Introduction

In typical weak-coupling theories of superconductivity, the effective attraction V between electrons is mediated by the exchange of bosons having a characteristic energy scale ωB, and superconductivity condenses below a transition temperature Tc parameterized as1:

$${T}_{c} \sim {\omega }_{{\rm{B}}}\exp \left(-\frac{1}{N({E}_{F})V}\right)={\omega }_{{\rm{B}}}\exp \left(-\frac{1+\lambda }{\lambda -{\mu }^{* }}\right),$$
(1)

where N(EF) is the density of states (DOS) near the Fermi level, λ is the electron–boson coupling strength, and μ* is the Coulomb pseudopotential that describes the residual Coulomb repulsion between quasiparticles2. For simplicity, we assume that all of the non-isotropic q- and k-dependencies that appear in a more realistic formulation of Cooper pairing have been averaged away. Note that within the range of validity of Eq. (1)—viz., 1 ≫ λ > μ*—increasing λ (increasing μ*) generally enhances (suppresses) Tc, respectively, assuming that superconductivity remains the dominant instability.

Experimental methods that boost Tc are highly desired from a practical perspective. Furthermore, by analyzing how these available knobs couple to the normal-state properties on the right side of Eq. (1), one can envisage engineering the electronic structure and electron–boson coupling to optimize Tc. For example, increasing N(EF) is a frequently suggested route towards realizing higher Tc, but how to achieve this for specific materials often remains unclear.

Historically, chemical doping and hydrostatic pressure have been the most common knobs used to manipulate superconductivity. Unfortunately, doping has the complication of explicitly introducing substitutional disorder, whereas pressure studies are incompatible with most probes of electronic structure. Moreover, because large pressures are usually required to appreciably increase Tc3, pressure-enhanced superconductivity exists transiently—oftentimes in different structural polymorphs than at ambient conditions—rendering it inaccessible for applications.

An alternative strategy for controlling superconductivity is epitaxial strain engineering. This approach is static, disorder-free, allows for the use of sophisticated experimental probes4, and enables integration with other materials in novel artificial interfaces and device structures5,6. To date, epitaxial strain has only been used to modulate Tc in known superconductors7,8,9,10,11,12. In this article, we describe the creation of a new superconductor through epitaxial strain, starting from a compound, RuO2, previously not known to be superconducting. By comparing the results of angle-resolved photoemission spectroscopy (ARPES) experiments with density functional theory (DFT) calculations, we show that splittings between the effective low-energy d orbital degrees of freedom in RuO2 respond sensitively to appropriate modes of strain, and we discuss how this approach may open the door to strain tuning of superconductivity in other materials.

Results

Electrical and structural characterization of RuO2 thin films

Bulk RuO2 crystallizes in the ideal tetragonal rutile structure (space group #136, P42/mnm) with lattice constants at 295 K of (a = 4.492 Å, c = 3.106 Å)13. RuO2 thin films in distinct epitaxial strain states were synthesized using oxide molecular-beam epitaxy (MBE) by employing different orientations of isostructural TiO2 substrates, (a = 4.594 Å, c = 2.959 Å)14. As shown in Fig. 1a, b, the surfaces of (101)-oriented substrates are spanned by the \([\overline{1}01]\) and [010] lattice vectors of TiO2, which ideally impart in-plane tensile strains on RuO2 (at 295 K) of  +0.04% and  +2.3%, respectively. On TiO2(110), the lattice mismatches with RuO2 are larger:  −4.7% along [001] and  +2.3% along \([1\overline{1}0]\).

