Main

Ferromagnets are typically metallic and exhibit strong magneto-electronic responses when their magnetization is reversed. Examples of these responses include the relativistic anomalous Hall effect (AHE) in ferromagnetic films1 and the non-relativistic giant magnetoresistance (GMR) in ferromagnetic multilayers2. Insulating rutile crystals, like FeF2, with a compensating antiparallel ordering of moments on neighbouring magnetic atoms, have been regarded as classic examples of antiferromagnets3,4,5, where—besides the insulating character—basic symmetry principles exclude the above magneto-electronic effects. In particular, when omitting the non-magnetic atoms in rutile crystals for brevity, the remaining lattice constructed by a periodic repetition of pairs of magnetic atoms with mutually compensated antiparallel moments has a symmetry combining real-space inversion (P) with time-reversal (T) transformations6,7. For PT symmetry of an antiferromagnetic crystal, Kramers theorem dictates that energy bands are spin degenerate, which excludes the magneto-electronic responses such as AHE or GMR6. This means that crystals with compensating antiparallel ordering of magnetic moments were—for a long time—largely ignored in magneto-electronic research, as well as in studies of spin-polarized magnetic topological phases and relativistic Berry-phase physics1,6.

Research into the AHE beyond ferromagnets was initiated in metallic, magnetically compensated crystals with three or four magnetic sublattices8,9,10,11,12,13,14. In this case, however, the magnetic order is compensated and breaks the PT symmetry at the expense of strong non-collinearity of the magnetic moment vectors, typically due to frustrated exchange interactions on the lattice6. Recently, an AHE arising from a compensated collinear magnetic order has been predicted in ruthenium dioxide (RuO2) (ref. 15). The material, whose antiparallel magnetic order was recently discovered16,17, has the classic rutile crystal structure15. Moreover, RuO2 is of interest in magneto-electronic research15,18,19,20,21,22,23 due to its metallic conduction and inferred Néel temperature exceeding 300 K (refs. 16,17).

In this Article, we report the experimental observation of an AHE in RuO2 with an anomalous Hall conductivity exceeding 1,000 Ω−1 cm−1. We measure epitaxial RuO2 films with different crystallographic orientations using vector magnetometry and magneto-transport measurements. We show that the AHE dominates over other Hall contributions, consistent with theoretical predictions.

Origin of T symmetry-breaking electronic structure in RuO2

The theoretically predicted large Berry-phase AHE in RuO2 is linked6,15 to spin splitting and T symmetry breaking in the band structure15,24,25,26,27. Spin splitting, whose strength is comparable to ferromagnets but whose sign alternates across the magnetically compensated Brillouin zone, and T symmetry breaking in the band structure have a strong non-relativistic origin15,24,25,26,27. Understanding the AHE and unconventional spin-polarized band structure of RuO2 has prompted a systematic and rigorous classification of collinear magnetic phases based on a non-relativistic spin-symmetry group formalism. Apart from ferromagnets and antiferromagnets, a third phase has emerged from this classification, which has been termed altermagnets. The three phases are exclusively distinct, and represent a complete spin-symmetry classification of collinear magnets. The magnetic rutiles, including RuO2, belong to the third altermagnetic class whose characteristic is an alternating spin polarization in both real-space crystal structure and momentum-space band structure26,27.

As well as AHE, collinear altermagnets are predicted to provide robust non-relativistic spin-polarized currents that could lead to counterparts of the GMR and other key non-relativistic phenomena employed for reading and writing information in ferromagnetic memories18,22,23. Recent transport experiments in RuO2 (refs. 19,20,21) support these predictions of spin-polarized currents.

In RuO2, the broken PT symmetry in the real-space crystal structure of rutiles (which enables the altermagnetic spin splitting and T symmetry breaking in the electronic band structure), as well as the resulting AHE or GMR magneto-electronic responses6,15,18,19,20,21,22,23,24,25,26,27,28,29, lies in the crystal field from the anisotropic arrangement of non-magnetic atoms15,26,27. The mechanism has, thus, been referred to as crystal PT symmetry breaking15. Instead of real-space inversion, the two crystal sublattices with opposite magnetic moments are connected by a real-space rotation transformation when the non-magnetic atoms are brought into the picture (Fig. 1a). The real-space rotation connecting opposite-spin sublattices is the defining symmetry of the altermagnetic phase. The symmetry protects the magnetic compensation and allows for T symmetry breaking and alternating spin polarization in the band structure26,27.

