Switching Properties: Basic Methods and Characteristics

  • Alexander K. Tagantsev
  • L. Eric Cross
  • Jan Fousek


We approach one of the main issues of the investigations of ferroics: their properties conditioned by dynamic domain phenomena. It is these properties that play the decisive role in many recent applications. But before entering this subject in Chap. 8, we wish to describe experimental methods used for obtaining integral data about such phenomena. We have in mind the data that reflect domain wall motion and other mechanisms involved in the processes in which a ferroic sample changes its domain state under the action of external forces. These mechanisms involve possible nucleation of new domains, growth of nucleated or of already existing domains, and their coalescence. Such integral data provide the basic information on characteristics of the switching process as a whole, like its speed, its dependence on the applied force, on the boundary conditions, or on temperature.


Hysteresis Loop Domain State Switching Process Triglycine Sulfate Pyroelectric Coefficient 

7.1 Introduction

We approach one of the main issues of the investigations of ferroics: their properties conditioned by dynamic domain phenomena. It is these properties that play the decisive role in many recent applications. But before entering this subject in Chap. 8, we wish to describe experimental methods used for obtaining integral data about such phenomena. We have in mind the data that reflect domain wall motion and other mechanisms involved in the processes in which a ferroic sample changes its domain state under the action of external forces. These mechanisms involve possible nucleation of new domains, growth of nucleated or of already existing domains, and their coalescence. Such integral data provide the basic information on characteristics of the switching process as a whole, like its speed, its dependence on the applied force, on the boundary conditions, or on temperature. In this chapter we describe the basic methods to obtain such data. We will mainly concentrate on the basic characteristics: P(E) dependences for ferroelectrics and ε(σ) dependences for ferroelastics or alternatively ε(E) dependences for ferroics which exhibit simultaneously ferroelastic and ferroelectric properties.

In some cases it may be difficult to measure the primary order parameter as a function of the conjugate force, such as the P(E) dependence for ferroelectrics. Then to obtain some information on the switching process one may rely on measuring other properties which can be believed to be linearly related to the order parameter. As an example we may mention the measurements of the pyroelectric coefficient dependence on the applied electric field, in the form of a hysteresis loop, for ultrathin films of ferroelectric polymers (Ducharme et al., 1997). There it was difficult to detect directly the small switched charge while the pyroelectric coefficient was of a large enough magnitude to be detected. This method of material characterization by measuring properties which are linearly coupled to the order parameter has become popular since the very beginning of the research of ferroics. Such data may be useful from the point of view of applications. In many cases it is relatively easy to measure macroscopic quantities which are coupled to a higher power of the order parameter η. If they are proportional to η2 we obtain the so-called butterfly hysteresis loops.

We stress that this chapter covers selected aspects of measuring switching characteristics of a considerable part or of the whole sample. Here we discuss neither “local switching” phenomena such as creating a new small domain by local application of electric field or mechanical stress nor methods for observing motion of individual domain walls.

7.2 Ferroelectric Hysteresis Loop

In Fig. 7.2.1 the basic circuit is shown for recording the switching process in a ferroelectric sample. It is often referred to as the Sawyer–Tower circuit, after the authors who were the first to use this system (Sawyer and Tower, 1930) when investigating polarization reversal in Rochelle salt crystals. Here the applied ac voltage U is divided between the sample and the capacitor C connected in series. On the horizontal axis of the oscilloscope we wish to record the magnitude of electric field applied to the sample alone. To fulfill this requirement the value of C has to be large compared to the effective capacitance (the ratio “the maximum charge over the maximum voltage applied”) of the sample. On the vertical axis, the recorded voltage corresponds to the instantaneous value of charge QP representing the dielectric displacement D = ε0E + P. In most ferroelectric materials, the first term is negligible compared to induced polarization. For this reason, the recorded hysteresis loop is interpreted as the dependence P(E) rather than D(E).
Fig. 7.2.1

The Sawyer–Tower circuit: 1, source of ac voltage; 2, oscilloscope; 3, capacitor with ferroelectric sample; 4, additional capacitor. \(R_1 > R_2\)—resistors of the voltage divider

It is obvious that if the sample is lossy, the conductive current flowing through it has a component which when integrated on the capacitance C produces charge Qcon such that an ellipse Qcon(E) is superimposed on the recorded hysteresis loop. This ellipse, in addition, is influenced (rotated in the charge–voltage coordinate system) by stray capacitances of the experimental setup. These effects may distort the recorded data on the D(E) loop of the ferroelectric and are unwanted; therefore, the ellipse of proper shape and orientation is to be subtracted from the total hysteretic response. Modified circuits which allow for such compensations were proposed by a number of authors (see, e.g., Roetschi, 1962; Gadkari et al., 1986; Sinha, 1965; Hatano et al., 1992; Diamant et al., 1957).

