Temperature dependence of the polarization, dielectric constant, damping constant and the relaxation time close to the ferroelectric-paraelectric phase transition in LiNbO3

Accepted Manuscript Title: Temperature dependence of the polarization, dielectric constant, damping constant and the relaxation time close to the ferroelectric-paraelectric phase transition in LiNbO3 Author: A. Kiraci H. Yurtseven PII: DOI: Reference:

S0030-4026(16)31570-4 http://dx.doi.org/doi:10.1016/j.ijleo.2016.12.020 IJLEO 58650

To appear in: Received date: Accepted date:

17-10-2016 7-12-2016

Please cite this article as: {http://dx.doi.org/ This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Temperature Dependence of the Polarization, Dielectric Constant, Damping Constant and the Relaxation Time Close to the Ferroelectric-Paraelectric phase transition in LiNbO3

A. Kiraci1,2 and H. Yurtseven1*

1

Department of Physics, Middle East Technical University,06531 Ankara-TURKEY

2

Inter-Curricular Courses Department, Physics Group, Cankaya University, 06790 AnkaraTURKEY

*

Corresponding author e-mail: [email protected]

ABSTRACT We calculate the order parameter (spontaneous polarization) and the inverse dielectric susceptibility at various temperatures in the ferroelectric phase of LiNbO3 for its ferroelectricparaelectric phase transition (TC= 1260 K) using the Landau phenomenological model. For this calculation, the Raman frequencies of the soft optic mode (TO1) are used as the order parameter and the fitting procedure is employed for both the order parameter and the inverse dielectric susceptibility by means of the observed data from the literature. The temperature dependences of the damping constant and the inverse relaxation time are also computed using the pseudospin-phonon coupled model and the energy fluctuation model for the ferroelectric phase of LiNbO3. The activation energy is deduced from the damping constant for both models studied and compared with the 𝑘𝐵 𝑇𝐶 value of LiNbO3. We find that the order parameter (Raman frequency of the TO1 mode) and the inverse dielectric susceptibility decrease with increasing temperature, as expected from the mean field model. We also find that the damping constant and the inverse relaxation time of this soft mode increases and decreases, respectively, with increasing temperature on the basis of the two models studied in the ferroelectric phase of LiNbO3. This indicates that our method of calculation is satisfactory to describe the observed behaviour of the ferroelectric-paraelectric phase transition in LiNbO3.

KEYWORDS: Mean field theory. Order parameter. Dielectric susceptibility. Damping constant. Relaxation time. LiNbO3.

1. INTRODUCTION The ferroelctric phase transition in lithum niobate has been studied extensively in the literature. Some years ago, various experimental techniques such as slow neutron scattering, DTA, X-ray diffraction were used to investigate its ferroelectric transition [1-5]. As a highTC ferroelectric material, it has many applications in nonlinear optics, optoelectronics, acoustics, optical waveguide, second harmonic generator and holographic storage [6,7]. The lattice dynamics of LiNbO3 has been investigated using the Raman [8-10] and IR reflectivity [11] spectroscopy in some earlier studies. Raman studies of phase transitions [12], in particular, the composition dependence of Raman scattering spectra in lithium niobate [13,14] have been reported in the literature. More recently, the temperature dependence of the Raman modes [15], Raman studies of ferroelectric domain walls [16] and micro-and nanoscale domain structures [17] in LiNbO3 have also been reported in the literature. Apart from those studies, defects in nonstoichiometric lithium niobate crystals [18], hexagonally poled lithium niobate [19] and the mechanism of the ferroelectric phase transition in this compound [20] have been investigated. The ferroelectric-paraelectric transition occurs in the temperature range of 1380 -1470 K (Curie temperature) or at about 1402 K according to composion [5] for the lithium niobate. Lattice structures in the ferroelectric and paraelectric phases have been 6 6 established with the unit cell symmetry R3c (𝐶3𝑣 ) and R3̅c (𝐷3𝑑 ), respectively, in LiNbO3 [2,3]. It has been pointed out that this compound undergoes phase transitions at high pressures and temperatures [5,6] with the NaIO3-type orthorhombic structure (Z=4) and with the structure of P63 or P63/m space group (Z=2) referred to as the room-temperature highpressure (RTHP) and high-temperature-high-pressure (HTHP), respectively, as also pointed out previously [21]. The Raman studies have revealed that the four optical phonons with A1 symmetry in LiNbO3 at 251, 273, 331 and 631 cm-1 correspond to a movement of the ions parallel to the optical axis of the crystal and are connected with the ferroelectric phase transition [7], as also considered in review articles [22-24].It has been shown that the order-disorder of the Li ions is not the driving mechanism for the ferroelectric instability in LiNbO3, implying that the oxygen order-disorder is the driving mechanism [25], as also pointed out previously [20]. Some studies including Rayleigh scattering, Raman spectroscopy and infrared reflectivity show a soft-mode behaviour for an A1(TO) optical phonons, which suggest a displacive

