Direct Imaging of Fractal-Dimensional Percolation in the 3D Cluster Dynamics of a Ferroelectric Super-Crystal

We use giant optical refraction (GR) to perform, for the first time, real-time imaging of ferroelectric volume cluster dynamics [1] . Experiments are carried out in biased KTN:Li kept below its room-temperature Curie point. The analysis of polarization transmission images through the sample versus electric field indicate a low-dimensional fractal percolation $\mathcal{D} = 1.65$ , followed by a second high-dimensional percolation $\mathcal{D} > 2$ , from a highly organized super-crystal (SC) state to a disordered ferroelectric cluster state ( Fig. 1 ). Results help shed light on a basic still open question: is there a general class of dynamics at the microscopic scale that characterizes the transition of a system from one macroscopic state to another? While continuous phase-transitions are macroscopically characterized by universal power laws that can be deduced from scale-invariance in their statistical field description, there is to date no evidence of an analogous universality in the microscopic details of how the transition actually occurs, i.e., in its underlying space-time dynamics. The issue has been previously investigated in ferroelectrics where the transition is from a non-polar or paraelectric crystal state to a polar ferroelectric one. One candidate microscopic mechanism driving the transition is polar cluster percolation [2] , a mechanism that, being self-similar, has intruiguing and hereto unexplored similarities to statistical scale-invariance. Percolation is triggered by changes in the average cluster size and can be activated by changes in temperature, pressure and external electric fields. Direct evidence of percolation dynamics in a ferroelectric transition in its full 3D geometry was previously unavailable. This is because while ferroelectric clusters can be optically imaged using crossed polarizers, percolation occurs for clusters whose size can range from tens of nanometers to hundreds of micrometers in a full macroscopic volume, so that direct wide-area detection is generally inaccessible. The discovery of a ferroelectric super-crystal state opens up an entirely new scenario [3] . The specific 3D geometry of the SC, with its spatially ordered and periodic birefringent polar clusters and its giant optical refraction [4] make direct crossed-polarizer high-resolution 3D orthographic projections possible ( Fig 1 ). Findings shed light on the nature and susceptibility of ferroelectric supercrystals. They suggest a specific role for organized 3D polarization structures below the Curie temperature which can have profound repercussions on the development of future energy and information storing technology [5] .