayleigh method has been widely used for analyzing the contribution of domain wall to dielectric and piezoelectric effects for ferroelectrics.[105, 178-184] With further increasing the applied electric field, ferroelectric phase transition will occur. The field level to induce the phase transition is closely related to the composition and temperature, lower electric field is required to induce phase transition with composition being on the proximity of MPB or at elevated temperature.
4.3.2 Cyclic Field Fatigue
In ferroelectric materials, the polarization fatigue is defined as the loss of switchable polarization with respect to cyclic electric field.[185-189] As shown in Fig. 35, after the loading of bipolar cycling, the switchable polarization decreases drastically while the coercive field increases in contrast. Lou has reviewed the impact factors to the fatigue phenomena for ferroelectric thin film and some bulk ferroelectric materials, including the influence from experimental conditions (electric field, temperature, frequency, oxygen partial pressure, optical, and thermal fatigue), and the influence of the electrodes and material modifications (conductive oxide electrodes, interface quality, interface layer, crystal microstructure, doping, processing condition, anisotropy). Although many investigations have suggested the degradation of the polarization (including both saturated and remnant polarization) in such a fatigue process, the change in the coercive field is subtler than that observed in the polarization, as observed in Fig. 36 for (K,Na)NbO3-based lead-free ceramics.
It is generally accepted that fatigue is a result of charge injection and accumulation of space charge that pins domain walls or retards the nucleation of reversion domain to permit switching. Many strategies have been developed to overcome this problem, mainly through three methods. (1) Doping the ferroelectrics with donor dopants to reduce the oxygen vacancies concentration.[192-194] (2) Using oxide electrodes for PZT thin film ferroelectrics.[195-197] (3) Searching for ferroelectric materials with fatigue-resistant characteristics, such as SrBiTa2O9 and SrBiNb2O9, and Bi3.25Sm0.75Ti3O12 sytems.[190, 198, 199]
Recently, the fatigue phenomena observed in relaxor-PT single crystals offer another approach to improve the fatigue resistance by means of so-called engineered domain structures. It was reported that there was no fatigue being observed in -oriented PZN–PT single crystal with rhombohedral phase, while severe fatigue occurred in - and -oriented crystals.[190, 200, 201] The polarization can be rejuvenate for -oriented crystals after annealing the sample above their Curie temperatures, while the fatigue in -oriented crystals are nonrecoverable due to the fact that the fatigue in -oriented crystals is induced by the strong anisotropic symmetry, which gives rise to microcracks under cyclic electric field.[202-204] It is inferred that an engineered domain state “4R” in relaxor-based ferroelectric crystal with spontaneous polarization inclined to the normal of the electrode is associated with negligible or no fatigue at room temperature.[190, 201] Although the mechanism by which engineered domain states mitigate fatigue is not well understood, one possibility is that the inclined polarization state redistribute the space charge accumulation and thereby reduce the fatigue rate at a given temperature and composition. It is also expected that the engineered domain structures with rhombohedral symmetry have a higher percentage of charged domain walls. These charged domain walls could act as sinks to the injected charge in ferroelectric systems. In fact, the mechanisms based on these models affect the fatigue phenomena concurrently. Therefore, it is not favorable to emphasize any individual model to interpret the fatigue.
It is interest to note that most fatigue phenomena have been studied in the ferroelectric materials without obvious aging effect. However, the study of the fatigue effect in hard ferroelectric perovskite materials (with strong aging effect and pinched hysteresis loop) would shed light on this point. Figure 37 shows the evaluation of the hysteresis loop of a hard 1.0 at.% Fe-doped PZT58/42 ceramics with respect to the number of the ac electric field cycle at 125°C. In contrast to previous reports on fatigue phenomena, both saturated and remnant polarizations are increased with respect to increasing the cycle number from 500 to 30000. As we have discussed in aging section, the domain wall motion is pinned by the ordered charged defects in hard ferroelectrics. Thus, not all the domain walls can response to the external electric field. However, with the disturbing by cyclic field, the ordered distribution of the charged defects gradually smear and their pinning effect to the domain wall disappear simultaneously. This trend is well depicted in Fig. 37, though such a deaging effect is more obvious at high temperature. On the other hand, for acceptor-doped relaxor-PT crystals, such as Mn-modified PIN–PMN–PT, it was reported that the crystals shown improved fatigue behavior when compared to their pure counterparts, due to the enhanced coercive field and the existence of internal bias.
