Transfer Matrices, Fresnel Relations, and Nonlocality
Graphene is a monolayer of carbon atoms tightly bound in a hexagonal honeycomb lattice. Recent research reports have shown that it is possibly an ideal material for conducting and processing electrical signals due to its high carrier mobility, surface-plasmon wave confinement, and excellent electrical tunability. As in the case of microwave and optical communication engineering, a reflection of plasmons when passing through different circuit elements is pivotal. There are theoretical and experimental researches of plasmon scattering in various types of graphene one-dimensional (1D) junctions. These studies imply that individual junctions allow extensions to the more complex structures with multiple junctions within the scattering and transfer-matrix approach. However, due to the nonlocality of the equations describing the charge carrier density dynamics in 2D conducting materials, plasmon scattered on their junction, besides reflected and transmitted waves, generates the evanescent waves, fading away at distances order of plasmon wavelengths λp1 and λp2 at both sides of the junction. These evanescent waves impose apparent limitations on the applicability of the transfer-matrix approach to the plasmonic junctions.
Moreover, considering plasmon scattering on two junctions, researchers from the University at Buffalo: Vyacheslav Semenenko and Professor Vasili Perebeinos, in collaboration with Professor Mengkun Liu at the Stony Brook University, have shown that the transfer-matrix approach in some cases brakes when the distance between the junctions is significantly longer than the total length of the regions between the junctions occupied by evanescent waves. The transfer matrix approach implacability happens when plasmon wavelengths outside the region between the junctions are significantly bigger than those in the region. The researchers explain this by the capacitive bound between the parts of plasmon-supporting material separated by the region between the junctions. Their work is currently published in the research journal, Physical Review Applied.
In their approach, the research team considered eigenmodes of one-dimensional quasistatic plasmons in periodic graphene structures with junctions of three different types. Further, based on Maxwell’s equations’ numerical solutions, the research team reconstructed the transmission and reflection coefficients for single junctions. Overall, they calculated reflections from the double-junction structures using their method and compared them with the semiphenomenological transfer-matrix approach based on the corresponding single-junction solutions.
The authors found that their solution to Maxwell’s equations was in perfect agreement with the available analytical results.
In summary, the study demonstrated that plasmon-scattering coefficients could be reconstructed directly from the numerical solutions of Maxwell equations for quasistatic plasmons with controllable accuracy. In fact, the study showed that the presented technique can be used to extract transmission coefficients from the numerical eigenvalue solutions for plasmons, including commercial software implementing the final-element methods for potentially more complex structures. In a statement to Advances in Engineering, Professor Vasili Perebeinos said the study might facilitate designing more advanced plasmonic resonators, topological waveguides, modulators, and photonic switches based on graphene and others conducting 2D materials.
Vyacheslav Semenenko, Mengkun Liu, Vasili Perebeinos. Scattering of Quasistatic Plasmons From One-Dimensional Junctions of Graphene: Transfer Matrices, Fresnel Relations, and Nonlocality. Physical Review Applied; volume 14, 024049.