A Mixed Finite Element Algorithm For Efficient Computation Of Graphene Waveguide Mode

Graphene,a newly emerging nano material,consists of a planar monoatomic layer of carbon arranged in a honeycomb lattice.Not only has graphene been proved of having numerous supreme parameters which can breakthrough bottlenecks in many scientific areas,but also it possesses great disruptive potential which can change many aspects of our lives.Thus,graphene has gained significant interest by scientists and researchers.Furthermore,the applications and devices of graphene electronics and photonics can be manufactured by using standard semiconductor technology,while those of other late-model nano materials still stay in theoretical stage.Especially in optoelectronics,compared with metal plasmons,graphene plasmons take the advantages of less transportation loss,tunable optical properties through electrostatic doping,large charge-carrier concentrations,and the capability to confine electromagnetic field at an extremely subwavelength scale in mid-infrared and terahertz(THz)regimes and so on.Based on those superiorities,many graphene-based plasmonic waveguides have been proposed and investigated.To well study and understand the propagation characteristics of electromagnetic wave in those graphene waveguides,Maxwell's equations need to be solved to obtain the distribution of electromagnetic field in waveguides.In electromagnetics computation,analytical method can only be employed in some classical waveguides,therefore,many numerical methods have been exploited.In this paper,we first introduce some traditional numerical method in electromagnetics computation,and then we put emphasis on the introduction of the fundamental principle and outline solving steps.In our work,we first present a surface current boundary condition(SCBC)in the mixed finite element method to solve the graphene plasmon modes.The second-order edge-based vector LT/QN basis functions are applied to expand the transverse components of the electric field and the nodal-based scalar basis functions are employed to discretize its longitudinal component.PML is used to truncate the computational domain.In the establishment of our method,SCBC is not only introduced into the vector Helmholtz equations,but also combined into the Gauss's law.Employing the mixed finite element method with SCBC(mixed FEM-SCBC)in the analysis of graphene plasmon modes can substantially reduce the calculation unknowns and save the consumption of memory and time.At last,several numerical examples of graphene-based waveguides are presented to verify the accuracy and efficiency of our method.