Abstract
Magnetic Weyl semimetals have novel transport phenomena related to pairs of Weyl nodes in the band structure. Although the existence of Weyl fermions is expected in various oxides, the evidence of Weyl fermions in oxide materials remains elusive. Here we show direct quantum transport evidence of Weyl fermions in an epitaxial 4d ferromagnetic oxide SrRuO_{3}. We employ machinelearningassisted molecular beam epitaxy to synthesize SrRuO_{3} films whose quality is sufficiently high to probe their intrinsic transport properties. Experimental observation of the five transport signatures of Weyl fermions—the linear positive magnetoresistance, chiralanomalyinduced negative magnetoresistance, π phase shift in a quantum oscillation, light cyclotron mass, and high quantum mobility of about 10,000 cm^{2}V^{−1}s^{−1}—combined with firstprinciples electronic structure calculations establishes SrRuO_{3} as a magnetic Weyl semimetal. We also clarify the disorder dependence of the transport of the Weyl fermions, which gives a clear guideline for accessing the topologically nontrivial transport phenomena.
Introduction
Weyl fermions in a magnetic material have novel transport phenomena related to pairs of Weyl nodes^{1,2,3,4,5}, and they are of both scientific and technological interest, with the potential for use in highperformance electronics, spintronics, and quantum computing. Although Weyl fermions have been predicted to exist in various oxides^{6,7,8}, evidence for their existence in oxide materials remains elusive^{9,10,11}. SrRuO_{3}, a 4d ferromagnetic metal often used as an epitaxial conducting layer in oxide heterostructures^{12,13,14,15}, provides a promising opportunity to seek the existence of Weyl fermions in a magnetic material. Stateoftheart oxide thin film growth technologies, augmented by machine learning techniques, may allow access to such topological matter. Here, we show direct quantum transport evidence of Weyl fermions in an epitaxial ferromagnetic oxide SrRuO_{3}: unsaturated linear positive magnetoresistance (MR)^{16,17,18,19,20}, chiralanomaly induced negative MR^{1,16,21}, π Berry phase accumulated along cyclotron orbits^{16,18,19,20}, light cyclotron masses^{16,17,18,19,20,22,23,24} and high quantum mobility of about 10,000 cm^{2} V^{−1} s^{−1} ^{16,17,22,23,24,25,26,27}. We employed machinelearningassisted molecular beam epitaxy (MBE)^{28} to synthesize SrRuO_{3} films whose quality is sufficiently high to probe their intrinsic quantum transport properties. We also clarified the disorder dependence of the transport of the Weyl fermions, and provided a brandnew diagram for the Weyl transport, which gives a clear guideline for accessing the topologically nontrivial transport phenomena. Our results establish SrRuO_{3} as a magnetic Weyl semimetal and topological oxide electronics as a new research field.
Weyl semimetals, which host Weyl fermions described by the Weyl Hamiltonian, have intriguing and fascinating transport phenomena based on the chiral anomaly and linear band dispersion with spinmomentum locking^{1,2,3,4,29}, such as chiralanomaly induced negative MR and high mobility^{16,17,21}. Compared with spaceinversionsymmetrybreaking Weyl semimetals^{30}, timereversalsymmetry (TRS)breaking ones are thought to be more suitable for spintronic applications^{3,31,32}. For example, since the distribution of Weyl nodes in magnets is determined by the spin texture^{1}, this distribution is expected to be controlled by the magnetization switching technique^{33,34}. Recent angleresolved photoemission spectroscopy (ARPES) studies have found experimental evidence for the electronic structure of magnetic Weyl semimetals Co_{3}Sn_{2}S_{2}^{2,3} and Co_{2}MnGa^{4}, such as the presence of bulk Weyl points with linear dispersions and surface Fermi arcs. Demonstrating the relevance of Weyl fermions in a magnetic material to spintronic and electronic applications requires information on quantum oscillations, which allows us to characterize transport properties of individual orbits in a magnetic Weyl semimetal. However, systematic and comprehensive measurements of quantum transport, including the quantum oscillations, have been hampered in magnetic Weyl semimetals because of the difficulty in achieving a quantum lifetime long enough to observe them in metallic systems. Since specimens in the form of epitaxial films are advantageous for future device applications of magnetic Weyl semimetals, it is urgently required to prepare singlecrystalline thin films^{35} of magnetic Weyl semimetals whose quality is sufficiently high to probe quantum transport properties.
Theoretically, the presence of Weyl fermions has been predicted for SrRuO_{3}, a 4d ferromagnetic material^{7}. SrRuO_{3} is widely used as an epitaxial conducting layer in oxide electronics and spintronics owing to the unique nature of ferromagnetic metal, compatibility with other perovskitestructured oxides, and chemical stability^{12,13,14,15,34}. Theoretical studies predicted that the electronic structure of SrRuO_{3} may include a large number of Weyl nodes caused by the TRS breaking and spin–orbit coupling (SOC) (Fig. 1a)^{7}, and suggested that the Berry phase from the Weyl nodes gives rise to an anomalous Hall effect (AHE) in it.^{7,10,36} However, a definitive conclusion on the presence of Weyl fermions near the Fermi energy (E_{F}) cannot be drawn from observations of the AHE alone^{7,9,10}, because the AHE reported so far for SrRuO_{3} can be well reproduced using a function composed of both intrinsic (Karplus–Luttinger (KL) mechanism) and extrinsic (sidejump scattering) terms^{37,38}.
In this study, we conducted comprehensive high field magnetotransport measurements (see Methods section “Magnetotransport measurements”) including quantum oscillations of resistivity (i.e., Shubnikovde Haas (SdH) oscillations), on an extraordinarily highquality SrRuO_{3} film (63nm thick) epitaxially grown on SrTiO_{3} (Fig. 1b, c and Supplementary Fig. 1a). Our firstprinciples electronic structure calculations predicted the presence of Weyl fermions within an energy range of −0.2 to 0.2 eV around the Fermi level in SrRuO_{3}. To probe the contribution of the Weyl fermions to the transport properties, it is necessary to identify the following five signatures from the magnetotransport data: (1) unsaturated linear positive MR^{16,17,18,19,20} (2) chiralanomaly induced negative MR^{1,16,21}, (3) π Berry phase accumulated along the cyclotron orbits^{16,18,19,20}, (4) light cyclotron mass^{16,18,19,20,22,23,24}, and (5) high quantum mobility^{16,17,22,23,24,25,26,27}. Although the light cyclotron mass, high mobility, and linear positive MR also exist in semiconductors with parabolic bands^{39,40,41}, we confirmed the existence of the Weyl fermions in SrRuO_{3} by observing all of the five signatures.
Results
Temperature dependence of resistivity
The residual resistivity ratio (RRR), which is defined as the ratio of the longitudinal resistivity ρ_{xx} at 300 K [ρ_{xx}(300 K)] and T\(\to\)0 K [ρ_{xx}(T\(\to\)0 K)] (T: temperature), is a good measure to gauge the purity of a metallic system, that is, the quality of singlecrystalline SrRuO_{3} thin films (see Methods section “Determination of the RRR in SrRuO_{3}”). In fact, high RRR values are indispensable for exploring intrinsic electronic states. More specifically, RRR values above 40 and 60 have enabled observations of sharp, dispersive quasiparticle peaks near the Fermi level by ARPES^{42} and quantum oscillations via electrical resistivity^{43}, respectively. To form SrRuO_{3} with quality exceeding current levels, we employed our recently developed machinelearningassisted MBE (see Methods section “Machinelearningassisted MBE”)^{28}.
