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The domains are of fundamental interest for engineering a ferroelectric material. The domain wall and its width control the ferroelectric behavior to a great extent. The stability of polarization in the context of Landau-Ginzburg free energy functional has been worked out in a previous work by a perturbation approach, where two limits of domain wall width were estimated within the stability zone and they were also found to correspond well with the data on lithium niobate and lithium tantalate. In the present work, it is shown that this model is valid for a wide range of ferroelectric materials and also for a given ferroelectric, such as lithium niobate with different levels of impurities, which are known to affect the domain wall width.

Below Curie temperature, the ferroelectricity is a very important property in solids that arises in certain crystals in terms of spontaneous polarization [_{0}) by taking the spatial variation of these parameters in terms of ordinary differential equations. This was based on a static soliton solution of Lines and Glass [_{c}) were found within the zone of stability through a linear Jacobian transformation. These values were estimated to lie between 1 and, which also showed the possibility of a large memory. The corresponding limits of half the domain wall width, i.e. higher limit at non-dimensional and lower limit at P_{c} = 1 (i.e. P = P_{s}) were also found in the case of lithium tantalate and lithium niobate ferroelectric crystals, whose switching and hysteresis behavior have been extensively studied [2,6]. While a lot of work has been done on domain wall and estimation of its half width, not much work has been done on the stability aspect [

In ferroelectrics literature, the width of the domain walls has been very controversial, with many authors claiming that the walls are very wide. The pioneering work of Zhirnov, Vanderbilt et al., Yacoby et al. and Roytburd are mentioned in Ref. [

It is quite pertinent to mention that the first principle calculations have not established a stability zone for polarization in a given ferroelectric, due to various assumptions involved in such study and uncertainty due to correct estimate of born effective charge, as pointed out by Vanderbilt [

Without going into a debate on the merits of first principle studies and different experimental data, it can be safely said that a theoretical model must have a wider validity in terms of its applicability for a wider class of ferroelectrics with various preparation techniques and even with different measurement procedures. Moreover, the motivation of the present effort is to emphasize on various uncertainties that can be tackled if we assume a perturbation model, i.e. fluctuations in different concerned variables that are used to describe ferroelectricity in terms of some so-called “material parameters” such as saturation polarization (P_{s}) in C/m^{2} and coercive field (E_{c}) in kV/cm.

The basic formulations of this paper can be found out in Ref. [

Solving the left hand and right hand inequalities for x_{0} gives the criterion for upper and lower limit of domain wall width as X_{L} and X_{U}. Thus, within the zone of stability, the lower and upper limits of half the domain wall width (X_{L} and X_{U}) were found as:

when the non-dimensional P_{c} values are 1 and respectively. The first value of P_{c} seems to refer to the most “stable situation” (i.e. an equilibrium situation) and the second one indicates the “limit of stability” in the P vs. E hysteresis diagram [_{1} is the first coefficient of the Landau-Ginzburg polynomial in m/F that is calculated from the dielectric data (e_{33}), and the other terms have their usual meaning.

It is noteworthy that after the zone of stability, the system becomes chaotic. The temporal chaotic dynamics has been studied in details in Ref. [_{c} [_{L} and X_{U}. As we are dealing with the system without any interaction, so the treatment done here is for a single domain, and then in other studies, as in Ref. [_{c}), which in turn is related to the impurity content [17,21].

Our preliminary estimate of the “half width” of such domain walls for only one crystal data of near-stoichiometric lithium niobate (LN) and lithium tantalate (LT) showed that the lower limit corresponds well with the result of Padilla et al. [_{c} values). Moreover, wherever the material parameters such as P_{s} and E_{c} are known from a voluminous literature on the subject, the above equations were also used for those materials for the same objective.

