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Abstract
Relaxor ferroelectrics have found significant applications in various functional devices in photonics and electronics because of their extraordinary optical and dielectric properties which are directly associated with the polarization relaxation dynamics. However, the related mechanism currently remains not clearly understood, especially for the polar domain reversal in nanostructured relaxor ferroelectrics. This paper reports an observation of intriguing spontaneous superlattices formation of self-ordered nanostructures in a ferroelectric perovskite of KTN:Cu by cooling the crystal below the temperature of its dielectric maximum. More importantly, the strong diffraction induced by the spontaneously formed superlattices can be switched with an AC electric field after a pre-polarization using a DC electric field. The experimental results indicate the switching effect originates from an asymmetric reversal process of the polarization in nanoscale domains. The study not only facilitates further understanding the physics of the ferroelectric relaxation process, but also have potential applications in optical switching, imaging and display.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Relaxor ferroelectrics have received special attention in recent years in both practical applications [1–4] and scientific research sectors [5–9] due to their giant piezoelectric coefficients [10,11] and unique electro-optic (EO) effects [12,13]. Compared to conventional ferroelectrics, the optical performances of the relaxor ferroelectrics depend prominently on the polarization relaxation dynamics rather than the structural phase transition [14]. At microscopic scales, the relaxation process is directly associated with the domain reversal property which can be characterized by a variety of approaches such as Raman scattering [15], infrared absorption [16], and strain-induced and stress-induced birefringence [17] under various electric fields or temperature fields [18,19]. However, the dynamics of the domain reversal is complicated, especially for the mechanism of individual nanostructured relaxor ferroelectrics which still remains unclear.
KTa1-xNbxO3 (KTN) is a relaxor ferroelectric and transparent in a wide range of wavelengths from 0.4 µm to 4 µm. In the vicinity of its dielectric peak (at temperature of Tm), the KTN possesses large permittivity and high EO coefficient due to its unique property of polarization relaxation [20]. As a solid solution of KTaO3 and KNbO3, the KTN has an adjustable Tm depending on its Ta-to-Nb concentration ratio. The KTN and its derivatives have shown a variety of potential applications in functional devices such as optical modulators [21,22], laser beam scanners [23,24] and emerging devices based on novel optical nonlinearities [13]. Recently, D. Pierangeli et al. report a super-crystal of KTN:Li with a 3D periodical structure leading to a unique X-ray-like optical diffraction [25]. In this paper, we report an observation of interesting optical switching effect of the spontaneously formed superlattices in a KTN:Cu crystal. We found the switching effect originates from an asymmetric reversal process of the polarization in nanoscale domains. Our study not only facilitates further understanding the physics of the ferroelectric phase transition and the relaxation process, but also have potential applications in functional devices such as optical switches, laser scanners, sensing and ladars. The unique switchable diffraction effects may also be applied for hyperspectral imaging, light structured high-resolution imaging, and holographic waveguides for near-eye display.
2. Method
2.1 Sample preparation and experimental setups
The KTa1−xNbxO3:Cu crystal, with x = 0.372, is grown by using the off-center TSSG method [26] with a grating period of 1.52 µm. The growth direction and the grating vector are both along the y axis. Tm = 14.5 °C is obtained based on the dielectric spectrum measured by using an LCR (Inductance, Capacitance, Resistance) meter with a sinusoidal signal (1-V peak amplitude and 1-kHz frequency). Cu ions with 0.1% (mol) is doped into the pure KTN crystal to improve the dielectric property by decreasing the resistance of the grains [27]. After growth, the crystal is cut into a cuboid shape in the size of 3.38(x)×1.89(y)×5.73(z) mm3 along the basic crystallographic directions with all faces optically polished. Silver colloid is coated on the two x×z faces as electrodes and the sample is connected to a signal source (Tektronix, AFG3102C) and a voltage amplifier (PINTECH, HA-405). The highest output voltage is limited to 400 Vpp by the amplifier. The temperature of the crystal is set by a Peltier linked to a temperature controller (TEC Source 5300) with 0.01 °C precision. The diffracted power during the switching is measured by a photodetector (ALPHALAS, UPD-500) connected to an oscilloscope (Tektronix, TBS1000B). The rising times of the amplifier and the photodetector are 1.580 µs and 0.5 ns, respectively.
