Room temperature phase superposition as origin of enhanced functional properties in BaTiO3 - based ceramics

Journal Pre-proof Room temperature phase superposition as origin of enhanced functional properties in BaTiO3 - based ceramics Nadejda Horchidan, Leontin Padurariu, Cristina E. Ciomaga, Lavinia Curecheriu, Mirela Airimioaei, Florica Doroftei, Florin Tufescu, Liliana Mitoseriu

PII:

S0955-2219(19)30844-1

DOI:

https://doi.org/10.1016/j.jeurceramsoc.2019.11.088

Reference:

JECS 12913

To appear in:

Journal of the European Ceramic Society

Received Date:

4 September 2019

Revised Date:

28 November 2019

Accepted Date:

29 November 2019

Please cite this article as: Horchidan N, Padurariu L, Ciomaga CE, Curecheriu L, Airimioaei M, Doroftei F, Tufescu F, Mitoseriu L, Room temperature phase superposition as origin of enhanced functional properties in BaTiO3 - based ceramics, Journal of the European Ceramic Society (2019), doi: https://doi.org/10.1016/j.jeurceramsoc.2019.11.088

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier.

Room temperature phase superposition as origin of enhanced functional properties in BaTiO3 - based ceramics Nadejda Horchidana,b*, Leontin Padurariua*, Cristina E. Ciomagaa,c, Lavinia Curecheriua, Mirela Airimioaeid, Florica Dorofteie, Florin Tufescua,b and Liliana Mitoseriua* a

Dielectrics, Ferroelectrics & Multiferroics Group, Faculty of Physics, “Al. I. Cuza” University,

Bv. Carol 11, Iasi, 700506, Romania GRADIENT SRL., Str. Codrescu, Nr.17, 700495, Iaşi, Romania

c

Research Department of the Faculty of Physics, “Al. I. Cuza” University, Bv. Carol 11, Iasi,

ro

700506, Romania

of

b

Faculty of Chemistry, “Al. I. Cuza” University, Bv. Carol 11, Iasi, 700506, Romania

e

Inst.of Macromolecular Chemistry “Petru Poni”, Aleea Grigore Ghica Voda 41A 700487 Iasi,

-p

d

Romania

Jo

ur na

lP

re

∗ Corresponding authors: [email protected], [email protected], [email protected]

Abstract

BaSnxTi1-xO3 (x = 0, 0.05) ceramics with orthorhombic/tetragonal phases at room temperature were comparatively investigated to understand the role of phase composition on their functional properties. With respect to the values of BaTiO3, the switching polarization, permittivity peak, tunability and piezoelectric coefficients are enhanced by doping with 5% Sn onto Ti4+ sites. The orthorhombic polymorph amount is larger in the doped ceramic and explains

of

its higher switching polarization. High field poling favors the orthorhombic phase in both compositions; this polymorph becomes predominant in the BaSn0.05Ti0.95O3 ceramic. Landaubased calculations developed for ceramics with randomly oriented grains predicted the stability

ro

of variable amounts of orthorhombic/tetragonal phases around room temperature and explain the field-induced predominant orthorhombic state, mostly in BaSn0.05Ti0.95O3. Due to the twelve

-p

allowed spontaneous polarization directions, the orthorhombic state is responsible for the

lP

with six polarization possible orientations.

re

enhanced polarization, tunability and piezoelectric properties with respect to the tetragonal state

Keywords: Barium titanate-based solid solutions, polymorph phase superposition, field-induced

Jo

ur na

phase transition, Landau-Devonshire theory

1. Introduction

Barium titanate, the prototype ferroelectric perovskite ABO3, is one of the most investigated oxides due to a high interest for applications in microelectronics related to its wellknown dielectric and ferro/pyro/piezoelectric properties. BaTiO3 (BT) presents three structural phase transitions: (i) rhombohedral (R) to orthorhombic (O) at −90 C, orthorhombic (O) to tetragonal (T) at 5 C and (iii) tetragonal (T) to cubic (C) at the Curie temperature TC = 120 C

of

[1]. The structural phase transitions are usually accompanied by enhancements of the electrical, elastic, thermal, electromechanical and thermoelectric properties [2,3]. Therefore, one way of

ro

engineering BaTiO3-based materials towards superior properties is to be used in the range of their structural transformations.

-p

The spontaneous polarization of BaTiO3 is dependent on the phase symmetry, as predicted by Landau-Devonshire theory [4,5] and experimentally demonstrated for single crystals [6]. At a fixed temperature, stress or electric field can induce phase transformations or

re

changes in the phase composition in barium titanate. For example, a T - O structural polymorph transition is induced at room temperature in single crystals by applying high fields along the

lP

[110] direction [6]. Such a phase transformation is also expected in ceramic grains favorable oriented with respect to the direction of the applied field, but its impact on the functional properties has not been yet reported.

ur na

As a general trend, the substitution with homovalent ions (e.g. Zr4+, Hf4+, Ce4+, Sn4+) onto the Ti4+ positions of BaTiO3 structure induces an upshift of the R - O and O - T transitions and a decrease of the Curie temperature, and for specific compositions, the coexistence of all the polymorphs can be reached [7-15]. For example, at the temperature of  40 C, BaSn0.12Ti0.82O3

Jo

presents a quadruple point [12-16] characterized by very high dielectric, ferroelectric and piezoelectric constants [16], due to the strongly reduced barrier energy separating the crystalline phases, as shown by Landau-Devonshire theory [17,18]. The phase diagram of BaSnxTi1-xO3 [19] also indicates phase superpositions in the range of 0.05 < x < 0.15. For such compositions, remarkable functional properties as electrocaloric effect [20] or strong nonlinear dielectric character [21] were reported. In our previous study concerning BaSnxTi1-xO3 solid solutions [2124] it was shown the possible existence of phase superpositions around room temperature, depending on the composition x. For example, the composition x = 0.15 presents a superposition

of C - T phases, while the composition x = 0.05 with remarkable room temperature dielectric properties was found as O by Raman analysis and T by XRD investigation [22-24]. This apparent disagreement gave the idea of possible polymorph superposition around room temperature in BaSn0.05Ti0.95O3 and motivated the interest to explore and understand the origin of the enhanced properties in this composition with respect to ones of BaTiO3 ceramics.

2. Sample preparation and experimental details

of

BaTiO3 and BaSn0.05Ti0.95O3 ceramics have been prepared by solid state reaction starting from p.a. grade oxides: TiO2 (Sigma Aldrich, purity > 99.5%), SnO2 (Merck, purity > 99%) and BaCO3 (Merck, purity > 99%). The precursors were homogenized in water inside polyethylene

ro

jars using zirconia media. After the calcination performed in air at 1100 C/2h, the mixed powders were hydrostatically pressed into pellets at 1400 bar for 30 s using an Isostatic Press

-p

P/N 2864 0000. The green body was further sintered in air at 1400 C/2h, with an increase/decrease temperature rate from room temperature to the maximum sintering temperature

re

of 3 C/min. X-ray diffraction (XRD) patterns were recorded using a Shimadzu LabX 6000 diffractometer (CuKα radiation, λ = 0.15406 nm) at room temperature, by using a scanning rate

lP

of 0.02 and counting time of 1 s/step over the 2θ = 20–80 range. The crystalline structure was determined by Rietveld refinement carried out using the General Structure Analysis System (GSAS) software package, developed by Larson & Von Dreele [25]. Both ceramic compositions

ur na

showed the formation of pure perovskite phase. A detailed structural analysis is presented in paragraph 3.3.

