Microstructure and electrical properties of Er-doped 0.67 Pb(Mg1/3Nb2/3)O3-0.33PbTiO3 ceramics with BaTiO3 templates

In this study, [001]-oriented Er-doped 0.67 Pb(Mg_1/3Nb_2/3)O₃-0.33PbTiO₃ (0.67PMN-0.33 PT) textured ceramics with different BaTiO₃ (BT) template concentrations were explored. The samples were prepared by tape-casting. Er³⁺ was added to modify the electrical properties of the polycrystalline ceramics, and the BT template was used to improve the texture of polycrystalline ceramics. The 0.67PMN-0.33 PT textured ceramics contained coexisting rhombohedral and tetragonal phases. The ceramics became increasingly textured as the sintering temperature increased up to 1250 °C. The piezoelectric coefficient of 0.67PMN-0.33 PT with 5 wt% BT was 634 pC/N, which is 1.2 times than that of randomly oriented 0.67PMN-0.33 PT. The strain of the ceramic with 5 wt% BT increased by 12.5% relative to a random control specimen. Analysis of the electrical properties and microstructure suggested that the enhancement of the piezoelectric coefficient and strain may be caused by the addition of Er³⁺ and the BT template. First, the directional growth of grains along the template affected the change-of-phase distribution of the system and formed a more adaptive phase. Second, Er³⁺ was substitutionally doped on the A-site of the perovskite to form local heterostructures. Finally, the relaxation components of the templates and Er³⁺ changed in the solid solution with the matrix. The solid solution of the BT templates and Er-doped-matrix powder changed the relaxation degree, which affected the interactions at the polar nanoregions and increased the piezoelectric coefficient of the ceramics.     

Double hysteresis loops and enhanced mechanical quality factor of Mn-doped 0.75PMN-0.25PT ceramics

Relaxor ferroelectric 0.75 Pb(Mg_1/3Nb_2/3)O₃-0.25PbTiO₃ (0.75PMN-0.25PT) ceramics with Mn-doping concentration of 0%, 1%, 2% and 3 mol% were prepared by two-step sintering. The phase composition, microstructure, dielectric property, ferroelectric property, piezoelectric property and electromechanical property were investigated. Results indicate that all ceramics exhibit pure perovskite phases and high density. Mn-doping induces some decrease in dielectric constant ε_r, dielectric loss tanδ, remnant polarization P_r, piezoelectric coefficient d₃₃, and electromechanical coupling coefficient k_p, while significant increase in mechanical quality factor Q_m and the figure of merit (FOM) of 0.75PMN-0.25PT ceramics. For 3 mol% Mn-doped 0.75PMN-0.25PT ceramics, Q_m enhances by 1449%, FOM increases by 923%, and tanδ decreases by 67%, which makes it more suitable for high power applications. Interestingly, double P-E loops are observed in 3 mol% Mn-doped 0.75PMN-0.25PT ceramics. The phenomenon was investigated by the symmetry-conforming principle of point defects and the internal bias field.     

Achieving large piezoelectric response and ultrahigh electrostriction in [001] textured BiScO3-PbTiO3 high-temperature piezoelectric ceramics

[001] textured 0.40BiScO₃-0.60PbTiO₃-0.125 mol%Nb⁵⁺ (BS-60PT-0.125Nb) high-temperature piezoelectric ceramics were synthesized using templated grain growth process. A high texture degree F₀₀₁ of 99% was obtained using 2 vol% BaTiO₃ (BT) templates. The piezoelectric charge constant d₃₃ and the unipolar strain under 40 kV cm⁻¹ at room temperature for the textured ceramics are 646 pC N⁻¹ and 0.36%, respectively, which is over two times as those for untextured ceramics (∼243 pC N⁻¹ and 0.17%). The electrostriction Q₃₃ value of the textured sample remarkably increased from 0.034 m⁴ C⁻² to 0.068 m⁴ C⁻² under 30 kV cm⁻¹, showing a twice higher than untextured. Compared with random ceramics, the improvement piezoelectric response of the textured ceramics is primarily attributed to the increase of the dielectric constant ε_r and electrostriction coefficient Q₃₃ along [001] orientation, which is originating from the anisotropy of piezoelectricity. The BS-60PT-0.125Nb textured ceramics have large piezoelectric response and ultrahigh electrostriction with high temperature stability (high depolarization temperature T_d of ∼360°C and high Curie temperature T_c of 421°C), showing great potential for the piezoelectric applications at high temperatures.     

