Flux method

The high-temperature solution growth (flux method) is the most widely used method for growing KNN-based single crystals [Fig. 3(f)]. This method is therefore described in detail in sections “The tricky choice of a suitable flux,” “Elemental segregation,” and “Different solution growth techniques”, while sections “Growth of KNN-based single crystals with heterogeneous flux” and “Growth of KNN-based single crystals with a self-flux” report the use of this method for KNN.

The flux method is a technique of growing crystals from a solution at its saturation temperature [Reference Elwell and Scheel44]. The solution consists of dissolved substances of the desired material in a solvent, also called the flux. The method is particularly suitable for materials with specific thermodynamic properties, such as KNN-based solid solutions, and is thus often used as an alternative to the conventional growth processes from the melt, such as Bridgman–Stockbarger and Czochralski. Main advantage of using a solvent is the reduction of the crystallization temperature of the solute to be grown, which is essential for materials that cannot be grown from the melt without major drawbacks, e.g.:

1. (i) Materials featuring incongruent melting, i.e., decomposition before their melting point. Here, growth from the liquid phase leads to crystals with a different stoichiometry from initial liquid composition (or other phase), as is usually encountered for KNN-based single crystals.

2. (ii) Materials that undergo solid–solid phase transitions, generally of the first order. This can lead to crystals with twins or cracks due to excessive deformation during phase transition. Also stabilization of an undesired phase can occur.

3. (iii) Materials having a very high vapor pressure or containing highly volatile elements, such as Li in KNN-based materials. This leads to nonstoichiometric crystal composition and/or secondary phase formation, such as the tetragonal tungsten bronze (TTB) structured phases in KNN (see section “Growth defects”).

4. (iv) Highly refractory compounds with very high melting points, such as CaTiO₃ or BaZrO₃ [Reference Xin, Veber, Guennou, Toulouse, Valle, Hatnean, Balakrishnan, Haumont, Saint Martin, Velazquez, Maillard, Rytz, Josse, Maglione and Kreisel53]. In this case, crystal growth requires complex and expensive techniques that may induce many defects, e.g., oxygen deficiencies or internal strains, and thus, post-growth annealing treatments are often needed.

Another advantage of high-temperature solution growth is that the crystal is not exposed to the temperature gradients and thermal fluctuations present in the furnace. The crystal can grow freely, without mechanical or thermal stresses, which allows the development of natural growth facets. In general, crystals obtained by this technique have an excellent crystalline quality and a low dislocation density, as compared to melt growth techniques. Among the drawbacks are the possibility for the incorporation of solvent ions into the crystal, macroscopic flux inclusions, impurities from the crucible, and composition gradients due to segregation phenomena. In addition, this method has a much slower pulling rate (few tenths of mm/h) than the Bridgman–Stockbarger or Czochralski techniques.

Elemental segregation

The difference between the elemental concentration in the initial liquid solution and that in the final crystal is described with the partition coefficient (k) of the elements, also referred to as the segregation coefficient. A partition coefficient k > 1 means that this element is preferentially incorporated into the crystal, while elements with k < 1 are rejected by the crystal. As a general note, high chemical homogeneity can be achieved in compositions with low number of different cations, which display partition coefficients close to 1. Several attempts to grow KNN perovskite–type crystals in quasi-ternary or quasi-quaternary systems have been performed with a self-flux, which enabled the determination of approximate trends for the partition coefficients. The partition coefficients calculated from the phase diagrams reported by Reisman et al. [Reference Reisman, Triebwasser and Holtzberg58, Reference Reisman and Banks59] are shown in Fig. 5. These can be considered as a first approximation for designing the composition of the starting materials. Interestingly, similar B-site partition coefficients as for alkali niobate perovskites were also reported for several niobates with the TTB structure [Reference Chani, Nagata, Kawaguchi, Imaeda and Fukuda60].

Figure 5: Effective partition (segregation) coefficients for selected elements, as a function of their molar contents in the initial liquid solution. The values were calculated using the phase diagrams from Reisman et al. [Reference Reisman, Triebwasser and Holtzberg58, Reference Reisman and Banks59].

In the following, we describe the practical use of Fig. 5 for selecting a starting composition to obtain the desired final composition of the crystal. This is done on the example of potassium, for which the effective partition coefficient is:

(2)$$k_{\rm{K}} = {{x_{{\rm{solid}}} \left( K \right)} \over {x_{{\rm{liquid}}} \left( K \right)}}\quad ,$$

where x _solid(K) and x _liquid(K) are the molar fractions of potassium in the solid and liquid phases, respectively. The k _K is plotted as a function of the x _liquid(K) in the initial liquid solution in Fig. 5, where we read that the calculated k _K is about 0.6 for x _liquid(K) = 0.8 (80%). The potassium content in the crystal at the early beginning of the growth from this solution will thus be:

(3)$$x_{{\rm{solid}}} \left( K \right) = x_{{\rm{liquid}}} \left( K \right) \cdot k_K = 0.6 \cdot 0.8 = 0.48\quad ,$$

resulting in a crystal with the composition K_0.48Na_0.52NbO₃. However, please note that a change in the potassium concentration in the solution during the growth will occur, which will modify the k _K and thus also the crystal composition throughout its entire volume. The above example is in good agreement with the report by Zheng et al. [Reference Zheng, Wang, Liu, Yang, Lu, Li, Huo, Lu, Yang and Cao61], who started with a potassium concentration of x _liquid(K) = 0.8 and grew a final crystal with the composition K_0.45Na_0.55NbO₃ (x _solid(K) = 0.45), resulting in a partition coefficient of k _K = 0.56.

