Optical Conductivity in Deformed Flat-Band α-T₃ Materials with Finite Band Gaps

We have derived closed-form analytical expressions for the optical conductivity of various types of α-T₃ materials with finite band gaps in their energy dispersion, at both zero and finite temperatures, and with near-zero or nonzero doping. In contrast to the α-T₃ lattice with a flat middle band, gapped α-T₃ materials exhibit several intriguing physical characteristics, including a curved middle band that may be partially or fully doped. The infinite degeneracy of the non-deformed flat band is completely lifted by the curvature, permitting various Fermi energy values within the band. As the flat band transitions into a curved one, two additional branch dispersion relations become nonlinear. This unique band structure emerges when an α-T₃ material is irradiated by an off-resonant, circularly polarized dressing field, interacting with the Dirac electrons. The optical conductivity of gapped α-T₃ materials exhibits all the known features of those with zero gaps, as well as those of silicene with two inequivalent band gaps. This distinguishes them from graphene, where zero-temperature conductivity in the frequency range ω < 2μ is zero, due to Pauli blocking that prevents carrier transfer between two occupied states. We demonstrate an additional Drude peak in the optical conductivity, arising from intraband transitions, which occurs in addition to the two successive steps, resulting from transitions between the valence and middle flat bands, and the valence and conduction bands observed in ungapped α-T₃ materials. These unique properties of the optical conductivity, along with its frequency dependence, hold promise for various device applications, including photodetectors, optical modulators, and metasurfaces.