A. Ceramic samples preparation

The ceramic samples were prepared using the standard solid-state reaction method (Xu, Reference Xu1991) from nominal composition Pb_0.88 Ln _0.08TiO₃, where Ln = La, Sm, Eu, and Dy. The stoichiometric mixture of high-purity oxides (PbO, TiO₂, and Ln₂O₃) was first ball-milled and then prefired at 900 °C in air for 2 h. The calcined powders were again ball-milled. The powders were pressed uniaxially under 200 MPa to form thick discs. Finally, sintering was carried out in air at 1200 °C for 2 h, in a well-covered platinum crucible in order to minimize the evaporation of reagents. The samples are hereafter labeled as PTLn8.

B. Characterization

Raman spectra were recorded at room temperature  using a Jobin Yvon T64000 spectrometer equipped, with a charge-coupled device, and polarized light from an Ar⁺ laser (λ = 514.5 nm). All spectra were obtained in backscattering geometry using a microprobe device that allows the incident light to be focused on the samples as a spot of about 2 µm in diameter. The spectra were corrected for the Bose–Einstein temperature factor.

The high-resolution XRD experiments were conducted using the XPD beam line at the Brazilian Synchrotron Light Laboratory (LNLS), Campinas, Brazil, with a Bragg–Brentano configuration, a crystal monochromator of Si (111) and a crystal analyzer of Ge (111). The measurements were made on powder samples at room temperature with a fixed counting time, a step of 0.02° and an incident wavelength of 1.770 93 Å. A LaB₆ sample was used as external standard. Micrographs using scanning electron microscopy (SEM) were taken from fractured polycrystalline pellets. The experiments were performed at the Brazilian National Nanotechnology Laboratory (LNNano). The field emission gun-scanning electron microscope analysis (SEM-FEG) was made using a high-resolution FEI Inspect F50 at 30 kV, which is equipped with an electron microprobe spectrometer. The energy-dispersive spectroscopy spectra were obtained with good resolution, showing little overlap of the signal coming from the different elements present in the samples. The statistic was good, reaching a counts number of about 15 000 s⁻¹ which means the possibility of achieving accuracy between 1 and 2% for the most of elements. In terms of at% it would represents an accuracy of up to 10⁻³ for the measurement.

A. SEM-FEG analysis

The nominal compositions of the doped PTLn8 samples were verified using the microchemical analysis by means of SEM-FEG. The obtained chemical stoichiometry for the studied samples is presented in Table I. All compositions are around the nominal composition Pb_0.88Ln_0.08TiO₃. The standard deviation of the weights % is <1%. However, the changes in composition appear in the second significant digit (Table I), which shows compositional changes as function of the lanthanides, in particular for the Pb²⁺ content. Moreover, a slight increase of the Pb²⁺ content for the Dy³⁺ doping can be observed.

The micrographs for all samples have shown well-defined particles with size of the order of 1 µm and with varying porosity. The PTDy8 sample shows a microstructure with large pores, while the porosity decreases with the increasing of the ionic radii (not shown).

B. Raman analysis

The room temperature Raman spectra for the 8 at%-doped PT samples are shown in Figure 1(a), within the frequency range of 30–900 cm⁻¹. All the Raman modes were assigned according to Foster et al. (Reference Foster, Li, Grimsditch, Chan and Lam1993).

Figure 1. (a) Raman spectra, at room temperature, for the PTLn8 ceramics; (b) low-frequency region of the Raman spectra.

Basically, the effect of Ln³⁺ ions doping is to soften the lowest transverse optical mode E (1TO) because of the increase of the ionic radius [Figure 1(b)]. This behavior suggests that the incorporation of lanthanide ions induces a change in the crystal structure of the PT. As the average grain sizes in these samples are approximately the same (~1 µm), its effect on the mode shifts is negligible. The value for undoped PT, used as a reference, was taken from Paris et al. (Reference Paris, Gurgel, Boschi, Joya, Pizani, Souza, Leite, Varela and Longo2008). It can be observed that the values increase from 5184 to 7056 cm⁻¹ with the decrease of the ionic radius of the rare-earth ions.

