**Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films**

N. D. Scarisoreanu, R. Birjega, A. Andrei, M. Dinescu, F. Craciun and C. Galassi

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/ 52395

### **1. Introduction**

Ferroelectric perovskites based on Na0.5Bi0.5TiO3 (NBT) are considered among the most promising lead-free candidate materials to substitute Pb(Zr*1-x*Ti*x*)O3 (PZT) in devices de‐ signed to respect standards and environmental laws. Taking into account the toxicity of lead-based systems, there are numerous lead-free piezoelectric materials under investigation in worldwide spread laboratories for replacing PZT in future devices. Constant efforts are made to find viable replacements for all these materials containing harmful elements.

Solid-solution systems based on lead-free perovskites like Na0.5K0.5NbO3 (NKN), BaTiO3 (BT), Na0.5Bi0.5TiO3 (NBT) or bismuth layered-structured SrBi2Ta2O9 (SBT), and SrBi2Nb2O9 (SBN) are considered as viable alternatives for replacing lead-based materials. For example, (K,Na)NbO3–LiTaO3–LiSbO3 alkaline niobate ceramics exhibit a *d <sup>33</sup>* piezoelectric coefficient up to 416 pC/N together with Curie temperature Tc around 526 K, as reported by Saito *et al* [1]. Sodium/bismuth titanate (NBT) belongs to the bismuth-based perovskites in which the A-site atom is replaced. The crystalline structure, phase transitions and physical properties have been intensively studied since the discovery of the material in 1960 by Smolensky *et al* [2]. NBT has a relatively high depolarization temperature, Td = 470 K, high remanent polari‐ zation, 38 *μ*C/cm2 and piezoelectric coefficient d33 = 125 pC/N [3]. However, owing to the high value of the coercive field and high electrical conductivity, NBT cannot be easily polar‐ ized, therefore different A-site substitutions have been attempted to avoid this drawback.

© 2013 Scarisoreanu et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

tion to the problem of avoiding leakage currents under high electric fields [12, 13]. Using pulsed laser deposition (PLD), Duclère *et al.* have reported the heteroepitaxial growth of NBT thin films on epitaxial platinum electrodes supported on a sapphire substrate [14]. More recently, M.Bousquet *et al* have described the electrical properties of (110)- oriented NBT thin films deposited by laser ablation on (110)Pt/(110) SrTiO3 substrates [15]. They re‐ ported the coexistence of two kinds of grains with different shapes in the films, flat and elongated grains corresponding to (100) and (110) oriented NBT crystallites. The effects of Bi- excess in target on the dielectric and ferroelectric properties of the films have been also presented; the reported values for relative permittivity and remnant polarization were

of (100)-oriented Na0.5Bi0.5TiO3-BiFeO3 thin films deposited by sol-gel have been reported by

However, despite the fact that ferroelectric materials with MPB have enhanced ferroelectric and piezoelectric properties, it is difficult to transpose them in thin films since MPB is limit‐ ed to a small composition range. Almost all the physical parameters involved in thin films deposition like the substrate type, the microstructure and stress have strong impact on their physical properties [17]. In some previous papers we have investigated the role of different deposition parameters on NBT-BT film growth and properties [18, 19]. In this chapter, we discuss the role of certain experimental conditions like deposition temperature and substrate type, as well as of the amount of BT present in the target on crystalline structure, microstruc‐ ture, dielectric properties, phase transition temperatures and stability limits of ferroelectric

Pulsed laser deposition (PLD) was used for the film growth. The targets with composition (NBT)1-*x*(BT)*<sup>x</sup>* (*x* = 0.06-0.08), further called NBT-BT6 and NBT-BT8, have been prepared fol‐

in crucibles with the sample surrounded with NBT pack, in order to avoid the loss of Na

ray diffraction analysis evidenced the obtaining of pure perovskite phase. The microstruc‐ ture of the sintered targets was investigated on polished and etched surfaces by scanning

For the film deposition, a Surelite II Nd:YAG pulsed laser with wavelength of 265 nm, pulse duration of 5 ns and frequency 10 Hz, has been employed. The laser fluence was set at 1.6

performed in flowing oxygen atmosphere (0.3-0.6 mbar) to favour the formation of perov‐ skite phase without oxygen vacancies. Chemical composition was checked via SIMS techni‐ que using a Hiden SIMS/SNMS system. The thickness of the thin films, evaluated by

. The films were grown on Nb:STO and Pt/TiO2/SiO2/Si substrates, placed at a distance of about 4.3 cm from the target. Different sets of films have been grown at different substrate

Qin et al, aiming to important applications such as photovoltaic devices [16].

, respectively. Furthermore, very recently, the electrical properties

Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films

http://dx.doi.org/10.5772/ 52395

353

C for 2 h. The sintering was performed

C; more details can be found in Ref.11. X-

C. Deposition and after-deposition cooling were

εr≈225-410 and 14 μC/cm2

phases in NBT-BT thin films produced by PLD.

lowing the mixed oxide route and sintered at 1150 o

electron microscopy. The observed grain sizes were 2-10 μm.

and Bi, which occurs at temperatures over 1000 o

temperatures, ranging between 650-730 o

spectroellipsometry, was between 300-500 nm.

**2. Experimental method**

J/cm2

**Figure 1.** The end-members of perovskite NBT-BT: rhombohedral NBT and tetragonal BT. Cations Na+/Bi3+ and Ba2+ occupy the A-sites while Ti4+ occupies B-sites (oxygen octahedra centers).

The solid solution with BaTiO3, (1-x) NBT-x BT shows a morphotropic phase boundary (MPB) between the rhombohedral and the tetragonal phase, at x between 0.06 and 0.07 for which the material properties are considerably improved. Indeed d33 values up to 450 pC/N, and huge electric field-induced strain have been reported [4, 5]. Figure 1 shows the crystal‐ line structures of the NBT and BT end members at room temperature. Perovskite structure deformations include oxygen octahedral rotations around different axis and cation shifts, therefore giving rise to a complex succession of ferroelastic and ferroelectric phase transfor‐ mations with temperature variation.

Due to this polymorphic structure, NBT and NBT-BT have been also intensively studied in order to clarify their complicated phase transitions, which still pose questions [6]*.*Structural and polar transformations in NBT-BT are more complicated than in other perovskite solid solutions, also due to the strong disorder of the A-sites occupied by Na+ , Bi3+ or Ba2+ ions, with different valence, mass and ionic radius. NBT transforms successively, from the high temperature cubic paraelectric into tetragonal antiferroelectric (or ferrielectric) and further into a rhombohedral ferroelectric phase [6]. In solid solution with BT, the ground ferroelec‐ tric phase changes from rhombohedral *R3c* to tetragonal ferroelectric *P4mm*, at the so-called morphotropic phase boundary (MPB) (*x* ≈ 0.06-0.07) [5, 7, 8]. The phase diagram of NBT-BT bulk material, mainly based on dielectric measurements, was completed by Cordero *et al* by performing direct anelastic measurements, the border between tetragonal and cubic phases being evidenced [9, 10, 11].

For NBT-BT thin films growth many techniques have been used. Guo *et al*. have investigated NBT-BT-based tri-layered films prepared by chemical solution deposition as a possible solu‐ tion to the problem of avoiding leakage currents under high electric fields [12, 13]. Using pulsed laser deposition (PLD), Duclère *et al.* have reported the heteroepitaxial growth of NBT thin films on epitaxial platinum electrodes supported on a sapphire substrate [14]. More recently, M.Bousquet *et al* have described the electrical properties of (110)- oriented NBT thin films deposited by laser ablation on (110)Pt/(110) SrTiO3 substrates [15]. They re‐ ported the coexistence of two kinds of grains with different shapes in the films, flat and elongated grains corresponding to (100) and (110) oriented NBT crystallites. The effects of Bi- excess in target on the dielectric and ferroelectric properties of the films have been also presented; the reported values for relative permittivity and remnant polarization were εr≈225-410 and 14 μC/cm2 , respectively. Furthermore, very recently, the electrical properties of (100)-oriented Na0.5Bi0.5TiO3-BiFeO3 thin films deposited by sol-gel have been reported by Qin et al, aiming to important applications such as photovoltaic devices [16].

However, despite the fact that ferroelectric materials with MPB have enhanced ferroelectric and piezoelectric properties, it is difficult to transpose them in thin films since MPB is limit‐ ed to a small composition range. Almost all the physical parameters involved in thin films deposition like the substrate type, the microstructure and stress have strong impact on their physical properties [17]. In some previous papers we have investigated the role of different deposition parameters on NBT-BT film growth and properties [18, 19]. In this chapter, we discuss the role of certain experimental conditions like deposition temperature and substrate type, as well as of the amount of BT present in the target on crystalline structure, microstruc‐ ture, dielectric properties, phase transition temperatures and stability limits of ferroelectric phases in NBT-BT thin films produced by PLD.

### **2. Experimental method**

**Figure 1.** The end-members of perovskite NBT-BT: rhombohedral NBT and tetragonal BT. Cations Na+/Bi3+ and Ba2+

The solid solution with BaTiO3, (1-x) NBT-x BT shows a morphotropic phase boundary (MPB) between the rhombohedral and the tetragonal phase, at x between 0.06 and 0.07 for which the material properties are considerably improved. Indeed d33 values up to 450 pC/N, and huge electric field-induced strain have been reported [4, 5]. Figure 1 shows the crystal‐ line structures of the NBT and BT end members at room temperature. Perovskite structure deformations include oxygen octahedral rotations around different axis and cation shifts, therefore giving rise to a complex succession of ferroelastic and ferroelectric phase transfor‐

Due to this polymorphic structure, NBT and NBT-BT have been also intensively studied in order to clarify their complicated phase transitions, which still pose questions [6]*.*Structural and polar transformations in NBT-BT are more complicated than in other perovskite solid

with different valence, mass and ionic radius. NBT transforms successively, from the high temperature cubic paraelectric into tetragonal antiferroelectric (or ferrielectric) and further into a rhombohedral ferroelectric phase [6]. In solid solution with BT, the ground ferroelec‐ tric phase changes from rhombohedral *R3c* to tetragonal ferroelectric *P4mm*, at the so-called morphotropic phase boundary (MPB) (*x* ≈ 0.06-0.07) [5, 7, 8]. The phase diagram of NBT-BT bulk material, mainly based on dielectric measurements, was completed by Cordero *et al* by performing direct anelastic measurements, the border between tetragonal and cubic phases

For NBT-BT thin films growth many techniques have been used. Guo *et al*. have investigated NBT-BT-based tri-layered films prepared by chemical solution deposition as a possible solu‐

, Bi3+ or Ba2+ ions,

solutions, also due to the strong disorder of the A-sites occupied by Na+

occupy the A-sites while Ti4+ occupies B-sites (oxygen octahedra centers).

mations with temperature variation.

352 Advances in Ferroelectrics

being evidenced [9, 10, 11].

Pulsed laser deposition (PLD) was used for the film growth. The targets with composition (NBT)1-*x*(BT)*<sup>x</sup>* (*x* = 0.06-0.08), further called NBT-BT6 and NBT-BT8, have been prepared fol‐ lowing the mixed oxide route and sintered at 1150 o C for 2 h. The sintering was performed in crucibles with the sample surrounded with NBT pack, in order to avoid the loss of Na and Bi, which occurs at temperatures over 1000 o C; more details can be found in Ref.11. Xray diffraction analysis evidenced the obtaining of pure perovskite phase. The microstruc‐ ture of the sintered targets was investigated on polished and etched surfaces by scanning electron microscopy. The observed grain sizes were 2-10 μm.

For the film deposition, a Surelite II Nd:YAG pulsed laser with wavelength of 265 nm, pulse duration of 5 ns and frequency 10 Hz, has been employed. The laser fluence was set at 1.6 J/cm2 . The films were grown on Nb:STO and Pt/TiO2/SiO2/Si substrates, placed at a distance of about 4.3 cm from the target. Different sets of films have been grown at different substrate temperatures, ranging between 650-730 o C. Deposition and after-deposition cooling were performed in flowing oxygen atmosphere (0.3-0.6 mbar) to favour the formation of perov‐ skite phase without oxygen vacancies. Chemical composition was checked via SIMS techni‐ que using a Hiden SIMS/SNMS system. The thickness of the thin films, evaluated by spectroellipsometry, was between 300-500 nm.

For the investigation of the crystalline structure of the targets and films, a PANalytical X'pert MRD diffractometer in Bragg-Brentano geometry was used. The measurements were performed with a step size of 0.020 and with a scanning time on step of 25 s or 250 s, depend‐ ing on the angular range.

The film surface morphology was examined by AFM (model XE100, Park Systems). Piezo‐ electric force microscopy measurements were performed with a PFM system which includes a lock-in amplifier SR-830 and a dc- high voltage amplifier WMA-280. Conductive all-metal Pt tips were employed for these measurements were the switching characteristcs of the films have been tested.

Several Au electrode dots with an area of about 0.22 mm2 have been evaporated through a mask on the films for electrical characterization. Polarization hysteresis was measured by using a Radiant Technology RT66A ferroelectric test system, in the virtual ground mode. The dielectric measurements were carried out in a frequency range between 200 Hz and 1 MHz using an HP 4194A impedance analyzer and an HP 4284A LCR meter with a four wire probe. The measurements were performed at 1.5 K/min between 300 and 570 K in a Delta Design climatic chamber model 9023 A (on targets) and in a Linkam variable-temperature stage (model HFS 600E) on films.

**Figure 2.** AFM images of NBT-BT6% films deposited on Nb:STO substrates at temperature of 650 oC. The displayed

Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films

http://dx.doi.org/10.5772/ 52395

355

**Figure 3.** AFM images of NBT-BT6% films deposited on Nb:STO substrates at 700oC. The displayed surfaces are 20x20

surfaces are 20x20 μm<sup>2</sup> (a) and 5x5 μm2 (b).

μm<sup>2</sup> (a) and 5x5 μm2 (b).

### **3. Results and discussion.**

### **3.1. Growth mode of NBT-BT thin films.**

The microstructure of ceramic thin films is one of the most important factors that influence their physical properties. Since the growth mode of thin films is strongly dependent on the substrate type, we investigated the deposition of NBT-BT thin films on two different types of substrates:


The AFM pictures obtained on the two sets of films show important microstructural differ‐ ences, mainly due to different growth mechanisms. In Figure 2 we show AFM images taken on a NBT-BT6 film deposited on Nb: STO monocrystalline substrate at 650 0 C. It can be ob‐ served that a first stage of growth resulting into a continuous layer stops when the critical thickness for misfit dislocations (probably a few tens of nm) is reached. After that, the growth continues in platelet-like form (see details in Fig. 2 b). If the deposition temperature is not sufficiently high to favor material exchange between platelets via surface migration, successive layers will grow on the top of the first islands and the growth will result into a discontinuous layer. This explains the platelet-like aspect of the film shown in Figure 2.

For the investigation of the crystalline structure of the targets and films, a PANalytical X'pert MRD diffractometer in Bragg-Brentano geometry was used. The measurements were

The film surface morphology was examined by AFM (model XE100, Park Systems). Piezo‐ electric force microscopy measurements were performed with a PFM system which includes a lock-in amplifier SR-830 and a dc- high voltage amplifier WMA-280. Conductive all-metal Pt tips were employed for these measurements were the switching characteristcs of the films

mask on the films for electrical characterization. Polarization hysteresis was measured by using a Radiant Technology RT66A ferroelectric test system, in the virtual ground mode. The dielectric measurements were carried out in a frequency range between 200 Hz and 1 MHz using an HP 4194A impedance analyzer and an HP 4284A LCR meter with a four wire probe. The measurements were performed at 1.5 K/min between 300 and 570 K in a Delta Design climatic chamber model 9023 A (on targets) and in a Linkam variable-temperature

The microstructure of ceramic thin films is one of the most important factors that influence their physical properties. Since the growth mode of thin films is strongly dependent on the substrate type, we investigated the deposition of NBT-BT thin films on two different types

The AFM pictures obtained on the two sets of films show important microstructural differ‐ ences, mainly due to different growth mechanisms. In Figure 2 we show AFM images taken

served that a first stage of growth resulting into a continuous layer stops when the critical thickness for misfit dislocations (probably a few tens of nm) is reached. After that, the growth continues in platelet-like form (see details in Fig. 2 b). If the deposition temperature is not sufficiently high to favor material exchange between platelets via surface migration, successive layers will grow on the top of the first islands and the growth will result into a discontinuous layer. This explains the platelet-like aspect of the film shown in Figure 2.

on a NBT-BT6 film deposited on Nb: STO monocrystalline substrate at 650 0

and with a scanning time on step of 25 s or 250 s, depend‐

have been evaporated through a

C. It can be ob‐

performed with a step size of 0.020

stage (model HFS 600E) on films.

**3. Results and discussion.**

of substrates:

**2.** Pt/TiO2/SiO2/Si.

**3.1. Growth mode of NBT-BT thin films.**

**1.** single crystal SrTiO3: Nb (Nb:STO) and

Several Au electrode dots with an area of about 0.22 mm2

ing on the angular range.

have been tested.

354 Advances in Ferroelectrics

**Figure 2.** AFM images of NBT-BT6% films deposited on Nb:STO substrates at temperature of 650 oC. The displayed surfaces are 20x20 μm<sup>2</sup> (a) and 5x5 μm2 (b).

**Figure 3.** AFM images of NBT-BT6% films deposited on Nb:STO substrates at 700oC. The displayed surfaces are 20x20 μm<sup>2</sup> (a) and 5x5 μm2 (b).

However, raising the substrate temperature to 700 0 C during the deposition of a second set of films while keeping constant all the other parameters, including the number of laser puls‐ es, produces a uniform layer of continuous platelets, on top of which new islands nucleate (Figure. 3).

A rather different morphology is displayed by NBT-BT films grown on Pt/TiO2/SiO2/Si (Fig. 4 and 5). In this case, the growth progresses from the beginning in island-like form since the polycrystalline Pt layer provide the nucleation sites for their formation. Moreover, these NBT-BT islands grow on the Pt layer without preserving a unique orientation, due to the same reason. Instead films grown on Nb:STO monocrystalline structures are uniaxially (001)-oriented, as it will be shown in the next section.

**Figure 5.** AFM images of NBT-BT8% films deposited on Pt/TiO2/SiO2/Si substrate at a temperature of 700 o C. The dis‐

Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films

http://dx.doi.org/10.5772/ 52395

357

The XRD spectrum of NBT-BT6 target corresponds to a mixture of rhombohedral R3c and tetragonal P4mm phases, as shown by the splitting of (111) and (200), (012) and (024) rhom‐ bohedral peaks in the bottom pattern in Fig. 6 [20]. The main Miller index of the rhombohe‐ dral phase are depicted horizontally on the bottom of the figures while those of the tetragonal phase vertically above. On the same graph, the pattern corresponding to the

BT6/Pt/TiO2/SiO2/Si film deposited in the same conditions but at a substrate temperature of

 exhibits similar features as we had reported and is not presented here [18].The as de‐ posited thin films exhibit pure perovskite phase with symmetry congruent with that of the target. The reflection peaks indicate a randomly oriented structure, consistent with the poly‐

C is given. The curve corresponding to NBT-

played surfaces are 20x20 μm<sup>2</sup> (a) and 5x5 μm2 (b).

NBT-BT6 film grown on Pt/TiO2/SiO2/Si at 700 0

**3.2. Crystalline structure**

crystalline nature of the films.

6500

**Figure 4.** AFM images of NBT-BT6% films deposited on Pt/TiO2/SiO2/Si substrate at a temperature of 700 oC. The dis‐ played surfaces are 20x20 μm<sup>2</sup> (a) and 5x5 μm2 (b).

A fine microstructure with grain size ranging from a few tens of nm up to a few hundred of nm is displayed by NBT-BT6 films (Fig. 4). We note the striking difference with bulk sam‐ ples microstructures (not shown here), which consists of crystallites of 1-10 μm size.

A similar fine microstructure is displayed by NBT-BT8% films grown on Pt/TiO2/SiO2/Si (Fig. 5 a). However, the enlarged AFM image displayed in Fig 5 b) reveales a somewhat dif‐ ferent aspect with triangular nanograins lying in plane.

**Figure 5.** AFM images of NBT-BT8% films deposited on Pt/TiO2/SiO2/Si substrate at a temperature of 700 o C. The dis‐ played surfaces are 20x20 μm<sup>2</sup> (a) and 5x5 μm2 (b).

### **3.2. Crystalline structure**

However, raising the substrate temperature to 700 0

(001)-oriented, as it will be shown in the next section.

played surfaces are 20x20 μm<sup>2</sup> (a) and 5x5 μm2 (b).

ferent aspect with triangular nanograins lying in plane.

(Figure. 3).

356 Advances in Ferroelectrics

of films while keeping constant all the other parameters, including the number of laser puls‐ es, produces a uniform layer of continuous platelets, on top of which new islands nucleate

A rather different morphology is displayed by NBT-BT films grown on Pt/TiO2/SiO2/Si (Fig. 4 and 5). In this case, the growth progresses from the beginning in island-like form since the polycrystalline Pt layer provide the nucleation sites for their formation. Moreover, these NBT-BT islands grow on the Pt layer without preserving a unique orientation, due to the same reason. Instead films grown on Nb:STO monocrystalline structures are uniaxially

**Figure 4.** AFM images of NBT-BT6% films deposited on Pt/TiO2/SiO2/Si substrate at a temperature of 700 oC. The dis‐

A fine microstructure with grain size ranging from a few tens of nm up to a few hundred of nm is displayed by NBT-BT6 films (Fig. 4). We note the striking difference with bulk sam‐

A similar fine microstructure is displayed by NBT-BT8% films grown on Pt/TiO2/SiO2/Si (Fig. 5 a). However, the enlarged AFM image displayed in Fig 5 b) reveales a somewhat dif‐

ples microstructures (not shown here), which consists of crystallites of 1-10 μm size.

C during the deposition of a second set

The XRD spectrum of NBT-BT6 target corresponds to a mixture of rhombohedral R3c and tetragonal P4mm phases, as shown by the splitting of (111) and (200), (012) and (024) rhom‐ bohedral peaks in the bottom pattern in Fig. 6 [20]. The main Miller index of the rhombohe‐ dral phase are depicted horizontally on the bottom of the figures while those of the tetragonal phase vertically above. On the same graph, the pattern corresponding to the NBT-BT6 film grown on Pt/TiO2/SiO2/Si at 700 0 C is given. The curve corresponding to NBT-BT6/Pt/TiO2/SiO2/Si film deposited in the same conditions but at a substrate temperature of 6500 exhibits similar features as we had reported and is not presented here [18].The as de‐ posited thin films exhibit pure perovskite phase with symmetry congruent with that of the target. The reflection peaks indicate a randomly oriented structure, consistent with the poly‐ crystalline nature of the films.

Figure 8 shows the XRD patterns of NBT-BT8 films grown on Pt/TiO2/SiO2/Si and Nb:STO

responds to the tetragonal P4mm symmetry. It can be observed that, similar to the previ‐ ous composition, the growth on single crystal Nb:STO substrate produces an epitaxial film, while the growth on Pt/TiO2/SiO2/Si substrate results into a polycrystalline randomly ori‐

**Figure 8.** XRD spectra of NBT-BT8% deposited on Pt/TiO2/SiO2/Si and on Nb:STONb:STO substrates

The dielectric and ferroelectric properties of NBT-BT thin films have been evaluated on ca‐ pacitors formed by evaporating through a mask an array of gold electrode dots with an area

The bottom electrode was formed by the Pt layer in the first case or by the Nb:STO substrate

The piezoresponce force microscopy results are presented in Figure 9. The full-Pt tips were brought in contact with the surface of the sample and then a *dc* bias and test *ac* bias were applied between the tip and the bottom electrode of the samples. The *dc* bias was generated by a high voltage amplifier and the *ac* bias was generated by a lock-in amplifier. The same lock-in amplifier was used to analyse the vertical deflection signal from the PSPD, in order to extract the amplitude and the phase of the cantilever oscillations induced by the local de‐

on the surface of films grown on Pt/TiO2/SiO2/Si and Nb:STO substrates.

**3.3. Dielectric and ferroelectric properties**

of about 0.22 mm2

itself in the second case.

C. The grey pattern represents the NBT-BT8 target spectrum, which cor‐

Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films

http://dx.doi.org/10.5772/ 52395

359

substrates at 700 0

ented film.

**Figure 6.** XRD spectra of NBT-BT6% deposited on Pt/TiO2/SiO2/Si substrate. The bottom pattern corresponds to the target

Figure 7 displays the XRD patterns of NBT-BT6 films deposited at two temperatures, 650 0 C and 700 0 C, around the (100)/(001) and (200)/(002) reflections of the Nb:STO substrate. The spectra indicate the epitaxial growth of NBT-BT6% films on the Nb:STO substrate at the two temperatures. This feature is congruent with the microstructure shown in the previous sec‐ tion (Fig. 2 and Fig. 3), consisting of large platelet-like crystallites which preserve the same axis of orientation with the monocrystalline substrate.

**Figure 7.** XRD spectra of NBT-BT6% /Nb:STONb:STO films deposited at 650 oC and 700 oC. The grey pattern represents the Nb:STONb:STO target reflection peaks.

Figure 8 shows the XRD patterns of NBT-BT8 films grown on Pt/TiO2/SiO2/Si and Nb:STO substrates at 700 0 C. The grey pattern represents the NBT-BT8 target spectrum, which cor‐ responds to the tetragonal P4mm symmetry. It can be observed that, similar to the previ‐ ous composition, the growth on single crystal Nb:STO substrate produces an epitaxial film, while the growth on Pt/TiO2/SiO2/Si substrate results into a polycrystalline randomly ori‐ ented film.

**Figure 8.** XRD spectra of NBT-BT8% deposited on Pt/TiO2/SiO2/Si and on Nb:STONb:STO substrates

### **3.3. Dielectric and ferroelectric properties**

**Figure 6.** XRD spectra of NBT-BT6% deposited on Pt/TiO2/SiO2/Si substrate. The bottom pattern corresponds to the

Figure 7 displays the XRD patterns of NBT-BT6 films deposited at two temperatures, 650 0

spectra indicate the epitaxial growth of NBT-BT6% films on the Nb:STO substrate at the two temperatures. This feature is congruent with the microstructure shown in the previous sec‐ tion (Fig. 2 and Fig. 3), consisting of large platelet-like crystallites which preserve the same

**Figure 7.** XRD spectra of NBT-BT6% /Nb:STONb:STO films deposited at 650 oC and 700 oC. The grey pattern represents

axis of orientation with the monocrystalline substrate.

the Nb:STONb:STO target reflection peaks.

C, around the (100)/(001) and (200)/(002) reflections of the Nb:STO substrate. The

C

target

and 700 0

358 Advances in Ferroelectrics

The dielectric and ferroelectric properties of NBT-BT thin films have been evaluated on ca‐ pacitors formed by evaporating through a mask an array of gold electrode dots with an area of about 0.22 mm2 on the surface of films grown on Pt/TiO2/SiO2/Si and Nb:STO substrates. The bottom electrode was formed by the Pt layer in the first case or by the Nb:STO substrate itself in the second case.

The piezoresponce force microscopy results are presented in Figure 9. The full-Pt tips were brought in contact with the surface of the sample and then a *dc* bias and test *ac* bias were applied between the tip and the bottom electrode of the samples. The *dc* bias was generated by a high voltage amplifier and the *ac* bias was generated by a lock-in amplifier. The same lock-in amplifier was used to analyse the vertical deflection signal from the PSPD, in order to extract the amplitude and the phase of the cantilever oscillations induced by the local de‐ formation of the sample due to the applied *dc* bias. The NBT-BT6/Pt/Si thin films show good switching behavior, the piezoelectric hysteresis and pronounced imprint (not showed here) confirming the piezoelectric and ferroelectric characteristics. The dependence of effective piezoelectric coefficient d33 eff on the applied electric field is given in Figure 9. The locally measured values with the highest being around d33 eff≈ 83 pm/V, are even higher then for previouslly reported values for pure NBT or lead-based thin films, such as Pb(ZrTi)O3 or PbTiO3 [21, 22]. However, these d33 eff are a bit smaller then NBT-BT6 ceramics which are re‐ ported to be more than 100 pm/V [21]. The reasons for these smaller values are related with the film's porosity, but also with the clamping effect which occurs because the PFM tip- ap‐ plied electric field will piezoelectrically deform only a small fraction of the film. The rest of the sample will restrict the relative deformation of this small fraction, resulting a lower val‐ ue for d33 eff [5, 22].

**Figure 10.** Room temperature dielectric constant ε' and loss tanδ variation with frequency for NBT-BT6% films depos‐

Figure 11 displays the room temperature dielectric constant and dielectric loss in the fre‐ quency range 100 Hz-1 MHz for NBT-BT8 films grown on Pt/TiO2/SiO2/Si at different tem‐

higher substrate temperatures was beneficial for the improvement of dielectric properties, at least in the frequency domain up to a few hundred kHz. Above this frequency there is a strong increase of dielectric loss. Since an increase is registered also in the dielectric con‐ stant, this could be caused by a relaxation mechanism which is active at room temperature

C. Unlike the previous composition, in this case growth at

Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films

http://dx.doi.org/10.5772/ 52395

361

ited at different temperatures on Pt/TiO2/SiO2/Si.

at these frequencies, like e.g. free charge relaxation.

peratures: 650, 700 and 730 0

**Figure 9.** The piezoresponse measurements performed on NBT-BT6 thin films.

In Fig. 10 the room temperature dielectric properties of NBT-BT6 films deposited on Pt/ TiO2/SiO2/Si at different substrate temperatures, 650 0 C and 730 0 C, have been compared in the frequency range 100 Hz-1 MHz. Films grown at 650 0 C show a higher dielectric constant (ε' ~ 1000), in the order of magnitude of the bulk values (ε'bulk ~ 1900), while films grown at 730 0 C show lower values (ε' ~ 700). The dielectric loss values are instead comparable in the two samples, and similar to bulk values.

formation of the sample due to the applied *dc* bias. The NBT-BT6/Pt/Si thin films show good switching behavior, the piezoelectric hysteresis and pronounced imprint (not showed here) confirming the piezoelectric and ferroelectric characteristics. The dependence of effective piezoelectric coefficient d33 eff on the applied electric field is given in Figure 9. The locally measured values with the highest being around d33 eff≈ 83 pm/V, are even higher then for previouslly reported values for pure NBT or lead-based thin films, such as Pb(ZrTi)O3 or PbTiO3 [21, 22]. However, these d33 eff are a bit smaller then NBT-BT6 ceramics which are re‐ ported to be more than 100 pm/V [21]. The reasons for these smaller values are related with the film's porosity, but also with the clamping effect which occurs because the PFM tip- ap‐ plied electric field will piezoelectrically deform only a small fraction of the film. The rest of the sample will restrict the relative deformation of this small fraction, resulting a lower val‐

ue for d33 eff [5, 22].

360 Advances in Ferroelectrics

730 0

**Figure 9.** The piezoresponse measurements performed on NBT-BT6 thin films.

TiO2/SiO2/Si at different substrate temperatures, 650 0

two samples, and similar to bulk values.

the frequency range 100 Hz-1 MHz. Films grown at 650 0

In Fig. 10 the room temperature dielectric properties of NBT-BT6 films deposited on Pt/

(ε' ~ 1000), in the order of magnitude of the bulk values (ε'bulk ~ 1900), while films grown at

C show lower values (ε' ~ 700). The dielectric loss values are instead comparable in the

C and 730 0

C, have been compared in

C show a higher dielectric constant

**Figure 10.** Room temperature dielectric constant ε' and loss tanδ variation with frequency for NBT-BT6% films depos‐ ited at different temperatures on Pt/TiO2/SiO2/Si.

Figure 11 displays the room temperature dielectric constant and dielectric loss in the fre‐ quency range 100 Hz-1 MHz for NBT-BT8 films grown on Pt/TiO2/SiO2/Si at different tem‐ peratures: 650, 700 and 730 0 C. Unlike the previous composition, in this case growth at higher substrate temperatures was beneficial for the improvement of dielectric properties, at least in the frequency domain up to a few hundred kHz. Above this frequency there is a strong increase of dielectric loss. Since an increase is registered also in the dielectric con‐ stant, this could be caused by a relaxation mechanism which is active at room temperature at these frequencies, like e.g. free charge relaxation.

here) evidenced good piezoelectric response, which indicates good intrinsic dielectric and

Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films

http://dx.doi.org/10.5772/ 52395

363

**Figure 12.** Polarization-electric field hysteresis loop measured on a NBT-BT6% film deposited on Pt/TiO2/SiO2/Si

Phase transitions in ferroelectric materials are accompanied by anomalies of complex dielec‐ tric permittivity variation with temperature, generally narrow peaks or steps, depending on the type of phase transformation. However NBT-BT compositions near the morphotropic phase boundary behave as relaxors, due to the cation disorder. This is evidenced in Fig. 13 for NBT-BT8 bulk material. The main characteristic of a relaxor ferroelectric is a broad dielectric peak at a temperature Tm which is not related to a structural transformation. This is due to a wide distribution of relaxation times which characterizes the dielectric re‐ sponse of polar nanoregions. This peak shifts with the increasing of the measurement fre‐ quency toward higher temperatures. Thus the dielectric maximum of NBT-BT8 shifts from about 497 K at 200 Hz to about 512 K at 100 kHz. A similar dependence is obeyed also by

In Fig. 14 the variation of dielectric constant and loss with temperature for a NBT-BT8 film deposited on Pt/TiO2/SiO2/Si is shown. The maximum of the dielectric constant occurs at about 485 K, not far from bulk Tm. However the anomaly is characteristic of a well-behaved phase transition, since the peak temperature Tm does not shift with frequency. A similar qualitative behavior was observed also on the dielectric permittivity variation with tempera‐

**3.4. Phase transitions**

the dielectric loss.

ture for NBT-BT6 films (not shown here).

ferroelectric properties, although quantitative values are difficult to extract.

**Figure 11.** Room temperature dielectric constant and loss variation with frequency for NBT-BT8% films deposited at different temperatures on Pt/TiO2/SiO2/Si

Polarization hysteresis measurements on NBT-BT6 films grown on Pt/TiO2/SiO2/Si are shown in Fig. 12. Spontaneous polarization was about 30 μC/cm2 and the remnant polariza‐ tion was about 10 μC/cm2 . The rather high value of coercive field (100 kV/cm) could be ex‐ plained by the presence of intrinsic strain and pinning defects.

Dielectric and ferroelectric properties measurements on films deposited on Nb:STO sub‐ strates have been less reliable, probably due to the presence of a non-ohmic contact at the NBT-BT film – semiconductor Nb:STO interface. However PFM measurements (not shown here) evidenced good piezoelectric response, which indicates good intrinsic dielectric and ferroelectric properties, although quantitative values are difficult to extract.

**Figure 12.** Polarization-electric field hysteresis loop measured on a NBT-BT6% film deposited on Pt/TiO2/SiO2/Si

### **3.4. Phase transitions**

**Figure 11.** Room temperature dielectric constant and loss variation with frequency for NBT-BT8% films deposited at

Polarization hysteresis measurements on NBT-BT6 films grown on Pt/TiO2/SiO2/Si are

Dielectric and ferroelectric properties measurements on films deposited on Nb:STO sub‐ strates have been less reliable, probably due to the presence of a non-ohmic contact at the NBT-BT film – semiconductor Nb:STO interface. However PFM measurements (not shown

. The rather high value of coercive field (100 kV/cm) could be ex‐

and the remnant polariza‐

shown in Fig. 12. Spontaneous polarization was about 30 μC/cm2

plained by the presence of intrinsic strain and pinning defects.

different temperatures on Pt/TiO2/SiO2/Si

tion was about 10 μC/cm2

362 Advances in Ferroelectrics

Phase transitions in ferroelectric materials are accompanied by anomalies of complex dielec‐ tric permittivity variation with temperature, generally narrow peaks or steps, depending on the type of phase transformation. However NBT-BT compositions near the morphotropic phase boundary behave as relaxors, due to the cation disorder. This is evidenced in Fig. 13 for NBT-BT8 bulk material. The main characteristic of a relaxor ferroelectric is a broad dielectric peak at a temperature Tm which is not related to a structural transformation. This is due to a wide distribution of relaxation times which characterizes the dielectric re‐ sponse of polar nanoregions. This peak shifts with the increasing of the measurement fre‐ quency toward higher temperatures. Thus the dielectric maximum of NBT-BT8 shifts from about 497 K at 200 Hz to about 512 K at 100 kHz. A similar dependence is obeyed also by the dielectric loss.

In Fig. 14 the variation of dielectric constant and loss with temperature for a NBT-BT8 film deposited on Pt/TiO2/SiO2/Si is shown. The maximum of the dielectric constant occurs at about 485 K, not far from bulk Tm. However the anomaly is characteristic of a well-behaved phase transition, since the peak temperature Tm does not shift with frequency. A similar qualitative behavior was observed also on the dielectric permittivity variation with tempera‐ ture for NBT-BT6 films (not shown here).

temperature of the peak intensity at Tm could be attributed to the non-polar layer contribu‐

Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films

http://dx.doi.org/10.5772/ 52395

365

The temperature Td where a strong increase of dielectric loss and dielectric constant occurs marks the ferroelectric – antiferroelectric phase transition which, in bulk samples with the same composition is visible only in the poled state. It is called also depolarization temperature.

**Figure 14.** The dielectric permittivity and loss variation with temperature measured on a NBT-BT8% thin film at differ‐

An apparent frequency dependence of the step increase in tanδ which marks Td is visible on the lower curves in Fig. 14. This could be attributed to a partial relaxor behavior due to mixed nanodomains-normal ferroelectric domains, which can be find also, but in proportion displaced to the nanodomain limit, in bulk samples. We must remark that the films were not

ent frequencies. The long arrows mark the increasing of frequency.

tion at higher temperatures due to conductivity variation.

**Figure 13.** The dielectric permittivity and loss variation with temperature measured on a NBT-BT8% bulk sample at different frequencies. The long arrows mark the increasing of frequency.

While the dielectric permittivity peak position Tm ~ 485 K does not shift with measuring fre‐ quency, peak height is strongly dependent on it, decreasing for higher frequencies. This can be attributed to the possible presence of a non-polar dielectric layer, which does not influ‐ ence the general behavior at phase transition, but can modify the value of the dielectric con‐ stant [23]. Generally these interface layers can have strong frequency-dependent dielectric properties which influence the overall properties of the heterostructures. However we stress again that the dependence on temperature and phase transition temperatures can be influ‐ enced only in the limits of a monotonous contribution, since the dielectric behaviour of nonpolar layers is free of temperature anomalies. Indeed the stronger variation with temperature of the peak intensity at Tm could be attributed to the non-polar layer contribu‐ tion at higher temperatures due to conductivity variation.

The temperature Td where a strong increase of dielectric loss and dielectric constant occurs marks the ferroelectric – antiferroelectric phase transition which, in bulk samples with the same composition is visible only in the poled state. It is called also depolarization temperature.

**Figure 14.** The dielectric permittivity and loss variation with temperature measured on a NBT-BT8% thin film at differ‐ ent frequencies. The long arrows mark the increasing of frequency.

**Figure 13.** The dielectric permittivity and loss variation with temperature measured on a NBT-BT8% bulk sample at

While the dielectric permittivity peak position Tm ~ 485 K does not shift with measuring fre‐ quency, peak height is strongly dependent on it, decreasing for higher frequencies. This can be attributed to the possible presence of a non-polar dielectric layer, which does not influ‐ ence the general behavior at phase transition, but can modify the value of the dielectric con‐ stant [23]. Generally these interface layers can have strong frequency-dependent dielectric properties which influence the overall properties of the heterostructures. However we stress again that the dependence on temperature and phase transition temperatures can be influ‐ enced only in the limits of a monotonous contribution, since the dielectric behaviour of nonpolar layers is free of temperature anomalies. Indeed the stronger variation with

different frequencies. The long arrows mark the increasing of frequency.

364 Advances in Ferroelectrics

An apparent frequency dependence of the step increase in tanδ which marks Td is visible on the lower curves in Fig. 14. This could be attributed to a partial relaxor behavior due to mixed nanodomains-normal ferroelectric domains, which can be find also, but in proportion displaced to the nanodomain limit, in bulk samples. We must remark that the films were not poled, since no bias electric field was applied. Therefore the occurrence of a ferroelectric ground state in the NBT-BT films, in striking contrast with relaxor bulk samples with the same composition, must be generated by some intrinsic differences between ceramic bulk samples and ceramic thin films. The first and most obvious reason could be related to the constraining stress of the substrate on the thin films. This is strong in epitaxially grown thin films with a thickness generally below 100 nm, but it should be almost absent in polycrystal‐ line films, randomly oriented and with a thickness of several hundreds of nm. This last one is the case for NBT-BT thin films deposited on Pt/TiO2/SiO2/Si substrates. The second reason could be related to the strong differences in the microstructures of ceramic thin films and bulk materials with the same composition. Therefore the occurrence of a ferroelectric ground state instead of a relaxor state in NBT-BT films, as well as the occurrence of a true ferroelectric phase transition could be due to the constrainig imposed by the nanograin boundaries on the ensemble of polar nanoregions.

**References**

*Lett*, 73, 3683.

Eng. *Mater*, 350, 93.

*Rev. B*, 82, 104112.

10, 929.

tallogr. *Sect. B: Struct. Sci*, 28, 3384.

[1] Saito, Y., Takao, H., Tani, T., Nonoyama, T., Takatori, K., Homma, T., Nagaya, T., &

Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films

http://dx.doi.org/10.5772/ 52395

367

[2] Smolenski, G. A., Isupov, V. A., Agranovskaya, A. I., & Krainik, N. N. (1961). New

[3] Chiang, Y. M., Farrey, G. W., & Soukhojak, A. N. (1998). Lead-free high-strain singlecrystal piezoelectrics in the alkaline-bismuth-titanate perovskite family. *Appl. Phys.*

[4] Takenaka, T., & Nagata, H. (1999). Present Status of Non-Lead-Based Piezoelectric‐

[5] Takenaka, T, Maruyama, K.-I., & Sakata, K. (1991). (Bi1/2Na1/2)TiO3-BaTiO3 System

[6] Jones, G. O., & Thomas, P. A. (2002). Investigation of the structure and phase transi‐ tions in the novel A-site substituted distorted perovskite compound Na0.5Bi0.5TiO3,

[7] Hiruma, Y., Watanabe, Y., Nagata, H., & Takenaka, T. (2007). Phase Transition Tem‐ peratures of Divalent and Trivalent Ions Substituted (Bi1/2Na1/2)TiO3 Ceramics, Key

[8] Glazer, A. M. (1972). The classification of tilted octahedra in perovskites, Acta Crys‐

[9] W.-van, B., Eerd, D., Damjanovic, N., Klein, N., Setter, , & Trodhal, J. (2010). Structur‐ al complexity of (Na0.5Bi0.5)TiOP3BaTiO3 as revealed by Raman spectroscopy. *Phys.*

[10] Ma, C., & Tan, X. (2010). Phase diagram of unpoled lead-free (1-x)(Bi/sub 1/2/Na/sub 1/2/)TiO/sub 3-x/BaTiO/sub 3/ ceramics. *Solid State Commun*, 150, 1497-1500.

[11] Cordero, F., Craciun, F., Trequattrini, F., Mercadelli, E., & Galassi, C. (2010). Phase transitions and phase diagram of the ferroelectric perovskite (Na0.5Bi0.5)1−xBaxTiO3

[12] Guo, Y. P., Akai, D. S., Sawada, K., & Ishida, M. (2008). Dielectric and ferroelectric properties of highly (100)-oriented (Na0.5Bi0.5)0.94Ba0.06TiO3 thin films grown on LaNiO3/γ-Al2O3/Si substrates by chemical solution deposition. *Solid State Sciences*,

[13] Guo, Y. P., Akai, D. S., Sawada, K., Ishida, M., & Gu, M. Y. (2009). Structure and elec‐ trical properties of trilayered BaTiO3/(Na0.5Bi0.5)TiO3BaTiO3/BaTiO3 thin films de‐

[14] Duclère, J.R., Cibert, C., Boulle, A., Dorcet, V., Marchet, P., Champeaux, C., Catheri‐ not, A., Députier, S., & Guilloux-Viry, M. (2008). Lead-free Na0.5Bi0.5TiO3 ferroelec‐

by anelastic and dielectric measurements. *Phys. Rev. B*, 81, 144124.

posited on Si substrate. *Solid State Communications*, 149, 14.

ferroelectrics of complex composition, Sov. *Phys. Solid State*, 2, 2651-196.

for Lead-Free Piezoelectric Ceramics. *Jpn. J. Appl. Phys*, 30, 2236-2239.

Nakamura, M. (2004). Lead-free piezoceramics. *Nature*, 432, 84.

Ceramics. *Key Engineering Materials*, 157/158, 57.

Acta Crystallogr. *Sect. B: Struct. Sci*, 58, 168 -178 .

### **4. Conclusions**

In summary, we have investigated the role of deposition temperature and substrate type as well as the amount of BT present in the target on crystalline structure, microstructure, die‐ lectric properties, phase transition temperatures and stability limits of ferroelectric phases in NBT-BT thin films grown by pulsed laser deposition. We have successfully deposited pure perovskite epitaxial films on single-crystal Nb:STO substrates. Successful growth of NBT-BT films on platinized silicon substrates has been achieved. Good dielectric and ferroelectric properties, comparable with bulk values, have been obtained. The NBT-BT6/Pt/Si thin films show a classic switching behavior, the piezoelectric hysteresis and pronounced imprint con‐ firming the piezoelectric and ferroelectric characteristics. The locally measured value of ef‐ fective piezoeletric coefficient d33 eff was around 83 pm/V, higher to the previouslly reported values for pure NBT or lead-based thin films. An enhanced stability of ferroelectric phase in thin films with respect to bulk has been observed and explained by their peculiar nanocrys‐ talline microstructure.

### **Author details**

N. D. Scarisoreanu1\*, R. Birjega1 , A. Andrei1 , M. Dinescu1 , F. Craciun2 and C. Galassi3

\*Address all correspondence to: snae@nipne.ro

1 NILPRP, National Institute for Laser, Plasma & Radiation Physics, Bucharest, Romania

2 CNR-ISC, Istituto dei Sistemi Complessi, Area della Ricerca Roma-Tor Vergata, Roma, Ita‐ ly

3 CNR-ISTEC, Istituto di Scienza e Tecnologia dei Materiali Ceramici, Faenza, Italy

### **References**

poled, since no bias electric field was applied. Therefore the occurrence of a ferroelectric ground state in the NBT-BT films, in striking contrast with relaxor bulk samples with the same composition, must be generated by some intrinsic differences between ceramic bulk samples and ceramic thin films. The first and most obvious reason could be related to the constraining stress of the substrate on the thin films. This is strong in epitaxially grown thin films with a thickness generally below 100 nm, but it should be almost absent in polycrystal‐ line films, randomly oriented and with a thickness of several hundreds of nm. This last one is the case for NBT-BT thin films deposited on Pt/TiO2/SiO2/Si substrates. The second reason could be related to the strong differences in the microstructures of ceramic thin films and bulk materials with the same composition. Therefore the occurrence of a ferroelectric ground state instead of a relaxor state in NBT-BT films, as well as the occurrence of a true ferroelectric phase transition could be due to the constrainig imposed by the nanograin

In summary, we have investigated the role of deposition temperature and substrate type as well as the amount of BT present in the target on crystalline structure, microstructure, die‐ lectric properties, phase transition temperatures and stability limits of ferroelectric phases in NBT-BT thin films grown by pulsed laser deposition. We have successfully deposited pure perovskite epitaxial films on single-crystal Nb:STO substrates. Successful growth of NBT-BT films on platinized silicon substrates has been achieved. Good dielectric and ferroelectric properties, comparable with bulk values, have been obtained. The NBT-BT6/Pt/Si thin films show a classic switching behavior, the piezoelectric hysteresis and pronounced imprint con‐ firming the piezoelectric and ferroelectric characteristics. The locally measured value of ef‐ fective piezoeletric coefficient d33 eff was around 83 pm/V, higher to the previouslly reported values for pure NBT or lead-based thin films. An enhanced stability of ferroelectric phase in thin films with respect to bulk has been observed and explained by their peculiar nanocrys‐

, A. Andrei1

, M. Dinescu1

1 NILPRP, National Institute for Laser, Plasma & Radiation Physics, Bucharest, Romania

3 CNR-ISTEC, Istituto di Scienza e Tecnologia dei Materiali Ceramici, Faenza, Italy

2 CNR-ISC, Istituto dei Sistemi Complessi, Area della Ricerca Roma-Tor Vergata, Roma, Ita‐

, F. Craciun2

and C. Galassi3

boundaries on the ensemble of polar nanoregions.

**4. Conclusions**

366 Advances in Ferroelectrics

talline microstructure.

N. D. Scarisoreanu1\*, R. Birjega1

\*Address all correspondence to: snae@nipne.ro

**Author details**

ly


tric thin films grown by Pulsed Laser Deposition on epitaxial platinum bottom electrodes. *Thin Solid Films*, 517, 592.

**Chapter 17**

**Thin-Film Process Technology**

Additional information is available at the end of the chapter

Recently thin-film ferroelectrics such as Pb(Zr, Ti)O3 (PZT) and (Ba, Sr)TiO3 (BST) have been utilized to form advanced semiconductor and electronic devices including Ferroelectric Ran‐ dom Access memory(FeRAM), actuators composing gyro meters, portable camera modules, and tunable devices for smart phone applications and so on. Processing technology of ferro‐ electric materials is one of the most important technologies to enable the above- mentioned

In this paper, we will report our development results of ferroelectric thin film processing tech‐ nologies including sputtering, MOCVD and plasma etching as well as manufacturing process‐ es for FeRAM, MEMS(actuators, tunable devices) and ultra-high density probe memory.

Ferroelectric Random Access Memory (FeRAM) is a main candidate of next generation nonvolatile random access memory. As its density has been continuously increasing, its applica‐ tions have spread from RFID card to Nonvolatile latch circuit with low power consumption. FeRAM has a ferroelectric thin films sandwiched by rare metal electrodes and we have estab‐ lished mass production capability on integrated FeRAM solution, consisting of electrode film deposition, PZT thin-film deposition, etching/ashing, anneal and passivation deposition.

We have been developing mass production technology for perovskite oxide thin-film such as PZT for a long period [1-5], since we consider these materials as the most promising can‐

and reproduction in any medium, provided the original work is properly cited.

© 2013 Suu; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

**2.1. Mass productive sputtering technology for perovskite oxide thin-film**

**for Ferroelectric Application**

Koukou Suu

**1. Introduction**

http://dx.doi.org/10.5772/54360

advanced devices and their productions.

**2. FeRAM technology**


**Chapter 17**

### **Thin-Film Process Technology for Ferroelectric Application**

Koukou Suu

tric thin films grown by Pulsed Laser Deposition on epitaxial platinum bottom

[15] Bousquet, M., , J., Ducle`re, R., Gautier, B., Boulle, A., Wu, A., De´, S., putier, D., Fas‐ quelle, F., Re´mondie`, re. D., Albertini, C., Champeaux, P., Marchet, M., Guilloux- , Viry, & Vilarinho, P. (2012). Electrical properties of (110) epitaxial lead-free ferroelec‐ tric Na0.5Bi0.5TiO3 thin films grown by pulsed laser deposition: Macroscopic and

[16] Qin, W., Guo, Y., Guo, B., & Gu, M. (2012). Dielectric and optical properties of Bi‐ FeO3-(Na0.5Bi0.5)TiO3 thin films deposited on Si substrate using LaNiO3 as buffer

[17] Shaw, T. M., Trolier Mc Kinstry, S., & Mc Intyre, P. C. (2000). The properties of ferro‐

[18] Scarisoreanu, N., Craciun, F., Chis, A., Birjega, R., Moldovan, A., Galassi, C., & Di‐ nescu, M. (2010). Lead-free ferroelectric thin films obtained by pulsed laser deposi‐

[19] Scarisoreanu, N., Craciun, F., Ion, V., Birjega, S., & Dinescu, M. (2007). Structural and electrical characterization of lead-free ferroelectric Na/sub 1/2/Bi/sub 1/2/TiO/sub 3/-

BaTiO/sub 3/ thin films obtained by PLD and RF-PLD. *Appl. Surf. Sci*, 254, 1292.

[20] Picht, G., Töpfer, J., & Henning, E. (2010). Structural properties of (Bi0.5Na0.5)1−xBaxTiO3 lead-free piezoelectric ceramics. *J. Eur. Ceram. Soc*, 30, 3445.

[21] Takenaka, T. (1999). Piezoelectric Properties of Some Lead-Free Ferroelectric Ceram‐

[22] Morelli, A., Sriram, Venkatesan, Palasantzas, , Palasantzas, Kooi, G, & De Hosson, J. Th. M. (2009). Piezoelectric properties of PbTiO3 thin films characterized with pie‐ zoresponse force and high resolution transmission electron microscopy. *J. Appl. Phys*,

[23] Craciun, F., & Dinescu, M. (2007). Piezoelectrics", in "Pulsed laser deposition of thin films: applications-led growth of functional materials". *edited by R. Eason (Wiley-Inter‐*

*science, John Wiley & Sons, Hoboken, New Jersey, U.S.A.,)*, 487-532.

electrodes. *Thin Solid Films*, 517, 592.

368 Advances in Ferroelectrics

nanoscale data. *J. Appl. Phys*, 111, 104 -106.

tion. *Appl. Phys A*, 101, 747 -751 .

ics. *Ferroelectrics*, 230, 87-98.

105, 064106.

layer for photovoltaic devices. *J. Alloys Compd*, 513, 154-158.

electric thin films at small dimensions. *Annu. Rev. Mater. Sci*, 30, 263.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54360

### **1. Introduction**

Recently thin-film ferroelectrics such as Pb(Zr, Ti)O3 (PZT) and (Ba, Sr)TiO3 (BST) have been utilized to form advanced semiconductor and electronic devices including Ferroelectric Ran‐ dom Access memory(FeRAM), actuators composing gyro meters, portable camera modules, and tunable devices for smart phone applications and so on. Processing technology of ferro‐ electric materials is one of the most important technologies to enable the above- mentioned advanced devices and their productions.

In this paper, we will report our development results of ferroelectric thin film processing tech‐ nologies including sputtering, MOCVD and plasma etching as well as manufacturing process‐ es for FeRAM, MEMS(actuators, tunable devices) and ultra-high density probe memory.

### **2. FeRAM technology**

Ferroelectric Random Access Memory (FeRAM) is a main candidate of next generation nonvolatile random access memory. As its density has been continuously increasing, its applica‐ tions have spread from RFID card to Nonvolatile latch circuit with low power consumption. FeRAM has a ferroelectric thin films sandwiched by rare metal electrodes and we have estab‐ lished mass production capability on integrated FeRAM solution, consisting of electrode film deposition, PZT thin-film deposition, etching/ashing, anneal and passivation deposition.

### **2.1. Mass productive sputtering technology for perovskite oxide thin-film**

We have been developing mass production technology for perovskite oxide thin-film such as PZT for a long period [1-5], since we consider these materials as the most promising can‐

© 2013 Suu; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

didate for ferroelectric material used in FeRAMs (Ferroelectric Random Access Memories) as a non-volatile memory device, piezoelectric MEMS (Micro Electro Mechanical System) devices due to its longer period of research, existence of actual production, manufacturing capability within the tolerable temperature range of general Si LSI technology.

producing electric factors (e.g. dielectric breakdown due to charge-up) and made the meas‐

We adapted the RF magnetron sputtering method for perovskite oxide thin-film sputtering. As for the PZT thin-films, sputtering methods included high-temperature deposition, where film deposition was made at substrate temperatures above 500°C [6, 7], and low tempera‐ ture deposition, where the films were deposited at room temperature and then crystallized by post-annealing process. The improvement in high-speed deposition, film composition

If the ceramic target with inferior thermal conductivity is used, application of high power leads to the destruction of the target. By adopting the backing plate with high cooling effi‐ ciency and high-density target, higher deposition rate by an increase in sputtering power is

The film composition control can be translated into the volatile element (e.g. Lead) control within film. Various factors that influence the volatile element within film are thought to ex‐ ist (Fig. 1) and we have investigated the influence of sputtering conditions, strength of mag‐

Zr O Pb O Pb O Zr Pb O Pb Zr

Ti Ti

Pb

Pb Zr

B. Controlling

**Figure 1.** Various factors that influence content with PZT Film. **Figure 1.** Various factors that influence content with PZT Film.

(Substrate temp., Plasma radiation)

Substrate Temperature

Pb O Incident energy (sputtering power) Number of Pb (or PbO)(Target surface) Binding energy (substrate condition etc.)

Pb re-sputtering

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

371

Substrate potential

Target Content

(Substrate temp.,Plasma radiation)

(or )

Pb

Thermal energy (Plasma radiation)

Ar+

Ti Ti Ti Ti

Pb

Magnetic field

**2.2. Optimization of sputtering processes for perovskite oxide thin-film**

control, and stability of sputtering processes is described in the following.

ures difficult to obtain.

achieved.

netic field and electric potential of substrate.

Shields etc.

Large-sized Shields

A. Introducing

Zr O Pb

Pb re-sputtering (Plasma radiation)

Ar Pressure

Ar+

The sputtering was selected for mass production technology of perovskite oxide thinfilms owing to the following factors: (1) Good compatibility with conventional Si LSI processes. (2) Superb controllability of film quality (e.g. film composition), which enabled relatively easy thin film deposition. (3) Better possibilities of obtaining uniform surfaces in large diameter substrates (e.g. 6-8 inch). (4) Sputtering was plasma processing which was promising for deposition and heat treatment at low temperature. (5) Feasibility of highspeed deposition. (6) Same deposition method as electrodes (Pt, Ir, Ru, etc.), which will facilitate in-situ integration. (7) Present difficulties and lack of future potentialities in oth‐ er technologies.

With emphasis on both mass production capability and advanced process capacity of perovskite oxide thin-film sputtering, we consider the following factors as important for development:

**1.** Throughput

Compared to the other processes (e.g. electrode deposition), the deposition speed of perov‐ skite oxide material sputtering was considerably slower and thereby limited the throughput. Though two ways for improving the throughput, high-speed deposition and thinner film deposition, are considered, the former is more promising for the improvement of through‐ put, while the latter is apt to cause deterioration of film characteristics.

**2.** Control of film composition

Film composition determines other film qualities (crystal structure, electric characteristics). Since volatile elements were included in the perovskite oxide materials, they were sensitive and easily fluctuated according to temperature or plasma status. Film composition control is the fundamental factor in this process.

**3.** Uniformity over large diameter substrates

Large diameter substrates up to 8 inch were expected to be used for mass production of per‐ ovskite oxide thin-film. Film thickness and uniformity of film quality were the keys to mass production.

**4.** Process stability / reliability

Under the circumstances where there were many unknown factors such as new materials, ceramic targets, insulator sputtering, etc., process stability / reliability was the more perti‐ nent factor. In fact, there was a problem in film composition that changed over time.

**5.** Prevention of particles

While the characteristics, such as ceramic targets or insulating thin film, increased the me‐ chanical factors (e.g. adherence, thermal expansion) for particle occurrence, they were also producing electric factors (e.g. dielectric breakdown due to charge-up) and made the meas‐ ures difficult to obtain.

### **2.2. Optimization of sputtering processes for perovskite oxide thin-film**

didate for ferroelectric material used in FeRAMs (Ferroelectric Random Access Memories) as a non-volatile memory device, piezoelectric MEMS (Micro Electro Mechanical System) devices due to its longer period of research, existence of actual production, manufacturing

The sputtering was selected for mass production technology of perovskite oxide thinfilms owing to the following factors: (1) Good compatibility with conventional Si LSI processes. (2) Superb controllability of film quality (e.g. film composition), which enabled relatively easy thin film deposition. (3) Better possibilities of obtaining uniform surfaces in large diameter substrates (e.g. 6-8 inch). (4) Sputtering was plasma processing which was promising for deposition and heat treatment at low temperature. (5) Feasibility of highspeed deposition. (6) Same deposition method as electrodes (Pt, Ir, Ru, etc.), which will facilitate in-situ integration. (7) Present difficulties and lack of future potentialities in oth‐

With emphasis on both mass production capability and advanced process capacity of perovskite oxide thin-film sputtering, we consider the following factors as important for development:

Compared to the other processes (e.g. electrode deposition), the deposition speed of perov‐ skite oxide material sputtering was considerably slower and thereby limited the throughput. Though two ways for improving the throughput, high-speed deposition and thinner film deposition, are considered, the former is more promising for the improvement of through‐

Film composition determines other film qualities (crystal structure, electric characteristics). Since volatile elements were included in the perovskite oxide materials, they were sensitive and easily fluctuated according to temperature or plasma status. Film composition control is

Large diameter substrates up to 8 inch were expected to be used for mass production of per‐ ovskite oxide thin-film. Film thickness and uniformity of film quality were the keys to mass

Under the circumstances where there were many unknown factors such as new materials, ceramic targets, insulator sputtering, etc., process stability / reliability was the more perti‐

While the characteristics, such as ceramic targets or insulating thin film, increased the me‐ chanical factors (e.g. adherence, thermal expansion) for particle occurrence, they were also

nent factor. In fact, there was a problem in film composition that changed over time.

put, while the latter is apt to cause deterioration of film characteristics.

capability within the tolerable temperature range of general Si LSI technology.

er technologies.

370 Advances in Ferroelectrics

**1.** Throughput

production.

**2.** Control of film composition

**4.** Process stability / reliability

**5.** Prevention of particles

the fundamental factor in this process.

**3.** Uniformity over large diameter substrates

We adapted the RF magnetron sputtering method for perovskite oxide thin-film sputtering. As for the PZT thin-films, sputtering methods included high-temperature deposition, where film deposition was made at substrate temperatures above 500°C [6, 7], and low tempera‐ ture deposition, where the films were deposited at room temperature and then crystallized by post-annealing process. The improvement in high-speed deposition, film composition control, and stability of sputtering processes is described in the following.

If the ceramic target with inferior thermal conductivity is used, application of high power leads to the destruction of the target. By adopting the backing plate with high cooling effi‐ ciency and high-density target, higher deposition rate by an increase in sputtering power is achieved.

The film composition control can be translated into the volatile element (e.g. Lead) control within film. Various factors that influence the volatile element within film are thought to ex‐ ist (Fig. 1) and we have investigated the influence of sputtering conditions, strength of mag‐ netic field and electric potential of substrate.

**Figure 1.** Various factors that influence content with PZT Film. **Figure 1.** Various factors that influence content with PZT Film.

In addition to the known characteristics in perovskite oxide thin-film sputtering that volatile elements easily fluctuate, it was confirmed that volatile elements within film were unstable even when the deposition was performed under identical conditions. The conceivable caus‐ es for that phenomenon were the instability of the target, fluctuation of temperature in the sputtering chamber, and variation of plasma status over time.

Main problem in perovskite oxide thin-film is the change in the volatile element content with the passage of sputtering time. As for the PZT, continuous sputtering of 2.0 kWh showed approximately a 30% decline in lead content compared to that at the beginning. It was determined that the reason for this problem was the change in plasma status. As shown in Fig. 2, when insulating PZT film adhered to the shields of the ground potential, charge-up occurred, the impedance of the system changed, plasma was pushed to the center of the chamber, exposure to plasma was enhanced, and as a consequence, Pb content within film is reduced. In order to stabilize the status of plasma, we installed a stable anode, that is, an anode that avoided charge-up due to the adhesion of insulating PZT film and maintained the role as an anode. Consequently, as can be seen in Fig. 3, stability of Pb content within film in continuous sputtering has been confirmed.

**Figure 3.** Stable transition of Pb content within film in continuous sputtering.

**2.3. Sputtered PZT thin-films for non-volatile memory application**

maintenance, short exhausting time, short down time, etc.

In this section, we introduce one of its achievements, the ferroelectric characteristics of PLZT capacitors for FeRAM. We used the multi chamber type mass production sputtering system equipped with an exclusive sputtering module for ferroelectric materials, CERAUS ZX-1000 from ULVAC. This system has the following features in addition to features such as easy

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

373

**Figure 4.** Uniformity in 8 inch area.

**Figure 2.** Change in plasma status which is responsible for Pb content variation.

Fig. 4 shows an example of thickness and Pb content uniformity in a PZT film on 8" sub‐ strate. Both thickness and Pb content uniformity varied according to the deposition condi‐ tion and was minimized around the sputtering pressure of 1.0 Pa. Thickness and Pb content uniformity represent good result as low as ±1.9% and ±1.1%, respectively. These uniformi‐ ties were also confirmed as stable, thereby satisfactorily meeting the requirements of mass production.

**Figure 3.** Stable transition of Pb content within film in continuous sputtering.

**Figure 4.** Uniformity in 8 inch area.

In addition to the known characteristics in perovskite oxide thin-film sputtering that volatile elements easily fluctuate, it was confirmed that volatile elements within film were unstable even when the deposition was performed under identical conditions. The conceivable caus‐ es for that phenomenon were the instability of the target, fluctuation of temperature in the

Main problem in perovskite oxide thin-film is the change in the volatile element content with the passage of sputtering time. As for the PZT, continuous sputtering of 2.0 kWh showed approximately a 30% decline in lead content compared to that at the beginning. It was determined that the reason for this problem was the change in plasma status. As shown in Fig. 2, when insulating PZT film adhered to the shields of the ground potential, charge-up occurred, the impedance of the system changed, plasma was pushed to the center of the chamber, exposure to plasma was enhanced, and as a consequence, Pb content within film is reduced. In order to stabilize the status of plasma, we installed a stable anode, that is, an anode that avoided charge-up due to the adhesion of insulating PZT film and maintained the role as an anode. Consequently, as can be seen in Fig. 3, stability of Pb content within

sputtering chamber, and variation of plasma status over time.

372 Advances in Ferroelectrics

film in continuous sputtering has been confirmed.

**Figure 2.** Change in plasma status which is responsible for Pb content variation.

production.

Fig. 4 shows an example of thickness and Pb content uniformity in a PZT film on 8" sub‐ strate. Both thickness and Pb content uniformity varied according to the deposition condi‐ tion and was minimized around the sputtering pressure of 1.0 Pa. Thickness and Pb content uniformity represent good result as low as ±1.9% and ±1.1%, respectively. These uniformi‐ ties were also confirmed as stable, thereby satisfactorily meeting the requirements of mass

### **2.3. Sputtered PZT thin-films for non-volatile memory application**

In this section, we introduce one of its achievements, the ferroelectric characteristics of PLZT capacitors for FeRAM. We used the multi chamber type mass production sputtering system equipped with an exclusive sputtering module for ferroelectric materials, CERAUS ZX-1000 from ULVAC. This system has the following features in addition to features such as easy maintenance, short exhausting time, short down time, etc.


Fig. 5 shows the transition of switching charge (QSW) and saturation characteristics of a Pt/ PZT(200 nm)/Pt capacitor measured at 5 V. QSW with 5 V applied was approximately 34 μC/cm2 and saturation voltage of 90% (V90%) was 3.1-3.2 V. The composition of PLZT film, which contained added Ca and Sr for the improvement of retention and imprint characteris‐ tics, was excellent.

**Figure 6.** Stability of Qsw in continuous sputtering of 1000 substrates.

**Figure 7.** Transition of switching charge of IrOx/PZT(80 nm)/Pt capacitor.

Further improvement of ferroelectric performancs is needed because scaled thinner capaci‐ tor (sub 100 nm) is demanded for next generation. Pb deficient surface layer is confirmed to be responsible for degradation of ferroelectric performance. Bottom electrode and PZT dep‐ osition process were modified to further improve the ferroelectric performance and achieve

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

375

**Figure 5.** Transition of switching charge of Pt/PZT(200 nm)/Pt capacitor.

Fig. 6 shows the stability of QSW in continuous sputtering of 1000 substrates. The result showed high stability sufficient for mass production. Also, reference data was the data be‐ fore chamber cleaning. The fact that it was equivalent to the data after chamber cleaning demonstrates the reliability of the system process.

**Figure 6.** Stability of Qsw in continuous sputtering of 1000 substrates.

**1.** Can mount 300 mm (12 inch) in diameter targets and process large diameter substrates of 200 mm. At present, deposition of PZT thin-films on 6 and 8 inch substrates are per‐

**2.** Including the heat chamber, this system has five process chambers, thereby achieving high flexibility. The system is presently executing the following in-situ processes as standard: pre-heating of substrate → substrate sputtering (e.g. Ti, TiN, Pt) → ferroelec‐

**3.** As a substrate heating mechanism, this system was capable of precise and rapid heating in a wide range from low to high temperatures, with the aid of an electrostatic chuck

Fig. 5 shows the transition of switching charge (QSW) and saturation characteristics of a Pt/ PZT(200 nm)/Pt capacitor measured at 5 V. QSW with 5 V applied was approximately 34

which contained added Ca and Sr for the improvement of retention and imprint characteris‐

Fig. 6 shows the stability of QSW in continuous sputtering of 1000 substrates. The result showed high stability sufficient for mass production. Also, reference data was the data be‐ fore chamber cleaning. The fact that it was equivalent to the data after chamber cleaning

and saturation voltage of 90% (V90%) was 3.1-3.2 V. The composition of PLZT film,

**4.** This system used RF sputtering for ferroelectric deposition and counters RF noises.

formed using 12 inch single ceramic target.

type hot plate, in addition to lamp heating.

**Figure 5.** Transition of switching charge of Pt/PZT(200 nm)/Pt capacitor.

demonstrates the reliability of the system process.

tric material sputtering.

μC/cm2

tics, was excellent.

374 Advances in Ferroelectrics

**Figure 7.** Transition of switching charge of IrOx/PZT(80 nm)/Pt capacitor.

Further improvement of ferroelectric performancs is needed because scaled thinner capaci‐ tor (sub 100 nm) is demanded for next generation. Pb deficient surface layer is confirmed to be responsible for degradation of ferroelectric performance. Bottom electrode and PZT dep‐ osition process were modified to further improve the ferroelectric performance and achieve thinner capacitor with good performances. [8]. Fig. 7 shows the QSW transition curve of a Ir‐ Ox/PZT(80 nm)/Pt capacitor measured at 3 V. QSW with 1.8 V applied was approximately 24 μC/cm2 and V90% was 1.4 V. Fig. 8 also shows the fatigue characteristics of voltage applica‐ tion at 2 V. QSW was not decreased even after switching in 109 cycles.

First, the features of MOCVD are briefly explained. In the method of MOCVD, raw material that was changed into organic metal with high vapor pressure is led to a substrate, and is thermally reacted with a reactant gas (such as oxygen) on the substrate to form a deposition. Methods for the gasification of organic metals are classified into two groups. One of them is the sublimation of solid raw materials (sublimation method). After a certain vapor pressure is obtained by heating a solid raw material using a heater, it is transported with carrier gas. In the second method, a liquid raw material or solid raw material is melted in an organic solvent. Then, after the solution is vaporized, it is transported with carrier gas (vaporization method). In the case of the former, the deposition rate is small. In the case of the latter, there is a problem of the instability of vaporization. For the target of mass-production system de‐ velopment, we adopted the latter vaporization method from the beginning of the develop‐

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

377

**1.** Vaporization and transport of multi-element raw materials In MOCVD using the vapor‐ ization of a ferroelectric solution, because a multi-element organic metal material is va‐ porized and transported, stable vaporization is difficult, and there is a possibility that

**2.** Mixture of gases whose molecular weights are mutually different It is difficult to uni‐ formly mix a reactant gas such as oxygen and an organic raw material whose molecular

**3.** Uniformity of large-diameter substrates For the film thickness and the composition of a multi-element oxide thin film PZT, in-plane uniformity in 8 inches is required.

precipitation or decomposition may occur in a pipe during transport.

ment. The problems that need to be resolved are as follows (see Fig. 9):

**Figure 9.** Features of MOCVD Module

weights are vastly different.

**Figure 8.** Fatigue characteristics of IrOx/PZT(80 nm)/Pt capacitor.

From above results, sputter-derived PZT capacitor was proved to be suitable for 0.18 μm technology node.

### **2.4. MOCVD technology for non-volatile memory application**

For next-generation FeRAMs beyond 0.18 μm, because it is necessary to achieve further larg‐ er packing densities and integration with logic devices, thinner films of ferroelectric associ‐ ated with shrinking of the thickness of ferroelectric capacitors and the three-dimensional structures associated with shrinking of the capacitor areas are demanded. In addition, it is necessary to achieve thinner films of ferroelectric in parallel with its higher quality in order to meet the demand of still lower voltage drives for device performance. It is said that MOCVD (Metal organic chemical vapor deposition) technology can meet these demands. While depositions are physically made by plasma collision in sputtering processes, deposi‐ tions are made on heated substrates by chemical reaction among source gases in MOCVD. Therefore, denser crystalline films are easily obtainable, and it is possible to achieve thinner films and higher quality. In addition, uniform deposition can be obtained also on three-di‐ mensional structure, and it is considered that good step coverage can be obtained. Because of these features, MOCVD is the prime candidate of deposition technologies for next-genera‐ tion FeRAMs in mass production, and its development is making progress.

First, the features of MOCVD are briefly explained. In the method of MOCVD, raw material that was changed into organic metal with high vapor pressure is led to a substrate, and is thermally reacted with a reactant gas (such as oxygen) on the substrate to form a deposition. Methods for the gasification of organic metals are classified into two groups. One of them is the sublimation of solid raw materials (sublimation method). After a certain vapor pressure is obtained by heating a solid raw material using a heater, it is transported with carrier gas. In the second method, a liquid raw material or solid raw material is melted in an organic solvent. Then, after the solution is vaporized, it is transported with carrier gas (vaporization method). In the case of the former, the deposition rate is small. In the case of the latter, there is a problem of the instability of vaporization. For the target of mass-production system de‐ velopment, we adopted the latter vaporization method from the beginning of the develop‐ ment. The problems that need to be resolved are as follows (see Fig. 9):

**Figure 9.** Features of MOCVD Module

thinner capacitor with good performances. [8]. Fig. 7 shows the QSW transition curve of a Ir‐ Ox/PZT(80 nm)/Pt capacitor measured at 3 V. QSW with 1.8 V applied was approximately 24

From above results, sputter-derived PZT capacitor was proved to be suitable for 0.18 μm

For next-generation FeRAMs beyond 0.18 μm, because it is necessary to achieve further larg‐ er packing densities and integration with logic devices, thinner films of ferroelectric associ‐ ated with shrinking of the thickness of ferroelectric capacitors and the three-dimensional structures associated with shrinking of the capacitor areas are demanded. In addition, it is necessary to achieve thinner films of ferroelectric in parallel with its higher quality in order to meet the demand of still lower voltage drives for device performance. It is said that MOCVD (Metal organic chemical vapor deposition) technology can meet these demands. While depositions are physically made by plasma collision in sputtering processes, deposi‐ tions are made on heated substrates by chemical reaction among source gases in MOCVD. Therefore, denser crystalline films are easily obtainable, and it is possible to achieve thinner films and higher quality. In addition, uniform deposition can be obtained also on three-di‐ mensional structure, and it is considered that good step coverage can be obtained. Because of these features, MOCVD is the prime candidate of deposition technologies for next-genera‐

tion at 2 V. QSW was not decreased even after switching in 109

**Figure 8.** Fatigue characteristics of IrOx/PZT(80 nm)/Pt capacitor.

**2.4. MOCVD technology for non-volatile memory application**

tion FeRAMs in mass production, and its development is making progress.

technology node.

and V90% was 1.4 V. Fig. 8 also shows the fatigue characteristics of voltage applica‐

cycles.

μC/cm2

376 Advances in Ferroelectrics


**4.** PTZ film properties Because of high-temperature processes and in-situ crystallization in the processes, the correlation between the kind of a raw material and a process is sensi‐ tive, and the control of a film composition and crystalline orientation is difficult. There‐ fore, necessary performance of films is difficult to obtain.

In order to establish MOCVD mass-production technology, it is absolutely necessary to re‐ solve the above problems in parallel, and advanced vaporization technology, mixing tech‐ nology, and reaction control technology are simultaneously required. We combined ferroelectric deposition technology for FeRAMs, module design technology, and CVD equipment technology, which were provided to users until now, and completed full-fledged MOCVD equipment for mass production. The major features include the following four points:

**1.** The reproducibility in the continuous operation of vaporization and the gas transport to a substrate that controls condensation and decomposition were achieved by accurate temperature control for each part of equipment and the optimization of vaporization conditions. Consequently, film thicknesses within ±2% and the reproducibility of the PZT composition were kept with no mechanical maintenance, and a running test for 1,000 substrates was successful as shown in Fig. 10. [9]

**Figure 11.** In-plane uniformity of PZT film thickness

**Figure 12.** In-plane uniformity of PZT composition

**3.** Because of the development of a new heater, including the optimization of the shape of the heater, the temperature distribution of 8 inch substrates can be continually control‐ led within ±3ºC. Consequently, in-plane uniformity within ±3% was achieved for both

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

379

film thicknesses and the composition as shown in Fig. 12. [10]

**Figure 10.** Reproducibility in MOCVD system (film thickness and PZT composition)

**2.** Because the design of the equipment was conducted in consideration of the flow of gas‐ es and the mixture between a raw material gas and a reactant gas on the basis of simu‐ lations, film thickness distribution within ±3% on 8 inch substrates was achieved as shown in Fig. 11.

**Figure 11.** In-plane uniformity of PZT film thickness

**4.** PTZ film properties Because of high-temperature processes and in-situ crystallization in the processes, the correlation between the kind of a raw material and a process is sensi‐ tive, and the control of a film composition and crystalline orientation is difficult. There‐

In order to establish MOCVD mass-production technology, it is absolutely necessary to re‐ solve the above problems in parallel, and advanced vaporization technology, mixing tech‐ nology, and reaction control technology are simultaneously required. We combined ferroelectric deposition technology for FeRAMs, module design technology, and CVD equipment technology, which were provided to users until now, and completed full-fledged MOCVD equipment for mass production. The major features include the following four

**1.** The reproducibility in the continuous operation of vaporization and the gas transport to a substrate that controls condensation and decomposition were achieved by accurate temperature control for each part of equipment and the optimization of vaporization conditions. Consequently, film thicknesses within ±2% and the reproducibility of the PZT composition were kept with no mechanical maintenance, and a running test for

fore, necessary performance of films is difficult to obtain.

1,000 substrates was successful as shown in Fig. 10. [9]

**Figure 10.** Reproducibility in MOCVD system (film thickness and PZT composition)

shown in Fig. 11.

**2.** Because the design of the equipment was conducted in consideration of the flow of gas‐ es and the mixture between a raw material gas and a reactant gas on the basis of simu‐ lations, film thickness distribution within ±3% on 8 inch substrates was achieved as

points:

378 Advances in Ferroelectrics

**3.** Because of the development of a new heater, including the optimization of the shape of the heater, the temperature distribution of 8 inch substrates can be continually control‐ led within ±3ºC. Consequently, in-plane uniformity within ±3% was achieved for both film thicknesses and the composition as shown in Fig. 12. [10]

**Figure 12.** In-plane uniformity of PZT composition

**4.** By taking advantage of the progress of the equipment hardware as described above, the optimization of processes was conducted, and PZT thin films can be controlled to en‐ sure preferential orientation in the <111> direction as shown in Fig. 13. Consequently, the formation of PZT films within 100 nm that have capacitor properties with a 1.5 V low voltage drive is achieved as shown in Fig. 14. [11,12] In addition excellent endur‐ ance properties which are over 1010cycles were obtained for 73nm-PZT in Fig 15.

**Figure 15.** Endurance properties of MOCVD-PZT films

*2.5.1. Issues of ferroelectric etching technology*

**2.5. Etching technology**

also MEMS productions:

Recently mass production tool for 300mm Si wafer was developed and excellent in-plane

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

381

Conventionally, the piezoelectric elements have been fabricated by chemical wet etching [13] or argon ion milling. With the miniaturization of MEMS, there have been increasing de‐ mands for dry etching with the excellent shape controllability as semiconductor technology. Recently dry etching technique for MEMS using PZT was reported. [14,15] The Pt, Ir and other rare metal electrodes and the PZT ferroelectric thin films that compose piezoelectric elements react poorly with halogen gases and their halides have low vapor pressures. For these reasons, these materials are called hard-to-etch materials. The following technical is‐ sues are important for dry etching of the PZT ferroelectric thin films for not only FeRAM but

A piezoelectric element film consists of PZT with a thickness of several micrometers and the rare metal electrodes with a thickness of about 100 nm. Generally, the bottom electrode is left after the PZT etching. Therefore, a low etching rate for the bottom electrode, the so-

The materials are hard to etch, and their etching products easily adhere to the pattern side‐ walls, and result in leaks between the top and bottom electrodes. What is worse, the pattern

uniformity less than 1.5% for thickness and PZT composition were obtained.