The resistivity ρ_{xx} vs. T curve of the SrRuO_{3} thin film shows a clear kink at the Curie temperature (T_{C}) of ∼152 K (Fig. 1d)^{12}, while the magnetization measurement at T = 10 K shows a typical ferromagnetic hysteresis loop (Fig. 1d, right inset). With a residual resistivity ρ_{xx} (T\(\to\)0 K) of 2.23 μΩ cm and an RRR of 84.3, SrRuO_{3} thin films grown by machinelearningassisted MBE are superior to those prepared by any other method^{12,28,43,44}. Below approximately 20 K, the T^{2} scattering rate (\(\rho _{xx} \propto T^2\)) expected in a Fermi liquid is observed (Fig. 1d, left inset)^{12,42}, indicating that the intrinsic transport phenomenon is seen below this temperature (hereafter called T_{F}).
Temperature dependence of magnetoresistance and Hall resistivity
In this Fermi liquid region [T < T_{F} (20 K)], a semimetallic behavior is seen in the MR (ρ_{xx}(B) − ρ_{xx}(0 T))/ρ_{xx}(0 T) (Fig. 1e) and Hall resistivity ρ_{xy}(B) (Fig. 1f) curves with the magnetic field B applied in the outofplane [001] direction of the SrTiO_{3} substrate. As shown in Fig. 1e, ρ_{xx} above T_{F} shows the negative MR because of the suppression of magnetic scattering^{12,45,46}, and the MR changes its sign below T_{F}. Importantly, the positive MR at 2 K shows no signature of saturation even up to 14 T, which is typical of a semimetal^{16,47} and also commonly seen in Weyl semimetals^{1,16,17,18,19,20}. Especially in the case of Weyl semimetals, linear energy dispersion of Weyl nodes is considered to be one of the most plausible origins of unsaturated linear positive MR^{48,49} (see Methods section “Excluding other possible origins of the positive MR in SrRuO_{3}”). In addition, as shown in Fig. 1f, the ρ_{xy}(B) curves below T_{F} are nonlinear, indicating the coexistence of multiple types of carriers (electrons and holes). We note that, below T_{F}, the AHE, which stems from the extrinsic sidejump scattering and intrinsic KL mechanisms in SrRuO_{3},^{37,38} is well suppressed due to the small residual resistivity of the SrRuO_{3} film with the RRR of 84.3, and thus the ρ_{xy}(B) curves below T_{F} are dominated by the ordinary Hall effect (see Methods section “Temperature dependence of the AHE in the SrRuO_{3} film with the RRR = 84.3”). Below 10 K, where the AHE is negligible, both the ρ_{xy}(B) values and the slopes of ρ_{xy}(B) change their signs in the highB region, signaling the possibility of the coexistence of highmobility electrons with lowmobility holes^{16}. Importantly, both the unsaturated linear positive MR and nonlinear Hall resistivity features start to appear simultaneously when the measurement temperature is decreased to the Fermi liquid range [T < T_{F} (20 K)]. This indicates that the unsaturated linear positive MR stems from the electron and holelike Weyl fermions.
Angle dependence of magnetoresistance
Next, we observed the chiralanomaly induced negative MR, which is an important signature of Weyl fermions^{1,16,21,50}. To clarify the anisotropic character of the chiralanomaly induced negative MR thoroughly and systematically, we measured ρ_{xx}(B) at B angles α, β, and γ in the xy, yz, and zx planes, respectively (Fig. 2a–c). The rotation angles α, β, and γ are defined in the insets of Fig. 2a–c. When B is applied perpendicular to the current I (B⊥I, α = 90° or β = 090° or γ = 90°), the unsaturated linear positive MR is observed. The unsaturated linear positive MR seen here is expected owing to the presence of the Weyl fermions, and those states are supposedly anisotropic because of the \(\sim\)0.5% compressive strain of SrRuO_{3} induced by the SrTiO_{3} substrates^{28}. This anisotropy is confirmed by varying β (Fig. 2b). In contrast, when B is rotated parallel to the current (B//I, α = 0° or γ = 0°), the MR turns negative and becomes linear above 8 T (Fig. 2a, c and Supplementary Fig. 4c). Theoretical calculations based on the semiclassical Boltzmann kinetic equation predict that TRSbreaking Weyl semimetals show a negative MR that is linear in B^{1,50}, in comparison with the quadratic dependence expected for spaceinversionsymmetrybreaking Weyl semimetals^{16,21}. Thus, the observed linear increase of the negative MR is consistent with a chiral anomaly in magnetic Weyl semimetals. The chiralanomaly induced negative MR can be understood from the violation of the conservation rules of chiral charges as shown in Fig. 2d^{50}, and it should be maximum when B is exactly parallel to I (α = 0°, 180°, or γ = 0°, 180°). As expected, this anisotropic feature of the negative MR is observed at α = 0°, 180° or γ = 0°, 180° under 14 T (Fig. 2e). In addition, the peak structures in the angle dependence of the negative MR at B//I (α = 0°, 180° or γ = 0°, 180°) are similar to those in previous observations of the chiral anomaly in other Weyl semimetals^{1,16,21,51,52}. These results confirm that this linear negative MR is induced by the chiral anomaly (see Methods section “Excluding other possible origins of the negative MR in SrRuO_{3}”). As a consequence of the contributions from the positive MR and negative MR, ρ_{xx}(B) is lower than the zero field resistivity ρ_{0} = ρ_{xx}(0 T) when α and γ are near 0° or 180°.
Quantum oscillations
The Berry phase has become an important concept in condensed matter physics over the past three decades, since it represents a topological classification of the system and it plays a fundamental role in various phenomena, such as electric polarization, orbital magnetism, etc.^{53}. However, revealing the π Berry phase, originating from a band touching point of the Weyl node^{16}, has been challenging in magnetic Weyl semimetals. Here, we detect the π Berry phase accumulation along the cyclotron orbits, for the first time in magnetic Weyl semimetals, by measurement techniques sensitive to the quantized energy levels, i.e., SdH oscillation. The nontrivial π Berry phase, which is acquired by a surface integral of the Berry curvature Ω over a closed surface containing a Weyl point in the kspace (Fig. 3a)^{16,18,19,20}, causes a phase shift of quantum oscillations. According to the Lifshitz–Kosevich (LK) theory, the magnitude of the SdH oscillation is described as^{16,18,19,20,54}
where Δσ_{xx} is the oscillation component of the longitudinal conductivity, A_{i} is the normalization factor, X_{i} = 2π^{2}k_{B}T/ħω_{ci}, k_{B} is the Boltzmann constant, ħ is the reduced Planck constant, ω_{ci} is the cyclotron frequency defined as eB/m_{i}^{*}, m_{i}^{*} is the cyclotron mass, T_{Di} is the Dingle temperature, F_{i} is the frequency of the SdH oscillation, and 2πβ_{Bi} is the phase shift caused by the Berry phase as mentioned above. The subscript i is the label of an orbit of carriers. Note that, even when a large number of carriers in 3D multiband systems pin the chemical potential, the linear dispersion of the Weyl fermions leads to an unconventional \(\pi\) phase shift in the SdH oscillation (see Methods section “Effect of the Fermilevel pinning on the phase shift in the SdH oscillation”). To extract the Berry phases in SrRuO_{3}, we used the LK formula [Eq. (1)] for two frequencies to fit the SdH oscillation data. Figure 3b shows the SdH oscillation data at 2 K and the fitting results by Eq. (1). (see Methods section “Data pretreatment for quantum oscillations”). The oscillation spectrum is considerably complex because of the contribution from several subbands^{7}. To reduce the fitting parameters, we first carried out fast Fourier transform of the SdH oscillation (Fig. 3c), and extracted the F_{1} (26 T), F_{2} (44 T), m_{1}^{*} (0.35m_{0}), and m_{2}^{*} (0.58m_{0}) (m_{0}, electron rest mass) values from the peak positions and the temperature dependence of F_{1} and F_{2} based on the LK theory (Fig. 3c, insets). Small masses are expected for Weyl fermions, which would fulfill the light cyclotron mass signature^{16,18,19,20,22,23,24}. From the fitting to the data at 0.07 T^{−1} < 1/B < 0.2 T^{−1} shown in Fig. 3b, we obtained T_{D1} = 0.63 K, T_{D2} = 0.34 K, β_{B1} = 0.27, and β_{B2} = 0.48. The 2πβ_{B2} value of 0.96π indicates the presence of the nontrivial π Berry phase arising from the massless dispersion of the Weyl fermions (see Methods section “Exclusion of other possible mechanisms of the phase shift in SrRuO_{3}”). Although the interpretations of phase shifts between 0 and π in topological materials are still controversial, the 2πβ_{B1} value of 0.54π implies that the energy dispersion of the F_{1} orbit has both quadratic (trivial) and linear (nontrivial) features^{55,56}. These results cannot be reproduced by fixed zero Berry phases, confirming the existence of a nonzero Berry phase (Supplementary Fig. 6a).
