Regression analysis of curve fittings for the numerical data shown in Fig. 4

## 1. Introduction

The characteristic property of ferroelectric materials, which is the reversal of polarization by an external electric field, is of technological importance in device applications, particularly in nonvolatile ferroelectric random access memories (NV-FeRAMs). These binary coded NV-FeRAMs can be fabricated by using ferroelectric materials in which the polarization direction can be switched between two stable states when a minimum electric field is applied. To fabricate good quality NV-FeRAMs to meet the demands of the current market, the ability to achieve low coercive field

From the theoretical perspective on this area of studies, a few models (Duiker and Beale, 1990; Orihara et al., 1994; Hashimoto et al., 1994; Shur et al., 1998) were proposed to study the switching properties of ferroelectric thin film based on the Kolmogorov-Avrami theory of crystallization kinetics (Avrami, 1939, 1940 and 1941). In these models, the authors focused on statistics of domain coalescence. Tagantsev et al. (2002) proposed another model based on the experimental work of a few groups (Lohse et al. 2001; Colla et al., 1998a; Ganpule et al., 2000). Their model also focuses on statistics of domains nucleation. Another approach, which is different from the classical nucleation reversal mechanism, is based on the Landau-type-free energy for inhomogeneous ferroelectric system as discrete lattices of electric dipoles (Ishibashi, 1990). However, all these models neglect the surface effect, which is shown to have influence on phase transitions of ferroelectric films, and as the films get thinner the surface effect becomes more significant.

The continuum Landau free energy for a ferroelectric film, extended by Tilley and Zeks (Tilley and Zeks, 1984), incorporates the surface parameter δ (extrapolation length). Positive δ models a decrease in local polarization at surface, and negative δ an increase, with a smaller absolute value of δ giving a stronger surface effect. This model has been used to elucidate phase transition and dielectric properties of FE thin films with great success (Wang C. L. et al., 1993; Zhong et al., 1994; Wang Y. G. et al., 1994; Wang C. L. and Smith, 1995; Ishibashi et al., 1998; Tan et al., 2000; Ong et al., 2001; Ishibashi et al., 2007) and the results of phase transition and dielectric properties of ferroelectric thin films obtained are well accepted. In this chapter, we outline the progress of theoretical and experimental work on switching phenomena of ferroelectric thin films, and the main focus is on the results of switching properties of ferroelectric thin film obtained from the Tilley – Zeks continuum Landau free energy and Landau-Khalatnikov (LK) dynamic equation (Ahmad et al., 2009; Ong and Ahmad, 2009; Ahmad and Ong, 2009; 2011a; 2011b). The surface effects, represented by _{3}) as electrodes, hence, we shall include the effects of misfit strain on switching phenomenon of epitaxial film of

## 2. Ferroelectric thin film and Tilley-Zeks model

The behaviour of ferroelectric thin films is significantly different from that of the bulk. The arrangement of atoms or molecules at the surface is different from that of the bulk material. Due to the process of surface assuming a different structure than that of the bulk, which is known as surface reconstruction, polarization at the surface is not the same as that in the bulk; and it affects the properties of the material. This so called surface effect may have little influence on the properties of material if the material is thick enough. However, when the material becomes thinner, the surface effect becomes significant. The demand by current technological applicants on material thickness of ferroelectric thin film is now in the range of nano-scale. Hence, surface effect in ferroelectric thin film is a significant phenomenon and how it can affect switching must be understood.

The Landau free energy for a ferroelectric film, extended by Tilley and Zeks (Tilley and Zeks, 1984), incorporates the surface parameter δ (also named the extrapolation length) and for convenient, we named it TZ model. Positive δ models a decrease in local polarization at the surface, and negative δ an increase; with a smaller absolute value of δ giving a stronger surface effect. This is important since both forms of behavior have been observed in experiments on different materials. With this surface parameter δ, the inherent material properties at the surfaces of a ferroelectric film which can be either of the two cases are explained. This Landau free energy is given by

where S is the area of the film with plane surfaces at

## 3. Phase transition in ferroelectric thin film

We (Ong et al., 2000; 2001) reinvestigated the TZ model and obtained much simpler expressions, compared with previous work (Tilley and Zeks, 1984), for the polarization profiles of ferroelectric thin films in Jacobian elliptic functions for both positive and negative δ. Variation of Eq. (1) about the equilibrium form P(z) shows that this satisfies the Euler-Lagrange equation (Ong et al., 2001)

with the following boundary conditions:

It follows from Eq. (3) that if the extrapolation length δ is positive, P(z) decreases near the surface, and if it is negative, P(z) increases. In consequence, the critical temperature

where G is the constant of integration. The extremum of P(z) is at z = 0 so that the central value P(0) is a solution of the quadratic equation corresponding to dP/dz = 0:

where the roots

The P integral resulting from Eq. (4) can be expressed by inverse elliptic functions so that ultimately P(z) is expressed in terms of an elliptic function. The detailed forms depend on the sign of δ.