Grain-orientation-engineered PMN-10PT ceramics for electrocaloric applications

The electrical properties of ferroelectric materials depend on their crystal orientations. The 〈001〉 and 〈111〉 textured 0.90Pb(Mg_1/3Nb_2/3)O₃–0.10PbTiO₃ (PMN-10PT) ceramics were synthesized using a template grain growth method and conventional solid sintering process. The Lotgering factors of the 〈001〉 and 〈111〉 textured PMN-10PT ceramics with 5% anisotropic BaTiO₃ (BT) precursors were 95% and 87%, respectively. The influence of grain orientation direction on dielectric properties was investigated, revealing that the permittivity of textured PMN-10PT ceramics (ε_r〈001〉 = 12200, ε_r〈111〉 = 11800) was lower than that of random ceramics (ε_r = 18700). Electrocaloric (EC) responses of the random and textured samples were analyzed using an indirect polarization-deduced method based on Maxwell's equations. When the applied field was 50 kV cm⁻¹, the 〈111〉 textured ceramics exhibited an enhanced adiabatic temperature change of 1.30 K, which was 20% higher than that of the random samples. In addition, the EC performance of the 〈111〉 oriented samples was significantly improved compared to that of the 〈001〉 oriented ones. This work characterizes the grain-orientation-dependent EC responses in the PMN-PT system, which would be a promising approach to EC response improvement in ferroelectric ceramics.     

Low‐temperature sintering and enhanced piezoelectric properties of random and textured PIN–PMN–PT ceramics with Li2CO3

Low‐temperature sintered random and textured 36PIN–30PMN–34PT piezoelectric ceramics were successfully synthesized at a temperature as low as 950°C using Li₂CO₃ as sintering aids. The effects of Li₂CO₃ addition on microstructure, dielectric, ferroelectric, and piezoelectric properties in 36PIN–30PMN–34PT ternary system were systematically investigated. The results showed that the grain size of the specimens increased with the addition of sintering aids. The optimum properties for the random samples were obtained at 0.5 wt% Li₂CO₃ addition, with piezoelectric constant d₃₃ of 450 pC/N, planar electromechanical coupling coefficient k_p of 49%, peak permittivity ε_max of 25 612, remanent polarization P_r of 36.3 μC/cm². Moreover, the low‐temperature‐sintered textured samples at 0.5 wt% Li₂CO₃ addition exhibited a higher piezoelectric constant d₃₃ of 560 pC/N. These results indicated that the low‐temperature‐sintered 36PIN–30PMN–34PT piezoelectric ceramics were very promising candidates for the multilayer piezoelectric applications.     

Revisiting the structural stability and electromechanical properties in lead zinc niobate-lead titanate-barium titanate (PZN-PT-BT) ternary system

Lead zinc niobate-lead titanate (PZN-PT) system is of particular interest for scientific researches and commercial applications due to the unique relaxation feature and superior electromechanical responses. However, it is difficult to prepare polycrystalline ceramics near the morphotropic phase boundary due to the stability of a competing lead niobate pyrochlore phase. BaTiO₃ (BT) was reported to be an effective perovskite phase stabilizer at a cost of reduction in Curie temperature and piezoelectric properties. Herein, the amount of BT in PZN-PT-BT system and the sintering conditions were optimized to simultaneously stabilize the perovskite phase and maintain high dielectric and piezoelectric response. An optimum piezoelectric coefficient d₃₃ ∼ 870 pC/N along with a T_m ∼ 133 °C and an electromechanical coupling coefficient k₃₃ ∼ 0.61 was obtained in the PZN-8PT-6BT ceramics sintered at 1100 °C. The low sintering temperature, wide processing window, and improved dielectric and piezoelectric properties make PZN-PT-BT ceramics potential candidates for piezoelectric devices.     