Partition coefficients of a specific element can substantially vary between different growth scenarios, depending on the number of ion species on A or B crystallographic sites in ABO₃ perovskite and depending on whether a self-flux is used or not [Reference Prakasam, Veber, Viraphong, Etienne, Lahaye, Pechev, Lebraud, Shimamura and Maglione54]. Especially sensitive is the segregation of Li; therefore, reliable partition coefficients for this element are difficult to obtain. Values reported for Li concentrations <10% are typically between 0.1 and 0.4 [Reference Hofmeister, Yariv and Agranat62, Reference Liu, Veber, Koruza, Rytz, Josse, Rödel and Maglione63]. Moreover, the segregation of Li strongly depends on the temperature and time of growth and therefore on the rate of volatilization of the solution, which makes crystal stoichiometry difficult to control [Reference Prakasam, Veber, Viraphong, Etienne, Lahaye, Pechev, Lebraud, Shimamura and Maglione54, Reference Veber, Benabdallah, Liu, Buse, Josse and Maglione55]. This was discussed in more detail by Sadel et al., who reported substantial Li volatilization above 1197 °C when growing Li_0.02Na_0.98NbO₃ [Reference Sadel, Von der Mühll, Ravez, Chaminade and Hagenmuller56]. Note that this is the typical temperature range for the growth of KNN-based crystals. The Li volatilization loss in (K,Na,Li)(Nb,Ta)O₃ was recently reported [Reference Liu, Veber, Koruza, Rytz, Josse, Rödel and Maglione63]. The molar amount of Li incorporated into the KNN perovskite matrix is substantially lower than what is thermodynamically expected. The calculated partition coefficient of Li is 0.25, whereas the real effective partition coefficient of Li has been estimated to be between 0.11 and 0.18. This difference was related to strong volatilization. The choice of the excess of initial oxides (or carbonates) constituting the solution and the self-flux for KNN-based crystal growth will thus depend on both the segregation and the experimental estimation of the volatilization during the growth.

Finally, it is noted that elemental segregation has to be considered in three dimensions. To obtain as homogeneous samples as possible, the as-grown boule should be cut along the steadiest chemical directions to achieve the lowest elemental variation in the longest lengths of the samples.

Growth of KNN-based single crystals with a self-flux

First research into the growth of KNN crystals by self-flux started in the 1950s by Cross [Reference Cross23]. Besides, growths of other alkali niobates, e.g., KNb_xTa_1–xO₃ (KNT) as storage medium for holographic memories [Reference Wilcox and Fullmer69], were performed in the late 1950s and 1960s. This growth research considerably improved the knowledge of the alkali niobate high-temperature phase diagrams, so that, since the 2000s, when the research on lead-free piezoelectrics regained interest, intensive self-flux growth has been performed by SSSG and TSSG methods (see references in Table II).

TABLE II: Summary of the main functional properties of the reported KNN-based piezoelectric crystals, along with their composition, growth method, and crystal size. Note that the crystals are ordered based on their chemical composition from pure KNN to more complex systems with several modifiers and/or dopants.

^a Unit of this d ₃₃ value is pm/V; n.a.: information not reported.

Growth with a self-flux can be performed by spontaneous nucleation either on platinum (crucible, spatula) or on an oriented single crystal seed with a composition similar to the solute to be grown. The seed should exhibit a melting or decomposition temperature above the saturation temperature of the KNN-based solution. A very suitable seed material is KTaO₃, since it exhibits the cubic perovskite structure with lattice parameters close to those of KNN, enabling an efficient epitaxy. Moreover, it has a high decomposition temperature, no phase transition from 0 K until its decomposition point, and features the same ions as those of the solute to be grown. Owing to the partition coefficients lower than 1 for Li, K, and Nb in KNN-based solid solutions, the self-flux is chosen with an excess of Li₂O (or Li₂CO₃), K₂O (or K₂CO₃), and Nb₂O₅ [Reference Liu, Koruza, Veber, Rytz, Maglione and Rödel46, Reference Prakasam, Veber, Viraphong, Etienne, Lahaye, Pechev, Lebraud, Shimamura and Maglione54, Reference Veber, Benabdallah, Liu, Buse, Josse and Maglione55, Reference Hofmeister, Yariv and Agranat62, Reference Liu, Veber, Koruza, Rytz, Josse, Rödel and Maglione63]. Example of a Mn-doped (K,Na,Li)(Nb,Ta)O₃ crystal grown using self-flux and a Pt spatula as the nucleation center is shown in Fig. 4(b). In some cases, Na₂O with K₂O is also reported to be used as self-flux [Reference Deng, Zhao, Zhang, Chen, Li, Lin, Ren, Jiao and Luo70], although the Na partition coefficient is higher than 1 in KNN crystals.

The largest reported KNN-based crystals grown by self-flux are few centimeter-wide [Reference Huo, Zheng, Zhang, Zhang, Liu, Wang, Yang, Cao and Shrout71, Reference Tian, Hu, Meng, Tan, Zhou, Li and Yang72] and exhibit brownish to white colors [Reference Prakasam, Veber, Viraphong, Etienne, Lahaye, Pechev, Lebraud, Shimamura and Maglione54], with clear and cloudy regions induced by the presence of numerous thin domains [Reference Deng, Zhang, Zhao, Chen, Wang, Li, Lin, Ren, Jiao and Luo73, Reference Liu, Veber, Zintler, Molina-Luna, Rytz, Maglione and Koruza74]. Crystals grown by SSSG exhibit natural growth faces predominantly along (001)_PC and (110)_PC orientations, whereas TSSG-grown crystals feature cylindrical shape with typical four symmetrical growth ridges at the top [Fig. 4(c)]. Saturation temperatures lie usually in the range of 1000–1200 °C, decreasing thermal ramps in the range of 0.1–0.5 °C/h, and pulling rates in the range of 0.1–0.5 mm/h with rotation speeds about 10–20 rpm. Since KNN-based materials are solid solutions, crystals exhibit chemical composition variations due to the segregation of elements.

For achieving the highest possible chemical homogeneity, a compromise has to be found for the setting of the highest growth rate, which allows a minimum defects concentration and is not detrimental to the crystal's performance. On the other hand, increasing the growth speed shortens the growth time and thus reduces the amount of the volatilized alkalis. Volatilization of alkalis was also demonstrated to be reduced by using oxygen atmosphere [Reference Popovič, Bencze, Koruza, Malič and Kosec75], which additionally lowers the oxygen vacancy concentration. Another approach to improve the homogeneity could be to control the growth of certain crystal faces and corners with particular crystallographic directions using engineering of the isotherms in the solution. This was reported for lead-based systems with 〈111〉 corners [Reference Lim and Rajan76]. Similarly in KNN-based crystals the natural 〈001〉 and 〈110〉 faces could be regarded.

Solid-state crystal growth

The solid-state crystal growth (SSCG) is a controlled case of exaggerated grain growth, whereby a single crystalline seed is growing into a polycrystalline powder matrix below the material's melting point [Fig. 3(c)]. Since this approach does not require melting of the whole sample, it is particularly suitable for incongruently melting materials with volatile species. Other benefits are low processing temperatures, use of conventional furnaces, and no need for temperature gradients. The main challenge is the growth of large single crystals due to the porosity and appearance of exaggerated grain growth in the matrix. The method was first applied to ferroelectric compositions by DeVries growing BaTiO₃ in 1964 [Reference DeVries77] and was later adopted to other systems. For more details on the SSCG method, please see Refs. Reference Benčan, Tchernychova, Uršič, Kosec, Fisher and Lallart78 and Reference Kang, Park, Ko and Lee79.

