In-situ four-tip STM investigation of the transition from 2D to 3D charge transport in SrTiO3

The electrical properties of SrTiO3(100) single crystals were investigated in-situ at different stages of thermal reduction by means of a 4-tip STM. Using the tips of the STM as electrical probes, distance-dependent four-point measurements were performed at the surface of the crystal at room temperature after reduction by thermal treatment. For annealing temperatures T ≤ 700 °C, charge transport is confined to a surface region <3 μm below the surface. For reduction at T ≥ 900 °C a transition from a conducting 2D sheet with insulating bulk to a system with dominant 3D bulk conductivity is found. At an intermediate reduction temperature of T = 800 °C, a regime with mixed 2D/3D contributions is observed in the distance-dependent resistance measurements. Describing the depth-dependent conductivity with an analytical N-layer model, this regime of mixed 2D/3D conductivity is evaluated quantitatively under the assumption of an exponentially decaying conductivity profile, correlated with the previously observed depth-dependent dislocation density in the sample. A non-monotonous temperature dependence of the 3D conductivity in the respective conducting layer is found and possible underlying mechanisms are discussed, particularly with regard to non-intrinsic material properties depending on details of the sample preparation.

In case of an infinitesimally thin conducting sheet, current injected from tips spreads cylindrically from the current-injecting electrodes and charge transport only takes place inplane. The corresponding four-point resistance can be calculated according to Eq. 1 of the main text.
How much the measured distance-dependent four-point resistance deviates from the limiting case of 2D behavior depends on the probe distance in relation to the thickness of the conducting layer. In case of a small sample thickness compared to the probe distances and , the current propagation is practically in-plane on the scale of the distance of the voltage-probing tips and thus, the four-point resistance of the thin finite layer follows the 2D resistance behavior. The corresponding observed 2D conductance of a sample with constant 3D specific conductivity σ 3 is related to the 3D specific conductivity via 2D = 3D .
In order to estimate for which sheet thickness a deviation from the 2D behavior occurs (for a particular measurement configuration), the four-point resistance of a conducting layer of finite thickness has to be calculated. We follow the procedure outlined in Just et al. 1 . For our calculations, we consider a system of two layers: one infinite sheet with thickness and constant 3D conductivity 3D on top of a half-infinite insulating bulk (Fig. S1). Solving Laplace's equation and obeying the boundary conditions given by the continuity of the electrical potential and the current density between the two layers, the four-point resistance on the surface is where 0 denotes the Bessel function of the first kind, while 1 and 2 are parameters containing the conductivity and the thickness of the infinite sheet.
The resulting four-point resistance of the two-layer system relative to the 2D resistance of an infinitely thin sheet (Eq. 1 of the main text) is shown in Fig. S2 for various ratios of sheet thickness to probe distance . As can be seen from the figure, the four-point resistance of the finite sheet is equal to the 2D resistance in Eq. 1 in the main text in the limit of large probe distances x/t ≫ 1. In the considered case of the non-equidistant probe-configuration ( < ), the probe distance / , at which deviations from the 2D limit can be detected for a layer of finite thickness, is found to depend on the probe distance . However, we can say that in the case of < , for sheets that are thinner than ≲ 0.7x, 3D transport within the finite layer cannot be distinguished from ideal 2D transport by means of four-point resistance measurements, i.e. 4 according to Eq. 1 is equal to 2 from Eq. 1 of the main text. With that in mind, we can say that whenever 2D behavior of the four-point resistance is found in our experiment, charge transport takes place within a depth of ≲ 0.7x below the surface.

Macroscopic four-probe measurements
In order to investigate the electrical properties of the sample during thermal reduction, macroscopic four-probe measurements were conducted. An as-received SrTiO3 single crystal (10 × 3 × 0.5 mm 3 ) was contacted with four electrodes that were fabricated by applying Pt paste. Subsequently, Pt wires (0.3 mm in diameter) were wrapped around the pasted electrodes.
The outer electrodes were connected to a voltage source and the inner electrodes were connected to sensitive electrometers. An AC voltage of 4 mV at a frequency of 171 Hz was used. This was done to prevent stoichiometry polarization within the sample that is known to occur when using DC voltage. The current through the outer electrodes and the potential at the inner electrodes were recorded using lock-in technique. The measurements were performed in situ under vacuum conditions ( < 10 -6 mbar). The sample was placed inside a quartz tube that was heated by an external tube furnace allowing for repeated measurements at different temperatures without changing the position of the electrodes. Here, we would like to note that due to the higher absolute pressure and the use of an external heater instead of a Si heater, as was used during the 4-tip STM measurements, the local oxygen partial pressure close to the sample was slightly higher but we still consider the experimental conditions as comparable. Figure S3. Resistivity of the SrTiO3 crystal during annealing at different temperatures and subsequent cooling to room temperature.
In analogy to the 4-tip STM measurements the sample was subsequently heated to different temperatures of 400°C, 600°C, 700°C, 800°, 900°C and 1000°C. The heating rate was approx. 4 °C/min. After the respective temperature was reached, it was held for 1 h before the sample was cooled down to room temperature again. Fig. S3 displays the resistivity of the sample measured during the treatment. Here c denotes the corrected time meaning that breaks at room temperature between the measurements are not shown. The beige-coloured areas denote the annealing periods at the respective temperatures. It can be seen that with increasing temperature, the resistivity of the sample is decreasing as function of the reduction. This is illustrated in Fig. S4 showing the resistivity values measured at room temperature after each reduction step at the respective annealing temperature A. A decrease of several orders of magnitude is observed. Additionally, regarding the cooling curves in Fig. S3, it can be seen that during cooling from 400°C and 600°C, the resistivity increased again while at all temperatures above 700°C the resistivity decreased during cooling. This indicates that an insulator-metal transition occurred during annealing at 700°C. Figure S4. Room temperature resistivity of the SrTiO3 crystal measured after each annealing step. The inset depicts the geometry of the setup.