The classical Sawyer–Tower technique and its modifications still offer very effective and inexpensive tools for examining polarization reversal processes in ac fields. In recent years, in addition to these classical methods, alternative electronic schemes have been developed. This is the so-called virtual ground method which, for the first time, was suggested by Glazer et al. (1984). Nowadays, this method is widely used in connection with the expanding activities in the area of ferroelectric thin film memories. As an example, we mention the setup used by the Radiant Technologies, Inc. Systems and represented schematically in Fig. 7.2.2. In this configuration, the transimpedance amplifier maintains the terminal A at a virtual ground potential. Thus the sample is effectively “grounded” during the switching process. All charge that flows through it as a result of the applied voltage is collected by an integrator circuit. The voltage generated on the output of the integrator is then measured and displayed as a function of the applied voltage. The system makes it possible to measure accurately a large range of capacitance values at a large range of speeds. The capacitor in series with the sample required in the Sawyer–Tower circuit is abolished and thus the effects of possible parasitic impedances are eliminated. Probably the most important advantage of the system becomes effective when only one period of ac field is applied. In this case, in the classical Sawyer–Tower system, after the voltage returns to zero the charge that has been collected in the sense capacitor generates a voltage Vback which is in fact applied to the sample in the direction opposite to the last applied voltage. This can lead to “backswitching”: In part of the sample polarization can return back to its previous orientation. In contrast, in the virtual ground measuring system the Vback voltage is not generated: In the interval between the two subsequent periods of applied voltage, typically several seconds long, the sample is virtually short circuited and backswitching could only be initiated by an internal bias in the sample.
Fig. 7.2.2

The virtual ground measuring system (Radiant Technologies, RT6000HVS)

It is now appropriate to specify the definitions of basic quantities used to characterize a ferroelectric hysteresis loop. A typical customarily observed P(E) loop is represented in Fig. 7.2.3a when driven by a continuous ac field. It defines maximum and remanent polarizations Pmax and Pr as well as the coercive field Ec which corresponds to the points where PD = 0. In general, the values of Pmax and Pr do not suffice to determine the value of spontaneous polarization PS. However, if the loop is saturated, i.e., the branches of the loop merge before the tip of the loop, and if the driving field is not too high, the intersection of a tangent of the loop taken at its tip yields the value of PS. The meaning of “not too high” is that the field does not result in appreciable nonlinearity of the lattice dielectric permittivity. This condition is not always met for characterization of thin films. There exist materials where PS can be determined directly from the hysteresis loop. This is usually the situation of high-quality single crystals when the P(E) loop is measured far below the phase transition. Such an “ideal loop” is shown in Fig. 7.2.3b and there is no doubt that the intersection with the vertical axis defines the value1 of PS which equals both Pmax and Pr. Connecting the extreme points (i.e. Pmax(Em)) of the curves taken at different field amplitudes, we obtain what is sometimes referred to, not quite logically, as the virgin curve.
Fig. 7.2.3

(a) Conventional P–E hysteresis loop of a ferroelectric (schematically); (b) ideal hysteresis loop; and (c) double hysteresis loop and its derivative

As an example, we refer to the delightful hysteresis loops taken for TGS crystals by Nakatani (1972) and shown in Fig. 7.2.4. As the amplitude Em of the applied field E = Em sin ωt increases, the shape of the P(E) dependence changes from an oblong-like dependence to an ideal hysteresis loop.
Fig. 7.2.4

Dependence of 60 Hz hysteresis loop shape of TGS on applied field, its amplitude being Em= 160, 320, 800, 1,600, and 3,200 V/cm, successively from the internal one. Temperature –24.5°C. After Nakatani (1972)

The classical Sawyer–Tower method and its analogies have been customarily used to record hysteresis loops in the frequency region between 1 Hz and 1 kHz (see, e.g., Campbell, 1957; Shil’nikov et al., 1999a). It may be of interest to perform measurements at even lower frequencies. Unruh (1965) obtained reliable data for hysteresis characteristics of Rochelle salt and triglycine sulfate, based essentially on the classical method, at frequencies down to 10–2Hz. In the same way, Shil’nikov et al. (1999b) measured hysteresis loops of TGS at frequencies between 0.05 and 90 Hz, for several field amplitudes; their data are reproduced in Fig. 7.2.5. At these frequencies, the experimentalists may face the problems connected with surface and bulk electrical conductivity of the specimen. Then the integral switching process at frequencies below 1 Hz may be investigated by measuring, instead of polarization itself, some quantities which are coupled to polarization in a known way and which are not affected by the electric current due to conductivity. As an example, we refer here to Abe’s (1964) measurements performed in ac fields of frequencies down to 4 mHz; the quantity measured as a function of applied field was the integrated intensity of polarized light passing through a sample. In the case of Rochelle salt this intensity can be shown to be proportional to the areas of reversed domains and thus to the average polarization.
Fig. 7.2.5

Hysteresis loops of TGS crystals at 18°C demonstrating the influence of field amplitude and frequency. Em= 35, 55, 74, and 92 V/cm (a)–(c) and Em= 140, 230, 370, and 550 V/cm (d)–(f). Horizontal axes units: 10 V/cm (a)–(c) and 100 V/cm (d)–(f). Vertical axes units: 10–4(a), 10–5(b,c) and 10–2 C/m2(d)–(f). Reprinted with permission from [Shil’nikov, A.V., Pozdnyakov, A.P., Nesterov, V.N., Fedorikhin, V.A., Uzakov, R.E., The analysis of domain boundaries dynamics of TGS single crystals under the ac-Fields of low and ultralow frequencies, Ferroelectrics, 223, 149 (1999))]. Copyright (1999), Taylor and Francis

Often, the registration of the P(E) dependence offers information not only on the basic ferroelectric switching process but also on some more involved phenomena. Deformation of the hysteresis loop may give evidence of internal biasing field and time changes of the loop shape demonstrate different kinds of ageing effects. Frequently the so-called double hysteresis loop is observed. It can be connected with the influence of lattice defects whose presence prefers a domain pattern with zero average polarization. As an example, Fig. 7.2.2c shows such loops for ceramic samples of BaTiO3 doped with Fe (Hagemann, 1978). Alternatively, double hysteresis P (E) loops are observed at temperatures just above TTR in ferroelectrics with the first-order phase transition; they arise from inducing the ferroelectric phase by ac biasing field (Merz, 1953, Hatano et al., 1985a). This is a mechanism analogue to that responsible for the double ferroelectric loops in antiferroelectrics; it is not defect related.