Fig. 1: Electrical transport behavior of bulk RuO2 single crystals and epitaxially strained RuO2 thin films.
figure 1

a, b Schematic diagrams of the crystal structures and in-plane lattice mismatches with TiO2 substrates of RuO2 thin films synthesized in (101)- and (110)-orientations. Gray and blue spheres represent Ru and O atoms, respectively. c Average resistivity versus temperature curves for 24.2 nm thick RuO2(110) and 18.6 nm thick RuO2(101) films, compared to results for bulk RuO2 single crystals from Ref. 15. For clarity the bulk RuO2 data have been rigidly shifted upward by 1 μΩ-cm (ρ0  ≈  0.3  μΩ-cm). d V(I) curve measured at 0.6 K on a 10 μm-wide resistivity bridge lithographically patterned on the RuO2(110) sample from (c) (as shown in the inset: scale bar  = 200  μm), which has the direction of current flow parallel to [001]rutile. Similarly large critical current densities Jc are obtained with \(I| | [1\overline{1}0]\) (Supplementary Note 1 and Supplementary Fig. 1). e, f Upper critical magnetic fields Hc⊥ versus superconducting Tcs extracted from magnetoresistance measurements for the RuO2(110) sample in (c) along with a characteristic R(H) sweep acquired at 0.45 K (inset in (f)). Superconducting Tcs are taken as the temperatures at which the resistance crosses 50% of its residual normal-state value R 4 K; error bars on these Tcs indicate where R crosses the 90% and 10% thresholds of R4 K, respectively (cf. the horizontal dashed lines in (e)).

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  Figure 1c shows electrical resistivity ρ(T) measurements for RuO2 films, along with results for bulk RuO2 single crystals from Ref. 15. To compare with bulk, for the thin-film samples we plot the geometric mean of the components of ρ along the two in-plane directions; the intrinsic resistive anisotropy is known to be small16, consistent with our findings (Supplementary Note 1 and Supplementary Fig. 1). ρ(T) data for the lightly strained RuO2/TiO2(101) sample—henceforth referred to as RuO2(101)—are nearly indistinguishable from bulk, exhibiting metallic behavior with a low residual resistivity ρ(0.4 K) < 2  μΩ-cm. In contrast, a clear superconducting transition is observed for the more heavily strained RuO2/TiO2(110) sample—referred to as RuO2(110)—at Tc = 2.0 ± 0.1  K.

Magnetoresistance measurements (Fig. 1e, f) with H⊥ applied along [110] (the out-of-plane direction) show a monotonic suppression of Tc with increasing fields and an extrapolated value of Hc⊥(T → 0 K) = 13.3 ± 1.5 kOe, corresponding to an average in-plane superconducting coherence length of ξ(T → 0 K) = 15.8 ± 0.9 nm (Supplementary Note 2 and Supplementary Fig. 2). In Fig. 1d, we show a V(I) curve measured on a lithographically patterned resistivity bridge at T/Tc = 0.3, from which we extract a critical current density Jc = (9.5 ± 1.2) × 104 A/cm2. This large value of Jc (over one order of magnitude larger than values reported on typical elemental superconductors with comparable Tcs) indicates that the superconductivity in RuO2(110) does not arise from a filamentary network, structural defects, minority phases, or from the substrate–film interface, which would all yield much smaller values of Jc.

In order to disentangle the effects of strain from other possible sources of superconductivity, we compare RuO2 films as functions of strain and film thickness, t. In Fig. 2a, we plot x-ray diffraction (XRD) data from similar-thickness films of RuO2(101) and RuO2(110), showing that the bulk-averaged crystal structures of the films are strained as expected along the out-of-plane direction based on their net in-plane lattice mismatches with TiO2. The primary 101 and 202 film peaks of RuO2(101) are shifted to larger angles than bulk RuO2, corresponding to a 1.1% compression of d101, while Nelson-Riley analysis of the primary 110, 220, and 330 (see, e.g., Supplementary Fig. 4) peak positions for RuO2(110) evidence a 2.0% expansion of d110 relative to bulk. In Fig. 2b, c, we plot resistivity data showing that reducing t in RuO2(110) decreases Tc, as is commonly observed in numerous families of thin-film superconductors17,18, with Tc dropping below our experimental threshold (0.4 K) between t = 11.5 and 5.8 nm. This suppression of Tc with thickness indicates superconductivity is not confined near the substrate–film interface, so possible interfacial modifications of the crystal structure19, carrier density20, substrate–film mode coupling21, and non-stoichiometry in the films or substrates22,23,24 can all be eliminated as potential causes of superconductivity. These conclusions are also supported by the facts that superconductivity is not observed in RuO2(101) films, nor in bare TiO2 substrates treated in an identical fashion to the RuO2 films. Finally, in Fig. 2d we include a scanning transmission electron microscopy (STEM) image of a superconducting RuO2(110) sample, which confirms uniform growth of the film over lateral length scales exceeding those expected to be relevant for superconductivity (e.g., ξ), and shows a chemically abrupt interface between RuO2 and TiO2 (Supplementary Fig. 5), with no evidence of minority phases.

