Disorder induced superconductor-insulator transition in epitaxial La$_{1.85}$Sr$_{0.15}$CuO$_{4}$ thin films

La$_{2-x}$Sr$_{x}$CuO$_{4\, }$is a well-known superconducting system showing various electronic properties as a function of Sr content. Especially, epitaxial thin layers of the compound show enormous increase of superconducting critical temperature ($T_{c})$ by a compressive strain. It has been reported that $T_{c}$ can be controlled by misfit strain, thickness, and oxygen annealing. In this study, we report structural and transport properties of high quality epitaxial La$_{1.85}$Sr$_{0.15}$CuO$_{4\, }$thin films. According to x-ray diffraction study, $c$-axis lattice parameter shows no significant change for various film thicknesses and the in-plane lattice parameters of the films are coherently matched with that of substrate. Electronic transport measurements show a clear superconductor-to-insulator transition (SIT), accompanying variation of $T_{c}$ depending on film thickness. These results are analyzed by using the McMillan equation to find the relation between the $T_{c}$ and a disorder correlating with film thickness. We have found the disorder exhibits an explicit power-law behavior with respect to film thickness in our La$_{1.85}$Sr$_{0.15}$CuO$_{4}$ thin films.