Effects of Periodic Polarization Profiles and Sample Size Commensurability in Graphene

Floquet engineering enables the control of material properties through external periodic driving. While most studies rely on homogeneous light sources where polarization, frequency, and intensity tune material properties, we investigate the effects of two monochromatic beams with distinct tilt angles and polarizations; irradiating a finite size zigzag terminatedgraphene sample. For fixed tilts, the beams’ interference produces a spatially varying polarization pattern whose periodicity can be adjusted by the beam parameters. Using Floquet theory in the high-frequency limit, we derive an analytic framework based on a generalization of the Jackiw-Rebbi problem. Numerical implementation of the model provides the bulk spectra and wavefunctions for arbitrary parameters. Our study reveals that the commensurationbetween the sample length and the light-imposed modulation period introduces unique modifications in the Diracspectrum and wavefunction structure. For perfect commensuration, the zero-energy band, characteristic of zigzagribbons, contains localized states at the interfaces where the polarization pattern changes sign. For any commensuration signatures of state localization at interfaces appear for higher energy states, with longer decay lengths.  Results from an effective analytic model of zigzag graphene edge states that incorporates incommensuration effects, displays a remarkable agreement with the numerical results.