We deposited $Al_2O_3$ thin films on GaN by remote plasma atomic layer deposition (RPALD) technique, trimethylaluminum(TMA) and oxygen were used as precursors, at fixed process condition, the number of cycle were changed. Growth rate per cycle was $1.2;{AA}$/cycle. and Growth rate was in proportion to a number of cycle, the GaN MIS capacitors that $Al_2O_3$ thin film were deposited above 12 nm, have excellent electrical properties, a low electrical leakage current density(${sim}10^{-10};A/cm^2$ at 1.5 MV), but below 12 nm, we can see the degradation of the leakage current density. After post deposition annealing, Dielectric constant was estimated by 1 MHz high-frequency C-V method, it was varied with the anealing temperature from 6.9 at no post anealed to 7.6 at $800^{circ}C$, and we can see a improvement of the leakage current density and breakdown voltage by post deposition anealing below $700^{circ}C$, but, after anealed at $800^{circ}C$, we can see the degradation of the leakage current density and breakdown voltage.