Solution-processed conformal coating of ferroelectric polymer film and its application to multi-bit memory device

Solution-processed conformal coating of ferroelectric polymer film and its application to multi-bit memory device

Microelectronic Engineering 160 (2016) 68–72 Contents lists available at ScienceDirect Microelectronic Engineering journal homepage: www.elsevier.co...

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Microelectronic Engineering 160 (2016) 68–72

Contents lists available at ScienceDirect

Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee

Solution-processed conformal coating of ferroelectric polymer film and its application to multi-bit memory device Woo Young Kim a,⁎, Hyun Bin Shim b, Gwang-Jae Jeon b, In-Ku Kang b, Hee Chul Lee b a b

Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea

a r t i c l e

i n f o

Article history: Received 24 November 2015 Received in revised form 2 March 2016 Accepted 20 March 2016 Available online 23 March 2016 Keywords: VDF-TrFE Solubility control Patterning Multilayer Multi-bit memory

a b s t r a c t Ferroelectric multi-bit storage memory which is fabricated by means of the patterning and double-coating of ferroelectric polymer film is demonstrated. The multi-bit memory device demonstrated here has two thicknesses in a capacitor. Therefore, ferroelectric switching at each thickness arises in different voltage range. The structured capacitor with two different thicknesses is realized by optimizing two processes, i.e., the photo-lithographical patterning of the ferroelectric film and a double-coating method for the formation of the multilayer structure. Not only photo-lithographical patterning but also the double-coating method of ferroelectric film was performed with a solubility-controlled ferroelectric polymer solution created by the addition of an insoluble solvent. From electrostatic force microscopy and displacement-voltage measurements, the fabricated multi-bit storage memory operated as predicted for a multi-bit memory scheme. The solubility-controlling method suggested here will offer additional promising routes to fabricate complex organic devices based on a solution process. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Recently, ferroelectric polymers based on vinylidene fluoride (VDF) have been attracting research attention owing to the simplicity of related processes and to their remarkable material features. In terms of the processes, ferroelectric film can be formed by various means, such as physical vapor deposition [1], chemical vapor deposition [2], the Langmuir-Blodgett method [3,4] or by solution processes [5–8]. By changing the molecular structures chemically with different moieties, their physical properties could also be modulated [9,10]. Mechanical flexibility and optical transparency in the visible range make ferroelectric polymers feasible for use in transparent conducting film applications [11]. Their additional environmental inertness offers the possibility of a nonvolatile memory medium [12]. Utilizing these properties of ferroelectric polymers, applications as multi-bit memory media have been reported. In one study [13], it was demonstrated that the morphology of ferroelectric polymer film was manipulated by simple indentation with a rigid molding, resulted from a capacitor consisting of two different thicknesses. This type of capacitor can operate as a multi-bit memory device if a top electrode covers the two areas with different thicknesses at the same time. This structure is described in Fig. 1. For a clear understanding of its operation, we assume that the region with a thin thickness and the relatively thick region are designated as capacitor A (CA) and capacitor B (CB), respectively. When the remanent polarization in ferroelectric film is initially ⁎ Corresponding author. E-mail address: [email protected] (W.Y. Kim).

http://dx.doi.org/10.1016/j.mee.2016.03.037 0167-9317/© 2016 Elsevier B.V. All rights reserved.

upward (i.e., a negative external voltage applied at the top electrode while bottom electrode is connected to the ground.), we define the logic state as 11. When external voltage (VA) is applied between the coercive voltage of CA (VCA) and the coercive voltage of CB (VCB), only the dipoles in CA are switched, resulting in logic state 10. Applying a VA value smaller than VCB cannot switch the dipoles in CB. When continuing to increase the value until it exceeds VCB, VA can switch the dipoles in CB, resulting in the 00 state. From the 00 state, a |−VA | value between |−VCA | and |−VCB | makes the dipoles in CA reverse again, resulting in the 01 state. Likewise, when |−VA | is larger than |−VCB |, the overall polarization state returns to the initial state, 11. Therefore, four logic states are addressable by tuning the amplitude and polarity of VA. The proposed scheme shows that the maximum operating voltage (VMAX) depends on the thickness of CB. Moreover, the thickness of CA must be scaled down for the distinct separation of all of the states. However, an indentation process is inappropriate when seeking to realize a thin film below 100 nm, though the fabrication of such a film can be achieved by means of one-step pressing. As a result, the thickness of CB needs to be thick enough for indentation. Compared to inorganic ferroelectric film such as lead zirconium titanate (PZT), the coercive field (EC) of the ferroelectric polymer film is 10 times higher than the EC of PZT. Therefore, there is no choice for a low-voltage operating ferroelectric system but to reduce the film thickness. An additional practical problem is expected in that the rigid molding for indentation may perish due to repetitive pressing. Another study [14] also demonstrated that the morphology of ferroelectric polymer film was manipulated by hybridized process of photolithographical patterning and transferring of ferroelectric polymer

