Abstract
Highmobility twodimensional carriers originating from surface Fermi arcs in magnetic Weyl semimetals are highly desired for accessing exotic quantum transport phenomena and for topological electronics applications. Here, we demonstrate highmobility twodimensional carriers that show quantum oscillations in magnetic Weyl semimetal SrRuO_{3} epitaxial films by systematic angledependent, highmagnetic field magnetotransport experiments. The exceptionally highquality SrRuO_{3} films were grown by stateoftheart oxide thin film growth technologies driven by machinelearning algorithm. The quantum oscillations for the 10nm SrRuO_{3} film show a high quantum mobility of 3.5 × 10^{3} cm^{2}/Vs, a light cyclotron mass, and twodimensional angular dependence, which possibly come from the surface Fermi arcs. The linear thickness dependence of the phase shift of the quantum oscillations provides evidence for the nontrivial nature of the quantum oscillations mediated by the surface Fermi arcs. In addition, at low temperatures and under magnetic fields of up to 52 T, the quantum limit of SrRuO_{3} manifests the chiral anomaly of the Weyl nodes. Emergence of the hitherto hidden twodimensional Weyl states in a ferromagnetic oxide paves the way to explore quantum transport phenomena for topological oxide electronics.
Introduction
Highmobility twodimensional carriers were first realized at semiconductor surfaces and interfaces, where they form the basis of the integer and fractional quantum Hall effects^{1,2,3,4}, twodimensional superconductivity^{5}, and practical applications such as highelectronmobility transistors^{6,7}. The twodimensional carriers originating from the surface Fermi arcs or the quantum confinement of bulk threedimensional Dirac fermions are reported for the Dirac semimetal Cd_{3}As_{2}^{8,9,10,11,12,13,14,15}. The highmobility twodimensional carriers are also pursued in Weyl semimetals and magnetic Weyl semimetals. Examples of Weyl semimetals and magnetic Weyl semimetals are TaAs^{16,17,18,19}, Na_{3}Bi^{20,21}, NbAs^{22,23}, Co_{3}Sn_{2}S_{2}^{24,25}, Co_{2}MnGa^{26}, and SrRuO_{3}^{27}. Since these states in topological semimetals are predicted to be robust against perturbations^{10,14,28,29}, they are of both scientific and technological interests, with potential for use in highperformance electronics, spintronics, and quantum computing. In particular, Weyl fermions in magnetic Weyl semimetals are thought to be more suitable for spintronic applications^{25,30,31}. Since the distribution of Weyl nodes in magnetic Weyl semimetals is determined by the spin texture^{27,32}, the highmobility twodimensional carriers originating from the Weyl nodes in magnetic Weyl semimetals are expected to be controllable by magnetization switching^{33} in addition to the electric field^{6,13}. To detect these highmobility twodimensional carriers in transport measurements, highquality epitaxial Weyl semimetal thin films, in which the contributions of surface states and/or quantumconfined bulk states are prominent, are needed. Conversely, the experimental detection of these Weyl states in transport has remained elusive due to the lack of highquality epitaxial Weyl semimetal films. Revealing the quantum transport properties of highmobility twodimensional 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 SrRuO_{3}^{27,32,34}, which is widely used as an epitaxial conducting layer in oxide heterostructures^{35,36,37,38,39,40,41,42,43,44,45,46}, 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 SrRuO_{3} has been reported in highquality epitaxial films^{27}, SrRuO_{3} provides a promising opportunity to realize highmobility twodimensional carriers in magnetic Weyl semimetals. However, their dimensional character remains unsolved due to the lack of angledependent quantum oscillations. Since the amplitude of the quantum oscillation becomes small for angles close to the inplane direction of the film, highmagnetic field transport measurements are indispensable to scrutinize the angular dependence of the quantum oscillations. Furthermore, highmagnetic field transport measurements will simultaneously provide information regarding the chiral anomaly in the quantum limit of Weyl fermions in SrRuO_{3}.
