Characteristic length scale for strain fields around impurity cations in perovskites

Lattice parameters have been determined from high-resolution neutron-powder diffraction data collected as a function of temperature for perovskites in the ${\text{Pr}}_{1-x}{\text{La}}_x{\text{AlO}}_3$ solid solution in order to determine the influence of La doping on the $Imma\leftrightarrow C2/m$ transition temperature, $T_{\text{c}}^{\ast }$. A plateau in $T_{\text{c}}^{\ast }$ at $\sim 150\text{K}$ extends to $x\approx 0.016\pm 0.002$, followed by a linear reduction with increasing ${\text{LaAlO}}_3$ content. The plateau is interpreted in terms of a saturation effect such that there is a composition limit below which the strain fields surrounding individual ${\text{La}}^{3+}$ ions are isolated in the ${\text{PrAlO}}_3$ matrix and above which they overlap. On this basis the strain fields are estimated to have diameters of $\sim 16–18\text{Å}$, which appears to be a characteristic length scale for structural relaxation around dopant atoms and other types of local clusters in oxide perovskites.

Figure 1

Subsolidus phase diagram for the ${\text{Pr}}_{1-x}{\text{La}}_x{\text{AlO}}_3$ solid solution. Filled triangles are transition temperatures for the $Pm\overline{3}m\leftrightarrow R\overline{3}c$ transition taken from Kennedy et al. (Ref. 32). Open circles represent transition temperatures delineating stability fields of $Imma$ and $C2/m$ structures taken from Glynn et al. (Ref. 31). Filled circles are transition temperatures from Carpenter et al. (Ref. 29) and this study. Glynn et al. (Ref. 31) described the $C2/m$ structure as being triclinic, but it is assumed here that the monoclinic structure is the stable phase at low temperatures almost across to pure $\text{La}{\text{AlO}}_3$.

Figure 2

Lattice parameter variations for the complete 2008 suite of samples.

Figure 3

Fitting of Eq. (4) to data for the 2008 suite, to obtain values of the $Imma\leftrightarrow C2/m$ transition temperature $\left(e_5^2\approx {\text{cos}}^2\beta \propto Q^2\right)$.

Figure 4

Variation in $T_{\text{c}}^{\ast }$ for the $Imma\leftrightarrow C2/m$ transition as a function of composition in the ${\text{Pr}}_{1-x}{\text{La}}_x{\text{AlO}}_3$ solid solution. The curve is a fit of Eq. (6) to the data, excluding the outlier at $x=0.25$.

Figure 5

Variations in ${\Theta }_{\text{s}}$ and $A/{\Theta }_{\text{s}}$ from fitting Eq. (4) to data for $e_5^2$ as a function of temperature.

Figure 6

Model for the variation in strain clouds around ${\text{La}}^{3+}$ ions as a function of composition in ${\text{Pr}}_{1-x}{\text{La}}_x{\text{AlO}}_3$, with $r$ representing the radius of strain clouds which just overlap at a given ${\text{LaAlO}}_3$ content, $x$. Data points are for successive coordination spheres of A cation sites round a single site. The curve is a power-law fit $\left(r=-15.5+13.3x^{-0.14}\text{Å}\right)$ for interpolation between data points.

Figure 7

Variation in $e_5^2$ $\left(\propto Q^2\right)$ as a function of composition at 8 K for the 2008 suite of samples. For coupling between composition and the order parameter of the form $\lambda X{Q}^2$, the variation in $Q^2$ is expected to show the same variation as $T_{\text{c}}^{\ast }$, as appears to be the case. The curve is a fit of Eq. (6) to all the data and has $X_{\text{s}}=0.013\pm 0.003$.