Fig. 1: Experimental method for detecting AHE in RuO2.
figure 1

a, Schematic of the arrangement of Ru atoms with overlaid ab initio spin density isosurfaces and opposite-spin sublattices depicted by green and magenta colours, respectively. The schematic of the arrangement of O atoms is shown by the black dots. The black double arrow and its label highlight that the two crystal sublattices with opposite magnetic moments are connected by a real-space fourfold rotation transformation. b–d, Schematic of the three sample orientations used in the Hall measurements. The blue, brown and black arrows correspond to the applied magnetic field vector, magnetization vector and Néel vector, respectively. e, Zero-field ab initio calculation of the normalized [001] component of magnetization, namely, \(M_{{{{\mathrm{[001]}}}}}^L\), and the normalized [110] component of the Hall vector, that is, \(h_{{{{\mathrm{[110]}}}}}^L\), corresponding to the Hall signal detected in the (110)-oriented sample, induced by the Néel vector rotated by angle α from the [001] axis in the (\(1\bar 10\)) plane. f, Cross-section transmission electron microscopy image of an optimized (110)-oriented RuO2/TiO2 film. g, X-ray diffraction spectrum of the RuO2/TiO2 film, indicating a highly ordered (110) orientation of the RuO2 film.

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In parallel to the non-equilibrium relativistic AHE6,15, additional equilibrium relativistic effects can arise from the antiparallel magnetic order of the rutile crystal30,31. The first of these effects is magneto-crystalline anisotropy, described by relativistic spin–orbit coupling terms in the thermodynamic potential that depend on the crystallographic orientation of the Néel vector, that is, L = M1 – M2, of the antiparallel magnetic order30,31. Here M1 and M2 are the moments on opposite magnetic sublattices. In the absence of an external magnetic field, the magneto-crystalline anisotropy in RuO2 tends to align the Néel vector along the c axis ([001]) of the tetragonal rutile crystal15,16,17. Incidentally, this is the singular crystal direction of the Néel vector for which the relativistic AHE arising from the antiparallel magnetic order is excluded by symmetry15.

The Néel vector in rutile-structured materials can be reoriented from the easy axis by applying a magnetic field in the orthogonal direction to the easy axis30. This is facilitated by the Zeeman term in the equilibrium thermodynamic potential, which couples the external magnetic field with net magnetization M = M1 + M2, and additional relativistic Dzyaloshinskii–Moriya interaction (DMI) terms, which couple the Néel vector with net magnetization30,31 (in tetragonal crystals like RuO2, the leading DMI terms are proportional to (lxmy + lymx), (l · m)2 and (l · m)lxly, where l = L/|L| and m = M/|M| (ref. 30)). We use this technique to experimentally induce the reorientation of the Néel vector from the zero-field easy axis in our RuO2 samples. The results of the accompanying Hall measurements are consistent with the theoretically predicted dominant AHE contribution from symmetry breaking by the antiparallel magnetic order on the rutile crystal (L-AHE), linked to the charge Hall transport by the relativistic spin–orbit coupling15. In contrast, we confirm the expected weak contributions from a Lorentz-force ordinary Hall effect (OHE) due to the applied out-of-plane magnetic field and a weak field-induced out-of-plane magnetization (M-AHE).

Néel-vector reorientation method and experimental samples

By considering the established vector magnetometry technique in rutiles30 and the symmetry of L-AHE15, we perform measurements in RuO2 films oriented in the (110), (001) and (100) planes. In all three sample geometries, the external magnetic field is applied normal to the film plane (Fig. 1b–d, schematic).

The earlier theoretical and experimental vector magnetometry studies in rutiles with the [001] easy axis of the Néel vector showed30 that a magnetic field applied along the [110] direction induces a continuous rotation of the Néel vector towards the [110] field direction in the (\(1\bar 10\)) plane by angle α (Fig. 1b), given approximately by sinα ≈ H/Hc. Here H is the applied field (the sense of rotation is given by the sign of H) and Hc is the scale of the field corresponding to the Néel vector reorientation into the [110] axis. Here Hc depends on the exchange, magneto-crystalline anisotropy and DMI terms of the thermodynamic potential30. It was shown30 that in rutiles, Hc can be weaker than the spin-flop reorientation field applied along the [001] easy axis. Note that this phenomenology contrasts with antiferromagnets lacking the DMI terms, in which case a field applied in the transverse direction to the easy axis does not reorient the Néel vector, but induces canting of the moments and—at large fields—eventually enforces their parallel alignment.

