Abstract
The quantum Hall effect is a macroscopic quantum phenomenon in a twodimensional electron system. The twodimensional electron system in SrTiO_{3} 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 Ladoped SrTiO_{3}twodimensional electron system, fabricated by metal organic molecularbeam epitaxy. The quantized Hall plateaus are found to be solely stemming from the low Landau levels with even integerfilling 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 delectron systems.
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Introduction
Conventional semiconductors such as Si, GaAs and ZnO are the main workhorses in the studies of integer and fractional quantum Hall effects (QHEs)^{1,2,3,4}. The mobile carriers in these materials are located in bands composed mainly of s and p orbitals. In contrast, the conduction band of perovskite transition metal oxides such as SrTiO_{3} (STO) is composed of 3d t_{2g} orbitals with a strong directional anisotropy^{5}. When confined into a twodimensional (2D) environment, these states can show very interesting properties^{6,7,8}. At sufficiently high carrier densities, the t_{2g} conduction band is quantized into a ladder of light and heavy subbands, whereas at low carrier densities resulting subbands are dominated by heavy orbitals, d_{yz}/d_{zx}^{6}. In the latter case, electron–phonon effects combined with the many body interactions could further modify the dispersion of subbands thereby leading to formation of unusual electron liquid states^{9}. Consequently, the STObased twodimensional electron system (2DES) can exhibit a variety of unconventional quantum effects^{10}. Moreover, since STO is a widely used substrate for epitaxial growth of versatile materials with exotic properties such as highT_{c} superconductivity, ferroelectricity, ferromagnetism and topological phases^{11,12}, one can potentially incorporate a STObased 2DES with high mobility into such systems to realize novel quantum effects.
Owing to the recent progress of thinfilm growth technique, the electron mobility of threedimensional carriers of STO has reached 53,000 cm^{2} V^{−1} s^{−1} in single crystalline films^{13}, which is larger than 22,000 cm^{2} V^{−1} s^{−1} of bulk single crystals^{14}. However, preserving a metallic state with reasonably high mobility has proven to be a challenge when the carriers are confined in two dimensions. While there is a large number of studies on STObased 2DES including LaAlO_{3}/STO (LAO/STO) interface and δdoped STO^{7,8,15,16,17,18,19,20}, the realization of the QHE in this class of systems has proven to be elusive and, thus, yet to be demonstrated. In particular, the low mobility of the doped carriers and their high concentration have hindered the successful demonstration of the QHE in STO. Any attempt to reduce the carrier density below 3 × 10^{13} cm^{−2} has turned out to result in a nonmetallic state^{15}. Although with the recent advancement in the growth of LAO/STO interfaces one can now realize 2DES’s with a relatively low carrier density (in the range of 10^{12} cm^{−2}) with maintaining a high mobility (nearly 10,000 cm^{2} V^{−1} s^{−1}), it is still practically impossible to reach Landau levels with filling factors ν≤10 (refs 17, 18, 19, 20). The realization of QHE at low enough filling factors in an easily accessible magnetic field (∼10 T) imposes the restriction on the carrier density, which favourably should be below 1 × 10^{12} cm^{−2}. At higher carrier densities, it is therefore practically only possible to observe the Shubnikovde Haas (SdH) effect, which is of course a different quantum phenomena from the QHE.
Here we employ molecularbeam epitaxy (MBE) at a very high temperature (1,200 °C) with metal organic (MO) precursors (MOMBE)^{21} to grow STO heterostructures confining the 2DES and reach electron mobility exceeding 20,000 cm^{2} V^{−1} s^{−1} at chargecarrier densities below 1 × 10^{12} cm^{−2}. With such a high mobilitylow carrier density heterostructure, we can successfully reach the quantum Hall regime in STO. The 2DES shows clear signatures of the QHE but there are peculiar features; the quantization occurs only at even integer states (ν=4, 6) and the ν=4 state disappears at low carrier concentration. To elucidate the origin for those, we performed first principles calculations and found that these features can be modelled if the spin susceptibility is small compared with the Landau level broadening and the crossover of hybridized two d orbitals made of d_{yz} and d_{zx} are taken into account.
