Overcoming Size Effects in Ferroelectric Thin Films

Ferroelectric thin films have recently received unprecedented attention due to the need to miniaturize electronic circuit devices. Synthesis and deposition processes along with theoretical calculations are improved remarkably to realize stable ferroelectric thin films up to nanometer thickness. However, even with technological advances, it is still difficult to overcome the size effect of ferroelectrics, so research is being conducted to achieve stable ferroelectricity in unit‐cell thicknesses thinner than the typical critical thicknesses. In this review, the size effects in ferroelectric thin films are described, and their importance and fundamental limitations are discussed. First, intrinsic and extrinsic factors affecting ferroelectricity are introduced based on the theoretical background of the size effects in ferroelectricity. Then, on understanding the size effects by considering complex interacting factors, the recent works showing ferroelectricity below the commonly known critical thicknesses in perovskite, fluorite oxides, and two‐dimensional (2D) ferroelectrics are introduced. Finally, the results of research efforts in scaling ferroelectric thin films with a future perspective are summarized.


Introduction
Since Joseph Valasek discovered ferroelectric materials (Rochelle salt) in 1920, [1] ferroelectric materials have been used in a variety of electronic devices including ferroelectric random access memory (FeRAM), ferroelectric tunnel junction (FTJ), and ferroelectric field effect transistor (FeFET). [2][3][4][5][6][7][8] These non-volatile DOI: 10.1002/apxr.202200096 memory devices use remnant polarization (P r ) of ferroelectrics and can quickly perform read/write operations and consume less operating power. [9] With the development of the semiconductor industry, where the role of memory devices is the mainstay, memory devices are becoming smaller and more integrated. The size effect must be considered for a ferroelectric material to be used in an integrated device. [10][11][12][13] It led to the concentration of research on the size effect related to variables that affect ultra-thin ferroelectric films. [14][15][16] When the ferroelectric material is thinner than its critical thickness, it loses ferroelectric characteristics. Under the critical thickness, the ferroelectric thin film has poor stability due to the intrinsic effect and depolarization field, which affects the long-term memory performance of memory devices. [14][15][16][17][18] Therefore, understanding the physics of size effect on the ferroelectric thin film is essential. Although the fundamental variables affecting the size effect of ferroelectric materials have been extensively investigated, their relationships have not been thoroughly examined. [16,18] Because a variety of factors, such as the intrinsic size effect, [16,18] surface characteristics, [18][19][20][21] substrate strain condition, [22,23] and chemical and electrical boundary conditions, [24][25][26][27][28][29][30] have complicated correlations on the ferroelectric characteristics in ultra-thin films.
Previous research on the size effect of ferroelectrics focused on intrinsic factors. The critical thickness was predicted to be hundreds of nanometers considering the depolarization field due to the alignment of dipoles. [16,18] The ferroelectrics were measured through the electrical measurement method, which is unreliable due to the leakage current of the ferroelectric film. [12,29,31] As the film thickness decreases, the influence of the charge at the interface and surface rises, degrading the film's ferroelectric characteristics. [18] Current studies additionally consider depolarization by bound charge at the interface related to various conditions. [14,15] Measurement using X-ray technology [12] has made it possible to independently measure the effect of the surface, [18][19][20][21] electro-ferroelectric interface, [24][25][26][27][28][29][30] and substrate strain. [22,23] It allowed for the independent analysis of interactions between variables like substrates, surfaces, and intrinsic sizes. Nevertheless, since various factors are complexly related to the size effect, research to identify this relationship is continuously being conducted. [32] The principle of operation of ferroelectric materials lies in the displacement of atoms that move by the external electric field. A perovskite oxide structure, represented by Pb(Zr x Ti 1−x )O 3 (PZT), Figure 1. Factors of size effect. Due to intrinsic and extrinsic factors, the ferroelectric materials of perovskite and fluorite structures can exhibit paraelectric properties. a) Even within the same material, the size effect appears differently depending on the synthesis or deposition method of the thin film. b) The grain boundary hinders domain growth and lowers the dielectric constant. c) The ferroelectric material in the free-standing state can reduce the effect of the substrate. Reproduced with permission. [46] Copyright 2022, The Author(s). d) Electrode screening results in a compensating surface charge. e) The inactive dead layer ( ) can reduce the actual ferroelectric thickness. f) The atomic structures of upward and downward-polarized ferroelectric phase with oxygen vacancies. Reproduced with permission. [47] Copyright 2019, Elsevier. g) A single domain has a smaller critical thickness. and a fluorite structure, represented by Hf 0.5 Zr 0.5 O 2 (HZO), comprise the basic crystal structure of ferroelectrics. The ABO 3 structure of PZT in perovskite oxide consists of Zr or Ti atoms at the B site and Pb at the A site. As the position of the Zr or Ti atom at site B varies, the polarization value changes. [32][33][34][35] HZO of the fluorite structure maintained the orthorhombic phase of the noncentrosymmetric structure even at room temperature by doping ZrO 2 on HfO 2 . Unlike tetragonal and monoclinic phases, HZO has ferroelectricity due to the generation of atomic displacement in orthorhombic phases. [13,36,37] Since the discovery of Si-doped HfO 2 in 2011, which retains ferroelectrics in a thinner thickness than conventional ferroelectrics, there has been a noticeable increase in the research into ultra-thin ferroelectrics. [13] The critical thickness of ferroelectrics recently reported in academia is only a few unit cells. [12,[38][39][40][41][42] The Hf 0.8 Zr 0.2 O 2 with a fluorite structure exhibits ferroelectricity even at a thickness of 1 nm under specific circumstances, allowing the implementation of a high-density ferroelectric device circuit. [43] It indicates that ferroelectric materials are promising materials to implement capacitors using ultra-thin films in the future. [29,31,44] As the thickness of the ferroelectrics decreases, the effect of intrinsic and extrinsic factors of the ferroelectrics increases physically and chemically. [23,45] According to the method of synthesizing a thin film (Figure 1a), grain size (Figure 1b), domain size (Figure 1g), and defect concentration of the thin film (Figure 1f) is determined. Furthermore, due to the capacitor characteristics of the electrode/ferroelectric material/electrode structure, the charge screening by the electrode (Figure 1d) and the formation of the dead layer that occurs at the interface of the ferroelectric material with the electrode (Figure 1e) could be made. Moreover, the degree of polarization is determined due to the strain applied to the ferroelectric film depending on the deposition conditions ( Figure 1c). In this review, we examine the physics of the size effect in ferroelectrics and what valuables influence the size effect under actual experimental conditions. www.advancedsciencenews.com www.advphysicsres.com

Landau-Ginzburg-Devonshire Theory
The size effect of ferroelectrics has been studied in various ways. In perovskite oxide, ferroelectricity is defined as a phenomenon resulting from the equilibrium of long-range Coulomb force and short-range covalent repulsion. [48] A theory based on soft mode phonon was introduced to explain the ferroelectric structural phase transition mechanism. At Curie temperature (T c ), the soft mode phonon has a zero frequency, and its atomic displacement is fixed where it has the largest dielectric constant value. At a temperature above T c , the soft mode phonon recovers the structural symmetry of the crystal. [49] Based on the Landau-Ginzburg-Devonshire (LGD) theory, which offers a model for phase transition as a function for T c , and a method of direct variation of one-parameter applied to a free energy function within a spatially constrained system, a phenomenological description of ferroelectricity is proposed. [50][51][52] From this explanation, the solution to the Euler-Lagrange boundary problem resulting from the minimization of the free energy function of LGD theory is obtained. When a weak external electric field is applied, the solution is consistent with the polarization distribution obtained from the paraelectric phase of a system where nonlinearity can be neglected. Next, the Euler-Lagrange boundary problem is linearized using the average value of polarization, and the equation for the average value and deviation of polarization that can be found through the standard method is obtained. The average solutions of this equation are used to calculate the average polarization. Compared to the direct variation of the one-parameter scheme, the direct variation of the two-parameter scheme produced by the technique mentioned above offers a simple solution with a wider valid range and more reliability. Calculating the size effect and other parameters of ferroelectric nanoparticles and thin films with a single domain is possible using the analytical results. [53] Based on these theoretical backgrounds, Hong et al. studied the size effect of BaTiO 3 (BTO) nanowires. It was demonstrated that the T c and polarization values decreased as the nanowire's diameter shrank. [54] In contrast, when the nanowire diameter was greater than 20 nm, the behavior resembled that of bulk ferroelectrics. This calculation result can be used to confirm the change in ferroelectricity caused by the nanowire's size effect and determine the size of the nanowire that will perform the best under the required operating conditions. [55] However, when it is scaled down to the nanoscale, there are some cases where the LGD theory describing the size effect in the existing bulk does not hold up. Because the interaction between the protons exposed on the surface differs from the inner protons' interaction in bulk, tunneling from one minimum value to the other minimum value occurs in the double-well potential curve. Proton's tunneling destroys the order on the lattice and makes the transition to the ferroelectric impossible. [56,57] To solve this problem, the case in a thin film was described by considering the surface strains and extrinsic mismatch effect together. [12,58] Nanoceramics size-dependent phase transition model was implemented using the LGD free energy co-efficient defined considering grain size. The critical size, phase transition T c , and dielectric constant are all accurately predicted by this model. [59]

