Universal Structural Influence on the 2D Electron Gas at SrTiO3 Surfaces

Abstract The 2‐dimensional electron gas (2DEG) found at the surface of SrTiO3 and related interfaces has attracted significant attention as a promising basis for oxide electronics. In order to utilize its full potential, the response of this 2DEG to structural changes and surface modification must be understood in detail. Here, a study of the detailed electronic structure evolution of the 2DEG as a function of sample temperature and surface step density is presented. By comparing the experimental results with ab initio calculations, it is shown that local structure relaxations cause a metal‐insulator transition of the system around 135 K. This study presents a new and simple way of tuning the 2DEG via surface vicinality and identifies how the operation of prospective devices will respond to changes in temperature.

shows the sheet resistance of Nb:STO crystals, measured with the standard four-point probe setup. From 8 to ∼ 80 K, the sample shows a metallic behavior. Close to the bulk structural phase transition the conductivity suddenly changes and becomes more insulating. At around 110 K, the sample turns metallic again, up to room temperature. This behavior may be the reason for the jump in E F and its subsequent recovery seen in Figures 1a and 1b of the main text.

B. Temperature ramp
The slow temperature ramp (∼0.1 K/min) displayed in Figure S2 was recorded with two diodes, one in contact with the cryostat (blue) and the other in close proximity to the sample (orange). The arrow indicates the start of the temperature ramp. Although the temperature ramps were performed very slowly, the error in the temperature reading is expected to be around 7%. This is partly due to the distance between the sample diode and the sample surface and partly due to the thermal lag between the sample and this diode, similar to the thermal lag between the sample diode and the cryostat illustrated by the offset between the orange and blue curves.  FIG. S2. Temperature evolution during the experiment with the flat STO crystal. The blue curve indicates the temperature of the cryostat and the orange curve the temperature of the diode closest to the sample.  Figure S3 shows the band dispersion maps acquired for the flat STO samples with hν=85 eV, with C + polarization around the Γ 10 point at 70 and 115 K. The dashed lines indicate the band bottoms at 141 and 91 meV, respectively. These values were used in Figure 5(c) of the main text, along with the information from the other band maps in Figure 1 of the main text.

D. Compositional analysis
The samples were obtained nominally TiO 2terminated from the supplier, after etching, as described in the sample preparation and further treated in-situ with annealing in 100 mbar of O 2 at 550 • C. This treatment, along with others shown in [S1], is a standard method to clean SrTiO 3 and create the 2DEG. In order to understand the effect of these treatments, we performed XPS measurements on unetched and etched SrTiO 3 samples, both of them non-annealed. In Figure S4 we compare these measurements with XPS data from the etched+annealed sample shown in the main text. Using the integrated intensity from the Sr 3d and Ti 3p core levels, the calculated Sr/Ti ratios are 1.14, 0.76 and 1.41 for the unetched, etched, and etched+annealed (reported in the manuscript) samples. With respect to the data shown in the supplementary information of [S2], these values fall in the categories "as-received STO wafer (∼ 1.18)", "TiO2-term. (∼ 0.8)", and "SrO-term. (∼ 1.5)" respectively. We thus propose that the excess of Sr on the surface is due to the etching process in conjunction with the last annealing step.  Figure S5 shows the LEED pattern recorded for the 5 •and 10 • -miscut STO samples, where we observe a (1×1) surface. The vertical streaking around the main spots in due to the step potential and is typical for stepped surfaces. For the 10 • -miscut surface the smaller step distance results in a better separation of the diffraction spots, although a detailed analysis would require a dedicated LEED set-up [S3]. Of main importance to the current work is that the average step density is maintained after sample preparation. No signature of the steps, nor of a surface reconstruction is observed in the Fermi surface in the k x -k z plane, shown in Figures ?? and Figures ??. As in the main text, an inner potential V 0 =14.5 eV was used in the conversion from hν to k z , and the the straight lines around the zone centres correspond to the non-dispersive, 2D light bands with d xy character. A similar conclusion can be drawn by considering the Fermi surfaces in the k x -k y plane, shown in Figures S8 and S9 spanning several Brillouin Zones. This data which resembles the Fermi surface from the flat STO sample, but with a smaller band filling.   Figure S11 shows the calculated band structure highlighting the contributions from the 3d xy,yz,xz originating from Ti atoms localized in either the bulk or the surface layer. As mentioned in the main text, this band structure reveals an electron-like band with Ti-d xy character almost entirely originating from the surface layer, as well as the d yz d xz bands with mixed bulk-surface origin, in good agreement with previous works [S1].

III. AB INITIO CALCULATIONS
In our slab calculations, the dispersion of conduction band in the direction perpendicular to the slab surface is replaced by two kinds of states. The first are a series of bands centered at different energies and composed from 3d orbitals from all Ti atoms, with their relative FIG. S11. Band structure of the SrO-terminated slab highlighting the surface-and bulk-projected dxy, dxz and dyz orbital character of the bands. contributions depending on each respective atomic layer. These states are greatly affected by the slab thickness and form the bulk bands in the limit of infinite unit cells. The second kind are states strongly localized in the surface layer, which are only weakly subject finite-size effects. Our surface-derived d xy states show a relatively small breaking of degeneracy of around 3 meV, which we assign to the finite size of the slab [S4]. As pointed out in the main text, we have also performed DFT calculations for a TiO 2 -terminated slab, whose band structure is shown on Figure S12, highlighting the contributions from the bulk and surface Ti 3d xy orbitals. The calculation shows that, in contrast with the SrO-terminated slab presented in Figure 5 of the main text, the lowest-lying bands have bulk character, and therefore are in direct contrast with surface character of the 2DEG.
Due to its symmetry, our cubic slab does not allow atomic displacements in the xy plane. To check whether this has an effect on the results, we also calculated a 1×1×3 STO slab in the AFD phase (please note that the unit cell is doubled with respect to the bulk one),  FIG. S13. Relative total energies calculated for each respective degree of distortion (relative to each relaxed structure) and value of ∆t2g, obtained for SrO-terminated, both cubic and AFD structures. The insets show details of the relaxed AFD slab and its band structure, highlighting ∆t2g.
which allows for such displacements. In the top panel of Figure S13 we show the total energies calculated for each respective degree of distortion, with respect to the relaxed structure, for SrO-terminated slabs in the cubic and AFD phases. In particular for the AFD calculations were performed for different AFD angles, ranging from 2 to 7 • . In the bottom panel of Figure S13 we show the AFD structure and highlight the AFD angle at the surface layer. The atoms of the central layer are fixed (bulk), while the ones at the outer unit cells (surface) are allowed to move. In this structure, the atoms are allowed to move in the xy-plane as well as in the z-direction. The inset further shows the the resulting band structure calculated for the relaxed structure (AFD angle of 7 • ) is shown in the bottom inset of Figure S13, indicating the splitting ∆ t2g . Please note that the AFD structure is rotated 45 • in-plane with respect to the cubic structure, which is why we show the band structure plotted in the ΓM direction in this case. In these calculations, due to limitations of our supercell size, there is a tiny lift of degeneracy regarding the surface d xy band of 1.5 meV. The main panel of Figure S13 compares the value of ∆ t2g obtained for both structures for different degrees of distortion, where we observe a similar trend.