In the case of

in standard notation for elliptic functions. The modulus λ is given by

In the case of

where

While for the temperature interval

Similarly, the modulus λ is given by

We have also presented new thermodynamic functions, the entropy and specific heat capacity, for ferroelectric films with both cases of

## 4. Formalism for switching in ferroelectric thin films

Theoretical and experimental work on switching phenomena of bulk ferroelectric began about half a century ago. The interest in this research area has further been extended to ferroelectric thin films; and the interest has not waned even up to these days because of the advancement in thin film fabrication technology, where higher quality and more reliable ferroelectric thin films can be fabricated; thus making the applications of ferroelectric thin films in microelectronic devices and memories (Uchida et al., 1977; Ganpule et al., 2000) more reliable. Current theoretical and experimental researches in polarization reversal in ferroelectric thin films are focused on phenomena related to effects of size and surface in thin films on switching time and coercive field (Ishibashi and Orihara, 1992; Wang and Smith, 1996; Chew et al., 2003).

From the literature, several theoretical models based on a Landau-typed phase transition have given good explanations on switching behaviours of mesoscopic ferroelectric structures (Ishibashi and Orihara, 1992; Wang and Smith, 1996); and some of the predictions concerning size on switching behaviours by Landau-typed models agree well with experimental observations. However, the detailed understanding of surface effect on ferroelectric films under the applied electric field is still inadequate, but understanding of surface effect is important for the overall understanding on the switching behaviours of ferroelectric films. Thus, we extended the TZ model for ferroelectric thin films given in Eq. (1) by adding in the energy expression a term due to electric field (–EP)

and minimization of Eq. (12) by variational method shows that the polarization satisfies the Euler Lagrange (EL) equation

with the following boundary conditions:

The Landau-Khalatnikov dynamic equation is used to study switching behaviours in ferroelectric thin films (Ahmad et al., 2009; Ong et al., 2008a; 2008b; 2009; Ahmad and Ong, 2011b), and it is simplified to the form as follow:

where γ is the coefficient of viscosity which causes a delay in domain motion and

where f (τ) is a usual step-function defined as

where

## 5. Polarization evolution in ferroelectric films

Surface condition due to δ and size of ferroelectric films affect the switching profiles of ferroelectric films. A ferroelectric film with zero

For thick films, surface switching takes place relatively faster than the interior of the films (Fig. 2); the reversal of polarization begins near the surfaces first, and then goes on to the centre, as shown in Figs. 2(a) and (b). This indicates that the domain wall is formed near the surfaces, followed by a domain wall movement towards the centre. Hence, when the film becomes thicker, the delay in switching at the centre of the film is more distinct compared with the delay nearer the film surfaces. In term of domain wall movement in the film, it obviously takes a longer time for a domain wall to move from the surface to the centre for a thick film than a thin film. However, the delay in switching at the centre is more remarkable in the zero-δ film as the film thickness increases. (Ong et al., 2008a; 2008b).

The general trend in polarization reversal of a film, irrespective of value of

## 6. Coercive field and switching time in ferroelectric films

The basic understanding of properties of thickness and surface dependence of switching time and coercive field in switching of ferroelectric materials is of great importance to the application of FE thin-films in non-volatile memories, for example the ferroelectric random access memory (FERAM). From the results of earlier work on switching behaviour of single crystal barium titanate (BaTiO3) (Merz, 1954; 1956; Miller and Savage, 1958; Stadler, 1958; Fatuzzo, 1962), a few empirical formulations which illustrate the dependence of switching time on applied electric field in switching of ferroelectric crystals are cited. For instant, in 1954 Merz reported that switching time _{3} is an exponential function_{3}.

A similar empirical formulation for domain wall velocity as an exponential function of applied electric field E was proposed by Miller and Savage (1958); and their formulation also does not imply a definite coercive field in the switching of ferroelectric crystals. From their formulations, we can deduced that when an electric field E, however small is applied to a sample, it is just a matter of time; the dipole moments in the sample will ultimately be switched. Around the same period of time, Stadler (1958) extended Merz’s work on a single crystal BaTiO_{3} for high applied field, ranging from 10 kV/cm to 100 kV/cm; and he found that switching time

On the other hand, evidence of a definite coercive field in the switching of FE materials are reported from experimental results, for example in sodium nobate (NaNbO_{3}) (Pulvari, 1960;. Miller et al., 1962). Another example is from Fousek and Brezina (1960; 1964), who reported that when the applied voltage on BaTiO_{3} is below a certain threshold value, no domain wall movement has been observed; but when the applied field is above a threshold field, domain wall movement is detected to be out of phase with the applied voltage. Further more, Fang and Fatuzzo (1962) also reported the occurrence of coercive field on bismuth titanate (Bi_{4}TiO_{12}). In later measurements on good single crystals by Pulvari (1962, 1964 cited in Fatuzzo and Merz, 1967) also indicated there was definite coercive field in the switching and it was confirmed by Cummins (1965). More recent evidence of definite coercive field observed experimentally was reported in the switching kinetics of ferroelectric Langmuir-Blodgett films of 70% vinylidenefluoride and 30% trifluoroethylene copolymer with thickness up to 15 nm (Vizdrik et al., 2003).

