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. First observed on ${\mathrm{SrTiO}}_3$ (001), it was then found in other compounds as diverse as anatase ${\mathrm{TiO}}_2$, ${\mathrm{KTaO}}_3$, ${\mathrm{BaTiO}}_3$, ZnO, and also on different surfaces of ${\mathrm{SrTiO}}_3$ (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. We present a simplified model aimed at describing the effect of its small spatial extension on measurements of its three-dimensional (3D) electronic structure by angular resolved photoemission spectroscopy. 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 $\left(k_z\right)$ 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, in light of our findings, we can get relevant information on the electronic wave functions and the nature of the confining potential.

Figure 1

Band structure and Fermi surfaces (amplitude squared of the electron wave functions) for a ${\mathrm{SrTiO}}_3$ slab taking a rectangular confinement potential of depth 0.35 eV and a five-unit-cell width. (a) Band structure along a $k_x$ path. Different energy levels indicate different amounts of doping (oxygen vacancies). Explicit values here are (b) $-0.28$ eV, (c) $-0.14$ eV, and (d) $-0.05$ eV. Zero of the energy axis is fixed to the lower conduction state for the bulk. (b)–(d) The corresponding Fermi surfaces in the $k_x/k_z$ plane.

Figure 2

Same as Fig. 1, but taking a wedge potential of depth 0.50 eV and, again, a five-unit-cell width. Explicit cutting energy levels are (b) $-0.28$ eV, (c) $-0.09$ eV, and (d) $-0.016$ eV.

Figure 3

(a) ARPES Fermi surface of $\mathrm{EuO}\left(1\mathrm{ML}\right)/{\mathrm{SrTiO}}_3$ along the $k_z$ axis, acquired by varying the photon energy between 30 and 97 eV in steps of 1 eV using linear vertical photon polarization. An inner potential of 16 eV was used in the calculation of the out-of-plane momentum [20]. The red discs and lines mark, respectively, the bulk $\mathrm{\Gamma }$ points and the border of the bulk Brillouin zone. (b) Zoom over the upper part of the Fermi surface in (a). All data in this figure were acquired at 15 K, and correspond (for better signal-to-background ratio) to the negative part of the second derivative of the raw data.

Figure 4

Band structures and Fermi surface (amplitude squared of the electron wave functions) for an anatase ${\mathrm{TiO}}_2$ slab with a rectangular confining potential of depth 0.35 eV and two different widths: (a) 5 ${\mathrm{TiO}}_2$ layers and (b) 10 ${\mathrm{TiO}}_2$ layers. (c) Corresponding Fermi surface for the case with 10 layers and $E_F=-0.10$ eV.

Figure 5

(a) Standard DFT bands for $\mathrm{SrTiO}{}_3$. The calculation method is DFT, using the code wien2k. Red bands below the Fermi level correspond to states with mostly oxygen character. Blue bands between the Fermi energy and $\approx 2.5$ eV, and green bands above 2.5 eV, correspond to the $t_{2g}$ and $e_g$ manifolds, respectively. (b) Wannier fitting to the $t_{2g}$ bands (black circles).

Figure 6

DFT band structures for anatase. Red bands below $-2$ eV: oxygen orbitals; blue bands between the Fermi level and $\approx 3$ eV: $t_{2g}$ orbitals; green bands above 3 eV: $e_g$ orbitals.

Figure 7

Effect of the damping parameter $\lambda$ [Eq. (1)] on the plot of Fig. 1 for (from left to right) $\lambda =5$ Å, 10 Å, and 50 Å.