Overall, these findings indicate that symmetry-modified SrNbO3 is a 3D correlated oxide Dirac semimetal entering the XQL, in which topology meets many-body physics, yielding fractional occupation of Landau levels. Thus, we think that SrNbO3 offers a promising platform for further exploration of exotic correlated quantum phases and behaviors that can provide innovative materials solutions for the next generation of quantum technologies.
MATERIALS AND METHODS
Charge accumulation at the interface from STEM-EELS analysis
Cross-sectional TEM specimens were prepared using low-energy ion milling at LN2 temperature after mechanical polishing. High-angle annular dark field (HAADF) STEM measurements were performed on a Nion UltraSTEM200 microscope operated at 200 kV. The microscope was equipped with a cold field-emission gun and a corrector of third- and fifth-order aberrations for sub-angstrom resolution. A convergence half-angle of 30 mrad was used, and the collection inner and outer half-angles for HAADF STEM were 65 and 240 mrad, respectively. A collection aperture of 5 mm was used for electron energy loss spectroscopy (EELS) measurement, and EELS spectrum imaging was performed at a speed of 30 frames per second.
Figure S2A shows a HAADF STEM image of a SrNbO3(d = 6.5 nm)/SrTiO3 sample viewed along the  zone axis. The HAADF STEM image confirms that the SrNbO3 thin film is epitaxially grown on the SrTiO3 substrate with a sharp interface. Note that the nonuniform background contrast in the HAADF STEM image can be attributed to the amorphized surface of the TEM specimen, which was formed by argon ion milling. Figure S2 (B and C) shows the integrated Ti-L2,3 and O-K edge EELS spectra across the interface. As shown in fig. S2B, the spectral features of the Ti-L2,3 edge broaden with the approach from the substrate to the interface. Note that the spectral change is most prominent at the first unit cell of the substrate, and the EELS spectrum fully returns to the spectrum of standard SrTiO3 from three unit cells below. This spectral change is attributed to the charge transfer from the SrNbO3 thin film.
Lattice symmetry and strain-induced octahedral rotation of SrNbO3
As shown in fig. S3 (A and B), we used x-ray Bragg rod diffraction L scans at four different H-K quadrants to check and determine the primary lattice symmetry of SrNbO3 as a function of film thickness. We present the fully strained 7.2-nm and fully relaxed 130-nm films as representative results (we also similarly measured 1.8-, 16-, and 20-nm SrNbO3 films). Both strained and relaxed SrNbO3 films displayed fourfold rotational symmetry. The 7.2-nm film showed a tetragonal-like lattice symmetry (c > a) as a result of the compressive strain, whereas the 130-nm film revealed a cubic-like lattice symmetry (c ≈ a).
OOR-induced perovskite lattice doubling produced a unique set of half-order superstructure Bragg peaks, which were used to determine the OOR pattern and quantify the rotation angles. For detailed OOR half-order peak investigations with very weak signals, we carried out synchrotron XRD measurements at room temperature at beamline 33-ID-D at the Advanced Photon Source at Argonne National Laboratory. Monochromatized x-rays with a wavelength of 0.61992 Å were used, and a Pilatus 100K photon-counting area detector was used to capture the weak half-order superstructure. To suppress the fluorescence signal of the SrTiO3 substrate, higher-energy x-rays (E = 20 keV, well above the Sr K absorption edge) were chosen. Scattering geometric corrections and background subtractions using a photon-counting area detector were conducted for all films. We surveyed all possible types of OOR half-order peaks to determine the rotation pattern with Glazer notation. The total absence of (odd/2, even/2, odd/2), (odd/2, odd/2, even/2), and (odd/2, even/2, even/2) types of peaks ruled out the existence of in-phase (+) rotation along either the a or c lattice axis or the existence of perovskite A-site cation off-symmetry point displacement (not shown in fig. S3). As shown in fig. S3C, the (H/2 K/2 L/2) (H = K) type peaks are also absent for all SrNbO3 films with different thicknesses and strain states, which suggests that there is no a− or b− type in-plane, out-of-phase rotation for any of the SrNbO3 films. As shown in fig. S3D, the (H/2 K/2 L/2) (H ≠ K) type peaks can be observed only in strained SrNbO3 films (including a partially strained 16-nm film).
We then attempted to quantify the (H/2 K/2 L/2) (H ≠ K) type half-order Bragg rod to estimate the rotation amplitude (γ) for each strained film, using the scattering structural factor calculation of the complete heterostructure with a confined overall scale factor [via the diffraction intensity at the (002) thin-film peak]. In fig. S3E, the simulated OOR half-order Bragg rods with different rotation γ angles are compared with the measured data. Figure S3F displays the thickness dependence of extracted c− octahedral rotation angle γ. More detailed information regarding the strain, symmetry, and octahedral rotation of SrNbO3 thin films with different thicknesses is summarized in table S1. Both γ angles for 7.2- and 16-nm strained films are close to 10°, which was used for the DFT calculations. The ultrathin 1.8-nm film exhibits a relatively reduced γ angle, probably because of the proximity effect of octahedral connectivity imposed by the underlying SrTiO3 substrate without any OOR (a0a0a0).