KNN-based crystals were first grown by the SSCG method in 2007 by Fisher et al. [Reference Fisher, Bencan, Holc, Kosec, Vernay and Rytz80, Reference Fisher, Bencan, Bernard, Holc, Kosec, Vernay and Rytz81]. The 〈001〉-oriented KTaO₃ seed was embedded into calcined KNN powder with the addition of the K₄CuNb₈O₂₃ liquid phase, compacted, packed in a closed crucible, and heated at 1100 °C for 10 h. The grown single-crystalline layer was up to 0.16 mm thick and exhibited high homogeneity. To reduce the porosity, hot pressing was applied either before or during the growth process. The largest reported size of pure KNN crystal was 4 mm, obtained after 100 h at 1100 °C with 50 MPa [Fig. 4(d)] [Reference Fisher, Bencan, Kosec, Vernay and Rytz49]. The reported dielectric and piezoelectric properties were comparable to the polycrystalline counterparts with the same composition [Reference Ursic, Bencan, Skarabot, Godec and Kosec82]. Recently, a faster crystal growth rate was reported for the 〈110〉 seed orientation [Reference Fujii, Ueno and Wada83].

In 2015, Jiang et al. reported a modification of the above method using the LiBiO₃ sintering aid and no seed [Reference Jiang, Randall, Guo, Rao, Tu, Gu, Cheng, Liu, Zhang and Li50]. This “Seed-Free” SSCG method (SFSSCG) resulted in 11 × 9 × 3 mm³ large crystals with low room-temperature dielectric losses (0.03) and a piezoelectric coefficient of about 205 pm/V [Fig. 4(e)]. Following this work, Yang et al. prepared (K_0.45Na_0.55)_0.96Li_0.04NbO₃ crystals using Li₂CO₃ and Bi₂O₃ as sintering aids [Reference Yang, Zhang, Yang, Liu, Li, Liu and Zhang84]. The powders were compacted into 25 mm diameter discs, isostatically pressed, and sintered at 1080 °C for 10 h. The method resulted in crystals up to 6 ×5 × 2 mm³ in size with excellent piezoelectric properties, d ₃₃ = 689 pC/N and d ₃₃* = 967 pm/V. High piezoelectric properties (d ₃₃ = 488 pC/N) and relatively large crystal sizes were also reported for CaZrO₃-modified KNN [Reference Song, Hao, Yan, Zhang, Li and Jiang85]. Recently, Jiang et al. used SFSSCG to grow KNN crystals co-doped with Mn, Li, and Bi [Reference Jiang, Zhang, Rao, Li, Randall, Li, Peng, Li, Gu, Liu and Huang25]. The crystals were grown at 1090 °C for 21 h to a final thickness of 0.6–0.8 mm. The small- and large-signal piezoelectric coefficients reached very high values of 1050 pC/N and 2290 pm/V, respectively, while the transition temperatures were T _O–T = 153 °C and T _C = 403 °C.

Bridgman–Stockbarger method

The Bridgman–Stockbarger technique involves translating the solid–liquid interface of a molten compound via the translation of a growth container (open crucible or sealed ampoule) within a fixed thermal gradient in a multi-zone furnace [Fig. 3(d)] [Reference Monberg and Hurle86]. The crystal growth is driven by supercooling and freezing the melt when passing through the temperature gradient zone. While the original technique of Bridgman used a roughly controlled thermal gradient located at the furnace exit [Reference Bridgman87], the Stockbarger's modification used a thermal diaphragm separating two coupled furnaces with temperatures above and below the melting point for a better control of the gradient at the solid–liquid crystallization interface [Reference Stockbarger88]. This method is typically applied for congruent-melting materials and allows the growth of relatively large crystals, but compositional inhomogeneities are often reported due to extensive segregation [Reference Benayad, Sebald, Lebrun, Guiffard, Pruvost, Guyomar and Beylat89]. The Bridgman–Stockbarger method is commonly used to produce lead-based single crystals [Reference Sun and Cao90].

First attempt to grow KNN-based single crystals by this method was reported by Chen et al. in 2007 [Reference Chen, Xu, Yang, Wang and Li91]. The calcined 0.95(K_0.5Na_0.5)NbO₃–0.05LiNbO₃ powder was pressed and sealed into a Pt crucible. After melting at 1330 °C for 10 h, the crucible was pulled down with a rate of 0.4–0.6 mm/h, while the temperature gradient near the liquid–solid interface was 30–50 °C/cm. Several plate-like samples with sizes 4 × 6 × 0.5 mm³ and orthorhombic structure could be obtained from the boule. The small-signal piezoelectric properties (d ₃₃ = 405 pC/N, k _t = 0.6) were comparable to ceramics with the same composition; however, the dielectric losses and leakage current were relatively high, which was related to a high concentration of oxygen vacancies. The same method was also used to prepare a Na-rich KNN composition, which also exhibited high losses [Fig. 4(f)] [Reference Lin, Li, Li, Cai, Liu and Zhang51]. In 2014, Liu et al. reported a growth of Li-modified KNN crystals with addition of MnO₂ using the Bridgman method and a KCl–K₂CO₃ flux [Reference Liu, Xu, Liu, Yang and Chen65]. The flux efficiently decreased the melting temperature down to 1200 °C, while the addition of Mn reduced the dielectric losses. The piezoelectric coefficient of 226 pC/N was lower than previous reports, but the crystals exhibited characteristic ferroelectric loops.

Reports for improving the chemical homogeneity of KNN-crystals are relatively scarce; however, Luo et al. have recently demonstrated that improved homogeneity can be achieved in PMN-PT crystals by the use of a modified Bridgman method and a careful optimization of growth parameters [Reference Luo, Xu, Xu, Wang and Yin92].

Floating zone method

The floating zone method (FZM), also referred to as zone leveling method, is generally used for the purification of materials and their crystal growth. The principle consists of locally melting the material, creating two liquid–solid interfaces, and subsequently moving the molten zone, so that the whole material undergoes a fusion/solidification transition [Fig. 3(e)]. The growth can be done horizontally or vertically and is generally initialized using an oriented seed. The melting can be achieved by high frequency induction, concentration of light radiation (image furnace, laser zone fusion), or resistive Joule heating effect. This crucible-free technique can feature high pulling speed and high temperature, which allows the growth of various materials without impurity contamination from the crucible.

