The Mystery of Dimensional Effects in Ferroelectricity

Abstract

The dimensional effect on ferroelectricity is a subject of long-understanding fundamental interest. While the low-dimensional finite ferroelectric structures are committed to the potential increase in electronics miniaturization, these anticipated benefits hinged on the existence of stable ferroelectric states in low-dimensional structures. This phenomenon can be understood from the point of basic physics. This chapter reviews the literature on the finite-size effects in ferroelectrics, emphasizing perovskite and polyvinylidene-based polymer ferroelectrics having technological importance. The reviewed data revealed that despite critical dimensionality being predicted in ferroelectrics, polarization switching phenomenon is possible in as thin as one monolayer film, at least in the case of P(VDF-TrFE) Langmuir–Blodgett thin film with stabilized functional properties. The roles of the depolarization field, electrode interfaces, domain wall motion, etc. in controlling the measured ferroelectric properties have been discussed. Further, the observed deviation from the bulk properties is explained based on both experimental and theoretical modeling.

1. Introduction

Ferroelectric materials have been recognized as one of the focal points in condensed matter physics and material science for over 50 years. This is the most exciting material used in the electronics industry possessing switchable spontaneous polarization with the direction of applied field stress. These ferroelectrics exhibit substantial piezoelectricity as well. Accordingly, these materials are widely exploited as ultrasonic devices, sensors, actuators, energy storage, memory components, and noticeably more consumer electronics products. At the next level up, modern electronics have taken the charge of electronics miniaturization with the nano-dimensional system including thin film and ultra-thin films precisely placed in the electronics circuit [1]. In the last few decades, the advancement in voltage-modulated scanning probe microscopy techniques, exemplified by piezoresponse force microscopy (PFM) and associated spectroscopies, opened a driveway to make use of ferroelectrics on a single-digit nanometer level. Current research in the United States and other nations is pushing the limits of miniaturization to the point that structures only hundreds of atom-thick will be commonly manufactured [2]. This high-precision microelectronics assembly is achieved by scaling down the materials in accord. Nevertheless, the performance of the ferroelectric material is related to the way they are structurally confined undoubtedly due to structure–property alliance. Whilst the dimensional downscaling of the ferroelectric materials from bulk to nanoscale boost the possibilities to endure the boxing up of increased numbers of components into single electronics integrated circuit, the functional properties are suppressed as the material goes down to the critical dimension. The theoretical studies on the nano-dimensional system including thin films and ultra-thin films have shown that ferroelectricity persists down to the nanoscale. However, the experimental approach at this scale revealed the disappearance of the ferroelectric switching phenomena as the critical size of the crystal in the ferroelectric system is reached. For example, 80% of the dielectric and piezoelectric properties of perovskite ceramics are suppressed compared to their bulk counterpart as the material is scaled down to ∼10 nm [3]. A bulk-like ferroelectricity with finite-size modifications has been observed in nanocrystals as thin as 25 Å crystalline ferroelectric polymer films [4, 5, 6], 100 Å perovskite films [7] and as small as 250 Å in diameter ultrafine nanoparticles [8]. These outcomes can be elucidated as the bulk ferroelectricity is stamped out by surface depolarization energies and inferred that the bulk transition is limited by minimum critical dimension. This is noted as the scaling effect. It occupies a prominent place in the research area as our limited intuition for the nanoworld and comprehensive knowledge of structure–property relations often lag behind technological advances. Since nanostructuring of ferroelectric materials ends up with the appearance of their critical size limit, below which the essential ferroelectric parameters cannot be sustained, a completely contrasting behavior has been observed in hafnium based thin films which displayed an unconventional form of ferroelectricity in thin films with a thickness of only a few nanometers. This allows the construction of nanometer-sized memories and logic devices. Until now, however, it is an unsolved mystery how ferroelectricity could turn-out at this scale. A study reported by scientists at the University of Groningen, Netherland revealed that migrating oxygen atoms (or vacancies) are supposed to be responsible for the distinguished polarization switching phenomena in a hafnium-based capacitor [9]. Likewise, Bune et al. [10] have reported the near-absence of finite-size effect in two monolayer crystalline Langmuir–Blodgett film of P(VDF-TrFE) ferroelectric polymer. This contrasting behavior of ferroelectrics increased the curiosity of the scientific community in this stream. Although, well-developed theories exist for bulk materials, the extrapolation of these theories to thin films and nanostructures is frequently ambiguous. Hence understanding the dimensional system and going into the issues with scaling and size effect is crucial and is the central challenge for the ferroelectrics-based electronics community.

