Superfluid-Insulator Transition and BEC-BCS Crossover in a Rashba Moat Band

We study the superconducting transition in a two-dimensional electron gas with strong Rashba spin-orbit coupling. We assume low electron density, such that only the majority spin band participates in the transition. The BEC side of the superfluid-insulator transition, pertaining to the case of chemical potential being below the bottom of the majority band, corresponds to the appearance of a dilute gas of tightly bound Cooper pairs in the system. It can be described in terms of a single-pair wave-function. We show that this transition can be strongly affected by the electrostatic effects of the Coulomb repulsion, which suppresses superfluidity in the system. We observe that in the case of strong screening, provided by a gate on top, the perturbatively small repulsion leads to renormalization of the critical parameters of the transition but does not change its nature. However, when the screening is very weak we show that there is a range of parameters for which the nature of transition changes dramatically and it becomes of the first order type.