To the best of our knowledge, only a few KNN-based crystals have been grown by the floating zone technique with a mirror furnace [Fig. 4(g)] [Reference Bah, Giovannelli, Retoux, Bustillo, Le Clezio and Monot-Laffez52, Reference Inagaki, Kakimoto and Kagomiya93, Reference Inagaki, Kakimoto and Kagomiya94, Reference Kimura, Tanahashi, Zhao, Maiwa, Cheng and Wang95]. Pure KNN and Mn-doped KNN crystal growth, performed by Inagaki et al. at 3 mm/h and 25 rpm, was difficult due the poor viscosity of the liquid and the molten zone instability. The centimeter-long and few millimeter-wide crystalline rods usually exhibit several grains at the beginning and millimeter-sized grains at the end of the growth. The maximum lengths of KNN and Mn-doped KNN crystals were 9.3 mm and 12.5 mm, respectively. However, when substituted by 5 mol.% Ta and 5 mol.% Mn, centimeter-sized crystals were obtained with 3 mm/h pulling speed and rotation of 20 rpm. In this case, k _t = 0.48 and d ₃₃ = 65–70 pC/N were measured for both KNN and Ta-doped crystals.

Growth defects

The complexity of the crystal growth process makes ideal control of all parameters virtually impossible, which inevitably results in the formation of growth defects in the crystals. The final habit of the crystal will be determined by the planes with the lowest growth velocities, the external thermal conditions, and the related growth rate. For KNN-based materials, as well as for several other perovskites, the lowest surface energy was reported for the {100}_C faces [Reference Fisher, Bencan, Holc, Kosec, Vernay and Rytz80]. The type of growth defects is related to the selected growth method. Some typical examples include growth twins (two intergrown crystals sharing a lattice plane), hoppers (terraced structures indicating 2D nucleation layer growth), striations (growth layers that appear due to the local fluctuations of the growth velocity related to the temperature fluctuations and convection of the solution), and stress cracks (due to the volume change at phase transitions or too rapid cooling) [Reference Luo, Li, Xia and Li97].

Unfortunately, detailed studies of the growth defects in KNN-based crystals are relatively scarce. Typical examples of defects during flux growth are growth twins [Fig. 6(a)]. They can be formed in the early growth stages by ions occupying geometrically imperfect positions or due to dislocations on the growth interfaces and due to rapid growth. Twinning due to strain relaxation in connection with structural phase transitions is also common in perovskites. Experimental approaches to prevent twin growth include reduction of thermal fluctuations in the vicinity of the growth interface, and the achievement of a low supersaturation of the solution via suitable slow growth rate (decreasing thermal ramp) with respect to the adequate pulling velocity.

Figure 6: A selection of typically growth defects observed in KNN crystals: (a) growth twins, (b) cracks, (c) flux inclusions and surface secondary phases (TTB), (d) growth line defects inducing regions with different domain structures (left: room temperature, right: 470 °C, i.e., above T _C) [Reference Liu, Veber, Zintler, Molina-Luna, Rytz, Maglione and Koruza74], (e) pores in SSCG-grown crystal (reprinted from Ref. Reference Fisher, Bencan, Bernard, Holc, Kosec, Vernay and Rytz81 with permission from Elsevier), and (f) periodic small-angle δ-boundaries in SSCG-grown KNN crystal (reprinted from Ref. Reference Fu, Yang, Yu, Wang, Lu, Yang, Xu and Li101 with permission from Elsevier).

As discussed in section “Elemental segregation,” the segregation phenomena during flux growth are very pronounced. This is particularly problematic for Li-modified KNN crystals. The very low partition coefficient of Li indicates the rejection of these ions by the growing crystal and thus an increase in Li concentration in the liquid solution during the growth process [Reference Hofmeister, Yariv and Agranat62, Reference Liu, Veber, Koruza, Rytz, Josse, Rödel and Maglione63]. A straightforward solution for this issue would be to increase the Li concentration in the starting liquid solution; however, unfortunately, this approach turned out to be detrimental as high Li concentration induces the formation of a TTB structured phase [Reference Scott, Giess, Olson, Burns, Smith and Okane98], for example, with the composition (K,Na)₃Li₂Nb₅O₁₅. The TTB structure can be avoided by keeping the K/Li ratio in the initial liquid above 5–6. The critical Li concentration in the liquid solution is also often achieved toward the end of the growth process and results in the formation of TTB crystals on the crucible walls or on surfaces of the perovskite KNN crystals [Fig. 6(c)]. Note that TTB structure formation was also reported in KNN-based polycrystalline ceramics with high Li concentrations, which was ascribed to the limited solubility of Li in the KNN lattice [Reference Guo, Kakimoto and Ohsato38].

Liu et al. investigated the influence of growth defects on the domain structure of KNN-based crystals grown by the SSSG method [Fig. 6(d)] [Reference Liu, Veber, Zintler, Molina-Luna, Rytz, Maglione and Koruza74]. The exact type of the observed line defects was not determined, but they are likely related to twin boundaries, growth twins, or transformation twins. Note that these line defects remained unchanged when the sample was heated above T _C. It was shown that they resulted in the formation of regions with different domain densities/sizes, which changed the optical properties of the crystal (appearance of transparent and cloudy regions). Interestingly, in the orthorhombic room temperature phase, the domain walls of the transparent regions were found along {110}_PC planes, while those of the cloudy regions were along {001}_PC planes. Such defects are also likely to be detrimental for electromechanical properties, as they may pin the ferroelectric domains walls and thus reduce the response.

The main growth defects during SSCG are pores entrapped inside the crystal [Fig. 6(e)]. These result in a nontransparent crystal appearance and can reduce the electromechanical properties by pinning the ferroelectric domain walls. Entrapped porosity is formed during the growth process due to the high mobility of the single crystal–matrix interface. This phenomenon is well-known from ceramic sintering and typically limits the densification process [Reference Brook99, Reference Koruza, Malič and Kosec100]. The entrapped pores are difficult to be removed due to the long diffusion distances and the high activation energy of lattice diffusion. Porosity can be reduced by reducing the porosity of the polycrystalline matrix or by applying external pressure during the growth process, which increases the driving force for pore shrinkage [Reference Benčan, Tchernychova, Uršič, Kosec, Fisher and Lallart78]. Another feature observed in some SSCG crystals are small-angle δ-boundaries [Fig. 6(f)], which were related to the stress relaxation and were demonstrated to pin the ferroelectric domain walls [Reference Fu, Yang, Yu, Wang, Lu, Yang, Xu and Li101].