The SdH oscillations not only give an insight into the topological nature, but also provide evidence of very high mobility of the Weyl fermions enclosing the Weyl points. We quantitatively determine the mobility of the charge carriers of the F_{1} and F_{2} orbits by calculating the quantum mobility, µ_{qi} = eħ/(2πk_{B}m_{i}^{*}T_{Di}). The obtained µ_{q1} and µ_{q2} values are 9.6 × 10^{3} and 1.1 × 10^{4} cm^{2} V^{−1} s^{−1}, respectively. In addition, assuming isotropic Weyl nodes, we can estimate the carrier concentrations n_{i} = \(\frac{1}{{6\pi ^2}}\left( {\frac{{2eF_i}}{\hbar }} \right)^{\frac{3}{2}}\) (n_{1} = 3.8\(\times\)10^{17} \({\mathrm{cm}}^{  3}\) and n_{2} = 8.3\(\times\)10^{17} \({\mathrm{cm}}^{  3}\)) and the chemical potentials µ_{i} = eħF_{i}/m_{i}^{*} (µ_{1} = 8.5 meV and µ_{2} = 8.8 meV).^{57} These results mean that F_{1} and F_{2} come from the highmobility and lowconcentration Weyl fermions enclosing the Weyl points.
In addition to the Weyl fermions as evidenced by the five signatures described above, there are trivial Ru 4d bands crossing the E_{F} in SrRuO_{3}^{7,12,42,43}. SdH experiments performed at 0.1 K (Fig. 3d) confirmed some of these trivial Fermi pockets with F_{3} = 300 T, F_{4} = 500 T, F_{5} = 3500 T, and F_{6} = 3850 T (see Methods section “SdH oscillations of trivial orbits”). These four orbits have heavier masses (>2.8m_{0}) than those of F_{1} and F_{2}, which are consistent with the reported values for SrRuO_{3}^{12,43}. The Fermi pocket areas of F_{3} (0.029 Å^{−2}) and F_{4} (0.048 Å^{−2}) are close to those of the 364 T (0.035 Å^{−2}) orbit reported in earlier de Haas–van Alphen measurements^{58}, and the Fermi pocket areas of F_{5} (0.334 Å^{−2}), and F_{6} (0.368 Å^{−2}) correspond to the \(\alpha\)_{1} band (0.33–0.37 Å^{−2}) observed by the ARPES and in earlier SdH measurements^{42,43}. Noteworthy is that the Fermi pocket areas of F_{1} (0.0025 Å^{−2}) and F_{2} (0.0042 Å^{−2}) are more than ten times smaller than those of the trivial orbits, indicating that F_{1} and F_{2} stem from the small Fermi pockets as a feature of the Weyl fermions.
Disorder dependence of transport properties
Observing the intrinsic transport signatures of the Weyl fermions requires a highquality SrRuO_{3} sample, since it is easily hindered by disorders such as defects and impurities. To show the disorder dependence of the transport phenomena in SrRuO_{3}, we investigated the RRR dependence of T_{C}, T_{F}, and the highest temperature where the linear positive MR, one of the clear features of Weyl fermion transport in SrRuO_{3}, remains (hereafter called T_{W}) (Fig. 4a–c) (see Methods section “RRR dependence of the ferromagnetism, Fermi liquid behavior, and Weyl behavior”). As shown in Fig. 4c, the ferromagnetism becomes weaker (T_{C} < 150 K) below RRR = 8.93, the Fermi liquid behavior remains regardless of the RRR values even with low RRR = 6.61, and the positive MR is observed over the RRR = 19.4. It is remarkable that the positive MR ratio at 9 T increases with increasing RRR (Fig. 4a, b), which indicates that the Weyl fermions become more dominant in the transport. In the RRR dependence of the Hall resistivity, ρ_{xy}(B), the nonlinear B dependence becomes more prominent with a sign change in dρ_{xy}/dB with increasing RRR (see details in Methods section “RRR dependence of the Hall resistivity”). Thus, a highquality SrRuO_{3} sample is essential for observing the intrinsic transport of the Weyl fermions, and the diagram presented in Fig. 4c will be an effective guideline for realizing topologically nontrivial transport phenomena of the Weyl fermions to connect magnetic Weyl semimetals to spintronic devices^{31,32}.
Electronic structure calculations
Thus far, the magnetotransport data of a highquality SrRuO_{3} shows all of the expected marks of a magnetic Weyl semimetal. For certainty and theoretical rigor, we performed firstprinciples electronic structure calculations (see Methods section “Computational details”) to analyze the energy dispersion of SrRuO_{3}. The calculated electronic structure for the orthorhombic Pbnm phase of SrRuO_{3} for the ferromagnetic ground state shows a halfmetallic behavior (Fig. 5a) which agrees with previous electronic structure calculations^{59}. The bands near the Fermi level are formed by the t_{2g} Ru states hybridized with the O 2p orbitals (Supplementary Fig. 8a). The calculated magnetic moment per Ru ion (~1.4 \(\mu _B\), as tabulated in Supplementary Table 2) is close to the experimental saturation magnetic moment of 1.25 \(\mu _B\)/Ru of our SrRuO_{3} films.^{28} We observe that in the ferromagnetic phase the Ru spins tend to align along the crystallographic b axis reducing the symmetry of the system from D_{2h} to C_{2h}. To identify the existence of the Weyl points (the band crossing points that carry \(\pm 1\) chiral charge) by evaluating the outward Berry flux enclosed in a small sphere, we examined each band crossing of two pairs of bands I and II, shown in Fig. 5b, near the Fermi level in the presence of SOC with the magnetization along the orthorhombic caxis. The resulting Weyl points are shown and listed in Fig. 5c and Supplementary Tables 3 and 4, respectively. We identified a total of 29 pairs of Weyl points with opposite chirality in the full Brillouin zone (BZ). An earlier theoretical study also made similar predictions of the existence of Weyl points in the case of a cubic structure of SrRuO_{3}.^{7} Most of these Weyl points are found to exist within an energy range of −0.2 to 0.2 eV around the Fermi level. Among them, \(\left {E  E_F} \right\) for WP_{z}6_{1–4} (8–16 meV) in Supplementary Table 4 is very close to the experimental chemical potentials of the Weyl fermions estimated from the SdH oscillations (µ_{1} = 8.5 meV and µ_{2} = 8.8 meV for the F_{1} and F_{2} orbitals, respectively). The Weyl points near the Fermi level are expected to contribute to all the observed quantum transport phenomena reported in this study. It is important to note that a small monoclinic distortion induced by the substrate does not significantly change the band structure of SrRuO_{3} and that Weyl points are robust with the distortion as long as the inversion symmetry is present (see Method section “Firstprinciples calculations of Weyl points”), leading to a congruence of the experimental findings with theoretical predictions.