The microstructure morphology of the uncoated samples was examined by using a Scanning Electron Microscope (SEM) type Quanta 200 (FEI), operating at 20 kV with secondary electrons in Low vacuum mode (LFD). The SEM images obtained in fresh fracture for the x = 0 and x = 0.05 presented in Figs. 1(a-b) show a relative uniform

Jo

compositions

microstructure with a slight tendency towards a bimodal distribution (mostly for x = 0). As result of coalescence, the grains in BaTiO3 present sometimes irregular shapes and undefined boundaries, with some exaggerated grown grains (Fig. 1(a)). All these features demonstrate that, in the mentioned conditions, the sintering process is still in progress. The addition of a small 5% Sn content tends to inhibit the exaggerated grain growth and produce a more uniform microstructure (Fig. 1(b)). A slight decrease of the overall average grain size can be noticed for x

= 0.05, of  19 m with respect to  31 m for BaTiO3 ceramic. Both compositions have relative

lP

re

-p

ro

of

densities of about 94%.

ur na

Fig. 1. SEM images of BaSnxTi1-xO3 ceramics sintered at 1400C/2h in fresh fracture: x = 0 (a) and x = 0.05 (b). The electrical measurements were performed on parallel-plate capacitor configuration after applying silver electrodes on the polished surfaces of the sintered ceramic disks. The complex impedance in the frequency domain (1 – 106) Hz and at temperatures of (−150 ÷ 150) C was determined by using a dielectric spectrometer CONCEPT 40 (Novocontrol Tehnologies,

Jo

Hundsagen, Germany) with heating rate 1 C/min. The high voltage dc-tunability measurements were performed at room temperature on the electroded ceramic disks immersed in transformer oil. The ceramics were subjected to high dc voltages (with a maximum applied dc field of 25 kV/cm) produced by a function generator coupled with a TREK 30/20A-H-CE amplifier (Trek Inc., Medina, NY) [26] while being tested by using a LCR - 8105G (GW Instek, New Taipei City, Taiwan) using a testing ac low field of 10 V/cm at various frequencies, in order to determine the permittivity and tangent loss vs. dc field. The macroscopic hysteresis P(E) loops at

room temperature were recorded by a Sawyer-Tower modified circuit fed by a sinusoidal waveform with an amplitude of E = 25 kV/cm and at a testing frequency of 10 Hz produced by a function generator (DS345, Stanford Res. Systems, Sunnyvale, California) coupled with a HighVoltage Amplifier (Trek 609E-6, Trek Inc., Medina, NY). Determination of piezoelectric coefficient d33 at room temperature was performed by using Piezometer PIEZOTEST PM300 (Piezotest Ltd, London, UK) on poled ceramics (at 25 kV/cm for 30 minutes at room temperature  23 C).

of

3. Results and discussions 3.1. Permittivity vs. temperature dependence

ro

The temperature dependences of the dielectric constant and tangent loss of the ceramic

Jo

ur na

lP

re

-p

compositions x = 0 and x = 0.05 at a fixed frequency (1 kHz) are shown in Figs. 2(a-b).

Fig. 2. Temperature dependence of the real part of permittivity (a) and of dielectric losses (b) for BaSnxTi1-xO3 ceramics (x = 0 and x = 0.05) at the frequency f = 1 kHz. Around room temperature permittivity is similar for both ceramic compositions ( 1500 at 20 C). Dielectric losses are below 5 % throughout the overall temperature range, thus

indicating that the ceramics have good dielectric character. The succession of structural phase transformations R – O – T – C is detected by anomalies of both permittivity and losses vs. temperature data and their corresponding temperatures are listed in Table 1. The highest peak in the (T) dependences indicates the ferroelectric-to-paraelectric Curie temperature (T-C structural transition). This maximum is higher in the solid solution, with a permittivity maximum of 7529 at its Curie temperature of 91 C with respect to a smaller one for pure BaTiO3, showing a permittivity maximum of 5102 at 130 C. These results are consistent with our previous data

of

[22-24] and with reports of other authors for similar compositions [9,27].

-p

x=0 -68 C 20 C 130 C 111 C 5102

x = 0.05 -7 C 35 C 91 C 79 C 7529

lP

re

Composition R - O transition temperature O - T transition temperature T - C transition (Curie) temperature (TC) Curie - Weiss temperature from linear fits (T0) Maximum value of permittivity at TC ( m)

ro

Table 1. Values of temperatures corresponding to the structural phase transitions (as detected from the dielectric anomalies), Curie Weiss temperature, and maximum value of permittivity (at TC).

As reported for other small homovalent substitutions onto Ti4+ site [7-10], the addition of

ur na

5% Sn causes a shift towards higher temperatures of the R - O and O - T structural transitions and a reducing of Curie (T - C) temperature to 91C with respect to the values corresponding to pure BaTiO3. According to the dielectric data (Table 1), for both compositions room temperature (23 C) is close to the range of O - T phase transformation, the difference being that BaTiO3 is in the T state (O - T transition at 20 C), while BaSn0.05Ti0.95O3 in its O state (O - T transition at 35

Jo

C). The structural analysis presented in paragraph 3.3 will clarify this aspect.

3.2. High field electrical properties 3.2.1 P(E) hysteresis loops To observe the role of Sn addition on the switching properties of BaTiO3 - based ceramics, polarization-field P(E) loops in dynamic ac regime have been recorded at room temperature. Both compositions show reproducible and stable high field responses with

symmetric P(E) loops due to their low losses (Fig. 3). At the maximum applied field of 25 kV/cm, the composition x = 0.05 shows a well-defined loop with rectangularity factor Pr/Ps = 0.53, where Pr and Ps are remanent and saturation polarizations, respectively, while pure BaTiO3 shows a more tilted P(E) loop with a slightly lower rectangularity of  0.35, at the same applied field. By using the symmetric minor loops, the first polarization curves have been determined and they are comparatively shown in the inset of Fig. 3. Noteworthy is the fact that the addition of only 5% Sn in BaTiO3 produces an increase of switching polarization at any given field. For

of

example, Ps has a 27% relative increase in BaSn0.05Ti0.95O3 with respect of BaTiO3 at 10 kV/cm. The difference in polarization between the two compositions is strongly reduced at higher fields when approaching the saturation. The loop area and coercivity are slightly higher for BaTiO3

ro

ceramic (Ec  2.8 kV/cm for BaTiO3 with respect to  2.1 kV/cm in BaSn0.05Ti0.95 O3) and they are similar to the values reported for other similar BT - based solid solutions [9,20,24,28].

-p

Excellent switching properties with higher polarization and significant electrocaloric properties

ur na

lP

re

were reported for BaSn0.05 Ti0.95O3 ceramics in ref. [20].

Jo

Fig. 3. Major hysteresis loops for BaTiO3 and BaSn0.05Ti0.95O3 ceramics, at 23 C. Inset: first polarization curves determined from the symmetric minor loops.

It is known that the mechanisms involved in the tunability response may be significantly

influenced not only by the Sn doping amount, but also by microstructural properties (density, grain size and grain size distribution) which would determine the structural phase composition and domain structure in such solid solutions. Taking into account this aspect, we comparatively investigated further the dc tunability properties to understand the modifications induced by 5% Sn addition in BaTiO3 ceramics in relationship with the structural properties.

3.2.2 Tunability properties Fig. 4 shows the dc tunability: n(E)=(0)/(E) and the corresponding dielectric losses vs. the applied dc field. Both ceramics sustained the applications of high dc fields with a maximum value of 25 kV/cm. For this field, tunability assumes values of  1.7 for pure BaTiO3 and  2.4 for BaSn0.05Ti0.95O3 (Fig. 4(a)). The dielectric losses remain below 4% for both types of ceramics, confirming their excellent dielectric properties even when subjected to high dc fields

Jo

ur na

lP

re

-p

ro

of

(Fig. 4(b)).

