## Introduction

The phenomena of quantum oscillations with magnetic field in matters were first discovered in bismuth more than 90 years ago1, and it was soon realized that such phenomena can be used as a measure for the Fermi surface in momentum space1,2. Quantitatively, quantum oscillatory phenomena are described by the general expression of ΔρDcos[$$2\pi (\frac{F}{{\mu }_{0}H}+\delta )$$], where D is the damping factor, F is the oscillation frequency, and δ is the phase shift of the oscillation associated with individual extremal areas of the Fermi surface in momentum space. In theory, the absolute value of δ is a consequence of the underlying band structure and dimensionality of the material under study in which the Berry phase is derived as 0 or π. From a modern point of view, the periodic oscillation of either ρ or magnetization with respect to the inverse of a magnetic field is a quantum effect deriving from the Bohr-Sommerfeld quantization rule on electrons’ cyclotron motions, where the requirement of the single-valued wave function gives rise to discrete energies for electrons3. The oscillation frequency (F) can be linked directly to the extremal cross-sectional area (Ae) of the Fermi surface by Onsager’s relation of F = Ae/2πe. In addition, the amplitude and phase of quantum oscillations encode the information about the electronic band topology at the Fermi surface2 that has been a major subject recently in the field of topological materials4,5.

In topological Dirac and Weyl semimetals, a nontrivial conical band with linear dispersion results in a π Berry phase that is picked up during the cyclotron motion and thus causes a phase shift by half a period in the quantum oscillation, which has been reported by several quantum oscillations experiments6,7. One practical approach for the determination of δ is to construct the Landau fan diagram, which plots the 1/μ0H values for peaks and valleys in the quantum oscillations (1/μ0Hn) as a function of the corresponding Landau level (LL) index n. δ can then be determined from the intercept (n0) as 1/μ0Hn approaches zero by extrapolating the curve of 1/μ0Hn versus n. As pointed out by several authors8,9, the peaks (valleys) in the ρ (conductivity) oscillations should be assigned with integer numbers of n for a three-dimensional (3D) system, and thus δ = n0 ≈ ±1/8 and ±5/8 for a conical band with π Berry phase and a trivial parabolic band with zero Berry phase, respectively.

On the other hand, Weyl semimetals (WSMs) are also known for their unusual Fermi-arc surface states10,11 that connect the projected pair of Weyl nodes on a surface. An unusual type of quantum oscillation due to the so-called Weyl-orbit effect (WOE) was first theoretically predicted12 and soon demonstrated by experiments in Dirac semimetal Cd3As213,14,15. The WOE results from an intriguing collaboration between the Fermi-arc surface states and the Weyl-node bulk states, where a nonlocal cyclotron orbit gives rise to a new type of quantum oscillation. One important consequence for WOE is the thickness dependence on the phase of the quantum oscillations, but, unfortunately, such a phase shift can be easily smeared out by a small density variation in different samples16. It is, therefore, important to grow high crystalline thin films of WSM, and the WOE can then be carefully investigated on WSM films with different thicknesses. While there are recent reports about possible topological and correlated phases in ruthenates17,18, the ferromagnetic oxide of SrRuO3 (SRO)19,20 appears to be another promising candidate. The nonmonotonic temperature (T) dependence of both anomalous Hall conductivity21 and spin wave gap in SRO22 suggest the presence of the Weyl nodes near the Fermi surface that coexists with other 3D bulk Fermi pockets, making SRO a candidate for the Weyl metal phase. In addition, quantum oscillations of ρ in SRO thin films on SrTiO3 (STO) (001) substrate also revealed an unusual F ≈ 30 T23,24 that is equivalent to a small Fermi pocket with a bulk density of only 9.3 × 1017 cm−3 for the 3D case. Such a small Fermi pocket seems to be in line with the Weyl metal phase in SRO, but its origin remains an open question. In this work, a series of highly crystalline and untwinned SRO films with different thicknesses (ts) ranging from about 7.7–35.3 nm were grown on miscut STO (001) substrates using adsorption-controlled growth by an oxide molecular beam epitaxy (MBE) system25,26, where pronounced quantum oscillations in ρ were observed. From detailed angle-dependent quantum oscillations with applied fields up to 35 T, we found that the small Fermi pocket of F ≈ 30 T behaved as a two-dimensional (2D) like Fermi pocket, and the corresponding Landau fan diagram showed a clear concave downward curvature for the curve of 1/μ0Hn versus n. Those results are discussed and compared to the simulated thickness-dependent quantum oscillations based on WOE.