**1.** Etching selectivity to resist mask and the bottom rare metal electrode

called high etching selectivity, is important as a PZT etching condition.

**2.** Adhesion of conductive deposit to the pattern sidewalls and damage to PZT

**Figure 13.** XRD spectrum for various thickness PZT

**Figure 14.** Characteristics of MOCVD-PZT film (applied voltage dependence of switching charge)

**Figure 15.** Endurance properties of MOCVD-PZT films

Recently mass production tool for 300mm Si wafer was developed and excellent in-plane uniformity less than 1.5% for thickness and PZT composition were obtained.

### **2.5. Etching technology**

**4.** By taking advantage of the progress of the equipment hardware as described above, the optimization of processes was conducted, and PZT thin films can be controlled to en‐ sure preferential orientation in the <111> direction as shown in Fig. 13. Consequently, the formation of PZT films within 100 nm that have capacitor properties with a 1.5 V low voltage drive is achieved as shown in Fig. 14. [11,12] In addition excellent endur‐

ance properties which are over 1010cycles were obtained for 73nm-PZT in Fig 15.

**Figure 14.** Characteristics of MOCVD-PZT film (applied voltage dependence of switching charge)

**Figure 13.** XRD spectrum for various thickness PZT

380 Advances in Ferroelectrics

### *2.5.1. Issues of ferroelectric etching technology*

Conventionally, the piezoelectric elements have been fabricated by chemical wet etching [13] or argon ion milling. With the miniaturization of MEMS, there have been increasing de‐ mands for dry etching with the excellent shape controllability as semiconductor technology. Recently dry etching technique for MEMS using PZT was reported. [14,15] The Pt, Ir and other rare metal electrodes and the PZT ferroelectric thin films that compose piezoelectric elements react poorly with halogen gases and their halides have low vapor pressures. For these reasons, these materials are called hard-to-etch materials. The following technical is‐ sues are important for dry etching of the PZT ferroelectric thin films for not only FeRAM but also MEMS productions:

**1.** Etching selectivity to resist mask and the bottom rare metal electrode

A piezoelectric element film consists of PZT with a thickness of several micrometers and the rare metal electrodes with a thickness of about 100 nm. Generally, the bottom electrode is left after the PZT etching. Therefore, a low etching rate for the bottom electrode, the socalled high etching selectivity, is important as a PZT etching condition.

**2.** Adhesion of conductive deposit to the pattern sidewalls and damage to PZT

The materials are hard to etch, and their etching products easily adhere to the pattern side‐ walls, and result in leaks between the top and bottom electrodes. What is worse, the pattern sidewalls are exposed to reactive gas plasma during etching, and tend to suffer lead and oxygen coming out and other damages.

**3.** Plasma stability during continuous processing

Adhesion of etching products to chamber walls, especially the RF introduction window that generates plasma, causes instabilities of plasma and deteriorates the etching rate and the shape reproducibility. Avoiding of adhesion of etching products to chamber walls is impor‐ tant for mass production.

**4.** Uniformity of etching rate within wafer

As in the case of (1), to stop the thick PZT at the thin bottom electrode after etching, the uni‐ formity of etching rate within wafer is important.

### *2.5.2. Ferroelectric etching systems and process for mass production*

As the piezoelectric PZT etching systems, this section explains about Apios NE series made by ULVAC, Inc. The etching module is equipped with the ISM (Inductively Super Magnetron) plasma source that can generate low-pressure and high-density plasma. Fig. 16 shows a draw‐ ing of the etching module. Table I shows the comparison between the normal ICP type plas‐ ma source and the ISM plasma source. The RF antenna is mounted in the upper part of the etching chamber, so that RF is introduced through the quartz window into the etching cham‐ ber to generate plasma. The uniformity of the etching rate within wafer can be easily opti‐ mized by positioning permanent magnets under the antenna. Fig. 17 shows the uniformity of PZT etching rate within 6 inch wafer. A high uniformity (<+/-5%) was realized by means of op‐ timization to permanent magnet layout. A STAR electrode is provided between the antenna and the quartz window to control adhesion of etching products to the quartz window by ap‐ plying RF to the STAR electrode. The substrate is held on the electrostatic chuck. The sub‐ strate temperature is controlled by introducing Helium to the backside of the substrate. The ion energy is controlled by applying RF power to the substrate. The materials to be etched are non-volatile, and the etching products adhere to the shield located in the chamber. The tem‐ perature of the shield is kept constant by heater, so process is high stability.

**Figure 16.** Etching chamber with inductively super magnetron (ISM) plasma performances

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

383

*2.5.3. Mass productive high temperature etching technology for high dense FeRAM*

Dry etching techniques are used for the patterning of FeRAM device. It is difficult to etch the material of FeRAM such as the noble metal and PZT, because these have low reactivity

**Figure 17.** PZT etching uniformity


**Table 1.** Inductively super magnetron (ISM) plasma performances.

**Figure 16.** Etching chamber with inductively super magnetron (ISM) plasma performances

**Figure 17.** PZT etching uniformity

sidewalls are exposed to reactive gas plasma during etching, and tend to suffer lead and

Adhesion of etching products to chamber walls, especially the RF introduction window that generates plasma, causes instabilities of plasma and deteriorates the etching rate and the shape reproducibility. Avoiding of adhesion of etching products to chamber walls is impor‐

As in the case of (1), to stop the thick PZT at the thin bottom electrode after etching, the uni‐

As the piezoelectric PZT etching systems, this section explains about Apios NE series made by ULVAC, Inc. The etching module is equipped with the ISM (Inductively Super Magnetron) plasma source that can generate low-pressure and high-density plasma. Fig. 16 shows a draw‐ ing of the etching module. Table I shows the comparison between the normal ICP type plas‐ ma source and the ISM plasma source. The RF antenna is mounted in the upper part of the etching chamber, so that RF is introduced through the quartz window into the etching cham‐ ber to generate plasma. The uniformity of the etching rate within wafer can be easily opti‐ mized by positioning permanent magnets under the antenna. Fig. 17 shows the uniformity of PZT etching rate within 6 inch wafer. A high uniformity (<+/-5%) was realized by means of op‐ timization to permanent magnet layout. A STAR electrode is provided between the antenna and the quartz window to control adhesion of etching products to the quartz window by ap‐ plying RF to the STAR electrode. The substrate is held on the electrostatic chuck. The sub‐ strate temperature is controlled by introducing Helium to the backside of the substrate. The ion energy is controlled by applying RF power to the substrate. The materials to be etched are non-volatile, and the etching products adhere to the shield located in the chamber. The tem‐

**ISM ICP**

High pressure process causes redeposition.

oxygen coming out and other damages.

**4.** Uniformity of etching rate within wafer

formity of etching rate within wafer is important.

*2.5.2. Ferroelectric etching systems and process for mass production*

perature of the shield is kept constant by heater, so process is high stability.

Damage Plasma density and substrate bias can be

Repeatability, stability Less re-deposition results in Low-pressure

Maintenance Chamber structure is simple for easy

**Table 1.** Inductively super magnetron (ISM) plasma performances.

Plasma density (cm-3) 1×1010 ~ 1×1011 5×109 ~ 5×1010 Operating pressure (Pa) 0.07<P<7 0.5<P<50

controlled independently.

etching -> better repeatability

maintenance.

Uniformity Optimized magnetic layout Determined by chamber structure

tant for mass production.

382 Advances in Ferroelectrics

**3.** Plasma stability during continuous processing

#### *2.5.3. Mass productive high temperature etching technology for high dense FeRAM*

Dry etching techniques are used for the patterning of FeRAM device. It is difficult to etch the material of FeRAM such as the noble metal and PZT, because these have low reactivity with halogen gas plasma and these halides have low vapor pressure. The photo-resist is used as etching mask for the patterning of FeRAM memory cell. But the etching selectivity to photo-resist is low. Therefore the etching profile becomes low taper angle. Since FeRAM device shrinks down recently(0.35-μm-design rule or lower), high temperature etching was developed for high density FeRAM device. Furthermore, FeRAM device changed to the stack structure from planer structure. At that time, the top electrode, PZT and the bottom electrode are etched in a series by using of hard-mask (Fig. 18). [16]

**Figure 19.** Repeatability of Ir etching time for 300-wafers running test

**Figure 20.** Stage temperature dependency of taper angle ( (a) 300°C, (b) 400°C)

**3.1. Sputtered PZT piezoelectric films for MEMS application**

**3. Technology for MEMS application**

ited on 6 inch diameter silicon substrates.

duction.

Usually, high density device is required for non-volatile memory. Therefore the high tem‐ perature etching technique contributes to realize the next generation FeRAM devices pro‐

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

385

A multi-chamber type mass production sputtering systems for electronic devices SME-200 equipped with an exclusive sputtering module described above. PZT films have been depos‐

**Figure 18.** Necessity of high-temperature etch

This section explains high temperature etching system ULHITE series made by ULVAC, Inc. This etching module is equipped with ISM plasma source. Most important feature is the novel electro-static-chuck (ESC) type hot plate stage at a temperature up to 450°C, and this stage can be supplied high bias power. The process chamber, variable conductance valve, pumps and gas exhaust are heated up. The deposition shields equipped in process chamber are also heated up to 200°C. The effect is reducing the deposition during the etch process, and high process stability is achieved.

Fig. 19 shows repeatability of Ir etching time for 300-wafers running test. The plasma clean‐ ing were carried out in every 25-wafers. Excellent stability was confirmed. For next genera‐ tion FeRAM devices, the continuous etching among the top electrode, ferroelectrics and bottom electrode by use of one mask is need. And the high taper angle and no sidewall dep‐ osition are needed for etching process.

Fig. 20 shows the results of FeRAM capacitor etching of stack structure dependence of the stage temperature. The halogen gases were used in etching process. Thus the higher profile can be obtained by higher stage temperature. Fig. 21 shows repeatability of etching profile for 300-wafers running test. Excellent repeatability of etching profile was confirmed.

**Figure 19.** Repeatability of Ir etching time for 300-wafers running test

with halogen gas plasma and these halides have low vapor pressure. The photo-resist is used as etching mask for the patterning of FeRAM memory cell. But the etching selectivity to photo-resist is low. Therefore the etching profile becomes low taper angle. Since FeRAM device shrinks down recently(0.35-μm-design rule or lower), high temperature etching was developed for high density FeRAM device. Furthermore, FeRAM device changed to the stack structure from planer structure. At that time, the top electrode, PZT and the bottom

This section explains high temperature etching system ULHITE series made by ULVAC, Inc. This etching module is equipped with ISM plasma source. Most important feature is the novel electro-static-chuck (ESC) type hot plate stage at a temperature up to 450°C, and this stage can be supplied high bias power. The process chamber, variable conductance valve, pumps and gas exhaust are heated up. The deposition shields equipped in process chamber are also heated up to 200°C. The effect is reducing the deposition during the etch process,

Fig. 19 shows repeatability of Ir etching time for 300-wafers running test. The plasma clean‐ ing were carried out in every 25-wafers. Excellent stability was confirmed. For next genera‐ tion FeRAM devices, the continuous etching among the top electrode, ferroelectrics and bottom electrode by use of one mask is need. And the high taper angle and no sidewall dep‐

Fig. 20 shows the results of FeRAM capacitor etching of stack structure dependence of the stage temperature. The halogen gases were used in etching process. Thus the higher profile can be obtained by higher stage temperature. Fig. 21 shows repeatability of etching profile

for 300-wafers running test. Excellent repeatability of etching profile was confirmed.

electrode are etched in a series by using of hard-mask (Fig. 18). [16]

**Figure 18.** Necessity of high-temperature etch

384 Advances in Ferroelectrics

and high process stability is achieved.

osition are needed for etching process.

**Figure 20.** Stage temperature dependency of taper angle ( (a) 300°C, (b) 400°C)

Usually, high density device is required for non-volatile memory. Therefore the high tem‐ perature etching technique contributes to realize the next generation FeRAM devices pro‐ duction.

### **3. Technology for MEMS application**

### **3.1. Sputtered PZT piezoelectric films for MEMS application**

A multi-chamber type mass production sputtering systems for electronic devices SME-200 equipped with an exclusive sputtering module described above. PZT films have been depos‐ ited on 6 inch diameter silicon substrates.

**Figure 21.** Repeatability of etching profile for 300-wafers running test

The platinum bottom electrodes, whose orientation is (111), were deposited on the substrate. The PZT films were deposited under Ar/O2 mixed gas atmosphere of 0.5 Pa. Substrate temper‐ ature was heated up to around 550°C. After the deposition, PZT films were conducted with no thermal treatments such as post-annealing. PZT films were deposited with relatively high growth rate about 2.1 μm/h and these thicknesses were from 0.5 to 3.0 μm in consideration of piezoelectric MEMS applications. The ceramic target with Zr/Ti ratio of 52/48, in which 30 mol % excess PbO was added for the compensation of the lead re-evaporation from the films, was used in order to obtain PZT films near the stoichiometric composition. After the PZT deposi‐ tion, top electrode 100-nm-thick Pt was deposited by the dc sputtering method.

**Figure 22.** Relationship between Pb composition, deposition rate and repeatability

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

387

**Figure 23.** Crystallization uniformity of PZT films Deposited on 6-inches substrate

For the measurement of the piezoelectric properties of the PZT films, Rectangular beams (cantilevers) with the size of about 30 mm (3 cm) × 3 mm were prepared. Polarization and displacement in these films were simultaneously observed using the laser doppler vibrome‐ ter (Graphtec AT-3600) and the laser interferometer (Graphtec AT-0023) which were attach‐ ed to a ferroelectric test system.

Fig. 22 shows relationship between Pb composition, deposition rate and repeatability. As a result, stable transition of Pb content within film in continuous sputtering has been con‐ firmed. As can be seen from the figure, the change in the deposition rate was 2.1 μm±1.4%, and changing Pb composition was 1.0±0.1% in the short running for 35 pieces (total thick‐ ness; 105 μm).

Fig. 23 shows XRD patterns of as-sputtered PZT film. No pyrochlore phase can be confirmed and the film appears to be almost perovskite phase with the preferred orientation to (001) or (100). It confirmed that the uniformity of the crystalline property were good in 6 inch area. In addition, the dielectric constant of these samples was measured as shown in Fig.24. As a result, excellent properties with the dielectric constant over 1000 and its uniformity of ±4.7% were confirmed.

**Figure 22.** Relationship between Pb composition, deposition rate and repeatability

**Figure 21.** Repeatability of etching profile for 300-wafers running test

ed to a ferroelectric test system.

ness; 105 μm).

386 Advances in Ferroelectrics

were confirmed.

The platinum bottom electrodes, whose orientation is (111), were deposited on the substrate. The PZT films were deposited under Ar/O2 mixed gas atmosphere of 0.5 Pa. Substrate temper‐ ature was heated up to around 550°C. After the deposition, PZT films were conducted with no thermal treatments such as post-annealing. PZT films were deposited with relatively high growth rate about 2.1 μm/h and these thicknesses were from 0.5 to 3.0 μm in consideration of piezoelectric MEMS applications. The ceramic target with Zr/Ti ratio of 52/48, in which 30 mol % excess PbO was added for the compensation of the lead re-evaporation from the films, was used in order to obtain PZT films near the stoichiometric composition. After the PZT deposi‐

For the measurement of the piezoelectric properties of the PZT films, Rectangular beams (cantilevers) with the size of about 30 mm (3 cm) × 3 mm were prepared. Polarization and displacement in these films were simultaneously observed using the laser doppler vibrome‐ ter (Graphtec AT-3600) and the laser interferometer (Graphtec AT-0023) which were attach‐

Fig. 22 shows relationship between Pb composition, deposition rate and repeatability. As a result, stable transition of Pb content within film in continuous sputtering has been con‐ firmed. As can be seen from the figure, the change in the deposition rate was 2.1 μm±1.4%, and changing Pb composition was 1.0±0.1% in the short running for 35 pieces (total thick‐

Fig. 23 shows XRD patterns of as-sputtered PZT film. No pyrochlore phase can be confirmed and the film appears to be almost perovskite phase with the preferred orientation to (001) or (100). It confirmed that the uniformity of the crystalline property were good in 6 inch area. In addition, the dielectric constant of these samples was measured as shown in Fig.24. As a result, excellent properties with the dielectric constant over 1000 and its uniformity of ±4.7%

tion, top electrode 100-nm-thick Pt was deposited by the dc sputtering method.

**Figure 23.** Crystallization uniformity of PZT films Deposited on 6-inches substrate

**3.2. Etching technology of PZT piezoelectric Films for MEMS application**

PZT over whole wafers, implying that Pt film is the role of an etch stop layer.

**Figure 26.** Etching rate and in-plane uniformity of Pt & PZT

**Figure 27.** SEM image of Pt/PZT etching profile

Additionally etching technology for mass production is developed. Since high etching rate is required in MEMS PZT process because of large PZT thickness, etching selectivity to the bottom electrode(Pt) and uniformity are important. In Fig. 26 we show etching rate profile of both electrode(Pt) and PZT film. It has been found that etching rate of Pt is smaller than

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

389

**Figure 24.** Dielectric constant of PZT films deposited on 6 inch substrate

Piezoelectric properties were finally confirmed for PZT films by checking the cantilever as shown in Fig. 25. Large piezoelectric coefficient from -60 to -120 pm/V was observed in our PZT films.

**Figure 25.** Relationship between piezoelectric coefficient and PZT film thickness

### **3.2. Etching technology of PZT piezoelectric Films for MEMS application**

Additionally etching technology for mass production is developed. Since high etching rate is required in MEMS PZT process because of large PZT thickness, etching selectivity to the bottom electrode(Pt) and uniformity are important. In Fig. 26 we show etching rate profile of both electrode(Pt) and PZT film. It has been found that etching rate of Pt is smaller than PZT over whole wafers, implying that Pt film is the role of an etch stop layer.

**Figure 26.** Etching rate and in-plane uniformity of Pt & PZT

**Figure 24.** Dielectric constant of PZT films deposited on 6 inch substrate

**Figure 25.** Relationship between piezoelectric coefficient and PZT film thickness

PZT films.

388 Advances in Ferroelectrics

Piezoelectric properties were finally confirmed for PZT films by checking the cantilever as shown in Fig. 25. Large piezoelectric coefficient from -60 to -120 pm/V was observed in our

**Figure 27.** SEM image of Pt/PZT etching profile

Fig. 27 shows the SEM image after the piezoelectric element was etched and the resist mask was removed. The film composition is Pt/PZT/Pt=100 nm/3 μm/100 nm, and the top Pt elec‐ trode and PZT were continuously etched by using a 5-μm-thick photo resist as a mask. Chlorine and fluorine mixed gases are used for PZT etching. The etching shape (taper angle) is about 65º, and nothing adhered to the pattern sidewalls. Despite 20% of over etching, the bottom Pt electrode was hardly etched. This indicates that a high etching selectivity to Pt was achieved. Fig. 28 shows the dependency of the etching rate and the taper angle on the bias power. As the bias power increases, the etching rate increases linearly. When 400 W was applied, the PZT etching rate of 190 nm/min. was achieved. The taper angle also in‐ creased gradually as the bias power increased.

Fig. 29 shows a 50-μm-diameter actuator element array fabricated by dry etching. Fig. 30 shows that the remanent polarization (Pr value) of the piezoelectric thin-film actuator with a

44.5-46.0 kV/cm at an applied voltage of 30 V, and the characteristics without the dependen‐ cy on element size (30-300 μm diameter) were obtained. The damage to the PZT piezoelec‐ tric thin-film actuator caused by the dry etching is considered to be negligible. The displacement of the PZT thin-film actuator was measured by contact-AFM. Fig.31 shows that a displacement of about 4 nm was obtained at 3-μm-thick PZT film, 30-μm-diameter el‐ ement size, and an applied electric field of 100 kV/cm. It was clarified that the processing of

**Figure 31.** Ferroelectric and displacement properties of Pt/PZT/Pt element with 30mm diameter etching rate and tap‐

PZT piezoelectric thin-film actuators by dry etching is very effective.

**Figure 30.** Dependence of ferroelectric properties on piezoelectric actuator size

er angle on the substrate bias power

and the coercive electric field (Ec value) was

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

391

3-μm-thick PZT film was 40.5-42.8 μC/cm2

**Figure 28.** Dependence of PZT etching rate and taper angle on the substrate bias power

**Figure 29.** Pt/PZT etching profile of ϕ50mm piezoelectric element array

Fig. 29 shows a 50-μm-diameter actuator element array fabricated by dry etching. Fig. 30 shows that the remanent polarization (Pr value) of the piezoelectric thin-film actuator with a 3-μm-thick PZT film was 40.5-42.8 μC/cm2 and the coercive electric field (Ec value) was 44.5-46.0 kV/cm at an applied voltage of 30 V, and the characteristics without the dependen‐ cy on element size (30-300 μm diameter) were obtained. The damage to the PZT piezoelec‐ tric thin-film actuator caused by the dry etching is considered to be negligible. The displacement of the PZT thin-film actuator was measured by contact-AFM. Fig.31 shows that a displacement of about 4 nm was obtained at 3-μm-thick PZT film, 30-μm-diameter el‐ ement size, and an applied electric field of 100 kV/cm. It was clarified that the processing of PZT piezoelectric thin-film actuators by dry etching is very effective.

**Figure 30.** Dependence of ferroelectric properties on piezoelectric actuator size

Fig. 27 shows the SEM image after the piezoelectric element was etched and the resist mask was removed. The film composition is Pt/PZT/Pt=100 nm/3 μm/100 nm, and the top Pt elec‐ trode and PZT were continuously etched by using a 5-μm-thick photo resist as a mask. Chlorine and fluorine mixed gases are used for PZT etching. The etching shape (taper angle) is about 65º, and nothing adhered to the pattern sidewalls. Despite 20% of over etching, the bottom Pt electrode was hardly etched. This indicates that a high etching selectivity to Pt was achieved. Fig. 28 shows the dependency of the etching rate and the taper angle on the bias power. As the bias power increases, the etching rate increases linearly. When 400 W was applied, the PZT etching rate of 190 nm/min. was achieved. The taper angle also in‐

creased gradually as the bias power increased.

390 Advances in Ferroelectrics

**Figure 28.** Dependence of PZT etching rate and taper angle on the substrate bias power

**Figure 29.** Pt/PZT etching profile of ϕ50mm piezoelectric element array

**Figure 31.** Ferroelectric and displacement properties of Pt/PZT/Pt element with 30mm diameter etching rate and tap‐ er angle on the substrate bias power

Good piezoelectric performance which is based on mass production technology including excellent in-plane uniformity and process rate of sputtering and etching was obtained in

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

393

(Ba,Sr)TiO3 (BST) is expected to use as the thin film capacitor, RF tunable component [18, 19] on-chip capacitor [20] for its excellent dielectric behavior. In these applications, decoupling, high permittivity and high tunability are required as dielectric characteristics. High permit‐ tivity is thought to be primarily important character as dielectric characteristics. BST thin films have been deposited by some deposition techniques. Among these techniques, RF magnetron sputtering is thought to be suitable for mass-production because of its stability and reproducibility as well as good film performances. In this section, BST films deposition by RF magnetron sputtering and an approach to deposit BST films with higher permittivity

BST films were also deposited by an RF magnetron sputtering method using sintered BST ceramic targets. Basic sputtering conditions such as RF power, deposition temperature and so on were varied. As a top electrode, Pt dot with 0.5 mm in diameter was deposited by DC magnetron sputtering. After top electrode deposition, BST capacitors were post-annealed for

Fundamental properties of BST thin films such as dielectric constant, crystalline quality (XRD), surface morphology (SEM) were investigated. Agilent 4284A LCR meter was used for dielectric properties measurement and measuring cycle and volts alternating current

The relationship between dielectric constant and deposition temperature was investigated as shown in Fig. 34. Ba/Sr ratio of BST target was 50/50. RF power was 1500 W. Ar/O2 flow

**3.4. Sputtered BST thin-films for capacitor and RF tunable devices**

1 hour in oxygen atmosphere under 1 atm using resistive heat furnace.

**Figure 34.** Relationship between dielectric constant and deposition temperature

above evaluation.

were described.

were 1 kHz and 1 Vrms, respectively.

**Figure 32.** XRD Patterns of PZT film before and after improvement on 8inch Pt/Ti/SiO2/Si Substrate

### **3.3. Further development for MEMS application**

We also note that post in-situ treatment [17] has been recently established, aiming to im‐ prove crystalline property and piezoelectric coefficient in our PZT films. Fig. 32 shows the results before and after improvement of crystalline properties. Peak intensity of a/c axis of PZT film deposited by improved process is approximately two times higher of peak intensi‐ ty of PZT film deposited by conventional process uniformity over 8 inch wafers. As a conse‐ quence good piezoelectric coefficient (12.9C/m2 ) and large breakdown voltage (68V) were obtained for PZT film deposited by improved sputtering process in Fig 33.

**Figure 33.** Improvement of piezoelectric coefficient(Left) and break down voltage(Right)

Good piezoelectric performance which is based on mass production technology including excellent in-plane uniformity and process rate of sputtering and etching was obtained in above evaluation.

### **3.4. Sputtered BST thin-films for capacitor and RF tunable devices**

**20 25 30 35 40 45 50**

**20 25 30 35 40 45 50**

Pt(111)

PZT(002)/(200)

Edge

Middle

Center

**2theta (deg.)**

) and large breakdown voltage (68V) were

**Intensity (a.u)**

(Conventional process) (Improved process)

We also note that post in-situ treatment [17] has been recently established, aiming to im‐ prove crystalline property and piezoelectric coefficient in our PZT films. Fig. 32 shows the results before and after improvement of crystalline properties. Peak intensity of a/c axis of PZT film deposited by improved process is approximately two times higher of peak intensi‐ ty of PZT film deposited by conventional process uniformity over 8 inch wafers. As a conse‐

**Figure 32.** XRD Patterns of PZT film before and after improvement on 8inch Pt/Ti/SiO2/Si Substrate

obtained for PZT film deposited by improved sputtering process in Fig 33.

**Figure 33.** Improvement of piezoelectric coefficient(Left) and break down voltage(Right)

PZT(002)/(200)

Pt(111)

PZT(001)/(100)

**2theta (deg.)**

**3.3. Further development for MEMS application**

quence good piezoelectric coefficient (12.9C/m2

**Intensity (a.u)**

392 Advances in Ferroelectrics

PZT(001)/(100)

(Ba,Sr)TiO3 (BST) is expected to use as the thin film capacitor, RF tunable component [18, 19] on-chip capacitor [20] for its excellent dielectric behavior. In these applications, decoupling, high permittivity and high tunability are required as dielectric characteristics. High permit‐ tivity is thought to be primarily important character as dielectric characteristics. BST thin films have been deposited by some deposition techniques. Among these techniques, RF magnetron sputtering is thought to be suitable for mass-production because of its stability and reproducibility as well as good film performances. In this section, BST films deposition by RF magnetron sputtering and an approach to deposit BST films with higher permittivity were described.

BST films were also deposited by an RF magnetron sputtering method using sintered BST ceramic targets. Basic sputtering conditions such as RF power, deposition temperature and so on were varied. As a top electrode, Pt dot with 0.5 mm in diameter was deposited by DC magnetron sputtering. After top electrode deposition, BST capacitors were post-annealed for 1 hour in oxygen atmosphere under 1 atm using resistive heat furnace.

Fundamental properties of BST thin films such as dielectric constant, crystalline quality (XRD), surface morphology (SEM) were investigated. Agilent 4284A LCR meter was used for dielectric properties measurement and measuring cycle and volts alternating current were 1 kHz and 1 Vrms, respectively.

**Figure 34.** Relationship between dielectric constant and deposition temperature

The relationship between dielectric constant and deposition temperature was investigated as shown in Fig. 34. Ba/Sr ratio of BST target was 50/50. RF power was 1500 W. Ar/O2 flow ratio was 3. Sputtering pressure was 2.0 Pa. BST film thickness was 100 nm. As a result, dep‐ osition rate was 4.8 nm/min and almost constant irrespective of deposition temperature. As can be seen in this figure, dielectric constant is strongly dependent on deposition tempera‐ ture and increasing with deposition temperature increasing. XRD patterns of these films are also shown in Fig. 35. We can see in this figure, BST grains are randomly orientated and XRD peak intensities from BST films are increasing with deposition temperature increasing. So, it is thought that this represents the relationship between dielectric constant and deposi‐ tion temperature.

The relationship between dielectric constant, deposition rate and RF power is shown in Fig. 36. Deposition temperature was 650°C in this relationship, while other conditions were not changed. We can see in this figure, there is trade-off relationship between dielectric constant and deposition rate and higher dielectric constant can be obtained at low sputtering power of 500 W in this experiment. BST morphology (SEM photographs) dependence on RF power is also shown in Fig. 37. As can be seen in this figure, BST film deposited at 500 W (b) has dense and uniform columnar structure. Possibly it represents some relationship between dielectric constant and RF power. It is speculated that there is a long time for atomic migration and the growth of BST crystal is encouraged under such low deposition rate around 1 nm/min.

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

395

**Figure 37.** BST morphology dependence on RF power

**Figure 38.** Relationship between dielectric constant and BST film thickness

Furthermore, the relationship between dielectric constant and BST film thickness is shown in Fig. 38. Ba/Sr ratio of BST target was 70/30 in this experiment. Deposition temperature was varied from 700°C to 800°C as a parameter, while other conditions were not changed.

**Figure 35.** Relationship between XRD pattern and deposition temperature

**Figure 36.** Relationship between dielectric constant, deposition rate and RF power temperature

The relationship between dielectric constant, deposition rate and RF power is shown in Fig. 36. Deposition temperature was 650°C in this relationship, while other conditions were not changed. We can see in this figure, there is trade-off relationship between dielectric constant and deposition rate and higher dielectric constant can be obtained at low sputtering power of 500 W in this experiment. BST morphology (SEM photographs) dependence on RF power is also shown in Fig. 37. As can be seen in this figure, BST film deposited at 500 W (b) has dense and uniform columnar structure. Possibly it represents some relationship between dielectric constant and RF power. It is speculated that there is a long time for atomic migration and the growth of BST crystal is encouraged under such low deposition rate around 1 nm/min.

**Figure 37.** BST morphology dependence on RF power

ratio was 3. Sputtering pressure was 2.0 Pa. BST film thickness was 100 nm. As a result, dep‐ osition rate was 4.8 nm/min and almost constant irrespective of deposition temperature. As can be seen in this figure, dielectric constant is strongly dependent on deposition tempera‐ ture and increasing with deposition temperature increasing. XRD patterns of these films are also shown in Fig. 35. We can see in this figure, BST grains are randomly orientated and XRD peak intensities from BST films are increasing with deposition temperature increasing. So, it is thought that this represents the relationship between dielectric constant and deposi‐

tion temperature.

394 Advances in Ferroelectrics

**Figure 35.** Relationship between XRD pattern and deposition temperature

**Figure 36.** Relationship between dielectric constant, deposition rate and RF power temperature

**Figure 38.** Relationship between dielectric constant and BST film thickness

Furthermore, the relationship between dielectric constant and BST film thickness is shown in Fig. 38. Ba/Sr ratio of BST target was 70/30 in this experiment. Deposition temperature was varied from 700°C to 800°C as a parameter, while other conditions were not changed. We can see in this figure, dielectric constant is also strongly and non-linearly dependent on BST film thickness.

If BST capacitor is assumed to be a simple series connected capacitor between transition layer near BST/Pt bottom electrode interface and main layer which represents BST film excluding transition layer, it is speculated that there have been the transition layer in this BST film. [21]

As a simple experiment, effects of gas flow sequence for BST deposition was investigated. Gas flow sequence, that is ON/OFF step was changed, while Ar/O2 flow ratio was not changed. Dielectric constant was improved by introducing of oxygen gas before BST deposi‐ tion. It is speculated in this experiment that BST/Pt interface was improved because the oxy‐ gen vacancies of BST in this region were reduced. Therefore, dielectric properties are noticeably influenced by gas flow sequencing variation.

**Figure 39.** AFM image of PZT at Pb flux of 0.160ml/min (Scanning range is 2um square)

**PZT 001**

**SRO 100**

**STO 100**

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

**XRD Intensity [cps.]**

**Figure 40.** XRD spectrum of PZT / SRO /STO

15 20 25 30 35 40 45 50 55

**PZT 002**

**SRO 200**

**STO 200**

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

397

**2theta [deg.]**

**Figure 41.** Writing Voltage & Writing Time Dependency of Dot diameter – [Small diameter (22nm) of recording-dot was obtained for 5V & 1ms writing condition. (PZT/SRO/c-STO) Estimated Recording density: 1.4Tbit/inch2. ]

### **4. Technology for ferroelectric probe memory**

Technology for ferroelectric probe memory was developed with the deposition technology for FeRAM manufacturing. Hard-disk using the magnetic recording is the one of the major storage device, but the recording density will reach the limit in the near future by exteriori‐ zation of the magnetization disappearance by the heat disturbance. Ferroelectric probe memory is ultrahigh-density memory applied Scanning Probe Microscope (SPM) and ferro‐ electric property.

Ferroelectric perovskite Pb(Zr, Ti)O3 (PZT) is polarized by electric field. For scanning probe memory device, ferroelectric layer required atomically-smooth surface and low leakage cur‐ rent to achieve large recording density by forming polarized domains as unit cell size. There‐ fore it is necessary to grow the epitaxial ferroelectric thin films oriented to *c*-axis direction.

In this time we prepared epitaxial growth PZT/ SrRuO3 (SRO) thin films on single crystal SrTiO3 (STO) substrate for probe memory device, and evaluated the film properties and the recording density with Piezoresponse force microscopy (PFM). The SRO films were deposit‐ ed on STO (100) single crystal substrate with DC sputtering. After that, the PZT film was deposited by MOCVD process.

Fig. 39 shows the AFM image of PZT / SRO / STO deposited at the optimal Pb flux. The sur‐ face morphology of PZT has step structure and smooth terraces similar to SRO/STO sub‐ strate before PZT deposition. This surface profile indicates layer-by-layer growth of PZT on SRO / STO substrate [22] [23] [24]. Fig. 40 shows XRD spectrum of PZT deposited at Pb flux of 0.160ml/min. There are only PZT, SRO and STO peaks. The fringe pattern appears around PZT (001) and (002) peaks because of the smooth interface between PZT and SRO. Fullwidth at half-maximum (FWHM) of PZT (002) with X-ray rocking curve is 0.129degree for optimal Pb flux of 0.160ml/min. Rms of surface roughness and PZT(002) FWHM of X-ray rocking curve have a similar trend relative to Pb flux.

**Figure 39.** AFM image of PZT at Pb flux of 0.160ml/min (Scanning range is 2um square)

**Figure 40.** XRD spectrum of PZT / SRO /STO

We can see in this figure, dielectric constant is also strongly and non-linearly dependent on

If BST capacitor is assumed to be a simple series connected capacitor between transition layer near BST/Pt bottom electrode interface and main layer which represents BST film excluding transition layer, it is speculated that there have been the transition layer in this BST film. [21]

As a simple experiment, effects of gas flow sequence for BST deposition was investigated. Gas flow sequence, that is ON/OFF step was changed, while Ar/O2 flow ratio was not changed. Dielectric constant was improved by introducing of oxygen gas before BST deposi‐ tion. It is speculated in this experiment that BST/Pt interface was improved because the oxy‐ gen vacancies of BST in this region were reduced. Therefore, dielectric properties are

Technology for ferroelectric probe memory was developed with the deposition technology for FeRAM manufacturing. Hard-disk using the magnetic recording is the one of the major storage device, but the recording density will reach the limit in the near future by exteriori‐ zation of the magnetization disappearance by the heat disturbance. Ferroelectric probe memory is ultrahigh-density memory applied Scanning Probe Microscope (SPM) and ferro‐

Ferroelectric perovskite Pb(Zr, Ti)O3 (PZT) is polarized by electric field. For scanning probe memory device, ferroelectric layer required atomically-smooth surface and low leakage cur‐ rent to achieve large recording density by forming polarized domains as unit cell size. There‐ fore it is necessary to grow the epitaxial ferroelectric thin films oriented to *c*-axis direction.

In this time we prepared epitaxial growth PZT/ SrRuO3 (SRO) thin films on single crystal SrTiO3 (STO) substrate for probe memory device, and evaluated the film properties and the recording density with Piezoresponse force microscopy (PFM). The SRO films were deposit‐ ed on STO (100) single crystal substrate with DC sputtering. After that, the PZT film was

Fig. 39 shows the AFM image of PZT / SRO / STO deposited at the optimal Pb flux. The sur‐ face morphology of PZT has step structure and smooth terraces similar to SRO/STO sub‐ strate before PZT deposition. This surface profile indicates layer-by-layer growth of PZT on SRO / STO substrate [22] [23] [24]. Fig. 40 shows XRD spectrum of PZT deposited at Pb flux of 0.160ml/min. There are only PZT, SRO and STO peaks. The fringe pattern appears around PZT (001) and (002) peaks because of the smooth interface between PZT and SRO. Fullwidth at half-maximum (FWHM) of PZT (002) with X-ray rocking curve is 0.129degree for optimal Pb flux of 0.160ml/min. Rms of surface roughness and PZT(002) FWHM of X-ray

noticeably influenced by gas flow sequencing variation.

**4. Technology for ferroelectric probe memory**

BST film thickness.

396 Advances in Ferroelectrics

electric property.

deposited by MOCVD process.

rocking curve have a similar trend relative to Pb flux.

**Figure 41.** Writing Voltage & Writing Time Dependency of Dot diameter – [Small diameter (22nm) of recording-dot was obtained for 5V & 1ms writing condition. (PZT/SRO/c-STO) Estimated Recording density: 1.4Tbit/inch2. ]

PFM measurement is carried out with various biases between 4V and 8V and pulse widths between 1msec and 100msec with PtIr5 coatedprobe (tip diameter is 25nm). Fig. 41 shows PFM write-condition and PFM image of the result. Polarization-inverted dot diameter has a minimum of 20nm at the bias of 5V and pulse width of 1msec. Recording density is estimat‐ ed on the assumption that this tiny dot is arrayed with pitch of 20nm. From the calculation result, device sample deposited under the optimal growth condition achieves large bit den‐ sity of 1.4Tb/inch2 .

[8] F. Chu, G. Fox, T. Davenport, Y. Miyaguchi and K. Suu: Integr. Ferroelectr. 48 (2002)

Thin-Film Process Technology for Ferroelectric Application

http://dx.doi.org/10.5772/54360

399

[9] T. Yamada, T. Masuda, M. Kajinuma, H. Uchida, M. Uematsu, K. Suu and M. Ishika‐

[10] T. Masuda, M. Kajinuma, T. Yamada, H. Uchida, M.Uematsu, K. Suu and M. Ishika‐

[11] Y. Nishioka, T. Jinbo, T. Yamada, T. Masuda, M. Kajinuma, M. Uematsu, K. Suu and

[12] Y. Nishioka, T. Masuda, M. Kajinuma, T. Yamada, M. Uematsu and K. Suu: MRS Fall

[14] Y. Kokaze, M. Endo, M. Ueda, and K. Suu: 17th Int. Symp. Integrated Ferroelectrics,

[15] Y. Kokaze, I Kimura, M. Endo, M. Ueda, S. Kikuchi, Y. Nishioka, and K. Suu: Jpn. J.

[16] M. Endo, M. Ueda, Y. Kokaze, M. Ozawa, T. Nakamura, K. Suu: SEMI Technology

[18] I. P. Koutsaroff, T. Bernacki, M. Zelner, A. Cervin-Layry, A. Kassam, P. Woo, L.

[19] X. H. Zhu, J. M. Zhu, S. H. Zhou, Z. G. Liu, N. B. Ming, S. G. Lu, H. L. W. Chen, C. L.

[20] M. C. Werner, I. Banerjee, P. C. McIntyre, N. Tani, M. Tanimura, Appl. Phys. Lett., 77

[21] T. Jimbo, I. Kimura, Y. Nishioka and K. Suu, Mat. Res. Soc. Symp. Proc. Vol. 784,

[22] P.-E. Janolin, B. Fraisse, F. Le Marrec, and B. Dkhil, Appl. Phys. Lett. 90, 212904

[23] Keisuke Saito, Toshiyuki Kurosawa, Takao Akai, Shintaro Yokoyama, Hitoshi Morio‐ ka, Takahiro Oikawa, and Hiroshi Funakubo, Mater. Res. Soc. Symp. Proc. 748,

[24] H. Hu, C. J. Peng and S. B. Krupanidhi, Thin Solid Films, 223 (1993) 327 333I. G. Baek, et al,: Tech. Dig. Int. Electron Devices Meet., San Francisco, 2004, 23, 6, p58

Woodward, A. Patel, Mat. Res. Soc. Symp. Proc. Vol. 762, C5.8.1 (2003).

161.

2005, 5-26-P.

Appl. Phys. 46 (2007) 282.

[17] Suu et.al in preparation

(2000) 1209.

C7.8.1 (2003).

U13.4.1-U13.4.6 (2003).

(2007)

wa : IFFF2002 abstract, 4 (2002) 37.

wa : Integr. Ferroelectr., 46 (2002) 66

Meeting Proceedings, 784-C7 (2003) 6

Symposium 2002; December 5, 2002

M. Ishikawa: Integr. Ferroelectr., 59 (2003) 1445

[13] K. Zheng, J. Lu, and J. Chu: Jpn. J. Appl. Phys. 43 (2004) 3934.

Choy, Journal of Electronic Materials, 32 (2003) 1125.

### **5. Conclusion**

We have been developing thin film process technologies for ferroelectric application of ad‐ vanced semiconductor and electronics usage for 20 years or more, and completed the ferro‐ electric thin film solutions (sputtering, MOCVD, and etching) that became a de facto standard. These technologies will support a wide variety of convenient energy-saving devi‐ ces such as FeRAM, MEMS production (actuators composing gyro meters, portable camera modules for smart phone applications, tunable devices and so on), and ultra-high density probe memory.

### **Author details**

### Koukou Suu

Institute of Semiconductor and Electronics Technologies, ULVAC, Inc., Shizuoka, Japan

### **References**


PFM measurement is carried out with various biases between 4V and 8V and pulse widths between 1msec and 100msec with PtIr5 coatedprobe (tip diameter is 25nm). Fig. 41 shows PFM write-condition and PFM image of the result. Polarization-inverted dot diameter has a minimum of 20nm at the bias of 5V and pulse width of 1msec. Recording density is estimat‐ ed on the assumption that this tiny dot is arrayed with pitch of 20nm. From the calculation result, device sample deposited under the optimal growth condition achieves large bit den‐

We have been developing thin film process technologies for ferroelectric application of ad‐ vanced semiconductor and electronics usage for 20 years or more, and completed the ferro‐ electric thin film solutions (sputtering, MOCVD, and etching) that became a de facto standard. These technologies will support a wide variety of convenient energy-saving devi‐ ces such as FeRAM, MEMS production (actuators composing gyro meters, portable camera modules for smart phone applications, tunable devices and so on), and ultra-high density

Institute of Semiconductor and Electronics Technologies, ULVAC, Inc., Shizuoka, Japan

Kamisawa and H. Takasu: Jpn. J. Appl. Phys. 35 (1996) 4967.

Kamisawa and H. Takasu: Integr. Ferroelectr. 14 (1997) 59.

[5] K. Suu: Proc. Semicon Korea Tech. Symp., 1998 p. 255.

[1] K. Suu, A. Osawa, N. Tani, M. Ishikawa, K. Nakamura, T. Ozawa, K. Sameshima, A.

[2] K. Suu, A. Osawa, N. Tani, M. Ishikawa, K. Nakamura, T. Ozawa, K. Sameshima, A.

[4] K. Suu, Y. Nishioka, A. Osawa and N. Tani: Oyo Buturi 65 (1996) 1248. (in Japanese)

[6] N. Inoue, Y. Maejima and Y. Hayashi: Int. Electron Device Meet. Tech. Dig., 1997 p.

[7] N. Inoue, T. Takeuchi and Y. Hayashi: Int. Electron Device Meet. Tech. Dig., 1998 p.

[3] K. Suu, A. Osawa, Y. Nishioka and N. Tani: Jpn. J. Appl. Phys. 36 (1997) 5789.

sity of 1.4Tb/inch2

398 Advances in Ferroelectrics

**5. Conclusion**

probe memory.

**Author details**

Koukou Suu

**References**

605.

819.

.


**Chapter 18**

**Provisional chapter**

**Ferroelectrics at the Nanoscale: A First Principle**

**Ferroelectrics at the Nanoscale: A First Principle**

The field of ferroelectric materials is driven by its possible use in various micro-electronic devices that take advantage of their multifunctional properties. The existence of a switchable spontaneous polarization (see Figure 1) is at the basis of the design of non volatile ferroelectric random access memories (FERAMs), where one bit of information can be stored by assigning one value of the Boolean algebra ("1" or "0") to each of the polarization states. Also, the high dielectric permitivity of ferroelectrics makes them possible candidates to replace silica as the gate dielectric in metal-oxide-semiconductor field effect transistors (MOSFETs). Their piezoelectric behavior enables them to convert mechanical energy in electrical energy and vice versa. Their pyroelectric properties are the basis for highly sensitive

The current miniaturization of microelectronic technology, imposed by the semiconductor industry, raises the question of possible size effects on the properties of the components. Except for the case of the gate dielectric problem in MOSFET's, the thickness of the films used in contemporary applications is still far away from the thickness range where size effects become a concern; therefore the question at the moment is merely academic. Nevertheless, it is very possible that the fundamental limits of materials might be reached in the future. Despite the efforts and advancement in the field, both theoretically and experimentally (see a recent review in Ref.[1]), many questions still remain open. The main reason for the poor understanding of some of the size effects on ferroelectricity is the vast amount of different effects that compete and might modify the delicate balance between long range dipole-dipole electrostatic interactions and the short range forces, whose subtle equilibrium is known to be

> ©2012 Núñez, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Núñez; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Approach**

Matías Núñez

**Approach**

Matías Núñez

**1. Introduction**

10.5772/52268

http://dx.doi.org/10.5772/52268

infrared room temperature detectors.

at the origin of the ferroelectric instability.

**Provisional chapter**

### **Ferroelectrics at the Nanoscale: A First Principle Approach Approach**

**Ferroelectrics at the Nanoscale: A First Principle**

Matías Núñez Additional information is available at the end of the chapter

Matías Núñez

Additional information is available at the end of the chapter 10.5772/52268

http://dx.doi.org/10.5772/52268

### **1. Introduction**

The field of ferroelectric materials is driven by its possible use in various micro-electronic devices that take advantage of their multifunctional properties. The existence of a switchable spontaneous polarization (see Figure 1) is at the basis of the design of non volatile ferroelectric random access memories (FERAMs), where one bit of information can be stored by assigning one value of the Boolean algebra ("1" or "0") to each of the polarization states. Also, the high dielectric permitivity of ferroelectrics makes them possible candidates to replace silica as the gate dielectric in metal-oxide-semiconductor field effect transistors (MOSFETs). Their piezoelectric behavior enables them to convert mechanical energy in electrical energy and vice versa. Their pyroelectric properties are the basis for highly sensitive infrared room temperature detectors.

The current miniaturization of microelectronic technology, imposed by the semiconductor industry, raises the question of possible size effects on the properties of the components. Except for the case of the gate dielectric problem in MOSFET's, the thickness of the films used in contemporary applications is still far away from the thickness range where size effects become a concern; therefore the question at the moment is merely academic. Nevertheless, it is very possible that the fundamental limits of materials might be reached in the future.

Despite the efforts and advancement in the field, both theoretically and experimentally (see a recent review in Ref.[1]), many questions still remain open. The main reason for the poor understanding of some of the size effects on ferroelectricity is the vast amount of different effects that compete and might modify the delicate balance between long range dipole-dipole electrostatic interactions and the short range forces, whose subtle equilibrium is known to be at the origin of the ferroelectric instability.

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Núñez; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Núñez, licensee InTech. This is an open access chapter distributed under the terms of the Creative

10.5772/52268

403

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

but also proposes a powerful algorithm for computing macroscopic polarizations from first

The modern theory can be equivalently reformulated using localized Wannier functions [12–15] instead of extended Bloch orbitals. The electronic contribution to the macroscopic polarization **P** is then expressed in terms of the dipole of the Wannier charge distribution associated with one unit cell. In this way, **P** is reformulated as a property of a localized charge distribution which goes back to the classic definition of polarization. However, one has to bear in mind that the phases of the Bloch orbitals are essential for building the Wannier functions. They are needed to specify how the periodic charge distribution should be decomposed into localized ones. The knowledge of the periodic charge distribution of the

Here I will review the central topics uncovered in the early 1990s, often known as the modern theory of polarization. Understanding this background is necessary in order follow this chapter. The general idea is to consider the change in polarization of a crystal as it undergoes a slow change and relate this to the current that flows during this adiabatic evolution of the system. These ideas will lead to an expression for the polarization that does not take the form of an expectation value of an operator, but takes the form of a Berry phase, which is a geometrical phase property of a closed manifold (the Brillouin zone) on which a set of

As a classical example of a Berry phase, lett's think of the paralel transport of a vector along a loop on a shpere ( for instance a compass needle carried in a car traveling on the surface of the Earth). After completing a closed path, or loop, the vector will be back at the original starting point but it will be rotated with respect to the direction it was pointing at when it started the trip. The reason for this rotation is purely geometrical topological and intrinsicaly connected to the curvature of the sphere and would not exist if the the vector would be parallel transported along a flat manyfold, like a plane, or cylinder. This rotation

Now lets see the quantum counterpart and its connection to the definition of Polarization in a solid. Let's assume that the crystal Hamiltonian *H<sup>λ</sup>* depends smoothly on parameter *<sup>λ</sup>* and has Bloch eigenvectors obeying *<sup>H</sup>λ*|*ψλ*,*n***k**� = *<sup>E</sup>λ*,*n***k**|*ψλ*,*n***k**� and that *<sup>λ</sup>* changes slowly

*dt* =

 *d***P** *dλ dλ dt*

**j**(*t*)*dt* (1)

we can write the

angle is related to the integral of the curvature on the surface bounded by the loop.

△**P** = 

and is phrased in terms of the current density that is physically flowing through the crystal as the systems traverses some adiabatic path. It can also be written in terms of the parameter

with time, so that it is correct to use the adiabatic approximation.

Since the spatially averaged current density is just **<sup>j</sup>** = *<sup>d</sup>***<sup>P</sup>**

change in polarization during some time interval as

polarized dielectric is not enough to determine the wannier functions.

principles.

**The main concepts**

*λ* as:

vectors, the occupied Bloch states, are defined.

**Figure 1.** Tetragonal ferroelectric structure of *BaTiO*3. Solid, shaded and empty circles represent Ba, Ti, O atoms, respectively. The arrows indicate the atomic displacements, exaggerated for clarity. The origin has been kept at the Ba site. Two structures, with polarization along [001], are shown. Application of a sufficiently large electric field causes the system to switch between the two states, reversing the polarization. in principle, we can assign one value of the Boolean algebra ("1" or "0") to each of the polarization states.

The next section will give an overview of the main concepts involved in the modern theory of polarization. A definition of a local polarization will be given in terms of the centers of the wannier functions associated with the band structure of the system. Next, some basic electrostatic notions related with ferroelectric films will be given, in particular the concept of depolarization field and screening by metal contacts. Finally, the results of this research work: applying the layer polarization concept, where we show the hidden structure of the polarization at the nanoscale.

### **2. Modern theory of polarization**

Classically, the macroscopic polarization in dielectric media is defined to be an intensive quantity that quantifies the electric dipole moment per unit volume [2–4]. Definitions along these lines work well for finite systems but have important conceptual problems when applied to periodic crystalline systems because there is no unique choice of cell boundaries [5].

In the early 1990s a new viewpoint emerged and lead to the development of a microscopic theory [6–8]. It starts by recognizing that the bulk macroscopic polarization cannot be determined, not even in principle, from a knowledge of the periodic charge distribution of the polarized crystalline material. This establishes a fundamental difference between finite systems (e.g., molecules, clusters, etc.) and infinite periodic ones. For the first case, the dipole moment can be easily expressed in terms of the charge distribution. While for periodic systems one focuses on differences in polarization between two states of the crystal that can be connected by an adiabatic switching process [6]. The polarization difference is then equal to the integrated transient macroscopic current that flows through the sample during the switching process.

Therefore, the macroscopic polarization of an extended system is, according to the modern viewpoint, a dynamical property of the current in the adiabatic limit. The charge density is a property connected with the square modulus of the wavefunction, while the current also has a dependence on the phase. Indeed, it turns out that in the modern theory of polarization [7–9], the polarization difference is related to the Berry phase [10, 11], defined over the manifold of Bloch orbitals. The theory not only defines what polarization really is, but also proposes a powerful algorithm for computing macroscopic polarizations from first principles.

The modern theory can be equivalently reformulated using localized Wannier functions [12–15] instead of extended Bloch orbitals. The electronic contribution to the macroscopic polarization **P** is then expressed in terms of the dipole of the Wannier charge distribution associated with one unit cell. In this way, **P** is reformulated as a property of a localized charge distribution which goes back to the classic definition of polarization. However, one has to bear in mind that the phases of the Bloch orbitals are essential for building the Wannier functions. They are needed to specify how the periodic charge distribution should be decomposed into localized ones. The knowledge of the periodic charge distribution of the polarized dielectric is not enough to determine the wannier functions.

### **The main concepts**

2 Advances in Ferroelectrics

the polarization states.

[5].

polarization at the nanoscale.

during the switching process.

**2. Modern theory of polarization**

**Figure 1.** Tetragonal ferroelectric structure of *BaTiO*3. Solid, shaded and empty circles represent Ba, Ti, O atoms, respectively. The arrows indicate the atomic displacements, exaggerated for clarity. The origin has been kept at the Ba site. Two structures, with polarization along [001], are shown. Application of a sufficiently large electric field causes the system to switch between the two states, reversing the polarization. in principle, we can assign one value of the Boolean algebra ("1" or "0") to each of

The next section will give an overview of the main concepts involved in the modern theory of polarization. A definition of a local polarization will be given in terms of the centers of the wannier functions associated with the band structure of the system. Next, some basic electrostatic notions related with ferroelectric films will be given, in particular the concept of depolarization field and screening by metal contacts. Finally, the results of this research work: applying the layer polarization concept, where we show the hidden structure of the

Classically, the macroscopic polarization in dielectric media is defined to be an intensive quantity that quantifies the electric dipole moment per unit volume [2–4]. Definitions along these lines work well for finite systems but have important conceptual problems when applied to periodic crystalline systems because there is no unique choice of cell boundaries

In the early 1990s a new viewpoint emerged and lead to the development of a microscopic theory [6–8]. It starts by recognizing that the bulk macroscopic polarization cannot be determined, not even in principle, from a knowledge of the periodic charge distribution of the polarized crystalline material. This establishes a fundamental difference between finite systems (e.g., molecules, clusters, etc.) and infinite periodic ones. For the first case, the dipole moment can be easily expressed in terms of the charge distribution. While for periodic systems one focuses on differences in polarization between two states of the crystal that can be connected by an adiabatic switching process [6]. The polarization difference is then equal to the integrated transient macroscopic current that flows through the sample

Therefore, the macroscopic polarization of an extended system is, according to the modern viewpoint, a dynamical property of the current in the adiabatic limit. The charge density is a property connected with the square modulus of the wavefunction, while the current also has a dependence on the phase. Indeed, it turns out that in the modern theory of polarization [7–9], the polarization difference is related to the Berry phase [10, 11], defined over the manifold of Bloch orbitals. The theory not only defines what polarization really is, Here I will review the central topics uncovered in the early 1990s, often known as the modern theory of polarization. Understanding this background is necessary in order follow this chapter. The general idea is to consider the change in polarization of a crystal as it undergoes a slow change and relate this to the current that flows during this adiabatic evolution of the system. These ideas will lead to an expression for the polarization that does not take the form of an expectation value of an operator, but takes the form of a Berry phase, which is a geometrical phase property of a closed manifold (the Brillouin zone) on which a set of vectors, the occupied Bloch states, are defined.

As a classical example of a Berry phase, lett's think of the paralel transport of a vector along a loop on a shpere ( for instance a compass needle carried in a car traveling on the surface of the Earth). After completing a closed path, or loop, the vector will be back at the original starting point but it will be rotated with respect to the direction it was pointing at when it started the trip. The reason for this rotation is purely geometrical topological and intrinsicaly connected to the curvature of the sphere and would not exist if the the vector would be parallel transported along a flat manyfold, like a plane, or cylinder. This rotation angle is related to the integral of the curvature on the surface bounded by the loop.

Now lets see the quantum counterpart and its connection to the definition of Polarization in a solid. Let's assume that the crystal Hamiltonian *H<sup>λ</sup>* depends smoothly on parameter *<sup>λ</sup>* and has Bloch eigenvectors obeying *<sup>H</sup>λ*|*ψλ*,*n***k**� = *<sup>E</sup>λ*,*n***k**|*ψλ*,*n***k**� and that *<sup>λ</sup>* changes slowly with time, so that it is correct to use the adiabatic approximation.

Since the spatially averaged current density is just **<sup>j</sup>** = *<sup>d</sup>***<sup>P</sup>** *dt* = *d***P** *dλ dλ dt* we can write the change in polarization during some time interval as

$$
\triangle \mathbf{P} = \int \mathbf{j}(t)dt\tag{1}
$$

and is phrased in terms of the current density that is physically flowing through the crystal as the systems traverses some adiabatic path. It can also be written in terms of the parameter *λ* as:

$$
\triangle \mathbf{P} = \int \frac{d\mathbf{P}}{d\lambda} d\lambda. \tag{2}
$$

10.5772/52268

405

http://dx.doi.org/10.5772/52268

**Figure 2.** From Ref.[18] Mapping of the distributed charge density onto the centers of charge of the Wannier functions.

set the stage for the crucial meaning of the expression in Eq.5.

<sup>Φ</sup>*nj* <sup>=</sup> <sup>−</sup> <sup>Ω</sup>

multiple bands, may be found in Refs.[7] and [8].

*Reformulation in terms of Wannier Functions.*

Fourier transform of the form

for band *n* in direction *j* is

Its eigenvalues turn out to give the average positions of an electron in different bands which coincide with the symmetry centers of the space group of the solid [17] These previous ideas

In three dimensions, the Brillouin zone can be regarded as a closed 3-torus obtained by the identifying points *un***<sup>k</sup>** = *un*,**k**+**G***<sup>j</sup>* where **<sup>G</sup>***<sup>j</sup>* are the three primitive reciprocal lattice vectors.

> <sup>2</sup>*π*<sup>Ω</sup> <sup>∑</sup> *j*

where **R***<sup>j</sup>* is the real space primitive translational corresponding to **G***j*, and the Berry phase

*BZ*

More details and discussion, including the equivalent of Eq.7 for the case of connected

We can rewrite Eq.5 in terms of the Wannier functions (WF's), which brings an alternative and more intuitive way of thinking about it. The WF's and Bloch functions can be regarded as two different orthonormal representations of the same occupied Hilbert space. The WF's are localized functions *wn***R**(**r**) that are labeled by band *n* and unit cell **R** and constructed by carrying out a unitary transformation of the Bloch states *ψn***k**. They are constructed via a

Φ*nj***R***<sup>j</sup>* (6)

Ferroelectrics at the Nanoscale: A First Principle Approach

*<sup>d</sup>*3*k*�*un***k**|**G***j*.∇**k**|*un***k**�. (7)

*<sup>i</sup>***k**.**R**|*ψn***k**� (8)

Then, the electronic polarization contribution of each band can be written as

**<sup>P</sup>***<sup>n</sup>* <sup>=</sup> <sup>−</sup>*<sup>e</sup>*

(2*π*)<sup>3</sup> *Im*

<sup>|</sup>*wn***R**� <sup>=</sup> <sup>Ω</sup>

(2*π*)<sup>3</sup>

 *BZ d***k***e*

Then, from Ref.[7]

$$\frac{d\mathbf{P}}{d\lambda} = \frac{ie}{(2\pi)^3} \sum\_{\mathbf{n}} \int\_{BZ} d\mathbf{k} \langle \nabla\_{\mathbf{k}} \mu\_{n\mathbf{k}} | \frac{d\mu\_{n\mathbf{k}}}{d\lambda} \rangle + c.c. \tag{3}$$

and the rate of change of polarization with *λ* is a property of the occupied bands only. This expression can be integrated with respect to *λ* to obtain

$$\mathbf{P}(\lambda) = \frac{ie}{(2\pi)^3} \sum\_{\mathbf{n}} \int\_{BZ} d\mathbf{k} \langle \boldsymbol{\mu}\_{\lambda, n\mathbf{k}} | \nabla\_{\mathbf{k}} | \boldsymbol{\mu}\_{\lambda, n\mathbf{k}} \rangle. \tag{4}$$

It can be verified by taking the *λ* derivative of both sides of Eq.4 and comparing with Eq.3. The result is independent of the particular path of *λ*(*t*) in time, and depends only on the final value of *λ* as long the change is adiabatically slow. We can associate the physical polarization of state *λ* with **P**(*λ*) and drop the *λ* label. Eq. 4 can be written then as

$$\mathbf{P} = \frac{e}{(2\pi)^3} Im \sum\_{\mathbf{n}} \int\_{BZ} d\mathbf{k} \langle u\_{n\mathbf{k}} | \nabla\_{\mathbf{k}} | u\_{n\mathbf{k}} \rangle , \tag{5}$$

and this is the electronic contribution to the polarization. To obtain the total polarization it must be added the nuclear (or ionic) contribution

$$\mathbf{P}\_{ion} = \frac{1}{\Omega} \sum\_{\mathbf{s}} Z\_{\mathbf{s}} \mathbf{r}\_{\mathbf{s}\prime}$$

where the sum is over atoms *s* having core charge *Zs* and spatial position **r***<sup>s</sup>* in the unit cell of volume Ω.

The equation 5 is the main result of the modern theory of polarization It states that the electronic contribution to the polarization of crystalline insulator may be expressed as a Brillouin zone integral of an operator *i*∇**k**. However, it is not a common quantum mechanical operator, the result of its action on the wavefunctions (*i*∇**k**|*un***k**�) depends on the relative phases of the Bloch functions at different **k**.

To understand the nature of the integrand in Eq.5 I want to recall the fundamental paper by Zak [16] in which he postulates the existence of a Berry phase associated with each one of the bands of a one dimensional crystalline solid. He shows how the variation of *k* over the entire Brillouin zone produces the appearance of a Berry phase. Moreover, he associates this phase, that is a characteristic feature of the whole band, with the band center operator [17].

**Figure 2.** From Ref.[18] Mapping of the distributed charge density onto the centers of charge of the Wannier functions.

Its eigenvalues turn out to give the average positions of an electron in different bands which coincide with the symmetry centers of the space group of the solid [17] These previous ideas set the stage for the crucial meaning of the expression in Eq.5.

In three dimensions, the Brillouin zone can be regarded as a closed 3-torus obtained by the identifying points *un***<sup>k</sup>** = *un*,**k**+**G***<sup>j</sup>* where **<sup>G</sup>***<sup>j</sup>* are the three primitive reciprocal lattice vectors. Then, the electronic polarization contribution of each band can be written as

$$\mathbf{P}\_{\rm ll} = \frac{-e}{2\pi\Omega} \sum\_{j} \Phi\_{\rm nj} \mathbf{R}\_{j} \tag{6}$$

where **R***<sup>j</sup>* is the real space primitive translational corresponding to **G***j*, and the Berry phase for band *n* in direction *j* is

$$\Phi\_{n\mathbf{j}} = -\frac{\Omega}{(2\pi)^3} \operatorname{Im} \int\_{BZ} d^3k \langle u\_{n\mathbf{k}} | \mathbf{G}\_{\mathbf{j}} \cdot \nabla\_{\mathbf{k}} | u\_{n\mathbf{k}} \rangle. \tag{7}$$

More details and discussion, including the equivalent of Eq.7 for the case of connected multiple bands, may be found in Refs.[7] and [8].

### *Reformulation in terms of Wannier Functions.*

4 Advances in Ferroelectrics

Then, from Ref.[7]

of volume Ω.

△**P** =

 *BZ*

(2*π*)<sup>3</sup> <sup>∑</sup>*<sup>n</sup>*

(2*π*)<sup>3</sup> <sup>∑</sup>*<sup>n</sup>*

of state *λ* with **P**(*λ*) and drop the *λ* label. Eq. 4 can be written then as

(2*π*)<sup>3</sup> *Im* <sup>∑</sup>*<sup>n</sup>*

*d***P** *<sup>d</sup><sup>λ</sup>* <sup>=</sup> *ie*

expression can be integrated with respect to *λ* to obtain

**<sup>P</sup>**(*λ*) = *ie*

**<sup>P</sup>** <sup>=</sup> *<sup>e</sup>*

must be added the nuclear (or ionic) contribution

phases of the Bloch functions at different **k**.

*d***P**

*d***k**�∇**k***un***k**|

and the rate of change of polarization with *λ* is a property of the occupied bands only. This

It can be verified by taking the *λ* derivative of both sides of Eq.4 and comparing with Eq.3. The result is independent of the particular path of *λ*(*t*) in time, and depends only on the final value of *λ* as long the change is adiabatically slow. We can associate the physical polarization

> *BZ*

and this is the electronic contribution to the polarization. To obtain the total polarization it

<sup>Ω</sup> <sup>∑</sup>*<sup>s</sup>*

where the sum is over atoms *s* having core charge *Zs* and spatial position **r***<sup>s</sup>* in the unit cell

The equation 5 is the main result of the modern theory of polarization It states that the electronic contribution to the polarization of crystalline insulator may be expressed as a Brillouin zone integral of an operator *i*∇**k**. However, it is not a common quantum mechanical operator, the result of its action on the wavefunctions (*i*∇**k**|*un***k**�) depends on the relative

To understand the nature of the integrand in Eq.5 I want to recall the fundamental paper by Zak [16] in which he postulates the existence of a Berry phase associated with each one of the bands of a one dimensional crystalline solid. He shows how the variation of *k* over the entire Brillouin zone produces the appearance of a Berry phase. Moreover, he associates this phase, that is a characteristic feature of the whole band, with the band center operator [17].

*Zs***r***s*,

**<sup>P</sup>***ion* <sup>=</sup> <sup>1</sup>

 *BZ* *dun***<sup>k</sup>**

*<sup>d</sup><sup>λ</sup> <sup>d</sup>λ*. (2)

*<sup>d</sup>***k**�*uλ*,*n***k**|∇**k**|*uλ*,*n***k**�. (4)

*d***k**�*un***k**|∇**k**|*un***k**�, (5)

*<sup>d</sup><sup>λ</sup>* � <sup>+</sup> *<sup>c</sup>*.*c*. (3)

We can rewrite Eq.5 in terms of the Wannier functions (WF's), which brings an alternative and more intuitive way of thinking about it. The WF's and Bloch functions can be regarded as two different orthonormal representations of the same occupied Hilbert space. The WF's are localized functions *wn***R**(**r**) that are labeled by band *n* and unit cell **R** and constructed by carrying out a unitary transformation of the Bloch states *ψn***k**. They are constructed via a Fourier transform of the form

$$|w\_{n\mathbf{R}}\rangle = \frac{\Omega}{(2\pi)^3} \int\_{BZ} d\mathbf{k} e^{i\mathbf{k}\cdot\mathbf{R}} |\psi\_{n\mathbf{k}}\rangle \tag{8}$$

where the Bloch states are normalized in one unit cell. There is some freedom in the choice of the these WF's, as the transformation is not unique. In particular, a set of Bloch functions

$$|\psi\_{n\mathbf{k}'}\rangle = e^{-i\beta\_n(\mathbf{k})}|\psi\_{n\mathbf{k}}\rangle \tag{9}$$

10.5772/52268

407

http://dx.doi.org/10.5772/52268

*zm*, (13)

Ferroelectrics at the Nanoscale: A First Principle Approach

*pj* (14)

where *s* and *m* run over ion cores (of charge *Qs* located at **R***s* ) and Wannier centers (of charge

They defined the Layer polarization (LP) along *z* for superlattices built from II-VI *ABO*<sup>3</sup> perovskites such as *BaTiO*3, *SrTiO*3, and *PbTiO*3. For these cases, they were able to decompose the system into neutral layers (that is *AO* and *BO*<sup>2</sup> subunits) and define a layer

*Qs***R***sz* <sup>−</sup> <sup>2</sup>*<sup>e</sup>*

in which the sums are restricted to entities belonging to layer *j*. *S* is the basal cell area and we are now focusing only on *z* components. The LP *pj* thus defined has units of dipole moment per unit area. The total polarization, with units of dipole moment per volume, is exactly

> *Pz* <sup>=</sup> <sup>1</sup> *c* ∑ *j*

They propose two conditions to be able to associate a physical meaning to the LP definition: (i) resolve the arbitrariness associated with the positions of the Wannier centers, and (ii)

As it will be shown in the next section, we applied this concept to thin ferroelectric films sandwiched between metal contacts. These kind of systems, in which metals and oxides coexist, bring new challenges. Condition (i) was satisfied using maximally localized wannier functions plus a disentanglement procedure (Ref.[21]) and (ii) was satisfied, even close to the

When dealing with ferroelectric thin films, a finite polarization normal to the surface will give rise to a depolarizing field and a huge amount of electrostatic energy, enough to be able to suppress the ferroelectric instability. In order to preserve ferroelectricity and minimize the total energy, the depolarization field must be screened, either by free charges coming from metallic electrodes or by breaking into a domain structure. I will go into more details about

In the case of a free standing slab of a ferroelectric material, a uniform polarization *P* with an out of plane direction will appear and originate a surface polarization charge density

associate without ambiguity the right number of Wannier centers to each layer.

<sup>Ω</sup> ∑ *m*∈*j*


*pj* <sup>=</sup> <sup>1</sup> *S* ∑ *s*∈*j*

*<sup>S</sup>* is the supercell lattice constant along *z*.

this, but first a basic electrostatic background will be given.

**Influence of the electrical boundary conditions**

polarization

where *<sup>c</sup>* = <sup>Ω</sup>

metal oxide interfaces.

**4. Depolarization field**

related to the sum of LP's via

results in WF's *wn***R**′ which are different from the *wn***R**. Usually, this "gauge" is set by some criterion that keeps the WF's well localized in real space, such as the minimum quadratic spread criterion introduced by Marzari and Vanderbilt [19]. However, it is expected that any physical quantity, such as the electronic polarization arising from band *n*, should be invariant with respect to the phase change *βn*(**k**).

Once we have obtained the WF's, we can locate the "wannier centers" **<sup>r</sup>***n***<sup>R</sup>** = �*wn***R**|**r**|*wn***R**�. It can be shown that

$$\mathbf{r}\_{\rm nR} = \mathbf{R} + \sum\_{j} \frac{\phi\_{\rm nj}}{2\pi} \mathbf{R}\_{j\prime} \tag{10}$$

where *φnj* is given by Eq.7. In simple words, the location of the *n*'th Wannier center in the unit cell is just given by the three Berry phases *φnj* of band *n* in the primitive lattice vector directions **R***j*. The key result, is that the polarization is just related to the Wannier centers by

$$\mathbf{P} = -\frac{\mathcal{e}}{\Omega} \sum\_{\mathcal{n}} \mathbf{r}\_{\mathcal{n}O}.\tag{11}$$

The Berry Phase theory can then be regarded as providing a mapping of the distributed quantum mechanical electronic charge density onto a lattice of negative point charges −*e* (see Fig.2).

### **3. Layer polarization definition**

One issue that has received much attention theoretically is how to quantify the concept of *local polarization*. This can be very useful in understanding the enhancement or suppression of spontaneous polarization; or even to find new spatial patterns of polarization that would remain hidden otherwise, as is the case that I will show in a following section. It can also be essential for characterizing and understanding interface contributions to such properties.

But first I will give a review of the concept introduced in 2006 by Wu.*et al.* [20] which is the one used here. As explained in the previous section, the modern theory of polarization establishes that the polarization of a crystal is expressed in the Wannier representation as the contribution of the ionic and electronic charge:

$$\mathbf{P} = \frac{1}{\Omega} \sum\_{\mathbf{s}} Q\_{\mathbf{s}} \mathbf{R}\_{\mathbf{s}} - \frac{2e}{\Omega} \sum\_{m} \mathbf{r}\_{m\prime} \tag{12}$$

where *s* and *m* run over ion cores (of charge *Qs* located at **R***s* ) and Wannier centers (of charge -2e located at **r***<sup>m</sup>* ), respectively, in the unit cell of volume Ω.

They defined the Layer polarization (LP) along *z* for superlattices built from II-VI *ABO*<sup>3</sup> perovskites such as *BaTiO*3, *SrTiO*3, and *PbTiO*3. For these cases, they were able to decompose the system into neutral layers (that is *AO* and *BO*<sup>2</sup> subunits) and define a layer polarization

$$p\_j = \frac{1}{S} \sum\_{s \in j} Q\_s \mathbf{R}\_{sz} - \frac{2e}{\Omega} \sum\_{m \in j} z\_{m\nu} \tag{13}$$

in which the sums are restricted to entities belonging to layer *j*. *S* is the basal cell area and we are now focusing only on *z* components. The LP *pj* thus defined has units of dipole moment per unit area. The total polarization, with units of dipole moment per volume, is exactly related to the sum of LP's via

$$P\_z = \frac{1}{c} \sum\_{j} p\_j \tag{14}$$

where *<sup>c</sup>* = <sup>Ω</sup> *<sup>S</sup>* is the supercell lattice constant along *z*.

They propose two conditions to be able to associate a physical meaning to the LP definition: (i) resolve the arbitrariness associated with the positions of the Wannier centers, and (ii) associate without ambiguity the right number of Wannier centers to each layer.

As it will be shown in the next section, we applied this concept to thin ferroelectric films sandwiched between metal contacts. These kind of systems, in which metals and oxides coexist, bring new challenges. Condition (i) was satisfied using maximally localized wannier functions plus a disentanglement procedure (Ref.[21]) and (ii) was satisfied, even close to the metal oxide interfaces.

### **4. Depolarization field**

6 Advances in Ferroelectrics

can be shown that

(see Fig.2).

with respect to the phase change *βn*(**k**).

**3. Layer polarization definition**

contribution of the ionic and electronic charge:

**<sup>P</sup>** <sup>=</sup> <sup>1</sup> Ω ∑*s*

where the Bloch states are normalized in one unit cell. There is some freedom in the choice of the these WF's, as the transformation is not unique. In particular, a set of Bloch functions

−*iβn*(**k**)

results in WF's *wn***R**′ which are different from the *wn***R**. Usually, this "gauge" is set by some criterion that keeps the WF's well localized in real space, such as the minimum quadratic spread criterion introduced by Marzari and Vanderbilt [19]. However, it is expected that any physical quantity, such as the electronic polarization arising from band *n*, should be invariant

Once we have obtained the WF's, we can locate the "wannier centers" **<sup>r</sup>***n***<sup>R</sup>** = �*wn***R**|**r**|*wn***R**�. It

*j*

where *φnj* is given by Eq.7. In simple words, the location of the *n*'th Wannier center in the unit cell is just given by the three Berry phases *φnj* of band *n* in the primitive lattice vector directions **R***j*. The key result, is that the polarization is just related to the Wannier centers by

Ω ∑*n*

The Berry Phase theory can then be regarded as providing a mapping of the distributed quantum mechanical electronic charge density onto a lattice of negative point charges −*e*

One issue that has received much attention theoretically is how to quantify the concept of *local polarization*. This can be very useful in understanding the enhancement or suppression of spontaneous polarization; or even to find new spatial patterns of polarization that would remain hidden otherwise, as is the case that I will show in a following section. It can also be essential for characterizing and understanding interface contributions to such properties. But first I will give a review of the concept introduced in 2006 by Wu.*et al.* [20] which is the one used here. As explained in the previous section, the modern theory of polarization establishes that the polarization of a crystal is expressed in the Wannier representation as the

*Qs***R***<sup>s</sup>* <sup>−</sup> <sup>2</sup>*<sup>e</sup>*

<sup>Ω</sup> ∑ *m*

*φnj* 2*π*

**<sup>r</sup>***n***<sup>R</sup>** <sup>=</sup> **<sup>R</sup>** <sup>+</sup> ∑

**<sup>P</sup>** <sup>=</sup> <sup>−</sup> *<sup>e</sup>*


**R***j*, (10)

**r***nO*. (11)

**r***m*, (12)

<sup>|</sup>*ψn***k**′� <sup>=</sup> *<sup>e</sup>*

When dealing with ferroelectric thin films, a finite polarization normal to the surface will give rise to a depolarizing field and a huge amount of electrostatic energy, enough to be able to suppress the ferroelectric instability. In order to preserve ferroelectricity and minimize the total energy, the depolarization field must be screened, either by free charges coming from metallic electrodes or by breaking into a domain structure. I will go into more details about this, but first a basic electrostatic background will be given.

### **Influence of the electrical boundary conditions**

In the case of a free standing slab of a ferroelectric material, a uniform polarization *P* with an out of plane direction will appear and originate a surface polarization charge density

**Figure 3.** Schematic representation of the planar averaged electrostatic potential (fill line) of a slab with polarization perpendicular to the surface under (a)*D* = 0 boundary conditions (vanishing external field) and b) a vanishing internal electric field *E* = 0. Plannar averages are taken on the (*x*, *y*) planes parallel to the surface. Dashed lines represent an average over unit cell of the planar averages. An estimate of the macroscopic internal field inside the slab can be obtained from their slope. *m* stands for the dipole moment parallel to the surface normal and *e* for the electron charge. From Ref.[23]

$$
\sigma\_{pol} = \mathbf{P}.\widehat{n}\_{\prime} \tag{15}
$$

10.5772/52268

409

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

**Figure 4.** Depolarization field for an isolated free standing slab in the absence of an external electric field (a), and for a ferroelectric capacitor with perfect screening (ideal metal) (b) and a real metallic electrode (c) under short circuit boundary

Another case, a vanishing internal electric field *E* = 0 corresponds to a case where no field opposes the polarization and a spontaneous polarization might arise (Fig.3b). In this case, it is the continuity of the normal component electric displacement vector that establishes the

These two cases illustrate how a polarization perpendicular to the surfaces can exist or not depending on the electrical boundary conditions. For simulating a real system, the mechanical boundary conditions should be included via atomic relaxations. These cases have been extensively analyzed [23–26] and have opened the ground for more realistic systems..

The next level of approximation to a real case,is the case of having the ferroelectric thin film sandwiched between metal contacts ( short circuit boundary conditions). In the case of a *perfect* metal, the surface charges *σpol* will be exactly compensated at the interfaces by the compensation charges *σcom* and we will have a null depolarization field (see Fig.4b). In this

electrode. However, in the case of *real electrodes*, the compensation of the polarization charges will be incomplete. In this case,the compensation charge *σcom* will be spread over a finite distance inside the electrode and will create an interface dipole density at each of the

*s* will lie in a sheet right at the interface between the metal and the ferroelectric

*Eext* = 4*πPs*. (17)

conditions. From Ref.[1]

value of it,

**Metal contacts**

case, both *<sup>σ</sup>*′

where *<sup>n</sup>* is a unit vector normal to the surface pointing outward. The charge density is proportional to the magnitude of the normal component of the polarization inside the material [22] and is positive one one side of the slab and negative on the other one. In order the determine the electric fields generated by the surface charge density, we need to know the electrical boundary conditions of the problem and solve the Poisson equation. For the general case, let's suppose that an external electric field *Eext*, perpendicular to the surface, is applied in the vacuum region. There, the electric displacement **D** = **E** + 4*π***P** equals the applied electrical field *D* = *Eext* as the polarization is zero. The normal component of *D* is conserved across the ferroelectric/vacuum interface,

$$E + 4\pi P = E\_{\text{ext}} \tag{16}$$

Now we can see two extreme cases. One of them is that of zero external electric field, *Eext* = 0, in which case the displacement vector is null while the internal field inside the slab is *E* = −4*πP* (see Fig.3a). The absence of an external electric field and the presence of the surface polarization charges produce an internal field that points in the direction opposite to that of the polarization. Therefore, it tends to restore the paraelectric configuration and that is the reason of its name, *Depolarization field*. The coupling between the polarization and this internal field is so big, that any atomic relaxation under these circumstances will end up with the atoms back in the centrosymmetric non polar configuration.

