Strain-stabilized superconductivity

Superconductivity is among the most fascinating and well-studied quantum states of matter. Despite over 100 years of research, a detailed understanding of how features of the normal-state electronic structure determine superconducting properties has remained elusive. For instance, the ability to deterministically enhance the superconducting transition temperature by design, rather than by serendipity, has been a long sought-after goal in condensed matter physics and materials science, but achieving this objective may require new tools, techniques and approaches. Here, we report the transmutation of a normal metal into a superconductor through the application of epitaxial strain. We demonstrate that synthesizing RuO2 thin films on (110)-oriented TiO2 substrates enhances the density of states near the Fermi level, which stabilizes superconductivity under strain, and suggests that a promising strategy to create new transition-metal superconductors is to apply judiciously chosen anisotropic strains that redistribute carriers within the low-energy manifold of d orbitals.


Supplementary Note 1: ELECTRICAL TRANSPORT MEASUREMENTS ON PATTERNED RESISTIVITY BRIDGES
In Supplementary Fig. 1, we show how the electrical transport properties of an RuO 2 /TiO 2 (110) sample depend on the direction of current flow in the film when it is confined to flow along the orthogonal in-plane crystallographic axes, [001] and [110]. Prior to lithographically patterning resistivity bridges on the film, we measured the resistance versus temperature of the entire 10 mm × 10 mm × 24.2 nm thick film by wire bonding four contacts directly to the surface of the sample in an inline contact geometry. Such a contact geometry probes the geometric mean of the two diagonal components of the in-plane resistivity tensor, i.e.
√ ρ 001 ρ 110 , neglecting small finite-size corrections that depend on how the contacts are oriented relative to the edges of the wafer [1]. The results of these measurements are shown by the blue traces in Supplementary Fig. 1a-b; these are the same data plotted on a logarithmic temperature scale in Fig. 1c of the main text.
Since RuO 2 has a tetragonal crystal structure in bulk (and orthorhombic or perhaps monoclinic in (110)oriented films), ρ 001 and ρ 110 are not guaranteed by symmetry to be equal. The intrinsic transport anisotropy in bulk RuO 2 is known to be small, with differences between ρ 100 and ρ 001 that are less than 10% at 300 K [2,3]; however, in heteroepitaxial thin films it is common for highly oriented structural defects-e.g., those nucleated at step edges on the substrate-to induce sizable extrinsic anisotropies between the different in-plane components of the resistivity tensor [4,5]. To investigate this possibility in this work, we used standard lithographic techniques to pattern the same RuO 2 /TiO 2 (110) sample into fourpoint resistivity bridges with dimensions 55 µm (length) × 10 µm (width) × 24.2 nm (thickness), where the direction of current flow is confined (via lithography) to be aligned with specific crystallographic directions. In the course of performing the lithography, we noticed that the TiO 2 substrates became mildly conducting, possibly due to oxygen vacancies formed during ion milling, as has been reported to occur for SrTiO 3 [6]. Therefore, we annealed the wafer containing the patterned resistivity bridges in air at elevated temperatures until the substrate again read open-circuit two-point resistances (> 100 MΩ); 2 hours at 500 • C was found to be sufficient.
The results of electrical measurements on these patterned resistivity bridges are shown by the green and orange traces in Supplementary Fig. 1. The temperature dependence of ρ(T ) is qualitatively consistent with the control measurements performed on the entire film before patterning, and the absolute magnitude of the resistivity anisotropies at 300 K and 4 K are both < 20%. Furthermore, the superconducting ρ(T ) and V (I) behavior does not depend strongly on the direction of current flow; this is contrary to what would be expected if the superconductivity arose purely from oriented structural defects.
In Supplementary Fig. 1b, we ascribe the substantial decrease in low-temperature resistivities observed in the patterned resistivity bridge data relative to the entire film data to the aforementioned annealing involved in preparing the bridges. We confirmed on other RuO 2 /TiO 2 (110) samples not containing bridges that post-growth annealing in air generically causes the low-temperature values of ρ to drop, by as much as a factor of four. Because of these complications and additional uncertainties involved in lithographically patterning resistivity bridges on films on TiO 2 substrates, all other electrical transport data presented in the main text and in the supplementary information were acquired by wire bonding directly to the surfaces of as-grown samples that were not subject to any post-growth annealing treatments.

