We analyzed the decomposition mechanisms of trimethylgallium (TMG) used for the gallium source of GaN fabrication based on first-principles calculations and thermodynamic analysis. We considered two conditions. One condition is under the total pressure of 1 atm and the other one is under metal organic vapor phase epitaxy (MOVPE) growth of GaN. Our calculated results show that H-2 is indispensable for TMG decomposition under both conditions. In GaN MOVPE, TMG with H2 spontaneously decomposes into Ga(CH3) and Ga(CH3) decomposes into Ga atom gas when temperature is higher than 440 K. From these calculations, we confirmed that TMG surely becomes Ga atom gas near the GaN substrate surfaces.