Investigation of dielectric and piezoelectric properties in aliovalent Eu3+-modified Pb(Mg1/3Nb2/3)O3-PbTiO3 ceramics

Rare earth (Eu³⁺)-modified Pb(Mg_1/3Nb_2/3)O₃-PbTiO₃ (PMN-PT) polycrystalline ferroelectric ceramics were fabricated by high-temperature solid-state sintering, the phase structure, dielectric and piezoelectric properties were investigated. Eu³⁺ addition was found to significantly improve dielectric and piezoelectric properties of PMN-PT, where the optimized properties were achieved for the composition of 2.5 mol%Eu: 0.72PMN-0.28PT, with the piezoelectric d₃₃ = 1420 pC/N, dielectric ε_r = 12 200 and electromechanical k₃₃ = 0.78, respectively. All these results indicate that the Eu³⁺-doped PMN-PT ceramics are promising candidates for high-performance room-temperature piezoelectric devices.     

Achieving high piezoelectric performances with enhanced domain-wall contributions in <001>-textured Sm-modified PMN-29PT ceramics

1% Sm substituted PMN-29PT ([Sm_0.01Pb_0.985][(Mg_1/3Nb_2/3)_0.71Ti_0.29]O₃) textured ceramics with a Lotgering factor of up to 80 % were successfully prepared. The dielectric, ferroelectric and piezoelectric properties have been investigated in detail for both textured and non-textured ceramics. The heterogeneous templates are found to be critical for the change in relaxor behaviors, leading to the reduced T_f, T_m, ε_max and P_r in textured ceramics. The textured ceramics show an enhanced d₃₃ of 810 pC/N, about 1.6 times greater than their randomly oriented ceramic counterparts. The domain structures and their responses to applied electric fields have been studied using piezoresponse force microscopy, signifying the enhanced extrinsic contributions by the increased domain wall densities and dynamics. The improved dielectric and piezoelectric properties are discussed and explained from the view point of extrinsic contributions in textured ceramics. The results demonstrate that the textured Sm-PMN-PT piezoelectric ceramics could be promising high performance piezoelectrics with low cost.     

Structural and piezoelectric properties of <001> textured PZT‐PZNN piezoelectric ceramics

Rhombohedral 0.69Pb(Zr_0.47Ti_0.53)‐0.31Pb(Zn_0.6Ni_0.4)NbO₃ (PZT‐PZNN) ceramics were textured using 10.0 vol. % BaTiO₃ (BT) platelets along the <001> direction at 950°C with a high Lotgering factor of 95.3%. BT platelets did not react with the PZT‐PZNN ceramics, and the textured PZT‐PZNN ceramic had a tetragonal structure. The PZT‐PZNN ceramics exhibited a strain of 0.174% with a piezoelectric strain constant (d*₃₃) of 580 pC/N at 3.0 kV/mm. The textured PZT‐PZNN ceramic showed an increased strain of 0.276% and d*₃₃ of 920 pC/N at 3.0 kV/mm, which can be explained by the domain rotation. However, the d₃₃ values of the textured specimens are smaller than those of the untextured specimens because of the small remanent polarization and relative dielectric constant of BT platelets. The textured PZT‐PZNN ceramic synthesized in this work can be used for piezoelectric multilayer actuators because of its large strain and low sintering temperature.     

Textured (K0.5Na0.5)NbO3 ceramics prepared by screen-printing multilayer grain growth technique

The screen-printing multilayer grain growth (MLGG) technique is successfully applied to alkaline niobate lead-free piezoelectric ceramics. Highly textured (K_0.5Na_0.5)NbO₃ (KNN) ceramics with 〈0 0 1〉 orientation (f = 93%) were fabricated by MLGG technique with plate-like NaNbO₃ templates. The influence of sintering temperature on grain orientation and microstructure was studied. The textured KNN ceramics showed very high piezoelectric constant d₃₃ = 133 pC/N, and high electromechanical coupling factor k_p = 0.54. These properties were superior to those of conventional randomly oriented ceramics, and reach the level of those of textured KNN ceramic prepared by tape-casting technique. Compared with other grain orientation techniques, screen-printing is a simple, inexpensive and effective method to fabricate grain oriented lead-free piezoelectric ceramics.     

