Introduction

High-mobility two-dimensional carriers were first realized at semiconductor surfaces and interfaces, where they form the basis of the integer and fractional quantum Hall effects1,2,3,4, two-dimensional superconductivity5, and practical applications such as high-electron-mobility transistors6,7. The two-dimensional carriers originating from the surface Fermi arcs or the quantum confinement of bulk three-dimensional Dirac fermions are reported for the Dirac semimetal Cd3As28,9,10,11,12,13,14,15. The high-mobility two-dimensional carriers are also pursued in Weyl semimetals and magnetic Weyl semimetals. Examples of Weyl semimetals and magnetic Weyl semimetals are TaAs16,17,18,19, Na3Bi20,21, NbAs22,23, Co3Sn2S224,25, Co2MnGa26, and SrRuO327. Since these states in topological semimetals are predicted to be robust against perturbations10,14,28,29, they are of both scientific and technological interests, with potential for use in high-performance electronics, spintronics, and quantum computing. In particular, Weyl fermions in magnetic Weyl semimetals are thought to be more suitable for spintronic applications25,30,31. Since the distribution of Weyl nodes in magnetic Weyl semimetals is determined by the spin texture27,32, the high-mobility two-dimensional carriers originating from the Weyl nodes in magnetic Weyl semimetals are expected to be controllable by magnetization switching33 in addition to the electric field6,13. To detect these high-mobility two-dimensional carriers in transport measurements, high-quality epitaxial Weyl semimetal thin films, in which the contributions of surface states and/or quantum-confined bulk states are prominent, are needed. Conversely, the experimental detection of these Weyl states in transport has remained elusive due to the lack of high-quality epitaxial Weyl semimetal films. Revealing the quantum transport properties of high-mobility two-dimensional carriers in magnetic Weyl semimetals is urgently required for a deep understanding of magnetic Weyl semimetals and demonstrating the relevance of Weyl fermions to spintronic and electronic applications.

The recent observation of the Weyl fermions in the itinerant 4d ferromagnetic perovskite SrRuO327,32,34, which is widely used as an epitaxial conducting layer in oxide heterostructures35,36,37,38,39,40,41,42,43,44,45,46, points to this material being an appropriate platform to integrate two emerging fields: topology in condensed matter and oxide electronics. Since the quantum transport of Weyl fermions in SrRuO3 has been reported in high-quality epitaxial films27, SrRuO3 provides a promising opportunity to realize high-mobility two-dimensional carriers in magnetic Weyl semimetals. However, their dimensional character remains unsolved due to the lack of angle-dependent quantum oscillations. Since the amplitude of the quantum oscillation becomes small for angles close to the in-plane direction of the film, high-magnetic field transport measurements are indispensable to scrutinize the angular dependence of the quantum oscillations. Furthermore, high-magnetic field transport measurements will simultaneously provide information regarding the chiral anomaly in the quantum limit of Weyl fermions in SrRuO3.

In this article, we present a systematic study of the angle-dependent magnetotransport, including quantum oscillations, up to 52 T in ultrahigh-quality SrRuO3 thin films with thicknesses of 10, 20, 40, 60, and 63 nm grown by machine-learning-assisted molecular beam epitaxy (MBE)47,48. The quantum oscillations in resistivity [i.e., Shubnikov-de Haas (SdH) oscillations] for the 10-nm film show the light cyclotron mass of 0.25m0 (m0: free electron mass in a vacuum), high quantum mobility of 3.5 × 103 cm2/Vs, and two-dimensional angular dependence, confirming the existence of high-mobility two-dimensional carriers in SrRuO3. The quantum oscillations for the 63-nm film, whose thickness is larger than the Fermi wavelength of the high-mobility carriers (~22.4 nm), also show the two-dimensional angular dependence, suggesting that the high-mobility two-dimensional carriers possibly come from the surface Fermi arcs, not the quantum confinement effect. The thickness-dependent phase shift establishes the non-trivial character of the quantum oscillations mediated from the surface Fermi arcs: The phase shift of the quantum oscillations shows a linear thickness dependence. We also observed the saturation of the chiral-anomaly-induced negative MR in the quantum limit, which had never been observed in magnetic Weyl semimetals. These findings of the high-mobility two-dimensional carriers and chiral anomaly in the quantum limit in an epitaxial ferromagnetic oxide provide an intriguing platform for topological oxide electronics and will stimulate further investigation of exotic quantum transport phenomena in epitaxial heterostructures.

Results and discussion

High-quality SrRuO3 thin films

We grew high-quality epitaxial SrRuO3 films with thicknesses t = 10, 20, 40, 60, and 63 nm on (001) SrTiO3 substrates (Fig. 1a) in a custom-designed molecular beam epitaxy (MBE) (see Methods section “Sample preparation” for details). The growth parameters, such as the growth temperature, supplied Ru/Sr flux ratio, and oxidation strength, were optimized by Bayesian optimization, a machine-learning technique for parameter optimization47,49,50, with which we achieved a high residual resistivity ratio (RRR) of 19.2 and 81.0 with t = 10 and 63 nm, respectively. The high crystalline quality and large coherent volume of the epitaxial single-crystalline films were confirmed by θ–2θ x-ray diffraction (XRD) and atomic force microscopy (AFM) (Fig. 1b–d; see Methods section “Sample preparation” for details). Epitaxial growth of the high-quality single-crystalline SrRuO3 film with an abrupt substrate/film interface is seen in the electron energy loss spectroscopy-cross sectional scanning transmission microscopy (EELS-STEM) images (Fig. 1e).

