Quantum interference effects of out-of-plane confinement on two-dimensional electron systems in oxides

It was recently discovered that a conductive, metallic state is formed on the surface of some insulating oxides. Firstly observed on SrTiO$_3$(001), it was then found in other compounds as diverse as anatase TiO$_2$, KTaO$_3$, BaTiO$_3$, ZnO, and also on different surfaces of SrTiO$_3$ (or other oxides) with different symmetries. The spatial extension of the wave function of this electronic state is of only a few atomic layers. Experiments indicate its existence is related to the presence of oxygen vacancies induced at or near the surface of the oxide. In this article we present a simplified model aimed at describing the effect of its small spatial extension on measurements of its 3D electronic structure by angular resolved photoemission spectroscopy (ARPES). For the sake of clarity, we base our discussion on a simple tight binding scheme plus a confining potential that is assumed to be induced by the oxygen vacancies. Our model parameters are, nevertheless, obtained from density functional calculations. With this methodology we can explain from a very simple concept of selective interference the wobbling, i.e., the photoemission intensity modulation and/or apparent dispersion of the Fermi surface and spectra along the out-of-plane ($k_z$) direction, and the mixed 2D/3D characteristics observed in some experiments. We conclude that the critical model parameters for such an effect are the relative strength of the electronic hopping of each band and the height/width aspect ratio of the surface confining potential. By considering recent photoemission measurements under the light of our findings, we can get relevant information on the electronic wave functions and of the nature of the confining potential.