Size determination and dynamic behavior of polar nanoregions in re-entrant relaxor 0.6Bi(Mg$_{1/2}$Ti$_{1/2}$)O$_3$-0.4PbTiO$_3$.

The existence of polar nanoregions is the most important characteristic of ferroelectric relaxors, however, the size determination and dynamic of PNRs remains uncertain. We reveal a re-entrant relaxor behavior and ferroelectric-paraelectric transition coexists in complex perovskite oxide 0.6Bi(Mg$_{1/2}$Ti$_{1/2}$)O$_3$-0.4PbTiO$_3$. Two dielectric anomalies (i) the low-temperature re-entrant relaxor transition and (ii) the high-temperature diffuse phase transition (DPT) were described by the phenomenological statistical model. The sizes of the two kinds of polar nanoregions (PNRs) corresponding to two ferroelectric states were obtained. The dynamic of PNRs were analyzed using isothermal electrical modulus, which shows three critical temperatures associated with the diffuse phase transition, the formation and freezing of PNRs, respectively. The temperature evolution of the PNRs evolution depends on the stoichiometry of bismuth. The results provide new insights into the dynamic behavior of PNRs and the modification way of re-entrant relaxor behavior.