Band structure calculation
Because the Dirac point at the P point is closer to the Fermi level, it can act as a source of a nontrivial Berry phase in the presence of a magnetic field. On the other hand, the Dirac points at the N point appear at ~0.7 eV above the Fermi level. Although these Dirac points at the N point are energetically unfavorable, oxide heterostructures would offer an avenue for designing a Dirac metallic phase by tuning the Fermi level closer to these Dirac points by strain or chemical substitution.
The Fermi velocity and effective mass near the Dirac point were estimated to be 7.07 × 107 m/s and 0.026 me, respectively. The high Fermi velocity near the P point was also expected to give rise to a high carrier mobility in strained SrNbO3 thin films.
Strain-tunable Dirac metallic state in SrNbO3
As observed in our lattice symmetry measurements and octahedral rotation-induced half-order superstructure diffraction measurements, the substrate strain in the thin-film limit stabilized SrNbO3 into a tetragonal crystalline symmetry (space group: I4/mcm) in which the NbO6 octahedra were rotated only in the x–y plane along the z axis. As discussed in the main text, Dirac points appeared at the P point and at the N point of the Brillouin zone in this tetragonal crystalline environment. Figure S5 shows the band dispersions at three different levels of the octahedral rotation. The Dirac point at the P point is found to remain closer to the Fermi energy in all three cases. The three Dirac points at the N point come closer to the Fermi energy with increasing octahedral rotation, as shown in fig. S5 (D to F), enabling them to be available in the electronic transport. The octahedral rotation, therefore, offers a route to tune the Dirac points in a controlled manner that can be efficiently engineered in oxide heterostructures by using the substrate strain.
Electrical transport measurements in this work were conducted with three measurement systems: the 14 T Physical Property Measurement System (Quantum Design), an 18 T dilution refrigerator at the University of Pittsburgh, and a 30 T bitter magnet with a He3 cryostat at the National High Magnetic Field Laboratory (Tallahassee, USA). Results from different systems and different samples were reproducible and consistent. Aluminum wire-bonded contacts with a Van de Pauw configuration were used for measuring magnetotransport properties.
Strain-tunable multiband nature of SrNbO3 thin films
with the restriction of zero field resistivity
where ne(nh) and μe(μh) are the carrier density and mobility, respectively, for electron(hole) type charge carriers. Figure S6B shows the magnetic field dependence of ρxy(H) (black dots) with fitting curves (colored solid lines). The convex-shaped ρxy(H) for a 6.4-nm thickness film was well explained by using two electron carriers, and the concave-shaped ρxy(H) found in the 12.4-nm-thickness film was captured by electron and hole carriers.
Figure S8 (A and B) displays the low-temperature MR and Hall resistivity of different SrNbO3 thin films with almost the same thickness, ~6.4 nm. All samples (S1 to S4) show consistent oscillations and the convex-shaped Hall effect that can be explained by the presence of two electron carriers, as previously discussed. Note that the oscillations are seen only in the films that have convex-shaped Hall effects. This finding supports the idea that one of the electron carriers, with low carrier density and high mobility, contributes to the oscillations. The oscillations are more pronounced in the second derivative curves (−d2ρ/dH2), as shown in fig. S8C. The minima of resistance are consistent with the minima of the second derivative, as displayed in fig. S9A.
By assigning the minima in oscillations to an integer Landau level index (N), we plotted the Landau fan diagram as illustrated in fig. S9C. The 1/H versus N deviates significantly from a conventional linear dependence. This unusual behavior cannot be explained even considering the strong Zeeman splitting with an enormous Lande g-factor. For the unconventional behavior to be explained as the effect of Zeeman splitting, at least the low–magnetic field region would have to show linear behavior, and its extrapolated line should be passing through near N = 0. However, we could not find any field region in which the Landau fan diagram showed liner behavior, even under a low magnetic field. Furthermore, the slope of the Landau fan diagram in a low–magnetic field region is too steep to pass N = 0. The deviation from conventional linear dependence is observed even in the low–magnetic field H < 3 T, at which the Zeeman effects are negligible (Zeeman energy < 0.5 meV). The unusual periodicity of oscillations, however, can be understood if the fractional Landau levels are taken into account, as discussed in the main text.
Strong mass enhancement at quantum limit
Origin of transport properties
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