Superfluidity of a rarefied gas of electron-hole pairs in a bilayer system

The conditions of stability of the superfluid phase in double layer systems with pairing of spatially separated electrons and holes in the low density limit are studied. The general expression for the collective excitation spectrum is obtained. It is shown that under increase in the distance $d$ between the layers the minimum emerges in the excitation spectrum. When d reaches the critical value the superfluid state becomes unstable relative to the formation of a kind of the Wigner crystal state. The same instability occurs at fixed d under increase in the density of carries. It is established that the critical distance and the critical density are related to each other by the inverse power function. The impact of the impurities on the temperature of the superfluid transition is investigated. The impact is found weak at the impurity concentration smaller than the density of the pairs. It is shown that in the rarefied system the critical temperature T_c = 100 K can be reached.