## Abstract

In this work, the temperature dependence of polarization degree of
ferroelectric phase barium titanate (BaTiO_{3}) 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 (*r*_{51}) is
modelled. Further, the correlation between the initial polarization
degree and the *r*_{51} 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 *r*_{51}, numerical simulations
are carried out. It was found that both the polarization degradation
of out-of-plane polarization (*P _{z}*) and the
electrooptic coefficient

*r*

_{51}of c-axis oriented crystal films exhibit significant nonlinear dependences on the initial fraction of in-plane polarization (

*P*) and the working temperature. Consequently, the temperature dependences of the measured results of

_{x}*r*

_{51}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 (BaTiO_{3})
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 BaTiO_{3} 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 BaTiO_{3} 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 BaTiO_{3}
[4]. For an analog tetragonal
crystal material, Pb(Zr,Ti)O_{3} (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
BaTiO_{3} 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 BaTiO_{3} 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 BaTiO_{3} 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 BaTiO_{3} 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 BaTiO_{3} film on the SrTiO_{3}-buffered Si
substrate to investigate the switching process of ferroelectric
polarization and an atomic layer deposition of epitaxial c-axis oriented
BaTiO_{3} 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
BaTiO_{3} domain orientations influence its EO properties by Hsu
et al. with spectroscopic ellipsometry [19] and the accurate measurement for the BaTiO_{3}
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 Ba_{1-x}Sr_{x}TiO_{3} 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
BaTiO_{3} 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 *r*_{51} 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 BaTiO_{3} 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, SrTiO_{3},
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 BaTiO_{3}
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
BaTiO_{3} 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 BaTiO_{3} 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]

*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
BaTiO_{3} 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*A*and

_{001}*A*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.

_{100}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 *i*th polarization component to the
*j*th 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 BaTiO_{3} 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 BaTiO_{3} 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]

_{3}crystal has EO coefficient ${{r}_{{51}}}$ showing strong dependence on the work temperature

*T*.

## 3. Simulations for the theoretical BaTiO_{3} crystal film

As a tetragonal material, a ferroelectric BaTiO_{3} 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 BaTiO_{3} crystal, the experimental value
for ${{G}_{{11}}}$ in Eq. (9b) has not been found
yet, so the values for PbTiO_{3} 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 BaTiO_{3} crystal film
are tabulated in Tables 2
and 3, respectively [32]_{.}

The wide studies of molecular dynamics have enabled to
accurately model the nucleation phenomenon of domain wall motion. For
BaTiO_{3} 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*=0*K*
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 BaTiO_{3} 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
BaTiO_{3} 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
(*P*_{z}/*P*_{s}) 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
(*P*_{z}/*P*_{s}) fraction.
Hence the in-plane polarization fraction and its effects on the
dielectric/ferroelectric properties are worthy of discussion.

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 ${{P}_{x}}{ = 0}{.1}$.

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 BaTiO_{3}
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 ${{f}_{{elec}}}$.

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 BaTiO_{3} 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 BaTiO_{3} 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 BaTiO_{3}
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.

As a perovskite crystal, in BaTiO_{3} 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 BaTiO_{3} 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 BaTiO_{3} 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 BaTiO_{3}
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
BaTiO_{3} 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.

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 BaTiO_{3} 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%.

## 4. Measurements for the electrooptic coefficient of BaTiO_{3}
crystal film

We designed an experimental setup for characterizing the EO coefficient and
birefringence of BaTiO_{3} crystal film as shown in
Fig. 7(a). For device
fabrication, a 450 nm thick BaTiO_{3} 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
BaTiO_{3} 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 BaTiO_{3} crystal film was set to be 4.0
µm, the electrode gap *G _{x}* 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 BaTiO

_{3}crystal film with a photoresist (PR) layer to form rib waveguide having a 100nm height and a 2∼4 µm width, (2) depositing a SiO

_{2}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 SiO

_{2}and continuing the etching into the BaTiO

_{3}crystal film by 100nm, (5) depositing a 1.0 µm Al film and fabricating the electrodes with lift-off technique. The BaTiO

_{3}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.

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 BaTiO_{3} 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]

*m*=1, 2 and 3. Then, we use (

*m*+

*p*)π to present the optical phases of a polarization modulations, so ${p = 0}$ indicates the states of the two linear polarizations and ${p ={\pm} (1/2)}$ indicates the states of the two ellipse polarizations shown in Fig. 7(c).

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.

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.

## 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
BaTiO_{3} 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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