2.2 Optical switching experiments
In the optical experiments, a continuous-wave 532 nm laser is polarized along the x direction and focused into the KTN:Cu crystal by a lens. The electric field is applied in the y direction. A photodetector is placed behind the sample to monitor the diffracted powers. Before conducting each experiment, the sample temperature is reduced from 34.5 °C to 11.5 °C and kept for 10 minutes to stabilize the crystal relaxation and the 90° domain structures. It should be noted that the input laser has only 300 µW to minimize the photorefractive effect of Cu ions [28]. The optical powers are 113 µW, 12 µW and 0.6 µW, respectively, corresponding to the zero (I0), the 1st (I1) and the 1'st (I1’) order diffracted beams as shown in Fig. 1(b). In the experiments, we monitored the lower right beam for I1 and the bottom beam for I1’, as illustrated in the experimental setup. The optical powers are normalized with 1.0 corresponding to an optical power of 12 µW. After polarization relaxation, the crystal is implemented with a -200 V DC electric field for pre-polarization. Then, the DC electric field is turned off and AC electric fields with different amplitudes and frequencies are applied for certain measurements.
Fig. 1. Electric-field-induced switchable superlattices in KTN:Cu crystals. (a) The diffraction of the intrinsic grating at Tm+20 °C (The red dash line circles indicate the positions of diffracted beams which are dim due to the weak intrinsic grating). (b) The strong diffraction induced by the spontaneously formed superlattices at Tm-3 °C. (c-d) Optical switching of diffraction due to the switchable superlattices under 1 Hz square electric fields. A video clip regarding the optical switching is supplied as supplementary materials (see Visualization 1). (e) Pre-polarization process by implementing a DC electric field of −200 V. The left and the right axes represent I0 and I1 respectively. The inset shows a schematic of the experimental setup. (f) Time-dependent curves of driving electric field and resultant optical diffraction. More details of the experimentation can be found in the experimental section.
3. Results
In the experiments, we initially keep the sample at 34.5 °C (well above Tm) to eliminate the spontaneous polarization. The crystal is transparent in its cubic phase and an incident laser beam propagating through the crystal is diffracted by the intrinsic grating-like structure as shown in Fig. 1(a). However, after decreasing the temperature to 11.5 °C, we observe a spontaneous formation process of a super-lattice nanostructure occurring in the crystal. At this moment when a laser beam is incident to the crystal, the transmitted beam shows a far-field diffraction pattern (Fig. 1(b)), which is indeed the Fourier transform of the spontaneously formed periodic nanostructure. Then, we implement a direct current electric field of -200 V on the crystal for pre-polarization after keeping the crystal at 11.5 °C for about 10 minutes. We found that the zero-order diffracted beam (I0) increases while the higher order diffraction (I1) decreases, as shown in Fig. 1(e). After this pre-polarization, the super-lattice nanostructure becomes switchable between two diffraction patterns (Fig. 1(c) and (d)) by using a square-wave electric field (Fig. 1(f)). D. Pierangeli et al. have also observed a similar X-ray-like diffraction in a KTN:Li crystal [25]. However, to the best of our knowledge, our study is the first work showing the spontaneously formed super-lattice nanostructure can be switched by using an electric field.