nature of the transition [11,26,27], in some other studies including neutron and Raman scattering experiments, no mode softening has been observed suggesting the order-disorder nature of the ferroelectric phase transition [4,28,29], as also pointed out previously [30]. Molecular dynamic (MD) simulations suggest that as for other ferroelectrics, the structural transition in LiNbO3 can have a displacive nature far from the transition temperature and an order-disorder type close to the Curie temperature 𝜃𝐶 [30]. The phase transition in lithium niobate and lithium tantalate has been studied as a second order transition by expanding the Ginzburg-Landau- Devonshire free energy per unit volume in terms of the polarization (order parameter) and coercive fields in those ferroelectrics have been calculated [31]. In this study, using the Landau phenomenological model we also expand the free energy in terms of the spontaneous polarization (P) and calculate the temperature dependence of P and the dielectric constant for the ferroelectric-paraelectric transition in LiNbO3. By relating the Raman frequency of the optic phonon A1(TO1) to the order parameter (spontaneous polarization), the temperature dependences of the order parameter and the dielectric constant which are derived from the free energy, are fitted to the observed data [15] for the Raman frequency (TO1) and for the dielectric constant in LiNbO3, as we have also studied previously for BaTiO3 [32]. Using the order parameter (Raman frequency) calculated, the temperature dependence of the damping constant Γ is predicted by means of the pseudospin-phonon coupled model and the energy fluctuation model in the ferroelectric phase of LiNbO3, as we calculated in our previous studies for BaTiO3 [33,34], PbTiO3 [35] and very recently PbZr1-xTixO3 (x=0.45) [36] , Cd2Nb2O7 [37] and SrZrO3 [38]. Our calculated Γ values are compared with the experimental linewidths [15] of the A1(TO1) Raman mode by a fitting procedure. From the predictions of the temperature dependence of the damping constant Γ from both models (pseudospin-phonon coupled model and the energy fluctuation model), activation energy of LiNbO3 is calculated. Finally, using the damping constant and the Raman frequency (TO1), the temperature dependence of the relaxation time is predicted for the ferroelectric-paraelectric phase transition in LiNbO3. Below, in section 2 we give an outline of the theory abd in section 3 we give our calculations and results. Sections 4 and 5 give discussion and conclusions, respectively.

2. THEORY The free energy of the ferroelectric phase can be expanded in terms of the polarization (order prameter) for the paraelectric-ferroelectric phase transition in LiNbO3 as 𝐹 = 𝑎0 + 𝑎2 𝑃 2 + 𝑎4 𝑃 4 + 𝑎6 𝑃 6

(2.1)

where the coefficient 𝑎2 is assumed to be the temperature dependent according to 𝑎2 =∝ (𝑇 − 𝑇𝐶 )

(2.2)

with ∝> 0 and the coeffiicients 𝑎0 , 𝑎4 and 𝑎6 are all constants. 𝑇𝐶 denotes the transition (Curie) temperature between the ferroelectric and paraelectric phases in LiNbO3. Using the free energy of the ferroelectric phase in this crystal, its polarization 𝑃 and the dielectric susceptibility 𝜒 can be obtained by means of the minimization of the free energy with respect to the polarization 𝑃 (𝜕𝐹 ⁄𝜕𝑃 = 0) as given below: 𝑎2 + 2𝑎4 𝑃2 + 3𝑎6 𝑃4 = 0

(2.3)

By solving this quadratic equation for 𝑃, we get 𝑎

1

𝑃2 = − 3𝑎4 ∓ 3𝑎 (𝑎42 − 3𝑎2 𝑎6 )1⁄2 6

(2.4)

6

In order to get a simplified form of the polarization 𝑃, we can take a negative solution of the root square in Eq. (4) under the ansatz 𝑎2 𝑎6 ⁄𝑎42 ≪ 1

(2.5)

which gives when expanded as (𝑎42 − 3𝑎2 𝑎6 )1⁄2 ≅ 𝑎4 −

3𝑎2 𝑎6 2𝑎4

(2.6)

By substituting Eq. (2.6) into Eq. (2.4), we then get ∝

2𝑎

𝑃2 = 2𝑎 (𝑇 − 𝑇𝐶 ) − 3𝑎4 4

6

(2.7)

through Eq. (2.2), the temperature dependence of the polarization in the ferroelectric phase of LiNbO3 for its ferroelectric-paraelectric phase transition.

𝜒

−1

We can also derive the temperature dependence of the inverse dielectric susceptibility from the free energy using the definition, 𝜒 −1 = (𝜕 2 𝐹/𝜕𝑃2 )

(2.8)

which gives

 1  2a2  12a4 P2  30a6 P4

(2.9)

When substituted Eq. (2.7) in Eq. (2.9), we then get the temperature dependence of the inverse susceptibility  1 as

 1 

15 2 16 a42 2 a  T  TC  . 62  12  T  TC   2 a4 3 a6

(2.10)

in the ferroelectric phase of LiNbO3 for the ferroelectric-paraelectric phase transition. The spontaneous polarization 𝑃 as an order parameter which measures the ordering in the ferroelectric phase can be associated with the soft mode behaviour of the A1(TO) Raman mode in LiNbO3. So, the Raman frequency of this soft mode can be considered as an order parameter, which can be related to the spontaneous polarization in the LiNbO3 crystal according to

 / max 

2

 c0  c1P 2

(2.11)

where the Raman frequency 𝜔 of the A1(TO) soft mode is normalized with respect to the maximum frequency since the order parameter (𝑃) varies between zero (paraelectric phase) and 1 (at low temperatures in the ferroelectric phase). In Eq. (2.11) 𝑐0 and 𝑐1 are constants. Thus, the temperature dependence of the Raman frequency of the A1(TO) soft mode can be predicted through Eq. (2.7) by determining the coefficients ∝, 𝑎4 and 𝑎6 in the ferroelectric phase of LiNbO3 using Eq. (2.11). This leads to predict the damping constant (linewidth) of this soft mode as derived for optic modes in general by Matsushita [39] who applied his extended model of an Ising pseudospin-phonon coupled system due to Yamada et al. [40]. According to the Matsushita’s model as derived by using the explicit form of the relaxation time of the order parameter by Lahajnar et al. [41], the damping constant Γ𝑆𝑃 due to the pseudospin-phonon (sp) coupling is given by Γ𝑆𝑃 ∝ (1 − P 2 )ln[𝑇−𝑇