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Fig. 1. Schematic of the ferroelectric multi-bit memory: (a) 3-D device structure of the multi-bit memory; (b) 2-D device structure of the multi-bit memory, where CA and CB refers to each capacitor with thicknesses of tA and tB, respectively. (c) Initial logic state, 11 (d) logic state, 01 (e) logic state, 00 (f) logic state, 10.

film. Compared with the indented case, the maximum operating voltage was reduced below 20 V. This reduction of the maximum operating voltage could be achieved by reducing the film thickness, which was fabricated by transferring of a spin-coated thin polymer film. Though the transferred film was also fabricated by spin-coating process and it could have superior dielectric properties before it is transferred, film uniformity is expected to be wrinkled, folded and torn after the transfer process is done as the transferred film is large. Such nonuniformity caused by microscale damage may confine the applicability of transfer-based process to small and rigid wafer-scale fabrication. Therefore, a breakthrough to minimize such nonuniformity in thin and largearea film is strongly required. Thus, approach to direct spin-coating process on a patterned ferroelectric film need to be considered as an alternative to transfer process. As is well-known, it is through the spin-coating method that the thickness of a polymer film is precisely and uniformly controlled. To control the film thickness, there are many variables, such as the spin speed, spin time, solute concentration, and solvent properties, which must be considered. So, spin-coating is considered to be a highly suitable method for the fabrication of ferroelectric devices which operate at low voltages in large and flexible substrate. Moreover, conformal coating of ferroelectric polymer film can be achieved if solvent is well selected. In this work, it is demonstrated that multi-bit memory can be fabricated by the patterning and double-coating of ferroelectric polymer film such that the multi-bit memory can also operate below 20 V. The essential technology involves a double-coating method to create a multi-layer structure. To hinder the dissolution of the pre-deposited ferroelectric film, the solubility of the solution to deposit upper film was controlled by the blending in of an insoluble solvent. 2. Experiment

nonlinear variables of the spin speed and the spin time of the spincoating process were fixed at 1500 rpm and 10 s, respectively. As a result, the thickness of the ferroelectric film could be controlled almost linearly according to the ferroelectric polymer concentration of the solution. At 1 wt% (we define wt% as (mass (gram) of solute per volume of solvent [unit: g/mL]) × 100), the thickness of the ferroelectric film was approximately 90 nm. After spin-coating, the sample was annealed at 130 °C on a hot plate for 1 h to remove any solvent residue and increase the crystalline β-phase. The thickness of the ferroelectric film was measured using an α-step profilometer (Dektak 6 M, Veeco Instruments, Inc.). 2.2. Patterning of ferroelectric film The ferroelectric polymer film (FPF) was patterned using conventional photo-lithography technology, which enables the edges of the patterned ferroelectric polymer film to be defined clearly. For the first procedure, FPF of 90 nm was spin-coated onto an Au-deposited SiO2/ Si wafer. A commercially-available g-line (436 nm)-sensitive photoresist (AZ 1512, AZ electronic materials) was deposited and soft-baked at 95 °C for 90 s according to the manufacturer's guidance. Next, the sample was exposed at the g-line (436 nm) for 20 s. The developing process was performed with a commercially-available photoresist developer (AZ 300 MIF, AZ electronic materials). To remove the revealed FPF, dry etching in ambient oxygen was conducted at an oxygen flow rate of 50 mL/min, a process pressure of 40 Pa, an RF-power of 100 W, and at a process time of 60 s. Photoresist stripping was performed with a commercially-available photoresist stripper (AZ 400 T, AZ electronic materials), but the original solution of AZ 400 T was diluted with deionized water because the main solvent of AZ 400 T, N-methyl-2-pyrrolidone, dissolves the FPF. Details of the dilution ratio are described in our previous publication [15].