The chapter is aspired to understand the fundamental mechanism underlying ferroelectric behavioral patterns in polymer and ceramics systems as it is scaled down to a critical dimensional range attractive for a variety of technological applications. This knowledge would be beneficial for the current ferroelectric materials as well as for designing new materials with even a cut above electroactive property. The chapter is divaricated into six sections. Section 1 introduces the topic of our discussion. Section 2 talks about the theoretical framework for the scaling effect in the ferroelectric system. Section 3 discusses about how material functional properties are depleted in nano-confined perovskite ferroelectric system including phase transition temperatures, spontaneous polarization, coercive field and piezoelectric coefficient. Next are the possible causes for the observed scaling effect. Section 5 explores the scaling effect in ferroelectric polymer thin films with special emphasis on PVDF and its copolymers. The fundamental ferroelectric polarization switching mechanism for nanostructures is introduced and the models for thin films at the nanoscale are reviewed in Section 6. The nucleation-limited-switching (NLS) model based on region-to-region switching kinetics for polymer thin films will be highlighted. Finally, the observed results will be summarized and the future outlook for ferroelectric nanostructures are discussed. We clarify here that the goal of this chapter is not to review all the work in the vast field of ferroelectrics but rather to provide a scholastic presentation for the readers through the use of select case studies and authors experience in the field.

2. Theoretical framework

The more is the challenge for developing nano-scaled devices, the more is the challenge to sustain their ferroelectricity at this scale. To capture the comprehensive knowledge in the versatility of ferroelectricity as the material is scaled down, particularly at the nanoscale, a theoretical framework is exceedingly advantageous. The first-principle density functional theory (DFT)-based modeling and simulations plays a significant role as the fundamental properties could be envisioned and act as guidelines in the design of ferroelectric nanostructures. For the last decade, it has been successfully implied to various ferroelectric bulk crystals as well as nanostructures. According to first-principle density functional theory, ferroelectricity is analyzed in two possible ways [11]: (a) calculation of total energy by solving ground state problem for a given potential, (b) computation of linear response (LR). This is done by discovering the lowest order changes in ground state energy as the potential changes. The former provides the knowledge about the parameters which is the first derivative of total energy such as stress or electric polarization while the latter computes the properties corresponding to the second and third derivatives of total energy such as phonons, dielectric, piezoelectric and other compliances. In the perovskite ferroelectrics oxides, the transition metal is in $d^0$ state, therefore the effect of electronic interaction is rather weak on the ground state electrons. Hence the first-principle calculation can be quite useful in their studies. In ferroelectric oxides, it is very unlikely to have electronic excitations due to the presence of large band insulators with unsettled d-states of transition metal (B). DFT calculation ascertains the crystal structure through energy minimization such as phonons, Raman tensors, dielectric, piezoelectric and other compliances. For example, DFT calculation provides subtle information about which structural distortions can destabilize the cubic structure in perovskite ferroelectrics [12]. Further, DFT calculation explains that the temperature dependence ferroelectricity arises from the phonon contribution and these operations hold sway over the interesting piezoelectric response as well. Even so, it has some limitations, firstly these simulations are relevant for the material properties at T = 0 K (or at low temperature). Secondly, DFT theory could simulate no more than 150 atoms (for a short time scale ∼ 100 ps) and have definable size errors in the approximation of thermodynamic properties of ferroelectrics. However, this dereliction is compensated in more intuitive way through an effective Hamiltonian methodology which dealt with finite temperatures along with large-scale simulations of ferroelectrics. This approach remains unaltered for the bulk ferroelectric but for thin film or at the nano-scale, effects of surrounding (appropriate boundary conditions) are captured as the estimated properties of nanostructures below the “critical dimension” depends on the length-scale measurement, that is on the ambient conditions not on the volume of a cluster [11]. Two important boundary conditions have been reported. First is the mechanical boundary condition specially for thin film developed epitaxially on a substrate. For this, the required in-plane strain component by the lattice constant of the substrate are frozen to constant value while in thick films, all the strain component are free to fluctuate. Second is the electrical boundary condition that creates depolarization field arising due to bound charges at the surface partially recompensated by free carriers assembled at the electrode. This interesting finding is summarized here in the context of BaTiO₃. Using the first-principle calculations, Junquera and Ghosez explained that a favorable polar state can be realized only for BaTiO₃ film as thin as six-unit cells (∼24 Å) and attributed this extraordinary ferroelectric stability to the depolarizing electrostatic field at the ferroelectric-metal electrode interface [13]. Since the depolarization field is responsible for diminishing ferroelectricity at the finite size, the author theoretically explained here that the electrons at the metal interface tend to screen the surface charge. As a result, the dipoles with the similar polarity appeared at the metal-ferroelectric interface and stabilized the ferroelectricity. The fusion of first-principle density functional (DFT) calculations with an effective Hamiltonian offers a multiscale driveway to analyze the various functional properties of ferroelectric oxides. It also provides the possibility to directly couple the properties to the atomic arrangements and the boundary conditions. Another important theory is Landau Devonshire theory which uses spatial inhomogeneity to show the smearing of phase transitions in ferroelectric nanostructures [14]. This theory explains that the inhomogeneity between a ferroelectric material and an electrode is a result of domain structure in ferroelectric thin films. A dead layer is formed between the film and the electrode. The reduced dead layer softens the domain structure contributing large dielectric response of the film. Landau–Devonshire theory is a free-energy-based phenomenological perspective for continuum mechanics ferroelectric functioning. This theory is very helpful in analyzing diverse phases in complex phase diagrams, microstructures as well as device simulations [15]. However, parameters obtained from Landau free energy are based on material-specific information. Therefore, it is highly desirable to link first-principle calculations with Landau-like theories at nonzero temperature so that analysis could be done at all length-scales fairly based on the information obtained from first principle calculations. The latter pushed the limit of fabrication of perovskite ferroelectrics below ∼ 15 nm [16] and as thin as 1 nm in ferroelectric polymer system [10].