Fig. 1: Sample characteristics and Hall bar device of SrRuO3.
figure 1

a Schematic of the sample and crystal structures of the SrRuO3 films on a SrTiO3 substrate105. In the schematic crystal image, yellow, blue, red, and purple spheres indicate strontium, oxygen, ruthenium, and titanium, respectively. b XRD θ-2θ scan of the SrRuO3 film with t = 63 nm. c AFM image of the SrRuO3 film with t = 63 nm. d Magnified image of c. e Cross-sectional high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) image of the SrRuO3 film with t = 63 nm along the [100] axis of the SrTiO3 substrate. EELS-STEM images (from left to right) for the Ru-M4,5- (red), Ti-L2,3- (green), Sr-M3- (blue), O-K-edge (orange), and a color overlay of the EELS-STEM images for Sr, Ru, and Ti. f Optical microscope image of a Hall bar device. The current flows along the x direction, and the longitudinal and Hall voltage were measured using the Vxx and Vxy terminals. The upper left inset shows the arrangement of the Hall bar device and the magnetic field B.

Full size image

Thickness-dependent magnetotransport

We performed magnetotransport measurements on the Hall bar devices of the SrRuO3 film (Fig. 1f) with t = 10 nm using the PPMS (see Methods section “Magnetotransport measurements” for details). In Fig. 2, the ρxx, MR (ρxx(B) − ρxx(0 T))/ρxx(0 T), and ρxy data for the SrRuO3 film with t = 63 nm reported in ref. 27. are also shown for comparison. The temperature T dependence of the longitudinal resistivity ρxx of the SrRuO3 films is shown in Fig. 2a. The ρxx of the films with t = 10 and 63 nm decreased with decreasing temperature, indicating that these films are metallic over the whole temperature range. The temperature dependence of the normalized resistivity for the SrRuO3 film with t = 10 nm is identical to that for the SrRuO3 film with t = 63 nm. This result indicates that bulk electronic conduction in electrical transport is dominant even in the 10-nm thick film (see the Sec. I of the supplementary information). The ρxx vs. T curves show clear kinks (arrows in Fig. 2a), at which the ferromagnetic transition occurs, and spin-dependent scattering is suppressed41. The magnetization measurement for the SrRuO3 film with t = 63 nm at T = 10 K shows a typical ferromagnetic hysteresis loop (left inset in Fig. 2a). As shown in the right inset in Fig. 2a, below 20 K, the SrRuO3 films showed a T2 scattering rate (ρ ∝ T2) that is expected for a Fermi liquid, in which electron-electron scattering dominates the transport and carriers are described as Landau quasiparticles27,41,51,52. Although the RRR decreases from 81.0 to 19.2 with decreasing t from 63 to 10 nm, the RRR of the 10-nm film is high enough to observe the intrinsic quantum transport of Weyl fermions in SrRuO327,53. The availability of such high-quality thin films allows obtaining insights into the dimensionality of the Weyl fermions in SrRuO3 via measurements of angle-dependent magnetotransport properties, as will be described in the “High-mobility two-dimensional carriers” section. The larger residual resistivity ρRes at 2 K for the SrRuO3 film with t = 10 nm suggests a disorder near the interface between SrRuO3 and the SrTiO3 substrate53. Figure 2b shows the MR ((ρxx(B) − ρxx(0T))/ρxx(0T)) for the SrRuO3 films with a magnetic field B applied in the out-of-plane [001] direction of the SrTiO3 substrate at 2 K. Importantly, the linear positive MR at 2 K showed no signature of saturation even up to 14 T, which is commonly seen in Weyl semimetals9,19,54,55,56,57 and is thought to stem from the linear energy dispersion of Weyl nodes58,59,60. In the Fermi liquid temperature range (T < 20 K), quantum lifetimes long enough to observe quantum oscillations were achieved, as evidenced by the observation of the SdH oscillations in both MR and ρxy (Fig. 2b, c). The SdH oscillations with the frequency F ~25 T for the thin film (t = 10 nm) is more prominent than that for the thick film (t = 63 nm), implying that the quantum oscillation has the surface origin. This is consistent with the fact that quantum oscillations with the frequency F ~25 T have only been observed in thin films27,51,53 and not in bulk samples61. As will be described in the “High-mobility two-dimensional carriers” section, the main component of the SdH oscillations for both films have the frequency F ~25 T, which was interpreted to be of the three-dimensional Weyl fermions with high mobility and light cyclotron mass in our previous study27. For clarifying the dimensionality of the SdH oscillations, we investigated the angular dependence of the magnetotransport properties in the SrRuO3 films. The angle-dependent magnetotransport measurements also provide experimental evidence of the chiral anomaly that Weyl fermions in SrRuO3 show.

Fig. 2: Thickness dependence of the transport properties.
figure 2

a Temperature dependence of longitudinal resistivity ρxx for the SrRuO3 films with t = 10 and 63 nm at zero magnetic field. The arrows indicate the ferromagnetic transition temperatures. The left inset shows the magnetization vs. magnetic field curve for the SrRuO3 film with t = 63 nm at 10 K with B applied in the out-of-plane [001] direction of the SrTiO3 substrate. The right inset shows the ρxx vs. T2 plot with the linear fitting (black dashed line). b Thickness t dependence of MR (ρxx(B)−ρxx(0 T))/ρxx(0 T) at 2 K with B applied in the out-of-plane [001] direction of the SrTiO3 substrate (θ = 90°). The inset shows the oscillation components obtained by subtracting a polynomial function up to the eighth order. c Thickness t dependence of transverse resistivity ρxy at 2 K with B applied in the out-of-plane [001] direction of the SrTiO3 substrate (θ = 90°). Here, the ρxx, MR, and ρxy data for the SrRuO3 film with t = 63 nm are the same data reported in ref. 27.

