Figure 1

(Color online) Pictorial description of the method employed in this work, where a unit cell of a ferroelectric material is represented by a square with an off-centered atom for simplicity. (a) Conventional Friedel pair data collection, in which the Bragg intensities $I\left(hkl\right)$ and $I\left(\overline{h}\overline{k}\overline{l}\right)$ are measured using x-ray energies slightly above an absorption edge of interest, where the imaginary dispersion correction to the corresponding atomic scattering factor is non-null. For noncentrosymmetric structures, $I\left(hkl\right)\ne I\left(\overline{h}\overline{k}\overline{l}\right)$. (b) In the more realistic case with the presence of ferroelectric domains with equal populations, the Friedel pair differences vanish. (c) With application of an external electric field, the ferroelectric domains are unbalanced or removed, and the electric polarization $P$ and the corresponding polar atomic displacements may be reversed. Measurement of $I\left(hkl\right)$ for both field polarities ($I^{+}$ and $I^{-}$) is equivalent to the Friedel pair measurement displayed in (a). (d) Scheme of the sample holder attached to the cold finger of a closed-cycle cryostat (see text).Reuse & Permissions

Figure 2

(Color online) Detection of the PFPD. (a) An axial $\left(\theta \text{−}2\theta \right)$ scan is performed for the chosen (3 6 2) reflection with applied fields along opposite polarities [$I^{+}$ and $I^{-}$, see Fig. 1c]. (b) The PFPD, $I^{+}-I^{-}$, is obtained as the filled area for a given energy. Observed (symbols) and simulated (line) energy dependencies of the (c) integrated intensity and (d) PFPD of the (3 6 2) reflection through the $\text{Mn}K$ edge, corrected by the absorption coefficient $\mu$. The vertical dashed line indicates the chosen condition for the subsequent measurements at fixed energy.Reuse & Permissions

Figure 3

(Color online) (a) Temperature dependence of the PFPD of the (3 6 2) reflection, $I^{+}-I^{-}$. The vertical dashed lines mark the transitions from the PE state to a ferroelectric state (FE1) at $T_c\sim 40\text{K}$ and between distinct ferroelectric states (FE2 and FE3) at $\sim 27$ and $\sim 14\text{K}$, respectively, such as reported in Refs. 37, 38. (b) Representation of the crystal structure of ${\text{DyMn}}_2{\text{O}}_5$, showing the polar displacements of the ${\text{Mn}}^{3+}$ ions along the $\mathbf{b}$ axis captured by the PFPD of the (3 6 2) reflection. The direction of the polar atomic displacements and the polarization $P$ can be switched by an external field [see also Fig. 1c].Reuse & Permissions