## Results

### Structure characterizations

The crystal structure of perovskite SRO is illustrated in the left panel of Fig. 1a, where the rotation and tilting of RuO6 octahedra give rise to an orthorhombic phase at room temperature. When growing on an STO (001) substrate, a compressive strain of about −0.4% in SRO films occurs and gives rise to a distorted orthorhombic phase with Pbnm space group and the crystalline axis of SRO [110]o nearly aligned along the STO [001]c, where the subscripts o and c refer to the orthorhombic- and cubic-phase, respectively19. The right panel of Fig. 1a shows a cross-sectional scanning transmission electron microscope (STEM) high-angle annular dark-field (HAADF) image of an SRO film we grew, indicating an atomically sharp interface between SRO and STO. High-resolution X-ray measurements were performed, and the left panel of Fig. 1b displays the STO (003)c crystal truncation rods (CTRs) for an SRO film with t ≈ 13.7 nm. The abscissa L is in unit of STO reciprocal lattice unit (r.l.u.), representing the momentum transfer along the surface normal. The SRO (330)o reflection appeared at a slightly low-L side of STO (003)c reflection25. The presence of pronounced thickness fringes around the SRO Bragg reflection indicates excellent film crystallinity and sharp interfaces. The oscillation period of about 0.031 r.l.u. gives a crystalline thickness of about 12.6 nm, which is close to t ≈ 13.7 nm as obtained from the X-ray low-angle reflectivity measurements. The right panel of Fig. 1b shows the CTR of SRO (221)o off-normal reflection. The observation of the orthorhombic specific SRO (221)o reflection with pronounced intensity oscillations confirmed the orthorhombic phase and also excellent thickness uniformity and crystallinity of the SRO film27. The corresponding azimuthal ϕ scan across the SRO (021)o reflection is shown in Fig. 1c, where two sets of equally spaced peaks with different peak intensities were observed. This indicates the presence of four orthorhombic twin domains28 as illustrated in Fig. 1d. Be aware that the [110]o direction of orthorhombic SRO can be considered as parallel to the normal of (110)o plane because of the tiny (<0.8%) difference between the length of its a- and b-axis that was highly exaggerated in Fig. 1d. From the peak intensity, the volume fractions for domains A–D are estimated to be about 94.1%, 1.7%, 2.0%, and 2.2%, respectively, which thus justifies the nearly single-domain and untwinned structure in our SRO films. Detailed investigations of the orthorhombic twin domains for SRO films grown on miscut STO (001) substrates have been reported in a separate paper26. On the other hand, the surface morphology of the SRO film with t ≈ 13.7 nm was measured by an atomic force microscope (AFM) as shown in Fig. 1e, where the lower panel shows the cross-section height profile across the red line located in the upper panel of Fig. 1e. The step heights at the terrace edges are about 1–2 unit cells of STO, which again supports for the high thickness uniformity of the SRO film. However, we do note the presence of well-isolated pits that are known to exist for the step-flow growth of SRO film19. By excluding those pits, the averaged roughness of the SRO film can be obtained from the AFM images, and the resulting t-dependence of the averaged roughness is shown in Fig. 2f. For t ≤ 35.3 nm, the average roughness of the film was determined to be less than about 0.3 nm (see Supplementary Note 1). We also note that the dominating domain is found to have SRO [001]o (the black arrow in Fig. 1e) along the terrace edge on an STO substrate. For the growth of SRO films in this work, all the STO substrates were chosen to have a small miscut angle of about 0.1o with the terrace edge being parallel to a lateral STO < 100 > direction to ensure the single-domain in our SRO films26. In addition, the bias current ($$\overrightarrow{{{{\bf{I}}}}}$$) for the transport measurements was set to be perpendicular to the terrace edge as demonstrated in Fig. 2a, i.e. $$\overrightarrow{{{{\bf{I}}}}}\perp$$ SRO [001]o, in order to avoid complication and inconsistency due to possible ρ anisotropy and domain dependence in SRO.

