**5. Scaling effect in ferroelectric polymers**

Ferroelectric polymers such as poly(vinylidene fluoride) and its copolymer systems have evinced the distinguishing properties in lower-dimensional structures. Their nanostructures are emphasized as electrospun nanofibers [44], anodic aluminum oxide-templated nanotubes [45] and the 2D Langmuir–Blodgett (LB) nanofilm [46]. Few reports have also described the PVDF-nanosphere [47]. For example, Zhengguo et al. [48] reported the formation of P(VDF-TrFE) nanoparticles with sizes of 60–100 nm using a solution method with the successful application in low bandgap polymer photovoltaic devices. Mostly, these polymers are analyzed in the form of thin films [49–51]. Unlike ferroelectric ceramics, the polymer ferroelectrics are semicrystalline (amorphous and crystal parts are intertwined) in nature, therefore the ferroelectricity in the polymer is strongly affected by the interaction between the crystalline and amorphous interface. This is known as the nanoconfinement effect [52], according to which the dipole switching in polymer ferroelectrics largely depends on the local electric field in the crystals. Definitely, these interactions are perturbed as the dimensionality of the polymer ferroelectrics goes down to the lowest possible range. As a consequence, the crystal orientations are varied that eventually influences the functional properties of the material. In the bulk form, P(VDF-TrFE, 70:30) exhibited the first-order ferroelectric to paraelectric phase transition temperature *Tc* � 100°C and a spontaneous polarization of Ps � 0.1 C/m<sup>2</sup> at room temperature [53]. A maximum polarization of 12 μC/cm2 at 4 V has been observed for 100 nm thick P(VDF-TrFE) film which is attributed to the presence of crystalline β-phase (a type of crystal orientation) [54]. Similarly, Xu et al. [55] suggested the preferential crystal orientation for the maximum polarization of 10 μC/cm2 and apparent coercive field �6 MV/m in 500 nm thick PVDF film at a very low switching voltage of 3 V. The study of ferroelectric polymer in their ultra-low dimensions were not possible until the discovery of Langmuir–Blodgett (LB) [56] technique of monolayer formation as the thin films constructed by the conventional route of synthesis such as uniaxial or biaxial drawing [57], solvent casting [58], uniaxial stretching [59] or spin coating limited the thickness as thin as � 60 nm only [50]. Langmuir–Blodgett (LB) monolayer transfer technique produces high-quality ferroelectric polymer ultrathin films which are few monolayers thick and can be switched at 1 V, permitting precise control of the film nanostructures [5]. In 1993 ferroelectricity was first discovered in 30 monolayers (15 nm) LB films of P(VDF-TrFE) random copolymer. Later, in 1998, using this method, Bune et al. [4, 10] reported the ultrathin ferroelectric film of PVDF-TrFE copolymer with a thickness of 1 nm. This film was prepared using a horizontal Langmuir–Blodgett (LB) technique, known as Langmuir–Schaefer (LS) technique. This gave the recognition of two-dimensional ferroelectric polymer thin film system implying that the state of ferroelectricity may be achieved by coupling only within the plane of the film and unlocked a new frontier in polarization switching development in ultrathin-single crystal films [4, 5, 60–62]. However, the larger interfacial effect may arrest the ferroelectric switching even in PVDF-based Langmuir– Blodgett (LB) nanofilms [63]. The P(VDF-TrFE) ferroelectric LB films displayed complete polarization reversal in samples for the thickness ranging from 30 to 100 monolayers. Also, the partial reversal has been observed at eight monolayers

thickness, the thinnest possible ferroelectric films made to date [64]. The 30-layer ferroelectric LB films (15 nm) exhibited the phase transition temperature (*Tc*) in the range 70–90°C lower than the typical values 90–110°C for spun films of P(VDF-TrFE) [65]. The decrease in *Tc* is typically attributed to the depolarization interfaces. Zhu et al. [66] demonstrated the lowering of the spontaneous polarization to 5 μC/cm<sup>2</sup> at a very high electric field of 700 MV/m for 18 nm thick P(VDF-TrFE) LB film even with 80% of crystallinity. The impression of reduced ferroelectric response reaches out to piezoelectric responses as well. The piezoelectric coefficient of |d33| = 5 pm/V for a 30-layer ferroelectric LB film was measured using an interferometric method as compared to the bulk P(VDF-TrFE) film 41 pm/V and pure PVDF film 26 pm/V. Further, a large coercive field of 1*:*2 0*:*3 V*=*m has been observed which is approximately 20 times larger than a bulk counterpart [64, 67]. To a great degree, the increase in coercive field as the film dimension is lowered is explained by power law (*<sup>E</sup> <sup>d</sup>*<sup>0</sup>*:*<sup>7</sup> , *d* is the thickness of the ferroelectric film) [68, 69]. Nevertheless, the advancement in characterization techniques for the nanostructures further decreases the dimensionality with stable ferroelectric state. Recently, the single monolayer (0.5 nm) of P(VDF-TrFE) LB film surprisingly exhibited the ferroelectric switching calculated theoretically by Fridkin [70] as shown in **Figure 2**. Earlier the near-absence of finite-size effect was reported for the P(VDF-TrFE) LB film as thin as 2 monolayer ( 10 Å) crystalline film [10]. A schematic representation showing the polarization switching in 1 and 10 monolayers of P(VDF-TrFE) thin films is delineated in **Figure 2**. Hence, it is noteworthy that there is no critical size thickness for exhibiting ferroelectric switching phenomena in ferroelectric