2.1. Ferroelectric film formation 2.3. Double-coating of the ferroelectric polymer For the ferroelectric polymer solution, poly(vinylidene fluoride-cotrifluoroethylene) (P(VDF-TrFE), 75/25 mol%, Elf Atochem, CAS No.: 28960-88-5) was purchased and dissolved in a solvent, methyl-ethylketone (MEK) at various concentrations. The solvent MEK was selected as main solvent because the vapor pressure (VP) of MEK is quite high (78 mm Hg) so the almost solvent can vaporize for spin-coating process, which enables ferroelectric film to be coated uniformly on the surface of substrate. For the case of a solvent with a low VP such as cyclohexanone (5 mmHg), in contrast, ferroelectric polymer solution can flow after spin-coating process and can change the final morphology of device. After complete dissolution with stirring, a polytetrafluoroethylene filter with a pore size of 0.22 μm was used to remove dust particles. The

Double-coating, in this work, means that a certain material was deposited by a solution-based process on the same material directly. Double-coating is possible if the solution for the ferroelectric polymer film coated secondly (the second FPF) does not dissolve the initiallycoated ferroelectric polymer film (the first FPF). However, this successive solution-based deposition process cannot be applied to FPF as the first FPF is clearly affected by exposure to the solvent in the solution of the second FPF. Therefore, the solubility of the solution used for the second FPF needs to be controlled. According to the literature [16], the solubility of a solvent can be varied by mixing a portion of an insoluble solvent into the original soluble solvent on the principle that the

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Table 1 Relationships between the concentration and the film thickness. For precise control of the film thickness, each film thickness was measured by calculating the average thickness from successive spin-coated films. Original concentration [wt%] Original thickness after one spin-coating [nm] No. of spin-coating after Et-OH mixing Total thickness after successive spin-coatings [nm] Average thickness of the Et-OH mixed solution [nm]

0.3 30 5 60 12

0.5 40 3 70 23

0.7 60 3 120 40

1.0 90 3 210 70

mixed insoluble solvent suppresses the dissolving power of the original soluble solvent. From our previous experiential results [17], the category of solvent included the liquid-phase solid-liquid mixture, the added insoluble solvent was selected to ethyl-alcohol (Et-OH) and it was used at a percentage of 20% with regard to its volume ratio in the overall mixed solution. Though various solvents were assessed for the formation of this multi-layer structure, Et-OH was selected because the coated film was flawless according to the naked eyes, with no defects found with an optical microscope as well. Moreover, Et-OH is not a danger to health compared to methyl-alcohol though the methyl-alcohol is effective solvent for double-coating. When Et-OH was used at a ratio of less than 20%, swelling and dissolving of the first FPF were observed by optical microscopy and α-step measurements. For percentages of Et-OH with exceeded 20%, multiple-coatings were feasible.

2.4. Fabrication of the multi-bit storage memory On the Au-deposited SiO2/Si wafer, the first FPF of 90 nm was spincoated and patterned. The width and the length of the final rectangular patterns were 500 μm and 1 cm, respectively. For the second FPF on the first FPF, the hybridized solution manufactured as described in Section 2.2 was prepared and spin-coated with the conditions for a target thickness of 70 nm, as listed in Table 1. Finally, Au of 50 nm was evaporated for the top electrode through a metallic shadow mask consisting periodic circular holes with area of 2.54 × 10−4 cm2 in an ultra-high vacuum at an evaporation rate of 0.3 Å/s. The final capacitor has two different thicknesses, 70 nm and 160 nm (=90 nm + 70 nm). Fig. 2 describes the overall process flow for the creation of the multi-bit memory device. To confirm the two different thicknesses, micro-scale cross-sectional image was observed with the conditions of accelerated voltage of 10 kV and magnification of 120,000 by field-emission scanning electron microscope (FE-SEM, Sirion, FEI). Electrostatic force microscope (EFM) measurements were performed with atomic force microscopy (AFM, XE-100, Park Systems Corp.). The EFM results were acquired with XEP data acquisition program and the results were processed by XEI image processing program. Both of them were provided by Park Systems Corp. The displacementvoltage relationships (D-V) were measured with a ferroelectric measurement system, RT-66 A (Radiant Technologies) after the Au was evaporated on the ferroelectric film as a top electrode.

