The Mystery of Dimensional Effects in Ferroelectricity

*Rolly Verma and Sanjeeb Kumar Rout*

### **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.

**Keywords:** perovskite ferroelectrics, boundary conditions, dimensional confinement, polarization switching kinetics

### **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 singledigit 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 bulklike ferroelectricity with finite-size modifications has been observed in nanocrystals as thin as 25 Å crystalline ferroelectric polymer films [4–6], 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 hafniumbased 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 firstprinciple 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*<sup>0</sup> 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* =0K (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 BaTiO3, PbTiO3, piezoelectric Pb Zr, Ti ð ÞO3, electrostrictive Pb Mg, Nb ð ÞO3, multiferroic BiFeO3 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

#### *The Mystery of Dimensional Effects in Ferroelectricity DOI: http://dx.doi.org/10.5772/intechopen.104435*

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 (*Tc*), suppression of remnant polarization (*Pr*), and increase in the coercive field (*Ec*). Few literature reports evidenced the shifting of *Tc* toward room temperature when the particle size is lowered down to 200 nm or below [18–20]. 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 *Tc* [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 (*Tc*) 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 *Tc* 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 *Tc* � 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 *Tc* 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 *Tc*, typically implied by the relation:

$$\delta T = \frac{[T\_\epsilon(\infty) - T\_\epsilon(d)]}{T\_\epsilon(\infty)} = Ad^{-\aleph} \tag{1}$$

where *Tc*ð Þ ∞ and *Tc*ð Þ *d* 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 (*Ec*), 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:

$$E\_c \propto d^{-2\_\circ} \tag{2}$$

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 PbZr0.2Ti0.8O3 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.

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

*The Mystery of Dimensional Effects in Ferroelectricity DOI: http://dx.doi.org/10.5772/intechopen.104435*

#### **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].*

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:

$$\frac{c}{a} \approx 1 - \exp^{-\sim d}, (where \, d \text{ is the grain size of the material}). \tag{3}$$

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 thinfilms. 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 substrateinduced 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.
