We use probes of three different length scales to examine symmetry of ($1-x$)Pb(Mg${}_{1/3}$Nb${}_{2/3}$)O${}_3$-$x$PbTiO${}_3$ (PMN-$x$PT) single crystals in the morphotropic phase boundary (MPB) region at composition $x$ $=$ 0.31 (PMN-31$\%$ PT). On the macroscopic scale, x-ray diffraction (XRD) shows a mixture of strong and weak diffraction peaks of different widths. The closest match to XRD peak data is made with monoclinic $Pm$ (M${}_{\mathrm{C}}$) symmetry. On the local scale of a few nanometers, convergent beam electron diffraction (CBED) studies, with a 1.6-nm electron probe, reveal no obvious symmetry. These CBED experimental patterns can be approximately matched with simulations based on monoclinic symmetry, which suggests locally distorted monoclinic structure. A monoclinic $Cm$ (M${}_{\mathrm{A}}$ or M${}_{\mathrm{B}}$)-like symmetry could also be obtained from certain regions of the crystal by using a larger electron probe size of several tens of nanometers in diameter. Thus the monoclinic symmetry of single crystal PMN-31$\%$PT is developed only in parts of the crystal by averaging over locally distorted structure on the scale of few tens of nanometers. The macroscopic symmetry observed by XRD is a result of averaging from the local structure in PMN-31$\%$PT single crystal. The lack of local symmetry at a few nanometers scale suggests that the polarization switching results from a change in local displacements, which are not restricted to specific symmetry planes or directions.

• Received 27 July 2012

DOI:https://doi.org/10.1103/PhysRevB.86.184113