𝑇𝐶 𝐶 (1−P

]

2)

(2.12)

in terms of the spontaneous polarization 𝑃 as an order parameter, as also given previously [42,43]. Eq. (2.12) can be expressed as 𝑇𝐶 Γ𝑆𝑃 = Γ́0 + Á(1 − P 2 )ln[𝑇−𝑇 (1−P 2 )] 𝐶

(2.13)

with the background damping constant (linewidth) Γ́0 and Á is a constant.

Using the temperature dependence of the polarization (Eq. 2.7) from the mean field model in Eq. (2.13) the damping constant can be predicted for the pseudospin-phonon coupled model. In particular, the temperature dependence of the Raman bandwidth of the soft mode A1(TO) can then be predicted using the Raman frequency of this mode in Eq. (2.13) with the background bandwith Γ́0 . The damping constant Γ𝑆𝑃 can also be obtained by the phonon frequency shifted which can be related to the variation in spontaneous polarization as obtained by Schaack and Winterfelt [44] according to T(1−P2 )

Γ𝑆𝑃 ∝ [𝑇−𝑇

2 𝐶 (1−P

]1⁄2

(2.14)

which has been given in an earlier work [43]. This can be expressed as T(1−P2 )

Γ𝑆𝑃 = Γ0 + A[𝑇−𝑇

2 𝐶 (1−P

]1⁄2

(2.15)

where Γ0 is the background damping constant (bandwidth) and A is a constant, as before. Because of this shift in the phonon frequency or correspondingly the variation in the order parameter (spontaneous polarization) which fluctuates, the damping constant Γ𝑆𝑃 (Eq. 2.14) can be considered in the energy-fluctuation model. Once we predict the damping constant (or linewidth) from both models, namely, pseudospin-phonon coupled model (Eq. 2.12) and the energy fluctuation model (Eq. 2.14), activation energy 𝑈 can be predicted for LiNbO3 crystal regarding the soft mode A1(TO) with its bandwidth in this crystal. Using the total linewidth as a function of temperature [45-47] Γ ≅ Γ𝑣𝑖𝑏 + 𝐶 exp(− 𝑈⁄𝑘𝐵 𝑇)

(2.16)

where 𝑘𝐵 is the Boltzmann constant and 𝐶 is a constant, and neglecting the linewidth due to the vibrational relaxation in the vicinity of 𝑇𝐶 (Γ𝑣𝑖𝑏 ≅ 0) we then obtain 𝑙𝑛Γ𝑆𝑃 ≅ − 𝑈⁄𝑘𝐵 𝑇

(2.17)

By plotting 𝑙𝑛Γ𝑆𝑃 against 1⁄𝑇 , the activation energy 𝑈 of the LiNbO3 crystal can be predicted as 𝑇𝐶 is approached from the ferroelectric phase. Finally, from the Raman frequency 𝜔𝑝ℎ (Eq. 2.7 through Eq. 2.11) and the damping constant (linewidth) Γ𝑆𝑃 of the phonon (ph) using both models (Eqs. 2.12 and 2.14), the temperature dependence of the inverse relaxation time (𝜏 −1) can be predicted for this soft phonon in LiNbO3 according to the relation 𝜏 −1 = 𝜔𝑝ℎ 2⁄Γ𝑆𝑃

(2.18)

3. CALCULATIONS AND RESULTS We firstly calculate the temperature dependence of the Raman frequency 𝜔1(TO1) and the dielectric constant 𝜀 using the experimental data [15] in the ferroelectric phase of LiNbO3. For this calculation, by relating the Raman frequency 𝜔1(TO1) in the normalized form (𝜔1,𝑚𝑎𝑥 = 252 𝑐𝑚−1) to the order parameter 𝑃 (spontaneous polarization) according to Eq. (2.11) and by fitting Eq. (2.7) to the observed Raman frequencies of this soft mode [15] in the form (𝜔1⁄𝜔1,𝑚𝑎𝑥 )2 ~𝑃2 = 𝑚(𝑇 − 𝑇𝐶 ) + 𝑛

(3.1)

where 𝑚 = ∝⁄2𝑎4 ,

𝑛 = − 2𝑎4⁄3𝑎6

(3.2)

the fitted parameters 𝑚 and 𝑛 were determined, as given in Table 1. Secondly, the inverse dielectric susceptibility 𝜒 −1 was calculated as a function of temperature using the experimental data for the dielectric constant (𝜀 = 1 + 𝜒) of LiNbO3 [15] according to Eq. (2.10). By expressing Eq. (2.10) in the form of 𝜒 −1 = 𝑎(𝑇 − 𝑇𝐶 )2 + 𝑏(𝑇 − 𝑇𝐶 ) + 𝑐

(3.3)

where 𝑎 = 15 ∝2 𝑎6 ⁄2𝑎42 𝑏 = −12 ∝ 𝑐 = 16𝑎42 ⁄3𝑎6

(3.4)

Eq. (3.3) was then fitted to the observed 𝜀 data [15] and the fitted parameters 𝑎, 𝑏 and 𝑐 were determined, as given in Table 1. Using the values of the parameters 𝑚 and 𝑛 (Eq. 3.2), and 𝑎, 𝑏 and 𝑐 (Eq. 3.4) the coefficients given in the free energy expansion of the ferroelectric phase (Eq. 2.1) were then determined (Table 1). Fig. 1 gives our plot of 𝜔1⁄𝜔1,𝑚𝑎𝑥 related to the order parameter 𝑃 (Eq. 2.11) as a function of 𝑇 − 𝑇𝐶 using the observed Raman frequency data [15] for the soft mode 𝜔1(TO1) according to Eq. (3.1) using Eq. (2.7) with the parameters determined (Table 1).