### Magnetotransport measurements

The T-dependent ρ exhibits a metallic behavior as shown in Fig. 2b for SRO films with different ts ranging from about 7.7–35.3 nm. The kink at T ≈ 150 K indicates the occurrence of a ferromagnetic transition, which does not vary significantly with t within the range of our interests. At lower temperatures, ρ follows a T2 dependence as predicted for a Fermi liquid system23, and ρ at T = 5 K (ρ(5 K)) shows a progressive increase from ρ(5 K) ≈ 3.8–12.7 μΩ cm as the SRO film thickness decreases from t ≈ 35.3–7.7 nm. ρ in ruthenates is known to show high sensitivity to disorder29, and thus the low ρ(5 K) values indicate the high crystallinity of our SRO films. A positive transverse magnetoresistance (MR) with an applied field $$\overrightarrow{{{{\bf{B}}}}}\parallel$$ SRO [110]o was observed for thicker SRO films with t ≥ 10 nm as shown in the upper panel of Fig. 2c for T = 2.5 K, and the MR, defined as [ρ(H)/ρ(H = 0)]−1, can be as large as +74 % at μ0H = 14 T for SRO film with t ≈ 17.6 nm. It gradually decreases with decreasing t, giving an MR of about +24% at μ0H = 14 T for SRO film with t ≈ 11.7 nm. For t ≈ 7.7 nm, the MR first decreases with increasing field to a value of −3% at μ0H ≈ 3.6 T above which increases with field instead, giving a small positive MR ≈ +4% at μ0H = 14 T. On the other hand, the field-dependent Hall resistivity (ρxy) shows a sign change in the Hall slope (dρxy/dμ0H) from negative to positive as μ0H goes from 0 to 14 T, which was plotted in the lower panel of Fig. 2c for T = 2.5 K. Pronounced quantum oscillations were clearly observed in both ρ and ρxy for μ0H ≥ 3 T, which is only possible for highly crystalline SRO films with low residual resistivity23,24. When tilting $$\overrightarrow{{{{\bf{B}}}}}$$ toward $$\overrightarrow{{{{\bf{I}}}}}$$ by an angle θ as illustrated in the inset cartoon of Fig. 2d, the MR changes from being positive to negative as θ goes from 0o to 90o, as shown in the upper panel and lower panel of Fig. 2d for SRO films with t ≈ 13.7 and 18.7 nm, respectively, at T ≈ 0.3 K. Remarkably, the transverse MR (θ = 0o) is practically H-linear without any tendency of saturation, giving a positive MR of about +158% and +117% for t ≈ 13.7 and 18.7 nm, respectively, at μ0H = 35 T. On the other hand, for θ = 90o with $$\overrightarrow{{{{\bf{B}}}}}\parallel$$$$\overrightarrow{{{{\bf{I}}}}}$$, apparent negative MRs were observed in the low field regime, giving a negative MR of about −20% and −5% at μ0H ~ 15 T for t ≈ 13.7 and 18.7 nm, respectively. The H-linear transverse MR and the negative longitudinal MR are reminiscent of the MR behaviors commonly observed in topological Dirac and Weyl semimetals5,24,30,31.

The resulting field-dependent conductivity ($$\sigma \equiv \rho /({\rho }^{2}+{\rho }_{{{{xy}}}}^{2})$$) for the three different ts ranging from 9.0 to 18.7 nm at T = 0.3 K is shown in the upper panel of Fig. 3a. In zero field, σ equals about 9.3 × 106 Ω−1 m−1 for t ≈ 9.0 nm, and it increases with t, giving a σ ≈ 22.6 × 106 Ω−1 m−1 for t ≈ 18.7 nm. σ drops with increasing fields up to 35 T, and the corresponding dσ/dH versus μ0H was plotted in the lower panel of Fig. 3a, where pronounced quantum oscillations were observed for μ0H ≥ 3.5 T. By taking the fast Fourier transform (FFT) on the curves of dσ/dH versus 1/μ0H, several distinct frequencies of Fs1 and F1−5 can be clearly identified from the FFT spectra as shown in Fig. 3b. The oscillations with F1−5 frequencies ranging from ≈ 300 to 7400 T appeared to show up when μ0H ≥ 12 T with the oscillation amplitudes grow with increasing film thickness t. On the other hand, the oscillation for Fs1 ≈ 30 T starts at a much lower field of ≈3.5 T, and it appears to vanish when the field is above a critical value of Bc ~ 15 T. Following the standard practice, the effective mass (m*) and Dingle temperature (TD) can be determined from the damping of the quantum oscillations1. From the temperature-dependent measurements, the effective mass was determined to be m* = (0.30 ± 0.01)me for Fs1, where me is the electron mass. On the other hand, the extracted m* values for the Fermi pockets with F ≈ 300 and 3680 T are more than tenfold larger with m* = (3.2 ± 0.3) and (5.5 ± 0.8)me, respectively (see Supplementary Note 2). A summary of the extracted parameters from quantum oscillation data is shown in Table 1. τq ≡ /2πkBTD is the quantum lifetime. The resulting q ≡ (kF/m*)τq is the quantum mean free path, and the Drude mean free path d was calculated using a bulk density of about 6.1 × 1021 cm−3 for all ts. At T = 2.5 K, q falls in a range from 31 to 62 nm with a relatively weak dependence on t as demonstrated in the middle panel of Fig. 3c. On the contrary, d at T = 2.5 K monotonically increases by nearly four-fold from 31 to 118 nm as t increases from 7.7 to 35.3 nm.

