Interface orientation dependent carrier distribution in LaAlO$_{3}$/SrTiO$_{3}$ heterostructures

Carrier distribution in widely studied LaAlO$_{3}$/SrTiO$_{3}$ heterostructures depends upon the directionality of the interface. From the first-principles calculations, we find that, for the (001) interface, most carriers are localized at the interface and have the $d_{xy}$ character, which decays away from the interface on the scale of 1 nm. The $d_{yz} $and$ d_{zx}$ carrier concentration increases, peaks at about three unit cells (1.15 nm), and then decays very slowly away from the interface. For the (111) interface, we find that the total carrier distribution is similar to that of $d_{yz}$ and $ d_{zx}$ in the (001) heterostructure. It has a maximum that lies at about four-bilayers (0.89 nm) away from interface and then decays very slowly producing a long-ranged tail. This difference in the carrier distribution is controlled by the $d$-orbital splitting which depends on the interface orientation. For the (001) interface, the tetragonal distortion brings the in-plane $d_{xy}$ orbital to about 0.5eV below the Fermi energy, creating a deep localized state, whereas the $d_{yz}$ and $ d_{zx}$ orbitals are much shallower and thence less localized. For the (111) interface, the rhombohedral distortion splits the $d-$orbitals by about 0.1 eV so that they are all located in vicinity of the Fermi energy and therefore less localized at the interface.