Depolarization Fields
In the physics of ferroelectrics, the depolarization field is significant. The ordering parameter of ferroelectric systems, spontaneous electric polarization, is prone to be destroyed by the depolarization field. [15,18,49,60,61] Therefore, in order to overcome the size effect at the ultra-thin scale, it is common to minimize the effect of the depolarization field generated in ferroelectrics. According to phenomenological theory, the single-domain ferroelectric film develops due to competition between surface energy, free energy, and depolarizing energy. The formation of domain structures and the presence of free carriers are major internal factors that decrease the effect of depolarization fields, helping to maintain ferroelectricity. The screening length of the electrodes in contact with ferroelectrics is also a factor affecting the depolarization field. In the case of an ideal metal electrode with a screening length of zero, the depolarization field is not induced due to the complete compensation of the charge in the electrode and the ferroelectric interface. However, the real metal electrode has a finite screening length, which results in a depolarization field because the accumulation of charges created at the ferroelectric and metal interfaces is not entirely compensated. [12,[62][63][64] The suppression of ferroelectricity and phase transition by the depolarization field generated from the incomplete electrode causes the size effect of the film to rise. Minimizing the depolarization field in the microstructure is considered a significant issue for maintaining ferroelectricity since it exists in inverse proportion to the film's thickness. The Landau-Ginzburg free energy equation was derived for a ferroelectric capacitor between two metal electrodes to confirm the effect of the depolarization field. It is assumed that the thickness of the electrode is thicker than that of the film, and the polarization gradient that occurs at the interface between the ferroelectric material and the electrode is ignored. Experimental findings agree with the calculation outcomes of this equation, which simplifies the parameters. [65] It was also confirmed that when a ferroelectric material is assumed to be a semiconductor material rather than an insulator, the effect of the depolarization field on ferroelectrics is reduced. [66] On the other hand, there is a result of enhancing ferroelectricity by using the depolarization field. Ferroelectricity was enhanced by inserting SrTiO 3 (STO) into ferroelectric PZT that developed epitaxially on La 1−x Sr x MnO 3 (LSMO)/STO. Between ferroelectric PZTs, a unit-cell thick STO insulator spacer is developed to modify the nanoscale domain structure of the PZT layer. In this structure, it was confirmed that when the thickness of the STO layer was increased, the depolarization field was strengthened. As the effective field that directly acts on the switching of polarization increases, the written domain size also increases. As a result, the depolarization field decreases polarization in the prepoled area. The sample with the larger depolarization field becomes easier since the energy needed to switch the decreased polarization is reduced. [67] Figure 2. The trend of experimental and simulated spontaneous/remnant polarization values for the thickness of perovskite oxide, [28,29,31,64,68] fluorite, [37,[69][70][71][72] and 2D ferroelectric materials. [7,[73][74][75]

Size Effects in Ferroelectric Materials
As research into thin-film ferroelectricity progressed, it became clear that ultra-thin ferroelectricity showed a different physical phenomenon from bulk ferroelectricity. As the surface-volume ratio of the ultra-thin ferroelectric material increases, the ferroelectricity is significantly affected by factors such as thickness, substrate, strain, and reaction to the electrode. In particular, ferroelectricity vanishes when it falls below the critical thickness. Figure 2 shows the trend of polarization values according to the thickness of perovskite oxide, fluorite, and 2D materials. In Figure 2, the polarization value falls as the thickness of the perovskite oxide decreases. On the other hand, the ferroelectric fluorite structure has the highest polarization value at a thickness between 8 and 10 nm. The polarization value tends to decrease as it decreases or increases from this thickness range. In the case of 2D materials, each material has a different tendency. Since experimental ferroelectricity measurements have not been made for a long time, it is necessary to clarify accurate characteristics from many studies in the future.

Processing Method and Stoichiometry
Even within the same material, changes in external factors, such as grain size, defects, vacancies, and impurities, occur depending on processing and annealing conditions, resulting in different ferroelectric properties. [76][77][78][79][80][81] Characteristics that depend on changes in external factors affect ferroelectrics' critical thickness and size effect. It was reported that the grain size and c/a ratio changed when the conditions of the sintering process, such as temperature, were changed in perovskite oxide. Depending on how the BTO nano-powder was processed, the grain size and c/a ratio appeared differently. [82,83] For example, when the grain size of BTO grows, the c/a ratio approaches 1.011 and the tetragonality of BTO is maintained. However, if nano-powder is fabri-cated using a method that makes the grain size smaller, the c/a ratio converges to 1. As a result, small grain sizes make a crystal structure comparable to the cubic one, weakening the tetragonality and ferroelectricity of BTO. It was also shown that when the material contains impurities depending on the processing conditions, the ferroelectricity of the material is affected. The processing condition in which the hydroxyl group is included during the fabrication includes cation vacancies to adjust charge neutrality. Charged defects, such as hydroxyls, can significantly interfere with the ferroelectricity in the lattice. As the concentration of hydroxyl groups is inversely proportional to the hydrothermal synthesis temperature, the nano-powder synthesized at a relatively low temperature contains many hydroxyl groups. [84,85] Incomplete removal and diffusion of impurities in the nano-powder are affected when the processing is carried out at a relatively low temperature, which diminishes the ferroelectricity. These impurities are responsible for the creation of microstructures in the crystal. Defects that induce weak ferroelectricity lead to a low dielectric constant, which shows that the processing temperature affects the material's dielectric constant, regardless of the grain size. [78,86] The ferroelectricity of HZO could be changed according to the ratio of Zr doped in HfO 2 . In general, since the paraelectric tetragonal phase of HfO 2 and the antiferroelectric monoclinic phase of ZrO 2 is thermally stable, the formation of orthorhombic phase exhibiting ferroelectricity in the fluorite structure was challenging at room temperature. However, if HfO 2 is doped with proper Zr concentration, the formation of the orthorhombic phase can be observed. Recently, although studies showing ferroelectricity even in undoped HfO 2 , [87][88][89][90] undoped ZrO 2 [91,92] and several doping ratios were reported, [93] most of the HZO research results have been investigated at the ratio of Hf and Zr 1:1 over the past 10 years. The most common technique for depositing HZO is atomic layer deposition (ALD), carried out at a temperature of about 300°C using a metal precursor and an oxygen source. The types of Hf and Zr precursors used in ALD could affect the ferroelectricity of HZO. Halide-based compounds, [94,95] www.advancedsciencenews.com www.advphysicsres.com alkylamides, [96][97][98] and heteroleptic alkylamide/cyclopentadienyl compounds [99,100] are typically employed as precursors. Numerous studies have been conducted on tetrakis(dimethylamino) (TDMA) and tetrakis(ethylmethylamino) (TEMA) precursors among the various precursors. Kim et al. investigated the ferroelectricity of HZO film deposited using two general precursors, TDMA-(Hf, Zr) and TEMA-(Hf, Zr). It preferred the formation of the orthorhombic phase of HZO when deposited utilizing the TDMA precursor compared to TEMA, alleviated the wake-up effect, and decreased the carbon concentration among impurities. [101] The types of oxygen sources also affect the ferroelectricity of HZO. Kim  was not significantly affected by the deposition temperature, but it was confirmed that the deposition temperature was very sensitive when deposited with O 3 . Additionally, in the case of employing O 3 , it was confirmed that ferroelectricity occurred even when the Hf and Zr ratio was 1:3 due to O 3 's ability to remove impurities. [103]

