However, the origin of the two states differs. Superparamagnetism reflects the balance of the anisotropy and thermal energies of small clusters leading to a temperature-dependent large susceptibility. But, as we will show in this paper, superparaelectricity reflects the incomplete ferroic long-range order in nanoclusters. While a ferroelectric state is an assembly of long-range-ordered polar structures yielding a sizable remnant polarization (P r) and a hysteric polarization response (P) to an external electric field (E) (Figure 1a,d), the superparaelectric state consists of short-range-ordered nanometer polar clusters (NPCs), thereby displaying a P r ∼ 0 and a nonhysterical P-E curve (Figure 1b,e). Both states show large dielectric constants ε r due to the strong capacitive responses of their polar regions. For a given material, its ferroelectric state shows a higher initial ε r than that of the superparaelectric state. However, the latter can sustain its large ε r in a broad field or frequency range (Figure 1h,k), as compared to a decreasing ε r of the former with an increasing field or frequency (Figure 1g,j). Such characteristics are ideal for high k dielectric and energy storage applications (Figure 1d-f). [6,7] A superparaelectric state will develop in a ferroelectric material when its grain/cluster size R falls in R cr < R < R c , R c is the correlation length of polarization fluctuations, R cr is the paraelectric limit (Figure 1b). [8] Paraelectric clusters of R < R cr Analogous to the superparamagnetic state, a superparaelectric state is both scientifically intriguing and technologically important. In this state, a ferroelectric material shows a high dielectric constant and a nonhysteretic response to an external electric field, ideal for high k and capacitive energy storage applications. Despite many explorative studies, only circumstantial evidences for its existence are reported. Here, the existence of a superparaelectric state is directly proven in engineered nanocrystalline perovskite films of (Ba 0.95 , Sr 0.05)(Zr 0.2 ,Ti 0.8)O 3 , which are grown on Si at a low temperature and consist of many well-dispersed, nanometer-size polar clusters. These nanoparticles are locally polar, and only lose their shortrange order after being heated up to ≈460 °C, above which they become paraelectrics. Furthermore, they collectively present a pseudo-linear, frequency-insensitive dielectric response, which is characterized by a closeto-zero remnant polarization and a large relative dielectric constant (≈65-70). Capacitors based on these superparaelectric films display high recyclable energy densities (≈100 J cm −3) and efficiencies (≈90%) at room temperature, as well as a reliable high field endurance and good frequency (up to ≈15 GHz) and temperature (up to ≈300 °C) stabilities in their dielectric properties.
