Introduction

Unconventional metallic states and the breakdown of the Landau Fermi liquid paradigm is a central topic in contemporary condensed matter science. A connection with high-temperature superconductivity is experimentally well established but the conditions under which these enigmatic metals form has remained perplexing1. One of the simplest routes to singular Fermi liquid behavior, at least conceptually, is through two-channel Kondo (2CK) physics2,3,4. Despite this long-standing interest, 2CK physics has thus far only been demonstrated to arise in carefully designed semiconductor nanodevices in narrow energy and temperature (T) ranges5,6,7,8, while claims of its observation in real quantum materials are contentious (see “Discussion” section for details). More recently, the interest in Dirac and Weyl fermions within a condensed matter framework has led to the exploration of the effects of strong spin-orbit coupling (SOC) and of topological states which are rooted in a combination of time-reversal, particle-hole, and space-group symmetries9,10. While there has been considerable progress in understanding weakly correlated topological metals, only a few materials have been identified as realizing topological phases driven by strong electron correlations, which includes the Weyl–Kondo semimetals11. This raises the question if the 2CK counterpart of such a Weyl–Kondo semimetal, featuring an entangled ground state of the low-energy excitations of the 2CK effect with band-structure enforced Dirac or Weyl excitations, could at least in principle be stabilized. Exploring such a possibility, however, hinges on whether the 2CK effect can be stabilized at all in native quantum matter.

In this work we establish that oxygen vacancies (VO’s) in the Dirac nodal line (DNL) materials IrO2 and RuO2 drive an orbital Kondo effect. VO’s are prevalent in transition-metal oxides, including, e.g., TiO2 and SrTiO3, and their properties and ramifications have become central research topics as they can lead to an intricate entanglement of spin, orbital, and charge degrees of freedom12,13,14,15. The active degree of freedom in the orbital Kondo effect is not a local spin moment but a ‘pseudospin’ formed by orbital degrees of freedom4. In IrO2 and RuO2, the orbital Kondo effect is symmetry stabilized by the space-group symmetries of the rutile structure (Fig. 1). Both materials have been characterized as topological metals which feature symmetry-protected DNLs in their Brillouin zones16,17. This provides a link between the formation of the orbital Kondo effect and the presence of DNLs. In IrO2 a nonmagnetic 2CK ground state ensues, while in RuO2 the absence of time-reversal symmetry results in an orbital one-channel Kondo (1CK) effect.

Fig. 1: Atomic arrangement around an oxygen vacancy in MO2 rutile structure.
figure 1

a Schematics for MO2 in the rutile structure. The olive and red spheres represent transition-metal ions M4+ and oxygen ions O2−, respectively. VO1 represents an oxygen vacancy. b The metal ions M1, M2, and M3 surrounding VO1 form an isosceles triangle. c The four oxygen ions surrounding M2, labeled O4, O9, O10, and O8, form an almost perfect planar square (while O1\({}^{\prime}\), O5\({}^{\prime}\), O3\({}^{\prime}\), and O4\({}^{\prime}\) only form a rectangle, cf. Supplementary Note 4 for details). d The dxz and dyz orbitals at M2 next to VO1, with \(\hat{z}\) perpendicular to the O4, O8, O10, and O9 plane, remain essentially degenerate as a result of mirror and C4 rotation symmetry around M2. (Due to the non-symmorphic rutile structure, \({\hat{z}}\)∦\({\hat{z}^{\prime}}\), where \({\hat{z}}^{\prime}\) is parallel to the C4 axis at M1.).