Grain Size and Grain Boundary Density
Many studies have been conducted to understand how grain size and boundaries affect films' ferroelectric properties. The density of the grain boundary in the thin film affects the distribution of domains, and it eventually determines the magnitude of polarization of the film. [104] As the thickness of the ferroelectric film decreases, the ratio of the grain boundary per volume generally increases, which increases the intrinsic effect on the film. The grain boundary forms a depolarization field around the boundary, reducing domain mobility.
In 1995, Cao et al. confirmed that when the grain size of the PZT was within the range of 1 to 10 μm, the traditional parabolic relationship between grain size and domain size was established. When the grain size is 10 μm or more, the exponent value is less than 1/2; when the grain size is 1 μm or less, the exponent value is greater than 1/2. [105] Liu et al. classified the grain size through the sintering process and studied the relationship between grain size and piezoelectricity of the (Na 0.5 Bi 0.5 )TiO 3 -xSrTiO 3 (NBT-xST) in the perovskite structure. As the grain size increased, the piezoelectric constant increased, and the critical grain size in the maximum piezoelectric constant was measured as 1 μm. [106] In addition, the dielectric and piezoelectric constant related to the ferroelectricity is affected by the grain size when high elastic clamping occurs within the grain rather than at the grain boundary. Mudinepalli et al. have shown that when the grain size is reduced to micro units, the dielectric/piezoelectric constant has the highest value. However, if the grain size is below 200 nm, the effect of the grain boundary increases exponentially, lowering the dielectric/piezoelectric constant value. [107] Proving the relationship between grain size and ferroelectricity is difficult. Because all elements are complexly involved in the polarization of ferroelectric films, it is valuable to examine the simulation results and the relationship between grain size and ferroelectricity, excluding other factors. Figure 3a,b is a graph representing the constant value obtained through phase-field simulation depending on the electric field loading frequency and grain size of the ferroelectric thin film. Tetragonal BTO was used for simulation as a representative ferroelectric material at room temperature. The grain size range was set from 10 to 170 nm, and the measurement frequency was from 10 to 2500 Hz. A constant value for a general BTO under the same conditions was used as the reference value. The electric displacement, electric field, and strain values were set to be 0.26 C m −2 (P 0 ), 218 kV cm −1 (E 0 ), and 0.82% ( 0 ), respectively. Figure 3a shows the ratio of remnant polarization, coercive electric field (E c ), and dielectric constant ( a ) to the initial value of the grain size (each ratio is expressed by P r /P 0 , E c /E 0 , and a / 0 ). Each value ratio decreases rapidly below the 50 nm grain size. In the case of Figure 3b, the differential value of the dielectric coefficient value (d 33 /d°3 3 ) and piezoelectric coefficient ( T /°) in the remnant state is the highest value on a 50 nm grain size. A grain size smaller than 50 nm is largely influenced by intrinsic because the proportion of the grain boundary is dominant. As a result, in the range where the intrinsic effect of grain size is dominant, the remnant polarization, piezoelectric efficiency, effective dielectric permittivity, and coercive field decrease as the grain size decreases. In white areas with a grain size of 50 nm or more, the intrinsic effect can be ignored, and the extrinsic effect mainly determines the ferroelectricity of the film. [108]

Substrate Clamping and Free-Standing Condition
The substrate clamping effect applied to the ferroelectric thin film affects the unique characteristics of the ferroelectrics. As the thickness of the ferroelectric film decreases, the clamping effect becomes more significant, impacting the ferroelectric film's size effect. In the case of a free-standing film, the mechanical interaction with the substrate is eliminated, affecting the domain wall motion and piezoelectric response of the ferroelectric materials. One of the ways to reduce the substrate clamping effect is to minimize the contact area between the ferroelectric material and the substrate. Wallace et al. increased ferroelectric domain reorientation by adjusting the substrate area using photolithography and etching. It was discovered that domain reorientation increased as the mechanical boundaries imposed on the film were removed. As a result, 26% of the film's 90°domain, about 75% released from the substrate, was reoriented compared to the 90°domain in bulk. [109] Nagarajan et al. used a focused ion beam (FIB) to pattern epitaxially grown PZT films into island forms. The reduction Figure 3. a) Trends of remnant polarization, coercive field, and axial actuation strain values by grain size from 10 to 170 nm for BaTiO 3 -based nanoceramics, b) each ratio of remnant state, coercive state, and the differential piezoelectric coefficient for reference values. When the grain size is less than 50 nm, the local property of the grain boundary dominates; in this case, the intrinsic factor has a dominant influence on the factor. Reproduced with permission. [108] Copyright 2017, American Physical Society. Out-of-plane (OOP) piezoresponse force microscopy (PFM) images of the c) as-grown 10 nm HZO on LSMO/STO (top) and HZO film transferred onto the Pt/SiO 2 (bottom). d) Cross-sectional STEM image showing a clear interface between 5 nm thick HZO and LSMO (left) and 4 nm thick HZO transferred to SiO 2 (right). e) Transfer of HZO film deposited on the LSMO buffer layer to the flexible substrate. An etchant dissolves the LSMO buffer layer, and the HZO film exists in a free-standing state. f) −2 XRD scans of as-grown and transferred HZO films with 10 and 5 nm thicknesses, respectively. Reproduced with permission. [117] Copyright 2022, Wiley-VCH GmbH. g) Depolarization field by the thickness of Hf x Zr 1−x O 2 as a ferroelectric material on TiN electrode. Reproduced with permission. [132] Copyright 2019, IEEE.
in substrate clamping made moving the 90°domains easier, leading to improved ferroelectricity and piezoelectric response. [110] Another method is to eliminate the cause of the clamping effect by introducing a free-standing layer. The 2D free-standing layer, which can be realized by selective etching of the sacrificial layer, [111,112] results in a more vulnerable defect formation due to the absence of 3D bonding, [113] but removes the substrate clamping from the bottom layer to lead to a change in size effect. It also enables the transition from the 3D to the 2D structure, making it possible to transfer to any substrate. Removing substrates and bottom layers, which are inevitably used in epitaxy growth, can change the factors affecting the critical thickness of the ferroelectric layer. Actually, the properties of the free-standing layer appear different before and after detaching. [114] The transition from as-grown film to free-standing film increases the proportion of the surface to the total volume, resulting in a structural change in the film. [112,115] The BiFeO 3 (BFO) free-standing film undergoes the phase transition from the rhombohedral phase to the tetragonal phase due to the removal of the clamping effect of the substrate. [112] While the rhombohedral phase of bulk BFO with a c/a ratio close to 1 has a polarization value of about 100 μC cm −2 , [116] the thickness of the 3-unit cells free-standing BFO with Adv. Physics Res. 2023, 2, 2200096 a phase transition to a tetragonal phase showing a c/a ratio of 1.22 has a polarization value of about 160 μC cm −2 beyond the polarization value in bulk. The first principle calculation showed that the displacement of the Fe atom at the center of the perovskite oxide unit cell is the source of these phenomena. [115] The conversion to free-standing also affects ferroelectric switching voltage and speed. The properties of the free-standing BFO layer grown on the LSMO sacrificial layer and the substrate-clamped BFO film were compared, and it was revealed that at an 80 nm scale, the switching voltage decreased by 40% and switching speed fell by 40%. In addition, the ferroelectric switching of the BFO freestanding film up to the thickness of 8 nm was confirmed by PFM. It was demonstrated that the change in these properties originated from the influence of dynamic elastic energy and oxygen octahedral tilts. [46] Zhong et al. confirmed changes in ferroelectricity and lattice constant before and after transfer. HZO was deposited on LSMO/STO using pulsed laser deposition (PLD). Afterward, the HZO film was detached with a free-standing membrane by selective etching of the sacrificial layer LSMO and then transferred to another substrate ( Figure 3e). Using PFM measurements, it was confirmed that the ferroelectricity of as-grown HZO film also appeared in the HZO membrane transferred to the Pt/SiO 2 substrate ( Figure 3c). Additionally, when the interface of transferred HZO to the SiO 2 substrate was analyzed, it showed a sharp interface similar to the HZO as-grown on LSMO (Figure 3d). Using X-ray diffraction (XRD), structural changes in 10 and 5 nm thick HZO films were observed before and after transfer to the Si (100) substrate (Figure 3f). For both 10 and 5 nm thick films, it was confirmed that the clamping effect of the bottom substrate was alleviated, and the main peak shifted to the right. For example, the lattice constant of 5 nm thick film decreased from 2.978 to 2.954 Å after transfer. [117] In this way, substrate clamping and the conversion to the free-standing structure could affect the critical thickness and the size effect by changing and removing factors that affect ferroelectricity.