Fig. 4. Dc field dependence of tunability (a) and dielectric losses (b) in the BaSnxTi1-xO3 solid solutions with compositions x = 0 and x = 0.05 determined at 50 kHz. The losses vs. field show a similar trend as permittivity vs. field dependence, with a

hysteretic behavior and a higher value up to 4% in the solid solution with respect to values below 2% for pure BaTiO3, thus confirming the excellent dielectric properties of the present ceramics even when being subjected to high dc fields (Fig. 4(b)). The dielectric loss might arise from two

causes, one due to the finite resistance of the ceramic material, which could cause a resistive energy loss and the other one a pure dielectric loss [29]. However, the leakage giving rise to resistive loss is usually significant at high voltage, while in this case the losses reduce when increasing the applied field (Fig. 4(b)), meaning that the total loss has a dielectric/ferroelectric origin and not a resistive one. While BaTiO3 has a hysteretic nonlinear tunability, the addition of 5% Sn transforms the tunability vs. field in an almost perfect linear dependence (Fig. 4(a)). As discussed in case of

of

other ferroelectrics [29-32], both the permittivity vs. field as well as the energy loss which is significant mainly at low fields should be related to the energy provided by small ac fields needed to release (unpin) the domain walls blocked by localized trapping centers created by

ro

electric or elastic fields [29] and giving rise to a Rayleigh-type subswitching polarization-field behavior. By analyzing the tunability response vs. field (Fig. 4(a)) it can be noticed that the

-p

lowest variation with field is shown by the undoped BaTiO3 ceramic. One reason to explain such a behavior is related to the fact that pure BaTiO3 has mainly 180 domains (and a reversal of

re

polarization direction by 180 would not produce a large modification in permittivity), besides some amount of 90 domains. By adding 5% BaSnO3 nonferroelectric cells, domain structure

lP

changes and lamellar domain structure with smaller thickness than in BaTiO3 assigned to orthorhombic regions were mostly observed [24]. Additionally, some new formed 90 domain walls which are easily re-orientable even at low fields may appear and they give a response with

ur na

a strong variation in permittivity, thus explaining the increased tunability in the solid solution with respect to one of the BaTiO3 parent phase. The apparent contradiction between the fact that high field ac P(E) loops are hysteretic for both compositions (Fig. 3) while the dc tunability shows almost linear and non-hysteretic field-dependence can be explained by the fact that in the two types of experiments different

Jo

mechanisms are involved: (i) reversible contribution to the macroscopic polarization as revealed by dc tunability experiments [29-32], mainly generated by reversible domain wall vibration with small amplitude in local free energy minima (Rayleigh) [33]; (ii) irreversible (switching) polarization between global potential (Landau free energy) minima as revealed from the dynamic P(E) responses. It results that the addition of small amount of 5%Sn in BaTiO3 completely changes the free energy environment for polarization switching and the difference of the switching polarization response is dependent on crystalline phase composition. Since both

compositions are at room temperature not far from their O - T structural transformation, this effect may originate in composition and field-induced modifications of the crystalline phase composition [34].

3.2.3. Piezoelectric properties In order to determine the piezoelectric response, the ceramic pellets were poled at room temperature of  23 C at 25 kV/cm for 30 minutes. The piezoelectric coefficient d33 determined

of

after poling assumes the values of 137 pC/N for x = 0 and 220 pC/N for 5%Sn addition, i.e. with 60% higher than in pure BaTiO3. Room temperature values for d33 higher than 200 pC/N have

ro

been reported for BaSnxTi1-xO3 dense ceramics in the compositional range x = 0.02 ÷ 0.12, with local higher values at O - T (250 pC/N) and O - R (325 pC/N) phase boundaries and with an exceptional value of 697 pC/N at the quadruple point (C - T - O - R phase coexistence, for x 

-p

0.11-0.12), but is worth to note that piezoelectric coefficients are also strongly dependent on the specific poling conditions [34]. In any case, such high values of piezoelectric constants are

re

related to the phase superposition (morphotropic phase boundary) allowing a minimisation of free energy barriers for both polarization increase and polarization rotation, thus leading to high

lP

values for permittivity and piezoelectric constants around the critical compositions [17,34]. The obtained results confirm other reports of properties in Sn - BaTiO3 ceramics, for which indeed the compositions around 5%Sn - BaTiO3 demonstrated a high inverse

ur na

electrocaloric effect and significant ferro-, pyro- or piezoelectric properties [20,35,36]. It was also demonstrated in literature [37] that an enhancement of piezoelectric response of BaTiO3 can be realised by crystallographically engineering single crystals through applied dc bias and this means that the nonlinear response is linked with electric-field-induced structural phase transitions in ferroelectrics. Polarization switching described by the P(E) dependence is another

Jo

closely related nonlinear phenomenon whose microscopic mechanism can be understood in the frame of the electric-field-temperature phase diagrams of ferroelectrics [34]. Therefore, it becomes natural to question if the high responses of BaSn0.05Ti0.95O3 composition are related to its O structure and to the proximity of the O - T transformation, whose balance easily may be changed by the application of high fields. A field-induced structural transformation explaining the high electrocaloric effect around the O - T phase transition was reported [35,36]. In-situ XRD and neutron diffraction analysis [38] probed field-induced alterations of (002)T and (200)T

tetragonal peaks and increase of O peak in between them for 4%Sn - BaTiO3 composition for fields in the range of (2 -4) kV/cm. Such effects were interpreted as related to a field-induced preferred orientation caused by ferroelectric/ferroelastic DWs motion in the T and O phases and/or to field-induced increase of O phase amount. The increase in the intensity of O peak persists after switching off the applied field, meaning that the transformation was irreversible. Since such a possible field-induced structural transformation in the favour of O state might have a fundamental role on the properties measured after poling, e.g. tunability and piezoelectric

of

properties) observed for our 5% Sn - BaTiO3 ceramics, we decided to test the possible irreversible (remanent) structural modifications induced by room temperature poling.

ro

3.3. Remanent field-induced modifications of crystalline symmetry

Room temperature XRD analysis of the two compositions was performed after two

-p

different types of thermal treatments: (i) after a thermal depoling (refreshing) at 500 C for 6 h followed by cooling down to the room temperature, and (ii) after the ceramic was subjected to a

re

poling field of 25 kV/cm at 23 C for 30 min. The XRD pattern collection have been performed after the field was removed and ceramic sample was short-circuited, i.e. the structural data

lP

describe a remanent structural situation. The corresponding experimental XRD patterns and the calculated ones obtained by using Rietveld refinement are comparatively shown in Fig. A1 Appendix A and in Figs. 5. For both compositions, a superposition of perovskites structures with

ur na

orthorhombic (PDF card no. 01-081-2196) and tetragonal symmetries (PDF card no. 00-0050626) with different weights was determined. Usually, BaTiO3 is tetragonal at room temperature, having the O - T phase transition in the range of (5-10) oC. In some cases, the mechanical stresses induced in the manufacturing process of ceramics by pressing or sintering or internal structural defects can shift or broaden this

Jo

transition temperature. Therefore, BaTiO3 ceramics with a superposition of O and T phases may be obtained, as reported in our previous work [39,40] or even other metastable symmetries like monoclinic or O besides the stable T ones may be induced by poling [41,42]. For the present ceramics, Rietveld refinement was carried out in order to determine the phase composition and unit cell parameters of BaTiO3 and 5%Sn-BaTiO3 ceramics before and after poling by using General Structure Analysis System (GSAS) software package [25] and taking into consideration a tetragonal (P4mm) and orthorhombic (Amm2) phase coexistence for both compositions. Fig.

A1 – Appendix A presents the results of these refinements: in each figures the experimental data, the calculated curves, the difference between the experimental profile and the calculated profile, the baseline, and the calculated Bragg positions are shown. The result of the refinements and the parameters which characterize the fitting quality (the goodness of fit χ2 and reliability factors Rp

ur na

lP

re

-p

ro

of

and Rwp) are listed in the Table A.1 - Appendix A.

Jo

Fig. 5. Experimental field-induced remanent structural modifications for BaSnxTi1-xO3 ceramics with x = 0 (a) and x = 0.05 (b) in the range of (002)-(200): 2  (44.5, 46.5) and (301)-(310): 2(74, 76) tetragonal reflections. The computed results show that in the fresh unpoled BaTiO3 the amount of T phase (52

wt.%) is slightly larger than of O one (48 wt.%). The coexistence of these two phases in unpoled BaTiO3 ceramics is also confirmed by the dielectric properties, where it was clearly detected an anomaly corresponding to the O - T transition around 21 C. Therefore, BaTiO3 is at room temperature within the range of its O - T phase transformation, with a slightly T predominance.

