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The ac-susceptibility of the electron doped single-layered manganiteLa$_{1.1}$Sr$_{0.9}$MnO$_4$ is analyzed in detail. A quasi two-dimensional(2$D$) antiferromagnetic (AFM) order with Ising anisotropy is stabilized below$T_N$ $\sim$ 80K. We show that below $T_N$, a rare 2$D$ spin-glass (SG)correlation develops with the same Ising anisotropy as the AFM state. Usingsimple scaling arguments of the droplet model, we derive a scaling form for theac-susceptibility data of a 2$D$ SG, which our experimental data follows fairlywell. Due to simplifications in this 2$D$ case, the proposed scaling form onlycontains two unknown variables $\psi\nu$ and $\tau_0$. Hence, the logarithmicgrowth law of the SG correlation predicted by the droplet model is convincinglyevidenced by the scaling of our experimental data. The origin and nature ofthis 2$D$ SG state is also discussed.

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