10.5772/52268

**Figure 4.** Depolarization field for an isolated free standing slab in the absence of an external electric field (a), and for a ferroelectric capacitor with perfect screening (ideal metal) (b) and a real metallic electrode (c) under short circuit boundary conditions. From Ref.[1]

Another case, a vanishing internal electric field *E* = 0 corresponds to a case where no field opposes the polarization and a spontaneous polarization might arise (Fig.3b). In this case, it is the continuity of the normal component electric displacement vector that establishes the value of it,

$$E\_{\rm ext} = 4\pi P\_{\rm s}.\tag{17}$$

These two cases illustrate how a polarization perpendicular to the surfaces can exist or not depending on the electrical boundary conditions. For simulating a real system, the mechanical boundary conditions should be included via atomic relaxations. These cases have been extensively analyzed [23–26] and have opened the ground for more realistic systems..

### **Metal contacts**

8 Advances in Ferroelectrics

**Figure 3.** Schematic representation of the planar averaged electrostatic potential (fill line) of a slab with polarization perpendicular to the surface under (a)*D* = 0 boundary conditions (vanishing external field) and b) a vanishing internal electric field *E* = 0. Plannar averages are taken on the (*x*, *y*) planes parallel to the surface. Dashed lines represent an average over unit cell of the planar averages. An estimate of the macroscopic internal field inside the slab can be obtained from their slope. *m*

where *<sup>n</sup>* is a unit vector normal to the surface pointing outward. The charge density is proportional to the magnitude of the normal component of the polarization inside the material [22] and is positive one one side of the slab and negative on the other one. In order the determine the electric fields generated by the surface charge density, we need to know the electrical boundary conditions of the problem and solve the Poisson equation. For the general case, let's suppose that an external electric field *Eext*, perpendicular to the surface, is applied in the vacuum region. There, the electric displacement **D** = **E** + 4*π***P** equals the applied electrical field *D* = *Eext* as the polarization is zero. The normal component of *D* is

Now we can see two extreme cases. One of them is that of zero external electric field, *Eext* = 0, in which case the displacement vector is null while the internal field inside the slab is *E* = −4*πP* (see Fig.3a). The absence of an external electric field and the presence of the surface polarization charges produce an internal field that points in the direction opposite to that of the polarization. Therefore, it tends to restore the paraelectric configuration and that is the reason of its name, *Depolarization field*. The coupling between the polarization and this internal field is so big, that any atomic relaxation under these circumstances will end up

*<sup>σ</sup>pol* <sup>=</sup> **<sup>P</sup>**.*<sup>n</sup>*, (15)

*E* + 4*πP* = *Eext* (16)

stands for the dipole moment parallel to the surface normal and *e* for the electron charge. From Ref.[23]

conserved across the ferroelectric/vacuum interface,

with the atoms back in the centrosymmetric non polar configuration.

The next level of approximation to a real case,is the case of having the ferroelectric thin film sandwiched between metal contacts ( short circuit boundary conditions). In the case of a *perfect* metal, the surface charges *σpol* will be exactly compensated at the interfaces by the compensation charges *σcom* and we will have a null depolarization field (see Fig.4b). In this case, both *<sup>σ</sup>*′ *s* will lie in a sheet right at the interface between the metal and the ferroelectric electrode. However, in the case of *real electrodes*, the compensation of the polarization charges will be incomplete. In this case,the compensation charge *σcom* will be spread over a finite distance inside the electrode and will create an interface dipole density at each of the ferroelectric/metal interfaces (see Fig.4c). Due to the short circuit boundary conditions, an amount of charge must flow from one interface to the other, creating a residual depolarization field inside the thin film. There are various models for expressing this field [24, 27–29], shown below is one described in Ref.[30]:

$$E\_{dep} = -\frac{8\pi P\lambda}{d} \tag{18}$$

10.5772/52268

411

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

**Figure 5.** Evolution of the *c*/*a* ratio with the film thickness for monodomain *PbTiO*<sup>3</sup> films grown epitaxially on top of Nb-doped *SrTiO*3. With circles and squares, the experimental results. The dashed line is the phenomenological theory prediction. Solid

**Figure 6.** Theoretical predictions of the thickness dependence of the normal average polarization *P*, the tetragonality *c*/*a*, and the out of plane piezoelectric constant *d*<sup>33</sup> at room temperature for *PbTiO*<sup>3</sup> thin films grown on a *SrRuO*3/*SrTiO*<sup>3</sup> substrate. Values of the quantities at the bulk level are represented by dashed lines for the unstrained configuration, and with dotted lines for a geometry under the strain imposed by the substrate. The evolution of the system, from a monodomain configuration at large thickness, (where the depolarization field *Ed* is small), to a 180◦ stripe domain structure in order to minimize the energy

first principle model Hamiltonian to thin films. It also has been used with the result of first principle calculations in Ref.[30]. In this work, we will use it as a starting point for a toy

There are two ways to minimize the energy due to the depolarization field , i) a reduction of the polarization, ii) a reduction of the depolarization field itself. This can be seen by

As an example of case i) it is shown in Fig.5 the experimental results for the tetragonality in function of the film thickness for monodomain *PbTiO*<sup>3</sup> epitaxially grown on Nb-doped *SrTiO*<sup>3</sup> [35]. The polarization is strongly coupled to strain in perovskites oxides [36], therefore a reduction of the polarization must be accompanied by a reduction of the tetragonality of the system. The incomplete screening of the depolarization field is the driving force for the reduction of the polarization. The structure remains in a monodomain

Case ii) might arise when there are no compensation charges provided by the electrodes or when the compensation charges do not provide an efficient enough screening of the polarization charges. In this case the system might break up into 180◦ stripe domains to

model that would mimic the result of our first principle calculations.

reduce the magnitude of the surface dipole density (see Fig.6).

lines correspond to the first principles model Hamiltonian results. From Ref. [35].

associated with *Ed*, is represented in the inset. From [37]

**4.1. Effects of the depolarization field**

inspecting the first term in Eq.21.

at all thickness.

where *d* is the film length, and *λ* is the effective screening length, proportional to the spatial spread of the interface dipole.

### **Electrostatic energy of the thin film**

In the presence of a residual depolarizing field the energy of the thin film *U* can be approximated by

$$E\left(P\_0\right) = \mathcal{U}\left(P\_0\right) + E\_{elec}\left(P\_0\right),\tag{19}$$

where *U* is the internal energy under zero field. If we assume that the interface effects are hidden in the screening length parameter, then they only appear through the depolarization field. This is a rough approximation but we will introduce later a model that will relax this condition, but let's keep it for now. Following this, then

$$
\mathcal{U}\left(P\_0\right) = \mathcal{U}\mathcal{U}\_{FE}\left(P\_0\right) \tag{20}
$$

where *m* stands for the number of unit cells of the film. *UFE* can be calculated from bulk DFT calculations. Its dependence on *P*<sup>0</sup> has typically the shape of a double well.

The electrostatic energy *Eelec* (*P*0) can be approximated by [1, 31]

$$E\_{elec}\left(P\_0\right) = m\Omega \left(-E\_{dep}.P\_0\right) + O\left(E\_{dep}\right)^2\tag{21}$$

The depolarization field can be calculated from first principles by taking the electrostatic average [32, 33] or can be expressed using Eq.18. In this case, Eq.19 can be rewritten as

$$E\left(P\_0\right) = \mathcal{U}\left(P\_0\right) + \frac{8\pi\lambda P\_0^2}{d} + \mathcal{O}\left(P\_0^4\right). \tag{22}$$

The electrostatic energy, the second term, is positive, meaning that the effect of the depolarization field is to suppress the ferroelectric instability by rescaling the quadratic term of the double well. This simple model has been implemented [34, 35] in order to extend the

10.5772/52268

**Figure 5.** Evolution of the *c*/*a* ratio with the film thickness for monodomain *PbTiO*<sup>3</sup> films grown epitaxially on top of Nb-doped *SrTiO*3. With circles and squares, the experimental results. The dashed line is the phenomenological theory prediction. Solid lines correspond to the first principles model Hamiltonian results. From Ref. [35].

**Figure 6.** Theoretical predictions of the thickness dependence of the normal average polarization *P*, the tetragonality *c*/*a*, and the out of plane piezoelectric constant *d*<sup>33</sup> at room temperature for *PbTiO*<sup>3</sup> thin films grown on a *SrRuO*3/*SrTiO*<sup>3</sup> substrate. Values of the quantities at the bulk level are represented by dashed lines for the unstrained configuration, and with dotted lines for a geometry under the strain imposed by the substrate. The evolution of the system, from a monodomain configuration at large thickness, (where the depolarization field *Ed* is small), to a 180◦ stripe domain structure in order to minimize the energy associated with *Ed*, is represented in the inset. From [37]

first principle model Hamiltonian to thin films. It also has been used with the result of first principle calculations in Ref.[30]. In this work, we will use it as a starting point for a toy model that would mimic the result of our first principle calculations.

### **4.1. Effects of the depolarization field**

10 Advances in Ferroelectrics

below is one described in Ref.[30]:

spread of the interface dipole.

approximated by

**Electrostatic energy of the thin film**

condition, but let's keep it for now. Following this, then

ferroelectric/metal interfaces (see Fig.4c). Due to the short circuit boundary conditions, an amount of charge must flow from one interface to the other, creating a residual depolarization field inside the thin film. There are various models for expressing this field [24, 27–29], shown

*Edep* <sup>=</sup> <sup>−</sup>8*πP<sup>λ</sup>*

where *d* is the film length, and *λ* is the effective screening length, proportional to the spatial

In the presence of a residual depolarizing field the energy of the thin film *U* can be

where *U* is the internal energy under zero field. If we assume that the interface effects are hidden in the screening length parameter, then they only appear through the depolarization field. This is a rough approximation but we will introduce later a model that will relax this

where *m* stands for the number of unit cells of the film. *UFE* can be calculated from bulk

−*Edep*.*P*<sup>0</sup>

8*πλP*<sup>2</sup> 0 *<sup>d</sup>* <sup>+</sup> *<sup>O</sup>*

 *P*4 0 

The depolarization field can be calculated from first principles by taking the electrostatic average [32, 33] or can be expressed using Eq.18. In this case, Eq.19 can be rewritten as

The electrostatic energy, the second term, is positive, meaning that the effect of the depolarization field is to suppress the ferroelectric instability by rescaling the quadratic term of the double well. This simple model has been implemented [34, 35] in order to extend the

 + *O Edep* 2

DFT calculations. Its dependence on *P*<sup>0</sup> has typically the shape of a double well.

The electrostatic energy *Eelec* (*P*0) can be approximated by [1, 31]

*Eelec* (*P*0) = *<sup>m</sup>*<sup>Ω</sup>

*E* (*P*0) = *U* (*P*0) +

*<sup>d</sup>* (18)

*E* (*P*0) = *U* (*P*0) + *Eelec* (*P*0), (19)

*U* (*P*0) = *mUFE* (*P*0) (20)

(21)

. (22)

There are two ways to minimize the energy due to the depolarization field , i) a reduction of the polarization, ii) a reduction of the depolarization field itself. This can be seen by inspecting the first term in Eq.21.

As an example of case i) it is shown in Fig.5 the experimental results for the tetragonality in function of the film thickness for monodomain *PbTiO*<sup>3</sup> epitaxially grown on Nb-doped *SrTiO*<sup>3</sup> [35]. The polarization is strongly coupled to strain in perovskites oxides [36], therefore a reduction of the polarization must be accompanied by a reduction of the tetragonality of the system. The incomplete screening of the depolarization field is the driving force for the reduction of the polarization. The structure remains in a monodomain at all thickness.

Case ii) might arise when there are no compensation charges provided by the electrodes or when the compensation charges do not provide an efficient enough screening of the polarization charges. In this case the system might break up into 180◦ stripe domains to reduce the magnitude of the surface dipole density (see Fig.6).


10.5772/52268

413

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

**Figure 8.** The thin film oxide develops a ferrielectric [47] pattern of polarization that exhibits antiferroelectric properties along the growth axis and with a net total polarization. This pattern would remain hidden by using more standard methods for

by the averaging procedure, or would have produced misleading results. In fact, by considering exclusively the ionic displacements (rumpling) as a measure of polarization, we do not account for the full electronic contributions to the local polarization and, depending on the system size, we might obtain results that would not describe accurately the physical

Here, we show that the ferroelectric structures associated with small length-scales are more complex than previously thought, mainly due to the redistribution of the electrons under the constraints imposed by the interfaces. In particular, the oxide develops a ferrielectric [47] pattern of polarization that exhibit antiferroelectric properties along the growth axis and

This unbalanced dipole structure is particularly evident in Fig.10A, where we show how positive rumpling on all the atomic layers of a thin BaTiO3 film correspond actually to layer dipoles of opposite signs. This is a consequence of the interplay between the orientation and magnitude of the dipoles with the DF, their mutual interaction and the nature of the interfaces. A simple analytic model of a ferroelectric thin film, where these effects are explicitly taken into account, can capture all these features and it will be discussed in detail

We have simulated thin films of BaTiO3 between Pt metal contacts in a (001) stack, where the oxide is terminated with a BaO plane at both interfaces. In this configuration, the metal atoms are directly bonded to the O atoms of the oxide plane.[30, 48] The different supercell constructed in this way can be labeled as Pt/(BaO-TiO2)m-BaO/Pt with m=1,2,4,6. Nine atomic planes of Pt have been found to be sufficient to simulate the contacts under short-circuit boundary conditions. The in plane lattice constant of the supercell is set equal to that of BaTiO3 at 0 K (3.991)[38], and kept it fixed in all calculations while all other geometrical parameters (atomic positions and intra-layer distances) were fully relaxed. All simulations have been performed using Density Functional Theory (DFT) within the Local Density Approximation, with ultrasoft pseudopotentials and a plane waves basis set.

calculating the local polarization.

behavior of the thin film.

at the end.

with a net total polarization (see Fig.8).

**Figure 7.** We simulated ferroelectric films at different thicknesses, sandwiched between metal contacts.

The reason why some systems remain in a monodomain configuration while other similar heterostructure break up into domains remains an open question that requires further clarification.

In a seminal theoretical paper, Junquera *et al.*[30] has suggested that the appearance of ferroelectricity in thin films of *BaTiO*3, with *SrRuO*<sup>3</sup> contacts, should be limited to a thicknesses of more than six unit cells. Below that limit, ferroelectricity would effectively be canceled by the strength of the depolarization field (DF) inside the oxide produced by unscreened charges at the interfaces, which, in that work, were calculated for the first time in a first principles framework. More recently this picture has been broadened, and a variety of studies have shown both experimentally and with more realistic ab initio simulations that thin oxide films can indeed maintain some ferroelectricity at thicknesses even smaller than six unit cells.[37–40]

### **5. The hidden nature of ferroelectricity at the nanoscale**

The size limit of ferroelectricity in ultrathin films can be understood using a simple electrostatics argument: as the thickness of the oxide film is reduced, the intensity of the DF is made stronger,[30] thus increasing the electrostatic energy of the system. In a ferroelectric crystal, this energy contribution can be minimized in two ways: either by breaking the polarization pattern in 180o stripe domains [41–44] to reduce the magnitude of the surface dipole density, or by a reduction of the ionic polarization while the system remains in a monodomain state.[35, 45] The final polarization pattern depends on the individual material and/or structure, and understanding which system modification will occur under what conditions is still a matter of debate.

At small dimensions, the spatial confinement and the interface details become extremely relevant since they determine the spatial localization of the electrons and thus influence directly the ferroelectric characteristics and, more in general, all the electronic properties of the system.[46] Therefore, to clarify the behavior of ferroelectric thin films in the nanometer regime it is crucial to obtain a precise description not only of the geometry and electronic properties of the film but first and foremost of the polarization profile inside the oxide.

In order to obtain a complete description of ferroelectricity at the nanoscale we have exploited the notion of layer polarization (LP). This idea, based on modern theory of polarization and the concept of maximally localized Wannier functions (see Sec.2), was recently introduced to describe the layer-by-layer modulation of polarization in ferroelectric superlattices (Sec.3) .

We have applied this method to a model metal-ferroelectric system (BaTiO3/Pt) (see Fig.7) and obtained a detailed spatial profile of the polarization in the oxide region in the direction of growth that accounts not only for the ionic displacements but also for the spatial rearrangement of the electron density. It is important to note that using more standard methods to evaluate macroscopic polarization, this information would have remained hidden

12 Advances in Ferroelectrics

clarification.

six unit cells.[37–40]

conditions is still a matter of debate.

**Figure 7.** We simulated ferroelectric films at different thicknesses, sandwiched between metal contacts.

**5. The hidden nature of ferroelectricity at the nanoscale**

The reason why some systems remain in a monodomain configuration while other similar heterostructure break up into domains remains an open question that requires further

In a seminal theoretical paper, Junquera *et al.*[30] has suggested that the appearance of ferroelectricity in thin films of *BaTiO*3, with *SrRuO*<sup>3</sup> contacts, should be limited to a thicknesses of more than six unit cells. Below that limit, ferroelectricity would effectively be canceled by the strength of the depolarization field (DF) inside the oxide produced by unscreened charges at the interfaces, which, in that work, were calculated for the first time in a first principles framework. More recently this picture has been broadened, and a variety of studies have shown both experimentally and with more realistic ab initio simulations that thin oxide films can indeed maintain some ferroelectricity at thicknesses even smaller than

The size limit of ferroelectricity in ultrathin films can be understood using a simple electrostatics argument: as the thickness of the oxide film is reduced, the intensity of the DF is made stronger,[30] thus increasing the electrostatic energy of the system. In a ferroelectric crystal, this energy contribution can be minimized in two ways: either by breaking the polarization pattern in 180o stripe domains [41–44] to reduce the magnitude of the surface dipole density, or by a reduction of the ionic polarization while the system remains in a monodomain state.[35, 45] The final polarization pattern depends on the individual material and/or structure, and understanding which system modification will occur under what

At small dimensions, the spatial confinement and the interface details become extremely relevant since they determine the spatial localization of the electrons and thus influence directly the ferroelectric characteristics and, more in general, all the electronic properties of the system.[46] Therefore, to clarify the behavior of ferroelectric thin films in the nanometer regime it is crucial to obtain a precise description not only of the geometry and electronic properties of the film but first and foremost of the polarization profile inside the oxide.

In order to obtain a complete description of ferroelectricity at the nanoscale we have exploited the notion of layer polarization (LP). This idea, based on modern theory of polarization and the concept of maximally localized Wannier functions (see Sec.2), was recently introduced to describe the layer-by-layer modulation of polarization in ferroelectric superlattices (Sec.3) . We have applied this method to a model metal-ferroelectric system (BaTiO3/Pt) (see Fig.7) and obtained a detailed spatial profile of the polarization in the oxide region in the direction of growth that accounts not only for the ionic displacements but also for the spatial rearrangement of the electron density. It is important to note that using more standard methods to evaluate macroscopic polarization, this information would have remained hidden

**Figure 8.** The thin film oxide develops a ferrielectric [47] pattern of polarization that exhibits antiferroelectric properties along the growth axis and with a net total polarization. This pattern would remain hidden by using more standard methods for calculating the local polarization.

by the averaging procedure, or would have produced misleading results. In fact, by considering exclusively the ionic displacements (rumpling) as a measure of polarization, we do not account for the full electronic contributions to the local polarization and, depending on the system size, we might obtain results that would not describe accurately the physical behavior of the thin film.

Here, we show that the ferroelectric structures associated with small length-scales are more complex than previously thought, mainly due to the redistribution of the electrons under the constraints imposed by the interfaces. In particular, the oxide develops a ferrielectric [47] pattern of polarization that exhibit antiferroelectric properties along the growth axis and with a net total polarization (see Fig.8).

This unbalanced dipole structure is particularly evident in Fig.10A, where we show how positive rumpling on all the atomic layers of a thin BaTiO3 film correspond actually to layer dipoles of opposite signs. This is a consequence of the interplay between the orientation and magnitude of the dipoles with the DF, their mutual interaction and the nature of the interfaces. A simple analytic model of a ferroelectric thin film, where these effects are explicitly taken into account, can capture all these features and it will be discussed in detail at the end.

We have simulated thin films of BaTiO3 between Pt metal contacts in a (001) stack, where the oxide is terminated with a BaO plane at both interfaces. In this configuration, the metal atoms are directly bonded to the O atoms of the oxide plane.[30, 48] The different supercell constructed in this way can be labeled as Pt/(BaO-TiO2)m-BaO/Pt with m=1,2,4,6. Nine atomic planes of Pt have been found to be sufficient to simulate the contacts under short-circuit boundary conditions. The in plane lattice constant of the supercell is set equal to that of BaTiO3 at 0 K (3.991)[38], and kept it fixed in all calculations while all other geometrical parameters (atomic positions and intra-layer distances) were fully relaxed. All simulations have been performed using Density Functional Theory (DFT) within the Local Density Approximation, with ultrasoft pseudopotentials and a plane waves basis set.

10.5772/52268

415

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

The LP profile calculated for the metal-oxide systems provides a detailed local description of the polarization, which is superior and even contradictory to the information obtained from empirical evaluations based solely on ionic displacements. As an illustration, in Fig.10A we display the rumpling profile (cation-oxygen normal distance within each atomic layer.) for a thin film comprised of six BaTiO3 unit cells (m6). From the ionic displacement profile one would be tempted to conclude that the film is in a ferroelectric domain configuration, since the rumpling values (and hence the *classical* local dipoles) associated to each plane are positive. Instead, if the correspondent LP profile is analyzed, a completely different picture emerges, where the main feature is an alternation of positive and negative values of the LP in the individual atomic layers of the oxide film. This is clearly shown in Fig.10B, where we plot the LPs values computed for m6. Here the central BaO planes (the BaO planes at the interface behave differently and will be discussed later) have positive dipoles while the TiO2 planes have negative ones. The first are aligned with the direction of the DF, while latter oppose it. A schematic illustration of the associated dipoles is drawn above the plot. This result implies that what could have been interpreted as a ferroelectric domain using standard geometrical information, in fact displays a much more complex spatial pattern of polarization. We believe that this effect is comparable to the onset of the formation of two-dimensional domain islands commonly observed both theoretically and experimentally in ferroelectric films.[37] In that case, the domains are composed of adjacent 180◦ domains in order to minimize the electrostatic energy associated with the DF. Here, our results show that the system can reduce its electrostatic energy in an alternative way: the energetic interplay between the dipoles, the DF and the interface bonding drives the individual atomic layer dipoles to arrange themselves in an uncompensated dipole pattern along the growth direction (see inset of Fig.10A), or, in other words, the system undergoes a ferroelectric-to-ferrielectric phase transition. Similarly to the formation of two-dimensional islands, we also predict the existence of a critical thickness (CT) above which the system exists in a true ferroelectric domain with all the dipoles pointing in the same direction (see Fig8B, where the LP's values

As the thickness is reduced, the magnitude of all the dipoles is diminished by the increase of the DF that opposes them. Due to the different physical properties of consecutive layers (BaO and TiO2), their associated dipoles reduce their magnitude at different rates. At the CT, the BaO layers will have null dipoles and eventually they will flip layer polarization direction, aligning with the DF in order to minimize the associated internal energy and creating a

The formation and characteristics of the FDP are determined by the bonding properties of the interfaces that pin the LP values of the outer layers of the oxide film. At small film thicknesses the interface effects become dominant and induce the overall spatial variation of the LP's that determines the orientation and magnitude of the layer dipoles along the structure. We can see how the FDP observed in the m6 geometry is increasingly influenced by the locality of the interfaces as the thickness is reduced to m4, m2 and m1 (see Fig.10C,D,E respectively). The middle crystal planes become more and more influenced by the interface environment, while the FDP becomes irregular, and finally disappear when the film is comprised by just 3 layers (i.e. one unit cell, m1), leaving instead a centro-symmetric paraelectric structure (Fig.10E). It is important to note that exclusively electronic effects drive this series of regime transitions. In fact, the ionic polarization, directly proportional to the ionic rumpling, remains almost

for bulk BaTiO3 are shown with triangles).

ferrielectric dipole pattern (FDP).

**Figure 9.** The set of MLWFs so obtained cluster around each atomic layer in sets containing the correct number of Wannier function needed to keep the layer charge neutral. Here we can see the *z* position (horizontal lines lines ) of the centers of the Wannier functions and the position of the ions (indicated with circles). To calculate the LP of each layer, we use Eq.12.

We used the expression of Eq.12 to calculate the LP for each plane of the thin film. A is the in plane area of the cell, and we are only interested on the z components (parallel to the growth direction). It is important to stress that in the definition of the LP we have explicitly included the full ionic and electronic information necessary to define unambiguously the local value of the polarization. The essential step for the evaluation of the layer polarizations *pj* is the determination of the Wannier centers *zm*. We used the maximally localized Wannier functions (MLWFs) algorithm originally proposed by Marzari and Vanderbilt[19] as implemented in the WanT code1. Since the valence bands of the oxide and the bands of the metal are mixed in the full supercell calculation, a disentanglement procedure (Ref.[21]) was applied before starting the localization algorithm[19, 49].

The set of MLWFs so obtained cluster around each atomic layer in sets containing the correct number of Wannier function needed to keep the layer charge neutral (Fig.9). This is a crucial condition for the validity of the definition of LP's in Eq.12 and might not be attainable in systems with more covalent nature.[20] In fact, the meaningful use of this concept is based on the condition of being able to decompose the system into neutral layers (TiO and BaO2 units in the case of perovskite BaTiO3) and resolve the arbitrariness associated with the positions of the Wannier centers. This allows us to associate each sets of Wannier centers to the corresponding crystal plane without ambiguity. By doing so we go back to the classic idea of defining dipoles in spatial neutral units, along the lines of the Clausius-Mossotti limit.v Moreover, we can associate each LP value *pj* with its correspondent (dimensionally correct) dipole *pjA*, which gives precise information about the orientation and magnitude of the atomic layer dipoles inside the crystal.

<sup>1</sup> code by A. Ferretti, B. Bonferroni, A. Calzolari, and M. Buongiorno Nardelli, (http://www.wannier-transport.org). In the non-self-consistent calculation for obtaining the MLWFs we used MP k meshes 2x2xN with N=16,14,12,10 for m=1,2,4,6 respectively. I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001). ote

The LP profile calculated for the metal-oxide systems provides a detailed local description of the polarization, which is superior and even contradictory to the information obtained from empirical evaluations based solely on ionic displacements. As an illustration, in Fig.10A we display the rumpling profile (cation-oxygen normal distance within each atomic layer.) for a thin film comprised of six BaTiO3 unit cells (m6). From the ionic displacement profile one would be tempted to conclude that the film is in a ferroelectric domain configuration, since the rumpling values (and hence the *classical* local dipoles) associated to each plane are positive. Instead, if the correspondent LP profile is analyzed, a completely different picture emerges, where the main feature is an alternation of positive and negative values of the LP in the individual atomic layers of the oxide film. This is clearly shown in Fig.10B, where we plot the LPs values computed for m6. Here the central BaO planes (the BaO planes at the interface behave differently and will be discussed later) have positive dipoles while the TiO2 planes have negative ones. The first are aligned with the direction of the DF, while latter oppose it. A schematic illustration of the associated dipoles is drawn above the plot. This result implies that what could have been interpreted as a ferroelectric domain using standard geometrical information, in fact displays a much more complex spatial pattern of polarization. We believe that this effect is comparable to the onset of the formation of two-dimensional domain islands commonly observed both theoretically and experimentally in ferroelectric films.[37] In that case, the domains are composed of adjacent 180◦ domains in order to minimize the electrostatic energy associated with the DF. Here, our results show that the system can reduce its electrostatic energy in an alternative way: the energetic interplay between the dipoles, the DF and the interface bonding drives the individual atomic layer dipoles to arrange themselves in an uncompensated dipole pattern along the growth direction (see inset of Fig.10A), or, in other words, the system undergoes a ferroelectric-to-ferrielectric phase transition. Similarly to the formation of two-dimensional islands, we also predict the existence of a critical thickness (CT) above which the system exists in a true ferroelectric domain with all the dipoles pointing in the same direction (see Fig8B, where the LP's values for bulk BaTiO3 are shown with triangles).

14 Advances in Ferroelectrics

**Figure 9.** The set of MLWFs so obtained cluster around each atomic layer in sets containing the correct number of Wannier function needed to keep the layer charge neutral. Here we can see the *z* position (horizontal lines lines ) of the centers of the Wannier functions and the position of the ions (indicated with circles). To calculate the LP of each layer, we use Eq.12.

We used the expression of Eq.12 to calculate the LP for each plane of the thin film. A is the in plane area of the cell, and we are only interested on the z components (parallel to the growth direction). It is important to stress that in the definition of the LP we have explicitly included the full ionic and electronic information necessary to define unambiguously the local value of the polarization. The essential step for the evaluation of the layer polarizations *pj* is the determination of the Wannier centers *zm*. We used the maximally localized Wannier functions (MLWFs) algorithm originally proposed by Marzari and Vanderbilt[19] as implemented in the WanT code1. Since the valence bands of the oxide and the bands of the metal are mixed in the full supercell calculation, a disentanglement procedure (Ref.[21]) was applied before

The set of MLWFs so obtained cluster around each atomic layer in sets containing the correct number of Wannier function needed to keep the layer charge neutral (Fig.9). This is a crucial condition for the validity of the definition of LP's in Eq.12 and might not be attainable in systems with more covalent nature.[20] In fact, the meaningful use of this concept is based on the condition of being able to decompose the system into neutral layers (TiO and BaO2 units in the case of perovskite BaTiO3) and resolve the arbitrariness associated with the positions of the Wannier centers. This allows us to associate each sets of Wannier centers to the corresponding crystal plane without ambiguity. By doing so we go back to the classic idea of defining dipoles in spatial neutral units, along the lines of the Clausius-Mossotti limit.v Moreover, we can associate each LP value *pj* with its correspondent (dimensionally correct) dipole *pjA*, which gives precise information about the orientation and magnitude of

<sup>1</sup> code by A. Ferretti, B. Bonferroni, A. Calzolari, and M. Buongiorno Nardelli, (http://www.wannier-transport.org). In the non-self-consistent calculation for obtaining the MLWFs we used MP k meshes 2x2xN with N=16,14,12,10 for

m=1,2,4,6 respectively. I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001). ote

starting the localization algorithm[19, 49].

the atomic layer dipoles inside the crystal.

As the thickness is reduced, the magnitude of all the dipoles is diminished by the increase of the DF that opposes them. Due to the different physical properties of consecutive layers (BaO and TiO2), their associated dipoles reduce their magnitude at different rates. At the CT, the BaO layers will have null dipoles and eventually they will flip layer polarization direction, aligning with the DF in order to minimize the associated internal energy and creating a ferrielectric dipole pattern (FDP).

The formation and characteristics of the FDP are determined by the bonding properties of the interfaces that pin the LP values of the outer layers of the oxide film. At small film thicknesses the interface effects become dominant and induce the overall spatial variation of the LP's that determines the orientation and magnitude of the layer dipoles along the structure. We can see how the FDP observed in the m6 geometry is increasingly influenced by the locality of the interfaces as the thickness is reduced to m4, m2 and m1 (see Fig.10C,D,E respectively). The middle crystal planes become more and more influenced by the interface environment, while the FDP becomes irregular, and finally disappear when the film is comprised by just 3 layers (i.e. one unit cell, m1), leaving instead a centro-symmetric paraelectric structure (Fig.10E). It is important to note that exclusively electronic effects drive this series of regime transitions. In fact, the ionic polarization, directly proportional to the ionic rumpling, remains almost

10.5772/52268

417

http://dx.doi.org/10.5772/52268

(24)

*M* )

**Figure 11.** Each individual crystal unit cell is comprised of two consecutive layers of BaO and TiO2. A *local polarization Pi* can be associated with it. It is calculated by adding both individual LP's and dividing by the lattice constant in the stacking direction, *c*. We can associate a dipole *pi A* to each plane *i*, separated from its next neighbors by distance *dij*. The full polarization profile of the film is completely specified by the full set of dipoles (*pL*, *p*2, ..., *p*2*M*<sup>−</sup>1, *pR*). The LP at the interfaces (*pL* and *pR*), are kept fixed. Two distinct interaction parameters *θ<sup>L</sup>* and *θR*, establish the locality of the interface environment.[50] The interaction

where each term of the first sum accounts for the internal energy of the individual unit cells under zero electric field. The parameters a and b are obtained from an ab initio calculation of the total energy of a BaTiO3 bulk for different ionic displacements along the soft mode.[51] The contribution from the depolarization field *E* is included in the second sum [22, 30] where Ω is the volume of the unit cell. Each one of its terms favors energetically the local dipoles that follow the same direction than the DF. The third sum represents the classical electrostatic

(−Ω**P***i*.**E**) + *Uint*(*pL*, *<sup>p</sup>*2, ..., *<sup>p</sup>*2*M*−1, *pR*), (23)

Ferroelectrics at the Nanoscale: A First Principle Approach

parameter *θij* for the internal dipoles is arbitrarily chosen to be unity.

(−*aP*<sup>2</sup>

*<sup>i</sup>* <sup>+</sup> *bP*<sup>4</sup>

energy for a series of dipoles separated by a distance *dij*,

*<sup>i</sup>* ) +

*M* ∑ *i*=1

*Uint* <sup>=</sup> <sup>−</sup> *<sup>A</sup>*<sup>2</sup>

8*πǫ*

The above sum can be separated in three contributions: two that account for the interaction between the interface dipoles with the internal ones, and another one that accounts for the mutual electrostatic interaction between the internal dipoles. We have chosen to assign a fixed magnitude and direction to the interface layer dipoles (*pL* and *pR*), with two distinct interaction parameters *θ<sup>L</sup>* and *θR*, thus establishing the locality of the interface environment.[50] The interaction parameter *θij* for the internal dipoles is arbitrarily chosen

With the constraint imposed by the simplicity of the model, we look for the energetically more favorable spatial polarization profile as a function of the film thickness. The spatial profile of polarization that characterize the film can be expressed as the vector (*pL*, *pTiO*<sup>2</sup> , *pBaO*, *pTiO*<sup>2</sup> , ..., *pBaO*, *pTiO*<sup>2</sup> , *pR*) where a ferroelectric domain and a FDP will have all dipoles parallel or antiparallel respectively. Note that the total energy per unit cell (*E*′ = *<sup>E</sup>*

an be expressed as function of only two single variables *pTiO*<sup>2</sup> and *pBaO* in the form:

2*M* ∑ *i*,*j*=1 *θij pj pi d*3 *ij*

*E* = *M* ∑ *i*=1

to be unity.

**Figure 10.** A. Rumpling (cation-oxygen perpendicular distance (in Å) within each atomic layer) profile for six unit cells (m6) of BaTiO<sup>3</sup> sandwiched between Pt contacts. (on the horizontal axis Ti and Ba indicate the position of the individual TiO2 and BaO planes). B. Solid circles (blue on line): layer polarization (LP) profile for the same system m6 in units of 10−10*C*/*m*. The values for the equivalent planes in bulk BaTiO<sup>3</sup> are shown with black triangles. The orientation of the individual layer dipoles is displayed by arrows in the bottom of the panel. C.D.E. LP profiles for m4, m2 and m1. As the thickness is reduced the FDP is modified by the increasingly dominant interface effects.

constant for all the thicknesses considered, and it is only the electronic polarization that changes its values in the different systems.

These results show that the ferroelectric response of a thin film is indeed critically influenced by the interface properties of the system and that the analysis of the local variation of layer polarization captures completely the physical characteristics of the ferroelectric. As the size of the film is increased, the DF decreases and the dipoles that are oriented along the DF direction get smaller, while the interface effects lose relevance. At the critical size, the dipoles flip direction and a ferroelectric domain structure is established again. Eventually, the dipoles will relax to the bulk structure, showed with triangles in Fig.10

### **Model**

We further illustrate these physical principles by developing a simple classical model of a ferroelectric thin film that reproduces the appearance of a FDP. We extended the model introduced in Ref. ([30]) by discretizing the thin film along the growth direction in each of its component unit cells in order to introduce a local spatial description of the polarization with the introduction of the *local polarization*, *Pi*. Each individual crystal unit cell is comprised of two consecutive layers of BaO and TiO2, therefore *Pi* could be obtained by adding each individual LP and dividing by the lattice constant in the stacking direction, *c*. The full polarization profile of the film is completely specified by the full set of LPs (*p*1, *p*2, ..., *p*2*M*−1, *p*2*M*) associated with each crystal plane of the oxide film (see Fig.11). We can write the total energy of the thin film composed by M unit cells as:

16 Advances in Ferroelectrics

modified by the increasingly dominant interface effects.

changes its values in the different systems.

**Model**

will relax to the bulk structure, showed with triangles in Fig.10

can write the total energy of the thin film composed by M unit cells as:

**Figure 10.** A. Rumpling (cation-oxygen perpendicular distance (in Å) within each atomic layer) profile for six unit cells (m6) of BaTiO<sup>3</sup> sandwiched between Pt contacts. (on the horizontal axis Ti and Ba indicate the position of the individual TiO2 and BaO planes). B. Solid circles (blue on line): layer polarization (LP) profile for the same system m6 in units of 10−10*C*/*m*. The values for the equivalent planes in bulk BaTiO<sup>3</sup> are shown with black triangles. The orientation of the individual layer dipoles is displayed by arrows in the bottom of the panel. C.D.E. LP profiles for m4, m2 and m1. As the thickness is reduced the FDP is

constant for all the thicknesses considered, and it is only the electronic polarization that

These results show that the ferroelectric response of a thin film is indeed critically influenced by the interface properties of the system and that the analysis of the local variation of layer polarization captures completely the physical characteristics of the ferroelectric. As the size of the film is increased, the DF decreases and the dipoles that are oriented along the DF direction get smaller, while the interface effects lose relevance. At the critical size, the dipoles flip direction and a ferroelectric domain structure is established again. Eventually, the dipoles

We further illustrate these physical principles by developing a simple classical model of a ferroelectric thin film that reproduces the appearance of a FDP. We extended the model introduced in Ref. ([30]) by discretizing the thin film along the growth direction in each of its component unit cells in order to introduce a local spatial description of the polarization with the introduction of the *local polarization*, *Pi*. Each individual crystal unit cell is comprised of two consecutive layers of BaO and TiO2, therefore *Pi* could be obtained by adding each individual LP and dividing by the lattice constant in the stacking direction, *c*. The full polarization profile of the film is completely specified by the full set of LPs (*p*1, *p*2, ..., *p*2*M*−1, *p*2*M*) associated with each crystal plane of the oxide film (see Fig.11). We

**Figure 11.** Each individual crystal unit cell is comprised of two consecutive layers of BaO and TiO2. A *local polarization Pi* can be associated with it. It is calculated by adding both individual LP's and dividing by the lattice constant in the stacking direction, *c*. We can associate a dipole *pi A* to each plane *i*, separated from its next neighbors by distance *dij*. The full polarization profile of the film is completely specified by the full set of dipoles (*pL*, *p*2, ..., *p*2*M*<sup>−</sup>1, *pR*). The LP at the interfaces (*pL* and *pR*), are kept fixed. Two distinct interaction parameters *θ<sup>L</sup>* and *θR*, establish the locality of the interface environment.[50] The interaction parameter *θij* for the internal dipoles is arbitrarily chosen to be unity.

$$E = \sum\_{i=1}^{M} (-aP\_i^2 + bP\_i^4) + \sum\_{i=1}^{M} (-\Omega \mathbf{P}\_i.\mathbf{E}) + \mathcal{U}\_{int}(p\_{L\prime}, p\_{2\prime}, \dots, p\_{2M-1\prime}, p\_{\mathcal{R}})\_{\prime} \tag{23}$$

where each term of the first sum accounts for the internal energy of the individual unit cells under zero electric field. The parameters a and b are obtained from an ab initio calculation of the total energy of a BaTiO3 bulk for different ionic displacements along the soft mode.[51] The contribution from the depolarization field *E* is included in the second sum [22, 30] where Ω is the volume of the unit cell. Each one of its terms favors energetically the local dipoles that follow the same direction than the DF. The third sum represents the classical electrostatic energy for a series of dipoles separated by a distance *dij*,

$$\mathcal{U}\_{int} = -\frac{A^2}{8\pi\varepsilon} \sum\_{i,j=1}^{2M} \frac{\theta\_{ij} p\_j p\_i}{d\_{ij}^3} \tag{24}$$

The above sum can be separated in three contributions: two that account for the interaction between the interface dipoles with the internal ones, and another one that accounts for the mutual electrostatic interaction between the internal dipoles. We have chosen to assign a fixed magnitude and direction to the interface layer dipoles (*pL* and *pR*), with two distinct interaction parameters *θ<sup>L</sup>* and *θR*, thus establishing the locality of the interface environment.[50] The interaction parameter *θij* for the internal dipoles is arbitrarily chosen to be unity.

With the constraint imposed by the simplicity of the model, we look for the energetically more favorable spatial polarization profile as a function of the film thickness. The spatial profile of polarization that characterize the film can be expressed as the vector (*pL*, *pTiO*<sup>2</sup> , *pBaO*, *pTiO*<sup>2</sup> , ..., *pBaO*, *pTiO*<sup>2</sup> , *pR*) where a ferroelectric domain and a FDP will have all dipoles parallel or antiparallel respectively. Note that the total energy per unit cell (*E*′ = *<sup>E</sup> M* ) an be expressed as function of only two single variables *pTiO*<sup>2</sup> and *pBaO* in the form:

$$E'(p\_{\rm TO2}, p\_{\rm BoO}) = -a \left(\frac{p\_{\rm TO2} + p\_{\rm BoO}}{c}\right)^2 + b \left(\frac{p\_{\rm TO2} + p\_{\rm BoO}}{c}\right)^4 + \Omega E \left(\frac{p\_{\rm TO2} + p\_{\rm BoO}}{c}\right)$$

$$-\frac{A^2}{4\pi\epsilon} \left(\frac{ap\_{\rm BoO}}{Mc^3} + \frac{p\_{\rm R}p\_L}{M(2M-1)c^3} + \frac{1}{M} \sum\_{i=3}^{2M-1} \frac{p\_{\rm L}p\_i}{d\_{\rm ij}^3} + \frac{1}{M} \sum\_{i=3}^{2M-1} \frac{p\_{\rm i}p\_R}{d\_{\rm iR}^3} + \frac{1}{2M} \sum\_{i,j=4}^{2M-2} \frac{p\_{i\}p\_j}{d\_{ij}^3}\right) \tag{25}$$

10.5772/52268

419

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

**Figure 12.** Energy landscape *<sup>E</sup>*'(*pTiO*<sup>2</sup> , *pBaO*) for different values of *<sup>θ</sup>*/*<sup>M</sup>* (film thickness). The interaction parameters for both interfaces were considered equal, *thetaR* = *θ<sup>L</sup>* = *θ* and were kept fixed as the film size *M* is varied. The remaining parameters of the model have been derived from our ab initio calculations. A. *θ*/*M* = 0, bulk limit where the usual double well potential produces two equivalent minima with all the layer dipoles parallel. B.*θ*/*M* = 3, as the thickness is reduced the interface effects gain relevance and start to modify the energy landscape. The layer dipoles associated with the BaO planes reduce their magnitude in order to minimize the total energy. C. At *θ*/*M* = 5 the system reaches a critical thickness where these dipoles

consequence of the complex energetic competition between the interface effects, the DF, and orientation and mutual interaction of the layer dipoles. The appearance of this particular ferrielectric state can be understood using a simple phenomenological model where the

These results suggest the possibility that such FDPs could be the normal state of a ferroelectric thin film at the nanoscale, even combined with the formation of two-dimensional island domains. It would be tempting to link the formation of such FDPs to the appearance or not of two-dimensional island below the critical thickness, and to understand the ultimate dependence upon the detailed interface structure and the nature of the metal contact. The

In this section we study the paradigmatic case of a BaTiO3 film between Pt contacts, already introduced in the previous section, where we modify the ferroelectric properties by a selective control of the chemical species present at the interface. In particular, by inserting a single layer of a different metal (Au,Cu) we demonstrate that we are able to tune the polarization of the ferroelectric film via the modification of the screening properties of the composite metal

We have simulated thin films of BaTiO3 between Pt metal contacts in a (001) supercell. The oxide is terminated with a BaO plane at both interfaces, with the metal atoms directly bonded to the O atoms of the oxide planei,ii. The supercells constructed in this way can be labeled as *Pt*/(*M*)/(*BaO* − *TiO*2)*<sup>m</sup>* − *BaO*/(*M*)/*Pt* with m=1,2,4,6 and M=(Pt, Au or Cu). Nine

become zero and eventually flip orientation. D. For *θ*/*M* = 10 the system display a ferrielectric dipole pattern.

interface effects are explicitly taken into account.

next section will explore in detail the former point.

**6.1. Methods and discussion**

**6. Tuning of polarization in metal-ferroelectric junctions**

contacts and through the change in the geometrical constraints at the interface.

where we used Eqs.2324 and we define an overall interface parameter *α* = (*θ<sup>R</sup> pR* + *θ<sup>L</sup> pL*) . Note that the thickness of the film is given by the number of unit cells M and directly affects the magnitude of the depolarization field *E*. The DF is calculated from an ab initio calculation for a supercell with the same size *M* and using a well-established averaging technique for the electrostatic potential inside the film.[50]

As *M* → ∞ , the terms in Eq.25 that describe the mutual interaction between the dipoles converge to constant values, the DF vanishes and only the first two terms remain in *E*?, recovering the bulk properties. Only for small *M* these terms contribute appreciably to the total energy of the system. This is true in particular for the interface term that contains the parameter (fourth term in Eq. 25) that carries the information that defines the interfaces. We can estimate the interface parameters *θR*, *θL*, *pL*, *pR* from our ab initio simulations<sup>2</sup> and we assume that their numerical values will not depend upon the film thickness (as observed in the actual calculations, at least to first order).

As we vary the thickness of the ferroelectric film, we observe a change of the total energy landscape consequence of the interplay between the different energetic contributions. This is clearly seen in Fig. 12, where we show contour plots of *E*'(*pTiO*<sup>2</sup> , *pBaO*) for different values of *M*. On the left, a diagram with the correspondent spatial distribution of dipoles is indicated. For thick films, the spatial distribution of dipoles that minimizes the total energy forms a domain (both LP's in the BaO and TiO2 planes have the same sign), as can be seen in the position of the minimum in the contour plot in Fig.12A. As the number of unit cells is lowered, the minimum shifts to lower values of *pBaO* until at a critical thickness (Fig. 12B) it becomes zero. Further reduction of the number of layers M flips only the dipoles belonging to BaO planes thus establishing a FDP (see Fig. 12D). In conclusion, this simple toy model captures the general physics of the thin film, and illustrates the intimate relation between the interface characteristics, the thickness of the film, and the existence of an FDP.

Thus, by exploiting the concept of layer polarization in the description of ferroelectric thin films between metal contacts, we have been able to obtain detailed information on the modulation of polarization at the nanoscale and to understand the constraining effects of the interfaces in the determination of the ferroelectric response of the system. Our results shows that when film thicknesses reach a critical value, the ferroelectric system responds via a transition from the bulk ferroelectric structure to a ferrielectric antidipole pattern, where individual atomic layers acquire uncompensated opposing dipoles. This state arises as

<sup>2</sup> The DF does not change much for the films thicknesses considered in this work. Therefore the value used as input for the model was calculated from the m6 averaged electrostatic potential (*E* = 2.141014*V*/*m*). The values for both interface layer dipoles used are (*pR* + *pL*)/*c* = 1.0*C*/*m*2). In the model we also chose *c*/*a* = 0.98, respecting the tetragonal structure of the internal cells of the oxide film.

**Figure 12.** Energy landscape *<sup>E</sup>*'(*pTiO*<sup>2</sup> , *pBaO*) for different values of *<sup>θ</sup>*/*<sup>M</sup>* (film thickness). The interaction parameters for both interfaces were considered equal, *thetaR* = *θ<sup>L</sup>* = *θ* and were kept fixed as the film size *M* is varied. The remaining parameters of the model have been derived from our ab initio calculations. A. *θ*/*M* = 0, bulk limit where the usual double well potential produces two equivalent minima with all the layer dipoles parallel. B.*θ*/*M* = 3, as the thickness is reduced the interface effects gain relevance and start to modify the energy landscape. The layer dipoles associated with the BaO planes reduce their magnitude in order to minimize the total energy. C. At *θ*/*M* = 5 the system reaches a critical thickness where these dipoles become zero and eventually flip orientation. D. For *θ*/*M* = 10 the system display a ferrielectric dipole pattern.

consequence of the complex energetic competition between the interface effects, the DF, and orientation and mutual interaction of the layer dipoles. The appearance of this particular ferrielectric state can be understood using a simple phenomenological model where the interface effects are explicitly taken into account.

These results suggest the possibility that such FDPs could be the normal state of a ferroelectric thin film at the nanoscale, even combined with the formation of two-dimensional island domains. It would be tempting to link the formation of such FDPs to the appearance or not of two-dimensional island below the critical thickness, and to understand the ultimate dependence upon the detailed interface structure and the nature of the metal contact. The next section will explore in detail the former point.

### **6. Tuning of polarization in metal-ferroelectric junctions**

In this section we study the paradigmatic case of a BaTiO3 film between Pt contacts, already introduced in the previous section, where we modify the ferroelectric properties by a selective control of the chemical species present at the interface. In particular, by inserting a single layer of a different metal (Au,Cu) we demonstrate that we are able to tune the polarization of the ferroelectric film via the modification of the screening properties of the composite metal contacts and through the change in the geometrical constraints at the interface.

### **6.1. Methods and discussion**

18 Advances in Ferroelectrics

 

(*pTiO*<sup>2</sup> , *pBaO*) = <sup>−</sup>*<sup>a</sup>*

*αpBaO*

*Mc*<sup>3</sup> <sup>+</sup> *pR pL*

the electrostatic potential inside the film.[50]

the actual calculations, at least to first order).

tetragonal structure of the internal cells of the oxide film.

� *pTiO*<sup>2</sup> + *pBaO c*

*<sup>M</sup>* (2*<sup>M</sup>* <sup>−</sup> <sup>1</sup>) *<sup>c</sup>*<sup>3</sup> <sup>+</sup>

�2 + *b*

> 2*M*−1 ∑ *i*=3

*pL pi d*3 *Lj* + 1 *M*

where we used Eqs.2324 and we define an overall interface parameter *α* = (*θ<sup>R</sup> pR* + *θ<sup>L</sup> pL*) . Note that the thickness of the film is given by the number of unit cells M and directly affects the magnitude of the depolarization field *E*. The DF is calculated from an ab initio calculation for a supercell with the same size *M* and using a well-established averaging technique for

As *M* → ∞ , the terms in Eq.25 that describe the mutual interaction between the dipoles converge to constant values, the DF vanishes and only the first two terms remain in *E*?, recovering the bulk properties. Only for small *M* these terms contribute appreciably to the total energy of the system. This is true in particular for the interface term that contains the parameter (fourth term in Eq. 25) that carries the information that defines the interfaces. We can estimate the interface parameters *θR*, *θL*, *pL*, *pR* from our ab initio simulations<sup>2</sup> and we assume that their numerical values will not depend upon the film thickness (as observed in

As we vary the thickness of the ferroelectric film, we observe a change of the total energy landscape consequence of the interplay between the different energetic contributions. This is clearly seen in Fig. 12, where we show contour plots of *E*'(*pTiO*<sup>2</sup> , *pBaO*) for different values of *M*. On the left, a diagram with the correspondent spatial distribution of dipoles is indicated. For thick films, the spatial distribution of dipoles that minimizes the total energy forms a domain (both LP's in the BaO and TiO2 planes have the same sign), as can be seen in the position of the minimum in the contour plot in Fig.12A. As the number of unit cells is lowered, the minimum shifts to lower values of *pBaO* until at a critical thickness (Fig. 12B) it becomes zero. Further reduction of the number of layers M flips only the dipoles belonging to BaO planes thus establishing a FDP (see Fig. 12D). In conclusion, this simple toy model captures the general physics of the thin film, and illustrates the intimate relation between the

Thus, by exploiting the concept of layer polarization in the description of ferroelectric thin films between metal contacts, we have been able to obtain detailed information on the modulation of polarization at the nanoscale and to understand the constraining effects of the interfaces in the determination of the ferroelectric response of the system. Our results shows that when film thicknesses reach a critical value, the ferroelectric system responds via a transition from the bulk ferroelectric structure to a ferrielectric antidipole pattern, where individual atomic layers acquire uncompensated opposing dipoles. This state arises as <sup>2</sup> The DF does not change much for the films thicknesses considered in this work. Therefore the value used as input for the model was calculated from the m6 averaged electrostatic potential (*E* = 2.141014*V*/*m*). The values for both interface layer dipoles used are (*pR* + *pL*)/*c* = 1.0*C*/*m*2). In the model we also chose *c*/*a* = 0.98, respecting the

interface characteristics, the thickness of the film, and the existence of an FDP.

1 *M* � *pTiO*<sup>2</sup> + *pBaO c*

�4

2*M*−1 ∑ *i*=3

+ Ω*E*

*pi pR d*3 *iR* + 1 2*M*

� *pTiO*<sup>2</sup> + *pBaO c*

> 2*M*−2 ∑ *i*,*j*=4

�

  (25)

*pi pj d*3 *ij*

*E*′

<sup>−</sup> *<sup>A</sup>*<sup>2</sup> 4*πǫ*

> We have simulated thin films of BaTiO3 between Pt metal contacts in a (001) supercell. The oxide is terminated with a BaO plane at both interfaces, with the metal atoms directly bonded to the O atoms of the oxide planei,ii. The supercells constructed in this way can be labeled as *Pt*/(*M*)/(*BaO* − *TiO*2)*<sup>m</sup>* − *BaO*/(*M*)/*Pt* with m=1,2,4,6 and M=(Pt, Au or Cu). Nine

10.5772/52268

421

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

**Figure 14.** Local Polarization for a unit cell of size m4. Only one layer of Cu at the interface betwen the BaTiO<sup>3</sup> and Pt is enough to drive the system from a ferroelectric structure (squared dots) into a paraelectric one (red triangular dots).

consecutive alternated signs along the direction perpendicular to the interface (see Section 5) From the behavior of the local polarization it is clear that while the pristine Pt (curve with black squares) interface display a distinct ferroelectric behavior, a single layer of Cu (red triangular dots) at the interface between BaTiO3 and Pt is sufficient to stabilize the system in a paraelectric (non polar) structure (the character of the structure correlates with the symmetry of the polarization profile: an asymmetric pattern correspond to a polar geometry and hence to a ferroelectric behavior; a centro-symmetric pattern on the contrary gives rise to a paraelectric behavior). If we exchange Cu for Au (curve with green triangles), the polarization is restored, although smaller that in the Pt case, and the whole polarization

These effects vary as the size of the film changes. In thinner films the local interface details have strong effects, while they are smoothed out in thicker systems. This can be seen in Fig.15 A-D where we have plotted the LP profile for different sizes of the oxide and different metal intralayers at the interfaces. In general we notice that the Pt contact maintain the system in a ferroelectric state for all the sizes although the polarization profiles vary greatly with the system size. The same effect is observed with the addition of an Au intralayer. These changes are more noticeable in the thinner systems (m1 and m2) while for the thicker samples both systems share a similar polarization profile with a higher overall polarization in the Au case. This behavior can be directly associated with the screening of the depolarization field <sup>4</sup> at the metallic contacts. We have estimated the depolarization field from the macroscopic average of the electrostatic potential.[30] Indeed we observed a decrease of the DF with the introduction of the Au layer. This clearly demonstrate that Au has better screening properties compared to Pt and in consequence a Au intralayer enhances the overall polarization of the system. A similar reasoning is more difficult to do in the smaller systems due to the strong asymmetry

<sup>4</sup> We used the maximally localized Wannier functions (N. Marzari, and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)) as implemented in the WanT code (A. Ferretti, B. Bonferroni, A. Calzolari, and M. Buongiorno Nardelli, http://www.wannier-transport.org) for the determination of the Wannier centers. The disentanglement procedure (I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001)) was applied before starting the localization algorithm, since the bands of the metal and the valence bands of the oxide are mixed in the full supercell calculation.

profile is substantially changed.

**Figure 13.** Normalized polarization for different number of oxide layers and interface phases. The local polarization is calculated for each oxide layer, added up and divided by the BaTiO<sup>3</sup> bulk polarization calculated in the same way with equivalent numbers of layers.

atomic metal planes (including the intralayer) have been found to be sufficient to simulate the contacts under short-circuit boundary conditions. We used the experimental in-plane lattice parameter of *BaTiO*<sup>3</sup> (3.99Å), and kept it fixed in all calculations. All simulations have been performed using Density Functional Theory (DFT) within the Local Density Approximation, using ultrasoft pseudopotentials and a plane waves basis set.[52]

A detailed analysis of the polarization at the nanoscale is critical for a complete description of the physical properties of the ferroelectric thin film. We have used Modern Theory of Polarization [6, 8, 18, 53, 54] and in particular the concept of layer polarization (LP)[20] to evaluate the polarization of the different structures and extract the information on the local profile of polarization at the nanoscale.

Turning to the results, in Fig.13 we show the total polarization (normalized to the bulk value) in BaTiO3 films of different thickness between composite metal contacts. These results clearly elucidate our claim: the polarization of the film is critically affected by the contact geometry at all the thicknesses we have considered and a residual polarization can be observed in films as thin as one unit cell.

In fact, a one unit cell thick BaTiO3 film between plain Pt contacts is still in a ferroelectric state with a polarization 10% of the ideal bulk, and the introduction of a Au intralayer enhances this value up to 70% of that! In contrast, thicker (6 unit cells) oxide films display similar ferroelectric characteristics with 50% of the bulk polarization, a clear indication of the rapid decay of the interface effects with the film thickness. It is worth to note that at variance with the previous cases, a Cu intralayer induces a paraelectric (or almost paraelectric) behavior for all thicknesses. We will come back to this point later in the discussion.

In order to understand better the behavior of ferroelectricity in such ultrathin films, we computed the layer-resolved spatial profile of the polarization, which is quantified by values of the layer polarization along <sup>3</sup> the structure. This is shown in Fig.14 for m4 films. When only Pt is present at the contact, the oxide develops a pattern of polarization composed of

<sup>3</sup> The local polarization is defined as *Pj* <sup>=</sup> *pj cj* for each layer *<sup>j</sup>* where *cj* is half the distance between neighboring cations and *pi* is the calculated layer polarization[20]. For the outermost layers, we had to make a somewhat arbitrary choice for *cj* and used the distance between the outermost cation and the opposite metal plus half the distance between the cation and the one belonging to the second oxide layer.

20 Advances in Ferroelectrics

of layers.

**Figure 13.** Normalized polarization for different number of oxide layers and interface phases. The local polarization is calculated for each oxide layer, added up and divided by the BaTiO<sup>3</sup> bulk polarization calculated in the same way with equivalent numbers

atomic metal planes (including the intralayer) have been found to be sufficient to simulate the contacts under short-circuit boundary conditions. We used the experimental in-plane lattice parameter of *BaTiO*<sup>3</sup> (3.99Å), and kept it fixed in all calculations. All simulations have been performed using Density Functional Theory (DFT) within the Local Density Approximation,

A detailed analysis of the polarization at the nanoscale is critical for a complete description of the physical properties of the ferroelectric thin film. We have used Modern Theory of Polarization [6, 8, 18, 53, 54] and in particular the concept of layer polarization (LP)[20] to evaluate the polarization of the different structures and extract the information on the local

Turning to the results, in Fig.13 we show the total polarization (normalized to the bulk value) in BaTiO3 films of different thickness between composite metal contacts. These results clearly elucidate our claim: the polarization of the film is critically affected by the contact geometry at all the thicknesses we have considered and a residual polarization can be observed in films

In fact, a one unit cell thick BaTiO3 film between plain Pt contacts is still in a ferroelectric state with a polarization 10% of the ideal bulk, and the introduction of a Au intralayer enhances this value up to 70% of that! In contrast, thicker (6 unit cells) oxide films display similar ferroelectric characteristics with 50% of the bulk polarization, a clear indication of the rapid decay of the interface effects with the film thickness. It is worth to note that at variance with the previous cases, a Cu intralayer induces a paraelectric (or almost paraelectric) behavior

In order to understand better the behavior of ferroelectricity in such ultrathin films, we computed the layer-resolved spatial profile of the polarization, which is quantified by values of the layer polarization along <sup>3</sup> the structure. This is shown in Fig.14 for m4 films. When only Pt is present at the contact, the oxide develops a pattern of polarization composed of

and *pi* is the calculated layer polarization[20]. For the outermost layers, we had to make a somewhat arbitrary choice for *cj* and used the distance between the outermost cation and the opposite metal plus half the distance between the

*cj* for each layer *<sup>j</sup>* where *cj* is half the distance between neighboring cations

for all thicknesses. We will come back to this point later in the discussion.

using ultrasoft pseudopotentials and a plane waves basis set.[52]

profile of polarization at the nanoscale.

<sup>3</sup> The local polarization is defined as *Pj* <sup>=</sup> *pj*

cation and the one belonging to the second oxide layer.

as thin as one unit cell.

**Figure 14.** Local Polarization for a unit cell of size m4. Only one layer of Cu at the interface betwen the BaTiO<sup>3</sup> and Pt is enough to drive the system from a ferroelectric structure (squared dots) into a paraelectric one (red triangular dots).

consecutive alternated signs along the direction perpendicular to the interface (see Section 5) From the behavior of the local polarization it is clear that while the pristine Pt (curve with black squares) interface display a distinct ferroelectric behavior, a single layer of Cu (red triangular dots) at the interface between BaTiO3 and Pt is sufficient to stabilize the system in a paraelectric (non polar) structure (the character of the structure correlates with the symmetry of the polarization profile: an asymmetric pattern correspond to a polar geometry and hence to a ferroelectric behavior; a centro-symmetric pattern on the contrary gives rise to a paraelectric behavior). If we exchange Cu for Au (curve with green triangles), the polarization is restored, although smaller that in the Pt case, and the whole polarization profile is substantially changed.

These effects vary as the size of the film changes. In thinner films the local interface details have strong effects, while they are smoothed out in thicker systems. This can be seen in Fig.15 A-D where we have plotted the LP profile for different sizes of the oxide and different metal intralayers at the interfaces. In general we notice that the Pt contact maintain the system in a ferroelectric state for all the sizes although the polarization profiles vary greatly with the system size. The same effect is observed with the addition of an Au intralayer. These changes are more noticeable in the thinner systems (m1 and m2) while for the thicker samples both systems share a similar polarization profile with a higher overall polarization in the Au case. This behavior can be directly associated with the screening of the depolarization field <sup>4</sup> at the metallic contacts. We have estimated the depolarization field from the macroscopic average of the electrostatic potential.[30] Indeed we observed a decrease of the DF with the introduction of the Au layer. This clearly demonstrate that Au has better screening properties compared to Pt and in consequence a Au intralayer enhances the overall polarization of the system. A similar reasoning is more difficult to do in the smaller systems due to the strong asymmetry

<sup>4</sup> We used the maximally localized Wannier functions (N. Marzari, and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)) as implemented in the WanT code (A. Ferretti, B. Bonferroni, A. Calzolari, and M. Buongiorno Nardelli, http://www.wannier-transport.org) for the determination of the Wannier centers. The disentanglement procedure (I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001)) was applied before starting the localization algorithm, since the bands of the metal and the valence bands of the oxide are mixed in the full supercell calculation.

10.5772/52268

423

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

between that BaO/metal interface and the BaTiO3/metal interface of this work. In both cases we considered a BaO terminated oxide slabs, that show very similar characteristics (as also indicated by the fact that the interfaces between BaO and BaTiO3 have almost zero valence

Indeed, we find similar behaviors for the band offsets of the interfaces BaO/M/Pd (see Ref. [52]) and BaTiO3/M/Pt when Au, Cu and Pt are used as interlayer M. Furthermore, the SBH for the BaTiO3/metal interfaces follow the same ascending order for each metal intralayer (Au (0.8 eV) < Pt (1.1 eV) < Cu (1.4 eV)) in both systems (values given correspond to the case of m4)5 This trend suggests a relation between the band offsets of the interfaces and the screening properties of the metals, that is to be expected given the strong correlation between the charge transfer at the interface and the SBH.[51] In particular, if a strong dipole is established at the interface, due to a high SBH, as in the case of Cu, fewer charges will be available for screening the depolarization field and the system will not support a ferroelectric distortion. Following Ref. [51] we can state the following phenomenological rule: metal intralayers that reduce the SBH at the interfaces will enhance the screening properties of the

We have used Density Functinal Theory and the layer polarization concept to analyze the effect of the interface structure on thin ferroelectric films between metal contacts. As the size of the film is reduced, the interface effects become strong and influence dramatically the spatial polarization profile of the system. We monitored these changes as different metal layers were introduced at the interface, modifying the local structure of the interface, the band alignment and in turn affecting the screening nature of the metal contact. In this way

we are able to tune the ferroelectric state by carefully choosing the metal interlayer.

Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina

Instituto de Ciencias Basicas, Universidad Nacional de Cuyo, Argentina Previously at North Carolina State University, Raeligh, North Carolina, USA

Nanostructures. *ArXiv Condensed Matter e-prints*, May 2006.

Centro Atomico Bariloche, U-A Tecnologia de Materiales y Dispositivos, Division Materiales

[1] P. Ghosez and J. Junquera. First-Principles Modeling of Ferroelectric Oxide

[2] L. D. Landau and E. M. Lifshitz. *Electrodynamics of Continuous Media*. Pergamon Press,

<sup>5</sup> We obtained the SBH values of the BaTiO3/metal interfaces by calculating the energy difference between the Fermi level and the upper edge of the valence band, by direct inspection of the LDOS of the systems projected on of each

[3] C. Kittel. *Introduction to Solid State Physics*. Wiley, New York, 7 edition, 1996.

band offset[52]).

**Author details**

Nuclkheares, Argentina

Oxford, 1984.

Matías Núñez

**References**

crystal plane.

contact and stabilize the ferroelectric distortion.

**Figure 15.** Layer polarization for different metal interfaces and films thicknesses A)m1, B)m2, C)m4, D)m6. As reference, the LPs for bulk BaTiO<sup>3</sup> are indicated with blue circles and dotted lines for each case. Metals intralayers used are Cu, Au, Pt. Case of m6 with Cu layer: this result is in fact an artificial effect due to the known LDA underestimation of the band gap. The Fermi level of the system overlaps the conduction band and starts filling states that should be empty. We expect the same issue for *m* > *m*6, or with the use of metals with a higher work function. The calculations for *m* < *m*6 are be correct under the DFT?LDA framework, as the Fermi level is safely below the conduction band.

introduced by the proximity of the interfaces and the uncertainty in the evaluation of the macroscopic averages. However, the qualitative behavior remains the same.

Contrary to Pt and Au, Cu intralayer do not stabilize any ferroelectric distortion at any thickness . This is a strong indication that Cu screening is weaker and not sufficient to reduce the depolarization field inside the oxide.

The above behavior is obviously correlated with the redistribution of charge across the interfaces in the different cases. The latter is quantified by the modification of the band alignment induced by the metal intralayer, and its influence on the screening properties of the metal contact. In fact, the knowledge of the band alignment, or Schottky barrier Height (SBH), allows us to define the properties of the interface phase between the metal and oxide. In a previous work,[51] we have demonstrated the correlation between the local structure of the interface composed by a crystalline oxide (BaO) and a *d* metal, and how we can tune the SBH by controlling the relative overlap of the local density of states of the different atoms close to the interface. Indeed we have found an almost complete similarity between that BaO/metal interface and the BaTiO3/metal interface of this work. In both cases we considered a BaO terminated oxide slabs, that show very similar characteristics (as also indicated by the fact that the interfaces between BaO and BaTiO3 have almost zero valence band offset[52]).

Indeed, we find similar behaviors for the band offsets of the interfaces BaO/M/Pd (see Ref. [52]) and BaTiO3/M/Pt when Au, Cu and Pt are used as interlayer M. Furthermore, the SBH for the BaTiO3/metal interfaces follow the same ascending order for each metal intralayer (Au (0.8 eV) < Pt (1.1 eV) < Cu (1.4 eV)) in both systems (values given correspond to the case of m4)5 This trend suggests a relation between the band offsets of the interfaces and the screening properties of the metals, that is to be expected given the strong correlation between the charge transfer at the interface and the SBH.[51] In particular, if a strong dipole is established at the interface, due to a high SBH, as in the case of Cu, fewer charges will be available for screening the depolarization field and the system will not support a ferroelectric distortion. Following Ref. [51] we can state the following phenomenological rule: metal intralayers that reduce the SBH at the interfaces will enhance the screening properties of the contact and stabilize the ferroelectric distortion.

We have used Density Functinal Theory and the layer polarization concept to analyze the effect of the interface structure on thin ferroelectric films between metal contacts. As the size of the film is reduced, the interface effects become strong and influence dramatically the spatial polarization profile of the system. We monitored these changes as different metal layers were introduced at the interface, modifying the local structure of the interface, the band alignment and in turn affecting the screening nature of the metal contact. In this way we are able to tune the ferroelectric state by carefully choosing the metal interlayer.

### **Author details**

### Matías Núñez

22 Advances in Ferroelectrics

**Figure 15.** Layer polarization for different metal interfaces and films thicknesses A)m1, B)m2, C)m4, D)m6. As reference, the LPs for bulk BaTiO<sup>3</sup> are indicated with blue circles and dotted lines for each case. Metals intralayers used are Cu, Au, Pt. Case of m6 with Cu layer: this result is in fact an artificial effect due to the known LDA underestimation of the band gap. The Fermi level of the system overlaps the conduction band and starts filling states that should be empty. We expect the same issue for *m* > *m*6, or with the use of metals with a higher work function. The calculations for *m* < *m*6 are be correct under the DFT?LDA

introduced by the proximity of the interfaces and the uncertainty in the evaluation of the

Contrary to Pt and Au, Cu intralayer do not stabilize any ferroelectric distortion at any thickness . This is a strong indication that Cu screening is weaker and not sufficient to

The above behavior is obviously correlated with the redistribution of charge across the interfaces in the different cases. The latter is quantified by the modification of the band alignment induced by the metal intralayer, and its influence on the screening properties of the metal contact. In fact, the knowledge of the band alignment, or Schottky barrier Height (SBH), allows us to define the properties of the interface phase between the metal and oxide. In a previous work,[51] we have demonstrated the correlation between the local structure of the interface composed by a crystalline oxide (BaO) and a *d* metal, and how we can tune the SBH by controlling the relative overlap of the local density of states of the different atoms close to the interface. Indeed we have found an almost complete similarity

macroscopic averages. However, the qualitative behavior remains the same.

framework, as the Fermi level is safely below the conduction band.

reduce the depolarization field inside the oxide.

Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina Centro Atomico Bariloche, U-A Tecnologia de Materiales y Dispositivos, Division Materiales Nuclkheares, Argentina Instituto de Ciencias Basicas, Universidad Nacional de Cuyo, Argentina

Previously at North Carolina State University, Raeligh, North Carolina, USA

### **References**


<sup>5</sup> We obtained the SBH values of the BaTiO3/metal interfaces by calculating the energy difference between the Fermi level and the upper edge of the valence band, by direct inspection of the LDOS of the systems projected on of each crystal plane.


10.5772/52268

425

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

[22] J.D. Jackson. *Classical electrodynamics*. Wiley, 3 edition, 1998.

electric fields. *Phys. Rev. B*, 63(20):205426, May 2001.

59(19):12301–12304, May 1999.

61(6):734–737, Aug 1988.

94(4):047603, 2005.

2006.

[36] R.E. Cohen. *Nature*, 358(136), 1992.

3] ultrathin films. *Applied Physics Letters*, 76(19):2767–2769, 2000.

[27] Batra I. P. and Silverman B. D. *Sol. State Comm.*, 11(291), 1972.

[30] J. Junquera and P. Ghosez. . *Nature*, 422(506), 2003.

*Phys. Rev. B*, 44(11):5572–5579, Sep 1991.

films with the dead layers. *Phys. Rev. B*, 63(13):132103, Mar 2001.

[23] B. Meyer and David Vanderbilt. Ab initio study of *batio*3 and *pbtio*3 surfaces in external

[24] Ph. Ghosez and K. M. Rabe. Microscopic model of ferroelectricity in stress-free pbtio[sub

[25] T. Maruyama, M. Saitoh, I. Sakai, T. Hidaka, Y. Yano, and T. Noguchi. Growth and characterization of 10-nm-thick c-axis oriented epitaxial pbzr[sub 0.25]ti[sub 0.75]o[sub

[26] Lennart Bengtsson. Dipole correction for surface supercell calculations. *Phys. Rev. B*,

[28] P.B.Littlewood M.Dawber1, P.Chandra and J.F. Scott. Depolarization corrections to the coercive field in thin-film ferroelectric. *J. Phys.: Condens. Matter*, 15:L393–L398, 2003.

[29] A. M. Bratkovsky and A. P. Levanyuk. Very large dielectric response of thin ferroelectric

[31] Na Sai, Karin M. Rabe, and David Vanderbilt. Theory of structural response to macroscopic electric fields in ferroelectric systems. *Phys. Rev. B*, 66(10):104108, Sep 2002.

[32] Alfonso Balderschi, Stefano Baroni, and Raffaele Resta. Band offsets in lattice-matched heterojunctions: A model and first-principles calculations for gaas/alas. *Phys. Rev. Lett.*,

[33] L. Colombo, R. Resta, and S. Baroni. Valence-band offsets at strained si/ge interfaces.

[34] Igor Kornev, Huaxiang Fu, and L. Bellaiche. Ultrathin films of ferroelectric solid solutions under a residual depolarizing field. *Phys. Rev. Lett.*, 93(19):196104, Nov 2004.

[35] Celine Lichtensteiger, Jean-Marc Triscone, Javier Junquera, and Philippe Ghosez. Ferroelectricity and tetragonality in ultrathin pbtio[sub 3] films. *Physical Review Letters*,

[37] V. Nagarajan, J. Junquera, J. Q. He, C. L. Jia, R. Waser, K. Lee, Y. K. Kim, S. Baik, T. Zhao, R. Ramesh, Ph. Ghosez, and K. M. Rabe. Scaling of structure and electrical properties in ultrathin epitaxial ferroelectric heterostructures. *Journal of Applied Physics*, 100(5):051609,

[38] C.-G. Duan, R.F. Sabirianov, W.-N. Mei, S.S. Jaswal, and E.Y. Tsymbal. Interface effect

on ferroelectricity at the nanoscale. *Nano Letters*, 6(3):483–487, 2006.

3] thin films on (100)si substrate. *Applied Physics Letters*, 73(24):3524–3526, 1998.


[22] J.D. Jackson. *Classical electrodynamics*. Wiley, 3 edition, 1998.

24 Advances in Ferroelectrics

*Rev. B*, 9(4):1998–1999, Feb 1974.

*Lett.*, 80(9):1800–1803, Mar 1998.

*Phys. Rev.*, 52(3):191–197, Aug 1937.

115(4):809–821, Aug 1959.

1989.

1982.

Netherlands, 2006.

*Phys. Rev. B*, 47(3):1651–1654, Jan 1993.

approach. *Rev. Mod. Phys.*, 66(3):899–915, Jul 1994.

*Journal of Physics: Condensed Matter*, 12(9):R107–R143, 2000.

[14] E.I. Blount. Formalism of band theory. *Solid State Physics*, 13(305), 1962.

[4] N. W. Ashcroft and N. D. Mermin. *Solid State Physics*. Saunders, Philadelphia, 1976.

[6] R. Resta. Theory of the electric polarization in crystals. *Ferroelectrics*, 136, 1992.

[5] Richard M. Martin. Comment on calculations of electric polarization in crystals. *Phys.*

[7] R.D. King-Smith and David Vanderbilt. Theory of polarization of crystalline solids.

[8] Raffaele Resta. Macroscopic polarization in crystalline dielectrics: the geometric phase

[9] Raffaele Resta. Quantum-mechanical position operator in extended systems. *Phys. Rev.*

[10] A. Shapere and F. Wilczek. *Geometric Phases in Physics*. World Scientific, Singapore, 1989.

[11] Raffaele Resta. Manifestations of berry's phase in molecules and condensed matter.

[12] Gregory H. Wannier. The structure of electronic excitation levels in insulating crystals.

[13] W. Kohn. Analytic properties of bloch waves and wannier functions. *Phys. Rev.*,

[15] Nicola Marzari and David Vanderbilt. Maximally localized generalized wannier functions for composite energy bands. *Phys. Rev. B*, 56(20):12847–12865, Nov 1997.

[16] J. Zak. Berry's phase for energy bands in solids. *Phys. Rev. Lett.*, 62(23):2747–2750, Jun

[17] J. Zak. Band center—a conserved quantity in solids. *Phys. Rev. Lett.*, 48(5):359–362, Feb

[18] D. Vanderbilt and R. Resta. *Conceptual foundations of materials properties: A standard model for calculation of ground- and excited-state properties.*, chapter Quantum electrostatics of insulators: Polarization, Wannier functions, and electric fields. Elsevier, The

[19] N. Marzari and D. Vanderbilt. Maximally localized generalized wannier functions for

[20] X. Wu, O. Dieguez, K. M. Rabe, and D. Vanderbilt. . *Phys. Rev. Lett.*, 97(107602), 2006.

[21] I. Souza, N. Marzari, and D. Vanderbilt. Maximally localized wannier functions for

composite energy bands. *Phys. Rev. B*, 56(20):12847–12865, 1997.

entangled energy bands. *Phys. Rev. B*, 65(035109), 2001.


[39] G. Gerra, A. K. Tagantsev, N. Setter, and K. Parlinski Parlinski. Ionic Polarizability of Conductive Metal Oxides and Critical Thickness for Ferroelectricity in BaTiO3. *Physical Review Letters*, 96(10):107603–+, March 2006.

10.5772/52268

427

http://dx.doi.org/10.5772/52268

Ferroelectrics at the Nanoscale: A First Principle Approach

[53] R. Raffaele. Macroscopic polarization in crystalline dielectrics: the geometric phase

[54] R.D. King-Smith and D. Vanderbilt. Theory of polarization of crystalline solids. *Phys.*

approach. *Rev. Mod. Phys.*, 66(3):899–915, Jul 1994.

*Rev. B*, 47(3):1651–1654, Jan 1993.


[53] R. Raffaele. Macroscopic polarization in crystalline dielectrics: the geometric phase approach. *Rev. Mod. Phys.*, 66(3):899–915, Jul 1994.

26 Advances in Ferroelectrics

[39] G. Gerra, A. K. Tagantsev, N. Setter, and K. Parlinski Parlinski. Ionic Polarizability of Conductive Metal Oxides and Critical Thickness for Ferroelectricity in BaTiO3. *Physical*

[41] Toshito Mitsui and Jiro Furuichi. Domain structure of rochelle salt and k*h*2p*o*4. *Phys.*

[42] Zhongqing Wu, Ningdong Huang, Zhirong Liu, Jian Wu, Wenhui Duan, Bing-Lin Gu, and Xiao-Wen Zhang. Ferroelectricity in pb ( zr0.5 ti0.5 ) o3 thin films: Critical thickness

[43] M. Sepliarsky, S. R. Phillpot, D. Wolf, M. G. Stachiotti, and R. L. Migoni. Atomic-level simulation of ferroelectricity in perovskite solid solutions. *Applied Physics Letters*,

[44] S. K. Streiffer, J. A. Eastman, D. D. Fong, Carol Thompson, A. Munkholm, M. V. Ramana Murty, O. Auciello, G. R. Bai, and G. B. Stephenson. Observation of nanoscale <sup>180</sup>◦ stripe domains in ferroelectric *pbtio*3 thin films. *Phys. Rev. Lett.*, 89(6):067601, Jul

[45] Y. S. Kim, D. H. Kim, J. D. Kim, Y. J. Chang, T. W. Noh, J. H. Kong, K. Char, Y. D. Park, S. D. Bu, J.-G. Yoon, and J.-S. Chung. Critical thickness of ultrathin ferroelectric

[46] R. A. McKee, F. J. Walker, M. Buongiorno Nardelli, W. A. Shelton, and G. M. Stocks. The Interface Phase and the Schottky Barrier for a Crystalline Dielectric on Silicon. *Science*,

[48] Nicola A. Spaldin. MATERIALS SCIENCE: Fundamental Size Limits in Ferroelectricity.

[49] K.S. Thygensen, L. B. Hansen, and K. W. Jacobsen. Partly Occupied Wannier Functions.

[50] S. Baroni, R. Resta, A. Baldereschi, and M. Peressi. *Spectroscopy of Semiconductor Microstructures*, chapter "Can we tune the band offset at semiconductor

[51] Matias Nunez and Marco Buongiorno Nardelli. First-principles theory of metal-alkaline earth oxide interfaces. *Physical Review B (Condensed Matter and Materials Physics)*,

[52] J. Junquera, M. Zimmerman, P. Orejon, and P. Ghosez. First-principles calculation of the band offset at BaO/BaTiO3 and SrO/SrTiO3 interfaces. *Phys. Rev. B*, 67(153327), 2003.

