Dynamic response of polar nanoregions under an electric field in a paraelectric KTa0.61Nb0.39O3 single crystal near the para-ferroelectric phase boundary

The dynamic response of polar nanoregions under an AC electric field was investigated by measuring the frequency dependence of the quadratic electro-optic (QEO) effect in a paraelectric KTa0.61Nb0.39O3 single crystal near the para-ferroelectric phase boundary (0 °C < T-Tc < 13 °C). The QEO coefficient R11 − R12 reached values as large as 5.96 × 10−15 m2/V2 at low frequency (500 Hz) and gradually decreased to a nearly stable value as the frequency increased to 300 kHz. Furthermore, a distortion of the QEO effect was observed at low frequency and gradually disappeared as R11 − R12 tended towards stability. The giant QEO effect in the KTa0.61Nb0.39O3 crystal was attributed to the dynamic rearrangement of polar nanoregions and its anomalous distortion can be explained by considering the asymmetric distribution of polar nanoregions.

site off-centered Nb ions become correlated. The PNRs represent dynamic polarization fluctuation with a finite lifetime shorter than the average acoustic phonon period, so they could not affect the dielectric property of the crystal at the low frequency. As the temperature further decreases to an intermediate temperature T * , the strengthened correlations increase the lifetime of PNRs, eventually stabilizing the dynamic PNRs into "static" PNRs (with permanent polarization fluctuation). The reorientation of the "static" PNRs gives rise to the relaxor behavior that starts at T * , which could be characterized by the deviation of the real part of the dielectric constant from a Curie-Weiss law 15 . Accordingly, at the temperature range T c < T < T * , the PNRs start to affect the QEO and electrostrictive effects of the crystal. Finally, when the temperature decreases to T c , the PNRs transform into large ferroelectric domains, indicating the transition into the tetragonal phase.
In this study, we investigated the dynamic response of PNRs to an electric field by measuring the frequency dependence (500 Hz− 300 kHz) of the QEO effect in a KTa 0.61 Nb 0.39 O 3 crystal at the temperature range 0 °C < T-T c < 13 °C (T c + 13 °C < T * ). The contribution of the PNRs to the QEO effect was estimated quantitatively. Additionally, a distortion of the QEO effect was observed at low frequency and gradually disappeared as the frequency increased. A phenomenological model is proposed to understand the mechanism with which PNRs contribute to the QEO effect.

Experimental
The KTN crystal was grown with an improved top-seeded solution growth method 16  The QEO coefficients R 11 − R 12 at the wavelength 632.8 nm were measured by the Senarmont compensator method 17 . The components were arranged as the schematic shown in Fig. 1. The output AC voltage signal 0 of a frequency f 0 from the lock-in amplifier (SR830 and SR844, Stanford Research Systems, USA) was amplified by the voltage amplifier (WMA-300, Falco, Netherlands), then applied to the crystal along the crystallographic [010] (2.20 mm) direction. The light intensity after the Analyzer was transformed to electric signal by the photo-detector (HCA-S-400 M-SI, Femto, Germany), then filtered by the lock-in amplifier to obtain the QEO signal. As the angle α changed from 0 to π , the light intensity after the Analyzer gradually changed in the range from I max to I min . Before the measurement, the angle α was set to certain angle to make the light intensity after the Analyzer equal (I max − I min )/2. Then the electric field was applied to induce birefringence between the x and z direction. The phase change Δ Φ induced by this birefringence led to the light intensity variation Δ I. The QEO coefficient R 11 − R 12 could be obtained by equation (1).
where λ is the wavelength and l is the length of the crystal along the [100] direction, n 0 is the refractive index of the crystal without the electric field E. Besides, the electric field and the light intensity after the Analyzer were monitored by the oscilloscope. More details about the Senarmont compensator method are discussed in Ref. 17.

