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In this study, density function theory was used to evaluate geometrical and electronic properties for pure and doped system as well as adsorption energy. Pure graphene nano-ribbon appeared in plane surface during adsorption energy to have low sensitivity to methane gas. Its energy gap changed only in distance 1A because of the chemical adsorption. Doping mechanism enhanced the proprieties of graphene nano-ribbon. In geometrical structure, copper (Cu) atom stretching the nano system and it is a sign of modification. Additionally, energy gap was decreasing by doped in transition metal atom and become opening. Adsorption energy of doped system was higher than pure nano-ribbon. It was noticed that the doped transition metal enhanced the sensitivity of the system 6 times greater than pure graphene nano-ribbon. Doping graphene nano-ribbon by copper atom revealed to be a key to design chemical and physical gas sensor for methane gas.