Electric response of edge bands and their decay property of phosphorene ribbons

Zigzag phosphorene ribbons can accommodate degenerate edge states that decay from edge to bulk. The electronic structure of phosphorene ribbons, both the bulk bands and the edge bands, can be tuned by normally applying an electric field. The electrical field enlarges the energy gap parabolically, which is a second order perturbation effect. The external field as a first order perturbation lifts the degeneration of the edge bands and changes their decay property. The localization of edge states is increased by the electric field and the decay length as a function of wavevector in the whole Brillouin zone can be obtained by measuring the electric response of the edge band extreme energies.