There are many electronic and magnetic properties exhibited by complex oxides. Electronic phase separation (EPS) is one of those, the presence of which can be linked to exotic behaviours, such as colossal magnetoresistance, metal–insulator transition and high-temperature superconductivity. A variety of new and unusual electronic phases at the interfaces between complex oxides, in particular between two non-magnetic insulators LaAlO₃ and SrTiO₃, have stimulated the oxide community. However, no EPS has been observed in this system despite a theoretical prediction. Here, we report an EPS state at the LaAlO₃/SrTiO₃ interface, where the interface charges are separated into regions of a quasi-two-dimensional electron gas, a ferromagnetic phase, which persists above room temperature, and a (superconductor like) diamagnetic/paramagnetic phase below 60 K. The EPS is due to the selective occupancy (in the form of 2D-nanoscopic metallic droplets) of interface sub-bands of the nearly degenerate Ti orbital in the SrTiO₃. The observation of this EPS demonstrates the electronic and magnetic phenomena that can emerge at the interface between complex oxides mediated by the Ti orbital.

Grain boundaries in high-$T_c$ superconductors have attracted wide interest for their potential in a variety of applications and in fundamental studies of high-$T_c$ superconductivity. Two recent experimental results provide a basis for a better understanding of the grain boundary properties, the mechanisms of which, despite their widespread use, are not yet completely understood. First, it is now well established that the order parameter in many high-$T_c$ cuprates has a predominant $d_{x^2-y^2}$ symmetry. Second, microscopy studies have revealed that practical grain boundaries are comprised of facets having various orientations and typical dimensions of the order of 10- 100 nm. We analyze the combined effects of faceting and $d_{x^2-y^2}$ symmetry on the transport properties of high-$T_c$ grain boundaries. It is found that these effects can partially account for the experimentally observed reduction of the critical current density $J_c$ with increasing grain boundary angle $\alpha$. The angular dependence of $J_c$ for individual grain boundary facets may deviate considerably from the $J_c\left(\alpha \right)$ dependence observed in standard measurements that employ macroscopic grain boundaries. This also holds for the product of $J_c$ and the normal state resistivity ${\rho }_n$. The $J_c{\rho }_n$ product measured for standard grain boundary junctions is therefore not a direct measure of the intrinsic barrier properties. The faceting and $d_{x^2-y^2}$ symmetry lead to an inhomogeneous current distribution in the grain boundary which is different for the superconducting and the normal states.