Ferroelectric hysteresis

Ferroelectric crystals exhibit spontaneous electric polarization and hysteresis effects in the relation between polarization and electric field, as shown in Figure 1. This behavior is usually observed in a limited temperature range, ie, usually below a transition temperature (10). 

Potassium Phosphates. The K2O—P20 —H2O system parallels the sodium system in many respects. In addition to the three simple phosphate salts obtained by successive replacement of the protons of phosphoric acid by potassium ions, the system contains a number of crystalline hydrates and double salts (Table 7). Monopotassium phosphate (MKP), known only as the anhydrous salt, is the least soluble of the potassium orthophosphates. Monopotassium phosphate has been studied extensively owing to its piezoelectric and ferroelectric properties (see Ferroelectrics). At ordinary temperatures, KH2PO4 is so far above its Curie point as to give piezoelectric effects in which the emf is proportional to the distorting force. There is virtually no hysteresis. 

Crystals with one of the ten polar point-group symmetries (Ci, C2, Cs, C2V, C4, C4V, C3, C3v, C(, Cgv) are called polar crystals. They display spontaneous polarization and form a family of ferroelectric materials. The main properties of ferroelectric materials include relatively high dielectric permittivity, ferroelectric-paraelectric phase transition that occurs at a certain temperature called the Curie temperature, piezoelectric effect, pyroelectric effect, nonlinear optic property - the ability to multiply frequencies, ferroelectric hysteresis loop, and electrostrictive, electro-optic and other properties [16, 388],... 

Hysteresis curve of a ferroelectric crystal, v = initial (virginal) curve, Pr = remanent polarization, Ps = spontaneous polarization, Ec = coercive field... 

There is considerable interest in developing new types of magnetic materials, with a particular hope that ferroelectric solids and polymers can be constructed— materials having spontaneous electric polarization that can be reversed by an electric field. Such materials could lead to new low-cost memory devices for computers. The fine control of dispersed magnetic nanostructures will take the storage and tunability of magnetic media to new levels, and novel tunneling microscopy approaches allow measurement of microscopic hysteresis effects in iron nanowires. 

Since P must remain normal to z and n, the polarization vector forms a helix, where P is everywhere normal to the helix axis. While locally a macroscopic dipole is present, globally this polarization averages to zero due to the presence of the SmC helix. Such a structure is sometimes termed a helical antiferroelectric. But, even with a helix of infinite pitch (i.e., no helix), which can happen in the SmC phase, bulk samples of SmC material still are not ferroelectric. A ferroelectric material must possess at least two degenerate states, or orientations of the polarization, which exist in distinct free-energy wells, and which can be interconverted by application of an electric field. In the case of a bulk SmC material with infinite pitch, all orientations of the director on the tilt cone are degenerate. In this case the polarization would simply line up parallel to an applied field oriented along any axis in the smectic layer plane, with no wells or barriers (and no hysteresis) associated with the reorientation of the polarization. While interesting, such behavior is not that of a true ferroelectric. 

Figure 3.32 Hysteresis behavior of the magnetization of a ferromagnetic crystal or polarization of a ferroelectric crystal with respect to the applied magnetic or electric switching field.

P, which can be switched by an electric field, E, as illustrated in the P vs. E hysteresis loops in Figure 2) in favor of the non-ferroelectric cubic and antiferroelectric (AFE) phases. At a 65/35 ratio of PbZr03 to PbTi03, a concentration of 9.5% lanthanum is sufficient to reduce the rhombohedral-cubic phase transi-... 

Electrooptic Properties, The electrooptic properties of the PLZT materials are intimately related to their ferroelectric properties. Consequently, varying the ferroelectric polarization with an electric field such as in a hysteresis loop, produces a change in the optical properties of the ceramic. In addition, the magnitude of the observed electrooptic effect is dependent on both the strength and direction of the electric field,... 

Polarization of a ferroelectric material varies nonlinearly with the applied electric field. The P-E behaviour is characterized by a hysteresis loop and observation of the hysteresis loop is the best evidence for the existence of ferroelectrcity in a material. The hysteresis loop has its origin in the rearrangement of domains under the influence of an applied elecric field. Generally, the domains are randomly distributed, giving a net zero polarization. Under an applied field or mechanical stress, favourably oriented domains... 

Fig. 27 indicates the apparent piezoelectric constant e of roll-drawn PVDF as a function of static bias field E0 (Oshiki and Fukada, 1972). The value of e at E0=0 represents the true piezoelectric constant e. The curve exhibits a hysteresis and the polarity of e changes according to the poling history. If the piezoelectricity in /)-form PVDF originates from the polarization charge due to spontaneous polarization, inversion of polarity of e would mean the inversion of the polarization by the external field and hence /S-form PVDF may be a ferroelectric material, as was first suggested by Nakamura and Wada (1971). 