4. Discussion
4.1 Asymmetric domain reversal and optical switching
The mechanism of the domain reversal is not yet fully understood. Two models, the Kolmogorov-Avrami-Ishibashi (KAI) [29–32] and the Nucleation [33–36], are often used to approximately explain the domain reversal dynamics. The KAI model is valid in some cases especially for single-crystalline bulk ferroelectrics because it assumes the crystal is infinite in dimension and allows domains nucleating at different locations to overlap each other. Thus, only the grown domains themselves affects the dynamic of domain reversal [37,38]. On the other hand, the Nucleation models are generally applied to polycrystalline ferroelectrics or ferroelectric thin films in which domain walls and grain boundaries play important roles in the domain reversals. Under the electric field, the domain reversals vary locally and individually at different stages of the nucleation process [33,39–41]. As a result, the reversal time for different regions obeys probabilistic distribution depending on the localized structures associated with 90° domain walls [35] or grain boundaries [39], which leads to the variation of the switching time [33]. In our study, we found the time response characteristics of the super-lattice switching agrees with the Nucleation model, which we consider as an asymmetric domain reversal due to its directional probabilistic distribution induced by the electric field.
For simplicity, we analyze the super-lattice structure in x-y plane since x and z directions are equivalent with the applied electric fields along the y direction. In the initial state without external field, the typical 180° and 90° domain configurations in perovskite ferroelectrics are confined by the periodical gratings in the KTN:Cu crystal, forming a periodic ferroelectric state with ordered polar domain walls (red lines in Fig. 2(a)). The probability that a nanoscale polar domain is aligned to y or -y direction equals, that is py0= p-y0. However, the alignment of nanoscale polar domains can be reversed by applying an electric field that exceeds the critical value Ec of the crystal [42]. The probability of the domains aligned to the direction of electric field is larger than that of opposite direction (i.e., py1> p-y1, as shown in Fig. 2(b)). As a result, the 90° domain configuration fades out. According to the Nucleation models [33,38], the strength of electric field and the applying time are two key factors affecting the domain reversal dynamics in certain materials. If the amplitude of the electric field keeps constant (106 V/mm in our case), the applying time determines the probability of the reversed domains. The relatively long period (∼80 s) in the pre-polarization causes the value of py1-p-y1 approaching to 100% [33,40]. In addition, the process also weakens the domain walls leading to the attenuation of the laser diffraction (Fig. 1(e)). However, the reversed domains always tend to return to their intrinsic ferroelectric state of the spontaneously formed super lattices. An inversed electric field can facilitate such process enabling the recovery of the “fade-out” 90° domain configurations induced by the pre-polarization. The probability of the reversed domains satisfies py1-p-y1 > py2-p-y2 > 0, as shown in Fig. 2(c). As such, an optical switching effect is obtained with a laser beam probing the electric-field switchable super-lattice structure (Fig. 2(d) and 2(f)).
Fig. 2. Mechanism of the switching effect in KTN:Cu crystal. (a-c) The domain walls (red lines) of nanoscale polar domain structures (blue-color ellipses, color gradient represents the polarization strength) leading to the X-ray-like optical diffraction with different polarization (a) py0 = p-y0 = 50%, (b) py1 - p-y1>0, (c) py1 - p-y1> py2 - p-y2 > 0, where py and p-y represent the probabilities of the polar domains reorienting to y and -y directions, respectively. Red-color lines with deeper color have relatively higher refractive index modulation. The ellipses with red contour imply polar domains oriented by external electric field. (d) The rising time of optical switching is 143 µs under a 1 Hz square-wave driving electric field. (e) Slow component of time response under a square-wave electric field with 50 s period (The red dashed arrow indicates the fast component corresponding to Fig. 2d). (f) Electric-field dependent amplitude (green line) and response time (blue line) of switching under 1 Hz frequency.
