High photo‐conversion efficiency in double‐graded Cu(In,Ga)(S,Se)2 thin film solar cells with two‐step sulfurization post‐treatment

Sulfur is extensively used to increase the bandgap of Cu(In,Ga)(S,Se)₂ (CIGSSe) solar cells and to improve the open circuit voltage (V_OC ) in order to optimize the characteristics of the devices. This study uses a sulfurization process to obtain a double‐graded bandgap profile. Selenization was carried out on Cu(In,Ga) precursors, followed by one sulfurization process or two consecutive sulfurization processes on top of the CIGSe absorber layer surface. The optimum two‐step sulfurization process provides an increase of V_OC of 0.05 V and an improvement of conversion efficiency of 1.17%. The efficiency of the 30 × 30 cm² monolithic module, which has 64 CIGS cells connected in series (aperture area: 878.6 cm²), is 15.85%. The optical and electrical properties of the phase and the work function distribution were investigated using the depth profiles of the absorber layer as a function of the sulfurization conditions. The CIGSSe thin film formed by two‐step sulfurization with a high sulfur concentration exhibits a single work function peak, better crystallinity, and higher conversion efficiency than those of the thin film formed by two‐step sulfurization at low sulfur concentration. In terms of the Raman spectra depth profile, the phase areas for the CIGSSe thin film that underwent the optimized high sulfur concentration two‐step‐sulfurization appeared to have less of Cu_2‐xSe phase than that with low sulfur concentration. Consequently, surface and interface phase analysis is an essential consideration to improve cell efficiency. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.     

Parameterized complex dielectric functions of CuIn1−xGaxSe2: applications in optical characterization of compositional non‐uniformities and depth profiles in materials and solar cells

In‐situ spectroscopic ellipsometry (SE) was employed to extract the complex dielectric functions ε = ε₁ + iε₂ over the spectral range of 0.75–6.5 eV for a set of polycrystalline CuIn_1−xGa_xSe₂ (CIGS) thin films with different alloy compositions x = [Ga]/{[In] + [Ga]}. For highest possible accuracy in ε for each CIGS thin film, specialized SE procedures were adopted including (i) deposition to a thickness of ~600 Å on smooth native oxide covered crystal silicon wafers, which minimizes the surface roughness on the film and thus the required corrections in data analysis, and (ii) measurement in‐situ, which minimizes ambient contamination and oxidation of the film surface. Assuming an analytical form for each of the ε spectra for these CIGS films, oscillator parameters were obtained in best fits, and these parameters were fit in turn to polynomials in x. With the resulting database of polynomial coefficients, the ε spectra for any composition of CIGS can be generated from the single parameter, x. In addition to enabling accurate contactless determination of bulk and surface roughness layer thicknesses of CIGS films by high speed multichannel SE, the database enables characterization of the composition and its profile with depth into these films, and even how the depth profile varies spatially within the plane of the films. In this study, depth profile parameters were found to correlate spatially with solar cell performance parameters. As a result, SE provides the capability of contactless compositional analysis of production‐scale CIGS photovoltaic modules at high speed. Copyright © 2016 John Wiley & Sons, Ltd.     

Effects of combined additional indium deposition and potassium fluoride post‐deposition treatments on Cu(In,Ga)Se2 thin film solar cells

The effect of additional indium on copper indium gallium selenide (CIGS) thin films and solar cells was investigated with respect to potassium fluoride post‐deposition treatment (KF‐PDT) using current‐voltage, external quantum efficiency, scanning electron microscopy, X‐ray photoelectron spectroscopy, time‐resolved photoluminescence and capacitance‐voltage measurements. The cell performance, particularly open‐circuit voltage (V _oc) improved drastically by the combined treatments of additional indium deposition after CIGS growth and subsequent KF‐PDT. A Cu deficient layer at the CIGS surface increased after both treatments rather than only KF‐PDT. Photoluminescence intensity, lifetime and net carrier concentration of KF‐untreated CIGS solar cells did not change significantly by only additional indium deposition. However, they improved because of the combined treatments. Copyright © 2017 John Wiley & Sons, Ltd.     

