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Lasing Threshold Conditions for Transversal Modes of Twin Graphene-Covered Circular Quantum Wires

The analysis of the electromagnetic field of a dimer of twin graphene-covered quantum wires (QW) made of a gain material is considered. We apply the Lasing Eigenvalue Problem (LEP) approach, which delivers the mode-specific eigenvalue pairs, namely the frequencies and the threshold values of the QW gain index for the plasmon and the wire modes of nanolaser. In our work, we use quantum Kubo formalism for the graphene conductivity and classical Maxwell boundary-value problem for the field functions. The technique involves the resistive boundary conditions, the separation of variables in the local coordinates, and the addition theorem for the cylindrical functions. We derive separate determinantal equations for four different classes of symmetry of the lasing supermodes and solve them numerically. Our investigation of the mode frequencies and thresholds versus the graphene and QW parameters shows that plasmon supermodes have lower frequencies and thresholds than the wire modes provided that the QW radius is smaller than 10 µm, however in thicker wires they are comparable. The plasmon-mode characteristics are tunable using the graphene chemical potential. These results open a way for building essentially single-mode plasmonic nanolasers and their arrays.