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In this paper, microscopic interactions of surface plasmons and carriers in the graphene layer are investigated and based on the second quantization formalism, rate equations of surface plasmons are presented. Plasmonic matrix element for surface plasmon-induced transitions in the graphene is also analytically studied. We have shown TE plasmonic modes, due to vanishing plasmonic matrix elements, cannot be amplified in the graphene layer and amplification is only possible for TM plasmonic modes. Using the rate equation and the plasmonic matrix element, we have investigated emission and absorption spectra of surface plasmons in graphene for thermal equilibrium, optical induced population inversion and carrier-injected inversion regimes in the graphene layer and demonstrated that only interband band processes are involved in the surface plasmon emission. We have shown optical pumping opens a gain window for large wavenumbers and the gain window shifts toward larger wavenumbers by increasing pump energy, while carrier injection provides gain for small wave numbers

Microscopic analysis of gain spectrum of surface plasmons in graphene