3. Critical dimensional range for perovskite ferroelectrics

On the edge of the ferroelectrics class is the ABO₃ oxides (where ‘A’ and ‘B’ are two cations, often of different sizes, and O is the oxygen atom that bonds to both ions) occurring in the perovskite structure. Typical materials that crystallize in the perovskite structure having technological importance are ferroelectric ${\text{BaTiO}}_3$, ${\text{PbTiO}}_3$, piezoelectric $\mathrm{Pb}\left(\mathrm{Zr},\mathrm{Ti}\right){\mathrm{O}}_3$, electrostrictive $\mathrm{Pb}\left(\mathrm{Mg},\mathrm{Nb}\right){\mathrm{O}}_3$, multiferroic ${\text{BiFeO}}_3$ etc. Ferroelectricity is a cooperative phenomenon of the orchestration of the charged dipoles within the crystal structure. In perovskite, it is governed by the existence of long-range ordering of elemental dipoles up to a distance ranging from millimeter to a few microns. The lower-dimensional confinement of the perovskite ferroelectric material especially in the nano to sub Å level, strongly perturbs the long-range ferroelectric order as the fraction of surface/interface atoms is increased. As the ferroelectric particle goes down to the nano-range, there is a greater probability of the arrangement of constituent atoms at the surface of the particles, thereby ratio of surface area to volume ratio is increased that changes the free energy of the crystal, triggering immense changes in the functional parameters of the material [17] such as the abnormal lowering of ferroelectric to paraelectric phase transition temperature (T_c), suppression of remnant polarization (P_r), and increase in the coercive field (E_c). Few literature reports evidenced the shifting of T_c toward room temperature when the particle size is lowered down to 200 nm or below [18, 19, 20]. It has been suggested that the surface charge layer and the depolarization field played a significant role in this scaling effect as the depolarization effect breaks the material into small domains of different polarization to minimize the macroscopic charge generated on the surface as it is cooled through T_c [21]. Ivan et al. [22] also mentioned the role of depolarizing field in nanoconfined perovskite material using an ab initio derived Hamiltonian. Daniel and his research colleague well-articulated the nature of ferroelectric phase transition temperature (T_c) on downscaling the barium titanate (BTO) nanocrystals using the surface plasmon technique [23]. They proposed that the behavior of surface ferroelectricity seems to be different from the volume ferroelectricity and is characterized by very long relaxation time scales. For nanoscale ferroelectrics, the surface and the volume of the crystals are well-tuned due to the dominance of the surface over the whole nanocrystals. Therefore, the volume T_c may probably close to the bulk-like but for nanocrystals, it decreases significantly relative to the bulk value. The BTO crystal size > 0.1 μm exhibited bulk like properties with a phase transition temperature T_c ∼ 130°C, while a continuous shift in the temperature range ∼50–90°C has been observed for the crystals with dimensions <50 nm. This behavior may be the consequence of barium titanate nanocrystalline size distribution [23]. For lead titanate (PbTiO₃) crystals, size effects were found to be applicable below 100 nm. The T_c decreases from 500 to 486°C as the particle size decreases from 80 nm to 30 nm respectively with a more diffused peak in the lower dimension and the phase transition peak completely disappeared after 26 nm [24]. This scaling effect on T_c, typically implied by the relation:

