We present a stability analysis on a driven-dissipative electron-hole condensate in the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover region. Extending the combined BCS-Leggett theory with the generalized random phase approximation to the nonequilibrium case by employing the Keldysh formalism, we show that the pumping and decay of carriers causes a depairing effect on excitons. This phenomenon gives rise to an attractive interaction between excitons in the BEC regime, as well as a supercurrent that anomalously flows antiparallel to θ(r) [where θ(r) is the phase of the condensate] in the BCS regime, both leading to dynamical instabilities of an exciton BEC. Our results suggest that a substantial region of the exciton-BEC phase in the phase diagram (in terms of the interaction strength and the decay rate) is unstable.
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