Influence of short-range and long-range order on the evolution of the morphotropic phase boundary in $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$

The local structures of rhombohedral, monoclinic, and tetragonal phases across the morphotropic phase boundary (MPB) of $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ are discussed in terms of the progression from short-range to long-range and back to short-range structural order. It is shown that, provided one considers the structures on a suitably small length scale, all three phases can be considered to be monoclinic at the local level and that there need not be any discrete phase boundaries across the MPB. Electron diffraction experiments showing significant variations in diffuse streaking, illustrating the progression from short-range to long-range order, are presented in support of this model. It is suggested that it is the change in the range of structural order that plays the most important role in the increase in electrical and electromechanical properties around the MPB.

Figure 1

The phase diagram of ${\mathrm{PbZr}}_{1-x}{\mathrm{Ti}}_x{\mathrm{O}}_3$ after Jaffe, Cooke, and Jaffe (Ref. 8).

Figure 2

Stereographic projection of pseudocubic perovskite. The special points marked $O$, $R$, and $T$ represent the directions of average $\mathrm{Pb}$ displacements for orthorhombic, rhombohedral, and tetragonal phases. A $\mathrm{Pb}$ displacement in the direction $M_A$ corresponds to the monoclinic $\mathrm{Cm}$ structure.

Figure 3

Part of the stereographic projection showing possible $\mathrm{Pb}$ displacements as the $\mathrm{Ti}$ concentration increases through the MPB composition. (a) Average $R$ phase with a single $\mathrm{Pb}$ displacement along $\left[111\right]$ showing a large flat ADP ellipsoid perpendicular to $\left[111\right]$. (b) Resolution of $\mathrm{Pb}$ displacements in the $R$ phase into three separate components disordered around $\left[111\right]$. The space group is still seen in diffraction experiments to be R3m. (c) Regions with one component of $\mathrm{Pb}$ displacement have now grown to a size exceeding the coherence length of the radiation used in a diffraction experiment. The structure is now seen to be monoclinic $M_A$. (d) The $\mathrm{Pb}$ displacement now is directed closer to the $\left[001\right]$ direction. The crystal structure is still seen to be monoclinic $M_A$. (e) Fourfold disordered arrangement of $\mathrm{Pb}$ displacements around $\left[001\right]$. The average structure is now seen to be tetragonal $\mathrm{P}4\mathrm{mm}$. (f) A single average $\mathrm{Pb}$ displacement along $\left[001\right]$ but with a large flat ADP perpendicular to $\left[001\right]$. The structure is now seen to be tetragonal P4mm.

Figure 4

Bond valence calculations for $\mathrm{Pb}\text{−}\mathrm{O}$ bonds as a function of $\mathrm{Ti}$ content, computed from the room-temperature data for the average crystal structure of Corker et al. (Ref. 30) ($x$ between 0.13 and 0.40) and from Frantti et al. (Ref. 52) $(x=0.46)$.

Figure 5

Electron diffraction patterns recorded along the zone axes $A$, $B$, and $C$ shown on the Kikuchi map (top center) corresponding to the rhombohedral $\left(R\right)$, monoclinic $\left(M_A\right)$, and tetragonal $\left(T\right)$ states, respectively of ${\mathrm{PbZr}}_{1-x}{\mathrm{Ti}}_x{\mathrm{O}}_3$. The compositions are shown on the top left corner of each image and the corresponding zone axis is indexed on the bottom right corner of each image. All the images were recorded on photographic films and care was taken to ensure that the background intensities in the negatives remained similar. The intensity differences depicted on the images are therefore reliable. The diffraction images are shown as negatives in order to make the diffuse scattering clearer to the eye.

Figure 6

Schematic diagram illustrating the relationship between the existence of diffuse planes of scattering and correlations between $\mathrm{Pb}$ displacements. In diffraction space, thin planes of diffuse scattering perpendicular to $⟨111⟩$ directions are observed passing through the Bragg points. In real space, this suggests $\mathrm{Pb}$ displacements that have little or no correlation between adjacent chains: In any one chain, $\mathrm{Pb}$ may be displaced in one of three possible directions perpendicular to the chain axis.