In rutiles, apart from magnetization along the applied [110] field, the reorientation of Néel vector within the (\(1\bar 10\)) plane away from the [001] easy axis is accompanied by the generation of a [001] component of magnetization, that is, a component along the Néel-vector easy axis and transverse to the applied field30 (Fig. 1b). In our vector magnetometry measurements, the detection of this magnetization component, thus, serves as an experimental signature of the Néel vector reorientation from the [001] axis.

Symmetry analysis and ab initio theory predict15 that the reorientation of Néel vector within the (\(1\bar 10\)) plane generates a strong L-AHE in the RuO2 film oriented in the (110) plane, and that opposite sense of the Néel vector rotation gives an opposite sign of L-AHE. This allows us to detect the L-AHE signal that is odd in the applied [110] field, and vanishes at zero field. In Fig. 1e, we use the (zero-field) ab initio theory of RuO2 (ref. 15) to further highlight the principles of our detection method, by plotting the dependence of the [001] component of magnetization and the [110] component of the Hall vector, corresponding to the L-AHE signal detected in the (110)-oriented sample, on the angle of the Néel vector in the (\(1\bar 10\)) plane (for more details on the ab initio calculations and the choice of experimentally relevant parameters, see ref. 15, Methods and Extended Data Fig. 1). We also emphasize that the sign of the field-induced magnetization vector as well as the Hall vector are determined by the sign of the applied [110] field, whereas they are independent of the initial sign of the Néel vector ([001] or [\(00\bar1\)]) at zero field. Therefore, the contributions to L-AHE from domains with opposite orientations of the Néel vector at zero field add up when the Néel vector is reoriented from the easy axis by the applied [110] field.

The RuO2 sample oriented in the (001) plane and the external magnetic field applied along the [001] axis serve as a reference (Fig. 1c). This sample and field geometry do not break the symmetry between opposite signs of the Néel-vector reorientation angle from the [001] easy axis30. Therefore, even if the applied field caused a reorientation of the Néel vector, it would not generate a non-zero average magnetization component transverse to the applied magnetic field30. Similarly, the geometry would not lead to an L-AHE signal that is odd in the applied field15. This geometry can, thus, serve as a measure of the expected weak contributions from the OHE and M-AHE.

For the (100)-oriented sample and field applied along the [100] axis, the Néel vector is not reoriented towards the field direction, which makes this experimental geometry distinct from the (110)-oriented sample and field applied along the [110] axis30. An additional distinction is that there is no magnetization component in the direction transverse to the magnetic field for the (100)-oriented sample and the field along the [100] axis30 (Fig. 1d). This applies even if the Néel vector was reoriented from the [001] easy axis (within the (100) plane in this experimental geometry)30. Since we do not detect a measurable L-AHE signal in this geometry, it is used to reconfirm that the OHE and M-AHE contributions are isotropic in RuO2, that is, they are independent of the film orientation, and reconfirm the quantitative measure of OHE and M-AHE contributions. It then allows us to extract the L-AHE contribution from the measured Hall signals in the (110)-oriented samples.

We prepared the (110)-, (001)- and (100)-oriented epitaxial films of rutile RuO2 by pulsed laser deposition on non-magnetic rutile TiO2 substrates, which are isostructural and closely lattice matched to RuO2, independent of the orientation of the growth plane. The three film orientations were simultaneously grown and therefore under identical conditions, to avoid differences due to unintentional variations in growth parameters. In a separate growth batch, we also prepared control samples on MgO and SrTiO3 substrates. Transmission electron microscopy image and X-ray diffraction pattern of optimized 27-nm-thick (110) RuO2/TiO2 are shown in Fig. 1f,g. Further details on the preparation and characterization of our samples are provided in Methods and Extended Data Figs. 2–9. We note that RuO2 films grown on these substrates were used in a recent study demonstrating the high-temperature antiparallel magnetic order on the RuO2 crystal17, as well as a parallel research on the non-relativistic magneto-electronic phenomena in RuO2 (refs. 19,20,21).

