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Abstract
In this work, the temperature dependence of polarization degree of ferroelectric phase barium titanate (BaTiO3) film is first modelled based on a conservation of the energy-focusing cost of a microstructural domain cell, and then the dual dependences on both temperature and initial polarization degree of an off-axis tensor electrooptic coefficient (r51) is modelled. Further, the correlation between the initial polarization degree and the r51 is investigated based on the correlative electrooptic and elasto-optic effects of a perovskite crystal. With the output of theoretical models for the attributes of r51, numerical simulations are carried out. It was found that both the polarization degradation of out-of-plane polarization (Pz) and the electrooptic coefficient r51 of c-axis oriented crystal films exhibit significant nonlinear dependences on the initial fraction of in-plane polarization (Px) and the working temperature. Consequently, the temperature dependences of the measured results of r51 with respect to the selected working temperatures in the range of 20°C∼100°C are consistent with the above numerical results, showing that the out-of-plane polarization degree presents the biggest degradation in the temperature range from 50°C to 70°C. This work provides a valuable reference to the research and development of high-speed electro-optic devices and applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The prestigious establishments of barium titanate (BaTiO3) crystal thin films in theoretical/technological research have been attracting much more enthusiasm owing to its extremely high electro-optic (EO) effect and advanced physical attributes. In fact, in the past decade, research and applications of BaTiO3 thin film material have shown its advancement in both the EO and ferroelectric properties [1–3]. Therefore, the investigation for the correlations among the ferroelectric, the electrooptic and the elasto-optic properties of BaTiO3 crystal thin films can be envisioned to have the academic and practical significances for the applications of high-speed modulation, high-density memory, etc. in the field of material performance [4–9].
The analyzing works on the relationships between the microstructure and the ferroelectric/electrooptic properties had been started from 1960s to present [4,5,10–20]. As early as 1960s, it was found by Miller and Weinreich that the mechanism of the domain motion is the sidewise motion of 180° domain wall in single crystal BaTiO3 [4]. For an analog tetragonal crystal material, Pb(Zr,Ti)O3 (PZT), the distortion property of material away from its parent perovskite structure was calculated and then the special properties of PZT were predicted by Grinberg et al [5]. The later works in the formation of crystal film were focused on a domain nucleation process that showed domains in one layer can have different polarizations from those of adjacent layers at the same position [6,7]. The major orientation of the domain is defined as the axis direction of the crystal, also known as the local polarization direction. For example, c-axis direction of crystal may have other axis orientation of domains, which is generally called polarization fraction. At that time, Watanabe in theory systematically studied the detailed polarization stability conditions [8].
Since 1990s, Wessels and co-workers made the analysis for the microstructure dependence of EO properties by investigating the optical response to the electric field and optical axis of crystal, then studied the second harmonic generation (SHG) of the a-axis oriented BaTiO3 crystal films, the electron beam induced poling effect and the poling/annealing effects on the EO coefficient and dielectric properties [9,10]. Around the beginning of this century, the Wessels’s team investigated the relationships between the EO properties of the epitaxial grown BaTiO3 crystal film and the growing technique conditions, and further observed the three-dimensional (3-D) crystalline structure and its stability [11–13]. Meanwhile, Reitze et al. carried out studies of the EO properties of single crystalline ferroelectric thin films [14], Choi et al. improved the ferroelectric transition temperature of BaTiO3 crystal films to over 500°C with a new technique – coherent epitaxy, which is applicable to both the nonvolatile memories and EO devices [15]. Until to 2011, Niu et al. epitaxially grown BaTiO3 crystal films by using the molecular beam epitaxy (MBE) to realize non-volatile memories [16]. Later in 2013–2014, Ngo et al. developed an epitaxial BaTiO3 film on the SrTiO3-buffered Si substrate to investigate the switching process of ferroelectric polarization and an atomic layer deposition of epitaxial c-axis oriented BaTiO3 film on Si(001) by using MBE technique and a vacuum annealing at 600°C, and then analyzed the crystalline structure by using XRD spectroscopy [17,18]. The latest studies were on how the BaTiO3 domain orientations influence its EO properties by Hsu et al. with spectroscopic ellipsometry [19] and the accurate measurement for the BaTiO3 crystalline orientations and EO properties by our team [20].