Full size image

Chiral anomaly in the quantum limit

We carried out the angle-dependent magnetotransport measurements for the SrRuO3 film with t = 10 nm (Fig. 3). Figure 3a shows the angular dependence of MR (ρxx(B) − ρxx(0 T))/ρxx(0 T) for the SrRuO3 film with t = 10 nm at 2 K measured by the PPMS (up to 14 T). Here, B is rotated from the out-of-plane [001] direction of the SrTiO3 substrate (θ = 90°) to the in-plane direction parallel to the current (θ = 0°). The rotation angle θ is defined in the inset of Fig. 3a. As has been already described in the “Thickness-dependent magnetotransport” section, the unsaturated linear positive MR is observed when B is applied perpendicular to the current I (θ = 90°); note that quantum oscillations are superimposed onto the data. With decreasing θ, the sign of the MR changes from positive to negative, and the negative MR ratio is enhanced. The negative MR at θ ≤ 30° becomes linear above 5 T (Fig. 3a). In ferromagnetic SrRuO3, the time-reversal-symmetry breaking lifts the spin degeneracy and leads to linear crossing of non-degenerate bands at many k points, resulting in a pair of Weyl nodes with opposite chiralities (L and R; Fig. 3c)27,32. In the presence of a magnetic field, the Landau quantization of a pair of Weyl nodes with opposite chiralities occurs (Fig. 3d, e). In Fig. 3d, e, BQL represents the magnetic field at which the quantum limit is reached and all the Weyl fermions occupy the zeroth Landau levels. When B < BQL (Fig. 3d), nonorthogonal electric and magnetic fields (E ∙ B ≠ 0) lead to the chiral charge transfer between the two Landau levels with opposite chiralities62,63. In the weak-B limit (B < BQL; Fig. 3d), calculations based on the semiclassical Boltzmann theory predict that time-reversal-symmetry-breaking Weyl semimetals (magnetic Weyl semimetals) show a negative MR that is linear in the projection component of B in the direction of I54,63,64, in contrast to the quadratic dependence expected for space-inversion symmetry breaking Weyl semimetals19,21,65. Thus, the linear negative MR observed for θ ≤ 30° and 5 ≤ B ≤ 14 T when rotating B from orthogonal to parallel to I (Fig. 3a) is consistent with a chiral anomaly in magnetic Weyl semimetals27. In our previous study27, similar chiral-anomaly-induced negative MR has been observed for the SrRuO3 film with t = 63 nm.

Fig. 3: Angle-dependent magnetotransport for the SrRuO3 film with t = 10 nm and Landau quantization of a pair of Weyl nodes.
figure 3

a Angular dependence of MR (ρxx(B)−ρxx(0 T))/ρxx(0 T) for the SrRuO3 film with t = 10 nm at 2 K, measured by the PPMS up to 14 T. b Angular dependence of MR (ρxx(B)−ρxx(0 T))/ρxx(0 T) for the SrRuO3 film with t = 10 nm at 0.7 K, measured by the mid-pulse magnet up to 52 T. c Schematic diagram of a pair of Weyl nodes with opposite chiralities (L and R) with B = 0. d, e Schematic diagram of Landau levels of a pair of Weyl nodes with B < BQL and B > BQL, respectively, where BQL is the magnetic field at which the quantum limit is reached. In b, beige and green areas indicate B < BQL and B > BQL range, respectively.

Full size image

To verify the negative MR stemming from the chiral anomaly, we carried out high-field angle-dependent magnetotransport measurements using the mid-pulse magnet up to 52 T. This is of importance because the negative MR at low field in Weyl semimetals can be also brought about by experimental artifacts and impurities65,66,67. Therefore, observing the behavior of the negative MR in the quantum limit (Fig. 3e), where the chiral-anomaly-induced negative MR is predicted to saturate21,65,68, is necessary. Figure 3b shows the angular dependence of the high-field magnetotransport for t = 10 nm at 0.7 K with B rotated from the out-of-plane [001] direction of the SrTiO3 substrate to the in-plane direction parallel to the current. As in the case of the PPMS measurements up to 14 T (Fig. 3a), the sign of the MR at high fields changes from positive to negative with decreasing θ (Fig. 3b). Importantly, the negative MR with θ = 5.3° saturates above 30 T (Fig. 3b), confirming that the negative MR originates from the chiral anomaly. In the quantum limit (Fig. 3e), the conductivity when B is parallel to I is expressed by

$$\sigma _{{{{\mathrm{xx}}}}} = N_{\mathrm{W}}\frac{{e^2v_{{{\mathrm{F}}}}}}{{4\pi \hbar l_{{{{B}}}}^2}}\tau _{{{{\mathrm{inter}}}}}\left( B \right),$$
(1)

where NW is the number of Weyl points in the Brillouin zone, e is the elementary charge, \(\hbar\) is the Dirac constant, \(v_{{{\mathrm{F}}}}\) is the Fermi velocity, \(l_{{{\mathrm{B}}}}\) = \(\sqrt {\left( {h/eB} \right)}\) is the magnetic length, and \(\tau _{{{{\mathrm{inter}}}}}\left( B \right)\) is the field-dependent inter-nodal scattering time62,68. In the quantum limit of Weyl semimetals, the scattering rate 1/\(\tau _{{{{\mathrm{inter}}}}}\left( B \right)\) is predicted to increase roughly in proportion to B under the assumption of short-range impurity scattering, and the scattering factor cancels the factor of \(l_{{{\mathrm{B}}}}^2\) in Eq. (1), resulting in the B-independent conductivity65,68,69. Indeed, the saturation of the chiral-anomaly-induced negative MR above the quantum limit (B > BQL) has been observed in typical Weyl semimetals Na3Bi21 and TaAs65 as an inherent property of Weyl fermions, although it has not been observed in magnetic Weyl semimetals. Nonetheless, our observation of the saturation of the negative MR when B is parallel to I is consistent with the theoretically predicted behavior for the chiral anomaly in magnetic Weyl semimetals.