Magnetization and magneto-transport measurements

In Fig. 2a–c, we summarize basic magnetic and electrical transport characteristics of the (110)-, (001)- and (100)-oriented RuO2/TiO2 films. All the samples show zero remanent magnetization, consistent with the compensated antiparallel magnetic order and [001] easy axis of the Néel vector. Furthermore, consistent with the expected magnetic ordering, the induced moment along the applied magnetic field is weak, reaching 0.2µB per Ru at 50 T, where µB stands for the Bohr magneton. The moments are similar in all the three sample orientations, with a slightly weaker susceptibility along the [001] easy axis of the Néel vector (in Extended Data Fig. 3, we illustrate the exchange bias effect of RuO2 on ferromagnetic CoFe deposited on top of RuO2).

Fig. 2: Magnetization and magneto-transport measurements.
figure 2

a, Magnetization M along the applied out-of-plane magnetic field µ0H for the three RuO2/TiO2 film orientations. b, Resistivity ρ versus temperature T for the three film orientations. The inset shows the temperature derivative of resistivity, highlighting the Néel-temperature transport anomaly. c, Longitudinal magnetoresistance (MR) in the applied out-of-plane magnetic field at different temperatures. d–f, Hall resistivity ρH at different temperatures for the three film orientations of (110) (d), (001) (e) and (100) (f).

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The temperature-dependent resistivity (Fig. 2b) confirms the metallic character of RuO2 films. It is practically identical in the three sample orientations, owing to our consistent sample preparation method, as well as in agreement with earlier reports32 on isotropic resistivities of bulk single crystals of RuO2. The cusp in the temperature-dependent resistivity curve32, highlighted by the singularity in the temperature derivative of resistivity (Fig. 2b, inset), is consistent with the Néel temperature above 300 K observed in earlier measurements on bulk and thin-film RuO2 samples16,17. The longitudinal magnetoresistance is again identical in the three sample orientations (Fig. 2c). Its weak quadratic field dependence is consistent with an ordinary (Lorentz-force) positive magnetoresistance origin.

The measured odd in the magnetic-field Hall signals is shown in Fig. 2d–f. At low temperatures, the field-dependent Hall resistivity of the (001)- and (100)-oriented samples is nearly identical, whereas the Hall signal of the (110)-oriented film is substantially stronger over the entire range of applied magnetic fields. Moreover, it shows a clear nonlinearity at higher fields, which is absent in the other two film orientations. Since the (001)- and (100)-oriented samples have (nearly) identical ordinary magnetoresistance, the magnetic moment induced along the field and Hall signal, and since L-AHE is excluded by symmetry in the (001)-oriented film, we can ascribe the Hall signals in these two sample orientations to the sum of OHE and M-AHE contributions. The independence of the OHE and M-AHE contributions on sample orientation is further confirmed by the Hall measurements at high temperatures approaching the Néel temperature, at which the Hall signals in all the three film orientations merge. We then attribute the additional contribution that dominates the lower-temperature Hall signals in the (110)-oriented film over the entire field range to the L-AHE. The observed large magnitude and high-field nonlinearity of the Hall signal in the (110)-oriented sample suggests a sizable reorientation in the Néel vector in the explored field range. Simultaneously, we recall our ab initio calculations (Fig. 1e) showing that L-AHE sharply increases from zero when the Néel vector is reoriented from the [001] easy axis towards the out-of-plane [110] direction, and that a large L-AHE signal can be expected well before the Néel vector is fully aligned with the [110] axis.

In Fig. 3a,b, by independent vector magnetometry measurements, we confirm that the Néel vector is reoriented in the (110) film from the [001] easy axis and that the reorientation is confined to the (\(1\bar 10\)) plane. The clearly detectable [001] magnetization component, which is transverse to the field applied along the [110] axis, is shown in Fig. 3a. As expected, it vanishes at low fields. As the field strength increases, its magnitude abruptly increases, and the field-dependence levels off at the highest experimentally available field of 9 T. Per symmetry and as confirmed by our ab initio calculations (Fig. 1e), the [001] magnetization component should again vanish when the Néel vector is fully aligned with the [110] axis; therefore, we conclude that the Néel-vector reorientation angle is sizable but far from complete at 9 T. It is worth emphasizing, both theoretically and experimentally, that there is no need for the misalignment of the field off the RuO2[110] crystallographic direction to detect the moment along the [001] direction. An additional confirmation of the expected phenomenology is shown in Fig. 3c,d. Consistent with earlier work30, we do not observe any transverse magnetization components in the (001)- and (100)-oriented samples.