CIGS Thin Films for Cd-Free Solar Cells by One-Step Sputtering Process

Cu(In_1?x Ga _x )Se₂ (CIGS) thin films were deposited by a one-step radio frequency (RF) magnetron sputtering process using a quaternary CIGS target. The influence of substrate temperature on the composition, structure, and optical properties of the CIGS films was investigated. All the CIGS films exhibited the chalcopyrite structure with a preferential orientation along the (112) direction. The CIGS film deposited at 623 K showed significant improvement in film crystallinity and surface morphology compared to films deposited at 523 and 573 K. To simplify the manufacturing procedure of solar cells and avoid the use of the toxic element Cd, the properties of ZnS films prepared by RF sputtering were also investigated. The results revealed that the sputtered ZnS film exhibits good lattice matching with the sputtered CIGS film with significantly lower optical absorption loss. Finally, all-sputtered Cd-free CIGS-based heterojunction solar cells with the structure SLG/Mo/CIGS/ZnS/AZO/Al grids were fabricated without post-selenization. Furthermore, the results demonstrated the feasibility of using a full sputtering process for the fabrication of Cd-free CIGS-based solar cell.     

Effect of Se flux on CuIn1‐xGaxSe2 film in reactive sputtering process

CuIn_1‐xGa_xSe₂ (CIGS) thin films were grown on Mo/soda lime glass using a reactive sputtering process in which a Se cracker was used to deliver reactive Se molecules. The Cu_0·6 Ga_0·4 and Cu_0·4In_0·6 targets were simultaneously sputtered under the delivery of reactive Se. The effects of Se flux on CIGS film deposition were investigated. The CIGS film growth rate decreased, and the surface roughness of a film increased as the Se flux increased. The [112] crystal orientation was dominant, and metallic crystal phases such as Cu₉Ga₄ and Cu₁₆In₉ in a film were disappearing with increasing Se flux. A solar cell fabricated using a 900‐nm CIGS film showed the power conversion efficiency of 8·6%, the highest value found in a sub‐micron thick CIGS solar cell related to a reactive sputtering process with metallic targets. Copyright © 2011 John Wiley & Sons, Ltd.     

Interface Analysis of Cu(In,Ga)Se2 and ZnS Formed Using Sulfur Thermal Cracker

We analyzed the interface characteristics of Zn‐based thin‐film buffer layers formed by a sulfur thermal cracker on a Cu(In,Ga)Se₂ (CIGS) light‐absorber layer. The analyzed Zn‐based thin‐film buffer layers are processed by a proposed method comprising two processes — Zn‐sputtering and cracker‐sulfurization. The processed buffer layers are then suitable to be used in the fabrication of highly efficient CIGS solar cells. Among the various Zn‐based film thicknesses, an 8 nm–thick Zn‐based film shows the highest power conversion efficiency for a solar cell. The band alignment of the buffer/CIGS was investigated by measuring the band‐gap energies and valence band levels across the depth direction. The conduction band difference between the near surface and interface in the buffer layer enables an efficient electron transport across the junction. We found the origin of the energy band structure by observing the chemical states. The fabricated buffer/CIGS layers have a structurally and chemically distinct interface with little elemental inter‐diffusion.     

Fabrication of CIGS Films by Electrodeposition Method for Photovoltaic Cells

Cu(InGa)Se₂ (CIGS) thin films were fabricated by electrochemical deposition in a single bath containing Cu, In, Ga, and Se ions. The electrolyte was prepared by dissolving CuCl₂, InCl₃, GaCl₃, H₂SeO₃, and LiCl in deionized water. The potentiostatic deposition process was achieved by applying a voltage ranging from ?0.5?V to ?0.8?V versus Ag/AgCl. The effects of different chemical bath concentrations on the film composition and morphology were investigated. Stoichiometric CIGS film composition could be achieved by controlling the chemical compositions of the bath and the voltage. Gelatin was added to the solution to improve the surface and microstructures of the CIGS film. The as-deposited films were annealed at 500°C in Ar atmosphere for crystallization. The structural, morphological, and compositional properties of the CIGS thin films before and after annealing were examined by x-ray diffraction, scanning electron microscopy, and energy-dispersive spectroscopy. This study showed that the composition of the CIGS films is dependent on the bath concentration, whereas the applied potential had relatively less effect on the CIGS film composition. In addition, the use of gelatin helped in the fabrication of crack-free CIGS thin films with greatly improved surface morphology.     