In conclusion, we have observed the emergence of Weyl fermions in epitaxial SrRuO_{3} film with the best crystal quality ever reported^{12}, which was grown by machinelearningassisted MBE. Experimental observation of the five important transport signatures of Weyl fermions—the linear positive MR, chiralanomaly induced negative MR, π phase shift in a quantum oscillation, light cyclotron mass, and high quantum mobility of about 10,000 cm^{2} V^{−1} s^{−1}—combined with firstprinciples electronic structure calculations establishes SrRuO_{3} as a magnetic Weyl semimetal. In addition, the RRR dependences of ferromagnetism, Fermi liquid behavior, and positive MR serve as a road map to merge two emerging fields: topology in condensed matter and oxide electronics. Our results will pave the way for topological oxide electronics.
Methods
Magnetotransport measurements
We first deposited the Ag electrodes on a SrRuO_{3} surface. Then, we patterned the samples into 200 × 350 μm^{2} Hall bar structures by photolithography and Ar ion milling. Resistivity was measured using the fourprobe method at 100 μA in a Physical Property Measurement System (PPMS) DynaCool sample chamber equipped with a rotating sample stage. Lownoise measurements below 1 K were performed by an AC analog lockin technique, and the sample was cooled down in a ^{3}He–^{4}He dilution refrigerator.
Determination of the RRR in SrRuO_{3}
The RRR was determined as the ratio of the longitudinal resistivity ρ_{xx} at 300 K [ρ_{xx}(300 K)] and T\(\to\)0 [ρ_{xx}(T\(\to\)0 K)]. SrRuO_{3} exhibits Fermi liquid behavior at a low temperature, which is characterized by a linear relationship between ρ_{xx} and T^{2}.^{12,42} Based on this relationship, we estimated ρ_{xx} (T\(\to\)0 K) by extrapolating the ρ_{xx} vs. T^{2} fitting line below 10 K (Supplementary Fig. 1b).
Machinelearningassisted MBE
We grew the highquality SrRuO_{3} films (63nm thick) on (001) SrTiO_{3} substrates in a customdesigned MBE setup equipped with multiple ebeam evaporators for Sr and Ru (Supplementary Fig. 2a). We precisely controlled the elemental fluxes, even those of elements with high melting points, e.g., Ru (2250 °C), by monitoring the flux rates with an electronimpactemissionspectroscopy sensor and feeding the results back to the power supplies for the ebeam evaporators. The oxidation during growth was carried out with ozone (O_{3}) gas. O_{3} gas (\(\sim\)15% O_{3} + 85% O_{2}) was introduced through an alumina nozzle pointed at the substrate. Further information about the MBE setup and preparation of the substrates is described elsewhere^{60}. The surface morphology of our SrRuO_{3} films is composed of atomically flat terraces and steps, as observed by atomic force microscopy^{28}. Together with Laue fringes in the θ–2θ Xray diffraction patterns^{28}, this indicates the high crystalline order and large coherent volume of our SrRuO_{3} films.
Finetuning of growth conditions is essential but challenging for highRRR SrRuO_{3} growth. Therefore, only a few papers have reported SrRuO_{3} films with RRRs over 50.^{28,43,44} While a conventional trialanderror approach may be a way to optimize the growth conditions, this is timeconsuming as well as costly, and the optimization efficiency largely depends on the skill and experience of individual researchers. To avoid such a timeconsuming approach and reduce experimental time and cost, we employed machinelearningassisted MBE, which we developed in previous research^{28}. Here, three important growth parameters [Ru flux rate, growth temperature, and nozzletosubstrate distance (Supplementary Fig. 2a)] were optimized by a Bayesian optimization (BO) algorithm, which is a sampleefficient approach for global optimization^{61}. This algorithm sequentially produces the three parameter values at which a high RRR value is predicted given past trials.
Supplementary Figure 2b shows the procedure for machinelearningassisted MBE growth of the SrRuO_{3} thin films based on the BO algorithm. Here, we optimized each parameter in turn using BO. We first chose one of the growth parameters to update and fixed the rest, ran the BO algorithm to search the growth parameter, and then switched to another growth parameter. This is because BO can be inefficient in large dimensions due to the difficulty of predicting the outcome value for unseen parameters. Here, RRR = S(x) is the target function specific to our SrRuO_{3} films, and x is the growth parameter (Ru flux rate, growth temperature, or nozzletosubstrate distance). BO constructs a model to predict the value of S(x) for unseen x using the result of past M trials \(\left\{ {x_m,\,{\mathrm{RRR}}_m} \right\}_{m \,=\, 1}^M\), where x_{m} is the mth growth parameter and RRR_{m} is the corresponding RRR value. Specifically, we use the Gaussian process regression (GPR) as a prediction model^{28,62}. GPR predicts S(x) as a Gaussian random variable following N (\(\mu\), \(\sigma ^2\)). This means \(\mu\) and \(\sigma ^2\) are calculated from x and the M data points. Subsequently, we choose the growth parameter x in the next run such that the expected improvement (EI)^{63} is maximized. EI balances the exploitation and exploration by using the predicted \(\mu\) and \(\sigma ^2\) at x. This measures the expectation of improvement over the best RRR observed so far. This routine is iterated until further improvement is no longer expected. In practice, we terminate the iteration when the number of trials reaches the predetermined budget. Here, we stopped the routine at 11 samplings per parameter. After completing 11 samplings for a certain parameter, we chose the value that gave the highest RRR and started the optimization of another parameter. In this optimization procedure, we used RRR (T = 4 K) instead of RRR (T\(\to\)0 K) for easy estimation. Further details about the implementation of machinelearningassisted MBE are described elsewhere^{28}.
In our previous study^{28}, we carried out the optimization of the Ru flux rate while keeping the other parameters unaltered. Subsequently, we tuned the growth temperature and the nozzletosubstrate distance. As a result, we obtained a highRRR (51.8) SrRuO_{3} film in only 24 MBE growth runs (Supplementary Fig 2c). Since the RRR was still lower than the highest value reported (\(\sim\)80) in the literature^{12,44}, we further carried out the reoptimization of the Ru flux rate and growth temperature with previously optimized parameters (the Ru flux = 0.42 Å s^{−1}, growth temperature = 721 °C, and nozzletosubstrate distance = 15 mm) as a starting point to find the globalbest point in the threedimensional parameter space. Supplementary Figure 2c shows the highest experimental RRR values plotted as a function of the total number of MBE growth runs. With the reoptimization of the Ru flux rate and growth temperature, the highest RRR (T = 4 K) value increased and reached 81 in only 44 MBE growth runs. The highest experimental RRR (T\(\to\)0 K), 84.3, was achieved at the Ru flux = 0.365 Å s^{−1}, growth temperature = 781 °C, and nozzletosubstrate distance = 15 mm. The availability of such highquality film allowed us to probe the intrinsic transport properties of SrRuO_{3}.
