Real Space Visualization of Order-Disorder Transition in BaTiO3

TL;DR

The paper investigates the ferroelectric-paraelectric transition in $BaTiO_{3}$, testing how order-disorder and displacive pictures relate. They use in situ STEM to map the local polar displacement $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{}}}}_{ ext{Ti}}$ and observe finite $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{}}}}_{ ext{Ti}}}$ in the PE phase with displacements aligned along $<$111$>$. They quantify real-space correlations using clustering of $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{}}}}_{ ext{Ti}}}$, the autocorrelation $A[oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{}}}}(oldsymbol{r})}]$, and the Fourier transform of Ti positions, showing anisotropic correlations in the FE phase and isotropic, weaker correlations in the PE phase, consistent with a transition near $T_c \approx 393$ K. Phase-field simulations with an eight-site BaTiO$_{3}$ model and $<$111$>$-type $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{}}}}_{ ext{Ti}}}$ and thermal fluctuations reproduce the R-O-T-C sequence and the temperature evolution of the displacement correlations and diffuse scattering. The findings provide direct atomistic support for the O-D mechanism in $BaTiO_{3}$ and connect real-space order-disorder fluctuations to reciprocal-space signatures, informing design and interpretation of ferroelectric perovskites.

Abstract

Ferroelectricity in BaTiO3 was observed nearly eighty years ago, but the mechanism underlying its ferroelectric-paraelectric phase transition remains elusive. The order-disorder transition has been recognized as playing a critical role, however, the precise nature of the order parameter still remains under scrutiny, including the local dipole direction and the correlations above and below the Curie temperature. Using in situ scanning transmission electron microscopy, we directly map polar displacements in BaTiO3 across the ferroelectric-paraelectric phase transition, providing atomistic insights into a order-disorder mechanism. Atomic tracking reveals finite polar Ti displacements in the paraelectric phase where they manifest as random polar nanoregions. The displacements align along <111> direction in both the ferroelectric and paraelectric phases. The paraelectric-ferroelectric transition emerges from real-space correlations of the <111> polar Ti displacements. Our direct visualizations provides atomic insights into the order-disorder mechanism in the ferroelectric-paraelectric transition of BaTiO3.

Figures (5)

• Figure 1: Ferroelectric (FE) - paraelectric (PE) phase transition of BaTiO$_{3}$ reproduced by in situ STEM. (A) Schematic graphs showing the displacive model (left panel) and order-disorder model (right panel) of the PE-PE phase transition. The arrow represents the Ti displacement. The transparent atom highlights the occupational distribution in order-disorder case. (B) The averaged lattice constant measured from single domain in ADF-STEM image of TEM lamella (upper panel) and X-ray diffraction (lower panel) of bulk crystal at different temperatures. The error bar comes from the mean absolute error of the normal distribution fitting. The dashed line and gray shadow determine the phase transition region. (C) Zoom-in diffraction patterns collected from FE (left panel) and PE (right panel) phases. The projection is [010] zone axis. The yellow arrows highlight the diffuse intensity.
• Figure 2: Finite $\mathbf{\Delta}_{\text{Ti}}$ in PE phase and evidence of $<$111$>$ displacements. (A) Left panel: Zoomed-in ADF-STEM image of BaTiO$_{3}$ overlapped with an atomic model showing Ba (blue) and Ti (green). The projection is along the [010] direction. Right panel: definition of the direction of $\mathbf{\Delta}_{\text{Ti}}$ ($\phi$). (B) Real-space map of $\mathbf{\Delta}_{\text{Ti}}$ in FE and PE phase. The color and transparent represents the direction and amplitude, respectively. (C) The polar histogram of $\mathbf{\Delta}_{\text{Ti}}$ in FE and PE phase. The radius of the polar histogram plot 40 pm. The inset shows zoomed-in ADF-STEM image overlapped with measured $\mathbf{\Delta}_{\text{Ti}}$ in the FE and PE phase. (D) Evolution of $\phi$ with temperature. The inset shows the schematic graph of the changes between FE and PE phase. The black dashed line and gray shadow highlight the phase transition region. The plot summarizes data from the region marked by a rectangle in (B) to exclude the effect of FE domain boundary. (E) Multislice image simulations and mapping of projected $\mathbf{\Delta}_{\text{Ti}}$ with increasing disorder (I-V) along the depth (electron beam imaging direction). The upper and lower panel represent $\Delta_{Ti}$ along the $<$111$>$ and $<$100$>$ direction, respectively. (F) $\phi$ of the projected displacements from multislice simulations.
• Figure 3: Real space correlations of polar displacements $\mathbf{\Delta}_{Ti}$. (A) K-means clustering of the region marked by rectangle in Fig. 2B, see details in SI notes. The color represents individual clusters of correlated displacements. The arrow represent $\mathbf{\Delta}_{Ti}$, with the size and length encoding the amplitude and direction, respectively (B) Autocorrelation function of $\mathbf{\Delta}_{Ti}$ ($A[\phi(\mathbf{r})]$) extracted along [100] (solid circle) and [001] (hollow circle) directions. (C) Fourier transform of 2D Ti positions in ferroelectric (left) and paraelectric (right). Anisotropy in the diffuse intensity is evident.
• Figure 4: Evolution of real space correlations of $\mathbf{\Delta}_{\text{Ti}}$ across temperatures. (A) K-means clusters, (B) autocorrelation function of $\mathbf{\Delta}_{\text{Ti}}$ and (C) Fourier transform result of 2D Ti positions measured across temperatures. (D) Standard deviation of displacement direction, $\sigma_{\phi}^{-1}$, at different temperatures. The inset shows the histogram at 293 K and 473 K, respectively. The mean value was set to 0. (E) Correlation length, ($\lambda$), extracted from $A[\phi(\mathbf{r})]$ at different temperatures. The hollow and solid circle represents the ($\lambda$) along [001] and [100] direction, respectively. (F) Temperature-dependent Diffuse intensity in the Fourier transform. The diffuse intensity are normalized to the the (200) and (002) Bragg peaks. The hollow and solid circle represents the [001] and [100] direction, respectively. The black dashed line and gray shadow highlights the phase transition region.
• Figure 5: Phase field simulations of a order-disorder mechanism in BaTiO$_{3}$. (A) Occupation of Ti displacement direction position at different temperatures collected from phase field simulations. The density is normalized from 0 to 1 for each temperature. (B) Distribution of eight $<$111$>$$\mathbf{\Delta}_{\text{Ti}}$ within a 32$\times$32$\times$32 super cell in PE and PE phase. (C) Autocorrelation function of yz-plane projected Ti displacements ($A[\phi(\mathbf{r})]$). The solid and hollow circle represents the profile extracted along the y (solid circle) and x (hollow circle) direction. (D) Fourier transform of 2-dimensional image reconstructed from Ti displacements projected on y-z plane. The raw data of projected Ti displacements and reconstructed 2D patterns are shown in Fig. S17.