[40] N. Sai, A. M. Kolpak, and A. M. Rappe. . *Phys. Rev.B*, 72(020101(R)), 2005.

and 180o stripe domains. *Phys. Rev. B*, 70(10):104108, Sep 2004.

batio[sub 3] films. *Applied Physics Letters*, 86(10):102907, 2005.

heterojunctions?", pages 251–271. Plenum, London, 1989.

*Review Letters*, 96(10):107603–+, March 2006.

*Rev.*, 90(2):193–202, Apr 1953.

76(26):3986–3988, 2000.

300(5626):1726–1730, 2003.

[47] C. F. Pulvari. *Phys. Rev.*, 120(1670), 1960.

*Science*, 304(5677):1606–1607, 2004.

*Phys. Rev. Lett.*, 94(026405), 2005.

73(23):235422, 2006.

2002.

[54] R.D. King-Smith and D. Vanderbilt. Theory of polarization of crystalline solids. *Phys. Rev. B*, 47(3):1651–1654, Jan 1993.

**Chapter 19**

**The Influence of Vanadium Doping on the Physical**

Recently, the various functional thin films were widely focused on the applications in nonvolatile random access memory (NvRAM), such as smart cards and portable electrical devi‐ ces utilizing excellent memory characteristics, high storage capacity, long retention cycles, low electric consumption, non-volatility, and high speed readout. Additionally, the various non-volatile random access memory devices such as, ferroelectric random access memory (FeRAM), magnetron memory (MRAM), resistance random access memory (RRAM), and flash memory were widely discussed and investigated [1-9]. However, the high volatile pol‐ lution elements and high fabrication cost of the complex composition material were serious difficult problems for applications in integrated circuit semiconductor processing. For this reason, the simple binary metal oxide materials such as ZnO, Al2O3, TiO2, and Ta2O5 were widely considered and investigated for the various functional electronic product applica‐

The (ABO3) pervoskite and bismuth layer structured ferroelectrics (BLSFs) were excellent candidate materials for ferroelectric random access memories (FeRAMs) such as in smart cards and portable electric devices utilizing their low electric consumption, nonvolatility, high speed readout. The ABO3 structure materials for ferroelectric oxide exhibit high rem‐ nant polarization and low coercive filed. Such as Pb(Zr,Ti)O3 (PZT), Sr2Bi2Ta2O9 (SBT), SrTiO3 (ST), Ba(Zr,Ti)O3 (BZ1T9), and (Ba,Sr)TiO3 (BST) were widely studied and discussed

> © 2013 Chen et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**and Electrical Properties of Non-Volatile Random**

**Access Memory Using the BTV, BLTV, and BNTV**

Kai-Huang Chen, Chien-Min Cheng, Sean Wu,

tions in resistance random access memory devices [10-12].

Additional information is available at the end of the chapter

Chin-Hsiung Liao and Jen-Hwan Tsai

**Oxide Thin Films**

http://dx.doi.org/10.5772/52203

**1. Introduction**

**The Influence of Vanadium Doping on the Physical and Electrical Properties of Non-Volatile Random Access Memory Using the BTV, BLTV, and BNTV Oxide Thin Films**

Kai-Huang Chen, Chien-Min Cheng, Sean Wu, Chin-Hsiung Liao and Jen-Hwan Tsai

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52203

**1. Introduction**

Recently, the various functional thin films were widely focused on the applications in nonvolatile random access memory (NvRAM), such as smart cards and portable electrical devi‐ ces utilizing excellent memory characteristics, high storage capacity, long retention cycles, low electric consumption, non-volatility, and high speed readout. Additionally, the various non-volatile random access memory devices such as, ferroelectric random access memory (FeRAM), magnetron memory (MRAM), resistance random access memory (RRAM), and flash memory were widely discussed and investigated [1-9]. However, the high volatile pol‐ lution elements and high fabrication cost of the complex composition material were serious difficult problems for applications in integrated circuit semiconductor processing. For this reason, the simple binary metal oxide materials such as ZnO, Al2O3, TiO2, and Ta2O5 were widely considered and investigated for the various functional electronic product applica‐ tions in resistance random access memory devices [10-12].

The (ABO3) pervoskite and bismuth layer structured ferroelectrics (BLSFs) were excellent candidate materials for ferroelectric random access memories (FeRAMs) such as in smart cards and portable electric devices utilizing their low electric consumption, nonvolatility, high speed readout. The ABO3 structure materials for ferroelectric oxide exhibit high rem‐ nant polarization and low coercive filed. Such as Pb(Zr,Ti)O3 (PZT), Sr2Bi2Ta2O9 (SBT), SrTiO3 (ST), Ba(Zr,Ti)O3 (BZ1T9), and (Ba,Sr)TiO3 (BST) were widely studied and discussed

© 2013 Chen et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

for large storage capacity FeRAM devices. The (Ba,Sr)TiO3 and Ba(Ti,Zr)O3 ferroelectric ma‐ terials were also expected to substitute the PZT or SBT memory materials and improve the environmental pollution because of their low pollution problem [9-15]. In addition, the high dielectric constant and low leakage current density of zirconium and strontium-doped Ba‐ TiO3 thin films were applied for the further application in the high density dynamic random access memory (DRAM) [16-20].

The polarization versus applied electrical field (*p-E*) curves of as-deposited BZ1T9 thin films were shown in Fig. 1(a). As the applied voltage increases, the remanent polarization of thin

BZ1T9 thin film deposited at high temperature exhibited high dielectric constant and high

The XRD patterns of as-deposited Bi4Ti3O12 thin films and ferroelectric thin films under 500~700 °C rapid thermal annealing (RTA) process were compared in Fig. 2(a). From the re‐ sults obtained, the (002) and (117) peaks of as-deposited Bi4Ti3O12 thin film under the opti‐ mal sputtering parameters were found. The strong intensity of XRD peaks of Bi4Ti3O12 thin film under the 700 °C RTA post-treatment were is found. They were (008), (006), (020) and (117) peaks, respectively. Compared the XRD patterns shown in Fig. 2, the crystalline inten‐ sity of (111) plane has no apparent increase as the as-deposited process is used and has ap‐ parent increase as the RTA-treated process was used. And a smaller full width at half maximum value (FWHM) is revealed in the RTA-treated Bi4Ti3O12 thin films under the 700 °C post-treatment. This result suggests that crystal structure of Bi4Ti3O12 thin films were im‐

**Figure 2.** a) XRD patterns of as-deposited Bi4Ti3O12 thin films, and (b)the SEM morphology of as-deposited Bi4Ti3O12

The surface morphology observations of as-deposited Bi4Ti3O12 thin films under the 700 °C RTA processes were shown in Fig. 2(b). For the as-deposited Bi4Ti3O12 thin films, the mor‐ phology reveals a smooth surface and the grain growth were not observed. The grain size and boundary of Bi4Ti3O12 thin films increased while the annealing temperature increased to 700 °C. In RTA annealed Bi4Ti3O12 thin films, the maximum grain size were about 200 nm and the average grain size is 100 nm. The thickness of annealed Bi4Ti3O12 thin films were calculated and found from the SEM cross-section images. The thickness of the depos‐ ited Bi4Ti3O12 thin films is about 800 nm and the deposited rate of Bi4Ti3O12 thin films is

leakage current density because of its polycrystalline structure [24].

**1.2. Bismuth Layer Ferroelectric Structure material system**

. In addition, the 2Pr and coercive field calculated and

The Influence of Vanadium Doping on the Physical and Electrical Properties

http://dx.doi.org/10.5772/52203

431

and 250 kV/cm, respectively. According to our previous study, the

films increases from 0.5 to 2.5 μC/cm2

proved in RTA-treated process.

films.

about 14 nm/mim.

were about 5 μC/cm2

Bismuth titanate system based materials were an important role for FeRAMs applications. The bismuth titanate system were given in a general formula of bismuth layer structure fer‐ roelectric, (Bi2O2) 2+(An-1BnO3n+1) 2- (A=Bi, B=Ti). The high leakage current, high dielectric loss and domain pinning of bismuth titanate system based materials were caused by defects, bis‐ muth vacancies and oxygen vacancies. These defects and oxygen vacancies were attributed from the volatilization of Bi2O3 of bismuth contents at elevated temperature [21-23].

### **1.1. ABO3 pervoskite structure material system**

For ABO3 pervoskite structure such as, BaTiO3 and BZ1T9, the excellent electrical and ferro‐ electric properties were obtained and found. For SOP concept, the ferroelectric BZ1T9 thin film on ITO substrate were investigated and discussed. For crystallization and grain grow of ferroelectric thin films, the crystal orientation and preferred phase of different substrates were important factors for ferroelectric thin films of MIM structures.

The XRD patterns of BZ1T9 thin films with 40% oxygen concentration on Pt/Ti/SiO2/Si sub‐ strates from our previous study were shown in Fig. 1 [24-25]. The (111) and (011) peaks of the BZ1T9 thin films on Pt/Ti/SiO2/Si substrates were compared with those on ITO sub‐ strates. The strongest and sharpest peak was observed along the Pt(111) crystal plane. This suggests that the BZ1T9 films grew epitaxially with the Pt(111) bottom electrode. However, the (111) peaks of BZ1T9 thin films were not observed for (400) and (440) ITO substrates. Therefore, we determined that the crystallinity and deposition rate of BZ1T9 thin films on ITO substrates differed from those in these study [24-27].

**Figure 1.** a) XRD patterns of as-deposited thin films on the ITO/glass and Pt substrates, and (b) *p-E* curves of thin films.

The polarization versus applied electrical field (*p-E*) curves of as-deposited BZ1T9 thin films were shown in Fig. 1(a). As the applied voltage increases, the remanent polarization of thin films increases from 0.5 to 2.5 μC/cm2 . In addition, the 2Pr and coercive field calculated and were about 5 μC/cm2 and 250 kV/cm, respectively. According to our previous study, the BZ1T9 thin film deposited at high temperature exhibited high dielectric constant and high leakage current density because of its polycrystalline structure [24].

### **1.2. Bismuth Layer Ferroelectric Structure material system**

for large storage capacity FeRAM devices. The (Ba,Sr)TiO3 and Ba(Ti,Zr)O3 ferroelectric ma‐ terials were also expected to substitute the PZT or SBT memory materials and improve the environmental pollution because of their low pollution problem [9-15]. In addition, the high dielectric constant and low leakage current density of zirconium and strontium-doped Ba‐ TiO3 thin films were applied for the further application in the high density dynamic random

Bismuth titanate system based materials were an important role for FeRAMs applications. The bismuth titanate system were given in a general formula of bismuth layer structure fer‐

and domain pinning of bismuth titanate system based materials were caused by defects, bis‐ muth vacancies and oxygen vacancies. These defects and oxygen vacancies were attributed

For ABO3 pervoskite structure such as, BaTiO3 and BZ1T9, the excellent electrical and ferro‐ electric properties were obtained and found. For SOP concept, the ferroelectric BZ1T9 thin film on ITO substrate were investigated and discussed. For crystallization and grain grow of ferroelectric thin films, the crystal orientation and preferred phase of different substrates

The XRD patterns of BZ1T9 thin films with 40% oxygen concentration on Pt/Ti/SiO2/Si sub‐ strates from our previous study were shown in Fig. 1 [24-25]. The (111) and (011) peaks of the BZ1T9 thin films on Pt/Ti/SiO2/Si substrates were compared with those on ITO sub‐ strates. The strongest and sharpest peak was observed along the Pt(111) crystal plane. This suggests that the BZ1T9 films grew epitaxially with the Pt(111) bottom electrode. However, the (111) peaks of BZ1T9 thin films were not observed for (400) and (440) ITO substrates. Therefore, we determined that the crystallinity and deposition rate of BZ1T9 thin films on

**Figure 1.** a) XRD patterns of as-deposited thin films on the ITO/glass and Pt substrates, and (b) *p-E* curves of thin films.

from the volatilization of Bi2O3 of bismuth contents at elevated temperature [21-23].

were important factors for ferroelectric thin films of MIM structures.

ITO substrates differed from those in these study [24-27].

2- (A=Bi, B=Ti). The high leakage current, high dielectric loss

access memory (DRAM) [16-20].

2+(An-1BnO3n+1)

**1.1. ABO3 pervoskite structure material system**

roelectric, (Bi2O2)

430 Advances in Ferroelectrics

The XRD patterns of as-deposited Bi4Ti3O12 thin films and ferroelectric thin films under 500~700 °C rapid thermal annealing (RTA) process were compared in Fig. 2(a). From the re‐ sults obtained, the (002) and (117) peaks of as-deposited Bi4Ti3O12 thin film under the opti‐ mal sputtering parameters were found. The strong intensity of XRD peaks of Bi4Ti3O12 thin film under the 700 °C RTA post-treatment were is found. They were (008), (006), (020) and (117) peaks, respectively. Compared the XRD patterns shown in Fig. 2, the crystalline inten‐ sity of (111) plane has no apparent increase as the as-deposited process is used and has ap‐ parent increase as the RTA-treated process was used. And a smaller full width at half maximum value (FWHM) is revealed in the RTA-treated Bi4Ti3O12 thin films under the 700 °C post-treatment. This result suggests that crystal structure of Bi4Ti3O12 thin films were im‐ proved in RTA-treated process.

**Figure 2.** a) XRD patterns of as-deposited Bi4Ti3O12 thin films, and (b)the SEM morphology of as-deposited Bi4Ti3O12 films.

The surface morphology observations of as-deposited Bi4Ti3O12 thin films under the 700 °C RTA processes were shown in Fig. 2(b). For the as-deposited Bi4Ti3O12 thin films, the mor‐ phology reveals a smooth surface and the grain growth were not observed. The grain size and boundary of Bi4Ti3O12 thin films increased while the annealing temperature increased to 700 °C. In RTA annealed Bi4Ti3O12 thin films, the maximum grain size were about 200 nm and the average grain size is 100 nm. The thickness of annealed Bi4Ti3O12 thin films were calculated and found from the SEM cross-section images. The thickness of the depos‐ ited Bi4Ti3O12 thin films is about 800 nm and the deposited rate of Bi4Ti3O12 thin films is about 14 nm/mim.

### **2. Experimental Detail**

S. Y. Wu firstly reported that an MFS transistor fabricated by using bismuth titanate in 1974 [28-29]. The first ferroelectric memory device was fabricated by replacing the gate oxide of a conventional metal-oxide-semiconductor (MOS) transistor with a ferroelectric material. However, the interface and interaction problem between the silicon substrate and ferroelec‐ tric films were very important factors during the high temperature processes in 1TC struc‐ ture. To overcome the interface and interaction problem, the silicon dioxide and silicon nitride films were used as the buffer layer. The low remnant polarization and high operation voltage of 1TC were also be induced by gate oxide structure with double-layer ferroelectric silicon dioxide thin films. Sugibuchi et al. provided a 50 nm silicon dioxide thin film be‐ tween the Bi4Ti3O12 layer and the silicon substrate [30].

**3. Results and Discussion**

under optimal parameters will be further developed.

thin films deposited using optimal parameters.

**3.1. Large memory window in the vanadium doped Bi4Ti3O12 (BTV) thin films**

The XRD pattern was used to identify the crystalline structures of as-deposited BTV thin films, labeled "vanadium doped at 550 °C," with various depositing parameters. From the XRD pattern, we found that the optimal deposition parameters of as-deposited BTV thin films were RF power of 130 W, chamber pressures of 10 mtorr and oxygen concentrations of 25%. The crystalline orientations of (117), (008) and (200) planes were apparently observed in the films. It was found that all of the films consisted of a single phase of a bismuth lay‐ ered structure showing the preferred (008) and (117) orientation. Both films were well c-axis oriented, but BTV thin film was more c-axis oriented than BIT, labeled "undoped at 550 °C". For the polycrystalline BTV thin films, the (117) peak was the strongest peak and the intensi‐ ty of the (008) peak was 10% of (117) peak intensity. An obvious change in the orientation due to the substitution was observed except for the degree of the (117) orientation for BTV films. In addition, the XRD patterns of the as-deposited BTV thin films deposited using opti‐ mal parameters at room and 550 °C substrate temperatures were observed in Fig. 2. This re‐ sult indicated that the crystalline characteristics of BTV thin films deposited at 550 °C were better than those of BTV thin films at room temperature. The crystalline and dielectric char‐ acteristics of as-deposited BTV thin films were influenced by substrate temperatures. The electrical characteristics of as-deposited BTV thin films at substrate temperatures of 550 °C

The Influence of Vanadium Doping on the Physical and Electrical Properties

http://dx.doi.org/10.5772/52203

433

**Figure 4.** The XRD patterns of (a) undoped at 550 °C (b) vanadium doped at R.T. and (c) vanadium doped at 550 °C

In Fig. 5, circular-like grains with 150 nm width were observed with scanning electron mi‐ croscopy (SEM) for as-deposited BTV thin films. From the cross-sectional SEM image, film thicknesses were measured to be 742 nm. As the depositing time increases from 30 and 60,

The ferroelectric ceramic target prepared, the raw materials were mixed and fabricated by solid state reaction method. After mixing and ball-milling, the mixture was dried, grounded, and calcined for some time. Then, the pressed ferroelectric ceramic target with a diameter of two inches was sintered in ambient air. The base pressure of the deposited chamber was brought down 1×10-7 mTorr prior to deposition. The target was placed away from the Pt/Ti/SiO2/Si and SiO2/Si substrate. For metal-ferroelectric-metal (MFM) capacitor structure, the Pt and the Ti were deposited by dc sputtering using pure argon plasma as bottom elec‐ trodes. The SiO2 thin films were prepared by dry oxidation technology. The metal-ferroelec‐ tric-insulator-semiconductor (MFIS) and metal-ferroelectric-metal (MFM) structures were shown in Fig. 3.

**Figure 3.** a) Metal-ferroelectric-insulator-semiconductor (MFIS) structure, and (b) Metal ferroelectric-metal (MFM) structure.

For the physical properties of ferroelectric thin films obtained, the thickness and surface morphology of ferroelectric thin films were observed by field effect scanning electron micro‐ scopy (FeSEM). The crystal structure of ferroelectric thin films were characterized by an Xray diffraction (XRD) measurement using a Ni-filtered CuKα radiation. The capacitancevoltage (C-V) properties were measured as a function of applied voltage by using a Hewlett-Packard (HP 4284A) impedance gain phase analyzer. The current curves versus the applied voltage (I-V characteristics) of the ferroelectric thin films were measured by a Hewlett-Pack‐ ard (HP 4156) semiconductor parameter analyzer.

### **3. Results and Discussion**

**2. Experimental Detail**

432 Advances in Ferroelectrics

shown in Fig. 3.

structure.

tween the Bi4Ti3O12 layer and the silicon substrate [30].

ard (HP 4156) semiconductor parameter analyzer.

S. Y. Wu firstly reported that an MFS transistor fabricated by using bismuth titanate in 1974 [28-29]. The first ferroelectric memory device was fabricated by replacing the gate oxide of a conventional metal-oxide-semiconductor (MOS) transistor with a ferroelectric material. However, the interface and interaction problem between the silicon substrate and ferroelec‐ tric films were very important factors during the high temperature processes in 1TC struc‐ ture. To overcome the interface and interaction problem, the silicon dioxide and silicon nitride films were used as the buffer layer. The low remnant polarization and high operation voltage of 1TC were also be induced by gate oxide structure with double-layer ferroelectric silicon dioxide thin films. Sugibuchi et al. provided a 50 nm silicon dioxide thin film be‐

The ferroelectric ceramic target prepared, the raw materials were mixed and fabricated by solid state reaction method. After mixing and ball-milling, the mixture was dried, grounded, and calcined for some time. Then, the pressed ferroelectric ceramic target with a diameter of two inches was sintered in ambient air. The base pressure of the deposited chamber was brought down 1×10-7 mTorr prior to deposition. The target was placed away from the Pt/Ti/SiO2/Si and SiO2/Si substrate. For metal-ferroelectric-metal (MFM) capacitor structure, the Pt and the Ti were deposited by dc sputtering using pure argon plasma as bottom elec‐ trodes. The SiO2 thin films were prepared by dry oxidation technology. The metal-ferroelec‐ tric-insulator-semiconductor (MFIS) and metal-ferroelectric-metal (MFM) structures were

**Figure 3.** a) Metal-ferroelectric-insulator-semiconductor (MFIS) structure, and (b) Metal ferroelectric-metal (MFM)

For the physical properties of ferroelectric thin films obtained, the thickness and surface morphology of ferroelectric thin films were observed by field effect scanning electron micro‐ scopy (FeSEM). The crystal structure of ferroelectric thin films were characterized by an Xray diffraction (XRD) measurement using a Ni-filtered CuKα radiation. The capacitancevoltage (C-V) properties were measured as a function of applied voltage by using a Hewlett-Packard (HP 4284A) impedance gain phase analyzer. The current curves versus the applied voltage (I-V characteristics) of the ferroelectric thin films were measured by a Hewlett-Pack‐

### **3.1. Large memory window in the vanadium doped Bi4Ti3O12 (BTV) thin films**

The XRD pattern was used to identify the crystalline structures of as-deposited BTV thin films, labeled "vanadium doped at 550 °C," with various depositing parameters. From the XRD pattern, we found that the optimal deposition parameters of as-deposited BTV thin films were RF power of 130 W, chamber pressures of 10 mtorr and oxygen concentrations of 25%. The crystalline orientations of (117), (008) and (200) planes were apparently observed in the films. It was found that all of the films consisted of a single phase of a bismuth lay‐ ered structure showing the preferred (008) and (117) orientation. Both films were well c-axis oriented, but BTV thin film was more c-axis oriented than BIT, labeled "undoped at 550 °C". For the polycrystalline BTV thin films, the (117) peak was the strongest peak and the intensi‐ ty of the (008) peak was 10% of (117) peak intensity. An obvious change in the orientation due to the substitution was observed except for the degree of the (117) orientation for BTV films. In addition, the XRD patterns of the as-deposited BTV thin films deposited using opti‐ mal parameters at room and 550 °C substrate temperatures were observed in Fig. 2. This re‐ sult indicated that the crystalline characteristics of BTV thin films deposited at 550 °C were better than those of BTV thin films at room temperature. The crystalline and dielectric char‐ acteristics of as-deposited BTV thin films were influenced by substrate temperatures. The electrical characteristics of as-deposited BTV thin films at substrate temperatures of 550 °C under optimal parameters will be further developed.

**Figure 4.** The XRD patterns of (a) undoped at 550 °C (b) vanadium doped at R.T. and (c) vanadium doped at 550 °C thin films deposited using optimal parameters.

In Fig. 5, circular-like grains with 150 nm width were observed with scanning electron mi‐ croscopy (SEM) for as-deposited BTV thin films. From the cross-sectional SEM image, film thicknesses were measured to be 742 nm. As the depositing time increases from 30 and 60, to 120 min, the thickness of as-deposited BTV thin films increases linearly from 197 and 386, to 742 nm, respectively, as the depositing rate decreases from 6.57 and 6.43, to 6.18 nm.

larization and coercive field were 23 μC/cm2 and 450 kV/cm. Comparing the vanadium dop‐ ed and undoped BIT thin films, the remanent polarization (2Pr) would be increased form 16μC/cm2 for undoped BIT thin films to 23 μC/cm2 for vanadium doped. However, the coer‐ cive field of as-deposited BTV thin films would be increased to 450 kV/cm. These results in‐ dicated that the substitution of vanadium was effective for the appearance of ferroelectricity at 550 °C. The 2Pr value and the Ec value were larger than those reported in Refs. [35-36], and the 2Pr value was smaller and the Ec value was larger than those reported in [37]. Based on above results, it was found that the simultaneous substitutions for B-site are effective to derive enough ferroelectricity by accelerating the domain nucleation and pinning relaxation

The leakage current density versus applied voltage curves of as-deposited BTV thin films for different depositing time on the MFIS structure were be found. We found that the leakage current density of undoped BIT thin films, labeled "undoped," were larger than those of va‐ nadium-doped BIT thin films. This result indicated that the substitution of B-site in ABO3 perovskite structure for BTV thin films was effective in lowering leakage current density. Besides, the thickness of BTV thin films has an apparent influence on the leakage current density of BTV thin films, and that will have an apparent influence on the other electrical characteristics of BTV thin films. At an electric field of 0.5 MV/cm, the leakage current densi‐ ty critically decreases from the 3.0 ×10-7 A/cm2 for 30 min-deposited BTV thin films to

Figure 7(a) show the capacitance versus applied voltage (*C-V*) curves of as-deposited vana‐ dium doped BTV and un-doped BIT thin films. The applied voltages, which are first changed from -20 to 20 V and then returned to -20 V, are used to measure the capacitance volt‐ age characteristics (*C-V*) of the MFIS structures. For the vanadium doped thin films, the mem‐ ory window of MFIS structure increased from 5 to 15 V, and the threshold voltage decreased from 7 to 3 V. This result demonstrated that the lower threshold voltage and decreased oxy‐ gen vacancy in undoped BIT thin films had been improved from the *C-V* curves measured.

**Figure 7.** a) The P-E characteristics of vanadium doped and undoped thin films, and (b) The normalization *C-V* curves

Figure 7(b) shows the *C-V* curves of 30 min-deposited BTV thin films, and Figure 7(b) com‐ pares the *C-V* curves of BTV thin films deposited at different depositing time. Figure 7(b) also shows that the capacitance at the applied voltage value of 0 V critically increases as the

for 60 and 120 min-deposited BTV thin films.

The Influence of Vanadium Doping on the Physical and Electrical Properties

http://dx.doi.org/10.5772/52203

435

caused by B-site substitution [31-37].

of vanadium doped and undoped thin films.

and 2×10-8 A/cm2

around 3×10-8A/cm2

**Figure 5.** The surface micro structure morphology of the BTV thin films deposited using optimal parameters.

Figure 6(a) compares the change in the capacitance versus the applied voltage (*C-V*) for the un-doped and vanadium doped thin films. Based on Fig. 4, the capacitances of the BIT thin films appear to increase due to the vanadium dopant. We found that the capacitances of BTV thin films increased from 1.3 to 4.5 nF. As suggested by Fig. 4, the improvement in the dielectric constants of the BTV thin films can be attributed to the compensation of the oxy‐ gen vacancy and the improvement in the B-site substitution of the ABO3 phase in the BTV thin films [31-34].

**Figure 6.** a) The P-E characteristics of vanadium doped and undoped thin films, and (b) The normalization C-V curves of vanadium doped and undoped thin films.

Figure 6 shows ferroelectric hysteresis loops of BIT and as-deposited BTV thin film capaci‐ tors measured with a ferroelectric tester (Radiant Technologies RT66A). The as-deposited BTV thin films, labeled "vanadium doped," clearly show ferroelectricity. The remanent po‐ larization and coercive field were 23 μC/cm2 and 450 kV/cm. Comparing the vanadium dop‐ ed and undoped BIT thin films, the remanent polarization (2Pr) would be increased form 16μC/cm2 for undoped BIT thin films to 23 μC/cm2 for vanadium doped. However, the coer‐ cive field of as-deposited BTV thin films would be increased to 450 kV/cm. These results in‐ dicated that the substitution of vanadium was effective for the appearance of ferroelectricity at 550 °C. The 2Pr value and the Ec value were larger than those reported in Refs. [35-36], and the 2Pr value was smaller and the Ec value was larger than those reported in [37]. Based on above results, it was found that the simultaneous substitutions for B-site are effective to derive enough ferroelectricity by accelerating the domain nucleation and pinning relaxation caused by B-site substitution [31-37].

to 120 min, the thickness of as-deposited BTV thin films increases linearly from 197 and 386, to 742 nm, respectively, as the depositing rate decreases from 6.57 and 6.43, to 6.18 nm.

**Figure 5.** The surface micro structure morphology of the BTV thin films deposited using optimal parameters.

thin films [31-34].

434 Advances in Ferroelectrics

of vanadium doped and undoped thin films.

Figure 6(a) compares the change in the capacitance versus the applied voltage (*C-V*) for the un-doped and vanadium doped thin films. Based on Fig. 4, the capacitances of the BIT thin films appear to increase due to the vanadium dopant. We found that the capacitances of BTV thin films increased from 1.3 to 4.5 nF. As suggested by Fig. 4, the improvement in the dielectric constants of the BTV thin films can be attributed to the compensation of the oxy‐ gen vacancy and the improvement in the B-site substitution of the ABO3 phase in the BTV

**Figure 6.** a) The P-E characteristics of vanadium doped and undoped thin films, and (b) The normalization C-V curves

Figure 6 shows ferroelectric hysteresis loops of BIT and as-deposited BTV thin film capaci‐ tors measured with a ferroelectric tester (Radiant Technologies RT66A). The as-deposited BTV thin films, labeled "vanadium doped," clearly show ferroelectricity. The remanent po‐

The leakage current density versus applied voltage curves of as-deposited BTV thin films for different depositing time on the MFIS structure were be found. We found that the leakage current density of undoped BIT thin films, labeled "undoped," were larger than those of va‐ nadium-doped BIT thin films. This result indicated that the substitution of B-site in ABO3 perovskite structure for BTV thin films was effective in lowering leakage current density. Besides, the thickness of BTV thin films has an apparent influence on the leakage current density of BTV thin films, and that will have an apparent influence on the other electrical characteristics of BTV thin films. At an electric field of 0.5 MV/cm, the leakage current densi‐ ty critically decreases from the 3.0 ×10-7 A/cm2 for 30 min-deposited BTV thin films to around 3×10-8A/cm2 and 2×10-8 A/cm2 for 60 and 120 min-deposited BTV thin films.

Figure 7(a) show the capacitance versus applied voltage (*C-V*) curves of as-deposited vana‐ dium doped BTV and un-doped BIT thin films. The applied voltages, which are first changed from -20 to 20 V and then returned to -20 V, are used to measure the capacitance volt‐ age characteristics (*C-V*) of the MFIS structures. For the vanadium doped thin films, the mem‐ ory window of MFIS structure increased from 5 to 15 V, and the threshold voltage decreased from 7 to 3 V. This result demonstrated that the lower threshold voltage and decreased oxy‐ gen vacancy in undoped BIT thin films had been improved from the *C-V* curves measured.

**Figure 7.** a) The P-E characteristics of vanadium doped and undoped thin films, and (b) The normalization *C-V* curves of vanadium doped and undoped thin films.

Figure 7(b) shows the *C-V* curves of 30 min-deposited BTV thin films, and Figure 7(b) com‐ pares the *C-V* curves of BTV thin films deposited at different depositing time. Figure 7(b) also shows that the capacitance at the applied voltage value of 0 V critically increases as the depositing time increases. The memory window would be decreased from 15.1 and 13.4, to 10.6 V as the depositing time increased. Two reasons may be the cause that the increases ca‐ pacitance of as-deposited BTV thin films in the MFIS structure for the different depositing time. First, the decrease leakage current density was attributed to the increase thickness of as-deposited BTV thin films in Fig. 4. Second, the different thickness and high dielectric con‐ stant of as-deposited BTV thin films are also an important factor. In Fig. 7, the capacitance of as-deposited BTV thin films for MFIS structure could be calculated from Eq. 1 and Eq. 2:

$$
\hat{\mathbf{c}}\_{i}^{\*} = \hat{\mathbf{c}}\_{i}^{\*} \hat{\mathbf{c}}\_{i}^{\*} / \frac{\hat{\mathbf{a}}\_{i}^{\*}}{|\hat{\mathbf{d}}|} \tag{1}
$$

**3.2. The Influence of Lanthanum Doping on the Physical and Electrical Properties of BTV**

The Influence of Vanadium Doping on the Physical and Electrical Properties

http://dx.doi.org/10.5772/52203

437

For MFM structures, the crystal orientation and preferred phase of ferroelectric thin films on Pt/Ti/SiO2/Si substrates was important factor. The x-ray diffraction (XRD) patterns of BLTV and BTV thin films prepared by rf magnetron sputtering were be found. From the XRD pat‐ tern, the BLTV and BTV thin film were polycrystalline structure. The (004), (006), (008), and (117) peaks were observed in the XRD pattern. All of thin films consisted of a single phase of a bismuth layered structure showing the preferred (117) orientation. All of thin films were exhibited well c axis orientation. The change in the orientation of BLTV thin films due to the substitution was not observed. The degree of the (117) orientation relative to the (001) orien‐

**Figure 9.** a)The *C-V* characteristics of as-deposited BTV and BLTV thin films. (b) The polarization versus electrical field

In Fig. 8a, rod-like and circular-board grains with 250 nm length and 150 nm width were observed with scanning electron microscopy (SEM) for as-deposited BTV films. The small grain was gold element in preparation for the SEM sample. However, the BLTV thin films exhibited a great quantity of 400 nm length and 100 nm width rod-like grain structure in Fig 8 b. The rod-like grain size of BLTV thin films was larger than those of BTV. We induced

**(BLTV) Ferroelectric Thin Films**

tation of BLTV thin films dominant was shown.

characteristics of as-deposited BTV and BLTV thin films.

**Figure 8.** The surface micro structure of as-deposited (a) BTV and (b) BLTV thin films.

$$|\dot{\varepsilon}\_{1\sim}| = \frac{(\dot{\varepsilon}\_1 \, d\_{21} + \dot{\varepsilon}\_{22} \, d\_{11})}{(\dot{d}\_{11} + \dot{d}\_{12})^{1-\kappa}}\tag{2}$$

The ε1 and d1 are the effective dielectric constant and the total thickness of silicon and SiO2 layer. The ε2 and d2 are the effective dielectric constant and the thickness of as-deposited BTV thin film layer. The relative dielectric constants of Si and SiO2 are 11.7 and 3.9. The εr value (ε1) of silicon and SiO2 layer is much smaller than that (ε2) of as-deposited BTV thin films. The ε1×d2 is the unchanged value and the ε2×d1 value increases with the increase of depositing time. In Eq. (2), the ε1×d2 increases more quickly as d2 increases. In Eq. (1), the C value will increase as the εr value increases.

For memory window characteristics at applied voltage of 0 volts, the upper and lower ca‐ pacitance values of as-deposited BTV thin films for 30 min depositing time were 0.056 and 0.033 nF, respectively. For 60 min and 120 min depositing time, they were 0.215~0.048 nF and 1.515 ~0.105 nF, respectively. The change ratios at zero voltage were defined in Eq.(3) from these experimental results:

$$
\langle \text{ratio} \rangle^{\text{base}} = \frac{\langle \langle \mathbf{C}\_{\text{in}} \rangle^{\text{no}} \mathbf{C}\_{\text{in}} \rangle}{\langle \mathbf{C}\_{\text{in}} \rangle^{\text{no}}} \tag{3}
$$

where Cu and Cl are the upper and lower capacitance values.

The capacitance change ratios of as-deposited BTV thin films for different depositing time were 41, 73 and 93%, respectively. From above statements, the good switching characteris‐ tics of ferroelectric polarization could be attributed to memory windows ratio and the thin‐ ner thickness of as-deposited BTV thin film for the depositing time of 30 min. These results indicted the upper and lower capacitance of memory window would be decreased by lower‐ ing the thickness of SiO2 layer.

### **3.2. The Influence of Lanthanum Doping on the Physical and Electrical Properties of BTV (BLTV) Ferroelectric Thin Films**

depositing time increases. The memory window would be decreased from 15.1 and 13.4, to 10.6 V as the depositing time increased. Two reasons may be the cause that the increases ca‐ pacitance of as-deposited BTV thin films in the MFIS structure for the different depositing time. First, the decrease leakage current density was attributed to the increase thickness of as-deposited BTV thin films in Fig. 4. Second, the different thickness and high dielectric con‐ stant of as-deposited BTV thin films are also an important factor. In Fig. 7, the capacitance of as-deposited BTV thin films for MFIS structure could be calculated from Eq. 1 and Eq. 2:

The ε1 and d1 are the effective dielectric constant and the total thickness of silicon and SiO2 layer. The ε2 and d2 are the effective dielectric constant and the thickness of as-deposited BTV thin film layer. The relative dielectric constants of Si and SiO2 are 11.7 and 3.9. The εr value (ε1) of silicon and SiO2 layer is much smaller than that (ε2) of as-deposited BTV thin films. The ε1×d2 is the unchanged value and the ε2×d1 value increases with the increase of depositing time. In Eq. (2), the ε1×d2 increases more quickly as d2 increases. In Eq. (1), the C

For memory window characteristics at applied voltage of 0 volts, the upper and lower ca‐ pacitance values of as-deposited BTV thin films for 30 min depositing time were 0.056 and 0.033 nF, respectively. For 60 min and 120 min depositing time, they were 0.215~0.048 nF and 1.515 ~0.105 nF, respectively. The change ratios at zero voltage were defined in Eq.(3)

The capacitance change ratios of as-deposited BTV thin films for different depositing time were 41, 73 and 93%, respectively. From above statements, the good switching characteris‐ tics of ferroelectric polarization could be attributed to memory windows ratio and the thin‐ ner thickness of as-deposited BTV thin film for the depositing time of 30 min. These results indicted the upper and lower capacitance of memory window would be decreased by lower‐

value will increase as the εr value increases.

where Cu and Cl are the upper and lower capacitance values.

from these experimental results:

436 Advances in Ferroelectrics

ing the thickness of SiO2 layer.

(1)

(2)

(3)

For MFM structures, the crystal orientation and preferred phase of ferroelectric thin films on Pt/Ti/SiO2/Si substrates was important factor. The x-ray diffraction (XRD) patterns of BLTV and BTV thin films prepared by rf magnetron sputtering were be found. From the XRD pat‐ tern, the BLTV and BTV thin film were polycrystalline structure. The (004), (006), (008), and (117) peaks were observed in the XRD pattern. All of thin films consisted of a single phase of a bismuth layered structure showing the preferred (117) orientation. All of thin films were exhibited well c axis orientation. The change in the orientation of BLTV thin films due to the substitution was not observed. The degree of the (117) orientation relative to the (001) orien‐ tation of BLTV thin films dominant was shown.

**Figure 8.** The surface micro structure of as-deposited (a) BTV and (b) BLTV thin films.

**Figure 9.** a)The *C-V* characteristics of as-deposited BTV and BLTV thin films. (b) The polarization versus electrical field characteristics of as-deposited BTV and BLTV thin films.

In Fig. 8a, rod-like and circular-board grains with 250 nm length and 150 nm width were observed with scanning electron microscopy (SEM) for as-deposited BTV films. The small grain was gold element in preparation for the SEM sample. However, the BLTV thin films exhibited a great quantity of 400 nm length and 100 nm width rod-like grain structure in Fig 8 b. The rod-like grain size of BLTV thin films was larger than those of BTV. We induced that the bismuth vacancies of BTV thin films compensate for lanthanum addition and microstructure were improved in BLTV thin films. From the cross-sectional SEM image, average thin film thicknesses for MFIS structure were about 610 nm. The average thickness of thin films for MFM structure was about 672 nm.

Figure 9(a) shows the change in the capacitance versus the applied voltage (*C-V*) of the BTV and BLTV thin films in MFM structure measured at 100 kHz. The applied voltages, which were first changed from -20 to 20 V and then returned to -20 V, were used to measure the capacitance voltage characteristics (CV). The BLTV thin films exhibited high capacitance than those of BTV thin films. We found that the capacitances of the lanthanum-doped BTV thin films were increased from 2.38 to 2.42 nF.

**Figure 11.** a) The (J/T2) and (J/E) versus E1/2 curves of as-deposited BIT, BTV, and BLTV thin films. (b) The normalization

The Influence of Vanadium Doping on the Physical and Electrical Properties

http://dx.doi.org/10.5772/52203

439

From the *J-E* curves, the conduction mechanism of as-deposited thin films was also proved the bismuth defect and oxygen vacancy results. For MFIS structure, the interface and inter‐ action problem between of the silicon and ferroelectric films was serious important. The ef‐ fect of the bound polarization charge from the carrier in the silicon substrate was observed. To overcome this problem, the insulator films of buffer layer were used for MFIS structure

Figure 11(a) shows the leakage current density versus electrical field (*J-E*) characteristics of as-deposited BTV and BLTV thin films for MFIS structure. The leakage current density of the as-deposited BLTV thin films were about one order of magnitude lower than those of the non-doped lanthanum thin films. However, the lanthanum and vanadium doped BIT thin films were lower than those of BIT thin films. We suggested that low leakage current density attributed to substituting a bismuth ion with a lanthanum ion at A-site for lanthanum doped BTV thin films. To discuss bismuth ion substituting by lanthanum ion effect, the leakage current versus electrical field curves of BLTV thin films were fitted to Schottky emission and Poole–Frankel transport models. The inset of Fig. 11(a) shows the JE characteristics for BIT, BTV, and BLTV thin films in terms of J/E as vertical axis and E1/2 as horizontal axis. The fit‐ ting curves were straight in this figure and a JE curve of thin films was the Poole−Frankel emission model. The high leakage current was attributed to the bismuth vacancies and oxy‐ gen vacancies of as-deposited BIT thin films [39−42]. Park et al. also suggested that the en‐ hanced stability of TiO6 octahedra against oxygen vacancies for fatigue resistance of

In a previous study, the low threshold voltage of ferroelectric thin films was attributed by bismuth and oxygen vacancy [9]. The threshold voltage for the lanthanum-doped BTV thin films of MFIS structure was improved from 5 to 3 V. The memory functional effect and de‐ pletion delay of the MFIS structure was caused by remanent polarization of ferroelectric thin films in CV curves. In this study, the memory window was increased from 15 to 18 V. The large memory window of lanthanum-doped BTV thin films was also proved by *p-E* curves in Fig. 11(b). As a result, the improvement in the capacitance of the BLTV thin films was at‐ tributed to the compensation of the oxygen vacancy of the BLSF structure. Additionally, the

capacitance versus applied voltage curves of as-deposited BTV and BLTV thin films.

[28-30, 37-38].

lanthanum doped BIT thin films [31].

Figure 9(b) shows the *p-E* curves of the different ferroelectric thin films under applied volt‐ age of 18V from the Sawyer−Tower circuits. The remanent polarization of non-doped, vana‐ dium-doped, and lanthanum-doped ferroelectric thin films linearly was increased from 5, 10 to 11 μC/cm2 , respectively. The coercive filed of non-doped, vanadium-doped, and lantha‐ num-doped ferroelectric thin films were about 300, 300, and 250 kV/cm, respectively. The ferroelectric properties of lanthanum-doped and vanadium-doped BIT thin films were im‐ proved and found.

**Figure 10.** a)The *C-V* characteristics of as-deposited BTV and BLTV thin films. (b) The polarization versus electrical field characteristics of as-deposited BTV and BLTV thin films.

The fatigue characteristics for ferroelectric thin films were the time dependent change of the polarization state. After a long time, the polarization loss of ferroelectric thin films was af‐ fected by oxygen vacancies, defect, and space charge in the memory device. Figure 10 shows the polarization versus electrical field (*p-E*) properties of the different ferroelectric thin films before and after the switching of 109 cycles. The remnant polarization loss of BLTV and BTV thin films were about 9% and 15% of initial polarization value, respectively. The remnant polarization of BLTV thin films were little changed after the switching cycles. The fatigue behavior and domain pinning were improved by lanthanum and vanadium addition in BIT thin films. To reduce bismuth defect and oxygen vacancy, the high-valence cation substitut‐ ed for the A-site of BLTV thin films were observed.

that the bismuth vacancies of BTV thin films compensate for lanthanum addition and microstructure were improved in BLTV thin films. From the cross-sectional SEM image, average thin film thicknesses for MFIS structure were about 610 nm. The average thickness of thin

Figure 9(a) shows the change in the capacitance versus the applied voltage (*C-V*) of the BTV and BLTV thin films in MFM structure measured at 100 kHz. The applied voltages, which were first changed from -20 to 20 V and then returned to -20 V, were used to measure the capacitance voltage characteristics (CV). The BLTV thin films exhibited high capacitance than those of BTV thin films. We found that the capacitances of the lanthanum-doped BTV

Figure 9(b) shows the *p-E* curves of the different ferroelectric thin films under applied volt‐ age of 18V from the Sawyer−Tower circuits. The remanent polarization of non-doped, vana‐ dium-doped, and lanthanum-doped ferroelectric thin films linearly was increased from 5, 10

num-doped ferroelectric thin films were about 300, 300, and 250 kV/cm, respectively. The ferroelectric properties of lanthanum-doped and vanadium-doped BIT thin films were im‐

**Figure 10.** a)The *C-V* characteristics of as-deposited BTV and BLTV thin films. (b) The polarization versus electrical field

The fatigue characteristics for ferroelectric thin films were the time dependent change of the polarization state. After a long time, the polarization loss of ferroelectric thin films was af‐ fected by oxygen vacancies, defect, and space charge in the memory device. Figure 10 shows the polarization versus electrical field (*p-E*) properties of the different ferroelectric thin films before and after the switching of 109 cycles. The remnant polarization loss of BLTV and BTV thin films were about 9% and 15% of initial polarization value, respectively. The remnant polarization of BLTV thin films were little changed after the switching cycles. The fatigue behavior and domain pinning were improved by lanthanum and vanadium addition in BIT thin films. To reduce bismuth defect and oxygen vacancy, the high-valence cation substitut‐

, respectively. The coercive filed of non-doped, vanadium-doped, and lantha‐

films for MFM structure was about 672 nm.

thin films were increased from 2.38 to 2.42 nF.

characteristics of as-deposited BTV and BLTV thin films.

ed for the A-site of BLTV thin films were observed.

to 11 μC/cm2

438 Advances in Ferroelectrics

proved and found.

**Figure 11.** a) The (J/T2) and (J/E) versus E1/2 curves of as-deposited BIT, BTV, and BLTV thin films. (b) The normalization capacitance versus applied voltage curves of as-deposited BTV and BLTV thin films.

From the *J-E* curves, the conduction mechanism of as-deposited thin films was also proved the bismuth defect and oxygen vacancy results. For MFIS structure, the interface and inter‐ action problem between of the silicon and ferroelectric films was serious important. The ef‐ fect of the bound polarization charge from the carrier in the silicon substrate was observed. To overcome this problem, the insulator films of buffer layer were used for MFIS structure [28-30, 37-38].

Figure 11(a) shows the leakage current density versus electrical field (*J-E*) characteristics of as-deposited BTV and BLTV thin films for MFIS structure. The leakage current density of the as-deposited BLTV thin films were about one order of magnitude lower than those of the non-doped lanthanum thin films. However, the lanthanum and vanadium doped BIT thin films were lower than those of BIT thin films. We suggested that low leakage current density attributed to substituting a bismuth ion with a lanthanum ion at A-site for lanthanum doped BTV thin films. To discuss bismuth ion substituting by lanthanum ion effect, the leakage current versus electrical field curves of BLTV thin films were fitted to Schottky emission and Poole–Frankel transport models. The inset of Fig. 11(a) shows the JE characteristics for BIT, BTV, and BLTV thin films in terms of J/E as vertical axis and E1/2 as horizontal axis. The fit‐ ting curves were straight in this figure and a JE curve of thin films was the Poole−Frankel emission model. The high leakage current was attributed to the bismuth vacancies and oxy‐ gen vacancies of as-deposited BIT thin films [39−42]. Park et al. also suggested that the en‐ hanced stability of TiO6 octahedra against oxygen vacancies for fatigue resistance of lanthanum doped BIT thin films [31].

In a previous study, the low threshold voltage of ferroelectric thin films was attributed by bismuth and oxygen vacancy [9]. The threshold voltage for the lanthanum-doped BTV thin films of MFIS structure was improved from 5 to 3 V. The memory functional effect and de‐ pletion delay of the MFIS structure was caused by remanent polarization of ferroelectric thin films in CV curves. In this study, the memory window was increased from 15 to 18 V. The large memory window of lanthanum-doped BTV thin films was also proved by *p-E* curves in Fig. 11(b). As a result, the improvement in the capacitance of the BLTV thin films was at‐ tributed to the compensation of the oxygen vacancy of the BLSF structure. Additionally, the memory window of ferroelectric thin films was changed by the sweeping speed. The ferro‐ electric capacitance and threshold voltage of MFIS structure slightly decreases as the sweep‐ ing speed increases. That was influenced by the mobile ions and charge injection between as-deposited ferroelectric thin films and metal electrode as the sweeping speed increased [9].

### **3.3. The Influence of Neodymium Doping on the Physical and Electrical Properties of BTV (BNTV) Ferroelectric Thin Films**

Figure 12(a) shows x-ray diffraction patterns of the as-deposited ferroelectric thin films for different oxygen concentration on ITO substrate. From the XRD patterns, we found that the ferroelectric thin films exhibited polycrystalline structure. In addition, the (117), (008), and (220) peaks were observed in the XRD pattern. The intensity of the (117) peak of the ferro‐ electric thin films increases linearly as the oxygen concentration increases from 0 to 40%. The intensity of the (117) peak of the as-deposited ferroelectric thin films decreases at oxy‐ gen concentration from 40 to 60%. As shown in Fig. 3, the (117) preferred phase and smallest full-width-half-magnitude (FWHM) value were exhibited by the as-deposited ferroelectric thin film with the 40% oxygen concentration. The polycrystalline structure of the as-deposit‐ ed (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film was optimal at 40% oxygen concentration.

**Figure 13.** The surface morphology of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film for 40% oxygen concentration. The surface micro-structure of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film for (a)

The Influence of Vanadium Doping on the Physical and Electrical Properties

http://dx.doi.org/10.5772/52203

441

From the SEM images in Fig. 13, the surface morphology and grain size of the as-deposit‐ ed (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films for 25 and 40% oxygen concentration were ob‐ served. The grain size of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films were about 110 nm and 50 nm, respectively. We deduced that grain size changed caused by the different

**Figure 14.** a) The capacitance versus applied voltage (C-V) properties of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 fer‐ roelectric thin film. (b) The polarization versus applied electrical field (p-E) properties of the as-deposited (Bi3.25Nd0.75)

Figure 14(a) shows the *C-V* characteristics measured with the MFM capacitor structure for the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films deposited under various oxygen concen‐ trations. The applied bias voltage is adjusted from -20 to 20V. The capacitance of ferroelec‐ tric thin films first increases with the increase of oxygen concentration and reaches the maximum value in the 40 % oxygen atmosphere. Then the capacitance apparently decreases in the further increase of oxygen to 60 %. This variation of capacitance has the similar results with the XRD patterns and the AFM images. The polarization versus applied electrical field (*p-E*) curves of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film at the frequen‐ cy of 1kHz were shown in fig. 14(b). From the *p-E* curves results, the remnant polarization of

remnant polarization and coercive filed of ferroelectric thin films for 40% oxygen concentra‐

, as the coercive filed of 220 kV/cm. In addition, the

and 300 kV/cm. From the experimental measurement, this result

25% and (b) 40% oxygen concentration.

oxygen xoncentration.

(Ti2.9V0.1)O12 ferroelectric thin film.

ferroelectric thin films were 11 μC/cm2

tion were about 10 μC/cm2

**Figure 12.** a) The x-ray diffraction patterns of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film for the different oxygen concentration. (b) The surface morphology of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film for 40% oxygen concentration.

Besides, the interface between an electrode and the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films was an important factor that seriously influences the physical and electrical properties of the MIM capacitor structure. Therefore, the surface roughness of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films for 40% oxygen concentration was absolutely determined and calculated. Figure 12(b) shows the surface roughness of the as-deposited (Bi3.25Nd0.75) (Ti2.9V0.1)O12 ferroelectric thin film from the AFM images. The roughness of the ferroelectric thin film were 4.278nm. The surface roughness of the as-deposited ferroelectric thin films in‐ creases with the oxygen concentration. Therefore, we assume that the surface roughness of the as-deposited ferroelectric thin films increases due to an increase in the crystallinity with oxygen concentration.

memory window of ferroelectric thin films was changed by the sweeping speed. The ferro‐ electric capacitance and threshold voltage of MFIS structure slightly decreases as the sweep‐ ing speed increases. That was influenced by the mobile ions and charge injection between as-deposited ferroelectric thin films and metal electrode as the sweeping speed increased [9].

**3.3. The Influence of Neodymium Doping on the Physical and Electrical Properties of**

Figure 12(a) shows x-ray diffraction patterns of the as-deposited ferroelectric thin films for different oxygen concentration on ITO substrate. From the XRD patterns, we found that the ferroelectric thin films exhibited polycrystalline structure. In addition, the (117), (008), and (220) peaks were observed in the XRD pattern. The intensity of the (117) peak of the ferro‐ electric thin films increases linearly as the oxygen concentration increases from 0 to 40%. The intensity of the (117) peak of the as-deposited ferroelectric thin films decreases at oxy‐ gen concentration from 40 to 60%. As shown in Fig. 3, the (117) preferred phase and smallest full-width-half-magnitude (FWHM) value were exhibited by the as-deposited ferroelectric thin film with the 40% oxygen concentration. The polycrystalline structure of the as-deposit‐ ed (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film was optimal at 40% oxygen concentration.

**Figure 12.** a) The x-ray diffraction patterns of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film for the different oxygen concentration. (b) The surface morphology of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric

Besides, the interface between an electrode and the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films was an important factor that seriously influences the physical and electrical properties of the MIM capacitor structure. Therefore, the surface roughness of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films for 40% oxygen concentration was absolutely determined and calculated. Figure 12(b) shows the surface roughness of the as-deposited (Bi3.25Nd0.75) (Ti2.9V0.1)O12 ferroelectric thin film from the AFM images. The roughness of the ferroelectric thin film were 4.278nm. The surface roughness of the as-deposited ferroelectric thin films in‐ creases with the oxygen concentration. Therefore, we assume that the surface roughness of the as-deposited ferroelectric thin films increases due to an increase in the crystallinity with

**BTV (BNTV) Ferroelectric Thin Films**

440 Advances in Ferroelectrics

thin film for 40% oxygen concentration.

oxygen concentration.

**Figure 13.** The surface morphology of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film for 40% oxygen concentration. The surface micro-structure of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film for (a) 25% and (b) 40% oxygen concentration.

From the SEM images in Fig. 13, the surface morphology and grain size of the as-deposit‐ ed (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films for 25 and 40% oxygen concentration were ob‐ served. The grain size of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films were about 110 nm and 50 nm, respectively. We deduced that grain size changed caused by the different oxygen xoncentration.

**Figure 14.** a) The capacitance versus applied voltage (C-V) properties of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 fer‐ roelectric thin film. (b) The polarization versus applied electrical field (p-E) properties of the as-deposited (Bi3.25Nd0.75) (Ti2.9V0.1)O12 ferroelectric thin film.

Figure 14(a) shows the *C-V* characteristics measured with the MFM capacitor structure for the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films deposited under various oxygen concen‐ trations. The applied bias voltage is adjusted from -20 to 20V. The capacitance of ferroelec‐ tric thin films first increases with the increase of oxygen concentration and reaches the maximum value in the 40 % oxygen atmosphere. Then the capacitance apparently decreases in the further increase of oxygen to 60 %. This variation of capacitance has the similar results with the XRD patterns and the AFM images. The polarization versus applied electrical field (*p-E*) curves of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelectric thin film at the frequen‐ cy of 1kHz were shown in fig. 14(b). From the *p-E* curves results, the remnant polarization of ferroelectric thin films were 11 μC/cm2 , as the coercive filed of 220 kV/cm. In addition, the remnant polarization and coercive filed of ferroelectric thin films for 40% oxygen concentra‐ tion were about 10 μC/cm2 and 300 kV/cm. From the experimental measurement, this result was attributed to the suitable oxygen concentration sample as compared to that of the asdeposited ferroelectric thin film. As the oxygen/argon mixtures were used as the depositing atmosphere, the defects and oxygen vacancies in ferroelectric thin films were filled and com‐ pensated by oxygen gas, and the leakage current density were decreased. The low leakage current density will reveal in the 40%-oxygen-deposited ferroelectric thin films. For that the capacitance will be increased and the leakage current density will be decreased. As the ap‐ plied voltage of 15V was used, the leakage current density of ferroelectric thin films deposit‐ ed at 40% oxygen concentration is about 1×10-9A/cm2 .

temperature increases from 400 to 550 °C. The intensity of the (110) peak of the as-deposited

(110) preferred phase and smallest full-width-half-magnitude (FWHM) value were exhibited by the as-deposited vanadium oxide thin film with the sintering temperature of 550 °C. The polycrystalline structure of the as-deposited vanadium oxide thin film was optimal at 550 °C

**Figure 16.** a) Metal-insulator-metal (MIM) structure using as-deposited vanadium oxide thin films. (b) The x-ray dif‐ fraction patterns of the as-deposited vanadium oxide thin films on ITO substrate for different sintering temperature.

The thickness of as-depoisted vandium oxide thin films for different sintering temperature was determined by SEM morphology. As the oxygen concentration increases from 0 to 60%, the thickness of as-deposited vandium oxide thin films linearly decreases. In addition, the deposition rate of as-deposited vanadium oxide thin films with 60% oxygen concentration was 2.62 nm/min. The decreases in the deposition rate and thickness of as-depoisted vanadi‐ um oxide thin films might be affected by the decrease in Ar/O2 ratio. The Ar/O2 ratio was adjusted using argon gas to generate the plasma on the surface of the as-depoisted vanadi‐ um oxide ceramic target during sputtering. Figure 17 shows the surface morphology for the as-deposited and 500 °C sintered vanadium oxide thin films. We found that the grain size of 500 °C sintered vanadium oxide thin films were larger than others. The better resistance

Figure 18 shows the current vsersus applied voltage (*I-V*) properties of vanadium oxide thin films for the different sintering temperature. After the staring forming process, the device reached a low resistance state (LRS) and high resistance state (HRS). By sweeping the bias to negative over the reset voltage, a gradual decrease of current was presented to switch the cells from LRS to HRS (reset process). Additionally, the cell turns back to LRS while applying a larg‐ er positive bias than the set voltage (set process). All of the vanadium oxide thin film were ex‐ hibit the bipolar behavior. The *I-V* properties of as-deposited vanadium oxide thin films of 60%

ing the rf sputtering deposition process, oxygen vacancies appear in the as-deposited vanadi‐ um oxide thin films. The defects and oxygen vacancies of as-deposited vanadium oxide thin films were filled and compensated for to different extents at different oxygen concentrations. In addition, the smallest leakage current density of as-deposited vanadium oxide thin films

when an applied electrical voltage of 0.1V. Dur‐

C. As shown in Fig. 16, the

http://dx.doi.org/10.5772/52203

443

The Influence of Vanadium Doping on the Physical and Electrical Properties

thin films decreases at sintering temperature from 550 to 600 o

sintering temperature.

properties might be caused by this reason.

oxygen concentration was about 1×10-4A/cm2

**Figure 15.** a) The leakage current density versus applied voltage (J-E) properties of the as-deposited (Bi3.25Nd0.75) (Ti2.9V0.1)O12 ferroelectric thin film.(b) The retention and fatigue properties of the as-deposited ferroelectric thin film corresponding hysteresis loop before and after fatigue test.

The retention and fatigue properties for the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelec‐ tric thin films were the time dependent change of the polarization state. After long time test‐ ing, the polarization loss of the as-deposited ferroelectric thin films was affected by oxygen vacancies, defect, and space charges in the memory device test. Figure 15 shows the polari‐ zation versus electrical field (*p-E*) properties of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 fer‐ roelectric thin films before and after the switching of 109 cycles. The remnant polarization loss of ferroelectric thin films was about 9% of initial polarization value, respectively. The remnant polarization of ferroelectric thin films was little changed after the test cycles. The fatigue behavior and domain pinning were improved by neodymium and vanadium addi‐ tion in BIT thin films. To improve bismuth and oxygen vacancy, the high-valence cation substituted for the A-site of (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films were observed.

### **3.4. Bipolar Resistive Switching Properties of Transparent Vanadium Oxide (V2O5) Resistive Random Access Memory**

Figure 16(b) shows x-ray diffraction patterns of the as-deposited vanadium oxide thin films for 60% oxygen concentration on ITO substrate prepared by different sintering temperature. From the XRD patterns, we found that the vanadium oxide thin films exhibited polycrystal‐ line structure. In addition, the (110), (222), and (400) peaks were observed in the XRD pat‐ tern. The intensity of the (110) peak of the thin films increases linearly as the sintering temperature increases from 400 to 550 °C. The intensity of the (110) peak of the as-deposited thin films decreases at sintering temperature from 550 to 600 o C. As shown in Fig. 16, the (110) preferred phase and smallest full-width-half-magnitude (FWHM) value were exhibited by the as-deposited vanadium oxide thin film with the sintering temperature of 550 °C. The polycrystalline structure of the as-deposited vanadium oxide thin film was optimal at 550 °C sintering temperature.

was attributed to the suitable oxygen concentration sample as compared to that of the asdeposited ferroelectric thin film. As the oxygen/argon mixtures were used as the depositing atmosphere, the defects and oxygen vacancies in ferroelectric thin films were filled and com‐ pensated by oxygen gas, and the leakage current density were decreased. The low leakage current density will reveal in the 40%-oxygen-deposited ferroelectric thin films. For that the capacitance will be increased and the leakage current density will be decreased. As the ap‐ plied voltage of 15V was used, the leakage current density of ferroelectric thin films deposit‐

**Figure 15.** a) The leakage current density versus applied voltage (J-E) properties of the as-deposited (Bi3.25Nd0.75) (Ti2.9V0.1)O12 ferroelectric thin film.(b) The retention and fatigue properties of the as-deposited ferroelectric thin film

The retention and fatigue properties for the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 ferroelec‐ tric thin films were the time dependent change of the polarization state. After long time test‐ ing, the polarization loss of the as-deposited ferroelectric thin films was affected by oxygen vacancies, defect, and space charges in the memory device test. Figure 15 shows the polari‐ zation versus electrical field (*p-E*) properties of the as-deposited (Bi3.25Nd0.75)(Ti2.9V0.1)O12 fer‐

loss of ferroelectric thin films was about 9% of initial polarization value, respectively. The remnant polarization of ferroelectric thin films was little changed after the test cycles. The fatigue behavior and domain pinning were improved by neodymium and vanadium addi‐ tion in BIT thin films. To improve bismuth and oxygen vacancy, the high-valence cation

substituted for the A-site of (Bi3.25Nd0.75)(Ti2.9V0.1)O12 thin films were observed.

**3.4. Bipolar Resistive Switching Properties of Transparent Vanadium Oxide (V2O5)**

Figure 16(b) shows x-ray diffraction patterns of the as-deposited vanadium oxide thin films for 60% oxygen concentration on ITO substrate prepared by different sintering temperature. From the XRD patterns, we found that the vanadium oxide thin films exhibited polycrystal‐ line structure. In addition, the (110), (222), and (400) peaks were observed in the XRD pat‐ tern. The intensity of the (110) peak of the thin films increases linearly as the sintering

cycles. The remnant polarization

.

ed at 40% oxygen concentration is about 1×10-9A/cm2

442 Advances in Ferroelectrics

corresponding hysteresis loop before and after fatigue test.

**Resistive Random Access Memory**

roelectric thin films before and after the switching of 109

**Figure 16.** a) Metal-insulator-metal (MIM) structure using as-deposited vanadium oxide thin films. (b) The x-ray dif‐ fraction patterns of the as-deposited vanadium oxide thin films on ITO substrate for different sintering temperature.

The thickness of as-depoisted vandium oxide thin films for different sintering temperature was determined by SEM morphology. As the oxygen concentration increases from 0 to 60%, the thickness of as-deposited vandium oxide thin films linearly decreases. In addition, the deposition rate of as-deposited vanadium oxide thin films with 60% oxygen concentration was 2.62 nm/min. The decreases in the deposition rate and thickness of as-depoisted vanadi‐ um oxide thin films might be affected by the decrease in Ar/O2 ratio. The Ar/O2 ratio was adjusted using argon gas to generate the plasma on the surface of the as-depoisted vanadi‐ um oxide ceramic target during sputtering. Figure 17 shows the surface morphology for the as-deposited and 500 °C sintered vanadium oxide thin films. We found that the grain size of 500 °C sintered vanadium oxide thin films were larger than others. The better resistance properties might be caused by this reason.

Figure 18 shows the current vsersus applied voltage (*I-V*) properties of vanadium oxide thin films for the different sintering temperature. After the staring forming process, the device reached a low resistance state (LRS) and high resistance state (HRS). By sweeping the bias to negative over the reset voltage, a gradual decrease of current was presented to switch the cells from LRS to HRS (reset process). Additionally, the cell turns back to LRS while applying a larg‐ er positive bias than the set voltage (set process). All of the vanadium oxide thin film were ex‐ hibit the bipolar behavior. The *I-V* properties of as-deposited vanadium oxide thin films of 60% oxygen concentration was about 1×10-4A/cm2 when an applied electrical voltage of 0.1V. Dur‐ ing the rf sputtering deposition process, oxygen vacancies appear in the as-deposited vanadi‐ um oxide thin films. The defects and oxygen vacancies of as-deposited vanadium oxide thin films were filled and compensated for to different extents at different oxygen concentrations. In addition, the smallest leakage current density of as-deposited vanadium oxide thin films was obtained at an oxygen concentration of 40%. As shown in Fig. 18, the high leakage current density and thin films of as-deposited vanadium oxide thin films for 60% oxygen concentra‐ tion were attributed to low argon sputtering gas concentration.

analysis. The BNTV, BLTV, and BTV shows clear ferroelectricity form the *p-E* curves. The remnant polarization properties of BLTV thin film decreased by 9%, while that of the BTV

electric capacitors was attributed to oxygen and bismuth vacancies. The leakage current density of as-deposited BLTV and BTV thin films were lower than those of BIT, which were were attributed to the decrease of oxygen and ismuth vacancies after vanadium and lantha‐ num addition. The conduction mechanism of as-deposited BTV and BLTV thin films were also proved these results in *J-E* curves. We indicated that small ions substitution for A and B site of BLSF structure was effective decreased the oxygen and bismuth vacancies. Finally, the low threshold voltage and memory window of BLTV thin films were improved from the *C-V* curves measured. In addition, the metal oxide thin films such as, TiO2, Ta2O5, Al2O3, and CuO were widely investigated and discussed for applications in nonvolatile resistive ran‐ dom access memory (RRAM) devices. The nonvolatile resistive random access memory (RRAM) devices were well developed and studied because of their structural simplicity,

1013) and other non-volatile advantages. Therefore, the electrical switching properties of the vanadium oxide thin films for nonvolatile resistive random access memory (RRAM) device

The authors will acknowledge to Prof. Ting-Chang Chang and Prof. Cheng-Fu Yang. Addi‐ tionally, this work will acknowledge the financial support of the National Science Council of

, Sean Wu3

1 Department of Electronics Engineering/Tung-Fang Design University/Taiwan,R.O.C., Tai‐

2 Department of Electronic Engineering/Southern Taiwan University of Science and Tech‐

3 Department of Electronics Engineering/Tung-Fang Design University/Taiwan,R.O.C., Tai‐

4 Department of Mathematics and Physics/Chinese Air Force Academy/Taiwan, R.O.C., Tai‐

the Republic of China (NSC 99-2221-E-272-003) and (NSC 100-2221-E-272-002).

switching cycles. Fatigue behavior in ferro‐

The Influence of Vanadium Doping on the Physical and Electrical Properties

ns), high operating cycles (>

http://dx.doi.org/10.5772/52203

445

and Jen-Hwan Tsai4

~ 103

, Chin-Hsiung Liao4

decreased by 15% after the fatigue test with 109

high density and low power, read / write speed (about 101

, Chien-Min Cheng2

were observed.

**Acknowledgements**

**Author details**

Kai-Huang Chen1

nology/Taiwan, R.O.C., Taiwan

wan

wan

wan

**Figure 17.** The surface morphology for (a) as-deposited (b) 500°C sintered vanadium oxide thin films.

**Figure 18.** a) Typical *I-V* characteristics of vanadium oxide thin films for the different oxygen concentration. (b) Typical *I-V* characteristics of the as-deposited vanadium oxide thin films for different sintering temperature.

In addition, The transport current of the vanadium oxide thin films decreases linearly as the sintering temperature increases from 450 to 500 °C. The transport current of the as-de‐ posited vanadium oxide thin films increases at sintering temperature from 500 to 550 °C. We found the as-deposited vanadium oxide thin films prepared by 500 °C sintering temper‐ ature were exhibited the large the on/off ratio resistance properties. In addition, the switch‐ ing cycling was measured another type of reliability and retention characteristics were observed. There was a slight flucation of resistance in the HRS and LRS states, and the sta‐ ble bipolar witching property was observed during 20 cycles. The results show remarka‐ ble reliability performance of the resistance random access memory devices for nonvolatile memory applications.

### **4. Conclusion**

In conclusion, BIT, BTV, BNTV, and BLTV thin films were prepared by rf magnetron sput‐ tering. We confirmed that all thin films on Pt/Ti/SiO2/Si substrate well crystallized by XRD analysis. The BNTV, BLTV, and BTV shows clear ferroelectricity form the *p-E* curves. The remnant polarization properties of BLTV thin film decreased by 9%, while that of the BTV decreased by 15% after the fatigue test with 109 switching cycles. Fatigue behavior in ferro‐ electric capacitors was attributed to oxygen and bismuth vacancies. The leakage current density of as-deposited BLTV and BTV thin films were lower than those of BIT, which were were attributed to the decrease of oxygen and ismuth vacancies after vanadium and lantha‐ num addition. The conduction mechanism of as-deposited BTV and BLTV thin films were also proved these results in *J-E* curves. We indicated that small ions substitution for A and B site of BLSF structure was effective decreased the oxygen and bismuth vacancies. Finally, the low threshold voltage and memory window of BLTV thin films were improved from the *C-V* curves measured. In addition, the metal oxide thin films such as, TiO2, Ta2O5, Al2O3, and CuO were widely investigated and discussed for applications in nonvolatile resistive ran‐ dom access memory (RRAM) devices. The nonvolatile resistive random access memory (RRAM) devices were well developed and studied because of their structural simplicity, high density and low power, read / write speed (about 101 ~ 103 ns), high operating cycles (> 1013) and other non-volatile advantages. Therefore, the electrical switching properties of the vanadium oxide thin films for nonvolatile resistive random access memory (RRAM) device were observed.

### **Acknowledgements**

was obtained at an oxygen concentration of 40%. As shown in Fig. 18, the high leakage current density and thin films of as-deposited vanadium oxide thin films for 60% oxygen concentra‐

**Figure 17.** The surface morphology for (a) as-deposited (b) 500°C sintered vanadium oxide thin films.

**Figure 18.** a) Typical *I-V* characteristics of vanadium oxide thin films for the different oxygen concentration. (b) Typical

In addition, The transport current of the vanadium oxide thin films decreases linearly as the sintering temperature increases from 450 to 500 °C. The transport current of the as-de‐ posited vanadium oxide thin films increases at sintering temperature from 500 to 550 °C. We found the as-deposited vanadium oxide thin films prepared by 500 °C sintering temper‐ ature were exhibited the large the on/off ratio resistance properties. In addition, the switch‐ ing cycling was measured another type of reliability and retention characteristics were observed. There was a slight flucation of resistance in the HRS and LRS states, and the sta‐ ble bipolar witching property was observed during 20 cycles. The results show remarka‐ ble reliability performance of the resistance random access memory devices for nonvolatile

In conclusion, BIT, BTV, BNTV, and BLTV thin films were prepared by rf magnetron sput‐ tering. We confirmed that all thin films on Pt/Ti/SiO2/Si substrate well crystallized by XRD

*I-V* characteristics of the as-deposited vanadium oxide thin films for different sintering temperature.

memory applications.

**4. Conclusion**

tion were attributed to low argon sputtering gas concentration.

444 Advances in Ferroelectrics

The authors will acknowledge to Prof. Ting-Chang Chang and Prof. Cheng-Fu Yang. Addi‐ tionally, this work will acknowledge the financial support of the National Science Council of the Republic of China (NSC 99-2221-E-272-003) and (NSC 100-2221-E-272-002).

### **Author details**

Kai-Huang Chen1 , Chien-Min Cheng2 , Sean Wu3 , Chin-Hsiung Liao4 and Jen-Hwan Tsai4

1 Department of Electronics Engineering/Tung-Fang Design University/Taiwan,R.O.C., Tai‐ wan

2 Department of Electronic Engineering/Southern Taiwan University of Science and Tech‐ nology/Taiwan, R.O.C., Taiwan

3 Department of Electronics Engineering/Tung-Fang Design University/Taiwan,R.O.C., Tai‐ wan

4 Department of Mathematics and Physics/Chinese Air Force Academy/Taiwan, R.O.C., Tai‐ wan

### **References**


[22] Kim, S. S., Song, T. K., Kim, J. K., & Kim, J. (2002). *J. Appl. Phys.*, 92 4.

[24] Velu, G., Legrand, C., Tharaud, O., Chapoton, A., Remiens, D., & Horowitz, G.

The Influence of Vanadium Doping on the Physical and Electrical Properties

http://dx.doi.org/10.5772/52203

447

[25] Chen, K. H., Yang, C. F., Chang, C. H., & Lin, Y. J. (2009). *Jpn. J. Appl. Phys.*, 48

[26] Miao, J., Yuan, J., Wu, H., Yang, S. B., Xu, B., Cao, L. X., & Zhao, B. R. (2001). *Appl.*

[27] Chen, K. H., Chen, Y. C., Chia, W. K., Chen, Z. S., Yang, C. F., & Chung, H. H. (2008).

[31] Park, B. H., Kang, B. S., Bu, S. D., Noh, T. W., Lee, L., & Joe, W. (1999). *Nature (Lon‐*

[32] Noguchi, Y., Miwa, I., Goshima, Y., & Miyayama, M. (2000). *Jpn. J. Appl. Phys*, 39

[34] Friessnegg, T., Aggarwal, S., Ramesh, R., Nielsen, B., Poindexter, E. H., & Keeble, D.

[35] Watanabe, T., Funakubo, H., Osada, M., Noguchi, Y., & Miyayama, M. (2002). *Appl.*

[39] Rost, T. A., Lin, H., Rabson, T. A., Baumann, R. C., & Callahan, D. C. (1991). IEEE

[23] Noguchi, Y., & Miyayama, M. (2001). *Appl. Phys. Lett.*, 78 13.

[28] Wu, S. Y. (1974). *IEEE Trans. Electron Devices*, 21 499.

[30] Sugibuchi, K., Kurogi, Y., & Endo, N. (1975). J. Appl. Phys. 46 2877. .

[33] Noguchi, Y., & Miyayama, M. (2001). *Appl. Phys. Lett.*, 78 1903.

[36] Kim, S. S., Song, T. K., Kim, J. K., & Kim, J. (2002). *J. Appl. Phys.*, 92, 4.

[38] Rost, T. A., Lin, H., & Rabson, T. A. (1991). *Appl. Phys. Lett.*, 59 3654.

[40] Fleischer, S., Lai, P. T., & Cheng, Y. C. (1994). *J. Appl. Phys.*, 73 8353.

[41] Mihara, T., & Watanabe, H. (1995). *Part I, Jpn. J. Appl. Phys.*, 34 5664.

[37] Noguchi, Y., & Miyayama, M. (2001). *Appl. Phys. Lett.*, 78 13.

Trans. Ultrason. Ferroelectr. Freq. Control, , 38

[42] Lin, Y. B., & Lee, J. Y. (2000). *J. Appl. Phys.*, 87 1841.

(2001). *Appl. Phys. Lett.*, 79 659.

*Phys. Lett.*, 90 022903.

*don)*, 401 682.

*Phys. Lett.*, 80 1.

L1259.

*Key Eng. Mater.*, 368-372.

[29] Wu, S. Y. (1976). *Ferroelectrics*, 11 379.

J. (2000). *Appl. Phys. Lett.*, 77 127.

091401.


**References**

446 Advances in Ferroelectrics

461.

*Process*, 90.

*Phys. Lett.*, 77-76.

*roelectr. Freq. Control*, 54.

*A-Mater. Sci. Process*, 89, 533.

*Phys. Lett.*, 100, 023109.

*Nature (London)*, 374, 627.

*roelectr. Freq. Control*, 54 1726.

401, 682.

89 533.

*Phys. Lett.*, 80 1.

[1] Shannigrahi, R., & Jang, H. M. (2001). *Appl. Phys. Lett.*, 79, 1051.

[3] Xiong, S. B., & Sakai, S. (1999). *Appl. Phys. Lett.*, 75.

[4] Kim, J. S., & Yoon, S. G. (2000). *Vac. Soc. Technol.*, B, 18, 216.

[5] Wu, T. B., Wu, C. M., & Chen, M. L. (1996). *Appl. Phys. Lett.*, 69, 2659.

[2] Hong, S. K., Suh, C. W., Lee, C. G., Lee, S. W., Hang, E. Y., & Kang, N. S. (2000). *Appl.*

[6] Chen, K. H., Chen, Y. C., Yang, C. F., & Chang, T. C. (2008). *J. Phys. Chem. Solids*, 69,

[7] Yang, C. F., Chen, K. H., Chen, Y. C., & Chang, T. C. (2007). *IEEE Trans. Ultrason. Fer‐*

[8] Yang, C. F., Chen, K. H., Chen, Y. C., & Chang, T. C. (2008). *Appl. Phys. A-Mater. Sci.*

[9] Chen, K. H., Chen, Y. C., Chen, Z. S., Yang, C. F., & Chang, T. C. (2007). *Appl. Phys.*

[11] Lee, S., Song, E. B., Kim, S., Seo, D. H., Seo, S., Won, K. T., & Wang, K. L. (2012). *Appl.*

[14] Araujo, C. A., Cuchiaro, J. D., Mc Millian, L. D., Scott, M. C., & Scott, J. F. (1995).

[15] Park, B. H., Kang, B. S., Bu, S. D., Noh, T. W., Lee, J., & Jo, W. (1999). *Nature (London)*,

[16] Leu, C. C., Yao, L. R., Hsu, C. P., & Hu, C. T. (2010). *J. Electrochem. Soc.*, 157, 3, G85. [17] Chen, K. H., Chen, Y. C., Yang, C. F., & Chang, T. C. (2008). *J. Phys. Chem. Solids*, 69. [18] Yang, C. F., Chen, K. H., Chen, Y. C., & Chang, T. C. (2007). *IEEE Trans. Ultrason. Fer‐*

[19] Yang, C. F., Chen, K. H., Chen, Y. C., & Chang, T. C. (2008). *Appl. Phys. A*, 90 329.

[20] Chen, K. H., Chen, Y. C., Chen, Z. S., Yang, C. F., & Chang, T. C. (2007). *Appl. Phys. A*,

[21] Watanabe, T., Funakubo, H., Osada, M., Noguchi, Y., & Miyayama, M. (2002). *Appl.*

[13] Scotta, J. F., Paz, C. A., & de Araujoa, B. M. (1994). *J. Alloys, Compounds*, 211 451.

[10] Wu, S., Lin, Z. X., Lee, M. S., & Ro, R. (2007). *Appl. Phys. Lett.*, 102, 084908.

[12] Yang, Y., Jin, L., & Yang, X. D. (2012). *Appl. Phys. Lett.*, 100, 031103.


**Chapter 20**

**Provisional chapter**

**Nanoscale Ferroelectric Films, Strips and Boxes**

**Nanoscale Ferroelectric Films, Strips and Boxes**

A major characteristic of ferroelectric materials is that they posses a non-zero polarization in the absence of any applied electric field provided that the temperature is below a certain critical temperature[1]; in the literature this is often called a spontaneous polarization, or an equilibrium polarization. Another important defining characteristic is that this polarization can be reversed by applying an electric field. This is the basis of a useful application of ferroelectrics in which binary information can be stored according to the polarization direction which can be switched between two states by an electric field. For example, a thin film of ferroelectric can be switched to different states at different regions by suitably

With the increasing miniaturization of devices it becomes important to investigate size-effects in ferroelectrics. Thin film geometries are of interest in which one spatial dimension is confined, as well as strip geometries in which two-dimensions are confined, and, what here we will call box geometry in which all three dimensions are confined. By confinement we mean that surfaces of the ferroelectric that intersect on at least one line going through the central region of the ferroelectric are sufficiently close to this region to make their presence cause a non-negligible effect on the ferroelectric as a whole so that the behaviour is different from what would be expected in a bulk region far from any surface. The geometries chosen are typical of the sort that can be fabricated in micro or nanoelectronics and fit conveniently

The aim of this chapter is to show how the equilibrium polarization can be calculated in general for a confined volume of ferroelectric and this is applied to the three aforementioned geometries. In fact such calculations for thin films are well established[3–7]; here they are included with the other geometries, which are not so well studied, for completeness, and because they fit logical into the theoretical framework to be presented. A general problem of a similar nature to the one of interest here, but applied to spheres and cylinders has been considered by Morison's et al.[8]. This work employed approximate analytical solutions to the problem of calculating the polarization and avoided a full three-dimensional application

> ©2012 Webb, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 F. Webb; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

patterned electrodes, thus creating a ferroelectric random access memory[2].

