Lithium ion battery (LiB) is widely used in portable electronic devices, such as cellular phones and laptop computers. Recently, needs for higher efficiency of the battery has been increasing rapidly due to the demand for electric and hybrid electric vehicle. At present, anode of LiB is mainly made of graphite. Recently, graphene has been attracting more attention as an alternative anode material. The effect of impurity doping to graphene has also been investigated recently. To optimize the materials combination for the anode applications microscopic mechanisms of such enhancement of electric capacity in LiB have to be identified. Using an ab-initio computational method, based on the density-functional theory, and its hybridized method with the classical molecular dynamics, we investigate dynamical behavior, such as diffusion processes, of Li atom in the carbon based materials with pristine, defected, and doped graphene.