The experimental results of time response suggest there are two mechanisms concurring in the switchable super lattices. We found that the diffracted signal rises quickly with a time constant of 143 µs (Fig. 2(d)) as soon as the electric field is switched from -200 V to 200 V. However, there also exists a steady mechanism showing the optical diffraction gradually enhanced with a much slow speed if the electric field is applied for an extended time period (Fig. 2(e)). In the perovskite-type KTN crystals, the cation sublattices can be easily shifted relative to the negatively charged oxygens by the external electric field, producing a polarization change. This process of polarization reversal is very fast especially in the nanoscale polarization regions that have less interaction with surrounding [33]. On the other hand, with applying continuously a constant electric field, any further changes have to overcome the potential barriers especially around grain boundaries [38,39] that significantly slow down the polarization reversal process. As other studies [33,38,39] have indicated, the grain boundaries limit the nucleation time of domains and affect the polarization reversal process depending on the size of domains and electric fields. For PZT films with a thickness of 135 nm, the nucleation time ranges broadly from several nanoseconds to seconds under electric fields. In fact, our crystal with periodic grating structure (the period of grating is calculated to be 1.52 µm based on the diffraction data shown in Fig. 1) is much similar to a pileup of ferroelectric films with a 1.52 µm period, rather than a bulk ferroelectrics. By considering the much larger interval (1.52 µm) of ferroelectric films in the Cu:KTN sample compared to that of PZT film (135 nm), we can anticipate the existence of a slow responding component of the nucleation process, which approximates to several ten seconds observed in the experiments. However, the fast component of the nucleation process dominates in the optical switching as shown in Fig. 2(e). Figure 2(f) shows the switching amplitude and time response as a function of the applied electric field. The switching amplitude increases but the response time decreases with increasing the electric field. We only plot the component of the fast time response since the slow one is less important for switching applications. The coercive field Ec is about 99 V/mm in our case, which is close to the values reported in Ref. [18] (<100 V/mm) and Ref. [25] (≈140 V/mm) in the vicinity of Tm. The above results agree with the presumption of Nucleation models [42].
4.2 Characterization of the polarization reversal current
To verify the above analysis and to further understand the mechanism of the polarization reversal, we investigate the response of the generated electric currents in the crystal, which is directly associated to the domain reversal dynamics [33,43,44]. The measurement procedure is similar to that used in Ref. [33]. A series of square wave voltages (Fig. 3(a)) are implemented to the sample after the pre-polarization process (t < 0.5s). At the front edge (t = 1.0 s) of the second pulse, the current response related to non-polarization reversal is obtained since the applied electric field has the same direction as the pre-polarization. We found that the current decays exponentially with a time constant of τ1 = 24.2µs (Fig. 3(b)). Then, we apply a square pulse of electric field with a direction opposing to the previous one. As such, the process of the polarization reversal occurs. At t = 3.0s, we apply the fourth pulse with the same electric-field direction as the first and the second ones. Simultaneously, we measure the current response (Fig. 3(c)) as well as the optical response (Fig. 3(d)) in the experiments. According to Refs. [31,41], the current responses in Fig. 3(c) includes two parts, the polarization reversal current and non-polarization reversal current. Thus, we fit the curve with two time constants, one is the same as that in Fig. 3(b) and the other is τ2 = 139.6 µs, which is very close to the optical response that has a time constant τ3 = 141.5µs. The experiment results further validate that the polarization reversal process contributes to the optical effects of the switchable supperlattices.
Fig. 3. Electric currents and optical response in polarization reversal process. (a) Sequence of electric-field pulses used in experiments. (b) Non-polarization reversal current with a single-exponential fitting. (c) Current response with two time constants. (d) Optical response with a single-exponential fitting.
4.3 Optical switching performances
In this section, we study the response of the optical switching on the driving frequency of the electric field after the pre-polarization. As shown in Fig. 4(a), the switching effect can be observed with a driving frequency up to 1 kHz. With further increasing the driving frequency, the switching contract of peak-to-valley value is attenuated due to the thermal effect induced by the dielectric loss in KTN crystals [45]. However, we believe the frequency response depends only on the time response of polarization domain reversal. The switching time can be much improved by eliminating the thermal effect such as using a thin film of the crystal [46]. In addition, by growing KTN crystals with smaller grating periods, the size of the polar domains can be significantly reduced, enabling the improvement of the time response due to the weak interaction between grain boundaries [42]. Due to the high contrast (about 19 dB) of the optical switching, it is possible to implement a multi-level switching by using our device as shown in Fig. 4(b). The multi-level technology is important for date transport and data storage. For example, the last technology of solid-state drive and flash memory exploits multi-level cells, which can store more than one bit per cell, enabling high data storage capacity in a very compact dimension with low cost. Similarly, the multi-level switching technique based on electro-optic effects facilitates implementation of high-density and high-speed data transport.
