Band structure and optical response of Kekulé-modulated α-T₃model

We study the electronic band structure and optical response of a hybrid model, a α−T3 model featuring a √3×√3 Kekulé pattern modulation. Such a hybrid system may result from the depositing of adatoms in a hexagonal lattice, where the two sublattices are displaced in the perpendicular direction, like in germanene and silicene. We derive analytical expressions for the energy dispersion and the eigenfunctions using a tight-binding approximation of nearest-neighbor hopping electrons. The energy spectrum consists of a double-cone structure with Dirac points at zero momentum caused by Brillouin zone folding and a doubly degenerate flat band resulting from destructive quantum interference effects. Furthermore, we study the spectrum of intraband and interband transitions through the joint density of states, the optical conductivity, and the Drude spectral weight. We find that the optical conductivity exhibits step-like features, reminiscent of a 5/2-pseudospin Dirac semimetal, which include a characteristic and tunable absorption window arising from intervalley response. These results might serve as a viable signature for detecting Kekulé periodicity in two-dimensional materials.