Effect of processing parameter on structural,optical and electrical properties of photovoltaic chalcogenide nanostructured RF magnetron sputtered thin absorbing films

The aim of this work was to develop high quality of CuIn_1−xGa_xSe₂ thin absorbing films with x (Ga/In+Ga)<0.3 by sputtering without selenization process. CuIn_0.8Ga_0.2Se₂ (CIGS) thin absorbing films were deposited on soda lime glass substrate by RF magnetron sputtering using single quaternary chalcogenide (CIGS) target. The effect of substrate temperature, sputtering power & working pressure on structural, morphological, optical and electrical properties of deposited films were studied. CIGS thin films were characterised by X-ray diffraction (XRD), Field emission scanning electron microscope (FE-SEM), Energy dispersive X-ray spectroscopy (EDAX), Atomic force microscopy (AFM), UV–vis–NIR spectroscopy and four probe methods. It was observed that microstructure, surface morphology, elemental composition, transmittance as well as conductivity of thin films were strongly dependent on deposition parameters. The optimum parameters for CIGS thin films were obtained at a power 100 W, pressure 5 mT and substrate temperature 500 °C. XRD revealed that thin film deposited at above said parameters was polycrystalline in nature with larger crystallite size (32 nm) and low dislocation density (0.97×10¹⁵ lines m⁻²). The deposited film also showed preferred orientation along (112) plane. The morphology of the film depicted by FE-SEM was compact and uniform without any micro cracks and pits. The deposited film exhibited good stoichiometry (Ga/In+Ga=0.19 and In/In+Ga=0.8) with desired Cu/In+Ga ratio (0.92), which is essential for high efficiency solar cells. Transmittance of deposited film was found to be very low (1.09%). The absorption coefficient of film was ~10⁵ cm⁻¹ for high energy photon. The band gap of CIGS thin film evaluated from transmission data was found to be 1.13 eV which is optimum for solar cell application. The electrical conductivity (7.87 Ω⁻¹ cm⁻¹) of deposited CIGS thin film at optimum parameters was also high enough for practical purpose.     

All‐Solution‐Processed Indium‐Free Transparent Composite Electrodes based on Ag Nanowire and Metal Oxide for Thin‐Film Solar Cells

Fully solution‐processed Al‐doped ZnO/silver nanowire (AgNW)/Al‐doped ZnO/ZnO multi‐stacked composite electrodes are introduced as a transparent, conductive window layer for thin‐film solar cells. Unlike conventional sol–gel synthetic pathways, a newly developed combustion reaction‐based sol–gel chemical approach allows dense and uniform composite electrodes at temperatures as low as 200 °C. The resulting composite layer exhibits high transmittance (93.4% at 550 nm) and low sheet resistance (11.3 Ω sq^‐1), which are far superior to those of other solution‐processed transparent electrodes and are comparable to their sputtered counterparts. Conductive atomic force microscopy reveals that the multi‐stacked metal‐oxide layers embedded with the AgNWs enhance the photocarrier collection efficiency by broadening the lateral conduction range. This as‐developed composite electrode is successfully applied in Cu(In_1‐x,Ga_x)S₂ (CIGS) thin‐film solar cells and exhibits a power conversion efficiency of 11.03%. The fully solution‐processed indium‐free composite films demonstrate not only good performance as transparent electrodes but also the potential for applications in various optoelectronic and photovoltaic devices as a cost‐effective and sustainable alternative electrode.     

Preferred orientation of Cu(In,Ga)Se2 thin film deposited on stainless steel substrate

The influences of process parameters and Fe diffusing into Cu(In,Ga)Se₂ (CIGS) films on the orientation of CIGS absorbers grown on the stainless steel (SS) foils are investigated. The structural properties, morphology, and elemental profiles are characterized using X‐ray diffraction, scanning electron microscopy, and second ion mass spectroscopy, respectively. The orientation of CIGS thin films on the SS substrates strongly depends on the texture of the (In,Ga)₂Se₃ precursor, determined by the substrate temperature at the first stage (Ts1) and the flux ratio of Se to (In + Ga). Among these factors, Ts1 is the prerequisite to achieve [300]‐oriented IGS layer, which will yield [200]‐oriented CIGS thin film in the later process. The results indicate that through the comparison of CIGS thin films on the Mo/SS substrates and on the Mo/ZnO/SS substrates and combined with simply calculation, Fe diffusing into the CIGS layer will hinder the growth of the CIGS grains along [112] orientation. The grazing‐incidence X‐ray diffraction results suggest that the surface of the [220]‐textured CIGS thin film on the SS substrate still has [220] predominance, whereas the surface texture of the [220]‐texture CIGS thin film on the Mo/soda‐lime glass substrate became [112] predominant, which is due to the different compensation ability between Fe and Na elements. Finally, the relations between the device parameters and the degrees of the preferred orientation of CIGS absorbers are investigated. Copyright © 2012 John Wiley & Sons, Ltd.     

