The electronic bandstructure of a solid is a collection of allowed bands separated by forbidden bands, revealing the geometric symmetry of the crystal structures. Comprehensive knowledge of the bandstructure with band parameters explains intrinsic physical, chemical and mechanical properties of the solid. Here we report the artificial polaritonic bandstructures of two-dimensional honeycomb lattices for microcavity exciton-polaritons using GaAs semiconductors in the wide-range detuning values, from cavity-photon-like (red-detuned) to exciton-like (blue-detuned) regimes. In order to understand the experimental bandstructures and their band parameters, such as gap energies, bandwidths, hopping integrals and density of states, we originally establish a polariton band theory within an augmented plane wave method with two-kind-bosons, cavity photons trapped at the lattice sites and freely moving excitons. In particular, this two-kind-band theory is absolutely essential to elucidate the exciton effect in the bandstructures of blue-detuned exciton-polaritons, where the flattened exciton-like dispersion appears at larger in-plane momentum values captured in our experimental access window. We reach an excellent agreement between theory and experiments in all detuning values.
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