Observation of superspin-glass behavior in Fe$_{3}$O$_{4}$ nanoparticles
The aging and memory effects of Fe$_{3}$O$_{4}$ nanoparticles have been studied using a series of zero-field cooled (ZFC) and field-cooled (FC) magnetization measurements at various aging protocols. The genuine ZFC magnetization after the ZFC procedure with a single stop and wait process shows an aging dip at the stop temperature on reheating. The depth of the aging dip is dependent on the wait time. The frequency dependence of the AC magnetic susceptibility is indicative of critical slowing down at a freezing temperature $T_{f}$ ($= 30.6 \pm 1.6$ K). The relaxation time $τ$ is described by a power law form with a dynamic critical exponent $x$ ($= 8.2 \pm 1.0$) and a microscopic relaxation time $τ_{0}$ [$=(1.33 \pm 0.05) \times 10^{-9}$ sec]. The ZFC-peak temperature decreases with increasing magnetic field ($H$), forming a critical line with an exponent $p = 1.78 \pm 0.26$, close to the de Almeida-Thouless exponent ($p = 3/2$). These results indicate that the superspin glass phase occurs below $T_{f}$.