It should be mentioned that the conventional solutions for the domains can also be tackled by variational problem by taking care of many terms in the Hamiltonian [

The coercive field data with different impurity contents for LN crystals have been collected from Ref. [

For both these cases, it is seen that irrespective of the impurity content (read, different E_{c} values), the lower limit of “half the domain wall width” (DWW) is remarkably constant at 0.74 - 0.77 nm, which is just about 1.5 times the lattice spacing of these crystals (i.e. 3 times the lattice spacing for the “full width”, (see Zhirnov et al. in Ref. [

The upper limit of half of DWW shows a large variation with the experimental impurity content or coercive field data. It is quite interesting to plot these values of X_{U} against impurity content, as shown in _{c} value equal to 40 kV/cm for LN crystals, and then it decreases asymptotically towards the congruent side. This is the most interesting observation in that Gopalan et al. found it relatively easier to work in this zone for the hysteresis study with an appropriate thickness of the sample [2,6,21].

The data for LT crystals with lower E_{c} values (1.61, 13.9, 17 and 210 kV/cm respectively) than those of LN crystals are not plotted here due to the paucity of the data in the intermediate range and also due to the lack of precision of the impurity contents, but they show the same trend as that of LN crystals when plotted against E_{c} values. Qualitatively speaking, the lower values of X_{U} for LT crystals (2213, 257, 211 and 19 nm respectively) show their relative usefulness as non-linear optical materials from the application point of view [

We have also analyzed 7 more ferroelectric crystals with a wider variety of chemical and physical characteristics with important implications for non-linear optical and other applications. However, it has been observed that here again the lower limit of DWW (X_{L}) is almost constant at 1.810 - 1.883 nm and it is equal to about 1.5 (1.454 - 1.499) times the lattice spacing (i.e. again the “full width” is found to be equal to 3 times the lattice spacing). At the outset, these data encompass quite a wide range of ferroelectric materials, where our perturbation model for estimating DWW seems to be valid, obviously assuming that the L-G equation is valid for all such crystals. However, the question can be raised about samarium and calcium doped lead titanates and barium titanate that are not included in our analysis since these data do not fall in the smooth curve of

These materials might contain different levels of impurities, even they are not known with precision. However, the plot of X_{U} vs. E_{c} (read, impurity content) is shown in _{U} value of 709 nm (i.e. E_{c} = 4 kV/cm) for KLN crystal up to a low value around 36 nm (i.e. E_{c} = 45 kV/cm) for SBiVN crystal, and then it remains almost constant towards higher E_{c} values (i.e. congruent side). It is to be noted that this near constant value is 19 - 38 nm, which is about to 1 order of magnitude higher than that found by Floquet and Valot by a combination of HRTEM images and XRD data [10,16], as mentioned in the Section 1. However, it is difficult to explain this anomaly at this stage, considering that both the preparation techniques (incorporating different levels of impurities) and measurement procedures (i.e. experimental errors) could be quite different from sample to sample.

A large variation of the X_{U} value seems to indicate a wide variation in the impurity levels. It is quite pertinent to mention that in our above model, no effect of the crystal strain is included, as done by Gopalan et al. [

This work shows that due to fluctuation in both the driving force and order parameter, the lower limit of DWW is remarkably constant showing its independence on the switching behavior or E_{c}, whereas the upper limit of DWW shows a relatively stronger functional dependence on the impurity content (i.e., on E_{c}) at the initial values before showing the asymptotic behavior that might indicate the beginning of a pinning effect. This signifies the need to work with near-stoichiometric samples containing some impurities for photonic and other applications [20-23]. It is pertinent to mention here that for lithium nobate system, after the third point from the left of

The perturbation or fluctuation model shows the lower limit of half the domain wall width to be remarkably constant due to some averaging mechanism taking place within a given crystal system during switching study for a wide variety of crystals and for a given crystal with different impurity contents, which signifies the most stable situation. The upper limit of half the width, after an initial decrease, shows an asymptotic behaviour with the coercive field (read, impurity content) after a particular value for both lithium niobate and lithium tantalate crystals respectively, whereas it is almost constant for other ferroelectrics towards the congruent side. An attempt will be made in future to use the “switching data” for estimating the temporal width of these domain walls from a dynamic system analysis, as done on such materials [

The authors would like to thank Professor V. Gopalan, Dept. of Materials Science, Penn State University (USA) for many helpful suggestions and comments.