Supplementary Note 2: FITTING AND EXTRAPOLATION OF SUPERCONDUCTING UPPER CRITICAL FIELDS VERSUS TEMPERATURE
In Supplementary Fig. 2, we present the results of magnetoresistance measurements for three RuO 2 /TiO 2 (110) samples with different film thicknesses; the data in Supplementary  (2). Tcs are taken as the temperatures at which R crosses 50% of R4 K (middle dashed lines in a, c, e); error bars in b, d, f indicate the temperatures at which R crosses the 90% and 10% thresholds of R4 K, respectively (top and bottom dashed lines in a, c, e). through the superconducting transitions. All resistances are normalized to their zero-field values at 4 K, well above the superconducting transitions; since the normal-state R(T, H) behavior of RuO 2 /TiO 2 (110) in the absence of superconductivity is negligible in this regime of low temperatures and fields, the choice of a single normalization factor R 4 K for all data does not appreciably affect any of the results that follow. Because percolation effects imply that resistive measurements of critical fields inherently contain some ambiguity about the definition and meaning of H c⊥ relative to truly bulk-sensitive measurements of superconductivity [7], here we adopt the same convention employed in the main text: the temperature at which R crosses 50% of R 4 K is taken as T c for the given H c⊥ , and the error bars on the extracted T c are the temperatures at which R crosses the 90% and 10% thresholds of R 4 K , respectively [8].
While there are considerable quantitative discrepancies in the values of H c⊥ and T c for the different-thickness samples shown in Supplementary Fig. 2, the H c⊥ (T c ) scaling behavior is remarkably linear for all samples, with no signs of H c⊥ saturation down to reduced temperatures T /T c ≈ 0.2 − 0.3, unlike what is expected in, e.g., Werthamer-Helfand-Hohenberg (WHH) theory [9]. For example, evaluating the right hand side of the WHH expression where Φ 0 is the superconducting flux quantum and µ 0 is the magnetic permeability of free space.
Notably, these values of ξ(T → 0 K) are less than values reported for traditional elemental superconductors with comparable T c s by almost an order of magnitude, corresponding to critical fields that are ≈ 1 − 2 orders of magnitude greater. While an explanation and understanding of these sizable critical field enhancements are beyond the scope of the present work, they are internally self-consistent with the large critical current densities noted in Fig. 1d of the main text and in Supplementary Fig. 1c. These results may motivate future realspace measurements of the superconducting condensate by scanning-probe techniques. In particular, an interesting question to address is whether the structural defects in RuO 2 (110) act as pinning sites for the vortices that form under applied fields, similar to what has been observed in numerous other thin-film superconductors [10], or whether the defects host regions of enhanced superfluid density that effectively act as barriers to vortex motion, akin to twin boundaries in bulk single crystals of iron-based superconductors [11,12]. In Supplementary Fig. 3, we include electrical characterization and more comprehensive lab-based x-ray diffraction (XRD) measurements for the RuO 2 (101) and RuO 2 (110) films of comparable thickness shown in Fig. 2a of the main text. Supplementary Fig. 3a,e show the zero-field ρ(T ) behavior for the two films: the 18.6 nm thick RuO 2 (101) film is non-superconducting down to < 0.4 K with a residual resistivity ρ 0 < 1.7 µΩ-cm, whereas the 14.2 nm thick RuO 2 (110) film is superconducting at T c = 0.92 ± 0.21 0.07 K with a residual resistivity ρ 0 < 32 µΩ-cm. Supplementary Fig. 3b,f show rocking curves for the films overlaid on rocking curves for the TiO 2 substrates they were synthesized on: in all cases the coherent components of the film peaks exhibit narrow full width at half maximum (FWHM) values that are limited by the underlying substrate FWHM, as expected for isostructural film growths. In our studies we found that the rocking curve shapes and widths of the TiO 2 substrates supplied by CrysTec, GmbH can vary significantly depending on how the in-plane momentum transfer q || is oriented relative to the crystal axes of a given wafer, which may be due to the Verneuil process used to synthesize the crystals; to give some idea of the magnitude of this asymmetric mosaic spread, we show scans with q || oriented along azimuths separated by 90 • for each sample.