In this article, we present a systematic study of the angledependent magnetotransport, including quantum oscillations, up to 52 T in ultrahighquality SrRuO_{3} thin films with thicknesses of 10, 20, 40, 60, and 63 nm grown by machinelearningassisted molecular beam epitaxy (MBE)^{47,48}. The quantum oscillations in resistivity [i.e., Shubnikovde Haas (SdH) oscillations] for the 10nm film show the light cyclotron mass of 0.25m_{0} (m_{0}: free electron mass in a vacuum), high quantum mobility of 3.5 × 10^{3} cm^{2}/Vs, and twodimensional angular dependence, confirming the existence of highmobility twodimensional carriers in SrRuO_{3}. The quantum oscillations for the 63nm film, whose thickness is larger than the Fermi wavelength of the highmobility carriers (~22.4 nm), also show the twodimensional angular dependence, suggesting that the highmobility twodimensional carriers possibly come from the surface Fermi arcs, not the quantum confinement effect. The thicknessdependent phase shift establishes the nontrivial 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 chiralanomalyinduced negative MR in the quantum limit, which had never been observed in magnetic Weyl semimetals. These findings of the highmobility twodimensional 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
Highquality SrRuO_{3} thin films
We grew highquality epitaxial SrRuO_{3} films with thicknesses t = 10, 20, 40, 60, and 63 nm on (001) SrTiO_{3} substrates (Fig. 1a) in a customdesigned 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 machinelearning technique for parameter optimization^{47,49,50}, 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 singlecrystalline films were confirmed by θ–2θ xray diffraction (XRD) and atomic force microscopy (AFM) (Fig. 1b–d; see Methods section “Sample preparation” for details). Epitaxial growth of the highquality singlecrystalline SrRuO_{3} film with an abrupt substrate/film interface is seen in the electron energy loss spectroscopycross sectional scanning transmission microscopy (EELSSTEM) images (Fig. 1e).
Thicknessdependent magnetotransport
We performed magnetotransport measurements on the Hall bar devices of the SrRuO_{3} film (Fig. 1f) with t = 10 nm using the PPMS (see Methods section “Magnetotransport measurements” for details). In Fig. 2, the ρ_{xx}, MR (ρ_{xx}(B) − ρ_{xx}(0 T))/ρ_{xx}(0 T), and ρ_{xy} data for the SrRuO_{3} film with t = 63 nm reported in ref. ^{27}. are also shown for comparison. The temperature T dependence of the longitudinal resistivity ρ_{xx} of the SrRuO_{3} films is shown in Fig. 2a. The ρ_{xx} 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 SrRuO_{3} film with t = 10 nm is identical to that for the SrRuO_{3} film with t = 63 nm. This result indicates that bulk electronic conduction in electrical transport is dominant even in the 10nm thick film (see the Sec. I of the supplementary information). The ρ_{xx} vs. T curves show clear kinks (arrows in Fig. 2a), at which the ferromagnetic transition occurs, and spindependent scattering is suppressed^{41}. The magnetization measurement for the SrRuO_{3} 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 SrRuO_{3} films showed a T^{2} scattering rate (ρ ∝ T^{2}) that is expected for a Fermi liquid, in which electronelectron scattering dominates the transport and carriers are described as Landau quasiparticles^{27,41,51,52}. Although the RRR decreases from 81.0 to 19.2 with decreasing t from 63 to 10 nm, the RRR of the 10nm film is high enough to observe the intrinsic quantum transport of Weyl fermions in SrRuO_{3}^{27,53}. The availability of such highquality thin films allows obtaining insights into the dimensionality of the Weyl fermions in SrRuO_{3} via measurements of angledependent magnetotransport properties, as will be described in the “Highmobility twodimensional carriers” section. The larger residual resistivity ρ_{Res} at 2 K for the SrRuO_{3} film with t = 10 nm suggests a disorder near the interface between SrRuO_{3} and the SrTiO_{3} substrate^{53}. Figure 2b shows the MR ((ρ_{xx}(B) − ρ_{xx}(0T))/ρ_{xx}(0T)) for the SrRuO_{3} films with a magnetic field B applied in the outofplane [001] direction of the SrTiO_{3} 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 semimetals^{9,19,54,55,56,57} and is thought to stem from the linear energy dispersion of Weyl nodes^{58,59,60}. 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 ρ_{xy} (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 films^{27,51,53} and not in bulk samples^{61}. As will be described in the “Highmobility twodimensional 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 threedimensional Weyl fermions with high mobility and light cyclotron mass in our previous study^{27}. For clarifying the dimensionality of the SdH oscillations, we investigated the angular dependence of the magnetotransport properties in the SrRuO_{3} films. The angledependent magnetotransport measurements also provide experimental evidence of the chiral anomaly that Weyl fermions in SrRuO_{3} show.