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Jeffrey F. Webb

**1. Introduction**

Jeffrey F. Webb

http://dx.doi.org/10.5772/52040

into a Cartesian coordinate system.

of the general theory.

**Provisional chapter**

### **Nanoscale Ferroelectric Films, Strips and Boxes Nanoscale Ferroelectric Films, Strips and Boxes**

Jeffrey F. Webb Jeffrey F. Webb

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52040

**1. Introduction**

A major characteristic of ferroelectric materials is that they posses a non-zero polarization in the absence of any applied electric field provided that the temperature is below a certain critical temperature[1]; in the literature this is often called a spontaneous polarization, or an equilibrium polarization. Another important defining characteristic is that this polarization can be reversed by applying an electric field. This is the basis of a useful application of ferroelectrics in which binary information can be stored according to the polarization direction which can be switched between two states by an electric field. For example, a thin film of ferroelectric can be switched to different states at different regions by suitably patterned electrodes, thus creating a ferroelectric random access memory[2].

With the increasing miniaturization of devices it becomes important to investigate size-effects in ferroelectrics. Thin film geometries are of interest in which one spatial dimension is confined, as well as strip geometries in which two-dimensions are confined, and, what here we will call box geometry in which all three dimensions are confined. By confinement we mean that surfaces of the ferroelectric that intersect on at least one line going through the central region of the ferroelectric are sufficiently close to this region to make their presence cause a non-negligible effect on the ferroelectric as a whole so that the behaviour is different from what would be expected in a bulk region far from any surface. The geometries chosen are typical of the sort that can be fabricated in micro or nanoelectronics and fit conveniently into a Cartesian coordinate system.

The aim of this chapter is to show how the equilibrium polarization can be calculated in general for a confined volume of ferroelectric and this is applied to the three aforementioned geometries. In fact such calculations for thin films are well established[3–7]; here they are included with the other geometries, which are not so well studied, for completeness, and because they fit logical into the theoretical framework to be presented. A general problem of a similar nature to the one of interest here, but applied to spheres and cylinders has been considered by Morison's et al.[8]. This work employed approximate analytical solutions to the problem of calculating the polarization and avoided a full three-dimensional application of the general theory.

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 F. Webb; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Webb, licensee InTech. This is an open access chapter distributed under the terms of the Creative

*T*

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

451

*TC*

In the simplest case, then, a ferroelectric phase transition can be described by a single parameter that in Fig. 1 would be describable by the displacement of the sublattice from its central position. Since the paraelectric phase is of higher symmetry than the ferroelectric phase we can consider it to be more orderly and so the parameter is sometimes referred to as an order parameter. Actually the atoms are ions with charge and the corner-atom lattice would have a charge opposite to that of the interior sub-lattice. From the macroscopic point of view the corner atoms can be considered to produce a focus of charge at the center of each rectangle of opposite sign to the atoms that are displaced during the phase transition (see Fig. 1 where the centers of positive and negative charge are marked for the ferroelectric phase). For this reason the order parameter is often taken to be the polarization, which is the dipole moment per unit volume. For the time being, however, we will stay with the displacement representation of the order parameter as this is more directly related to the structural transition. If the displacement is denoted by *η* then it is clear that in the

The description above implies that each center ion is displaced in the same direction and this is the case for what are termed displacive ferroelectrics. There is another possibility, however, in which the displacement is not the same for every unit cell. For example, in some cells the displacement could be up while in others it could be down. Macroscopically ferroelectric and paraelectric phases can still result but now the ferroelectric phase occurs when statistically there are more displaced in one direction than in the other; and the paraelectric phase corresponds to the number of atoms displaced upwards is statistically equal to the number displaced downwards. In this case the ferroelectric is called an order-disorder ferroelectric. In terms of the macroscopic crystal symmetry however displacive and order-disorder ferroelectrics are equivalent and both can be described in general by Landau-Devonshire theory. The crystal structure of order-disorder ferroelectrics tends to be more complicated

The basis of the Landau-Devonshire theory is that in the phase transition the ferroelectric phase may be represented by a distorted symmetrical phase. As outlined above this is a macroscopic view and the theory is phenomenological, as will be brought out further below. It turns out that the symmetry elements lost by the crystal at the transition temperature is sufficient information for a description of the anomalies of practically all

*ǫ*

ferroelectric phase *η* �= 0 and in the paraelectric phase *η* = 0.

than for the displacive type[9, 10].

of the thermodynamic properties of the crystal[10].

**Figure 2.** A peak in the dielectric function *ǫ* occurs as *T* → *TC*.

**Figure 1.** A two-dimensional representation of how a crystal structure changes symmetry when passing between the paraelectric (*T* > *TC*) and ferroelectric (*T* < *TC*) phases

### **2. Foundations of the Landau-Devonshire theory of ferroelectrics**

### **2.1. Bulk ferroelectrics**

Since Landau-Devonshire theory is the theoretical basis for the work in this chapter it will be introduced here. Further details can be found elsewhere such as in Refs. [1, 9, 10]. In this section we consider the case of a bulk ferroelectric; the effect of surfaces will be introduced after that.

It is common to state that the starting point for Landau-Devonshire theory is the free energy per unit volume of a bulk ferroelectric expressed as an expansion in powers of the polarization *P*. The observed equilibrium polarization *P*<sup>0</sup> is then given by the value of *P* that minimizes the free energy. Here, however, following Strukov and Lenanyuk[10], we outline how this expansion comes from a statistical thermodynamic treatment involving an incomplete Gibbs potential, since this gives more insight into the theory. First a few more details about the nature of ferroelectrics.

When a ferroelectric crystal is above the critical temperature *TC* there is no spontaneous polarization; this is the paraelectric or high symmetry phase in which the crystal structure is of higher symmetry than the ferroelectric phase below *TC* since the appearance of the spontaneous polarization lowers the symmetry, as is evident from the two-dimensional representation in Fig. 1. Thus we see that during the transition from the paraelectric to ferroelectric state, a structural phase transition occurs. A consequence of this is that as the temperature approaches *TC* the structure becomes easy to distort giving rise to anomalous peaks in some of the properties such as the dielectric function *ǫ*, as illustrated in Fig. 2. In fact, in the ferroelectric phase, a crystal may have different domains such that the polarization in adjacent domains is not in the same direction[9], however it is possible to produce single-domain crystals—by cooling through *TC* in the presence of an electric field in one of the possible polarization directions for instance—and that is what we consider here.

**Figure 2.** A peak in the dielectric function *ǫ* occurs as *T* → *TC*.

2 Advances in Ferroelectrics

paraelectric (*T* > *TC*) and ferroelectric (*T* < *TC*) phases

details about the nature of ferroelectrics.

**2.1. Bulk ferroelectrics**

after that.

*T* > *TC T* < *TC*

**Figure 1.** A two-dimensional representation of how a crystal structure changes symmetry when passing between the

Since Landau-Devonshire theory is the theoretical basis for the work in this chapter it will be introduced here. Further details can be found elsewhere such as in Refs. [1, 9, 10]. In this section we consider the case of a bulk ferroelectric; the effect of surfaces will be introduced

It is common to state that the starting point for Landau-Devonshire theory is the free energy per unit volume of a bulk ferroelectric expressed as an expansion in powers of the polarization *P*. The observed equilibrium polarization *P*<sup>0</sup> is then given by the value of *P* that minimizes the free energy. Here, however, following Strukov and Lenanyuk[10], we outline how this expansion comes from a statistical thermodynamic treatment involving an incomplete Gibbs potential, since this gives more insight into the theory. First a few more

When a ferroelectric crystal is above the critical temperature *TC* there is no spontaneous polarization; this is the paraelectric or high symmetry phase in which the crystal structure is of higher symmetry than the ferroelectric phase below *TC* since the appearance of the spontaneous polarization lowers the symmetry, as is evident from the two-dimensional representation in Fig. 1. Thus we see that during the transition from the paraelectric to ferroelectric state, a structural phase transition occurs. A consequence of this is that as the temperature approaches *TC* the structure becomes easy to distort giving rise to anomalous peaks in some of the properties such as the dielectric function *ǫ*, as illustrated in Fig. 2. In fact, in the ferroelectric phase, a crystal may have different domains such that the polarization in adjacent domains is not in the same direction[9], however it is possible to produce single-domain crystals—by cooling through *TC* in the presence of an electric field in one of the possible polarization directions for instance—and that is what we consider here.

**2. Foundations of the Landau-Devonshire theory of ferroelectrics**

In the simplest case, then, a ferroelectric phase transition can be described by a single parameter that in Fig. 1 would be describable by the displacement of the sublattice from its central position. Since the paraelectric phase is of higher symmetry than the ferroelectric phase we can consider it to be more orderly and so the parameter is sometimes referred to as an order parameter. Actually the atoms are ions with charge and the corner-atom lattice would have a charge opposite to that of the interior sub-lattice. From the macroscopic point of view the corner atoms can be considered to produce a focus of charge at the center of each rectangle of opposite sign to the atoms that are displaced during the phase transition (see Fig. 1 where the centers of positive and negative charge are marked for the ferroelectric phase). For this reason the order parameter is often taken to be the polarization, which is the dipole moment per unit volume. For the time being, however, we will stay with the displacement representation of the order parameter as this is more directly related to the structural transition. If the displacement is denoted by *η* then it is clear that in the ferroelectric phase *η* �= 0 and in the paraelectric phase *η* = 0.

The description above implies that each center ion is displaced in the same direction and this is the case for what are termed displacive ferroelectrics. There is another possibility, however, in which the displacement is not the same for every unit cell. For example, in some cells the displacement could be up while in others it could be down. Macroscopically ferroelectric and paraelectric phases can still result but now the ferroelectric phase occurs when statistically there are more displaced in one direction than in the other; and the paraelectric phase corresponds to the number of atoms displaced upwards is statistically equal to the number displaced downwards. In this case the ferroelectric is called an order-disorder ferroelectric. In terms of the macroscopic crystal symmetry however displacive and order-disorder ferroelectrics are equivalent and both can be described in general by Landau-Devonshire theory. The crystal structure of order-disorder ferroelectrics tends to be more complicated than for the displacive type[9, 10].

The basis of the Landau-Devonshire theory is that in the phase transition the ferroelectric phase may be represented by a distorted symmetrical phase. As outlined above this is a macroscopic view and the theory is phenomenological, as will be brought out further below. It turns out that the symmetry elements lost by the crystal at the transition temperature is sufficient information for a description of the anomalies of practically all of the thermodynamic properties of the crystal[10].

To make further progress consider the crystal as a system of *N* interacting particles with potential energy *U*(**r**1,...,**r***N*). According to Gibbs[11, Chapter 8], the equilibrium thermodynamic potential at pressure *p* and temperature *T* associated with the potential energy of interaction of the particles is given by

$$
\Phi(p, T) = -k\_B T \ln Z,\tag{1}
$$

*dw*(*η*) = *dη*

and

 <sup>∞</sup> −∞ exp

<sup>Φ</sup>(*p*, *<sup>T</sup>*, *<sup>η</sup>*) = <sup>−</sup>*kBT* ln <sup>∞</sup>

The thermodynamic potential is now also a function of *η*, and we write

−∞ exp

*dw*(*η*) = exp

<sup>Φ</sup>(*p*, *<sup>T</sup>*) = <sup>−</sup>*kBT* ln <sup>∞</sup>

Φ(*p*, *T*, *η*)—and expand Φ(*p*, *T*, *η*) in a series about the point *η* = *η*0:

<sup>Φ</sup>(*p*, *<sup>T</sup>*, *<sup>η</sup>*) = <sup>Φ</sup>(*p*, *<sup>T</sup>*, *<sup>η</sup>*0) + ∆Φ(*p*, *<sup>T</sup>*, *<sup>η</sup>* <sup>−</sup> *<sup>η</sup>*0) = <sup>Φ</sup>(*p*, *<sup>T</sup>*, *<sup>η</sup>*0) + <sup>1</sup>

<sup>Φ</sup>(*p*, *<sup>T</sup>*) = <sup>Φ</sup>(*p*, *<sup>T</sup>*, *<sup>η</sup>*0) <sup>−</sup> *kBT* ln <sup>∞</sup>

are very small and the second term can be neglected to give

Using Eq. (10), Φ(*p*, *T*) can be written as

[Φ(*p*, *<sup>T</sup>*) − *<sup>U</sup>*(*η*, *<sup>ξ</sup>*2,..., *<sup>ξ</sup>*3*N*)] /(*kBT*)

[Φ(*p*, *<sup>T</sup>*) − <sup>Φ</sup>(*p*, *<sup>T</sup>*, *<sup>η</sup>*)] /(*kBT*)

Here it can be seen that the value of *η* that minimizes Φ(*p*, *T*, *η*) will maximize the distribution function and so this value corresponds to the equilibrium value of *η*. If Φ(*p*, *T*, *η*)

is known, the usual equilibrium function Φ(*p*, *T*) can be found, using Eq. (9), from

−∞ exp

−∞ exp

In Eq. (12) the first term is the minimum of the potential as a function of *p*, *T* and *η*, and we are interested in the minimizing value *η* = *η*<sup>0</sup> for given values of *p* and *T*. The second term is the contribution to the thermodynamic potential of fluctuations in *η* with order of magnitude given by the thermal energy per degree of freedom *KBT*/2. However since we are dealing with a large number of particles in a macroscopic sample (∼ 1023) the fluctuations

Now, let the equilibrium value of *η* be *η*0—which corresponds to the minimum value of

[−*U*(*η*, *<sup>ξ</sup>*2,..., *<sup>ξ</sup>*3*N*)] /(*kBT*)

[−Φ(*η*)] /(*kBT*)

Nanoscale Ferroelectric Films, Strips and Boxes

[−∆Φ(*p*, *<sup>T</sup>*, *<sup>η</sup>* − *<sup>η</sup>*0)] /(*kBT*)

Φ(*p*, *T*) = Φ(*p*, *T*, *η*0). (13)

*<sup>d</sup>ξ*<sup>2</sup> ··· *<sup>ξ</sup>*3*N*. (7)

http://dx.doi.org/10.5772/52040

453

*<sup>d</sup>ξ*<sup>2</sup> ··· *<sup>ξ</sup>*3*N*, (8)

*dη*. (9)

*dη*. (10)

<sup>2</sup> *<sup>A</sup>*(*p*, *<sup>T</sup>*)(*<sup>η</sup>* <sup>−</sup> *<sup>η</sup>*0)<sup>2</sup> <sup>+</sup> ··· . (11)

*dη*. (12)

where *kB* is Boltzmann's constant and

$$Z = \int\_{-\infty}^{\infty} \exp\left[-\mathcal{U}(\mathbf{r}\_1, \dots, \mathbf{r}\_N) / (k\_B T)\right] \prod\_i d\mathbf{r}\_i. \tag{2}$$

The expression for *Z* in Eq. (2) comes from the probability that the value of the radius vector of the first particle lies between **r**<sup>1</sup> and **r**<sup>1</sup> + *d***r**<sup>1</sup> and similarly for the other vectors, being given by

$$dw = \mathbb{C} \exp\left[-\mathcal{U}(\mathbf{r}\_1, \dots, \mathbf{r}\_N) / (k\_B T)\right] \prod\_i d\mathbf{r}\_i. \tag{3}$$

Since integration of all of the variables must yield unity, *C* = 1/*Z*. The probability distribution can therefore be written as

$$dw = \exp\left\{ \left[ \Phi(p, T) - \mathcal{U}(\mathbf{r}\_1, \dots, \mathbf{r}\_N) \right] / (k\_B T) \right\} \prod\_i d\mathbf{r}\_i. \tag{4}$$

A Gibbs thermodynamic potential for nonequilibrium states can also be formulated by using a larger number of variables than only *p* and *T*. The new variables are introduced via a linear transformation:

$$
\mathbf{r}\_{1\prime} \dots \mathbf{r}\_{N} \to \mathfrak{J}\_{1\prime} \dots \mathfrak{J}\_{3N} . \tag{5}
$$

We choose one of these variables to be *η*, which describes the nonequilibrium state of interest as explained above, by setting *ξ*<sup>1</sup> = *η*. Now

$$d\boldsymbol{w} = \exp\left\{ \left[ \Phi(\boldsymbol{p}, \boldsymbol{T}) - \boldsymbol{U}(\boldsymbol{\eta}, \boldsymbol{\xi}\_{2}, \dots, \boldsymbol{\xi}\_{3N}) \right] / (k\_B \boldsymbol{T}) \right\} d\boldsymbol{\eta} d\boldsymbol{\xi}\_{2} \cdots \boldsymbol{\xi}\_{3N} \tag{6}$$

the probability that the variables will lie in the ranges *η* to *η* + *dη*, *ξ*<sup>2</sup> to *ξ*<sup>2</sup> + *dξ*2, and so on. Focusing on the variable of interest *η* the probability of finding the system in a state in which this variable is in the range *η* + *dη* is

$$dw(\eta) = d\eta \int\_{-\infty}^{\infty} \exp\left\{ \left[ \Phi(p, T) - \mathcal{U}(\eta, \mathfrak{f}\_{2\prime}, \dots, \mathfrak{f}\_{3N}) \right] / (k\_B T) \right\} d\mathfrak{f}\_2 \cdots \mathcal{G}\_{3N}.\tag{7}$$

The thermodynamic potential is now also a function of *η*, and we write

$$\Phi(p,T,\eta) = -k\_B T \ln \int\_{-\infty}^{\infty} \exp\left\{ \left[ -\mathcal{U}(\eta, \mathfrak{f}\_{2\prime}, \dots, \mathfrak{f}\_{3N}) \right] / (k\_B T) \right\} d\_{\mathfrak{F}2}^{\mathfrak{x}} \cdots \mathfrak{f}\_{3N\prime} \tag{8}$$

and

4 Advances in Ferroelectrics

by

transformation:

energy of interaction of the particles is given by

*Z* =

 <sup>∞</sup> −∞

where *kB* is Boltzmann's constant and

distribution can therefore be written as

as explained above, by setting *ξ*<sup>1</sup> = *η*. Now

*dw* = exp

this variable is in the range *η* + *dη* is

*dw* = exp

To make further progress consider the crystal as a system of *N* interacting particles with potential energy *U*(**r**1,...,**r***N*). According to Gibbs[11, Chapter 8], the equilibrium thermodynamic potential at pressure *p* and temperature *T* associated with the potential

exp [−*U*(**r**1,...,**r***N*)/(*kBT*)]∏

The expression for *Z* in Eq. (2) comes from the probability that the value of the radius vector of the first particle lies between **r**<sup>1</sup> and **r**<sup>1</sup> + *d***r**<sup>1</sup> and similarly for the other vectors, being given

*dw* <sup>=</sup> *<sup>C</sup>* exp [−*U*(**r**1,...,**r***N*)/(*kBT*)]∏

Since integration of all of the variables must yield unity, *C* = 1/*Z*. The probability

[Φ(*p*, *<sup>T</sup>*) − *<sup>U</sup>*(**r**1,...,**r***N*)] /(*kBT*)

A Gibbs thermodynamic potential for nonequilibrium states can also be formulated by using a larger number of variables than only *p* and *T*. The new variables are introduced via a linear

We choose one of these variables to be *η*, which describes the nonequilibrium state of interest

[Φ(*p*, *<sup>T</sup>*) − *<sup>U</sup>*(*η*, *<sup>ξ</sup>*<sup>2</sup> ..., *<sup>ξ</sup>*3*N*)] /(*kBT*)

the probability that the variables will lie in the ranges *η* to *η* + *dη*, *ξ*<sup>2</sup> to *ξ*<sup>2</sup> + *dξ*2, and so on. Focusing on the variable of interest *η* the probability of finding the system in a state in which

<sup>Φ</sup>(*p*, *<sup>T</sup>*) = −*kBT* ln *<sup>Z</sup>*, (1)

*i*

*i*

 ∏ *i*

**<sup>r</sup>**1,...,**r***<sup>N</sup>* → *<sup>ξ</sup>*1,..., *<sup>ξ</sup>*3*N*. (5)

*d***r***i*. (2)

*d***r***i*. (3)

*d***r***i*. (4)

*<sup>d</sup>ηdξ*<sup>2</sup> ··· *<sup>ξ</sup>*3*N*, (6)

$$d\mathfrak{w}(\eta) = \exp\left\{ \left[ \Phi(p, T) - \Phi(p, T, \eta) \right] / (k\_B T) \right\} d\eta. \tag{9}$$

Here it can be seen that the value of *η* that minimizes Φ(*p*, *T*, *η*) will maximize the distribution function and so this value corresponds to the equilibrium value of *η*. If Φ(*p*, *T*, *η*) is known, the usual equilibrium function Φ(*p*, *T*) can be found, using Eq. (9), from

$$\Phi(p,T) = -k\_B T \ln \int\_{-\infty}^{\infty} \exp\left\{ \left[ -\Phi(\eta) \right] / (k\_B T) \right\} d\eta. \tag{10}$$

Now, let the equilibrium value of *η* be *η*0—which corresponds to the minimum value of Φ(*p*, *T*, *η*)—and expand Φ(*p*, *T*, *η*) in a series about the point *η* = *η*0:

$$\Phi(p, T, \eta) = \Phi(p, T, \eta\_0) + \Delta\Phi(p, T, \eta - \eta\_0) = \Phi(p, T, \eta\_0) + \frac{1}{2}A(p, T)(\eta - \eta\_0)^2 + \cdots \cdot \tag{11}$$

Using Eq. (10), Φ(*p*, *T*) can be written as

$$\Phi(p,T) = \Phi(p,T,\eta\_0) - k\_B T \ln \int\_{-\infty}^{\infty} \exp\left\{ \left[ -\Delta\Phi(p,T,\eta - \eta\_0) \right] / (k\_B T) \right\} d\eta. \tag{12}$$

In Eq. (12) the first term is the minimum of the potential as a function of *p*, *T* and *η*, and we are interested in the minimizing value *η* = *η*<sup>0</sup> for given values of *p* and *T*. The second term is the contribution to the thermodynamic potential of fluctuations in *η* with order of magnitude given by the thermal energy per degree of freedom *KBT*/2. However since we are dealing with a large number of particles in a macroscopic sample (∼ 1023) the fluctuations are very small and the second term can be neglected to give

$$
\Phi(p, T) = \Phi(p, T, \eta\_0). \tag{13}
$$

Note that if the total number of degrees of freedom (left unintegrated as the single degree *η* was) required to describe the system is comparable to *N*, then the neglect of the second term is no longer valid and Eq. (10) must be used in place of Eq. (13). We will not be concerned with such situations here, but very close to the transition temperature the Landau-Devonshire theory breaks down and the fluctuations must be considered, which can be done with the aid of Eq. (10), as discussed in Ref. [10].

In general thermodynamic functions which contain extra variables that remain unintegrated as *η* was above, are nonequilibrium thermodynamic functions known as incomplete functions. The general form is

$$\Phi(p, T, \eta\_1, \eta\_2, \dots, \eta\_n) = \Phi\_0(p, T) + \Phi\_1(p, T, \eta\_1, \eta\_2, \dots, \eta\_n). \tag{14}$$

The equilibrium values of the *η<sup>i</sup>* are found by minimizing Φ(*p*, *T*, *η*1, *η*2,...,, *ηn*) according to

$$\frac{\partial \Phi}{\partial \eta\_1} = 0, \dots, \frac{\partial \Phi}{\partial \eta\_n} = 0,\tag{15}$$

*η*

parameters, and we obtain

*TC*

(a)

it is a first-order transition close to a second order transition. at *T* = *TC*.

<sup>Φ</sup>(*p*, *<sup>T</sup>*, *<sup>η</sup>*) = <sup>Φ</sup>(*p*, *<sup>T</sup>*, 0) + <sup>Φ</sup>′

polarization (this point is discussed in more detail in Refs.[9, 10, 12]).

*T*

**Figure 3.** (a) Second order phase transition in which the order parameter decreases continuously to zero. (b) First order phase transition in which the order parameter displays a discontinuity. There is a continuous decrease in *η* before, which means that

minima<sup>1</sup> at *η* �= 0 in the nonsymmetric phase (*T* < *TC*). The function Φ(*p*, *T*, *η*) must be consistent with this as *T* changes through the transition temperature so that it is determined for both the symmetric and nonsymmetric phases. These requirements place restrictions on the dependence of the potential Φ on *η*: Φ is a scalar function characterizing the physical properties of ferroelectric crystals, and as such must be invariant under any symmetry operations on the symmetric phase consistent with the crystal symmetry of this phase.

Before exploring further the form of Φ that describes the phase transition the difference between first and second order phase changes will be discussed. In a second order transition *η* decreases continuously to zero as the temperature is lowered through *TC*; in a first order transition, on the other hand, there is a discontinuous jump of *η* to zero at *T* = *TC*. Both situations are illustrated in Fig. 3. The first order transition shown in this figure is in fact close to a second order transition and so there is a continuous decrease in *η* before the jump and the jump is not very large. Crystals can exhibit first or second order transitions depending on the crystal structure[9, 10]. Either case can be treated by Landau-Devonshire theory, as long as the first order case is close to a second order transition. Turning back now to the development of an expression for Φ, we see that in the vicinity of a second order transition it is permissible to deal only with small lattice distortions, that is, small *η*. Hence the thermodynamic potential can be expanded into a series in *η*, with *p* and *T* treated as

(*p*, *T*, 0)*η* +

To represent a given crystal this expression must be invariant under symmetry operations that correspond to the symmetry elements of the crystal. A ferroelectric phase transition resulting in a spontaneous polarization occurs as long as the order parameter transforms as a vector component. Only crystal symmetries for which this is true are ferroelectric and then the order parameter can be considered to be equivalent to a component of the spontaneous

<sup>1</sup> Part of the definition of a ferroelectric is that it has a reversible polarization in the nonsymmetric phase so that at

least two minimum are required; otherwise the reverse direction would not be an equilibrium state.

1 2 Φ′

*η*

*TC*

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

455

(*p*, *T*, 0)*η*<sup>2</sup> + ··· . (17)

(b)

*T*

which determine a set of minimizing values *ηoi*(*p*, *T*). Substituting these values into Eq. (14) determines the equilibrium thermodynamic function

$$\Phi(p,T) = \Phi\_0(p,T) + \Phi\_1(p,T,\eta\_{01},\eta\_{02},\dots,\eta\_{0n}).\tag{16}$$

This does not take fluctuations of the *η*0*<sup>i</sup>* into account but the corresponding error will be small provided that *n* ≪ 3*N*.

Although we are only considering a ferroelectric phase transition of the simplest type describable by introducing a single extra variable *η* into the thermodynamic potential the general form shows how to treat more complicated cases for which the polarization may occur in more ways than along one line, as we will see later. Polarization in three-dimensionss can be treated with three parameters, *η*<sup>1</sup> to *η*3.

Returning to a one-component case we now show how the form of the thermodynamic potential can be worked out. As explained, in a ferroelectric phase transition we are dealing with displacements of certain atoms or groups of atoms; the structure of the nonsymmetric phase can be obtained from the structure of the symmetric phase by small displacements. Although the same lowering of symmetry in a phase transition can occur with different types of ordering implying that the choice of the order parameter is ambiguous, it turns out that the character of the anomalies of the physical properties in the transitions can be elucidated for any particular relationship between the order parameter and the displacements as long at the appearance of the order parameter leads to a symmetry change corresponding to that of the crystal's[10, 12].

In view of this we can take the order parameter as the extra variable of the incomplete thermodynamic potential. The phase transition can then be described at a given pressure if Φ(*p*, *T*, *η*) has a minimum at *η* = 0 in the symmetric phase (*T* > *TC*), and at least two

6 Advances in Ferroelectrics

aid of Eq. (10), as discussed in Ref. [10].

functions. The general form is

small provided that *n* ≪ 3*N*.

of the crystal's[10, 12].

to

Note that if the total number of degrees of freedom (left unintegrated as the single degree *η* was) required to describe the system is comparable to *N*, then the neglect of the second term is no longer valid and Eq. (10) must be used in place of Eq. (13). We will not be concerned with such situations here, but very close to the transition temperature the Landau-Devonshire theory breaks down and the fluctuations must be considered, which can be done with the

In general thermodynamic functions which contain extra variables that remain unintegrated as *η* was above, are nonequilibrium thermodynamic functions known as incomplete

The equilibrium values of the *η<sup>i</sup>* are found by minimizing Φ(*p*, *T*, *η*1, *η*2,...,, *ηn*) according

<sup>=</sup> 0, . . . , *<sup>∂</sup>*<sup>Φ</sup>

which determine a set of minimizing values *ηoi*(*p*, *T*). Substituting these values into Eq. (14)

This does not take fluctuations of the *η*0*<sup>i</sup>* into account but the corresponding error will be

Although we are only considering a ferroelectric phase transition of the simplest type describable by introducing a single extra variable *η* into the thermodynamic potential the general form shows how to treat more complicated cases for which the polarization may occur in more ways than along one line, as we will see later. Polarization in three-dimensionss

Returning to a one-component case we now show how the form of the thermodynamic potential can be worked out. As explained, in a ferroelectric phase transition we are dealing with displacements of certain atoms or groups of atoms; the structure of the nonsymmetric phase can be obtained from the structure of the symmetric phase by small displacements. Although the same lowering of symmetry in a phase transition can occur with different types of ordering implying that the choice of the order parameter is ambiguous, it turns out that the character of the anomalies of the physical properties in the transitions can be elucidated for any particular relationship between the order parameter and the displacements as long at the appearance of the order parameter leads to a symmetry change corresponding to that

In view of this we can take the order parameter as the extra variable of the incomplete thermodynamic potential. The phase transition can then be described at a given pressure if Φ(*p*, *T*, *η*) has a minimum at *η* = 0 in the symmetric phase (*T* > *TC*), and at least two

*∂η<sup>n</sup>*

*∂*Φ *∂η*<sup>1</sup>

determines the equilibrium thermodynamic function

can be treated with three parameters, *η*<sup>1</sup> to *η*3.

Φ(*p*, *T*, *η*1, *η*2,...,, *ηn*) = Φ0(*p*, *T*) + Φ1(*p*, *T*, *η*1, *η*2,...,, *ηn*). (14)

Φ(*p*, *T*) = Φ0(*p*, *T*) + Φ1(*p*, *T*, *η*01, *η*02,...,, *η*0*n*). (16)

= 0, (15)

**Figure 3.** (a) Second order phase transition in which the order parameter decreases continuously to zero. (b) First order phase transition in which the order parameter displays a discontinuity. There is a continuous decrease in *η* before, which means that it is a first-order transition close to a second order transition. at *T* = *TC*.

minima<sup>1</sup> at *η* �= 0 in the nonsymmetric phase (*T* < *TC*). The function Φ(*p*, *T*, *η*) must be consistent with this as *T* changes through the transition temperature so that it is determined for both the symmetric and nonsymmetric phases. These requirements place restrictions on the dependence of the potential Φ on *η*: Φ is a scalar function characterizing the physical properties of ferroelectric crystals, and as such must be invariant under any symmetry operations on the symmetric phase consistent with the crystal symmetry of this phase.

Before exploring further the form of Φ that describes the phase transition the difference between first and second order phase changes will be discussed. In a second order transition *η* decreases continuously to zero as the temperature is lowered through *TC*; in a first order transition, on the other hand, there is a discontinuous jump of *η* to zero at *T* = *TC*. Both situations are illustrated in Fig. 3. The first order transition shown in this figure is in fact close to a second order transition and so there is a continuous decrease in *η* before the jump and the jump is not very large. Crystals can exhibit first or second order transitions depending on the crystal structure[9, 10]. Either case can be treated by Landau-Devonshire theory, as long as the first order case is close to a second order transition. Turning back now to the development of an expression for Φ, we see that in the vicinity of a second order transition it is permissible to deal only with small lattice distortions, that is, small *η*. Hence the thermodynamic potential can be expanded into a series in *η*, with *p* and *T* treated as parameters, and we obtain

$$\Phi(p,T,\eta) = \Phi(p,T,0) + \Phi'(p,T,0)\eta + \frac{1}{2}\Phi'(p,T,0)\eta^2 + \cdots \tag{17}$$

To represent a given crystal this expression must be invariant under symmetry operations that correspond to the symmetry elements of the crystal. A ferroelectric phase transition resulting in a spontaneous polarization occurs as long as the order parameter transforms as a vector component. Only crystal symmetries for which this is true are ferroelectric and then the order parameter can be considered to be equivalent to a component of the spontaneous polarization (this point is discussed in more detail in Refs.[9, 10, 12]).

<sup>1</sup> Part of the definition of a ferroelectric is that it has a reversible polarization in the nonsymmetric phase so that at least two minimum are required; otherwise the reverse direction would not be an equilibrium state.


Generally the exact form of the temperature dependence may be difficult to find. However near the phase transition temperature a series expansion in powers of *T* − *TC* (at a given

Taking into account the above mentioned properties: *A*(*TC*) = 0 and *B*(*TC*) > 0, the simplest

4

The numerical factors are introduced for convenience, as will be clear shortly. We thus obtain,

2

Usually it is sufficient to consider the energy due to the presence of the spontaneous polarization per unit volume (we are dealing with a bulk sample assumed uniform of the

2

2

*a*(*T* − *TC*)*P*<sup>2</sup> +

*α*(*T* − *TC*)*P*<sup>2</sup> +

*<sup>∂</sup>P*<sup>2</sup> <sup>&</sup>gt; 0.

0 for *T* > *TC*,

1 4

> 1 4

*<sup>β</sup>* for *<sup>T</sup>* <sup>&</sup>lt; *TC*. (25)

*<sup>A</sup>*(*T*) = *<sup>A</sup>*1(*<sup>T</sup>* <sup>−</sup> *TC*) = <sup>1</sup>

*<sup>B</sup>*(*T*) = *<sup>B</sup>*(*TC*) = <sup>1</sup>

<sup>Φ</sup>(*T*, *<sup>P</sup>*) = <sup>Φ</sup>0(*T*) + <sup>1</sup>

*<sup>F</sup>* <sup>=</sup> <sup>Φ</sup>(*T*, *<sup>P</sup>*) <sup>−</sup> <sup>Φ</sup>0(*T*)

*∂F*

*P*2 <sup>0</sup> = 

crystal volume) so that it is convenient to write, for a volume *v* of ferroelectric,

*<sup>v</sup>* <sup>=</sup> <sup>1</sup>

where *α* = *a*/*v* > 0, *β* = *b*/*v* > 0, and we refer to *F* as the Gibbs free energy density. The equilibrium polarization *P*<sup>0</sup> is now found by minimizing *F* from the conditions

*<sup>∂</sup><sup>P</sup>* <sup>=</sup> 0, and *<sup>∂</sup>*2*<sup>F</sup>*

−*α*(*T*−*TC*)

Here we see that it is necessary to differentiate to find *P*<sup>0</sup> which explains why the numerical coefficients 1/2 and 1/4 were introduced into the expansion of the thermodynamic potential. The form of *F* in Eq. (24) is sufficient to describe a second order phase transition since *P*<sup>0</sup> → 0 continuously as *T* → *TC*. For second order transitions in which there is a discontinuous jump, Eq. (24) is insufficient. However a first-order transitions near to a second-order transition the

*A*(*T*) = *A*(*TC*) + *A*1(*T* − *TC*)*A*2(*T* − *TC*)<sup>2</sup> + ··· (19) *B*(*T*) = *B*(*TC*) + *B*1(*T* − *TC*)*B*2(*T* − *TC*)<sup>2</sup> + ··· . (20)

*a*(*T* − *TC*) (21)

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

457

*bP*4. (23)

*βP*4, (24)

*b*. (22)

pressure) can be used to give

for a given pressure,

From this we find

forms retaining the essential non-zero first terms gives

**Table 1.** Transformation of the order parameter

**Figure 4.** The form of Φ − Φ<sup>0</sup> that satisfies the basic requirements for a ferroelectric phase transition. When *T* > *TC* there is single minimum at *P* = 0 and there is no spontaneous polarization; for *T* < *TC* a spontaneous polarization can exist at one of the minima at *P* = ±*P*0.

To make further progress we consider a definite example of a symmetry which allows the order parameter to transform as a vector and so corresponds to a polarization having a single component; to reflect this correspondence we also make the notation change *η* → *P* where *P* is the polarization component.2 Consider a second-order phase transition such that the change into the ferroelectric state at the critical temperature is accompanied by a symmetry change from the group 2/*m* (symmetry elements: 1, 1, ¯ *m*, 2) to 2 (symmetry elements: 1, 2). The ferroelectric triglycine sulphate shows this symmetry change during its phase transition[13, 14]. During the transition to the nonsymmetric phase the crystal loses the inversion center and the plane of symmetry, 1 and ¯ *m*. These disappearing symmetry elements correspond to the symmetry operations that reverse the sign of the order parameter *η* = *P*, as is shown in Table 1. Since Φ must be invariant under any transformations of the symmetrical phase it is clear from this that it cannot include terms linear in *P* or odd powers of *P*, since such terms would be changed by the operations *m* and 1. ¯

Therefore the first *η* = *P* dependent term in Eq. (17) will be of the form *A*(*p*, *T*)*P*2. For simplicity we now assume that the pressure is at some fixed value and discuss the temperature dependence. The expression for Φ must reflect the fact that at *T* > *TC* there is a minimum at *P* = 0 so that the equilibrium state (represented by the minimum) is one in which *P* = 0, and when *T* < *TC* there will be two minima for which *P* �= 0 so that the possible equilibrium states occur at nonzero values of *P*. This is satisfied if *A*(*T*) passes continuously from *A*(*T* < *TC*) < 0 to *A*(*T* > *TC*) > 0 when *T* > *TC*, with *A*(*T* = *TC*) = 0, provided that the next allowable term *B*(*T*)*P*<sup>4</sup> (remembering that odd terms have the wrong symmetry) is included with *B*(*T*) > 0, as can be seen from Fig. 4. This results in a thermodynamic potential of the form

$$
\Phi = \Phi\_0 + A(T)P^2 + B(T)P^4. \tag{18}
$$

<sup>2</sup> The general case in which the polarization has three components would be covered by three order parameters and the correspondence would be (*η*<sup>1</sup> → *P*1, *η*<sup>2</sup> → *P*2, *η*<sup>3</sup> → *P*3), with **P** = (*P*1, *P*2, *P*3).

Generally the exact form of the temperature dependence may be difficult to find. However near the phase transition temperature a series expansion in powers of *T* − *TC* (at a given pressure) can be used to give

$$A(T) = A(T\_{\mathbb{C}}) + A\_1(T - T\_{\mathbb{C}})A\_2(T - T\_{\mathbb{C}})^2 + \cdots \tag{19}$$

$$B(T) = B(T\_{\mathbb{C}}) + B\_1(T - T\_{\mathbb{C}})B\_2(T - T\_{\mathbb{C}})^2 + \cdots \ . \tag{20}$$

Taking into account the above mentioned properties: *A*(*TC*) = 0 and *B*(*TC*) > 0, the simplest forms retaining the essential non-zero first terms gives

$$A(T) = A\_1(T - T\_\odot) = \frac{1}{2}a(T - T\_\odot) \tag{21}$$

$$B(T) = B(T\_{\mathbb{C}}) = \frac{1}{4}b.\tag{22}$$

The numerical factors are introduced for convenience, as will be clear shortly. We thus obtain, for a given pressure,

$$
\Phi(T\_\prime P) = \Phi\_0(T) + \frac{1}{2}a(T - T\_\odot)P^2 + \frac{1}{4}bP^4. \tag{23}
$$

Usually it is sufficient to consider the energy due to the presence of the spontaneous polarization per unit volume (we are dealing with a bulk sample assumed uniform of the crystal volume) so that it is convenient to write, for a volume *v* of ferroelectric,

$$F = \frac{\Phi(T, P) - \Phi\_0(T)}{v} = \frac{1}{2}\mathfrak{a}(T - T\_{\mathbb{C}})P^2 + \frac{1}{4}\beta P^4. \tag{24}$$

where *α* = *a*/*v* > 0, *β* = *b*/*v* > 0, and we refer to *F* as the Gibbs free energy density.

The equilibrium polarization *P*<sup>0</sup> is now found by minimizing *F* from the conditions

$$
\frac{\partial F}{\partial P} = 0, \quad \text{and} \quad \frac{\partial^2 F}{\partial P^2} > 0.
$$

From this we find

8 Advances in Ferroelectrics

the minima at *P* = ±*P*0.

potential of the form

**Table 1.** Transformation of the order parameter

Φ − Φ<sup>0</sup>

*T* > *TC*

symmetry change from the group 2/*m* (symmetry elements: 1, 1,

of *P*, since such terms would be changed by the operations *m* and 1.

the correspondence would be (*η*<sup>1</sup> → *P*1, *η*<sup>2</sup> → *P*2, *η*<sup>3</sup> → *P*3), with **P** = (*P*1, *P*2, *P*3).

2/*m* 1 2 *m* 1¯ *η* = *PPP* −*P* −*P*

Φ − Φ<sup>0</sup>

¯

Φ = Φ<sup>0</sup> + *A*(*T*)*P*<sup>2</sup> + *B*(*T*)*P*4. (18)

*<sup>P</sup>* <sup>−</sup>*P*<sup>0</sup> *<sup>P</sup>*<sup>0</sup>

*T* < *TC*

¯ *m*, 2) to 2 (symmetry

*P*

**Figure 4.** The form of Φ − Φ<sup>0</sup> that satisfies the basic requirements for a ferroelectric phase transition. When *T* > *TC* there is single minimum at *P* = 0 and there is no spontaneous polarization; for *T* < *TC* a spontaneous polarization can exist at one of

To make further progress we consider a definite example of a symmetry which allows the order parameter to transform as a vector and so corresponds to a polarization having a single component; to reflect this correspondence we also make the notation change *η* → *P* where *P* is the polarization component.2 Consider a second-order phase transition such that the change into the ferroelectric state at the critical temperature is accompanied by a

elements: 1, 2). The ferroelectric triglycine sulphate shows this symmetry change during its phase transition[13, 14]. During the transition to the nonsymmetric phase the crystal loses the inversion center and the plane of symmetry, 1 and ¯ *m*. These disappearing symmetry elements correspond to the symmetry operations that reverse the sign of the order parameter *η* = *P*, as is shown in Table 1. Since Φ must be invariant under any transformations of the symmetrical phase it is clear from this that it cannot include terms linear in *P* or odd powers

Therefore the first *η* = *P* dependent term in Eq. (17) will be of the form *A*(*p*, *T*)*P*2. For simplicity we now assume that the pressure is at some fixed value and discuss the temperature dependence. The expression for Φ must reflect the fact that at *T* > *TC* there is a minimum at *P* = 0 so that the equilibrium state (represented by the minimum) is one in which *P* = 0, and when *T* < *TC* there will be two minima for which *P* �= 0 so that the possible equilibrium states occur at nonzero values of *P*. This is satisfied if *A*(*T*) passes continuously from *A*(*T* < *TC*) < 0 to *A*(*T* > *TC*) > 0 when *T* > *TC*, with *A*(*T* = *TC*) = 0, provided that the next allowable term *B*(*T*)*P*<sup>4</sup> (remembering that odd terms have the wrong symmetry) is included with *B*(*T*) > 0, as can be seen from Fig. 4. This results in a thermodynamic

<sup>2</sup> The general case in which the polarization has three components would be covered by three order parameters and

$$P\_0^2 = \begin{cases} 0 & \text{for } T > T\_{\mathbb{C}\prime} \\ -\frac{a(T - T\_{\mathbb{C}})}{\beta} & \text{for } T < T\_{\mathbb{C}} \end{cases} \tag{25}$$

Here we see that it is necessary to differentiate to find *P*<sup>0</sup> which explains why the numerical coefficients 1/2 and 1/4 were introduced into the expansion of the thermodynamic potential.

The form of *F* in Eq. (24) is sufficient to describe a second order phase transition since *P*<sup>0</sup> → 0 continuously as *T* → *TC*. For second order transitions in which there is a discontinuous jump, Eq. (24) is insufficient. However a first-order transitions near to a second-order transition the

**Figure 5.** A component of polarization that is directed along a normal to the plane of the film when *T* < *TC*. The resulting surface charges with opposite sides having signs of opposite charge, creates an electric field which in the opposite direction to the polarization.

phase change can be described by adding a further term in *P*<sup>6</sup> (a term in *P*5, since it is an odd power, does not fulfill the symmetry requirements) which results in an energy density given by

$$F = \frac{1}{2}a(T - T\_{\mathbb{C}})P^2 + \frac{1}{4}\beta P^4 + \frac{1}{6}\gamma P^6. \tag{26}$$

*G* =

in which

*<sup>F</sup>* <sup>=</sup> *<sup>G</sup> <sup>S</sup>* <sup>=</sup>  *<sup>L</sup>* 0

With boundary conditions

and

*dz* 1 <sup>2</sup> *AP*<sup>2</sup> <sup>+</sup>

*dxdy <sup>L</sup>*

0

*<sup>f</sup>*(*P*, *dP*/*dz*) = <sup>1</sup>

1 4 *BP*<sup>4</sup> + 1 6 *CP*<sup>6</sup> + 1 2 *D dP dz* 2 + 1 2 *D P*2(0) *<sup>d</sup>* <sup>+</sup>

brackets in Eq. (29) would be replaced by *P*(0)/*d*<sup>1</sup> + *P*(*L*)/*d*2.

boundary conditions for the equilibrium polarization in the film:

*<sup>D</sup> <sup>d</sup>*2*P*<sup>0</sup>

*dP*<sup>0</sup> *dz* <sup>−</sup> <sup>1</sup> *d*

*dP*<sup>0</sup> *dz* <sup>+</sup> 1 *d*

*dz*<sup>2</sup> <sup>−</sup> *AP*<sup>0</sup> <sup>−</sup> *BP*<sup>3</sup>

<sup>0</sup> <sup>−</sup> *CP*<sup>5</sup>

*f*(*P*, *dP*/*dz*) *dz* +

<sup>2</sup> *AP*<sup>2</sup> <sup>+</sup>

this areas is *S*, then *dxdy* = *S*, and we can write a free energy per unit area as

1 4 *BP*<sup>4</sup> + 1 6 *CP*<sup>6</sup> + 1 2 *D dP dz* 2

the temperature dependence is in *<sup>A</sup>* through *<sup>A</sup>* = *<sup>a</sup>*(*<sup>T</sup>* − *TC*), and the integrals over *<sup>x</sup>* and *<sup>y</sup>* are factored out of the three dimensional integral over a volume of the film because *P* does not vary in these directions. Thus any surface area can be chosen to integrate over so that, if

Here since the film is free standing with identical surface properties for both surfaces the surface terms at *z* = 0 and *z* = *L* involve the same factor, 1/*d*, which simplifies the problem. If the film was not free standing, with for example, one surface an interface with a substrate, then this could be modeled by introducing different factors[6] so that the last term in square

The equilibrium polarization is still the *P* that minimizes *F*, but now the problem is to find the function *P*(*z*) = *P*0(*z*) that minimizes *F* rather than a finite set of values. So we are dealing with the minimization of a functional and the classical methods from the calculus of variations[16, Chapter 4] may be used, which cast the problem into the form of a differential equation known as the Euler-Lagrange equation and the function that minimizes *F* is the solution to this equation for a set of boundary conditions that also need to be specified. In this case, as will be shown below the boundary conditions used are implicit in Eq. (29) and are equivalent to allowing the polarization to be free at the film surfaces rather than fixed. More details on the method for the general three dimensional case are given in Section 4. Here we simply state that the classical methods lead to the following Euler-Lagrange equation and

1 2 *D d*

*dx dy*

*P*2(0) + *P*2(*L*)

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

, (28)

*P*2(*L*) *d*

<sup>0</sup> = 0. (30)

*P*<sup>0</sup> = 0, at *z* = 0, (31)

*P*<sup>0</sup> = 0, at *z* = *L*. (32)

. (29)

, (27)

459

But, whereas *β* > 0 for a second order transition, for *F* to exhibit a discontinuous jump it is necessary to choose *β* < 0 with *γ* > 0. More on this can be found in Lines and Glass[9].

We have thus shown in some detail how it is that a relatively simple free energy expression can account for the main characteristics of a ferroelectric, with the observed equilibrium polarization found by minimizing the free energy. Other terms can be added to the free energy to account for various external influences. For example an external electric field with component *E* along the direction of polarization can be accounted by first substituting the equilibrium polarization *P*<sup>0</sup> into *F* and adding a term *EP*0. The corresponding susceptibility is then given by *<sup>χ</sup>* = *<sup>∂</sup>*2*<sup>F</sup> <sup>∂</sup>E*<sup>2</sup> .

Note that Landau-Devonshire theory is a phenomenological theory of the macroscopic properties of the ferroelectric: the coefficients, *α*, *β*, etc., are not derivable from it; instead they must be found from experiment, or, in some cases, can be found from first-principle calculations[1].

### **2.2. Extension to thin films**

Here a free-standing thin film is considered. If the equilibrium polarization below *TC* has a component that is aligned with a normal to the plane of the film, then depolarization effects due to the space charge that appears at the surfaces, as illustrated in Fig. 5, must be taken into account. At first we avoid this complication by assuming a polarization that is in-plane. Also to remain with a one dimensional treatment for now we assume that this this is along the *x* direction of a Cartesian coordinate system and that the plane surfaces of the film are normal to the *z* axis at *z* = 0 and *z* = *L*, where *L* is the thickness of the film. The effect of the surfaces is such that the polarization near them may differ from the bulk value. For the case just described this implies that *P* = *P*(*z*). It can be shown[1, 3, 15] that this can be accounted for in the free energy by adding surface terms involving *P*<sup>2</sup> integrated over the surfaces, and a gradient term |*dp*/*dz*|. The free energy is now given by

$$\mathcal{G} = \left( \iint d\mathbf{x} dy \right) \int\_0^L f(\mathbf{P}, d\mathbf{P}/d\mathbf{z}) \, d\mathbf{z} + \frac{1}{2} \frac{D}{d} \left( \iint d\mathbf{x} \, dy \right) \left[ P^2(\mathbf{0}) + P^2(\mathbf{L}) \right],\tag{27}$$

in which

10 Advances in Ferroelectrics

the polarization.

given by

is then given by *<sup>χ</sup>* = *<sup>∂</sup>*2*<sup>F</sup>*

**2.2. Extension to thin films**

calculations[1].

*<sup>∂</sup>E*<sup>2</sup> .

a gradient term |*dp*/*dz*|. The free energy is now given by

**Figure 5.** A component of polarization that is directed along a normal to the plane of the film when *T* < *TC*. The resulting surface charges with opposite sides having signs of opposite charge, creates an electric field which in the opposite direction to

phase change can be described by adding a further term in *P*<sup>6</sup> (a term in *P*5, since it is an odd power, does not fulfill the symmetry requirements) which results in an energy density

But, whereas *β* > 0 for a second order transition, for *F* to exhibit a discontinuous jump it is necessary to choose *β* < 0 with *γ* > 0. More on this can be found in Lines and Glass[9].

We have thus shown in some detail how it is that a relatively simple free energy expression can account for the main characteristics of a ferroelectric, with the observed equilibrium polarization found by minimizing the free energy. Other terms can be added to the free energy to account for various external influences. For example an external electric field with component *E* along the direction of polarization can be accounted by first substituting the equilibrium polarization *P*<sup>0</sup> into *F* and adding a term *EP*0. The corresponding susceptibility

Note that Landau-Devonshire theory is a phenomenological theory of the macroscopic properties of the ferroelectric: the coefficients, *α*, *β*, etc., are not derivable from it; instead they must be found from experiment, or, in some cases, can be found from first-principle

Here a free-standing thin film is considered. If the equilibrium polarization below *TC* has a component that is aligned with a normal to the plane of the film, then depolarization effects due to the space charge that appears at the surfaces, as illustrated in Fig. 5, must be taken into account. At first we avoid this complication by assuming a polarization that is in-plane. Also to remain with a one dimensional treatment for now we assume that this this is along the *x* direction of a Cartesian coordinate system and that the plane surfaces of the film are normal to the *z* axis at *z* = 0 and *z* = *L*, where *L* is the thickness of the film. The effect of the surfaces is such that the polarization near them may differ from the bulk value. For the case just described this implies that *P* = *P*(*z*). It can be shown[1, 3, 15] that this can be accounted for in the free energy by adding surface terms involving *P*<sup>2</sup> integrated over the surfaces, and

1 4 *βP*<sup>4</sup> + 1 6

*γP*6. (26)

*<sup>α</sup>*(*<sup>T</sup>* − *TC*)*P*<sup>2</sup> +

*<sup>F</sup>* <sup>=</sup> <sup>1</sup> 2

$$f(P, dP/dz) = \frac{1}{2}AP^2 + \frac{1}{4}BP^4 + \frac{1}{6}CP^6 + \frac{1}{2}D\left(\frac{dP}{dz}\right)^2,\tag{28}$$

the temperature dependence is in *<sup>A</sup>* through *<sup>A</sup>* = *<sup>a</sup>*(*<sup>T</sup>* − *TC*), and the integrals over *<sup>x</sup>* and *<sup>y</sup>* are factored out of the three dimensional integral over a volume of the film because *P* does not vary in these directions. Thus any surface area can be chosen to integrate over so that, if this areas is *S*, then *dxdy* = *S*, and we can write a free energy per unit area as

$$F = \frac{G}{S} = \int\_0^L dz \left[ \frac{1}{2} AP^2 + \frac{1}{4} BP^4 + \frac{1}{6} CP^6 + \frac{1}{2} D \left( \frac{dP}{dz} \right)^2 \right] + \frac{1}{2} D \left[ \frac{P^2(0)}{d} + \frac{P^2(L)}{d} \right]. \tag{29}$$

Here since the film is free standing with identical surface properties for both surfaces the surface terms at *z* = 0 and *z* = *L* involve the same factor, 1/*d*, which simplifies the problem. If the film was not free standing, with for example, one surface an interface with a substrate, then this could be modeled by introducing different factors[6] so that the last term in square brackets in Eq. (29) would be replaced by *P*(0)/*d*<sup>1</sup> + *P*(*L*)/*d*2.

The equilibrium polarization is still the *P* that minimizes *F*, but now the problem is to find the function *P*(*z*) = *P*0(*z*) that minimizes *F* rather than a finite set of values. So we are dealing with the minimization of a functional and the classical methods from the calculus of variations[16, Chapter 4] may be used, which cast the problem into the form of a differential equation known as the Euler-Lagrange equation and the function that minimizes *F* is the solution to this equation for a set of boundary conditions that also need to be specified. In this case, as will be shown below the boundary conditions used are implicit in Eq. (29) and are equivalent to allowing the polarization to be free at the film surfaces rather than fixed. More details on the method for the general three dimensional case are given in Section 4. Here we simply state that the classical methods lead to the following Euler-Lagrange equation and boundary conditions for the equilibrium polarization in the film:

$$D\frac{d^2P\_0}{dz^2} - AP\_0 - BP\_0^3 - CP\_0^5 = 0.\tag{30}$$

With boundary conditions

$$\frac{dP\_0}{dz} - \frac{1}{d}P\_0 = 0, \quad \text{at } z = 0,\tag{31}$$

and

$$\frac{dP\_0}{dz} + \frac{1}{d}P\_0 = 0, \quad \text{at } z = L. \tag{32}$$

**Figure 6.** The extrapolation length *d* for one of the film surfaces at *z* = 0. For *d* < 0 the polarization turns down as the surface is approached; for *d* > 0 it turns up; this is also true for the surface at *z* = *L* (not shown). Here the extrapolated gradient line crosses the *z*-axis at *z* = 0 + *d*; for the surface at *z* = *L* it would cross at *z* = *L* + *d*.

It can be seen from the boundary conditions that *d* can be interpreted as an extrapolation length, as illustrated in Fig. 6. For *d* < 0 the polarization turns upwards as it approaches a surface; for *d* > 0 it turns down as a surface is approached.

Now the task is to solve the differential equation—Eq. (30) which is nonlinear—subject to the boundary conditions. For first order transitions in which *C* �= 0 the equation must be solved numerically[17, 18]. However, for the simpler case of second-order transitions (*C* = 0) even though the equation is still nonlinear, an analytical solution can be found in terms of elliptic functions[3, 19, 20] and is given by

$$P\_0(z) = P\_1 \operatorname{sn} \left[ K(\lambda) - \frac{z - L/2}{\mathbb{Q}}, \lambda \right], \quad \text{for } d < 0,\tag{33}$$

where sn is the Jacobian elliptic function with modulus *λ* = *P*1/*P*2, *K*(*λ*) is the complete elliptic integral of the first kind (see Ref. [21] for more on elliptic functions and integrals),

$$P\_1^2 = -\frac{A}{B} - \sqrt{\frac{A^2}{B^2} - \frac{4G}{B}},\tag{34}$$

0 0.25 0.5 0.75 1

**Figure 7.** Polarization in a thin film plotted from Eq. (33) and the boundary conditions in Eqs. (31) and (32). Dimensionless variables and parameters used are: *P*0<sup>∞</sup> = (*aTC*/*B*)1/2 (the spontaneous polarization for a bulk ferroelectric), *ζ*<sup>0</sup> = [2*D*/(*aTC*)]1/2, *<sup>ζ</sup>*<sup>1</sup> = *<sup>P</sup>*0∞/*P*<sup>2</sup> (*P*<sup>2</sup> is given by Eq. (35)), <sup>∆</sup>*T*′ = (*<sup>T</sup>* − *TC*)/*TC* = −0.4, *<sup>L</sup>*′ = *<sup>L</sup>*/*ζ*<sup>0</sup> = <sup>1</sup>, *<sup>d</sup>*′ = <sup>4</sup>*L*′

The case *d* > 0 is complicated by the appearance of a surface term between *TC* and a lower temperature *TC*<sup>0</sup> for which the regions near the film surfaces have a spontaneous polarization but the polarization in the interior is still zero. Only below *TC*<sup>0</sup> is there a nonzero polarization in the interior of the film. The form of the solution, although still expressible in terms of an elliptic function is somewhat different for the surface state, as is discussed elsewhere[3, 19, 20] The above was for in-plane polarization which, as has been explained, avoids the need to consider a depolarization field induced by the polarization. In general, however, there may be an out-of-plane component that would give rise to a depolarization field. Theoretical descriptions of the depolarization field have been considered by several authors[5, 7, 22, 23]. For the case in which there are no free charges at the surfaces it can be shown[5, 22] from

> 1 2*ǫ*0*ǫ*<sup>∞</sup> *P P* − �*P*�

�*P*� <sup>=</sup> <sup>1</sup> *L <sup>L</sup>* 0

added to the energy density in Eq. (28) accounts for the effect of the depolarization field provided that *P* is now taken to be directed along a normal to the plane of the the film (as it is in Fig. 5). In Eq. (37) *ǫ*<sup>0</sup> is the permittivity of free space and *ǫ*<sup>∞</sup> is the background dielectric

**3. The free energy density appropriate for a three-dimensional treatment**

The general form of the free energy density for the interior of a ferroelectric crystal in three-dimensions can be written as series expansion in components of the polarization,

*z*/*ζ*<sup>0</sup>

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

*P dz*, (38)

,

461

(37)

0.2

Maxwell's equations that a term

constant of the film[5, 24], and

**of a ferroelectric crystal**

which is the average value of *P*(*z*) in the film.

*G*′ = 4*GB*/(*a*/*TC*)<sup>2</sup> = 0.105 (found from the boundary conditions).

0.3

0.4

*P*0(*z*/*ζ*1)/*P*0<sup>∞</sup>

$$P\_2^2 = -\frac{A}{B} + \sqrt{\frac{A^2}{B^2} - \frac{4G}{B}},\tag{35}$$

$$
\zeta = \frac{1}{P\_2} \sqrt{\frac{2D}{B}}.\tag{36}
$$

and *G* is found by substituting it into the boundary conditions and solving the resulting transcendental equations numerically.

A plot of the polarization profile according to Eq. (33) is given in Fig. 7 for parameter values given in the figure caption. The polarization throughout the film decreases continuously to zero as the temperature increases and approaches *TC*. So there are two states: a ferroelectric state below *TC* for which there is a polarization everywhere in the film and a paraelectric state above *TC* for which the polarization is zero everywhere.

**Figure 7.** Polarization in a thin film plotted from Eq. (33) and the boundary conditions in Eqs. (31) and (32). Dimensionless variables and parameters used are: *P*0<sup>∞</sup> = (*aTC*/*B*)1/2 (the spontaneous polarization for a bulk ferroelectric), *ζ*<sup>0</sup> = [2*D*/(*aTC*)]1/2, *<sup>ζ</sup>*<sup>1</sup> = *<sup>P</sup>*0∞/*P*<sup>2</sup> (*P*<sup>2</sup> is given by Eq. (35)), <sup>∆</sup>*T*′ = (*<sup>T</sup>* − *TC*)/*TC* = −0.4, *<sup>L</sup>*′ = *<sup>L</sup>*/*ζ*<sup>0</sup> = <sup>1</sup>, *<sup>d</sup>*′ = <sup>4</sup>*L*′ , *G*′ = 4*GB*/(*a*/*TC*)<sup>2</sup> = 0.105 (found from the boundary conditions).

The case *d* > 0 is complicated by the appearance of a surface term between *TC* and a lower temperature *TC*<sup>0</sup> for which the regions near the film surfaces have a spontaneous polarization but the polarization in the interior is still zero. Only below *TC*<sup>0</sup> is there a nonzero polarization in the interior of the film. The form of the solution, although still expressible in terms of an elliptic function is somewhat different for the surface state, as is discussed elsewhere[3, 19, 20]

The above was for in-plane polarization which, as has been explained, avoids the need to consider a depolarization field induced by the polarization. In general, however, there may be an out-of-plane component that would give rise to a depolarization field. Theoretical descriptions of the depolarization field have been considered by several authors[5, 7, 22, 23]. For the case in which there are no free charges at the surfaces it can be shown[5, 22] from Maxwell's equations that a term

$$\frac{1}{2\epsilon\_0\epsilon\_\infty} \left[ P\left(P - \left< P \right>\right) \right] \tag{37}$$

added to the energy density in Eq. (28) accounts for the effect of the depolarization field provided that *P* is now taken to be directed along a normal to the plane of the the film (as it is in Fig. 5). In Eq. (37) *ǫ*<sup>0</sup> is the permittivity of free space and *ǫ*<sup>∞</sup> is the background dielectric constant of the film[5, 24], and

$$
\langle P \rangle = \frac{1}{L} \int\_0^L P \, d\mathbf{z},\tag{38}
$$

which is the average value of *P*(*z*) in the film.

12 Advances in Ferroelectrics

*P*0(*z*)

*d* < 0

crosses the *z*-axis at *z* = 0 + *d*; for the surface at *z* = *L* it would cross at *z* = *L* + *d*.

surface; for *d* > 0 it turns down as a surface is approached.

*P*0(*z*) = *P*<sup>1</sup> sn

*P*2 <sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>A</sup>*

*P*2 <sup>2</sup> <sup>=</sup> <sup>−</sup> *<sup>A</sup>*

state above *TC* for which the polarization is zero everywhere.

*z*

**Figure 6.** The extrapolation length *d* for one of the film surfaces at *z* = 0. For *d* < 0 the polarization turns down as the surface is approached; for *d* > 0 it turns up; this is also true for the surface at *z* = *L* (not shown). Here the extrapolated gradient line

It can be seen from the boundary conditions that *d* can be interpreted as an extrapolation length, as illustrated in Fig. 6. For *d* < 0 the polarization turns upwards as it approaches a

Now the task is to solve the differential equation—Eq. (30) which is nonlinear—subject to the boundary conditions. For first order transitions in which *C* �= 0 the equation must be solved numerically[17, 18]. However, for the simpler case of second-order transitions (*C* = 0) even though the equation is still nonlinear, an analytical solution can be found in terms of elliptic

*<sup>K</sup>*(*λ*) <sup>−</sup> *<sup>z</sup>* <sup>−</sup> *<sup>L</sup>*/2

where sn is the Jacobian elliptic function with modulus *λ* = *P*1/*P*2, *K*(*λ*) is the complete elliptic integral of the first kind (see Ref. [21] for more on elliptic functions and integrals),

> *A*2 *<sup>B</sup>*<sup>2</sup> <sup>−</sup> <sup>4</sup>*<sup>G</sup>*

> *A*2 *<sup>B</sup>*<sup>2</sup> <sup>−</sup> <sup>4</sup>*<sup>G</sup>*

and *G* is found by substituting it into the boundary conditions and solving the resulting

A plot of the polarization profile according to Eq. (33) is given in Fig. 7 for parameter values given in the figure caption. The polarization throughout the film decreases continuously to zero as the temperature increases and approaches *TC*. So there are two states: a ferroelectric state below *TC* for which there is a polarization everywhere in the film and a paraelectric

2*D*

*<sup>B</sup>* <sup>−</sup>

*<sup>B</sup>* <sup>+</sup>

*<sup>ζ</sup>* <sup>=</sup> <sup>1</sup> *P*2 *<sup>ζ</sup>* , *<sup>λ</sup>* 

*P*0(*z*)

*d*

*d* > 0

*z*

, for *d* < 0, (33)

*<sup>B</sup>* , (34)

*<sup>B</sup>* , (35)

*<sup>B</sup>* , (36)

*d*

functions[3, 19, 20] and is given by

transcendental equations numerically.

### **3. The free energy density appropriate for a three-dimensional treatment of a ferroelectric crystal**

The general form of the free energy density for the interior of a ferroelectric crystal in three-dimensions can be written as series expansion in components of the polarization, together with terms taking into account the influence of surfaces and depolarization fields. The region occupied by the interior of the ferroelectric (the surface energy will appear later as a surface energy density integrated over the surface), is taken to be the interior of a rectangular box defined by

$$V = \{ \mathbf{x} \mid \mathbf{x}\_i \in (l\_i^-, l\_i^+), i = 1, 2, 3 \}, \tag{39}$$

– a practice which will be followed here. Nonetheless, in the future it may be interesting to

Finally *f*dep is the term due to the depolarization field for the general case in which the

 + *P*<sup>2</sup> 

Having dealt with the energy density for the interior of the ferroelectric the remaining energy density is the surface energy density at any point on the surface of the ferroelectric. This will be proportional to *P*2, where *P* = |**P**| and, as will be evident below, it is convenient to write

*<sup>P</sup>*<sup>2</sup> − �*P*2�

+ <sup>2</sup> <sup>∪</sup> *Sx*3=*<sup>l</sup>*

*σ*(*x*1),..., *σ*(*x*3)

(*i*) <sup>∈</sup> *l* − *<sup>i</sup>* , *l* + *i*   + *P*<sup>3</sup> 

− <sup>3</sup> <sup>∪</sup> *Sx*3=*<sup>l</sup>*

, *j* = 1, 2

*<sup>σ</sup>*(*xi*) = *xj*<sup>+</sup><sup>1</sup> (mod 3); (48)

*P*2(**x**) for **x** ∈ *S*, (45)

+

<sup>3</sup> (46)

, (47)

. (49)

*<sup>P</sup>*<sup>3</sup> − �*P*3�

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

. (44)

463

*<sup>P</sup>*<sup>1</sup> − �*P*1�

<sup>∪</sup> *Sx*2=*<sup>l</sup>* − <sup>2</sup> <sup>∪</sup> *Sx*2=*<sup>l</sup>*

*<sup>i</sup>* are given by

∓ *<sup>i</sup>* , *xσ<sup>j</sup>*

where we have introduced the cyclic operator *<sup>σ</sup>* which performs the operation *<sup>x</sup>*<sup>1</sup> �→ *<sup>x</sup>*<sup>2</sup> �→

also *σ* applied *n* times where *n* 0 is denoted by *σn*, and the definition of the operator is

It is easy to see that many of the terms in the free energy can be written in a shortened form with this notation because successive terms can often be generated by cyclic permutations of a starting term; later this operator notation will be useful when the free energy for the box is

Showing the form of the free energy densities for the ferroelectric box helps pave a way for dealing with the general problem of finding the minimum of the free energy density for an

 = *h* 

∓

**<sup>x</sup>** | *xi* = *<sup>l</sup>*

extended to when the *xi* are arguments of a function *h* such that

*h*(*x*1,..., *x*3)

*σ* 

inserted into the Euler-Lagrange equations in Section 5.

arbitrary volume of ferroelectric, which will be discussed next.

study such terms to extended the formalism given here.

*f*dep

it as

**x**, **P**,�**P**� <sup>=</sup> <sup>1</sup> 2*ǫ*0*ǫ*<sup>∞</sup> *P*1 

polarization has components along *x*1, *x*<sup>2</sup> and *x*3. It is given by

*f*surf **x**, **P** <sup>=</sup> *<sup>δ</sup>* 2*d*(**x**)

where *S* is the entire surface of the ferroelectric box given by

− <sup>1</sup> <sup>∪</sup> *Sx*1=*<sup>l</sup>* + 1

*<sup>S</sup>* <sup>=</sup> *Sx*1=*<sup>l</sup>*

*Sxj*<sup>=</sup>*<sup>l</sup>* ∓ *j* =

where the sides of the box at *xi* = *l*

*<sup>x</sup>*<sup>3</sup> �→ *<sup>x</sup>*1, so that

so that the sides of the box are given by *li* = *l* + *<sup>i</sup>* − *<sup>l</sup>* − *<sup>i</sup>* , *i* = 1, 2, 3. Extending the formalism in the previous sections to three dimensions, the energy density can be written

$$f\left(\mathbf{x}, \mathbf{P}, \mathbf{P}\_{\mathbf{x}\_{1}}, \mathbf{P}\_{\mathbf{x}\_{2}}, \mathbf{P}\_{\mathbf{x}\_{3}}, \left<\mathbf{P}\right>\right) = f\_{\text{series}}\left(\mathbf{x}, \mathbf{P}\right) + f\_{\text{grad}}\left(\mathbf{x}, \mathbf{P}\_{\mathbf{x}\_{1}}, \mathbf{P}\_{\mathbf{x}\_{2}}, \mathbf{P}\_{\mathbf{x}\_{3}}\right) + f\_{\text{dep}}\left(\mathbf{x}, \mathbf{P}\_{\mathbf{}}\left<\mathbf{P}\right>\right) \tag{40}$$

The notation used for the arguments, which will be useful when applying calculus of variations methods in the next section, is as follows: **x** = (*x*1, *x*2, *x*3), for a position in the crystal in Cartesian coordinates; **P** = (*P*1, *P*2, *P*3), the polarization vector; *Pi* = *Pi*(*x*1, *x*2, *x*3), *i* = 1, 2, 3; **P***xi* = *∂***P**/*∂xi*, *i* = 1, 2, 3, and the averages vector is �**P**� = �*P*1�,�*P*2�,�*P*3� in which

$$
\langle P\_{\dot{l}} \rangle = \frac{1}{l\_{\dot{l}}} \int\_{l\_{\dot{l}}^{-}}^{l\_{\dot{l}}^{+}} P\_{\dot{l}} \, d\mathfrak{x}\_{\dot{l}}.\tag{41}
$$

The terms on the right of Eq. (40) are given next. The first of these terms *f*series is the series expansion part of the free energy (due to the polarization, without a constant first term) in terms of the components of the polarization. A general series expansion in these terms can be written as

$$f\_{\text{series}}(\mathbf{x}, \mathbf{P}) = \sum\_{i=1}^{\infty} \frac{1}{i} \left( \sum\_{a\_1 \geqslant \cdots \geqslant a\_i} A\_{a\_1 \dotsc a\_i} P\_{a\_1} \dotsc \cdot P\_{a\_i} \right). \tag{42}$$

Here each *<sup>α</sup><sup>i</sup>* runs from *<sup>x</sup>*<sup>1</sup> to *<sup>x</sup>*<sup>3</sup> and the notation *<sup>α</sup>*<sup>1</sup> ··· *<sup>α</sup><sup>i</sup>* indicates that all permutations of *<sup>P</sup>α*<sup>1</sup> ··· *<sup>P</sup>α<sup>i</sup>* of the *<sup>i</sup>*th term, which would otherwise appear as separate, have already been summed: the inequalities insure that only one permutation is present for the *i*th term with coefficient *Aα*1···*α<sup>i</sup>* . However in practice, depending on the crystal symmetry, terms which are not separately invariant under the symmetry transformations of the symmetry group of the crystal, will not appear.

The term *f*grad, a gradient term, is given by

$$f\_{\rm grad} \left( \mathbf{x}, \mathbf{P}\_{\mathbf{x}1}, \mathbf{P}\_{\mathbf{x}2}, \mathbf{P}\_{\mathbf{x}3} \right) = \frac{1}{2} \delta \left[ |\nabla P\_1|^2 + |\nabla P\_2|^2 + |\nabla P\_3|^2 \right] = \frac{1}{2} \delta \left[ \left| \frac{\partial \mathbf{P}}{\partial \mathbf{x}\_1} \right|^2 + \left| \frac{\partial \mathbf{P}}{\partial \mathbf{x}\_2} \right|^2 + \left| \frac{\partial \mathbf{P}}{\partial \mathbf{x}\_3} \right|^2 \right].\tag{43}$$

It is easy to show that <sup>1</sup> 2 *δ* |∇*P*1| <sup>2</sup> + |∇*P*2| <sup>2</sup> + |∇*P*3| <sup>2</sup> is equal to the right hand side of Eq. (43). Both forms are shown so that it can be seen that *f*grad involves gradient terms. However the other form is sometimes convenient to work with. Note that in general there are other symmetry allowed terms such as *∂***P** *∂x*1 2 *∂***P** *∂x*2 2 . However usually such terms are ignored – a practice which will be followed here. Nonetheless, in the future it may be interesting to study such terms to extended the formalism given here.

Finally *f*dep is the term due to the depolarization field for the general case in which the polarization has components along *x*1, *x*<sup>2</sup> and *x*3. It is given by

$$f\_{\rm dep}\left(\mathbf{x},\mathbf{P}\_{\prime}\left(\mathbf{P}\right)\right) = \frac{1}{2\epsilon\_{0}\epsilon\_{\infty}} \left[ P\_{1}\left(P\_{1} - \left\right) + P\_{2}\left(P\_{2} - \left\right) + P\_{3}\left(P\_{3} - \left\right) \right]. \tag{44}$$

Having dealt with the energy density for the interior of the ferroelectric the remaining energy density is the surface energy density at any point on the surface of the ferroelectric. This will be proportional to *P*2, where *P* = |**P**| and, as will be evident below, it is convenient to write it as

$$f\_{\rm surf}(\mathbf{x}, \mathbf{P}) = \frac{\delta}{2d(\mathbf{x})} P^2(\mathbf{x}) \quad \text{for} \quad \mathbf{x} \in \mathcal{S}\_{\prime} \tag{45}$$

where *S* is the entire surface of the ferroelectric box given by

$$\mathcal{S} = \mathcal{S}\_{\mathbf{x}\_1 = l\_1^{-}} \cup \mathcal{S}\_{\mathbf{x}\_{1-l\_1^{+}}} \cup \mathcal{S}\_{\mathbf{x}\_2 = l\_2^{-}} \cup \mathcal{S}\_{\mathbf{x}\_2 = l\_2^{+}} \cup \mathcal{S}\_{\mathbf{x}\_3 = l\_3^{-}} \cup \mathcal{S}\_{\mathbf{x}\_3 = l\_3^{+}} \tag{46}$$

where the sides of the box at *xi* = *l* ∓ *<sup>i</sup>* are given by

14 Advances in Ferroelectrics

*f* 

which

be written as

coefficient *Aα*1···*α<sup>i</sup>*

*f*grad 

crystal, will not appear.

**x**, **P***x*<sup>1</sup> , **P***x*<sup>2</sup> , **P***x*<sup>3</sup>

It is easy to show that <sup>1</sup>

rectangular box defined by

together with terms taking into account the influence of surfaces and depolarization fields. The region occupied by the interior of the ferroelectric (the surface energy will appear later as a surface energy density integrated over the surface), is taken to be the interior of a

> + *<sup>i</sup>* − *<sup>l</sup>* −

The notation used for the arguments, which will be useful when applying calculus of variations methods in the next section, is as follows: **x** = (*x*1, *x*2, *x*3), for a position in the crystal in Cartesian coordinates; **P** = (*P*1, *P*2, *P*3), the polarization vector; *Pi* = *Pi*(*x*1, *x*2, *x*3),

The terms on the right of Eq. (40) are given next. The first of these terms *f*series is the series expansion part of the free energy (due to the polarization, without a constant first term) in terms of the components of the polarization. A general series expansion in these terms can

> ∑*α*<sup>1</sup>···*α<sup>i</sup>*

Here each *<sup>α</sup><sup>i</sup>* runs from *<sup>x</sup>*<sup>1</sup> to *<sup>x</sup>*<sup>3</sup> and the notation *<sup>α</sup>*<sup>1</sup> ··· *<sup>α</sup><sup>i</sup>* indicates that all permutations of *<sup>P</sup>α*<sup>1</sup> ··· *<sup>P</sup>α<sup>i</sup>* of the *<sup>i</sup>*th term, which would otherwise appear as separate, have already been summed: the inequalities insure that only one permutation is present for the *i*th term with

not separately invariant under the symmetry transformations of the symmetry group of the

<sup>2</sup> + |∇*P*3|

<sup>2</sup> + |∇*P*3|

Eq. (43). Both forms are shown so that it can be seen that *f*grad involves gradient terms. However the other form is sometimes convenient to work with. Note that in general there are

, *i* = 1, 2, 3

**x**, **P***x*<sup>1</sup> , **P***x*<sup>2</sup> , **P***x*<sup>3</sup>

*<sup>A</sup>α*1···*α<sup>i</sup> <sup>P</sup>α*<sup>1</sup> ··· *<sup>P</sup>α<sup>i</sup>*

. However in practice, depending on the crystal symmetry, terms which are

2 <sup>=</sup> <sup>1</sup> 2 *δ*   + *f*dep **x**, **P**,�**P**�

, (39)

(40)

in

�*P*1�,�*P*2�,�*P*3�

. (42)

*<sup>i</sup>* , *i* = 1, 2, 3. Extending the formalism in

*Pi dxi*. (41)

*∂***P** *∂x*<sup>1</sup> 

2 + 

*∂***P** *∂x*<sup>2</sup> 

<sup>2</sup> is equal to the right hand side of

. However usually such terms are ignored

2 + 

*∂***P** *∂x*<sup>3</sup> 

2 .

(43)

*V* =

= *f*series **x**, **P** + *f*grad 

so that the sides of the box are given by *li* = *l*

*f*series **x**, **P** = ∞ ∑ *i*=1

The term *f*grad, a gradient term, is given by

 <sup>=</sup> <sup>1</sup> 2 *δ* |∇*P*1|

other symmetry allowed terms such as

2 *δ* |∇*P*1|

**<sup>x</sup>**, **<sup>P</sup>**, **<sup>P</sup>***x*<sup>1</sup> , **<sup>P</sup>***x*<sup>2</sup> , **<sup>P</sup>***x*<sup>3</sup> ,�**P**�

**x** | *xi* ∈ *l* − *<sup>i</sup>* , *l* + *i* 

the previous sections to three dimensions, the energy density can be written

*i* = 1, 2, 3; **P***xi* = *∂***P**/*∂xi*, *i* = 1, 2, 3, and the averages vector is �**P**� =

�*Pi*� <sup>=</sup> <sup>1</sup> *li l* + *i l* − *i*

> 1 *i*

<sup>2</sup> + |∇*P*2|

<sup>2</sup> + |∇*P*2|

 *∂***P** *∂x*1 2 *∂***P** *∂x*2 2 

$$\mathcal{S}\_{\mathbf{x}\_{\rangle} = l\_{\rangle}^{\mp} = \{ \mathbf{x} \mid \mathbf{x}\_{\mathbf{i}} = l\_{\mathbf{i}}^{\mp}, \mathbf{x}\_{\sigma^{\dagger}(\mathbf{i})} \in \left[ l\_{\mathbf{i}}^{-}, l\_{\mathbf{i}}^{+} \right], j = 1, 2 \}, \tag{47}$$

where we have introduced the cyclic operator *<sup>σ</sup>* which performs the operation *<sup>x</sup>*<sup>1</sup> �→ *<sup>x</sup>*<sup>2</sup> �→ *<sup>x</sup>*<sup>3</sup> �→ *<sup>x</sup>*1, so that

$$
\sigma(\mathbf{x}\_i) = \mathbf{x}\_{j+1 \pmod{3}};\tag{48}
$$

also *σ* applied *n* times where *n* 0 is denoted by *σn*, and the definition of the operator is extended to when the *xi* are arguments of a function *h* such that

$$
\sigma\left(h(\mathbf{x}\_1, \dots, \mathbf{x}\_3)\right) = h\left(\sigma(\mathbf{x}\_1), \dots, \sigma(\mathbf{x}\_3)\right). \tag{49}
$$

It is easy to see that many of the terms in the free energy can be written in a shortened form with this notation because successive terms can often be generated by cyclic permutations of a starting term; later this operator notation will be useful when the free energy for the box is inserted into the Euler-Lagrange equations in Section 5.

