Phase separation and long wave-length charge instabilities in spin-orbit coupled systems

We investigate a two-dimensional electron model with Rashba spin-orbit interaction where the coupling constant $g=g(n)$ depends on the electronic density. It is shown that this dependence may drive the system unstable towards a long-wave length charge density wave (CDW) where the associated second order instability occurs in close vicinity to global phase separation. For very low electron densities the CDW instability is nesting-induced and the modulation follows the Fermi momentum $k_F$. At higher density the instability criterion becomes independent of $k_F$ and the system may become unstable in a broad momentum range. Finally, upon filling the upper spin-orbit split band, finite momentum instabilities disappear in favor of phase separation alone. We discuss our results with regard to the inhomogeneous phases observed at the LaAlO$_3$/SrTiO$_3$ or LaTiO$_3$/SrTiO$_3$ interfaces.