Fig. 2: Structural characterization of epitaxially strained RuO2 thin films, and film-thickness-dependent superconductivity for RuO2(110).
figure 2

a XRD data acquired with Cu-Kα radiation along the specular crystal truncation rods for 18.6 nm thick RuO2(101) and 14.2 nm thick RuO2(110) films. Bragg peaks arising from the TiO2 substrates are marked with asterisks, and the peak positions that would be expected for unstrained bulk RuO2 are indicated by dashed red lines13. Insets display rocking curves with FWHMs  < 0.01° acquired at the 2θ values corresponding to the primary 220 and 202 film peaks. Here q∣∣ is aligned with TiO\({\,}_{2}[1\overline{1}0]\) for the (110)-oriented sample, and with TiO\({\,}_{2}[\overline{1}01]\) for the (101)-oriented sample. b Resistance versus temperature curves for RuO2(110) samples with different film thicknesses t, normalized to their values at 300 K. c Superconducting Tcs and residual resistivities ρ0 plotted versus film thickness for the RuO2(110) samples from (b). Error bars on Tcs have the same meaning as in Fig. 1. The horizontal dashed line represents the base temperature attainable in our refrigerator, 0.4 K. d STEM image of the same 14.2 nm thick RuO2(110) sample from (a–c) (scale bar = 5 nm). More comprehensive structural and electrical characterization of the samples shown here are included in Supplementary Notes 3, 4 and Supplementary Figs. 3–10.

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We believe the thickness dependence of Tc results primarily from the competition between: (i) an intrinsic strain-induced enhancement of Tc that should be maximized for thinner, commensurately strained RuO2(110) films, versus (ii) disorder-induced suppressions of Tc that become amplified in the ultrathin limit (see, e.g., ρ0 versus t in Fig. 2c). While the thinnest films experience the largest substrate-imposed strains, stronger disorder scattering (likely from interfacial defects) reduces Tc below our detection threshold. Films of intermediate thickness (t  ≈  10–30 nm) have lower residual resistivities and higher Tcs, but do exhibit signatures of partial strain relaxation. Nevertheless, a detailed analysis of misfit dislocations by STEM and XRD reciprocal-space mapping (Supplementary Notes 3, 4 and Supplementary Figs. 8–10) indicates that these films are largely structurally homogeneous and, on average, much closer to commensurately strained than fully relaxed. Finally, in much thicker samples (e.g., t  =  48 nm) where a more significant volume fraction of the film should be relaxed, the strain is further released by oriented micro-cracks that make such samples spatially inhomogeneous and cause severely anisotropic distributions of current flow, preventing reliable resistivity measurements (Supplementary Fig. 11).