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The rutile structure type possesses mirror reflection, inversion, and a fourfold rotation (C4) symmetry which enforce the presence of DNLs in the band structure of rutile oxides10. Some of these DNLs are protected from gapping out due to large SOC by the non-symmorphic symmetry of the rutile structure18,19. For IrO2 and RuO2 this has been recently confirmed by angle-resolved photoemission spectroscopy and band structure studies16,17,19. In the vicinity of VO’s, this set of symmetries promotes the formation of the orbital 1CK and 2CK effect. The emergent Majorana zero mode that accompanies the formation of the 2CK effect is reflected in a singular excitation spectrum above the ground state which generates a \(\sqrt{T}\)-dependence of the resistivity ρ(T) below a low-T energy scale20, the Kondo temperature TK. This requires a well-balanced competition of two otherwise independent and degenerate screening channels and makes the 2CK effect extremely difficult to realize, especially in a natural quantum material4,21,22. If one channel dominates over the other, the low-T behavior will be that of conventional fermions. If the 2CK state arises out of orbital Kondo scattering, magnetic-field (B) independence is expected for field strengths well above TK as long as gμBB ≪ W, where g is the Landé factor, μB is the Bohr magneton, and W is the conduction electron half-bandwidth. Our study is based on rutile (MO2, M =  Ir, Ru) nanowires (NWs) which allow us to combine a high degree of sample characterization with an exceptional measurement sensitivity while probing material properties in the regime where the characteristic sample dimension is much larger than the elastic electron mean free path (cf. Supplementary Note 3). That is, we are concerned with weakly disordered, diffusive metals which are three-dimensional (3D) with respect to the Boltzmann transport, whereas strong correlation effect causes a resistivity anomaly at low T. Table 1 lists the relevant parameters for the NWs studied in this work.

Table 1 Relevant parameters for MO2 NWs.
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Results

Oxygen vacancies in transition-metal rutiles MO2

In Fig. 1a, the vicinity of an VO, denoted VO1, is shown. The metal ions surrounding VO1, labeled M1, M2, and M3, form an isosceles triangle (Fig. 1b). For the sites M1 and M2, an almost perfect C4ν symmetry exists which implies a corresponding degeneracy associated with the two-dimensional irreducible representation of C4ν, see Fig. 1c and Supplementary Note 4. In the pristine system, the metal ions are surrounded by oxygen octahedra anchored around the center and the corners of the tetragonal unit cell. The π/2 angle between adjacent octahedra leads to a fourfold screw axis symmetry. This non-symmorphic symmetry not only protects DNLs in IrO2 against SOC-induced splitting17,19. It has also been linked to the high electrical conductivity of IrO2 (ref. 10) and, as we find, is in line with the strong tendency to localize electrons near VO’s required for the formation of orbital Kondo correlations. Moreover, the fourfold screw axis symmetry ensures that the C4 rotation axes centered at the sites M1 and M2 near VO1 are not parallel (\({\hat{z}}^{\prime}\)∦\({\hat{z}}\), see Fig. 1d). This enhances the phase space for the orbital Kondo effect over orbital order linking sites M1 and M2 (see also Supplementary Note 5).

Experimental signatures of orbital 2CK effect in IrO2 NWs

Now we turn to our experimental results which, to the best of our knowledge, demonstrate the most convincing realization of the long searched orbital 2CK effect in a solid. Fig. 2 demonstrates the formation of an orbital 2CK effect in IrO2 NWs. We find that as T decreases from room temperature to approximately  a few Kelvin, ρ(T) decreases in all IrO2 NWs, as expected for typical metallic behavior (cf. Supplementary Note 2). However, below T ~ 20 K, ρ(T) displays a \(\sqrt{T}\) increase of the ρ(T) upon lowering T over almost two decades in T(!), until a deviation sets in at  ~0.5 K. We performed systematic thermal annealing studies to adjust the oxygen contents in the NWs, which indicate that the anomalous low-T transport properties are driven by the presence of VO’s (ref. 23 and Supplementary Note 1). This is exemplified in Fig. 2. The top left inset shows a scanning electron microscopy image of NW A. In the oxygenated NW 3 which should contain a negligible amount of VO’s, ρ(T) decreases monotonically with decreasing T, revealing a residual resistivity, ρB0, below  ~4 K (top right inset). In contrast, in NWs A, B1 and B2 which contain large amounts of VO’s, ρ(T) increases with decreasing T, manifesting a robust \(\rho \propto \sqrt{T}\) law between  ~0.5 and  ~20 K. The slope of NW B2 is smaller than that of NW B1, which indicates a decrease in the number density of oxygen vacancies (n\({}_{{{V}}_{{\rm{O}}}}\)) due to prolonged aging (for about 5 months) in the atmosphere. The data explicitly demonstrate that the \(\rho \propto \sqrt{T}\) behavior is independent of B up to at least 9 T. The observed behavior is consistent with the 2CK effect as indicated by the straight solid lines which are linear fits to the 2CK effect calculated within the dynamical large-N method (cf. Supplementary Note 5), with n\({}_{{{V}}_{{\rm{O}}}}\) as an adjustable parameter (see Table 1 for the extracted values and Supplementary Notes 5 and  6 for the extraction method).