Figure 3c summarizes the thickness dependence of residual resistivity ratio (RRR ≡ ρ(300 K)/ρ(5 K)) (upper panel), mean free path (middle panel), and FFT amplitudes (lower panel). As t increases from 7.7 to 35.3 nm, the RRR follows an increasing trend from 14.4 to 51.6, and the corresponding ρ(2.5 K) decreases from 12.3 to 3.2 μΩ cm. The observed values of Fs1, Fs2, F3, and F4 show relatively weak t-dependence, but the FFT amplitude for Fs1 appears to show nonmonotonic t dependence. In particular, for 10 ≤ t ≤ 20 nm, the FFT amplitude for Fs1 is maximal with a weak t dependence, regardless of a nearly 50% variation in RRR as shown in the upper panel of Fig. 3c. This is in sharp contrast to the nearly t linear dependence of FFT amplitudes for F3 and F4 as expected for 3D bulk pockets with increased RRR and t up to about 20 nm, and, in principle, it shall increase monotonically with larger t for t ≥ 20 nm that calls for more high field measurements up to 35 T. Nevertheless, from the current data shown in the lower panel of Fig. 3c, such a nonmonotonic t dependence on the FFT amplitude indicates that the quantum oscillation for Fs1 is unusual and more likely to be the surface origin.

### Electronic band calculations

We have calculated electronic structures using first-principles methods32,33,34,35,36. For U = 3.4 eV and J = 0.68 eV, we obtain a ferromagnetic (FM) ground state with the magnetic moment on Ru sites close to 1.20μB, which agrees well with our experiments and previous DFT study37. Moreover, we find that the ground state energy of the system with the magnetic moment oriented along [110]o direction is slightly lower than that along [001]o direction ($${E}_{110}^{{{{\rm{FM}}}}}-{E}_{001}^{{{{\rm{FM}}}}}\approx -0.546$$ meV per f.u.), which is consistent with the observed magnetic easy axis along [110]o in our SRO films. The calculated band structures are shown in Fig. 4a, and a number of Weyl nodes were clearly identified when searching in four subband pairs near the Fermi energy (see Supplementary Note 3). In order to identify the Fermi pockets for the observed quantum oscillations shown in Fig. 3, the kx and ky in the original momentum space were transformed into k and k with k lying in the (110)o plane. The calculated Fermi surface (FS) sliced along kkz plane with k = 0 is shown in Fig. 4b with the center being the Γ point. We identified three major pockets of A–C with areas of 48, 72, and 53 nm−2, respectively. The corresponding frequencies are 5077, 7531, and 5517 T, which are relatively close to the observed F3 ≈ 3680 T, F5 ≈ 7400 T, and F4 ≈ 4000 T, respectively, as shown in the lower panel of Fig. 3b. The A and B pockets are well consistent with the observed FS from an earlier angle-resolved photo-emission experiment38. We notice several smaller pockets between the A and B, but, unlike the three major pockets, they changed dramatically as k varies across the Γ point (see Supplementary Note 4). Therefore, those smaller pockets may not contribute coherently to give observable quantum oscillations. We further moved on to look for possible nonoverlapping Weyl nodes when projecting on (110)o plane. The energy dependence of the Weyl nodes at (k, kz) within a window of EEF ≤ 50 meV is shown in Fig. 4c, where the squares and triangles are Weyl nodes from subband pair II and subband pair III, respectively, with the red (blue) color being the chirality of +1 (−1). A number of nonoverlapping Weyl nodes projected on kkz plane are clearly observed. By selecting the projected Weyl nodes at similar energies with opposite chiralities, four Weyl-node pairs are identified and shown in Fig. 4d, where the black arrows represent wave vectors connecting Weyl-node pairs from +1 to −1. The corresponding Fermi-arc lengths fall in a range from k0 ≈ 0.8–6.5 nm−1. Those band calculation results support the presence of Fermi-arc surface states on (110)o plane, which is essential for the occurrence of WOE. However, we remark that, due to the complex band crossings near the Fermi energy, the Weyl node locations are highly sensitive to several parameters used for the band calculations. Nevertheless, nonoverlapping Weyl-node pairs projected on (110)o plane are always present for different U values ranging from 2 to 3.4 eV (see Supplementary Note 3).