Results
Transport properties of δdoped SrTiO_{3}
The δdoped STO, studied here, is composed of a 100nmthick bottom STO buffer layer, a 10nmthick STO doped with 3 × 10^{19} cm^{−3} La, and a 100nmthick top STO capping layer epitaxially grown on a (001) STO singlecrystal substrate. The device is sketched in Fig. 1a and its corresponding optical microscope image is shown in Fig. 1b. The sample is scratched by a needle to pattern a van der Pauw type device with size 500 × 500 μm^{2}. At the four edges of the scratched square, aluminium wire is ultrasonically bonded to make an Ohmic contact with the 2D layer. The longitudinal resistance R_{xx} and Hall resistance R_{xy} showing below are all raw data of one configuration in van der Pauw geometry without calculating the average with the orthogonal configuration value of R_{xx} and R_{xy}. The sample is fixed with silver epoxy on a chip carrier with a metallic surface. Due to a large dielectric constant of STO at low temperature^{22}, the chip carrier surface acts as a global back gate for STObased 2DES and thus enables in situ tuning of carrier density. First, the asgrown samples, that is, without applying the backgate voltage (V_{G}), are characterized at 2 K. Figure 1b shows the relation between the ‘2D’ mobility and the carrier density at 2 K for various δdoped STO samples and STObased heterostructures that indicate pure electron conduction. The data are picked up from previous reports showing clear evidence of 2D conduction with low carrier density such as SdH oscillations in tilted magnetic field^{15,16,17,18,19,20}. The quantum transport measurements are performed at dilution refrigerator temperatures and in magnetic fields up to 14 T employing a lowfrequency (7–9 Hz) lockin technique with a low excitation current of 100 nA to suppress heating.
The best demonstration of the QHE is achieved using a sample which becomes insulating below 1 K on floating the backgate electrode. The application of a positive V_{G} accumulates the charge carriers in the δdoped region and the device becomes conducting for V_{G}>4.3 V. Figure 2a,b shows V_{G} dependence of carrier density and mobility of the device at 50 mK. As V_{G} increases, the carrier density, as determined from the Hall effect measurement, increases from 7.7 × 10^{11} to 1.2 × 10^{12} cm^{−2} (See Supplementary Note 1), roughly following the linear relationship. The slope of 2.8 × 10^{11} cm^{−2} V^{−1} depicted as broken line corresponds to the model of a planeparallel capacitor assuming a dielectric constant ɛ=20,000 for the gate insulator STO^{22} and a substrate thickness of 500 μm. The mobility shows a maximum value of 18,000 cm^{2} V^{−1} s^{−1} at a carrier density of 1.0 × 10^{12} cm^{−2} (see Fig. 2b) and thus is comparable with the value obtained in metallic asgrown devices shown in Fig. 1b. Figure 2c,d shows an example of the R_{xx} and R_{xy} at 50 mK for V_{G}=5.0 V (1.2 × 10^{12} cm^{−2}). First, one recognizes instantly the strong oscillations of R_{xx}, whose welldeveloped R_{xx} minima coincide with the Hall plateau structures of R_{xy}. Second, the plateaus at the negative field axis can be clearly assigned to Landau level filling factors ν=4 and 6 in R_{xy}=h/νe^{2}. At the positive field axis, R_{xy} is also well quantized for ν=4, whereas ν=6 deviates from the exact quantized value. Furthermore, Supplementary Note 2 shows that the R_{xx} minima at integerfilling factors show thermaly activated behaviour. Given the current stateoftheart for STO heterostructures, of which quality is doubtless lower than that of the wellestablished highmobility semiconductor heterostructures^{2}, the slight asymmetry in magnetotransport with respect to the magneticfield direction is not very surprising. Despite all imperfections that the current structure may suffer from, for example, disorder, chargecarrier inhomogeneity, the all metrics mentioned above strongly suggests the realization of QHE in the δdoped STO.
