Local structure, pseudosymmetry, and phase transitions in Na${}_{1/2}$Bi${}_{1/2}$TiO${}_3$–K${}_{1/2}$Bi${}_{1/2}$TiO${}_3$ ceramics

The structural behavior of ceramic solid solutions (1−$x$)Na${}_{1/2}$Bi${}_{1/2}$TiO${}_3$–$x$K${}_{1/2}$Bi${}_{1/2}$TiO${}_3$ (NBT-KBT) was studied using high-resolution powder diffraction and transmission electron microscopy. A temperature-independent morphotropic phase boundary (MPB) separating NBT-like pseudorhombohedral (R) and KBT-like pseudotetragonal (T) phases was observed at $x$ ≈ 0.2. For $x<0.2$, both local and average room-temperature structures are similar to those in NBT. Simultaneous long-range antiphase and short-range in-phase octahedral rotations average, resulting in effective antiphase $a^{-}{a}^{-}{c}^{-}$ tilting, which yields monoclinic symmetry when probed by x-ray diffraction (XRD). For these compositions, polar ordering is coupled to antiphase octahedral rotations so that tilting and ferroelectric (FE) domains coincide. Compositions with $x>0.2$ exhibit a tetragonal-like distortion; however, complex splitting of reflections in XRD patterns suggests that the actual symmetry is lower than tetragonal. For $0.2\le x\le 0.5$, in-phase octahedral tilting $a^0{b}^{+}{a}^0$ (or $a^{+}{b}^0{b}^0$) is present but confined to the nanoscale, while for $x>0.5$ the structure becomes untilted. In-phase tilting evolves above the ferroelectric transition and occurs around a nonpolar ($a$ or $b$) axis of the average T structure. The onset of polar order has no significant effect on the coherence length of in-phase tilting, which suggests only weak coupling between the two phenomena. The average symmetry of the T phase is determined by the effective symmetry (Imm2) of assemblages of coherent in-phase tilted nanodomains. Near the MPB, the coexistence of extended R- and T-like regions is observed, but lattice distortions within each phase are small, yielding narrow peaks with a pseudocubic appearance in XRD. The temperature of the FE phase transition exhibits a minimum at the MPB. The structured diffuse scattering observed in electron diffraction patterns for all the compositions suggests that polar order in NBT-KBT solid solutions is modulated away from the average displacements refined using powder diffraction data.

Figure 1

Traces of 200, 111, and 110 x-ray diffraction reflections (synchrotron data) for several (1−$x$)NBT–$x$KBT compositions. Peak splitting is consistent with pseudorhombohedral and pseudotetragonal symmetries for $x<0.2$ and $x>0.2$, respectively. The profiles for $x$ = 0.1 are similar to those for $x$ = 0 and therefore were omitted from this figure. The lattice distortion is minimal at the MPB ($x$ = 0.2), which creates the appearance of a cubic structure.

Figure 2

Representative ⟨310⟩ selected area electron diffraction patterns for a series of (1−$x$)NBT–$x$KBT compositions. The superlattice reflections change from 1/2$\{$ooo$\}$ for $x<0.2$ to 1/2$\{$ooe$\}$ for $x>0.2$. For $x$ = 0.2, both types of reflections are present though the dominant type varies from grain to grain. For $x<0.2$, the diffuse spots at 1/2$\{$ooo$\}$, which are streaked along ⟨100⟩ directions, signify short-range ordered in-phase tilting. In contrast, for $x$ ≥ 0.2, the round-shaped diffuse spots at 1/2$\{$ooo$\}$ arise from intersections of several ⟨100⟩ diffuse rods associated with 1/2$\{$ooe$\}$ reflections, as discussed in the text. The reciprocal lattices for both cases are illustrated in the figure.