Fig. 3: Measurements of magnetization M components transverse to the applied magnetic field.
figure 3

a,b, The [001] (a) and [1\(\bar 1\)0] (b) components of magnetization for the field applied along the [110] axis in the (110)-oriented sample. c, The [100] and [010] components of magnetization for the field applied along the [001] axis in the (001)-oriented sample. d, The [001] and [010] components of magnetization for the field applied along the [100] axis in the (100)-oriented sample. Data are presented as mean values ± standard deviation, where the statistics have been derived from the measurements of three repeating samples.

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Figure 4a,b summarize our experimental demonstration of L-AHE by subtracting the measured Hall signals in the (001)- and (100)-oriented samples, respectively, from the Hall signal of the (110)-oriented film. Consistent with the theory prediction for the Berry-phase AHE mechanism15 (Extended Data Fig. 1), we observe L-AHE resistivities that are comparable to metallic ferromagnets or non-collinear antiferromagnets, where AHE also dominates the OHE1,6.

Fig. 4: Anomalous Hall resistivity from symmetry breaking by the antiparallel magnetic order on rutile crystal (ρL-AHE).
figure 4

a,b, The ρL-AHE value obtained by subtracting the measured Hall resistivity ρH in the (001)-oriented (a) and (100)-oriented (b) samples from Hall resistivity ρH of the (110)-oriented sample at different temperatures. c, Magnetization M along the applied out-of-plane magnetic field in the (110)-oriented RuO2/MgO and (100)-oriented RuO2/SrTiO3 samples. The top inset shows the longitudinal magnetoresistance (MR) in the applied out-of-plane magnetic field at different temperatures. The bottom inset shows the temperature derivative of resistivity, highlighting the Néel-temperature transport anomaly. d, Hall resistivity ρH in the (110)-oriented RuO2/MgO and (100)-oriented RuO2/SrTiO3 samples, and ρL-AHE obtained by subtracting the measured Hall resistivity in the (100)-oriented sample from the Hall resistivity of the (110)-oriented sample at 10 K. The inset shows the Hall resistivity ρH of the two samples at 340 K. The shaded area highlights the difference in Hall resistivities between the (110)-oriented RuO2/MgO and (100)-oriented RuO2/SrTiO3 samples.

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To confirm that our results are not specific to the RuO2 films grown on a TiO2 substrate, we performed control experiments on the (110)-oriented RuO2/MgO and (100)-oriented RuO2/SrTiO3 samples. The measurements (Fig. 4c,d) show identical phenomenology and are in good quantitative agreement with the measurements on the RuO2/TiO2 films (Extended Data Figs. 6–8).

Consistency with theoretical prediction

Because we do not observe signatures of a full reorientation of the Néel vector into the (001) hard plane in our RuO2 films, we cannot quantitatively infer the parameters of thermodynamic potential from the experiment and the corresponding dependence of the Néel-vector reorientation angle on the applied magnetic field30. We can, however, make the following estimates. The scale of the critical field applied along the [001] easy axis, corresponding to the abrupt spin-flop reorientation, is given by \(H_{{{{\mathrm{AE}}}}} = \sqrt {H_\mathrm{A}H_\mathrm{E}}\), where HA is the anisotropy and HE is the exchange field. In the mean-field approximation and considering the Néel temperature inferred from the resistivity measurements in our RuO2 films (Fig. 2b), we estimate that HE ≈ 350 T. Our ab initio calculations (Extended Data Fig. 1) indicate that HA can reach an ~50 T scale due to the strong relativistic spin–orbit coupling in RuO2, resulting in a spin-flop field exceeding 100 T. This is consistent with the experimentally observed order of magnitude weaker spin-flop field in rutile CoF2, whose Néel temperature and corresponding HE are about ten times smaller than in RuO2, and whose weaker spin–orbit coupling also gives an order of magnitude smaller HA (ref. 30). It also explains why we do not observe a spin-flop signature in our measured magnetization curves up to the maximum applied magnetic field of 50 T.