As early as 2000, Li et al. adopted a metrology of direct-current/alternating-current birefringence EO measurement technique for measuring the composition dependence of effective EO coefficient of a tetragonal crystal Ba1-xSrxTiO3 by changing its composition x (i.e., BTO/STO) to study the effective EO coefficient [21]. From 2013 to present, Abel and co-workers investigated the Pockels effect of BaTiO3 crystal thin films grown on semiconductor wafer Si (001) using the MBE and Kormondy et al. further characterized the structure of the crystalline using high energy electron diffraction and XRD, consequently at this phase, the EO coefficients of about 148 pm/V were measured [22,23]. Recently, Abel and co-workers reported some progresses in the measurements of EO coefficients. Illustratively, in 2019 the EO coefficient r51 of 923 pm/V were measured in which the test precision up to ±23% was reached [24]. In 2014, Tang et al. carried out the measurements of the effective EO coefficient and the advancement analysis of BaTiO3 crystal thin-films on silicon-on-insulator (SOI) platform, reported an EO coefficient of 213 pm/V [25]. In our recent work, the precision measurements of the EO coefficients were recently carried out by Sun and co-workers and the ${{r}_{{51}}}$ values of 400–650pm/V were obtained [20,26]. Ortmann et al. studied the properties and performance potential of the other perovskite crystal, SrTiO3, then made an important establishment [27].
By analyzing the above actions and establishments we find that no adequate efforts have been made to investigate the correlations among the motion energy conversion of crystal domains, the polarization movements, the elasto-optic coefficient and electrooptic coefficient of BaTiO3 crystal thin films despite these processes can lead to complex temperature dependences and then have critical effects on the research and development of functional devices. Thus, in this work, in Sec. II, a conservative relation of all the energies within the crystal domain are found first, then the theoretical models for defining both the temperature dependence of the polarization degradation and the temperature dependence of an off-axis tensor EO coefficient ${{r}_{{51}}}$ are derived. Further in Sec. III, we measure the surface profiles and the initial polarization fractions of two BaTiO3 crystal films using the atomic force microscopy (AFM) and XRD techniques, respectively, and then systematically simulate the temperature dependences of both polarization degradation and off-axis tensor EO coefficient ${{r}_{{51}}}$. In Sec. IV, with a fabricated straight waveguide based EO phase modulator and by selecting several temperature values in the range from 20°C-120°C, the corresponding values of ${{r}_{{51}}}$ are measured and then the temperature dependence of measured ${{r}_{{51}}}$ values are discussed. Finally, in Sec. V, the conclusions are given.
2. Theoretical models
During deposition of BaTiO3 crystal film, the molecule domains are generally separated by an interface of two categories of crystal domains under the effect of energy conservation process in the crystalline system [6–8]. In terms of molecular thermodynamics, the energy conversion process must give rise to a thermal effect in the crystal structure, which generally influences the physical property of an EO crystal material. Illustratively, in a recent publication on the evaluation method for an elasto-optic coefficient of EO crystal, the temperature dependance of genuine EO coefficients was taken into account [28]. With Landau-Ginzburg-Devonshire (LGD) theory, the energy taken by the general polarization P(x,y,z) can be defined as [29]
where the ratio of two coefficients A and B determines the spontaneous polarization as ${P}_{s}^{2}{ = -A/B}$. The coefficient A is determined by the difference of working temperature T and Curie temperature ${{T}_{c}}$ as ${A = }{{A}_{0}}{(T}-{{T}_{c}}{)}$, where ${A_{0}>0}$, and ${B>0}$. During the formation of the tetragonal crystalline, the energy difference between the relaxed tetragonal ferroelectric phase and the highly symmetric paraelectric phase of the tetragonal crystal in a domain unit plays a key role in the formation of spontaneous polarization ${{P}_{s}}$ . Further, based on the energy conservation of the growing process of a crystal domain, the total residual energy ${U_{loc}(P_{z})\; }$ in a single-crystalline single-domain equals the total energy of a domain unit subtracted the nucleation energy cost for ${P_z}$ polarization. Accordingly, the relationship between the polarization degree $\; P_z/P_s\; $ and ${U_{loc}(}{{P}_{z}}{)}$ is defined by [29,30]:In previous research, we find that the total residual energy ${U_{loc}(}{{P}_{z}}{)}$ is the sum of the total local kinetic energy ${{W}_{{dwf}}}$ and the free potential energy ${{F}_{{crys}}}$. Then, based on the energy conservation law in a crystal domain, we have an energy conservative equation as [29]