Figure 3

(a) A dark field image of a single grain $x$ = 0.1 reveals antiphase domain boundaries associated with octahedral tilting. (b) A dark-field image of another single grain in the same sample, which features both twin and antiphase domain boundaries. The twin boundaries reside largely on $\{$100$\}$ planes. Both images were recorded with 1/2$\{$ooo$\}$ reflections strongly excited near the ⟨310⟩ orientation.

Figure 4

(a) and (b) Dark-field images of a single grain $x$ = 0.15 which exhibits a coexistence of regions that feature strong 1/2$\{$ooo$\}$ and weak 1/2$\{$ooe$\}$ superlattice reflections, as indicated in (c) the corresponding selected area electron diffraction pattern. These images were recorded using (a) 1/2$\{$ooo$\}$ and (b) 1/2$\{$ooe$\}$ reflections strongly excited near the ⟨310⟩ zone-axis orientation. Image (a) reveals a twin-domain structure typical for a pseudorhombohedral structure with domain interfaces on both $\{$100$\}$ and $\{$110$\}$ planes. In fact, most grains in this sample exhibited exclusively 1/2$\{$ooo$\}$ reflections with a similar twin-domain structure. Image (b) highlights an area with a nanoscale domain texture that is typical for a pseudotetragonal phase encountered for $x>0.2$. An interface between the R- and T-like phases is irregular without any obvious crystallographic orientation; yet, this interface is likely to be coherent. Image (d) is a superposition of the images (a) and (b) obtained using an overlay feature in Photoshop.

Figure 5

(a) and (b) Representative dark-field images and the corresponding selected area electron diffraction patterns recorded from single grains, in $x$ = 0.2. Image (a), which was obtained using a $\text{½}$310-type reflection (c) reveals a ⟨100⟩ nanoscale domain texture typical for the pseudotetragonal phase. Image (b), recorded using 1/2$\{$ooo$\}$ reflections (d), highlights relatively large antiphase domains typical for a pseudorhombohedral structure with antiphase octahedral tilting. Some grains exclusively exhibit one structure or another, whereas other grains contain both phases separated by irregular interfaces as imaged in Fig. 2.

Figure 6

Representative selected area electron diffraction patterns $x$ = 0.3 which exhibit well-structured diffuse scattering similar to that observed in NBT. The diffuse scattering contours are interpreted as traces of $\{$111$\}$${}^{*}$ and $\{$100$\}$${}^{*}$ diffuse sheets. Significant enhancement of diffuse intensity at the intersections of these traces is observed. All patterns display 1/2$\{$ooe$\}$ superlattice reflections. The diffuse spots at 1/2$\{$ooo$\}$ locations seen in [130] orientation arise from intersection of this reciprocal-lattice section with ⟨100⟩ diffuse rods (Fig. 2), as inferred from the presence of other characteristic extra spots (indicated using white arrows) and streaking of 1/2$\{$ooo$\}$ spots (circled).

Figure 7

Many-beam bright-field images of typical twin domain structures in samples with $x>0.2$.

Figure 8

A pair of dark-field images of the same area in a single grain $x$ = 0.3 recorded using $\text{½}$310 and $\text{½}$301 reflections. The domains are textured along (a) [001] and (b) [010] directions. These images confirm that the same area contains at least two of the domain variants associated with 1/2$\{$ooe$\}$ reflections.

Figure 9

A colored overlay of the two ⟨111⟩ HRTEM images of the same T1 domain ($x$ = 0.3), each obtained by Fourier filtering a raw image using a distinct set of $1/2$110-type reflections visible in the FFT pattern (inset). A given set of these superlattice reflections corresponds to one of the two domains (red or blue), which appear as spatially separate regions.

Figure 10

A bright-field image and the corresponding ⟨111⟩ zone-axis electron diffraction pattern ($x$ = 0.3) recorded from a single T1 domain (circle 1), $x$ = 0.3. The diffraction pattern exhibits two variants of superlattice reflections; the third variant is missing. No changes in the orientation and number of variants were observed by recording a diffraction pattern from the area that encompassed two adjacent T1 domains (circle 2).