The tilting of the Weyl nodes is an important feature that characterizes Weyl semimetals. The calculated band structures for a set of selected Weyl nodes in the vicinity of the Fermi level are shown in Fig. 4 (see Methods section “First-principles calculations” for details). According to our calculations, the tilt of the linear band crossings is small and preserves an elliptical Weyl cone with the point-like Fermi surface. As shown in previous theoretical studies64, tilting of the Weyl nodes can result in a one-dimensional chiral anomaly, which in turn shows linear negative MR. Therefore, our electronic structure calculations suggest that the observed linear negative MR below the quantum limit originates from the one-dimensional chiral anomaly as a distinct feature of time-reversal symmetry broken (e.g., magnetic) Weyl semimetals.

Fig. 4: Selected Weyl nodes (A-D) in the vicinity of the Fermi level in orthorhombic SrRuO3.
figure 4

a Position of the Weyl nodes in the irreducible part of the Brillouin zone. Vertical yellow, red, pink, and blue lines indicate the in-plane (Γ–X–S–Y plane) positions of the Weyl nodes for easy viewing. The Weyl nodes are given in fractional coordinates as A = (0, 0, 0.332), B = (0, 0.307, 0.357), C = (0, 0.464, 0.4), and D = (0.233, 0.306, 0.483). b–d Band structures for the ferromagnetic ground state with spin–orbit coupling as obtained from GGA + U calculations with U = 2.6 eV and J = 0.6 eV. In b–d, the intersection of the straight lines represents the Weyl nodes. EF stands for the Fermi level.

Full size image

In principle, the chiral anomaly induced negative MR should be accompanied by SdH oscillations originating from the bulk three-dimensional Weyl fermions, because the chiral anomaly stems from the Landau quantization of the bulk three-dimensional Weyl fermions. However, oscillations originating from the bulk three-dimensional Weyl fermions were not clear in the SrRuO3 samples studied here, as will be described in the “High-mobility two-dimensional carriers” section. Since the Landau levels could be broadened by impurity scattering, disorder in the crystals, if any, readily hampers the observation of clear quantum oscillations70,71. In fact, in Dirac semimetal Na3Bi and zero-gap semiconductor GdPtBi, chiral-anomaly-induced MR without quantum oscillations of the bulk three-dimensional Weyl fermions has been reported72. Accordingly, further improvement in the crystallinity of SrRuO3 is required to observe clear SdH oscillations of the bulk three-dimensional Weyl fermions, which will allow for detailed characterization of the Fermi surface. For instance, using a substrate having smaller lattice mismatch with SrRuO3, e.g. DyScO3 and Sr1−xBaxTiO3, appears to be a promising means since the SrRuO3 film on the DyScO3 substrate has fewer defects owing to the smaller lattice mismatch73.

High-mobility two-dimensional carriers

To study the dimensionality of the SdH oscillations, we investigated their angular dependence in the SrRuO3 film with t = 10 nm (Fig. 5; see Methods section “Fourier transformation analysis for SdH oscillations” for details). The SdH oscillations are observed from θ = 90° to 30°, but not for θ ranging from 20° to 0° (Fig. 5a). This behavior of weakened oscillations with decreasing θ down to the in-plane direction is a typical feature of two-dimensional carriers6,12,74. As shown in Fig. 5b, the peak frequency F of the FFT spectra gradually shifts from ~25 T to a high frequency with B rotated from θ = 90° to 30°. The frequency is well described by a 1/cos(90°−θ) dependence, as shown later (Fig. 6c), indicating that the field component perpendicular to the surface is relevant for the cyclotron orbit. This angular dependence is a hallmark of two-dimensional carriers6,10,12,74. As shown in Fig. 5d, e, we also measured the temperature dependence of the SdH oscillations and estimated the effective cyclotron mass m* according to the Lifshitz-Kosevich theory at each angle as follows:9,27,75,76

$${{\Delta }}\sigma _{xx} \propto \frac{T}{{{{{\mathrm{sinh}}}}\left( {\alpha T} \right)}},$$
(2)

where \(\alpha = \frac{{2\pi ^2k_{{{\mathrm{B}}}}m^ \ast }}{{\hbar e\overline B }}\), \(k_{{{\mathrm{B}}}}\) is the Boltzmann constant, and \(\bar B\) is defined as the average inverse field of the FFT interval. The estimated m* at θ = 90° is 0.25m0 (m0: the free electron mass in a vacuum) (Fig. 5c). We also determined the Dingle temperature TD from the Lifshitz-Kosevich theory (see the Sec. II of the supplementary information) and obtained the quantum mobility \(\mu _{{{\mathrm{q}}}} = e\hbar /\left( {2\pi k_{{{\mathrm{B}}}}m^ \ast T_{{{\mathrm{D}}}}} \right)\) = 3.5 × 103 cm2/Vs. These results confirm the existence of the high-mobility two-dimensional carriers with a light cyclotron mass in the SrRuO3 film with t = 10 nm. The F and m* values (F ~25 T and m* = 0.25m0) of the SrRuO3 film with t = 10 nm are, respectively, the same as and comparable to those of the main component of the SdH oscillations in the SrRuO3 film with t = 63 nm (F ~26 T and m* = 0.35m0) reported previously27. Notably, the \(\mu _{{{\mathrm{q}}}}\) value of the SrRuO3 film with t = 10 nm is about one third of that in the SrRuO3 film with t = 63 nm (\(\mu _{{{\mathrm{q}}}} =\) 9.6 × 103 cm2/Vs). The reduced \(\mu _{{{\mathrm{q}}}}\) value in the former film suggests that disorder near the interface between SrRuO3 and SrTiO3 substrate is the dominant scattering mechanism, which shortens the quantum lifetime of the high mobility two-dimensional carriers in the SrRuO3 film with t = 10 nm. This disorder may be the origin of the short-range scattering mechanism postulated in the theory of the saturation of the negative MR in the quantum limit65,68,69.