Fig. 2: Orbital 2CK resistivity of IrO2 NWs.
figure 2

ρ versus \(\sqrt{T}\) for IrO2 NWs A, B1 and B2 in magnetic fields B = 0, 6, and 9 T, as indicated. For clarity, the data of NWs B1 and B2 are shifted by 34.7 and 33.6 μΩ cm, respectively. A \(\rho \propto \sqrt{T}\) law, which is B independent, is observed between  ~0.5 and  ~20 K in all three NWs. The straight solid lines are linear fits to the 2CK resistivities calculated by the dynamical large-N method (see text). Top left inset: a scanning electron microscopy image of NW A. The scale bar is 1 μm. Top right inset: Low-T ρ(T) curves of NW A and a reference, oxygenated NW 3 (diameter d = 330 nm, ρ(300 K) = 124 μΩ cm).

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Ruling out the 3D electron–electron interaction (EEI) effect

To complicate matters, the EEI effect in 3D weakly disordered metals generically leads to a \(\sqrt{T}\) term in ρ(T) at low T (refs. 24,25). Unambiguously establishing that \(\rho (T) \sim \sqrt{T}\) indeed originates from 2CK physics thus requires a proper analysis of the EEI effect of the charge carriers. For example, for the NW B1 with ρB0 = 74 μΩ cm and the electron diffusion constant D ≃  6.2 cm2 s−1, the 3D EEI effect would predict a largest possible resistance increase of Δρ/ρ ≃ 2.8 × 10−4 as T is cooled from 20 to 1 K. Experimentally, we have observed a much larger resistance increase of 5.1 × 10−3. Furthermore, the 3D EEI effect would predict similar values for the magnitude of the low-T resistivity increase in NWs B1 and B2 to within  ≈3%, due to their ρB0 values differing by  ≈1% (Table 1). This is definitely incompatible with our observation of a  ≈50% difference. In addition, we find a deviation from the \(\sqrt{T}\) behavior at  ~0.5 K. If the \(\sqrt{T}\) anomaly were caused by the EEI effect, no such deviation should occur (see Supplementary Note 3 for an in-depth analysis of the EEI effect and its 3D dimensionality in our MO2 NWs).

VO-driven orbital Kondo scattering in MO2

For IrO2 the valency of the transition-metal ion M is close to the nominal valence of +IV in MO2 (ref. 26). Each VO generates two defect electrons due to charge neutrality. To minimize Coulomb interaction, the defect electrons will tend to localize at different M ions in the vicinity of the VO. In IrO2 this results in a nonmagnetic 5d6 ground state configuration of the Ir ions. For the electron localizing on ion M2 or M1 (Fig. 1a), the symmetry of the effective potential implies the almost perfect degeneracy of the orbitals dxz and dyz as defined in Fig. 1d. It is this orbital degeneracy that drives the orbital 2CK effect in IrO2 where the dxz and dyz form a local pseudospin basis, while the spin-degenerate conduction electrons act as two independent screening channels. Group theoretical arguments ensure that the exchange scattering processes between conduction electrons and pseudospin degree of freedom have a form compatible with the Kondo interaction27 (cf. Supplementary Note 5). Deviations from perfect symmetry which act as a pseudo-magnetic field are expected to become visible at lowest T. This explains the deviations from the \(\sqrt{T}\) behavior observed below  ~0.5 K in Fig. 2. If the two defect electrons localize at sites M1 and M2, a two-impurity problem might be expected which could lead to inter-site orbital order between the two defect electrons28. The non-symmorphic rutile structure, however, ensures that the C4 rotation axes centered at the sites M1 and M2 are not parallel. This together with the local nature of the decomposition provided in Supplementary Eq. (3) (see Supplementary Note 5) favor local orbital Kondo screening in line with our observation. These conclusions are further corroborated by demonstrating tunability of the orbital 2CK effect to its 1CK counterpart.