Chiral anomaly in the quantum limit
We carried out the angledependent magnetotransport measurements for the SrRuO_{3} film with t = 10 nm (Fig. 3). Figure 3a shows the angular dependence of MR (ρ_{xx}(B) − ρ_{xx}(0 T))/ρ_{xx}(0 T) for the SrRuO_{3} film with t = 10 nm at 2 K measured by the PPMS (up to 14 T). Here, B is rotated from the outofplane [001] direction of the SrTiO_{3} substrate (θ = 90°) to the inplane 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 “Thicknessdependent 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 SrRuO_{3}, the timereversalsymmetry breaking lifts the spin degeneracy and leads to linear crossing of nondegenerate 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, B_{QL} represents the magnetic field at which the quantum limit is reached and all the Weyl fermions occupy the zeroth Landau levels. When B < B_{QL} (Fig. 3d), nonorthogonal electric and magnetic fields (E ∙ B ≠ 0) lead to the chiral charge transfer between the two Landau levels with opposite chiralities^{62,63}. In the weakB limit (B < B_{QL}; Fig. 3d), calculations based on the semiclassical Boltzmann theory predict that timereversalsymmetrybreaking Weyl semimetals (magnetic Weyl semimetals) show a negative MR that is linear in the projection component of B in the direction of I^{54,63,64}, in contrast to the quadratic dependence expected for spaceinversion symmetry breaking Weyl semimetals^{19,21,65}. 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 semimetals^{27}. In our previous study^{27}, similar chiralanomalyinduced negative MR has been observed for the SrRuO_{3} film with t = 63 nm.
To verify the negative MR stemming from the chiral anomaly, we carried out highfield angledependent magnetotransport measurements using the midpulse 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 impurities^{65,66,67}. Therefore, observing the behavior of the negative MR in the quantum limit (Fig. 3e), where the chiralanomalyinduced negative MR is predicted to saturate^{21,65,68}, is necessary. Figure 3b shows the angular dependence of the highfield magnetotransport for t = 10 nm at 0.7 K with B rotated from the outofplane [001] direction of the SrTiO_{3} substrate to the inplane 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
where N_{W} 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 fielddependent internodal scattering time^{62,68}. 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 shortrange impurity scattering, and the scattering factor cancels the factor of \(l_{{{\mathrm{B}}}}^2\) in Eq. (1), resulting in the Bindependent conductivity^{65,68,69}. Indeed, the saturation of the chiralanomalyinduced negative MR above the quantum limit (B > B_{QL}) has been observed in typical Weyl semimetals Na_{3}Bi^{21} and TaAs^{65} 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 “Firstprinciples calculations” for details). According to our calculations, the tilt of the linear band crossings is small and preserves an elliptical Weyl cone with the pointlike Fermi surface. As shown in previous theoretical studies^{64}, tilting of the Weyl nodes can result in a onedimensional 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 onedimensional chiral anomaly as a distinct feature of timereversal symmetry broken (e.g., magnetic) Weyl semimetals.
In principle, the chiral anomaly induced negative MR should be accompanied by SdH oscillations originating from the bulk threedimensional Weyl fermions, because the chiral anomaly stems from the Landau quantization of the bulk threedimensional Weyl fermions. However, oscillations originating from the bulk threedimensional Weyl fermions were not clear in the SrRuO_{3} samples studied here, as will be described in the “Highmobility twodimensional 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 oscillations^{70,71}. In fact, in Dirac semimetal Na_{3}Bi and zerogap semiconductor GdPtBi, chiralanomalyinduced MR without quantum oscillations of the bulk threedimensional Weyl fermions has been reported^{72}. Accordingly, further improvement in the crystallinity of SrRuO_{3} is required to observe clear SdH oscillations of the bulk threedimensional Weyl fermions, which will allow for detailed characterization of the Fermi surface. For instance, using a substrate having smaller lattice mismatch with SrRuO_{3}, e.g. DyScO_{3} and Sr_{1−x}Ba_{x}TiO_{3}, appears to be a promising means since the SrRuO_{3} film on the DyScO_{3} substrate has fewer defects owing to the smaller lattice mismatch^{73}.