DFT calculations and ARPES measurements

Having established the strain-induced nature of the superconductivity in RuO2(110), we now explore its underlying origin using a combination of DFT and ARPES. In Fig. 3a, we present the electronic structure of commensurately strained RuO2(110) calculated by DFT + U (U = 2  eV), following the methods of Berlijn et al.13. Despite being constructed of RuO6 octahedra having the same 4d4 electronic configuration as in (Ca,Sr,Ba)RuO3, the electronic structure of RuO2 is markedly different from that of perovskite-based ruthenates. These distinctions arise from a sizable ligand-field splitting of the t2g orbitals, such that the most natural description of the low-energy electronic structure is in terms of states derived from two distinct types of orbitals: d∣∣ and (dxz, dyz), as illustrated by plots of Wannier functions in Fig. 3b25,26. Viewed in the band basis in Fig. 3a, the differentiation in k-space between these orbitals becomes apparent: the near-EF d∣∣ states (yellow-orange) form mostly flat bands concentrated around the k001 = π/c (i.e., Z-R-A) plane, whereas the (dxz, dyz) states (purple) form more isotropically dispersing bands distributed uniformly throughout the Brillouin zone.

Fig. 3: Electronic structure of RuO2.
figure 3

a Non-magnetic band structure of RuO2(110) according to DFT, calculated within the generalized gradient approximation (GGA) including spin–orbit coupling (SOC) and a static  + U  =  2  eV correction on the Ru sites. The color scale indicates the magnitudes of projections of the Kohn-Sham eigenstates at each k onto Ru-centered Wannier functions with d∣∣ and (dxz, dyz) orbital characters, which are constructed from the manifold of self-consistent eigenstates spanning EF and are plotted in drawings of the crystal structure in (b). Ru (O) atoms are colored gray (blue), as in Fig. 1a, b. Green and orange surfaces in (b) represent isosurfaces of the Wannier functions that have equal absolute magnitudes, but opposite (i.e., positive and negative) signs, respectively. c Brillouin zone schematic defining the coordinate system utilized for describing ARPES measurements of the electronic structure on (110)-oriented surfaces: kx ∣∣ [001]rutile, \({k}_{y}\ | | \ {[1\overline{1}0]}_{{\rm{rutile}}}\), and kz ∣∣ [110]rutile. The Brillouin zone of the parent tetragonal rutile structure is outlined in purple, the high-symmetry contour for the spaghetti plot from (a) is colored red, and the region probed on (110)-oriented surfaces with He-Iα photons (21.2 eV) is shaded green (Supplementary Note 6 and Supplementary Fig. 13). d Slice through the Fermi surface experimentally measured for a 7 nm thick RuO2(110) film (left), compared to the Fermi surface from DFT + U simulations (right) projected onto the region of the Brillouin zone colored green in (c). E(k) spectra acquired along the one-dimensional cuts indicated by dashed white lines in (d) show: e flat bands with d∣∣ orbital character and f more dispersive bands with (dxz, dyz) character, both consistent with DFT + U expectations (solid white lines). The intensities of the experimental data shown in (d–f) and of the DFT simulations shown in (d) are plotted in arbitrary units where we define 0 (1) to be the minimum (maximum) value, respectively, of the given data set. Only relative changes in intensity within a given panel (as visualized by the false color scales) are meaningful.