For antiferroelectric materials, similar degradation of the polarization after fatigue has been reported, as shown in Fig. 38. However, compared to ferroelectrics, antiferroelectrics normally show higher fatigue resistance under bipolar electric cycling. Lou attributes this phenomenon to the lower depolarization field, lower local injected power density, and lower local phase decomposition probability of antiferroelectrics at the phase nuclear sites. Furthermore, microcracks are also reported in antiferroelectrics after fatigue. The microcracks actually pin the domain wall motion, resulting in the lower switchable polarization. By comparing with bipolar and unipolar fatigue processes, Lou and Wang suggested a scenario that polarization fatigue is mainly caused by the switching induced charge injection other than the charge injection during the stable/quasi-stable leakage stage.
It is often observed that the shape of the hysteresis loops is strongly affected by the measuring frequency. Generally, a square hysteresis loop transformers to a slanted one with increasing frequency, where the Ec is found to increase, while the Pr maintains similar value. These phenomena have been observed in ferroelectric materials, including ceramics, single crystals, and thin films.[185, 209, 210] As shown in Fig. 39(a), the Ec of PbZr0.2Ti0.8O3 thin film does increase gradually as a function of the frequency, with the enhancement being as much as 80% with the frequency increasing from 50 to 2000 Hz.
It was discussed earlier that the domains played an important role in the polarization reversal process for a ferroelectric materials. With subjected to external electric field, the volume of the favored domain patterns increases by the nucleation and subsequent growth of the domains. Therefore, the morphologies of the hysteresis loops are strongly affected by the nucleation and domain wall motion, leading to the fact that the P–E loops are frequency dependent, as nucleation and domain wall motion are time dependent. There are two models based on the domain dynamics being generally used to interpret the frequency-dependent hysteresis loop. A phenomenological model was developed on considering the domain growth process; the time-dependent fractional volume of the reversed domains was calculated based on the extended Avrami theory, suggesting that the Ec follows a simple power law relationship Ec ∝ fβ. Another model assumed that the limiting step of the frequency-dependent Ec is the nucleation process. According to these models, the Ec is expected to increase with increasing frequency, being consistent with the experimental observations. A careful analysis of the characteristic parameters of the PZT thin films shown in Fig. 39(a) suggests that there are two scaling regimes, which are well depicted in the log–log plot of Ec and f, as shown in Fig. 39(b).
In bulk materials, such as soft PZT ceramics, -orientated KNN and BT single crystals, only one scaling regime was reported. It should be noted, however, due to the limitation of the voltage amplifier power, the varying frequency range is much narrower for bulk materials than that for thin films. More than one regime is expected if a broader frequency window can be applied to the bulk materials. Due to the fact that the domain reversal process is composed of two stages, that is, nucleation and domain growth, the dynamic process of these two stages will be different as reflected in the log–log plot of Ec and f.
The influence of temperature on hysteresis loops is an inevitable topic from both fundamental and application points of view. Here, we only discuss the temperature effect on ferroelectric materials far below the ferroelectric-to-paraelectric phase transition temperature TC, above which, the macroscopic net polarization vanishes completely, with the PE loop being transformed from normal square shape through slim loop (2S loop for first-order ferroelectrics), and finally to a linear nonhysteresis response between the P and E, as discussed in the above section, as shown in Fig. 40.