Epitaxial Strain
The epitaxial growth of the film can eliminate the grain boundary or external effects of the ferroelectricity, which is a beneficial experimental condition for studying the size effect. Estandía et al. applied the epitaxial strain to the film of HZO by varying the lattice constants of the substrate and putting the LSMO between the substrate and HZO for the bottom electrode. The LSMO layer could change the lattice parameter of HZO from −2% to +2% and transfer the lattice parameter of the substrate to the ferroelectric film as it is. The lattice parameter of the substrate is in the range of 3.71 to 4.21 Å, and the orthorhombic phase is formed when the tensile stain of the substrate is applied to the film of HZO. In this article, the authors measured the scanning transmission electron microscopy (STEM) images of the substrates of (LaAlO 3 ) 0.3 (Sr 2 TaAlO 6 ) 0.7 (LSAT) (3.868 Å), STO (3.905 Å), and GdScO 3 (GSO) (3.97 Å) and confirmed that the monoclinic, monoclinic/orthorhombic, and orthorhombic phase was formed in order. As a result, it was confirmed that the orthorhombic phase was induced when the lattice parameter was 3.87 Å or more, and the monoclinic phase was formed below that. [118] The tensile strain applied to the HZO film due to the lattice mismatch of the substrate spreads from the interface between the electrode and the HZO layer. The tensile strength due to lattice mismatch decreases as the distance from the interface of HZO increases. It was reported that the P r value of the HZO film was thickness dependent, and the P r value increased only up to 8 nm in the measurement of the Pt/HZO/LSMO/STO. [71] The external strain also affects the ferroelectricity of perovskite-based on epitaxial growth. Zhang et al. simulated the effect of strain on the epitaxial growth of ferroelectric materials with a perovskite structure. The ferroelectricity of the perovskite film by the epitaxial strain is greatly affected when the strain applied to the substrate is −0.1% and −1.74% in high-strength electric field frequency (0.1-100 kHz). Based on the phasefield model of the epitaxially grown film with the nano-unit thickness of BTO, a representative perovskite material, the P r and the E c were calculated. Phase-field theory is based on the time-dependent Ginzburg-Landau kinetic equation and the main variables are epitaxial strain, film thickness, and electric field frequency. The range of compression and tensile strain is from −3.2% to 1%, and as a result, when the compression strain acts on the substrate, the P r and the E c increase. The film thickness range decreased from 100 to 5 nm, while the P r increased, and the dielectric constant and piezoelectric constant decreased. [119,120]

Mismatch in Thermal Expansion Coefficients
Amorphous HZO deposited with ALD is crystallized through the annealing process after deposition. [121][122][123] In general, HZO crystallizes into a thermally stable monoclinic phase at room temperature and atmospheric pressure. [124,125] Utilizing the coefficient of thermal expansion (CTE) difference between the top or bottom electrode and HZO, the method known as the capping layer effect is used to crystallize orthorhombic phases with ferroelectric characteristics. Kim et al. investigated how the TiN top electrode affected the crystallization of 10 nm thick HZO. When annealing was done after depositing the TiN top electrode, the formation of the orthorhombic phase could be verified. However, it was found that the film's ferroelectricity was reduced if annealing was conducted without the deposition of a TiN top electrode because the monoclinic phase and orthorhombic phase are simultaneously crystallized. Additionally, the crystallization ratio to the orthorhombic phase increased as the TiN top electrode's thickness increased. It shows that the in-plane (IP) tensile strain applied to the HZO film during the annealing process is related to ferroelectricity. [126] Cheema et al. investigated the crystallization of the ferroelectric orthorhombic phase dependent on the presence or absence of the top electrode during annealing in a Hf 0.8 Zr 0.2 O 2 (1 nm)/SiO 2 (2 nm)/Si structure. Using PFM, the ferroelectricity of HZO was verified for both the electrode-capped and electrode-uncapped regions of HZO during annealing. The capped region showed a perfect phase contrast of 180°, but the uncapped region could not be checked. Also, butterfly-shaped amplitude loops were visible only in the capped HZO. [43] It was verified using substrates with different thermal expansion coefficients and how the difference in thermal expansion coefficient affected the ferroelectricity of HZO. [37,[127][128][129][130][131] www.advancedsciencenews.com www.advphysicsres.com The thermal expansion coefficient of the substrate also affects the ferroelectricity of HZO. On a Pt/SiO 2 , Si, and CaF 2 substrate with thermal expansion coefficients of 0.47 × 10 −6 , 4.5×10 −6 , and 22×10 −6°C−1 , respectively, a 17 nm-thick HZO layer was deposited and crystallized through the annealing process. In the case of SiO 2 and Si, the tensile strain was applied to the HZO film to create ferroelectricity of HZO, but in the case of CaF 2 , the compressive strain was applied, so the ferroelectricity of HZO could not be seen. Furthermore, it was shown that HZO's ferroelectricity increased as the substrate's CTE decreased. [131]

Charge Screening: Electrode Dependence
Electrode screening compensates for the surface charge between the ferroelectric and the electrodes. The electrode screening effect is specifically associated with the stability of maintaining the ferroelectric phase. The change in surface charge accompanies a polarization reaction in the ferroelectric capacitor structure, and this change is to be balanced through the accumulation of countercharge in an electrode. The ferroelectric polarization and the depolarization field are only partially screened in actual metals because the charge occupies finite physical thickness.
The depolarization field is affected by the crystal structure. If the film has a crystal structure with an ideal single domain, this film will have spontaneous polarization regardless of thickness and the charge compensation of the surface. When the thickness decreases, the surface charge induced by the ferroelectric material remains the same, the depolarization field increases, and the effect of the depolarization field on the entire ferroelectric material increases. As a result, the depolarization field created by the electrode's surface reduces net spontaneous polarization, aids in domain formation, and destabilizes the ferroelectric phase. The influence of the depolarization field increases depending on ferroelectric thickness due to the electrode charge screening, and the depolarization field varies depending on the ferroelectric characteristics. [15] The first principle calculation can quantify the degree to which electrical boundary conditions affect ferroelectric materials. For example, in order to accurately calculate the depolarization value produced by the electrode, Junqera et al. simulated the condition of the capacitor made with the SrRuO 3 (SRO)-BTO-SRO structure on an STO substrate. The depolarization field increased as the 2.4 nm 6-unit cells were reduced. In this case, the ferroelectric phase below 15 nm is not maintained because of the destabilization caused by the electrostatics of the depolarization field by incomplete charge screening in the metallic electrodes of SRO. [64] Compared to the perovskite structure (e.g., PZT, BTO), HZO with a fluorite structure has a relative permittivity of about 10 times lower (thickness (d FE ) ≈10 nm, relative permittivity ( FE ) ≈30). Since the depolarization field due to perfect screening is determined by the ratio of d FE / FE , both perovskite and fluorite structures have significant depolarization size effects. However, in the case of the coercive field, the fluorite structure HZO has a higher value than the perovskite structure, which means that the ferroelectric layer can resist the higher depolarization field. Figure 3g is the result of calculating the depolarization field of HZO with a typical FE to 30 value. It was drawn as a graph of thickness with different P r values. A TiN electrode with a screen-ing length of 0.83 Å was considered, and as the ferroelectric film thickness decreased, the depolarization field value increased. According to the simulation results, the higher the remnant polarization value, the greater the effect of the depolarization field as the thickness of the film decreases. Therefore, it can adversely affect data retention and endurance when a capacitor device is manufactured. To improve the retention and endurance of the capacitor, there is a method of inserting the interlayer between HZO and the electrode or changing the doping element of HZO. [132][133][134][135][136][137][138][139][140][141][142]