Results and Discussions
The temperature dependence of the dielectric constant ε r of the crystal at 10 kHz, 100 kHz and 300 kHz is shown in Fig. 2(a). The peaks of the curves indicate that the T c of the sample was 21.0 °C and its composition was KTa 0.61 Nb 0.39 O 3 , which was determined by the empirical formula presented in Ref. 18. The maximum dielectric constant was 18000, which confirmed the high quality of the crystal. The sharp peaks appear in the same position for all the examined frequencies, which is why the para-ferroelectric phase transition of the KTN crystal has been regarded as non-relaxation in the past decades. However, the relaxation factor γ, defined by equation (2) 19,20 , was found to be approximately 1.16, according to the fitting results shown in the inset of Fig. 2(a).
where C´ is the modified Curie-Weiss constant and ε max is the maximum dielectric constant. According to R. Clarke and J. C. Burfoot 20 , 1 < γ < 2 indicates that the para-ferroelectric phase transition of KTN actually corresponds to a slight relaxation, which implies the existence of PNRs above T c . To determine the temperature range T c < T < T * , at which the PNRs contribute to the QEO effect, the intermediate temperature T * = 58 °C was defined as the point where the dielectric constant deviates from the Curie-Weiss equation (γ = 1), as shown in Fig. 2(b).
The birefringence Δ n induced by the QEO effect can be expressed by equation (3).
The contribution of PNRs to the QEO effect at the temperature range 0 °C < T-T c < 13 °C was investigated using the frequency dependence (500 Hz− 300 kHz) of R 11 − R 12 , shown in Fig. 3. The coefficient R 11 − R 12 reached 5.96 × 10 −15 m 2 /V 2 at low frequency (500 Hz, 21.3 °C) and gradually decreased to a nearly stable value as the frequency increased to 300 kHz. Owing to the essentially local composition fluctuation of KTN, the size distribution of PNRs in the crystal varies near T c . As the frequency increases at a constant temperature, PNRs of relatively large size cannot respond to the electric field; consequently, the QEO coefficient decreases gradually to reach a stable value until none of the PNRs contribute to the QEO effect. The contribution of the PNRs to the QEO effect was estimated by calculating the attenuation factor β 11 12 max 11 12 11 12 where (R 11 − R 12 ) max is the maximum of R 11 − R 12 and (R 11 − R 12 ) stab is the stable value of R 11 − R 12 at high frequencies.
The attenuation factors at different temperatures were listed in Table 1. It was found that higher temperatures corresponded to lower attenuation factors; this was mainly because as the temperature increased, the average size of the PNRs decreased and fewer PNRs remained to contribute to the QEO effect. This was evidenced by the evolution trend of the PNRs with the temperature change, which was observed with a polarizing microscope and was shown in Fig. 4. In the Fig. 4(a-d), the light transmitted through the Polarizer, crystal and Analyzer in sequence. As shown by the inset of Fig. 4(a), the directions of the Polarizer and Analyzer were perpendicular to each   Fig. 4(a-d), with the increasing temperature, the darker and darker brightness of the image means that the extinction became more complete and the isotropy of PNRs densities increased gradually, i.e., the average size of PNRs decreased. Furthermore, a distortion phenomenon of the QEO effect corresponding to the asymmetric distribution of PNRs was also observed at the low frequency of the electric field. Figure 5 shows the phase change Δ Φ induced by the QEO effect under electric fields of different frequencies at 21.3 °C. It is well known that the phase change induced by the QEO effect is independent of the direction of the electric field in the paraelectric phase. Therefore, the induced phase change Δ Φ should have the same peak value for both electric field maximum ± E max along the position and negative directions. However, Fig. 5 shows that the peaks at ± E max differ. This is attributed to an overlay of linear EO signals induced by the asymmetric distribution of PNRs along the electric field directions. This phenomenon is relatively are the densities of PNRs along the six directions of spontaneous polarization and g 11 and g 12 are the polarization-optic coefficients. The electric field is applied along the x direction. The polarization along the y and z directions is perpendicular to the electric field and responds equally to the positive and negative electric field; therefore, it does not contribute to the distortion of the QEO effect. For simplification, we consider only the asymmetric polarization distribution along the x direction, namely k x + ≠ k x − and k y + = k y − = k z + = k z − = k. The polarization is redistributed under the electric field. The polarization perpendicular to the electric field is oriented along the direction of the electric field to reduce the potential energy, while the thermal kinetic energy tends to retain the initial disordered polarization distribution. When the system reaches equilibrium, the PNRs density k ⊥ along the y and z directions can be expressed as: where k B is the Boltzmann constant, W is the driving energy which is proportional to the product of the electric field E and the average moment p, α is the ratio constant. The moment p along the x direction would increase a variation when its direction is along the direction of the electric field or would decrease a variation when its direction is reverse to the direction of the electric field. Then, the refractive indices along the x and y directions become