Chemical origin of nanoscale polar domains in ${\mathrm{PbZn}}_{1∕3}{\mathrm{Nb}}_{2∕3}{\mathrm{O}}_3$

We describe the development of an atom-based Monte Carlo simulation model which gives rise to a nanoscale polar domain structure as envisaged to occur in $\mathrm{Pb}\left({\mathrm{Zn}}_{1∕3}{\mathrm{Nb}}_{2∕3}\right){\mathrm{O}}_3$ (PZN) and similar relaxor ferroelectric materials. Individual domains are essentially thin platelike domains normal to each of the six $⟨110⟩$ directions. Calculated diffuse scattering patterns have been obtained from the simulations, and these are in good agreement with observed neutron scattering data. Nanoscale domain formation is driven by the need for the Pb atoms to satisfy their valence requirements; within a planar domain, the Pb atoms are displaced in a concerted fashion away from the center of their 12-fold coordination polyhedra with an in-plane displacement along $⟨110⟩$ towards one of the coordinating O atoms. The $B$-site cations Zn and Nb display a strong tendency to alternate in the $⟨100⟩$ directions but complete order is frustrated by the 2:1 stoichiometry. No diffraction evidence has been found that this $B$-site ordering is directly linked to the nanoscale polar domain ordering. Such a linkage cannot be completely ruled out, but if it does exist, its effect on the diffraction pattern must be quite subtle. The $B$-site ordering does play an indirect role in establishing the average cell dimension, which in turn dictates the magnitude of the Pb displacements. The effect of applying an external electric field is modeled, and the results are found to be consistent with experiment.

Figure 1

Three sections of neutron diffuse scattering taken from a complete 3D volume recorded on the SXD time-of-flight instrument at ISIS [see Welberry et al. (Ref. 9)]. Darker gray scale indicates higher scattered intensity.

Figure 2

The envisaged displacement patterns for the different ions in the nanoscale polar domains in PZN [see Welberry et al. (Ref. 9)].

Figure 3

(Color online) (a) View down $\left[1\overline{1}0\right]$ of the 12-fold coordination of the Pb ion. Black arrows indicate the proposed off-center displacements. (b) The Monte Carlo energies $E_1$ (connecting neighbors in the [001] direction) and $E_2$ (connecting neighbors in the [110] direction), which promote the formation of domains in which neighboring displacements in the [110] and [001] directions tend to be parallel.

Figure 4

(Color online) Distribution of Pb displacement vectors within one layer normal to $\left[1\overline{1}0\right]$ of the simulated structure. [001] is vertical and [110] is horizontal. The black (blue) and dark gray (pink) arrows are in-plane vectors directed along [110] and $\left[\overline{1}\overline{1}0\right]$, respectively. Light gray arrows are out-of-plane $⟨110⟩$ vectors, while small circles are vectors normal to the plane. (a) and (b) are two layers separated by $\sqrt{2}{a}_0$ along $\left[1\overline{1}0\right]$. Note that the correlated domains of black (blue) or dark gray (pink) arrows do not persist in successive layers, indicating that they are essentially thin platelike nanoscale domains.

Figure 5

The effect of the different ions on the calculated diffraction patterns. (a), (b) were computed using Pb only; (c), (d) were computed using Pb and $\mathrm{Nb}∕\mathrm{Zn}$; (e), (f) were computed using Pb, $\mathrm{Nb}∕\mathrm{Zn}$, and O. (a), (c), and (e) are of the $hk0$ section; (b), (d), and (f) are of the $hk1$ section.

Figure 6

(Color online) The effect of nanoscale domain thickness on the intensity distribution within the diffuse rods. The $hk0$ section diffraction patterns and a single layer of the real-space distribution viewed down [001], for models including (a) only in-plane interactions, (b) nearest-neighbor out-of-plane correlation, and (c) a tendency for double layers. [100] is vertical and [010] is horizontal.

Figure 7

Calculated diffuse neutron scattering patterns for three sections of PZN obtained from the final Monte Carlo model.

Figure 8

(Color online) (a), (b) Distribution of Pb displacement vectors within two layers normal to $\left[1\overline{1}0\right]$ of the simulated structure in an electric field. The two layers are separated by $\sqrt{2}{a}_0$ along $\left[1\overline{1}0\right]$. [001] is vertical and [110] is horizontal. The black (blue) and dark gray (pink) arrows are in-plane vectors directed along [110] and $\left[\overline{1}\overline{1}0\right]$, respectively. Light gray arrows are out-of-plane $⟨110⟩$ vectors, while small circles are vectors normal to the plane. (c) A comparable layer normal to [110] from the same distribution. In this layer [001] is vertical and $\left[1\overline{1}0\right]$ is horizontal. Note the much smaller nanoscale domains. (d) The $hk0$ section of the diffraction pattern for this example. Features $A$, $B$, $C$, and $D$ are described in the text.