Point defects

Point defects are inevitable constituents of all perovskite (and other) materials and strongly influence dielectric and ferroelectric properties [Reference Raymond and Smyth102]. This predominantly occurs by increasing the conductivity and interacting with the ferroelectric/ferroelastic domain walls. Moreover, the rearrangement of charged defects often results in aging of ferroelectric/piezoelectric properties. Point defects are introduced during the high-temperature growth process, by impurities from raw materials or crucibles, and, in some cases, by intentional chemical doping [Reference Maglione, Philippot, Levasseur, Payan, Aymonier and Elissalde103]. Especially in case of oxides, there is a high sensitivity toward changes in the oxygen partial pressure during growth. The introduced defects are usually described by the Kröger–Vink notation, whereby the symbol denotes the defect type; the subscript, the lattice site occupied by this defect; and the superscript, the charge of the defect site [Reference Kröger and Vink104]. Typical examples are negatively charged cation vacancies (predominantly on the perovskite A-site, e.g., $V_{{\rm{Na}}}^\prime$), positively charged oxygen vacancies $\left( {V_{\rm{O}}^{{\rm{ \bullet \bullet }}} } \right)$, substitutional ions $\left( {{\rm{Sb}}_{{\rm{Nb}}}^{\prime \prime } } \right)$, free electrons (e′), and free holes (h^•). It should be noted that certain defects are also prone to combine into charged defect complexes, which are particularly important for the pinning of ferroelectric domains walls [Reference Carl and Härdtl105]. Typical example is the defect complex between acceptor ion and oxygen vacancy, e.g., $\left( {{\rm{Cu}}_{{\rm{Nb}}}^{\prime \prime \prime } - V_{\rm{O}}^{{\rm{ \bullet \bullet }}} } \right)^\prime$.

The defect chemistry of KNN-based single crystals is rather complex, and many experimental observations remain without proper explanation. The following origins of point defects have been discussed so far. The volatility of alkalis at high temperatures results in the formation of A-site vacancies [Reference Kizaki, Noguchi and Miyayama106, Reference Popovič, Bencze, Koruza and Malič107]. These generate acceptor states in the band gap, and thus, undoped KNN is predicted to exhibit p-type conductivity (conduction due to holes, h^•). For the case of the more volatile K [Reference Popovič, Bencze, Koruza and Malič107], the defect formation process can be written as follows:

(4)$$K_{\rm{K}}^x \to V_{\rm{K}}^\prime + {\rm{h}}^ \bullet \quad .$$

Note that A-site vacancies can also form as charge-compensating defects on donor doping. Another common species in perovskite ferroelectrics are oxygen vacancies. These can form due to oxygen loss during high temperature processing [intrinsic mechanism, Eq. (5)] or to compensate the charge of the formed A-site vacancies or the acceptor dopants/impurities (extrinsic mechanism) [Reference Chan, Sharma and Smyth108, Reference Nowotny and Rekas109].

(5)$${\rm{O}}_{\rm{O}}^x \to V_{\rm{O}}^{ \bullet \bullet } + {1 \over 2}{\rm{O}}_2^x + 2{\rm{e'}}\quad {\rm{.}}$$

Compositions containing elements with multi-oxidation states (e.g., manganese, antimony, niobium, iron) can additionally form substitutional defect centers. This is accompanied by the formation of charge-compensating holes and additionally contributes to the increase of oxygen vacancy concentration. Example of such process is the reduction of Nb⁵⁺ to Nb⁴⁺:

(6)$${\rm{Nb}}\left( {{\rm{IV}}} \right)\buildrel {{\rm{B &#x2010; site}}\left( {\rm{V}} \right)} \over \scale 177%\longrightarrow {\rm{Nb}}\left( {{\rm{IV}}} \right)_{\rm{B}}^\prime + {\rm{h}}^ \bullet \quad .$$

It should be noted that experimental determination of the point defects is difficult and requires a combination of several complementary techniques, for example, impedance spectroscopy, electron paramagnetic resonance, positron annihilation spectroscopy, Mössbauer spectroscopy, and X-ray photoelectron spectroscopy. Moreover, the exact charge-compensating mechanisms will strongly depend on the processing atmosphere, temperature, and chemical composition. While the doping and defect chemistry of KNN-based materials have been discussed in several previous publications [Reference Wu, Xiao and Zhu14, Reference Li, Wang, Zhu, Cheng and Yao33, Reference Malič, Koruza, Hreščak, Bernard, Wang, Fisher and Benčan34, Reference Rafiq, Tkach, Costa and Vilarinho110], the goal of the following section is to highlight some examples of single crystals.

Among the parameters that most decisively influence the defect state is the atmosphere. The as-grown KNN crystals are often reported to have a blue/dark color, which was related to the presence of Nb⁴⁺ states [Reference Matthias and Remeika64, Reference Kizaki, Noguchi and Miyayama106]. Note that some later reports suggested that the color originates from the higher concentration of oxygen vacancies [Reference Gupta and Priya111]. Similar phenomena are also observed in other materials, e.g., reduced LiNbO₃, where both mechanisms are being discussed as the origin [Reference Ketchum, Sweeney, Halliburton and Armington112]. Annealing of such KNN crystals in air at 1100 °C resulted in the oxidation of Nb⁴⁺ to Nb⁵⁺ and a white crystal color. This induced a large decrease in the leakage current and considerably improved ferroelectric properties [Reference Kizaki, Noguchi and Miyayama106].

Similarly, oxygen annealing was demonstrated to be beneficial to improve the properties of more complex compositions, such as (K,Na,Li)(Nb,Ta,Sb)O₃ [Reference Liu, Veber, Rödel, Rytz, Fabritchnyi, Afanasov, Patterson, Frömling, Maglione and Koruza113]. Annealing of the crystals in oxygen for one week at 900 °C dramatically improved the ferroelectric behavior and increased the room temperature piezoelectric coefficient from 381 pC/N to 732 pC/N [Fig. 7(a)]. This was related to a decreased oxygen vacancy concentration and partial oxidation of Sb³⁺ to Sb⁵⁺, as evidenced by impedance spectroscopy and Mössbauer spectroscopy, respectively. It should be noted that the annealing time in this study was chosen arbitrarily and the exact equilibrium conditions were not determined. More recently, Xue et al. reported that oxygen annealing of Er-doped KNN crystals at 800 °C for 10 h could enhance the upconversion photoluminescence intensity by 20 times, which was also ascribed to the Nb valence change and the reduced concentration of oxygen vacancies [Reference Xue, Deng, Xie, Hu, Yan, Zhao, Wang, Zhang, Luo, Deng, He, Lin, Li, Wang and Luo114].