Dead Layer: Interfacial Relaxation and Intermixing
There are many causes of the dead layer, and they may be classified into two categories: intermixing, which alters the chemical composition of the ferroelectric film, and interfacial relaxation, which occurs when the electrode and ferroelectric film relax together. Dead layers are regions with polarization or permittivity different from that of the bulk part of the film. In 2005, Li et al., the dead layer with a thickness of about 9 Å was measured using angle-resolved X-ray spectroscopy at an interface between Pt/BTO. This dead layer was not generated by interdiffusion between BTO and Pt but by interface-induced relaxation. In this dead layer, Ba 3d binding energy in bulk BTO was higher than in other kinds of electrodes. [143] In addition, many authors investigated the relaxation of the free surface of the ferroelectric perovskite structure; Ishikawa et al. evaluated the size of the surface relaxation of 3.8% and eight layers as BTO dead layers. [144] Craciun et al. measured the same dead layer on the Ba x Sr 1−x TiO 3 surface. [145] In this paper, the dead layer existed in surface-induced relaxation Pt/BTO interfaces, and the relaxation layer present at this interface had different properties from bulk BTO.
The dead layer is also formed by chemical intermixing between the electrode and the ferroelectric film. Considering the HZO ferroelectric film, if its thickness is decreased below 5 nm, orthorhombic formation is suppressed, so the device's performance is decreased sharply. As the chemical reaction at the interface varies depending on the electrode type, the degree of formation of a dead layer also varies. Also, the dead layer could affect the film's overall phase and electrical properties. Considering the interface's interaction, selecting an appropriate electrode to implement the ultra-thin HZO film is important. When the electrode TiN or W metal is used in the HZO layer, an oxygendeficient dead layer is present between the ferroelectric material and the electrode, generating a high leakage current. When considering the dead layer formation mechanism, the most appropriate electrode is Pt. Additionally, since TiN and W electrodes have less work function than Pt, the barrier between the ferroelectric film and the electrode is smaller. In this paper, the authors compared the capacitance (C) value of TiN/HZO/TiN, W/HZO/W, and W/HZO/Pt structures. As a result, the C values of the dead layer increase as the total thickness of the ferroelectric film decreases due to the formation of the dead layer. Comparing the wake-up process and switching-up current in the capacitor structure with different electrodes, the Pt electrode's polarizationelectric field (P-E) curve, which hardly forms a dead layer, is the same with or without a wake-up process. On the other hand, in the W/HZO/W structure, it is necessary to remove the dead Figure 4. a) The depolarization field for each thickness of a ferroelectric material with a dead layer of 1 nm on the dielectric surface. Reproduced with permission. [132] Copyright 2019, IEEE. b) Schematic illustration of the basic structure of perovskite oxide and the type and location of defects present inside. Reproduced with permission. [176] Copyright 2021, The Royal Society of Chemistry. Schematic illustration of the c) upward and d) downward polarization state of HfO 2 in which the oxygen vacancies exist. Red dotted circles indicate oxygen vacancy. e) Under the various oxygen vacancy concentrations, the graph shows the relative total energy of each phase compared to the m phase. f) Simulation results for the spontaneous polarization (black) and the energy barrier (red) value of ferroelectric HfO 2 according to various oxygen vacancy concentrations. Reproduced with permission. [47] Copyright 2019, Elsevier.
layer because it interferes with the formation of the orthorhombic phase and increases ferroelectricity's effect in the ultra-thin HZO film. In the case of the W electrode, an additional wake-up sequence is required to eliminate the dead layer by redistribution of oxygen vacancy and phase transition. [146] In Figure 4a, for PZT and HfO 2 , the 1 nm-thick dead layer of the interface and the relative permittivity of 14 and 50, respectively, were assumed. The graph shows that HfO 2 is unstable to a thickness of 10 nm or less, whereas the perovskite structure is unstable to 50 nm or less. The fluorite structure (HfO 2 ) could have a thinner dead layer (5 nm or less) because the chemical formula is simpler than perovskite, and the thinner dead layer does not additionally cause depolarization, so the fluorite structure has a thinner critical thickness than the perovskite structure. The authors addition-ally measured the capacitor device's cycling stability and data retention by inserting a dielectric layer of Al 2 O 3 between a top metal electrode and a ferroelectric and adjusting the depolarization field. [132]

Point Defects and Dopants
Generally, perovskite oxides could have defects in the A and B sites and oxygen sites. The perovskite oxide thin film has a relatively low concentration of defects, but low defect concentration can also significantly impact the film's properties. In particular, defects generated at the A and B sites of the perovskite oxide appear differently depending on the ion size and the oxidation state of the dopant atom. Also, defective dipoles formed in perovskite oxides cause ion vacancies or affect the oxidation state of surrounding ions. These effects can cause the ferroelectricity of the perovskite oxide. The ferroelectric polarization of ultra-thin ferroelectrics is significantly influenced by the interfacial screening charge produced to compensate for the electric field of the ferroelectric thin film. The metal electrode generally compensates for the interfacial screening charge, but oxygen vacancies also compensate for the interfacial screening charge. In the PZT/SRO interface where screening by SRO electrode affects, electrostatic energy can be mitigated by combining metal screening, ion screening, and domain. [27] However, at the PZT/STO interface, where there is no metal electrode, the charge compensation offered by the oxygen vacancy reduces the screening effect of the metal electrode, stabilizing the ferroelectric state. Because oxygen vacancies offered charge compensation, ferroelectricity was verified in PZT/STO of 3-unit cells [27] and 1.5-unit cells. [68] In order to demonstrate the effect of doping, Liu et al. presented a model that suppresses dipoles associated with the lattice where the Fe doped in BTO is located and the neighboring lattice. According to the model's simulation results, the creation of defect dipoles was considered a mechanism to account for the effects of doping. [147] In the fluorite structure, the defects alter the relative free energies of different crystals, which impacts ferroelectricity, and cause charge trap sites, which influence electrical conductivity and reliability. Recent experimental studies have shown that oxygen vacancies play an important role in stabilizing the ferroelectric orthorhombic phase in HfO 2 -based thin films. Figure 4c,d shows the structure of HfO 2 , which changes with the upward and downward polarization in the presence of oxygen vacancies. The total energy difference of each of the tetragonal (t), ferroelectric orthorhombic (f), and non-ferroelectric orthorhombic (o) phases were calculated compared to monoclinic (m) phase according to the concentration of oxygen vacancy per formula unit (f.u.) (Figure 4e). One f.u. contains one Hf atom and two O atoms. When oxygen vacancy concentration is 12.5 f.u.%, the energy differences of t, f, and o phases with respect to the m phase are calculated as 4.15, 1.21, and 0.57 eV, respectively. The values at this oxygen vacancy concentration are lower than those of 5.29, 2.63, and 2.18, which are the values when the oxygen vacancy concentration is zero. Although the energy difference of the f phase is still higher than that of the thermally stable m phase, it was confirmed that the oxygen vacancy reduces the energy difference to help form the f phase. The spontaneous polarization (left y-axis, black) and energy barrier of HfO 2 (right y-axis, red) according to the oxygen vacancy were calculated (Figure 4f). Spontaneous polarization increases as the oxygen vacancy concentration increases, and the energy barrier has a minimum value when the concentration is 3.125 f.u.%. [47] According to Mittmann et al., oxygen vacancies might serve as a defect nucleation site. A low oxygen flow rate during sputtering increased the number of nucleation sites by introducing more oxygen vacancies into the lattice. The presence of many nuclei made the average grain size smaller than the film with low oxygen vacancy concentration, resulting in more orthorhombic phase formation under the size effect. [90] As a result, the concentration of oxygen vacancies affects the size effect of the ferroelectrics and changes the degree of stabilization of each phase.
The proper mixture of Hf and Zr, or the doping of other elements, may cause ferroelectricity in the HZO film. Böscke et al. first discovered the ferroelectricity of the HfO 2 film at a Si-doped concentration of 2.6 to 3.1 mol% in 2011. [13,148] Similar results to Si doping could be obtained even in 4.8 mol% Al doping. [13,149] In small-sized dopants, such as Si and Al, paraelectric-ferroelectric-antiferroelectric properties appear in order as the dopant concentration increases. This change is because as the small-sized dopant concentration increases, the phase transition occurs in monoclinic-orthorhombic-tetragonalcubic order. [150] The tetragonal phase showing antiferroelectric has four relatively short Hf─O bonds and four relatively long Hf─O bonds. Since the small-sized dopant easily forms a short bond, the formation of the tetragonal phase is preferred when the small-sized dopant concentration increases. [150,151] In the case of large-sized dopants such as Y, [69,152] Gd, [153] and La, [154] characteristic changes occur in the order of paraelectric-ferroelectricparaelectric, unlike small-sized dopants. Tian et al. confirmed the change in the switching polarization value of 3 nm HfO 2 according to the Y dopant concentration. It can be seen that the switching polarization value increases according to the concentration up to 1.51 at% and then decreases again, which can confirm the change in the characteristics of the paraelectric-ferroelectricparaelectric. [69] It has been reported that when the relatively largesized dopant is doped on HZO, remnant polarization is about four times larger than that of the small-sized dopant. The relatively large remnant polarization is obtained because when the electric field is applied, the large dopant, along with the oxygen vacancy, facilitates the transition from cubic to orthorhombic phases. [155]

