  We propose a model for the two-dimensional electron gas formed at the
interface of oxide heterostructures that includes a Rashba spin-orbit coupling
proportional to an electric field oriented perpendicularly to the interface.
Taking into account the electron density dependence of this electric field
confining the electron gas at the interface, we report the occurrence of a
phase separation instability (signaled by a negative compressibility) for
realistic values of the spin-orbit coupling and of the electronic
band-structure parameters at zero temperature. We extend the analysis to finite
temperatures and in the presence of an in-plane magnetic field, thereby
obtaining two phase diagrams which exhibit a phase separation dome. By varying
the gating potential the phase separation dome may shrink and vanish at zero
temperature into a quantum critical point where the charge fluctuates
dynamically. Similarly the phase separation may be spoiled by a planar magnetic
field even at zero temperature leading to a line of quantum critical points.