Experimental signatures of orbital 1CK effect in RuO2 NWs

RuO2 is also a DNL metal with the same non-symmorphic symmetry group as IrO2 but weaker SOC. In contrast to IrO2, it lacks time-reversal symmetry19,29. Based on the analysis for IrO2, we expect that VO’s in RuO2 will drive an orbital 1CK effect. This is indeed borne out by our transport data on RuO2 NWs. Fig. 3a shows the T dependence of the time-averaged Kondo resistivity 〈ρK〉 for NW C, where ρK(T) = ρ(T) − ρB0, and 〈…〉 denotes averaging. (RuO2 NWs often demonstrate temporal ρ fluctuations. Details can be found in Supplementary Note 2.) At low T, 〈ρK〉 follows the 1CK form30. The inset demonstrates the recovery of a Fermi-liquid ground state with its characteristic 〈ρK〉 ∝ T2 behavior below  ~12 K and unambiguously rules out the 3D EEI effect. Fig. 3b shows ρ(T) of NW E in B = 0 and 4 T. For clarity, the B = 0 data (black symbols) are averaged over time, while the B = 4 T data (red symbols) are non-averaged to demonstrate the temporal fluctuations of the low-T ρ(T) (ref. 31). Note that, apart from the aforementioned much smaller resistance increase as would be predicted by the 3D EEI effect compared with the experimental results in Fig. 3a, b, no \(\sqrt{T}\) dependence is detected here. In fact, the low-T resistivity anomalies conform very well to the 1CK scaling form for three decades in T/TK (Fig. 4a). Thus, the 3D EEI effect can be safely ruled out as the root of the observed low-T resistivity anomalies in RuO2 NWs.

Fig. 3: Orbital 1CK resistivity of RuO2 NWs.
figure 3

a Time-averaged Kondo resistivity 〈ρK〉 versus \(\mathrm{log}\,\it T\) for NW C. The straight line in the inset, which shows a low-T zoom-in, is a guide to the eye. b ρ versus \(\mathrm{log}\,\it T\) in B = 0 and 4 T for NW E. For clarity, the B = 0 data are time-averaged, while the 4-T data are non-averaged to demonstrate the temporal resistivity fluctuations at low T. The inset shows the time-averaged B = 4 T data (red symbols), which closely overlap the B = 0 data. c ρ versus \(\mathrm{log}\,\it T\) for NW A in B = 0, 3, and 5 T. Occasional resistivity jumps, or random telegraph noise, are observed. The dash-dotted curves depict the magnetoresistance predicted by the spin-\(\frac{1}{2}\) Kondo impurity model (see text). Note that the experimental data are independent of B. Inset: Low-T ρ(T) curves of NW A and a reference, oxygenated NW 4 (d = 150 nm, ρ(300 K) = 336 μΩ cm). In a–c, the solid curve shows the B = 0 numerical renormalization group result for 1CK effect59.

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Fig. 4: Comparison of 2CK and 1CK resistivities.
figure 4

a Normalized Kondo resistivity 〈ρK〉/ρK0 versus T/TK for RuO2 NWs A–E manifests the 1CK scaling form (solid curve) for over three decades of reduced temperature. b 〈ρK〉/ρK0 versus \(\sqrt{T/{T}_{{\rm{K}}}}\) for IrO2 NW A and RuO2 NWs B–E. The data of IrO2 NW A obeys a \(\sqrt{T}\) law between 0.39 and 21 K. For clarity, the experimental data points for RuO2 NWs are plotted with small open symbols. c 〈ρK〉/ρK0 of IrO2 NW B1 obeys a \(\sqrt{T/{T}_{{\rm{K}}}}\) law between 0.66 and 22 K, distinctively deviating from the 1CK function. d Results for the resistivity of a diluted system of 2CK impurities in a metallic host evaluated using a dynamical large-N limit (black symbols), which follows a \(\sqrt{T/{T}_{{\rm{K}}}}\) law at low T (see text and Supplementary Note 5). The ordinate is plotted in unit of half-bandwidth W = 4 eV (ref. 60).