It is well known that there is a direct correspondence of the soft mode with the tetragonal distortion [(c/a) − 1 relation] (Dobal and Katiyar, Reference Dobal and Katiyar2002). Previous XRD experiments have shown an increase of the [(c/a) − 1] values with the decrease of the ionic radii of the rare earths (Peláiz-Barranco et al., Reference Peláiz-Barranco, Mendez-González, Arnold, Saint-Grégoire and Keeble2012). It could be expected an increase of [E (1TO)]² as well (Figure 2). The increase of the [(c/a) − 1] values has been explained previously considering the partial occupation of Ln³⁺ ions at both A and B sites of the perovskite structure (Mendez-González et al., Reference Mendez-González, Peláiz-Barranco, Calderón-Piñar and Castellanos-Guzmán2010, Reference Mendez-González, Pentón-Madrigal, Peláiz-Barranco, Figueroa, de Oliveira and Concepción-Rosabal2014; Peláiz-Barranco et al., Reference Peláiz-Barranco, Mendez-González, Calderón-Piñar, Arnold, Keeble and Guerra2010, Reference Peláiz-Barranco, Mendez-González, Arnold, Saint-Grégoire and Keeble2012). For the initial structural model it could be considered that the observed [E (1TO)]² behavior as a function of the tetragonal distortion, is also an experimental evidence of the partial occupation of Ln³⁺ ions at both crystallographic sites of the ABO₃ structure.

Figure 2. Squared frequency of the E (1TO) soft mode as function of [(c/a) − 1]. The data for the undoped lead titanate were taken from the previous papers (Paris et al., Reference Paris, Gurgel, Boschi, Joya, Pizani, Souza, Leite, Varela and Longo2008; Peláiz-Barranco et al., Reference Peláiz-Barranco, Mendez-González, Arnold, Saint-Grégoire and Keeble2012).

C. Rietveld refinements

The refinement of the structure for each composition was carried out using the Rietveld method, which is available in the FullProf_Suit software (Rodríguez-Carvajal, Reference Rodríguez-Carvajal1990). The initial structural model for the calculation of the diffraction patterns corresponds to the PT structure with a non-centrosymmetric space group P4mm (SPG No. 99). The structure data were taken from the “Inorganic Crystal Structure Database” (ICSD) (FIZ Karlsruhe, 2014). In the initial structural model, the experimental evidence from the Raman spectra and the results of the previous structural studies suggesting the incorporation of the lanthanide ions at both A and B sites have been considered (Mackie et al., Reference Mackie, Peláiz-Barranco and Keeble2010; Mendez-González et al., Reference Mendez-González, Peláiz-Barranco, Calderón-Piñar and Castellanos-Guzmán2010, Reference Mendez-González, Pentón-Madrigal, Peláiz-Barranco, Figueroa, de Oliveira and Concepción-Rosabal2014; Peláiz-Barranco et al., Reference Peláiz-Barranco, Mendez-González, Calderón-Piñar, Arnold, Keeble and Guerra2010, Reference Peláiz-Barranco, Mendez-González, Arnold, Saint-Grégoire and Keeble2012). The general nominal composition was Pb_0.94Ln_0.04Ti_0.96Ln_0.04O_2.98 taking into account that the initial occupancy of Ln³⁺ at B site should be in correspondence with the calculated tolerance factor value for these compositions (Mendez-González et al., Reference Mendez-González, Peláiz-Barranco, Calderón-Piñar and Castellanos-Guzmán2010, Reference Mendez-González, Pentón-Madrigal, Peláiz-Barranco, Figueroa, de Oliveira and Concepción-Rosabal2014; Peláiz-Barranco et al., Reference Peláiz-Barranco, Mendez-González, Arnold, Saint-Grégoire and Keeble2012). The unit-cell origin was defined by fixing the Pb/Ln atoms at the (0, 0, 0) positions. The position of Ti/Ln was taken at [1/2, 1/2, z _Ti/Ln], and the O(I) and O(II) positions were taken at [1/2, 1/2, z _O(I)] and [1/2, 0, z _O(II)], respectively.