After poling, the O state is favored, with a relative weight of 63 wt.% (O) with respect to 37 wt.% (T). In the 5%Sn - BaTiO3 composition, for which at zero field the O phase is predominant, the same trend is observed, with an increase of the O symmetry content after the application and removal of the poling field from 68 wt.% (refreshed) to 73 wt.% (refreshed & poled). The structural modifications in the two compositions after the application and removal of the dc poling field are observed in detail in the range of 2(44.5, 46.5) and  (74, 76), where (002)-(200) and respectively (301)-(310) tetragonal reflections are found (Figs. 5).

of

The high field-induced transformation of T into O state which easy takes place in the vicinity of phase transformation when the free energy minimum is almost flat has a direct

ro

consequence on the macroscopic properties. The presence of 5% Sn ions favors the domination of O phase. A larger ratio of this phase produces a higher ferroelectric polarization response due to the fact that in O state polarization is higher than in T phase. In addition, since samples have a

-p

higher O content after poling (Fig. 5), a higher piezoelectric response is obtained (Section 3.2.3), due to the easy reorientation of polarization which can assume 12 possible orientations in O

re

phase with respect to only 6 in the T state. This also may explain the higher polarization in 5%Sn-BaTiO3 ceramic with respect to BaTiO3, as revealed from the P(E) dependences (Fig. 3).

lP

In order to demonstrate the phase coexistence around room temperature in our compounds and the prevalence of O state after high field poling and its major role on the superior properties of 5%Sn-BaTiO3 ceramics, Landau-Devonshire calculations developed for

ur na

polycrystalline ceramics with randomly oriented grains have been performed.

3.4 Modeling of the field-induced phase transition by Landau -Devonshire theory The model is based on a classical Landau-Devonshire free energy, in a similar way as proposed in refs. [3-6,17-18], where it was employed to describe single crystals only. The

Jo

novelty of the present approach consists in the use of such free energy at local scale to describe ceramic single grains and then to perform statistical averaging in order to derive the overall ceramic properties. More specifically, a sixth order Landau free energy was used [3-6,17-18]: 𝐺 = 𝛼1 (𝑃12 + 𝑃22 + 𝑃32 ) + 𝛼11 (𝑃14 + 𝑃24 + 𝑃34 ) + 𝛼111 (𝑃16 + 𝑃26 + 𝑃36 ) + 𝛼12 (𝑃12 𝑃22 + 𝑃22 𝑃32 + 𝑃12 𝑃32 )+𝛼112 (𝑃12 (𝑃24 + 𝑃34 ) + 𝑃22 (𝑃14 + 𝑃34 ) + 𝑃32 (𝑃14 + 𝑃24 )) + 𝛼123 𝑃12 𝑃22 𝑃32 −𝑃1 𝐸1 − 𝑃2 𝐸2 − 𝑃3 𝐸3 ,

(1)

where P1, P2, P3, and E1, E2, E3 are the components of the polarization vector on [100], [010], and [001] axis of a single crystal or a grain (in the case of a ceramic). The 𝛼1 , 𝛼11 , 𝛼111 , 𝛼12 , 𝛼112 , 𝛼123 parameters are dependent on the temperature (T) and on the substitution degree with Sn4+ onto Ti4+ positions (x) in BaSnxTi1-xO3 solid solutions. Some important factors in ceramics as elastic stresses, polarization variation within grains, surface effect etc. were not explicitly introduced in Eq. (1), but this simplification does not completely restrict the

of

application of such free energy in the case of real ceramics. There are several reports in literature which show that a free energy expansion with additional terms corresponding to mechanical contribution (shear or flexoelectric stresses) or depolarization factor can be written in a

ro

simplified form, similar to Eq. (1), with renormalized values for α parameters (usually α 1 and α12) [43,44]. However, it is important to state from the beginning that the numerical description

-p

of structural phase transitions based on the simplified Eq. (1) can be performed at a qualitative level only. For a more complete theoretical description one should include 3D domain structures,

re

domain walls dynamics (phase field or Monte Carlo simulations) and to consider the distributions of local strains and electric fields (e.g. Finite Element calculations), which is a

lP

difficult and challenging approach from computational point of view. For x = 0, Landau parameters should reduce to those of undoped BaTiO3 and describe correctly all the three structural phase transitions. The Landau coefficients proposed by Bell and Cross in Refs. [4,5]

ur na

for BaTiO3 single crystals represent excellent reference parameters in our attempt to develop Landau coefficients for BaSnxTi1-xO3 solid solutions. For x = 0.12, the coefficients should describe the rhombohedral state below the Curie temperature and the cubic phase above it. At the Curie temperature of 40 C, the peculiar composition BaSn0.12Ti0.88O3 presents a quadruple point characterized by the coexistence of all its polymorphs: R, O, T and C phases [18].

Jo

In order to accomplish all these conditions, the Landau coefficients were calculated as a

linear combination between the coefficients for BaTiO3 and BaSn0.12Ti0.88O3. The values for the Landau coefficients further employed in our simulations are given by Eqs. (B.1) - Appendix B. The critical temperatures for BaTiO3 were taken in accordance with the calculated values from the dielectric measurements presented in Table 1: 𝑇𝐶𝐵𝑇 = 130 C and 𝑇0𝐵𝑇 = 111 C. For the 𝑞𝑢

composition BaSn0.12Ti0.88O3, the following critical temperatures were chosen: 𝑇0 = 35 C and 𝑞𝑢

𝑇𝐶 = 40 C, as reported in literature [16,18].

The free energy for BaTiO3 and BaSn0.05Ti0.95O3 single grain calculated at T = 23 C as a function on P1 and P2 (the third component of polarization P3 being considered null), in the absence of an external electric field are represented in Figs. 6(a) and 6(d). As expected, in the case of BaTiO3 the free energy minimum is established in the T phase. However, it can be noticed that the O state also has a stable minimum of energy. The doping with small amounts of tin in the composition BaSn0.05Ti0.95O3 induces a more pronounced minimum energy for the O phase, while the T symmetry state is still allowed by the existence of a stable minimum of free

of

energy corresponding to this polymorph. Therefore, the present calculations demonstrate that near the O - T phase transition both O and T stable phases may be present with a certain

ro

probability. Since a ceramic statistically contains many grains with their own polarizations, it is reasonable to state, based on the present Landau-Devonshire calculations, that near the O - T

Jo

ur na

lP

re

instead of a neat symmetry phase modification.

-p

phase transformation a coexistence of phases would occur in a certain range of temperature,

Fig. 6. Diagrams of the free energy calculated at T = 23 C for BaTiO3 (a-c) and BaSn0.05Ti0.95O3 (d-f) single grains: (a, d) at zero external field, (b, e) under an electric field of 25 kV/cm oriented along the [110] direction and (c, f) under an electric field of 25 kV/cm applied along the [100] direction. For simplicity, the third components of the polarization and electric field (P3 and E3) were considered equal to zero in these representations.

In order to describe the amounts of phases in a ferroelectric ceramic which was not subjected to high fields before (virgin state), a Metropolis-like algorithm was developed for computing single grain polarization [45]. The algorithm starts with a randomly generated polarization for a given ceramic grain, which is most probably not in the stable minimum of the free energy. The algorithm consists in performing small and random modifications of the polarization (which can be attributed to small thermal agitation fluctuations) and to evaluate the difference between the free energy of the successive steps (∆𝐺). The modified state is accepted if the polarization change leads to a decrease of the free energy ( ∆𝐺 < 0) . Otherwise, the

of

modification can be accepted but with a small probability proportional to 𝑒 −∆𝐺/𝑘𝑇 , where k is the

ro

Boltzmann constant and T is the absolute temperature. By this procedure, the blocking of polarization into an unstable minimum of its free energy (e.g., P = 0) is avoided. The iteration stops when the polarization does not modify anymore with respect to the average value at the

-p

previous steps and the ceramic grain is in a stable minimum of free energy. This algorithm was performed for 10000 grains in order to derive statistically the weight of the phases in the

re

ferroelectric ceramic.

