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Abstract
Spatially analyzing non-uniform distributions of electric phenomena such as electric field and permittivity in ferroelectric devices is very challenging. In this study, we apply an optical beam deflection method to map the non-uniform electric phenomena in relaxor ferroelectric potassium tantalate niobate (KTN) crystals. To adequately correlate the physical parameters and their spatial distributions in KTN crystals, a general model that describes the giant electro-optic response and associated beam deflection is derived. The proposed model is in good agreement with the experimental results and is envisioned to be useful for analyzing electric field-induced phenomena in non-linear dielectric materials and devices.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Relaxor ferroelectrics have been investigated extensively because of their unique electro-mechanical (EM) and electro-optic (EO) properties that can be applied to the fields of piezo actuators and optical modulators [1–8]. Applied voltage has been considered as a main factor to control the ferroelectric devices. However, the actual performances of relaxor ferroelectric devices can be easily affected by different electric phenomena including spatially varying permittivity and non-uniform electric field [9–13].
One important feature in relaxor ferroelectrics is polar nanoregions (PNRs) formed by off-centering B-site ions from the central-symmetric position in a non-polar cubic matrix in the temperature range near and above Curie temperature ${T_C}$ [14–17]. PNRs are highly polarizable, and their B-site rattling ions can be moved further by applying a sufficiently high electric field [18]. At temperature near and above ${T_C}$, the permittivity of relaxor ferroelectrics can be drastically increased with increasing the applied electric field, and become a function of electric field [19,20].
The field dependence of dielectric permittivity exists widely in the operating regimes of many ferroelectric devices but has yet to be completely studied. Modeling such dependence is therefore of great importance for the associated material analysis and device design [21–24]. There are several techniques that could be used to characterize the spatial distributions of electric field and permittivity, such as piezo force microscopy and transmission electron microscopy [12,22,25]. However, these methods are only suitable for nano- or micro-scale measurement. Since the change in permittivity by the applied electric field can be substantially large, in order to optimize the performances of EM and EO ferroelectric devices, a characterization technique capable of performing analyses with dimensions at the device level is desired but challenging.
In this paper, we demonstrate an optical beam deflection method to resolve the spatial distributions of electric field and permittivity in potassium tantalate niobate KTa1-xNbxO3 (KTN, $x = 0.39$) crystals at a temperature above and near Curie temperature ${T_C}$. KTN, a perovskite (ABO3) relaxor ferroelectric material, has been widely studied in the field of EO. As a solid solution of potassium tantalate and potassium niobate, by adjusting the composition ratio of tantalum to niobium, the ${T_C}$ of KTN can be obtained at room temperature, which makes the experiment easier for studying physics in the temperature range near ${T_C}$ [26]. Both theoretical and experimental approaches to study the spatial information of electric phenomena are discussed. The electric field distribution in KTN is verified by solving Poisson’s equation and taking light scattering images.
2. Theoretical analyses
2.1 Relationship between permittivity and electric field
To extract the relationship between permittivity and electric field of KTN, a KTN single crystal with a Curie temperature ${T_C}$ near $21.3^{\circ}\textrm{C}$, ordered from NTT, was used. The KTN crystal was cut to a dimension of $2.5 \times 2.5 \times 3\textrm{m}{\textrm{m}^3}$. All six faces of the sample were polished to optical grade. Platinum (Pt) was deposited onto the two side faces of the KTN crystal for forming Schottky contacts to block most charge injection [27]. The distance between the electrodes is $3\textrm{mm}$. Subsequently, the sample was attached to a Peltier module for temperature adjustment.