An important feature of the observed diffraction profiles, e.g. (00h) and (h00) reflections, is the particular asymmetry that they exhibit. Peaks of the form (00h) show asymmetry on their right side; (h00) peaks show it on their left side. The described asymmetry has been studied and explained by considering the formation of a ferroelectric domains microstructure (Floquet et al., Reference Floquet, Valot, Mesnier, Niepce, Normand, Thorel and Kilaas1997; Boysen, Reference Boysen2005; Daniels et al., Reference Daniels, Jones and Finlayson2006; Heywang et al., Reference Heywang, Lubitz and Wersing2008) and it is a consequence of the X-ray scattering from the distorted interplanar distance values (d) within the 90° domain walls (Boysen, Reference Boysen2005). The asymmetric effect cannot be attributed to instrumental factors because it was not observed for the reflections of the LaB₆ standard sample. The reflections which are affected by this type of asymmetry do not provide satisfactory fits if an incorrect model is proposed. In our case, it has been considered a model, which assumes the occurrence of two phases for the same compound but with different unit-cell parameters (Boysen, Reference Boysen2005). One of the phases describes ferroelectric domains, whereas the second one describes domain wall regions, which are the source of the asymmetric effects on the profiles.

The (00h) diffraction peaks exhibit a strong anisotropic broadening, which is related with anisotropic microstrains as a result of the spontaneous polarization acting along the [001] crystallographic direction. This effect is described considering a phenomenological model (Stephens, Reference Stephens1999), which is implemented in the FullProf software (Rodríguez-Carvajal, Reference Rodríguez-Carvajal1990) as well. Both, the two phases’ model and the phenomenological model, have been used previously for the crystal structure refinement of the Eu³⁺ doping PT system (Mendez-González et al., Reference Mendez-González, Pentón-Madrigal, Peláiz-Barranco, Figueroa, de Oliveira and Concepción-Rosabal2014).

Figure 3 shows the Rietveld refinement of the X-ray powder diffraction patterns for the PTLa8, PTSm8, and PTDy8 samples taking into account the above considerations. The experimental data are represented with dots, and the corresponding calculated profiles with a continuous line. The vertical marks below the patterns correspond to the calculated Bragg reflections for both phases. The difference between the experimental and calculated patterns is also plotted at the bottom of the graphs.

Figure 3. Rietveld refinement plots for the PTLn8 ceramics. The experimental patterns (dots), the calculated profiles (continuous line), and the difference between them (below de patterns) are shown. The calculated Bragg positions, for both phases, are also shown (vertical lines).

To illustrate how the effects of the asymmetry and the anisotropy on the diffraction profiles have been managed using the proposed models, peaks of the forms (h00) and (00h) have been represented in Figure 4. It can be seen a good agreement between the calculated and experimental patterns, supporting that the used models are consistent with the proposed physical origin of the asymmetry and the anisotropic broadening.

Figure 4. Rietveld refinement plots for (h00) and (00h) peaks of PTLn8 samples taking into account the models for the anisotropic and the asymmetric effects on diffraction profiles.

Table II shows the new nominal compositions and the goodness-of-fit parameters [R _p (%) and R _wp (%)] for both phases. Compositions were calculated considering the refined occupations factors for each atom (Table A1), except for oxygen, while thermal displacement parameters were not refined. The final oxygen compositions were calculated indirectly taking into account the refined occupation factors for the lanthanide ions at the B site using the equation: Pb_1−3x′/2Ln _x′Ti_1−y Ln _y O_3−y/2, where x′ + y = 0.08.

The uncertainty estimation on the refined parameters has been managed taking into account the possible contributions of systematic errors while measuring the external standard sample (Table A2). The most common contributions, which are associated with this type of error in the measured data, have been evaluated. Random errors are usually the most difficult to estimate. We have tried to minimize those ensuring good statistical experimental data, which in our case is guaranteed with the use of synchrotron radiation and measuring all samples under the same experimental conditions (Table A2). As a result, it has been assumed that the occupation factors of the lanthanides at both A and B sites of the perovskite structure are reliable if the corresponding values of the standard deviations do not exceed an equivalent value to 1 at% of the refined occupation factor parameter, in particular for the rare-earth elements (Table A1).

