Reconciling Local Structure Disorder and the Relaxor State in (Bi1/2Na1/2)TiO3-BaTiO3

Lead-based relaxor ferroelectrics are key functional materials indispensable for the production of multilayer ceramic capacitors and piezoelectric transducers. Currently there are strong efforts to develop novel environmentally benign lead-free relaxor materials. The structural origins of the relaxor state and the role of composition modifications in these lead-free materials are still not well understood. In the present contribution, the solid-solution (100-x)(Bi1/2Na1/2)TiO3-xBaTiO3 (BNT-xBT), a prototypic lead-free relaxor is studied by the combination of solid-state nuclear magnetic resonance (NMR) spectroscopy, dielectric measurements and ab-initio density functional theory (DFT). For the first time it is shown that the peculiar composition dependence of the EFG distribution width (ΔQISwidth) correlates strongly to the dispersion in dielectric permittivity, a fingerprint of the relaxor state. Significant disorder is found in the local structure of BNT-xBT, as indicated by the analysis of the electric field gradient (EFG) in 23Na 3QMAS NMR spectra. Aided by DFT calculations, this disorder is attributed to a continuous unimodal distribution of octahedral tilting. These results contrast strongly to the previously proposed coexistence of two octahedral tilt systems in BNT-xBT. Based on these results, we propose that considerable octahedral tilt disorder may be a general feature of these oxides and essential for their relaxor properties.


ESI1. Interpretation of 23 Na NMR 3QMAS spectra
3QMAS NMR spectra provide two pieces of information, represented by the diagonal lines in Figure 1ESI. The first of these is the chemical shift (CS), a diagonal line with slope equal to 1, along which the chemical shift interaction is present. The second is the quadrupolar induced shift (QIS Line -displayed in green), which reveals the effects of the second order quadrupolar interaction on the apparent chemical shift. This interaction results in a shift of the signal's center of gravity away from the CS line. This shift occurs along the quadrupolar induced shift axis (QIS), a line with the slope of -10/17 in a sheared two-dimensional spectrum. [1] The farther the signal is located below the CS line, the larger the corresponding EFG at the nuclear site will be. Figure 1ESI: Example of a 23 Na 3QMAS spectrum of BNT-xBT materials highlighting its relevant features concerning the quadrupolar coupling (δQIS and ΔQISwidth).
The magnitude of the components of the EFG tensor for 23 Na (a nucleus with a I=3/2 nuclear spin), can be determined from the magnitude of δQIS,(in ppm) as described by equation 1, [2] where νL stands for the Larmor frequency, η for the asymmetry parameter of the EFG and CQ for the quadrupolar coupling constant: Eq. 2.
The quadrupolar induced shift δQIS is computed on the MAS dimension as the distance between the center of gravity of the signal (δCG) and the isotropic value of the chemical shift (δIso) (see Eq. 3).

δQIS = δCG -δISO
The δISO can be determined as the point where the CS axis is intersected by the QIS line that passes through the signal's center of gravity, with its value read on the MAS dimension. Both δISO and δCG are highlighted in Figure 1ESI as vertical dashed lines. Their separation amounts to δQIS, which is displayed by the blue double arrow line.
A further feature of the 23 Na 3QMAS NMR spectra of BNT-xBT concerns the signal width along the QIS line (ΔQISwidth). This parameter is obtained by measuring the full width at half height (fwhm) along the QIS line and is depicted as a orange double arrow in Figure 1 ESI. In the spectra of materials with a well-defined local structure, this width amounts to less than a half ppm (approximately 70 Hz under the experimental conditions reported here) and is therefore negligible. Contrasting to that, the signal width along the QIS line is very pronounced in spectra of BNT-xBT samples. Based on the overall signal's shape, and especially along the QIS line, this width (ΔQISwidth) can be considered as a measure of the distribution of quadrupolar coupling constants (CQ), and hence the distribution of the magnitude of the main component of the EFG tensor (Vzz).

ESI2. Chemical configurations employed in DFT calculations
Thirteen different arrangements of the A-site cations have been considered in the DFT calculations reported in this work. They are displayed in Figure 2ESI below. Figure 3ESI below presents the EFG values calculated for each of the thirteen chemical configurations displayed in figure 2ESI. Values calculated for chemical configurations based in pure BNT are displayed in blue, whereas those from barium containing models are displayed in red. No significant differences can be observed in the range of calculated values of models with or without barium, as both cases result in Vzz values ranging from 0 to 3 V/A 2 . ESI3. Single pulse excitation 23 Na NMR spectrum of BNT-BT.
The magnitude and the presence of a distribution of the EFG can be evaluated both from the central transition of 23 Na in 3QMAS spectra or from its satellite transitions in one-dimensional spectra recorded with single pulse excitation. Figure 4ESI compares the 23 Na NMR spectrum of BNT-6BT to that of NaNO2, where only the spinning sidebands of the satellite transitions are shown for clarity. The usual shape for a powder sample with a well defined EFG tensor is observed for NaNO2, with clear maxima and singularities. Contrastingly, the BNT-6BT sample exhibits only a featureless Gaussian shaped envelope of spinning sidebands, what indicates a distribution of EFG components and, hence, a distribution of local environments for sodium in this material. Figure 4ESI: One-dimensional single pulse 23 Na NMR spectrum of (a) NaNO2 (for comparison of a material with a unique, well-defined EFG) and (b) BNT-6BT, recorded under magic angle spinning at 10 kHz in a 14.1T magnet. Only the satellite transitions are presented in order to highlight the shape caused by a single EFG tensor (a) and the distribution of EFGs (b).