Determining the effects of BaTiO3 template alignment on template grain growth of Pb(Mg1/3Nb2/3)O3 –PbTiO3 and effects on piezoelectric properties

Crystallographic texturing of ferroelectric ceramics is an established method of inducing single crystal-like properties in a ceramic material via epitaxial grain growth according to the template used, otherwise known as Templated Grain Growth (TGG). The piezoelectric enhancement is dependent on the degree of TGG throughout the ceramic, which is closely linked to the tape casting parameters and sintering conditions. Pb(Mg1/3Nb2/3)O₃-PbTiO₃ (PMN-PT) perovskite ferroelectrics with morphotropic phase boundary compositions using 3 vol% BaTiO₃ templates were used for analysis. The blade gap and template size have the greatest impact on the overall grain alignment. The varying degrees of template alignment is related to the different enhancements of TGG and the correlating piezoelectric d₃₃ and d*₃₃. The highest piezoelectric property was demonstrated in PMN-31PT with Lotgering factor of 93% where d₃₃ = 1020 pC/N and d*₃₃ = 1420 pm/V were achieved.     

3D printing orientation controlled PMN-PT piezoelectric ceramics

Piezoelectric textured ceramics have drawn increasing research and industry interests by balancing the production cost and material performances. A new approach to realize the texture in piezoelectric ceramics is developed based on 3D printing stereolithography (SL) technique and successfully applied in the preparation of < 001 > -textured 0.71(Sm_0.01Pb_0.985)(Mg_1/3Nb_2/3)O₃-0.29(Sm_0.01Pb_0.985)TiO₃ (1 %Sm-PMN-29PT) ceramics in this work. As a critical process in texture ceramic fabrication, the alignment of BaTiO₃ templates along the horizontal direction is achieved by the shear force produced from the relative motion between the resin container and the blade during SL. The textured ceramics with obvious grain orientation features are successfully obtained. The enhanced piezoelectric properties of d₃₃ ≈ 652 pC N^?1 and d₃₃* ≈ 800 pm V^?1 are achieved in the 3D printed textured ceramic, which are about 60 % and 40 %, respectively, higher than their non-textured counterparts. Moreover, the textured sample shows a significant improvement on thermal stability of d₃₃*_T, which varies by less than ± 6 % from RT to 110 °C. Furthermore, the introduction of 3D printing into the synthesis of textured piezoelectric ceramics shows great advantages over the traditional tape-casting method. This work is expected to provide a promising way for the future design of textured piezoelectric functional materials.     

Low sintering temperature for lead-free BiFeO3-BaTiO3 ceramics with high piezoelectric performance

Through modification of the heat-treatment process using a higher heating rate and a lower binder burnout temperature, the piezoelectric performance of water-quenched 0.67Bi_1.05FeO₃-0.33BaTiO₃ (BF33BT) lead-free piezoelectric ceramics was improved. The observed physical properties of BF33BT ceramics were very sensitive to the process temperatures. The sintering temperature (T_S) was changed within a narrow temperature range, and its effects were investigated. The largest rhombohedral distortion (90°-α_R = 0.14°) and tetragonality (c_T/a_T = 1.022) were observed for the ceramic sintered at 980°C, and its Curie temperature was 476°C. This ceramic showed good piezoelectric properties and large grains; the piezoelectric sensor charge coefficient (d₃₃) was 352 pC/N, and the piezoelectric actuator charge coefficient () was 270 pm/V. The high piezoelectric performance and low T_S of BF33BT ceramics indicate their potential as new low-cost eco-friendly lead-free piezoceramics.     

Investigations on electric properties and domain structures of Nd-doped 0.70Pb(Mg1/3Nb2/3)O3–0.30PbTiO3 relaxor ferroelectric ceramics with high piezoelectric properties

Although rare earth neodymium (Nd) doping is common in Pb(Mg1/3Nb2/3)O₃–PbTiO₃ (PMN–PT) single crystals, it is rarely reported in PMN–PT ceramics. To explore the effect of Nd doping on PMN–PT ceramics, PMN–30PT:xNd³⁺ (x = 0%, 1%, 2%, and 3%) relaxor ferroelectric ceramics were fabricated using a solid-state method via two-step sintering. An enhanced piezoelectric charge coefficient (d₃₃) of ∼870 pC/N and a high piezoelectric strain coefficient (d₃₃*) of ∼1025 pm/V were achieved for x = 2%. Through Rayleigh analysis of polarization–electric field (P–E) hysteresis loops under small electric fields, it was found that the dielectric property was mainly influenced by the intrinsic contribution (local lattice distortion). Furthermore, by investigating domain configurations, high piezoelectric properties were found to be associated with the domain size reduction and local structural heterogeneity. The results indicate that the PMN–30PT:xNd³⁺ ceramics is a promising material for electronic devices, and that rare earth Nd doping is an efficient strategy for improving the electronic performance of Pb-based relaxor ferroelectrics.     