To inspect the QHE in more detail, we acquired the magnetoresistance traces of R_{xx} and R_{xy} at various V_{G}’s and display them in Fig. 3. In accordance with expectations, when V_{G} increases (that is, the carrier density increases), the positions of valleys and peaks in R_{xx} (Fig. 3a) systematically shift to higher magnetic fields. However, a distinct behaviour is found for filling factors ν=4 and ν=6. While the state at ν=6 with a welldeveloped plateau at or close to R_{xy}=h/6e^{2} (Fig. 3b) and R_{xx} minima is observed for all V_{G}’s, the quantum Hall state at ν=4 strikingly vanishes, that is, R_{xy} deviates from the quantized value and R_{xx} minimum disappears, when V_{G} is lowered. To visualize the quantization behaviour, Fig. 3c replots the data in the plane of σ_{xx} (B) and σ_{xy} (B) with B being the parameter for various V_{G}’s. Such representation demonstrates that the curves seem to converge towards (σ_{xy}, σ_{xx})=(±6e^{2}/h, 0), which seems to be a stable point for all V_{G}’s, while (±4e^{2}/h, 0) forms only at high V_{G} (high carrier density). It should be noted that such conversions can be observed only when R_{xy} plateau and R_{xx} minima are realized simultaneously. It is quite evident that the observed quantization is imperfect since the σ_{xy} deviates from its exact quantization value and σ_{xx} does not reach zero. Such behaviour might be caused by an additional conduction channel or some final bulk conductance remaining even in the regime of the QHE, which does not show localization behaviour in the magnetic field. We believe that improving the sample quality and gaining more knowledge on the origin for the disorder in the heterostructure (among the suspects is the inactive La dopants) will result eventually in the exact quantization of σ_{xy} (νe^{2}/h) concomitant with σ_{xx} reaching zero. Finally, we note that a conspicuous absence of odd filling factors at all V_{G}’s indicates either the spin degeneracy or the orbital degeneracy at high magnetic fields. Since our density functional theory (DFT) calculation presented below rules in (out) the former (latter), we expect that the Landau levels at high magnetic field are spin degenerate. This expectation may also be valid when the Zeeman spin splitting (gμ_{B}B, where g is the electron gfactor) is smaller than the other energy scales such as that arising from the disorder.
Electronic structures of δdoped SrTiO_{3}
To shed light on the mechanism of quantum oscillations, we have calculated the electronic structure of a δdoped STO thinfilm sandwiched between two sufficiently thick undoped STO slabs using realistic tightbinding supercell calculations, incorporating the bandbending potential in the δdoped region (see Methods). To be consistent with the design of our experimental heterostructure, the thickness of the quantum well (QW) is considered to be 10 nm, as schematically shown in Fig. 4e. We have assumed a square potential well (Fig. 4f) and varied its depth until the total amount of carrier density confined inside, and in the vicinity of QW, becomes n=1.0 × 10^{12} cm^{−2}. Under these conditions, the QW formed at the δdoped region confines two subbands. As shown in Fig. 4h,i by the false colour scale, both subbands are dominantly made of heavy d_{xz/yz} orbtials at the Fermi level E_{F}, whereas the d_{xy} orbital is the main contributor at the bottom of the lowest subband. It is to be noted that these subbands are distinct from the subbands previously observed at the surface of STO and KTaO_{3} (refs 6, 23). In those systems, the near surface bandbending potentials are much deeper but effectively confined within a much narrower region (that is, a few STO units). Consequently, d_{xy} orbitals contribute dominantly to the lowest subbands thereby making them highly dispersive, whereas the d_{xz/yz} can only contribute to the heavy subbands at much higher energies near the Fermi level. In the present system, on the other hand, due to the large width of the δdoped STO QW (10 nm), both the inplanar d_{xy} orbital and outofplaner d_{yz}/d_{xz} orbitals can be comparably confined within the QW region, thereby causing each subband to have complicated orbital characters. Moreover, the shallowness of the QW potential (which is due to the low carrier density) combined with the spin–orbit coupling can further complicate the orbital character of the subbands. In fact, the confined carrier density here is so low that the corresponding QW potential is expected not to be deeper than a few meV. This value is much smaller than the energy scale of the spin–orbit coupling between Ti t_{2g} states (∼36 meV). Therefore, the resulting subbands below Fermi level are subject to a strong orbital mixing, as shown in Fig. 4.
The resulting Fermi surface (FS) is composed of two spindegenerated pockets, which are coaxially centred at the Γ point (see Fig. 4i). The outer FS has a starshaped geometry and encloses an area of A_{OFS}=0.00138 Å^{−2}. The inner FS, on the other hand, has a less distorted shape with an enclosed area of A_{IFS}=0.00077 Å^{−2}, almost half of A_{OFS}. For both subbands denoted by EB_{1} for the energy band of outer FS and EB_{2} for that of inner FS in Fig. 4g, the corresponding carrier densities are mainly distributed inside the QW. However, they also have a fading tail reaching up to 15 nm beyond the δdoped region. Using the Onsager relation F=(Φ_{0}/2π^{2})A, where Φ_{0} is the flux quantum, the frequency of SdH oscillations (or the slope of fan diagram) corresponding to outer and inner FS’s are found to be 12.7 and 7.1 T, respectively.