Next, we give the inverse dielectric susceptibility 𝜒 −1 plotted in Fig. 2 as a function of 𝑇 − 𝑇𝐶 according to Eq. (3.3) with the values of the fitted parameters and 𝑎, 𝑏 and 𝑐 and using the values of the parameters ∝, 𝑎4 and 𝑎6 (Eq. 3.4) as given in Table 1 according to Eq. (2.10) in the ferroelectric phase of LiNbO3. In this figure, the experimental data from the measurements of the dielectric constant at 10 MHz [15] are also given. On the basis of the temperature dependence of the Raman frequency 𝜔1 (TO1) in the normalized form, which is related to the order parameter 𝑃 (spontaneous polarization) according to Eq. (3.1), the damping constant Γ𝑆𝑃 due to the pseudospin-phonon interactions of this soft mode was calculated using the pseudospin-phonon coupled model (Eq. 2.12) and the energy fluctuation model (Eq. 2.13) for LiNbO3. By fitting Eqs. (2.12) and (2.13) to the experimental data for the damping constant 𝛾1(TO1) of the Raman soft mode [15] and using our calculated values of the order parameter 𝑃 (Eq. 2.7) in those Γ𝑆𝑃 relations (Eqs. 2.12 and 2.13), we were able to compute Γ𝑆𝑃 as a function of temperature in the two temperature intervals of the ferroelectric phase of LiNbO3.The fitted parameters of the background damping constant (linewidth) Γ́0 (Γ0 ) and the amplitude Á (𝐴) for both models are given in Table 2. Fig. 3 gives our calculated damping constant (Γ𝑆𝑃 ) as a function of temperature for both models (pseudospin-phonon coupled model and the energy fluctuation model) for the TO1 soft Raman mode of LiNbO3. The observed damping (𝛾𝑖 ) data for this mode [15] are also given in Fig. 3. The activation energy 𝑈 can be deduced from the damping constant Γ𝑆𝑃 for the ferroelectric phase of LiNbO3 according to Eq. (2.17) using our calculated values of Γ𝑆𝑃 for both models (pseudospin-phonon coupled model and the energy fluctuation model). Using the Γ𝑆𝑃 values at various temperatures and by plotting lnΓ𝑆𝑃 against 1⁄𝑇, we extracted values of the activation energy 𝑈 (Table 3). In Fig. 4 we give as an example, a linear plot of lnΓ𝑆𝑃 against 1⁄𝑇 for the pseudospin-phonon coupled model using the calculated Γ𝑆𝑃 (Eq. 2.13), according to lnΓ𝑆𝑃 = 𝑎́ + 𝑏́𝑇

(3.5)

where the slope is defined as 𝑏́ = −𝑈/𝑘𝐵

(3.6)

Similarly, a linear plot of lnΓ𝑆𝑃 against 1⁄𝑇 is given using the calculated Γ𝑆𝑃 (Eq. 2.15) for the energy fluctuation model in Fig. 4. Values of the parameters 𝑎́ and 𝑏́ for both models are given in Table 3. Finally, we calculated the temperature dependence of the inverse relaxation time (𝜏 −1 ) using the experimental data for 𝜏 −1 of the Raman soft mode (TO1) according to Eq. (2.18) for the pseudospin-phonon coupled model (Eq. 2.13) and the energy fluctuation model (Eq. 2.15) with the Raman frequencies 𝜔𝑝ℎ (TO1) of this soft mode in the ferroelectric phase of LiNbO3. For this prediction of 𝜏 −1 for both models, we fitted Eq. (2.18) to the observed 𝜏 −1 data [15] according to the relation

𝜏 −1 = 𝑏𝑜 + 𝑏1 𝜅 + 𝑏2 𝜅 2

(3.7)

where 𝜅 ≡ 𝜔𝑝ℎ 2⁄Γ𝑆𝑃

(3.8)

which was defined as the calculated inverse relaxation time for the iteration procedure since it was found from our analysis that our calculated 𝜏 −1 is related to the observed 𝜏 −1 nonlinearly. As a consequence of this, our calculated 𝜅 for both models was fitted to the observed 𝜏 −1 [15] by means of Eq. (3.7) with the values of the fitted parameters of 𝑏𝑜 , 𝑏1 and 𝑏2 , as given in Table 4. Fig. 5 gives our predicted 𝜏 −1(relaxation time) as a function of 𝑇 − 𝑇𝐶 (Eq. 3.7) using Γ𝑆𝑃 (Eq. 2.13) and Γ𝑆𝑃 (Eq. 2.15) for the pseudospin-phonon model and the energy fluctuation model, respectively, for the Raman TO1 soft mode in the ferroelectric phase of LiNbO3. The observed 𝜏 −1 data [15] are also given in this figure.