Fig. 5: Angular dependence of the SdH oscillations for the SrRuO3 film with t = 10 nm using the PPMS.
figure 5

a Angular dependence of the background-subtracted SdH oscillations at 2 K with B (5 T < B < 14 T) for the SrRuO3 film with t = 10 nm. b Angular dependence of the FFT spectra of SdH oscillations at 2 K for the film with t = 10 nm. The dashed line is a guide for the peak shift. c Angular dependence of the FFT frequency (red filled circles) and the cyclotron mass (blue open circles). d Temperature dependence of the Fourier transform spectra of SdH oscillations at θ = 90° for the SrRuO3 film with t = 10 nm. e Temperature dependence of the FFT amplitude from θ = 90–40° for mass estimations according to the Lifshitz-Kosevich theory.

Full size image
Fig. 6: Angular dependence of the SdH oscillations for the SrRuO3 film with t = 10 nm using the mid-pulse magnets.
figure 6

a Angular dependence of the background-subtracted SdH oscillations at 0.7 K with B (10 T < B < 52 T) for the SrRuO3 film with t = 10 nm, measured by the mid-pulse magnet. These spectra were smoothed (see the Sec. III of the supplementary information). The arrows indicate the quantum limit, which is the inverse value of the FFT frequency in c. b Angular dependence of the FFT spectra of SdH oscillations at 0.7 K for the film with t = 10 nm. The dashed line is a guide for the peak shift. c Angular dependence of the FFT frequency. Here, the data obtained by the long-pulse magnet are also shown. The dashed line is the fitting of a two-dimensional angular dependence (~1/cos(90°−θ)).

Full size image

As confirmed in Fig. 5a, the SdH oscillation amplitude up to 14 T (=0.071 T−1) becomes small at lower angles, and it is difficult to follow the frequency. To further scrutinize the quantum oscillations, especially at lower angles (θ ≤ 30°), we analyzed the SdH oscillations using the mid-pulse magnet up to 52 T in the SrRuO3 film with t = 10 nm (Fig. 6). Except for 5.3°, the SdH oscillations can be clearly seen at each angle. At θ = 90.8° and 80.8°, the SdH oscillations are distinct between the two 1/B ranges (Fig. 6a): the high 1/B range from 0.035 to 0.1 T−1 and the low 1/B range from 0.018 to 0.035 T−1. The SdH oscillation at θ = 90.8° in the high 1/B range has the frequency F ~25 T, which corresponds to the Weyl fermions with the high \(\mu _{{{\mathrm{q}}}}\) and light m* observed in Fig. 5. The other oscillation has a high frequency of F ~300 T and the cyclotron mass m* of 2.4m0 (see the Sec. IV of the supplementary information). These F and m* values are consistent with those of the trivial Ru 4d band that crosses EF, which were reported in an early de Haas–van Alphen measurement61 and an SdH measurement using a standard lock-in technique at 0.1 K up to 14 T27. Since our focus is not on the trivial Ru 4d orbit in the low 1/B range but on the Weyl fermions, the data in the low 1/B range was excluded from the data for θ = 90.8° and 80.8° for the FFT analysis. As in the case of the PPMS measurements, the SdH peak frequency gradually shifts to a high frequency with decreasing θ from 90.8° to 15.6° in Fig. 6b, c. In Fig. 6a, the quantum limit, which is the inverse value of the FFT frequency, is indicated by black arrows. Since the FFT frequencies at high angles (θ ≥ 61.6°) are below the highest applied magnetic field of 52 T, we can observe the quantum limit as evidenced by the disappearance of the SdH oscillations of the two-dimensional carriers in the high-B region above the SdH frequencies. The frequency obtained by the high-field measurements is also well fitted by 1/cos(90°−θ; Fig. 6c), as is the case in the PPMS measurements.

We also measured the SdH oscillations for the Weyl fermions in the SrRuO3 film with t = 10 nm at 0.7 K using a long-pulse magnet up to 40 T (Supplementary Fig. 6), which allows for a more accurate measurement of resistivity because measurement time is longer than that of the mid-pulse magnet up to 52 T. Since the SdH oscillations are clearly observed from low magnetic fields by improved measurement accuracy, the spectra of the Fourier transform have sharper peaks than those obtained by the PPMS and the mid-pulse magnet measurements, as shown in Supplementary Fig. 6b. All the results from PPMS, mid-, and long-pulse magnet measurements converge and are well fitted with an angular-dependence curve for two-dimensional carriers (Fig. 6c), strongly indicating the existence of the two-dimensional Weyl fermions in the SrRuO3 film with t = 10 nm. Altogether, the existence of high-mobility two-dimensional carriers, which had never been observed in ferromagnets, is established in the epitaxial ferromagnetic oxide SrRuO3.