Buffered template strategy for improving texture quality and piezoelectric properties of heterogeneous templated grain growth (K,Na)NbO3-based ceramics through interface engineering

Notwithstanding the advances in improving piezoelectric properties through templated grain growth, insights into the mechanical stress and microstructure of such materials are necessary. This is because the properties can be significantly varied depending on these parameters. Constructing heterostructure templates (²D–⁰D), in which the 0D nanoparticles have compositions similar to that of matrix powders, has significant potential in improving the above aspects. Here, BaTiO₃ (BT) templates and elements-doped (K,Na)NbO₃ (KNN) matrix powders were selected. Heterostructure BT (h-BT) templates were prepared by nucleation and growth of dopant-free KNN nanoparticles on bare BT. h-BT enabled a reduction in the mechanical stress at the interfaces within textured ceramics (h-BT-KNN), leading to larger grain growth and a higher texturing degree than those of textured ceramics (BT-KNN) fabricated using bare BT. The h-BT-KNN exhibited the enhanced piezoelectric constant (d₃₃) by ～170% and ～600% compared to those of the BT-KNN and nontextured ceramics, respectively.     

Lead-reduced Bi(Ni2/3Ta1/3)O3-PbTiO3 perovskite ceramics with high Curie temperature and performance

With increasing demand of high-temperature piezoelectric devices and growing concern over environment protection, a feasible reduction in lead from lead-based high Curie temperature piezoelectric materials are desperately needed. Herein, a new system of lead-reduced Bi(Ni_2/3Ta_1/3)O₃-PbTiO₃ (BNT-PT) ferroelectric ceramics is fabricated by a conventional solid-state sintering process. The phase transition behaviors as a function of composition and temperature, electrical properties, as well as the domain configurations from a microscopic level have been investigated in detail. The results indicate that crystal structures, phase transition behaviors, and electric properties of BNT-PT ceramics can be affected significantly by the content of BNT counterpart. Dielectric measurements show that xBNT-(1−x)PT ceramics transfer from the normal ferroelectrics to the relaxor ferroelectrics at compositions of x = 0.3-0.35. The BNT-PT ceramics exhibit high Curie temperature T_C ranging from 474 to 185°C with the variation in BNT content. The relative dielectric tunability n_r also rises from only 0.65% for 0.10BNT-0.90PT to 50.23% for 0.40BNT-0.60PT with increasing BNT content. The tetragonal-rich composition 0.30BNT-0.70PT ceramic possesses the maximum remnant polarization of P_r ~ 34.9 μC/cm². Meanwhile, a highest piezoelectric coefficient of d₃₃ ~ 271 pC/N and a high field piezoelectric strain coefficient of  ~ 560 pm/V are achieved at morphotropic phase boundary (MPB) composition of 0.38BNT-0.62PT. The maximum value of strain ~0.31% is obtained in the 0.36BNT-0.64PT ceramic. The largest electromechanical coupling coefficient k_p is 44.5% for 0.37BNT-0.63PT ceramic. These findings demonstrate that BNT-PT ceramics are a system of high-performance Pb-reduced ferro/piezoelectrics, which will be very promising materials for piezoelectric devices. This study offers an approach to developing and exploring new lead-reduced ferroelectric ceramics with high performances.     

Large electric field induced strain of Bi(Mg1/2Ti1/2)O3-Pb(Mg1/3Nb2/3)O3-PbTiO3 ceramics textured by Template Grain Growth