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In many other d4 ruthenates (such as Sr2RuO4 and Ca2RuO4), static mean-field electronic structure calculations (such as DFT + U) often predict quantitatively incorrect effective masses27,28,29,30,31—and sometimes even qualitatively incorrect ground states32—because these approaches neglect local (atomic-like) dynamical spin-orbital correlations (driven by Hund’s rules) that strongly renormalize the low-energy quasiparticle excitations. Therefore, it is imperative to compare DFT calculations for RuO2 with experimental data, to establish the reliability of any theoretically predicted dependence of the electronic structure on strain. The left half of Fig. 3d shows the Fermi surface of RuO2(110) measured with He-Iα (21.2  eV) photons at 17 K, which agrees well with a non-magnetic DFT + U simulation of the Fermi surface at a reduced out-of-plane momentum of k110 = −0.2 ± 0.2 π/d110 (right half of Fig. 3d). In Fig. 3e, f, we plot energy versus momentum spectra acquired along the white dashed lines in Fig. 3d: in Fig. 3e, the spectrum is dominated by the flat d∣∣ bands centered around a binding energy of 300 meV, whereas in Fig. 3f the (dxz, dyz)-derived bands are steeply dispersing and can be tracked down to several hundred meV below EF, both of which are well reproduced by DFT + U calculations. The reasonable agreement between the experimentally measured and DFT band velocities is consistent with recent ARPES studies of Ir-doped RuO2 single crystals33 and with earlier specific heat measurements of the Sommerfeld coefficient in bulk RuO2, which suggested a modest momentum-averaged quasiparticle mass renormalization of γexp. = 1.45γDFT34,35. The fact that the true electronic structure of RuO2 can be well accounted for by DFT + U allows us to utilize such calculations to understand how epitaxial strains can be employed to engineer features of the electronic structure to enhance the instability towards superconductivity.

Evolution of electronic structure under strain

In Fig. 4a, we show the strain dependence of the DFT-computed band structure and DOS for RuO2(110), RuO2(101), and bulk RuO2. While the results for RuO2(101) are almost identical to bulk, the results for RuO2(110) exhibit significant differences: the large d∣∣-derived peak in the DOS (centered around a binding energy of 800 meV for bulk) is split into multiple peaks for RuO2(110), several of which are shifted closer to the Fermi level, thereby increasing N(EF). In our studies, we found that this strain-dependent trend was robust against details of the DFT calculations, such as whether U was finite (Supplementary Note 5 and Supplementary Fig. 12). In order to determine whether this strain dependence of N(EF) is realized in experiment, we compared the electronic structure of a thin (7 nm) highly strained RuO2(110) film with a much thicker (48  nm) partially strain-relaxed RuO2(110) film. The surface lattice constants of the 48 nm thick film were closer to bulk RuO2 than the 7 nm thick film (Supplementary Note 7 and Supplementary Fig. 14), so we expect that the surface electronic structure probed by ARPES of the thicker film to be more representative of bulk RuO2. Comparisons between the RuO2(110) and RuO2(101) surfaces are less straightforward, since different parts of the three-dimensional Brillouin zone are sampled by ARPES (Supplementary Note 8 and Supplementary Fig. 15). Figure 4b shows E(k) spectra side by side for the 7 nm (left) and 48 nm (right) films of RuO2(110) along the same cut through k-space from Fig. 3e where the photoemission intensity is dominated by d∣∣ initial states. The higher levels of strain present at the film surface for the 7 nm thick sample cause a substantial shift of the flat bands towards EF by 120 ± 20 meV relative to the more strain-relaxed 48 nm thick sample. Integrating the ARPES data over the full measured region of k-space for both samples gives the average energy distribution curves plotted in Fig. 4c, which show that spectral weight near EF is enhanced as the d∣∣ states move towards EF, in qualitative agreement with the trend predicted by DFT. Our results indicate that the primary electronic effect of the epitaxial strains in RuO2(110) is to alter the relative occupancies of the d∣∣ and (dxz, dyz) orbitals as compared with bulk, and to push a large number of states with d∣∣ character closer to EF, which enhances N(EF) and likely Tc.

Fig. 4: Strain-induced changes to the electronic structure of RuO2.
figure 4

a DFT + U (U = 2 eV) band structures and corresponding density of states (DOS) traces for bulk RuO2 and epitaxially strained RuO2(110) and RuO2(101) thin films. The RuO2(101) results are omitted from the spaghetti plot for clarity since they are very similar to bulk. b Comparison of E(k) spectra along the cut shown in Fig. 3e for two different RuO2(110) samples: a highly strained 7 nm thick film (left), and a partially strain-relaxed 48 nm thick film (right). The false color scale used to visualize the intensities in each spectrum is defined and normalized in the same way as in Fig. 3. c As an approximate proxy of the total DOS, for these samples we plot the energy distribution curves of photoemission intensity averaged over the entire region of k-space probed experimentally with 21.2 eV photons (cf. Fig. 3c), which demonstrate that the epitaxial strains imposed by TiO2(110) substrates shift d∣∣ states towards EF and thereby increase N(EF).