Fig. 4. (a) Optical switching results with various frequencies of 400 Vpp square electric fields; (b) Multi-level optical switching results driven by different electric fields (-200 V,0,+200 V).
4.4 Exclusion of EO effect
The EO effect is eliminated in our study because the electric field is applied in the y direction that is orthogonal to the polarization direction of the incident laser. According to Ref. [25], when the applied electric field is parallel to the light polarization, the electro-optic coefficient, g11, is 0.16 m4C−2 for KLTN crystal. On the other hand, when the electric field is orthogonal to the light polarization, the coefficient, g12, is only 0.02 m4C−2 that is much smaller than g11. In our experiments, however, the contrast of the optical switching is nearly 19 dB with an orthogonally-polarized light filed. Therefore, the EO-induced diffraction effect in Ref. [25] does not exist in our experiments.
5. Conclusion
We have studied the spontaneous formation of superlattices in a ferroelectric perovskite of KTN:Cu by reducing the temperature below Tm. The spontaneously formed periodic nanostructure of superlattices induces a highly-efficient laser diffraction which is switchable under the electric field. The experimental results suggest two mechanisms concur in the intriguing observation of switchable superlattices. The fast reaction may originate from the spatial shifts of charged sublattices by the electric field in the nanoscale polar regions that have less interaction with surroundings. After this initial process, overcoming potential barriers especially around grain boundaries further enhances the polarization reversal. Our analysis agrees well with the experimental characterization of the polarization reversal currents generated in the crystal, which is directly associated to the domain reversal dynamics. Currently, the domain reversal dynamics in relaxor ferroelectrics has been mainly studied in Pb(Mg1/3Nb2/3)O3, Pb(Zn1/3Nb2/3)O3 or BaTiO3 films [8–11]. To the best of our knowledge, this is the first work showing the spontaneously formed super-lattice nanostructure can be switched by using an electric field. It should be emphasized that the study in Ref. [25] focused on the EO effect and the resultant X-ray-like diffraction under static electric field. In this paper, we exploit an alternative electric field to obtain an interesting optical switching effect that originates from the mechanism of asymmetric polarization reversal process rather than the EO effect. The related study using KTN crystals will be an important complement in the research of ferroelectric materials. The switchable device can be further engineered by controlling the grating period and the composition of the KTN crystals, providing the flexibility in a variety of device applications. The mechanisms involved in the switchable spontaneously formed superlattices are significant for not only better understanding the physics of the relaxor ferroelectrics but also the development of novel functional components for practical applications and emerging technologies.
Funding
National Natural Science Foundation of China (61575097, 11704201, 51672164, 51975192); Natural Science Foundation of Tianjin City (17JCQNJC01600); Fundamental Research Funds for the Central Universities; Natural Science Foundation of Shandong Province (2016ZRC01087, ZR2017MEM016); Project of Hubei University of Arts and Science (XK2019055); Doctoral Research Foundation Project of Hubei University of Arts and Science (2059039).