Fig. 2. Process flow chart: (a) Au-deposited SiO2/Si substrate; (b) ferroelectric polymer, P(VDF-TrFE), spin-coating; the chemical structure of P(VDF-TrFE) is depicted and the ratio of m and n is 0.75:0.25. (c) patterning of the ferroelectric polymer film; (d) conformal double-coating of the ferroelectric polymer film; (e) cross-sectional SEM image of sample after process (d); (f) deposition of the Au top electrode; (g) photographic image of the final sample; (h) microscopic image of the final sample. The white scale bar represents a distance of 500 μm and the area of the top electrode (circle) is 2.54 × 10−4 cm2.

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Fig. 3. EFM images of the final sample: (a) 3-D topological image within 10 μm × 10 μm in size (area A). (b) 2-D image of area A; the histogram shows the thickness difference between CA and CB. (c) Amplitude response of area A after a writing process was performed within an area 9 μm × 8 μm is size (area B); (d) Phase response of area A after a writing process was performed within an area 9 μm × 8 μm in size (area B).

3. Results and discussion 3.1. Film thickness control

bit memory described in Fig. 1 must be discarded. Next, a reset pulse of − 10 V was applied in the range of 9 μm × 8 μm (area B) within area A. To write with a variable voltage ranging from 1 V to 10 V, an

The thickness of each diluted solution was extracted by dividing the total thickness by the number of successive spin-coating passes. For instance, for the original concentration of 0.3 wt%, five spin-coating and annealing were performed, as listed in Table 1. After 5 passes of spin coating and annealing were performed, the final film thickness was 60 nm. As a result, the average thickness became 12 nm for each spincoating pass. For various concentrations, each average thickness is listed in Table 1. Compared to the spin-coating results for the original concentration, the solution diluted with Et-OH reduced the total thickness by approximately 20 nm. To fabricate a multi-bit memory capacitor, two solutions of the original 1.0 wt% for a thickness of 90 nm and the diluted 1.0 wt% solution for a thickness of 70 nm were used. 3.2. EFM measurements To observe the multi-bit memory operations, electrostatic force microscope (EFM) measurements were taken. Fig. 3a shows the topology image in the range of 10 μm × 10 μm (area A). As designed in the experimental section, the thickness difference between the two layers is 90 nm, as determined from a histogram. It is worthwhile to note that the border between two different-thick films is not ambiguous but quite clear, which means the second FPF was coated on the first FPF conformally. If the border is not clear, the operation scheme of multi-

Fig. 4. Displacement-voltage relationships of CA100, CA30 and CA0. To observe the coercive positions in CA30, a derivative of CA30 is plotted as (CA30)′ by the blue solid line.

