A new transfer matrix for investigation of surface plasmon modes in multilayer structures containing anisotropic graphene layers

Abstract

In this paper, we derive a new transfer matrix for optical calculations of anisotropic graphene layers in the presence of an external magnetic field. We employ this method to calculate the reflectance of a configuration which can be used for the optical excitation of surface plasmon modes in a magnetostatically biased single graphene layer surrounded by two dielectric media under illumination of a transverse magnetic (TM) polarized wave. Due to the external magnetic field, both transverse electric (TE) and magnetic polarizations exist in the reflected wave. There is a dip in the reflectance spectrum of the reflected TM polarized wave which corresponds to the excitation of graphene plasmons in the terahertz region. The frequency at which reflectance becomes minimum increases with increasing the external magnetic field. However, the reflectance of reflected TE polarized wave becomes maximum when graphene plasmons are excited. We use the proposed transfer matrix method to obtain the dispersion relation of hybrid TE-TM polarized surface waves supported by the air-graphene-SiO2 structure at different external magnetic fields. Our results exactly coincide with the previous plasmon dispersion relation. Next, we consider a configuration for the excitation of surface plasmons in a multilayered structure containing three magnetostatically graphene layers. The reflectance spectrum of TM reflected waves for this structure is calculated at different external magnetic fields and for each value of magnetic field three dips appear in the reflectance which shift to higher frequencies with increase of the magnetic field. Finally, the plasmon dispersion curve corresponding to three graphene layers separated with different dielectric layers is calculated at different magnetic fields. For each value of magnetic field, there are three branches in the dispersion curve corresponding to three plasmon modes. At the same frequency, the wave number of each plasmon mode decreases with increasing the magnetic field leading to its easier optical excitation from air.