Ferroelectric nanoscale domains and the $905\mathrm{K}$ phase transition in $\mathrm{Sr}\mathrm{Sn}{\mathrm{O}}_3$: A neutron total-scattering study

The $905\mathrm{K}$ $Pnma–Imma$ phase transition in $\mathrm{Sr}\mathrm{Sn}{\mathrm{O}}_3$ is studied here using a combination of variable-temperature neutron total scattering together with the reverse Monte Carlo (RMC) refinement method. The real-space RMC configurations obtained are analyzed in terms of bond distance and bond-angle distributions, and a geometric algebra approach is used to quantify the associated octahedral-tilting distributions. What emerges from this analysis is that the transition is displacive in nature, in contrast to the results of a recent average-structure investigation in which an order-disorder model was proposed [E. H. Mountstevens et al., Phys. Rev. B 71, 220102(R) (2005)]. Three-dimensional diffuse scattering patterns calculated from the same RMC configurations reveal the existence of an additional disorder mechanism which persists across the $Pnma\text{−}Imma$ transition. The “reflection conditions” of this diffuse scattering, together with displacement correlation calculations, point to the existence of ferroelectric nanoscale domains within the configurations, which are found to extend across planar regions of approximately $10–15\mathrm{\AA }$ in diameter.

Figure 1

(Color online) Representations of the $Pnma$ (left) and $Imma$ (right) crystal structures of $\mathrm{Sr}\mathrm{Sn}{\mathrm{O}}_3$, viewed down the crystallographic $\mathbf{b}$ (top) and $\mathbf{a}$ (bottom) axes. The transition from $Pnma$ to $Imma$ involves a loss of the in-phase octahedral-tilt system along $\mathbf{b}$. The Sn positions (centers of the octahedra) remain unchanged throughout the transition.

Figure 2

Comparison of experimental $D\left(r\right)$ data (points) and refined fits (lines) for $\mathrm{Sr}\mathrm{Sn}{\mathrm{O}}_3$ RMC configurations. Plots for successive temperature points have been raised by intervals of three units for ease of representation.

Figure 3

Thermal variation in average Sn–O bond lengths as calculated from the $\mathrm{Sr}\mathrm{Sn}{\mathrm{O}}_3$ RMC configurations. Open circles correspond to Sn–O1 bonds, closed circles to Sn–O2 bonds, and crosses to their weighted averages. The $Pnma–Imma$ transition temperature is indicated by a dashed vertical line.

Figure 4

(Color online) Upper panels: (a) Sn–O and (b) Sr–O nearest-neighbor distributions in the partial pair distribution functions. Lower panels: Sn–O–Sn bond-angle distributions as (c) raw probabilities $f\left(\theta \right)$ and (d) sine-weighted probabilities $f\left(\theta \right)∕\sin \theta$. In all panels, fine and bold lines correspond to functions involving O1 and O2 atoms, respectively.

Figure 5

(Color online) $\mathrm{Sn}{\mathrm{O}}_6$ octahedral-tilt angle distributions calculated for rotations about the crystallographic $\mathbf{b}$ axis using GASP (Ref. 23).

Figure 6

Average octahedral-tilt angles $\varphi$ calculated from the RMC configurations relative to the $\mathbf{a}$ (large open circles) and $\mathbf{b}$ (large filled circles) crystallographic axes. Values calculated from the anisotropic average-structure refinement described in Sec. 5 are shown as small points, and the guide-to-the-eye fit obtained using Eq. (7) is shown as a solid line. The dashed vertical line indicates the $Pnma–Imma$ transition temperature, and the inset shows the temperature dependence of the fourth power of the in-phase tilt angle.

Figure 7

Four representative diffuse scattering patterns calculated as described in the text from the $500\mathrm{K}$ $\mathrm{Sr}\mathrm{Sn}{\mathrm{O}}_3$ RMC configuration. The basic form of these patterns was found to be almost entirely temperature independent.

Figure 8

(Color online) Comparison of $\left[\mathrm{Sn}{\mathrm{O}}_6\right]$ octahedral geometries in (left) disordered $Pnma$ and (right) ordered $Imma$ average structural models of $\mathrm{Sr}\mathrm{Sn}{\mathrm{O}}_3$, obtained by Rietveld refinement against the $923\mathrm{K}$ data of Ref. 8. Anisotropic thermal ellipsoids are drawn at 50% probability level.

Figure 9

(Color online) Sn displacement correlation functions $\eta \left(r\right)$, calculated for (top) RMC configurations and (center) disordered configurations as described in the text. The raw values have been offset by various constants in this representation. The difference between RMC and disordered $\eta \left(r\right)$ functions is shown in the bottom set of curves, where the values have been multiplied by a factor of 2 in order to aid visibility.

Figure 10

(Color online) Ferroelectric nanoscale domains in our $\mathrm{Sr}\mathrm{Sn}{\mathrm{O}}_3$ configurations perpendicular to the (left to right) $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ crystal axes. $\left[\mathrm{Sn}{\mathrm{O}}_6\right]$ octahedra in which the Sn displacement is strongly correlated with that of its neighbors are given in either solid red or striped blue representations, depending on the direction of the displacements.