By setting ${{A}_{{loc}}}{ = (1/4B)[}{{A}_{0}}{(T}-{{T}_{c}}{)}{{]}^{2}}$, then from Eqs. (1)-(4), we obtain the polarization ratio $\; (P_z/P_s$) in a single-crystalline single-domain of BaTiO3 crystal at c-axis orientation as
From the LGD model, if ${{W}_{{loc}}}$ stands for the local energy, and ${{W}_{{gx}}}$ and ${{W}_{{gz}}}$ are the energy gradients of the electric dipole-dipole interaction along the x-axis and z-axis, respectively, the local kinetic energy in a single-domain is the arithmetic sum of three energy sources as [29]
where where A001 and A100 are, respectively, the domain areas at c-axis and a-axis, ${{g}_{x}}$ and ${{g}_{z}}$ are the gradient coefficients at the in-plane and the out-of-plane directions, respectively.After the forming process of crystal axis orientation reaches the stable state, an equilibrium potential energy inside the crystal domain, the total free potential energy ${{F}_{{crys}}}$ in one crystal domain unit can be calculated from the integration of the arithmetic sum of four energy density components - the elastic energy density ${{f}_{{elas}}}$, the gradient energy density ${{f}_{{grad}}}$, the electrostatic energy density ${{f}_{{elec}}}$, and the film energy density ${{f}_{{film}}}$ [29,31,32], which is obtained from the bulk state [31–33]. Thus, the total free energy in a volume V is defined by
WithWhere $\alpha_{1}$, $\alpha_{{11}}$, $\alpha_{{12}}$, $\alpha_{{111}}$, $\alpha_{{112}}$, $\alpha_{{123}}$ are the thermodynamic expansion coefficients of the crystal film, while their transformation parameters are defined as the transformation parameters of expansion coefficients: ${{s}_{{11}}}$, ${{s}_{{12}}}$, ${{s}_{{44}}}$, and the electro-strictive coefficients: ${{Q}_{{11}}}$, ${{Q}_{{12}}},\; {{Q}_{{44}}}$:
In Eq. (9a), ${{C}_{{ijkl}}}$ is the elastic stiffness tensor, ${{e}_{{ij}}}{ = }{\varepsilon_{{ij}}}-\varepsilon _{{ij}}^{0}$ is the elastic strain, ${\varepsilon_{{ij}}}$ is the total strain of the crystal compared to the parent paraelectric phase, and $\varepsilon_{{ij}}^{0}$ is the stress-free strain. In Eq. (9b), ${{G}_{{11}}}$ is the gradient energy coefficient and ${{P}_{{i,j}}}{ = }\partial {{P}_{i}}{/}\partial {{x}_{j}}$ denotes the spatial differentiation of the ith polarization component to the jth coordinate ${{x}_{j}}$. In Eq. (9c), $E_{i, \text { dipole }},\, E_{i, \text { appel }}, \,\textrm{and} \,E_{i, \text { depol }}$ denote the dipole-dipole interaction caused electric field, the applied external electric field and the surface charges caused inside depolarized electric field (also called reversed electric field), respectively. The Eq. (9d) is the definition of the potential energy of a ferroelectric BaTiO3 thin film on a thick substrate [32], in which ${{u}_{m}}$ is a uniform misfit strain at the film/substrate interface under the condition of the lattice matching between crystal film and crystal substrate defined by ${{u}_{m}}{ = (b}-{{a}_{0}}{)/b}$ with the lattice constants of the film and substrate, b and ${{a}_{0}}$, respectively. In Eqs. (9b) and (9c), ${{P}_{i}}$, (i = 1, 2 and 3) are the components of polarization at the crystal axis directions, ${{P}_{1}}{ + }{{P}_{2}}{ + }{{P}_{3}}{ = 1}$.
Energy conservation of the nucleation cost of a crystal domain and the local kinetic/potential domain energies of BaTiO3 crystal films based on the molecular thermodynamic theory leads to the dual dependences on the temperature and the initial polarization degree.
In our previous work, the excellent role of an off-axis component of EO coefficient tensor, ${{r}_{{51}}}$, is demonstrated in realizing highly efficient electrooptic modulation functions due to its ultrahigh values in both the visible and infrared wavelength regimes [20]. In the last century, the theoretical research of Bernasconi’s team showed that both the EO and elasto-optic effects of a perovskite crystal have the scalar dependences on the working temperature T. As a result, a relation among the EO coefficient ${{r}_{{ij}}}$ and the elasto-optic coefficient ${{g}_{{ij}}}$ was given as [30]
where ${{g}_{{ij}}}$ and ${{P}_{s}}$ stand for the elasto-optic coefficient with respect to EO coefficient ${{r}_{{ij}}}$ and the spontaneous polarization, respectively, ${{C}_{2}}$ and ${{T}_{2}}$ are two parameters of material that were determined by fitting the published data [30,34]. As a perovskite, BaTiO3 crystal has EO coefficient ${{r}_{{51}}}$ showing strong dependence on the work temperature T.3. Simulations for the theoretical BaTiO3 crystal film