Virtually Bare Nanocrystal Surfaces: Significantly Enhanced Electrical Transport in CuInSe2 and CuIn1−xGaxSe2 Thin Films upon Ligand Exchange with Thermally Degradable 1‐Ethyl‐5‐Thiotetrazole

A facile and safe ligand exchange method for readily synthesized CuInSe₂ (CIS) and CuIn_1‐xGa_xSe₂ (CIGS) nanocrystals (NCs) from oleylamine to 1‐ethyl‐5‐thiotetrazole, preserving the colloidal stability of the chalcopyrite structure, is presented. 1‐Ethyl‐5‐thiotetrazole as thermally degradable ligand is adapted for the first time for trigonal pyramidal CIS (18 nm), elongated CIS (9 nm) and CIGS NCs (6 nm). Exchanged NC solutions are processed onto gold electrodes yielding ordered thin films. These films are thermally annealed at 260 °C to completely remove 1‐ethyl‐5‐thiotetrazol leaving individual closely assembled NCs with virtually bare surfaces. The current–voltage characteristics of the NC solids are measured prior to ligand thermolysis in the dark and under illumination and after ligand thermolysis in the same manner. The conductivity of trigonal pyramidal CIS increases by four orders of magnitude (1.4 × 10^?9 S cm^?1 (dark) to 1.4 × 10^?5 S cm^?1 (illuminated)) for ligand‐free NC films. Elongated CIS NC films show a three orders of magnitude conductivity increase and CIGS NC films exhibit improved conductivity by two orders of magnitude. Conductivity enhancement thereby depends on the NC size accentuating the role of trap‐states and internal grain boundaries in ligand‐free NC solids for electrical transport. This approach for the first time offers the possibility to address chalcopyrite materials’ electrical properties in a virtually ligand‐free state.     

A stacked chalcopyrite thin‐film tandem solar cell with 1.2 V open‐circuit voltage

CuGaSe₂ (CGS) thin films were prepared on tin‐doped indium oxide (ITO) coated soda‐lime glass substrates by thermal co‐evaporation to fabricate transparent solar cells. The films consisted of columnar grains with a diameter of approximately 1 μm. Some deterioration of the transparency of the ITO was observed after deposition of the CGS film. The CGS solar cells were electrically connected in series with Cu(In,Ga)Se₂ (CIGS) solar cells and mechanically stacked on the CIGS cells to construct tandem cells. The tandem solar cell with the CGS cell as the top cell showed an efficiency of 7.4% and an open‐circuit voltage of 1.18 V (AM 1.5, total area). Copyright © 2003 John Wiley & Sons, Ltd.     

Non‐destructive optical analysis of band gap profile,crystalline phase,and grain size for Cu(In,Ga)Se2 solar cells deposited by 1‐stage, 2‐stage,and 3‐stage co‐evaporation

Cu(In,Ga)Se₂ (CIGS) thin films co‐evaporated by 1‐stage, 2‐stage, and 3‐stage processes have been studied by spectroscopic ellipsometry (SE). The disappearance of a Cu_2‐xSe optical signature, detected by real time SE during multistage CIGS, has enabled precise endpoint control. Band gap energies determined by SE as depth averages show little process variation for fixed [Ga]/([In] + [Ga]) atomic ratio, whereas their broadening parameters decrease with increasing number of stages, identifying successive grain size enhancements. Refined SE analysis has revealed band gap profiling only for 3‐stage CIGS. Solar cells incorporating these absorbers have yielded increased efficiencies in correlation with phase control, grain size, and band gap profiling. Copyright © 2013 John Wiley & Sons, Ltd.     