Figure 7: (a) Increase in the piezoelectric coefficient in (K,Na,Li)(Nb,Ta,Sb)O₃ crystals after oxygen annealing (reprinted from Ref. Reference Liu, Veber, Rödel, Rytz, Fabritchnyi, Afanasov, Patterson, Frömling, Maglione and Koruza113 with permission from Elsevier). (b) Dependence of the leakage current on the oxidation states in KNN and Mn-doped KNN single crystals (reprinted from Ref. Reference Kizaki, Noguchi and Miyayama106 with permission from AIP Publishing). (c) Influence of Mn doping and subsequent oxygen annealing at 900 °C on the ferroelectric hysteresis loop (1 Hz) and unipolar strain (2 Hz) of (K,Na,Li)(Nb,Ta)O₃ single crystals.

Kizaki et al. were among the first to propose reducing the leakage current of KNN crystals by Mn doping [Reference Kizaki, Noguchi and Miyayama106]. Note that this approach was previously used in other perovskite ferroelectrics, e.g., BaTiO₃. The samples were grown by the self-flux method and had a final size of 2 × 2 × 2 mm³. The as-grown KNN crystals showed a high leakage current, related to electrical conduction through 4d electrons of Nb⁴⁺ [Fig. 7(b)]. On the other hand, Mn-doped crystals annealed in air had a lower leakage current due to the oxidation of Nb⁴⁺ to Nb⁵⁺. This was also accompanied by a color change. Further oxidation resulted in the formation of h^•, which would increase the leakage current in pure KNN crystals; however, the present Mn²⁺ absorbed the generated h^• by valence increase and thus preserved a low leakage current over a broad oxidation range. Due to the observed lattice expansion and the electron paramagnetic resonance (EPR) results, the authors concluded that Mn is incorporated into the Nb site [Reference Kizaki, Noguchi and Miyayama106, Reference Lin, Li, Zhang, Xu and Yao115]. It should be noted that some other authors reported that Mn predominantly occupies the A-site [Reference Yao, Zhang, Wang, Zhou, Chen, Xu, Li, Shen, Zhang, Gu, Zhang and Li116]. The occupancy is presumably strongly dependent on the atmosphere, concentration of Mn, and chemical composition of the material. Nevertheless, Mn doping was subsequently extensively used in KNN-based single crystals to reduce the losses, improve poling behavior, and strongly enhance the electromechanical properties [Reference Huo, Zhang, Zheng, Zhang, Wang, Wang, Sang, Yang and Cao26, Reference Liu, Koruza, Veber, Rytz, Maglione and Rödel46, Reference Liu, Xu, Liu, Yang and Chen65, Reference Lin, Li, Zhang, Xu and Yao115].

The influence of Mn doping and oxygen annealing on the ferroelectric properties of (K,Na,Li)(Nb,Ta)O₃ crystals is exemplified in Fig. 7(c). The domains in the undoped crystal are strongly clamped and polarization switching is absent (also dielectric losses are high). Mn-doped sample exhibits a characteristic polarization hysteresis loop with a minor contribution from the leakage current. While the unipolar strain increases, the absence of hysteresis in the unipolar strain loop indicates a low contribution from domain walls. Upon oxygen annealing at 900 °C, the resistivity of the Mn-doped sample further increases. The unipolar strain and the strain hysteresis increase, which indicates increased mobility of the domain walls, as compared with the non-annealed sample.

Domains and domain configurations

Physical properties of ferroelectrics largely depend on their domain structure [Reference Tagantsev, Cross and Fousek117, Reference Damjanovic118]. Cooling a perovskite ferroelectric below the Curie point results in the displacement of the B-site cations along one of the polar directions and a consequent formation of spontaneous polarization and strain. This process induces large depolarizing electric fields and mechanical stresses, which get compensated by the formation of ferroelectric domains, i.e., coherent regions with uniform polarization orientations. The resulting domain walls can be separated into ferroelectric 180° domain walls, minimizing the electrostatic energy, and ferroelectric/ferroelastic non-180° domain walls, additionally minimizing the mechanical energy. The final domain configuration will be strongly dependent on the local electrical and mechanical boundary conditions. It should be noted that some ferroelectric domain walls can also be charged, which was recently observed in KNN materials as well [Reference Esin, Alikin, Turygin, Abramov, Hreščak, Walker, Rojac, Benčan, Malič, Kholkin and Shur119, Reference Rubio-Marcos, Del Campo, Rojas-Hernandez, Ramirez, Parra, Ichikawa, Ramajo, Bausa and Fernandez120].

The domain configuration will be determined by the structure and phase transitions of KNN, which are described in section “K_xNa_1–xNbO₃ system”. The cubic–tetragonal phase transition (T _C) results in the formation of 90° and 180° domain walls. Due to 6 possible 〈100〉_PC spontaneous polarization directions of the tetragonal structure [Fig. 8(a)], 90° domain walls are formed by twinning the {110}_PC planes, while 180° walls appear in 〈100〉_PC directions. It should be noted that the actual angles of non-180° domain walls slightly deviate from the values used for their notation. For example, the angle α at the 90° domain wall can be calculated from the cell parameters a and c by:

(7)$$\alpha = 2 \cdot \tan ^{ - 1} \left( {{a \mathord{\left/ {\vphantom {a c}} \right. \kern-\nulldelimiterspace} c}} \right)\quad .$$

Figure 8: Domain configurations of tetragonal and orthorhombic phases under various conditions: (a, e) initial state and E-field applied along (b, f) [001]_PC, (c, g) [011]_PC, and (d, h) [111]_PC direction.