In connection with the development of ferroelectric thin films it becomes usual to use just one period of a triangular ac voltage wave for switching characterization. Figure 7.2.6 shows an example of this kind of data recorded with an RT-6600S setup (Radiant Technology Inc.). It portraits the switching processes in PLZT thin films, annealed at different temperatures (Hirano et al., 1999). Here the triangular-shaped voltage consists of a number of short intervals during which the voltage is constant and the charge is integrated and displayed in form of points. The discontinuous jump at zero voltage images the process taking place during the period between two subsequent cycles. Typically, the duration of the triangular pulse is 10 ms (corresponding to the frequency of 100 Hz) with 200 sampling points.

In the classical Sawyer–Tower circuit the capacitor C can be replaced by a resistor R. In this configuration, what is detected as a function of applied ac field E = Em sin ωt is the electric current
$$i = \frac{{{\rm{d}}D}}{{{\rm{d}}t}} = \frac{{\partial D}}{{\partial E}}\frac{{\partial E}}{{\partial t}} = \omega \frac{{\partial D}}{{\partial E}}\sqrt {E_{\,\rm{m}}^{\,2} - E^2 }$$
Here the slope ∂D/∂E of the hysteresis loop is multiplied by the function \(\omega \sqrt {E_{\,\rm{m}}^{\,2} - E^{\,2} } \), which is an ellipse. Thus, depending on the shape of the hysteresis loop and on the applied amplitude, the maximum current may not correspond to the maximum ∂D/∂E. If the data are taken applying a triangular voltage wave so that within one half-period the derivative ∂E/∂t is constant, then the curve of i(E) corresponds to the real derivative of hysteresis loop. In this case and for a typical unconstricted loop the positions of its maxima are sometimes considered to represent the coercive fields. Since the slope ∂D/∂E may not reach its maximum exactly at P = 0, the value of the coercive field defined in this way can slightly differ from that defined above (Fig. 7.2.3).
Fig. 7.2.6

Hysteresis loop of PLZT thin films obtained with the system shown in Fig. 7.2.2. After Harano et al. 1999

It is useful to point out that the area of the hysteresis loop D(E) determines the effective dielectric losses. Obviously, the density of energy lost in the sample during one cycle of period T is
$$\int\limits_0^T {iE} \,{\rm{d}}t = \int\limits_0^T {\frac{{{\rm{d}}D}}{{{\rm{d}}t}}E\,{\rm{d}}t = \oint {E\,{\rm{d}}D} }$$
and the energy lost (or heat developed) in 1 s equals
$$Q = f \oint {E\,{\rm{d}}D} .$$
At the same time, this loss of energy can be conveniently written in terms of the imaginary part κ″ of permittivity as
$$Q = \pi f\kappa "E_{\rm{m}}^2 ,$$
where Em is the amplitude of ac field of frequency f = 1/T. Thus the imaginary part of effective permittivity can be related to the area of the loop:
$$\kappa " = (1/\pi E_{\rm{m}}^2 )\oint {E\,{\rm{d}}D} .$$

This equation implies the possibility of a cross-check of the hysteresis loop itself and the dielectric loss data.

At the end of this section we wish to discuss several artifacts and possible factors not related to the intrinsic properties of the ferroelectric material itself.

First, the surface conditions can seriously influence the obtained data. Thus the polarization reversal process can strongly depend on the coupling of the sample with electrodes. Janovec et al. (1960) showed that a BaTiO3 crystal plate with two identical liquid electrodes showed a symmetric hysteresis loop with fast switching. With two identical indium electrodes the loop was also symmetric but switching was slower. The use of different materials for different electrodes led to an asymmetric loop, showing that a liquid electrode provided more favorable conditions for switching starting at that electrode. The existence of a surface layer located between the homogeneous sample and electrode has a tremendous effect on the switching properties, as demonstrated by Brezina and Fotchenkov (1964) and discussed by Drougard and Landauer (1959). Rosenman and Kugel (1994) showed experimentally that a vacuum gap or a thin teflon layer located between the sample and the electrode can seriously influence or fully suppress the switching process. These and similar factors have to be considered when interpreting experimental data on switching and we will come back to this problem in Chaps. 8 and 9.

Second, in a specific device, it may happen that the source of the applied voltage may not be able to provide the current required for the switching process to proceed fast enough as determined by its natural domain processes. This situation can be modeled by a resistor in series with the ac power supply. As a result, the large slope ∂D/∂E may be reduced so that the hysteresis loop is deformed.