The calculated dependences of the weights of all the four polymorphs (R, O, T, and C) on

lP

temperature are represented in Fig. 7 for three compositions: BaTiO3, BaSn0.05Ti0.95O3 and the reference one BaSn0.12Ti0.88O3. The calculations show that fresh BaTiO3-based ceramics are characterized by superpositions of the phases near O - T and R - O phase transitions. At -80 C,

ur na

the R and O are equally distributed and correspond to the R - O polymorph transition temperature. The O - T phase transition in BaTiO3 occurs at  21 C, which is very close to the experimental value (Table 1). The computed Curie temperature is  130 C, which is also in accordance with the experimental observations. As mentioned before, the Landau parameters proposed in Eqs. B1 – Appendix B were inspired from refs. [4,5,17], but they were adjusted to

Jo

describe the phase transitions observed in the present compositions. Although the BaSn0.12Ti0.88O3 was not experimentally investigated in this work, this composition was introduced in Fig. 7(c) as a reference in order to show that it corresponds to the quadruple point around its Curie temperature, i.e. the system is in R phase below TC  40 C, at quadruple point at TC and in C phase above TC. Such critical composition is the key system in the attempt to describe ferroelectric properties of any BaMxTi1-xO3 solid solutions by Landau-Devonshire theory. The present simulations describe correctly, at least at qualitative level, the modification

induced by the substitution of 5% Sn on the Ti4+ site on the transition temperatures with respect with undoped BaTiO3: the R - O transition temperature shows an increase from -80 C to -20 C, the O - T transition temperature increases from 21 C to 27 C and the T - C transition temperature decreases from 130 C down to 72 C. The computed concentrations for the O phase at room temperature (23 C) are very close to the experimentally determined values: 48% in

Jo

ur na

lP

re

-p

ro

of

BaTiO3 and  68% in BaSn0.05Ti0.95O3.

Fig. 7. The weight of the polymorph phases (R, O, T and C phases) vs. temperature calculated for three BaSnxTi1-xO3 ceramic compositions: (a) x = 0, (b) x = 0.05 and (c) x = 0.12.

Further, the room temperature poling experiments were simulated, in order to check the role of a dc field on the phase composition in BaSnxTi1-xO3 ceramics. When applying an external electric field, the free energy diagrams become no longer symmetric (Figs. 6(b,c,e,f)), which means that the field favors one of the polarization orientations. If the electric field is high enough, the energy barrier between O and T phases can be completely removed and this effect corresponds to a field-induced structural phase transformation. For example, in Figs. 6(b,e) it is shown that an applied field of 25 kV/cm in the [110] direction favors a structural phase transition

of

from T to O in both BaTiO3 and BaSn0.05Ti0.95O3 ceramics, while if the same field is applied along the [100] direction the transition from O to T phase is favored. Both situations may occur in a real ceramic with randomly oriented grains, depending on the orientation of the

ro

crystallographic axes of an individual grain (considered as single-crystalline) with respect to the direction of the electric field. If Ox, Oy, Oz are the orthogonal axes of the ceramic (fixed

-p

reference system) and the electric field is applied on Oz direction, the crystallographic axes of a

,

(3)

lP

𝑢 ⃗ 1 = 𝑢1𝑥 𝑖 + 𝑢1𝑦 𝑗 + 𝑢1𝑧 𝑘⃗ {𝑢 ⃗ 2 = 𝑢2𝑥 𝑖 + 𝑢2𝑦 𝑗 + 𝑢2𝑧 𝑘⃗

re

single grain are described by the following vectors with respect to the fixed reference system:

𝑢 ⃗ 3 = 𝑢3𝑥 𝑖 + 𝑢3𝑦 𝑗 + 𝑢3𝑧 𝑘⃗

ur na

which can be used to calculate the components of the electric field along the crystallographic axes by using the Eq.:

𝑢1𝑦 𝑢2𝑦 𝑢3𝑦

𝑢1𝑧 0 𝑢2𝑧 ] ∙ { 0 } 𝑢3𝑧 𝐸𝑎𝑝𝑝

(4)

Jo

𝑢1𝑥 𝐸1 {𝐸2 } = [𝑢2𝑥 𝑢3𝑥 𝐸3

The components of the electric field along the crystallographic axes can be introduced in Eq. (1) and used in the Metropolis procedure in order to describe the phase transition of a ceramic grain subjected to its own field orientation. After the minimum of energy for a grain is established, the components of the polarization in the fixed reference system are calculated from the components of the polarization on the crystallographic axes:

𝑢1𝑥 𝑃𝑥 {𝑃𝑦 } = [𝑢1𝑦 𝑢1𝑧 𝑃𝑧

𝑢2𝑥 𝑢2𝑦 𝑢2𝑧

𝑢3𝑥 𝑃1 𝑢3𝑦 ] ∙ {𝑃2 }. 𝑢3𝑧 𝑃3

(5)

In order to describe the structural phase transition at room temperature (T = 23 C) under a high electric field, we developed a procedure that consists of the following steps: (i) 10000 grains with random orientations of the crystallographic axis (𝑢 ⃗ 1, 𝑢 ⃗ 2 and 𝑢 ⃗3) with respect to the reference axes (Oxyz) are generated;

of

(ii) The Metropolis algorithm is employed in the absence of external electric field in order to derive the initial weights of the O and T phases for the unpoled (fresh) ceramics;

ro

(iii) The components of the electric field along the crystallographic axes are computed for each grain according to Eq. (4) for different applied field in the range of (0 - 25) kV/cm;

-p

(iv) The Metropolis-like algorithm is employed for each single grain in order to derive the new minimum state of the free energy and the corresponding polarization components (𝑃1 , 𝑃2

re

and 𝑃3 );

(v) The polymorph state is identified for each grain in order to derive statistically the

lP

weight for the O and T phases;

(vi) The component of the polarization along the applied field direction is computed for

Jo

ur na

each grain from Eq. (5) in order to derive the average polarization of the system.

Fig. 8. Simulated polarization vs. field dependence during poling in BaSnxTi1-xO3 (x = 0 and x = 0.05) ceramics. The percentage of O phase is indicated at a few selected fields.

The first polarization vs. field curve computed for the two compositions following the described procedure are comparatively shown in Fig. 8, in which the percentage of the O phase is indicated at some selected field values. The zero field phase compositions (O weight of  48% in BaTiO3 and  68% in BaSn0.05Ti0.95O3) as determined from Fig. 7(a,b) are in perfect agreement with the results of Rietveld calculations. The application of an electric field modifies the O/T phase amounts by favoring the O state for values above 15 kV/cm. The prevalence of O state in the BaSn0.05Ti0.95O3 causes a higher polarization with respect to one of BaTiO3 ceramic (Fig. 8)

of

for all the applied fields. The computed polarization vs. field curves tend to merge when approaching the saturation field, as experimentally noticed (see Inset of Fig. 3). As also observed

ro

experimentally, the composition BaSn0.05Ti0.95O3 reaches the saturation faster than BaTiO3 ceramic.

Although the representations in Fig. 6 show that both T - O and O - T phase transitions

-p

are possible during poling, the statistical calculation for different randomly rotated grains in ceramics demonstrates that the predominance of the O phase over the T one is favored at high

re

electric fields. This change in ratio of the phases is explained by the fact the O phase has 12 possible polarization orientations by comparisons with the T phase in which polarization may

lP

assume only 6 possible orientations. In other words, the probability that the electric field is applied along a [110] - like direction is very high in a random ceramic. At the maximum applied field (25 kV/cm) the computed weight of O phase reaches 92% in BaSn0.05Ti0.95O3 and 65% in

ur na

BaTiO3 ceramics. These values are higher than the experimental ones determined by Rietveld refinement on the poled ceramics, i.e.  63% in BaTiO3 and  73% in BaSn0.05Ti0.95O3 ceramics (Table A.I). This difference between the computed and experimental results is mainly related to the fact that Landau-based approach allows the calculation of in-field polarization, while the experimental phase analysis after poling was realized in a remanent state. In addition, the

