Logarithmic growth law in the two-dimensional Ising spin glass state resulting from the electron doping in single-layered manganites

The ac susceptibility of the electron-doped single-layered manganite ${\mathrm{La}}_{1.1}{\mathrm{Sr}}_{0.9}\mathrm{Mn}{\mathrm{O}}_4$ is analyzed in detail. A quasi-two-dimensional antiferromagnetic (AFM) order with Ising anisotropy is stabilized below $T_N\sim 80\mathrm{K}$. We show that below $T_N$, a rare two-dimensional (2D) spin-glass (SG) correlation develops with the same Ising anisotropy as the AFM state. Using simple scaling arguments of the droplet model, we derive a scaling form for the ac susceptibility data of a 2D SG, which our experimental data follow fairly well. Due to simplifications in this 2D case, the proposed scaling form only contains two unknown variables, $\psi \nu$ and ${\tau }_0$. Hence, the logarithmic growth law of the SG correlation predicted by the droplet model is convincingly evidenced by the scaling of our experimental data. The origin and nature of this 2D SG state are also discussed.