For ferroelectric materials, temperature increasing is associated with the increase in thermal fluctuation, which will disrupt the long-range order of the polarization and weaken the coupling stabilizing effect between the charged defects and domains. Therefore, both Ec and Pr are generally decreased with increasing the temperature, as shown in Fig. 41.
Generally speaking, the Ec is decreased with respect to temperature increasing. A linear relationship between Ec and T, that is, (Ec0 − Ec) ∝ T (Ec0 is the coercivity at T ~ 0 K), was reported for soft PZT ceramics, Pb(Mn1/3Sb2/3)–Pb(Zr,Ti)O3 ceramics, (K,Na)NbO3 thin films, and Zr-rich PZT ceramics. However, this trend is often disturbed by the PPT. In the vicinity of the ferroelectric phase transition temperature, such a linear relationship is invalid, where an abnormal increase in the Ec with increasing temperature was reported, as shown in Fig. 42(a).
The abnormal Ec observed on the proximity of PPT transition can be explained by the coexistence of different ferroelectric phases. It is interesting to note that for a poled Mn-doped PMN–PZT ceramic, the asymmetric P–E hysteresis loop was found to gradually transform to symmetric shape with increasing temperature. Similar results can also be observed in Mn-modified PIN–PMN–PT crystals, where the internal bias field Ei and Ec follow a similar linear relationship with respect to temperature, as shown in Fig. 42(b). This is due to the fact that the aligned defect dipoles decoupled gradually by increasing temperature and consequently, an almost symmetric square P–E hysteresis loop was observed at elevated temperature.
The change in Pr as a function of temperature is more subtle compared to Ec. Yimnirum et al. reported the effect of temperature on Pr for soft and hard PZT ceramics, suggesting a power law relationship, that is, Pr ∝ Tβ, with a negative β.[217, 222] In other report, however, the Pr was found to maintain similar values in the temperature range 298–473 K for poled Mn-doped PMN–PZT ceramics, which may be due to the existence of PPT in the studied temperature range. In a Zr-rich PZT ceramic, Chen et al. found that the Pr–T relationship could only be well described by two power law fittings, and the β changed from −0.62 at low T to −0.31 at high T. The critical temperature separating these two regions is around 313 K, corresponding to a low-temperature ferroelectric rhombohedral FR(LT) to high-temperature ferroelectric rhombohedral FR(HT) phase transition point. It is interesting to note that the FR(LT) − FR(HT) phase transition cannot be detected by the dielectric properties as a function of temperature, but very sensitive to the polarization variation, where the β shows distinguished values below and above the transition point. Similar results were also reported for KNN thin film, where there are two power law regions for Pr measured from 100 to 340 K, being expressed as Pr ∝ T−0.77 for T < 245 K and Pr ∝ T0.88 for T > 245 K, respectively. Meanwhile, a rhombohedral-to-orthorhombic phase transition was found to occur at 245 K, corresponding to the inversion temperature of the polarization. Although the power law was valid at both regions, the β became positive at high-temperature region, which means the polarization was increased with increasing the temperature.
It is important to point out that there are very limited studies on the hysteresis loop in cryogenic temperature region, which may shed more light on the understanding of the domain behavior as a function of temperature and electric field, especially near the freezing temperature range.
Ferroelectric materials are frequently subjected to external stress for practical applications. According to the loading methods, three kinds of stress effects on the hysteresis loops will be discussed in this section.