Four different domain angles can be formed in the orthorhombic phase, namely, 60°, 90°, 120°, and 180° [Reference Wiesendanger121]. The orientation depends on the 12 possible spontaneous polarization directions of the orthorhombic phase [〈110〉; Fig. 8(e)] and on the mechanical compatibility based on the spontaneous strain tensor [Reference Fousek and Janovec122]. 90° domain walls lie on {001}_PC pseudocubic or {110}_O orthorhombic planes [Reference Deshmukh and Ingle123]. 180° domain walls can appear at the same planes as 90° walls, but because antiparallel domains have identical strain tensors, they are mechanically not limited to certain planes [Reference Wiesendanger121]. 60° and 120° domain walls lie on the pseudocubic {110}_PC or orthorhombic {111}_O planes [Reference Deshmukh and Ingle123]. For the differentiation between 60° and 120° walls, the spontaneous polarization configuration (head-to-head, head-to-tail, etc.), the charges, and the permissible domain wall criteria have to be considered carefully [Reference Wiesendanger121]. Examples of 90° and 60° domain walls for the orthorhombic system and 90° and 180° for the tetragonal system are shown in Fig. 9.

Figure 9: Schematics for exemplary domain walls appearing in perovskite ferroelectrics with tetragonal structure: (a) 90° domain wall on (101) plane, (b) 180° domain wall and orthorhombic structure (c) 60° domain wall on (111)_O plane, and (d) 90° domain wall on (110)_O plane. Redrawn from Refs. Reference Wiesendanger121 and Reference Deshmukh and Ingle123.

Domain engineering

The functional properties of the as-grown piezoelectric crystals are enhanced by adjusting their domain structures. This process is denoted as domain engineering [Reference Zhang and Li4, Reference Park and Shrout7, Reference Bell124, Reference Davis, Damjanovic, Hayem and Setter125]. While this term is today often used to describe any kind of changes in the crystals' domain structure, it was originally defined as the process of poling along one of the crystal's poling axis that is different from the zero-field polar axis, creating a set of domains in which polarization vectors have a minimized angle to the poling direction [Reference Bell124]. Figure 8 shows domain configurations of a crystal with electric field aligned along [001]_PC, [011]_PC, and [111]_PC directions for both the orthorhombic and tetragonal phases. The orthorhombic configurations are labeled as “4O,” “1O,” and “3O” and the tetragonal ones as “1T,” “2T,” and “3T”. “1O” and “1T” domain configurations are single domain states, while others are polydomain.

Which of the above configurations will result in enhancement of the electromechanical response depends on material's intrinsic anisotropy and the vicinity of phase transitions [Reference Budimir, Damjanovic and Setter126, Reference Davis, Budimir, Damjanovic and Setter127]. Strictly speaking, the domain walls in domain-engineered materials are not expected to move when electric fields are applied along the poling direction, thus the electromechanical response is expected to be predominantly intrinsic in nature (for detailed description of intrinsic and extrinsic contributions, see “Electromechanical properties” section). The anisotropy can be estimated by comparing the longitudinal (d ₃₃), transverse (d ₃₁), and shear (d ₁₅) piezoelectric coefficients. Materials with large anisotropy [e.g., where (d ₁₅ + d ₃₁)/d ₃₃ > 1] exhibit the largest longitudinal piezoelectric response along nonpolar directions, which is related to facilitated polarization rotation. On the other hand, in materials with (d ₁₅ + d ₃₁)/d ₃₃ below a critical value the highest response is obtained along the polar direction (polarization extension mechanism). The piezoelectric coefficients are directly related to the dielectric anisotropy; therefore, the above relations can also be described by the ratio of the transverse and the longitudinal dielectric susceptibilities (χ₁₁/χ₃₃). Note that the dielectric anisotropy is often very large in the vicinity of phase transitions [Reference Budimir, Damjanovic and Setter126, Reference Liang, Li, Hu, Chen and Lu128], which can strongly increase the piezoelectric response. Interestingly, while orthorhombic KNbO₃ shows a dielectric anisotropy of χ₁₁/χ₃₃ = 3.5, higher values were reported for some of the KNN-based crystals [Reference Hu, Tian, Meng, Shi, Cao and Zhou129].

Domain engineering of orthorhombic [001]_C-oriented KNbO₃ crystals was studied in detail by Wada et al. [Reference Wada, Muraoka, Kakemoto, Tsurumi and Kumagai130]. Calculated and measured piezoelectric properties for single-domain crystals were compared and different engineered polydomain configurations were prepared. The latter had a much higher response as compared with the single-domain state and the properties increased with decreasing the domain size. The individual contributions of the different domain variants to the longitudinal and transverse piezoelectric coefficients were later calculated by Davis et al. [Reference Davis, Damjanovic, Hayem and Setter125], while their temperature dependence and the influence of phase transitions were reported by Liang et al. [Reference Liang, Li, Hu, Chen and Lu128]. Highest electromechanical properties were demonstrated for [001]_C-oriented KNbO₃ crystals rotated by 45° away from the polar axis [Reference Davis, Klein, Damjanovic, Setter, Gross, Wesemann, Vernay and Rytz131]. Reports of domain-engineered KNN-based single crystals are relatively rare; some examples will be presented in the following section.

Modifications of the domain structures in KNN crystals

The ferroelectric domain structures in KNN-based single crystals were investigated by several authors using polarized light microscopy (PLM) [Reference Zheng, Wang, Huo, Wang, Sang, Li, Zheng and Cao48, Reference Lin, Li, Li, Cai, Liu and Zhang51, Reference Deng, Zhang, Zhao, Chen, Wang, Li, Lin, Ren, Jiao and Luo73, Reference Lin, Li, Xu and Yao132, Reference Inagaki, Kakimoto and Kagomiya133, Reference Lin, Zhang, Cai and Liu134, Reference Wang, Zheng, Lu, Yang, Yang, Zhang, Lv and Cao135], transmission electron microscopy (TEM) [Reference Jiang, Randall, Guo, Rao, Tu, Gu, Cheng, Liu, Zhang and Li50, Reference Liu, Veber, Zintler, Molina-Luna, Rytz, Maglione and Koruza74], and piezoresponse force microscopy (PFM) [Reference Fu, Yang, Yu, Wang, Lu, Yang, Xu and Li101, Reference Rafiq, Costa and Vilarinho136, Reference Bah, Alyabyeva, Retoux, Giovannelli, Zaghrioui, Ruyter, Delorme and Monot-Laffez137, Reference Cui, Jiang, Zhang, Xu, Xu, Chen, Hu and Chu138]. Unlike the complicated zigzag or herringbone-like structures reported for KNN polycrystalline ceramics [Reference Qin, Zhang, Yao, Wang and Zhang139], the domains in single crystals arrange into less-complex configurations and are generally larger in size. This is related to the different local boundary conditions, e.g., reduced internal stresses due to the absence of a microstructure with randomly oriented grains. Below, we list a series of reported approaches for adjusting the domain structure of KNN single crystals.