The SdH oscillations are also analysed experimentally. To properly determine the positions of peaks and valleys in R_{xx} for the small amplitude oscillations at low fields, we take d^{2}R_{xx}/dB^{2} and plot it as a function of 1/B at V_{G}=4.7 V (see Fig. 4a). We then assign the integer indices to the d^{2}R_{xx}/dB^{2} peaks (valley positions of R_{xx}), denoted by the closed circles in Fig. 4b and the half integer indices to the d^{2}R_{xx}/dB^{2} valleys (R_{xx} peaks), indicated by the open circles in Fig. 4b. The frequency is deduced from the slope of indices versus 1/B. The change of the slope at 1/B=0.3 T^{−1} (B=3.3 T) from 12.9 T at high field to 6.4 T at lowfield region signals two transport regimes.
Discussion
These two values agree quite well with the respective band calculation data as shown above. While the fact that the ratio of slope change is close to two may be interpreted as the spin degeneracy lifting at high field, this change turns out to be due to the peculiar subband structure of the present STO 2DES (see Supplementary Note 3). Taking the spin degeneracy into account, the total carrier density extracted from these two SdH frequencies (6.4 and 12.9 T) is found to be 9.0 × 10^{11} cm^{−2} (3.0 × 10^{11} cm^{−2} for 6.4 T and 6.0 × 10^{11} cm^{−2} for 12.9 T); slightly lower than the carrier density estimated from the Hall effect, 1.0 × 10^{12} cm^{−2}. This deviation is much smaller than that of previous reports^{8,16} and is likely due to a minor contribution from an additional conduction channel as mentioned above, which can only affect the Hall effect (and not the SdH oscillations). This may accordingly explain why R_{xx} shows nonvanishing values in its minima. For the two peculiar oscillations, we expect that the outer FS contributes to the oscillations at highmagneticfield region and the inner FS dominates the observed oscillations at low fields. This is due to the fact that at low fields the amplitude of oscillations originating from the high index Landau levels of outer FS are much weaker than that of low index of Landau levels of inner FS. At 3.3 T, the inner FS is expected to reach its quantum limit, and thus can no longer contribute to the oscillatory part of R_{xx}. On the other hand, the outer FS, due to its larger area, is still far from its quantum limit. Therefore, the oscillations observed at higher fields, are merely from the outer FS.
We have also calculated the cyclotron effective mass for each FS ( and : the cyclotron effective mass for outer and inner FS) using the relation and obtained and . To compare these values with the experiment, we have deduced m* from the temperature dependence of the SdH oscillations after subtracting the nonoscillating background (ΔR_{xx}, Fig. 4c). To be consistent with our calculations, we consider the R_{xx} oscillations for n=1.0 × 10^{12} cm^{−2} corresponding to V_{G}=4.7 V. m* is then determined at each ΔR_{xx} extremum, as denoted by the dashed lines in Fig. 4c,d (additional information is provided in the Supplementary Note 4). Because of the weighted contribution of large and small FSs to R_{xx} oscillations, the electron mass at low fields is clearly smaller than that at high fields despite the uncertainty in estimated values of m* (indicated by the error bars in Fig. 4d). This tendency is in accordance with our calculations of and , predicting a smaller (larger) m* for the inner (outer) FS as indicated by horizontal lines.