$$\mathit{\delta T}=\frac{\left[T_c\left(\infty \right)-T_c\left(d\right)\right]}{T_c\left(\infty \right)}=A{d}^{-\mathrm{ƛ}}$$E1

where $T_c\left(\infty \right)$ and $T_c\left(d\right)$ are phase transition temperature of bulk crystal and thin film of thickness ‘d’ respectively, too deviates at the ultralow-dimensional scale as reported by Emad et al. [25]. Genesta et al. [26] reported the disappearance of ferroelectric switching in barium titanate nanowire below a critical size of about 1.2 nm. The author explained that the global contraction of the unit cell at the wire surface is attributed to the disappearance of ferroelectricity. Vincenzo and Randall [17] have provided a very good discussion about the size and scaling effect in the barium titanate ferroelectric system. Even though discrepancies on size limit still persist as ferroelectricity not only depends on the absolute critical size of the material but the preparation route to achieve the limit. For example, Ishikawa et al. illustrated that sol–gel-prepared PbTiO₃ nanoparticles exhibited a critical dimensional limit of ∼10 nm at 300 K which was later defied by Fong et al. [27] who suggested the stable ferroelectric phase in PbTiO₃ thin films down to the thickness of 3-unit cells (1.2 nm) at room temperature. Recently Hao et al. [28] demonstrated the structural and polarization switching behavior of 4.5 nm BaTiO₃ ultrafine nanoparticles. The author attributed the switchable polarization to the presence of local spatial coherent asymmetric nanoparticles with discernable Ti-distortion and paved the way for the construction of high-density memory devices. This finding evidenced that the absence of ferroelectricity reported literature may not be inherent to the system. The abnormal response of phase transition temperature on downscaling the perovskite ferroelectrics extends to other ordered parameters as well. Daopei et al. [29] theoretically demonstrated three types of equilibrium polarization patterns based on various sizes and material parameters combination, i.e., monodomain, vortex-like, and multidomain, in isolated BaTiO₃ or PbTiO₃ octahedral nanoparticles embedded in a dielectric medium, like SrTiO₃ (ST, high dielectric permittivity) and amorphous silica (a-SiO₂, low dielectric permittivity) using a time-dependent Landau–Ginzburg method with coupled-physics finite-element-method-based simulations. The author further discussed the existence of. The critical particle size below which ferroelectricity vanishes in their calculations was 2.5 and 3.6 nm for PbTiO₃ octahedral nanoparticles for high- and low-permittivity matrix materials respectively. However, this size was unalike for BaTiO₃ octahedral nanoparticles (∼3.6 nm) for all that of the matrix materials. Yan et al. [30] synthesized barium titanate nanoparticle by high-gravity reactive precipitation (HGRP) method and found that crystal with the size of 30 nm exhibited a completely paraelectric cubic phase which changes to tetragonal ferroelectric phase at 70 nm confirmed by XRD and Raman spectral analysis. Nuraje et al. [31] confirmed the tetragonal BaTiO₃ nanoparticles (∼6–12 nm) at room temperature by electrostatic force microscopy (EFM). Besides, coercive field (E_c), the field of negligible polarization, an important functional parameter pertains to the scaling effect in perovskite ferroelectrics as well. According to Janovec–Ka–Dunn (JKD) law, the scaling dimension (∼thickness ‘d’) of ferroelectric thin-film and the coercive field is given by semiempirical relation:

