The transverse electric plasmonic modes in a negative refractive index black phosphorene waveguide structure

We consider the transverse electric (TE) plasmonic modes supported by black phosphorene (BP) in a parallel waveguide structure with left-handed material (LHM) instead of the conventional right-handed dielectric material. The existence condition of the TE BP surface plasmon polariton (SPP) is $\mathrm{Im}\sigma>0$. When an electric field is polarized along one of the two orthogonal crystal axes, the anisotropic symmetric and anti-symmetric plasmonic modes depend on the incident optical energy, the chemical potential, and the distance between two BP sheets can be observed. The symmetric mode has a more extensive effective refractive index, which possesses stronger field confinement. With a decreasing distance $d$ between two BP sheets, the coupling strength between the two separate BPSPP waves increases. When $d$ is small enough, the anti-symmetric mode root does not exist. LHMs can be used to realize a TE BPSPP mode to enhance the localization of the BPSPP, which is a practical method in optoelectronic devices based on black phosphorene.