Jo

computed polarizations are overestimated with respect to the experimental ones. The experimental polarization is reduced first of all as a “dilution” effect due to the field inhomogeneity (in orientation and magnitude) related to porosity, charge defects, local compositional inhomogeneity or grain boundary phenomena [46-50]. A very interesting fact revealed by the Landau-based calculations of P(E) responses in the present compositions (Fig. 8) is the fact that O phase amount does not show a monotonous increase on the expense of T polymorph when increasing the applied field. In fact, it can be

noticed at intermediate fields that O - T transitions instead of predominant T - O take place in BaTiO3, as revealed by the reducing amount of O state in the range of (5 - 15) kV/cm. Contrary, the weight of O polymorph of BaSn0.05Ti0.95O3 reaches a maximum of  94% in the same field range. Such peculiar effect causing some polarization fluctuation were not detected experimentally and they deserve a further detailed analysis, which is beyond the aim of this paper. Although the main experimental features induced by the small amount of Sn addition in

of

BaTiO3 ceramics have been qualitatively described by the present Landau-Devonshire approach developed for ceramics with randomly distributed grains, some limits of the employed model

ro

should be mentioned. The differences between the experimental and computed polarization and phase composition after poling may be understood by considering that the following approximations have been used in the modeling approach: (i) the ceramic grain was considered

-p

as being single-crystalline with single domain structure, characterized by either an O or T phase, while in fact, a real ceramic with large grains has a complex domain structure and possible

re

compositional and phase inhomogeneity at single grain level; (ii) the same applied electric field was considered in the model for each grain, while a real ceramic is characterized by a certain

lP

level of field inhomogeneity introduced by grain boundaries, pores or electrostatic interactions between grains [46-50], which may be different in the two ceramic compositions and cannot be estimated; (iii) in the model the switching is produced uniformly within a single grain as a

ur na

transition between different energy minima, while in real ferroelectric ceramics, an important role is attributed to defects which act as nucleation or pinning centers and switching proceeds trough domain walls displacements. Despite of the mentioned simplifications, the Landau-based approach was able to describe at a qualitative level a few important features observed in the present BaTiO3-based ceramics: the polymorph superposition around the O - T phase transition

Jo

in unpoled ceramics and the increase of the weight of O phase at high field poling on the expense of the T one. The phase composition and its modification under fields have a very important impact on the functional properties (polarization, dielectric and piezoelectric coefficients), which are superior in the BaSn0.05Ti0.95O3 composition due to the predominance of its as-grown and field-induced amount of O phase over the T one.

4. Conclusions A small homovalent substitution of BaTiO3 ceramics with 5% Sn onto the Ti4+ positions resulted in remarkable dielectric and piezoelectric properties enhancement and higher switching polarization. This effect was discussed in terms of O - T phase superposition in the range of room temperature, by considering the field-induced polymorph modification supported by structural investigations of BaTiO3 and BaSn0.05Ti0.95O3 ceramics in their unpoled and remanent state (after high field poling at 25 kV/cm). The simulations based on Landau theory implemented

of

to describe polycrystalline ceramics with randomly oriented grains predict the existence of a higher relative amount of O phase in the doped ceramics in their fresh state and a field-induced increase of O polymorph amount for both compositions, at high fields. The existence of a higher

ro

amount of O polymorph after poling, in particular in the BaSn0.05Ti0.95O3 ceramic in which it becomes predominant over the T state, explains the higher polarization and enhanced functional

-p

properties experimentally observed. The present calculations may be also adapted for other BaMxTi1-xO3 solid solutions with homovalent substitutions (M = Zr4+, Hf4+, Ce4+, etc.) to explain

re

their enhanced properties as reported in refs. [9,20-22,24,35,36,49,50] and can lead to future opportunities to improve the functional properties of BaTiO3 - based ceramics through tailoring

lP

the phase superposition.

ur na

Acknowledgements: This work was supported by the UEFISCDI Romanian grants PN-III-P4-

Jo

IDPCE-2016-0817 and PN-III-P1-1.1-2016-1069.

lP

re

-p

ro

of

Appendix A. Rietveld refinement

ur na

Fig. A.1. Experimental and calculated X-ray diffraction patterns obtained by using Rietveld refinement for BaTiO3 ceramics (a - refreshed and b - poled) and for BaSn0.05Ti0.95O3 ceramics (c - refreshed and d - poled). Table A.1 The results of Rietveld refinement. Parameters characterizing the refinement quality

Phase composition

lattice parameters (Å)

Jo

Samples

BaTiO3 (refreshed, unpoled)

O

T

O

a = 4.0016; b = 5.6999; c = 5.7162 a = b = 4.0016; c = 4.0455; c/a = 1.0109 a = 3.9998;

V (Å3)

Wt. %

130.38 48.44

Theoretical density (g/cm3)

Χ2

Rp

5.941 5.12

64.78

Rwp

51.56

5.978

130.09 63.05

5.954

27.35 18.41

O BaSn0.05Ti0.95O3 (refreshed, unpoled)

T

O BaSn0.05Ti0.95O3 (refreshed & poled)

T

64.54

36.95

6.00

130.06 68.35

6.046

65.45

31.65

6.007

130.24 73.02

6.024

28.09 20.06

2.12

26.08 18.38

3.46

65.26

26.98

27.79 19.91

6.037

ro

T

4.56

of

b = 5.6977; c = 5.7083 a = b = 3.9967; c = 4.0405; c/a = 1.0109 a = 4.0082; b = 5.6916; c = 5.7010 a = b = 4.02; c = 4.0503; c/a = 1.0075 a = 4.0111; b = 5.6910; c = 5.7053 a = b = 4.0134; c = 4.0517; c/a = 1.0095

-p

BaTiO3 (refreshed & poled)

Appendix B. Parameters in LD calculations

0.12−𝑥

𝛼11 =

0.12

0.12−𝑥 0.12

𝛼111 = 𝛼12 =

∙ 2.91 ∙ 105 (𝑇 − 𝑇0𝐵𝑇 ) +

0.12

0.12−𝑥

0.12−𝑥

𝛼112 =

0.12−𝑥

0.12

𝑥

0.12

∙ 3.89 ∙ 109 + ∙ 4.27 ∙ 109 +

Jo

0.12

𝑥

𝑞𝑢

0.12

∙ [−5.28 ∙ 107 (𝑇 − 𝑇𝐶𝐵𝑇 ) + 1.68 ∙ 109 ] +

∙ 2.81 ∙ 108 −

𝛼112 =

𝑞𝑢

∙ 106 (𝑇 − 𝑇0 ) VmC-1

∙ [3.47 ∙ 106 (𝑇 − 𝑇𝐶𝐵𝑇 ) − 1.93 ∙ 108 ] +

0.12−𝑥

0.12

𝑥

0.12

ur na

𝛼1 =

lP

re

The Landau coefficients employed in simulations were calculated as a linear combination between the coefficients for BaTiO3 and BaSn0.12Ti0.88O3. Their values are given by the following equations:

∙ [−6 ∙ 105 (𝑇 − 𝑇𝐶 ) − 2 ∙ 108 ] Vm5C-1 𝑥

0.12

𝑞𝑢

∙ [−5 ∙ 106 (𝑇 − 𝑇𝐶 ) + 2 ∙ 109 ] Vm9C-5

∙ 4 ∙ 108 Vm5C-1

𝑥

0.12 𝑥

0.12

∙ 6 ∙ 109 Vm9C-5 ∙ 1.2 ∙ 1010 Vm9C-5,

(B.1) 𝑞𝑢

𝑞𝑢

where 𝑇0𝐵𝑇 and 𝑇𝐶𝐵𝑇 are the Curie-Weiss and Curie temperatures of BaTiO3, and 𝑇0 and 𝑇𝐶 are the Curie-Weiss and Curie temperatures of BaSn0.12Ti0.88O3. As mentioned before, the α parameters for BaTiO3 (x=0 in Eq. (B.1)) are inspired from the parameters reported by Bell and Cross in Refs. [4,5]. However, it is important to state that these are not the only parameters proposed in literature for BaTiO3. As presented by Lu et. al. in Ref. [51], there are at least 6 sets of Landau parameters proposed by various authors which describe satisfactory the phase transitions in BaTiO3. The values of Landau parameters also depend on the number of terms considered in the free energy expansion (6th or 8th order terms).