As a tetragonal material, a ferroelectric BaTiO3 crystal film only has its three independent elastic constants ${{C}_{{11}}}$, ${{C}_{{12}}}$ and ${{C}_{{44}}}$ in the Voigt’s notation, therefore only three permittivity components: ${\varepsilon_{{11}}}$, ${\varepsilon _{{22}}}$, ${\varepsilon_{{33}}}$ can be taken into account, while all the other components ${\varepsilon_{{12}}}$, ${\varepsilon_{{21}}}$, ${\varepsilon_{{23}}}$, ${\varepsilon_{{32}}}$, ${\varepsilon_{{13}}}$ and ${\varepsilon_{{31}}}$ can be taken as zero. The permittivity components at low frequency regions are taken as ${\varepsilon_{{11}}}{ = }{\varepsilon_{{22}}}{ = 2200}$, ${\varepsilon_{{33}}}{ = 56}$. For the stress-free state of the c-axis oriented ferroelectric BaTiO3 crystal, the experimental value for ${{G}_{{11}}}$ in Eq. (9b) has not been found yet, so the values for PbTiO3 of ${{G}_{{110}}}{ = 7}{.12 \times 1}{{0}^{{ - 10}}}C^{-2}{{m}^{4}}{N}$ and ${{G}_{{11}}}{/}{{G}_{{110}}}{ = 0}{.6}$ are assumed, and the crystal film energy density ${{f}_{{film}}}$ in Eq. (9d) can be defined with the Legendre transformation formula of the elastic Gibbs function of a quasi-cubic style ferroelectric crystal where the values of all the parameters are expressed and tabulated into Table 1 [31]. What need to be clarified on the values of the thermodynamic expansion coefficients cover all the one-, two- and three-dimensional states in the film, so they have different units as presented in Table 1. The values of all the transformation parameters of expansion coefficients: ${{s}_{{11}}}$, ${{s}_{{12}}}$, ${{s}_{{44}}}$ and the electro-strictive coefficients: ${{Q}_{{11}}}$, ${{Q}_{{12}}},\; {{Q}_{{44}}}$ for the BaTiO3 crystal film are tabulated in Tables 2 and 3, respectively [32].
Table 1. Thermodynamic expansion parameters of BaTiO3 crystal film.
Table 2. Transformation parameters of expansion coefficients for BaTiO3 crystal film.
Table 3. Transformation parameters of electro-strictive coefficients for BaTiO3 crystal film.
The wide studies of molecular dynamics have enabled to accurately model the nucleation phenomenon of domain wall motion. For BaTiO3 crystal films, the constants of domain density as a=3.900 Å and c=4.150 Å, the calculated values for the two parameters of crystal at T=0K are ${{P}_{0}}{ = 0}{.36(C/}{{m}^{2}}{)}$ and ${{T}_{1}}{ = 436(K)}$ [5,30], and the two energy gradient coefficients of the electric dipoles along the x- and z-axis in Eq. (7) are taken as ${{g}_{x}}{ = 0}{.63 \times 1}{{0}^{{ - 11}}} m^{3}F^{-1}$ and ${{g}_z}{ = 1}{.07 \times 1}{{0}^{{ - 11}}}m^3F^{-1}$ [29].
Numerous research works showed that it is seldom a 100% <001> or <100> crystalline film grown by the pulsed laser deposition (PLD) technique [6,7,16,17,35], so the values of the components ${{P}_{1}}$, ${{P}_{2}}$ and ${{P}_{3}}$ of the crystal axis in three directions are determined by the specific conditions of the crystal film. In this article, two BaTiO3 crystal thin films are grown on a magnesium oxide (MgO) crystal substrate by the PLD technique, and then straight waveguide EO phase modulators are fabricated for carrying out the experiments of EO modulations. Figure 1 shows the atomic force microscopic (AFM) image of the surface profile and the XRD spectrum of the 450 nm thick epitaxial BaTiO3 crystal films grown on the MgO crystal substrates, where (a) and (b) are for the first film and the (c) and (d) for the second film. One can find that even though these two crystal films are both c-axis (<001>) oriented, the second film is obviously not 100% <001> crystalline. Note from Fig. 1(b) that an extremely high diffraction peak appears at both <001> and <002>, implying an expectable c-axis crystal for film-1, so resulting in an extremely high (Pz/Ps) fraction. In contrast, note from Fig. 1(d) that a diffraction peak appears at <101> for film-2, implying a co-existence of a- and c-axis domains, so resulting in a relatively low (Pz/Ps) fraction. Hence the in-plane polarization fraction and its effects on the dielectric/ferroelectric properties are worthy of discussion.
Fig. 1. BaTiO3 crystal film and the structural measurements: (a) and (b) are, respectively, the AFM image and the XRD spectrum of an extremely high-ratio c-axis crystal film a <100>MgO crystal substrate, and (c) and (d) are, respectively, the AFM image and the XRD spectrum of a low-ratio c-axis crystal film grown on a <110>MgO crystal substrate.