For the in-plane hard-axis field along the [110] axis, the continuous field-induced reorientation is given approximately by sinα ≈ H/Hc, where the reorientation-field scale, namely, \(H_\mathrm{C} = (H_{{{{{AE}^2}}}} - H_\mathrm{d}^2)/H_\mathrm{d}\), was experimentally inferred in CoF2 to be about 3/4 of the easy-axis spin-flop field30. Here Hd is the relativistic DMI field30. Assuming that the relativistic DMI field in RuO2 is again an order of magnitude6 larger than in CoF2, the ratio between the easy-axis spin-flop field and hard-axis reorientation field scale can be expected to be comparable in the two rutiles. This puts the estimated [110] reorientation-field scale in RuO2 above 50 T, consistent with a sizable but not complete reorientation in our measurements performed up to 50 T.

In analogy to the AHE in ferromagnets, the L-AHE can, in principle, originate from scattering-independent intrinsic and side-jump mechanisms, as well as from the extrinsic skew-scattering mechanism1. Regarding the skew-scattering contribution, previous studies in ferromagnets have shown that it becomes important only in samples with conductivities above 106 Ω−1 cm−1, which is much higher than the conductivity of our RuO2 films. The side jump was shown in ferromagnets to be only a small correction of the scattering-independent contribution33. As a result, theoretical studies of the scattering-independent contribution to the AHE conductivity have focused on the intrinsic Berry-curvature mechanism1,6. Ab initio calculations of the intrinsic Berry-curvature AHE tend to give a semiquantitative prediction of the AHE conductivity, but often quantitatively disagree with the measured values. For example, theory overestimates or underestimates (by up to a factor of three) the measured AHE conductivity in Ni or Fe, respectively1.

In our case, the measured L-AHE conductivity exceeds 1,000 Ω−1 cm−1 at 50 T. The ab initio magnitudes near the Fermi level peak at a value above 300 Ω−1 cm−1 (ref. 15), which gives a comparable level of a semiquantitative agreement between theory and experiment as in conventional ferromagnets. In our case, the strong magnetic field used in the experiment can be an additional source of quantitative discrepancy between the measured magnitude of anomalous Hall conductivity and ab initio Hall transport calculations performed at zero magnetic field15. In Extended Data Fig. 10, we compare the ab initio equilibrium band structure of RuO2 at zero field and at 50 T. The strong magnetic field only generates a weak magnetization on the scale of ~0.1µB per unit cell, consistent with experiment. Moreover, the overall character of the spin-split T symmetry-broken band structures is similar in the zero-field and 50 T cases, supporting the expectation that the L-AHE origin is not principally affected by the applied magnetic field. Quantitatively, however, the small field-induced shifts in bands around the band (anti)-crossings, where the subtle relativistic Berry curvature tends to have the largest contributions15, can quantitatively modify the resulting anomalous Hall conductivity.

Finally, we comment on our ab initio calculations of the dependence of anomalous Hall conductivity in RuO2 on the Hubbard correlation parameter U for the Néel vector along the [110] axis (Extended Data Fig. 1). Values of U ≈ 1.6–2.0 eV correspond to the magnitude of the sublattice magnetic moment consistent with previous ab initio studies15,16,17,24. The anomalous Hall conductivity peaks at U ≈ 1.6 eV and vanishes at small values of U due to the vanishing magnetic order (for large values of U, the anomalous Hall conductivity vanishes due to the opening of an insulating bandgap). Since decreasing U in the ab initio calculations mimics the effect of an increasing temperature on the magnetic order, the steeply decreasing theoretical Hall conductivity with decreasing U is consistent with the experimentally observed drop in L-AHE with increasing temperatures.