Such a mixed subband contribution strongly affects the appearance of the QHE. Taking into account the relative positions of the subbands and their different effective masses, a schematic fan diagram for the spindegenerate Landau levels stemming from each subband is depicted in Fig. 3d; green and purple lines correspond to the Landau levels (N) of outer (EB_{1}) and inner (EB_{2}) subbands, respectively. This diagram can explain both the disappearance of ν=4 and the stability of ν=6. As pointed out in refs 24, 25, an even integerfilling factor is suppressed if two conditions are fulfilled: odd filling factors should occur simultaneously in each subband and two Landau levels, each from one of the subbands, should be degenerate at the chemical potential μ. Figure 3e visualizes a particular arrangement of Landau levels for which the quantum Hall state (QHS) ν=4 is suppressed. Here the chemical potential, denoted as μ_{1}, is located at the crossing between N=1 of EB_{1} and N=0 of EB_{2}. In this situation, EB_{1} is at filling factor ν=3 and EB_{2} is at ν=1, so that the total filling factor becomes 4. However, this does not lead to the QHE, since μ_{1} is not in a gap. Changing the chargecarrier density, and correspondingly relative population of the subbands, can shift the chemical potential into the gap between the Landau levels and thus lead to the formation of QHS at ν=4 as illustrated in Fig. 3f. In the same manner, we can explain the stability of QHS at ν=6. Considering the fact that EB_{1} and EB_{2} have different energies, the QHS at ν=6 can be suppressed if the filling factors of these subbands are ν=5 and ν=1, respectively. This, however, imposes a large imbalance on the chargecarrier densities of subbands, which cannot practically be realized in our STO structure. Thus, the QHS ν=6 is found to be stable in our experiment. In addition, one can find that slightly higher chemical potential than μ_{2} makes ν=4 and 8 quantum Hall plateau more stable. However, much higher magnetic field will be needed to observe the ν=2 plateau in that case.
In conclusion, we have observed QHE in δdoped STO grown at high temperature by MOMBE. This is the first observation of the QHE in perovskite oxides. The Hall conductance is quantized at even integerfilling factors. The absence of odd integerfilling factors is proposed to be due to the small gfactor of electrons. Using sophisticated electronic structure calculations, the peculiar behaviour of the QHE is attributed to the strong orbital anisotropy of subbands in the QW formed at the δdoped STO region. For ν=4, the corresponding plateau disappears at certain low carrier densities due to a crossing of the two subbands. The realization of such a unique QHE system opens up a new route to explore the unknown aspects of quantum transport and their functionalities.
Methods
MBE growth
All films were grown by metal organic gas source molecularbeam epitaxy (MOMBE) at a high temperature^{13,16,21,26}. In this method, Sr and La flux were evaporated from a conventional effusion cell with a pure elemental source, where La atoms act as dopants by substituting the Sr sites. For the Ti source, Titanium tetra isopropoxide (TTIP) (99.9999 %) was kept around 100 °C for thermal evaporation from a MO container without any carrier gas. Sr flux was kept at a beam equivalent pressure (BEP) of 8 × 10^{−8} torr and TTIP was varied to optimize the TTIP/Sr ratio (see Supplementary Note 5). La flux was controlled by the temperature of the effusion cell calibrated by a quartz crystal microbalance thickness monitor as described in Supplementary Note 6. Although the sheet La concentration in all samples is set to 3 × 10^{13} cm^{−2}, the measured sheet carrier density at 2 K varies widely between 1 × 10^{12} and 3 × 10^{13} cm^{−2} as shown in Fig. 1b. Such a variation might be caused by several reasons. As shown in Supplementary Note 6, Supplementary Fig. 6, and Supplementary Fig. 7, one is the experimental uncertainty of actual La beam flux and activation ratio of dopant. In fact, the carrier density at room temperature is found to vary between 1.5 × 10^{13} and 4.5 × 10^{13} cm^{−2}. Another reason is a partial freezing of charge carriers while lowering the temperature. We found this freezing is more pronounced for samples with smaller carrier density at room temperature. This expands the variation of carrier density at 2 K towards smaller carrier density side. The reason of partial freezing is not clear but such a behaviour has been commonly observed in a number of previous studies on STObased 2DESs^{8,18}. Taking into account the fact that thick single crystalline films with comparable or smaller La concentration do not show such a behaviour^{14}, we presume that the chargecarrier freezing is related to the localization of carriers due to disorder effect pronounced by confinement. Distilled pure ozone as oxidizing agent was generated and supplied from MPOG104A1R, MEIDENSHA Co. to the chamber at a pressure of 5 × 10^{−7} torr. The films were grown at a substrate temperature of 1,200 °C, which can be achieved with a semiconductorlaser heating system^{27} and is much higher than that used in previous MOMBE^{13,16,21}. Despite such a hightemperature growth, the depth profile measurement of La density revealed that La diffusion is absent as shown in Supplementary Note 7. To fill the oxygen vacancies formed during the growth, the samples were annealed in the growth chamber at 600 °C in P_{ozone}=1 × 10^{−6} torr for 1 h after deposition. In contrast to previous reports^{21}, the growth window of stoichiometric films is much wider, across a BEP ratio of TTIP/Sr=25–140, due to higher growth temperature by laser heating as described in Supplementary Note 5. The lattice constant is constant at 3.905 Å, which is same with that of stoichiometric single crystals. The mobility at low temperatures exceeds 53,000 cm^{2} V^{−1} s^{−1} for the thick homogeneous Ladoped STO film, which is higher than the record value of bulk single crystal of STO and comparable to the record of electron doped STO film reported previously^{13}.