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EFM tip scanned on every horizontal line with an interval of 0.5 μm within area B. Finally, the entire area A was scanned with a read voltage of +1 V. As shown in the amplitude and phase responses in Fig. 3, polarization switching started to arise from +4 V in CA. In CB, the indication of switching appears at +9 V. In our experiment, the EFM equipment has a limited magnitude of applied voltage of 10 V; therefore, the maximum voltage of 10 V does not appear to be sufficient for the full switching of CB, with some dimness appearing in the CB region. However, it was clearly observed that switching started at +9 V in the phase response. The area out of area B exhibits that strong positive bias was previously applied, which was likely stemming from unintentional phenomena such as a self-polarization or an internal bias field [18–20]. In conclusion, the EFM measurement shows that the device scheme of one capacitor with two different thicknesses can operate as multi-bit memory. In the structure with the two thicknesses of 70 nm and 160 nm, two types of voltages are required: a middle voltage between +4 V and 8 V and a full switching voltage that exceeds +9 V. 3.3. Displacement-voltage relationship By depositing Au as a top electrode, a multi-bit memory capacitor was fabricated. It has an area ratio of CA:CB = 30:70. For comparison, two reference capacitors (CA:CB = 100:0, and CA:CB = 0:100) were also measured in terms of the displacement - voltage relationships. We assign CA100 (CA:CB = 100:0), CA0 (CA:CB = 0:100), and CA30 (CA:CB = 30:70). The value of VMAX was set to 20 V, as its corresponding electric field (20 V/160 nm = 1.25 MV/cm) was enough to reverse all of the dipoles in a ferroelectric capacitor [21]. In Fig. 4, all remanent polarizations of CA100, CA30 and CA0 are identical to 7.75 μC/cm2, as all dipole in CA and CB contribute to the overall polarization switching at 1.25 MV/ cm. At VMAX, the magnitude of displacement (D) tends toward the inequality of D(CA100) N D(CA30) N D(CA0). As the displacement was calculated by integrating the current through a capacitor, the portion of CA dominantly determines the total current flowing through the capacitor. The trace of D(CA30) matches D(CA100) below + 8 V and D(CA0) over +8 V. As expected, CA contributes to the total polarization only when the applied voltage is between VCA and VCB, and the CB capacitor contributes more than VCB. By differentiating D(CA30) with regard to the voltage, the coercive voltage can be determined quantitatively. To model a ferroelectric capacitor, a hyperbolic tangent function, tanh(x), is used for simplicity [22]. The result of the differentiation of the hyperbolic tangent function is the sech2(x) function, which reaches its maximum value at the coercive voltage. Fig. 4 shows that the peaks of the derivative of D(CA30) are positioned at the individual VCA and VCB values of CA and CB, which also supports that the ferroelectric multi-bit memory consists of two individually-operating ferroelectric capacitors. 4. Summary and conclusion In this work, capacitor-type ferroelectric multi-bit memory was demonstrated, which was realized by the hybridized process of the patterning and double-coating of a ferroelectric polymer based on a solution process. The demonstrated device for multi-bit memory operation consists of two different thicknesses in a capacitor, which enables the ferroelectric dipoles at each thickness to switch on different voltage ranges. For the patterning and double-coating of the ferroelectric polymer film without any degradation, the solubility of the solution for the upper-coated ferroelectric film was controlled by blending ethylalcohol as an insoluble solvent on the principle that a small amount of insoluble solvent added to a soluble solvent suppresses the power of the soluble solvent to dissolve a polymer solute. From 12 nm to 70 nm of the ferroelectric polymer film, ferroelectric polymer films with various thicknesses could be deposited by the above-mentioned solubility-controlling principle. According to electrostatic force