Since both ${{P}_{1}}$ and ${{P}_{2}}$ are the polarization fractions of a-axis orientation, and ${{P}_{3}}$ is the polarization fraction of c-axis orientation, we set ${{P}_{x}}{ = }{{P}_{1}}{ = }{{P}_{2}}$, ${{P}_{z}}{ = }{{P}_{3}}$. All the information is called the initial states of crystal films and determined by the growth technique and can be measured with the method shown in Fig. 1. Then, by selecting two initial in-plane polarization fractions:$\; {{P}_{x}}{ = 0}$ and ${{P}_{x}}{ = 0}{.1}$ and with Eqs. (6) through (10), we obtain the temperature dependences of the total local kinetic energy ${{W}_{{dwf}}}$ of a single-crystalline single-domain as shown in Fig. 2. Note that the total local kinetic energy ${{W}_{{dwf}}}$ linearly increases with temperature and decreases with the polarization fraction of ${{P}_{x}}$. Illustratively, for the case of ${{P}_{x}}{ = 0}$, when the temperature increases from 20°C to 100°C, ${{W}_{{dwf}}}$ increases from 840 pJ to 1360 pJ, while for the case of ${{P}_{x}}{ = 0}{.1}$, it increases from 680 pJ to 1100 pJ, so a drastic drop happens when the initial polarization changes from ${{P}_{x}}{ = 0}$ to ${{P}_{x}}{ = 0}{.1}$.
Fig. 2. Numerical simulation for the temperature dependence of the total local energy in a domain of c-axis BaTiO3 crystal grown on MgO crystal with respect to two values of initial polarization fraction: Px=0 and Px=0.1.
For the free potential energy of a domain ${{F}_{{crys}}}$, with Eqs. (9a) through (9d) and the values of the material parameters provided in Tables 1 through 3, for the two cases of the initial in-plane polarization fractions: ${{P}_{x}}{ = 0}$ to ${{P}_{x}}{ = 0}{.1}$, we obtain the numerical simulation results of the temperature dependences of all the four sources of the free potential energy ${{F}_{{crys}}}$ in a crystal domain of BaTiO3 crystal film as shown in Figs. 3(a) and 3(b), respectively. First note from Fig. 3(a) that two energy sources in ${{F}_{{crys}}}$: the film-structural energy ${{f}_{{film}}}$ and the elastic energy ${{f}_{{elas}}}$ have the dominant contributions compared with the other ones: the gradient energy ${{f}_{{grad}}}$ and the electrostatic energy ${{f}_{{elec}}}$.
Fig. 3. Numerical simulation for the temperature dependence of the free energy in a domain of c-axis BaTiO3 crystal with respect to four energy sources for two initial in-plane polarization states. (a) Px=0; (b) Px=0.1.
Further, note from Fig. 3(b) that only the free potential energy of film plays the dominant roles compared with all the other energy sources. So, the difference between (a) and (b) implies that a small polarization fraction change can create a large impact upon the energy distributions of a crystalline domain, resulting in a complicated nonlinear function.
By comparing Fig. 3(a) with Fig. 3(b), we also find that when the initial in-plane polarization fraction is changed from 0.0 to 0.1, the film-structural energy ${{f}_{{film}}}$ drastically increases by one order. Illustratively, for the case of ${{P}_{x}}{ = 0}$, when the temperature increases from 20°C to 100°C, ${{f}_{{film}}}$ increases from 28 pJ to 75 pJ, while for the case of ${{P}_{x}}{ = 0}{.1}$, in the same temperature change range, it increases from 700 pJ to 3000 pJ. So, it turns out that the property of the free potential energy in a domain of BaTiO3 crystal film is of importance. The distributions of the local kinetic energy and the free potential energy shown in Figs. 2 and 3, respectively, show that the total local kinetic energy ${{W}_{{dwf}}}$ is averagely higher than the free potential energy ${{F}_{{crys}}}$ in a crystal domain by one order for the case of ${{P}_{x}}{ = 0}$. Thus, at the expectable initial polarization fraction ${{P}_{x}}{ = 0}$, the polarization ratio degradation process in Eq. (5) is only dependent on the total local kinetic energy ${{W}_{{dwf}}}$. In contrast, for the case of ${{P}_{x}}{ = 0}{.1}$, the total free potential energy ${{F}_{{crys}}}$ quickly becomes higher than the local kinetic energy ${{W}_{{dwf}}}$, so at this situation, both ${{W}_{{dwf}}}$ and ${{F}_{{crys}}}$ can have comparable shares of the total residual energy in the crystal domains, resulting in the important dual dependences of the polarization degradation on both the temperature and the initial polarization fraction. Accordingly, both ${{W}_{{dwf}}}$ and ${{F}_{{crys}}}$ would have more complicated dependences on the working temperature. It turns out that the free potential energy ${{F}_{{crys}}}$ is specifically dependent of ${{P}_{x}}$.