Electronic structure calculation
To calculate the interface band structure, we initially performed a DFT calculation using the Perdew–Burke–Ernzerhof exchangecorrelation functional, modified by Becke–Johnson potential as implemented in the WIEN2K program^{28}. Relativistic effects, including spin–orbit coupling, were fully included. The muffin–tin radius of each atom R_{MT} was chosen such that its product with the maximum modulus of reciprocal vectors K_{max} become R_{MT}K_{max}=7.0. The Brillouin zone was sampled by a 15 × 15 × 15 kmesh. The resulting DFT Hamiltonian was then downfolded using maximally localized Wannier functions^{29,30,31} to generate a 200 unit cell tight binding supercell stacking along [001] direction with additional onsite terms, accounting for the QW potential. The same method has been already applied and successfully reproduced the results of ARPES data of the 2DEG confined at the surface of STO^{6}. Assuming a 10nmthick interface, the depth of QW was varied until the total amount of carrier density confined at, and in the vicinity of, the interface became n=1.0 × 10^{12} cm^{−2}.
We emphasize that in our tightbinding supercell Hamiltonian, there is no adjustable parameter other than an onsite potential term, representing the bandbending potential in the QW. Even for this bending potential we consider the same width as that realized in our experiment. The only variable parameter in our calculation, as mentioned above, is the depth of the potential that is chosen such that it yields the same confined chargecarrier density as that observed in our experiment (1 × 10^{12} cm^{−2}).
Data availability
The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information.
Additional information
How to cite this article: Matsubara, Y. et al. Observation of the quantum Hall effect in δdoped SrTiO_{3}. Nat. Commun. 7:11631 doi: 10.1038/ncomms11631 (2016).
References
von Klitzing, K., Dorda, G. & Pepper, M. New Method for Highaccuracy determination of the finestructure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).
Sarma, S. D. & Pinczuk, A. Perspectives in Quantum Hall Effects: Novel Quantum Liquids in LowDimensional Semiconductor Structures Wiley (2008).
Ezawa, Z. F. Quantum Hall Effects: Field Theorectical Approach and Related Topics 2nd edn World Scientific Publishing Co. Pte. Ltd. (2008).
Tsukazaki, A. et al. Observation of the fractional quantum Hall effect in an oxide. Nat. Mater. 9, 889–893 (2010).
Mattheiss, L. F. Effect of the 110 K phase transition on the SrTiO3 conduction bands. Phys. Rev. B 6, 4740–4753 (1972).
King, P. D. C. et al. Quasiparticle dynamics and spinorbital texture of the SrTiO3 twodimensional electron gas. Nat. Commun. 5, 3414 (2014).
Ohtomo, A. & Hwang, H. Y. A highmobility electron gas at the LaAlO3/SrTiO3 heterointerface. Nature 427, 423–426 (2004).
Kozuka, Y. et al. Twodimensional normalstate quantum oscillations in a superconducting heterostructure. Nature 462, 487–490 (2009).
Wang, Z. et al. Tailoring the nature and strength of electronphonon interactions in the SrTiO3(001) twodimensional electron liquid. Preprint at http://arxiv.org/abs/1506.01191 (2015).
Li, L., Richter, C., Mannhart, J. & Ashoori, R. C. Coexistence of magnetic order and twodimensional superconductivity at LaAlO3/SrTiO3 interfaces. Nat. Phys. 7, 762–766 (2011).
Hwang, H. Y. et al. Emergent phenomena at oxide interfaces. Nat. Mater. 11, 103–113 (2012).
Chang, C.Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).
Cain, T. A., Kajdos, A. P. & Stemmer, S. Ladoped SrTiO3 films with large cryogenic thermoelectric power factors. Appl. Phys. Lett. 102, 182101 (2013).