The crystals grown in this manner are in the form of clear sheets. The symmetry of the crystals is pseudotetragonal with a cell size of a0 = 3.841 A. and c0 = 32.83 A. Electrodes can be evaporated, or indium amalgam can be applied to the flat surfaces of the crystals, to produce samples for measurements. The d.c. resistance of the crystals is about 1012 fi-cm. They exhibit ferroelectric hysteresis loops up to the Curie temperature of 643°C. 

If it is possible to reorient the spontaneous polarization of a material between crystal-lographically equivalent configurations by an external electric field, then in analogy to ferromagnetics one speaks about ferroelectrics. Thus, it is not the existence of spontaneous polarization alone, but the switchability by an external field which defines a ferroelectric material. Figure 1.2 displays a characteristic hysteresis loop occurring during the reversal of the polarization in a ferroelectric. 

The dielectric susceptibility x is related to the relative dielectric constant er by x = er — 1 Equations (1.4) are only valid for small fields. Large amplitudes of the ac field lead to strong non-linearities in dielectrics, and to sub-loops of the hysteresis in ferroelectrics. Furthermore, the dielectric response depends on the bias fields as shown in Figure 1.4. From the device point of view this effect achieves the potential of a tunable dielectric behavior, e. g. for varactors. 

Figure 1.6 Ferroelectric hysteresis of single domain single crystal (dashed line) and polycrystalline sample (full line)...

There is a small thermal hysteresis of the transition temperature, which depends on many parameters such as the rate of temperature change, mechanical stresses or crystal imperfections. From a crystal chemical view, the Ba-O framework evokes an interstitial for the central Ti4+ ion which is larger than the actual size of the Ti4+ ion. As a result, the serie of phase transformations takes place to reduce the Ti cavity size. Certainly, the radii of the ions involved impact the propensity for forming ferroelectric phases thus both PbTi03 and BaTi03 have ferroelectric phases, while CaTi03 and SrTi03 do not [5]. 

The ferroelectric hysteresis originates from the existence of irreversible polarization processes by polarization reversals of a single ferroelectric lattice cell (see Section 1.4.1). However, the exact interplay between this fundamental process, domain walls, defects and the overall appearance of the ferroelectric hysteresis is still not precisely known. The separation of the total polarization into reversible and irreversible contributions might facilitate the understanding of ferroelectric polarization mechanisms. Especially, the irreversible processes would be important for ferroelectric memory devices, since the reversible processes cannot be used to store information. 

Based on these assumptions the measurement of the large signal ferroelectric hysteresis with additional measurements of the small signal capacitance at different bias voltages are interpreted in terms of reversible and irreversible parts of the polarization. As shown for ferroelectric thin films in Figure 1.24, the separation is done by substracting from the total polarization the reversible part, i. e. the integrated C(V)-curve [18]. 

It is also important to realize that piezoelectricity implies a linear coupling between dielectric displacement and strain, for example. However, in many ferroelectric materials, this response is linear only over a relatively limited field range (See for example, Figure 2.2). Non-linearity is especially important in ferroelectric materials which show a strong extrinsic contribution to the piezoelectric response [5], In addition, it is quite common for the response to be hysteretic. The amount of hysteresis that is observed depends strongly on the measurement conditions. Larger amplitude excitations often result in larger extrinsic contributions to the coefficients, and more non-linearity and hysteresis in the response. 

Figure 2.4 Strain-field curves for < 001 > oriented 0.91PbZn1/3Nb2/303-0.09PbTi03 single crystals. The sample in (a) was poled at room temperature, where the resulting domain state is unstable (due to induction of tetragonal material associated with the curved morphotropic phase boundary), yielding substantial hysteresis. In (b) the crystal was poled at low temperatures to keep it in the rhombohedral phase. When measured at room temperature, the piezoelectric response is much more linear and non-hysteretic, due to the improved stability of the ferroelectric domain state. Data courtesy of S. E. Park.

With respect to the equivalent circuit in Figure 3.3, an evaluation of the known methods for hysteresis measurements will be given, in view of the effective parasitic capacitance and the influence of reflection. Well known methods to record the hysteresis loop of ferroelectric capacitors by measuring the current response are Sawyer Tower, Virtual Ground, and Shunt measurement as shown in Figure 3.4. 

Table 3.1 Comparison of different measurement methods for hysteresis measurements of ferroelectrics.

To obtain the dynamic hysteresis loop of a ferroelectric capacitor the polarization is measured versus the applied voltage. Since the hysteresis is neither a linear nor a time invariant property, the hysteresis loop is dependent on the sample history and on the measurement method. To have a standardized and comparable hysteresis loop, certain parameters are commonly fixed. One is the absolute position of the loop on the polarization axis, since the initial (virgin) state of the polarization is unknown in almost all cases, the hysteresis loop is balanced to a reference value. Most commonly the positive and negative saturation polarization are set to... 

Furthermore, the history of a hysteresis loop plays an important role in the determination of lifetime and reliability of ferroelectric capacitors, especially for applications in ferroelectric memories. Three main effects are characterized in particular as changes in the hysteresis loop under various conditions, which are described later in this chapter as fatigue, retention, and imprint with the corresponding ways to measure these effects. 

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