Domain Walls
The domain is an area where polarization moves in the same direction. The ferroelectricity is identified based on the mobility of the domain wall. The domain wall is influenced by various physical factors along with the angle of the neighboring domain. Among the domains, the grain boundary serves as a pin for the mobility of the domain wall. Therefore, as the grain size decreases, the movement of the domain wall decreases. [156] Only the intrinsic effect could be considered for monodomain in film. However, the extrinsic effect should be considered since the actual ferroelectric film is in a multi-domain state. Understanding the structure and properties of the domain wall is essential to implement devices using ultra-thin ferroelectric layers. The domain wall is crucial in the implementation of nanoelectronics utilizing the ferroelectric material because it improves conductivity, [157][158][159][160][161] photovoltages, [162][163][164][165][166][167][168] and current rectification [169] in the ferroelectric material. Ultra-thin BFO film can be used to comprehend the relationship between the domain wall and the ferroelectricity of the film. The domain wall is classified according to the polarization direction of the neighboring domain, and for BFO, it is divided into 71°, 109°, and 180°. Since the phase transition occurs to lower free energy at the domain's boundary destroying the symmetric structure, the domain wall can be found below the curie temperature. [170][171][172] The conventional BFO is a rhombohedral structure in a bulk state with a pseudo-cubic perovskite unit cell. The domain distribution varies  [156,[173][174][175] depending on the crystal state of BFO. The distribution of the domain of BFO in each direction is adjusted to the thickness of the bottom electrode, and when the SRO thickness is 5 nm or less, the 109°domain is grown. In order to form the 71°domain, a BFO film is formed to a thickness of 25 nm or more. The domain structure becomes unstable if the bound charge in the ferroelectric film creates a depolarization field, and the free charge cannot cancel it out. The 109°domain wall pattern is dominantly distributed if the surface area charge of the BFO on the thick bottom electrode is not compensated with free energy. The direction of the ferroelectric domain changes depending on the thickness of the film. Chen et al. reported that 180°domains are directly deposited in a thin layer in the direction of GSO (010). Then, 71°domains are not distributed, lowering the electrostatic energy or depolarization field and promoting the dominance of 180°d omains. [173] The vortex domain boundary leading to the nonuniform depolarization field can occur on the STO and TbScO 3 (TSO) substrate. A new vortex domain is created as the top and bottom electrodes when the thickness of BFO decreases. BFO has an oxygen atom octahedral structure in a pseudo-cubic unit cell rhombohedral (R)-like structure, and Fe cations have a separate placement in their face center and body center structure, respectively, with a spontaneous polarization value of about 100 μC cm −2 in the <111> p direction. On the other hand, polarization in the R-like direction may be 71°domain, 109°domain, and 180°d omain. In the tetragonal (T)-like unit cell structure, five oxygen occurs in the body center structure, and the polarization is 150 μC cm −2 .
Many studies have been conducted to indicate the correlation between the crystal structure represented by the grain structure and the domain. According to a study on the relationship between grain size and domain size, the tendency of dielectric constant for grain size is divided by the boundary with the highest value around a specific thickness as a boundary. Arlt et al. suggested that the domain size was influenced by grain size, which is inversely proportional to the parabolic scaling (m) constant. The m value is consistently 0.5 in experimental BTO and PZT ceramics with intermediate grain sizes of 1 to 10 μm. [105,174] As the grain size decreases, the density of the domain wall with an angle of 90°i ncreases proportionally to the relative permittivity, which greatly influences the external permittivity. When the grain size is less than 1 μm, explaining that the dielectric permittivity value decreases, the grain boundary interferes with the dielectric reaction and changes the domain boundary density and mobility. Ghosh et al. confirmed the most appropriate for rearranging the 90°domain wall in 2 μm grain size. [175] Mobility or the density of the domain wall decreases when the grain size is reduced to less than 1 μm. It results in a functional connection with extrinsic factors, such as a reduction in mobility or a reduction in the density of the domain wall, which frequently lowers the complexity of the domain structure and decreases the number of stable external variables. Generation of a new domain region becomes more difficult in each electric field, and the polarization process is more difficult when the grain size is reduced to a single domain size.
The causes and solutions of the size effect mentioned in Section 3 are summarized in Table 1.

Perovskite Structure-Based Ferroelectrics
A thickness limit of ferroelectrics has been anticipated in a perovskite structure. Initially, the critical thickness of ferroelectric films was expected to be hundreds of nanometers. However, research on the extrinsic effect affecting ferroelectric films progressed, and ferroelectricity can be implemented even with a thinner film thickness. In 1999, Tybell et al. deposited PZT using sputtering, and stable ferroelectric polarization of a film at a thin thickness of 40 Å (10-unit cells) for 140 h was confirmed through electrostatic force microscope (EFM) and piezoelectric microscopy. [38] Since then, there have been continuous reports of decreasing critical thickness by adjusting the extrinsic effect in ferroelectric film formation. Fong et al. confirmed the stable ferroelectric phase of PTO in 2004 at a thickness of 3-unit cells (1.2 nm), which meant that there was no limit to the thickness of the actual device fabrication by the intrinsic ferroelectric size effect. [12] Schlom et al. measured T c of each thickness of the BTO film on the STO substrate, ferroelectricity was confirmed at a thin thickness of 1.6 nm, and T c was measured at a lower temperature than 925 K. When the thickness of ferroelectrics was 10 nm, the T c value was 75 K. [39] Chu et al. deposited an epitaxial BFO film on a single crystal substrate STO (001) and DyScO 3 (DSO) (110) and measured using atomic force microscopy (AFM), transmission electron microscopy (TEM), and XRD to confirm that the polarization direction was the closest to the <111> direction and the film thickness could be at least 2 nm, [40] and extrinsic factors were also studied by local remnant piezo response at the thickness of 200 to 4-unit cells. [42] Junquera et al. broke the conventional notion that 10-unit cells (40 Å) were a critical thickness in 2003 and measured stable ferroelectricity at 6-unit cells (24 Å) below critical thickness by placing BTO between the SRO. [64] In 2013, Wen et al. used a semiconductor with high conductivity as an electrode instead of metal and manufactured an FTJ device with Pt/BTO/Nb:STO structure. The on/off ratio showed a higher 10 4 than other FTJ devices, and BTO at this time was very thin about 7-unit cells, which significantly improved tunneling electronics by using semiconductors as the electrode. This semiconductor electrode not only improves endurance but also enables non-destructive reading operation in non-volatile memory. [41] In 2013, Gao et al. experimented that polarization appeared below the critical thickness of the typical PZT, roughly 4 nm for 10unit cells. Furthermore, the polarization did not disappear even at a thinner thickness, and polarization could be confirmed up to 1.5-unit cells (0.6 nm) using the STO substrate, and using an SRO electrode, remnant polarization was measured at 22 μC cm −2 in a film with a thickness of 2-unit cells, which was enabled practically by the presence of a Pb─O bond. If only the appropriate electrode is used, the ferroelectricity can be measured even in a single unit cell. Polarization was measured even at a thickness that was too thin to define critical thickness. In the case of the PZT used in this paper, the ferroelectricity could not be lost in a very thin layer by using the strong displacement of the Pb─O layer. The thickness of the transition zone is measured in the 3unit cells range, and it can be considered an interfacial dead layer. Even within this range, there is spontaneous polarization because the position of Pb for element O has a displacement. Figure 5a shows a stable polarization value in the range from 1-to 10-unit cells or more, and the degree of polarization gradually decreased at the thickness of 3-to 10-unit cells. The polarization value was also measured in the thickness of the 3-unit cell. Figure 5b compares the value with the previous experimental results, and the polarization charge was effectively compensated for the stable polarization in the ultra-thin PZT film. Polarization screening could be formed from an external-internal charge. The external screening effect refers to the compensation of charges occurring at the boundary of all ferroelectrics (PZT/STO or PZT/SRO) including the interface between electrodes and ferroelectrics, and the internal screening effect refers to the screening effect occurring within the material structure by the free carrier or ionic charge position. Based on the thickness of the 2-unit cells, the calculated depolarization field and the density functional theory (DFT) calculation results were combined. The depolarization field value for the thickness is shown in the structure as named ( Figure 5c). As observed, polarization occurs even in an ultra-thin film condition because the displacement of the Pb─O bond is maintained as the thickness lowers (Figure 5d). It could be confirmed at 1.5-unit cell thickness in the annular bright-field (ABF) image ( Figure 5e).
In addition to actual ferroelectric thin film synthesis, research has been conducted on theoretical, critical thickness limitations. Ghosez et al. published in 2000, identified the Hamiltonian constant as critical thickness in the direction of (001) obtained using the first principle. The 3-unit cell thickness had a significant synergistic effect on the surface, and the ferroelectric instability was consistent with what has been reported. It showed theoretical foundations. [177] In Figure 5f-i, polarization values for two directions other than general OOP directions were calculated through DFT calculation, and the thickness of the ultra-thin film state was assumed to be in the range of 1-unit cell to 5-unit cells. The values for the [100] direction and [110] direction were considered, respectively, and the values were calculated as DFT in a structure where each termination ends in a BO 2 structure. There were three types of FE states from different micro-mechanisms in the in-plane direction. The appropriate ferroelectric state was a polarization value generated in the [110] direction due to the empty d orbital of B ions related to the secondary Jahn-Teller effect. The second was the most powerful ferroelectric state induced by the surface effect acting in the [100] direction. Third, a suitable ferroelectric polarization state was the direction [110] induced by trilinear coupling between rotational mode and a-site disposition and two rotational modes in a suitable ferroelectric state. [178]