### Quantum oscillations analyses

For better revelations of the quantum oscillations in σ without subtracting any artificial background, we plot $$-{{{{\rm{d}}}}}^{2}\sigma /{{{\rm{d}}}}{(1/{\mu }_{0}H)}^{2}$$ as a function of 1/μ0H as shown in Fig. 5a for SRO films with different ts ranging from about 7.7 to 35.3 nm. The peak and valley locations in the oscillation turn out to show systematic phase shifts to lower 1/μ0H values as t increases, indicated by long dashed lines in Fig. 5a, and an integer number of LL index (n) was assigned to the minimum of −d2σ/d(1/H)25,8. The corresponding FFT spectra for SRO films with different t values are shown in Fig. 5b. A dominant frequency of Fs1 ≈ 30 T was clearly observed, and another frequency of Fs2 ≈ 50 T was also observable with a much smaller FFT amplitude and also broader distribution in the FFT spectra. Therefore, the quantum oscillations for μ0H ≤ 14 T as shown in Fig. 5a are dominated by an unusually small Fermi pocket associated with Fs1, which turns out to show the largest magnitude for t ≈ 17.6 nm. A decrease of the FFT amplitude occurs for t > 20 nm and t < 10 nm as is evident from the results in Fig. 5a. In order to further probe the Fermi pocket associated with Fs1, detailed angle dependence of the quantum oscillations was performed with γ-rotation and θ-rotation setups as illustrated in the inset of Fig. 5e, where $$\overrightarrow{{{{\bf{B}}}}}$$ was tilted toward $$\overrightarrow{{{{\bf{I}}}}}$$ and SRO [001]o for θ-rotation and γ-rotation, respectively. θ and γ values are the angles between $$\overrightarrow{{{{\bf{B}}}}}$$ and the surface normal of SRO films, i.e. SRO [110]o that appears to be the magnetic easy axis with the lowest coercive field (see Supplementary Note 5). The resulting FFT spectra of γ-rotation and θ-rotation for SRO film with t ≈ 17.6 nm and fields up to 14 T at T = 2.5 K are shown in the upper panel and lower panel, respectively, of Fig. 5c for angles ranging from 0o to 60o. As the angle increases from 0o to 60o, Fs1 gradually shifts to higher frequencies for both θ-rotation and γ-rotation. Figure 5d shows the θ and γ dependent quantum oscillations of dρ/d(μ0H) versus μ0H for SRO thin film with t ≈ 18.7 nm and fields up to 35 T at T = 0.3 K. For angles larger than about 30o, the bulk frequencies of F1–5 damp out rapidly as expected in thin films with increased surface scatterings. The extracted Fs1 as a function of γ and θ are summarized in Fig. 5e and f, respectively, where Fs1(5.5, 5) and Fs1(5.5, 4.5) are extracted from the oscillation periods of (1/μ0H5.5−1/μ0H5) and (1/μ0H5.5−1/μ0H4.5), respectively. We included the results for the SRO film with t ≈ 18.7 nm shown as diamond and downward-triangle symbols in Fig. 5e, f, which was measured with applied fields up to 35 T. However, we do note that, at some angles of around 30o, Fs1 in the FFT spectra seems to split into two frequencies but then merged into a single peak again at higher angles. Nevertheless, the extracted Fs1 values for all SRO films with different ts tend to follow the 1/cosθ or 1/cosγ dependence (red-dashed lines in Fig. 5e, f), indicating a 2D-like nature of the Fermi pocket for Fs1 (see Supplementary Note 6).

By taking the oscillation period of adjacent peaks and valleys in Fig. 5a for θ = 0o, the extracted Fs1 values as a function of the LL index n are shown in the upper panel of Fig. 6a for SRO films with t ≈ 13.7 and 17.6 nm. Here, n was determined by setting Bc as the field above which the quantum limit occurs. Fs1 marches down from about 34.5 T for n = 8.5–28 T for n = 4, and this behavior is consistent with the observed downward bending of the 1/μ0Hnn curve in the Landau fan diagram shown in the lower panel of Fig. 6a. Surprisingly, the intercept n0, obtained by linearly extrapolating the high n data (black-dashed line), gives an unusual large phase shift of δ ≈ −2.0.

## Discussions

The surface nature of Fs1 ≈ 30 T with a small effective mass m* = (0.30 ± 0.01)me was strongly supported by the 2D-like angle dependence of Fs1 and nonmonotonic t dependence of oscillation amplitude as shown in Figs. 5e, f and 3c, respectively. Considering a conventional 2D parabolic band with a frequency of 30 T, the field required for the quantum limit (Bq) can be calculated, giving Bq ≈ 30 and 15 T for spin-degenerate bands and single spin subband, respectively. We also remark that such a 2D Fermi pocket is not likely coming from the SRO/STO interface conduction due to band bending or oxygen loss in STO, which typically gives a larger effective mass of m* > 1.0me39,40. One intriguing possibility for such a 2D-like small pocket of Fs1 ≈ 30 T is then the unusual quantum oscillation due to WOE as illustrated in Fig. 6b, which is supported by the existence of nonoverlapping Weyl-node pairs near the Fermi energy from our band calculations (Fig. 4) and also several earlier reports20,24. The dashed circles around the projected Weyl nodes with a length scale of inverse magnetic length ($${\ell }_{{{{\rm{B}}}}}^{-1}$$) are boundaries where the transition from the Fermi-arc surface states to the bulk states happens before reaching the ends of the Fermi-arcs, giving rise to a reduced effective area for the Weyl-orbit and thus the non-adiabatic correction effect12. The quantum oscillation due to WOE can be described as12