Showing the form of the free energy densities for the ferroelectric box helps pave a way for dealing with the general problem of finding the minimum of the free energy density for an arbitrary volume of ferroelectric, which will be discussed next.

### **4. The calculus of variations applied to the polarization of a ferroelectric of arbitrary shape in three dimensions**

The next step towards working out the equilibrium polarization for ferroelectric boxes and strips—an extension of the theory for thin films—is to work out in general the free energy for an arbitrary volume of ferroelectric. The minimization of this, as for the thin film, involves the minimization of a functional, which involves the calculus of variations and will be dealt with using classical methods.

It can be seen from the previous section that the general form of the free energy for a ferroelectric in the region *V* in **R**<sup>3</sup> bounded by the closed surface *S* is given by

$$\mathbf{G} = \int\_{V} f\left(\mathbf{x}, \mathbf{P}, \mathbf{P}\_{\mathbf{x}\_{1}}, \mathbf{P}\_{\mathbf{x}\_{2}}, \mathbf{P}\_{\mathbf{x}\_{3}}, \langle \mathbf{P} \rangle\right) dV + \int\_{S} f\_{\text{surf}}\left(\mathbf{x}, \mathbf{P}\right) d\mathbf{S}.\tag{50}$$

Applying this condition to Eq. (52), and using the chain rule when differentiating the

(∇**P**<sup>0</sup> *<sup>f</sup>*surf) · <sup>η</sup> *dS* = 0, (54)

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

465

*B*<sup>3</sup> (55)

(∇**P**<sup>0</sup> *<sup>f</sup>*surf) · <sup>η</sup> *dS* = 0, (56)

· η*x*<sup>3</sup> , (57)

**x**ˆ3, (60)

integrands, we find

in which the notation

has been introduced.

*<sup>V</sup>* η · **Λ**<sup>1</sup> +

(∇**P**<sup>0</sup> *<sup>f</sup>*) · <sup>η</sup> +

To facilitate this we start by writing Eq. (54) as

I<sup>1</sup> = <sup>∇</sup>**P**0*x*<sup>1</sup> *<sup>f</sup>* · η*x*<sup>1</sup> + <sup>∇</sup>**P**0*x*<sup>2</sup> *<sup>f</sup>* · η*x*<sup>2</sup> + <sup>∇</sup>**P**0*x*<sup>3</sup> *<sup>f</sup>* 

I<sup>2</sup> = <sup>∇</sup>**<sup>P</sup>**<sup>0</sup> *<sup>f</sup>* 

in which the following notation has been used:

 *∂A*<sup>1</sup> *∂x*<sup>1</sup> , *∂A*<sup>2</sup> *∂x*<sup>2</sup> , *∂A*<sup>3</sup> *∂x*<sup>3</sup> 

∇**A** =

*xi* axes of a Cartesian coordinate system. Also,

*dG dǫ ǫ*=0 = *V* 

show that it can be rewritten as

 <sup>∇</sup>**P**0*x*<sup>1</sup> *<sup>f</sup>* · η*x*<sup>1</sup> + <sup>∇</sup>**P**0*x*<sup>2</sup> *<sup>f</sup>* · η*x*<sup>2</sup> + <sup>∇</sup>**P**0*x*<sup>3</sup> *<sup>f</sup>* · η*x*<sup>3</sup>

(∇**<sup>A</sup>** *<sup>f</sup>*) · **<sup>B</sup>** <sup>=</sup> *<sup>∂</sup> <sup>f</sup>*

+ <sup>∇</sup>**<sup>P</sup>**<sup>0</sup> *<sup>f</sup>* · η *dV* + *S*

*∂A*<sup>1</sup>

(∇**P**<sup>0</sup> *<sup>f</sup>*) · <sup>η</sup> <sup>+</sup> <sup>I</sup><sup>1</sup> <sup>+</sup> <sup>I</sup><sup>2</sup>

Now we want to find **<sup>Λ</sup>**<sup>1</sup> and **<sup>Λ</sup>**<sup>2</sup> such that I<sup>1</sup> = <sup>η</sup> · **<sup>Λ</sup>**<sup>1</sup> and I<sup>2</sup> = <sup>η</sup> · **<sup>Λ</sup>**2.

*B*<sup>1</sup> +

If all terms in the integrands in Eq. (54) could be written as a dot product with η, as is the case for the first and last terms on the right of Eq. (54), then η would appear as a single factor in the volume and surface integrands. Eq. (54) could then be written in the form

equations and boundary conditions will then follow by setting **Λ**<sup>1</sup> = **Λ**<sup>2</sup> = 0, which will satisfy Eq. (54). Therefore we now proceed to express Eq. (54) in the η factored form.

Starting with I1, by expanding the dot products and re-grouping the terms, it is easy to

<sup>=</sup> *<sup>∂</sup>A*<sup>1</sup> *∂x*<sup>1</sup>

the normal gradient operation, where **x**ˆ*i*, *i* = 1, 2, 3, are unit vectors along the corresponding

<sup>I</sup><sup>1</sup> <sup>=</sup> (∇*Po*1**<sup>x</sup>** *<sup>f</sup>*) · (∇*η*1) <sup>+</sup> (∇*Po*2**<sup>x</sup>** *<sup>f</sup>*) · (∇*η*2) <sup>+</sup> (∇*Po*3**<sup>x</sup>** *<sup>f</sup>*) · (∇*η*3), (59)

**x**ˆ1 +

*∂A*<sup>2</sup> *∂x*<sup>2</sup>

**x**ˆ2 +

*∂A*<sup>3</sup> *∂x*<sup>3</sup>

*∂ f ∂A*<sup>2</sup>

*<sup>S</sup>* η · **Λ**2, where **Λ**<sup>1</sup> and **Λ**<sup>2</sup> do not involve η. As shown below, the Euler-Lagrange

 *dV* + *S*

*B*<sup>2</sup> +

*∂ f ∂A*<sup>3</sup>

· η. (58)

*dG dǫ ǫ*=0 = *V* 

where

Here *G* depends on whatever function **P** is, so the domain of *G* is a function space and the equilibrium polarization distribution is that function **P**<sup>0</sup> in the space which minimizes *G*. At this point certain boundary conditions could be imposed on **P** according to the physical situation. For a ferroelectric crystal no such conditions are imposed which implies that **P** is free at the boundaries. However natural boundary conditions (that do not fix **P**) will emerge from the minimization as we will see.

The minimization of the functional *G* in Eq. (50) will be carried out using a classical technique due to Euler[16] which involves considering a variation around **P**<sup>0</sup> = (*P*01, *P*02, *P*03) in function space. To this end we write

$$\mathbf{P}(\mathbf{x}) = \mathbf{P}\_0(\mathbf{x}) + \epsilon \eta(\mathbf{x}),\tag{51}$$

in which *ǫ* is a variable parameter and η = (*η*1, *η*2, *η*3) is an arbitrary vector function. *G* can now be expressed as

$$G(\boldsymbol{\varepsilon}) = \int\_{V} f(\mathbf{x}, \mathbf{P}\_0 + \boldsymbol{\varepsilon}\boldsymbol{\eta}, \mathbf{P}\_{0\mathbf{x}\_1} + \boldsymbol{\varepsilon}\boldsymbol{\eta}\_{\mathbf{x}\_1\prime}\mathbf{P}\_{0\mathbf{x}\_2} + \boldsymbol{\varepsilon}\boldsymbol{\eta}\_{\mathbf{x}\_2\prime}\mathbf{P}\_{0\mathbf{x}\_3} + \boldsymbol{\varepsilon}\boldsymbol{\eta}\_{\mathbf{x}\_3\prime}\langle \mathbf{P}\_0 + \boldsymbol{\varepsilon}\boldsymbol{\eta}\rangle),\tag{52}$$

where

$$\mathbf{P}\_{0\mathbf{x}\_{i}} = \frac{\partial \mathbf{P}\_{0}}{\partial \mathbf{x}\_{i}}.\tag{53}$$

The key idea for finding the minimizing function **P**<sup>0</sup> is that a necessary condition for its existence is

$$\left. \frac{d\mathcal{G}}{d\varepsilon} \right|\_{\varepsilon=0} = 0.$$

Applying this condition to Eq. (52), and using the chain rule when differentiating the integrands, we find

$$\frac{d\mathbb{G}}{d\varepsilon}\Big|\_{\varepsilon=0} = \int\_{V} \left\{ (\nabla\_{\mathbb{P}\_{0}} f) \cdot \eta + \left(\nabla\_{\mathbb{P}\_{01\_{1}}} f\right) \cdot \eta\_{\mathbb{X}\_{1}} + \left(\nabla\_{\mathbb{P}\_{01\_{2}}} f\right) \cdot \eta\_{\mathbb{X}\_{2}} + \left(\nabla\_{\mathbb{P}\_{01\_{3}}} f\right) \cdot \eta\_{\mathbb{X}\_{3}} \right\} \, d\mathbb{M}$$

$$+ \left(\nabla\_{\langle\mathbb{P}\_{0}\rangle} f\right) \cdot \langle\eta\rangle\, dV + \int\_{S} \left(\nabla\_{\mathbb{P}\_{0}} f\_{\text{surf}}\right) \cdot \eta\, d\mathbb{S} = 0,\tag{54}$$

in which the notation

16 Advances in Ferroelectrics

with using classical methods.

**of arbitrary shape in three dimensions**

*G* = *V f* 

from the minimization as we will see.

function space. To this end we write

now be expressed as

where

existence is

*G*(*ǫ*) =

 *V f* 

**4. The calculus of variations applied to the polarization of a ferroelectric**

The next step towards working out the equilibrium polarization for ferroelectric boxes and strips—an extension of the theory for thin films—is to work out in general the free energy for an arbitrary volume of ferroelectric. The minimization of this, as for the thin film, involves the minimization of a functional, which involves the calculus of variations and will be dealt

It can be seen from the previous section that the general form of the free energy for a

Here *G* depends on whatever function **P** is, so the domain of *G* is a function space and the equilibrium polarization distribution is that function **P**<sup>0</sup> in the space which minimizes *G*. At this point certain boundary conditions could be imposed on **P** according to the physical situation. For a ferroelectric crystal no such conditions are imposed which implies that **P** is free at the boundaries. However natural boundary conditions (that do not fix **P**) will emerge

The minimization of the functional *G* in Eq. (50) will be carried out using a classical technique due to Euler[16] which involves considering a variation around **P**<sup>0</sup> = (*P*01, *P*02, *P*03) in

in which *ǫ* is a variable parameter and η = (*η*1, *η*2, *η*3) is an arbitrary vector function. *G* can

**<sup>P</sup>**0*xi* <sup>=</sup> *<sup>∂</sup>***P**<sup>0</sup> *∂xi*

The key idea for finding the minimizing function **P**<sup>0</sup> is that a necessary condition for its

= 0.

*dG dǫ ǫ*=0

**<sup>x</sup>**, **<sup>P</sup>**<sup>0</sup> <sup>+</sup> *<sup>ǫ</sup>*η, **<sup>P</sup>**0*x*<sup>1</sup> <sup>+</sup> *<sup>ǫ</sup>*η*x*<sup>1</sup> , **<sup>P</sup>**0*x*<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*η*x*<sup>2</sup> , **<sup>P</sup>**0*x*<sup>3</sup> <sup>+</sup> *<sup>ǫ</sup>*η*x*<sup>3</sup> ,**<sup>P</sup>**<sup>0</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>η</sup>

**P**(**x**) = **P**0(**x**) + *ǫ*η(**x**), (51)

 *dV* + *S f*surf **x**, **P** 

*dS*. (50)

. (53)

, (52)

ferroelectric in the region *V* in **R**<sup>3</sup> bounded by the closed surface *S* is given by

**<sup>x</sup>**, **<sup>P</sup>**, **<sup>P</sup>***x*<sup>1</sup> , **<sup>P</sup>***x*<sup>2</sup> , **<sup>P</sup>***x*<sup>3</sup> ,**<sup>P</sup>**

$$(\nabla\_{\mathbf{A}}f) \cdot \mathbf{B} = \frac{\partial f}{\partial A\_1}B\_1 + \frac{\partial f}{\partial A\_2}B\_2 + \frac{\partial f}{\partial A\_3}B\_3 \tag{55}$$

has been introduced.

If all terms in the integrands in Eq. (54) could be written as a dot product with η, as is the case for the first and last terms on the right of Eq. (54), then η would appear as a single factor in the volume and surface integrands. Eq. (54) could then be written in the form *<sup>V</sup>* η · **Λ**<sup>1</sup> + *<sup>S</sup>* η · **Λ**2, where **Λ**<sup>1</sup> and **Λ**<sup>2</sup> do not involve η. As shown below, the Euler-Lagrange equations and boundary conditions will then follow by setting **Λ**<sup>1</sup> = **Λ**<sup>2</sup> = 0, which will satisfy Eq. (54). Therefore we now proceed to express Eq. (54) in the η factored form.

To facilitate this we start by writing Eq. (54) as

$$\left. \frac{dG}{d\varepsilon} \right|\_{\varepsilon=0} = \int\_{V} \left\{ (\nabla \mathbf{p}\_0 f) \cdot \boldsymbol{\eta} + \mathcal{Z}\_1 + \mathcal{Z}\_2 \right\} dV + \int\_{S} (\nabla \mathbf{p}\_0 f\_{\text{surf}}) \cdot \boldsymbol{\eta} \, d\mathbf{S} = 0,\tag{56}$$

where

$$\mathcal{L}\_1 = \left(\nabla\_{\mathbf{P}\_{0\chi\_1}} f\right) \cdot \eta\_{\chi\_1} + \left(\nabla\_{\mathbf{P}\_{0\chi\_2}} f\right) \cdot \eta\_{\chi\_2} + \left(\nabla\_{\mathbf{P}\_{0\chi\_3}} f\right) \cdot \eta\_{\chi\_3} \tag{57}$$

$$\mathcal{L}\_2 = \left(\nabla\_{\langle \mathbf{P}\_0 \rangle} f\right) \cdot \langle \boldsymbol{\eta} \rangle. \tag{58}$$

Now we want to find **<sup>Λ</sup>**<sup>1</sup> and **<sup>Λ</sup>**<sup>2</sup> such that I<sup>1</sup> = <sup>η</sup> · **<sup>Λ</sup>**<sup>1</sup> and I<sup>2</sup> = <sup>η</sup> · **<sup>Λ</sup>**2.

Starting with I1, by expanding the dot products and re-grouping the terms, it is easy to show that it can be rewritten as

$$\mathcal{L}\_1 = (\nabla\_{P\_{\rm o1}\mathbf{x}} f) \cdot (\nabla \eta\_1) + (\nabla\_{P\_{\rm o2}\mathbf{x}} f) \cdot (\nabla \eta\_2) + (\nabla\_{P\_{\rm o3}\mathbf{x}} f) \cdot (\nabla \eta\_3) \, , \tag{59}$$

in which the following notation has been used:

$$\nabla \mathbf{A} = \left(\frac{\partial A\_1}{\partial \mathbf{x}\_1}, \frac{\partial A\_2}{\partial \mathbf{x}\_2}, \frac{\partial A\_3}{\partial \mathbf{x}\_3}\right) = \frac{\partial A\_1}{\partial \mathbf{x}\_1}\mathbf{\hat{x}}\_1 + \frac{\partial A\_2}{\partial \mathbf{x}\_2}\mathbf{\hat{x}}\_2 + \frac{\partial A\_3}{\partial \mathbf{x}\_3}\mathbf{\hat{x}}\_3 \tag{60}$$

the normal gradient operation, where **x**ˆ*i*, *i* = 1, 2, 3, are unit vectors along the corresponding *xi* axes of a Cartesian coordinate system. Also,

$$\nabla\_{A\_{\rm{ix}}} = \left(\frac{\partial}{\partial A\_{\rm{ix}\_1}}, \frac{\partial}{\partial A\_{\rm{ix}\_2}}, \frac{\partial}{\partial A\_{\rm{ix}\_3}}\right), \tag{61}$$

 *V* η · 

<sup>∇</sup>**P**<sup>0</sup> *<sup>f</sup>* <sup>−</sup>

3 ∑ *i*=1

∇ ·

surface integral in Eq. (67) vanishes resulting in

 *V* η · 

Bois-Reymond[25], that

which can be rewritten as

<sup>∇</sup>*P*0*i***<sup>x</sup>** *<sup>f</sup>* **x**ˆ*<sup>i</sup>* + <sup>∇</sup>�**P**0� *<sup>f</sup>*

<sup>∇</sup>**P**<sup>0</sup> *<sup>f</sup>* <sup>−</sup>

<sup>∇</sup>**P**<sup>0</sup> *<sup>f</sup>* <sup>−</sup>

− ∇ ·

to the surface integral term. This implies the boundary conditions

(∇*Pi***<sup>x</sup>** *f*) · **n**ˆ +

prescribed condition), and the second term is due to the surface energy.

for the case of a ferroelectric exhibiting a symmetric phase of cubic symmetry.

*∂ f ∂P*0*<sup>i</sup>*

3 ∑ *i*=1

3 ∑ *i*=1

∇ ·

<sup>∇</sup>*P*0*i***<sup>x</sup>** *<sup>f</sup>* + *∂ f <sup>∂</sup>*�*P*0*i*�

∇ ·

+ *S* η · 3 ∑ *i*=1 <sup>∇</sup>*P*0*i***<sup>x</sup>** *<sup>f</sup>* · **n**ˆ 

 *dV*

However, at first we consider a restricted function space such that η is zero on *S*; then the

<sup>∇</sup>*P*0*i***<sup>x</sup>** *<sup>f</sup>* **x**ˆ*<sup>i</sup>* + <sup>∇</sup>�**P**0� *<sup>f</sup>*

For this to be zero for any η in the restricted space, it follows from the lemma of du

These form the Euler-Lagrange equations that we seek. It is clear that the lifting of the restriction on η so that it is not necessarily zero on *S*, means that the Euler-Lagrange equations still result from Eq. (67), provided the lemma of du Bois-Reymond is applied

> *∂ f*surf *∂Pi*

For any *i* the first term is a natural boundary condition arising from the minimization (not a

Thus we have shown that the equilibrium polarization for a ferroelectric of arbitrary shape in the region *V* is found by solving the Euler-Lagrange equations in Eq. (70), subject to the boundary conditions in Eq. (71). This formulation can be applied to finding the polarization in ferroelectrics with the geometries, box, strip and film, as will be shown in the next section

<sup>∇</sup>*P*0*i***<sup>x</sup>** *<sup>f</sup>* **x**ˆ*<sup>i</sup>* + <sup>∇</sup>�**P**0� *<sup>f</sup>*  **<sup>x</sup>**ˆ*<sup>i</sup>* <sup>+</sup> <sup>∇</sup>**P**<sup>0</sup> *<sup>f</sup>*surf

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

*dS* = 0. (67)

467

*dV* = 0. (68)

= 0, (69)

= 0, *i* = 1, 2, 3. (70)

on *S*, *i* = 1, 2, 3. (71)

with *Aixj* = *<sup>∂</sup>Ai*/*∂xj*.

Now, we utilize the vector identity

$$(\nabla \phi) \cdot \mathbf{A} \equiv \nabla \cdot (\phi \mathbf{A}) - \phi \left(\nabla \cdot \mathbf{A}\right) \,, \tag{62}$$

together with the divergence theorem

$$\int\_{V} \nabla \cdot \mathbf{A} \,dV = \oint\_{S} \mathbf{A} \cdot (\mathbf{\hat{n}} \,d\mathbf{S})\,\prime \,\tag{63}$$

where at each point on the surface *S*, **n**ˆ = **n**ˆ **x**|*S* is a unit normal vector directed outwards (from the enclosed volume). Using Eqs. (62) and (63), it follows that

$$\int\_{V} \mathcal{D}\_{1} \, dV = -\int\_{V} \eta \cdot \left[ \sum\_{i=1}^{3} \nabla \cdot \left( \nabla\_{\text{Pux}} f \right) \hat{\mathbf{x}}\_{i} \right] \, dV + \int\_{S} \eta \cdot \left[ \sum\_{i=1}^{3} \left\{ \left( \nabla\_{\text{Pux}} f \right) \cdot \hat{\mathbf{n}} \right\} \hat{\mathbf{x}}\_{i} \right] \, dS. \tag{64}$$

Next the I<sup>2</sup> term is dealt with. But here is is better to look at it together with the integral from the outset, by considering

$$\int\_{V} \mathcal{D}\_2 \, dV = \int\_{V} \left( \nabla\_{\langle \mathbf{P}\_0 \rangle} f \right) \cdot \langle \eta \rangle \, dV = \int\_{V} \left( \sum\_{i=1}^3 \frac{\partial f}{\partial \langle P\_i \rangle} \langle \eta\_1 \rangle \right) dV. \tag{65}$$

The reason for considering I<sup>2</sup> inside the integral is that, remembering that �*ηi*� is proportional to *<sup>l</sup>* + *i l* − *i η<sup>i</sup> dxi*, Eq. (65) can be written as

$$\int\_{V} \mathcal{D}\_2 \, dV = \int\_{V} \left( \sum\_{i=1}^3 \eta\_i \left< \frac{\partial f}{\partial P\_{0i}} \right> \right) dV = \int\_{V} \eta \cdot \left< \nabla\_{\left< \mathbf{P}\_0 \right>} f \right> dV \, \tag{66}$$

by changing the order of integration. In doing this we have used the fact that *dV* ≡ *dx*1*dx*2*dx*<sup>3</sup> and *<sup>V</sup>* <sup>≡</sup> *x*1 *x*2 *x*3 .

The goal of factoring out η involving a dot product has now been achieved and Eq. (54) can be written, using Eqs. (64) and (66), as

$$\begin{split} \int\_{V} \eta \cdot \left[ \nabla\_{\mathbf{P}\_{0}} f - \sum\_{i=1}^{3} \nabla \cdot \left( \nabla\_{P\_{0} \mathbf{x}} f \right) \hat{\mathbf{x}}\_{i} + \left\langle \nabla\_{\left( \mathbf{P}\_{0} \right)} f \right\rangle \right] dV \\ &+ \int\_{S} \eta \cdot \left[ \sum\_{i=1}^{3} \left\{ \left( \nabla\_{P\_{0} \mathbf{x}} f \right) \cdot \hat{\mathbf{n}} \right\} \hat{\mathbf{x}}\_{i} + \nabla\_{\mathbf{P}\_{0}} f\_{\text{surf}} \right] d\mathbf{S} = 0. \end{split} \tag{67}$$

However, at first we consider a restricted function space such that η is zero on *S*; then the surface integral in Eq. (67) vanishes resulting in

$$\int\_{V} \eta \cdot \left[ \nabla\_{\mathbf{P}\_{0}} f - \sum\_{i=1}^{3} \nabla \cdot \left( \nabla\_{P\_{0} \mathbf{x}} f \right) \hat{\mathbf{x}}\_{i} + \left\langle \nabla\_{\left( \mathbf{P}\_{0} \right)} f \right\rangle \right] dV = 0. \tag{68}$$

For this to be zero for any η in the restricted space, it follows from the lemma of du Bois-Reymond[25], that

$$\left(\nabla\_{\mathbf{P}\_0} f - \sum\_{i=1}^3 \nabla \cdot \left(\nabla\_{P\_0 \mathbf{x}} f\right) \right) \mathbf{\hat{x}}\_i + \left\langle \nabla\_{\left< \mathbf{P}\_0 \right>} f \right\rangle = 0,\tag{69}$$

which can be rewritten as

18 Advances in Ferroelectrics

with *Aixj* = *<sup>∂</sup>Ai*/*∂xj*.

 *V*

proportional to *<sup>l</sup>*

*<sup>V</sup>* <sup>≡</sup> *x*1 *x*2 *x*3 .

and

I<sup>1</sup> *dV* = −

from the outset, by considering

 *V*

> + *i l* − *i*

> > *V*

be written, using Eqs. (64) and (66), as

I<sup>2</sup> *dV* =

I<sup>2</sup> *dV* =

 *V* <sup>∇</sup>�**P**0� *<sup>f</sup>* 

 *V*

*η<sup>i</sup> dxi*, Eq. (65) can be written as

 3 ∑ *i*=1 *ηi ∂ f ∂P*0*<sup>i</sup>*

Now, we utilize the vector identity

together with the divergence theorem

where at each point on the surface *S*, **n**ˆ = **n**ˆ

 *V* η · 3 ∑ *i*=1

<sup>∇</sup>*Ai***<sup>x</sup>** =

 *V*

(from the enclosed volume). Using Eqs. (62) and (63), it follows that

∇ ·

<sup>∇</sup>*P*0*i***<sup>x</sup>** *<sup>f</sup>* **x**ˆ*i dV* + *S* η · 3 ∑ *i*=1 <sup>∇</sup>*P*0*i***<sup>x</sup>** *<sup>f</sup>* · **n**ˆ **x**ˆ*i* 

 *∂ <sup>∂</sup>Aix*<sup>1</sup>

∇ · **A** *dV* =

 *S*

 **x**|*S* 

Next the I<sup>2</sup> term is dealt with. But here is is better to look at it together with the integral

· �η� *dV* =

The reason for considering I<sup>2</sup> inside the integral is that, remembering that �*ηi*� is

by changing the order of integration. In doing this we have used the fact that *dV* ≡ *dx*1*dx*2*dx*<sup>3</sup>

The goal of factoring out η involving a dot product has now been achieved and Eq. (54) can

*dV* = *V* η · <sup>∇</sup>�**P**0� *<sup>f</sup>* 

 *V*  3 ∑ *i*=1

*∂ f <sup>∂</sup>*�*Pi*� �*η*1� 

, *<sup>∂</sup> <sup>∂</sup>Aix*<sup>2</sup> , *<sup>∂</sup> <sup>∂</sup>Aix*<sup>3</sup>

(∇*φ*) · **A** ≡ ∇ · (*φ***A**) − *φ* (∇ · **A**), (62)

, (61)

**A** · (**n**ˆ *dS*), (63)

is a unit normal vector directed outwards

*dS*. (64)

*dV*. (65)

*dV*, (66)

$$
\frac{\partial f}{\partial P\_{0i}} - \nabla \cdot \left(\nabla\_{P\_{0i}} f\right) + \left\langle \frac{\partial f}{\partial \langle P\_{0i} \rangle} \right\rangle = 0, \quad i = 1, 2, 3. \tag{70}
$$

These form the Euler-Lagrange equations that we seek. It is clear that the lifting of the restriction on η so that it is not necessarily zero on *S*, means that the Euler-Lagrange equations still result from Eq. (67), provided the lemma of du Bois-Reymond is applied to the surface integral term. This implies the boundary conditions

$$(\nabla\_{P\_{\rm li}} f) \cdot \mathbf{\hat{n}} + \frac{\partial f\_{\rm surf}}{\partial P\_{\rm li}} \quad \text{on } \mathbb{S}, \quad i = 1, 2, 3. \tag{71}$$

For any *i* the first term is a natural boundary condition arising from the minimization (not a prescribed condition), and the second term is due to the surface energy.

Thus we have shown that the equilibrium polarization for a ferroelectric of arbitrary shape in the region *V* is found by solving the Euler-Lagrange equations in Eq. (70), subject to the boundary conditions in Eq. (71). This formulation can be applied to finding the polarization in ferroelectrics with the geometries, box, strip and film, as will be shown in the next section for the case of a ferroelectric exhibiting a symmetric phase of cubic symmetry.

The Euler-Lagrange equations that need to be solved to find the equilibrium polarization in the nano-box are then found from Eq. (70) with *f* = *f*cubic. The steps will not be shown here but the cyclic operator can make the working easier and it is straightforward to show that

+ *<sup>δ</sup>*∇<sup>2</sup>*Pi* +

∓ *j*

*<sup>i</sup>* <sup>=</sup> *<sup>d</sup>* <sup>∀</sup>*<sup>i</sup>* <sup>=</sup> 1, 2, 3.

The surface term is given by Eq. (45), but here we assume that *<sup>d</sup>*(**x**|*S*) is constant over each surface of the box. The boundary conditions in Eq. (71) can then be shown to reduce to

<sup>=</sup> <sup>0</sup> <sup>∀</sup>**<sup>x</sup>** <sup>∈</sup> *Sxi*<sup>=</sup>*<sup>l</sup>*

The solution of Eqs. (73) and (74) has not yet been obtained but progress is being made on it and will be reported later. Eq. (73) is a nonlinear partial differential equation, and as such it is unlikely that an analytical solution can be found. Instead, a numerical approach or an approximate solution using trial functions with adjustable parameters could be employed. The box geometry is likely to be amenable to a finite difference numerical solution. Dealing with a simpler symmetry such as that discussed in the introduction would make the problem less computationally intensive but still not amenable to an analytical solution, as will now be

If Eq. (72) were reduced to a simpler form by putting *α*<sup>3</sup> = *β*<sup>2</sup> = *β*<sup>3</sup> = *γ*<sup>3</sup> = 0, neglecting the depolarization terms, and arranging for the polarization to be aligned along only one coordinate axis; the problem would then be something like a three-dimensional form of the thin-film case discussed in Section 2.2. It might then be tempting to try a separation of variables approach to solving the Euler-Lagrange equations with the hope that the solution would be of the form *P*(*x*1)*P*(*x*2)*P*(*x*3) in which each factor is a solution of the form of the thin film solution given in Section 2.2. However any attempt to do this would fail to produce an exact solution to the three-dimensional problem due to the nonlinearity of a term cubic in the polarization. However it might be useful to treat *P*(*x*1)*P*(*x*2)*P*(*x*3) as an approximate function and introduce into it some variable parameters that could be optimized to find the best approximation. This idea is currently being explored but as yet there are no final

The case of a strip in which confinement along two of the axes can be thought of as a special case of the box in which one side, *<sup>l</sup>*<sup>1</sup> say, approaches infinity such that **<sup>P</sup>**(**x**) = **<sup>P</sup>**(*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3) →

1 *ǫ*0*ǫ*<sup>∞</sup>

(*Pi* − �*Pi*�), *<sup>i</sup>* = 1, 2, 3. (73)

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

469

*i* = 1, 2, 3. (74)

2 ∑ *j*=1 *σj P*2 *j* 

the required Euler-Lagrange equations are given by

∓ *∂Pi ∂xi* + 1 *d*∓ *i*

For a free standing box it is reasonable to have *<sup>d</sup>*<sup>∓</sup>

**5.2. Polarization for a ferroelectric nano-strip**

*<sup>P</sup>*<sup>2</sup>*<sup>j</sup>*−<sup>1</sup> <sup>+</sup> *<sup>β</sup>*2*Pi*

3 ∑ *j*=2 *αj* 

*<sup>α</sup>*1(*<sup>T</sup>* − *TC*)*Pi* + *Pi*

+ *β*3*Pi* 2*P*<sup>2</sup> *i* 2 ∑ *j*=1 *σj P*2 *i* + 2 ∑ *j*=2 *σj P*4 *j* 

discussed.

conclusions.

**Figure 8.** A structural unit for barium titanate, BaTiO3.

### **5. The equilibrium Polarization in ferroelectric boxes, strips and films for cubic symmetry**

Cubic symmetry is chosen because it is the symmetry of the popular perovskite ferroelectrics such as barium titanate (BaTiO3) and lead titanate (PbTiO3). A structural unit for barium titanate is represented in Fig. 8

The interest is in nanoscale boxes, strips and films. The formalism itself could easily be applied to thicker films, however for confinement in the nanoscale (in all three dimensions for boxes, in two dimensions in strips and in one for films) it is expected that the influence of the change in the polarization as it approaches a surface will have a more significant effect that at lager scales where the bulk properties would be more dominant, and so is sometimes referred to as a size effect.

### **5.1. Polarization for a ferroelectric nano-box**

For a cubic crystal only those terms in *f* given by Eq. (40) that are invariant under the symmetry 16 operations of the cubic symmetry group[26] are allowed.<sup>3</sup> It can be shown[10, 12] that these are given by

$$\begin{split} f\_{\text{cubic}}\left(\mathbf{x}, \mathbf{P}, \mathbf{P}\_{\mathbf{X}\_{1}}, \mathbf{P}\_{\mathbf{X}\_{2}}, \mathbf{P}\_{\mathbf{X}\_{3}}, \left(\mathbf{P}\right)\right) &= a\_{1} (T - T\_{\complement}) P^{2} + \sum\_{i=2}^{3} \frac{1}{2i} a\_{i} \left(P^{2}\right)^{i} \\ &+ \frac{1}{2} \beta\_{2} \sum\_{i=0}^{2} \sigma^{i} \left(P\_{1}^{2} P\_{2}^{2}\right) + \frac{1}{2} \beta\_{3} \sum\_{i=0}^{2} \sigma^{i} \left(P\_{1}^{4} \left(P\_{2}^{2} + P\_{3}^{2}\right)\right) + \frac{1}{2} \gamma\_{3} P\_{1}^{2} P\_{2}^{2} P\_{3}^{2} \\ &+ \frac{1}{2} \delta \sum\_{i=1}^{3} \left|\frac{\partial \mathbf{P}}{\partial x\_{i}}\right|^{2} + \frac{1}{2\varepsilon\_{0} \varepsilon\_{\infty}} \sum\_{i=1}^{3} P\_{i} \left(P\_{i} - \left\langle P\_{i} \right\rangle\right), \end{split} \tag{72}$$

where the cyclic operator defined by Eqs. (48) and (49) is being used and, to lighten the notation a little we make the replacement **<sup>P</sup>**<sup>0</sup> → **<sup>P</sup>** ⇒ *Poi* → *Pi*, although the subscript that distinguishes the equilibrium polarization from the general function space remains implied.

<sup>3</sup> The operations are: 1, 1, 4, 4 ¯ <sup>−</sup>1, 4, ¯ 4¯<sup>−</sup>1, 2′ , 2′′, 2*x*, 2*y*, 2*<sup>z</sup>* , *<sup>m</sup>*1, *<sup>m</sup>*2, *<sup>m</sup>*3, *<sup>m</sup>*4, *<sup>m</sup>*5. The Euler-Lagrange equations that need to be solved to find the equilibrium polarization in the nano-box are then found from Eq. (70) with *f* = *f*cubic. The steps will not be shown here but the cyclic operator can make the working easier and it is straightforward to show that the required Euler-Lagrange equations are given by

$$\begin{aligned} a\_1(T - T\_\mathcal{C})P\_i + P\_i \sum\_{j=2}^3 a\_j(P^2)^{j-1} + \beta\_2 P\_i \sum\_{j=1}^2 \sigma^j(P\_j^2) \\ + \beta\_3 P\_i \Big[ 2P\_i^2 \sum\_{j=1}^2 \sigma^j(P\_i^2) + \sum\_{j=2}^2 \sigma^j(P\_j^4) \Big] + \delta \nabla^2 P\_i + \frac{1}{\epsilon\_0 \epsilon\_\infty} \left( P\_i - \langle P\_i \rangle \right), \quad i = 1, 2, 3. \end{aligned} \tag{73}$$

The surface term is given by Eq. (45), but here we assume that *<sup>d</sup>*(**x**|*S*) is constant over each surface of the box. The boundary conditions in Eq. (71) can then be shown to reduce to

$$\mp \frac{\partial P\_i}{\partial \mathbf{x}\_i} + \frac{1}{d\_i^{\mp}} = \mathbf{0} \,\,\forall \mathbf{x} \in \mathbb{S}\_{\mathbf{x}\_i = I\_j^{\mp}} \quad i = 1, 2, 3. \tag{74}$$

For a free standing box it is reasonable to have *<sup>d</sup>*<sup>∓</sup> *<sup>i</sup>* <sup>=</sup> *<sup>d</sup>* <sup>∀</sup>*<sup>i</sup>* <sup>=</sup> 1, 2, 3.

20 Advances in Ferroelectrics

**cubic symmetry**

titanate is represented in Fig. 8

referred to as a size effect.

12] that these are given by

**<sup>x</sup>**, **<sup>P</sup>**, **<sup>P</sup>***x*<sup>1</sup> , **<sup>P</sup>***x*<sup>2</sup> , **<sup>P</sup>***x*<sup>3</sup> ,�**P**�

+ 1 2 *β*2 2 ∑ *i*=0 *σi P*2 <sup>1</sup> *<sup>P</sup>*<sup>2</sup> 2 + 1 2 *β*3 2 ∑ *i*=0 *σi P*4 1 *P*2 <sup>2</sup> <sup>+</sup> *<sup>P</sup>*<sup>2</sup> 3 + 1 2 *γ*3*P*<sup>2</sup> <sup>1</sup> *<sup>P</sup>*<sup>2</sup> <sup>2</sup> *<sup>P</sup>*<sup>2</sup> 3

<sup>3</sup> The operations are: 1, 1, 4, 4 ¯ <sup>−</sup>1, 4,

*f*cubic 

**5.1. Polarization for a ferroelectric nano-box**

¯ 4¯<sup>−</sup>1, 2′

**Figure 8.** A structural unit for barium titanate, BaTiO3.

Ba2<sup>+</sup>

<sup>O</sup>2<sup>−</sup>

Ti4<sup>+</sup>

**5. The equilibrium Polarization in ferroelectric boxes, strips and films for**

Cubic symmetry is chosen because it is the symmetry of the popular perovskite ferroelectrics such as barium titanate (BaTiO3) and lead titanate (PbTiO3). A structural unit for barium

The interest is in nanoscale boxes, strips and films. The formalism itself could easily be applied to thicker films, however for confinement in the nanoscale (in all three dimensions for boxes, in two dimensions in strips and in one for films) it is expected that the influence of the change in the polarization as it approaches a surface will have a more significant effect that at lager scales where the bulk properties would be more dominant, and so is sometimes

For a cubic crystal only those terms in *f* given by Eq. (40) that are invariant under the symmetry 16 operations of the cubic symmetry group[26] are allowed.<sup>3</sup> It can be shown[10,

> + 1 2 *δ* 3 ∑ *i*=1

where the cyclic operator defined by Eqs. (48) and (49) is being used and, to lighten the notation a little we make the replacement **<sup>P</sup>**<sup>0</sup> → **<sup>P</sup>** ⇒ *Poi* → *Pi*, although the subscript that distinguishes the equilibrium polarization from the general function space remains implied.

, 2′′, 2*x*, 2*y*, 2*<sup>z</sup>* , *<sup>m</sup>*1, *<sup>m</sup>*2, *<sup>m</sup>*3, *<sup>m</sup>*4, *<sup>m</sup>*5.

3 ∑ *i*=2

1 2*i αi P*<sup>2</sup>*<sup>i</sup>*

> 

*∂***P** *∂xi* 

2 +

1 2*ǫ*0*ǫ*<sup>∞</sup>

3 ∑ *i*=1 *Pi* 

*Pi* − �*Pi*�

 , (72)

= *<sup>α</sup>*1(*<sup>T</sup>* − *TC*)*P*<sup>2</sup> +

The solution of Eqs. (73) and (74) has not yet been obtained but progress is being made on it and will be reported later. Eq. (73) is a nonlinear partial differential equation, and as such it is unlikely that an analytical solution can be found. Instead, a numerical approach or an approximate solution using trial functions with adjustable parameters could be employed. The box geometry is likely to be amenable to a finite difference numerical solution. Dealing with a simpler symmetry such as that discussed in the introduction would make the problem less computationally intensive but still not amenable to an analytical solution, as will now be discussed.

If Eq. (72) were reduced to a simpler form by putting *α*<sup>3</sup> = *β*<sup>2</sup> = *β*<sup>3</sup> = *γ*<sup>3</sup> = 0, neglecting the depolarization terms, and arranging for the polarization to be aligned along only one coordinate axis; the problem would then be something like a three-dimensional form of the thin-film case discussed in Section 2.2. It might then be tempting to try a separation of variables approach to solving the Euler-Lagrange equations with the hope that the solution would be of the form *P*(*x*1)*P*(*x*2)*P*(*x*3) in which each factor is a solution of the form of the thin film solution given in Section 2.2. However any attempt to do this would fail to produce an exact solution to the three-dimensional problem due to the nonlinearity of a term cubic in the polarization. However it might be useful to treat *P*(*x*1)*P*(*x*2)*P*(*x*3) as an approximate function and introduce into it some variable parameters that could be optimized to find the best approximation. This idea is currently being explored but as yet there are no final conclusions.

### **5.2. Polarization for a ferroelectric nano-strip**

The case of a strip in which confinement along two of the axes can be thought of as a special case of the box in which one side, *<sup>l</sup>*<sup>1</sup> say, approaches infinity such that **<sup>P</sup>**(**x**) = **<sup>P</sup>**(*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3) →

**P**(*x*2, *x*3), and is constant with respect to *x*<sup>3</sup> since the boundaries in this direction have no influence, which also implies that the depolarization terms only involve �**P**� = (�*P*2�,�*P*3�).

Now the relevant quantity is a free energy density per unit length, since for a section of the strip between *x*<sup>1</sup> = 0 and *x*<sup>1</sup> = *l* ′ <sup>1</sup> the free energy is

$$\mathbf{G} = \int\_{0}^{l\_1'} d\mathbf{x}\_1 \int\_{A} f\left(\mathbf{x}, \mathbf{P}, \mathbf{P}\_{\mathbf{x}\_2}, \mathbf{P}\_{\mathbf{x}\_3}, \langle \mathbf{P} \rangle\right) dA + \int\_{0}^{l\_1'} d\mathbf{x}\_1 \int\_{\Gamma} f\_{\text{surf}}(\mathbf{x}, \mathbf{P}) d\Gamma,\tag{75}$$

**6. Conclusion**

the symmetry group are allowed.

on several of the ideas presented in this chapter.

Sarawak Campus, Kuching, Sarawak, Malaysia

*Ferroelectr. Rev.*, 1:1, 1998.

*Commun.*, 49:823, 1984.

*Physics*, page 69. Springer, Heidelberg, 2007.

**Acknowledgements**

**Author details** Jeffrey F. Webb

**References**

The foundations of Landau-Devonshire theory have been introduced and it has been used to develop a general formulation for the calculation of the spontaneous polarization in a ferroelectric nano-box which is confined to the nanoscale in all three dimensions, and for which the influence of the surfaces is expected to be more pronounced than for large scales. From the formalism for the box calculations it has been shown how to deal with a nano-strip which is confined to the nanoscale in two dimensions and to a nano-film with such confinement in only one dimension. The thin film case has been fairly well studied, but much less work has been done on the other geometries, box and strip. Such work is timely given the increasing use of nanoscale structures in electronic devices which include

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

471

A particular example of how the formalism can applied to a ferroelectric crystal of cubic symmetry has been given for all three geometries. Many ferroelectric materials have this symmetry. However the general formalism presented is not restricted to this symmetry, any symmetry can be handled through knowledge of which terms in the free energy needed to be dropped, the criterion being that only terms invariant under the symmetry operations of

Future work will involve investigating the numerical or approximate function solutions to the Euler-Lagrange equations which need to be solved in order to be able to plot the spontaneous

The author would like to thank Manas Kumar Halder for useful and enlightening discussions

Faculty of Engineering, Computing and Science, Swinburne University of Technology,

[1] P. Chandra and P. B. Littlewood. A Landau primer for ferroelectrics. In K. Rabe, Ch. H. Ahn, and J. M. Triscone, editors, *Physics of Ferroelectrics*, volume 105 of *Topics in Applied*

[2] J. F. Scott. The physics of ferroelectric ceramic thin films for memory applications.

[3] D. R. Tilley and B. Zeks. Landau theory of phase transtions in thick films. *Solid State*

[4] D. R. Tilley. Phase transitions in thin films. In N. Setter and E. L. Colla, editors,

*Ferroelectric Ceramics*, page 163. Birkhäuser Verlag, Berlin, 1993.

ferroelectric materials, for example ferroelectric random access memories.

polarization. Such work is in progress and will be reported in due course.

leading to an energy density

$$F = \frac{G}{l\_1'} = \int\_A f\left(\mathbf{x}, \mathbf{P}, \mathbf{P}\_{\mathbf{x}\_{2\prime}}, \mathbf{P}\_{\mathbf{x}\_{3\prime}} \left< \mathbf{P} \right>\right) dA + \int\_\Gamma f\_{\text{surf}}(\mathbf{x}, \mathbf{P}) d\Gamma\_\prime \tag{76}$$

where <sup>Γ</sup> is a line integral around the path Γ that traces a rectangular cross section of the strip normal to *x*1, *<sup>A</sup>* <sup>≡</sup> *<sup>l</sup>* + 2 *l* − 2 *l* + 3 *l* − 3 , and *dA* = *dx*2*dx*3.

For *f* = *f*cubic (Eq. (72)), the Euler-Lagrange equations are similar to Eq. (73) but *P*<sup>1</sup> is constant, and so ∇<sup>2</sup> = *<sup>∂</sup>*<sup>2</sup> *∂x*2 2 + *<sup>∂</sup>*<sup>2</sup> *∂x*2 3 , and the depolarization term for *i* = 1 reduces to zero. Also the boundary conditions are given by Eq. (74), but with *i* = 2, 3. The solution is again best approached by numerical or other approximate methods. Work on this is in progress.

The comments made in at the end of the previous section for the ferroelectric nano-box with regard to simplifying it to a three dimensional form of the problem in Section 2.2 also apply here: for the same reason it is not possible to express an exact solution of the form *P*(*x*2)*P*(*x*3), where *P* has the form of the solution given in Section 2.2. Again it may be possible to utilize such a solution as an approximate function. This needs to be investigated further.

### **5.3. Polarization for a ferroelectric nano-film**

For a film confined in the *x*<sup>3</sup> direction, a similar argument to the one given for a nano-strip yields **P** = **P**(*x*3), �**P**� = �*P*3�**x**ˆ3, and a free energy density

$$F = \frac{G}{l\_1' l\_2'} = \int\_{l\_3^-}^{l\_3^+} f\left(\chi\_3, \mathbf{P}\_\prime \mathbf{P}\_{\chi\_3\prime} \left< \mathbf{P}\_3 \right>\right) d\mathbf{x}\_3. \tag{77}$$

With *f* = *f*cubic given by Eq. (72), the Euler-Lagrange equations again take the form of Eq. (73); now with *P*<sup>1</sup> and *P*<sup>2</sup> constant so that the depolarization terms reduce to zero for *i* = 1, 2, and ∇<sup>2</sup> → *<sup>∂</sup>*<sup>2</sup> *∂x*2 3 . The boundary conditions are given by Eq. (74), with *i* = 3 and *∂ <sup>∂</sup>x*<sup>3</sup> <sup>→</sup> *<sup>d</sup> dx*<sup>3</sup> . An analytical solution, even for this one-dimensional case is still not likely to exist unless *f* is reduced to the simpler higher symmetry form for second order transitions and in-plane polarization discussed in Section 2.2. However it is amenable to numerical solution or other approximate methods. A similar thin film problem, but involving strain in the ferroelectric as well, which we do no consider, has been solved numerically using a finite difference method by Wang and Zhang[7].

### **6. Conclusion**

22 Advances in Ferroelectrics

strip between *x*<sup>1</sup> = 0 and *x*<sup>1</sup> = *l*

*G* = *l* ′ 1 0 *dx*1 *A f* 

> *<sup>F</sup>* <sup>=</sup> *<sup>G</sup> l* ′ 1 = *A f*

> > *∂x*2 2 + *<sup>∂</sup>*<sup>2</sup> *∂x*2 3

**5.3. Polarization for a ferroelectric nano-film**

*∂x*2 3

difference method by Wang and Zhang[7].

yields **P** = **P**(*x*3), �**P**� = �*P*3�**x**ˆ3, and a free energy density

*<sup>F</sup>* <sup>=</sup> *<sup>G</sup> l* ′ 1*l* ′ 2 = *l* + 3 *l* − 3 *f* 

 *<sup>A</sup>* <sup>≡</sup> *<sup>l</sup>* + 2 *l* − 2 *l* + 3 *l* − 3

leading to an energy density

where

further.

*∂ <sup>∂</sup>x*<sup>3</sup> <sup>→</sup> *<sup>d</sup> dx*<sup>3</sup>

strip normal to *x*1,

constant, and so ∇<sup>2</sup> = *<sup>∂</sup>*<sup>2</sup>

*i* = 1, 2, and ∇<sup>2</sup> → *<sup>∂</sup>*<sup>2</sup>

**P**(*x*2, *x*3), and is constant with respect to *x*<sup>3</sup> since the boundaries in this direction have no influence, which also implies that the depolarization terms only involve �**P**� = (�*P*2�,�*P*3�). Now the relevant quantity is a free energy density per unit length, since for a section of the

> *dA* + *l* ′ 1 0 *dx*1 Γ

> > *dA* + Γ

<sup>Γ</sup> is a line integral around the path Γ that traces a rectangular cross section of the

, and the depolarization term for *i* = 1 reduces to zero.

*f*surf(**x**, **P**)*d*Γ, (75)

*f*surf(**x**, **P**)*d*Γ, (76)

*dx*3. (77)

<sup>1</sup> the free energy is

**<sup>x</sup>**, **<sup>P</sup>**, **<sup>P</sup>***x*<sup>2</sup> , **<sup>P</sup>***x*<sup>3</sup> ,�**P**�

**<sup>x</sup>**, **<sup>P</sup>**, **<sup>P</sup>***x*<sup>2</sup> , **<sup>P</sup>***x*<sup>3</sup> ,�**P**�

, and *dA* = *dx*2*dx*3.

For *f* = *f*cubic (Eq. (72)), the Euler-Lagrange equations are similar to Eq. (73) but *P*<sup>1</sup> is

Also the boundary conditions are given by Eq. (74), but with *i* = 2, 3. The solution is again best approached by numerical or other approximate methods. Work on this is in progress. The comments made in at the end of the previous section for the ferroelectric nano-box with regard to simplifying it to a three dimensional form of the problem in Section 2.2 also apply here: for the same reason it is not possible to express an exact solution of the form *P*(*x*2)*P*(*x*3), where *P* has the form of the solution given in Section 2.2. Again it may be possible to utilize such a solution as an approximate function. This needs to be investigated

For a film confined in the *x*<sup>3</sup> direction, a similar argument to the one given for a nano-strip

With *f* = *f*cubic given by Eq. (72), the Euler-Lagrange equations again take the form of Eq. (73); now with *P*<sup>1</sup> and *P*<sup>2</sup> constant so that the depolarization terms reduce to zero for

exist unless *f* is reduced to the simpler higher symmetry form for second order transitions and in-plane polarization discussed in Section 2.2. However it is amenable to numerical solution or other approximate methods. A similar thin film problem, but involving strain in the ferroelectric as well, which we do no consider, has been solved numerically using a finite

*<sup>x</sup>*3, **<sup>P</sup>**, **<sup>P</sup>***x*<sup>3</sup> ,�*P*3�

. An analytical solution, even for this one-dimensional case is still not likely to

. The boundary conditions are given by Eq. (74), with *i* = 3 and

′

The foundations of Landau-Devonshire theory have been introduced and it has been used to develop a general formulation for the calculation of the spontaneous polarization in a ferroelectric nano-box which is confined to the nanoscale in all three dimensions, and for which the influence of the surfaces is expected to be more pronounced than for large scales. From the formalism for the box calculations it has been shown how to deal with a nano-strip which is confined to the nanoscale in two dimensions and to a nano-film with such confinement in only one dimension. The thin film case has been fairly well studied, but much less work has been done on the other geometries, box and strip. Such work is timely given the increasing use of nanoscale structures in electronic devices which include ferroelectric materials, for example ferroelectric random access memories.

A particular example of how the formalism can applied to a ferroelectric crystal of cubic symmetry has been given for all three geometries. Many ferroelectric materials have this symmetry. However the general formalism presented is not restricted to this symmetry, any symmetry can be handled through knowledge of which terms in the free energy needed to be dropped, the criterion being that only terms invariant under the symmetry operations of the symmetry group are allowed.

Future work will involve investigating the numerical or approximate function solutions to the Euler-Lagrange equations which need to be solved in order to be able to plot the spontaneous polarization. Such work is in progress and will be reported in due course.

### **Acknowledgements**

The author would like to thank Manas Kumar Halder for useful and enlightening discussions on several of the ideas presented in this chapter.

### **Author details**

Jeffrey F. Webb

Faculty of Engineering, Computing and Science, Swinburne University of Technology, Sarawak Campus, Kuching, Sarawak, Malaysia

### **References**


[5] D. R. Tilley. Finite-size effects on phase transitions in ferroelectrics. In Carlos Paz de Araujo, J. F. Scott, and G. W. Taylor, editors, *Ferroelectric Thin Films: Synthesis and Basic Properties*, Integrated Ferroelectric Devices and Technologies, page 11. Gordon and Breach, Amsterdam, 1996.

[21] M. Abramowitz and I. A. Stegun, editors. *Handbook of Mathematical Functions*. Dover,

Nanoscale Ferroelectric Films, Strips and Boxes

http://dx.doi.org/10.5772/52040

473

[22] R. Kretschmer and K. Binder. Surface effects on phase transitions in ferroelectrics and

[23] M.D. Glinchuk, E.A. Eliseev, and V.A. Stephanovich. Ferroelectric thin film properties—depolarization field and renormalization of a "bulk" free energy

[25] E. W. Honson. On the fundamental lemma of the calculus of variations and on some related theorems. In *Proceedings of the London Mathematical Society*, volume s2-11, pages

[26] J. Nye. *Physical Properties of Crystals. Their Representation by Tensors and Matrices*. Oxford

dipolar magnets. *Phys. Rev. B*, 20:1065–1076, 1979.

coefficients. *J. Appl. Phys.*, 93:1150–1159, 2003.

[24] D. L. Mills. *Nonlinear Optics*. Springer, Berlin, second edition, 1998.

New York, 1972.

17–28, 1913.

University Press, Oxford, 1964.


24 Advances in Ferroelectrics

Breach, Amsterdam, 1996.

*J. Electroceram.*, 16:463, 2006.

014102–13, 2007.

Singapore, 1987.

York, 1953.

2011.

*Phys C*, 17:1793–1823, 1984.

PhD thesis, University of Colorado, 1990.

*Solid State Communications*, 116:61–65, 2000.

1998.

*B*, 77:014104–1 to 014104–7, 2008.

*Materials*. Clarendon, Oxford, UK, 1977.

sulfate. *Physical Review Letters*, 124:1036–1038, 1961.

[5] D. R. Tilley. Finite-size effects on phase transitions in ferroelectrics. In Carlos Paz de Araujo, J. F. Scott, and G. W. Taylor, editors, *Ferroelectric Thin Films: Synthesis and Basic Properties*, Integrated Ferroelectric Devices and Technologies, page 11. Gordon and

[6] J. F. Webb. Theory of size effects in ferroelectric ceramic thin films on metal substrates.

[7] J. Wang and T. Y. Zhang. Influence of depolarization field on polarization states in epitaxial ferroelectric thin films with nonequally biaxial misfit strains. *Physical Review*

[8] Anna N. Morozovska, Maya D. Glinchuk, and Eugene A. Eliseev. Phase transitions induced by confinement of ferroic nanoparticles. *Physical Review B*, 76:014102–1 to

[9] M. E. Lines and A. M. Glass. *Principles and Applications of Ferroelectrics and Related*

[10] B. A. Strukov and A. P. Lenanyuk. *Ferroelectric Phenomena in Crystals*. Springer, Berlin,

[11] R. C. Tolman. *The Principles of Statistical Thermodynamics*. Clarendon Press, Oxford, 1938.

[12] J. C. Toledano and P. Toledano. *The Landau Theory of Phase Transitions*. World Scientific,

[13] R. Blinc, S. Detoni, and M. Pintar. Nature of the ferroelectric transition in triglycine

[14] K. Itoh and T. Mitsui. Studies of the crystal structure of triglycine sulfate in connection

[15] M. G. Cottam, D. R. Tilley, and B. Zeks. Theory of surface modes in ferroelectrics. *J.*

[16] R. Courant and D. Hilbert. *Methods of Mathematical Physics*, volume 1. Interscience, New

[17] X. Gerbaux and A. Hadni. *Static and dynamic properties of ferroelectric thin film memories*.

[18] E. K. Tan, J. Osman, and D. R. Tilley. First-order phase transitions in ferroelectric films.

[19] K. H. Chew, L. H. Ong, J. Osman, and D. R. Tilley. Theory of far-infrared reflection and

[20] J. F. Webb. Harmonic generation in nanoscale ferroelectric films. In Mickaël Lallart, editor, *Ferroelectrics - Characterization and Modeling*, chapter 26. Intech, Rijeka, Croatia,

transmission by ferroelectric thin films. *J. Opt. Soc. Am B*, 18:1512, 2001.

with its ferroelectric phase transition. *Ferroelectrics*, 5:235–251, 1973.


**Chapter 21**

**Emerging Applications of Ferroelectric Nanoparticles in**

Consider an insulating system with non-zero spontaneous polarization *Ps* (dielectric di‐ pole moment per unit volume). If an applied external electric field *E* that is greater than the so-called coercive field *Ec* can reverse *Ps* then our system is a ferroelectric system. Ferroelectricity has a long and exciting history described in [1,2]. In the beginning of its historical development (the Rochelle salt period) ferroelectricity was considered an aca‐ demic curiosity with no practical applications. There was little theoretical interest due to the quality of the ferroelectric materials (very fragile and water-soluble) existing at that time. The discovery of ferroelectricity in robust ceramic materials (barium titanate) dur‐ ing World War II launched a new era of rapid progress in the field. The structural sim‐ plicity of barium titanate stimulated numerous theoretical works, while its physical properties were utilized in many devices. Since that time, ferroelectric response has been found in a wide range of materials, including inorganic, organic, and biological species. According to [3] there are 72 families of ferroelectrics presented in Landolt–Börnstein‐ Vol.III*/*36 (LB III*/*36). Forty-nine of these families are inorganic crystals (19 families of ox‐ ides + 30 families of crystals other than oxides), and 23 families are organic crystals,

The enormously broad range of materials exhibiting ferroelectricity and the variety of their physical properties result in numerous applications of bulk ferroelectrics [4]. Table 1 shows the connections between different physical effects exhibited by bulk ferroelectrics and their

and reproduction in any medium, provided the original work is properly cited.

© 2013 Garbovskiy et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Materials Technologies, Biology and Medicine**

Yuriy Garbovskiy, Olena Zribi and

Additional information is available at the end of the chapter

Anatoliy Glushchenko

http://dx.doi.org/10.5772/52516

liquid crystals, and polymers.

applications.

**1. Introduction**

## **Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine**

Yuriy Garbovskiy, Olena Zribi and Anatoliy Glushchenko

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52516

### **1. Introduction**

Consider an insulating system with non-zero spontaneous polarization *Ps* (dielectric di‐ pole moment per unit volume). If an applied external electric field *E* that is greater than the so-called coercive field *Ec* can reverse *Ps* then our system is a ferroelectric system. Ferroelectricity has a long and exciting history described in [1,2]. In the beginning of its historical development (the Rochelle salt period) ferroelectricity was considered an aca‐ demic curiosity with no practical applications. There was little theoretical interest due to the quality of the ferroelectric materials (very fragile and water-soluble) existing at that time. The discovery of ferroelectricity in robust ceramic materials (barium titanate) dur‐ ing World War II launched a new era of rapid progress in the field. The structural sim‐ plicity of barium titanate stimulated numerous theoretical works, while its physical properties were utilized in many devices. Since that time, ferroelectric response has been found in a wide range of materials, including inorganic, organic, and biological species. According to [3] there are 72 families of ferroelectrics presented in Landolt–Börnstein‐ Vol.III*/*36 (LB III*/*36). Forty-nine of these families are inorganic crystals (19 families of ox‐ ides + 30 families of crystals other than oxides), and 23 families are organic crystals, liquid crystals, and polymers.

The enormously broad range of materials exhibiting ferroelectricity and the variety of their physical properties result in numerous applications of bulk ferroelectrics [4]. Table 1 shows the connections between different physical effects exhibited by bulk ferroelectrics and their applications.

© 2013 Garbovskiy et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recent advances in nanotechnologies, especially in nanoinstrumentation (for example, scanning probe microscopy [5]) and materials nanofabrication [6], allowed the direct prob‐ ing of ferroelectricity at the nanoscale. The new and unexplored world of nanoscale ferro‐ electrics (nanoparticles of different shapes and sizes, nanofilms, nanopatterned structures, etc.) raised fundamental questions and stimulated very active research in both academic and industrial sectors [7]. As a result, a new era of nanoscale ferroelectrics was launched. Novel effects, associated with reduced dimensions and found in nanoscaleferroelectrics, highlighted exciting possibilities for new applications reviewed recently in [8]. Almost all of the attention for the mentioned review [8] was devoted to the *thin film* nanoscale de‐ vice structures (which can be easily integrated with a Si chip) with focus on ultrafast switching, electrocaloric coolers for computers, phase-array radar, three-dimensional trenched capacitors for dynamic random access memories, room temperature magnetic field detectors, and miniature X-ray and neutron sources. So far, we have not found a co‐ herent review summarizing the actual and possible applications of ferroelectric *nanoparti‐ cles*. Our book chapter is an attempt to describe and analyze the state of the field of applications of ferroelectric nanoparticles with focus on materials technologies, medicine, and biology.

**2. Methods of ferroelectric nanoparticles production: Nano-powders and**

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

http://dx.doi.org/10.5772/52516

477

The existing methods that produce ferroelectric nanoparticles are numerous and can be clas‐ sified as physical, chemical, and biological (Fig. 1). The primary goal of each method is to control size, shape, morphology, and crystallinity of nanoparticles to produce a desirable ef‐ fect. This task represents a real challenge, and, as a result, there are no well-defined bounda‐ ries between physical, chemical, or biological methods. Moreover, in many cases, a combination of at least two methods (for example, physical and chemical methods, i.e. a physical-chemical approach) is required in order to fabricate good quality particles (small

The most widely used chemical methods for the synthesis of ferroelectric nanoparticles are: 1) solid-state reaction; 2) sol-gel technique; 3) solvothermal method; 4) hydrothermal meth‐

Table 2 shows selected examples of these methods applied to synthesis of ferroelectric nano‐ particles of BaTiO3. As can be seen from Table 2, the ultra-fine ferroelectric nanoparticles (<10 nm) in almost all cases except [16] are synthesized in a cubic phase which is not ferro‐ electric. The tetragonal phase (with ferroelectric response) is possible for relatively large par‐ ticles (~50-70 nm). This fact can be critical for certain types of applications, and will be

**nano-colloids**

sizes, narrow size distribution, ferroelectric phase).

**Figure 1.** Methods used to fabricate ferroelectric nanoparticles

**2.1. Chemical methods**

discussed later.

od; and 5) molten salt method.


**Table 1.** Applications of bulk ferroelectrics

### **2. Methods of ferroelectric nanoparticles production: Nano-powders and nano-colloids**

The existing methods that produce ferroelectric nanoparticles are numerous and can be clas‐ sified as physical, chemical, and biological (Fig. 1). The primary goal of each method is to control size, shape, morphology, and crystallinity of nanoparticles to produce a desirable ef‐ fect. This task represents a real challenge, and, as a result, there are no well-defined bounda‐ ries between physical, chemical, or biological methods. Moreover, in many cases, a combination of at least two methods (for example, physical and chemical methods, i.e. a physical-chemical approach) is required in order to fabricate good quality particles (small sizes, narrow size distribution, ferroelectric phase).

**Figure 1.** Methods used to fabricate ferroelectric nanoparticles

### **2.1. Chemical methods**

Recent advances in nanotechnologies, especially in nanoinstrumentation (for example, scanning probe microscopy [5]) and materials nanofabrication [6], allowed the direct prob‐ ing of ferroelectricity at the nanoscale. The new and unexplored world of nanoscale ferro‐ electrics (nanoparticles of different shapes and sizes, nanofilms, nanopatterned structures, etc.) raised fundamental questions and stimulated very active research in both academic and industrial sectors [7]. As a result, a new era of nanoscale ferroelectrics was launched. Novel effects, associated with reduced dimensions and found in nanoscaleferroelectrics, highlighted exciting possibilities for new applications reviewed recently in [8]. Almost all of the attention for the mentioned review [8] was devoted to the *thin film* nanoscale de‐ vice structures (which can be easily integrated with a Si chip) with focus on ultrafast switching, electrocaloric coolers for computers, phase-array radar, three-dimensional trenched capacitors for dynamic random access memories, room temperature magnetic field detectors, and miniature X-ray and neutron sources. So far, we have not found a co‐ herent review summarizing the actual and possible applications of ferroelectric *nanoparti‐ cles*. Our book chapter is an attempt to describe and analyze the state of the field of applications of ferroelectric nanoparticles with focus on materials technologies, medicine,

**Physical effect/property Applications**

thousands) Capacitors

Ferroelectric hysteresis Nonvolatile computer information storage

Direct piezoelectric effect Sensors (microphones, accelerometers, hydrophones, etc.)

Electro-optic effects Laser Q-switches, optical shutters and integrated optical (photonic)

Nonlinear optical effects Laser frequency doubling, optical mixing, including four-wave mixing and

devices

holographic information storage

Electric-motor overload-protection devices and self-stabilizing ceramic heating elements

Converse piezoelectric effect Actuators, ultrasonic generators, resonators, filters

Pyroelectric effect Uncooled infra-red detectors

Coupling between stress and birefringence Radar signal processing

and biology.

476 Advances in Ferroelectrics

High relative permittivities (several

Positive temperature coefficient of resistance (PTCR)

**Table 1.** Applications of bulk ferroelectrics

The most widely used chemical methods for the synthesis of ferroelectric nanoparticles are: 1) solid-state reaction; 2) sol-gel technique; 3) solvothermal method; 4) hydrothermal meth‐ od; and 5) molten salt method.

Table 2 shows selected examples of these methods applied to synthesis of ferroelectric nano‐ particles of BaTiO3. As can be seen from Table 2, the ultra-fine ferroelectric nanoparticles (<10 nm) in almost all cases except [16] are synthesized in a cubic phase which is not ferro‐ electric. The tetragonal phase (with ferroelectric response) is possible for relatively large par‐ ticles (~50-70 nm). This fact can be critical for certain types of applications, and will be discussed later.


**Shape of nanoparticles; Ref.**

Single crystalline nanoparticles; [14]

Nanoparticles; [15]

Nanopowders; [9]

Nanocrystals; [16]

**Table 2.** Chemical methods

**2.2. Physical methods**

**Synthesis method; Raw materials**

Solvothermal method; Ba(CH3COO)2 and Ti(OC4H9)4 ; the autoclave was maintained at 200 °C for 12 hr

Hydrothermal method; (Ba,Sr)(OiPr)2, Ti(OiPr), EtOH, 330–400 °C, 16-30 MPa

Combined wet-chemical and rapid calcination process; BaCO3–TiO2 precursors

One-step solvothermal route; Tetra-n-butyl titanate, Barium hydroxide octahydrate, solvent (Diethylene glycol), surfactant (Polyvinyl Pyrrolidone)

Dry and wet mechanical grindings are methods of choice for inexpensive nanoparticle prepa‐ ration. For technological applications, wet grinding is preferable because it allows more op‐ tions to control the size of the nanoparticles. During the last few years, substantial progress was made in the field of ferroelectric nanoparticle preparation by means of wet mechanical grinding [17]. Generally, three components are needed: raw material to grind (micron-sized powders of BaTiO3); surfactant (which covers the particle surface and prevents their aggrega‐ tion and overheating during grinding; oleic acid is a good choice for BaTiO3); and fluid carrier (both raw material and surfactant are mixed with fluid carrier; heptane is widely used to grind BaTiO3). An extensive list of references, as well as recent achievements in the field, are dis‐ cussed in the review [18]. Table 3 shows how particle sizes depend upon grinding time [19].

As is seen from Table 3, wet mechanical grinding can produce 9 nm nanoparticles, and such small particles are still in ferroelectric phase. It is important to remember that BaTiO3 nano‐ particles of the same sizes synthesized by the majority of chemical methods are not ferro‐

**Relevant parameters (size, morphology, crystallinity, spontaneous polarization etc.); Measurement and characterization methods**

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

5-20 nm, *cubic phase*; XRD, TEM, SAED, FTIR

15-50 nm,*cubic or tetragonal phase*; XRD, TEM

125 nm, *tetragonal phase*, remnant polarization Pr = 1.64 µC/cm2, the coercive field, Ec = 4.91 kV/cm; (cigar-like loop);surface area 7.96 m2/g XRD, TEM, TGA, SMPS, SAED,dielectric spectroscopy

5 nm, *tetragonal phase*; XRD, TEM, HRTEM, Raman spectroscopy

**Possible applications & comments**

479

http://dx.doi.org/10.5772/52516

Dense bulk nanocomposites

Dense bulk nanocomposites

Multilayer ceramic capacitor

Multilayer ceramic capacitor


**Table 2.** Chemical methods

**Shape of nanoparticles; Ref.**

478 Advances in Ferroelectrics

Nanotubes; [10]

Nanocrystals; [11]

Dense polycrystalline aggregates (~80 nm) of nanocrystals (~30 nm); [12]

Polyhedral with hexagonal outline in shape; [13]

**Synthesis method; Raw materials**

Wet chemical route at low temperature (50°C); H2TiO3 nanotubes; ethanol/water mixture with 25% ethanol by volume; Ba(OH)2·8H2O

Sol-gel technique; Barium titanium ethyl hexanoisopropoxide, BaTi(O2CC7H15) (OCH(CH3)2)5 ; a mixture of diphenyl ether, (C5H5)2O; stabilizing agent oleic acid, CH3(CH2)7CHdCH(CH2)7- CO2H, at 140 °C, under argon or nitrogen

Solid-state reaction as a function of temperature (400–8000C), time (1–24 hr); Nanocrystalline TiO2, ultrafine BaCO3 and submicrometer BaCO3 were intensively mixed in an aqueous suspension for 24 hr using polyethylene jars and zirconia media. The polymer (ammonium polyacrylate) was required for the formation of a monolayer on the particle surface

Molten salt method; Hydroxide octahydrate (Ba(OH)2·8H2O), titanium dioxide (TiO2), and the eutectic salts (NaCl–KCl), 600-900 °C

**Relevant parameters (size, morphology, crystallinity, spontaneous polarization etc.); Measurement and characterization methods**

Uniform BaTiO3 nanotubes, a *cubic phase*, average diameter = 10 nm and wall thickness = 3 nm at room temperature; Powder X-ray diffraction (XRD), field-emission scanning electron microscopy (FE-SEM), transmission electron microscopy (TEM), Raman spectroscopy, and X-ray photoelectron spectroscopy

Monodisperse nanoparticles with diameters ranging from 6 to 12 nm, *cubic phase;* XRD,TEM

Nanoparticle size is 70 nm. Specific surface area up to ~15m2/g, *tetragonal phase*; TG and DTA analysis, XRD, SEM

50 nm, *cubic phase*; XRD, Fourier transform infrared spectrometry, UV–Vis diffuse reflectance spectra, and field emission SEM

**Possible applications & comments**

> Promising microwaveabsorbing materials

Multilayer ceramic capacitors

Multilayer ceramic capacitors

Nanoparticles are well dispersed

### **2.2. Physical methods**

Dry and wet mechanical grindings are methods of choice for inexpensive nanoparticle prepa‐ ration. For technological applications, wet grinding is preferable because it allows more op‐ tions to control the size of the nanoparticles. During the last few years, substantial progress was made in the field of ferroelectric nanoparticle preparation by means of wet mechanical grinding [17]. Generally, three components are needed: raw material to grind (micron-sized powders of BaTiO3); surfactant (which covers the particle surface and prevents their aggrega‐ tion and overheating during grinding; oleic acid is a good choice for BaTiO3); and fluid carrier (both raw material and surfactant are mixed with fluid carrier; heptane is widely used to grind BaTiO3). An extensive list of references, as well as recent achievements in the field, are dis‐ cussed in the review [18]. Table 3 shows how particle sizes depend upon grinding time [19].

As is seen from Table 3, wet mechanical grinding can produce 9 nm nanoparticles, and such small particles are still in ferroelectric phase. It is important to remember that BaTiO3 nano‐ particles of the same sizes synthesized by the majority of chemical methods are not ferro‐ electric – the method used to fabricate nanoparticles really does matter. The ability to produce ferroelectric nanoparticles of very small sizes is a defining characteristic of this physical method (see [18] for more references).

As can be seen from Table 4, chemical reactions under the presence of an external driver (microwave, ultrasonic, milling, heat, pressure) are able to produce very fine particles (~5-10

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

**Relevant parameters (size, morphology, crystallinity, spontaneous polarization etc.); Measurement and characterization methods**

Cross section 70 ± 9 nm in [100] projection; *tetragonal structure*; remnant polarization is 15.5 µC/cm2; saturation polarization is 19.3 *µ*C/cm2; XRD, high resolution TEM, impedance spectroscopy

Sizes range from 50 to 75nm; transition of BaTiO3 from *cubic* to a *pseudotetragonal phase* with the increase ofthe reaction temperature from 80 to 220 °C; remnant polarization Pr = 2.2 C/cm2; the coercive field, Ec = 3.2 kV/cm (cigar-like loop); XRD, TEM, Raman and dielectric spectroscopy; energy-dispersive X-ray spectroscopy; AFM

> 7 nm, *tetragonal structure*; XRD, TEM, Raman spectroscopy

100 nm aggregates, *cubic phase*; XRD, TEM, DLS, SAED

**Possible applications & comments**

http://dx.doi.org/10.5772/52516

481

Charge storage devices

Oxygen atmosphere decreases sizes of the nanoparticles (for example , at 200 °C sizes decrease from 72.54 nm to 49.54 nm)

Chemical reaction during milling

Multilayer ceramic capacitors

nm) with tetragonal structures.

**Synthesis method; Raw materials**

Microwave (2.45 GHz, output power ~ 800W) hydrothermal; High-purity reagents barium nitrate (Sigma Aldrich, purity ≥99.99%), titanium isoproxide (Sigma Aldrich, purity ≥97%), nitric acid (ACS reagent 70%), ammonium hydroxide (Sigma Aldrich), and glycine (Sigma-Aldrich, purity ≥99%)

Sol–gel-hydrothermal method under an oxygen (partial pressure is ~60 bar); TiCl4; HCl; BaCl2·2H2O; deionized water, stirring and N2 bubbling, NaOH

Direct synthesis from solution (DSS) – mechanochemical synthesis; Anhydrous Ba(OH)2 and tetrabutyl titanate [Ti (OC4H9)4] Mechanical milling at the rate of 200 rpm, room temperature

> Sonochemical synthesis; BaCl2 ; TiCl4; NaOH

**Shape of nanoparticles; Ref.**

Truncated nanocubes; [22]

> Nearly spherical nanoparticles; [23]

Nanocrystals; [24, 25]

Large aggregates of 5–10 nm small nanocrystals; [26]


**Table 3.** BaTiO3 particle sizes as a function of grinding time (liquid carrier – heptane; two-station PM200 planetary ball mill from Retsch)

The net dipole moment of ferroelectric nanoparticles allows them to be harvested by using an inhomogeneous electric field. The harvesting concept proposed in [20] is based on the fact that dipoles experience a translational force only when exposed to a field gradient. For a given linear field gradient and assuming a single ferroelectric domain, the net translational force on a dipole scales proportionally with the particle size. The Brownian motion effects also become progressively more pronounced at smaller particle sizes and so the required field strength for successful separation scales nonlinearly as the particle size is reduced.

Both gas-phase and liquid-phase harvesting methods were proposed and tested successfully, and single ferroelectric monodomain nanoparticles as small as 9 nm from mechanically ground nanoparticles were selectively harvested [20]. In contrast to many reports on the lack of ferroelectricity for nanoparticles below 10 nm (see Table 2), the harvested nanoparticles do maintain ferroelectricity. The ferroelectric response of such tiny nanoparticles was attributed to the existence of an induced surface strain as a result of the grinding process [20]. The lack of a mechanically induced strain in similarly sized but chemically produced nanoparticles ac‐ counts for the absence of ferroelectricity in these materials. The concept of stress-induced fer‐ roelectricity was verified in further experiments, which are reviewed in [18]. It was found [21] that the spontaneous polarization of the nanoparticles is four to five times larger than the spontaneous polarization of the bulk raw materials. To obtain this result the following two conditions must be satisfied: (1) a nonpolar solvent for the nanoparticle suspension and (2) nonaggregated nanoparticles. Under these conditions, for 9 nm nanoparticles, the values of 100–120 μC/cm2 and 8.9 10-23C cm have been measured for the spontaneous polarization and dipole moment, respectively. The aggregation of ferroelectric nanoparticles masks the ferro‐ electric response due to the partial compensation of the dipole moments of the individual par‐ ticles. Finally, we can conclude that recent advances in the production of uniform, monodomain, highly ferroelectric nanoparticles indicate that this field has reached its maturi‐ ty. The particles can now be reliably prepared to certain specifications and characteristics [18].

### **2.3. Physical-chemical methods**

In order to prepare ferroelectric nanoparticles with controllable sizes and shapes a combina‐ tion of chemical methods with external physical factors (for example, a chemical reaction in the presence of electromagnetic fields or mechanical milling) is needed. Table 4 shows sever‐ al examples of these physical-chemical methods for the case of BaTiO3.

As can be seen from Table 4, chemical reactions under the presence of an external driver (microwave, ultrasonic, milling, heat, pressure) are able to produce very fine particles (~5-10 nm) with tetragonal structures.

electric – the method used to fabricate nanoparticles really does matter. The ability to produce ferroelectric nanoparticles of very small sizes is a defining characteristic of this

**Grinding time, hrs** 3 7 10 16 20 24

**Average particle diameter, nm** 26 14.9 11.6 9.5 9.2 9

**Table 3.** BaTiO3 particle sizes as a function of grinding time (liquid carrier – heptane; two-station PM200 planetary ball

The net dipole moment of ferroelectric nanoparticles allows them to be harvested by using an inhomogeneous electric field. The harvesting concept proposed in [20] is based on the fact that dipoles experience a translational force only when exposed to a field gradient. For a given linear field gradient and assuming a single ferroelectric domain, the net translational force on a dipole scales proportionally with the particle size. The Brownian motion effects also become progressively more pronounced at smaller particle sizes and so the required field strength for successful separation scales nonlinearly as the particle size is reduced.

Both gas-phase and liquid-phase harvesting methods were proposed and tested successfully, and single ferroelectric monodomain nanoparticles as small as 9 nm from mechanically ground nanoparticles were selectively harvested [20]. In contrast to many reports on the lack of ferroelectricity for nanoparticles below 10 nm (see Table 2), the harvested nanoparticles do maintain ferroelectricity. The ferroelectric response of such tiny nanoparticles was attributed to the existence of an induced surface strain as a result of the grinding process [20]. The lack of a mechanically induced strain in similarly sized but chemically produced nanoparticles ac‐ counts for the absence of ferroelectricity in these materials. The concept of stress-induced fer‐ roelectricity was verified in further experiments, which are reviewed in [18]. It was found [21] that the spontaneous polarization of the nanoparticles is four to five times larger than the spontaneous polarization of the bulk raw materials. To obtain this result the following two conditions must be satisfied: (1) a nonpolar solvent for the nanoparticle suspension and (2) nonaggregated nanoparticles. Under these conditions, for 9 nm nanoparticles, the values of

dipole moment, respectively. The aggregation of ferroelectric nanoparticles masks the ferro‐ electric response due to the partial compensation of the dipole moments of the individual par‐ ticles. Finally, we can conclude that recent advances in the production of uniform, monodomain, highly ferroelectric nanoparticles indicate that this field has reached its maturi‐ ty. The particles can now be reliably prepared to certain specifications and characteristics [18].

