Investigating the stability of an explicit ADE-FDTD scheme for modeling graphene: Avoiding erroneous conclusions

Abstract

In recent years, an explicit auxiliary differential equation finite difference time domain (ADE-FDTD) scheme has been frequently used in graphene simulations. In this respect, a differencing scheme in which the magnetic field and the current density are collocated in time was developed and it has been recently stated that this explicit ADE-FDTD implementation is conditionally stable with the maximum time-step size bounded by the common Courant- Friedrichs-Lewy (CFL) limit. In this communication, the stability of this explicit ADE-FDTD scheme is revisited and it is shown that the conventional CFL constraint is not retained, and a time-step less than the CFL limit must be used in all domain grids to insure the stability in the whole system. Although this time-step stringent can be of little effect for some graphene applications, and in order to avoid drawing erroneous conclusions, it is shown that when the given ADE-FDTD scheme is used for simulating graphene structures that include noble and other plasmonic materials the derived time-step stringent can be of significant effect. To retain the CFL stability constraint, alternative formulation based on the Runge-Kutta (RK) time-differencing scheme is also presented in this communication. Contrary to the other existing CFL-stable ADE-FDTD approaches, the presented RK-FDTD method is equally applicable to both magnetized and unmagnetized graphene structures without incurring additional computational cost. Furthermore, the RK-FDTD scheme is extended for modeling the complex-frequency -shifted perfectly matched layer (CFS-PML) mesh truncating technique. Finally, the stability and accuracy of the explicit ADE/RK-FDTD schemes are demonstrated through numerical examples.