$$\frac{1}{{\mu }_{0}{H}_{n}}=\frac{e}{\hslash {k}_{0}^{{\prime} }}\left[\frac{\pi \hslash \upsilon n}{\mu }-t\right],$$
(1)

where n is the LL index, μ is the chemical potential, and υ is the Fermi velocity. $${k}_{0}^{{\prime} }\equiv {k}_{0}(1-4\alpha /{k}_{0}{\ell }_{{{{\rm{B}}}}})$$ is defined as the effective wave vector connecting the projected Weyl-node pair on the surface, and k0 is the length of the Fermi-arc. $${\ell }_{{{{\rm{B}}}}}\equiv \sqrt{\hslash /e{\mu }_{0}{H}_{n}}$$ is the magnetic length, and α is the parameter governing the non-adiabatic correction12. The corresponding frequency can then be calculated using Fs1 = 1/μ0Hn−1/μ0Hn−1. One important consequence of Eq. (1) is the thickness dependence of the phase in the oscillation. The 1/μ0Hn value will progressively decrease with increasing t values, and the amount of phase shift should be independent of n, which is in accordance with our observation shown in Fig. 5a. We remark that the phase shift due to trivial minor carrier density variations is expected to be linearly proportional to n, which is thus strikingly different from the n-independent phase shift due to WOE (see Supplementary Note 7). By solving for 1/μ0Hn in Eq. (1), the resulting simulated Fs1n and 1/μ0Hnn curves are shown as solid lines in the upper panel and lower panel of Fig. 6a, respectively, where the gradual change of line color represents the progressive increase of α parameter from 0 to 2. Here, we used m* = 0.3me, t = 13.7 nm, and Fs1 = 40 T for α = 0.

For α = 0, Fs1 is independent of n, and the 1/μ0Hnn curve is strictly linear with an intercept of n0 ≈ 0.76. When taking into account the non-adiabatic correction effect (α ≠ 0), several notable variations occur. First, the overall Fs1 values shift to lower values, and the resulting Fs1n curves show a concave downward feature, where Fs1 values decrease with decreasing n. Secondly, the 1/μ0Hnn curve in the Landau fan diagram also shows gradual upward-shifting with increasing concave downward curvature as α increases. We note that the concave downward curvature in the 1/μ0Hnn curve is distinct from the behavior due to the Zeeman energy contribution that causes a concave upward curvature instead13. The experimental data of Fs1 and 1/μ0Hn are shown as downward triangles and upward triangles in Fig. 6a for t ≈ 13.7 and 17.6 nm, respectively, which turns out to agree well with the simulated curves with α ≈ 1.3. The amount of deviation δ(1/μ0Hn) from linearity with respect to that for n ≥ 8 is shown in Fig. 6c, where the slope of the linear background derives from the 1/μ0Hn curve for α = 0. The simulated curve of δ(1/μ0Hn)−n (red dashed line in Fig. 6c) with α = 1.3 shows quantitative agreements with the experimental data for SRO films with t ≈ 13.7 and 17.6 nm. We note that the simulated curves for different α were calculated solely from Eq. (1) without imposing additional scaling parameters (see Supplementary Note 7). Once α = 1.3 is determined, the corresponding WOE parameters can then be calculated, giving k0 = 1.09 nm−1, μ = 15.43 meV, and υ = 1.34 × 105 m s−1. The extracted k0 value falls in the same order of magnitude as the calculated k0 as shown in Fig. 4d. We also note that there can be multiple Weyl-node pairs on (110)o plane that support for multiple Weyl-orbits with different frequencies, such as Fs2. However, due to the small oscillation amplitude for Fs2, it is not possible to resolve its intrinsic mechanism with current data. In particular, we remark that the non-adiabatic correction effect, i.e. a finite α, can, in principle, shift the 1/μ0Hnn curve in the Landau fan diagram, resulting in a significant change in the intercept n0. Therefore, an unusually large phase shift of δ ≈ −2.0 is possible for the WOE with the non-adiabatic correction, which can be another unusual feature for quantum oscillations deriving from WOE.