All dielectric experiments of KTN were measured in a cooling process. The temperature dependent permittivity was measured by LCR meter at 1 kHz is shown in the inset of Fig. 1. It should be mentioned that as a relaxor ferroelectric material, the ${T_C}$ of KTN can be slightly different depended on different characterization techniques [28]. Instead, ${T_m}$. in the inset of Fig. 1 was defined as a reference point where maximum relative dielectric constant occurs, and does not shift at different frequencies according to previous studies on KTN and other relaxors such as $({1 - x} )\textrm{Pb}({\textrm{M}{\textrm{g}_{1/3}}\textrm{N}{\textrm{b}_{2/3}}} ){\textrm{O}_3}-{-}x\textrm{PbTi}{\textrm{O}_3}$ with $x = 0.5$ [29,30]. Typically, the ${T_m}$ is slightly higher than the ${T_C}$ in the case of KTN. We define any temperature as T, and the difference between T and ${T_m}$ as $\Delta T = T - {T_m}$. In order to investigate the relationship between the permittivity and the electric field, the electric field dependent permittivity of KTN crystal was measured by using the charge-voltage conversion method at different temperatures, as shown in Fig. 1. Since the permittivity was measured at 1 kHz, which is low frequency response, complex permittivity is not concerned in this study as the dissipation factor within theange of ${10^{ - 3}}$ to ${10^{ - 4}}$. Figure 1 shows that, at $32^{\circ}\textrm{C}$ which is $10^{\circ}\textrm{C}$ above the ${T_m}$ ($\Delta T = 10^\circ{\textrm{C}}$), the permittivity slightly increases from 10800 to 11600 with increasing the applied electric field to $483 \textrm{V}/\textrm{mm}$. As the temperature decreases to $23^{\circ}\textrm{C}$, which is $1^{\circ}\textrm{C}$ above the ${T_m}$ ($\Delta T = 1^\circ{\textrm{C}}$), the permittivity become more sensitive to the applied electric field. At $23^{\circ}\textrm{C}$ ($\Delta T = 1^\circ{\textrm{C}}$), the permittivity greatly increases by 5 times from 19700 to 98400 as the applied electric field increases to $158\textrm{V}/\textrm{mm}$. The significant increase in the permittivity can be explained by the formation of PNRs and the existence of electric field induced phase transition in relaxor ferroelectric materials [18–20].
As shown in Fig. 1, the electric field dependent permittivity of KTN at a certain temperature can be defined by
where ${\varepsilon _r}[{E(x )} ]$, ${\varepsilon _{r0}}$, and $\Delta {\varepsilon _r}[{E(x )} ]$ are the total permittivity, the permittivity of KTN without an applied voltage, and the increment of the permittivity of KTN by the applied electric field, respectively.At each temperature point, the critical field strength (${E_C}$), the onset of partial or complete transition from paraelectric to ferroelectric phase induced by electric field, can be found from the peak of the permittivity. It is known that three factors including temperature, electric field, and stress can induce a phase transition. The original Curie−Weiss (C−W) law is to describe the permittivity of a ferroelectric material above ${E_C}$. In the case of this study, we derive an approximate equation by modifying Curie−Weiss law, named by modified C−W law in the form of electric field, to describe the increase in the permittivity of a relaxor ferroelectric material with an applied electric field below ${E_C}$ at a temperature above ${T_m}$, as expressed by
2.2 Resolving non-uniform electric field using optical beam deflection
For some relaxor ferroelectric devices without the need of charge injection such as actuators, the only modification in the equation of strain is the permittivity function when the operation temperature is near and above ${T_m}$ [5]. For other devices with unavoidable charge injection by the applied voltage, the impact on the electric field distribution is rather complicated and needed to be further investigated. Here, we used the optical beam deflection method to resolve the distributions of electric field and permittivity in a KTN crystal from its cathode to anode, and studied the impact caused by the electric field dependent permittivity.
