**2.2. Aging**

be switched from one to another, giving rise to symmetrical polarization-electric field (P-E), strain-stress (S-σ) and magnetization-magnetic field (M-H) hysteresis loops. It is often the case where two or more order parameters are coupled within the same phase and a change in one parameter produces a variation in another, as it occurs in multiferroic systems. The free energy profile can be influenced by many factors, such as compositional and structural inhomogene‐ ity, defects, internal fields, thermal and loading history. These can induce preferential states of the order parameters, leading to the appearance of deformed and asymmetric hysteresis loops. The understanding of these biasing effects and the study of their possible advantages/ disadvantages for specific applications, with the relative elaboration of strategies to induce/ avoid them, is of crucial importance to maximizing the performance of functional devices

The present chapter aims to provide an overview of different biasing effects which can occur in different types of ferroic materials, with particular focus on the phenomenology and on the underlying microscopic mechanisms of the anomalies observed in hysteresis loops. The chapter is organized in three main parts. The first part describes the distortions of P-E and S-E hysteresis loops in ferroelectric/ferroelastic materials. The two main phenomena, which cause biased hysteresis loops, namely aging and fatigue, are comprehensively reviewed by describing the most important features in bulk systems and the inherent microscopic mecha‐ nisms. Attention is also given to imprint phenomena in thin films, with description of the most important models developed to explain the shifting of the polarization hysteresis loops. The second part is focused on biasing effects in ferroelectric/ferroelastic materials under mechan‐ ical stress, with detailed description of the influence of composition, poling state and temper‐ ature on the stress-strain curves of perovskite systems. Additionally, biased stress-strain loops in shape memory alloys are also briefly reviewed, highlighting the different mechanical behavior of the martensite and austenite phases. This section is concluded with a summary on Bauschinger effect observed in ferroelastic materials during cyclic mechanical loading. The third and last part describes the biasing processes occurring in ferromagnetic materials, with the main focus on the asymmetric M-H loops caused by the exchange bias effect, and by the

coexistence of different magnetic phases in inhomogeneous magnetic systems.

**2.1. Deformations of hysteresis loop shape of ferroelectrics: pinching and asymmetries**

The study of the hysteresis loops, namely current-electric field (I-E), polarization-electric field (P-E) and strain-electric field (S-E), is one of the most important tools to investigate the behavior and to assess the properties of ferroelectric/ferroelastic materials.[1-3] These curves usually show certain symmetry properties with respect to one or both the two axes, but they are susceptible to substantial modifications of their shapes, frequently observed in certain structures and compositions, which have undergone a certain thermal and electrical history. The most relevant examples of shape modifications of hysteresis loops are represented by *pinched loops* (constricted P-E loops in the region E ≈ 0, remanent polarization approaching to zero), *asymmetric loops* (shift of the P-E loops along the E- and P-axis, suppression of left or

based on ferroic systems.

206 Ferroelectric Materials – Synthesis and Characterization

**2. Ferroelectric materials**

### *2.2.1. Definition and introduction*

The term *aging* in ferroelectric/ferroelastic systems usually refers to the process that produces variations of dielectric, piezoelectric and ferroelectric properties over time and in absence of an external field. These variations are usually undesired and the understanding of the controlling mechanism is crucial to maximize the performance of materials in most electronic applications. Although the prevailing microscopic mechanisms of aging are still under debate, there is a general agreement that aging is due to the stabilization of charged species or defect complexes in certain locations and configurations. Referring to the most common perovskites (ABO3 chemical formula), aging is absent or weak in materials with high purity and in compounds in which the A- and B-site ions are partially substituted by donor species (i.e. soft compositions). On the other hand, aging phenomena are much more pronounced when the ions are partly replaced by acceptor dopants with a lower valence (i.e. hard compositions).[4]

The effect of aging often results in the presence of pinched P-E hysteresis loops schematically shown in Fig. 1a, and/or in one of the two possible cases of asymmetric P-E loops schematically shown in Figs. 1b and 1c. The internal bias field *Eint* can be estimated in each case using the position of the current peaks (Fig. 1) as [4]:

$$E\_{\rm int} = \frac{E\_c^\circ + E\_c^-}{\mathfrak{D}} \tag{1}$$

where *Ec* + and *Ec* − are the fields corresponding to positive and negative current neighboring peaks in the current-electric field (I-E) curve.

### *2.2.2. Effect of thermal and electrical history on the hysteresis loops*

The presence of pinched and asymmetric hysteresis loops depends on the specific thermal and electrical history of the system. Fig. 2 shows the aging characteristics of CuO-modified BaTiO3 ceramics.[5] It can be seen that unpoled aged ceramics display pinched P-E and S-E loops, while poled ceramics display asymmetric P-E and S-E hysteresis curves.

F

Fig. 2. Hysteres

sis loops compa

**pic mechanisms** croscopic mecha *in effect*, and c) sm is the main c hese mechanisms

e volume effect e stabilized in p e polarization alo model was propo *rming property*  l structures [7]. xygen vacancies he crystal struc g. 3). During co es into a lower sy l arranged in th wer compared t cture, the presen

**2.2.3 Microscop** Three main mic *effect*, b) *domai* which mechanis fact that often th

**(a) Volume effe** According to th the system, are switching of the volume effect m *symmetry-confo* different crystal defects (e.g., ox symmetry of th polarization (Fig abruptly change defects are still migration is slo perovskite struc

**ect** 

**2** T *e* w f

**(** A t s v *s* d d s p a d m p

**Figure 1.** Definition of the internal bias field, Eint, in (a) pinched and (b, c) asymmetric P-E loops. After [4].

p c a t s prepared ceram corresponds to a aged ceramics. T the blue curve w S-E loops of th subsequently ag mics and aged c a poled non-age The red curve w was obtained wit he as-prepared a ed ceramics (po ceramics (aging ed sample, red an was generated by th a positive ele and aged cerami sitive field paral g at room temp nd blue curves a y applying a pos ectric field antipa ics; (d) S-E loo llel to poling). A perature for 24h are relative to p itive field parall arallel to the pre ps of poled non After [5]. h); (b) the blac poled and subseq lel to the poling evious poling fie n-aged, and pol ck line quently g, while eld; (c) led and **Figure 2.** Hysteresis loops comparison of CuO-modified BaTiO3 ceramics: (a) P-E loops of the as-prepared ceramics and aged ceramics (aging at room temperature for 24h); (b) the black line corresponds to a poled non-aged sample, red and blue curves are relative to poled and subsequently aged ceramics. The red curve was generated by applying a pos‐ itive field parallel to the poling, while the blue curve was obtained with a positive electric field antiparallel to the pre‐ vious poling field; (c) S-E loops of the as-prepared and aged ceramics; (d) S-E loops of poled non-aged, and poled and subsequently aged ceramics (positive field parallel to poling). After [5].

modified BaTiO

O3 ceramics: (a)

) P-E loops of t

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verlap.

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4