Highmobility twodimensional carriers
To study the dimensionality of the SdH oscillations, we investigated their angular dependence in the SrRuO_{3} 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 inplane direction is a typical feature of twodimensional carriers^{6,12,74}. 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 twodimensional carriers^{6,10,12,74}. 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 LifshitzKosevich theory at each angle as follows:^{9,27,75,76}
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 0.25m_{0} (m_{0}: the free electron mass in a vacuum) (Fig. 5c). We also determined the Dingle temperature T_{D} from the LifshitzKosevich 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 × 10^{3} cm^{2}/Vs. These results confirm the existence of the highmobility twodimensional carriers with a light cyclotron mass in the SrRuO_{3} film with t = 10 nm. The F and m^{*} values (F ~25 T and m^{*} = 0.25m_{0}) of the SrRuO_{3} film with t = 10 nm are, respectively, the same as and comparable to those of the main component of the SdH oscillations in the SrRuO_{3} film with t = 63 nm (F ~26 T and m^{*} = 0.35m_{0}) reported previously^{27}. Notably, the \(\mu _{{{\mathrm{q}}}}\) value of the SrRuO_{3} film with t = 10 nm is about one third of that in the SrRuO_{3} film with t = 63 nm (\(\mu _{{{\mathrm{q}}}} =\) 9.6 × 10^{3} cm^{2}/Vs). The reduced \(\mu _{{{\mathrm{q}}}}\) value in the former film suggests that disorder near the interface between SrRuO_{3} and SrTiO_{3} substrate is the dominant scattering mechanism, which shortens the quantum lifetime of the high mobility twodimensional carriers in the SrRuO_{3} film with t = 10 nm. This disorder may be the origin of the shortrange scattering mechanism postulated in the theory of the saturation of the negative MR in the quantum limit^{65,68,69}.
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 midpulse magnet up to 52 T in the SrRuO_{3} 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 2.4m_{0} (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 E_{F}, which were reported in an early de Haas–van Alphen measurement^{61} and an SdH measurement using a standard lockin technique at 0.1 K up to 14 T^{27}. 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 twodimensional carriers in the highB region above the SdH frequencies. The frequency obtained by the highfield 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 SrRuO_{3} film with t = 10 nm at 0.7 K using a longpulse 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 midpulse 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 midpulse magnet measurements, as shown in Supplementary Fig. 6b. All the results from PPMS, mid, and longpulse magnet measurements converge and are well fitted with an angulardependence curve for twodimensional carriers (Fig. 6c), strongly indicating the existence of the twodimensional Weyl fermions in the SrRuO_{3} film with t = 10 nm. Altogether, the existence of highmobility twodimensional carriers, which had never been observed in ferromagnets, is established in the epitaxial ferromagnetic oxide SrRuO_{3}.
Origin of the twodimensional carriers: surface Fermi arcs
Possible origins of the highmobility twodimensional carriers in Weyl semimetals are the surface Fermi arcs (Fig. 7) and the quantum confinement of the threedimensional Weyl fermions^{8,9,10,11,12,13,14,15,77}. 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, respectively^{6,12,77}. 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 highmobility twodimensional carriers for the SrRuO_{3} 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 SrRuO_{3} 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, respectively^{27}. To determine the origin of the twodimensional quantum transport, we also studied the angular dependence of the SdH oscillations in the SrRuO_{3} 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 Cd_{3}As_{2}, suppression of backscattering, which results in a transport lifetime 10^{4} times longer than the quantum lifetime, has been reported^{78}. Therefore, the mean free path estimated by quantum lifetime may also be underestimated in SrRuO_{3}. Figure 8 shows the angular dependence of the SdH oscillations in the SrRuO_{3} film with t = 63 nm. As in the case of the SrRuO_{3} 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 twodimensional 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 twodimensional high mobility carriers. This result suggests that the high mobility twodimensional 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 twodimensional trend, indicating that these SdH oscillations are different from those of the high mobility twodimensional 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 threedimensional Weyl fermions. Since the SdH oscillations from the surface Fermi arcs are connected via the Landau quantization of the bulk threedimensional Weyl fermions^{77}, further improvement in the crystallinity of SrRuO_{3} will allow for detailed characterization of the SdH of the bulk Weyl fermions and their relevance to the surface Fermi arcs.