Third, we wish to note that in exceptional cases, a hysteresis curve of the typical shape may be observed and yet the material could turn out not to be ferroelectric at all. If the dielectric response of a material is strongly nonlinear but non-hysteretic and at the same time the sample is lossy, one obtains the D(E) curve strongly resembling a hysteresis loop. This was, e.g., the case of LiN2H5SO4 where seemingly properly shaped loops (Schmidt and Parker, 1972) were explained in terms of intrinsic protonic conductivity of this material. A similar situation was recognized for ceramic samples of TlTaO3 and some other materials (Le Bihan et al., 1978). As pointed out by Scott et al. (1993), another example of “false” hysteresis curves is provided by polymeric electrets. Charges originating in mobile ions diffusing under applied voltage can slowly accumulate at the surfaces of plate-like samples of electrets, resulting in a bistable state whose orientation depends on the polarity of applied voltage. Thus seemingly the electrets can also be switched in polarity; however, the processes resulting in a measurable hysteresis curve are very slow (Sessler et al., 1980).

Fourth, one has to realize that, when studying the frequency dependences of the mentioned quantities EC, Pmax, etc., at higher frequencies, the experimentalist may face the problem of self-heating. This is discussed in some detail in the following section.

7.3 TANDEL Effect

When studying the frequency dependence of coercive field, it was observed (see, e.g., Campbell, 1957) that Ec first increases but then, with further increasing frequency, it starts to decrease again. The explanation was based on the assumption that due to hysteresis losses (cf. Eq. 7.2.3) the sample heats up and approaches the Curie point so that the area of the transversed loop decreases. Later it was found by Shuvalov (1960) that at some critical frequency of the field applied to the Y-cut of triglycine sulfate crystal the sample increases its temperature with a jump.

Glanc et al. (1964) studied these effects in detail and found that when an ac voltage of high enough frequency is applied to the Y-cut of a TGS crystal, the specimen is heated to a temperature TS close to TC and the value of TS is stabilized with respect to the ambient temperature TA of the surroundings. This was explained by Dvorak et al. (1964) in general terms as the consequence of negative temperature coefficient of losses. The crystal is in a state of temperature autostabilization and since it reveals nonlinear properties it can be referred to as “temperature autostabilized nonlinear dielectric element” (TANDEL). In addition to TGS, the effect was later observed in a number of other ferroelectrics. We give a schematic insight into the phenomenon.

We know (Eq. (7.2.3)) that the rate of heat production Q1 in the ferroelectric is proportional to the area of the DE hysteresis loop. It is also known the both height and width of the loop decrease on increasing temperature so that the loop area and Q1 are decreasing functions of temperature. In the stationary state, the rate heat production Q1 should be balanced by the heat dissipation into the ambient Q2, which is proportional to the difference TSTA. The cycling of the sample leads to an increase in its temperature but the negative temperature coefficient of losses (decrease of the loop area with increasing temperature) will result in temperature stabilization. The final temperature of the system is given by a solution to the equation satisfying the stability condition (Dvorak et al., 1964)
$$Q_1 = Q_2 ,\quad \partial Q_1 /\partial T < \partial Q_2 /\partial T.$$

This phenomenon was simulated by Fousek (1965a) in the approximation of rectangular polarization loop. In this approximation, the rate heat production \(Q_1 \propto P_0 E_{\rm{c}}\) where P0 and Ec are the half-height and half-width of the loop, respectively. It was also assumed that depending on the amplitude of the driving field E0 three regimes are possible: (i) for small fields (\(E_0 < E_{0k}\)), the loss is negligibly small and one sets \(P_0 = 0\); (ii) for intermediate fields (\(E_{0\,k} < E_0 < E_{0{\rm{c}}} \)), \(P_0 \propto E_0 - E_{0\,k}\) and \(E_{\rm{c}} = E_0 \); and (iii) for large fields (\(E_{0{\rm{c}}} < E_0 \)), \(P_0 = P_{\rm{s}} \) and \(E_{\rm{c}} = E_{0{\rm{c}}} + g(E_0 - E_{0{\rm{c}}} )\), where Ps is the spontaneous polarization. The simulation has been performed for the experimental situation close to that in TGS. This defines the choice of the temperature dependences of the parameters controlling the problem: \(E_{0\,k} \propto E_{0{\rm{c}}} \propto g \propto \sqrt {T_{\rm{C}} - T}\) where TC is the transition temperature.

Novák and Hrdlička (1968) confirmed the validity of this model experimentally, for TGS samples. The TANDEL effect was observed in a number of ferroelectric materials (Malek et al., 1964) as well as in glass ceramics (Lawless, 1987). It represents some danger when data on hysteresis loops are interpreted without taking the process of self-heating into account.

7.4 Pulse Switching

An ac voltage applied to a ferroelectric sample drives a switching process whose time development is in some correspondence with the time-dependent magnitude of the applied field. It is obvious that when the applied field during the whole operation remains constant, we obtain more straightforward information about the polarization reversal process. This was for the first time realized by Merz (1956) who investigated the switching process in barium titanate crystals by applying rectangular voltage pulses of alternate polarity and detecting the switching current flowing through the ferroelectric sample. Often a pause with zero field is inserted between the pulses of opposite polarity. Figure 7.4.1 shows the basic scheme of the circuit employed and also the typical profiles of the current i(t) monitored. The first sharp current peak corresponds to the linear capacitance of the sample, not connected to any domain phenomena. When the polarization reversal takes place in the whole volume of the sample, obviously
$$\int\limits_0^\infty {i(t)\,{\rm{d}}t} = 2P_{\rm{s}}$$
Fig. 7.4.1