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Discussion

Observations of Fermi-liquid-like quasiparticles near EF34,36,37,38 that scatter at higher energies primarily via their interaction with phonons16,35, along with the fact that superconductivity in RuO2(110) persists in the dirty limit (Supplementary Note 4 and Supplementary Fig. 9), are both consistent with conventional Cooper pairing, suggesting that calculations assuming an electron–phonon mechanism may be enlightening. We performed DFT-based Migdal-Eliashberg calculations of Tc for bulk RuO2 and commensurately strained RuO2(110) that indeed indicate epitaxial strain can enhance Tc by several orders of magnitude. For bulk RuO2, we find that the empirical Coulomb pseudopotential must satisfy μ* > 0.30 to be compatible with the experimentally measured least upper bound on Tc (Tc < 0.3  K15). For this range of μ*, Tc for RuO2(110) can be as high as 7 K (Supplementary Note 9 and Supplementary Fig. 16). A robust strain-induced enhancement of the electron–phonon coupling λel−ph boosts Tc by a factor of 20 (for μ* = 0.30), and this ratio becomes even larger for higher values of μ*—e.g., for μ* = 0.37, Tc(110)/Tc(bulk) = 5 K/5 mK). Although these estimations of Tc are broadly consistent with our experimental findings, conventional superconductivity in RuO2 remains a working hypothesis until measurements of the order parameter are possible.

In principle, assuming that all Fermi liquids are eventually unstable towards some channel(s) of Cooper pairing at sufficiently low temperatures and magnetic fields (including internal fields arising from magnetic impurities), the strain-stabilized superconductivity observed here in RuO2 is not strictly a change in the ground state of the system. For our purposes, however, extremely low temperatures and fields below what are experimentally achievable can be regarded as effectively zero, justifying our use of phrases such as strain-induced superconductivity interchangeably with huge enhancement of critical temperature. If we limit the scope of this semantic discussion to conventional, non-sign-changing (s-wave) order parameters, we note that in the presence of Coulomb repulsion and other effects, an instability towards s-wave superconductivity is not present in every system; the electron–phonon coupling generally must exceed some finite critical value. In the present context, the effects of strain reported in this article might be boosting the electron–phonon coupling above the critical value appropriate for RuO2, thus inducing a new s-wave state that is absent (even in theory) for the unstrained material.

We believe our results demonstrate that a promising strategy to create new transition-metal superconductors is to apply judiciously chosen anisotropic strains that modulate degeneracies among d orbitals near EF. Many classic studies of conventional superconductors that have nearly-free-electron states spanning EF derived from (s, p) orbitals actually show decreases in Tc under hydrostatic pressure39, due to lattice stiffening dominating over any pressure-induced changes to the Hopfield parameter40. In a limited number of elemental metals where Tc monotonically increases under pressure (such as vanadium41), pressure-induced electron transfer between s → d orbitals has been suggested as a likely cause of the enhanced transition temperatures3; a drawback of this approach, however, is that large pressures of ⪸ 10  GPa are typically required to, e.g., double Tc. More recently, measurements on single crystals of the unconventional superconductor Sr2RuO4 have shown that appropriately oriented uniaxial pressures of only  ≈1 GPa can boost Tc by more than a factor of two42. Independent of the underlying mechanism, it appears that anisotropic strains may prove to be significantly more efficacious than hydrostatic pressure for tuning superconductivity in multi-orbital systems, as shown here for RuO2, as well as in Sr2RuO4.