Origin of the two-dimensional carriers: surface Fermi arcs

Possible origins of the high-mobility two-dimensional carriers in Weyl semimetals are the surface Fermi arcs (Fig. 7) and the quantum confinement of the three-dimensional Weyl fermions8,9,10,11,12,13,14,15,77. The SdH oscillations mediated by the surface Fermi arcs and the quantum confinement appear when the film is thinner than the mean free path and the Fermi wavelength of carriers, respectively6,12,77. The mean free path \(\hbar ^2k_{{{\mathrm{F}}}}/\left( {2\pi k_{{{\mathrm{B}}}}m^ \ast T_{{{\mathrm{D}}}}} \right)\) and the Fermi wavelength \(2\pi /k_{{{\mathrm{F}}}}\) obtained by the SdH oscillations of the high-mobility two-dimensional carriers for the SrRuO3 film with t = 10 nm in Fig. 5a are ~65.4 and ~22.4 nm, respectively. Here, \(k_{{{\mathrm{F}}}}\) is the Fermi wave number. Hence, the film thickness of 10 nm satisfies both conditions. In the SrRuO3 film with t = 63 nm, the mean free path and the Fermi wavelength of the main component of the SdH oscillations are ~179 and ~22.4 nm, respectively27. To determine the origin of the two-dimensional quantum transport, we also studied the angular dependence of the SdH oscillations in the SrRuO3 film with t = 63 nm, which is thinner than the mean free path but thicker than the Fermi wavelength. We note that, in Dirac semimetal Cd3As2, suppression of backscattering, which results in a transport lifetime 104 times longer than the quantum lifetime, has been reported78. Therefore, the mean free path estimated by quantum lifetime may also be underestimated in SrRuO3. Figure 8 shows the angular dependence of the SdH oscillations in the SrRuO3 film with t = 63 nm. As in the case of the SrRuO3 thin film with t = 10 nm, the peak frequency of the FFT spectra gradually shifts to a higher frequency with decreasing θ from 90° to 10° in Fig. 8b. The frequency also shows the two-dimensional 1/cos(90°−θ) dependence, with the exception of that at θ = 0° (Fig. 8c), indicating that the SdH oscillations at θ from 90° to 10° come from the two-dimensional high mobility carriers. This result suggests that the high mobility two-dimensional carriers possibly come from the surface Fermi arcs, not from the quantum confinement effect. In contrast, the SdH peak frequency of 64 T at θ = 0° is outside the above two-dimensional trend, indicating that these SdH oscillations are different from those of the high mobility two-dimensional carriers. Additionally, for θ ≥ 40°, small FFT peaks are also discernible, e.g., at ~44 T for θ = 90° (black solid arrows in Fig. 8b). These peaks and their weak angular dependencies possibly come from the orbit of the bulk three-dimensional Weyl fermions. Since the SdH oscillations from the surface Fermi arcs are connected via the Landau quantization of the bulk three-dimensional Weyl fermions77, further improvement in the crystallinity of SrRuO3 will allow for detailed characterization of the SdH of the bulk Weyl fermions and their relevance to the surface Fermi arcs.

Fig. 7: Schematic diagram of a pair of Weyl nodes with opposite chiralities (L and R) and surface Fermi arcs in SrRuO3.
figure 7

The pink curves represent surface Fermi arcs connecting the projections of the bulk Weyl nodes.

Full size image
Fig. 8: Angular dependence of the SdH oscillations for the SrRuO3 film with t = 63 nm.
figure 8

a Angular dependence of the background-subtracted SdH oscillations at 2 K with B (5 T < B < 14 T) for the SrRuO3 film with t = 63 nm. b Angular dependence of the FFT spectra of SdH oscillations at 2 K for the film with t = 63 nm. The dashed line is a guide for the peak shift of the two-dimensional Weyl fermions. Black solid arrows indicate the peak positions of the three-dimensional Weyl fermions. c Angular dependence of the FFT frequency. The dashed line is the fitting of a two-dimensional angular dependence (~1/cos(90°−θ)).

Full size image

The thickness-dependent phase shift of the SdH oscillations will provide essential evidence for the non-trivial nature of the SdH oscillations mediated by the surface Fermi arcs. According to the semiclassical analysis of the closed magnetic orbit, called Weyl orbit79,80,81,82, formed by connecting the surface Fermi arcs via the Weyl nodes in bulk10,77, the thickness t dependence of the position of the Nth conductivity maximum BN is expressed by

$$B_N^{ - 1} = ek_0^{ - 1}\left( {\frac{{(N + \gamma )\pi v_{{{\mathrm{F}}}}}}{{E_{{{\mathrm{F}}}}}} - \frac{t}{\hbar }} \right),$$
(3)

where \(k_0\), γ, \(v_{{{\mathrm{F}}}}\), and \(E_{{{\mathrm{F}}}}\) are the length of the surface Fermi arc in reciprocal space, the sum of a constant quantum offset and the Berry phase, the Fermi velocity, and the Fermi energy. We note that it is assumed that identical Fermi arcs are on the top and bottom surfaces in Eq. (3)83. Equation (3) describes that the inverse of the SdH oscillation frequency 1/F = \(B_N^{ - 1} - B_{N - 1}^{ - 1}\) and the thickness-dependent phase shift \(\beta (t)\) of the SdH oscillation are given by \(ek_0^{ - 1}\left( {\frac{{\pi v_{{{\mathrm{F}}}}}}{{E_{{{\mathrm{F}}}}}}} \right)\) and \(\frac{{E_{{{\mathrm{F}}}}t}}{{\pi v_{{{\mathrm{F}}}}\hbar }}\), respectively. Thus, the SdH oscillation originating from Weyl orbits can be expressed by

$$\Delta \sigma _{xx} \propto \cos \left[ {2\pi \left( {\frac{F}{B} + \gamma + \beta (t)} \right)} \right]$$
(4)