Conclusions

By combining vector magnetometry and magneto-transport measurements of epitaxial RuO2 films of different crystallographic orientations, we have reported an AHE due to the compensated collinear magnetic order on a classic rutile crystal structure. RuO2 is only one example of a broad family of altermagnetic materials that have compensated antiparallel magnetic order on the crystal structure and strong spin polarization and T symmetry breaking in the band structure. Thus, more materials can be expected to show similar spin physics and magneto-electronics behaviour6,15,18,19,20,21,22,23,24,25,26,27,28,29,34,35,36,37,38,39,40,41. Our work illustrates that T symmetry-breaking magneto-electronic effects are not limited to ferromagnets or complex non-collinear magnets, and could lead to further studies of magnetic topological insulators, axion insulators or quantum AHE systems6,42,43 in the abundant class of collinear compensated magnets, many of which have high ordering temperatures and are composed of common elements.

Methods

Sample fabrication

Systematic thin-film growth for optimizing the conductivity of RuO2 films indicates that the optimal conditions are 550 °C and 10−3 torr oxygen pressure (Extended Data Fig. 2). RuO2 thin films were grown onto different single-crystal substrates at an oxygen pressure of 10−3 torr and 550 °C by a pulsed laser deposition system with a base pressure of 1.5 × 10−8 torr (Shenyang Baijujie Scientific Instrument). The target–substrate distance was 60 mm. The laser fluence was ~1.6 J cm–2 and the repetition rate was kept at 10 Hz during deposition. The ramp rate was 20 °C min–1 for heating and 10 °C min–1 for cooling.

The experimental consistency of the characteristics of the films deposited on different substrates was maximized using the following method:

  1. (1)

    All the three RuO2/TiO2 films (measurements on which are reported in the main text) were simultaneously grown under identical conditions. Specifically, the three TiO2 single-crystal substrates with different crystallographic orientations were pasted onto the central region of a substrate holder of our pulsed laser deposition system by silver paint.

  2. (2)

    To avoid any possible non-uniform deposition of RuO2 films onto the three different TiO2 substrates owing to the limited uniform area of the pulsed laser plume, the substrate holder was kept at a fixed rotation speed of 25 rounds per minute. In this way, the thin-film deposition onto the three different TiO2 substrates was made uniform. As a result, the three RuO2/TiO2 films show very similar longitudinal resistivities (Fig. 2b).

  3. (3)

    We also note that the two films grown on the two different substrates, namely, RuO2/SrTiO3 and RuO2/MgO, were simultaneously fabricated under the same deposition conditions. Despite the different substrates, they show very similar temperature-dependent resistivities with only a small difference in the ratio of room-temperature versus low-temperature resistivities (2.52 and 2.49, respectively).

  4. (4)

    The room-temperature resistivity of the RuO2/SrTiO3 and RuO2/MgO films is ~64 μΩ cm, whereas the room-temperature resistivity of the RuO2/TiO2 films is ~53 μΩ cm. We ascribe the difference in resistivity of these two batches of RuO2 films to variations in the deposition conditions, including the control of the oxygen partial pressure, laser fluence, condition of heaters, substrate–target distance and so on. Although we nominally utilized the same conditions for thin-film deposition, the experimental conditions could unavoidably vary in time, and the batch on TiO2 substrates was grown nearly two years after the batch on SrTiO3 and MgO substrates. We also point out that our systematic growth experiments for optimizing the conductivity of RuO2 thin films (Extended Data Fig. 2) have shown a high sensitivity of conductivity to the growth conditions such as the oxygen partial pressure and deposition temperature, especially around the optimized deposition conditions (550 °C and 10−3 torr oxygen pressure).

The room-temperature growth of ferromagnetic Co90Fe10 and the capping Pt thin films was carried out by a d.c. sputtering system with a base pressure of 7.5 × 10−9 torr. For Co90Fe10 deposition, the d.c. sputtering power was 90 W and the Ar pressure was 3 mtorr. The growth rate was ~0.11 Å s–1. For Pt sputtering, the power was 30 W and Ar pressure was 3 mtorr. The growth rate was determined to be 0.5 Å s–1. No chemical treatment was taken for the substrates before deposition.

We have also more systematically explored why the conductivity of RuO2 films fabricated above 600 °C drops dramatically (Extended Data Fig. 2). With surface topography characterization, we have found that there is a gradual transition for the thin-film growth mode with increasing temperature, that is, from a quasi-two-dimensional growth at low temperatures to high-temperature three-dimensional island growth. The difference in the surface energies of RuO2 and oxide substrates becomes larger at higher temperatures, which results in the wetting issue and consequently leads to island growth. As a result, to form a continuous film at higher growth temperatures, one would need to deposit much thicker films.