The thicknessdependent phase shift of the SdH oscillations will provide essential evidence for the nontrivial nature of the SdH oscillations mediated by the surface Fermi arcs. According to the semiclassical analysis of the closed magnetic orbit, called Weyl orbit^{79,80,81,82}, formed by connecting the surface Fermi arcs via the Weyl nodes in bulk^{10,77}, the thickness t dependence of the position of the Nth conductivity maximum B_{N} is expressed by
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 thicknessdependent 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
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 nontrivial 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 backgroundsubtracted SdH oscillations at 2 K with B (3.3 T < B < 14 T) applied in the outofplane [001] direction of the SrTiO_{3} substrate for the SrRuO_{3} 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 nontrivial 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 highmobility twodimensional 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.
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 SrRuO_{3}, our findings will stimulate a challenge for its observation by angleresolved 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 highquality SrRuO_{3} films and successfully detected highmobility twodimensional 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 twodimensional highmobility carriers and chiralanomalyinduced negative MR in the quantum limit in magnetic Weyl semimetals and magnetic oxides. The interface with the SrTiO_{3} substrate and the surface of the SrRuO_{3} films might be subject to slight structural differences, such as octahedral distortion and rotation^{84,85,86,87}. However, our previous band structure calculations showed that a slight distortion induced by the substrate does not significantly change the band structure of SrRuO_{3} and that Weyl points are robust with the distortion as long as the band inversion is present^{27}. In addition, generally, magnetic anisotropy in magnetic Weyl semimetals should affect angledependent MR originating from the quantum transport of Weyl fermions since the distribution of Weyl nodes in magnetic Weyl semimetals is determined by the magnetic arrangement^{27}. Especially in SrRuO_{3} films, the magnetic anisotropy can be controlled via epitaxial strain, and therefore, SrRuO_{3} will be a platform for exploring the relationship between magnetic anisotropy, quantum transport properties of Weyl fermions, and chiralanomalyinduced 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 SrRuO_{3} include those with various physical properties such as insulators (e.g. SrTiO_{3})^{41}, metals (e.g. Nb:SrTiO_{3})^{41}, ferroelectrics (e.g. BaTiO_{3})^{41}, and superconductors (e.g. Sr_{2}RuO_{4})^{88}. Utilizing magnetic Weyl states in these epitaxial heterostructures may enable functionalities^{89,90} and thus widen the accessible oxide electronics devices. In addition, the emergence of the hitherto hidden twodimensional 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 preprint^{91}, which was uploaded later than the preprint of this article (arXiv:2106.03292), that report similar 2Dlike 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 highquality epitaxial SrRuO_{3} films with thicknesses t = 10, 20, 40, 60, and 63 nm on (001) SrTiO_{3} substrates (Fig. 1a) in a customdesigned MBE setup equipped with multiple ebeam evaporators for Sr and Ru^{27,47}. The SrRuO_{3} films were prepared at the Ru flux = 0.365 Ås^{−1}, growth temperature = 781 °C, and O_{3}nozzletosubstrate 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 machinelearning technique for parameter optimization^{47,49,50}, 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 electronimpactemissionspectroscopy sensor, and the stoichiometric Sr/Ru ratio of a SrRuO_{3} film was confirmed using energy dispersive xray spectroscopy^{92}. Figure 1b shows the θ2θ XRD patterns around the (002) pseudocubic diffractions of the SrRuO_{3} film with t = 63 nm. Laue fringes in the θ2θ XRD pattern indicate high crystalline quality and a large coherent volume of the film. The outofplane lattice constant, estimated from the NelsonRiley extrapolation method^{93}, was 3.950 Å, ∼0.5% larger than the pseudocubic lattice constant of the bulk specimens (3.93 Å)^{41}. This indicates that the SrRuO_{3} film is compressively strained due to the lattice constant of the SrTiO_{3} substrate (3.905 Å) being ∼0.6% smaller than the pseudocubic lattice constant of SrRuO_{3}. 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 rootmeansquare roughness is 0.17 nm, indicating that the SrRuO_{3} film has an atomically smooth surface. The SrRuO_{3} film with t = 63 nm has an orthorhombic structure with twofold inplane symmetry, as evidenced by the XRD ϕ scans in our previous study^{73}. Epitaxial growth of the highquality singlecrystalline SrRuO_{3} film with an abrupt substrate/film interface is seen in EELSSTEM images (Fig. 1e). Further information about the MBE setup, preparation of the substrates, and sample properties are described elsewhere^{27,53,94,95,96}.