Left: Basic scheme for pulse switching; the source S applies voltage pulses of prescribed polarity. Right: The curve “1” shows the switching current density i which is typically characterized by the values of imax, tmax, and ts. The curve “0” shows the response when the applied field is parallel to spontaneous polarization; it corresponds to linear capacitance of the sample

In reality, the value of polarization determined in this way may be smaller than PS because of certain backswitching after the previous polarization reversal process. It is essential that the output impedance of the current source is low enough so that the voltage would not drop even when the switching current reaches its maximum value imax.2

Three quantities characterizing the switching process are defined unambiguously, namely, the applied field E, the maximum value imax of the current, and the time tmax at which it occurs. For practical purposes, the time required to complete the switching process is an important attribute. In customary measurements, the length tappl of the applied field pulse should be long enough to virtually complete the switching process so that integral (7.4.1) taken from 0 to tappl would be very close to 2PS. Otherwise the switching process is completed only partially. As another characteristic of switching a notion of switching time ts is introduced. This time is often defined as that necessary to reverse PS in a certain fraction, e.g., 95%, of the sample volume (Fatuzzo and Merz, 1966). For convenience, however, the switching time ts is defined in another way, as the time necessary for the switching current to drop to a certain fraction, e.g., 5% of its maximum value imax.

It is this method, when the processes proceed at constant applied field, that allows for well formulated theoretical discussions. The dependences ts(E) and imax(E) for BaTiO3 and other materials provide the core information on which theoretical interpretations of switching are based. It is obvious that to obtain reliable results, again the power of the current source is of importance to provide the rising time of the applied pulse considerably shorter than tmax. In practice, it is required that the value of tmax should be independent of the capacitance (area of electrodes, in fact) of the sample.

Apart from the classical transient current method just discussed above an alternative technique, which can be called poling back technique, has been suggested by the group of Waser (Grossmann et al., 2000). This technique comprises an application of the pulse sequence shown in Fig. 7.4.2. In this sequence, the first, the second, and the fourth pulses have the same amplitudes and durations large enough to perform the full polarization reversal whereas the amplitude, V3, and length, t3, of the third pulse are variable. The amount of the polarization switched by the third pulse is determined by switching it back by the fourth pulse. This method was shown to have a clear advantage compared to the classical one in the case where the switching is stretched for many decades in time. The reason for that is an increasing difficulty with reliable monitoring of very small currents that are typical for the stretched switching. Usually, the transient current method enables us to cover no more than two decades in time (Merz 1956; DeVilbiss and DeVilbiss, 1999; Song et al., 1997), whereas the poling back technique can readily cover a six to eight decade interval (Lohse et al., 2001; Tagantsev et al., 2002a,b). An example of the plot of time dependence of switched polarization for PZT thin films is shown in Fig. 7.4.3.

It is worth mentioning that the application of both techniques requires considerable precaution in order not to take the RC-controlled dynamics of the measuring setup for a manifestation of the real switching dynamics of the material (Larsen et al., 1991; DeVilbiss and DeVilbiss, 1999; Seike et al., 2000). A simple reliable test excluding the RC artifact is to check that the switching current kinetics is independent of the capacitor area.
Fig. 7.4.2

Sequence of voltage pulses used for measurements of the switching polarization with the poling back technique

Fig. 7.4.3

Switching polarization as a function of the switching time, for different voltages measured with the poling back technique on a capacitor containing 135 nm thick film of PZT. After Tagantsev et al. (2002b)

The switching times, understandably, depend strongly on the applied field. To give an example of their magnitude, for bulk ferroelectric crystals, in which 180° polarization reversal process is not accompanied by a change of spontaneous strain (BaTiO3, TGS), typical switching times are of the order of 10–0.1 μs in fields of several kilovolts per centimeter. However, we shall see in Chap. 9 that in thin ferroelectric films switching times below 1 ns have been reached, though for much higher fields.

Concerning ferroelectric switching produced by short- and low-voltage pulses the observations of Fatuzzo and Merz (1959) are worth mentioning. They showed that if a series of voltage pulses much shorter than ts are applied to a TGS crystal, there is no net reversal of PS. There exists a critical pulse length t* at which the crystal begins to switch and if a series of pulses, each longer than t*, is applied, PS will be completely reversed. Both phenomena are represented schematically in Fig. 7.4.4. This kind of behavior is called “t* effect.” It is interesting that in the latter case the shapes of the individual small current pulses fit together yielding the “regular” pulse, except for the initial peaks A, B, C, D, E, F. The critical time t* depends on the amplitude of the applied pulse; for very low fields it is a very small fraction of ts while for high fields it can approach the magnitude of ts. Taylor (1965) pointed out that the “t* effect” could be used for non-destructive readout from a memory matrix as well as for its successive addressing. He also showed (Taylor, 1966) that the value of t* depends on the applied field and sample thickness and in particular that the ratio t*/ts depends on the quality of the sample surface. While in this chapter we pay attention to the methods only, it can be noted in passing that partial switching phenomena have not received corresponding attention of theorist. Here the work by Burfoot (1959) can only be mentioned.
Fig. 7.4.4