In order to prepare ferroelectric nanoparticles with controllable sizes and shapes a combina‐ tion of chemical methods with external physical factors (for example, a chemical reaction in the presence of electromagnetic fields or mechanical milling) is needed. Table 4 shows sever‐

al examples of these physical-chemical methods for the case of BaTiO3.

and 8.9 10-23C cm have been measured for the spontaneous polarization and

physical method (see [18] for more references).

mill from Retsch)

480 Advances in Ferroelectrics

100–120 μC/cm2

**2.3. Physical-chemical methods**



**Shape of nanoparticles; Ref.**

Aggregates (~500 nm) of nanocrystals(~ 50 nm); [37]

> Quasi-spherical nanoparticles; [38]

Spherical nanoparticles; [39]

Spherical nanoparticles; [40]

Spherical nanoparticles; [41]

Spherical nanoparticles; [42]

**Table 5.** Biological methods

**Synthesis method; Raw materials**

The peptides BT1 and BT2induce the room-temperature *precipitation* of BaTiO3; Aqueous precursor solution composed of barium acetate (Ba(OOCCH3)2) potassium bis(oxalato) oxotitanate(IV) (K2[TiO(C2O4)2]2H2O); pH 6.8

*Lactobacillus* - assisted biosynthesis; BaCO3 and TiO2 solid state synthesis BaTiO3; Slush of micron-sized BaTiO3 particles+ Lactic acid Bacillus spore tablets

*(Fusarium oxysporum*) fungus-assisted biosynthesis of BaTiO3 nanoparticles; (CH3COO)2Ba and K2TiF6 + fungal micelles of *Fusarium oxysporum*

Peptide nanorings used as templates; BaTi(O2CC7-H15)[OCH(CH3)2]5

(*Saccharomyces cerevisiae*) baker's yeast-assisted biosynthesis of BaTiO3 nanoparticles; BaCO3 and TiO2 solid state synthesis BaTiO3, Slurry of large BaTiO3 + yeast culture

Kinetically controlled vapor diffusion Single source, bimetallic alkoxide with the vapor diffusion of a hydrolytic catalyst (H2O)

**Relevant parameters (size, morphology, crystallinity, spontaneous polarization etc.); Measurement and characterization methods**

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

50-100 nm; *tetragonal phase* remnant polarization Pr ~ 2-3 C/cm2; the coercive field, Ec = 5-6 kV/cm (cigar-like loop); SEM, TEM, SAED and XRD analysis, dielectric measurements

> 20-80 nm, *single-phase tetragonal structure*; XRD, TEM

4-5 nm, *tetragonal structure*; XRD, DCS, TEM, SAED, XPS, SPM

6-12 nm, *tetragonal structure*;

8-21 nm, s*ingle-phase hexagonal structure*; XRD, TEM

6-8 nm, *cubic phase*;

XRD, TEM ~250 g.

AFM, TEM, XRD, EFM, Raman 4 days

**Comments**

483

http://dx.doi.org/10.5772/52516

Rapid roomtemperature synthesis of ferroelectric (tetragonal) BaTiO3 within 2 hr

> Extracellular synthesis

> Extracellular synthesis

> Extracellular synthesis

**Table 4.** Physical-chemical methods

Another interesting subject is the possible application of ceramic nanofibers produced by the method of electrospinning. These possible applications include: nanofiber-based sup‐ ports for catalysts, nanofiber-based photocatalysts, nanofiber-based membrane for filtration, nanofiber-based sensors, nanofiber-based photoelectrodes for photovoltaic cells, nanofiberbased electrodes for lithium batteries, nanofiber-based electrode-supports for fuel cells, and materials for implants [28, 29, 30].

### **2.4. Biological methods**

The proper combination of chemical and physical methods (Table 4) allows the produc‐ tion of very small ferroelectric nanoparticles (5-10 nm). However, toxic chemicals and/or high temperatures/pressures are needed in most cases. These requirements substantially limit the possible biomedical applications of ferroelectric nanoparticles. Biological meth‐ ods were proposed as eco-friendly "green" alternatives to existing chemical and physical methods. The biosynthesis of different types of nanoparticles was reviewed recently in a few papers [32, 33, 34, 35], but biological methods that produce specifically ferroelectric nanomaterials are not very numerous. For example, the first review [32] lists nearly a hundred variations for the synthesis of nanoparticles by microorganisms, and only a few of them produce ferroelectric nanoparticles. The biological methods applied to synthesize the nanoparticles of the most widely known ferroelectric BaTiO3 are summarized in Table 5 (papers published before 2008 were discussed in [36]). The references listed in this table show that most of the biological methods mentioned employ some chemical or physical steps as well.


**Table 5.** Biological methods

**Shape of nanoparticles; Ref.**

482 Advances in Ferroelectrics

Single nanocrystals; [27]

> Nanofibers; [28, 29, 30]

Nanopowders; [31]

**Table 4.** Physical-chemical methods

materials for implants [28, 29, 30].

**2.4. Biological methods**

steps as well.

**Synthesis method; Raw materials**

RF-Plasma chemical vapor deposition (CVD); iso-propoxide (Ti(OiPr)4, bisdipivaloylmethanate (Ba(DPM)2

Electrospinning, sol-gel; barium acetate, PVP, titanium isopropoxide, ethanol, acetic acid

Flame spray pyrolysis; barium carbonate and titanium tetra-*iso*-propoxide, citric acid

**Relevant parameters (size, morphology, crystallinity, spontaneous polarization etc.); Measurement and characterization methods**

> 12.2-15.4 nm; XRD, TEM, SAED, EDX

80-190 nm in diameter, 0.1 mm in length; XRD, TEM, SAED

70 nm, *tetragonal structure*; XRD, TEM, SEM

Another interesting subject is the possible application of ceramic nanofibers produced by the method of electrospinning. These possible applications include: nanofiber-based sup‐ ports for catalysts, nanofiber-based photocatalysts, nanofiber-based membrane for filtration, nanofiber-based sensors, nanofiber-based photoelectrodes for photovoltaic cells, nanofiberbased electrodes for lithium batteries, nanofiber-based electrode-supports for fuel cells, and

The proper combination of chemical and physical methods (Table 4) allows the produc‐ tion of very small ferroelectric nanoparticles (5-10 nm). However, toxic chemicals and/or high temperatures/pressures are needed in most cases. These requirements substantially limit the possible biomedical applications of ferroelectric nanoparticles. Biological meth‐ ods were proposed as eco-friendly "green" alternatives to existing chemical and physical methods. The biosynthesis of different types of nanoparticles was reviewed recently in a few papers [32, 33, 34, 35], but biological methods that produce specifically ferroelectric nanomaterials are not very numerous. For example, the first review [32] lists nearly a hundred variations for the synthesis of nanoparticles by microorganisms, and only a few of them produce ferroelectric nanoparticles. The biological methods applied to synthesize the nanoparticles of the most widely known ferroelectric BaTiO3 are summarized in Table 5 (papers published before 2008 were discussed in [36]). The references listed in this table show that most of the biological methods mentioned employ some chemical or physical

**Possible applications & comments**

Multilayer ceramic capacitors

Piezo, microwave material, sensors, capacitors

Multilayer ceramic capacitors

One can note from Table 5 that these biological methods can be roughly grouped into two categories: the synthesis of ferroelectric nanoparticles using a number of seed chemicals and some type of biological or bio-inspired system, and the creation of nanoparticles using large particles of ready ferroelectric material and some type of biological system. Most of the methods that are currently available fall into the first category; the methods in the second category have appeared only very recently.

**3. Applications of ferroelectric nanoparticles in materials technologies**

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

http://dx.doi.org/10.5772/52516

485

As was previously mentioned in the introduction, the era of applied ferroelectricity was launched after the first reliable ferroelectric material – ceramic BaTiO3 – was discovered [1]. The very high dielectric constants (~1000) of bulk ferroelectric materials were applied in the manufacturing of discrete and multilayered ceramic capacitors. Advances in the production of ferroelectric nanoparticles broadened this application to a large extent. Currently, nanosized BaTiO3 powders are successfully used for manufacturing miniatur‐ ized and highly volume-efficient multilayer ceramic capacitors (MLCCs, for more details

Typical MLCCs utilize pellets of nanopowders sintered at high temperatures, produc‐ ing structures with high rigidity that can impose certain limitations. Nanocomposites made of flexible polymers and ferroelectric nanoparticles overcome the restrictions caused by rigidity of sintered structures. Polymeric materials doped with ferroelectric nanoparticles are of considerable interest as a solution for processable high permittivity materials for various electronic applications: volume efficient multilayer capacitors, highenergy-density capacitors, embedded capacitors, and gate insulators in organic field ef‐

Both components (polymer and ferroelectric nanoparticle) of such nanocomposites are es‐ sential. The polymeric material brings flexibility and the ability to cover a large surface area, while nanoparticles share high values of dielectric permittivity with the matrix-enhancing effective dielectric permittivity of the composite. It's not easy to achieve homogeneous dis‐ persion of ferroelectric nanoparticles in a polymeric matrix because of the high surface ener‐ gy of nanoparticles, which usually leads to aggregation and phase separation, resulting in poor processability of the films and a high defect density. In order to make high-quality nanocomposites (good solution processability, low leakage current, high permittivity, and high dielectric strength), surface modification of nanoparticles is needed. The surface modi‐ fication of nanoparticles decreases attractive forces between them, thus preventing aggrega‐ tion. In addition, the proper design of a surface agent can increase interactions between the nanoparticles and the polymeric matrix. As a result, a quite homogeneous distribution of the nanoparticles in the polymeric matrix can be achieved. However, there are several obstacles that prevent nanocomposites "polymer/ferroelectric nanoparticle" from mass commerciali‐ zation. The most common are poor temperature stability of dielectric constants and large di‐ electric losses. The proper design of both the nanoparticles (material size, shape, surface modification) and the polymer matrix can overcome these problems. For example, Table 6 shows short descriptions of the relatively successful (in terms of material performance) nanocomposites "polymer/BaTiO3". All of them are optically transparent, have high dielec‐ tric permittivity (25-30), and moderate losses (loss tangent is in the range 0.01-0.05 for differ‐

**3.1. Multilayered capacitors and nanocomposites**

see review [45]).

fect transistors [46].

ent materials).

The earliest discoveries of nanoparticle synthesis by microorganisms (fungus, *Lactobacil‐ lus* and yeast) involve metal, alloy, and metal oxide nanoparticles. With a lag of one to two years, this research was followed by attempts at ferroelectric nanoparticle synthesis by the same living systems [32]. Fungus (*Fusarium oxysporum*, commonly found in soil) has been shown to synthesize extracellularly ferroelectric nanoparticles of barium tita‐ nate (4-5 nm of average size) [39, 43] at room temperature, producing small ferroelectric nanoparticles on a scale that has been previously inaccessible. This method falls into the first category, since fungus produces barium titanate from more than one seed chemical. *Lactobacillus* (bacteria commonly used to curdle milk and produce buttermilk) has been shown to synthesize ferroelectric nanoparticles (with sizes ranging from 20 to 80 nm) from the slush of large barium titanate particles [38]. Baker's yeast (*Saccharomyces cerevi‐ siae*) has also been attempted as a biosynthesis agent; the barium titanate nanoparticles received were on average ~10nm [41]. Both the yeast- and lactobacillus-mediated produc‐ tion of barium titanate nanoparticles fall into the second category: they start with large particles of ready barium titanate in water and produce small (under 100 nm) nanoparti‐ cles extracellularly.

Peptide nanorings provide another interesting biomimetic route to the template-mediated synthesis of BaTiO3 and SrTiO3 nanoparticles [40]. Developed in 2006, it was the very first method of obtaining ferroelectric barium titanate nanoparticles at room temperature (many previous room-temperature methods had trouble growing barium titanate not in the cubic phase).

In this category of biological methods, one has to mention bioinspired methods of dis‐ persing nanoparticles and producing stable colloids. Take for examplethe production of stable nanoparticle suspensions using microbial-derived surfactants [44] where usual gar‐ den variety surfactants are replaced with bio-derived materials that are significantly smaller in size and that help prevent aggregation of multiple nanoparticles during solgel synthesis. This process relies on some physical/chemical steps, just like the majority of other biological methods of ferroelectric nanoparticle production that have been devel‐ oped to date.

These bio-inspired methods show great promise because they produce relatively small fer‐ roelectric nanoparticles at room (or low) temperature compared to conventional physical and chemical methods, many of which require lengthy processes, use of high temperatures, harsh chemicals, etc.

### **3. Applications of ferroelectric nanoparticles in materials technologies**

### **3.1. Multilayered capacitors and nanocomposites**

One can note from Table 5 that these biological methods can be roughly grouped into two categories: the synthesis of ferroelectric nanoparticles using a number of seed chemicals and some type of biological or bio-inspired system, and the creation of nanoparticles using large particles of ready ferroelectric material and some type of biological system. Most of the methods that are currently available fall into the first category; the methods in the second

The earliest discoveries of nanoparticle synthesis by microorganisms (fungus, *Lactobacil‐ lus* and yeast) involve metal, alloy, and metal oxide nanoparticles. With a lag of one to two years, this research was followed by attempts at ferroelectric nanoparticle synthesis by the same living systems [32]. Fungus (*Fusarium oxysporum*, commonly found in soil) has been shown to synthesize extracellularly ferroelectric nanoparticles of barium tita‐ nate (4-5 nm of average size) [39, 43] at room temperature, producing small ferroelectric nanoparticles on a scale that has been previously inaccessible. This method falls into the first category, since fungus produces barium titanate from more than one seed chemical. *Lactobacillus* (bacteria commonly used to curdle milk and produce buttermilk) has been shown to synthesize ferroelectric nanoparticles (with sizes ranging from 20 to 80 nm) from the slush of large barium titanate particles [38]. Baker's yeast (*Saccharomyces cerevi‐ siae*) has also been attempted as a biosynthesis agent; the barium titanate nanoparticles received were on average ~10nm [41]. Both the yeast- and lactobacillus-mediated produc‐ tion of barium titanate nanoparticles fall into the second category: they start with large particles of ready barium titanate in water and produce small (under 100 nm) nanoparti‐

Peptide nanorings provide another interesting biomimetic route to the template-mediated synthesis of BaTiO3 and SrTiO3 nanoparticles [40]. Developed in 2006, it was the very first method of obtaining ferroelectric barium titanate nanoparticles at room temperature (many previous room-temperature methods had trouble growing barium titanate not in the cubic

In this category of biological methods, one has to mention bioinspired methods of dis‐ persing nanoparticles and producing stable colloids. Take for examplethe production of stable nanoparticle suspensions using microbial-derived surfactants [44] where usual gar‐ den variety surfactants are replaced with bio-derived materials that are significantly smaller in size and that help prevent aggregation of multiple nanoparticles during solgel synthesis. This process relies on some physical/chemical steps, just like the majority of other biological methods of ferroelectric nanoparticle production that have been devel‐

These bio-inspired methods show great promise because they produce relatively small fer‐ roelectric nanoparticles at room (or low) temperature compared to conventional physical and chemical methods, many of which require lengthy processes, use of high temperatures,

category have appeared only very recently.

cles extracellularly.

484 Advances in Ferroelectrics

phase).

oped to date.

harsh chemicals, etc.

As was previously mentioned in the introduction, the era of applied ferroelectricity was launched after the first reliable ferroelectric material – ceramic BaTiO3 – was discovered [1]. The very high dielectric constants (~1000) of bulk ferroelectric materials were applied in the manufacturing of discrete and multilayered ceramic capacitors. Advances in the production of ferroelectric nanoparticles broadened this application to a large extent. Currently, nanosized BaTiO3 powders are successfully used for manufacturing miniatur‐ ized and highly volume-efficient multilayer ceramic capacitors (MLCCs, for more details see review [45]).

Typical MLCCs utilize pellets of nanopowders sintered at high temperatures, produc‐ ing structures with high rigidity that can impose certain limitations. Nanocomposites made of flexible polymers and ferroelectric nanoparticles overcome the restrictions caused by rigidity of sintered structures. Polymeric materials doped with ferroelectric nanoparticles are of considerable interest as a solution for processable high permittivity materials for various electronic applications: volume efficient multilayer capacitors, highenergy-density capacitors, embedded capacitors, and gate insulators in organic field ef‐ fect transistors [46].

Both components (polymer and ferroelectric nanoparticle) of such nanocomposites are es‐ sential. The polymeric material brings flexibility and the ability to cover a large surface area, while nanoparticles share high values of dielectric permittivity with the matrix-enhancing effective dielectric permittivity of the composite. It's not easy to achieve homogeneous dis‐ persion of ferroelectric nanoparticles in a polymeric matrix because of the high surface ener‐ gy of nanoparticles, which usually leads to aggregation and phase separation, resulting in poor processability of the films and a high defect density. In order to make high-quality nanocomposites (good solution processability, low leakage current, high permittivity, and high dielectric strength), surface modification of nanoparticles is needed. The surface modi‐ fication of nanoparticles decreases attractive forces between them, thus preventing aggrega‐ tion. In addition, the proper design of a surface agent can increase interactions between the nanoparticles and the polymeric matrix. As a result, a quite homogeneous distribution of the nanoparticles in the polymeric matrix can be achieved. However, there are several obstacles that prevent nanocomposites "polymer/ferroelectric nanoparticle" from mass commerciali‐ zation. The most common are poor temperature stability of dielectric constants and large di‐ electric losses. The proper design of both the nanoparticles (material size, shape, surface modification) and the polymer matrix can overcome these problems. For example, Table 6 shows short descriptions of the relatively successful (in terms of material performance) nanocomposites "polymer/BaTiO3". All of them are optically transparent, have high dielec‐ tric permittivity (25-30), and moderate losses (loss tangent is in the range 0.01-0.05 for differ‐ ent materials).


The concept of ferroelectric colloids in liquid crystals was created by Yu. Reznikov and co‐ authors [53] – liquid crystal material was doped with stabilized ferroelectric nanoparticles of low concentration (~0.3 %). Ferroelectric nanoparticles share their intrinsic properties with the liquid crystals matrix due to the alignment within the liquid crystal and interactions be‐ tween mesogenic molecules. The low concentration of nanoparticles and their stabilization with surfactant (oleic acid) provided the stability of such systems. This classic paper report‐ ed on the main features of such colloids: (1) a lower threshold voltage by a factor of 1.7; (2) an enhanced dielectric anisotropy by a factor of 2; (3) a linear electro-optical response (the sensitivity to the sign of an applied electric field). It should be pointed out that pure nemat‐ ics are not sensitive to the sign of an electric field – this property is intrinsic to ferroelectric liquid crystals rather than to nematics. The first experimental results stimulated a very ac‐ tive global interest in the field. More than 100 papers were published during the last decade, and, in the last few years, several review papers summarized the most important results [18, 54, 55]. The review paper [54] published this year by the founder of this field is of special interest since it comprehensively discusses the past, the present, and the future of the liquid

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

http://dx.doi.org/10.5772/52516

487

Ferroelectric nanoparticles embedded into a liquid crystal host strongly interact with the surrounding mesogenic molecules, due to the strong permanent electric field of this particle. These interactions can affect the basic physical parameters of liquid crystals: birefringence, dielectric permittivity, elastic constants, viscosity, electrical and thermal conductivity, tem‐ peratures of phase transitions, etc. There are two basic mechanisms responsible for the ob‐ served effects: (1) the increase of the orientation coupling between mesogenic molecules; and (2) the direct contribution of the permanent polarization of the particle [54]. Experimen‐ tal data suggests that the first scenario (the increase of the orientation coupling between mesogenic molecules) is more likely to happen in the case of single component liquid crys‐ tals [54]. The second mechanism (associated with the direct contribution of the permanent polarization of the ferroelectric nanoparticle to the physical properties of liquid crystals) is

So far most of the reported experimental data describes the properties of multi-component nematics: liquid crystals doped with ferroelectric nanoparticles. The electric field generated by ferroelectric nanoparticles can cause micro- and/or nano-separation of such mixturesan‐ daffect the macroscopic properties of the liquid crystal colloid. As a result, in some cases, the data found for single component liquid crystals (5CB) is different from the data obtained for multi-component mixtures. Nevertheless, an analysis of the existing literature [18] shows that most of the published experimental data report that ferroelectric nanoparticles embed‐ ded in liquid crystals at low concentrations: (1) enhance dielectric permittivity, dielectric anisotropy, and optical birefringence; (2) lower the switching voltage Uth for the Freeder‐ icksz transition; (3) increase the orientational order parameter *S* and the isotropic–nematic transition temperature *TNI*; and (4) reduce the switching times needed to reorient liquid crys‐ tals by an external electric field. It should be pointed out that in all of these cases, wet grind‐ ing was used to prepare the ferroelectric nanoparticles. Few papers claim results opposite to the ones (1)-(4) listed above; however, nanoparticles used to make a liquid crystal colloid in

the primary factor in the case of multi-component liquid crystal mixtures [54].

crystal colloids.

**Table 6.** Nanocomposites of polymer/ferroelectric nanoparticles

### **3.2. Ferroelectric nanoparticles and liquid crystals: Display and non-display applications**

It is an established fact that modern technologies require novel and highly advanced materi‐ als. During the last decade, nanochemistry has developed a dozen conceptually different pathways of synthesis to satisfy the constantly growing demand for new materials [52]. However, chemical methods are elaborate, time consuming, and expensive. Although they are generally accepted as universal, nanochemistry methods do not work perfectly in certain cases. For example, it's very difficult (but not impossible) to produce 5-10 nm ferroelectric nanoparticles with tetragonal structures through chemical methods (see Table 2). In many cases, the chemical methods used to produce novel materials can be efficiently supplement‐ ed or even replaced with non-synthetic ones. The addition of ferroelectric nanoparticles to different materials (polymers; see Table 6, liquids, liquid crystals etc.) is a good example of a non-synthetic method used to develop materials with improved properties. This subsection will discuss how ferroelectric nanoparticles can modify properties of liquid crystals – mate‐ rials that found numerous applications in the display industry, photonics, optical process‐ ing, and the biotech industry.

The concept of ferroelectric colloids in liquid crystals was created by Yu. Reznikov and co‐ authors [53] – liquid crystal material was doped with stabilized ferroelectric nanoparticles of low concentration (~0.3 %). Ferroelectric nanoparticles share their intrinsic properties with the liquid crystals matrix due to the alignment within the liquid crystal and interactions be‐ tween mesogenic molecules. The low concentration of nanoparticles and their stabilization with surfactant (oleic acid) provided the stability of such systems. This classic paper report‐ ed on the main features of such colloids: (1) a lower threshold voltage by a factor of 1.7; (2) an enhanced dielectric anisotropy by a factor of 2; (3) a linear electro-optical response (the sensitivity to the sign of an applied electric field). It should be pointed out that pure nemat‐ ics are not sensitive to the sign of an electric field – this property is intrinsic to ferroelectric liquid crystals rather than to nematics. The first experimental results stimulated a very ac‐ tive global interest in the field. More than 100 papers were published during the last decade, and, in the last few years, several review papers summarized the most important results [18, 54, 55]. The review paper [54] published this year by the founder of this field is of special interest since it comprehensively discusses the past, the present, and the future of the liquid crystal colloids.

**Composition (polymer/ nanoparticle)**

486 Advances in Ferroelectrics

Barium titanate / poly(vinylidene fluoride-*co*hexafluoro propylene), volume concentration < 80%

Barium titanate/polyimide (BaTiO3/PI); volume concentration < 40%

Poly(vinylidene fluoride)/ Barium titanate (PVDF/ BaTiO3); volume concentration 10%- 30%

Barium titanate/polyimide (BaTiO3/PI); volume concentration < 50%

Barium titanate/poly(methyl methacrylate) (BaTiO3/PMMA); volume concentration 53%

Barium titanate/polyimide (BaTiO3/PI); volume concentration 59%

ing, and the biotech industry.

**Table 6.** Nanocomposites of polymer/ferroelectric nanoparticles

**Particle surface functionalization**

Phosphonic acid surface-modified BaTiO3 nanoparticles

Core-shell structure; polyamic acid (PAA) was used to cover particle

BaTiO3 nanoparticles were prepared by the alkoxide route

BaTiO3 nanoparticles were modified with a silane coupling agent (3 methacryloxypropyltrimethoxysilane)

BaTiO3 nanoparticles are surface modified by phthalimide with the aid of a silane coupling agent as a scaffold

**3.2. Ferroelectric nanoparticles and liquid crystals: Display and non-display applications**

It is an established fact that modern technologies require novel and highly advanced materi‐ als. During the last decade, nanochemistry has developed a dozen conceptually different pathways of synthesis to satisfy the constantly growing demand for new materials [52]. However, chemical methods are elaborate, time consuming, and expensive. Although they are generally accepted as universal, nanochemistry methods do not work perfectly in certain cases. For example, it's very difficult (but not impossible) to produce 5-10 nm ferroelectric nanoparticles with tetragonal structures through chemical methods (see Table 2). In many cases, the chemical methods used to produce novel materials can be efficiently supplement‐ ed or even replaced with non-synthetic ones. The addition of ferroelectric nanoparticles to different materials (polymers; see Table 6, liquids, liquid crystals etc.) is a good example of a non-synthetic method used to develop materials with improved properties. This subsection will discuss how ferroelectric nanoparticles can modify properties of liquid crystals – mate‐ rials that found numerous applications in the display industry, photonics, optical process‐

**Dielectric permitti-vity of nanocom-posite**

35

20

30

Surface hydroxylated BaTiO3 18-25 85-100 nm [48]

**Particle size &breakdown electric field**

30-50 nm >164 V/µm

70-100 nm 67 MV/m

30-50 nm 50-80 nm

36 16-21 nm [50]

37 20 nm [51]

**Ref.**

[46]

[47]

[49]

Ferroelectric nanoparticles embedded into a liquid crystal host strongly interact with the surrounding mesogenic molecules, due to the strong permanent electric field of this particle. These interactions can affect the basic physical parameters of liquid crystals: birefringence, dielectric permittivity, elastic constants, viscosity, electrical and thermal conductivity, tem‐ peratures of phase transitions, etc. There are two basic mechanisms responsible for the ob‐ served effects: (1) the increase of the orientation coupling between mesogenic molecules; and (2) the direct contribution of the permanent polarization of the particle [54]. Experimen‐ tal data suggests that the first scenario (the increase of the orientation coupling between mesogenic molecules) is more likely to happen in the case of single component liquid crys‐ tals [54]. The second mechanism (associated with the direct contribution of the permanent polarization of the ferroelectric nanoparticle to the physical properties of liquid crystals) is the primary factor in the case of multi-component liquid crystal mixtures [54].

So far most of the reported experimental data describes the properties of multi-component nematics: liquid crystals doped with ferroelectric nanoparticles. The electric field generated by ferroelectric nanoparticles can cause micro- and/or nano-separation of such mixturesan‐ daffect the macroscopic properties of the liquid crystal colloid. As a result, in some cases, the data found for single component liquid crystals (5CB) is different from the data obtained for multi-component mixtures. Nevertheless, an analysis of the existing literature [18] shows that most of the published experimental data report that ferroelectric nanoparticles embed‐ ded in liquid crystals at low concentrations: (1) enhance dielectric permittivity, dielectric anisotropy, and optical birefringence; (2) lower the switching voltage Uth for the Freeder‐ icksz transition; (3) increase the orientational order parameter *S* and the isotropic–nematic transition temperature *TNI*; and (4) reduce the switching times needed to reorient liquid crys‐ tals by an external electric field. It should be pointed out that in all of these cases, wet grind‐ ing was used to prepare the ferroelectric nanoparticles. Few papers claim results opposite to the ones (1)-(4) listed above; however, nanoparticles used to make a liquid crystal colloid in this case were not ferroelectric [18]. The important conclusion to take from these findings is that a strong ferroelectric response of ferroelectric nanoparticles is a key factor leading to all "positive" effects (1)-(4).

tion very frequently adds a thick surface layer, resulting in particles that are a few hun‐ dred nanometers in size. This poses a challenge which a few researchers are attempting to overcome through bio-inspired surfactants [44]. The surface functionalization of nanopar‐ ticles can be crucial for a desired effect in a biosystem – the response is often cell-specif‐ ic, and biocompatibilty has to be experimentally tested for each new combination of nanoparticle and cell [61]. For example, a recent study [62] indicates that larger (over 200 nm) barium titanate nanoparticles have been successfully used with and without surface

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

http://dx.doi.org/10.5772/52516

489

Current efforts in this area concentrate on carrying these findings through to the systems of cells and tissue in vivo [63, 64]. Such biocompatible nanoparticles can be used in a variety of ways: as imaging agents [58, 59, 60, 64, 65], biocompatible nanoprobes [62, 66, 67, 68], cell proliferation agents [65], etc. Ferroelectric nanoparticles are currently being widely explored for applications in the area of medical imaging, e.g. for enhancing contrast in second har‐ monic generating methods for imaging deep tissue in vitro and in vivo [64, 69] for such im‐

**5. Ferroelectric nanoparticles applications in medicine and medical**

As mentioned above, ferroelectric nanoparticles have become widely used in material sci‐ ence and engineering applications, but biological and medical applications of these fascinat‐ ing nanoparticles has only begun to be explored in the last decade. It has been recently discovered that they can be used for cell imaging or the detection of malignancies in lung cancer studies, or they can be functionalized to induce cell proliferation. Ferroelectric mate‐ rials have a non-centrosymmetric crystalline structure, and are thus capable of generating a second harmonic of light [67]. This distinctive feature of ferroelectrics is the basis for a grow‐ ing number of applications of ferroelectric nanoparticles as imaging/diagnostic agents and

Optical imaging and detection methods are the most widespread among biological and medical communities. For example, second harmonic generation imaging has been success‐ fully used for detection of *osteogenesis imperfecta* in biopsies of human skin [71] and lung can‐ cer [72]. To improve contrast, many of the imaging methods rely on imaging probes, such as fluorescent markers or quantum dots [73]. This is the case for the SHG imaging technique: the SHG signal from biological cell components is often weak, so novel nanoprobes have been introduced to enhance contrast. SHG probes are photo-stable, and do not bleach or blink unlike conventional fluorescent probes. By definition, second harmonic generating nanoprobes (such as ferroelectric nanoparticles) are capable of converting two photons of light into one photon of half the incident wavelength [74]. This second-harmonic light can be detected using methods of nonlinear optical microscopy. Nonlinear optical properties of fer‐ roelectric nanomaterials can be used for optical phase conjugation [75] and nonlinear micro‐ scopy [62, 76] – these properties have allowed them to spread to the area of medical sensors.

modification and they cause no toxic response in endothelial cell cultures.

portant applications as screening for genetic diseases or cancer[70].

**engineering**

nanoprobes in optical imaging [64].

The stability of liquid crystal colloids is the most challenging problem that prevents the tran‐ sition of liquid crystal colloids of ferroelectric nanoparticles from the academic sector to the industrial domain. During the last few years, the general focus of the research has shifted toward this problem, and the proper surface functionalization of nanoparticles is now being considered as one of the most important factors affecting the stability of liquid crystal col‐ loids. The shape and size of nanoparticles are also of the utmost importance – it was found that 10-20 nm ferroelectric nanoparticles affect the properties of a liquid crystal host much morethan the same but larger nanoparticles(~100 nm).

In summary, ferroelectric nanoparticles can modify the intrinsic properties of liquid crystal materials without time-consuming and expensive chemical synthesis. Experimental data on the enhancement of electro-optical, optical, and nonlinear-optical responses of such materi‐ als shows the strong potential of ferroelectric nanoparticles for improving the ''practical'' properties of liquid crystals, especially for those materials where the method of chemical synthesis has reached its limit. Such modified materials are very attractive and suitable for use in switchable lenses, displays, and beam steering, as well as other light-controlling devi‐ ces (spatial light modulators, tunable filters etc.) [18].

### **4. Ferroelectric nanoparticles in biology**

In the last decade, ferroelectric nanoparticle areas of application veered toward biology and medicine. The first challenge to address when introducing nanoparticles to biological and medical systems is how to make them stable in media that is bio-compatible, namely in aqueous solutions. If the ferroelectric nanoparticles are not coated with a stabilizing agent, the intrinsic properties of such nanoparticles very often lead to unwanted effects such as aggregation and precipitation, as well as the leaching of some ions (leading to change in particle properties) in aqueous solutions. There has been a tremendous amount of effort in the last twenty years to create many species of ferroelectric nanoparticles with different surface coatings [56and references within]. For example, such stabilizing agents as polyacrylic acid, polymethacrylic acid, polyaspartic acid, (aminomethyl) phosphonic acid and poly-l-lysine were tried on barium titanate nanoparticles. A large variety of pos‐ sible coatings yields different surface properties of said nanoparticles – different thick‐ ness and charge of the surface layer, different strength of stabilizing agent adhesion, different resulting particle size – all of it leading to very different interactions with biolog‐ ical objects such as cells and cellular components. Such water-soluble ferroelectric nano‐ particles can be further functionalized with fluorescent markers or antibodies [57], and they have been recently observed within a number of mammalian cells [58, 59, 60] in vi‐ tro. The majority of ferroelectric nanoparticles that have been successfully functionalized are in the middle of nano size range (~100 nm or more), and the surface functionaliza‐ tion very frequently adds a thick surface layer, resulting in particles that are a few hun‐ dred nanometers in size. This poses a challenge which a few researchers are attempting to overcome through bio-inspired surfactants [44]. The surface functionalization of nanopar‐ ticles can be crucial for a desired effect in a biosystem – the response is often cell-specif‐ ic, and biocompatibilty has to be experimentally tested for each new combination of nanoparticle and cell [61]. For example, a recent study [62] indicates that larger (over 200 nm) barium titanate nanoparticles have been successfully used with and without surface modification and they cause no toxic response in endothelial cell cultures.

this case were not ferroelectric [18]. The important conclusion to take from these findings is that a strong ferroelectric response of ferroelectric nanoparticles is a key factor leading to all

The stability of liquid crystal colloids is the most challenging problem that prevents the tran‐ sition of liquid crystal colloids of ferroelectric nanoparticles from the academic sector to the industrial domain. During the last few years, the general focus of the research has shifted toward this problem, and the proper surface functionalization of nanoparticles is now being considered as one of the most important factors affecting the stability of liquid crystal col‐ loids. The shape and size of nanoparticles are also of the utmost importance – it was found that 10-20 nm ferroelectric nanoparticles affect the properties of a liquid crystal host much

In summary, ferroelectric nanoparticles can modify the intrinsic properties of liquid crystal materials without time-consuming and expensive chemical synthesis. Experimental data on the enhancement of electro-optical, optical, and nonlinear-optical responses of such materi‐ als shows the strong potential of ferroelectric nanoparticles for improving the ''practical'' properties of liquid crystals, especially for those materials where the method of chemical synthesis has reached its limit. Such modified materials are very attractive and suitable for use in switchable lenses, displays, and beam steering, as well as other light-controlling devi‐

In the last decade, ferroelectric nanoparticle areas of application veered toward biology and medicine. The first challenge to address when introducing nanoparticles to biological and medical systems is how to make them stable in media that is bio-compatible, namely in aqueous solutions. If the ferroelectric nanoparticles are not coated with a stabilizing agent, the intrinsic properties of such nanoparticles very often lead to unwanted effects such as aggregation and precipitation, as well as the leaching of some ions (leading to change in particle properties) in aqueous solutions. There has been a tremendous amount of effort in the last twenty years to create many species of ferroelectric nanoparticles with different surface coatings [56and references within]. For example, such stabilizing agents as polyacrylic acid, polymethacrylic acid, polyaspartic acid, (aminomethyl) phosphonic acid and poly-l-lysine were tried on barium titanate nanoparticles. A large variety of pos‐ sible coatings yields different surface properties of said nanoparticles – different thick‐ ness and charge of the surface layer, different strength of stabilizing agent adhesion, different resulting particle size – all of it leading to very different interactions with biolog‐ ical objects such as cells and cellular components. Such water-soluble ferroelectric nano‐ particles can be further functionalized with fluorescent markers or antibodies [57], and they have been recently observed within a number of mammalian cells [58, 59, 60] in vi‐ tro. The majority of ferroelectric nanoparticles that have been successfully functionalized are in the middle of nano size range (~100 nm or more), and the surface functionaliza‐

"positive" effects (1)-(4).

488 Advances in Ferroelectrics

morethan the same but larger nanoparticles(~100 nm).

ces (spatial light modulators, tunable filters etc.) [18].

**4. Ferroelectric nanoparticles in biology**

Current efforts in this area concentrate on carrying these findings through to the systems of cells and tissue in vivo [63, 64]. Such biocompatible nanoparticles can be used in a variety of ways: as imaging agents [58, 59, 60, 64, 65], biocompatible nanoprobes [62, 66, 67, 68], cell proliferation agents [65], etc. Ferroelectric nanoparticles are currently being widely explored for applications in the area of medical imaging, e.g. for enhancing contrast in second har‐ monic generating methods for imaging deep tissue in vitro and in vivo [64, 69] for such im‐ portant applications as screening for genetic diseases or cancer[70].

### **5. Ferroelectric nanoparticles applications in medicine and medical engineering**

As mentioned above, ferroelectric nanoparticles have become widely used in material sci‐ ence and engineering applications, but biological and medical applications of these fascinat‐ ing nanoparticles has only begun to be explored in the last decade. It has been recently discovered that they can be used for cell imaging or the detection of malignancies in lung cancer studies, or they can be functionalized to induce cell proliferation. Ferroelectric mate‐ rials have a non-centrosymmetric crystalline structure, and are thus capable of generating a second harmonic of light [67]. This distinctive feature of ferroelectrics is the basis for a grow‐ ing number of applications of ferroelectric nanoparticles as imaging/diagnostic agents and nanoprobes in optical imaging [64].

Optical imaging and detection methods are the most widespread among biological and medical communities. For example, second harmonic generation imaging has been success‐ fully used for detection of *osteogenesis imperfecta* in biopsies of human skin [71] and lung can‐ cer [72]. To improve contrast, many of the imaging methods rely on imaging probes, such as fluorescent markers or quantum dots [73]. This is the case for the SHG imaging technique: the SHG signal from biological cell components is often weak, so novel nanoprobes have been introduced to enhance contrast. SHG probes are photo-stable, and do not bleach or blink unlike conventional fluorescent probes. By definition, second harmonic generating nanoprobes (such as ferroelectric nanoparticles) are capable of converting two photons of light into one photon of half the incident wavelength [74]. This second-harmonic light can be detected using methods of nonlinear optical microscopy. Nonlinear optical properties of fer‐ roelectric nanomaterials can be used for optical phase conjugation [75] and nonlinear micro‐ scopy [62, 76] – these properties have allowed them to spread to the area of medical sensors. This also gives the ferroelectric SHG particle an edge in cell and tissue imaging in vivo. For example, recent advances in this area include imaging a tail of a living mouse with the aid of barium titanate nanoparticles [69].

tric nanoparticles can extend the area of biomedical application even more, since these

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

http://dx.doi.org/10.5772/52516

491

The applied science of ferroelectric nanoparticles is in the beginning of its development. Many fundamental and applied problems still need to be solved before the potential appli‐ cations will be converted into actual ones. Nevertheless, despite the fact that the emerging problems in applying nanoparticles in material technologies and biomedicine are still nu‐ merous [80], we can conclude that a good balance between purely academic and purely ap‐ plied foci of research is the key toward mass commercialization of ferroelectric nanoparticles

The authors are very grateful to all coauthors of the papers related to the topic of ferroelec‐ tric nanoparticles. We acknowledge also the support of the UCCS Center for the Biofrontiers Institute, University of Colorado at Colorado Springs. A great portion of the work described in this review has been supported by the grants from the NSF #1102332 "Liquid Crystal Sig‐ nal Processing Devices for Microwave and Millimeter Wave Operation" and # 1010508 "STTR Phase 1: Design, Fabrication, and Characterization of Ferroelectric Nanoparticle Dop‐

nanoparticles can be used as both passive and active nanoprobes.

in all applied fields mentioned above.

ed Liquid Crystal / Polymer Composites."

**Acknowledgements**

**List of abbreviations**

AFM atomic force microscopy

DLS dynamic light scattering

DTA differential thermal analysis

EFM electrostatic force microscopy

MLCC multilayer ceramic capacitors

PMMA poly(methyl methacrylate)

PAA polyamic acid

DCS differential scanning calorimetry

EDX energy dispersive X-rayspectroscopy

FTIR Fourier transform infrared spectroscopy

FE-SEM field-emission scanning electron microscopy

PTCR positive temperature coefficient of resistance

HRTEM high resolution transmission electron microscopy

The intrinsic large values of the dielectric permittivities of ferroelectric nanoparticles suggest their use to enhance the dielectric contrast of materials, such as polymers [77 and references within] and biological tissue [70]. These unique properties of ferroelectric nanoparticles lead to their novel use as contrast-enhancing agents for microwave tomography, which is a meth‐ od of non-invasive assessment and diagnostics of soft tissues (such as detecting malignan‐ cies)[70].Recently, ferroelectric electro spun nanofibers also emerged in various biomedical areas including medical prostheses, tissue engineering, wound dressing, and drug delivery [28,29,30].

In conclusion, ferroelectric materials found a wide variety of biomedical applications in the last decade – and the list is constantly growing [78]. The ferroelectric material (e.g. ba‐ rium titanate) used in medical implants has been known to accelerate osteogenesis [79], and the same material in nanoparticle form works both as an SHG probe to detect Osteo‐ genesis Imperfecta [71] and, through microwave tomography, to detect lung cancer [70]. We expect more applications will become possible if other effects, such as piezoelectrici‐ ty, ferroelectric hysteresis or stress-birefringence coupling, are exploited with biology and medicine in mind.

### **6. Conclusions**

A review of recently published research papers allows us to conclude that the unique prop‐ erties of ferroelectric nanoparticles offer an enormous range of applications, especially in material technologies, biology, and medicine. However, since such a conclusion has become an academic cliché, we would like to make just a few realistic and optimistic comments on the subject.

Discussing any application, we have to be more specific and distinguish between potential applications and actual (i.e. mass commercialized) ones. So far, the "applied" field of ferro‐ electric nanoparticles is at the stage of high potential for commercialization rather than real mass-commercialization. However, after the very first proof-of-concept research studies were completed just recently, a serious shift toward real applications was initiated. The re‐ search community realized that the controllable preparation of nanoparticles (size, shape), their proper surface functionalization, and the homogeneous dispersion into host material (liquid crystals, polymers, biological species) are among the most critical steps on the way to mass-commercialization. Many applications (especially bio-medical) require a large-scale preparation of mono-dispersed, very fine (5-10 nm) ferroelectric nanoparticles in the tetrago‐ nal phase – and only recently has substantial progress been made in this direction (see sec‐ tion 2). However, such ultra-fine nanoparticles were mostly tested in material technologies (ferroelectric liquid crystal colloids), while bio-medical methods utilized relatively large fer‐ roelectric nanoparticles (~50-100 nm) (sections 4-5). The bioconjugation of 5-10 nm ferroelec‐ tric nanoparticles can extend the area of biomedical application even more, since these nanoparticles can be used as both passive and active nanoprobes.

The applied science of ferroelectric nanoparticles is in the beginning of its development. Many fundamental and applied problems still need to be solved before the potential appli‐ cations will be converted into actual ones. Nevertheless, despite the fact that the emerging problems in applying nanoparticles in material technologies and biomedicine are still nu‐ merous [80], we can conclude that a good balance between purely academic and purely ap‐ plied foci of research is the key toward mass commercialization of ferroelectric nanoparticles in all applied fields mentioned above.

### **Acknowledgements**

This also gives the ferroelectric SHG particle an edge in cell and tissue imaging in vivo. For example, recent advances in this area include imaging a tail of a living mouse with the aid of

The intrinsic large values of the dielectric permittivities of ferroelectric nanoparticles suggest their use to enhance the dielectric contrast of materials, such as polymers [77 and references within] and biological tissue [70]. These unique properties of ferroelectric nanoparticles lead to their novel use as contrast-enhancing agents for microwave tomography, which is a meth‐ od of non-invasive assessment and diagnostics of soft tissues (such as detecting malignan‐ cies)[70].Recently, ferroelectric electro spun nanofibers also emerged in various biomedical areas including medical prostheses, tissue engineering, wound dressing, and drug delivery

In conclusion, ferroelectric materials found a wide variety of biomedical applications in the last decade – and the list is constantly growing [78]. The ferroelectric material (e.g. ba‐ rium titanate) used in medical implants has been known to accelerate osteogenesis [79], and the same material in nanoparticle form works both as an SHG probe to detect Osteo‐ genesis Imperfecta [71] and, through microwave tomography, to detect lung cancer [70]. We expect more applications will become possible if other effects, such as piezoelectrici‐ ty, ferroelectric hysteresis or stress-birefringence coupling, are exploited with biology and

A review of recently published research papers allows us to conclude that the unique prop‐ erties of ferroelectric nanoparticles offer an enormous range of applications, especially in material technologies, biology, and medicine. However, since such a conclusion has become an academic cliché, we would like to make just a few realistic and optimistic comments on

Discussing any application, we have to be more specific and distinguish between potential applications and actual (i.e. mass commercialized) ones. So far, the "applied" field of ferro‐ electric nanoparticles is at the stage of high potential for commercialization rather than real mass-commercialization. However, after the very first proof-of-concept research studies were completed just recently, a serious shift toward real applications was initiated. The re‐ search community realized that the controllable preparation of nanoparticles (size, shape), their proper surface functionalization, and the homogeneous dispersion into host material (liquid crystals, polymers, biological species) are among the most critical steps on the way to mass-commercialization. Many applications (especially bio-medical) require a large-scale preparation of mono-dispersed, very fine (5-10 nm) ferroelectric nanoparticles in the tetrago‐ nal phase – and only recently has substantial progress been made in this direction (see sec‐ tion 2). However, such ultra-fine nanoparticles were mostly tested in material technologies (ferroelectric liquid crystal colloids), while bio-medical methods utilized relatively large fer‐ roelectric nanoparticles (~50-100 nm) (sections 4-5). The bioconjugation of 5-10 nm ferroelec‐

barium titanate nanoparticles [69].

[28,29,30].

490 Advances in Ferroelectrics

medicine in mind.

**6. Conclusions**

the subject.

The authors are very grateful to all coauthors of the papers related to the topic of ferroelec‐ tric nanoparticles. We acknowledge also the support of the UCCS Center for the Biofrontiers Institute, University of Colorado at Colorado Springs. A great portion of the work described in this review has been supported by the grants from the NSF #1102332 "Liquid Crystal Sig‐ nal Processing Devices for Microwave and Millimeter Wave Operation" and # 1010508 "STTR Phase 1: Design, Fabrication, and Characterization of Ferroelectric Nanoparticle Dop‐ ed Liquid Crystal / Polymer Composites."

### **List of abbreviations**

AFM atomic force microscopy DCS differential scanning calorimetry DLS dynamic light scattering DTA differential thermal analysis EDX energy dispersive X-rayspectroscopy EFM electrostatic force microscopy FE-SEM field-emission scanning electron microscopy FTIR Fourier transform infrared spectroscopy HRTEM high resolution transmission electron microscopy MLCC multilayer ceramic capacitors PAA polyamic acid PTCR positive temperature coefficient of resistance PMMA poly(methyl methacrylate)

PVDF poly(vinylidene fluoride) SAED selected area electron diffraction SEM scanning electron microscopy SHG probe second harmonic generating probe SMPS scanning mobility particle size SPM scanning probe microscopy TEM transmission electron microscopy TG or TGA thermo gravimetric analysis XPSX-ray photoelectron spectroscopy XRDX-ray diffraction

### **Author details**

Yuriy Garbovskiy, Olena Zribi and Anatoliy Glushchenko

UCCS Center for the Biofrontiers Institute, Department of Physics, University of Colorado at Colorado Springs, Colorado Springs, Colorado, USA

[7] Ahn, C.H., Rabe, K.M. & Triscone, J.M. Ferroelectricity at the nanoscale: local polari‐

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

http://dx.doi.org/10.5772/52516

493

[9] Sarkar, D. Synthesis and Properties of BaTiO3 Nanopowders. *Journal of the American*

[10] Zhu, Y.-F., Zhang, L., Natsuki, T., Fu, Y.-Q. & Ni, Q.-Q. Facile Synthesis of BaTiO3 Nanotubes and Their Microwave Absorption Properties. *ACS Applied Materials & In‐*

[11] O'Brien, S., Brus, L. & Murray, C.B. Synthesis of Monodisperse Nanoparticles of Bari‐ um Titanate: Toward a Generalized Strategy of Oxide Nanoparticle Synthesis. *Jour‐*

[12] Buscaglia, M.T., Bassoli, M., Buscaglia, V. & Vormberg, R. Solid-State Synthesis of Nanocrystalline BaTiO3: Reaction Kinetics and Powder Properties. *Journal of the*

[13] Zhang, Y., Wang, L. & Xue, D. Molten salt route of well dispersive barium titanate

[14] Wei, X. et al. Synthesis of Highly Dispersed Barium Titanate Nanoparticles by a Nov‐ el Solvothermal Method. *Journal of the American Ceramic Society*91, 315-318 (2008). [15] Hayashi, H. & Hakuta, Y. Hydrothermal Synthesis of Metal Oxide Nanoparticles in

[16] Zhang, H. et al. Fabrication of Monodispersed 5-nm BaTiO3 Nanocrystals with Nar‐ row Size Distribution via One-Step Solvothermal Route. *Journal of the American Ce‐*

[17] Atkuri, H. et al. Preparation of ferroelectric nanoparticles for their use in liquid crys‐ talline colloids. *Journal of Optics A: Pure and Applied Optics*11, 024006 (2009).

[18] Garbovskiy, Y. & Glushchenko, A. in Solid State Physics (ed. Camley, R.E.) 1 – 74

[19] Cook, G. et al. Nanoparticle doped organic-inorganic hybridphotorefractives. *Optics*

[20] Cook, G. et al. Harvesting single ferroelectric domain stressed nanoparticles for opti‐

[21] Basun, S.A., Cook, G., Reshetnyak, V.Y., Glushchenko, A.V. & Evans, D.R. Dipole moment and spontaneous polarization of ferroelectric nanoparticles in a nonpolar

[22] Swaminathan, V. et al. Microwave synthesis of noncentrosymmetric BaTiO3 truncat‐ ed nanocubes for charge storage applications. *ACS Appl Mater Interfaces*2, 3037-42

cal and ferroic applications. *Journal of Applied Physics*108, 064309-4 (2010).

zation in oxide thin films and heterostructures. *Science*303, 488-91 (2004).

[8] Scott, J.F. Applications of modern ferroelectrics. *Science*315, 954-9 (2007).

*nal of the American Chemical Society*123, 12085-12086 (2001).

*American Ceramic Society*91, 2862-2869 (2008).

nanoparticles. *Powder Technology*217, 629-633 (2012).

Supercritical Water. *Materials*3, 3794-3817 (2010).

fluid suspension. *Physical Review B*84, 024105 (2011).

*ramic Society*94, 3220-3222 (2011).

*Express*16, 4015-4022 (2008).

(2010).

(2010).

*Ceramic Society*94, 106-110 (2011).

*terfaces*4, 2101-2106 (2012).

### **References**


PVDF poly(vinylidene fluoride)

492 Advances in Ferroelectrics

SAED selected area electron diffraction

SHG probe second harmonic generating probe

SEM scanning electron microscopy

SMPS scanning mobility particle size

TEM transmission electron microscopy

TG or TGA thermo gravimetric analysis

Yuriy Garbovskiy, Olena Zribi and Anatoliy Glushchenko

Colorado Springs, Colorado Springs, Colorado, USA

*Ceramic Society*82, 797-818 (1999).

*Research*37, 189-238 (2007).

UCCS Center for the Biofrontiers Institute, Department of Physics, University of Colorado at

[1] Haertling, G.H. Ferroelectric ceramics: history and technology. *Journal of the American*

[2] Cross, L.E. & Newnham, R.E. in *High-Technology Ceramics—Past, Present,and Future*

[3] Mitsui, T. in Springer Handbook of Condensed Matter and Materials Data (ed. Warli‐

[4] Whatmore, R. in Springer Handbook of Electronic and Photonic Materials (ed. Cap‐

[5] Kalinin, S.V. et al. Nanoscale Electromechanics of Ferroelectric and Biological Sys‐ tems: A New Dimension in Scanning Probe Microscopy. *Annual Review of Materials*

[6] Gruverman, A. & Kholkin, A. Nanoscale ferroelectrics: processing, characterization

and future trends. *Reports on Progress in Physics*69, 2443-74 (2006).

289–305 (American Ceramic Society, Westerville, OH, 1987).

mont, W.M.a.H.) 903 – 938 (Springer, Berlin, 2006).

per, S.K.a.P.) 597 – 623 (Springer, Berlin, 2006).

XPSX-ray photoelectron spectroscopy

XRDX-ray diffraction

**Author details**

**References**

SPM scanning probe microscopy


[23] Fuentes, S. et al. Synthesis and characterization of BaTiO3 nanoparticles in oxygenat‐ mosphere. 505, 568–572 (2010).

[39] Bansal, V., Poddar, P., Ahmad, A. & Sastry, M. Room-Temperature Biosynthesis of Ferroelectric Barium Titanate Nanoparticles. *Journal of the American Chemical Soci‐*

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

http://dx.doi.org/10.5772/52516

495

[40] Nuraje, N. et al. Room Temperature Synthesis of Ferroelectric Barium Titanate Nano‐ particles Using Peptide Nanorings as Templates. *Advanced Materials*18, 807-811

[41] Jha, A.K. & Prasad, K. Synthesis of BaTiO3 nanoparticles: A new sustainable green

[42] Ould-Ely, T. et al. Large-scale engineered synthesis of BaTiO3 nanoparticles using

[43] Bansal, V., Bharde, A., Ramanathan, R. & Bhargava, S.K. Inorganic materials using

[44] Kamiya, H. et al. Preparation of Highly Dispersed Ultrafine Barium Titanate Powder by Using Microbial‐Derived Surfactant. *Journal of the American Ceramic Society*86,

[45] Pithan, C., Hennings, D. & Waser, R. Progress in the Synthesis of Nanocrystalline Ba‐ TiO3 Powders for MLCC. *International Journal of Applied Ceramic Technology*2, 1-14

[46] Kim, P. et al. High Energy Density Nanocomposites Based on Surface-Modified Ba‐

[47] Dang, Z.-M. et al. Fabrication and Dielectric Characterization of Advanced BaTiO3/ Polyimide Nanocomposite Films with High Thermal Stability. *Advanced Functional*

[48] Zhou, T. et al. Improving Dielectric Properties of BaTiO3/Ferroelectric Polymer Com‐ posites by Employing Surface Hydroxylated BaTiO3 Nanoparticles. *ACS Applied Ma‐*

[49] Fan, B.-H., Zha, J.-W., Wang, D.-R., Zhao, J. & Dang, Z.-M. Experimental study and theoretical prediction of dielectric permittivity in BaTiO3/polyimide nanocomposite

[50] Nagao, D., Kinoshita, T., Watanabe, A. & Konno, M. Fabrication of highly refractive, transparent BaTiO3/poly(methyl methacrylate) composite films with high permittivi‐

[51] Abe, K., Nagao, D., Watanabe, A. & Konno, M. Fabrication of highly refractive bari‐ um-titanate-incorporated polyimide nanocomposite films with high permittivity and

[52] Balzani, V. Nanochemistry: A Chemical Approach to Nanomaterials. Geoffrey A.

low-temperature bioinspired principles. *Nature Protocols*6, 97-104 (2011).

'unusual' microorganisms. *Advances in Colloid and Interface Science*.

TiO3 and a Ferroelectric Polymer. *ACS Nano*3, 2581-2592 (2009).

*ety*128, 11958-11963 (2006).

approach. *Integrated Ferroelectrics*117, 49-54 (2010).

(2006).

2011-2018 (2012).

*Materials*18, 1509-1517 (2008).

*terials & Interfaces*3, 2184-2188 (2011).

films. *Applied Physics Letters*100, 092903-4 (2012).

ties. *Polymer International*60, 1180-1184 (2011).

thermal stability. *Polymer International*, n/a-n/a (2012).

Ozin and André C. Arsenault. *Small*2, 678-679 (2006).

(2005).


[39] Bansal, V., Poddar, P., Ahmad, A. & Sastry, M. Room-Temperature Biosynthesis of Ferroelectric Barium Titanate Nanoparticles. *Journal of the American Chemical Soci‐ ety*128, 11958-11963 (2006).

[23] Fuentes, S. et al. Synthesis and characterization of BaTiO3 nanoparticles in oxygenat‐

[24] Qi, J.Q. et al. Direct synthesis of ultrafine tetragonal BaTiO3 nanoparticles at room

[25] Kong, L.B., Zhang, T.S., Ma, J. & Boey, F. Progress in synthesis of ferroelectriccera‐ micmaterialsviahigh-energymechanochemicaltechnique. 53, 207–322 (2008).

[26] Dang, F. et al. Aneweffect of ultrasonication on the formation of BaTiO3 nanoparti‐

[27] Suzuki, K. & Kijima, K. Well-crystallized barium titanate nanoparticles prepared by

[28] Ramaseshan, R., Sundarrajan, S., Jose, R. & Ramakrishna, S. Nanostructured ceramics

[29] He, Y. et al. Humidity sensing properties of BaTiO3 nanofiber prepared via electro‐

[30] Dai, Y., Liu, W., Formo, E., Sun, Y. & Xia, Y. Ceramic nanofibers fabricated by elec‐ trospinning and their applications in catalysis, environmental science, and energy

[31] Jung, D.S., Hong, S.K., Cho, J.S. & Kang, Y.C. Nano-sizedbariumtitanatepowders with tetragonalcrystalstructureprepared by flamespraypyrolysis. 28, 109–115 (2008).

[32] Li, X., Xu, H., Chen, Z.-S. & Chen, G. Biosynthesis of Nanoparticles by Microorgan‐

[33] Chen, C.-L. & Rosi, N.L. Peptide-Based Methods for the Preparation of Nanostruc‐ tured Inorganic Materials. *Angewandte Chemie International Edition*49, 1924-1942

[34] Briggs, B.D. & Knecht, M.R. Nanotechnology Meets Biology: Peptide-based Methods for the Fabrication of Functional Materials. *The Journal of Physical Chemistry Letters*3,

[35] Dhillon, G.S., Brar, S.K., Kaur, S. & Verma, M. Green approach for nanoparticle bio‐ synthesis by fungi: current trends and applications. *Critical Reviews in Biotechnolo‐*

[36] Beier, C.W., Cuevas, M.A. & Brutchey, R.L. Room-Temperature Synthetic Pathways

[37] Ahmad, G. et al. Rapid Bioenabled Formation of Ferroelectric BaTiO3 at Room Tem‐ perature from an Aqueous Salt Solution at Near Neutral pH. *Journal of the American*

[38] Jha, A.K. & Prasad, K. Ferroelectric BaTiO3 nanoparticles: Biosynthesis and character‐

mosphere. 505, 568–572 (2010).

494 Advances in Ferroelectrics

cles. 17, 310–314 (2010).

(2010).

405-418 (2012).

*gy*32, 49-73 (2011).

*Chemical Society*130, 4-5 (2007).

temperature. *Nanoscale Research Letters*6, 466 (2011).

plasma chemical vapor deposition. 58, 1650–1654 (2004).

by electrospinning. *Journal of Applied Physics*102, 111101-17 (2007).

spinning. *Sensors and Actuators B: Chemical*146, 98-102 (2010).

technology. *Polymers for Advanced Technologies*22, 326-338 (2011).

isms and Their Applications. *Journal of Nanomaterials*2011 (2011).

to Barium Titanate Nanocrystals. *Small*4, 2102-2106 (2008).

ization. *Colloids and Surfaces B: Biointerfaces*75, 330-334 (2010).


[53] Reznikov, Y. et al. Ferroelectric nematic suspension. *Applied Physics Letters*82, 1917-1919 (2003).

[68] Hsieh, C.-L., Grange, R., Pu, Y. & Psaltis, D. (eds. Mohseni, H. & Razeghi, M.)

Emerging Applications of Ferroelectric Nanoparticles in Materials Technologies, Biology and Medicine

http://dx.doi.org/10.5772/52516

497

[69] Hsieh, C.-L., Lanvin, T., Grange, R., Pu, Y. & Psaltis, D. in Conference on Lasers and

[70] Semenov, S., Pham, N. & Egot-Lemaire, S. in World Congress on Medical Physics and Biomedical Engineering (eds. Dössel, O. & Schlegel, W.C.) 311-313 (Springer Ber‐

[71] Adur, J. et al. (eds. Periasamy, A., Konig, K. & So, P.T.C.) 82263P-7 (SPIE, San Fran‐

[72] Wang, C.-C. et al. Differentiation of normal and cancerous lung tissues by multipho‐

[73] Chan, W.C.W. in Advances in Experimental Medicine and Biology 208 (Springer Sci‐

[74] Dempsey, W.P., Fraser, S.E. & Pantazis, P. SHG nanoprobes: Advancing harmonic

[75] Yust, B.G., Sardar, D.K. & Tsin, A. (eds. Cartwright, A.N. & Nicolau, D.V.) 79080G-7

[76] Ganeev, R.A., Suzuki, M., Baba, M., Ichihara, M. & Kuroda, H. Low- and high-order nonlinear optical properties of BaTiO3 and SrTiO3 nanoparticles. *Journal of the Optical*

[77] Lai, K.T., Nair, B.G. & Semenov, S. Optical and microwave studies of ferroelectric nanoparticles for application in biomedical imaging. *Microwave and Optical Technolo‐*

[79] Furuya, K., Morita, Y., Tanaka, K., Katayama, T. & Nakamachi, E. (eds. Martin-Pal‐ ma, R.J. & Lakhtakia, A.) 79750U-6 (SPIE, San Diego, California, USA, 2011).

[80] Florence, A.T. "Targeting" nanoparticles: The constraints of physical laws and physi‐

77590T-6 (SPIE, San Diego, California, USA, 2010).

ton imaging. *Journal of Biomedical Optics*14, 044034-4 (2009).

Electro-Optics (CLEO), 2011 1-2 (2011).

lin Heidelberg, Munich, Germany, 2009).

cisco, California, USA, 2012).

ence+Business Media, LLC, 2007).

imaging in biology. *BioEssays*34, 351-360 (2012).

(SPIE, San Francisco, California, USA, 2011).

[78] Ciofani, G. et al. 987-990 (KINTEX, Korea, 2010).

*Society of America B*25, 325-333 (2008).

*gy Letters*54, 11-13 (2012).

cal barriers. (2012).


[53] Reznikov, Y. et al. Ferroelectric nematic suspension. *Applied Physics Letters*82,

[54] Reznikov, Y. in Liquid Crystals Beyond Displays: Chemistry, Physics, and Applica‐

[55] Liang, H.-H. & Lee, J.-Y. Enhanced Electro-Optical Properties of Liquid Crystals De‐ vices by Doping with Ferroelectric Nanoparticles. *Ferroelectrics - Material Aspects*,

[56] Ciofani, G. et al. Preparation of stable dispersion of barium titanate nanoparticles: Potential applications in biomedicine. *Colloids and Surfaces B: Biointerfaces*76, 535–543

[57] Hsieh, C.-L., Grange, R., Pu, Y. & Psaltis, D. Bioconjugation of barium titanate nano‐ crystals with immunoglobulin G antibody for second harmonic radiation imaging

[58] Hsieh, C.-L., Grange, R., Pu, Y. & Psaltis, D. (eds. Campagnola, P.J., Stelzer, E.H.K. &

[59] Hsieh, C.-L., Grange, R., Pu, Y. & Psaltis, D. Three-dimensional harmonic holograph‐ ic microcopy using nanoparticles as probes for cell imaging. *Optics Express*17,

[60] Grange, R., Hsieh, C.L., Pu, Y. & Psaltis, D. in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Eu‐

[61] Ciofani, G. et al. in Piezoelectric Nanomaterials for Biomedical Applications 213-238

[62] Yust, B.G., Razavi, N., Pedraza, F. & Sardar, D.K. (eds. Cartwright, A.N. & Nicolau,

[63] Zipfel, W.R. et al. Live tissue intrinsic emission microscopy using multiphoton-excit‐ ed native fluorescence and second harmonic generation. *Proceedings of the National*

[64] Pantazis, P., Maloney, J., Wu, D. & Fraser, S.E. Second harmonic generating (SHG) nanoprobes for in vivo imaging. *Proceedings of the National Academy of Sciences*107,

[65] Ciofani, G., Ricotti, L. & Mattoli, V. Preparation, characterization and in vitro testing of poly(lactic-co-glycolic) acid/barium titanate nanoparticle composites for enhanced

[66] Pantazis, P., Pu, Y., Psaltis, D. & Fraser, S. (eds. Periasamy, A. & So, P.T.C.) 71831P-5

[67] Horiuchi, N. Imaging: Second-harmonic nanoprobes. *Nature Photonics*5, 7-7 (2011).

tions (ed. Li, Q.) 403-426 (John Wiley & Sons, Inc., 2012).

von Bally, G.) 73670D-6 (SPIE, Munich, Germany, 2009).

D.V.) 82310H-6 (SPIE, San Francisco, California, USA, 2012).

cellular proliferation. *Biomedical Microdevices*13, 255-266 (2011).

probes. *Biomaterials*31, 2272–2277 (2010).

1917-1919 (2003).

496 Advances in Ferroelectrics

193-210 (2011).

2880-2891 (2009).

14535-14540 (2010).

rope - EQEC 2009. 1-1 (2009).

(Springer Berlin Heidelberg, 2012).

*Academy of Sciences*100, 7075-7080 (2003).

(SPIE, San Jose, CA, USA, 2009).

(2010).


**Chapter 22**

**Photorefractive Effect in Ferroelectric Liquid Crystals**

Ferroelectric liquid crystals have been attracting great interest for their application in photo‐ refractive devices. The photorefractive effect is a phenomenon by which a change in refrac‐ tive index is induced by the interference of laser beams [1, 2]. Dynamic holograms are easily realized by the photorefractive effect. Holograms generate three-dimensional images of ob‐ jects. They are produced by recording interference fringes generated by light reflected from an object and a reference light (Figure 1). A hologram diffracts incident light to produce a three-dimensional images of an object. 3D displays are expected to be widely used as nextgeneration displays. However, current 3D displays are essentially stereograms. However,

The photorefractive effect has the potential to realize dynamic holograms by recording holo‐ grams as a change in the refractive index of the medium [1, 2]. The photorefractive effect induces a change in the refractive index by a mechanism involving both photovoltaic and electro-optic effects (Figure 2). When two laser beams interfere in an organic photorefractive material, a charge generation occurs at the bright positions of the interference fringes. The generated charges diffuse or drift within the material. Since the mobilities of positive and negative charges are different in most organic materials, a charge separated state is formed. The charge with the higher mobility diffuses over a longer distance than the charge with lower mobility, such that while the low mobility charge stays in the bright areas, the high mobility charge moves to the dark areas. The bright and dark positions of the interference fringes are thus charged with opposite polarities and an internal electric field (space-charge field) is generated in the area between the bright and dark positions. The refractive index of this area between the bright and dark positions is changed through the electro-optic effect. Thus, a refractive index grating (or hologram) is formed. One material class that exhibits high photorefractivity is that of glassy photoconductive polymers doped with high concen‐ trations of D-π-A chromophores, in which donor and acceptor groups are attached to a π-

> © 2013 Sasaki; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

holographic displays that can realize natural 3D images are anticipated.

Takeo Sasaki

**1. Introduction**

http://dx.doi.org/10.5772/52921

Additional information is available at the end of the chapter

### **Photorefractive Effect in Ferroelectric Liquid Crystals**

Takeo Sasaki

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52921

### **1. Introduction**

Ferroelectric liquid crystals have been attracting great interest for their application in photo‐ refractive devices. The photorefractive effect is a phenomenon by which a change in refrac‐ tive index is induced by the interference of laser beams [1, 2]. Dynamic holograms are easily realized by the photorefractive effect. Holograms generate three-dimensional images of ob‐ jects. They are produced by recording interference fringes generated by light reflected from an object and a reference light (Figure 1). A hologram diffracts incident light to produce a three-dimensional images of an object. 3D displays are expected to be widely used as nextgeneration displays. However, current 3D displays are essentially stereograms. However, holographic displays that can realize natural 3D images are anticipated.

The photorefractive effect has the potential to realize dynamic holograms by recording holo‐ grams as a change in the refractive index of the medium [1, 2]. The photorefractive effect induces a change in the refractive index by a mechanism involving both photovoltaic and electro-optic effects (Figure 2). When two laser beams interfere in an organic photorefractive material, a charge generation occurs at the bright positions of the interference fringes. The generated charges diffuse or drift within the material. Since the mobilities of positive and negative charges are different in most organic materials, a charge separated state is formed. The charge with the higher mobility diffuses over a longer distance than the charge with lower mobility, such that while the low mobility charge stays in the bright areas, the high mobility charge moves to the dark areas. The bright and dark positions of the interference fringes are thus charged with opposite polarities and an internal electric field (space-charge field) is generated in the area between the bright and dark positions. The refractive index of this area between the bright and dark positions is changed through the electro-optic effect. Thus, a refractive index grating (or hologram) is formed. One material class that exhibits high photorefractivity is that of glassy photoconductive polymers doped with high concen‐ trations of D-π-A chromophores, in which donor and acceptor groups are attached to a π-

© 2013 Sasaki; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

conjugate system [3-5]. In order to obtain photorefractivity in polymer materials, a high electric field of 10–50 V/μm is usually applied to a polymer film [6-8]. This electric field is necessary to increase charge generation efficiency. Significantly, the photorefractive effect permits two-beam coupling. It can be used to coherently amplify signal beams, and so has the potential to be used in a wide range of optical technologies as a transistor in electric cir‐ cuits. With the aim of developing a 3D display, a multiplex hologram has been demonstrat‐ ed using a photorefractive polymer film [9,10]. Clear 3D images were recorded in the film. However, the slow response (~100 ms) and the high electric field (30–50 V/μm) required to activate the photorefractive effect in the polymer materials both need to be improved.

**Figure 2.** Schematic illustration of the mechanism of the photorefractive effect. (a) Two laser beams interfere in the photorefractive material; (b) charge generation occurs at the light areas of the interference fringes; (c) electrons are trapped at the trap sites in the light areas and holes migrate by diffusion or drift in the presence of an external electric field and generate an internal electric field between the light and dark positions; (d) the refractive index of the corre‐

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

501

Nematic liquid crystals are used in LC displays. On the other hand, smectic liquid crystals are very viscous and, hence, are not utilized in any practical applications. Ferroelectric liq‐ uid crystals (FLCs) belong to the class of smectic liquid crystals that have a layered structure (Figure 3). The molecular structure of a typical FLC contains a chiral unit, a carbonyl group, a central core - which is a rigid rod-like structure, such as biphenyl, phenylpyrimidine or phenylbenzoate - and a flexible alkyl chain [15]. Thus, the dipole moment of an FLC mole‐ cule is perpendicular to the molecular long axis. FLCs exhibit a chiral smectic C phase (SmC\*) that possesses a helical structure. Compared to nematic LCs, FLCs are more crystal than liquid and the preparation of fine FLC films requires several sophisticated techniques. Obtaining a uniformly aligned, defect-free, surface-stabilized FLC (SS-FLC, Figure 3) using a single FLC compound is very difficult, and mixtures of several LC compounds are usually

sponding area is altered by the internal electric field generated.

used to obtain fine SS-FLC films.

**Figure 1.** Hologram recording and image reconstruction. (a) A light reflected from an object is interfered with by a reference light in a photo-sensitive material, such as a photopolymer. (b) The interference fringe is recorded in a pho‐ tosensitive material (hologram). (c) A readout beam is irradiated by the hologram. (d) The readout beam is diffracted by the hologram and the object image is reconstructed.

Ferroelectric liquid crystals are expected to be used as high performance photorefractive ma‐ terials [11-13]. A photorefractive ferroelectric liquid crystal with a fast response of 5 ms has been reported [14]. The photorefractive effect has been reported in surface-stabilized ferro‐ electric liquid crystals (SS-FLCs) doped with a photoconductive compound. Liquid crystals are classified into several groups. The most well known are nematic liquid crystals and smectic liquid crystals.

conjugate system [3-5]. In order to obtain photorefractivity in polymer materials, a high electric field of 10–50 V/μm is usually applied to a polymer film [6-8]. This electric field is necessary to increase charge generation efficiency. Significantly, the photorefractive effect permits two-beam coupling. It can be used to coherently amplify signal beams, and so has the potential to be used in a wide range of optical technologies as a transistor in electric cir‐ cuits. With the aim of developing a 3D display, a multiplex hologram has been demonstrat‐ ed using a photorefractive polymer film [9,10]. Clear 3D images were recorded in the film. However, the slow response (~100 ms) and the high electric field (30–50 V/μm) required to activate the photorefractive effect in the polymer materials both need to be improved.

**Figure 1.** Hologram recording and image reconstruction. (a) A light reflected from an object is interfered with by a reference light in a photo-sensitive material, such as a photopolymer. (b) The interference fringe is recorded in a pho‐ tosensitive material (hologram). (c) A readout beam is irradiated by the hologram. (d) The readout beam is diffracted

Ferroelectric liquid crystals are expected to be used as high performance photorefractive ma‐ terials [11-13]. A photorefractive ferroelectric liquid crystal with a fast response of 5 ms has been reported [14]. The photorefractive effect has been reported in surface-stabilized ferro‐ electric liquid crystals (SS-FLCs) doped with a photoconductive compound. Liquid crystals are classified into several groups. The most well known are nematic liquid crystals and

by the hologram and the object image is reconstructed.

smectic liquid crystals.

500 Advances in Ferroelectrics

**Figure 2.** Schematic illustration of the mechanism of the photorefractive effect. (a) Two laser beams interfere in the photorefractive material; (b) charge generation occurs at the light areas of the interference fringes; (c) electrons are trapped at the trap sites in the light areas and holes migrate by diffusion or drift in the presence of an external electric field and generate an internal electric field between the light and dark positions; (d) the refractive index of the corre‐ sponding area is altered by the internal electric field generated.

Nematic liquid crystals are used in LC displays. On the other hand, smectic liquid crystals are very viscous and, hence, are not utilized in any practical applications. Ferroelectric liq‐ uid crystals (FLCs) belong to the class of smectic liquid crystals that have a layered structure (Figure 3). The molecular structure of a typical FLC contains a chiral unit, a carbonyl group, a central core - which is a rigid rod-like structure, such as biphenyl, phenylpyrimidine or phenylbenzoate - and a flexible alkyl chain [15]. Thus, the dipole moment of an FLC mole‐ cule is perpendicular to the molecular long axis. FLCs exhibit a chiral smectic C phase (SmC\*) that possesses a helical structure. Compared to nematic LCs, FLCs are more crystal than liquid and the preparation of fine FLC films requires several sophisticated techniques. Obtaining a uniformly aligned, defect-free, surface-stabilized FLC (SS-FLC, Figure 3) using a single FLC compound is very difficult, and mixtures of several LC compounds are usually used to obtain fine SS-FLC films.

Figure 5 shows a schematic illustration of the mechanism of the photorefractive effect in FLCs. When laser beams interfere in a mixture of an FLC and a photoconductive compound, charge separation occurs between bright and dark positions and an internal electric field is produced. The internal electric field alters the direction of spontaneous polarization in the area between the bright and dark positions of the interference fringes, which induces a peri‐ odic change in the orientation of the FLC molecules. This is different from the processes that occur in other photorefractive materials in that the molecular dipole rather than the bulk po‐ larization responds to the internal electric field. Since the switching of FLC molecules is due

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

503

**Figure 5.** Schematic illustration of the mechanism of the photorefractive effect in FLCs. (a) Two laser beams interfere in the surface-stabilized state of the FLC/photoconductive compound mixture; (b) charge generation occurs at the bright areas of the interference fringes; (c) electrons are trapped at the trap sites in the bright areas, holes migrate by diffusion or drift in the presence of an external electric field to generate an internal electric field between the bright and dark positions; (d) the orientation of the spontaneous polarization vector (i.e., the orientation of mesogens in the

Since a change in the refractive index via the photorefractive effect occurs in the areas be‐ tween the bright and dark positions of the interference fringe, the phase of the resulting in‐ dex grating is shifted from the interference fringe. This is characteristic of the photorefractive effect that the phase of the refractive index grating is π/2-shifted from the interference fringe. When the material is photochemically active and is not photorefractive,

FLCs) is altered by the internal electric field.

**2. Characteristics of the photorefractive effect**

to the response of bulk polarization, the switching is extremely fast.

**Figure 3.** Structures of the smectic phase and the surface stabilized smectic-C phase (SS-FLC).

The FLC mixtures are composed of the base LC - which is also a mixture of several LC com‐ pounds - and a chiral compound. The chiral compound introduces a helical structure into the LC phase through supramolecular interactions. It should be mentioned here that in or‐ der to observe ferroelectricity in these materials, the ferroelectric liquid crystals must be formed into thin films. The thickness of the film must be within a few micrometers. When an FLC is sandwiched between glass plates to form a film a few micrometers thick, the helical structure of the smectic C phase uncoils and a surface-stabilized state (SS-state) is formed in which spontaneous polarization (Ps) appears. For display applications, the thickness of the film is usually 2 μm. In such thin films, FLC molecules can align in only two directions. This state is called a surface-stabilized state (SS-state). The alignment direction of the FLC mole‐ cules changes according to the direction of the spontaneous polarization (Figure 4). The di‐ rection of the spontaneous polarization is governed by the photoinduced internal electric field, giving rise to a refractive index grating with properties dependent on the direction of polarization.

**Figure 4.** Electro-optical switching in the surface-stabilized state of FLCs.

Figure 5 shows a schematic illustration of the mechanism of the photorefractive effect in FLCs. When laser beams interfere in a mixture of an FLC and a photoconductive compound, charge separation occurs between bright and dark positions and an internal electric field is produced. The internal electric field alters the direction of spontaneous polarization in the area between the bright and dark positions of the interference fringes, which induces a peri‐ odic change in the orientation of the FLC molecules. This is different from the processes that occur in other photorefractive materials in that the molecular dipole rather than the bulk po‐ larization responds to the internal electric field. Since the switching of FLC molecules is due to the response of bulk polarization, the switching is extremely fast.

**Figure 5.** Schematic illustration of the mechanism of the photorefractive effect in FLCs. (a) Two laser beams interfere in the surface-stabilized state of the FLC/photoconductive compound mixture; (b) charge generation occurs at the bright areas of the interference fringes; (c) electrons are trapped at the trap sites in the bright areas, holes migrate by diffusion or drift in the presence of an external electric field to generate an internal electric field between the bright and dark positions; (d) the orientation of the spontaneous polarization vector (i.e., the orientation of mesogens in the FLCs) is altered by the internal electric field.

### **2. Characteristics of the photorefractive effect**

Sm e c tic SS-FLC

The FLC mixtures are composed of the base LC - which is also a mixture of several LC com‐ pounds - and a chiral compound. The chiral compound introduces a helical structure into the LC phase through supramolecular interactions. It should be mentioned here that in or‐ der to observe ferroelectricity in these materials, the ferroelectric liquid crystals must be formed into thin films. The thickness of the film must be within a few micrometers. When an FLC is sandwiched between glass plates to form a film a few micrometers thick, the helical structure of the smectic C phase uncoils and a surface-stabilized state (SS-state) is formed in which spontaneous polarization (Ps) appears. For display applications, the thickness of the film is usually 2 μm. In such thin films, FLC molecules can align in only two directions. This state is called a surface-stabilized state (SS-state). The alignment direction of the FLC mole‐ cules changes according to the direction of the spontaneous polarization (Figure 4). The di‐ rection of the spontaneous polarization is governed by the photoinduced internal electric field, giving rise to a refractive index grating with properties dependent on the direction of

Electro-optical switching of FLC

**Figure 4.** Electro-optical switching in the surface-stabilized state of FLCs.

electric field electric field

**Figure 3.** Structures of the smectic phase and the surface stabilized smectic-C phase (SS-FLC).

polarization.

502 Advances in Ferroelectrics

FLC m olecule

Since a change in the refractive index via the photorefractive effect occurs in the areas be‐ tween the bright and dark positions of the interference fringe, the phase of the resulting in‐ dex grating is shifted from the interference fringe. This is characteristic of the photorefractive effect that the phase of the refractive index grating is π/2-shifted from the interference fringe. When the material is photochemically active and is not photorefractive, a photochemical reaction takes place at the bright areas and a refractive index grating with the same phase as that of the interference fringe is formed (Figure 6(a)).

diffraction condition is within the Bragg regime or within the Raman-Nath regime. These

beam 1 beam 2

**Figure 7.** Schematic illustrations of the experimental set-up for the (a) two-beam coupling, and (b) four-wave-mixing

Q > 1 is defined as the Bragg regime of optical diffraction. In this regime, multiple scattering is not permitted, and only one order of diffraction is produced. Conversely, Q < 1 is defined as the Raman-Nath regime of optical diffraction. In this regime, many orders of diffraction can be observed. Usually, Q > 10 is required to guarantee that the diffraction is entirely with‐ in the Bragg regime. When the diffraction is in the Bragg diffraction regime, the two-beam

coupling gain coefficient Γ (cm-1) is calculated according to the following equation:

*<sup>D</sup>* ln( *gm*

where D = L/cos(θ) is the interaction path for the signal beam (L=sample thickness, θ =prop‐ agation angle of the signal beam in the sample), g is the ratio of the intensities of the signal beam behind the sample with and without a pump beam, and m is the ratio of the beam

A schematic illustration of the experimental setup used for the four-wave mixing experi‐ ment is shown in Figure 7(b). S-polarized writing beams are interfered in the sample film and the diffraction of a p-polarized probe beam, counter-propagating to one of the writing beams, is measured. The diffracted beam intensity is typically measured as a function of time, applied (external) electric field and writing beam intensities, etc. The diffraction effi‐ ciency is defined as the ratio of the intensity of the diffracted beam and the intensity of the probe beam that is transmitted when no grating is present in the sample due to the writing beams. In probing the grating, it is important that beam 3 does not affect the grating or inter‐ act with the writing beams. This can be ensured by making the probe beam much weaker than the writing beams and by having the probe beam polarized orthogonal to the writing

*<sup>Γ</sup>* <sup>=</sup> <sup>1</sup>

intensities (pump/signal) in front of the sample.