In general, the phase shift of the ρ oscillations in a 2D system can be expressed as δ = $$\frac{1}{2}+\frac{{\phi }_{{{{\rm{B}}}}}}{2\pi }$$ (modulo 1), where ϕB is the Berry phase, and the magnitude of δ is not likely to exceed 1.0. On the other hand, the phase shift of quantum oscillations from a simple 3D bulk Weyl-node should give a δ close to ±1/85,8,9. It was pointed out theoretically9 that a large anomalous phase shift can occur in a 3D topological WSM when the Fermi level is close to the Lifshitz point at which both the linear and parabolic bands are important. According to theory9, the model Hamiltonian can be expressed as $${{{\mathcal{H}}}}$$ = A(kxσx + kyσy)+M ($${k}_{{{{\rm{w}}}}}^{2}-{k}^{2}$$), where A and M are the coefficients for the linear and parabolic band, respectively. kw is the separation of the Weyl-node pair. Two energy scales of EA ≡ Akw/2 and $${E}_{{{{\rm{M}}}}}\equiv M{k}_{{{{\rm{w}}}}}^{2}/4$$ govern the phase shift behavior, where an anomalous large phase shift appears for EA ≤ EM. By using the parameters from our quantum oscillation data of Fs1, EA, and EM are estimated to be about 48.5 and 2.3 meV, respectively, where we used kw = k0 and an effective mass of 5me for the parabolic band. Therefore, the observed quantum oscillations in our SRO films do not fall in the condition for the large anomalous phase shift due to the coexistence of the linear and parabolic bands.

In a Weyl metal, the observed quantum oscillations can derive from complex competitions among several different mechanisms, such as the WOE, 3D Weyl nodes, and 3D bulk pockets. In our SRO thin films, the quantum oscillation of Fs1 we observed for t ≤ 40 nm is not likely dominated by 3D Weyl-node bulk states, since its amplitude shows nonmonotonic variation with t (lower-panel of Fig. 3c), and also its angle dependence behaves closer to a 2D-like Fermi surface (Fig. 5e, f). On the other hand, the occurrence of quantum oscillations due to WOE requires a high uniformity of film thickness and also td, which are both well satisfied in our SRO films as shown in Fig. 1 and Table 1. Phenomenologically, the oscillation amplitude of WOE is expected to be proportional to exp(−t/d)13. The ratio of d/t versus t in SRO thin films is well above 3 for t < 40 nm and attends maximum values for 10 ≤ t ≤ 20 nm as shown in Fig. 6d, inferring an optimum thickness range for a maximum oscillation amplitude due to WOE. In addition, the condition of d/t ≥ 3 also supports for observed rapid damping of the bulk frequencies of F1–5 due to the increased surface scattering for angles larger than 30o as shown in Fig. 5d. On the other hand, we also simulated the t dependence of 1/μ0Hnn curves with α = 1.3 as demonstrated in Fig. 6e, where an unusually large phase shift of n0 ~ −2 can only be observed for t ≤ 20 nm. As t increases to 50 nm or larger, the 1/μ0Hn-n curve shifts downward to smaller values of 1/μ0Hn, and thus a much higher field strength is required to access the low n states, posing an inevitable limitation on the lowest n that can be reached by available field strength. In other words, for t > 50 nm, the WOE amplitude and field accessible range in n are both reduced, and the contribution from 3D bulk pockets may become dominant instead, making it difficult to identify the quantum oscillations due to WOE.

At last, the observed Bc ~ 15 T is not the saturation field (Bsat ≡ (kFk0/πe)) for the WOE. For μ0H ≥ Bsat, the Weyl orbit mostly happens within the bulk states, and thus the associated quantum oscillations vanish12,13. Using the extracted parameters in our SRO films, Bsat is estimated to be about 80 T and >Bc. In a recently reported quantum oscillations data in SRO films41, a similar 2D-like small Fermi pocket with a frequency of 30 T was reported and attributed to Fermi-arc surface states. Nevertheless, the discussions in our work are focused on the untwinned SRO film thickness regime with t < d, particularly for t in a range from 10 to 20 nm, and the effect of non-adiabatic correction turns out to be an important ingredient to explain the observed features in Weyl-orbit quantum oscillations. However, precise identification of the quantum limit regime for Fs1 is challenging and may call for further investigations, which requires a careful and precise extraction of Fs1 contribution from a large oscillating background (F1−5) that grows rapidly with field strength as shown in Fig. 6b (see Supplementary Note 8).