Pulses of applied electric field and of the switching current in a TGS crystal (a) for field pulses shorter than a critical duration (tpulse< t*) no switching was observed and (b) for a series of longer pulses (t* < tpulse< ts) the switching process is completed. Reprinted with permission from [Fatuzzo, E., Merz, W.J., Phys. Rev., 116, 61 (1959)]. Copyright (1959) by the American Physical Society

7.5 Ferroelastic Hysteresis Loops

Ferroelastic hysteresis loops show the strain induced by an applied mechanical stress of alternating polarity, i.e., the ε(σ) dependence. In principle, what is to be measured—in the simplest correspondence to the symmetry change induced by the phase transition—is the deformation of the sample subjected to a homogeneous mechanical stress. In ferroelastics which are simultaneously ferroelectric, we can also get information about the behavior of strain by applying an electric field, i.e., from the ε(E) dependence. Alternatively, we could also measure the P(σ) as well as the P(E) dependences, which all would have the characteristics of hysteresis loops. It might appear possible that these dependences could be easily converted into each other since we know the P(ε) or ε(P) relations “dictated” by the symmetry of the parent phase. However, in general, different kinds of loops do not bring identical information and such conversions should include a number of rather complicated factors; for instance, the shape of the P(σ) hysteretic dependence may not conform with the shape of the P(E) hysteresis loop since customarily the two are taken at different boundary conditions. We remind the reader that except for 180° nonferroelastic ferroelectric switching, ferroelastic aspects are always present even in ferroelectric hysteresis loops.

In this section we concentrate on the basic ferroelastic ε(σ) hysteresis loops. Their main characteristics are defined in analogy with ferroelectric loops. The meanings of spontaneous strain, remanent strain, maximum strain, and coercive stress are self-evident and these concepts are routinely used.

Basically, we are interested in measuring a strain component as a function of the conjugated stress component. In principle, the application of “axial” stress, i.e., tension and compression, is required for a properly oriented sample. Ferroelastic hysteresis loops have been investigated in several laboratories for a number of ferroelastics, but the methods employed significantly differ. We mention several examples. Pakulski et al. (1987) used an apparatus shown in Fig. 7.5.1 which allows one to measure directly ε(σ) loops, revealing components of the spontaneous strain. Its notable features are that the sample can be simultaneously observed in a microscope and the whole setup allows the measurements at low temperatures. The specimen has the form of a rectangular bar. Its lateral faces are polished to allow optical observations and then it is glued into the apparatus. The force is transmitted to it through a ceramic rod; it is proportional to the current flowing through the coil placed in a magnetic field, with the proportionality coefficient p = 5 N/A. The deformation induced by this force is monitored with a displacement sensor. The sample slot makes it possible to measure capacitance of the sample, however, with gaps 0.2 mm thick between the electrodes and the sample. Figure 7.5.2a gives examples of data obtained with this apparatus. Ferroelastic loops ε12(σ12) were measured for the LiCsO4 crystal, representing the species mmmεs–2/m, by monitoring the elongation and contraction of a z-45° cut with dimensions 10 × 2 × 0.5 mm3. Loops of permittivity vs. stress, also shown in the figure, demonstrate the ferrobielectric features of the phase transition in this material. The change in the polarity of the κ(σ) loop, when passing from 197 to 172 K is surprising and we include this data to demonstrate what interesting features such studies can offer. Several explanations have been proposed (Pakulski et al., 1987) for this effect.
Fig. 7.5.1

Stress apparatus (Pakulski et al., 1987): 1, sample; 2, electrodes for permittivity measurement; 3, capacitor displacement sensor; 4, ceramic rod; 5, cold finger; 6, electrical feedthrough; 7, teflon ring; 8, coil; 9, permanent magnet; 10, glass window; A–A optical axes of the microscope

Fig. 7.5.2

(a) Stress–strain and stress–permittivity loops of LiCsO4; the stress frequency is 0.05 Hz (Pakulski et al., 1987). (b) Ferroelastic hysteresis loop of KFe(MoO4)2 recorded using the current proportional to the light flux through the sample as a measure of the strain (Krainyuk et al., 1983b). Elastic analogues of Barkhausen jumps are seen

In some cases “double loops” were observed (Pakulski et al., 1983; Shuvalov et al., 1984). Such loops are probably connected with a strong tendency of the sample for backswitching and after repeated cycling they often change into standard hysteresis loops (Kudryash et al., 1989).

Often tensile testing commercial apparatus is used to demonstrate ferroelastic hysteresis, such as Instron-type machines constructed for testing mechanical properties of metals. Prasad and Subbarao (1977) and Tsunekawa and Takei (1976) employed this technique for BaTiO3 and LaNbO4, respectively. In this case, only a partial loop corresponding to one sign of the stress is available. This is a strong disadvantage since no information about the shape of the full switching curve is available. The reliable information is just one quarter of the loop corresponding to the backswitching.

Information on ferroelastic switching can also be obtained in more involved geometries. Some authors offer data obtained for the torsion stress which involve twisting of the sample. In the torsion geometry, a rod-shaped sample is fixed at one end and a torque is applied to its face at the opposite side, with the rotation axis parallel to the rod. If z is the rod axis, the strain components which are induced are εxz and εyz. They are roughly independent of z but inhomogeneous along the radius of the rod. Alternative experiments have been performed in the bending geometry. Here again the bar-like sample is fixed at one end and force is applied to the opposite end, directing along x, perpendicular to the bar. The sample bends and the induced strain εzz is inhomogeneous, changing linearly with x, passing through zero in the middle of the bar. Torsion and bending can be used to obtain some information about the hysteretic behavior of strain but generally do not provide data about the order parameter dependence on the conjugate force. No complete analysis of response of samples of materials representing different ferroelastic species to such applied forces seems to be available.

