General argument supporting Bose–Einstein condensate of dark excitons in single and double quantum wells

We show through a novel general field theory argument that for the very same reason that excitons are bright, i.e. emitting photons, they have a higher energy than dark excitons, whatever the carrier spatial configurations is, i.e., even in stressed geometry or for electrons well separated from holes as in a double quantum well structure. Indeed, the same channel which produces the necessary finite electron–hole effective overlap to make them bright, allows for Coulomb interband exchange processes, which are nothing but a sequence of virtual recombination and creation of one electron–hole pair, a fact known in relativistic quantum field theory but never extended to semiconductor physics. The repulsive electron–hole Coulomb exchange interaction, which exists for bright excitons, but not for dark excitons, pushes the bright exciton energy up. If we now remember that dark excitons with spins ± 2 are formed in a natural way through carrier exchange between opposite spin bright excitons, we here predict that in a double quantum well sample with one parabolic trap–a configuration quite appropriate to get a high density–exciton Bose–Einstein condensation should appear, when cooling down the sample, as a dark spot made of ( ± 2 ) excitons at the center of the trap. In this paper, we also suggest a possible link between the observed ring structure in a double quantum well and the formation of dark exciton condensate.