References
1. H. Takenaka, I. Grinberg, S. Liu, and A. M. Rappe, “Slush-like polar structures in single-crystal relaxors,” Nature 546(7658), 391–395 (2017). [CrossRef]
2. E. DelRe, F. Di Mei, J. Parravicini, G. Parravicini, A. J. Agranat, and C. Conti, “Subwavelength anti-diffracting beams propagating over more than 1,000 Rayleigh lengths,” Nat. Photonics 9(4), 228–232 (2015). [CrossRef]
3. F. D. Mei, P. Caramazza, D. Pierangeli, G. D. Domenico, H. Ilan, A. J. Agranat, P. Di Porto, and E. DelRe, “Intrinsic negative mass from nonlinearity,” Phys. Rev. Lett. 116(15), 153902 (2016). [CrossRef]
4. D. Wang, A. A. Bokov, Z. G. Ye, J. Hlinka, and L. Bellaiche, “Subterahertz dielectric relaxation in lead-free Ba(Zr,Ti)O3 relaxor ferroelectrics,” Nat. Commun. 7(1), 11014 (2016). [CrossRef]
5. F. Di Mei, L. Falsi, M. Flammini, D. Pierangeli, P. D. Porto, A. J. Agranat, and E. DelRe, “Giant broadband refraction in the visible in a ferroelectric perovskite,” Nat. Photonics 12(12), 734–738 (2018). [CrossRef]
6. J. Parravicini, E. DelRe, A. J. Agranatc, and G. Parravicinid, “Liquid-solid directional composites and anisotropic dipolar phases of polar nanoregions in disordered perovskite,” Nanoscale 9(27), 9572–9580 (2017). [CrossRef]
7. T. P. Dougherty, G. P. Wiederrecht, K. A. Nelson, M. H. Garrett, H. P. Jensen, and C. Warde, “Femtosecond resolution of soft mode dynamics in structural phase transitions,” Science 258(5083), 770–774 (1992). [CrossRef]
8. G. Y. Xu, J. S. Wen, C. Stock, and P. M. Gehring, “Phase instability induced by polar nanoregions in a relaxor ferroelectric system,” Nat. Mater. 7(7), 562–566 (2008). [CrossRef]
9. W. Dmowski, S. B. Vakhrushev, I. K. Jeong, M. P. Hehlen, F. Trouw, and T. Egami, “Local lattice dynamics and the origin of the relaxor ferroelectric behavior,” Phys. Rev. Lett. 100(13), 137602 (2008). [CrossRef]
10. H. X. Fu and R. E. Cohen, “Polarization rotation mechanism for ultrahigh electromechanical response in single-crystal piezoelectrics,” Nature 403(6767), 281–283 (2000). [CrossRef]
11. X. B. Ren, “Large electric-field-induced strain in ferroelectric crystals by point-defect-mediated reversible domain switching,” Nat. Mater. 3(2), 91–94 (2004). [CrossRef]
12. M. Sawaki and H. Motai, “Successful preparation of KTN crystals with the highest reported electro-optic effect and the potential for providing a great improvement in optical device performance,” NTT Tech. Rev. 1(9), 56–58 (2003).
13. E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011). [CrossRef]
14. B. Dkhil and J. M. Kiat, “Electric-field-induced polarization in the ergodic and nonergodic states of PbMg1/3Nb2/3O3 relaxor,” J. Appl. Phys. 90(9), 4676–4681 (2001). [CrossRef]
15. J. Toulouse, P. DiAntonio, B. E. Vugmeister, X. M. Wang, and L. A. Knauss, “Precursor effects and ferroelectric macroregions in KTa1-xNbxO3 and K1-yLiyTaO3,” Phys. Rev. Lett. 68(2), 232–235 (1992). [CrossRef]
16. A. Pashkin, V. Železný, J. Petzelt, and J. Phys, “Infrared spectroscopy of KTa1-xNbxO3 crystals,” J. Phys.: Condens. Matter 17(25), L265–L270 (2005). [CrossRef]
