Quantitative investigation of polar nanoregion size effects in relaxor ferroelectrics

Understanding the size effects of polar nanoregion (PNR) in relaxor ferroelectrics is vital for a deep understanding mechanism to improve piezoelectric/dielectric constants. In this work, we theoretically predict that, below a critical size, the polar nanoregions possess the collinear state with matrix phase, which leads to the enhancement of the piezoelectric effect. More significantly, the critical size quantitatively tends to increase with a reduced free energy barrier between the matrix and PNR phases. In contrast, it tends to decrease with gradient energy coefficient. This work promotes an understanding of the mechanism of the relaxor ferroelectric system and guides the design of high piezoelectric materials with a wide temperature range.