Figure 2 illustrates the configuration of the optical beam deflection measurement. After the laser beam passes through the polarizer, the polarization state of the incoming laser beam is parallel with the direction of the applied electric field $ {E(x} )$ along the x-axis. Therefore, the polarized light can only experience the EO effect induced by the electric field dependent permittivity and the applied electric field along x-axis. The corresponding refractive index modulation, $\Delta n(x )$, driven by the applied electric field can be expressed by
where n is the original refractive index of KTN, and ${s_{11}}$ is the EO Kerr coefficient, as defined by3. Experiment and discussion
To resolve the electric field distribution that is affected by the electric field dependent permittivity in a KTN crystal with electron injection, the mapping of optical beam deflection measurement was conducted, as illustrated in Fig. 2. For the optical beam deflection experiment, another KTN crystal with a ${T_m}$ of $20.9^{\circ}\textrm{C}$ was cut to a dimension of $2.5 \times 3 \times 4\;\textrm{m}{\textrm{m}^3}$. Following the polishing process to all faces of the KTN crystal, titanium films were deposited on both cathode and anode for electron injection. As illustrated in Fig. 2, the distance, W, between cathode and anode is $4\;\textrm{mm}$, and the optical interaction length, L, in the KTN crystal is $3\;\textrm{mm}$. The light source is He-Ne laser with output wavelength of $632.8\textrm{nm}$, and the corresponding original refractive index n and QEO coefficient in the polar form ${g_{11}}$ of KTN are 2.31 and $0.136\,{\textrm{m}^4}/{\textrm{C}^2}$, respectively. The diameter of the laser beam was collimated to $200\mathrm{\mu}\textrm{m}$, and propagated along z-axis. The KTN crystal on the Peltier module was moved by a translation stage along x-axis to map the electric field distribution from its cathode to anode.
The optical beam deflection angle was measured from the cathode to anode of the KTN crystal with the different applied voltages at different temperatures, as shown Fig. 3(a). Each voltage was applied to the KTN crystal for 10 minutes prior to the measurement of optical beam deflection to inject the electrons to the crystal. Firstly, the optical beam deflection of the KTN crystal with the applied voltage of 900V is mapped at a temperature $10^{\circ}\textrm{C}$ above the ${T_m}$ ($\Delta T = 10^{\circ}\textrm{C}$), as plotted in Fig. 3(a). Note that the direction of internal electric field formed by the injected electrons is opposite to the direction of the applied electric field. When the internal field is equally strong as the applied electric field, the net electric would be zero so that $|{\theta (x )} |= 0$ near the cathode. At $\Delta T = 10^{\circ}\textrm{C}$, since the relative permittivity of KTN crystal is barely dependent on the applied electric field, by applying the derived Eq. (1) and (7) based on the result from the dielectric experiment to the mapping result of the optical beam deflection (Fig. 3(a)), the distributions of electric field and relative permittivity in KTN crystal with the applied voltage of 900V are resolved, as shown in Figs. 3(b) and (c), respectively. This linear distribution of the electric field in the KTN crystal can be easily predicted by Maxwell's equations under the assumption of fixed permittivity ($\Delta \varepsilon _r(E )\approx 0$) and uniform charge density.
Fig. 3. (a) The mapping of optical beam deflection, (b) electric field distribution, and (c) relative permittivity distribution.
Secondly, the optical beam deflection of the KTN crystal with the applied voltage of $600\textrm{V}$ is mapped at a temperature $4^{\circ}\textrm{C}$ above the ${T_m}$ ($\Delta T = 4^{\circ}\textrm{C}$), as shown in Fig. 3(a). However, at $\Delta T = 4^{\circ}\textrm{C}$, the relative permittivity of KTN has a certain electric field dependency ($\Delta \varepsilon _r(E )\ne 0$). Again, by applying the derived Eq. (1) and (7) based on the result from the dielectric experiment to the mapping result of the optical beam deflection at $\Delta T = 4^{\circ}\textrm{C}$ (Fig. 3(a)), one can see that both electric field and permittivity increase upon approaching the anode of the KTN crystal, as plotted in Figs. 3(b) and (c), respectively. It is noteworthy that, at $\Delta T = 4^{\circ}\textrm{C}$, the electric field distribution tends to be saturated in Fig. 3(b), and the permittivity is substantially increased by the electric field in Fig. 3(c). To verify the electric field distribution in the KTN crystal, the potential is calculated by integrating the electric field from the cathode to anode. The calculated potentials are 879 V and 573 V for the cases of optical beam deflection of KTN at $\Delta T = 10^{\circ}\textrm{C}$ and $\Delta T = 4^{\circ}\textrm{C}$, respectively, which indicate an accuracy around 95% corresponding to the actual applied voltages of 900V and 600V.
