Observation of the quantum Hall effect in δ-doped SrTiO3

The quantum Hall effect is a macroscopic quantum phenomenon in a two-dimensional electron system. The two-dimensional electron system in SrTiO3 has sparked a great deal of interest, mainly because of the strong electron correlation effects expected from the 3d orbitals. Here we report the observation of the quantum Hall effect in a dilute La-doped SrTiO3-two-dimensional electron system, fabricated by metal organic molecular-beam epitaxy. The quantized Hall plateaus are found to be solely stemming from the low Landau levels with even integer-filling factors, ν=4 and 6 without any contribution from odd ν's. For ν=4, the corresponding plateau disappears on decreasing the carrier density. Such peculiar behaviours are proposed to be due to the crossing between the Landau levels originating from the two subbands composed of d orbitals with different effective masses. Our findings pave a way to explore unprecedented quantum phenomena in d-electron systems.

In Fig. 2c, 2d, 3a, and 3b, one can find an asymmetry in Rxx and Rxy with respect to the magnetic field direction which is seen in quantum transport measurements of many low dimensional systems, even in the extreme high mobility GaAs heterostructures [1]. Even a small inhomogeneity damps the amplitude of the oscillations. The fact, that the amplitude of SdH in our sample is different for positive and negative field polarity, means that the charge carrier density inhomogeneity plays a role. Another cause for the asymmetry might be the van der Pauw configuration in sub millimeter scale. For precise estimation, the carrier density in Fig. 2a is calculated from the slope of Rxy (B) between -2 T and 2 T after its symmetrization as shown in Supplementary Fig. 1.

Supplementary Note 2 | Activation behaviors of quantum Hall states
To determine the activation energies () of quantum Hall states, the temperature dependence of resistance is analyzed for VG = 4.7 V which is shown in the inset of Supplementary Fig. 2. Supplementary Figure 2 shows the Arrhenius plots of the resistance minima for = 6 (at B = 6.7 T) and = 4 (at B = 11.2 T) denoted by arrows in the inset. We determine the value of  according to the formula where kB is the Boltzmann constant. Except for lowest temperature data points at 50 mK, the data obey the activation formula. The large difference of activation energy for two minima indicates that = 6 state is almost quantized while the = 4 state suffers from a delocalization.

Supplementary Note 3 | Possibility of another interpretation for Landau level formation with spin split bands
The present study concludes that the presence of two Fermi surface is important for the The relative arrangement of these levels on the energy axis will depend on both the quantum well confinement potential and the relation between the cyclotron and the Zeeman energies. The test of such a potential scenario will require both the samples with a much improved quality, i.e. low disorder, and the application of higher magnetic fields.

Supplementary Note 5 | TTIP/Sr ratio dependence of the mobility at low temperatures
Crystal quality and mobility in electron doped STO strongly depends on the stoichiometry, that is, the Sr/Ti ratio [6]. The c-axis lattice constant of nonstoichiometric homoepitaxial STO is known to be expanded [7]. In the case of MOMBE growth, a wide growth window, where the c-axis lattice constant of the films is independent of the beam flux ratio of TTIP to Sr and is identical with that of bulk STO crystal (0.3905 nm), has been found in previous reports [8]. In this study, a TTIP/Sr ratio was optimized not only by the structure analysis but also by the transport properties for the -doped structures. In order to obtain high quality -doped structure, high temperature growth using laser heating system was employed [9]. Supplementary Figure 4 shows the out of plane lattice constants of the films grown at various temperatures as a function of TTIP/Sr ratio. AFM images are also given for the films grown at a certain TTIP/Sr ratio inside the growth window (denoted by arrows). As increasing the growth temperature, the growth window of TTIP/Sr ratio becomes wider. In AFM images, though step-and-terrace surface structures are observed for all films in the growth windows, the step edges become dramatically smoother at higher growth temperature. Note that all films grown at 1200°C within a very wide range of TTIP/Sr ratio (from 25 to 140) turned out to have identical lattice constant to that of the substrate. This flux ratio window is much wider than that reported previously for lower substrate temperatures [8].
Though all films grown at 1200°C are identical from the view point of XRD measurement and AFM surface topography measurement, we found a significant dependence of mobility at 2 K on the TTIP/Sr ratio for -doped structures. Supplementary Figure 5 shows the mobility of confined 2DES as a function of TTIP/Sr. Around TTIP/Sr=60, a sharp peak structure is seen. We consider that the really optimized condition, which could not be identified by XRD measurement, is the flux ratio at around 60, resulting in maximum of the mobility. With such optimized growth condition, the

Supplementary Note 6| La flux control
Since we need to control La concentration to very low level, La flux could not be directly monitored by a beam flux monitor (quartz crystal microbalance). Supplementary Supplementary Figure 7a shows the relation of carrier density at room temperature and that at 2 K. The former varies between 1.5  10 13 and 4.5  10 13 cm -2 . If we assume that all dopants are activated at room temperature, this variation should be attributed to the variation in actual La concentration. Then, this corresponds to the variation in La cell temperature of about 50°C around 1035°C referring the calibration curve in Supplementary Fig. 6. However, this uncertainty in temperature may be too large.
Therefore, we presume that there is uncontrollable variation both in La flux and in activation ratio of dopants at room temperature. Supplementary Figure 7b shows the temperature dependence of carrier density for a -doped STO sample and a thick Ladoped STO film. All the -doped STO samples show such a carrier freezing behavior. As shown in Supplementary Fig. 7a, there is a clear superlinear trend, indicating carrier freezing is more pronounced for the samples with small carrier density at room temperature.

Supplementary Note 7 | La density depth profile in -doped STO
To examine the possible diffusion of La ions in the -doped STO, the depth profile of La density was measured by a secondary ion mass spectroscopy (SIMS). The sample was prepared under the same growth conditions with a similar structure: a 120-nm-thick bottom STO buffer layer, a 12-nm-thick STO doped with 'nominal' 310 19 cm -3 La, and a 120-nm-thick top STO capping layer grown on STO single-crystal substrate. As shown in Supplementary Fig. 8, the SIMS measurement reveals that we can judge the appreciable La diffusion is absent, taking into account a spatial resolution of about 5 nm.