Figure 11

⟨310⟩ zone-axis patterns recorded from a single grain, $x$ = 0.4, at (a) room temperature, (b) 100 °C, and (c) 150 °C above the temperature of disappearance of the T1 domains. The 1/2$\{$ooe$\}$ reflections still remain visible after the T1 domains disappear, even though the correlation length of the underlying in-phase tilting becomes significantly reduced relative to the room temperature structure.

Figure 12

A diffusionless phase diagram for the (1−$x$)NBT–$x$KBT system determined from XRD, TEM measurements, and dielectric measurements. The asterisks (orange) represent experimental points determined from the XRD/TEM data, whereas open (green) circles correspond to the temperatures of the FE anomaly estimated from the dielectric data. Octahedral tilting and ferroelectric displacements in different phase fields are indicated using the notation introduced by Stokes et al.[28] Local- and average-distortion patterns are indicated using plain (in parentheses) and bold letters, respectively. The solid (black) line corresponds to the transition temperatures determined by fitting a sigmoidal function to the XRD data; light-gray regions associated with this line reflect the width of the transitions as determined by the Δ$T$ values obtained from the sigmoidal fits. The line (dashed/blue) of the in-phase tilting transition should be considered as approximate. The transition line (dashed/black) that delineates a stability field for the high-temperature NBT-like tetragonal phase is hypothetical and has yet to be established experimentally. The coherence length for the in-phase tilting is limited to the nanoscale outside of this field. The dashed green line signifies the ferroelectric transition, which for $0\le x\le 0.1$ appears to occur below the tilting transition.

Figure 13

Temperature dependence for widths of characteristic XRD peaks (laboratory data) used to estimate temperatures of phase transitions. The diffraction data for $x$ = 0.15 and $x$ = 0.25 were measured using a laboratory diffractometer, whereas the data for $x$ = 0.2 were obtained using synchrotron radiation. In all cases, peaks were fitted using a four-parameter Pearson VII function. The error bars correspond to standard uncertainties due to profile fitting. For $x$ = 0.15 and $x$ = 0.25, the uncertainties are smaller than the symbol size. For $x$ = 0.2 an integral breadth was used instead of a full-width at half-maximum (FWHM) to reduce the noise in the data. For $x$ = 0.15 and $x$ = 0.25, the results for both heating (red) and cooling (blue) are shown. In both cases, a small but reproducible hysteresis is observed. The dashed lines passing through red symbols in the plots for $x$ = 0.15 and $x$ = 0.25 represent fits using a sigmoidal function. The arrow in the plot for $x$ = 0.2 indicates an anomaly in the slope, which is attributed to the phase transition.

Figure 14

Temperature dependence for (a) widths of 110 and 200, (b) asymmetry of 111 peak, and (c) integrated intensities of $\text{½}$311 and $\text{½}$310 superlattice reflections in NBT, as measured using XRD (laboratory data). Temperatures of the structural phase transitions are indicated using dashed gray lines. The depolarization temperature, determined from dielectric measurements, is indicated using a dashed orange line.

Figure 15

Temperature dependence of dielectric constant and losses for a series of unpoled NBT-KBT compositions. Different measurement frequencies are indicated using color: black is 0.1 kHz, red is 1 kHz, blue is 10 kHz, orange is 100 kHz, and green is 1000 kHz. The direction of increasing frequency is additionally indicated using a dashed-line arrow in the plot for $x$ = 0. All samples exhibit a broad maximum around 300 °C and another anomaly at lower temperatures. The lower temperature anomaly (indicated using a dashed gray line in the plot for $x$ = 0.1) has been attributed to a ferroelectric transition. The spread of this anomaly correlates with a spread of the corresponding phase transition seen in Fig. 12. The structural origin of the high-temperature dielectric peak remains unclear.