Q = 2πlL/n*Λ* <sup>2</sup> (1)

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

505

FLC sample

<sup>1</sup> <sup>+</sup> *<sup>m</sup>*<sup>−</sup> *<sup>g</sup>* ) (2)

detector

beam 3 beam 4

diffraction conditions are distinguished by a dimensionless parameter Q.

FLC sample

beam 2

techniques.

beams.

detectors beam 1

(a) (b)

**Figure 6.** (a) Photochromic grating, and (b) photorefractive grating.

The interfering laser beams are diffracted by this grating; however, the apparent transmitted intensities of the laser beams do not change because the diffraction is symmetric. Beam 1 is diffracted in the direction of beam 2 and beam 2 is diffracted in the direction of beam 1. However, if the material is photorefractive, the phase of the refractive index grating is shift‐ ed from that of the interference fringes, and this affects the propagation of the two beams. Beam 1 is energetically coupled with beam 2 for the two laser beams. Consequently, the ap‐ parent transmitted intensity of beam 1 increases and that of beam 2 decreases (Figure 6(b)). This phenomenon is termed 'asymmetric energy exchange' in the two-beam coupling ex‐ periment. The photorefractivity of a material is confirmed by the occurrence of this asym‐ metric energy exchange.

### **3. Measurement of photorefractivity**

The photorefractive effect is evaluated by a two-beam coupling method and by a four-wave mixing experiment. Figure 7(a) shows a schematic illustration of the experimental setup used for the two-beam coupling method. A p-polarized beam from a laser is divided into two beams by a beam splitter and the beams are interfered within the sample film. An elec‐ tric field is applied to the sample using a high voltage supply unit. This external electric field is applied in order to increase the efficiency of charge generation in the film. The change in the transmitted beam intensity is monitored. If a material is photorefractive, an asymmetric energy exchange is observed. The magnitude of photorefractivity is evaluated using a parameter called the gain coefficient, which is calculated from the change in the transmitted intensity of the laser beams induced through the two-beam coupling [1]. In or‐ der to calculate the two-beam coupling gain coefficient, it must be determined whether the diffraction condition is within the Bragg regime or within the Raman-Nath regime. These diffraction conditions are distinguished by a dimensionless parameter Q.

a photochemical reaction takes place at the bright areas and a refractive index grating with

The interfering laser beams are diffracted by this grating; however, the apparent transmitted intensities of the laser beams do not change because the diffraction is symmetric. Beam 1 is diffracted in the direction of beam 2 and beam 2 is diffracted in the direction of beam 1. However, if the material is photorefractive, the phase of the refractive index grating is shift‐ ed from that of the interference fringes, and this affects the propagation of the two beams. Beam 1 is energetically coupled with beam 2 for the two laser beams. Consequently, the ap‐ parent transmitted intensity of beam 1 increases and that of beam 2 decreases (Figure 6(b)). This phenomenon is termed 'asymmetric energy exchange' in the two-beam coupling ex‐ periment. The photorefractivity of a material is confirmed by the occurrence of this asym‐

The photorefractive effect is evaluated by a two-beam coupling method and by a four-wave mixing experiment. Figure 7(a) shows a schematic illustration of the experimental setup used for the two-beam coupling method. A p-polarized beam from a laser is divided into two beams by a beam splitter and the beams are interfered within the sample film. An elec‐ tric field is applied to the sample using a high voltage supply unit. This external electric field is applied in order to increase the efficiency of charge generation in the film. The change in the transmitted beam intensity is monitored. If a material is photorefractive, an asymmetric energy exchange is observed. The magnitude of photorefractivity is evaluated using a parameter called the gain coefficient, which is calculated from the change in the transmitted intensity of the laser beams induced through the two-beam coupling [1]. In or‐ der to calculate the two-beam coupling gain coefficient, it must be determined whether the

Asymm etric Energy Exchange

the same phase as that of the interference fringe is formed (Figure 6(a)).

**Figure 6.** (a) Photochromic grating, and (b) photorefractive grating.

**3. Measurement of photorefractivity**

metric energy exchange.

504 Advances in Ferroelectrics

$$\mathbf{Q} = 2\pi \text{lL/n}\Lambda^2 \tag{1}$$

**Figure 7.** Schematic illustrations of the experimental set-up for the (a) two-beam coupling, and (b) four-wave-mixing techniques.

Q > 1 is defined as the Bragg regime of optical diffraction. In this regime, multiple scattering is not permitted, and only one order of diffraction is produced. Conversely, Q < 1 is defined as the Raman-Nath regime of optical diffraction. In this regime, many orders of diffraction can be observed. Usually, Q > 10 is required to guarantee that the diffraction is entirely with‐ in the Bragg regime. When the diffraction is in the Bragg diffraction regime, the two-beam coupling gain coefficient Γ (cm-1) is calculated according to the following equation:

$$I = \frac{1}{D} \ln \left( \frac{gm}{1 + m - g} \right) \tag{2}$$

where D = L/cos(θ) is the interaction path for the signal beam (L=sample thickness, θ =prop‐ agation angle of the signal beam in the sample), g is the ratio of the intensities of the signal beam behind the sample with and without a pump beam, and m is the ratio of the beam intensities (pump/signal) in front of the sample.

A schematic illustration of the experimental setup used for the four-wave mixing experi‐ ment is shown in Figure 7(b). S-polarized writing beams are interfered in the sample film and the diffraction of a p-polarized probe beam, counter-propagating to one of the writing beams, is measured. The diffracted beam intensity is typically measured as a function of time, applied (external) electric field and writing beam intensities, etc. The diffraction effi‐ ciency is defined as the ratio of the intensity of the diffracted beam and the intensity of the probe beam that is transmitted when no grating is present in the sample due to the writing beams. In probing the grating, it is important that beam 3 does not affect the grating or inter‐ act with the writing beams. This can be ensured by making the probe beam much weaker than the writing beams and by having the probe beam polarized orthogonal to the writing beams.

### **4. Photorefractive effect of FLCs**

### **4.1. Two-beam coupling experiments on FLCs**

The photorefractive effect in an FLC was first reported by Wasielewsky et al. in 2000 [11]. Since then, details of photorefractivity in FLC materials have been further investigated by Sasaki et al. and Golemme et al. [12-14]. The photorefractive effect in a mixture of an FLC and a photoconductive compound was measured in a two-beam coupling experiment using a 488 nm Ar+ laser. The structures of the photoconductive compounds used are shown in Figure 8. A commercially available FLC, SCE8 (Clariant), was used. CDH was used as a pho‐ toconductive compound and TNF was used as a sensitizer. The concentrations of CDH and TNF were 2 wt% and 0.1 wt% respectively. The samples were injected into a 10-μm-gap glass cell equipped with 1 cm2 ITO electrodes and a polyimide alignment layer (Figure 9).

Figure 10 shows a typical example of asymmetric energy exchange observed in the FLC(SCE8)/CDH/TNF sample under an applied DC electric field of 0.1 V/μm [12]. The inter‐ ference of the divided beams in the sample resulted in the increased transmittance of one beam and the decreased transmittance of the other. The change in the transmitted intensities of the two beams is completely symmetric, as can be seen in Figure 10. This indicates that the phase of the refractive index grating is shifted from that of the interference fringes. The grating formation was within the Bragg diffraction regime and no higher-order diffraction

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

507

The temperature dependence of the gain coefficient of SCE8 doped with 2 wt% CDH and 0.1 wt% TNF is shown in Figure 11(a). Asymmetric energy exchange was observed only at tem‐ peratures below 46°C. The spontaneous polarization of the identical sample is plotted as a

**Figure 10.** A typical example of asymmetric energy exchange observed in an FLC (SCE8) mixed with 2 wt% CDH and

0.1 wt% TNF. An electric field of +0.3 V/μm was applied to the sample.

was observed under the conditions used.

function of temperature in Figure 11(b).

**Figure 8.** Structures of the photoconductive compound CDH, ECz and the sensitizer TNF

**Figure 9.** Laser beam incidence condition and the structure of the LC cell.

Figure 10 shows a typical example of asymmetric energy exchange observed in the FLC(SCE8)/CDH/TNF sample under an applied DC electric field of 0.1 V/μm [12]. The inter‐ ference of the divided beams in the sample resulted in the increased transmittance of one beam and the decreased transmittance of the other. The change in the transmitted intensities of the two beams is completely symmetric, as can be seen in Figure 10. This indicates that the phase of the refractive index grating is shifted from that of the interference fringes. The grating formation was within the Bragg diffraction regime and no higher-order diffraction was observed under the conditions used.

**4. Photorefractive effect of FLCs**

a 488 nm Ar+

506 Advances in Ferroelectrics

glass cell equipped with 1 cm2

N

**4.1. Two-beam coupling experiments on FLCs**

N N

N

**ECz**

**Figure 9.** Laser beam incidence condition and the structure of the LC cell.

**Figure 8.** Structures of the photoconductive compound CDH, ECz and the sensitizer TNF

The photorefractive effect in an FLC was first reported by Wasielewsky et al. in 2000 [11]. Since then, details of photorefractivity in FLC materials have been further investigated by Sasaki et al. and Golemme et al. [12-14]. The photorefractive effect in a mixture of an FLC and a photoconductive compound was measured in a two-beam coupling experiment using

Figure 8. A commercially available FLC, SCE8 (Clariant), was used. CDH was used as a pho‐ toconductive compound and TNF was used as a sensitizer. The concentrations of CDH and TNF were 2 wt% and 0.1 wt% respectively. The samples were injected into a 10-μm-gap

**CDH TNF**

laser. The structures of the photoconductive compounds used are shown in

ITO electrodes and a polyimide alignment layer (Figure 9).

NO2

O2N NO2

O

The temperature dependence of the gain coefficient of SCE8 doped with 2 wt% CDH and 0.1 wt% TNF is shown in Figure 11(a). Asymmetric energy exchange was observed only at tem‐ peratures below 46°C. The spontaneous polarization of the identical sample is plotted as a function of temperature in Figure 11(b).

**Figure 10.** A typical example of asymmetric energy exchange observed in an FLC (SCE8) mixed with 2 wt% CDH and 0.1 wt% TNF. An electric field of +0.3 V/μm was applied to the sample.

SCE8 doped with 2 wt% CDH decreased when the external electric field exceeded 0.4 V/μm. The same tendency was observed for M4851/050 as well. The formation of an orientational grating is enhanced when the external electric field is increased from 0 to 0.2 V/μm as a re‐ sult of induced charge separation under a higher external electric field. However, when the external electric field exceeded 0.2 V/μm, a number of zigzag defects appeared in the sur‐ face-stabilized state. These defects cause light scattering and result in a decrease in the gain

2.0wt%

0.5wt%

0

γ(t) – 1 = (γ– 1)[1 – exp(–t/τ)]<sup>2</sup> (3)

**Figure 12.** Electric field dependence of the gain coefficient of SCE8 and M4851/050 mixed with several concentra‐

The formation of a refractive index grating involves charge separation and reorientation. The index grating formation time is affected by these two processes and both may be ratedetermining steps. The refractive index grating formation times in SCE8 and M4851/050 were determined based on the simplest single-carrier model of photorefractivity [1,2], wherein the gain transient is exponential. The rising signal of the diffracted beam was fitted

Here, γ(t) represents the transmitted beam intensity at time t divided by the initial intensity (γ(t) = I(t)/I0) and τ is the formation time. The grating formation time in SCE8/CDH/TNF is plotted as a function of the strength of the external electric field in Figure 13(a). The grating formation time decreased with increasing electric field strength due to the increased efficien‐ cy of charge generation. The formation time was shorter at higher temperatures, corre‐ sponding to a decrease in the viscosity of the FLC with increasing temperature. The formation time for SCE8 was found to be 20 ms at 30°C. As shown in Figure 13(b), the for‐

5

10

15

20

25

<sup>M</sup> <sup>4851</sup> /05 <sup>0</sup> (b)

Electric field (V/mm)

0.5wt%

1.0wt%

0 0 .0 5 0.10 0 .1 5 0.20 0 .2 5 0.30 0 .3 5

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

509

1.0wt%

Electric field (V/mm )

0 0.2 0.4 0.6 0 .8 1.0 1.2

tions of CDH and 0.1 wt% TNF in a 10 μm-gap cell measured at 30 °C.

using a single exponential function, shown in equation (3).

**4.3. Refractive index grating formation time**

coefficient.

0

5

1 0

1 5

2 0

2 5

3 0

SC E8 (a)

**Figure 11.** Temperature dependence of (a) the gain coefficient, and (b) the spontaneous polarization of an FLC (SCE8) mixed with 2 wt% CDH and 0.1 wt% TNF. For two-beam coupling experiments, an electric field of 0.1 V/μm was ap‐ plied to the sample.

Similarly, the spontaneous polarization vanished when the temperature was raised above 46°C. Thus, asymmetric energy exchange was observed only in the temperature range in which the sample exhibited ferroelectric properties; in other words, the SmC\* phase. Since the molecular dipole moment of the FLCs is small and the dipole moment is aligned perpen‐ dicular to the molecular axis, large changes in the orientation of the molecular axis cannot be induced by an internal electric field in the SmA or N\* phase of the FLCs. However, in the SmC\* phase, reorientation associated with spontaneous polarization occurs due to the inter‐ nal electric field. The spontaneous polarization also causes the orientation of FLC molecules in the corresponding area to change accordingly. A maximum resolution of 0.8 μm was ob‐ tained in this sample.

### **4.2. Effect of the magnitude of the applied electric field**

In polymeric photorefractive materials, the strength of the externally applied electric field is a very important factor. The external electric field is necessary to increase the charge separa‐ tion efficiency sufficiently to induce a photorefractive effect. In other words, the photorefrac‐ tivity of the polymer is obtained only with the application of a few V/μm electric fields. The thickness of the polymeric photorefractive material commonly reported is about 100 μm, so the voltage necessary to induce the photorefractive effect is a few kV. On the other hand, the photorefractive effect in FLCs can be induced by applying a very weak external electric field. The maximum gain coefficient for the FLC (SCE8) sample was obtained using an elec‐ tric field strength of only 0.2–0.4 V/μm. The thickness of the FLC sample is typically 10 μm, so that the voltage necessary to induce the photorefractive effect is only a few V. The de‐ pendence of the gain coefficient of a mixture of FLC (SCE8)/CDH/TNF on the strength of the electric field is shown in Figure 12. The gain coefficient of SCE8 doped with 0.5–1 wt% CDH increased with the strength of the external electric field. However, the gain coefficient of SCE8 doped with 2 wt% CDH decreased when the external electric field exceeded 0.4 V/μm. The same tendency was observed for M4851/050 as well. The formation of an orientational grating is enhanced when the external electric field is increased from 0 to 0.2 V/μm as a re‐ sult of induced charge separation under a higher external electric field. However, when the external electric field exceeded 0.2 V/μm, a number of zigzag defects appeared in the sur‐ face-stabilized state. These defects cause light scattering and result in a decrease in the gain coefficient.

**Figure 12.** Electric field dependence of the gain coefficient of SCE8 and M4851/050 mixed with several concentra‐ tions of CDH and 0.1 wt% TNF in a 10 μm-gap cell measured at 30 °C.

### **4.3. Refractive index grating formation time**

0

plied to the sample.

tained in this sample.

25 30 35 40 45 50 55 60 65

Te m p era ture (°C )

**4.2. Effect of the magnitude of the applied electric field**

25 30 35 40 45 50 55 60 65

Te m p e ratu re (°C )

SCE8 CDH 2.0wt%

0

**Figure 11.** Temperature dependence of (a) the gain coefficient, and (b) the spontaneous polarization of an FLC (SCE8) mixed with 2 wt% CDH and 0.1 wt% TNF. For two-beam coupling experiments, an electric field of 0.1 V/μm was ap‐

Similarly, the spontaneous polarization vanished when the temperature was raised above 46°C. Thus, asymmetric energy exchange was observed only in the temperature range in which the sample exhibited ferroelectric properties; in other words, the SmC\* phase. Since the molecular dipole moment of the FLCs is small and the dipole moment is aligned perpen‐ dicular to the molecular axis, large changes in the orientation of the molecular axis cannot be induced by an internal electric field in the SmA or N\* phase of the FLCs. However, in the SmC\* phase, reorientation associated with spontaneous polarization occurs due to the inter‐ nal electric field. The spontaneous polarization also causes the orientation of FLC molecules in the corresponding area to change accordingly. A maximum resolution of 0.8 μm was ob‐

In polymeric photorefractive materials, the strength of the externally applied electric field is a very important factor. The external electric field is necessary to increase the charge separa‐ tion efficiency sufficiently to induce a photorefractive effect. In other words, the photorefrac‐ tivity of the polymer is obtained only with the application of a few V/μm electric fields. The thickness of the polymeric photorefractive material commonly reported is about 100 μm, so the voltage necessary to induce the photorefractive effect is a few kV. On the other hand, the photorefractive effect in FLCs can be induced by applying a very weak external electric field. The maximum gain coefficient for the FLC (SCE8) sample was obtained using an elec‐ tric field strength of only 0.2–0.4 V/μm. The thickness of the FLC sample is typically 10 μm, so that the voltage necessary to induce the photorefractive effect is only a few V. The de‐ pendence of the gain coefficient of a mixture of FLC (SCE8)/CDH/TNF on the strength of the electric field is shown in Figure 12. The gain coefficient of SCE8 doped with 0.5–1 wt% CDH increased with the strength of the external electric field. However, the gain coefficient of

1

2

3

4

5

6

(b)

SCE8 CDH2.0wt%

5

1 0

1 5

2 0

508 Advances in Ferroelectrics

(a)

The formation of a refractive index grating involves charge separation and reorientation. The index grating formation time is affected by these two processes and both may be ratedetermining steps. The refractive index grating formation times in SCE8 and M4851/050 were determined based on the simplest single-carrier model of photorefractivity [1,2], wherein the gain transient is exponential. The rising signal of the diffracted beam was fitted using a single exponential function, shown in equation (3).

$$\gamma(\mathbf{t}) - 1 = (\gamma - 1)[1 - \exp(-\mathbf{t}/\tau)]^2 \tag{3}$$

Here, γ(t) represents the transmitted beam intensity at time t divided by the initial intensity (γ(t) = I(t)/I0) and τ is the formation time. The grating formation time in SCE8/CDH/TNF is plotted as a function of the strength of the external electric field in Figure 13(a). The grating formation time decreased with increasing electric field strength due to the increased efficien‐ cy of charge generation. The formation time was shorter at higher temperatures, corre‐ sponding to a decrease in the viscosity of the FLC with increasing temperature. The formation time for SCE8 was found to be 20 ms at 30°C. As shown in Figure 13(b), the for‐ mation time for M4851/050 was found to be independent of the magnitude of the external electric field, with a time of 80-90 ms for M4851/050 doped with 1 wt% CDH and 0.1 wt% TNF. This is slower than for SCE8, although the spontaneous polarization of M4851/050 (-14 nC/cm2 ) is larger than that of SCE8 (-4.5 nC/cm2 ) and the response time of the electro-optical switching (the flipping of spontaneous polarization) to an electric field (±10 V in a 2 μm cell) is shorter for M4851/050. The slower formation of the refractive index grating in M4851/050 is likely due to the poor homogeneity of the SS-state and charge mobility.

Electric field (V/mm)

field is independent of the concentration of ionic species.

*4.5.1. Photoconductive chiral dopants*

TN F con cen tra tion

(b ) E C z

0

**Figure 14.** Dependence of the TNF concentration on the gain coefficients of an FLC doped with photoconductive dopants. (a) SCE8 doped with 2 wt% CDH, and (b) SCE8 doped with 2 wt% ECz. An electric field of ± 0.5 V/μm, 100 Hz

These findings suggest that ionic conduction plays a major role in the formation of the space-charge field. The mobility of the CDH cation is smaller than that of the TNF anion, and this difference in mobility is thought to be the origin of the charge separation. In this case, the magnitude of the internal electric field is dominated by the concentration of the ionic species. On the other hand, the difference in the mobilities of ECz and TNF is small and, thus, less effective charge separation is induced, indicating that the internal electric

**4.5. Photorefractive effect in FLC mixtures containing photoconductive chiral compounds**

The photorefractive effect of FLC mixtures containing photoconductive chiral dopants has been investigated [15]. The structures of the LC compounds, the electron acceptor trinitro‐ fluorenone (TNF) and the photoconductive chiral compounds are shown in Figure 15. The mixing ratio of 8PP8 and 8PP10 was set to 1:1 because the 1:1 mixture exhibits the SmC phase over the widest temperature range. Hereafter, the 1:1 mixture of 8PP8 and 8PP10 is referred to as the base LC. The concentration of TNF was 0.1 wt%. Four photoconductive chiral compounds with the terthiophene chromophore (3T-2MB, 3T-2OC, 3T-OXO, and 3T-CF3) were synthesized. The base LC, TNF and a photoconductive chiral compound were dissolved in dichloroethane and the solvent was evaporated. The mixture was then dried in a vacuum at room temperature for one week. The samples were subsequently injected into a 10-μm-gap glass cell equipped with 1-cm2 ITO electrodes and a polyimide align‐ ment layer for the measurements. The base LC, which is a 1:1 mixture of 8PP8 and 8PP10, was mixed with the photoconductive chiral dopants and the electron acceptor (TNF). The concentration of TNF was set to 0.1 wt%. The terthiophene chiral dopants showed high

0.2 0 .4 0.6 0.8 1.0

0.1w t%

TN F con ce ntratio n

0.5w t% 0.3w t% 511

http://dx.doi.org/10.5772/52921

Electric field (V/mm )

Photorefractive Effect in Ferroelectric Liquid Crystals

(a) CD H

was applied.

**Figure 13.** Electric field dependence of the index grating formation time. (a) SCE8 mixed with 2 wt% CDH and 0.1 wt % TNF in the two-beam coupling experiment. , measured at 30 °C (T/TSmC\*-SmA=0.95); , measured at 36 °C (T/TSmC\*- SmA=0.97). (b) M4851/050 mixed with 1 wt% CDH and 0.1 wt% TNF in a two-beam coupling experiment. , measured at 42 °C (T/TSmC\*-SmA=0.95); , measured at 49 °C (T/TSmC\*-SmA=0.97).

### **4.4. Formation mechanism of the internal electric field in FLCs**

Since the photorefractive effect is induced by the photoinduced internal electric field, the mechanism of the formation of the space-charge field in the FLC medium is important. The two-beam coupling gain coefficients of mixtures of FLC (SCE8) and photoconductive com‐ pounds under a DC field were investigated as a function of the concentration of TNF (elec‐ tron acceptor). The photoconductive compounds - CDH, ECz and TNF (Figure 8) - were used in this examination. When an electron donor with a large molecular size (CDH) rela‐ tive to the TNF was used as the photoconductive compound, the gain coefficient was strong‐ ly affected by the concentration of TNF (Figure 14(a)). However, when ethylcarbazole (ECz) - the molecular size of which is almost the same as that of TNF - was used, the gain coeffi‐ cient was less affected by the TNF concentration (Figure 14(b)).

**Figure 14.** Dependence of the TNF concentration on the gain coefficients of an FLC doped with photoconductive dopants. (a) SCE8 doped with 2 wt% CDH, and (b) SCE8 doped with 2 wt% ECz. An electric field of ± 0.5 V/μm, 100 Hz was applied.

These findings suggest that ionic conduction plays a major role in the formation of the space-charge field. The mobility of the CDH cation is smaller than that of the TNF anion, and this difference in mobility is thought to be the origin of the charge separation. In this case, the magnitude of the internal electric field is dominated by the concentration of the ionic species. On the other hand, the difference in the mobilities of ECz and TNF is small and, thus, less effective charge separation is induced, indicating that the internal electric field is independent of the concentration of ionic species.

### **4.5. Photorefractive effect in FLC mixtures containing photoconductive chiral compounds**

### *4.5.1. Photoconductive chiral dopants*

mation time for M4851/050 was found to be independent of the magnitude of the external electric field, with a time of 80-90 ms for M4851/050 doped with 1 wt% CDH and 0.1 wt% TNF. This is slower than for SCE8, although the spontaneous polarization of M4851/050 (-14

switching (the flipping of spontaneous polarization) to an electric field (±10 V in a 2 μm cell) is shorter for M4851/050. The slower formation of the refractive index grating in M4851/050

0

**Figure 13.** Electric field dependence of the index grating formation time. (a) SCE8 mixed with 2 wt% CDH and 0.1 wt % TNF in the two-beam coupling experiment. , measured at 30 °C (T/TSmC\*-SmA=0.95); , measured at 36 °C (T/TSmC\*- SmA=0.97). (b) M4851/050 mixed with 1 wt% CDH and 0.1 wt% TNF in a two-beam coupling experiment. , measured

Since the photorefractive effect is induced by the photoinduced internal electric field, the mechanism of the formation of the space-charge field in the FLC medium is important. The two-beam coupling gain coefficients of mixtures of FLC (SCE8) and photoconductive com‐ pounds under a DC field were investigated as a function of the concentration of TNF (elec‐ tron acceptor). The photoconductive compounds - CDH, ECz and TNF (Figure 8) - were used in this examination. When an electron donor with a large molecular size (CDH) rela‐ tive to the TNF was used as the photoconductive compound, the gain coefficient was strong‐ ly affected by the concentration of TNF (Figure 14(a)). However, when ethylcarbazole (ECz) - the molecular size of which is almost the same as that of TNF - was used, the gain coeffi‐

5 0

1 00

1 50

2 00

(b)

) and the response time of the electro-optical

Electric field (V/mm)

0 0 .5 1 1.5 2 2 .5 3 3 .5

M4851/050 CDH1.0wt%

42°C

49°C

) is larger than that of SCE8 (-4.5 nC/cm2

is likely due to the poor homogeneity of the SS-state and charge mobility.

SCE8 CDH2.0wt%

0 0.2 0 .4 0.6 0.8 1 .0 1.2 1.4 1.6 Electric field (V/mm)

at 42 °C (T/TSmC\*-SmA=0.95); , measured at 49 °C (T/TSmC\*-SmA=0.97).

**4.4. Formation mechanism of the internal electric field in FLCs**

cient was less affected by the TNF concentration (Figure 14(b)).

nC/cm2

510 Advances in Ferroelectrics

37°C

30°C

0

50

10 0

15 0

20 0

(a)

The photorefractive effect of FLC mixtures containing photoconductive chiral dopants has been investigated [15]. The structures of the LC compounds, the electron acceptor trinitro‐ fluorenone (TNF) and the photoconductive chiral compounds are shown in Figure 15. The mixing ratio of 8PP8 and 8PP10 was set to 1:1 because the 1:1 mixture exhibits the SmC phase over the widest temperature range. Hereafter, the 1:1 mixture of 8PP8 and 8PP10 is referred to as the base LC. The concentration of TNF was 0.1 wt%. Four photoconductive chiral compounds with the terthiophene chromophore (3T-2MB, 3T-2OC, 3T-OXO, and 3T-CF3) were synthesized. The base LC, TNF and a photoconductive chiral compound were dissolved in dichloroethane and the solvent was evaporated. The mixture was then dried in a vacuum at room temperature for one week. The samples were subsequently injected into a 10-μm-gap glass cell equipped with 1-cm2 ITO electrodes and a polyimide align‐ ment layer for the measurements. The base LC, which is a 1:1 mixture of 8PP8 and 8PP10, was mixed with the photoconductive chiral dopants and the electron acceptor (TNF). The concentration of TNF was set to 0.1 wt%. The terthiophene chiral dopants showed high miscibility with the phenyl pyrimidine-type smectic LC. The chiral smectic C (SmC\*) phase appeared in all of the mixtures of the base LC and the chiral dopants. With the in‐ crease of the concentration of the chiral dopants, the temperature range of the SmC\* phase and the chiral nematic (N\*) phase were reduced, whereas that of the smectic A (SmA) phase was enhanced. The miscibility of the 3T-CF3 with the base LC was the low‐ est among the four chiral dopants. It was considered that the dipole moment of the tri‐ fluoromethyl substituted group is large, so that the 3T-CF3 molecules tend to aggregate. All the samples exhibited absorption maxima at 394 nm. Absorption at 488 nm (wave‐ length of the laser used) was small. The absorption spectra were not changed when TNF (1.0 × 10-5 mol/L) was added to the solution. The small absorption at the laser wavelength is advantageous for minimizing the optical loss. The photocurrents in the mixtures of the base LC, photoconductive chiral dopants and TNF were measured. As shown in Figure 16, the samples were good insulators in the dark. When a 488 nm laser irradiated the samples, photocurrents were clearly observed. The magnitudes of the photocurrents were slightly different in the four samples. The only difference in the molecular structures of these compounds is the chiral substituent. Thus, the difference in the photocurrent cannot be attributed to the difference in the molecular structure. It was considered that the misci‐ bility of the photoconductive chiral compounds to the LC and the homogeneity of the LC phase affected the magnitude of the photocurrent.

a) b)

140 120

80 100

> 60 40

> > 0

140 120

80 100

> 60 40

> > 0

laser (488 nm, continuous wave) was divided into two by a beam

**Figure 16.** Magnitudes of light-current and dark-current of mixtures of the base LC, photoconductive chiral com‐ pound and TNF measured in a 10-μm-gap LC cell as a function of the external electric field. An electric field of 0.1

The photorefractive effect was measured in a two-beam coupling experiment. A linearly po‐

splitter, each of which was then interfered in the sample film. A p-polarized beam was used. The laser intensity was 2.5 mW for each beam (1 mm diameter). The incident beam angles to the glass plane were 30° and 50°. Each interval of the interference fringe was 1.87 μm. Figure 17 shows typical examples of the asymmetric energy exchange observed in a mixture of the base LC, 3T-2MB and TNF at 30 ºC with the application of an electric field of 0.2–0.4 V/μm.

V/μm was applied. A 488 nm Ar+ laser (10 mW/cm2, 1 mm diameter) was used as the irradiation source.

*4.5.2. Two-beam coupling experiment on photoconductive FLC mixtures*

c) d)

0 0.2 0.4 0.6 0.8 1.0 1.2 E le c tric fie ld (V /mm )

0 0.2 0.4 0.6 0.8 1.0 1.2 E le c tric fie ld (V /mm )

<sup>20</sup> <sup>D</sup> ark cu rre nt

L ig ht cu rre n t

3T-O XO 4 w t%

<sup>20</sup> <sup>D</sup> <sup>a</sup> rk cu rre <sup>n</sup> <sup>t</sup>

L ig ht cu rre n t

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

513

3T-2O C 4 w t%

0 0.2 0.4 0.6 0.8 1.0 1.2 E lec tric fie ld (V /mm )

0 0.2 0.4 0.6 0.8 1.0 1.2 E lec tric fie ld (V /mm )

<sup>20</sup> <sup>D</sup> ark cu rren <sup>t</sup>

Ligh t curre nt

3T-C F 3 3w t%

<sup>20</sup> <sup>D</sup> <sup>a</sup> rk curre <sup>n</sup> <sup>t</sup>

3T-2 M B 4w t%

Ligh t curre nt

140 120

80 100

> 60 40

> > 0

140 120

80 100

> 60 40

> > 0

larized beam from an Ar+

**Figure 15.** Structures of the smectic LCs (8PP8 and 8PP10), photoconductive chiral dopants (3T-2MB, 3T-2OC, 3T-OXO and 3T-CF3) and the sensitizer TNF used in this work.

**Figure 16.** Magnitudes of light-current and dark-current of mixtures of the base LC, photoconductive chiral com‐ pound and TNF measured in a 10-μm-gap LC cell as a function of the external electric field. An electric field of 0.1 V/μm was applied. A 488 nm Ar+ laser (10 mW/cm2, 1 mm diameter) was used as the irradiation source.

### *4.5.2. Two-beam coupling experiment on photoconductive FLC mixtures*

miscibility with the phenyl pyrimidine-type smectic LC. The chiral smectic C (SmC\*) phase appeared in all of the mixtures of the base LC and the chiral dopants. With the in‐ crease of the concentration of the chiral dopants, the temperature range of the SmC\* phase and the chiral nematic (N\*) phase were reduced, whereas that of the smectic A (SmA) phase was enhanced. The miscibility of the 3T-CF3 with the base LC was the low‐ est among the four chiral dopants. It was considered that the dipole moment of the tri‐ fluoromethyl substituted group is large, so that the 3T-CF3 molecules tend to aggregate. All the samples exhibited absorption maxima at 394 nm. Absorption at 488 nm (wave‐ length of the laser used) was small. The absorption spectra were not changed when TNF (1.0 × 10-5 mol/L) was added to the solution. The small absorption at the laser wavelength is advantageous for minimizing the optical loss. The photocurrents in the mixtures of the base LC, photoconductive chiral dopants and TNF were measured. As shown in Figure 16, the samples were good insulators in the dark. When a 488 nm laser irradiated the samples, photocurrents were clearly observed. The magnitudes of the photocurrents were slightly different in the four samples. The only difference in the molecular structures of these compounds is the chiral substituent. Thus, the difference in the photocurrent cannot be attributed to the difference in the molecular structure. It was considered that the misci‐ bility of the photoconductive chiral compounds to the LC and the homogeneity of the LC

O

TN F

S

3T-C F3

<sup>8</sup><sup>H</sup> <sup>1</sup> <sup>7</sup> <sup>O</sup>

S

<sup>8</sup><sup>H</sup> <sup>1</sup> <sup>7</sup> <sup>O</sup>

O

O

C6 H1 <sup>3</sup>

O

M e

C F3

O

M e

C S S

C S S

O2N NO2

NO2

phase affected the magnitude of the photocurrent.

C <sup>8</sup>H1 <sup>7</sup> OC <sup>8</sup>H1 <sup>7</sup> N

8PP8

8PP10

N

N

N C8 H1 <sup>7</sup> OC <sup>1</sup> <sup>0</sup>H <sup>2</sup> <sup>1</sup>

S

3T-2O C

and 3T-CF3) and the sensitizer TNF used in this work.

<sup>8</sup>H1 <sup>7</sup> <sup>O</sup>

S <sup>C</sup> <sup>S</sup> <sup>S</sup> <sup>8</sup><sup>H</sup> <sup>1</sup> <sup>7</sup> <sup>O</sup>

O

O

C6H1 <sup>3</sup>

3T-2M B 3T-OXO

**Figure 15.** Structures of the smectic LCs (8PP8 and 8PP10), photoconductive chiral dopants (3T-2MB, 3T-2OC, 3T-OXO

C S S

512 Advances in Ferroelectrics

The photorefractive effect was measured in a two-beam coupling experiment. A linearly po‐ larized beam from an Ar+ laser (488 nm, continuous wave) was divided into two by a beam splitter, each of which was then interfered in the sample film. A p-polarized beam was used. The laser intensity was 2.5 mW for each beam (1 mm diameter). The incident beam angles to the glass plane were 30° and 50°. Each interval of the interference fringe was 1.87 μm. Figure 17 shows typical examples of the asymmetric energy exchange observed in a mixture of the base LC, 3T-2MB and TNF at 30 ºC with the application of an electric field of 0.2–0.4 V/μm.

2wt%

60

8 0

1 00

a)

12 0

40

20

0

6 0

80

1 00

b)

12 0

4 0

2 0

0

0 0.2 0 .4 0.6 0.8 1 1.2 Elec tric field (V/mm )

2wt%

0 0.2 0 .4 0 .6 0 .8 1 1 .2 Electric fie ld (V/mm )

compounds and TNF. (a) 3T-2MB; (b) 3T-2OC; (c) 3T-OXO; (d) 3T-CF3.

1w t%

6w t%

4w t%

3T-2OC

4wt%

8wt%

3T-2M B

6 0

80

10 0

d)

1 20

4 0

2 0

0

**Figure 18.** Electric field dependence of the gain coefficients of the mixtures of the base LC, photoconductive chiral

The difference in the gain coefficients in mixtures of the base LC, photoconductive chiral dopants (3T-2MB, 3T-2OC, 3T-OXO, and 3T-CF3) and TNF was investigated. All the sam‐ ples formed a finely aligned SS-state in 10-μm-gap cells with a LX-1400 polyimide alignment layer and exhibited clear photorefractivity in the ferroelectric phase. The asymmetric energy exchange was only observed in the temperature range in which the sample exhibits ferro‐ electric properties (SmC\* phase). The gain coefficients of the samples are plotted as a func‐ tion of the magnitude of the external electric field in Figure 18. The gain coefficient increased with the strength of the external electric field up to 0.2-0.6 V/μm and then decreased with the strength of the external electric field. As the concentration of the photoconductive chiral dopants increased, so did the gain coefficient. This may be due to the increased density of charge carriers in the FLC medium and an increase in the magnitude of Ps. All the samples exhibited relatively large photorefractivity. A gain coefficient higher than 100 cm-1 was ob‐ tained in the 3T-2OC (6 wt%) sample with the application of only 0.2 V/μm. This means that a voltage of only 2 V is needed to obtain the gain coefficient of 100 cm-1 in a 10 μm FLC sam‐

70 60 c)

8 0

8wt%

3T-OXO

515

6wt%

4wt%

3T-C F3

2wt%

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

0 0.2 0.4 0 .6 0 .8 1 1 .2 E lectric field (V /mm )

1wt%

0 0 .2 0 .4 0.6 0.8 1 1 .2 E lectric field (V /mm )

2wt%

The interference of the divided beams in the sample resulted in the increased transmittance of one of the beams and the decreased transmittance of the other beam. These transmittance characteristics were reversed when the polarity of the applied electric field was reversed. Asymmetric energy exchange was only observed when an electric field was applied, indicat‐ ing that beam coupling was not caused by a thermal grating. With an increased concentration of 3T-2MB, the magnitude of the gain coefficient also increased. In order to calculate the twobeam coupling gain coefficient, the diffraction condition needs to be correctly identified.

**Figure 18.** Electric field dependence of the gain coefficients of the mixtures of the base LC, photoconductive chiral compounds and TNF. (a) 3T-2MB; (b) 3T-2OC; (c) 3T-OXO; (d) 3T-CF3.

The difference in the gain coefficients in mixtures of the base LC, photoconductive chiral dopants (3T-2MB, 3T-2OC, 3T-OXO, and 3T-CF3) and TNF was investigated. All the sam‐ ples formed a finely aligned SS-state in 10-μm-gap cells with a LX-1400 polyimide alignment layer and exhibited clear photorefractivity in the ferroelectric phase. The asymmetric energy exchange was only observed in the temperature range in which the sample exhibits ferro‐ electric properties (SmC\* phase). The gain coefficients of the samples are plotted as a func‐ tion of the magnitude of the external electric field in Figure 18. The gain coefficient increased with the strength of the external electric field up to 0.2-0.6 V/μm and then decreased with the strength of the external electric field. As the concentration of the photoconductive chiral dopants increased, so did the gain coefficient. This may be due to the increased density of charge carriers in the FLC medium and an increase in the magnitude of Ps. All the samples exhibited relatively large photorefractivity. A gain coefficient higher than 100 cm-1 was ob‐ tained in the 3T-2OC (6 wt%) sample with the application of only 0.2 V/μm. This means that a voltage of only 2 V is needed to obtain the gain coefficient of 100 cm-1 in a 10 μm FLC sam‐

**Figure 17.** Examples of the results of two-beam coupling experiments for mixtures of the base LC, 3T-2MB and TNF

The interference of the divided beams in the sample resulted in the increased transmittance of one of the beams and the decreased transmittance of the other beam. These transmittance characteristics were reversed when the polarity of the applied electric field was reversed. Asymmetric energy exchange was only observed when an electric field was applied, indicat‐ ing that beam coupling was not caused by a thermal grating. With an increased concentration of 3T-2MB, the magnitude of the gain coefficient also increased. In order to calculate the twobeam coupling gain coefficient, the diffraction condition needs to be correctly identified.

measured at 30 °C.

514 Advances in Ferroelectrics

ple. In the FLCs reported previously, gain coefficients of only 50-60 cm-1 were obtained with the application of a 1 V/μm electric field. A gain coefficient higher than 100 cm-1 was also obtained in the 3T-2MB sample with an applied electric field of 0.5 V/μm. In order to obtain photorefractivity in polymer materials, the application of a high electric field of 10-50 V/μm to the polymer film is typically required. The small electric field necessary for the activation of the photorefractive effect in FLCs is thus a great advantage for their use in photorefrac‐ tive devices. The miscibility of 3T-CF3 with the base LC was low and could be mixed with the base LC at concentrations lower than 2 wt%. The grating formation time in the mixtures of the base LC, photoconductive chiral dopants and TNF is plotted as a function of the strength of the external electric field in Figure 19. The grating formation time decreased with an increased electric field strength due to the increased efficiency of charge generation. The shortest formation time was obtained as 5–8 ms with a 1 V/μm external electric field in all chiral compounds. The 3T-CF3 sample exhibited the fastest response. This was because of the larger polarity of the 3T-CF3.

*4.5.3. Photorefractive effect in a ternary mixture of a SmC liquid crystal doped with the*

The photorefractive effect of the mixture shown in Figure 20 was investigated. The gain co‐ efficients of the samples were measured as a function of the applied electric field strength (Figure 21(a)). The gain coefficient was calculated to be higher than 800 cm–1 in the 10 wt% sample with the application of only 1 V/μm. This gain coefficient is eight times higher than that of the FLCs described in the previous section. It was considered that a higher transpar‐ ency of the ternary LC mixture contributed to the high gain coefficient. The small electric field required to activate the photorefractive effect in FLCs is a great advantage for photore‐ fractive devices. The response time decreased with an increased electric field strength due to the increased charge separation efficiency (Figure 21(b)). The shortest formation time ob‐ tained was 8 ms for an external electric field of 1.5 V/μm. The large gain and fast response are advantageous for realizing optical devices, such as real-time image amplifiers and accu‐

base LC

sensitizer / electron trap

3T-2MB

TNF

**Figure 20.** Photorefractive FLC mixture containing a ternary mixture of smectic LCs.

photoconductive

chiral dopant

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

517

*photoconductive chiral compound*

rate measurement devices.

7

**Figure 19.** Electric field dependence of the index grating formation times of mixtures of the base LC, photoconductive chiral compounds and TNF measured at 30 °C. (a) 3T-2MB; (b) 3T-2OC; (c) 3T-OXO; (d) 3T-CF3.

### *4.5.3. Photorefractive effect in a ternary mixture of a SmC liquid crystal doped with the photoconductive chiral compound*

ple. In the FLCs reported previously, gain coefficients of only 50-60 cm-1 were obtained with the application of a 1 V/μm electric field. A gain coefficient higher than 100 cm-1 was also obtained in the 3T-2MB sample with an applied electric field of 0.5 V/μm. In order to obtain photorefractivity in polymer materials, the application of a high electric field of 10-50 V/μm to the polymer film is typically required. The small electric field necessary for the activation of the photorefractive effect in FLCs is thus a great advantage for their use in photorefrac‐ tive devices. The miscibility of 3T-CF3 with the base LC was low and could be mixed with the base LC at concentrations lower than 2 wt%. The grating formation time in the mixtures of the base LC, photoconductive chiral dopants and TNF is plotted as a function of the strength of the external electric field in Figure 19. The grating formation time decreased with an increased electric field strength due to the increased efficiency of charge generation. The shortest formation time was obtained as 5–8 ms with a 1 V/μm external electric field in all chiral compounds. The 3T-CF3 sample exhibited the fastest response. This was because of

the larger polarity of the 3T-CF3.

50

516 Advances in Ferroelectrics

a)

40

30

20

10

0

50

b)

40

30

20

10

0

8wt%

2wt%

0 0.2 0.4 0.6 0.8 1 1.2 Electric fie ld (V/mm)

0 0.2 0.4 0.6 0.8 1 1.2 Electric field (V/mm )

4wt%

4wt%

3T-2OC

2wt% 6wt%

1wt%

chiral compounds and TNF measured at 30 °C. (a) 3T-2MB; (b) 3T-2OC; (c) 3T-OXO; (d) 3T-CF3.

3T-2M B

25

d)

30

20

15

10

5

0

**Figure 19.** Electric field dependence of the index grating formation times of mixtures of the base LC, photoconductive

60 70

c)

6wt%

0 0.2 0.4 0.6 0.8 1 1.2 Electric field (V/mm )

1wt%

0 0.2 0.4 0.6 0.8 1 1.2 Electric field (V/mm)

2wt%

8wt%

3T-CF3

3T-OXO

2wt%

4wt%

The photorefractive effect of the mixture shown in Figure 20 was investigated. The gain co‐ efficients of the samples were measured as a function of the applied electric field strength (Figure 21(a)). The gain coefficient was calculated to be higher than 800 cm–1 in the 10 wt% sample with the application of only 1 V/μm. This gain coefficient is eight times higher than that of the FLCs described in the previous section. It was considered that a higher transpar‐ ency of the ternary LC mixture contributed to the high gain coefficient. The small electric field required to activate the photorefractive effect in FLCs is a great advantage for photore‐ fractive devices. The response time decreased with an increased electric field strength due to the increased charge separation efficiency (Figure 21(b)). The shortest formation time ob‐ tained was 8 ms for an external electric field of 1.5 V/μm. The large gain and fast response are advantageous for realizing optical devices, such as real-time image amplifiers and accu‐ rate measurement devices.

**Figure 20.** Photorefractive FLC mixture containing a ternary mixture of smectic LCs.

PB S: p o larizin g b e a m sp litter

PBS

l/2

M irro r

l/2

L e ns

**Figure 22.** Dynamic hologram formation experiment on an FLC sample. A computer-generated animation was dis‐ played on the SLM. The SLM modulated the object beam (488 nm), which was irradiated on the FLC sample and inter‐ fered with the reference beam. The readout beam (633 nm) was irradiated on the FLC and diffraction was observed.

Vo lta ge su p ply

L en s

l/2

Sa m p le

SL <sup>M</sup> PBS

H e-N e la se r (63 3 n m )

M irro r

http://dx.doi.org/10.5772/52921

519

Photorefractive Effect in Ferroelectric Liquid Crystals

PC

l/2 : h alf-w a ve p la te

D P SS la se r (48 8 n m )

**Figure 21.** (a) Electric field dependence of gain coefficients of mixtures of the base LC, 3T-2MB and TNF (0.1 wt%) measured at 30 °C. The 3T-2MB concentration was within the range 2–10 wt%. (b) Refractive index grating formation time (response time) of mixtures of the base LC, 3T-2MB and TNF (0.1 wt%) measured at 30 °C. The 3T-2MB concen‐ tration was within the range 2–10 wt%.

#### *4.5.4. Formation of dynamic holograms in FLC mixtures*

The formation of a dynamic hologram was demonstrated. A computer-generated animation was displayed on a spatial light modulator (SLM). A 488 nm beam from a DPSS laser was irradiated on the SLM and the reflected beam was incident on the FLC sample. A reference beam interfered with the beam from the SLM in the FLC sample. The refractive index gra‐ ting formed was within the Raman–Nath regime, in which multiple scattering is allowed. A 633 nm beam from a He–Ne laser was irradiated on the FLC sample and the diffraction was observed. A moving image of the animation on the computer monitor was observed in the diffracted beam (Figure 22). No image retention was observed, which means that the holo‐ gram image (refractive index grating) formed in the FLC was rewritten sufficiently rapidly to project the reproduction of a smooth holographic movie.

**Figure 21.** (a) Electric field dependence of gain coefficients of mixtures of the base LC, 3T-2MB and TNF (0.1 wt%) measured at 30 °C. The 3T-2MB concentration was within the range 2–10 wt%. (b) Refractive index grating formation time (response time) of mixtures of the base LC, 3T-2MB and TNF (0.1 wt%) measured at 30 °C. The 3T-2MB concen‐

The formation of a dynamic hologram was demonstrated. A computer-generated animation was displayed on a spatial light modulator (SLM). A 488 nm beam from a DPSS laser was irradiated on the SLM and the reflected beam was incident on the FLC sample. A reference beam interfered with the beam from the SLM in the FLC sample. The refractive index gra‐ ting formed was within the Raman–Nath regime, in which multiple scattering is allowed. A 633 nm beam from a He–Ne laser was irradiated on the FLC sample and the diffraction was observed. A moving image of the animation on the computer monitor was observed in the diffracted beam (Figure 22). No image retention was observed, which means that the holo‐ gram image (refractive index grating) formed in the FLC was rewritten sufficiently rapidly

tration was within the range 2–10 wt%.

518 Advances in Ferroelectrics

*4.5.4. Formation of dynamic holograms in FLC mixtures*

to project the reproduction of a smooth holographic movie.

**Figure 22.** Dynamic hologram formation experiment on an FLC sample. A computer-generated animation was dis‐ played on the SLM. The SLM modulated the object beam (488 nm), which was irradiated on the FLC sample and inter‐ fered with the reference beam. The readout beam (633 nm) was irradiated on the FLC and diffraction was observed.

### **5. Photorefractive effect in FLCs with the application of an AC field**

### **5.1. Formation of dynamic holograms based on the spatial modulation of the molecular motion of FLCs**

The formation of dynamic holograms based on the spatial modulation of the molecular mo‐ tion of ferroelectric liquid crystals (FLCs) was demonstrated [16, 17]. The switching move‐ ment of an FLC molecule is essentially a rotational motion along a conical surface. When an alternating triangular-waveform voltage is applied, the FLC molecules uniformly go through a consecutive rotational switching motion along a conical surface (Figure 23(a)). If a photoconductive FLC material is used, this rotational motion can be modulated by illumi‐ nating the material with interfering laser beams, as shown in Figure 23. The internal electric field vector is directed along the interference fringe wave vector and, in many cases, it dif‐ fers from the direction of the applied alternating electric field. Thus, the total electric field at each moment on the FLC molecules is altered by the presence of the internal electric field. The consecutive rotational motion of the FLC molecules in the areas between the light and dark positions of the interference fringes is biased by the internal electric field. Consequent‐ ly, a grating based on the spatial difference in the rotational motion (or switching motion) of the FLC molecules is created. This grating is different from those currently used for holo‐ grams, wherein changes in the static properties of a medium - such as absorbance, transpar‐ ency, film thickness and molecular orientation - are induced by photochemical reactions.

The formation of a motion-mode grating was examined using a SCE8/CDH/TNF mixture in a 10 μm-gap-cell. The formation of a holographic grating and the occurrence of a phase shift be‐ tween the formed grating and the interference fringes were examined in this experiment. An al‐ ternating triangular-waveform electric field (0 to ±1 V/μm, and 1 kHz to 3 MHz) was applied to the sample. Under the effect of an alternating triangular-waveform electric field of ±0.5 V/μm - 100 kHz in the two-beam coupling experiment - the FLC molecules exhibited a consecutive switching motion. This switching was confirmed using a polarizing microscope equipped with a photodetector Figure 24. Figure 25(a) shows the transmitted intensities of the laser beams through the FLC/CDH/TNF mixture upon the application of an alternating electric field as a function of time. The interference of the divided beams in the sample resulted in the increased transmittance of one of the beams and the decreased transmittance of the other. The incident beam conditions were the same as those used in the DC experiment.

Although the transmitted intensity of the laser beam oscillates due to the switching motion of the FLC molecules, the average intensities of the beams were symmetrically changed, as shown in Figure 25. This indicates that a diffraction grating based on the spatial difference in the rotational switching motion of FLC molecules was formed. The symmetric change in the transmittance of the two beams proves that the phase of the motion-mode grating shift‐ ed from that of the interference fringe. No higher-order diffraction was observed under this condition. The spacing of the grating was calculated to be 0.9 μm (1,100 lines/mm). Figure 26 shows the temperature dependence of the gain coefficient obtained under an AC electric field. Asymmetric energy exchange was observed only at those temperatures at which the FLC/CDH/TNF mixture exhibits a ferroelectric phase. This suggests that ferroelectricity or the switching movement of SmC\* is necessary for beam coupling under an AC electric field.

**Figure 23.** (a) Electro-optical switching of an FLC. (b) The rotational motion of FLC molecules under the application of an alternating electric field. (c) Positive and negative charges appear at the light positions of the interference fringe. (d) An internal electric field develops in the area between the light and dark positions of the interference fringes. The

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

521

rotational motion of the FLC molecules in the corresponding area is biased by the internal electric field.

**5. Photorefractive effect in FLCs with the application of an AC field**

**motion of FLCs**

520 Advances in Ferroelectrics

**5.1. Formation of dynamic holograms based on the spatial modulation of the molecular**

The formation of dynamic holograms based on the spatial modulation of the molecular mo‐ tion of ferroelectric liquid crystals (FLCs) was demonstrated [16, 17]. The switching move‐ ment of an FLC molecule is essentially a rotational motion along a conical surface. When an alternating triangular-waveform voltage is applied, the FLC molecules uniformly go through a consecutive rotational switching motion along a conical surface (Figure 23(a)). If a photoconductive FLC material is used, this rotational motion can be modulated by illumi‐ nating the material with interfering laser beams, as shown in Figure 23. The internal electric field vector is directed along the interference fringe wave vector and, in many cases, it dif‐ fers from the direction of the applied alternating electric field. Thus, the total electric field at each moment on the FLC molecules is altered by the presence of the internal electric field. The consecutive rotational motion of the FLC molecules in the areas between the light and dark positions of the interference fringes is biased by the internal electric field. Consequent‐ ly, a grating based on the spatial difference in the rotational motion (or switching motion) of the FLC molecules is created. This grating is different from those currently used for holo‐ grams, wherein changes in the static properties of a medium - such as absorbance, transpar‐ ency, film thickness and molecular orientation - are induced by photochemical reactions.

The formation of a motion-mode grating was examined using a SCE8/CDH/TNF mixture in a 10 μm-gap-cell. The formation of a holographic grating and the occurrence of a phase shift be‐ tween the formed grating and the interference fringes were examined in this experiment. An al‐ ternating triangular-waveform electric field (0 to ±1 V/μm, and 1 kHz to 3 MHz) was applied to the sample. Under the effect of an alternating triangular-waveform electric field of ±0.5 V/μm - 100 kHz in the two-beam coupling experiment - the FLC molecules exhibited a consecutive switching motion. This switching was confirmed using a polarizing microscope equipped with a photodetector Figure 24. Figure 25(a) shows the transmitted intensities of the laser beams through the FLC/CDH/TNF mixture upon the application of an alternating electric field as a function of time. The interference of the divided beams in the sample resulted in the increased transmittance of one of the beams and the decreased transmittance of the other. The incident

Although the transmitted intensity of the laser beam oscillates due to the switching motion of the FLC molecules, the average intensities of the beams were symmetrically changed, as shown in Figure 25. This indicates that a diffraction grating based on the spatial difference in the rotational switching motion of FLC molecules was formed. The symmetric change in the transmittance of the two beams proves that the phase of the motion-mode grating shift‐ ed from that of the interference fringe. No higher-order diffraction was observed under this condition. The spacing of the grating was calculated to be 0.9 μm (1,100 lines/mm). Figure 26 shows the temperature dependence of the gain coefficient obtained under an AC electric field. Asymmetric energy exchange was observed only at those temperatures at which the FLC/CDH/TNF mixture exhibits a ferroelectric phase. This suggests that ferroelectricity or the switching movement of SmC\* is necessary for beam coupling under an AC electric field.

beam conditions were the same as those used in the DC experiment.

**Figure 23.** (a) Electro-optical switching of an FLC. (b) The rotational motion of FLC molecules under the application of an alternating electric field. (c) Positive and negative charges appear at the light positions of the interference fringe. (d) An internal electric field develops in the area between the light and dark positions of the interference fringes. The rotational motion of the FLC molecules in the corresponding area is biased by the internal electric field.

**Figure 25.** a) Typical example of the asymmetric energy exchange observed in two-beam coupling experiments. A trian‐ gular waveform AC electric field of ± 0.5 Vm/μm, 100 Hz was applied in this case. The sample angle α was 50° and the in‐ tersection angle ϕ was 20°. The shutter was opened at t = 0 s and closed at t = 2 s. (b) An example of the curve fit.

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

523

**30 35 40 45**

**Figure 26.** Gain coefficient of the two-beam coupling under an AC electric field as a function of temperature. An AC

**0**

electric field of ± 0.5 V/μm, 100 Hz was applied.

**5**

**10**

**15**

**20**

**Figure 24.** Switching behaviour of SCE8 mixed with 2 wt% CDH and 0.1 wt% TNF under a 100 Hz, ± 0.5 Vm/μm trian‐ gular wave electric field. (a) Transmittance of light through the FLC under a polarizing microscope. (b) Magnitude of the applied electric field.

**Figure 25.** a) Typical example of the asymmetric energy exchange observed in two-beam coupling experiments. A trian‐ gular waveform AC electric field of ± 0.5 Vm/μm, 100 Hz was applied in this case. The sample angle α was 50° and the in‐ tersection angle ϕ was 20°. The shutter was opened at t = 0 s and closed at t = 2 s. (b) An example of the curve fit.

**Figure 26.** Gain coefficient of the two-beam coupling under an AC electric field as a function of temperature. An AC electric field of ± 0.5 V/μm, 100 Hz was applied.

**Figure 24.** Switching behaviour of SCE8 mixed with 2 wt% CDH and 0.1 wt% TNF under a 100 Hz, ± 0.5 Vm/μm trian‐ gular wave electric field. (a) Transmittance of light through the FLC under a polarizing microscope. (b) Magnitude of

the applied electric field.

522 Advances in Ferroelectrics

Figure 27 shows the gain coefficient plotted as a function of the electric field strength. Asym‐ metric energy exchange was observed at electric field strengths higher than ±0.1 V/μm. Fur‐ thermore, no asymmetric energy exchange was observed without an external electric field. This eliminates the possibility that beam coupling resulted from either a thermal grating or from a photochemically-formed grating. The gain coefficient was independent of the exter‐ nal electric field strength for fields higher than ±0.1 V/μm. This behaviour differs from that reported previously for the photorefractive effect of FLCs under an applied DC electric field, wherein the gain coefficient decreased with the increasing strength of the external DC elec‐ tric field. The refractive index grating formation time was measured based on the simplest single-carrier model for photorefractivity, whereby the gain transient is exponential. The grating formation time τ was obtained from the fitted curve.

**0**

tion time was determined to be 30-40 ms in the present case.

100 Hz.

tric field in Figure 29.

**0 ±0.2 ±0.4 ±0.6 ±0.8 ±1.0 ±1.2 ±1.4**

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

525

**Figure 28.** Grating formation time as a function of the AC electric field strength. The frequency of the AC field was

As seen in Figure 28, the grating formation time decreased with an increased applied electric field strength as a result of the increased efficiency of charge generation. The grating forma‐

The gain coefficients are plotted as a function of the frequency of the applied external elec‐

The gain coefficient was observed to increase with an increase in frequency within the range of 1 Hz to 100 Hz. The FLC exhibited switching based on polarization reversal at a frequen‐ cy lower than 500 Hz. The gain coefficient reached a maximum at frequencies within the range of 60 Hz to 500 Hz. However, the magnitude of the gain coefficient decreased as the frequency was increased to higher than 500 Hz. This is thought to be because the FLC can‐ not perform a complete switching motion at frequencies higher than 500 Hz; therefore, the FLC molecules showed vibrational motion. Moreover, at frequencies from 500 Hz to 2000 Hz, some of the FLC molecules go through a rotational switching motion, whereas others undergo vibrational motion. When the frequency exceeded 2000 Hz, the magnitude of the gain coefficient reaches a constant value, regardless of the frequency. Thus, the value of the

**5.2. Frequency dependence of the gain coefficient and the grating formation time**

**20**

**40**

**60**

**80**

**100**

**Figure 27.** Gain coefficient of the two-beam coupling as a function of the applied AC electric field strength. The fre‐ quency of the field was 100 Hz.

Figure 27 shows the gain coefficient plotted as a function of the electric field strength. Asym‐ metric energy exchange was observed at electric field strengths higher than ±0.1 V/μm. Fur‐ thermore, no asymmetric energy exchange was observed without an external electric field. This eliminates the possibility that beam coupling resulted from either a thermal grating or from a photochemically-formed grating. The gain coefficient was independent of the exter‐ nal electric field strength for fields higher than ±0.1 V/μm. This behaviour differs from that reported previously for the photorefractive effect of FLCs under an applied DC electric field, wherein the gain coefficient decreased with the increasing strength of the external DC elec‐ tric field. The refractive index grating formation time was measured based on the simplest single-carrier model for photorefractivity, whereby the gain transient is exponential. The

**0 ±0.2 ±0.4 ±0.6 ±0.8 ±1.0 ±1.2 ±1.4**

**Figure 27.** Gain coefficient of the two-beam coupling as a function of the applied AC electric field strength. The fre‐

grating formation time τ was obtained from the fitted curve.

**0**

quency of the field was 100 Hz.

**5**

**10**

**15**

**20**

**25**

**30**

524 Advances in Ferroelectrics

**Figure 28.** Grating formation time as a function of the AC electric field strength. The frequency of the AC field was 100 Hz.

As seen in Figure 28, the grating formation time decreased with an increased applied electric field strength as a result of the increased efficiency of charge generation. The grating forma‐ tion time was determined to be 30-40 ms in the present case.

### **5.2. Frequency dependence of the gain coefficient and the grating formation time**

The gain coefficients are plotted as a function of the frequency of the applied external elec‐ tric field in Figure 29.

The gain coefficient was observed to increase with an increase in frequency within the range of 1 Hz to 100 Hz. The FLC exhibited switching based on polarization reversal at a frequen‐ cy lower than 500 Hz. The gain coefficient reached a maximum at frequencies within the range of 60 Hz to 500 Hz. However, the magnitude of the gain coefficient decreased as the frequency was increased to higher than 500 Hz. This is thought to be because the FLC can‐ not perform a complete switching motion at frequencies higher than 500 Hz; therefore, the FLC molecules showed vibrational motion. Moreover, at frequencies from 500 Hz to 2000 Hz, some of the FLC molecules go through a rotational switching motion, whereas others undergo vibrational motion. When the frequency exceeded 2000 Hz, the magnitude of the gain coefficient reaches a constant value, regardless of the frequency. Thus, the value of the gain coefficient at frequencies lower than 500 Hz can be attributed to the photorefractive ef‐ fect based on the complete switching of FLC molecules and, at frequencies higher than 2000 Hz, this represents the photorefractive effect based on the vibrational motion of FLC mole‐ cules caused by an alternating electric field. Figure 30 shows the frequency dependence of the grating formation time.

**1000 2000 3000 4000**

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

527

**Frequency (Hz)**

**Figure 30.** Grating formation time as a function of the frequency of the applied electric field. The frequency was var‐

The grating formation time was shortened as the frequency increased. In addition, the fre‐ quency dependence of the formation time was not correlated with the dependence of the gain coefficient. As the frequency of the applied electric field increased, the switching and vibrational motion of the FLC molecules is thought to be accelerated and the affect on the internal electric field becomes apparent. Asymmetric energy exchange in FLCs under a static DC field is dominated by the direction of the DC field. The increase/decrease in the beam intensity switches when the direction of the field is reversed. This originates from the direc‐ tion of charge separation. Thus, an internal electric field is thought to be difficult to form under an AC electric field. However, asymmetric energy exchange was observed with good reproducibility, suggesting that an internal electric field is formed even under an applied AC electric field. In addition, no energy exchange was observed in any FLC lacking a photo‐ conductive dopant under an AC electric field (Figure 31). This observation indicates that

photoconductivity is necessary for two-beam coupling under an AC field.

**20**

**0**

ied from 1 Hz to 4000 Hz. The strength of the field was ± 0.5 V/μm.

**30**

**40**

**50**

**60**

**70**

**80**

**Figure 29.** Gain coefficient as a function of the frequency of the applied electric field. The frequency was varied from 1 to 5000 Hz. The strength of the field was ± 0.5 V/μm.

gain coefficient at frequencies lower than 500 Hz can be attributed to the photorefractive ef‐ fect based on the complete switching of FLC molecules and, at frequencies higher than 2000 Hz, this represents the photorefractive effect based on the vibrational motion of FLC mole‐ cules caused by an alternating electric field. Figure 30 shows the frequency dependence of

**0 1000 2000 3000 4000 5000**

**Frequency (Hz)**

**Figure 29.** Gain coefficient as a function of the frequency of the applied electric field. The frequency was varied from

the grating formation time.

526 Advances in Ferroelectrics

**0**

1 to 5000 Hz. The strength of the field was ± 0.5 V/μm.

**2**

**4**

**6**

**8**

**10**

**Figure 30.** Grating formation time as a function of the frequency of the applied electric field. The frequency was var‐ ied from 1 Hz to 4000 Hz. The strength of the field was ± 0.5 V/μm.

The grating formation time was shortened as the frequency increased. In addition, the fre‐ quency dependence of the formation time was not correlated with the dependence of the gain coefficient. As the frequency of the applied electric field increased, the switching and vibrational motion of the FLC molecules is thought to be accelerated and the affect on the internal electric field becomes apparent. Asymmetric energy exchange in FLCs under a static DC field is dominated by the direction of the DC field. The increase/decrease in the beam intensity switches when the direction of the field is reversed. This originates from the direc‐ tion of charge separation. Thus, an internal electric field is thought to be difficult to form under an AC electric field. However, asymmetric energy exchange was observed with good reproducibility, suggesting that an internal electric field is formed even under an applied AC electric field. In addition, no energy exchange was observed in any FLC lacking a photo‐ conductive dopant under an AC electric field (Figure 31). This observation indicates that photoconductivity is necessary for two-beam coupling under an AC field.

is accomplished through charge generation and diffusion. The application of an AC field to an FLC results in a very stable photorefractive response. However, it is obvious that an AC field is not advantageous for charge separation. Thus, the effect of a biased AC electric field on the photorefractivity of SS-FLCs was investigated. Figure 32 shows the concept of apply‐

0V

**Figure 32.** Concept of a biased AC field application. (a) Alternating electric field. (b) Biased alternating electric field.

switching motion of FLC molecules was formed and acted as a diffraction grating.

**Figure 33.** FigAsymmetric energy exchange with a square waveform of 2.0 Vp-pμm-1, 100 Hz (a) without bias, and (b)

with a bias of 0.5 V μm-1. The intersection angle ϕ was 20° and the sample angle α was 50°.

A typical example of the two-beam coupling signal is observed with the application of a 2.0 Vp-pμm-1, 100 Hz square waveform electric field without a bias field (Eb = 0), as shown in Figure 33(a). The transmitted intensities of the laser beams through the FLC/CDH/TNF mix‐ ture upon the application of an alternating electric field as a function of time is shown in the figure. The interference of the divided beams in the sample resulted in the increased trans‐ mittance of one of the beams and the decreased transmittance of the other. Although the transmitted intensity of the laser beam oscillates due to the switching motion of the FLC molecules, the average intensities of the beams were symmetrically changed, as shown in Figure 33(a). This indicates that a grating based on the spatial difference in the rotational

Eb

**Tim e**

Ep-p

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

529

**E**

(b)

ing a biased AC field.

**E**

(a)

0V

**Tim e**

**Figure 31.** An example of the results of the two-beam coupling experiment under an AC electric field for SCE8 with‐ out a photoconductive dopant. An AC electric field of ± 0.5 V/μm, 100 Hz was applied. The shutter was opened at t = 0 s and closed at t = 2 s.

It is necessary to consider both hole transport by a hopping mechanism as well as ionic con‐ duction in order to explain the formation of the space-charge field under an AC field. In ad‐ dition, a number of experimental results support the large contribution of ionic conduction to the FLC medium. In addition, the photorefractive effect in SCE8 doped with 2 wt% ECz under an AC field was very small. If ionic conduction is the major contributor to the forma‐ tion of the space-charge field, the anisotropic mobility of ionic species in the LC medium may affect the formation of the field. As shown in Figure 3, interference fringes are formed across the smectic layer so that migration of ionic species occurs in the inter-layer direction. The asymmetric structure of the surface stabilized state of the FLC may lead to the asym‐ metric mobility of cations and anions, which is thought to be one possible model explaining the charge separation under an AC electric field. However, the mechanism of the formation of the space-charge field under an AC field requires further investigation.

### **5.3. Two beam coupling experiment with the application of a biased AC field**

An asymmetric energy exchange was observed upon the application of an AC field. This grating was interpreted as being based on the spatial difference in the molecular motion of the FLC molecules. The response time was on the order of a few tens of milliseconds and was dominated by the formation of the internal electric field. The photorefractivity in FLCs is accomplished through charge generation and diffusion. The application of an AC field to an FLC results in a very stable photorefractive response. However, it is obvious that an AC field is not advantageous for charge separation. Thus, the effect of a biased AC electric field on the photorefractivity of SS-FLCs was investigated. Figure 32 shows the concept of apply‐ ing a biased AC field.

**Figure 32.** Concept of a biased AC field application. (a) Alternating electric field. (b) Biased alternating electric field.

A typical example of the two-beam coupling signal is observed with the application of a 2.0 Vp-pμm-1, 100 Hz square waveform electric field without a bias field (Eb = 0), as shown in Figure 33(a). The transmitted intensities of the laser beams through the FLC/CDH/TNF mix‐ ture upon the application of an alternating electric field as a function of time is shown in the figure. The interference of the divided beams in the sample resulted in the increased trans‐ mittance of one of the beams and the decreased transmittance of the other. Although the transmitted intensity of the laser beam oscillates due to the switching motion of the FLC molecules, the average intensities of the beams were symmetrically changed, as shown in Figure 33(a). This indicates that a grating based on the spatial difference in the rotational switching motion of FLC molecules was formed and acted as a diffraction grating.

**Figure 31.** An example of the results of the two-beam coupling experiment under an AC electric field for SCE8 with‐ out a photoconductive dopant. An AC electric field of ± 0.5 V/μm, 100 Hz was applied. The shutter was opened at t =

It is necessary to consider both hole transport by a hopping mechanism as well as ionic con‐ duction in order to explain the formation of the space-charge field under an AC field. In ad‐ dition, a number of experimental results support the large contribution of ionic conduction to the FLC medium. In addition, the photorefractive effect in SCE8 doped with 2 wt% ECz under an AC field was very small. If ionic conduction is the major contributor to the forma‐ tion of the space-charge field, the anisotropic mobility of ionic species in the LC medium may affect the formation of the field. As shown in Figure 3, interference fringes are formed across the smectic layer so that migration of ionic species occurs in the inter-layer direction. The asymmetric structure of the surface stabilized state of the FLC may lead to the asym‐ metric mobility of cations and anions, which is thought to be one possible model explaining the charge separation under an AC electric field. However, the mechanism of the formation

of the space-charge field under an AC field requires further investigation.

**5.3. Two beam coupling experiment with the application of a biased AC field**

An asymmetric energy exchange was observed upon the application of an AC field. This grating was interpreted as being based on the spatial difference in the molecular motion of the FLC molecules. The response time was on the order of a few tens of milliseconds and was dominated by the formation of the internal electric field. The photorefractivity in FLCs

0 s and closed at t = 2 s.

528 Advances in Ferroelectrics

**Figure 33.** FigAsymmetric energy exchange with a square waveform of 2.0 Vp-pμm-1, 100 Hz (a) without bias, and (b) with a bias of 0.5 V μm-1. The intersection angle ϕ was 20° and the sample angle α was 50°.

The asymmetric energy exchange between the signal and the pump beams proves the for‐ mation of a diffraction grating phase-shifted from the interference fringe. The grating is con‐ sidered to be based on the spatial difference in the rotational switching motion of FLC molecules. When the biased electric field was set to Eb = 0.5 V μm-1, the two-beam coupling signal was dramatically enhanced (Figure 33(b)). The gain coefficient increased from 10 cm-1 (Eb = 0) to 70 cm-1 (Eb = 0.5 V μm-1). Obviously, the biased field contributes to the formation of the diffraction grating. The internal electric field is thought to be difficult to form under an AC electric field. It has been reported that ionic conduction is the major contributor to the formation of the space-charge field in the photorefractive effect of an FLC. The anisotropic mobility of ionic species in an LC medium affects the formation of the field. In the experi‐ mental conditions reported, the interference fringe is formed across the smectic layer so that migration of ionic species occurs in the inter-layer direction. The asymmetric structure of the surface stabilized state of the FLC may lead to the asymmetric mobility of cations and anions. It was considered that the charge separated state is more effectively formed under a biased AC field.

**Author details**

Tokyo University of Science, Japan

refractive Materials; Oxford: New York, 1996.

L.; Peyghambarian, N. Science 1998, 279, 54-56.

ics Applications II, Springer, 2002, 87-156.

[8] Sasaki, T. Polymer Journal, 2005, 37, 797-812.

Peyghambarian, N. Nature, 2008, 451, 694-698.

[13] Talarico, M.; Goelemme, A. Nature Mater., 2006, 5, 185-188.

[3] Moerner, W. E.; Silence, S. M. Chem. Rev. 1994, 94, 127-155.

[1] Yeh, P. Introduction to Photorefractive Nonlinear Optics; John Wiley: New York,

Photorefractive Effect in Ferroelectric Liquid Crystals

http://dx.doi.org/10.5772/52921

531

[2] Solymar, L.; Webb, J. D.; Grunnet-Jepsen, A. The Physics and Applications of Photo‐

[4] Meerholz, K.; Volodin, B. L.; Kippelen, B.; Peyghambarian, N. Nature 1994, 371,

[5] Kippelen, B.; Marder, S. R.; Hendrickx, E.; Maldonado, J. L.; Guillemet, G.; Volodin, B. L.; Steele, D. D.; Enami, Y.; Sandalphon; Yao, Y. J.; Wang, J. F.; Röckel, H.; Erskine,

[6] Kippelen, B.; Peyghambarian, N. Advances in Polymer Science, Polymers for Photon‐

[9] Tay, S.; Blanche, P. A.; Voorakaranam, R.; Tunc, A. V.; Lin, W.; Rokutanda, S.; Gu, T.; Flores, D.; Wang, P.; Li, G.; Hilarie, P.; Thomas, J.; Norwood, R. A.; Yamamoto, M.;

[10] Blanche, P. A.; Bablumian, A.; Voorakaranam, R.; Christenson, C.; Lin, W.; Gu, T.; Flores, D.; Wang, P.; Hsieh, W. Y.; Kathaperumal, M.; Rachwal, B.; Siddiqui, O.; Tho‐ mas, J.; Norwood, R. A.; Yamamoto, M.; Peyghambarian, N. Nature, 2010, 468, 80-83.

[11] Wiederrecht, G. P., Yoon, B. A., Wasielewski, M. R. Adv. Materials 2000, 12,

[12] Sasaki, T.; Katsuragi, A.; Mochizuki, O.; Nakazawa, Y. J. Phys. Chem. B, 2003, 107,

[14] Sasaki, T; Miyazaki, D.; Akaike, K.; Ikegami, M.; Naka, Y. J. Mater. Chem., 2011, 21,

[7] Ostroverkhova, O.; Moerner, W. E. Chem. Rev., 2004, 104, 3267-3314.

Takeo Sasaki

**References**

1993.

497-500.

1533-1536.

7659 -7665.

8678-8686.

### **6. Conclusion**

The reorientational photorefractive effect based on the response of bulk polarization was observed in dye-doped FLC samples. Photorefractivity was observed only in the ferroelec‐ tric phase of these samples and the refractive index formation time was found to be short‐ er than that of nematic LCs. The response time was in the order of ms and was dominated by the formation of the internal electric field. These results indicate that the mechanism responsible for refractive index grating formation in FLCs is different from that for non-ferroelectric materials and that it is clearly related to the ferroelectric proper‐ ties of the material. The photorefractivity of FLCs was strongly affected by the properties of the FLCs themselves. Besides, with properties such as spontaneous polarization, viscos‐ ity and phase transition temperature, the homogeneity of the SS-state was also found to be a major factor. The gain coefficient, refractive index grating formation time (response time) and stability of the two-beam coupling signal were all strongly affected by the ho‐ mogeneity of the SS-state. Therefore, a highly homogeneous SS-state is necessary to create a photorefractive device. The techniques employed recently in the development of fine LC display panels will be utilized in the future in the fabrication of photorefractive devices.

### **Acknowledgment**

The authors would like to thank the Japan Science and Technology Agency (JST) S-innova‐ tion and the Canon Foundation for support.

### **Author details**

Takeo Sasaki

The asymmetric energy exchange between the signal and the pump beams proves the for‐ mation of a diffraction grating phase-shifted from the interference fringe. The grating is con‐ sidered to be based on the spatial difference in the rotational switching motion of FLC molecules. When the biased electric field was set to Eb = 0.5 V μm-1, the two-beam coupling signal was dramatically enhanced (Figure 33(b)). The gain coefficient increased from 10 cm-1 (Eb = 0) to 70 cm-1 (Eb = 0.5 V μm-1). Obviously, the biased field contributes to the formation of the diffraction grating. The internal electric field is thought to be difficult to form under an AC electric field. It has been reported that ionic conduction is the major contributor to the formation of the space-charge field in the photorefractive effect of an FLC. The anisotropic mobility of ionic species in an LC medium affects the formation of the field. In the experi‐ mental conditions reported, the interference fringe is formed across the smectic layer so that migration of ionic species occurs in the inter-layer direction. The asymmetric structure of the surface stabilized state of the FLC may lead to the asymmetric mobility of cations and anions. It was considered that the charge separated state is more effectively formed under a

The reorientational photorefractive effect based on the response of bulk polarization was observed in dye-doped FLC samples. Photorefractivity was observed only in the ferroelec‐ tric phase of these samples and the refractive index formation time was found to be short‐ er than that of nematic LCs. The response time was in the order of ms and was dominated by the formation of the internal electric field. These results indicate that the mechanism responsible for refractive index grating formation in FLCs is different from that for non-ferroelectric materials and that it is clearly related to the ferroelectric proper‐ ties of the material. The photorefractivity of FLCs was strongly affected by the properties of the FLCs themselves. Besides, with properties such as spontaneous polarization, viscos‐ ity and phase transition temperature, the homogeneity of the SS-state was also found to be a major factor. The gain coefficient, refractive index grating formation time (response time) and stability of the two-beam coupling signal were all strongly affected by the ho‐ mogeneity of the SS-state. Therefore, a highly homogeneous SS-state is necessary to create a photorefractive device. The techniques employed recently in the development of fine LC display panels will be utilized in the future in the fabrication of photorefractive devices.

The authors would like to thank the Japan Science and Technology Agency (JST) S-innova‐

biased AC field.

530 Advances in Ferroelectrics

**6. Conclusion**

**Acknowledgment**

tion and the Canon Foundation for support.

Tokyo University of Science, Japan

### **References**


[15] Skarp, K.; Handschy, M. A. Mol. Cryst. Liq. Cryst. 1988, 165, 439-569.

Appl. Phys. Lett., 2004, 85, 1329-1331.

17083-17088.

532 Advances in Ferroelectrics

[16] Sasaki, T.; Mochizuki, O.; Nakazawa, N.; Fukunaga, G.; Nakamura, T.; Noborio, K.

[17] Sasaki, T.; Mochizuki, O.; Nakazawa, Y.; Noborio, K. J. Phys. Chem. B, 2004, 108,

### *Edited by Aimé Peláiz Barranco*

Ferroelectricity is one of the most studied phenomena in the scientific community due the importance of ferroelectric materials in a wide range of applications including high dielectric constant capacitors, pyroelectric devices, transducers for medical diagnostic, piezoelectric sonars, electrooptic light valves, electromechanical transducers and ferroelectric random access memories. Actually the ferroelectricity at nanoscale receives a great attention to the development of new technologies. The demand for ferroelectric systems with specific applications enforced the in-depth research in addition to the improvement of processing and characterization techniques. This book contains twenty two chapters and offers an up-to-date view of recent research into ferroelectricity. The chapters cover various formulations, their forms (bulk, thin films, ferroelectric liquid crystals), fabrication, properties, theoretical topics and ferroelectricity at nanoscale.

Advances in Ferroelectrics

Advances in Ferroelectrics

*Edited by Aimé Peláiz Barranco*

Photo by Yurkoman / iStock