Atomic-scale characterization of oxygen vacancy ordering in La$_{0.5}$Sr$_{0.5}$CoO$_{3-\delta \, }$thin films on SrTiO$_{3}$ via in-situ cooling experiments

La$_{1-x}$Sr$_{x}$CoO$_{3-\delta }$ thin films have been studied extensively due to their rich magnetic phase behavior and interesting interplay between strain, defects, and magnetism. At x $=$ 0, ferromagnetic order is observed under tensile strain (in sharp contrast to bulk), whereas at higher x strain can stabilize oxygen vacancy ordering, with profound effects on transport and magnetism. For La$_{0.5}$Sr$_{0.5}$CoO$_{3-\delta }$ films grown on SrTiO$_{3\, }$substrates, the phase transition of SrTiO$_{3}$ from cubic to tetragonal at around 105 K can further modify the magnetic/transport properties of the films due to the coupling of the out-of-phase tilt of TiO$_{6}$ octahedral with the CoO$_{6\, }$octahedral network. In this contribution, we utilize atomic-resolution imaging and spectroscopies in a scanning transmission electron microscope to study the coupling between the TiO$_{6}$ octahedral tilt in the SrTiO$_{3}$ substrate and the oxygen vacancy ordering in the La$_{0.5}$Sr$_{0.5}$CoO$_{3-\delta }$ thin films using in-situ cooling experiments. The structural analysis will be compared to films grown on LaAlO$_{3}$, and the formation of magnetic domains will be studied using angular resolved electron energy-loss spectroscopy.