It is apparent from these equations that the phase shift \(\beta (t)\) of the SdH oscillations mediated by the surface Fermi arcs has a linear relation with t. To examine this non-trivial phase shift, we carried out the Landau level (LL) fan diagram analyses for the conductivity maximum of the SdH oscillations with various thicknesses. Figure 9 shows the background-subtracted SdH oscillations at 2 K with B (3.3 T < B < 14 T) applied in the out-of-plane [001] direction of the SrTiO3 substrate for the SrRuO3 films with t = 10, 20, 40, and 60 nm. The FFT frequencies F (= 27, 26, 25, and 34 T for t = 10, 20, 40, and 60 nm, respectively) are determined from the peak positions of the FFT spectra of the SdH oscillations from 5 to 14 T. We extracted the magnetic field \(B_N\) of the Nth conductivity maximum, and estimated the phase shift \(\gamma + \beta (t)\) from the LL fan diagram for the conductivity maximum with a fixed slope of F (Fig. 10). As expected from Eqs. (3) and (4), we can see the linear relation between the \(\beta (t)\) and t (Fig. 10e) as a distinguished non-trivial nature of the SdH oscillations mediated by the surface Fermi arcs. The linear thickness dependence of the shift of the SdH oscillations verifies that the high-mobility two-dimensional carriers come from the surface Fermi arcs (Fig. 7), not from the trivial origins such as quantum confinement effect, trivial 2D gas on the surface of the film, anisotropic bulk Fermi surface, etc.

Fig. 9: Background-subtracted SdH oscillations at 2 K with B (3.3 T < B < 14 T).
figure 9

The magnetic field was applied in the out-of-plane [001] direction of the SrTiO3 substrates for the SrRuO3 films with t = 10, 20, 40, and 60 nm. The black triangles show peak positions.

Full size image
Fig. 10: Thickness t dependence of the LL fan diagram of the conductivity maximum for the SrRuO3 films.
figure 10

a t = 10 nm, b 20 nm, c 40 nm, and d 60 nm. The fixed slopes of the linear fittings are obtained by the frequency F. All plotted data points were used in the linear fittings. \(\gamma + \beta (t)\) is the phase shift in Eq. (4). e Thickness t dependence of the \(\gamma + \beta (t)\). The red line represents the linear fitting. The error bars are defined as the standard deviation of the linear fittings in a–d.

Full size image

From the 1/F = \(ek_0^{ - 1}\left( {\frac{{\pi v_{{{\mathrm{F}}}}}}{{E_{{{\mathrm{F}}}}}}} \right)\) value and the slope of the phase shift \(\frac{{E_{{{\mathrm{F}}}}}}{{\pi v_{{{\mathrm{F}}}}\hbar }}\) of the SdH oscillation of the Weyl orbit, we can determine the \(\frac{{E_{{{\mathrm{F}}}}}}{{v_{{{\mathrm{F}}}}}}\) and \(k_0\) values. The estimated \(\frac{{E_{{{\mathrm{F}}}}}}{{v_{{{\mathrm{F}}}}}}\) and \(k_0\) values are 2.92 × 10−8 eVs/m and 2.8 nm−1, respectively. Since the length of the surface Fermi arc is not much smaller than the size of the pseudocubic Brillouin zone \(\frac{{2\pi }}{a}\) = 15.98 nm−1 of SrRuO3, our findings will stimulate a challenge for its observation by angle-resolved photoemission spectroscopy. In addition, we can determine the γ value from the extrapolation of the t dependence of the \(\gamma + \beta\)(t) to t = 0. The obtained γ value is −0.16.

Discussion

We have systematically studied quantum oscillations observed in high-quality SrRuO3 films and successfully detected high-mobility two-dimensional carriers with light cyclotron mass, which possibly comes from the surface Fermi arcs. We also observed the saturation of the negative MR in the quantum limit, which is strong evidence of the chiral anomaly. These are the first observations of two-dimensional high-mobility carriers and chiral-anomaly-induced negative MR in the quantum limit in magnetic Weyl semimetals and magnetic oxides. The interface with the SrTiO3 substrate and the surface of the SrRuO3 films might be subject to slight structural differences, such as octahedral distortion and rotation84,85,86,87. However, our previous band structure calculations showed that a slight distortion induced by the substrate does not significantly change the band structure of SrRuO3 and that Weyl points are robust with the distortion as long as the band inversion is present27. In addition, generally, magnetic anisotropy in magnetic Weyl semimetals should affect angle-dependent MR originating from the quantum transport of Weyl fermions since the distribution of Weyl nodes in magnetic Weyl semimetals is determined by the magnetic arrangement27. Especially in SrRuO3 films, the magnetic anisotropy can be controlled via epitaxial strain, and therefore, SrRuO3 will be a platform for exploring the relationship between magnetic anisotropy, quantum transport properties of Weyl fermions, and chiral-anomaly-induced negative MR. The knowledge gained through such study will be helpful for controlling those properties by magnetization switching in future.

Our findings provide a foundation for studying exotic quantum transport phenomena in magnetic Weyl semimetals embedded in oxide heterostructures. Perovskite oxides that can be epitaxially grown with SrRuO3 include those with various physical properties such as insulators (e.g. SrTiO3)41, metals (e.g. Nb:SrTiO3)41, ferroelectrics (e.g. BaTiO3)41, and superconductors (e.g. Sr2RuO4)88. Utilizing magnetic Weyl states in these epitaxial heterostructures may enable functionalities89,90 and thus widen the accessible oxide electronics devices. In addition, the emergence of the hitherto hidden two-dimensional Weyl states in a ferromagnet is an essential step toward realizing the intriguing quantum Hall effect, which should be controllable through magnetization switching.

Note added: we became aware of a recent preprint91, which was uploaded later than the preprint of this article (arXiv:2106.03292), that report similar 2D-like SdH oscillations with a frequency of ~30 T. From the nonmonotonic thickness dependence on the FFT amplitude, they also argued the SdH oscillation with a frequency of ~30 T is more likely to be surface origin.