In accordance with the above simulation results and analyses, we know that the total local kinetic energy ${{W}_{{dwf}}}$ is averagely higher than the free potential energy ${{F}_{{crys}}}$ in a crystalline domain by one order for the case of ${{P}_{x}}{ = 0}$. Thus, the polarization ratio ${(P_z/P_s}$) defined by Eq. (5) is only dependent on the total local kinetic energy ${{W}_{{dwf}}}$. In contrast, for the case of ${{P}_{x}}{ = 0}{.1}$, the total free potential energy ${{F}_{{crys}}}$ is close to and higher than the local kinetic energy ${{W}_{{dwf}}}$. Thus, at this situation, both ${{W}_{{dwf}}}$ and ${{F}_{{crys}}}$ can have comparable shares of the total residual energy in one domain structure of BaTiO3 crystal film, resulting in the dual influences on ${P_z/P_s}$. It can be predicted that both ${{W}_{{dwf}}}$ and ${{F}_{{crys}}}$ would have more complicated dependences on the working temperature.
By referring to the above simulations, with Eqs. (1)-(5), in the range of ${{P}_{x}}$ from 0 to 0.2, we obtain the numerical simulation results for the dual dependences of ${P_z/P_s}$ on both the temperature T and the initial polarization fraction ${{P}_{x}}$ as shown in Fig. 4. We can immediately find from Fig. 4 that the polarization ratio is that a nonlinear degradation process with both the temperature and the initial in-plane polarization fraction. For instance, at ${{P}_{x}}{ = 0}{.2}$, when the temperature changes from 20°C to 70°C, the polarization ratio slowly degrades from 64% to 56% and then quickly goes back to 64%. For ${{P}_{x}}{ = 0}$, in a temperature range 20–100°C, the change of polarization ratio is linear and very small, which is maintained within the range of 96%±1%. For a real epitaxial BaTiO3 crystal film, the initial in-plane polarization is always nonzero, so the out-of-plane polarization degree has the biggest degradation from 50°C to 70°C.
Fig. 4. Numerical simulation for the dual dependences of the polarization degradation of c-axis BaTiO3 crystal domains on both temperature and initial polarization fraction: Px=0∼0.2.
As a perovskite crystal, in BaTiO3 crystal films there are only three valid elasto-optic coefficients: ${{g}_{{11}}}$, ${{g}_{{22}}}$ and ${{g}_{{55}}}$ [30]. Based on the above theoretical model defined by Eqs. (11) and (3) for the temperature dependence of EO coefficient, the characteristics of BaTiO3 crystal films, the off-axis tensor EO coefficient, ${{r}_{{51}}} = {{r}_{{42}}}$, is response to the electric field imposed in the a-axis, ${E(x)}$, then activates the material parameters close to the values associated with the elasto-optic coefficients ${{g}_{{55}}}$, while ${{g}_{{55}}} = {{g}_{{44}}}$. Thereby, we take ${{g}_{{55}}}{ = 7}{.0 \times 1}{{0}^{{ - 2}}}{(}{{m}^{4}}{/}{{C}^{2}}{)}$ [30]. Further, in this model ${{C}_{2}}{ = 22}{.3 \times 1}{{0}^{4}}$ and ${{T}_{2}}{ = } - {80(K)}$ are the fitting values based on the measured values. However, as ${{r}_{{51}}}$ needs to be derived from the off-axis tensor EO coefficient equations so that the deviation of the measured value is very big. In terms of the theoretical outcomes, the value of ${{C}_{2}}$ for bulky BaTiO3 tetragonal crystal is in the range of $1.0 \sim {22}{.3 \times 1}{{0}^{4}}$, then by setting ${{C}_{2}}$ five different values as ${22}{.3 \times 1}{{0}^{4}}$, ${18}{.3 \times 1}{{0}^{4}}$, ${14}{.3 \times 1}{{0}^{4}}$, ${10}{.3 \times 1}{{0}^{4}}$ and ${6}{.3 \times 1}{{0}^{4}}$, we obtain the temperature dependence of ${{r}_{{51}}}$ on the working temperature with respect to the five values of ${{C}_{2}}$ as shown in Fig. 5 where (a) and (b) are for the cases of ${{P}_{x}}{ = 0\; {\textrm{and}}\; }{{P}_{x}}{ = 0}{.1}$ [30]. Note that if the polarization degree of BaTiO3 crystal film is not taken into account, namely, the expectable c-axis oriented crystal film, the theoretical off-axis tensor EO coefficient ${{r}_{{51}}}$ is linearly dependent on the working temperature, namely, ${{r}_{{51}}}$ quickly decreases with temperature, but it strongly depends on the value of ${{C}_{2}}$. By referring our previous work, the values in the ${{C}_{2}}$ in the range of ${14}{.3 \times 1}{{0}^{4}}\sim {22}{.3 \times 1}{{0}^{4}}$ are close to the real cases. However, as shown in the analyses in the above sections, even though in theory once a BaTiO3 crystal film is grown on MgO crystal substrate with PLD technique, it should be a c-axis oriented at the out-of-plane direction, in practice it still has a small ratio of a-axis component at the in-plane direction, resulting in a temperature dependent polarization degradation.