Excluding other possible origins of the positive MR in SrRuO_{3}
Since SrRuO_{3} has complex Fermi surfaces with both topologically trivial and nontrivial bands, the contribution from the trivial bands to the MR, such as orbital MR (∝B^{2})^{64}, anisotropic MR (AMR) (∝ relative angle between the electric current and the magnetization), and weak antilocalization (WAL) (∝B^{−1/2})^{65}, will appear in the MR in our SrRuO_{3} films. In fact, the AMR feature is clearly observed in the nearzero field region for the SrRuO_{3} film with the RRR of 84.3 as shown in Supplementary Fig. 3a. While it is difficult to distinguish each contribution on the MR in a lowfield region (<0.5 T), the MR in a high field region (>0.5 T) is clearly dominated by the unsaturated linear positive MR. The MR curve at 2 K for the SrRuO_{3} film with the RRR of 84.3 with B applied in the outofplane [001] direction of the SrTiO_{3} substrate (Supplementary Fig. 3b) is linear above 0.5 T. This means that the magnetic field dependences of the WAL (∝B^{−1/2}) and of the orbital MR (∝B^{2}) provide only a very limited contribution to the net MR response and that the WAL and the orbital MR are negligibly small above 0.5 T. In addition, the contribution of the AMR above 0.5 T is also negligible since the magnetization of the SrRuO_{3} films saturates at about 0.5 T (see Supplementary Fig. 3c).
Next, the linear positive MR caused by carrier fluctuations^{39,48,66,67,68,69,70}, originating from disorders and/or nonuniformity of dopants, can be excluded in the SrRuO_{3} film with the RRR of 84.3, since its Hall resistivity is not a linear function of B as shown in Supplementary Fig. 3d. The carrier fluctuations can cause the linear positive MR by an admixture of a component of the Hall resistivity to the longitudinal resistivity since the mobility fluctuation makes the Hall voltage contribute to the longitudinal voltage. In ref. ^{39}, the admixture of a component of the Hall resistivity to the longitudinal resistivity ρ_{xx} due to the carrier fluctuations is expressed as
where ρ_{xy} is the Hall resistivity, and B is the external magnetic field. Therefore, ρ_{xx} is proportional to B when dρ_{xy}/dB is constant. For example, the GaAs 2DEG system shows clear linear MR reflecting its highlinearity of ρ_{xy} in the whole measurement range (0–33 T)^{39}. In our case, however, the Hall resistivity shows nonlinear behavior reflecting its semimetallic feature (Supplementary Fig. 3d). In fact, the \(d\rho _{xy}/dB \times B\) curve calculated by the ρ_{xy} curve (Supplementary Fig. 3e) does not agree with the measured MR (ρ_{xx}(B)−ρ_{xx}(0 T))/ρ_{xx}(0 T) in the SrRuO_{3} film, especially in the lowmagneticfield range (0 T < B < 4 T), where the nonlinearity of the Hall resistivity (Supplementary Fig. 3d) is prominent. This means that carrier fluctuations are not the origin of the linear positive MR in our SrRuO_{3} films.
The admixture of a component of the Hall resistivity to the longitudinal resistivity is also ruled out by taking the small Hall resistivity of SrRuO_{3} into consideration. For the admixture of the Hall voltage, to cause the large positive MR over 100%, a large Hall voltage due to a small carrier density is necessary; only semiconducting materials can show this type of large MR. For example, the GaAs 2DEG system with the small sheet carrier density of 3 × 10^{11} cm^{−2} shows by far larger linear MR (~10^{5}%)^{39} than that of Bi_{2}Se_{3} (~1%) with the large sheet carrier density of 1.7 × 10^{16} cm^{−2}.^{68} In contrast, since the SrRuO_{3} film with the RRR of 84.3 is metallic, the Hall resistivity (=0.36 μΩ cm) at 2 K and 14 T is more than ten times smaller than the longitudinal resistivity (=3.79 μΩ cm) at 2 K and 14 T. This means that the longitudinal voltage in SrRuO_{3} is less affected by the Hall voltage.
Therefore, we concluded that the observed unsaturated linear positive MR above 0.5 T originates from the linear energy dispersion around the Weyl nodes and that the contribution from the conventional bands on the MR is negligible in the SrRuO_{3} film with the RRR of 84.3.
Temperature dependence of the AHE in the SrRuO_{3} film with the RRR = 84.3
It is known that the AHE in SrRuO_{3} is caused by an extrinsic factor (sidejump scattering) in addition to an intrinsic factor (KL mechanism)^{12,37,38}. To determine the contributions of the intrinsic and extrinsic factors to the AHE, we investigated the temperaturedependent scaling of the AHE in the SrRuO_{3} film with the RRR = 84.3.
The Hall resistivity ρ_{xy}(B) in SrRuO_{3} is described as the summation of the ordinary \(\rho _{xy}^{{\mathrm{OHE}}}\left( B \right)\) and anomalous \(\rho _{xy}^{{\mathrm{AHE}}}\left( B \right)\) components of Hall effects^{12,37,38}
\(\rho _{xy}^{{\mathrm{AHE}}}\left( B \right)\) is proportional to the perpendicular magnetization component M_{⊥}: \(\rho _{xy}^{{\mathrm{AHE}}}\left( B \right)\) = cρ_{s}M_{⊥}(c; constant). Here, the proportional coefficient ρ_{s} differs depending on the origin of the AHE, which can be of intrinsic (KL mechanism) and extrinsic (side jump scattering) origin^{12,37,38}. As shown in Supplementary Fig. 4a, above T_{F} (~20 K), clear AHE, which is proportional to the magnetization hysteresis curve (e.g., Fig. 1d, right inset), is discernible, and it dominates Hall resistivity in a nearzero magnetic field. On the other hand, below T_{F}, the AHE components are negligibly small.
The temperature dependence of the AHE in SrRuO_{3} has been well reproduced by a model where both the intrinsic KL mechanism and the extrinsic sidejump scattering terms are taken into account^{38}. In this model, the relationship between ρ_{s} and ρ_{xx} is described as
where a_{1}–a_{3}, and Δ are the fitting parameters, which are associated with the band structure^{38}. The first term describes a contribution from the offdiagonal matrix elements of the velocity operators, called the KL term in the model^{38}, and the second term is the contribution from the sidejump scattering. In this model, the KL term is obtained by incorporating the finite scattering rate, which is inversely proportional to \(\rho _{xx}\), into Kubo’s formula^{71}, and the constant a_{1} in Eq. (4) is expressed as
where b is a constant and \(A\) is expressed as
Here, \(A\) can be associated with a Berry phase^{72}. The v_{y} and v_{x} are the velocity operators in the direction of y and x axes, respectively, \({\Bbb K}\) is a set of quasimomentum of the states producing the dominant contribution to \(A\), and \(k \in {\Bbb K}\). Here, “1” and “2” are indices to denote two individual bands: the former crosses the Fermi level, while the latter is fully occupied. When the system has multiple bands crossing the E_{F}, the \(a_1\) value is expressed by the sum of the \(A\) values for each band. Therefore, the relationship between \(\rho _s\) and \(\rho _{xx}\) (Eq. (4)) does not even change in the system with multiple bands. Accordingly, this model can be applied to analyze the temperature dependence of the AHE of SrRuO_{3}, although detailed information about the electronic bands near E_{F} in SrRuO_{3} is required to assign the contribution of each band to the intrinsic AHE.
Supplementary Figure 4b shows the ρ_{s} vs. ρ_{xx} plot of the SrRuO_{3} film with the RRR = 84.3 and the fitting result of Eq. (4). The fitting curve reproduces the AHE sufficiently, indicating that the AHE in the SrRuO_{3} film arises from the intrinsic KL mechanisms along with the extrinsic sidejump scattering. The important point in Eq. (4) is that the AHE is asymptotic to zero when ρ_{xx}\(\to\)0. Accordingly, for the SrRuO_{3} film with such a very high RRR (84.3), equivalently with a very small residual ρ_{xx}, AHE becomes negligibly small at low temperatures. Therefore, ρ_{xy}(B) curves below T_{F} in our data are dominated by the ordinary Hall effect.