There are a couple of theoretical models proposed to study the switching behaviours of FE films; the Kolmogorov-Avrami-Ishibashi theory (Ishibashi and Orihara, 1992a; 1992b; Ishibashi, 1993) which is originated from a model of crystal growth (Kolmogorov, 1937; Avrami, 1939, 1940, 1941) and the Landau-typed model (Ishibashi, 1990; 1992; Wang and Smith, 1996). In the later model, one of the authors (Ishibashi, 1992) fitted his numerical data by the formula

There are several definitions of switching time in the literature (Fatuzzo and Merz, 1967; Ishibashi, 1990; 1992; Nagaya and Ishibashi, 1991; Omura and Ishibashi, 1992; Katayama et al., 1993); however, in our case, the switching time ^{32}. The variation of switching time

three chosen formulations are drawn to fit the data. The solid-line is drawn based on the formulation by Merz (1954)

where e is the applied field and k and

where

where

Formulations | Regression factor, | Standard error | Values of coefficients | |

0.9550 | 0.9119 | 1.5573 | = 0.433 | |

0.9891 | 0.9783 | 0.7753 | = 0.014= 0.577 | |

0.9305 | 0.8658 | 1.9225 |

Based on the regression factors obtained, we find that our simulated data are best fitted by Eq. (22) with the highest regression factor of 0.99. To ascertain this fact, a plot of

Figs. 5 (a), (b) and (c) show the plots of calculated data for switching time

In order to illustrate how switching time

Formulations | Regression factor, | Standard error | Values of coefficients | |

0.9711 | 0.9430 | 1.9225 | k = 1.9439; | |

0.9931 | 0.9863 | 0.9458 | a =0.1883 b =1.5405 | |

0.8947 | 0.8005 | 3.5973 |

In negative

Figs. 8 and 9 show the thickness dependence of coercive field for positive

While for negative

It is worth giving some comments on Figs. 8 and 9. For positive

In contrary to a film with surface effect of positive

## 7. Hysteresis loops of ferroelectric films

Another experimental method in the elucidation of switching behaviours of a ferroelectric film is by applying a sinusoidal electric field to the film and observing the hysteresis loops. From the hysteresis loop obtained, one can find out the coercive field and the remnant polarization; hence the switching characteristics of the film. We use the TZ model, which is described in Section 2, to simulate the hysteresis loops of a ferroelectric film in the application of sinusoidal electric field. The influence of the sinusoidal electric field strength

The effect of the temperature t on the hysteresis loop for positive and negative

The effects of thickness l on hysteresis loops are illustrated in Figs. 12 (a) and (b) for positive and negative

Surface effects of FE films are represented via the effects of positive and negative extrapolation lengths on hysteresis loop are shown in Figs. 13 (a) and (b) for positive and negative

## 8. Summary and future work

From the analytical calculations using the TZ model for ferroelectric films, we discovered that the minimum thickness of a ferroelectric film is dependent on critical temperature and surface parameter δ; and these parameters are material dependent. In our study of the intrinsic switching phenomena of a ferroelectric film, an exponential function, proposed in Eq. (22), for switching time dependence of applied electric field reveals that a ferroelectric film has an intrinsic coercive field

On the whole, surface parameter δ is a crucial parameter to consider when making the choice of material for thin film fabrication in memory devices. However, it is beyond the scope of a phenomenological model to predict the value of δ from material properties; the type of surface parameter δ in a material could be determined theoretically, for example, by comparison of the film critical temperature with the bulk value. When the film critical temperature is lower than the bulk critical temperature, then the material is said to be of positive-δ material; and for the negative-δ material, the opposite is true.

In our current work, we have restricted attention to materials in which the bulk phase transition is second order and the extension of our analysis to first-order materials is straightforward. The model presented here is concerned with single domain switching, as might occur in a sample of small lateral dimension. Lastly, since in reality, ferroelectric thin films are fabricated on conductive materials (such as SrRuO_{3}) as electrodes, we have included the effects of misfit strain in the study of phase transition of epitaxial film of BaTiO_{3} (Ahmad and Ong, 2011a). The results showed that the order of transition is modified and the transition temperature has also been increased when the misfit strain is high. The extension of the model to the study of strain effect on switching phenomena is in our immediate plan and we anticipate the results from this study will provide more hints to resolve the current problems in memory device application.

## Acknowledgments

The work is funded by the FRGS grant, Malaysian Ministry of Higher Learning (Grant No.: 305/PFIZIK/613605).