4. DISCUSSION Using the Landau phenomenological model, the temperature dependence of the order parameter and the dielectric susceptibility was calculated, as plotted in Figs. (1) and (2), respectively, in the ferroelectric phase of LiNbO3 (TC= 1260 K). The Raman frequency of the soft mode A1(TO1) was considered as the order parameter (spontaneous polarization) of this ferroelectric material on the basis of the experimental frequency data [15] in the ferroelectric phase. By assuming a linear relationship between the Raman frequency (𝜔) and the spontaneous polarization (𝑃) equations (2.7) and (3.1) were fitted to the observed Raman frequencies [15] of this soft mode in the ferroelectric phase, as stated above. Since a softening of the phonon mode A1(TO1) was observed by the Raman-scattering measurements in the whole temperature range up to the Curie temperature [15,28,29], the order parameter (spontaneous polarization) is also expected to decrease as the transition temperature (TC) is approached from the ferroelectric phase (Fig. 1). This soft-mode picture has been argued by showing that the lowest frequency phonon is nearly constant in frequency and highly damped but it is coupled with a quasielastic scattering [48]. This, however, indicates that the ferroelectric-paraelectric transition in LiNbO3 is not an order-disorder type but it is rather a displacive type [48], as also pointed out previously [15]. As observed experimentally, the temperature dependence of the A1(TO) spectrum shows the coexistance of a large phonon softening and a quasielastic scattering [15]. The lowest mode A1(TO1) exhibits a large softening with increasing temperature from the value 252 cm-1 at room temperature down to 160 cm-1 at 1100 K [15]. On the other hand, the A1(TO2) phonon softens more rapidly than the A1(TO1) mode and the ionic motions associated with those two lower frequency modes are transferred to each other [15]. The temperature dependent broadening and asymmetry of the Raman lines of the optical phonons with the A1 symmetry, has been interpreted as the transitions between higher levels in the anharmonic potential for the ionic motion and the decay of the optical phonon into two acoustic phonons [7]. Using the parameters determined from the temperature dependence of the order parameter (Eq. 3.1), the inverse dielectric susceptibility (𝜒 −1 ) as given by Eq. (3.3) was also fitted to the observed dielectric constant [15] (𝜖 = 1 + 𝜒). Since both observed data [15] for the Raman frequency and the dielectric constant were used for the fitting procedure to determine the coefficients accordingly, agreement that has been achieved between observed and the calculated 𝜒 −1 was reasonably good at the temperatures closer to the TC (Fig.2) for the ferroelectric-paraelectric transition in LiNbO3. Above 𝑇 − 𝑇𝐶 ≅ 45𝑂 𝐾, our calculated values are much higher than the observed 𝜒 −1 values [15]. Also, the inverse susceptibility

(𝜒 −1) decreases or the dielectric susceptibility 𝜒 increases as the temperature increases toward the TC from the ferroelectric phase, as expected. This dielectric permittivity has also been calculated previously as arising from the simple superposition of damped harmonic ascillators [15]. It has been found that by taking into account the relaxation mode contribution, the reciprocal dielectric constant shows a linear decrease as −1 𝜖(𝑐𝑎𝑙) ∝ 𝐶 (𝑇0∗ − 𝑇)

(4.1)

with 𝑇0∗ = 1260 K [15]. It has been pointed out that the relaxation mode exhibiting a critical slowing down is characteristic of an order-disorder phase transition in LiNbO3 [15]. Once we calculated the order parameter (spontaneous polarization) from the Raman frequencies of the soft mode (TO1), we were then able to calculate the temperature dependence of the damping constant Γ using the pseudospin-phonon coupled model (Eq. 2.13) and the energy fluctuation model (Eq. 2.15) for the ferroelectric phase of LiNbO3, as plotted in Fig. 3. This calculation was performed by fitting Eqs. (2.13) and (2.15) to the observed bandwidth data [15] for the TO1 mode in the two temperature intervals (Table 2), as stated above. The damping constant (bandwidth) increases considerably as the TC is approached (Fig. 3) and it diverges at T=TC. The temperature dependence of the phonon damping 𝛾 of crystals [13], in particular for the A1(z) modes of 𝐴11 (256 cm-1) and 𝐴13 (322 cm-1) in LiNbO3 have been calculated [14] using 𝛾 = 𝐴 + 𝐵𝑇 + 𝐷𝑇 2

(4.2)

where 𝐴, 𝐵 and 𝐷 are constants. In Eq. (4.2), the first term denotes damping due to scattering at defects, which is temperature independent whereas the second and third terms denote damping caused by scattering due to third- and-fourth- order anharmonicity, respectively [14]. The damping constant 𝛾 increases with the temperature for different compositions of optical phonons in LiNbO3 [14]. Regarding asymmetric Raman lines caused by an anharmonic lattice potential, a model has been introduced for undoped and doped lithium niobate [7]. We also extracted the values of the activation energy 𝑈 according to Eq. (2.17) as plotted in Fig. 4 from both models (pseudospin-phonon coupled model and the energy fluctuation model) in the ferroelectric phase, which are quite close to the 𝑘𝐵 𝑇𝐶 value (109 meV) of LiNbO3 (Table 3). This also indicates that both models studied, are adequate to describe the observed behaviour of the Raman linewidths of the soft mode TO1 in the ferroelectric phase of LiNbO3. Finally, the temperature dependence of the inverse relaxation time (𝜏 −1 ) was calculated (Eq. 2.18). In order to compare our calculated 𝜏 −1 with the onserved data [15], a nonlinear fit (Eq. 3.7) was carried out due to the fluctuations occuring in the ferroelectric phase for the ferroelectric-paraelectric transition in LiNbO3. The inverse relaxation time (𝜏 −1 ) decreases as the temperature increases toward the TC in the ferroelectric phase of LiNbO3 (Fig. 5). This is because of the decreasing of the spontaneous polarization 2 squared (𝜔𝑝ℎ ) with respect to the incraesing of the damping constant (Γ𝑠𝑝 ), as given in Figs. (1) and (3) respectively, with increasing temperature toward the TC in the ferroelectric phase