$$E_c\propto d^{\raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$$E2

Following the JKD scaling theory, Xu et al. [32] investigated the ferroelectric properties in 20–330 nm of (0 0 1)- and (1 1 1)-oriented PbZr_0.2Ti_0.8O₃ ceramics system. The change in the spontaneous polarization and the coercive field by lowering the dimension of thin PZT thin-film is delineated in Figure 1. Likewise, Venkata et al. [33], Hong et al. [34] also confirmed the falling of field-induced polarization behavior with the downscaling in perovskite polycrystals and ferroelectric nano-thin films respectively (Figure 1). It has been observed that (0 0 1)-oriented PZT film followed the JKD scaling while (1 1 1)-oriented heterostructures (∼<165 nm) deviated from the expected scaling. The first principle DFT calculation attributed this deviation to the formation of a lower energy barrier phase for switching which eventually reduces the domain-wall energy and exacerbates the deviation.

Figure 1.

Variation of (a) perovskite polycrystals [33] (open access) (b) ferroelectric nano-thin films [34] (c) polarization hysteresis, (d) Coercive field, (e) Remnant polarization (Pr) for (001)- and (111)- oriented PZT thin-films [32].

However, defying the general hypothesis on the scaling effect in perovskite ceramics, an increase in long-range ferroelectric order is observed in NaNbO₃ by Juriji et al. [35] in 2017 as the material was scaled down below 0.27 μm which was attributed to the existence of intra-granular stresses induced during the formation of non-180° domain walls as the grain dimension is reduced. Recently, Lorenzo et al. [36] successfully developed an unusual ferroelectric orthorhombic phase (Pmma) in 24 nm crystal of NaNbO₃ using a microwave synthesis route. Further, the exceptional property of ferroelectricity’s appearance in antiferroelectric PbZrO₃ ceramics as the material attained its critical dimension ∼400–500 nm [37]. This unique result provided the possibility among the research community to stabilize ferroelectricity in lower dimensions which was not observed in other ferroic-system. The disappearance of ferroelectricity below the critical nano-dimension was long thought of the past. In recent years, advanced characterization techniques enabled the fundamental size effect at the sub-Å level (∼6 unit cells) in perovskite ferroelectric systems [13] that lowered the critical dimension for the existence of ferroelectricity in thin films by orders of magnitude.

4. Genesis of scaling effect in perovskite ferroelectrics

Ferroelectric instability is a consequence of a delicate balance between short-range and long-range dipolar interactions. These interactions are definitely perturbed in nanostructures. With the downscaling of ferroelectrics to nanoscale, the surface to volume ratio is changed, the short-range forces are altered at the surfaces and interfaces while long-range character is influenced by the limitation in finite sizes of the material. One of the critical issues in downscaling the perovskite ferroelectrics is the distortion of the ferroelectric phase such as orthorhombic or tetragonality (c/a) in crystals. It is noted that tetragonality in PbTiO₃ crystals were rapidly decreased to 1 as it was scaled down to 7 nm [24] following the relation:

$$\frac{c}{a}\approx 1-{\mathit{exp}}^{-\sim d},\left(\mathit{\text{where}}d\mathit{\text{is the grain size of the material}}\right).$$E3