X-ray diffraction

The X-ray spectra were collected using an SmartLab XRD diffractometer. The Cu Kα X-ray wavelength was 1.541882 Å.

Transmission electron microscopy

The focused ion beam technique was used to fabricate the cross-section samples. Afterwards, the transmission electron microscopy characterization was conducted on a ZEISS LIBRA 200 MC STEM system at 200 kV.

Electrical measurements

For electrical transport measurements, Hall bar structures with a channel width of 20 μm and electrode pad size of 200 μm were achieved by optical lithography and Ar ion milling. Electrical contacts were fabricated by wire bonding through Al wires with a diameter of 25 μm. The linear four-probe geometry was utilized for resistivity measurements of RuO2 thin films, which were carried out in a Quantum Design physical property measurement system with a measuring current of 1 mA.

Pulsed high-field Hall measurements

The conventional Hall geometry was established by Cu wires at Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology. The magnetic field was applied along the out-of-plane direction of the thin-film samples. The amplitude of the a.c. current was 3 mA and frequency was 100 kHz. The Hall voltage collection was performed by a National Instruments PXIe 5105 oscilloscope at a sampling frequency of 4 MHz.

Pulsed high-field magnetic moment measurements

Pulsed high-field magnetic moment signals were collected by a pickup coil coaxial with the pulsed magnetic field and calibrated by a low-field magnetic moment obtained by a Quantum Design superconducting quantum interference device. To obtain the high-field magnetic moment of the RuO2 thin films, the substrates with the same dimensions were measured, and the difference in magnetic moments of the heterostructure and substrate was subsequently extracted to represent the thin-film signal.

Vector magnetometry measurements

A modified vector magnetometer system, which combines a high-field vector magnet with a high-sensitivity vector sensor setup, is able to collect the transverse moment signals relative to the external magnetic field. The alignment accuracy of the magnetic field to the crystallographic direction is less than 5°, which includes the miscut angles of the single-crystal oxide substrates purchased from the vendors.

Density functional theory calculations

We performed the density functional theory (DFT) calculations employing the projector augmented-wave method44 implemented in the Vienna ab initio simulation package code and we used the spherically symmetric Dudarev DFT+U (ref. 45). We set the energy cut-off of the plane-wave basis to 520 eV, used the Perdew–Burke–Ernzerhof exchange–correlation functional46 and the crystal momentum grid was 16 × 16 × 24. We used DFT-relaxed lattice parameters as a = b = 4.5337 Å and c = 3.1240 Å. We constructed the maximally localized Wannier functions in the wannier90 code47 and we calculated the intrinsic Hall conductivity by employing the Berry-curvature formula. We used a fine mesh of 321 × 321 × 321 Brillouin-zone sampling points. We have studied the effect of the Zeeman coupling of the external magnetic field on energy bands by DFT calculations using the Elk code (https://elk.sourceforge.io/).

Note that a quantitative determination of the sublattice magnetic moment in RuO2 remains an open question. DFT calculations16, subsequently confirmed by us and other groups15,24,48, gave a moment per Ru of approximately 1µB. Unpolarized neutron diffraction16 gave undistorted rutile crystal structure and moment of 0.23µB. Polarized neutron scattering16 gave a moment of ~0.05µB. Simultaneously, the dominant contribution to the polarized neutron scattering signal was ascribed to a structural distortion whose origin, however, was unidentified.

We also note that the sizable moments per Ru found in the DFT studies corroborate the robust high-Néel-temperature ordering in RuO2, as observed in the neutron and X-ray measurements. Moreover, a recent theory work from a physical–chemistry field48 emphasizes that sizable moments on Ru atoms in the ordered antiparallel magnetic phase of RuO2 play a decisive role in making RuO2 the prime catalyst for the oxygen evolution reaction (we recall that the utility of RuO2 in catalysis is a chemistry field established long before the spin physics in RuO2 has been brought to the attention of physicists).

Finally, we note that in our DFT calculations (Extended Data Fig. 1), we can control the ordered sublattice moment per Ru by changing the Hubbard U parameter. The corresponding calculations of L-AHE conductivity show that it remains sizable even for a substantially reduced moment on Ru.