Sizable coupling between the lattice and electronic degrees of freedom in rutile-like crystal structures has been well established both theoretically26 and experimentally in VO2, where strain-induced variations in the orbital occupancies can be used to modify the metal-insulator transition temperature by δTMIT ≈ 70 K43,44. Therefore, it may be promising to explore other less strongly correlated (i.e., 4d and 5d) rutile compounds such as MoO2 for strain-stabilized superconductivity, instead of employing chemical doping45,46,47. Finally, since RuO2/TiO2(110) is the first known stoichiometric superconductor within the rutile family, further optimization of the superconductivity may enable the creation of structures that integrate superconductivity with other functional properties that have been extensively studied in other rutile compounds, such as high photocatalytic efficiency, half-metallic ferromagnetism, and large spin Hall conductivities.

Methods

Film synthesis

Epitaxial thin films of RuO2 were synthesized on various orientations of rutile TiO2 substrates using a GEN10 reactive oxide MBE system (Veeco Instruments). Prior to growth, TiO2 substrates (Crystec, GmbH) were cleaned with organic solvents, etched in acid, and annealed in air to produce starting surfaces with step-terrace morphology, following the methods in Ref. 48. Elemental ruthenium (99.99% purity, ESPI Metals) was evaporated using an electron-beam evaporator in background oxidant partial pressures of 1 × 10−6 − 5 × 10−6  Torr of distilled ozone (≈80% O3 + 20% O2) at substrate temperatures of 250–400 °C, as measured by a thermocouple. Reflection high-energy electron diffraction was used to monitor the surface crystallinity of the films in situ and showed characteristic oscillations in intensity during most of the Ru deposition, indicating a layer-by-layer growth mode following the initial nucleation of several-monolayer-thick RuO2 islands49.

Film characterization

The crystal structures of all RuO2 thin-film samples were characterized via lab-based x-ray diffraction (XRD) measurements with Cu-Kα radiation (Rigaku SmartLab and Malvern Panalytical Empyrean diffractometers). Four-point-probe electrical transport measurements were conducted from 300 K down to a base temperature of 0.4 K using a Physical Properties Measurement System equipped with a He-3 refrigerator (Quantum Design). All RuO2/TiO2(110) samples were superconducting with Tcs ranging from 0.5 to  2.4 K, except for ultrathin films with residual resistivities ρ0 ⪸  40 μΩ-cm, as shown in Fig. 2 and Supplementary Fig. 9.

A subset of films studied by XRD and transport were also characterized in situ by ARPES and low-energy electron diffraction (LEED). For these measurements, films were transferred under ultrahigh vacuum immediately following growth to an analysis chamber with a base pressure of 5 × 10−11 Torr equipped with a helium plasma discharge lamp, a hemispherical electron analyzer (VG Scienta R4000), and a four-grid LEED optics (SPECS ErLEED 150).

A subset of films studied by XRD and transport were also imaged using cross-sectional STEM. Cross-sectional specimens were prepared using the standard focused ion beam (FIB) lift-out process on a Thermo Scientific Helios G4 X FIB. High-angle annular dark-field STEM (HAADF-STEM) images were acquired on an aberration-corrected FEI Titan Themis at 300 keV with a probe convergence semi-angle of 21.4 mrad and inner and outer collection angles of 68 and 340  mrad.

Electronic structure calculations

Non-magnetic DFT calculations for the electronic structure of RuO2 were performed using the Quantum ESPRESSO software package50,51 with fully relativistic ultrasoft pseudopotentials for Ru and O52. We represented the Kohn-Sham wavefunctions in a basis set of plane waves extending up to a kinetic energy cutoff of 60  Ry, and used a cutoff of 400 Ry for representing the charge density. Brillouin zone integrations were carried out on an 8 × 8 × 12 k-mesh with 70 meV of Gaussian smearing. Perdew, Burke, and Ernzerhof’s parametrization of the generalized gradient approximation was employed as the exchange-correlation functional53, supplemented by an on-site correction of  +Ueff = U − J = 2 eV within spheres surrounding the Ru sites, following Ref. 13.