microscopy and displacement-voltage measurements of a device with two thicknesses of 70 nm and 160 nm, multi-bit memory operations for 4 states were observed. This solubility-controlling principle will be a very useful route for complex organic devices based on solution processes. Acknowledgments This work was based on the main author's Ph. D dissertation entitled of “A study on the fabrication and characterization for multi-bit memory device using multilayer structured ferroelectric polymer thin film” in Korea Advanced Institute of Science and Technology (KAIST). Also this work was partially supported by the Preparatory Project (N10110074) in KAIST. References [1] S. Horie, K. Noda, H. Yamada, K. Matshshige, K. Ishida, S. Kuwajima, Flexible programmable logic gate using organic ferroelectric multilayer, Appl. Phys. Lett. 91 (2007) 193506. [2] A.C. Rastogi, S.B. Desu, Ferroelectric poly(vinylidene fluoride) thin films grown by low-pressure chemical vapor polymerization, Chem. Vap. Depos. 12 (2006) 742–750. [3] A. Buno, S. Ducharme, V.M. Fridkin, L. Blinov, S. Palto, N. Petukhova, S. Yudin, Novel switching phenomena in ferroelectric Langmuir-Blodgett films, Appl. Phys. Lett. 67 (1995) 3975–3977. [4] A.V. Buno, V.M. Fridkin, S. Ducharme, L.M. Blinov, S.P. Palto, A.V. Sorokin, S.G. Yudin, A. Zlatkin, Two-dimensional ferroelectric films, Nature 391 (1998) 874–877. [5] R.C.G. Naber, C. Tanase, P.W.M. Blom, G.H. Gelinck, A.W. Marsem, F.J. Touwslager, S. Setayesh, D.M. de Leeuw, High-performance solution-processed polymer ferroelectric field-effect transistors, Nat. Mater. 4 (2005) 243–248. [6] H. Xu, X. Fang, X. Liu, S. Wu, Y. Gu, X. Meng, J. Sun, J. Chu, Fabrication and properties of solution processed all polymer thin-film ferroelectric device, J. Appl. Polym. Sci. 120 (2010) 1510–1513. [7] J. Li, Z. Sun, F. Yan, Solution processable low-voltage organic thin film transistors with high-k relaxor ferroelectric polymer as gate insulator, Adv. Mater. 24 (2012) 88–93. [8] S.-H. Bae, O. Kahya, B.K. Sharma, J. Kwon, H.J. Cho, B. Ozyilmaz, J.-H. Ahn, GrapheneP(VDF-TrFE) multilayer film for flexible applications, ACS Nano 7 (2013) 3130–3138. [9] X. Chen, L. Liu, S.-Z. Liu, Y.-S. Cui, X.-Z. Chen, H.-X. Ge, Q.-D. Shen, P(VDF-TrFE-CFE) terpolymer thin-film for high performance nonvolatile memory, Appl. Phys. Lett. 102 (2013) 063103. [10] J.R. Kim, S.W. Choi, S.M. Jo, W.S. Lee, B.C. Kim, Characterization and properties of P(VDF-HFP)-based fibrous polymer electrolyte membrane prepared by electrospinning, J. Electrochem. Soc. 152 (2005) A295–A300. [11] G.-X. Ni, Y. Zheng, S. Bae, C.Y. Tan, O. Kahya, J. Wu, B.H. Hong, K. Yao, B. Ozyilmaz, Graphene? Ferroelectric hybrid structure for flexible transparent electrodes, ACS Nano 6 (2012) 3935–3942. [12] M. Poulsen, S. Ducharme, Why ferroelectric polyvinylidene fluoride is special, IEEE Trans. Dielectr. Electr. Insul. 17 (2010) 1028–1035. [13] A.K. Tripathi, A.J.J.M. van Breemen, J. Shen, Q. Gao, M.G. Ivan, K. Reimann, E.R. Meinders, G.H. Gelinck, Multilevel information storage in ferroelectric polymer memories, Adv. Mater. 23 (2011) 4146–4151. [14] W.Y. Kim, H.C. Lee, Low-voltage nonvolatile multi-bit memory fabricated by the patterning and transferring of ferroelectric polymer film, Org. Electron. 19 (2015) 1–6. [15] W.Y. Kim, H.C. Lee, Development of manipulation technology of ferroelectric polymer film: photo-lithographic patterning and multilayer formation, Microelectron. Eng. 88 (2011) 1576–1581. [16] S.V.R. Gowariker, N.V. Viswanathan, J. Sreedhar, Polymer ScienceISBN: 0-85226307-4, Chapter 13 1986 377. [17] W.Y. Kim, H.C. Lee, J.-H. Bae, Fabrication and characterization of ferroelectric multilayered films fabricated by using solvent blending, J. Korean Phys. Soc. 61 (2012) 1518–1522. [18] A.L. Kholkin, K.G. Brooks, D.V. Taylor, S. Hiboux, N. Setter, Self-polarization effect in Pb(Zr,Ti)O3 thin films, Integr. Ferroelectr. 22 (1998) 525–533. [19] M. Park, S. Hong, J. Kim, Y. Kim, S. Bühlmann, Y.K. Kim, K. No, Piezoresponse force microscopy studies of PbTiO3 thin films grown via layer-by-layer gas phase reaction, Appl. Phys. Lett. 94 (2009) 092901. [20] Y. Kim, W. Kim, H. Choi, S. Hong, H. Ko, H. Lee, K. No, Nanoscale domain growth dynamics of ferroelectric poly(vinylidene fluoride-co-trifluoroethylene) thin films, Appl. Phys. Lett. 96 (2010) 012908. [21] T. Nakajima, R. Abe, Y. Takahashi, T. Furukawa, Intrinsic switching characteristics of ferroelectric ultrathin vinylidene fluoride/trifluoroethylene copolymer films revealed using Au electrode, Jpn. J. Appl. Phys. 44 (2005) L1385–L1388. [22] S.L. Miller, R.D. Nasby, J.R. Schwank, M.S. Rodgers, P.V. Dressendorfer, Device modeling of ferroelectric capacitors, J. Appl. Phys. 68 (1990) 6463–6471.