Fig. 5. Numerical simulation for the temperature dependence of the EO coefficient ${{r}_{{51}}}$ of c-axis BaTiO3 crystal grown on MgO crystal with respect to five values of ${{C}_{2}}$ for two initial in-plane polarization states. (a) the ${{P}_{x}}{ = 0}$, (b) ${{P}_{x}}{ = 0}{.1}$.
Based on the above simulation results of the temperature dependences of both ${P_z/P_s}$ and ${{r}_{{51}}}$, after introducing the temperature dependence of the polarization degradation shown in Fig. 5 into Eq. (11) and by selecting two different values of ${{C}_{2}}$ as ${22}{.3 \times 1}{{0}^{4}}$ and ${14}{.3 \times 1}{{0}^{4}}$, we obtain the numerical simulation results of the dual dependences of ${{r}_{{51}}}$ on the working temperature and the in-plane initial polarization as shown in Figs. 6(a) and 6(b), respectively. Note that, after the polarization degradation of BaTiO3 crystal film is introduced into the theoretical model for defining the temperature dependence, the theoretical off-axis tensor EO coefficient ${{r}_{{51}}}$ has presented the obvious dual dependences on the initial polarization degree and the working temperature, with the temperature has stronger impact on the ${{r}_{{51}}}$ value than the initial polarization. ${{C}_{2}}{ = 14}{.3 \times 1}{{0}^{4}}$ and ${{C}_{2}}{ = 22}{.3 \times 1}{{0}^{4}}$ have the similar temperature dependences of ${{r}_{{51}}}$. Illustratively, for the case of ${{C}_{2}}{ = 22}{.3 \times 1}{{0}^{4}}$ and a variation of T from 20°C to 120°C, at ${{P}_{x}}{ = 0}$, ${{r}_{{51}}}$ decreases from 804 pm/V to 478 pm/V, a degradation of 40%; at ${{P}_{x}}{ = 0}{.1}$, ${{r}_{{51}}}$ decreases from 669 pm/V to 420 pm/V, a degradation of 37%; and at ${{P}_{x}}{ = 0}{.2}$, ${{r}_{{51}}}$ decreases from 532 pm/V to 368 pm/V, a degradation of 31%. For the case of ${{C}_{2}}{ = 14}{.3 \times 1}{{0}^{4}}$ and a variation of T from 20°C to 120°C, at ${{P}_{x}}{ = 0}$, ${{r}_{{51}}}$ decreases from 516 pm/V to 307 pm/V, a degradation of 40%; at ${{P}_{x}}{ = 0}{.1}$, ${{r}_{{51}}}$ decreases from 429 pm/V to 269 pm/V, a degradation of 37%; and at ${{P}_{x}}{ = 0}{.2}$, ${{r}_{{51}}}$ decreases from 341 pm/V to 236 pm/V, a degradation of 30%.
Fig. 6. Numerical simulation for the dual dependences of the polarization degradation of c-axis BaTiO3 crystal domains on both temperature and initial polarization fraction: Px=0∼0.2. (a) ${{C}_{2}}{ = 22}{.3 \times 1}{{0}^{4}}$ and (b) ${{C}_{2}}{ = 14}{.3 \times 1}{{0}^{4}}$.
4. Measurements for the electrooptic coefficient of BaTiO3 crystal film
We designed an experimental setup for characterizing the EO coefficient and birefringence of BaTiO3 crystal film as shown in Fig. 7(a). For device fabrication, a 450 nm thick BaTiO3 crystal film shown in Fig. 1(c) was selected by referring to its XRD spectrum shown in Fig. 1(d) since it stands for an undesirable c-axis BaTiO3 crystal film. Then in-plane, at the waveguide/electrodes direction, the light-wave propagates at b-axis direction and the electric field at a-axis direction. we fabricated the straight waveguides and the embedded electrodes to form an EO modulator as shown in Fig. 7(b) where the left image is the sample of device wired on a print-circuit board (PCB) and the right photo is the amplified part of the waveguides and electrodes. In this device regime, the rib etched on BaTiO3 crystal film was set to be 4.0 µm, the electrode gap Gx and the rib width W meet a relation as ${{G}_{x}}{ - W \ge 2}{.0\;\mathrm{\mu} {\rm m}}$. The fabricating procedure includes (1) etching the BaTiO3 crystal film with a photoresist (PR) layer to form rib waveguide having a 100nm height and a 2∼4 µm width, (2) depositing a SiO2 film to form a top cladding layer above the rib waveguide, (3) operating a photolithography with PR for the embedded trenches, (4) etching through the top cladding layer of SiO2 and continuing the etching into the BaTiO3 crystal film by 100nm, (5) depositing a 1.0 µm Al film and fabricating the electrodes with lift-off technique. The BaTiO3 crystal film rib waveguides and the metallic electrodes are not at the same plane, i.e., an embedded configuration is formed. In the experiments of EO modulations for measuring the correlative solutions of ${{b}_{{eo}}}$ and ${{r}_{{51}}}$, two linear polarizations and two ellipse polarizations are selected as shown in Fig. 7(c) as the typical optical phase values in EO modulations where the ±45° linear polarization stand for 0 and ±π optical phases and the two ellipses stand for ±π/2 or ±3π/2 optical phases.