Any other set of parameters could replace the parameters chosen in this work, if they describe correctly all the three structural phase transitions for BaTiO3. A key aspect in developing the Landau parameters for BaSn0.12Ti0.88O3 composition is to describe the quadruple point at 40 oC. The easiest way to derive the relation between the Landau coefficients at the quadruple point is to particularize the free energy expansion (Eq. (1)) for polarizations oriented along [100], [110] and [111] directions, which correspond to the tetragonal, orthorhombic and rhombohedral symmetry, respectively [51]:

3

of

𝐺𝑇 = 𝛼1 𝑃𝑇2 + 𝛼11 𝑃𝑇4 + 𝛼111 𝑃𝑇6 = 𝛼 𝑇 𝑃𝑇2 + 𝛽𝑇 𝑃𝑇4 + 𝛾𝑇 𝑃𝑇6 2𝛼11 + 𝛼12 4 𝛼111 + 𝛼112 6 𝐺𝑂 = 𝛼1 𝑃𝑂2 + 𝑃𝑂 + 𝑃𝑂 = 𝛼𝑂 𝑃𝑂2 + 𝛽𝑂 𝑃𝑂4 + 𝛾𝑂 𝑃𝑂6 4 4 𝛼 +𝛼 3𝛼 +6𝛼 +𝛼 𝐺𝑅 = 𝛼1 𝑃𝑅2 + 11 12 𝑃𝑅4 + 111 112 123 𝑃𝑅6 = 𝛼𝑅 𝑃𝑅2 + 𝛽𝑅 𝑃𝑅4 + 𝛾𝑅 𝑃𝑅6 , 27

(B.2)

𝛼123 =

3(𝛼11 +𝛼12 )2 4𝛼1

lP

𝛼112

− 3𝛼111 − 6𝛼112

ur na

𝛼111

2 𝛼11 = 4𝛼1 (2𝛼11 + 𝛼12 )2 = − 𝛼111 16𝛼1

re

-p

ro

where the following expressions for the polarizations were considered: (i) 𝑃1 = 𝑃𝑇 and 𝑃2 = 𝑃3 = 0 for T phase, (ii) 𝑃1 = 𝑃2 = 𝑃𝑂 /√2 and 𝑃3 = 0 for O phase and (iii) 𝑃1 = 𝑃2 = 𝑃3 = 𝑃𝑅 /√3 for R phase. At the quadruple point, all the three equations should describe a ferroelectric-paraelectric phase transitions and the Landau parameters should satisfy the general condition: 𝛽2 = 4𝛼𝛾. If we assume that α1, α11 and α12 are independent, at the quadruple point, the other 3 parameters will be described by the following relations:

(B.3)

Jo

Among all the possible values of Landau parameters that satisfy these equations, we have chosen the simplest case in which α12 is twice the value of α11. Therefore, in our simulations we considered only α1 and α11 as being independent, while the other 4 parameters were calculated accordingly to the following simplified equations: 𝛼12 = 2𝛼11 2 𝛼11 𝛼111 = 4𝛼1 2 3𝛼11 𝛼112 = 4𝛼1 𝛼123 =

2 6𝛼11

4𝛼1

(B.4)

The advantage of the chosen parameters is that the free energy expansion (Eq.(1)) can be rewritten in a more simplified form: 𝐺 = 𝛼1 𝑃2 + 𝛼11 𝑃4 + 𝛼111 𝑃6 , where 𝑃2 = 𝑃12 + 𝑃22 + 𝑃23 ,

(B.5)

-p

ro

of

in which the free energy does no longer depend on the orientation of the polarization, but only on its magnitude. This equation was also proposed to describe quadruple points by Yao et. al. in Ref. [17]. It is important to mention that for such parameters (Eqs. (B.5)) there is no energy barrier between T, O and R states and their corresponding minimum of energy is not stable, which might be considered a rough simplification. However, the aim of the presented work is not to accurately describe the quadruple point, but only to use it as a reference system and to qualitatively show how the Landau parameters should be modified when increasing the Sn content in Ba(Sn,Ti)O3 ceramics with respect to those describing pure BaTiO3. The simulations of the field induced phase transitions at room temperature (Figs. 6 and 8) considered only stable minimum of energy in T and O phases and the general trends presented in this work are expected to remain similar irrespective the parameters choice for BaSn0.12Ti0.88O3 system, as long as they describe a quadruple point at a certain temperature.

re

References

Jo

ur na

lP

[1] W. J. Merz, The Electric and Optical Behavior of BaTiO3 Single-Domain Crystals, Phys. Rev. 76 (1949) 1221. [2] M.E. Lines, A.M. Glass, Principles and applications of ferroelectrics and related materials, Clarendon Press, Oxford (1977). [3] F. Devonshire, Theory of ferroelectrics, Adv. Phys. 3 (1954) 85. [4] A.J. Bell, L.E. Cross, A phenomenological Gibbs function for BaTiO3 giving correct E-field dependence of all ferroelectric phase changes, Ferroelectrics 59 (1984) 197. [5] A.J. Bell, Phenomenologically derived electric field-temperature phase diagrams and piezoelectric coefficients for single crystal barium titanate under fields along different axes, Journal of Applied Physics 89 (2001) 3907. [6] D.J. Franzbach, Y.J. Gu, L.Q. Chen, K.G. Webber, Electric field-induced tetragonal to orthorhombic phase transitions in [110]c-oriented BaTiO3 single crystals, Appl. Phys. Lett. 101 (2012) 232904. [7] T. Maiti, R. Guo, A.S. Bhalla, Structure-property phase diagram of BaZrxTi1-xO3 system, J. Am. Ceram. Soc. 91 (2008) 1769. [8] M. Acosta, N. Novak, V. Rojas, S. Patel, R. Vaish, J. Koruza, G.A. Rossetti Jr., J. Rodel, BaTiO3-based piezoelectrics: fundamentals, current status, and perspectives, Appl. Phys. Rev. 4 (2017) 041305. [9] A.K. Kalyani, K. Brajesh, A. Senyshyn, R. Ranjan, Orthorhombic-tetragonal phase coexistence and enhanced piezo-response at room temperature in Zr, Sn, and Hf modified BaTiO3, Appl. Phys. Lett. 104 (2014) 252906.

Jo

ur na

lP

re

-p

ro

of

[10] G. Canu, G. Confalonieri, M. Deluca, L. Curecheriu, M.T. Buscaglia, M. Asandulesa, N. Horchidan, M. Dapiaggi, L. Mitoseriu, V. Buscaglia, Structure-property correlations and origin of relaxor behaviour in BaCexTi1-xO3, Acta Mater. 152 (2018) 258-268. [11] L. Padurariu, C. Enachescu, L. Mitoseriu, Monte Carlo simulations for describing the ferroelectric-relaxor crossover in BaTiO₃-based solid solutions, J. Phys.: Condens. Matter 23 (2011) 325901. [12] G.A. Smolensky, Physical phenomena in ferroelectrics with diffused phase transition, J. Phys. Soc. Jpn. 28 (1970) 26-37. [13] X. Wei, Y.J. Feng, X. Yao, Dielectric relaxation behavior in barium stannate titanate ferroelectric ceramics with diffused phase transition, Appl. Phys. Lett. 83 (2003) 2031-2033. [14] V.V. Shvartsman, J. Dec, Z.K. Xu, J. Banys, P. Keburis, W. Kleemann, Crossover from ferroelectric to relaxor behavior in BaTi1-xSnxO3 solid solutions, Phase Transitions 81 (2008) 1013. [15] T. Shi, L. Xie, L. Gu, J. Zhu, Why Sn doping significantly enhances the dielectric properties of Ba(Ti1-xSnx)O3, Sci. Rep. 5 (2015) 8606. [16] C. Lei, A.A. Bokov, Z.-G. Ye, Ferroelectric to relaxor crossover and dielectric phase diagram in the BaTiO3–BaSnO3 system, J. Appl. Phys. 101 (2007) 084105. [17] Y. Yao, C. Zhou, D. Lv, D. Wang, H. Wu, Y. Yang, X. Ren, Large piezoelectricity and dielectric permittivity in BaTiO3- xBaSnO3 system: The role of phase coexisting, EPL, 98 (2012) 27008. [18] W. Liu, J. Wang, X. Ke, S. Li, Large piezoelectric performance of Sn doped BaTiO3 ceramics deviating from quadruple point, J. Alloys Compd. 712 (2017) 1-6. [19] X. Wei, X. Yao, Preparation, structure and dielectric property of barium stannate titanate ceramics, Mater. Sci. Eng. B 137 (2007) 184. [20] S.K. Upadhyay, V.R. Reddy, P. Bag, R. Rawat, S.M. Gupta, A. Gupta, Electro-caloric effect in lead-free Sn doped BaTiO3 ceramics at room temperature and low applied fields, Appl. Phys. Lett. 105 (2014) 112907. [21] M.V. Pop, N. Horchidan, A.C. Ianculescu, L. Mitoseriu, Frequency mixing application based on BaTiO3 ceramic: Design and functional characterization, Int. J. Appl. Ceram. Technol. 14 (2017) 593–603. [22] N. Horchidan, A.C. Ianculescu, L.P. Curecheriu, F. Tudorache, V. Musteata, S. Stoleriu, N. Dragan, D. Crisan, S. Tascu, L. Mitoseriu, Preparation and characterization of barium titanate stannate solid solutions, J. Alloys Compd. 509 (2011) 4731. [23] M. Deluca, L. Stoleriu, L.P. Curecheriu, N. Horchidan, A.C. Ianculescu, C. Galassi, L. Mitoseriu, High-field dielectric properties and Raman spectroscopic investigation of the ferroelectric-to-relaxor crossover in BaSnxTi1-xO3 ceramics, J. Appl. Phys. 111 (2012) 084102. [24] N. Horchidan, A.C. Ianculescu, C.A. Vasilescu, M. Deluca, V. Musteata, H.Ursic, R. Frunza, B. Malic, L. Mitoseriu, Multiscale study of ferroelectric–relaxor crossover in BaSnxTi1−xO3 ceramics, J. Eur. Ceram. Soc. 34 (2014) 3661–3674.

