Graphene is a single atomic layer of carbon atoms forming a dense honeycomb crystal lattice. Now tremendous results of two dimensional (2D) graphene have been obtained recently in the electronic properties both experimentally and theoretically due to the massless energy dispersion relation of electrons and holes with zero (or close to zero) bandgap. In addition, through the process of stimulated emission in population inverted graphene layers, the coupling of the plasmons to interband electron-hole transitions can lead to plasmon amplification. Recently, research results have also shown that at moderate carrier densities (109-1011/cm2), the frequencies of plasma waves in graphene are in the terahertz range.In this paper, based on the Maxwell's equations and material constitutive equation, the gain characteristics of the surface plasmon in graphene are theoretically studied in the terahertz range. In the simulations process we assume a nonequilibrium situation in graphene, where the densities of the electron and the hole are equal. And the gain characteristics for different carrier concentrations, graphene temperature and the momentum relaxation time are calculated. The calculated results show that the peak gain positions shift towards the higher frequencies with the increase of the quasi Fermi level of electron and hole associated with electron-hole concentrations. The reason may be that the change rate of the electron quasi Fermi level is higher than the hole's and thus the distributions of electrons and holes in energy are broader, resulting in the peak gain frequency shifting towards higher frequencies. However, the results also indicate that the temperature of the graphene has little effect on both the peak gain value and the peak gain position of the plasmon. It is maybe because in the simulation process the temperature is taken to be less than 50 K, which is corresponding to the energy of the 1 THz. However the calculated results show that the frequencies of the gain peak positions are all larger than 1 THz, hence, the effects of the temperature on the peak gain value and peak position both could be neglected. Moreover, it is obviously seen that the peak gain value is a function of momentum relaxation time in graphene. This is because when the momentum relaxation time increases, more electrons will be excited, and this will increase the plasmon gain probability in graphene. However, the momentum relaxation time has no effect on the position of the gain peak. It is maybe because the momentum relaxation time has little effect on radiation frequency in the whole momentum relaxation period.