Excluding other possible origins of the negative MR in SrRuO_{3}
Here, we exclude other possible mechanisms as the origin of the anisotropic linear negative MR at 2 K for the SrRuO_{3} film with the RRR of 84.3. As described in the Methods section “Excluding other possible origins of the positive MR in SrRuO_{3}”, the contribution from the trivial bands to the MR, such as orbital MR^{64}, AMR, and WAL^{65}, are negligible above 0.5 T.
The electronmagnon scattering^{73} is also excluded as an origin of the linear negative MR (Fig. 2). The measurement temperature (2 K) of the anisotropic linear negative MR is too low to excite magnons in SrRuO_{3}. A recent inelastic neutron scattering study on singlecrystalline SrRuO_{3}^{74} reveals that the magnon gap of SrRuO_{3} is equal to 11.5 K. Therefore, the contribution from magnon scattering to the negative MR is negligible.
Finally, the boundary scattering effect is also excluded. The values of mean free paths l_{m} for the Weyl fermions (F_{1} (=26 T) and F_{2} (=44 T) orbits), estimated from the SdH oscillation in Fig. 3b, are 89 and 132 nm, respectively. Here, the \(l_m\) values are estimated by \(l_m = \hbar v_F/(2\pi k_BT_D)\) (where v_{F} is Fermi velocity, k_{B} is Boltzmann’s constant, and T_{D} is the Dingle temperature). The mean free paths of other orbits (F_{3}–F_{6}) are smaller than these two orbits (F_{1} and F_{2}). Since the mean free paths of the Weyl fermions are larger than the thickness of the SrRuO_{3} films (\(d_{{\mathrm{SrRuO}}_3}\) = 63 nm), the boundary scattering may contribute to the negative MR. However, we can exclude its contribution as described as follows. The boundaryscatteringinduced MR should saturate in a highmagneticfield region^{75}, since boundary scattering does not occur when cyclotron diameter d_{c} (=ℏk_{F}/(πeB); k_{F}, Fermi wave number) is smaller than the film thickness (\(d_{{\mathrm{SrRuO}}_3}\) = 63 nm). If the MR had been induced by boundary scattering, it would have saturated when B is larger than 0.93 and 1.2 T for the F_{1} and F_{2} orbits, respectively, at which d_{c} becomes comparable to \(d_{{\mathrm{SrRuO}}_3}\). On the contrary, the observed negative MR (Fig. 2a, c) does not show saturation behavior with B. Note that the coherent lengths of the other orbits (F_{3}–F_{6}) are not long enough to travel a full cyclotron trajectory at the measurement temperature (=2 K), where only the F_{1} and F_{2} SdH oscillation peaks are detected; as shown in Fig. 3d, the SdH oscillation peaks of F_{3}–F_{6} are observed at 0.1 K. Therefore, the contribution of the boundary scattering to the negative MR (Fig. 2a, c) is negligibly small.
Effect of the Fermilevel pinning on the phase shift in the SdH oscillation
A large number of carriers in 3D multiband systems pin the chemical potential. In 2018, Kuntsevich et al. phenomenologically explained that, even when a large number of carriers in normal bands pin the chemical potential, the SdH oscillation of the Dirac (linear) dispersion should show the π phase shift^{76}. Here, we will explain why the chemical potential pinning does not affect the phase shift in the SdH oscillations, along with ref. ^{76}.
We will begin with the general explanation of SdH oscillations without the Fermilevel (chemical potential) pinning. In the SdH oscillation, the minimal conductivity is obtained when the chemical potential μ_{c} is located in the midgap between the Landau levels (Supplementary Fig. 5a). For this μ_{c}, the Nth Landau level beneath the chemical potential is fully occupied and the density of states shows the minimal value. Reflecting this situation, in the LK theory^{77}, the magnetic field at which the magnitude of the SdH oscillation takes a minimal value is expressed as
Here, F_{i} is the frequency of the SdH oscillation, \(B_N^{\rm{min}}\) is the magnetic field giving the minimal value of the conductivity at the Nth Landau level, and 2πβ_{Bi} is the phase shift. The important point of Eq. (7) is that the \(1/B_N^{{\mathrm{min}}}\) can be expressed as a linear function of N. This ensures the validity of using Eq. (7) for the determination of the phase shift caused by the Berry phase as we did in the main text.
Next, we consider the situation where a large number of carriers in normal bands pin the chemical potential, i.e. \(\mu _c\left( B \right) = const.\) The Nth Landau levels of the quadratic energy band \({\it{\epsilon }}_{N,Q}\) and the linear energy band \({\it{\epsilon }}_{N,D}\) are expressed as^{76,78}
and
respectively, where m is an effective mass, v is the velocity of electrons in the linear band, and N is the Landau level index. The conductivity becomes maximum at \(B\) with which the Landau level crosses μ_{c}, since the density at μ_{c} becomes maximum when μ_{c} is located at the center of the Landau level (Supplementary Fig. 5b). Therefore, by solving the equation \({\it{\epsilon }}_{N,Q(D)} = \mu _c\), we can obtain the relationships between N and the magnetic field \(B_{N,Q(D)}^{{\mathrm{max}}}\) at which the conductivity becomes maximum as
and
for the quadratic and linear bands, respectively. Since \(B_{N,Q(D)}^{\rm{min}}\) is located roughly at the middle point between \(B_{N  1,Q(D)}^{{\mathrm{max}}}\) and \(B_{NQ(D)}^{{\mathrm{max}}}\), we can obtain the relationship between \(B_{NQ(D)}^{{\mathrm{min}}}\)and N by shifting N in Eqs. (10) and (11) by a halfinteger as
and
respectively. Then, by using the relationship between the \(\mu _c\) and the Fermi wave number k_{F} for the quadratic and linear band, \(\mu _c = \left( {\hbar k_F} \right)^2/2m\) and \(\mu _c = \hbar vk_F\), respectively, and the relationship between the Fermi surface area S (\(= \pi k_F^2\)) and F_{i}, \(S = 2\pi eF_i/\hbar\), Eqs. (12) and (13) can be simply expressed as
and
Equations (14) and (15) mean that the trivial (quadratic) and nontrivial (linear) dispersion provides 0 and π (=2π × 1/2) phase shifts, respectively, even when a large number of carriers pin the chemical potential. Therefore, whether the chemical potential is pinned or not in our SrRuO_{3} film, we can estimate the phase shift by using the LK theory [Eq. (1)]. Physically, the difference in the phase shifts in Eqs. (14) and (15) comes from the difference in the Berry phases of quadratic and linear dispersions^{76}.
In a 3D system with B applied in the k_{z} direction, a twodimensional cyclotron motion occurs in the k_{x}–k_{y} plane in 3D kspace at the k_{z} position where the area of the Fermi surface takes an extremal value^{79}. This cyclotron orbit is called an “extremal orbit”. As in the case of 2DEG Rashba systems, graphene, and topological surface states, observation of the π Berry phase, which originates from a band touching point of the Weyl node and accumulates along the extremal orbit, is one of the important signatures of Weyl fermions^{20,23,24}. Therefore, we think that the observed π Berry phase in the SdH oscillations is acquired by a surface integral of the Berry curvature Ω over a closed surface containing a Weyl point in kspace (Fig. 3a).