Fluorite Structure-Based Ferroelectrics
Generally, the thin thickness of perovskite oxide prefers a paraelectric cubic phase with high symmetry to a low symmetric tetragonal phase showing ferroelectric characteristics. [179] For this reason, stable ferroelectricity at thin thickness is typically challenging to maintain. In contrast, the non-centrosymmetric orthorhombic phase in a thin fluorite structure has a higher degree of symmetry than the bulk-stable centrosymmetric monoclinic phase. [180] In the 2D limit, the surface energy facilitates inversion symmetric breaking and enables HZO to maintain  [68] Copyright 2017, The Author(s). ΔE value (pink line) and polarization calculation value according to the thickness of each thin film of f) BaTiO 3 , g) SrTiO 3 , h) SrSiO 3 , and i) CaSiO 3 according to proper ferroelectric state in [110] direction (FE 110-P) and ferroelectric state in [100] direction by surface effect (FE 100-S). Reproduced with permission. [178] Copyright 2017, The Author(s). ferroelectricity even in low thicknesses. [180,181] However, in the ultra-thin structure, it is not easy to maintain stable ferroelectricity not only in the perovskite oxide but also in the fluorite structure. Recently, studies have shown stable ferroelectricity in ultrathin HZO by overcoming fundamental limitations using various analytical methods. [43,69,182,183] Tian et al. deposited Y-doped HfO 2 in a thickness range of 3 to 30 nm by sputtering and confirmed ferroelectricity. It was discovered that the switching polarization (P sw ) gradually increased as thickness decreased from 30 to 8 nm, then rapidly decreased in the thickness range of 8 nm or less. This decrease in P sw is the same outcome as the high symmetric phase (orthorhombic, tetragonal, and cubic) ratio calculated using the  [43] Copyright 2020, The Author(s), under exclusive license to Springer Nature Limited. f) Remnant piezoresponse values as measured by PFM and g) 1D in-plane GIXRD of the HZO films with thicknesses from 2 to 10 nm. h) HAADF STEM image and i) simulated image with ferroelectric orthorhombic phase's Hf/Zr atoms located along the [112] zone axis at 2 nm HZO. j) Schematic illustration of the atomic arrangement of the ferroelectric HfO 2 along the [112] zone axis. k) The relative polarization ratio of each direction compared to the [001] direction with the maximum polarization value. Reproduced with permission. [183] Copyright 2021, American Chemical Society.
peaks of each phase's XRD intensity that rapidly decrease based on 8 nm. Optimizing the Y-doped concentration and annealing process improved the low P sw seen at 3 nm, and the P sw value was raised from 2.8 to 10 μC cm −2 . This result demonstrated that optimizing the doping and annealing conditions could minimize the size effect. [69] Lee et al. proved that ferroelectricity remained even at a 2 nm scale through analysis of the microstructure and crystal graphical orientation. PFM was measured for HZO films with thicknesses of 2 to 10 nm to verify ferroelectricity (Figure 6f). Although the reduction in remnant piezoresponse is confirmed as the thickness decreases, remnant piezoresponse that still appears even at 2 nm thick HZO proves ferroelectricity. In order to confirm the crystal structure of the film, it was measured along the in-plane direction using synchrotron grazing-incidence X-ray diffraction (GIXRD) (Figure 6g). A film with a thickness of 4 to 10 nm shows a randomly oriented polycrystalline structure with peaks of the monoclinic and orthorhombic phases. However, there was no monoclinic phase for films below 3 nm thickness, and only the orthorhombic/tetragonal phase was identified. [183] Because surface energy contributes more to phase formation as film thickness falls, tetragonal phase growth-which has lower surface energy than the monoclinic phase-is emphasized. [181] A crystal structure image was obtained using aberration-corrected high-angle annular dark-field (HAADF) STEM to observe crystals in grains of HZO directly. The image obtained by measuring along the [112] zone axis of 2 nm thick HZO (Figure 6h) and the image obtained using the orthorhombic HfO 2 atomic structure model (Figure 6i) are found to be identical. The (112)-oriented grain of a 2 nm film was confirmed to exhibit a 62% greater polarization value than the grain in the random direction observed at a film of 4 nm or more (Figure 6k). It was confirmed that the decrease in ferroelectricity due to the size effect of the 2 nm thickness film was compensated by forming a highly (112)-oriented crystal structure. [183] Cheema et al. deposited the 1 nm thick Hf 0.8 Zr 0.2 O 2 on the heavily p-doped Si (100) substrate ( Figure 6a). Figure 6b shows the change of fluorite structure in the polarization state in the upward and downward directions. The annealing process was carried out to stabilize the polar orthorhombic phase following the deposition of the W capping layer. On the other hand, weak ferroelectricity was observed in the region where annealing was done without the presence of the metal capping layer. This difference refers to the significance of ultra-thin confinement and the strain imposed by the capping layer. The 1 nm thick Hf 0.8 Zr 0.2 O 2 shows ferroelectricity, as demonstrated by the butterfly-shaped capacitance-voltage (C-V) graph in scanning capacitance microscopy (SCM) spectroscopy measurements (Figure 6e) and the phase and amplitude loops. Ferroelectricity is also demonstrated apparently on PFM images (Figure 6d). To demonstrate that write/erase is feasible, subsequent polling was taken in the same location but in the opposite direction. It was also confirmed that the positively polarized and unpolled regions are measured in the same phase. These PFM images prove that the 1 nm Hf 0.8 Zr 0.2 O 2 film exhibits spontaneous polarization without wake-up. Diffraction markers such as d 111 lattice spacing and structural aspect ratio, as well as orbital polarization and crystal field splitting results, indicate that Figure 7. a) In the 3D crystal structure of the layered In 2 Se 3 , the In atom is represented by blue, the Se atom is represented by red, and a black dashed square represents the QL. Reproduced with permission. [192] Copyright 2017, The Author(s). b) IP and OOP ferroelectric switching of 6 nm In 2 Se 3 flakes. c) OOP and d) corresponding IP image of a 6 nm-thick In 2 Se 3 flake were obtained by continuously applying voltages of −7 and +6 V. For 6 nm In 2 Se 3 flakes, PFM amplitude and phase switching loops show the ferroelectric characteristics e) without and f) with an Au top electrode. Reproduced with permission. [194] Copyright 2018, American Chemical Society. g) Top and side view of P 2 O 3 -I, P 2 O 3 -II, and P 2 O 3 -III. Each structure shows OOP electric polarization, IP electric polarization, and paraelectric. Reproduced with permission. [202] Copyright 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Both side and top views of the h) CIPS bulk and i) 2D monolayer. j) The major polar displacements of Cu and In ions in the ground state of ferroelectric CIPS. Simulated polarization-energy double well potential curve and l) P-E curve. Dots represent the calculation results, and the line fits the calculation results using the Landau model. Reproduced with permission. [7] Copyright 2020, The Royal Society of Chemistry.
this experimental result strengthens polar distortion at the ultrathin film limit. It has been demonstrated that polar distortion at the ultra-thin limit overcomes the size effect and renders ferroelectricity visible. [43] Lee et al. found a flat band of polar phonons that induces scale-free ferroelectricity in HfO 2 and the localized dipoles. The researchers discovered a 2D polar layer that, at the ultimate limit of 2.7 Å thickness, retains a stable and switchable ferroelectric domain beyond the 3D structure of HfO 2 . The flat phonon band with ferroelectric HfO 2 allows ferroelectric switching at the irreducible sub-nanometer size. By separating the polar layer from a nonpolar spacer layer of the size of a half-unit cell (2.5 Å), the vertical dipole is laterally localized within the halfunit cell size. Unlike conventional ferroelectrics, whose ferroelectricity disappears below the critical thickness, localized dipoles, which are stable and switchable below the sub-nanometer scale, allow bits to be stored in the local domain of angstrom-size. In addition, the researchers explained the causes of the large coercive field [184,185] and slow domain propagation [186] of HfO 2 compared to ferroelectric perovskite oxide, [187,188] eliminating the interaction between ferroelectric dipoles. [182] Several investigations have been conducted since the discovery of HZO's ferroelectricity to understand and overcome the size impact. Although numerous strategies have been proposed to identify and overcome the reason, more research is still needed.