Fig. 7. Measurements for the electrooptic coefficient and birefringence of BaTiO3 crystal film waveguide: (a) the experimental setup where the thick and thin lines are wires and fibers, respectively, PA – polarization analyzer; (b) the photo images of the wired device and the amplified part of waveguide/electrodes; (c) the polarization states obtained with PA, two linear and two ellipse polarizations to indicate the optical phases of 0, π, π/2 and 3π/2, respectively.
During the EO modulation experiments, a laser beam is set as a −45° linear polarization (at the second/fourth quadrants), then it is launched into the waveguide and three voltages are imposed onto it to create three different birefringence modulations in the waveguide. In this BaTiO3 crystal film EO modulator, an embedded waveguide/electrode scheme can form a 2-D optic-electrical interaction efficiency ${\mathrm{\Gamma }_{\textrm{2D}}}$. If a drive voltage ${{V}_{d}}{(m)}$ is imposed between anode and cathode, and the activation voltage of the system is ${{V}_{{act}}}$, the circle refractive index ${{n}_{o}}$ of the o-ray and the ellipse refractive index ${{n}_{e}}$ of the e-ray. As a result, the EO modulation cause optical phases at three typical output polarization states shown Fig. 7(c): (i) from the second to the first; (ii) from the second to the third; and (iii) from the second to the fourth, corresponding to the three values of drive voltage (${{V}_{m}}$) meet Eq. (12) as [20,26]
During the experiments of EO modulations, a plate heater is used to create four temperature values of 25°C, 55°C, 70°C and 85°C. At each temperature, with the three values of ${{V}_{d}}{(m)}$ corresponding to three modulated optical phases in the polarization characterizations [20,26], we obtain four curves of ${{r}_{{51}}}{ - }{{b}_{{eo}}}$ relation with Eq. (12) as shown in Figs. 8(a), 8(b), 8(c) and 8(d) where the activation voltage ${{V}_{{act}}}$ is used to cancel the initial birefringence caused optical phase and the intrinsic electric field in the crystal film [20,26]. Thus, the overlapped coordinates of the three lines are the measured values of ${{b}_{{eo}}}$ and ${{r}_{{51}}}$. Note that on each solution figure, taking Fig. 8(a) as an example, the first and third lines of m=1 and 3 are completely overlapped since they have an optical phase difference of 2π, so the third line (the red line) is shadowed by the blue one.
Fig. 8. Simulation drawings for ${(}{{r}_{{51}}}{,}{{b}_{{eo}}}{)}$, where (a), (b), (c) and (d) are under the temperatures of 25°C, 55°C, 70°C and 85°C, respectively.
In the solutions of ${{b}_{{eo}}}$ and ${{r}_{{51}}}$, ${{b}_{{eo}}}$ is dependent of the two polarization states in an EO modulation, while ${{r}_{{51}}}$ is not. Finally, we tabulate all the solutions of ${{r}_{{51}}}$ at all the four temperatures as listed in Table 4. It turns out from Table 4 that the change trend of the measured values with the temperature is in accord with the simulation results shown in Fig. 6.
Table 4. Measured values of the EO coefficient ${{r}_{{51}}}$.
5. Conclusions
Starting with the thermodynamic energy equilibrium among the potential energy, the kinetic energy, and the energy-focusing cost inside the crystal domain, a model for defining the temperature dependence of c-axis oriented polarization degree is developed, and the numerical calculations predict a complicated temperature dependence of the polarization degradation. In theory, the temperature dependence of ${{r}_{{51}}}$ is the combination of polarization based on an energy conservation in a crystalline domain defined by Eq. (5) and the thermal properties of the electrooptic coefficient defined by Eq. (11). In practice, the principle of the above measurements of ${{r}_{{51}}}$ is based on the nonlinear EO modulation relation defined by Eq. (12). The theoretical predicted and the measured results of ${{r}_{{51}}}$ dependance on temperature are in good agreement. This work and its outcomes obtained are very sustainable to research of the characteristics of tetragonal crystal film of BaTiO3 and development of nonvolatile memories and EO functional device.
Funding
Department of Science and Technology of Jilin Province (20180101223JC); Jilin Provincial Human Resources and Social Security (634190874002); National Natural Science Foundation of China (NSFC) (51725203, 51721001).
Acknowledgments
Authors acknowledge that Profs. Mei Kong, Dan Fang, Lun Jiang and postgraduate Jian Cui from Changchun University of Science and Technology gave helps to this work in experiments.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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