In most cases, experimental data on ferroelastic hysteresis are usually based on measurements performed with frequencies in the range from 10 to 10–4 Hz. These rates are much lower than in the case of ferroelectric switching, including that in ferroelastic ferroelectrics. This fact is connected with technical aspects of the experiments rather than with the speed of ferroelastic switching: We know that polarization reversal in ferroelectric ferroelastics, when driven by electric fields, can be quite fast and ferroelectric hysteresis loops in these materials are often taken at frequencies above 102 Hz.

For the above-mentioned torsion and bending geometry, Krainyuk et al. (1983a,b) used an apparatus in which a rod-shaped sample is fixed at one end and provided with a magnet clamped to the other end. Helmholtz coils excite magnetic field of chosen orientation which acts on the magnet and the developed force results in deformation of the sample. A light beam reflects at a mirror attached to the magnet and its reflection angle is a measure for strain at the end of the rod. The setup allows also for simultaneous optical observations of the sample. Depending on the direction of magnetic field excited by Helmholtz coils, bending or torsion can be induced; this is obvious from Fig. 7.5.3a,b which shows the sample, S and N poles of the magnet, direction of the magnetic field H, direction of the mechanical moment M, as well as the small mirror and reflected beam; in (a) the bending and in (b) the torsion are induced. Data obtained by this apparatus clearly demonstrate nonlinear and hysteretic elastic responses of the sample. However, it may be difficult to obtain information about spontaneous strain and basic switching properties because of the inhomogeneity of strain in the sample.
Fig. 7.5.3

Basic geometry of the apparatus for measuring bending (a) and torsion (b) (Krainyuk et al., 1983a,b). The detections of the applied magnetic field (H) and angular momentum produced (M) are shown

Gridnev and co-workers used a method with several similar features for studying strains of prevailingly torsion character (Gridnev and Shuvalov, 1983). The sample is glued into the metallic main axis of rotational pendulum; the torsion of the axis is determined by torsion deformation of the sample and detected by capacitor detectors mounted on a rod fixed perpendicular to the main axis. A number of ferroelastic materials were investigated by this method. Figure 7.5.4 shows ferroelastic loops of KH3(SeO3)2, at different temperatures and amplitudes, where the hysteretic dependence of ε5 is plotted vs. applied stress (Gridnev et al., 1979).
Fig. 7.5.4

Ferroelastic hysteresis in KH3(SeO3)2 at different temperatures (Gridnev et al., 1979): (a) –178°C; (b) –100°C; (c) –67,5°C; (d) –62,4°C; (e) –59,6°C; and (f) shows the loops for different stress amplitudes at T = –178°C

All methods for studying macroscopic characteristics of ferroelastic switching mentioned above were based on quasistatic processes: the applied mechanical forces were changing slowly, in correspondence with the frequency range specified above. Vagin et al. (1979) offered data for switching times measured when applying mechanical pulses, in analogy with the ferroelectric pulse methods; however, no technical details seem to have been offered. For single crystals of Pb3(PO4)2 and applied stresses between 2 × 102 and 9 × 102 N/cm2, reported switching times amounted to 3–0.3 ms.

At the end of this section it is to be pointed out that here we have concentrated mainly on the methods of measurements of “true” ferroelastic switching phenomena. There are, however, many other methods to obtain indirect information on ferroelastic switching. For Rochelle salt crystals, Abe (1958) concluded that the average rotation angle of polarized light is nearly proportional to the change of average strain. This method, instead of recording true strain–stress hysteresis loops, can give more easily information about the frequency and temperature dependences of the coercive field (Abe, 1964). Salje and Hoppmann (1976) (see also Salje, 1990) studied ferroelastic loops in Pb3(PxV1–xO4)2 crystals using two approaches: direct information about the average value of strain measured by the angle of laser beam reflected from the surface of strained sample and indirect information obtained by measuring the averaged birefringence. The shapes of the two loops were found slightly different and this was attributed to the nonlinear relationship between strain and birefringence. Similar approach, namely, recording the change in light flux through the sample located between crossed polarizers, was used for detecting ferroelastic loops of KFe(MoO4)2 (Krainyuk et al., 1983), obtained under rigorous uniaxial loading of proper orientation. With the appropriate orientation of the crystal, the photosensor signal is proportional to the relative volume of one of the orientational states in the sample. One of the observed loops is shown in Fig. 7.5.2b. Here the elastic analogy of huge Barkhausen jumps is evident, an effect very often accompanying ferroelastic switching (see data on BaTiO3 from Sect. 8.6).

7.6 More Involved Methods

In the last section of this chapter we wish to mention, in passing, switching characteristics which can be achieved by methods other than those mentioned above or which are important for ferroics with more involved switching properties.

In ferroelectrics, one method could be found useful to characterize some features of polarization reversal in ac fields, namely, Fourier analysis of the switching current. Little attention seems to have been paid to this possibility. Karpov and Poplavko (1984) offered Fourier analysis of the hysteresis loop in TGS. In addition to the information itself, this experimental approach, when proper characteristics would be assigned to the spectrum, could provide a tool for specifying some particular aspects of switching, e.g., basic features of the ageing processes without observing the whole loop.