#### **Figure 2.**

*The schematic representation of polarization switching in one and ten monolayer of P(VDF-TrFE) LB film (replotted taking the Ref. [10, 71]).*

polymer P(VDF-TrFE) thin films. These outstanding results vitalized the search for the critical dimensions in other ferroelectrics.

### **6. Polarization switching kinetics for nanoscale ferroelectrics**

Electric polarization is the first-order framework of ferroelectric transitions, whose non-zero value apprehends the ferroelectric phase from the paraelectric one. The phenomenon of macroscopic polarization reversal with the external field stress is termed polarization switching. The kinetics for the same was contrasting for the lower-dimensional system compared to its bulk counterpart. Ferroelectric materials including bulk ceramics, spin-coated epitaxial oxide thin film or the Langmuir– Blodgett polymer thin films, consist of widely distributed domains. Earlier studies have shown that polarization switching is a complex inhomogeneous phenomenon involving domain nucleation and growth. This process can be realized in terms of Kolmogorov–Avrami framework of inhomogeneous phase transformation, [72] where polarization is associated with the lower energy phase. At the macroscopic level, typically two frameworks have been observed in partially polarized ferroelectric materials: (a) the whole material may experience an identical polarization or (b) the presence of spatial inhomogeneous polarization. The second situation is practically observed in ferroelectric P(VDF-TrFE) thin films. Devonshire was the first scientist to develop a theory on polarization switching on barium titanate ceramics system based on Landau mean-field phase transition [73]. Later on, the theory was improved with the consideration of Ginzburg spatial inhomogeneity framework and termed as Landua–Ginzburg–Devonshire (LGD) theory [5, 74]. According to this theory, the free energy for macroscopic polarization which is considered as order parameter is expanded as Eq. (4).

$$F(P) = \frac{1}{2}ap^2 + \frac{1}{2}\beta p^4 + \frac{1}{6}\gamma p^6 - EP \tag{4}$$

where α, β and γ are the Landau coefficients and *E* is the electric field within the ferroelectric material. The term EP of Eq. (1) defines the polarization alignment in the direction of the field to lower the free energy. The calculated P–E relation for P(VDF-TrFE) using Landau–Devonshire theory is shown in **Figure 3**.

In the computed P–E relation (**Figure 3**), the author theoretically explained an unstable region between point a and b and proposed that the polarization switching as a consequence of lowering the free energy of the system. Nevertheless, a gap always persists between the theoretical and experimental values. For example, the field for minimal polarization was computed in the order of magnitude in GV/m while it is typically 50 MV/m, as verified experimentally. The explanation of polarization switching based on nucleation and multidomain [75, 76], is labeled as *extrinsic switching*. This process involves the recasting of free energy of the crystal system due to the presence of sporadic dipolar defects, thereby lowering the energy barrier for local dipole reversal, thus creates a nucleation center for emerging ferroelectric switching domains. Likewise, the Monte-Carlo simulations unduly confirmed the non-collective polarization switching phenomenon mediated by the formation and development of domains as well.