2D Ferroelectrics: Intrinsically No Size Effects
In recent years, interest in 2D ferroelectric materials has persisted. 2D ferroelectric materials not affected by size effect without finite size limitations have a stable layer structure and lower surface energy. [189,190] The properties of 2D ferroelectric materials open the way for the fabrication of ultra-thin ferroelectric materials, which have been researched for fabrication in conventional perovskite oxide or fluorite. Theory and experiments have supported the existence of ferroelectricity in several materials, including oxide, which has been explored for its potential to exist in the 2D structure. [74,191] In this section, we introduce promising 2D materials among the candidate group and discuss the ferroelectricity of materials.
Using the first principle DFT based on III 2 -VI 3 compounds, Ding et al. reported in 2017 that the ground state structure of -In 2 Se 3 quintuple layer (QL) could have OOP and IP electric fields (Figure 7a). The alternating covalent bond between the Se and In atomic layers constitutes a single layer of -In2Se3. The OOP and IP electric polarization of -In 2 Se 3 originate from the movement of the central Se atom (Figure 7b). By modulating the interlayer distance between the central and outer layers, -In 2 Se 3 successfully breaks centrosymmetry and shows spontaneous OOP electrical polarization. On the other hand, IP electric polarization is caused by in-plane centrosymmetry breaking. [192] In order to check the effect of the depolarization field, which is the main cause of inhibiting ferroelectricity in conventional ultrathin ferroelectric materials, [12,[62][63][64] on -In 2 Se 3 , a calculation was performed on the ferroelectric phase and nonpolar facecentered cubic (FCC) phase. According to the calculation, the depolarization field lessens the energy difference between the two phases, but the ferroelectric phase still has lower energy and exhibits realistic 2D ferroelectricity. [192] In the same year, Zhou et al. confirmed OOP electric polarization of 10 nm thick -In 2 Se 3 . [193] In the following year, Cui et al. checked the OOP and IP electric polarization of 6 nm thick -In 2 Se 3 using PFM. They confirmed that the IP polarization was changed simultaneously under the influence of OOP polarization (Figure 7c,d).
The graphs of 180-degree phase change and butterfly-shaped amplitude loops produced by the applied voltage demonstrate -In 2 Se 3 's ferroelectricity (Figure 7e). [194] Similar hysteresis loops can be seen due to the electrostatic effect between tip and surface even when not ferroelectric, [195] so ferroelectric hysteresis was identified by inducing the generation of uniform electric fields by depositing Au top electrode (Figure 7f). [194] Xiao et al. confirmed that the OOP ferroelectricity of -In 2 Se 3 had a curie temperature of 700 K and confirmed the ferroelectricity at 3 nm thick. This discovery presents a novel covalent bond reconstruction-based polarization switching mechanism and offers a distinctive framework for studying 2D ferroelectric physics. [196] Next, the ferroelectricity of P 2 O 3 predicted by 2D ferroelectric oxide will be described. Phosphorene, 2D material comprised of black P atoms, [197,198] has novel features, but its properties are easily degraded in a humid environment with oxygen. [199][200][201] Using particle swarm optimization (PSO) technology, Luo et al. predicted a stable structure for 2D P x O y , an oxide of phosphorene. Three different P 2 O 3 structure types were discovered among different structures with various oxygen concentrations, and two of these P 2 O 3 kinds (P 2 O 3 -I and P 2 O 3 -II) were reported to display ferroelectricity ( Figure 7g). P 2 O 3 -I, where three oxygen atoms surround one P atom, has the lowest energy structure at a thickness of less than 1.4 Å. All of the P atoms are in the top layer, while all the O atoms are in the bottom layer. Non-zero electrical polarization occurs in the OOP direction due to structures located in different planes. Unlike the P 2 O 3 -I structure, P 2 O 3 -II, where P atoms do not exist in the same plane, has the lowest energy at a 3.2 Å or less thickness. O atoms do not exist in the same plane as P atoms. As a result, OOP polarization does not appear. However, IP polarization was found due to collective oxygen displacement along the y-axis. [202] Finally, we introduce 2D ferroelectric material CuInP 2 S 6 (CIPS), which demonstrates stable OOP polarization. CIPS has a structure in which P-P pairs form a triangular pattern in the sulfur framework, and octahedral oxygen voids are filled with Cu and In atoms. [203] Bulk CIPS forms a structure in which all monolayers are stacked along the OOP direction (Figure 7h). [7] Due to the weak van der Waals interaction between layers, CIPS can easily exfoliate from the bulk state to form 2D monolayer nanoflakes. [74,[204][205][206] Figure 7i shows the crystal structure of the 2D monolayer CIPS. The OOP directional movement of Cu and In cations present inside the S octahedra induces the ferroelectricity of the CIPS 2D monolayer (Figure 7j). [7] In d1T-MoTe 2 0.8 OOP polarization [212] 2015, Belianinov et al. confirmed ferroelectricity in 50 nm thick 2D CIPS flakes. [74] In the same year, ferroelectricity was confirmed at 20 nm thickness by changing the Cu and In composition ratio. [207] In 2016, the size effect and switching characteristic were discussed with the ferroelectricity of the 10 nm thick CIPS. In order to explain the size effect and switching of nanoscale CIPS, ionic and ferroelectric degrees of freedom and disorder were considered simultaneously. [208] Additionally, structural phase changes, including the transition from ferroelectric to paraelectric at T c , were observed by second harmonic generation (SHG) measurements. [206] Zhao et al. investigated the interaction between OOP polarization and the depolarization field through the first principles calculation. The calculation results proved that the CIPS of the 2D structure could be switched up to 3.4 Å thickness. Applying polarization and total energy to the polar displacement of the CIPS monolayer to the Landau model yields a standard polarization-energy double-well potential curve ( Figure 7k). Additionally, a hysteresis polarization-electric field graph showing a typical ferroelectric was obtained using the Landau model set up (Figure 7l). The comparatively weak polarization of CIPS and the deep potential well are the sources of the stable OOP polarization observed in the ultra-thin scale. [7] In addition, there are several 2D ferroelectric materials such as SnTe, [209] WTe 2 , [210] and BA 2 PbCl 4 , [211] which have been proven to be ferroelectric through experiments. It is expected that via further research on 2D ferroelectric materials, more diverse and stable 2D ferroelectric materials will be discovered, and a breakthrough will be achieved to overcome the intrinsic size effect of current ferroelectric materials.
In Section 4, we reviewed studies on perovskite oxide, fluorite, and 2D material that show ferroelectricity beyond the commonly known critical thickness and summarized these results in Table 2.

Summary and Perspective
In this review article, we introduced the size effect and factors that affect the size effect of ferroelectric oxide materials such as perovskite oxide and fluorite. In addition, we reviewed recent studies to overcome the size effect. Based on the LGD theory introduced to explain ferroelectricity, we introduce the influence of the depolarization field. Factors affecting the size effect include 1) process conditions for synthesizing ferroelectrics, 2) domain changes due to grain size and grain boundary density, 3) intrinsic and extrinsic strain, 4) charge screening and damage caused by electrodes with a capacitor structure, 5) dead layer between electrodes and ferroelectrics, 6) intrinsic defect and extrinsic dopants, and 7) mobility of the domain wall according to the crystal structure. Several studies have been conducted to effectively control the size effect of ferroelectrics and realize ferroelectricity up to the sub-nanometer scale, and the results suggest ways to reduce the intrinsic effect and depolarization field of ferroelectrics. Furthermore, by calculating and proving that ferroelectricity can exist in several 2D materials and observing it in a particular material, we present another way to overcome the intrinsic size effect of ferroelectrics.
In the past 20 years, the theoretical understanding of the size effect has been continuously improved, and as the utilization of ferroelectric materials in electronic device equipment is increasing, the experiment on the size effect is becoming more important. Based on several simulation results predicting ferroelectric's theoretical critical thickness is thinner than current technology. It is anticipated that the size effect of ferroelectrics will be overcome more successfully in the future due to the improvement of precise calculation, synthesis and deposition techniques, and measurement techniques. Ferroelectrics could be applied in memory devices and various fields such as sensors, actuators, and energy harvesting. Next-generation semiconductor-related devices are implemented in a highly integrated circuit with a few nanometers of thickness. In this condition, it is important to control the size effect factors of ferroelectrics to maintain ferroelectric properties even in a few unit cells. Even the ferroelectric thin films manufactured so far need low leakage current, high cycling stability, and thermal performance to be applied to ferroelectric devices. Accurate understanding and overcoming the size effect will be a key strategy for further advancement and application of the technology.