After obtaining self-consistent Kohn-Sham eigenstates via DFT, we used the pw2wannier and Wannier90 codes54 to construct 20 Wannier functions spanning the manifold of eigenstates surrounding EF (20 = 10 d-orbitals per Ru atom  ×  2 Ru atoms per unit cell). Following Ref. 55, to account for the non-symmorphic space group symmetries of rutile crystal structures, we referenced the trial orbitals employed in the Wannierisation routine to locally rotated coordinate systems centered on the two Ru sites within each unit cell. Orbital designations employed in the main text such as d∣∣ and (dxz, dyz) refer to projections onto this basis of Wannier functions. The more computationally efficient Wannier basis was used to calculate quantities that required dense k meshes to be properly converged, such as the projected Fermi surface in Fig. 3d (51 × 51 × 51 k-mesh) and the near-EF density of states traces in Fig. 4a (32 × 32 × 48 k-meshes).

Because the RuO2 samples studied in this work are thin films subject to biaxial epitaxial strains imposed by differently oriented rutile TiO2 substrates, we performed DFT + Wannier calculations of the electronic structure for several different crystal structures of RuO2 as described in Supplementary Note 5 and Supplementary Table 1. We used the ISOTROPY software package56 to study distortions of the parent tetragonal rutile crystal structure that are induced in biaxially strained thin films. Crystal structures and Wannier functions were visualized using the VESTA software package57.

Electron–phonon coupling calculations

To generate the inputs required for the electron–phonon coupling calculations described below, first-principles electronic structure and phonon calculations were performed using the Quantum ESPRESSO software package with norm-conserving pseudopotentials and plane-wave basis sets50,51. Here we employed a kinetic energy cutoff of 160 Ry, an electronic momentum k-point mesh of 16 × 16 × 24, 20  meV of Methfessel-Paxton smearing for the occupation of the electronic states, and a tolerance of 10−10 eV for the total energy convergence. The generalized gradient approximation as implemented in the PBEsol functional58 was employed as the exchange-correlation functional. For the Wannier interpolation, we used an interpolating electron-momentum mesh of 8 × 8 × 12 and a phonon-momentum mesh of 2 × 2 × 3. Results for bulk RuO2 were calculated using the crystal structure that minimizes the DFT-computed total energy with the PBEsol functional: (a = 4.464 Å, c = 3.093 Å) and xoxygen = 0.3062. Results for strained RuO2(110) were calculated by changing the lattice constants of this simulated bulk crystal structure by  +2.3% along \([1\overline{1}0]\),  −4.7% along [001],  +2.2% along [110], and setting xoxygen = yoxygen = 0.2996. The lattice parameter along [110] and internal coordinates of this simulated RuO2(110) structure were determined by allowing the structure to relax so as to (locally) minimize the DFT-computed total energy.

Electron–phonon coupling calculations were performed using the EPW code59, using an interpolated electron-momentum mesh of 32 × 32 × 48 and an interpolated phonon-momentum mesh of 8 × 8 × 12. The isotropic Eliashberg spectral function α2F(ω) and total electron–phonon coupling constant λel−ph (integrated over all phonon modes and wavevectors) were calculated with a phonon smearing of 0.2 meV. From the calculated α2F(ω) and λel−ph, we estimated the superconducting transition temperature using the semi-empirical McMillan-Allen-Dynes formula60,61:

$${T}_{c}=\frac{{\omega }_{\mathrm{log}\,}}{1.2}\exp \left[-\frac{1.04(1+{\lambda }_{{\rm{el}}-{\rm{ph}}})}{{\lambda }_{{\rm{el}}-{\rm{ph}}}-{\mu }^{* }(1+0.62{\lambda }_{{\rm{el}}-{\rm{ph}}})}\right]$$
(2)