The dynamics of magnetization in phase separated manganite around half doping: a case study for Pr$_{0.5}$Sr$_{0.5}$Mn$_{0.925}$Ga$_{0.075}$O$_{3}$

We investigate the dynamics of magnetization in the phase separated (PS) state after introducing the quenched disorder at the Mn-site of a manganite around half doping. The compound, Pr$_{0.5}$Sr$_{0.5}$Mn$_{0.925}$Ga$_{0.075}$O$_{3}$, exhibits PS with the coexistence of ferromagnetic (FM) and antiferromagnetic (AFM) clusters where the size of the FM clusters is substantially reduced due to the disorder introduced by nonmagnetic Ga substitution. At low temperature, the system develops a new magnetic anomaly, which is marked by a peak in the zero field cooled magnetization. Detailed study of linear as well as nonlinear ac susceptibilities coupled with dc magnetization indicates that this peak arises due to the thermal blocking of nanometer size FM clusters demonstrating superparamagnetic behavior. The system, however, exhibits slow magnetic relaxation, aging effect, memory effect in both field cooled and zero field cooled magnetization below the blocking temperature. These imply the presence of collective behavior induced by the interaction between the clusters. Moreover, the magnetic relaxation measured with positive and negative temperature excursions exhibits asymmetric response suggesting that the dynamics in this phase separated system is accounted by the hierarchical model rather than the droplet model which are commonly used to describe the similar collective dynamics in glassy system.