of LiNbO3. The relaxation times of the higher-energy levels are relatively shorter with a larger linewidth (damping constant) as also stated previously [7]. Regarding the ferroelectric phase transition in LiNbO3 on the basis of the calculations of molecular dynamics within density functional theory, the structural phase transition occurs as a continous process over a range of about 100 K in this compound [31]. It has been pointed out that because of the different behaviour of the Li and Nb sublattices, the ferroelectric transition displays both displacive and order-disorder character in LiNbO3 [31]. Some theoretical studies support an order-disorder model for the oxygen atoms as the driving mechanism for the ferroelectric instability in LiNbO3 [48.49]. Using the molecular dynamics within the shell model [50], the ferroelectric phase transition has been characterized as showing coupled displacive and order-disorder dynamics [51], as also pointed out previously [31]. It has been suggested that the phase transition in LiNbO3 is a two-step process involving a displacive transition of the Nb ions in the oxygen octhohedral cages at a temperature below 𝜃𝐶 (Curie temperature) and an order-disorder transition in the Li-O planes, which is completed at 𝜃𝐶 [31,51]. Also, it has been stated on the basis of the calculations that in the paraelectric phase Li ions are distributed randomly above and below the oxygen planes with an avarage zero net polarization so that the spontaneous polarization in the ferroelectric phase changes gradually over a temperature range of 100 K toward the Curie temperature [31]. This is also an indication of the order-disorder transition in LiNbO3. Composition dependence of optical phonon damping in lithium niobate crystals shows a marked variation [13,14]. Composition (x) dependence of the Raman modes of A1(TO) can be studied in LiNbO3 when the experimental data are available in the literature to investigate the mechanism of the ferroelectric-paraelectric phase transition in these compounds. Also, the temperature dependence of the Raman frequency, damping constant (linewidth), activation energy and the relaxation time can be calculated for the A1(TO2) mode of LiNbO3 as we calculated for the A1(TO1) Raman mode in this study.

5. CONCLUSIONS

Order parameter (spontaneous polarization) and the inverse dielectric susceptibility were calculated as a function of temperature in the ferroelectric phase (T < TC) of LiNbO3 by the mean field theory. Expressions derived from the free energy were fitted to the experimental data from the literature for the Raman frequencies of the soft mode (TO1) and the dielectric constant in the ferroelectric phase of this ferroelectric material. Using the Raman frequencies of this soft mode as the order parameter, the temperature dependence of the damping constant was predicted using the pseudospin-phonon coupled model and the energy fluctuation model in the ferroelectric phase of LiNbO3. From the damping constant, the values of the activation energy were extracted for both models and also the inverse relaxation time was computed as a function of temperature in the ferroelectric phase of LiNbO3. Our results show that the mean field theory explains adequately the observed behaviour of the Raman frequencies of the soft mode and the dielectric constant close to the ferroelectric-paraelectric transition in LiNbO3 (TC= 1260 K). Also, both models (the pseudospin-phonon coupled model and the energy fluctuation model) are adequate to describe the temperature dependence of the damping constant (bandwidth) and the inverse relaxation time for this soft mode in the ferroelectric phase of LiNbO3, as observed experimentally. Values of the activation energy as deduced from the damping constant, are acceptable when compared with the 𝑘𝐵 𝑇𝐶 value at the transition temperature for the ferroelectric-paraelectric transition in LiNbO3. This method of calculating the order parameter, inverse dielectric susceptibility, damping constant and the inverse relaxation time can be applied to some other ferroelectric materials.

REFERENCES [1] A. Smolonski, N. N. Kranik, N. P. Khuchua, V. V. Zhdanova and I. E. Mylnkova, Phys. Stat. Solidi 13 (1966) 309. [2] S. C. Abrahams, E. Buechler, W. C. Hamilton and S. J. Laplaca, J. Phys. Chem. Solids 34 (1973) 521. [3] J. Samuelsen and A. P. Grande, Z. Phys. B 24 (1973) 521. [4] M. R. Chowdhury, G. E. Peckham and D. H. Saunderson, J. Phys. C 11 (1978) 1671. [5] P. K. Gallagher and H. M. O’Brian Jr, J. Am. Ceram. Soc. 68 (1985) 147. [6] D. Psaltis and F. Mok, Sci. Am. 273 (1995) 70. [7] U. T. Schwartz and M. Maier, Phys. Rev. B 55 (1997) 11041. [8] R. F. Schauufele and M. J. Weber, Phys. Rev. 152 (1966) 705. [9] I. P. Kaminow and W. D. Johnston, Phys. Rev. 160 (1967) 519. [10] R. Claus, G. Borstel, E. Wiesendanger and I. Steffan, Z. Naturfa 27 (1972) 1187. [11] J. L. Servoin and F. Gervais, Solid State Commun. 31 (1979)n387. [12] Yu. K. Voronka, A. B. Kudryavtsev, V. V. Osiko, A. A.A Sobol and E. V. Sorokin, Sov. Phys. Solid State 29 (1987) 771. [13] B. A. Scott and G. Burns, J. Am. Ceram. Soc. 55 (1972) 225. [14] S. Koijima, Jpn. J. Phys. 32 (1993) 4373. [15] A. Ridah, M. D. Fontana and P. Boursen, Phys. Rev. B 56 (1997) 5967. [16] P. Capek, G. Stone, V. Dierolf, C. Althouse and V. Gopolan, Phys. Stat. Sol. (c) 4 (2007) 830. [17] P. S. Zelenovskiy, M. D. Fontana, V. Y. Shur, P. Bourson and D. K. Kuzentsov, App. Phys. A 99 (2010) 741. [18] Y. Kong, J. Xu, X. Chen, C. Zhang, W. Zhang and G. Zhang, J. App. Phys. 87 (2000)