Extremely low surface recombination velocities on low‐resistivity n‐type and p‐type crystalline silicon using dynamically deposited remote plasma silicon nitride films

Extremely low upper‐limit effective surface recombination velocities (S_eff.max) of 5.6 and 7.4 cm/s, respectively, are obtained on ~1.5 Ω cm n‐type and p‐type silicon wafers, using silicon nitride (SiN_x) films dynamically deposited in an industrial inline plasma‐enhanced chemical vapour deposition (PECVD) reactor. SiN_x films with optimised antireflective properties in air provide an excellent S_eff.max of 9.5 cm/s after high‐temperature (>800 °C) industrial firing. Such low S_eff.max values were previously only attainable for SiN_x films deposited statically in laboratory reactors or after optimised annealing; however, in our case, the SiN_x films were dynamically deposited onto large‐area c‐Si wafers using a fully industrial reactor and provide excellent surface passivation results both in the as‐deposited condition and after industrial‐firing, which is a widely used process in the photovoltaic industry. Contactless corona‐voltage measurements reveal that these SiN_x films contain a relatively high positive charge of (4–8) × 10¹² cm⁻² combined with a relatively low interface defect density of ~5 × 10¹¹ eV⁻¹ cm⁻². Copyright © 2012 John Wiley & Sons, Ltd.     

Gallium gradients in Cu(In,Ga)Se2 thin‐film solar cells

The gallium gradient in Cu(In,Ga)Se₂ (CIGS) layers, which forms during the two industrially relevant deposition routes, the sequential and co‐evaporation processes, plays a key role in the device performance of CIGS thin‐film modules. In this contribution, we present a comprehensive study on the formation, nature, and consequences of gallium gradients in CIGS solar cells. The formation of gallium gradients is analyzed in real time during a rapid selenization process by in situ X‐ray measurements. In addition, the gallium grading of a CIGS layer grown with an in‐line co‐evaporation process is analyzed by means of depth profiling with mass spectrometry. This gallium gradient of a real solar cell served as input data for device simulations. Depth‐dependent occurrence of lateral inhomogeneities on the µm scale in CIGS deposited by the co‐evaporation process was investigated by highly spatially resolved luminescence measurements on etched CIGS samples, which revealed a dependence of the optical bandgap, the quasi‐Fermi level splitting, transition levels, and the vertical gallium gradient. Transmission electron microscopy analyses of CIGS cross‐sections point to a difference in gallium content in the near surface region of neighboring grains. Migration barriers for a copper‐vacancy‐mediated indium and gallium diffusion in CuInSe₂ and CuGaSe₂ were calculated using density functional theory. The migration barrier for the In_Cu antisite in CuGaSe₂ is significantly lower compared with the Ga_Cu antisite in CuInSe₂, which is in accordance with the experimentally observed Ga gradients in CIGS layers grown by co‐evaporation and selenization processes. Copyright © 2014 John Wiley & Sons, Ltd.     

Impact of Cu-rich growth on the Cu2ZnSnSe4 surface morphology and related solar cells behavior

In order to study the influence of Cu-rich growth on the performance of the Cu₂ZnSnSe₄ (CZTSe) thin film solar cells, a multi-stage co-evaporation process is applied. The CZTSe films are grown at a lower substrate temperature to reduce the existence time of Cu_xSe_y at the first period caused by the volatility of SnSe_x. This study examines the surface morphology and device performance in Cu-rich growth and close-to-stoichiometric growth. Although the grain size of Cu-rich growth film increases a little, the difference was not dramatic as the results of CIGS reported previously. A model based on the grain boundary migration theory is proposed to explain the experimental results. The mechanisms of Cu-rich growth between CZTSe and CIGS might be different.     

