Picometer polar atomic displacements in strontium titanate determined by resonant X-ray diffraction

Physical properties of crystalline materials often manifest themselves as atomic displacements either away from symmetry positions or driven by external fields. Especially the origin of multiferroic or magnetoelectric effects may be hard to ascertain as the related displacements can reach the detection limit. Here we present a resonant X-ray crystal structure analysis technique that shows enhanced sensitivity to minute atomic displacements. It is applied to a recently found crystalline modification of strontium titanate that forms in single crystals under electric field due to oxygen vacancy migration. The phase has demonstrated unexpected properties, including piezoelectricity and pyroelectricity, which can only exist in non-centrosymmetric crystals. Apart from that, the atomic structure has remained elusive and could not be obtained by standard methods. Using resonant X-ray diffraction, we determine atomic displacements with sub-picometer precision and show that the modified structure of strontium titanate corresponds to that of well-known ferroelectrics such as lead titanate.

the Sr-K edge, on the other hand, no changes were observed since no pre-edge features are expressed which, in contrast to the Ti d-states, is due to a lack of local density of states at the conduction band minimum. The very small effect of formation of the MFP phase on the Ti-K XANES can be explained by the small atomic displacements that are involved. XANES is typically used to study displacements that are more than 10 times larger 3;2 .

Supplementary Note 2: Range of application
From the methodological point of view, the question arises whether the method based on resonant suppression of X-ray diffraction can be applied to other crystal structures, in particular, which prerequisites it entails and how likely it is to find suitable reflections. The amount of available reflections grows with higher photon energies, larger unit cells and lower symmetry. On the other hand, high energies require heavy atoms to allow resonant measurements and too large unit cells lead to a large number of independent atoms which could render a full refinement of all parameters exceedingly difficult. Nevertheless, there is a large set of structures remaining that fits the conditions for an application of the method, among which each material needs to be assessed individually. To give an outlook on the probability to find suitable reflections and perform similar measurements, we analyzed a large random set of structures taken from the Crystallography Open Database 4 (COD) having the following characteristics: a unit cell smaller than 2000 Å 3 , a limited number of four different elements and at least one element heavier than potassium. For this set we compared calculated RXD spectra in the 5 . . . 100 keV range of all reflections having a momentum transfer of up to 2 sin θ/λ ≤ 3 Å wide range. It is clear that the method cannot be applied routinely to all structures fulfilling the conditions mentioned above. However, SrTiO 3 is only an average representative of the described set -almost half of the processed samples have reflections that show a larger response to the dynamic displacement. This observation is very promising for future studies that could answer open questions in structure analysis or reveal structure dynamics. It should be highlighted that the presented development directly complements existing methods of crystal structure determination. Spectroscopic techniques, such as diffraction anomalous fine structure (DAFS) or X-ray absorption fine structure (XAFS), do not discriminate whether distance changes are a result of lattice strain or molecule deformation. Moreover, XAFS does not even allow for lattice parameter selective probing. These techniques can therefore not provide the sub-picometer precision for atomic displacement we aimed for. The routine methods of X-ray crystallography, based on pure Bragg diffraction, including the multi-wavelength anomalous dispersion (MAD) only reach picometer resolution of atomic positions by measuring a very high number of isolated Bragg reflections and require very homogeneous samples.
The analysis of crystal truncation rods (CTRs) in reciprocal space bears similarity to the presented method, since in both cases the substantial variations in the signal are observed at the slopes, away from intensity maxima. In the body centered unit cell, a selection rule exists for the set of Bragg reflections that was in focus for our analysis (one odd miller index). This means they would be precisely zero if the two metal cations (Sr, Ti) would have the same scattering amplitude. This can also be referred to as anti-Bragg condition where surface contributions (CTR) to the scattered intensity become important. When we use the energy dependence to reduce the Sr scattering amplitude, we come close to this situation and, in principle, CTR contributions may play a role. However, the fundamental difference is that we maintain the Bragg condition for each selected crystalline region in our case of resonantly suppressed diffraction (RSD): The enormous sensitivity to positional changes of atoms is achieved through the analysis of energy dependencies near destructive interference effects while we reside at a fixed position of reciprocal space. Therefore, the CTR would only result in a constant offset of the whole spectrum. In our case it was it was not necessary to take such contribution into account, because even though the Bragg intensity at the minimum is reduced by several orders of magnitude, it is still much stronger than the surface scattering. This assessment can be made based on the clear appearance of a Bragg maximum from cubic SrTiO 3 even at the energy of destructive interference (see black crosses in Fig. 6 of the main text).
Since it relies on thin layers having a well defined truncation (interface), the CTR method would not be suitable for structure determination of the MFP phase which is likely to be distributed inhomogeneously as the formation takes place along vacancy migration paths. With our approach, on the other hand, we can determine the structure of layers that are potentially buried, selecting each individual layer according to its lattice constants. The probed regions may be inhomogeneous and knowledge of the morphology is not necessary.
Finally, methods like Grazing incidence diffraction (GID) or reciprocal space mapping (RSM) utilize interference effects to investigate thickness and strain in layered or 3D structured systems, but are usually not sensitive enough to detect such small atomic displacements. In fact, a combination of these techniques with the RSD approach can be a powerful tool for investigation of complex objects.