However, the nanosized polymer ferroelectric P(VDF-TrFE) LB thin films (within the critical thickness) exhibited a critical behavior, a homogeneous non-domain

**Figure 3.** *The computed P-E relation for P(VDF-TrFE) using Landau-Devonshire theory [75, 76].*

switching of polarization is observed [5]. Gaynutdinov et al. [71] demonstrated that polarization switching kinetics for 54 nm thick film of P(VDF-TrFE) copolymer were subjectively different from the 18 nm thick film. While bulk-like properties exhibited the nucleation and domain growth as the cause of polarization switching, 18 nm thick film exhibited purely intrinsic switching kinetics with a true threshold field. Vizdrik et al. [76] simulated the switching kinetics in P(VDF-TrFE) LB film with thickness of 30 monolayer. It was observed that the film experienced a pronounced slowing of polarization switching over six orders of magnitude in close proximity of coercive field which is distinct from the extrinsic switching that lacks true coercive field with increased field or temperature. The extrinsic switching is associated with the activation of nucleation and is a function of frequency. If the nucleation is non-existing, a very high coercive field is required to obtain the uniform polarization in ferroelectric crystal ideally, typically known as intrinsic switching and the associated threshold field is known as the intrinsic coercive field. Also, the intrinsic switching is not possible below the intrinsic coercive field as the constituent crystal dipoles are exceedingly harmonized and they tend to switch coherently or not at all. This type of switching is specifically observed in ultrathin P(VDF-TrFE) LB films. The reduced thickness of LB films apparently takes the edge off nucleation volume and therefore prohibits the occurrence of extrinsic switching. Notably, intrinsic switching process takes larger time (>1 s) as compared to extrinsic switching (works in microseconds) observed in thicker films and at lower field. Paramonova et al. [77] validated the intrinsic homogenous switching in PVDF/PVDF-TrFE Langmuir–Blodgett (LB) films using the molecular dynamic simulation method. Further, the intrinsic coercive field is independent of film thickness in PVDF-based LB film below �15 nm, evincing the absence of finite size scaling below 15 nm [78, 79]. However, critical thickness for the intrinsic switching may vary in different polymer films because of diverse molecular structures. Theoretical modeling is a constructing way in guiding research for the dimensional effects in ferroelectricity. The nanoscale ferroelectrics constituted the switching kinetics contesting between extrinsic and intrinsic switching mechanism.

These mechanisms are associated with the film thickness, as the film thickness increases, domain mechanism carry the way, else the nucleation-independent switching mechanism is endured [80].

## **7. Summary and future outlook**

Ferroelectrics with reduced dimension has exciting applications in modern electronics system, especially in medical engineering and material technologies [81]. The first challenge conveyed by nanoscale ferroelectrics for device application is the stability of ferroelectric properties at the desired ultralow-dimensional range. For the last few decades, tremendous effort, both theoretically and experimentally have been implied for finding stable ferroelectricity in nanoparticles at their maximum reduced dimensions. However, setting aside the academic cliché, the real scenario probably deals with the lacking of crucial steps toward the real-mass commercialization of nanoscale ferroelectrics. The science and technology of nano and ultra-nanoscale ferroelectrics is in infant stage. Numerous fundamental issues are still unsolved hampering the real-mass commercialization. It is expected that with the proper selection of material-system, minimizing intrinsic and extrinsic effects and the advancement in nanoscale characterization techniques, the possibility of scaling and size-effects could be minimized.

This chapter dealt with the ferroelectric phenomena emphasizing important functional parameters, such as phase transition temperature (*Tc*), polarization switching, coercive field (*Ec*), etc., taking the frame of reference of finite-size and scaling effect. The existence of critical dimensional range for ferroelectricity is limited by the experimental conditions, shape of the nanoparticles and the characterization techniques. Further, the theoretical analysis revealed that the rich set of complexities in the lowerdimensional scale of ferroelectrics were sensitively hung on structural, electrical and mechanical nature in their circumjacent. The pushing limit for perovskite ferroelectric crystal is as small as �15 nm and the thinnest possible films were �200 Å. Unlike nanoscaled ferroelectric ceramics system, the lower-dimensional polymer ferroelectric thin films are out of the way from the scaling effect. Langmuir–Blodgett deposition technique has produced high quality of ultrathin ferroelectric films of one monolayer thickness (�10 Å) of P(VDF-TrFE) ferroelectric polymer. Their long chain nature and the conformational variability countermanded the quantum confinement effect. This technique has opened a new frontier of finite-size effects on the atomic scale. Further, LB films also exhibited the two-dimensional properties of ferroelectrics by demonstrating that there is no supposed critical thickness in polymer ferroelectrics as films of only two monolayers (�1 nm) are ferroelectric with a transition temperature near that of the bulk material. However, the long-range cooperative ferroelectric interactions among dipoles are debilitated in otherwise customary ferroelectrics.

### **Acknowledgements**

All authors gratefully acknowledge the financial support from the KIRAN Division, Ministry of Science and Technology, Department of Science and Technology (DST), Government of India through Project No. SR/WOS-A/PM-75/2018 (G) and Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India through Project No. EMR/2016/005281.