Methods

Sample preparation

We grew high-quality epitaxial SrRuO3 films with thicknesses t = 10, 20, 40, 60, and 63 nm on (001) SrTiO3 substrates (Fig. 1a) in a custom-designed MBE setup equipped with multiple e-beam evaporators for Sr and Ru27,47. The SrRuO3 films were prepared at the Ru flux = 0.365 Ås−1, growth temperature = 781 °C, and O3-nozzle-to-substrate distance = 15 mm. These conditions are the same as those in ref. 27. The growth parameters, such as the growth temperature, supplied Ru/Sr flux ratio, and oxidation strength, were optimized by Bayesian optimization, a machine-learning technique for parameter optimization47,49,50, with which we achieved a high residual resistivity ratio (RRR) of 19.2 and 81.0 with t = 10 and 63 nm, respectively. We precisely controlled the elemental fluxes by monitoring the flux rates with an electron-impact-emission-spectroscopy sensor, and the stoichiometric Sr/Ru ratio of a SrRuO3 film was confirmed using energy dispersive x-ray spectroscopy92. Figure 1b shows the θ-2θ XRD patterns around the (002) pseudocubic diffractions of the SrRuO3 film with t = 63 nm. Laue fringes in the θ-2θ XRD pattern indicate high crystalline quality and a large coherent volume of the film. The out-of-plane lattice constant, estimated from the Nelson-Riley extrapolation method93, was 3.950 Å, ∼0.5% larger than the pseudocubic lattice constant of the bulk specimens (3.93 Å)41. This indicates that the SrRuO3 film is compressively strained due to the lattice constant of the SrTiO3 substrate (3.905 Å) being ∼0.6% smaller than the pseudocubic lattice constant of SrRuO3. The AFM images show the surface morphology composed of flat terraces and molecular steps with height of ∼0.4 nm, corresponding to a single unit cell thickness (Fig. 1c, d). The root-mean-square roughness is 0.17 nm, indicating that the SrRuO3 film has an atomically smooth surface. The SrRuO3 film with t = 63 nm has an orthorhombic structure with two-fold in-plane symmetry, as evidenced by the XRD ϕ scans in our previous study73. Epitaxial growth of the high-quality single-crystalline SrRuO3 film with an abrupt substrate/film interface is seen in EELS-STEM images (Fig. 1e). Further information about the MBE setup, preparation of the substrates, and sample properties are described elsewhere27,53,94,95,96.

Magnetotransport measurements

We fabricated the Hall bar structures (length l = 350 μm, width w = 200 μm; Fig. 1f) by photolithography and Ar ion milling. We deposited Ag electrodes on a SrRuO3 surface before making the Hall bar structure. Magnetotransport up to 14 T was measured in a DynaCool physical property measurement system (PPMS) sample chamber equipped with a rotating sample stage. High-field magnetotransport measurements up to 52 T and 40 T were performed using a non-destructive mid-pulse magnet with a pulse duration of 36 ms and long-pulse magnet with a pulse duration of 1.2 s, respectively, at the International MegaGauss Science Laboratory at the Institute for Solid State Physics, the University of Tokyo97.

First-principles calculations

Electronic structure calculations were performed within density functional theory by using generalized gradient approximation98 for the exchange correlation functional in the projector-augmented plane wave formalism99 as implemented in the Vienna Ab-initio Simulation Package100. The energy cutoff was set to 500 eV, the Brillouin zone was sampled by an 8×8×6 Monkhorst-Pack mesh101, and the convergence criterion for electronic density was put to 10–8 eV. Experimental crystal structure of orthorhombic SrRuO3 was adopted for all calculations (the Pbnm space group, a = 5.567 Å, b = 5.5304 Å, c = 7.8446 Å)41. The calculations were performed for the ferromagnetic state with spin-orbit coupling. The effect of electronic correlations in the Ru 4d shell was considered by using the rotationally invariant GGA + U scheme102 with U = 2.6 eV and J = 0.6 eV.

For numerical identification of the Weyl nodes, the electronic band structure was interpolated with the maximally localized Wannier functions by projecting the bands near the Fermi level onto the Ru d atomic orbitals, as implemented in the wannier90 package103. Band crossings in the reciprocal space were calculated by steepest-descent optimization of the gap function, \({{\Delta }} = \left( {E_{n + 1,{{{\boldsymbol{k}}}}} - E_{n,{{{\boldsymbol{k}}}}}} \right)^2\), on a uniform mesh for the Brillouin zone up to \(25 \times 25 \times 25\)104. The two bands are considered degenerate when the gap is below the threshold of 10−5 eV. The results of such electronic structure calculations and the full set of the Weyl points in the vicinity of the Fermi level were reported in our previous study27.

Fourier transformation analysis for SdH oscillations

We measured the longitudinal resistance Rxx and Hall resistance Rxy of the Hall bar patterned SrRuO3 films (width w = 200 μm, length l = 350 μm). The longitudinal resistivity ρxx = Rxxwt/l and Hall resistivity ρxy = Rxyt were obtained from the Rxx and Rxy values. From ρxx and ρxy, we obtained the longitudinal conductivity σxx = ρxx/(ρxx2 + ρxy2) of the conductivity tensor. We subtracted the background of the conductivity from the σxx data (see the Sec. VI of the supplementary information) using a polynomial function up to the eighth order and extracted the oscillation components. For the Fourier transformation, we interpolated the data in Figs. 5a, 6a, 8a, and 9, and Supplementary Fig. 6a to prepare equally spaced x-axis (1/B) points. Then, we multiplied the data with a Hanning window function to obtain the periodicity of the experimental data. Finally, we conducted a fast Fourier transform (FFT) on the data. The FFT frequencies F are determined from the peak positions of the FFT spectra obtained by the above procedure. The error bars of the FFT frequencies are defined as the full width at half maximum of the FFT peaks.

Data avairability

The data that support the findings of this study are available from the corresponding author upon reasonable request.