In summary, the revelation of quantum oscillations due to WOE in Dirac and Weyl systems is challenging due to the mixing of quantum oscillations from 3D bulk Fermi pockets. Combining quantum oscillation measurements in SRO thin films and WOE simulations, we identified an optimum thickness of t ~ 10−20 nm for achieving a dominant WOE contribution, where several unusual features associated with WOE were clearly observed. Starting from the growth of a series of untwinned SRO films on miscut STO (001) substrates with different ts by using an adsorption-controlled growth technique with an oxide MBE, the RRR (ρ(2.5 K)) shows a progressive increase (decrease) with film thicknesses ranging from 7.7 to 35.3 nm, and the condition of t < d is always satisfied to favor the bulk tunneling and thus WOE, where the measurement geometry was kept the same with respect to the SRO orthorhombic crystalline direction. The calculated band structure of SRO shows complex band crossings and supports for the existence of nonoverlapping Weyl nodes when projecting on the film surface plane. The observed H-linear transverse MR with fields up to 35 T and also negative longitudinal MR in our SRO films further support its topological Weyl metal phase. From the rigorous angle and thickness-dependent measurements of quantum oscillations with fields up to 35 T, we revealed an unusual quantum oscillation with a small frequency of Fs1 ≈ 30 T, which behaves like a 2D Fermi pocket with a small effective mass of about 0.3me. Its oscillation amplitude shows nonmonotonic t-dependence and attends a maximum when t is in the range of 10–20 nm, where a systematic phase shift of the oscillation with t was observed. In addition, the extracted 1/μ0Hnn curve in the Landau fan diagram shows an unusual concave downward curvature. Those observations agree well with the WOE formalism with non-adiabatic corrections, corresponding to a Fermi-arc length (k0) of 1.09 nm−1 and a non-adiabatic correction parameter (α) of about 1.3. With excellent control on the growth of highly crystalline and untwinned oxide thin films, the topological Weyl metal SRO can be an ideal platform to explore the exotic phenomena associated with the bulk Weyl nodes and Fermi-arc surface states.

## Methods

### SRO thin film growths

The adsorption-controlled growth of SRO thin films was carried out using an oxide MBE system. The detailed growth conditions and structural characterization were reported in an earlier paper26. For SRO films with t ≈ 9, 13.7, and 18.7 nm, the magnetotransport measurements with applied fields up to 35 T were carried out at High Field Magnet Laboratory in Nijmegen. For other ts and another t ≈ 13.7 nm of the same batch, the magnetotransport measurements were performed using a superconducting magnet with fields up to 14 T. The high-resolution X-ray measurements were performed at TPS 09A and TLS 07A of the NSRRC in Taiwan, where the orthorhombic phase, film thicknesses, and orientation of the SRO films were carefully examined (see Supplementary Note 1). The surface roughness of SRO films and the STO substrate miscut orientation were measured by an AFM. The SRO films presented in this work were carefully selected to have the same STO miscut orientation with respect to the direction of the probing currents for the magnetotransport measurements, i.e. the current direction is perpendicular to the terrace edge of the STO (001) substrate.

### Band calculations

The DFT + U calculations were performed with full potential linear augmented plane waves plus local orbitals (FP-LAPW + lo) and the Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) provided in the WIEN2k code.32,33 For the treatment of Hubbard U terms, we have adopted the self-interaction corrections developed by Anisimov et al.34 which is available as a SIC scheme in the WIEN2k package. We have used the crystal lattice constants of a0 = 5.584 Å, b0 = 5.540 Å, and c0 = 7.810 Å, which were determined by high-resolution X-ray measurements on our samples, and a k-mesh of 22 × 22 × 15 was used to sample the Brillouin zone. We have performed calculations with several different values of U with J fixed to be J = 0.2U. All the DFT + U calculations were done with the inclusion of spin-orbit coupling on the heavy atoms of Sr and Ru.

The tight-binding model employed for the calculation of Weyl nodes was composed of the d-orbitals of the Ru atoms as well as the p orbitals of the O atoms, and the hopping parameters are determined by fitting the DFT band structure using the Wannier9035. We focused on five bands near the Fermi energy, as shown in Fig. 6a of the main text. We searched for all the band crossings residing in four pairs of bands out of these five bands. These four pairs are defined as: pair I → green band and the one below; pair II → (green, red); pair III → (red, blue); pair IV → (blue, pink). Finally, we confirmed the Weyl nodes by calculating the chirality of each band crossing using the WannierTools36.

The Fermi surfaces are plotted using the spectral function which is defined as

$$A(\overrightarrow{{{{\bf{k}}}}},E)=\frac{1}{\pi }{{{\rm{Im}}}}\left\{{{{\rm{Tr}}}}\left[\frac{1}{E-{\hat{H}}_{{{{\rm{TB}}}}}(\overrightarrow{{{{\bf{k}}}}})-{{{\rm{i}}}}\epsilon }\right]\right\},$$
(2)

where $${\hat{H}}_{{{{\rm{TB}}}}}(\overrightarrow{{{{\bf{k}}}}})$$ is the tight-binding model fitted by the Wannier90, and we have used the codes provided by WannierTools36 for evaluating the spectral function.