17. W. Kleemann, V. Schönknecht, D. Sommer, and D. Rytz, “Dissipative quantum tunneling and absence of quadrupolar freezing in glassy K0.989Li0.011TaO3,” Phys. Rev. Lett. 66(6), 762–765 (1991). [CrossRef]
18. P. Tan, H. Tian, C. P. Hu, X. D. Meng, C. Y. Mao, F. Huang, G. Shi, and Z. X. Zhou, “Temperature field driven polar nanoregions in KTa1-xNbxO3,” Appl. Phys. Lett. 109(25), 252904 (2016). [CrossRef]
19. X. Zhang, S. He, Z. Zhao, P. F. Wu, X. P. Wang, and H. L. Liu, “Abnormal optical anisotropy in correlated disorder KTa1-xNbxO3:Cu with refractive index gradient,” Sci. Rep. 8(1), 2892 (2018). [CrossRef]
20. T. Imai, S. Toyoda, J. Miyazu, J. Kobayashi, and S. Kojima, “Permittivity changes induced by injected electrons and field-induced phase transition in KTa1-xNbxO3 optical beam deflectors,” Jpn. J. Appl. Phys. 53(9S), 09PB02 (2014). [CrossRef]
21. Y. C. Jia, M. Winkler, C. Cheng, F. Chen, L. Kirste, V. Cimalla, A. Žukauskaitė, J. Szabados, I. Breunig, and K. Buse, “Pulsed laser deposition of ferroelectric potassium tantalate-niobate optical waveguiding thin films,” Opt. Mater. Express 8(3), 541–548 (2018). [CrossRef]
22. Y. C. Chang, C. Wang, S. Z. Yin, R. C. Hoffman, and A. G. Mott, “Kovacs effect enhanced broadband large field of view electro-optic modulators in nanodisordered KTN crystals,” Opt. Express 21(15), 17760–17768 (2013). [CrossRef]
23. K. Nakamura, J. Miyazu, Y. Sasaki, T. Imai, M. Sasaura, and K. Fujiura, “Space-charge-controlled electro-optic effect: Optical beam deflection by electro-optic effect and space-charge-controlled electrical conduction,” J. Appl. Phys. 104(1), 013105 (2008). [CrossRef]
24. X. P. Wang, B. Liu, Y. G. Yang, Y. Y. Zhang, X. S. Lv, G. L. Hong, R. Shu, H. H. Yu, and J. Y. Wang, “Anomalous laser deflection phenomenon based on the interaction of electro-optic and graded refractivity effects in Cu-doped KTa1-xNbxO3 crystal,” Appl. Phys. Lett. 105(5), 051910 (2014). [CrossRef]
25. D. Pierangeli, M. Ferraro, F. D. Mei, G. D. Domenico, C. E. M. de Oliveira, A. J. Agranat, and E. DelRe, “Super-crystals in composite ferroelectrics,” Nat. Commun. 7(1), 10674 (2016). [CrossRef]
26. V. Bermudez, M. D. Serrano, and E. Dieguez, “Bulk periodic poled lithium niobate crystals doped with Er and Yb,” J. Cryst. Growth 200(1-2), 185–190 (1999). [CrossRef]
27. X. P. Wang, Q. G. Li, Y. G. Yang, Y. Y. Zhang, X. S. Lv, L. Wei, B. Liu, J. H. Xu, L. Ma, and J. Y. Wang, “Optical, dielectric and ferroelectric properties of KTa0.63Nb0.37O3 and Cu doped KTa0.63Nb0.37O3 single crystals,” J. Mater. Sci.: Mater. Electron. 27(12), 13075–13079 (2016). [CrossRef]
28. Y. Kabessa, A. Yativ, H. E. Ilan, and A. J. Agranat, “Electro-optical modulation with immunity to optical damage by bipolar operation in potassium lithium tantalate niobite,” Opt. Express 23(4), 4348–4356 (2015). [CrossRef]
29. Y. Ishibashi and Y. Takagi, “Ferroelectric domain switching,” J. Phys. Soc. Jpn. 31(2), 506–510 (1971). [CrossRef]
30. H. Orihara, S. Hashimoto, and Y. Ishibashi, “A Theory of D-E Hysteresis Loop Based on the Avrami Model,” J. Phys. Soc. Jpn. 63(3), 1031–1035 (1994). [CrossRef]
31. A. Kolmogorov, “Zur Statistik der Kristallisationsvorgänge in Metallen,” Izv. Akad. Nauk. Ser. Math. 1(3), 355–359 (1937).
