Excitonic condensation for the surface states of topological insulator bilayers

We propose a generic topological insulator bilayer (TIB) system to study the excitonic condensation with self-consistent mean-field (SCMF) theory. We show that the TIB system presents the crossover behavior from the Bardeen-Cooper-Schrieffer (BCS) limit to Bose-Einstein condensation (BEC) limit. Moreover, by comparison with traditional semiconductor systems, we find that for the present system the superfluid property in the BEC phase is more sensitive to electron-hole density imbalance and the BCS phase is more robust. Applying this TIB model into Bi$_{2}$Se$_{3}$-family material, we find that the BEC phase is most probable to be observed in experiment. We also calculate the critical temperature for Bi$_{2}$Se$_{3}$-family TIB system, which is $\mathtt{\sim}100$ K. More interestingly, we can expect this relative high-temperature excitonic condensation since our calculated SCMF critical temperature is approximately equal to the Kosterlitz-Thouless transition temperature.