Substrate‐Dependent Spin–Orbit Coupling in Hybrid Perovskite Thin Films

Solution‐processing hybrid metal halide perovskites are promising materials for developing flexible thin‐film devices. This work reports the substrate effects on the spin–orbit coupling (SOC) in perovskite films through thermal expansion under thermal annealing. X‐ray diffraction (XRD) measurements show that using a flexible polyethylene naphthalate (PEN) substrate introduces a smaller mechanical strain in perovskite MAPbI_3?xCl_x films, as compared to conventional glass substrates. Interestingly, the linear/circular photoexcitation‐modulated photocurrent studies find that decreasing mechanical strain gives rise to a weaker orbit–orbit interaction toward decreasing the SOC in the MAPbI_3?xCl_x films prepared on flexible PEN substrates relative to rigid glass substrates. Simultaneously, decreasing the mechanical strain causes a reduction in the internal magnetic parameter inside the MAPbI_3?xCl_x films, providing further evidence to show that introducing mechanical strain can affect the SOC in hybrid perovskite films upon using flexible substrates toward developing flexible perovskite thin‐film devices. Furthermore, thermal admittance spectroscopy indicates that the trap states are increased in the perovskite films prepared on flexible PEN substrates as compared to glass substrates. Consequently, PEN and rigid glass substrates lead to shorter and longer photoluminescence lifetimes, respectively. Clearly, these findings provide an insightful understanding on substrate effects on optoelectronic properties in flexible perovskite thin‐film devices.     

Reactive Sputtering Process for CuIn1‐xGaxSe2 Thin Film Solar Cells

CuIn_1‐xGa_xSe₂ (CIGS) thin films are grown on Mo/soda lime glass using a reactive sputtering process in which a Se cracker is used to deliver reactive Se molecules. The Cu and (In_0.7Ga_0.3)₂Se₃ targets are simultaneously sputtered under the delivery of reactive Se. The effects of Se flux on film composition are investigated. The Cu/(In+Ga) composition ratio increases as the Se flux increases at a plasma power of less than 30 W for the Cu target. The (112) crystal orientation becomes dominant, and crystal grain size is larger with Se flux. The power conversion efficiency of a solar cell fabricated using an 800‐nm CIGS film is 8.5%.     

Accelerated development of CuSbS2 thin film photovoltaic device prototypes

Development of alternative thin film photovoltaic technologies is an important research topic because of the potential of low‐cost, high‐efficiency solar cells to produce terawatt levels of clean power. However, this development of unexplored yet promising absorbers can be hindered by complications that arise during solar cell fabrication. Here, a high‐throughput combinatorial method is applied to accelerate development of photovoltaic devices, in this case, using the novel CuSbS₂ absorber via a newly developed three‐stage self‐regulated growth process to control absorber purity and orientation. Photovoltaic performance of the absorber, using the typical substrate CuIn_xGa_{1 − x}Se₂ (CIGS) device architecture, is explored as a function of absorber quality and thickness using a variety of back contacts. This study yields CuSbS₂ device prototypes with ~1% conversion efficiency, suggesting that the optimal CuSbS₂ device fabrication parameters and contact selection criteria are quite different than for CIGS, despite the similarity of these two absorbers. The CuSbS₂ device efficiency is at present limited by low short‐circuit current because of bulk recombination related to defects, and a small open‐circuit voltage because of a theoretically predicted cliff‐type conduction band offset between CuSbS₂ and CdS. Overall, these results illustrate both the potential and limits of combinatorial methods to accelerate the development of thin film photovoltaic devices using novel absorbers. Copyright © 2016 John Wiley & Sons, Ltd.     

Engineering of Self‐Assembled Domain Architectures with Ultra‐high Piezoelectric Response in Epitaxial Ferroelectric Films

Substrate clamping and inter‐domain pinning limit movement of non‐180° domain walls in ferroelectric epitaxial films thereby reducing the resulting piezoelectric response of ferroelectric layers. Our theoretical calculations and experimental studies of the epitaxial PbZr_xTi_1–xO₃ films grown on single crystal SrTiO₃ demonstrate that for film compositions near the morphotropic phase boundary it is possible to obtain mobile two‐domain architectures by selecting the appropriate substrate orientation. Transmission electron microscopy, X‐ray diffraction analysis, and piezoelectric force microscopy revealed that the PbZr_0.52Ti_0.48O₃ films grown on (101) SrTiO₃ substrates feature self‐assembled two‐domain structures, consisting of two tetragonal domain variants. For these films, the low‐field piezoelectric coefficient measured in the direction normal to the film surface (d₃₃) is 200 pm V^–1, which agrees well with the theoretical predictions. Under external AC electric fields of about 30 kV cm^–1, the (101) films exhibit reversible longitudinal strains as high as 0.35 %, which correspond to the effective piezoelectric coefficients in the order of 1000 pm V^–1 and can be explained by elastic softening of the PbZr_xTi_1–xO₃ ferroelectrics near the morphotropic phase boundary.