A well-known method for modifying the domain structure is chemical doping. For example, Lin et al. demonstrated a reduction in the domain size by about 50% after Mn doping [Fig. 10(A)] [Reference Lin, Li, Zhang, Xu and Yao115]. The doped crystal exhibited a 69% and 200% increase in the piezoelectric coefficient and dielectric permittivity, respectively. Note that Mn doping resulted in a minor decrease in the T _O–T by 15 °C. Interestingly, Huo et al. reported that Mn doping of (K,Na,Li)(Nb,Ta)O₃ crystals resulted in reduced extrinsic contributions [Reference Huo, Zhang, Zheng, Zhang, Wang, Wang, Sang, Yang and Cao26, Reference Huo, Zheng, Zhang, Wang, Wang, Sang, Wang, Yang and Cao140]. The observed increase in the piezoelectric coefficient from 420 pC/N in undoped to 630 pC/N in Mn-doped crystals was thus related to enhanced intrinsic contribution by Mn doping.

Figure 10: Examples of different domain configurations reported in KNN-based single crystals. (A) Reduction of domain size by Mn doping (reprinted from Ref. Reference Lin, Li, Zhang, Xu and Yao115 with permission from Wiley). (B) Reduction of domain size by changing the high-temperature poling conditions: (a) 9 µm after 120 °C, 20 kV/cm, (b) 6.5 µm after 205 °C, 6 kV/mm, (c) 2.5 µm after 205 °C, 12 kV/cm, and (d) 2 µm after 205 °C, 18 kV/cm (reprinted from Ref. Reference Lin, Zhang, Cai and Liu134 with permission from AIP Publishing). Note the change in domain pattern from (a, b) laminar to (c, d) twinned. (C) Temperature-dependent domain evolution in [001]_PC-oriented KNLTN single crystal: (a) 25 °C, (b) 100 °C, (c) 250 °C, and (d) 450 °C [Reference Liu, Veber, Zintler, Molina-Luna, Rytz, Maglione and Koruza74]. Pictures in (A) and (B) were obtained by PLM, while (C) with TEM.

Inagaki et al. investigated the dependence of domain patterns in [001]_PC-oriented Mn-doped KNN crystal on the cooling rate of the flux growth method [Reference Inagaki, Kakimoto and Kagomiya133]. Upon rapid cooling into the orthorhombic phase at 1.25 °C/min, only 90° domain walls were detected. On the other hand, both 90° and 60° domain walls were observed if the cooling rate was reduced to 0.25 °C/min, which resulted in crystals with higher polarization, higher piezoelectric response, and a lower leakage current density. However, it should be noted that the chemical composition of the crystals was not investigated and may have been altered by the different segregation due to the varying cooling rates.

Early attempts to domain engineer the Li-modified KNN crystals were reported by Davis et al., who demonstrated large and temperature-stable thickness coupling coefficients on a [001]_C-poled crystal [Reference Davis, Klein, Damjanovic, Setter, Gross, Wesemann, Vernay and Rytz131]. The influence of the poling electric field and temperature on the domain structure of [001]_PC-oriented 0.5%MnO₂-doped (K_0.5Na_0.5)NbO₃ was systematically studied by Lin et al. [Reference Lin, Zhang, Cai and Liu134]. Increasing the poling conditions from 120 °C and 20 kV/cm to 205 °C and 18 kV/cm resulted in a domain size decrease from 9 µm to 2 µm [Fig. 10(B)] and a change in the domain pattern form laminar to twinned. This was accompanied by an increase of the piezoelectric coefficient from 270 pC/N to 350 pC/N and relative permittivity from 730 to 850. Influence of poling on the [011]_C-oriented tetragonal (K,Na,Li)(Nb,Sb,Ta)O₃ crystal was studied by Wang et al. [Reference Wang, Zheng, Lu, Yang, Yang, Zhang, Lv and Cao135]. Applying 4–5 kV/cm along the [011]_C direction switched the polarization vectors into the nearby 〈100〉_C directions and the “2T” engineered domain structure [Fig. 8(c)] was formed. Highest properties were obtained after poling with 6 kV/mm, while higher fields resulted in overpoling. The same group has previously also attempted to obtain monodomain state in [001]_C-oriented tetragonal (K,Na,Li)(Nb,Sb,Ta)O₃ crystals [Reference Wang, Zheng, Yang, Luo, Lu, Liu, Zhang, Lv and Cao141], i.e., the “1T” configuration shown in Fig. 8(b). However, this was prevented by the large shear elastic stiffness constant c₄₄, which was reported to be more than 5 times larger for KNN crystals (7.8 × 10¹⁰ N/m²) than for typical lead-based crystals (1.28 × 10⁻¹⁰ N/m²).

Another important factor is the thermal stability of the domain structure, which is directly reflected in the temperature stability of the functional properties. This includes the changes in the domain structure between two phase transitions, as well as the transformation between different domain configurations at phase transition (e.g., from orthorhombic to tetragonal). Temperature-dependent domain evolution in KNN-based crystals was investigated by Liu et al. using TEM [Fig. 10(C)] [Reference Liu, Veber, Zintler, Molina-Luna, Rytz, Maglione and Koruza74], Zheng et al. using PLM [Reference Zheng, Wang, Huo, Wang, Sang, Li, Zheng and Cao48], and Cui et al. using PFM and Raman spectroscopy [Reference Cui, Jiang, Zhang, Xu, Xu, Chen, Hu and Chu138]. Increasing the temperature in the orthorhombic phase resulted in a slight decrease in the 60° domain wall density, which was ascribed to the higher elastic energy of 60° domain walls as compared with 90° domain walls. The domain transition in the vicinity of the orthorhombic–tetragonal phase boundary was initiated at locations where stresses are easily released and was accompanied by depolarization of the orthorhombic phase and the formation of the tetragonal phase [Reference Zheng, Wang, Huo, Wang, Sang, Li, Zheng and Cao48]. However, contradictive results were reported for the domain sizes; while Zheng et al. observed smaller domains in the tetragonal phase, Liu et al. reported smaller domains in the orthorhombic phase compared with the tetragonal one. Cui et al. tested the stability of the domain structure by thermal cycling and reported increase in the amount of 60° domain wall and decrease in 90° domain wall after continuous cycles, which was explained by the higher internal stress and bounding strength induced by thermal fields.