Inverse Melting of Polar Order in Chemically Substituted BaTiO3

TL;DR

The paper reports inverse melting of polar order in a chemically doped ferroelectric oxide BaTiO3, focusing on BaZr0.2Ti0.8O3 (BZTO) using in situ aberration-corrected STEM to map atomic-scale polarization. They observe a sequence where high-temperature thermal fluctuations give way to increased local order at intermediate temperatures, followed by a re-entrant disordered state at low temperature, i.e., inverse melting driven by dopant-induced randomness. A minimal 2D spin-1 random-field Ising model with $P_i \in \{0, \pm 1\}$ and Hamiltonian $H = -J \\sum_{\\langle ij \\rangle} P_i P_j - \\sum_i h_i P_i$, with $J>0$, captures the mechanism: dopant-generated random fields $h_i$ create a rough energy landscape that pins local polarization and induces low-temperature disorder. Atomic-scale maps of polar displacement angles $\\phi(\\mathbf{r})$, their standard deviation $\\sigma_{\\phi}$, and autocorrelation $A[\\phi(\\mathbf{r})]$ reveal non-monotonic ordering with temperature, and the Fourier-based diffuse intensity and $g(r)$ analysis indicate inhomogeneous strain and disorder in the low-temperature phase. The results suggest the inverse-melting phenomenon may be general to chemically doped ferroelectrics and could enable tunable thermal expansion and electrocaloric responses through controlled disorder.

Abstract

In many condensed matter systems, long range order emerges at low temperatures as thermal fluctuations subside. In the presence of competing interactions or quenched disorder, however, some systems can show unusual configurations that become more disordered at low temperature, a rare phenomenon known as "inverse melting". Here, we discover an inverse melting of the polar order in a ferroelectric oxide with quenched chemical disorder (BaTi1-xZrxO3) through direct atomicscale visualization using in situ scanning transmission electron microscopy. In contrast to the clean BaTiO3 parent system in which long range order tracks lower temperatures, we observe in the doped system BaTi1-xZrxO3 that thermally driven fluctuations at high temperature give way to a more ordered state and then to a re-entrant disordered configuration at even lower temperature. Such an inverse melting of the polar order is likely linked to the random field generated by Zr dopants, which modulates the energy landscape arising from the competition between thermal fluctuations and random field pinning potential. These visualizations highlight a rich landscape of order and disorder in materials with quenched disorder, which may be key to understanding their advanced functionalities.

Figures (4)

• Figure 1: Inverse melting in the presence of a random field. (a, b) Evolution of the order parameter (solid circle) and energy (hollow circle) as a function of temperature in a two-dimensional spin-1 random field Ising model. The (a) and (b) represents the evolution in the absence and presence of random field, respectively. (c, d) left panels: Snapshots of real-space configurations extracted at different temperatures in the absence (c) and presence (d) of a random field. The orange, blue and yellow represents that order parameter equal to 1 (polar up), -1 (polar down) and 0 (non polar), respectively. Middle panels: Fourier transformation of 2-dimensional pattern constructed from real-space configurations. Right panels: 2D difference plot after subtraction with Fourier transformation of ordered 2D pattern, which is used to highlight the diffuse intensity. The presence of diffuse intensity reflects the degree of disorder.
• Figure 2: Atomic-scale visualizations of polar order across normal melting in BaTiO$_{3}$. (a) Upper panel: the ADF-STEM image of [001]-projected BaTiO${_3}$ collected at 104 K. The atomic model of Ba and Ti is overlaid on the image. Lower panel: the measured polar displacement of Ti deviating from the central position determined by four neighboring Ba atoms The color represents the angle of polar displacement. (b) Real space maps of polar displacement angles at three temperatures, 473 K, 300 K and 104 K. The color and transparency represents the direction and amplitude of polar displacement. The left corner is the polar histogram plot of polar displacements at the three temperatures. The radius is from 0 to 40 pm. (c) Fourier transform of the real space Ti positions at three temperatures. The left corner shows the cropped real space Ti positions (raw data is shown in Fig. S4a). The amplitude of polar displacements is magnified 10 times to highlight the difference in positional disorder across the three temperatures. (d) The polar displacement angle deviations ($\sigma_{\phi}^{-1}$). (e) Auto correlation function of polar displacement angle $A[\phi(\textit{r})]$ at three temperatures.
• Figure 3: Atomic-scale mapping of chemically quenched disorder and accompanied random field in BaZr${_{0.2}}$Ti${_{0.8}}$O${_3}$. (a) The schematic shows the generation of random field by Zr-doping. The Zr dopant locally expands the lattice and disturbs ferroelectric order. (b) Atomic-resolution EDS chemical mapping of Ba, Zr, and Ti signals collected at 300 K. The white circles highlight representative positions with negative correlation between Zr and Ti. (c) Radial pair distribution function $g(\textit{r})$ of BaTiO${_3}$ and BaZr${_{0.2}}$Ti${_{0.8}}$O${_3}$ at 104 K. The right panel shows an enlarged view of the next-nearest atomic spacing. The gray line shows a Gaussian fit to the peaks. The arrow helps distinguish the broaden of the peak in Ba(Ti, Zr)O${_3}$. (d) ADF-STEM image overlaid with polar displacements in BaZr${_{0.2}}$Ti${_{0.8}}$O${_3}$ at 104 K. The projection is along the [001] zone axis. The scale bar is 5 nm.
• Figure 4: Inverse melting of polar order in BaZr${_{0.2}}$Ti${_{0.8}}$O${_3}$. (a) Real space maps of polar displacement angles at three temperatures, 473 K, 300 K and 104 K. The color and transparency represents the direction and amplitude of polar displacement. The left corner is polar histogram plot of polar displacement at three temperatures. The radius is from 0 to 40 pm. (b) Fourier transform of the real space Ti positions with normalized displacement at three temperatures. The left corner shows the cropped real space Ti positions (raw data in Fig. S4b). The amplitude of polar displacement is magnified 10 times to highlight the difference among three temperatures. (c) The polar displacement angle deviation ($\sigma_{\phi}^{-1}$) measured at three temperatures. (d) Auto correlation function of polar displacement angle $A[\phi(\textit{r})]$ at three temperatures.