Data pretreatment for quantum oscillations
Pretreatments of the SdH oscillation data are crucial for deconvoluting quantum oscillation spectra, since magnetoconductivity data generally contain not only oscillation components but also other magnetoresistive components as background signals^{43}. In particular, SdH oscillations in SrRuO_{3} are subject to being masked by large nonsaturated positive MR (Supplementary Fig. 5c)^{43}. Here, we subtracted the background using a polynomial function up to the fifth order and extracted the oscillation components as shown in Supplementary Fig. 5d. Then, we carried out the wellestablished pretreatment procedure for Fourier transform of quantum oscillations^{80,81}. First, we interpolated the backgroundsubtracted data to prepare an equally spaced data set as a function of 1/B. Then, we multiplied the Hanning window function to obtain the periodicity of the experimental data. Finally, we performed fast Fourier transform on the treated data set.
Exclusion of other possible mechanisms of the phase shift in SrRuO_{3}
Here, we exclude other possible nontopological effects that could cause the phase shift in SdH oscillations, which are the mosaic effect, magnetic breakdown, and Zeeman splitting.
The phase shift and deviation of the SdH oscillations from the conventional LK theory, which occur at crystal grains and magnetic domains, are collectively called the mosaic effect^{79,82}. This effect may occur in samples having multiple crystal domains, such as polycrystals, or in ferromagnets having magnetic domain structures. However, it should be negligible in our samples, because they are highquality singlecrystalline thin films as shown by a STEM image (Fig. 1c, Supplementary Fig. 1a, and Supplementary Fig. 5e) and they are free from magnetic domain structures as shown in Supplementary Fig. 3c where magnetization is saturated at about 0.5 T.
Next, the possibility of magnetic breakdown is also ruled out since there is no magnetic breakdown orbits in the SdH oscillations for the SrRuO_{3} film with the RRR of 84.3 (Fig. 3). Magnetic breakdown occurs when different orbits approach each other closely in kspace under the presence of large magnetic fields, resulting in a new orbit (a magnetic breakdown orbit) whose frequency is given by the sum of the frequencies of the original orbits^{83}. If magnetic breakdown had occurred in our measurement field range (B < 14 T), we would have observed magnetic breakdown orbits whose frequencies are F_{1} + F_{2} (=70 T), F_{1} + F_{3} (=326 T), and so on. However, we did not find such frequencies in our quantum oscillation analysis as shown in Supplementary Table 1. In particular, F_{1} and F_{2}, which are responsible for the nontrivial phase shifts, cannot be produced by the sum of the other orbitals’ frequencies. Therefore, magnetic breakdown is not the origin of the phase shift.
Finally, we focus on the effect of Zeeman splitting on the phase shift in a magnetic Weyl semimetal. The condition where the quantum oscillation takes minimal or maximum values is expressed as^{78}
Here, F is a frequency, H is a magnetic field, n is an integer value, γ indicates the phase shift caused by the Berry phase, and S is the spinsplitting parameter from Zeeman effect. The ± sign indicates the up/down spins in each Landau level. Due to the Zeeman effect, every Landau level is splitoff by the magnetic field, and finally it affects the phase shift γ through the change of the S value. In fact, this Zeeman splitting of Landau levels changes SdH spectra in Weyl semimetals and magnetic Dirac materials^{20,84}, in which Landau levels are degenerate, and this effect has to be taken into account to deduce the phase shift γ from the experimental data. By contrast, in SrRuO_{3}, all Landau levels are not degenerate since the ferromagnetic exchange coupling lifts the spin degeneracy of all the electronic bands crossing the E_{F}^{12,42,43,85}. Altogether, since the S value of Eq. (16) is zero in SrRuO_{3}, we can simply estimate the phase shift γ by the LK theory^{77}, in which S is not taken into account, and assign the phase shift in the SdH oscillations to the Berry phase accumulation along the cyclotron orbits.
SdH oscillations of trivial orbits
Together with SdH oscillations from the nontrivial orbits having low frequencies (F_{1} and F_{2}) (Fig. 3b, c), we observed SdH oscillations from the trivial orbits having high frequencies (F_{3}–F_{6}) at 0.07 K < T < 0.75 K and 12.5 T < B < 14 T (Fig. 3d and Supplementary Fig. 6b). Since the carriers in the trivial orbits in SrRuO_{3} are expected to have larger effective masses than those in the F_{1} and F_{2} orbits^{12,43,58}, measurements of the former oscillations should be carried out in relatively lowT and highB regions. We estimated the cyclotron masses of the carriers in F_{3}–F_{6} orbits from the temperature dependences of the respective peaks based on the LK theory (Supplementary Fig. 6c–f). In the LK theory for the mass estimation, B is determined as the interval value in the magnetic field range. Supplementary Table 1 shows the estimated cyclotron masses for F_{1}–F_{6}. The cyclotron masses in the F_{3}–F_{6} orbits are relatively high (>2.8m_{0}), reflecting the trivial band structure (energy dispersions) as their origin.
RRR dependence of the ferromagnetism, Fermi liquid behavior, and Weyl behavior
In Fig. 4a–c, we investigated the RRR dependence of the ferromagnetism, Fermi liquid behavior, and Weyl behavior in SrRuO_{3}. For the ferromagnetism, T_{C} values are estimated as the position of the kinks in ρ_{xx} vs. T curves as shown in Supplementary Fig. 7a. For the Fermi liquid behavior, we defined the Fermi liquid region (T < T_{F}) as the temperature range where the experimental ρ_{xx} and the linear fitting line in ρ_{xx} vs. T^{2} are close enough to each other (<0.1 μΩ cm) as shown in Supplementary Fig. 7b. The upper limit temperature for measuring Weyl behavior in SrRuO_{3} (T_{W}) is defined as the highest temperature at which the resistivity at zero field is lower than that at 9 T (ρ_{xx}(0 T) < ρ_{xx}(9 T)).
RRR dependence of the Hall resistivity
Supplementary Figure 7c shows the Hall resistivity ρ_{xy}(B) curves of the different RRR samples at 2 or 2.3 K with B applied in the outofplane [001] direction of the SrTiO_{3} substrate. As we explained in the main paper, the ρ_{xy}(B) curves of the SrRuO_{3} film with the RRR of 84.3 at 2 K is nonlinear, indicating the coexistence of multiple types of the Weyl fermions from which the unsaturated linear positive MR stems. Notably, as shown in Supplementary Fig. 7c, dρ_{xy}/dB changes its sign from negative to positive with decreasing RRR. In the SrRuO_{3} film with the RRR of 8.93, clear AHE is observed near zero magnetic field due to its large residual resistivity of 20.2 μΩ cm, and the ρ_{xy}(B) curve shows the linear dependence on B above 5 T as highlighted in Supplementary Fig. 7c. The carrier concentration and the mobility of the holes of the SrRuO_{3} film with the RRR of 8.93, which are estimated from the slope of the ρ_{xy}(B) above 5 T, are 4.04 × 10^{22} cm^{−3} and 7.65 cm^{2} V^{−1} s^{−1}, respectively. The carrier concentration and the mobility are consistent with the reported values for the trivial Ru 4d bands crossing the E_{F} in SrRuO_{3}^{44,86}. These results mean that the Weyl fermions become more dominant in the transport properties with increasing RRR and that the contribution of the Weyl fermions on the Hall resistivity is negligibly small when the RRR is 8.93.
As described in the main text, the unsaturated linear positive MR also becomes more prominent with increasing RRR and decreasing temperature below T_{F}, indicating again that the nonlinear ρ_{xy}(B) is a hallmark of the existence of the Weyl fermions in SrRuO_{3} and that the Weyl fermions become dominant in the transport when scatterings from impurities and phonons are sufficiently suppressed.