32. M. Avrami, “Kinetics of phase change. II transformation-time relations for random distribution of nuclei,” J. Chem. Phys. 8(2), 212–224 (1940). [CrossRef]
33. A. K. Tagantsev, I. Stolichnov, N. Setter, J. S. Cross, and M. Tsukada, “Non-Kolmogorov-Avrami switching kinetics in ferroelectric thin films,” Phys. Rev. B 66(21), 214109 (2002). [CrossRef]
34. I. Stolichnov, A. Tagantsev, N. Setter, J. S. Cross, and M. Tsukada, “Crossover between nucleation-controlled kinetics and domain wall motion kinetics of polarization reversal in ferroelectric films,” Appl. Phys. Lett. 83(16), 3362–3364 (2003). [CrossRef]
35. W. Li and M. Alexe, “Investigation on switching kinetics in epitaxial Pb(Zr0.2Ti0.8)O3 ferroelectric thin films: Role of the 90° domain walls,” Appl. Phys. Lett. 91(26), 262903 (2007). [CrossRef]
36. J. Y. Jo, H. S. Han, J. G. Yoon, T. K. Song, S. H. Kim, and T. W. Noh, “Domain switching kinetics in disordered ferroelectric thin films,” Phys. Rev. Lett. 99(26), 267602 (2007). [CrossRef]
37. D. J. Jung, M. Dawber, J. F. Scott, L. J. Sinnamon, and J. M. Gregg, “Switching dynamics in ferroelectric thin films: an experimental survey,” Integr. Ferroelectr. 48(1), 59–68 (2002). [CrossRef]
38. X. J. Lou, “Statistical switching kinetics of ferroelectrics,” J. Phys.: Condens. Matter 21(1), 012207 (2009). [CrossRef]
39. I. Stolichnov, L. Malin, E. Colla, and A. K. Tagantsev, “Microscopic aspects of the region-by-region polarization reversal kinetics of polycrystalline ferroelectric Pb(Zr,Ti)O3 films,” Appl. Phys. Lett. 86(1), 012902 (2005). [CrossRef]
40. O. Lohse, M. Grossmann, U. Boettger, D. Bolten, and R. Waser, “Relaxation mechanism of ferroelectric switching in Pb(Zr,Ti)O3 thin films,” J. Appl. Phys. 89(4), 2332–2336 (2001). [CrossRef]
41. T. Takaaki, S. M. Num, Y. B. Kil, and S. Wada, “High frequency measurements of P-E hysteresis curves of pzt thin films,” Ferroelectrics 259(1), 43–48 (2001). [CrossRef]
42. S. Wan and K. Bowman, “Modeling of electric field induced texture in lead zirconate titanate ceramics,” J. Mater. Res. 16(8), 2306–2313 (2001). [CrossRef]
43. M. Grossmann, D. Bolten, O. Lohse, and U. Boettger, “Correlation between switching and fatigue in PbZr0.3Ti0.7O3 thin films,” Appl. Phys. Lett. 77(12), 1894 (2000). [CrossRef]
44. W. J. Merz, “Switching time in ferroelectric BaTiO3 and its dependence on crystal thickness,” J. Appl. Phys. 27(8), 938–943 (1956). [CrossRef]
45. X. Zhang, H. L. Liu, Z. Zhao, X. P. Wang, and P. F. Wu, “Electric-field control of the ferro-paraelectric phase transition in Cu: KTN crystals,” Opt. Express 25(23), 28776–28782 (2017). [CrossRef]
46. S. Tatsumi, Y. Sasaki, S. Toyoda, T. Imai, J. Kobayashi, and T. Sakamoto, “700 kHz beam scanning using electro-optic KTN planar optical deflector,” Proc. SPIE 9744, 97440L (2016). [CrossRef]