In Supplementary Fig. 3c,d and Supplementary  Fig. 3g,h we show off-specular (q || , q ⊥ ) reciprocal space maps (RSMs) for both samples in regions surrounding HKL Bragg peaks that have q || purely aligned with the crystallographic directions indicated in the labels on the horizontal axes. For reference, the peak positions that would be expected for the lattice parameters of bulk RuO 2 and bulk TiO 2 at 295 K [13,14] are shown as red and white squares, respectively; the orange squares represent the central peak positions expected for commensurately strained RuO 2 thin films calculated using appropriately constrained density functional theory structural relaxations. To give a more quantitative sense of the logarithmic false color scale used here, the solid white lines overlaid on each plot represent the scattered intensity along the crystal truncation rods (CTRs)-i.e., the one-dimensional line cuts through the RSMs with q || equal to that of the substrate Bragg peaks. These results show that the 18.6 nm thick RuO 2 (101) film is coherently strained to the substrate along both in-plane directions, within the ≈ 0.1% resolution of the measurements. The variable widths of the CTRs versus q || in different RSMs are an artifact of instrumental resolution effects-namely, the "tall" incident beam profile convolved with the scattering geometries used to measure each RSM-which we do not attempt to correct for in this work. On the other hand, the 14 observed in RuO 2 /TiO 2 (110) samples, we measured RSMs around several Bragg peaks along the specular (q || = 0) CTR. Supplementary Fig. 4 summarizes the results of such measurements for the same 14.2 nm thick RuO 2 (110) sample for which off-specular RSMs are shown in Supplementary Fig. 3, which was also characterized by XRD and scanning transmission electron microscopy in Fig. 2 of the main text. By taking line cuts averaged over the dashed boxes-which span ranges of q ⊥ where the measured intensities are predominantly due to scattering from the film-we obtained the three rocking curves plotted in Supplementary Fig. 4d. Each rocking curve shows a sharp central peak that is resolutionlimited in width (or substrate-limited, cf. Supplementary  Fig. 3b,f), superimposed on a much broader, nearly Lorentzian (FWHM = 0.003 − 0.005Å −1 ), component of the scattering that is also centered at q || = 0. Furthermore, the integrated intensity of the former coherent component of the scattering decays relative to that of the diffuse component as the magnitude of |q| = q ⊥ increases in progressing from cut (a) to cut (c).
The non-vanishing intensity of the diffuse component in the film rocking curves, and the scaling behavior of how the total integrated intensity is distributed between the coherent and diffuse components as |q| is varied, are both completely consistent with published data for numerous epitaxial thin films grown on lattice-mismatched substrates where the films are thick enough to exhibit some form of strain relaxation [15][16][17][18][19][20]. In principle, by analyzing the diffuse scattering profiles around multi-ple Bragg peaks with q that project differently onto the Burgers vectors of the relevant misfit dislocations that relax the strain, one can obtain quantitative information on the types of dislocations that exist, the dislocation densities, etc. [21,22]. We leave a more systematic analysis of this type to future synchrotron XRD studies, where the measurement noise floor is significantly lower and the strongly q-dependent instrumental resolution effects observed here are mitigated by having a more point-like incident beam profile. We note, however, that the similar FWHM values of the diffuse scattering versus q || around the 110, 220, and 330 peaks imply that the structural defects responsible for this scattering are more translational in nature than rotational (which in typical mosaic crystals, produce rocking curves of constant angular widths) [15,16]. Whether the inverses of these FWHM values for the fitted Lorentzians can be directly interpreted as the Fourier transform of a real-space correlation length (200 − 300Å) depends on whether the film is in the limit of weak (structural) disorder, in the formalism of Refs. [16,21]. Supplementary Fig. 5b) confirms that an abrupt interface exists between TiO 2 and RuO 2 . In particular, a sharp transition from lower intensity peaks in the substrate (Ti: Z = 22) to higher intensity peaks in the film (Ru: Z = 44) occurs over a region thinner than 1 nm surrounding the black arrow at the substrate-film interface; this indicates that any Ti/Ru chemical interdiffusion is minimal and cannot be the cause of the enhanced su-perconductivity observed in RuO 2 (110). At the lattice scale, we find that different regions from the same film exhibit varying degrees of crystalline coherence under the epitaxial strain applied by the TiO 2 substrate. The lateral in-plane direction imaged in Supplementary Fig. 5 is the [110] axis of the RuO 2 film, subject to +2.3% tensile strain from the TiO 2 substrate. Some regions, such as the one shown in Supplementary Fig. 5b, exhibit ex-Supplementary Figure 7. HAADF-STEM structural characterization of a non-superconducting RuO2(101) sample. a, As in the superconducting RuO2(110) samples, continuous film growth is observed across the entire length of the STEM lamella. b, Epitaxial growth between the RuO2 film and TiO2 substrate is again confirmed. Here, however, the observed contrast is comparatively smooth across the film, without the clear signs of high strain observed in the RuO2(110) sample. c, Atomic-resolution STEM image demonstrating the high crystalline quality of the RuO2(101) sample. Inset shows the expected structure for this projection (not to scale).