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As a further demonstration of the B-field independence, we present in Fig. 3c ρ(T) data for NW A in magnetic fields of strength B = 0, 3, and 5 T. With \({T}_{{\rm{K}}}^{{\rm{A}}}\) = 3 K, NW A has the lowest TK among NWs A–E (Table 1). The data between 50 mK and 10 K, corresponding to T/TK = 0.017–3.3, can be well described by the 1CK function (solid curve). The dash-dotted curves depict the magnetoresistance predicted by the spin-\(\frac{1}{2}\) Kondo impurity model30 with gμBB/kBTK = 1.0, 2.0, and 4.1, as indicated, where kB is the Boltzmann constant. Our experimental data clearly demonstrate B independence, ruling out a magnetic origin of this phenomenon.

We remark on the relation between the residual resistivity ρB0 and the concentration of orbital Kondo scatterers n\({}_{{{V}}_{{\rm{O}}}}\) extracted from ρK0, the Kondo contribution to the ρ(T → 0) (see Supplementary Note 6), for RuO2 NWs. With the exception of NW B, our data indicate an approximately linear relation between n\({}_{{{V}}_{{\rm{O}}}}\) and ρB0 (Table 1 and Supplementary Fig. 3). It is not unexpected that the approximately linear relation between ρB0 and n\({}_{{{V}}_{{\rm{O}}}}\) holds for larger impurity concentrations, corresponding to larger values of ρB0 as all defects, screened dynamic and static defects, contribute to ρB0. This relation strongly demonstrates that the low-T resistivity anomalies are indeed due to VO-driven orbital Kondo effect. (We focus on RuO2 NWs because of the larger number of samples with a larger variation of ρB0 values compared with IrO2 NWs).

Comparison of 2CK and 1CK ρ(T) curves

  Figure 4a demonstrates that 〈ρK〉/ρK0 for RuO2 NWs follow the universal 1CK scaling over three decades in T/TK while TK ranges from 3 to 80 K! To further substantiate the subtle but distinct differences between the \(\sqrt{T}\) dependence of the 2CK behavior in IrO2 NWs from the 1CK scaling form, we plot 〈ρK〉/ρK0 as a function of \(\sqrt{T/{T}_{{\rm{K}}}}\) for IrO2 NWs A and B1, together with RuO2 NWs B–E and the 1CK function, in Fig. 4b, c, respectively. (The value for ρK0 was identified with the maximum values of the measured ρK(T) anomalies.) Fig. 4d illustrates that a dilute system of 2CK scattering centers immersed in a metallic host indeed displays a \(\sqrt{T}\) term in its low-T ρ(T). This \(\sqrt{T}\) power-law behavior is determined by the leading irrelevant operator near the 2CK fixed point32 and captured by the dynamical large-N method33,34,35.

Discussion

Despite the ubiquitous appearance of magnetic Kondo scattering in real quantum materials36, no convincing demonstration of the orbital Kondo effect37 or the 2CK effect22,38 exists. Many claims rest on a model of two-level systems immersed in a metallic host as a possible route to 2CK physics3,4. Theoretical arguments have, however, made it clear that this is not a viable route to nonmagnetic Kondo scattering22,38. Moreover, the creation of scattering centers in a real quantum material necessarily places the system in the weakly disordered regime where a conductance anomaly, the Altshuler–Aronov correction, occurs whose T dependence can be mistaken for a 2CK signature, see, e.g., refs. 39,40,41,42. Dilution studies on common Kondo lattice systems43,44, on the other hand, typically create disorder distributions of Kondo temperatures that may result in a behavior of observables, which can easily be mistaken for that of a generic non-Fermi liquid45.