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 SrRuO_{3} 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. Highfield magnetotransport measurements up to 52 T and 40 T were performed using a nondestructive midpulse magnet with a pulse duration of 36 ms and longpulse 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 Tokyo^{97}.
Firstprinciples calculations
Electronic structure calculations were performed within density functional theory by using generalized gradient approximation^{98} for the exchange correlation functional in the projectoraugmented plane wave formalism^{99} as implemented in the Vienna Abinitio Simulation Package^{100}. The energy cutoff was set to 500 eV, the Brillouin zone was sampled by an 8×8×6 MonkhorstPack mesh^{101}, and the convergence criterion for electronic density was put to 10^{–8} eV. Experimental crystal structure of orthorhombic SrRuO_{3} 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 spinorbit coupling. The effect of electronic correlations in the Ru 4d shell was considered by using the rotationally invariant GGA + U scheme^{102} 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 package^{103}. Band crossings in the reciprocal space were calculated by steepestdescent 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 study^{27}.
Fourier transformation analysis for SdH oscillations
We measured the longitudinal resistance R_{xx} and Hall resistance R_{xy} of the Hall bar patterned SrRuO_{3} films (width w = 200 μm, length l = 350 μm). The longitudinal resistivity ρ_{xx} = R_{xx}wt/l and Hall resistivity ρ_{xy} = R_{xy}t were obtained from the R_{xx} and R_{xy} values. From ρ_{xx} and ρ_{xy}, we obtained the longitudinal conductivity σ_{xx} = ρ_{xx}/(ρ_{xx}^{2} + ρ_{xy}^{2}) of the conductivity tensor. We subtracted the background of the conductivity from the σ_{xx} 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 xaxis (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.
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Acknowledgements
S.K.T. acknowledges the support from the Japan Society for the Promotion of Science (JSPS) Fellowships for Young Scientists. H.D. acknowledges the support from GrantsinAid for Scientific Research No. JP19K05246 and No. JP19H05625 from the Japan Society for the Promotion of Science (JSPS).
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Y.K.W. conceived the idea, designed the experiments, and directed and supervised the project. Y.K.W. and Y.Kro. grew the samples. S.K.T. and Y.K.W. carried out the sample characterizations. S.K.T., Y.K.W., and H.I. fabricated the Hall bar structures. S.K.T., Y.K.W., and K.T carried out the magnetotransport measurements. S.K.T., Y.K.W., T.N., and Y.Koh. performed the highfield magnetotransport measurements. S.K.T. and Y.K.W. analyzed and interpreted the data. S.A.N. and H.D. carried out the electronicstructure calculations. S.K.T. and Y.K.W. cowrote the paper with input from all authors.
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KanetaTakada, S., Wakabayashi, Y.K., Krockenberger, Y. et al. Highmobility twodimensional carriers from surface Fermi arcs in magnetic Weyl semimetal films. npj Quantum Mater. 7, 102 (2022). https://doi.org/10.1038/s41535022005110
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DOI: https://doi.org/10.1038/s41535022005110
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