In the above sections we concentrated on ferroelectric and ferroelastic switching processes. Have hysteresis loops been observed also for higher order ferroics, ferroelastoelectrics, and ferrobielastics in particular? Many attempts have been made along this line, which unambiguously demonstrated domain reversal processes; however, the resulting data have not been presented in the form of hysteresis loops. In ferroelastoelectrics, switching should be driven by applying simultaneously electric field and elastic stress, based on the energy term d ijk E i σ jk with the value of d ijk differing in different domain states. An obvious candidate for this phenomenon is quartz, representing the species 622–ds–32. Its two domain states differ in sign of the piezoelectric coefficients d111= –d122= – (1/2)d212, but also in sign of the elastic compliances s1123= –s2223= (1/2)s1213. Thus when an electric field with nonzero components E1 and E2 is applied together with the stress of nonzero components σ11, σ22, σ12, σ13, and σ23, the free energy densities of the two domain states differ by
$$\Delta \Phi = 2d_{111} (E_1 \sigma _{11} - E_1 \sigma _{22} - 2E_2 \sigma _{12} ) \hskip 1pt + \hskip 1pt 4s_{1123} (\sigma _{11} \sigma _{23} - \sigma _{22} \sigma _{23} \hskip 1pt + \hskip 1pt 2\sigma _{12} \sigma _{13} ).$$

The first term in this formula provides the driving force for elastoelectric switching: Concurrent application of E1 and σ11 or of E1 and σ22 or of E2 and σ12 makes favorable one of the two domain states and any of these couples of applied fields could be used as the switching force. Careful experiments were performed by Laughner et al. (1979) which at temperatures close to TC resulted in domain state reorientation in limited volume parts of a quartz crystal. No changes in the spatially averaged piezoelectric response of the whole sample were recorded which would remind a hysteretic dependence of, e.g., d111 on the product E1σ11. Ammonium chloride (\(m\bar 3m - d - \bar 43m\)) provides another alternative for the observation of d() hysteresis loops. Mohler and Pitka (1974) applied uniaxial stress along the [011] axis, together with an electric field along the axis [100]. Since the two domain states differ in sign of the d123 coefficient, the domain pattern was really strongly influenced as demonstrated by the change of d123 from zero to a maximum value of 3 × 10–12 C/N. Another candidate for ferroelastoelectric switching is CsCuCl3 (6/mmmd–622) but attempts to induce ferroelastoelectric switching were not successful (Fousek et al., 1980).

In ferrobielastics, the switching process would be driven by the difference in the s ijkl σ ij σ kl energy terms for different domain states. In other words, the anisotropy of elastic response is the driving force for the phenomenon. The ferrobielastic hysteresis curve would then be represented by the dependence of s ijkl on the product σ ij σ kl of the applied stress components.3 As in the previous case, quartz crystals are good candidates for observing this phenomenon. In fact this field of investigations was driven by the fact that quartz plays the leading role in the area of utilization of crystalline piezoelectrics and domains (Dauphiné twinning), since they differ in the sign of piezoelectric coefficients d111, d122, and d212, representing an obstruction in applications. The source of driving force for the ferrobielastic switching in quartz is the last term in Eq. (7.6.1). Wooster and Wooster (1946) were probably the first to use this effect for controlling twins. Applying torque they succeeded in bringing multidomain samples into single-domain states and could even propose a spatial diagram demonstrating the magnitudes of differently oriented torques required to complete the switching process. Bertagnolli et al. (1979) investigated ferrobielastic switching in quartz by applying uniaxial stress in the [011] direction, which leads to nonzero stress components σ22= σ33= σ23. A part of the hysteresis loop (for compressive stress) was detected by recording the piezoelectric charge, instead of measuring the quantity which defines ferrobielasticity, namely, elastic compliances. Shiau et al. (1984) described ferrobielastic switching in quartz in detail: Compressive stress was applied onto faces of a bar-shaped sample whose normal made angles of 90° and 55° with the crystallographic axes x1 and x3, respectively. Again, instead of measuring one of the components s1123, s2223, or s1213 which define the ferrobielasticity of quartz, the density of charge on the (001) faces of the sample was recorded. The ferrobielastic switching was unambiguously registered; however, using an instrument with unipolar loading (the Instron machine), the complete hysteresis loop with both signs of applied stress was not recorded.


  1. 1.

    Or its projection, if the normal of a plate-like sample is not parallel to the ferroelectric axis.

  2. 2.

    A number of researchers constructed their own pulse generators; as an example we may mention the ‘economical’ apparatus designed by Ravi et al. (1980): a bipolar square pulse generator with a low output impedance, short rise time, variable pulse amplitude, and repetition frequency. A number of convenient sources are now commercially available.

  3. 3.

    We may note that in the original paper of Newnham and Cross (1974b), ferrobielastics were characterized by the dependence of strain vs. stress in form of a ‘butterfly’ loop.

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Alexander K. Tagantsev
    • 1
  • L. Eric Cross
    • 2
  • Jan Fousek
    • 3
  1. 1.EPFLLausanneSwitzerland
  2. 2.Department of Electrical EngineeringPennsylvania State UniversityUniversity ParkUSA
  3. 3.Department of PhysicsTechnical University of LiberecLiberec 1Czech Republic

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