However, this explanation was not appropriate as the theoretical calculations pushed the limit of fabrication of perovskite ferroelectrics as thin as ∼15 nm [10, 16]. The origination of scaling and size effect is still not realized although two important explanations were suggested: (a) the distinctive intrinsic properties of nanoparticles smaller than critical dimension, (b) generation of local depolarization field due to the surface ions arresting the ferroelectric phase. The development of the depolarization field is a consequence of extrinsic effects such as electrical boundary conditions and electrode screening effect. It is the key issue in analyzing the ferroelectric domain structures, Further, the strain and the electrical polarization in ferroelectrics are coupled phenomena, therefore any misfit strain affairs change the spontaneous polarization of the material. Hence the materials are responsive to mechanical boundary conditions as well. The boundary conditions cognate with the contact situation between the surface of the ferroelectric film and the electrode, play a prominent role in the scaling effect of thin-films. The suppression of spontaneous polarization by instigating the surface and interfacial charges offsetting the normal component of the polarization, creates a depolarization field [38]. In a few cases, the depolarization electrical energy guided the retention of polar crystals by electrode screening effect [39]. The latter is associated with the perfect screening of the electrode and depolarization phase, thereby stabilizing the ferroelectric phase and its resulting properties [13]. While in other cases, it is completely considered for destabilizing the ferroelectric domains [17, 21, 40]. The size of the ferroelectric crystals strongly influences the magnitude of the depolarization field. The scaling of ferroelectrics to their critical dimensional range, being the surface charge remains constant, increases the voltage developed per unit length which induces the depolarization-field-induced scaling effect. The latter is eminent in thin films, when present strongly influences the ferroelectric domains. Further, with the reduced film thickness, rational growth is promoted that leads to strong mechanical boundary conditions, contributes to the scaling effect in ferroelectrics. Factors such as lattice mismatch in epitaxial grown thin films, the difference in the properties of the substrate and the ferroelectric film or growth-related strain generated during the fabrication process creates mechanical boundary conditions. It is associated with the substrate-induced stress/strain that is not only coupled with the spontaneous polarization but strongly influences the array of ferroelastic domains, if present. For example, if the polarization vector switches ferroelastically between [0 0 1] and [1 0 0] directions, then biaxial compression perpendicular to the polar axis will stabilize that orientation and increases the phase transition temperature. However, when these strain effects are overlaid on the scaling effect, the process is supposed to be reversed [41, 42]. Therefore, it is notable that mechanical boundary condition functions along with the intrinsic scaling effect [3]. In bulk ceramics, mechanical boundary conditions are created at the grain boundaries and developed a spontaneous dipole. Apart from surface/ferroelectric film interfaces, the other factors that strongly influence the scaling effect in ferroelectrics are the volume of domain walls and grain boundaries in the lower-dimensional scale of ferroelectric system. The extreme reduction in thin-films/grain size lessen the number of stable domain configurations and eventually mobility of domain boundaries decreases which resulted in low permittivity of the system [3]. Besides, crystal imperfections, doping effect, grain boundaries, microstructures, etc., are interlinked to the processing condition [43] may influence the scaling effect in perovskite ferroelectrics and requires independent assessment.

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 band-gap polymer photovoltaic devices. Mostly, these polymers are analyzed in the form of thin films [49, 50, 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 T_c ∼ 100°C and a spontaneous polarization of P_s ∼ 0.1 C/m² 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, 61, 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 (T_c) 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 T_c is typically attributed to the depolarization interfaces. Zhu et al. [66] demonstrated the lowering of the spontaneous polarization to 5 μC/cm² 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 $1.2\pm 0.3\mathrm{V}/\mathrm{m}$ 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 ($E\sim d^{-0.7}$, 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 polymer P(VDF-TrFE) thin films. These outstanding results vitalized the search for the critical dimensions in other ferroelectrics.

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]).

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).

$$F\left(P\right)=\frac{1}{2}{\mathit{\alpha p}}^2+\frac{1}{2}{\mathit{\beta p}}^4+\frac{1}{6}{\mathit{\gamma p}}^6-\mathit{EP}$$E4

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.

Figure 3.

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

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 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 (T_c), polarization switching, coercive field (E_c), 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 lower-dimensional 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.

Acknowledgments

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.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be constructed as a potential conflict of interest.