Jo

ur na

lP

re

-p

ro

of

[25] A.C. Larson, R.B. Von Dreele, General Structure Analysis System, Los Alamos National Laboratory, LAUR, (2000) 86–748. [26] F.M. Tufescu, L.P. Curecheriu, A. Ianculescu, C.E. Ciomaga, L. Mitoseriu, High-voltage tunability measurements of the BaZrxTi1-xO3 ferroelectric ceramics, J. Optoel. Adv. Mater. 10 (2008) 1894–1897. [27] W. Cai, Y. Fan, J. Gao, Microstructure, dielectric properties and diffuse phase transition of barium stannate titanate ceramics, J. Mater. Sci.: Mater. Electron. 22 (2011) 265–272. [28] L. Geske, V. Muller, H. Beige, Influence of the Poling Conditions on the Ferroelectric Properties of Orthorhombic Ba(Ti,Sn)O3 Ceramics, Ferroelectrics 361 (2007) 37–44. [29] S. Bhattacharyya, S. Saha, S.B. Krupanidhi, Investigation of reversible and irreversible polarizations in thin films of SrBi2(Ta0.5Nb0.5)2O9, Thin Solid Films 422 (2002) 155–160. [30] D. Bolten, O. Lohse, M. Grossmann, R. Waser, Reversible and Irreversible Domain Wall Contributions to the Polarization in Ferroelectric Thin Films, Ferroelectrics 221 (1999) 251-257. [31] D. Bolten, U. Bottger, R. Waser, Reversible and irreversible polarization processes in ferroelectric ceramics and thin films, J. Appl. Phys. 93 (2003) 1735-1742. [32] P. Gerber, C. Kugeler, U. Bottger, R. Waser, Effects of reversible and irreversible ferroelectric switchings on the piezoelectric large-signal response of lead zirconate titanate thin films, J. Appl. Phys. 98 (2005) 124101. [33] N. Bassiri-Gharb, I. Fujii, E. Hong, S. Trolier-McKinstry, D.V. Taylor, D. Damjanovic, Domain wall contributions to the properties of piezoelectric thin films, J. Electroceram 19 (2007) 47–65. [34] J. Paul, T. Nishimatsu, Y. Kawazoe, U.V. Waghmare, Polarization rotation, switching, and electric-field–temperature phase diagrams of ferroelectric BaTiO3: A molecular dynamics study, Phys. Rev. B 80 (2009) 024107. [35] Y. Bai, K. Ding, G.P. Zheng, S.Q. Shi, J.L. Cao, L. Qiao, The electrocaloric effect around the orthorhombic tetragonal first-order phase transition in BaTiO3, AIP Adv. 2 (2012) 022162. [36] Y. Bai, X. Han, K. Ding, L. Qiao, Electrocaloric Refrigeration Cycles with Large Cooling Capacity in Barium Titanate Ceramics Near Room Temperature, Energy Technol. 5 (2017) 703707. [37] S.-E. Park, S. Wada, L. E. Cross, T. R. Shrout, Crystallographically engineered BaTiO3 single crystals for high-performance piezoelectrics, J. Appl. Phys. 86 (1999) 2746. [38] A.K. Kalyani, H. Krishnan, A. Sen, A. Senyshyn, R. Ranjan, Polarization switching and high piezoelectric response in Sn-modified BaTiO3, Phys. Rev. B 91 (2015) 024101. [39] N. Ma, B.P. Zhang, W.G. Yang, D. Guo, Phase structure and nano-domain in high performance of BaTiO3 piezoelectric ceramics, J. Eur. Ceram. Soc. 32 (2012) 1059–1066. [40] I. Turcan, V.A. Lukacs, L. Curecheriu, L. Padurariu, C.E. Ciomaga, M. Airimioaei, G. Stoian, N. Lupu, L. Mitoseriu, Microstructure and dielectric properties of Ag-BaTiO3 composite ceramics, J. Eur. Ceram. Soc. 38 (2018) 5420-5429. [41] A.K. Kalyani, D.K. Khatua, B. Loukya, R. Datta, A.N. Fitch, A. Senyshyn, R. Ranjan, Metastable monoclinic and orthorhombic phases and electric field induced irreversible phase

Jo

ur na

lP

re

-p

ro

of

transformation at room temperature in the lead-free classical ferroelectric BaTiO3, Phys. Rev. B 91 (2015) 104104. [42] C. Eisenschmidt, H.T. Langhammer, R. Steinhausen, G. Schmidt, Tetragonal-Orthorhombic Phase Transition in Barium Titanate via Monoclinic MC Type Symmetry, Ferroelectrics 432 (2012) 103–116. [43] S. Lin, T. Lu, C. Jin, X. Wang, Size effect on the dielectric properties of BaTiO3 nanoceramics in a modified Ginsburg-Landau-Devonshire thermodynamic theory, Phys. Rev. B, 74 (2006) 134115. [44] A.N. Morozovska, M.D. Glinchuk, Flexo-chemo effect in nanoferroics as a source of critical size disappearance at size-induced phase transitions, J. Appl. Phys. 119 (2016) 094106. [45] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [46] L. Padurariu, L.P. Curecheriu, L. Mitoseriu, Nonlinear dielectric properties of paraelectricdielectric composites described by a 3D finite element method based on Landau-Devonshire theory, Acta Mater. 103 (2016) 724–734. [47] J. Schultheiss, J.I. Roscow, J. Koruza, Orienting anisometric pores in ferroelectrics: Piezoelectric property engineering through local electric field distributions, Phys. Rev. Mater. 3 (2019) 084408. [48] L. Padurariu, L. Curecheriu, V. Buscaglia, L. Mitoseriu, Field-dependent permittivity in nanostructured BaTiO3 ceramics: Modeling and experimental verification, Phys. Rev. B 85 (2012) 224111. [49] S. Zhukov, Y. A. Genenko, O. Hirsch, J. Glaum, T. Granzow, and H. von Seggern, Dynamics of Polarization Reversal in Virgin and Fatigued Ferroelectric Ceramics by Inhomogeneous Field Mechanism, Phys. Rev. B. 82 (2010) 014109. [50] R. Khachaturyan, J. Wehner, Y. A. Genenko, Correlated polarization-switching kinetics in bulk polycrystalline ferroelectrics: A self-consistent mesoscopic switching model, Phys. Rev. B, 95 (2017) 054113. [51] X. Lu, H. Li, W. Cao, Landau expansion parameters for BaTiO3, J. Appl. Phys. 114 (2013) 224106.