4410. [19] N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson and D. C. Hanna, Phys. Rev. L ett 84 (2000) 4345. [20] Y. Ohkubo, Y. Murakami, T. Saito, A. Yokoyama, S. Vehara and Y. Kawase, Phys. Rev. B 65 (2002) 052107. [21] K. Nakamura, S. Higuchi and T. Ohnuma, J. Appl. Phys. 111 (2012) 033522. [22] A. Rauber, in Current Topics in Materials Science, edited by E. Kaldis (North Holland, New York, 1978), Vol. 1, Chap. 1, p. 481. [23] J. F. Scott, in Properties of Lithium Niobate, edited by J. Sears (INSPEC, London, 1988) Chap. 9. P. 174. [24] O. F. Schirmer, O. Thiemann and M. Wohlecke, J. Phys. Chem. Solids 52 (1991) 185. [25] I. Inbar and R. E. Cohen, Phys. Rev. B 53 (1996) 1193. [26] A. S. Barker, Jr. AndR. Loudon, Phys. Rev. 158 (1967) 433. [27] W. D. Johnston, Jr. and I. P. Kaminow, Phys. Rev. 168 (1968) 1045. [28] M. R. Chowdhury, G. E. Peckham, R. T. Ross and D. H. Saunderson, J. Phys. C, 7 (1974) L99. [29] S. Kojima, Ferroelectrics 223 (1999) 63. [30] S. Sanna and W. G. Schmidt, IEEE Trans 59 (2012) 1925. [31] S. Kim, V. Gopalan and A. Gruverman, App. Phys. Lett. 80 (2002) 2740. [32] H. Yurtseven and A. Kiraci, J. Mol. Model. 19 (2013) 3925. [33] A. Kiraci and H. Yurtseven, Ferroelectrics 432 (2012) 14. [34] H. Yurtseven and A. Kiraci, Ferroelectrics 437 (2012) 137. [35] A. Kiraci and H. Yurtseven, IEEE Int. Symp. ISAF/PFM 53 (2013) [36] H. Yurtseven and A. Kiraci, IEEE Trans. 63 (2016) 1647. [37] A. Kiraci and H. Yurtseven, Optik 127 (2016) 11497. [38] H. Yurtseven and A. Kiraci, J. Mol. Struc. 1128 (2017) 51. [39] M. Matsushita, J. Chem. Phys. 65 (1976) 23. [40] Y. Yamada, N. Mori, and Y. Noda,J. Phys. Soc. Jpn, 32 (1972) 1565. [41] G. Lahajnar, R. Blinc, S. Zumer, Phys. Con. Matt. 18 (1974) 301.

[42] I. Laulicht and N. Luknar, Chem. Phys. Lett. 47 (1977) 237. [43] I. Laulicht, J. Phys. Chem. Solids 39 (1978) 901. [44] G. Schaack, and V. Winterfeldt, Ferroelectrics, 15 (1977) 35. [45] A.V. Rakov, Opt. Spectrosc. 7 (1959) 128. [46] F. J. Bartoli, T. A Litovitz, J. Chem. Phys. 56 (1972) 413. [47] M.A. Fahim, Thermochimica Acta. 363 (2000) 121. [48] Y. Okomoto, P. C. Wang and J. F. Scott, Phys Rev. B 32 (1985) 6787. [49] J. Yu and K. Park, Physica B 237 (1997) 341. [50] S. M. Tomlison, C. V. Freman, C. R. A Catlow, H. Donnerberg and M. Leslie, J. Chem. Soc. Faraday Trans. II 85 (1989) 367. [51] S. R. Philipst and V. Gopalan, Appl. Phys. Lett. 84 (2004) 1916.