The quantitative analysis shows that Eu³⁺and Dy³⁺ ions can be incorporated at both A and B sites of the PT perovskite structure, particularly for the phase 1 of the model. For the phase 2, their B-site occupation is within the error limits of the refinement (1 at%), so it is assumed to be zero. Additionally, the contribution of the phase 2 to the integral intensity of the diffraction peaks is low, so any information should be extracted carefully. In the PTLa8 and PTSm8 samples, for phase 1, La³⁺and Sm³⁺ ions preferably substitute at the A site. A model considering the possible incorporation of both La³⁺ and Sm³⁺ ions at both sites of the perovskite structure has shown unrealistic values for the fractional occupancies.

The results have suggested a higher incorporation of Ln³⁺ ions at the A site concerning the B site of the ABO₃ structure. Moreover, the B-site occupation increases with the decreasing of the Ln³⁺ ions size. The final results for the occupation factors refinement are in correspondence with the chemical stoichiometry, which has been obtained using SEM (Table I). A slightly increase of the occupation factor of the rare earths at B sites should favor an increase of the Pb²⁺ content at A sites as shown in Table I for the PTDy8 case.

The occupancy of the lanthanides at B sites does not exceed the tolerance factors, which were previously calculated for these samples (Mendez-González et al., Reference Mendez-González, Peláiz-Barranco, Calderón-Piñar and Castellanos-Guzmán2010; Peláiz-Barranco et al., Reference Peláiz-Barranco, Mendez-González, Arnold, Saint-Grégoire and Keeble2012). The XRD refinements have shown the existence of both lead and oxygen vacancies, which are formed in the perovskite structure to keep the charge neutrality because of the replacement of Pb²⁺ and Ti⁴⁺ by Ln³⁺ ions.

The R _p(%) and R _wp(%) values, in some cases, are distant of 10%. However, the refined structural parameters, e.g. fractional occupancies, can be considered a good approximation taking into account the previous theoretical and experimental studies which have been used to implement the initial structural models for each composition. In absence of a better model, which can describe more appropriately the effects of the ferroelectric domain microstructure on the diffraction profiles, it could be concluded that the present model is appropriated to obtain reliable results, once the quality of the experimental data is guaranteed.

The recovering of the [(c/a) − 1] value of the undoped PT when the Ln³⁺ ionic radii decreases (Figure 2, Table A1), can be related to the probability of a fraction of Ln³⁺ ions incorporating at the B site of the perovskite structure. The possibility of having a different structure within the domain walls as result of smaller c/a ratios compared with the corresponding ones of ferroelectric domains (Table A1), in particular for larger ionic radii, was excluded. Attempts to include a second cubic or orthorhombic phase in the model yielded no satisfactory results.

The behavior of [2arctan(a/c)], which is the angle between the polarization directions in adjacent 90° domains (Floquet et al., Reference Floquet, Valot, Mesnier, Niepce, Normand, Thorel and Kilaas1997; Heywang et al., Reference Heywang, Lubitz and Wersing2008), has been also reported in A1. This parameter decreases with the increase of the tetragonal distortion, approaching the angle of fully relaxed bulk PT. For the pure PT sample an angle of 86.4° was obtained, which is in correspondence with previously reported values (Catalan et al., Reference Catalan, Lubk, Vlooswijk, Snoeck, Magen, Janssens, Rispens, Rijnders, Blank and Noheda2011). Adjacent domains between their polarization directions angles near to 90° could favor a higher mismatch between the crystal lattices of each domain, due to disruption of the lattice periodicity, whereby the stress field in the domain wall and in areas near the domain wall should also increase. This could favor the asymmetry that has been observed in the diffraction profiles. While for the PTLa8 sample a high asymmetry is observed, it decreases for smaller Ln³⁺ ions (Figure 4).

The disruption of the lattice periodicity within the domain wall, in particular 90° domain walls, has been also related to the concept of domain pair analysis (Janovec, Reference Janovec1976, Reference Janovec1981). Using the space of the order parameters, the 90° domain walls could be described as a region with a structure, which varies smoothly accommodating the adjacent domains. Changes of the unit-cell parameters in adjacent domains, as a result of the Ln³⁺ substitution, seem to affect how the regions on both sides of the central plane (off-central layers) of the domain wall (twin) are built in order to reduce the mismatch between adjacent ferroelectric domains. For this analysis, it has been assumed that the contributions to the peak profiles owing to other types of crystal defects in the ferroelectric domain, such as vacancies or dislocations, or even 180° domain walls, are of symmetrical nature.