Relaxation of dynamically disordered tetragonal platelets in the relaxor ferroelectric $0.964{\mathrm{Na}}_{1/2}{\mathrm{Bi}}_{1/2}{\mathrm{TiO}}_3-0.036{\mathrm{BaTiO}}_3$

The local dynamics of the lead-free relaxor $0.964{\mathrm{Na}}_{1/2}{\mathrm{Bi}}_{1/2}{\mathrm{TiO}}_3-0.036{\mathrm{BaTiO}}_3$ (NBT-3.6BT) have been investigated by a combination of quasielastic neutron-scattering (QENS) and ab initio molecular dynamics simulations. In a previous paper, we were able to show that the tetragonal platelets in the microstructure are crucial for understanding the dielectric properties of NBT-3.6BT [Pforr et al., Phys. Rev. B 94, 014105 (2016)]. To investigate their dynamics, ab initio molecular dynamics simulations were carried out using ${\mathrm{Na}}_{1/2}{\mathrm{Bi}}_{1/2}{\mathrm{TiO}}_3$ with 001 cation order as a simple model system for the tetragonal platelets in NBT-3.6BT. Similarly, 111-ordered ${\mathrm{Na}}_{1/2}{\mathrm{Bi}}_{1/2}{\mathrm{TiO}}_3$ was used as a model for the rhombohedral matrix. The measured single-crystal QENS spectra could be reproduced by a linear combination of calculated spectra. We find that the relaxational dynamics of NBT-3.6BT are concentrated in the tetragonal platelets. Chaotic stages, during which the local tilt order changes incessantly on the time scale of several picoseconds, cause the most significant contribution to the quasielastic intensity. They can be regarded as an excited state of tetragonal platelets, whose relaxation back into a quasistable state might explain the frequency dependence of the dielectric properties of NBT-3.6BT in the 100 GHz to THz range. This substantiates the assumption that the relaxor properties of NBT-3.6BT originate from the tetragonal platelets.

Figure 1

Ground-state structures of NBT with 111 cation order viewed along the $a$ axis (left) and with 001 cation order viewed along the $a$ and $c$ axes (middle and right, respectively). The structures were drawn using vesta [93] with the following color scheme: ${\mathrm{Na}}^{+}$: yellow; ${\mathrm{Bi}}^{3+}$: violet; ${\mathrm{Ti}}^{4+}$: light blue; ${\mathrm{O}}^{2-}$: red.

Figure 2

Comparison of experimental and modeled QENS spectra at $\frac{1}{2}\left(310\right)$ above room temperature. The quasielastic component is reproduced, but not the elastic component. Each model spectrum was synthesized by linear combination of four data sets with different cation orders and temperatures. See the text for details. To compensate for the effects of barium doping in the measured single crystal, the temperature scale was empirically shifted according to Eq. (1).

Figure 3

Comparison of experimental [117] (left) and modeled (right) QENS spectra at $\frac{1}{2}\left(310\right)$ below room temperature. The $T_{\exp }$ values given on the right-hand side were extrapolated using Eq. (1). The label “low T” is used in place of the calculated value of 0 K due to the uncertainty associated with the extrapolation. The pronounced oscillations of the calculated DCSF are mainly due to the finite simulation time and size of the simulation box.

Figure 4

Location of the dynamic effects in the nanostructure of NBT-3.6BT. The tetragonal regions (black) appear to be much more dynamic than the rhombohedral ones (light red), so that we suspect the most pronounced dynamics in the center of the tetragonal platelets. The white regions represent the cubic intermediate phase as in Ref. [63]. The color scheme for the atoms in the enlarged part corresponds to that in Fig. 1. The added ${\mathrm{Ba}}^{2+}$ is shown in light green. Since the actual cation arrangement is not known, a random distribution is shown instead.

Figure 5

Snapshot structures of 001-ordered NBT at the simulation temperature $T_{\mathrm{sim}}=700\mathrm{K}$, viewed along the $b$ axis. Snapshots were taken at $40900$ fs (left), $41130$ fs (middle), and $41320$ fs (right), and drawn using vesta [93] (same color scheme as in Fig. 1). Rapid changes of the local $b$-axis tilting from antiphase via in-phase back to antiphase are clearly seen. Concurrently, the tilt angle about the $c$ axis increases between $40900$ fs and $41130$ fs. Subsequently, the $a$-axis tilting changes from nearly tilt free at $41130$ fs to strongly antiphase at $41320$ fs.

Figure 6

Development of the octahedral tilt structure in the ab initio MD simulation with 001 cation order at $T_{\mathrm{sim}}=700\mathrm{K}$ (corresponding to $T_{\exp }=360\mathrm{K}$). Top: RMSD of the oxygen and bismuth atoms. Vertical lines indicate stages separated by sudden jumps of the RMSD. Bottom: SCSF indicate the degree of in-phase $[\frac{1}{2}\left(310\right)$; scaled up by a factor of 3] and antiphase $\left[\frac{1}{2}\left(311\right)\right]$ tilt order. SCSF data were obtained on a frame-by-frame basis and subsequently smoothed using an FFT filter to reduce high-frequency oscillations. Also indicated are tilt systems in stages with stable tilt symmetry, as determined from one snapshot per stage. “m” indicates mixed tilting. The dotted ellipse highlights the period during which the snapshots in Fig. 5 were taken.

Figure 7

Time dependence of the SCSF (abscissa) condensed into histograms. Left: $\frac{1}{2}\left(311\right)$, right: $\frac{1}{2}\left(310\right)$; top: 111 order, bottom: 001 order. Temperatures are given as $T_{\exp }$, calculated using Eq. (1).

Figure 8

DCSF at $T_{\mathrm{sim}}=700\mathrm{K}$ ($T_{\exp }=360\mathrm{K}$) with 001 cation order (left) and at $T_{\mathrm{sim}}=600\mathrm{K}$ ($T_{\exp }=180\mathrm{K}$) with 111 order (right), at $\frac{1}{2}\left(310\right)$. The total DCSF, which was used in the reproduction of the measured QENS (compare Fig. 2), is shown as a black line. Additionally, the individual contributions from chaotic and quasistable stages, as identified in Fig. 6, are shown as lines with red squares and green diamonds, respectively (dark and light grey in the printed version).

Figure 9

Proposed fast switching behavior of an ensemble of tetragonal platelets (left) and an example platelet (right), marked by an asterisk, upon application of an electric field at $t=0$. Initially, most platelets are in a stable structural configuration, as indicated by the black color. Light red and white regions represent rhombohedral and cubic regions as in Fig. 4. The application of an electric field changes their free-energy landscapes, so that their configurations become metastable. Subsequently, the thermal energy is sufficient for inducing chaotic configuration changes in the excited state of the platelets (red). With time, more and more platelets encounter a configuration that is stabilized by the applied electric field (hatched black and blue). They carry a nonzero polarization that is consistent with the external field. The ensemble average time for one platelet to reach a stable configuration through a chaotic stage thus defines the relaxation time of the macroscopic polarization.