  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.

Mesoscale and Nanoscale Physics;Quantum Gases