(a) Uniaxial Stress Domain switching occurs when the applied electric field exceeds the coercive field, being determined by the crystalline structure and symmetry. If an uniaxial compressive stress is applied to the sample with the direction parallel to the electric field, there are two factors need to be considered, being associated with intrinsic and extrinsic contributions.[49, 52] One is the crystal lattice distortion or spontaneous deformation, which is suppressed by such an external mechanical stress to a certain extent. As a consequence, the intrinsic contribution by the crystal lattices to the polarizations, which can be reflected by Ps and Pr, are reduced. The other one is the domain wall motion (especially the non-180° ferroelastic domain walls), which is clamped or constrained when subjected to compressive stress. In this case, less domains will be involved in the movement responding to the external electric driving field, leading to a decreased contribution to the polarization. Furthermore, even the electric field exceeds the coercive field, the domain wall motion is not easy to occur under the stress, and thus higher field is required to switch the domain walls. Both the intrinsic and extrinsic contributions account for the hysteresis loops being transformed from a square loop to a slanted one. If the stress is large enough, domain wall motions, especially non-180° ferroelastic domains, will be frozen and a nonhysteresis loop is expected. This was confirmed by the investigation on the electromechanical properties of 8/65/35 PLZT ceramics under uniaxial compressive stress, where both Ps and Pr were found to decrease with respect to increasing stress. Later, this observation was also verified in soft PZT, Fe3+/Nb5+ BaTiO3, Pb(In0.5Nb0.5)O3–PT, and PZT-885. A typical evolution of the hysteresis loops with respect to the uniaxial stress is given in Fig. 43(a). However, the effect of the uniaxial stress on the Ec is controversial based on the results reported by different research groups. In most cases, Ec was found to decrease as the stress increases [Fig. 43(c)], with the exception that the Ec was reported to be insensitive to the external compressive stress in soft ceramics.
On the contrary, a stable polar phase is preferred with the uniaxial compressive stress applied perpendicular to the electric field direction. Figure 44 shows the evolution of the hysteresis loops for 0.9PMN–0.1PT relaxor ferroelectrics, with increasing perpendicular uniaxial stress from 0 to 130 MPa. It is clear that there is a transition from a slim loop under the stress-free condition to a near square hysteresis loop under a high compressive stress, suggesting that the macroscopic polar phase induced by the electric field becomes more stable with the aid of the perpendicular stress. This will benefit the practical application, especially for shear vibration mode, where the driving electric field is perpendicular to the poling direction. Li et al. reported that for application of thickness shear crystals, the main drawback is the low allowable ac electric field, being less than half of its coercive field.[89, 228] The shear piezoelectric response was found to drastically decrease with unexpected interference by other vibration modes once the ac field exceeds the allowable field. With the prestress perpendicular to the poling direction, the polar state was sustained effectively; in addition, it will induce the ferroelectric phase transition if the stress is large enough. As shown in Fig. 45, the hysteresis properties of PIN–PMN–PT crystals were drastically reduced by applying the perpendicular stress to the crystals, due to the fact that the domain wall motions are stabilized by the stress. As a consequence, the allowable drive field is increased by more than 50%.
(b) Radial Stress It was reported that the domain wall motion of PZT ceramics became facilitated with the aid of a radial mechanical load, leading to the increased Ps and Pr and decreased Ec.[230, 231] However, this mechanical confinement effect seems more obvious in relaxor ferroelectric single crystals. Marsilius et al. suggested that the radial compressive stress broaden the hysteresis loops for both rhombohedral and tetragonal PMN–PT single crystals, as shown in Fig. 46. A drastic increase of 200% then followed by a saturation plateau of the Ec was reported, with the Pr initially increased and then decreased with increasing the radial stress. These observations are interpreted in terms of direct/converse piezoelectric effect and possible phase transition in rhombohedral crystals and by the ferroelastic domain switching in tetragonal crystals. It is noted that due to the ultrahigh piezoelectric response, the coupling between the mechanical and electrical field is much larger than that in polycrystalline ceramics. This may be the reason why the radial stress effect in ferroelectric single crystals is more significant than that in ceramics.