<001> oriented xBi(Mg_1/2Ti_1/2)O₃-(0.7-x)Pb(Mg_1/3Nb_2/3)O₃-0.3PbTiO₃ (BMT-PMN-PT) textured ceramics are successfully fabricated by the template grain growth method using BaTiO₃ platelets as template. BMT-PMN-PT textured ceramics with different BMT contents are studied in terms of crystal structure, microstructures, dielectric and ferroelectric properties, and electric field induced strain. The as-fabricated BMT-PMN-PT textured ceramics were found to have a strong orientation along <001> direction. The frequency dispersion of dielectric constant of BMT-PMN-PT textured ceramics increases gradually and its relaxability becomes stronger with increasing BMT content. A large electric-field induced strain (0.42 % at 4 kV/mm) is obtained in 0.25BMT-0.45PMN-0.3PT textured ceramics with Lotgering factor 0.94, which is about 83 % enhancement than that of the randomly oriented ceramics (0.23 % at 4 kV/mm). The strain of 0.25BMT-0.45PMN-0.3PT textured ceramics have a relatively high thermal stability, with a slight decrease from 0.42 % to 0.28 % in the temperature range of 20−100 °C. Our research suggests that 0.25BMT-0.45PMN-0.3PT textured ceramics have a greatly potential for actuator devices applications owing to its advantages of large electric field induced strain response.     

Enhanced piezoelectric properties in BF-BT based lead-free ferroelectric ceramics for high-temperature devices

Due to the high Curie temperature (T_C), BiFeO₃–BaTiO₃ (BF-BT) ceramics have been broadly investigated in high-temperature piezoelectric devices. The piezoelectric constant is one of the most significant factors in determining the sensitivity and reliability of piezoelectric functional components. However, the poor piezoelectric constant (d₃₃) of BF-BT ceramic has prevented the practical application of the material. In this work, we innovatively introduce the 0.93Bi_0.5Na_0.5TiO₃-0.07BaTiO₃ (0.93NBT-0.07BT) component to 0.7BF-0.3BT ceramic, to build a morphotropic phase boundary (MPB) for enhancing d₃₃. The XRD analysis shows that the (0.7BF-0.3BT)-x(0.93NBT-0.07BT) ceramics are still in the MPB region with R–PC phases coexistence, and exhibits a homogeneous solid solution. Moreover, the introduction of 0.93NBT-0.07BT ceramic suppresses the generation of defects and facilitates grain growth, thus enhancing piezoelectric property. In consequence, an optimum piezoelectricity d₃₃ = 213 pC/N along with T_c～450 °C was obtained in (0.7BF-0.3BT)-0.01(0.93NBT-0.07BT). This research provides a new idea for the application of BF-BT ceramics in high-temperature piezoelectric devices.     

Small Reduction of the Piezoelectric d33 Response in Potassium Sodium Niobate Thick Films

We have investigated the electromechanical response of potassium sodium niobate (K_0.5Na_0.5NbO₃ or KNN) thick films. The high‐field strain hysteresis loops and weak‐field converse piezoelectric d₃₃ coefficient of the films were measured and compared with those of KNN bulk ceramics under the same electric field conditions. The converse d₃₃ values of the thick films and bulk ceramics were equal to 82.5 and 138 pm/V, respectively, at 0.4 kV/mm. The fundamental difference between the piezoelectric response of the KNN films and the ceramics was studied in terms of the effective (“clamped”) piezoelectric d₃₃ coefficient. The reduction in the piezoelectric d₃₃ coefficient of the KNN films, resulting from the clamping by the substrate, was compared to lead‐based ferroelectric thick films, including Pb(Zr,Ti)O₃ (PZT) and (1 ? x)Pb(Mg_1/3Nb_2/3)O₃?xPbTiO₃ (PMN‐PT). We propose a possible explanation, based on the particular elastic properties of KNN, for the small relative difference observed between the “clamped” and “unclamped” (“bulk”) d₃₃ of KNN, in comparison with lead‐based systems.     

Pb(Mg1/3Nb2/3)O3–PbTiO3 Textured Ceramics with High Piezoelectric Response by a Novel Templated Grain Growth Approach

Pb(Mg_1/3Nb_2/3)O₃–PbTiO₃ is used as a model system of perovskite solid solutions with very high piezoelectric response at tailored morphotropic phase boundaries to demonstrate the processing of textured ceramics by ceramic‐only technology. A novel homogeneous templated grain growth approach that uses conventional ceramic procedures and a single‐source nanocrystalline powder for the matrix and also for obtaining the templates is described. Two batches of (100) faceted cube‐shaped microcrystals with average sizes of 27 and 10 μm were successfully used as templates, and aligned by tape casting for the processing of <001>‐textured Pb(Mg_1/3Nb_2/3)O₃–PbTiO₃ piezoelectric ceramics. Materials with effective piezoelectric coefficients up to 1000 pC/N and ferroelectric properties approaching those of single crystals are obtained.