Effects of High Frequency Irradiation and Commensurability on Finite Graphene Ribbons

Dirac materials such as graphene; host electrons with linear energy-momentum dispersion and exhibit tunable spectra under external fields, making them an active platform for engineered band structures. In this work, we study the continuum Hamiltonian model for graphene under a spatially periodic, polarized-light vector potential. The time dependence is described using Floquet theory with the Van Vleck high-frequency approximation. We derive an effective Hamiltonian for a graphene ribbon with zigzag termination. We implement a finite difference numerical approach to solve the Hamiltonian and obtain its eigenstates. We analyze the wavefunctions, dispersion, and symmetries of the resulting edge states under varying beam parameters; intensity, polarization, and incidence angle. For finite ribbon sizes, we find that commensuration between the sample length and the light-imposed modulation period introduces distinct modifications to the Dirac spectrum and wavefunction structure. For perfect commensuration, the zero-energy band, characteristic of zigzag ribbons, contains Jackiw-Rebbi-like modulations that form at interfaces where the polarization pattern changes sign. Signatures of interface localization appear in states closer to the K and K′ points. In the high-intensity limit, the states at these points localize completely at these interfaces. We develop an effective analytic model that incorporates incommensuration effects and shows remarkable agreement with the numerical results.