Furthermore, the ${E_C}$ plays an important role that determines the onset of the electric field induced phase transition from the paraelectric to ferroelectric phase in relaxor ferroelectrics. If the non-uniform phase transition is induced by a non-uniform electric field in a relaxor ferroelectric crystal, the crystal may be easily damaged because of the lattice mismatch between cubic and tetragonal crystal structures. In the optical application, the phase transition from paraelectric phase to ferroelectric in relaxor ferroelectric materials can also cause light scattering by the ferroelectric domain structures. Therefore, we carefully verify the saturation effect in the increasing electric field from the cathode to the anode of the KTN crystal with the electron injection by the applied voltage using light scattering effect.
The optical beam deflection of KTN crystal with different applied voltages of 450V and 700V at a temperature $2^{\circ}\textrm{C}$ above the ${T_m}$ ($\Delta T = 2^{\circ}\textrm{C}$) was measured and mapped upon the same procedure as mentioned above, as plotted in Fig. 4(a). Based on the mapping result of the optical beam deflection, the derived Eq. (1) and Eq. (7) are applied to resolve the electric field distribution of the KTN crystal, as shown in Fig. 4(b). The critical field strength ${E_C}$ of the KTN crystal at $\Delta T = 2^{\circ}\textrm{C}$ is $200\textrm{V}/\textrm{mm}$ approximately. The blue dots in Fig. 4(b) show that the electric field in KTN crystal at 450V is gradually saturated upon approaching anode, and reaches $174.1\,\textrm{V}/\textrm{mm}$ near the anode. For the case of KTN crystal with the applied voltage of 700V at $\Delta T = 2^{\circ}\textrm{C}$, the green dots in Fig. 4(b) show that the electric field eventually reaches $193.3\,\textrm{V}/\textrm{mm}$, which is pretty close to the ${E_C}$ at the position around 3.4 $\textrm{mm}$ away from the cathode. The arrows A and B in Fig. 4(b) point out the position where the laser passing through the KTN crystal near the anode at 450V and 700V, and the corresponding images of light transmission are shown in Figs. 4(c) and (d), respectively. One can see that the image of light transmission is homogenous in Fig. 4(c) but scattered in Fig. 4(d). Due to the existence of electric field induced phase transition in ferroelectrics at paraelectric phase, the long-range order ferroelectric domain structures in the KTN crystal can be formed by applying a sufficiently high electric field at the temperature above ${T_C}$. The scattering laser beam in the image of light transmission verifies that the electric field is near the ${E_C}$ to induce the transition from paraelectric to ferroelectric phase. The proposed method can perform an accuracy around 95% to resolve the electric field distribution in relaxor ferroelectrics at paraelectric phase. The error range maybe resulted from the imperfect optical alignment and temperature control in the experiment. The result in this study shows a good agreement between the derived equations and the experimental results. Such optical characterization method is non-disruptive and suitable for spatially analyzing non-uniform electric phenomena in ferroelectric devices.
Fig. 4. (a) The mapping of optical beam deflection, (b) the electric field distribution in KTN crystal, the images of output laser beam of KTN crystal with the applied voltages (c) 450 V, and (d) 700 V, respectively, at $\Delta T = 2^{\circ}\textrm{C}$.
4. Conclusion
In conclusion, this work has comprehensively studied spatially non-uniform electric field and relative permittivity in KTN crystal at paraelectric phase and near the phase transition point. with the applied voltage using the measurement of optical beam deflection. A mathematical model based on the spatially dependent optical beam deflection angle was derived to describe the spatial change in the electric field from the cathode to anode of the KTN. According to the mapping of the optical beam deflection angle, the electric field distribution was successfully resolved and verified by the experimental parameters and critical field strength ${E_C}$. The result in this study provides an approach to accurately determine the electric field in relaxor ferroelectric materials and devices.
Funding
Joint Directed Energy Transition Office (DE-JTO) via Office of Naval Research (N00014-17-1-2571).
Acknowledgment
This research was sponsored and partially supported by the Office of Naval Research (ONR) under Grant Number N00014-17-1-2571. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.
Disclosures
The authors declare no conflicts of interest.
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