Bose–Einstein Condensation of Three‐Dimensional Exciton‐Polaritons

We develop a band‐structure‐based theory of exciton‐polaritons in a three‐dimensional inverse‐opal photonic crystal doped with semiconductor quantum dots. Starting from a symmetry‐selected bright photonic branch near the photonic gap edge, we construct an exciton‐photon Hamiltonian and obtain a lower‐polariton band with a W‐point global minimum and a nearby X‐point van‐Hove‐enhanced density of states. We show that the W valleys determine the equilibrium Bose–Einstein condensation threshold, while the X‐point saddle provides a finite excited‐state capacity that renormalizes the critical temperature when the W–X offset is thermally accessible. By tuning the exciton resonance and the light‐matter coupling, the relative W–X ordering can be reconstructed, leading to a strong variation of the critical temperature. We further formulate a momentum‐resolved Boltzmann model for driven‐dissipative kinetics. Under non‐resonant pumping, reservoir feeding, radiative decay, and inter‐sector relaxation can produce either W‐dominated condensation, a mixed W–X regime, or an X‐dominated nonequilibrium coherent state. Our results establish three‐dimensional photonic‐crystal polaritons as a platform where condensation is controlled not only by the band minimum but also by valley geometry, van‐Hove‐enhanced phase space, and relaxation pathways.