(c) Hydrostatic Stress The hydrostatic stress effect on the ferroelectric properties for PLZT ceramics with MPB compositions was studied. For the ferroelectric PLZT 2/90/10, both Ps and Pr show a small decrease with increasing the hydrostatic stress up to 200 MPa. However, a drastic decrease in Pr and pronounced broadening of the switching region were observed at 300 MPa, suggesting a transformation form ferroelectric-to-antiferroelectric phase. In contrast, for PLZT4/90/10, typical antiferroelectric double hysteresis loops were obtained up to 200 MPa. Both the forward and backward switching fields increased significantly, following by a rapid decrease of Pmax, as Ps cannot be induced with a limited electric field. Importantly, for PLZT3/90/10 sample, a clear ferroelectric-to-antiferroelectric phase transitions occurs when the pressure exceeded 100 MPa. Based on the above, the antiferroelectric phase is more preferred with respect to the hydrostatic stress, due to its smaller volume compared to its corresponding ferroelectric phase. It is interesting to note that for antiferroelectric ceramics, even uniaxial and radial stresses prefer the antiferroelectric phase. Similar observations as in PLZT4/90/10 were also reported by Tan et al.
It should be noted that the stress effect on the P–E characteristics of thin film is depended on the film thickness, which was discussed in the Section IV. 1(E).
5 Summary and Future Perspective
In this paper, the hysteresis loop which is the critical characteristic of ferroic materials has been discussed for ferroelectric-related materials. The ferroelectric loops are divided into four groups according to their morphology features, that is, classical ferroelectric loops, double hysteresis loops, asymmetric hysteresis loops, and slim hysteresis loops. To clarify the comprehension of the loops in ferroelectrics, we discussed the effects of materials, aging, and measuring conditions on hysteresis loops in terms of recent developments of ferroelectrics. By analyzing various examples, the general rules of the evolution of the hysteresis loops were summarized in this study and the hysteresis phenomena in ferroelectric could be understood thoroughly. In the future perspective, it is expected that more information of ferroelectric materials, including macroscopic properties and microscopic structures, can be “read” from the hysteresis loop, by combining the modeling and simulation of materials.
The authors gratefully thank Prof. Dragan Damjanovic for his original inspiration at ISAF-PFM-2011 conference to compose this review article and Prof. Thomas R. Shrout for his valuable comments. This work was supported by the National Nature Science Foundation of China (grant nos. 51102193, 51202183, and 51372196), the China Postdoctoral Science Foundation and the Fundamental Research Funds for the Central Universities.
Li Jin: Li Jin is associate professor at school of electronic and information engineering, Xi'an Jiaotong University, Xi'an, China. He received B.E. and M.E. degrees in Electronics Science and Technology from Xi'an Jiaotong University, in 2003 and 2006, respectively. From September 2006 to March 2011, he studied at the Swiss Federal Institute of Technology-EPFL, Lausanne, Switzerland, and received his Ph.D. in Materials Science and Engineering in 2011. Prior to joining Xi'an Jiaotong University in 2012, he was a postdoctoral research fellow in Ceramics Laboratory of EPFL. His research interests are in the ferroelectric/piezoelectric materials and the related characterization techniques.
Fei Li: Fei Li was born in Shannxi, China, in 1983. He received B.E. and Ph.D. degrees in electronics science and electronic materials from Xi'an Jiaotong University, Xi'an, China, in 2006 and 2012, respectively. From September 2009 to September 2010, he worked in the Material Research Institute of The Pennsylvania State University as visiting scholar. He is now a faculty member at Xi'an Jiaotong University and his research interests are in the field of piezoelectric and ferroelectric single crystals and ceramics.
Shujun Zhang: Shujun Zhang received Ph.D. from Shandong University, China, in 2000. He is Senior Research Associate at Materials Research Institute and Associate Professor at Materials Science and Engineering Department of The Pennsylvania State University. He is associate editor for Journal of the American Ceramic Society, IEEE Transaction on UFFC and Journal of Electronic Materials. He was a recipient of the Ferroelectrics Young Investigator Award of IEEE UFFC Society in 2011. He is a Senior Member of IEEE and Member of the American Ceramic Society. He holds three patents and has authored/coauthored more than 270 papers in refereed journals. He is now focusing on the structure- property- performance relationship of high temperature and high performance dielectric and ferroelectric/piezoelectric crystals and ceramics, for sensor and transducer applications.