Computational details
Electronic structure calculations were performed within the density functional theory and generalized gradient approximation (GGA, Perdew–Burke–Ernzerhof)^{87} for the exchange correlation functional in the projectoraugmented plane wave (PAW) formalism^{88} as implemented in the Vienna abinitio Simulation package^{89}. The energy cutoff was set to 500 eV, the Brillouin zone was sampled by an 8 × 8 × 6 Monkhorst–Pack mesh^{90}, and the convergence criterion for the electronic density was defined as 10^{−8} eV. The effect of electronic correlations in the Ru d shell (4d^{4} for Ru^{4+}) was taken into account by using the rotationally invariant GGA + U scheme^{91} with U = 2.6 eV and J = 0.6 eV. The choice of parameters is justified by early estimations^{92} and is in agreement with other studies^{93,94}.
Firstprinciples calculations of Weyl points
The orthorhombic phase of SrRuO_{3} has the Pbnm (#62) space group, which corresponds to the D_{2h} point group with symmetries of inversion (I), three mirror planes (m_{x}, m_{y}, m_{z}), and 180° rotations around the orthorhombic axes (C_{x}, C_{y}, C_{z}). The crystal structure parameters considered in the present study are a = 5.5670 Å, b = 5.5304 Å, c = 7.8446 Å,^{12} and the atomic Wyckoff positions in fractional coordinates are given as 4c (0.5027, 0.5157, 0.25) for Sr, 4b (0.25, 0, 0) for Ru, and 8d (0.7248, 0.2764, 0.0278) for O.
The results of electronic structure calculations with and without SOC are shown in Supplementary Fig. 8. One can see that the bands close to the Fermi level are formed by the Ru 4d states hybridized with the O 2p states and the electronic spectrum reveals halfmetallicity (Supplementary Fig. 8a) in agreement with previous electronic structure calculations^{59}. The ferromagnetic alignment was found to be the ground state configuration with the easy axis along the orthorhombic b axis (\(E_{010}  E_{001} =\)−2.17 meV/f.u. and \(E_{010}  E_{100} =\)−0.35 meV/f.u.), and the calculated magnetic moments per Ru ion (Supplementary Table 2) are close to the experimental saturation magnetic moment of 1.25 \(\mu _B\)/Ru.^{28} The ferromagnetic state reduces the symmetry to the C_{2h} point group with one mirror plane and one rotation axis symmetry, perpendicular and parallel to the magnetization direction, respectively.
For numerical identification of the Weyl points, one needs to have a band structure with high resolution in the reciprocal space. To interpolate the resulting electronic spectrum, we employed maximally localized Wannier functions as implemented in the wannier90 package^{95}. The wannierization was carried out by projecting the bands corresponding to the Ru e_{g} and t_{2g} states onto the atomic d orbitals in the local coordinate frame at each Ru site.
To locate the points of degeneracy between the bands in the reciprocal space, we performed a steepestdescent minimization of the gap function \({\mathrm{{\Delta}}} = (E_{n + 1,{\boldsymbol{k}}}  E_{n,{\boldsymbol{k}}})^2\)on a uniform grid of up to 31 × 31 × 31 covering the Brillouin zone, where the bands are considered degenerate when the gap is below the threshold of 10^{−5} eV^{96}. To eliminate accidental crossings, the corresponding chirality at each identified point is determined by evaluating the outward Berry flux enclosed in a small sphere. The calculated chiralities \(\chi\) should obey the following symmetry properties: \(\chi\) does not change its sign under 180° rotations around the magnetization axis and changes its sign under mirror reflection and inversion. In this study, we only consider the points with \(\chi = \pm 1\).
We have selected two pairs of bands I and II for the cases when the magnetization is along the orthorhombic c and b axes, as shown in Supplementary Fig. 9 and Supplementary Fig. 10, respectively. The corresponding gap function \({\Delta} \le 0.1\) eV is calculated to demonstrate the proximity of the selected bands. From the number of the calculated band crossings, the Weyl points were identified as the ones that have close energy positions and kpoint coordinates and whose chiralities obey the symmetry properties in the full Brillouin zone. The resulting Weyl points are listed in Supplementary Tables 3–6. Numerical differences in the coordinates of the Weyl points can be attributed to small spin canting (see Supplementary Table 2), which slightly breaks inversion symmetry (the reflection and rotation symmetries along to the magnetization are intact). From Supplementary Tables 3–6, most of the Weyl points are found to exist within an energy range of −0.2 to 0.2 eV around the Fermi level. In particular, \(\left {E  {E}_{\mathrm{F}}} \right\) for WP_{z}6_{1–4} (8–16 meV), which are located near the Y–T line in the Z–Γ–Y–T plane as shown in Fig. 5c, is very close to the experimental chemical potentials of the Weyl fermions estimated from the SdH oscillations (µ_{1} = 8.5 meV and µ_{2} = 8.8 meV for the F_{1} and F_{2} orbitals, respectively).
From the obtained results, one can clearly see the presence of the Weyl fermions below and above the Fermi level coexisting with trivial halfmetallic bands. However, it is worth commenting on another scenario. According to previous theoretical studies^{93,94}, there is no clear consensus on whether the electronic spectrum of orthorhombic SrRuO_{3} in the ferromagnetic state is halfmetallic or not, and the result turns out to depend on the details of electronic structure calculations. While our theoretical results are in good agreement with the present experiments, we do not rule out the possibility of a nonhalfmetallic behavior with both spin channels crossing close to the Fermi level. Assuming that the spinup states can also lie at the Fermi level, there will be an extra set of band crossings in addition to the Weyl points already defined in a halfmetallic scenario.
Finally, it is worth noting that a small monoclinic distortion induced by the SrTiO_{3} substrate breaks the orthorhombic D_{2h} symmetry. The reported crystal structure parameters of epitaxial SrRuO_{3} on SrTiO_{3} are a = 5.5290 Å, b = 5.5770 Å, c = 7.8100 Å, \(\alpha\) = 89.41°.^{12} According to our electronic structure calculations within GGA + U with U = 2.6 eV and J = 0.6 eV for the monoclinic SrRuO_{3} on SrTiO_{3} (Supplementary Fig. 11), the electronic spectrum does not reveal any qualitative changes from that for the orthorhombic D_{2h} symmetry, while additional band crossings may occur due to the lowered crystal symmetry.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Yuichi Onuma for valuable discussions. A part of this work was conducted at Advanced Characterization Nanotechnology Platform of the University of Tokyo, supported by “Nanotechnology Platform” of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. K.T. acknowledges the support from the Japan Society for the Promotion of Science (JSPS) Fellowships for Young Scientists and the Material Education program for the future leaders in Research, Industry, and Technology (MERIT).
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Y.K.W. conceived the idea, designed the experiments, and led the project. Y.K.W. and Y.K. designed and built the MBE system. Y.K.W., T.O., and H.S. implemented the Bayesian optimization algorithm for the sample growth. Y.K.W. grew the samples. Y.K.W. and K.T. carried out the sample characterizations. K.T., H.I., and Y.K.W. fabricated the Hall bar structures and carried out transport measurements. K.T. and Y.K.W. analyzed and interpreted the data. S.A.N. and H.D. carried out the electronicstructure calculations. K.T., Y.K.W., H.I., Y.K., T.O., H.S., S.A.N., H.D., M.T., Y.T., and H.Y. contributed to the discussion of the data. K.T. and Y.K.W. cowrote the paper with input from all authors.
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Takiguchi, K., Wakabayashi, Y.K., Irie, H. et al. Quantum transport evidence of Weyl fermions in an epitaxial ferromagnetic oxide. Nat Commun 11, 4969 (2020). https://doi.org/10.1038/s41467020186468
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