We have shown that the low-T resistivity anomaly in the transition-metal rutile IrO2 is caused by VO’s, demonstrating key signatures of an orbital 2CK effect and ruling out alternative explanations due to, e.g., the EEI effect. The most convincing argument in favor of 2CK physics would be the demonstration of direct tunability of 2CK physics to 1CK physics upon breaking the channel degeneracy. This is difficult, as the channel degeneracy is protected by time-reversal symmetry. A perhaps less direct, yet complementary, demonstration of this tunability is provided by our results for RuO2 NWs which develop an orbital 1CK effect. In RuO2, the antiferromagnetic order breaks the channel degeneracy. Our analysis also indicates that the underlying symmetries which support the existence of DNLs in the Brillouin zones of both transition-metal rutiles also aid the formation of orbital 2CK and 1CK physics.

Materials condensing in the rutile structure type and its derivatives form an abundant and important class that has helped shaping our understanding of correlated matter. The metal-insulator transition in VO2, e.g., has been known for 60 years46, yet its dynamics is still not fully understood47. The demonstration that the non-symmorphic rutile space group supports a VO-driven orbital Kondo effect in MO2 holds promise for the realization of novel states of matter. The potential richness of orbital Kondo physics, e.g., on superconducting pairing, was recently pointed out in ref. 37 but may be even richer when considering the possibility of its interplay with topological band structures. Specifically, we envision the creation of a 2CK non-symmorphic superlattice of VO’s in IrO2 where the 2CK Majorana modes entangle with the band structure-enforced Dirac excitations forming a strongly correlated topological non-Fermi liquid state. Understanding its properties will foster deeper insights into the interplay of topology with strong correlations beyond the usual mean field treatment. The theoretical approach to this non-symmorphic superlattice is reminiscent of the topologically garnished strong-coupling fixed-point pioneered in the context of Weyl–Kondo semimetals11,48, suitably generalized to capture the intermediate coupling physics of the 2CK effect and its low-T excitations. The fabrication of superlattices of Kondo scattering centers has already been demonstrated49 while defect engineering of vacancy networks, including VO networks is currently explored in a range of materials50,51. The specifics of this unique state and its manufacturing are currently being explored.

Methods

NW growth

IrO2 NWs were grown by the metal-organic chemical vapor deposition method, using (MeCp)Ir(COD) supplied by Strem Chemicals as the source reagent. Both the precursor reservoir and the transport line were controlled in the temperature range of 100–130 °C to avoid precursor condensation during the vapor-phase transport. High purity O2, with a flow rate of 100 sccm, was used as the carrier gas and reactive gas. During the deposition, the substrate temperature was kept at  ≈350 °C and the chamber pressure was held at  ≈17 torr to grow NWs52,53. Selected-area electron diffraction patterns52 and X-ray diffraction (XRD) patterns54 revealed a single-crystalline rutile structure.

RuO2 NWs were grown by the thermal evaporation method based on the vapor-liquid-solid mechanism, with Au nanoparticles as catalyst. A quartz tube was inserted in a furnace. A source material of stoichiometric RuO2 powder (Aldrich, 99.9%) was placed in the center of the quartz tube and heated to 920–960 °C. During the NW growth, an O2 gas was introduced into the quartz tube and the chamber was maintained at a constant pressure of  ≈2 torr. Silicon wafer substrates were loaded at the downstream end of the quartz tube, where the temperature was kept at 450–670 °C (ref. 55). The morphology and lattice structure of the NWs were studied using XRD and high-resolution transmission electron microscopy (HR-TEM). The XRD patterns demonstrated a rutile structure55, and the HR-TEM images revealed a polycrystalline lattice structure56.

Electrical measurements

Submicron Cr/Au (10/100 nm) electrodes for 4-probe ρ(T) measurements were fabricated by the standard electron-beam lithography technique. The electrode fabrication was done after the thermal treatment (annealing and/or oxygenation) of each NW was completed. To avoid electron overheating, the condition for equilibrium, eVs ≪ kBT, was assured in all resistance measurements57, where e is the electronic charge, and Vs is the applied voltage across the energy relaxation length. The electrical-transport measurements were performed on a BlueFors LD-400 dilution refrigerator equipped with room-temperature and low-temperature low-pass filters. The electron temperature was calibrated down to ≲50 mK. In several cases (RuO2 NWs B–E), the measurements were performed on an Oxford Heliox 3He cryostat with a base temperature of ≃250 mK. The magnetic fields were supplied by superconducting magnets and applied perpendicular to the NW axis in all cases.