FIGURE CAPTIONS Figure 1 Order parameter (squared) as the observed (𝜔⁄𝜔𝑚𝑎𝑥 )2 [15] and the spontaneous polarization (𝑃2 ) calculated (Eq. 2.7) from the mean field model at various temperatures according to Eq. (2.11) using the experimental Raman frequencies 𝜔1 of the soft mode (TO1) in the ferroelectric phase for the ferroelectric-paraelectric phase transition (TC = 1260 K) in LiNbO3. Figure 2 Temperature dependence of the inverse dielectric susceptbility (𝜒 −1 ) calculated (Eq. 3.3) using the observed [15] Raman mode TO1 (Eq. 3.1) in the ferroelectric phase of LiNbO3 for the ferroelectric-paraelectric phase transition (TC = 1260 K). Experimental data [] at 10 MHz are also shown here. Figure 3 Temperature dependence of the damping constant Γ𝑆𝑃 due to the pseudospin-phonon interactions for the Raman soft mode (TO1) using the pseudospin-phonon coupled model (Eq. 2.13) and the energy fluctuation model (Eq. 2.15) in the ferroelectric phase of LiNbO3 (TC = 1260 K). The observed damping 𝛾𝑖 [15] are also given here. Figure 4 Damping constant Γ𝑆𝑃 (in the logarithmic form) calculated for the Raman soft mode (TO1) as a function of temperature to extract the activation energy 𝑈 according to Eq. (3.5) using the pseudospin-phonon coupled model (Eq. 2.13) in the ferroelectric phase for the ferroelectric-paraelectric phase transition (TC = 1260 K) in LiNbO3. Figure 5 Temperature dependence of the inverse relaxation time (𝜏 −1 ) calculated from the observed Raman frequencies [15] of the soft mode (TO1) and the damping constant Γ𝑆𝑃 due to the pseudospin-phonon coupled model (Eq. 2.13) and the energy fluctuation model (Eq. 2.15) according to Eq. (3.7) in the ferroelectric phase of LiNbO3 for the ferroelectric-paraelectric phase transition (TC = 1260 K). The observed 𝜏 −1 for this soft mode [15] are also shown here.

FIGURES Figure 1

1,0

(Order parameter)

2

0,8

0,6

0,4

2

Observed (max) [15] 2

Calculated P ( Eqs. 2.7 and 2.11) -900

-600

-300

T-TC (K)

Figure 2

Calculated (Eq. 3.3) Observed [15]

Inverse dielectric -1 susceptbility ( )

0,04

0,02

0,00 -900

-600

-300

T-TC (K)

Figure 3

-1

cm 

150

Calculated (Eq.2.13) Calculated (Eq.2.15) Observed [15]

100

50

-900

-600

-300

T-TC

Figure 4

ln SP

4,8

4,2

3,6 10

15

20 -1

1/T (K )

Figure 5

-1

 (cm )

30

-1

20

10

Calculated using SP(Eq. 2.13) Calculated using SP(Eq. 2.15) 

Observed   -900

-600

-300

T-TC

Table 1 Values of the coefficients 𝑚, 𝑛 (Eq. 3.1), 𝑎, 𝑏, 𝑐 (Eq. 3.3), ∝, 𝑎4 and 𝑎6 (Eq. 3.4) using the Raman frequencies of the soft mode (TO1) in the ferroelectric phase of LiNbO3 (TC = 1260 K).

Raman Mode TO1

𝑚 𝑛 −4 𝑥10 (𝐾 −1 ) -7.46 0.35

𝑎 𝑏 𝑐 ∝ 𝑎4 𝑎6 𝑐0 −4 −6 −6 −3 −5 𝑥10−4 𝑥10 𝑥10 𝑥10 𝑥10 𝑥10 (𝐾 −2 ) (𝐾 −1 ) (𝐾 −1 ) 1.81 5.35 4.40 4.46 -3.82 1.77 -0.10

𝑐1 1.19

Temperature Range (𝐾) 301< <1103

T

Table 2 Values of the background damping constant (linewidth) Γ́0 (Γ0 ) and the amplitude 𝐴́ (𝐴) for the damping constant Γ𝑆𝑃 of the Raman mode (TO1) according to the models within the temperature intervals indicated in the ferroelectric phase of LiNbO3. Pseudospin-phonon coupled model (Eq. 2.13) Raman Mode Γ́0 (cm-1) 𝐴́ (cm-1) 1.50 146.74 Γ𝑆𝑃 57.71 78.94 LiNbO3

Energy fluctuation model (Eq. 2.15) Γ0 (cm-1) 𝐴 (cm-1) 22.13 291.07 86.08 107.03

Temperature Interval (K) 479< T <857 920< T <1103

Table 3 Values of the activation energy 𝑈 (Eq. 2.17) using the damping models with the parameters 𝑎́ and 𝑏́ (Eq. 3.5) within the temperature interval indicated in the ferroelectric phase of LiNbO3. 𝑘𝐵 𝑇𝐶 value for this crystal is also given here (TC = 1260 K). LiNbO3

Damping Models

Raman Mode (TO1)

Pseudosoin-phonon coupled model (Eq. 2.13) Energy fluctuation model (Eq. 2.15)

Activation Energy (meV)

-𝑎́

Temperature 𝑏́ (𝐾 −1 ) Interval (K)

104

0.121

6.157 479< T <1103

107

0.124

6.187

𝑘𝐵 𝑇𝐶 (meV)

109

Table 4 Values of the parameters 𝑏𝑜 , 𝑏1 and 𝑏2 of the inverse relaxation time (𝜏 −1) according to Eq. (3.7) for the Raman mode (TO1) using the damping models within the temperature interval indicated in the ferroelectric phase of LiNbO3. LiNbO3

Raman Mode (TO1)

Inverse Relaxation time

−1

𝜏 (Eq.2.18)

Damping Models Pseudosoin phonon coupled model Energy fluctuation model

Γ

𝑏𝑜 (cm-1)

Eq. (2.13)

0.99

Eq. (2.15)

1.05

𝑏1

3108.36

−𝑏2 (cm-1)

Temperature Interval (K)

73967.64 479< <1103

3077.68

71412.89

T