Tufte, O. N. & Chapman, P. W. Electron mobility in semiconducting strontium titanate. Phys. Rev. 155, 796–802 (1967).
Kozuka, Y. et al. Enhancing the electron mobility via deltadoping in SrTiO3 . Appl. Phys. Lett. 97, 222115 (2010).
Jalan, B. & Stemmer, S. Twodimensional electron gas in δdoped SrTiO3 . Phys. Rev. B 82, 081103(R) (2010).
Xie, Y. et al. Quantum longitudinal and Hall transport at the LaAlO3/SrTiO3 interface at low electron densities. Solid State Commun. 197, 25–29 (2014).
Caviglia, A. D. et al. Twodimensional quantum oscillations of the conductance at LaAlO3/SrTiO3 interfaces. Phys. Rev. Lett. 105, 236802 (2010).
Chen, Y. Z. et al. Extreme mobility enhancement of twodimensional electron gases at oxide interfaces by chargetransferinduced modulation doping. Nat. Mater. 14, 801 (2015).
Chen, Y. Z. et al. A highmobility twodimensional electron gas at the spinel/perovskite interface of γAl2O3/SrTiO3 . Nat. Commun. 4, 1371 (2013).
Jalan, B., Moetakef, P. & Stemmer, S. Molecular beam epitaxy of SrTiO3 with a growth window. Appl. Phys. Lett. 95, 032906 (2009).
Müller, K. A. & Burkard, H. SrTiO3: An intrinsic quantum paraelectric below 4 K. Phys. Rev. B 19, 3593–3602 (1979).
King, P. D. C. et al. Subband structure of a twodimensional electron gas formed at the polar surface of the strong spinorbit perovskite KTaO3 . Phys. Rev. Lett. 108, 117602 (2012).
Ensslin, K. et al. Singleparticle subband spectroscopy in a parabolic quantum well via transport experiments. Phys. Rev. B 47, 1366–1378 (1993).
Guldner, Y. et al. Quantum Hall effect in In0.53Ga0.47AsInP heterojunctions with two populated electric subbands. Phys. Rev. B 33, 3990–3993 (1986).
Matsubara, Y., Takahashi, K. S., Tokura, Y. & Kawasaki, M. Singlecrystalline BaTiO3 films grown by gassource molecular beam epitaxy. Appl. Phys. Express 7, 125502 (2014).
Ohashi, S. et al. Compact laser molecular beam epitaxy system using laser heating of substrate for oxide film growth. Rev. Sci. Inst. 70, 178–183 (1999).
Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka, & Luitz, D. J. WIEN2K, An Augmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties Techn. Univ. Wien (2001).
Souza, I., Marzari, N. & Vanderbilt, D. Maximally localized Wannier functions for entangled energy bands. Phys. Rev. B 65, 035109 (2001).
Mostofi, A. A. et al. Wannier90: a tool for obtaining maximally localized Wannier functions. Comp. Phys. Commun. 178, 685–699 (2008).
Kuneš, J. et al. Wien2wannier: From linearized augmented plane waves to maximally localized Wannier functions. Comp. Phys. Commun. 181, 1888–1895 (2010).
Acknowledgements
We are grateful to H.Y. Hwang and N. Nagaosa for fruitful discussions. This work was partly supported by the ‘Funding Program for WorldLeading Innovative R&D on Science and Technology (FIRST)’ of the Japan Society for the Promotion of Science (JSPS) initiated by the Council for Science and Technology Policy, by JSPS Grantsin Aid for Scientific Research, No. 24226002, and by PRESTOJST ‘Innovative nanoelectronics through interdisciplinary collaboration among material, device and system layers’. Y.M. is supported by the RIKEN Junior Research Associate Program.
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Y.M. and K.S.T. grew and characterized the films. Y.M., Y.K. and J.F. performed the lowtemperature measurements. M.S.B. performed the electronic structure calculations. Y.M. and K.S.T. analysed the data. D.M., A.T., Y.T. and M.K. contributed to discussion of the results and guided the project. Y.M., K.S.T., M.S.B. and D.M. wrote the manuscript with contributions from all authors.
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Matsubara, Y., Takahashi, K., Bahramy, M. et al. Observation of the quantum Hall effect in δdoped SrTiO_{3}. Nat Commun 7, 11631 (2016). https://doi.org/10.1038/ncomms11631
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DOI: https://doi.org/10.1038/ncomms11631
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