Strongly coupled modes in bi-waveguides based on graphene

We theoretically study the confined (guided) subbands (modes) in strongly coupled double waveguides based on graphene. We find the spectra of the two-dimensional (2D) Dirac-like Hamiltonian with a quasi-mass parameter β, which is incorporated into the Hamiltonian in an ad hoc manner. Such a term can represent phenomenologically the spin-orbit interaction strength in the graphene sample. We found strongly coupled modes as the wavevector ky increased. We show that different choices for the parameter β in both waveguides induce asymmetry effects on the guided modes, reflecting anomalous effects on their probability density functions. We also find the same spectra for a single waveguide of width 50 nm. We plot the guided modes up to the seventh order in the E-θ phase space. By showing them, we completely map out the values of the reflection angle θ for which the guidance takes place with the most probability.