ceptionally clean crystalline quality: all of the atomic columns of Ru stack uniformly in the projection of the STEM image to produce highly ordered atomic contrast. In other regions of the same film, strain gradients in the film distort the RuO 2 lattice such that the columns of Ru atoms are slightly misaligned to the electron beam projection. This local misalignment of the lattice causes the apparent blurring and more mottled contrast of the STEM image seen in Supplementary Fig. 5c.
In Supplementary Fig. 6, the same sample is studied with HAADF-STEM imaging in the orthogonal projection direction. This orientation allows us to assess the crystalline response of the RuO 2 film along the [001] in-plane direction, which is subject to a larger lattice mismatch with the TiO 2 substrate, −4.7% compressive strain. Again, Supplementary Fig. 6a confirms the continuous and epitaxial growth of the RuO 2 (110) thin film over the mesoscopic and macroscopic length scales relevant for interpreting the electrical transport data shown elsewhere in the manuscript. Effects of the large compressive strain along the in-plane direction of this projection are apparent in Supplementary Fig. 6a-b as characteristic V-shaped contrast in the RuO 2 film. Contributions from electron channeling in ADF-STEM imaging produces this bright/dark contrast in regions of local crystallographic strain; such contrast is a common signature of epitaxial lattice strain in many other oxide systems. Supplementary Fig. 6c shows the same structural response at atomic resolution, where-similar to Supplementary Fig. 5cthe apparent blurring of atomic columns arises from regions where the film lattice has been locally distorted.
Finally, for completeness we also performed HAADF-STEM measurements on the same non-superconducting 18.6 nm thick RuO 2 (101) film characterized in Fig. 1c and  Supplementary Fig. 7: the film is comparably continuous and epitaxial as the superconducting RuO 2 (110) films we have studied, without any signatures of extended defects or secondary phase inclusions that might otherwise alter its electrical properties. In good agreement with the XRD and electrical transport data shown in Supplementary Fig. 3, the RuO 2 (101) film exhibits more coherent crystalline order than the more drastically strained superconducting RuO 2 (110) films, even over relatively large fields of view as shown in Supplementary Fig. 7b. Supplementary Fig. 7c shows that the lattice remains largely defect-free down to the atomic scale.
The data presented in Supplementary Figs. 3-6 indicate that the crystal structures of superconducting RuO 2 (110) films are not commensurately strained to the TiO 2 substrates. To better visualize how this partial strain relaxation manifests in real space, we employed Fourier filtering of STEM images to find edge dislocations, following the techniques described in Ref. [23]. Specifically, in Supplementary Fig. 8a Supplementary Fig. 8a  (Supplementary Fig. 8c), and then taking the inverse FFT with the contributions of all spatial frequencies smoothly masked out except for those in a narrow region of q-space surrounding the noted ±q || . The horizontal dashed lines in Supplementary Fig. 8b,d represent the boundaries of the film along the out-of-plane direction. In this representation, edge dislocations appear as topological defects in the otherwise continuous vertical streaks appearing in Supplementary Fig. 8b,d. These vertical streaks are formally the lattice points of the film and substrate crystal structures, blurred into streaks along the out-of-plane direction because we discard any highspatial-frequency information about out-of-plane correlations of the electron density (i.e., q ⊥ ≈ 0) when computing the inverse FFTs. Hereafter we loosely refer to these streaks as atomic columns, since the contrast in these HAADF-STEM data arises predominantly from electron scattering by the atomic cores with larger Z (i.e., Ti and Ru). Dislocations indicated by green markers add one atomic column to the number of columns that exist in layers beneath it (thus relaxing tensile strain in the lateral direction), whereas dislocations indicated by orange markers remove one atomic column to the number of columns that exist in layers beneath it (thus relaxing compressive strain in the lateral direction). Therefore, a fully strain-relaxed film of RuO 2 /TiO 2 (110) would show a collection of only green (only orange) dislocations accumulated at the substrate-film interface in Supplementary  Fig. 8b (Supplementary Fig. 8d, respectively). The dislocation densities expected in this fully strain-relaxed scenario would be 1 per every 1/0.023 ≈ 43 vertical streaks for green dislocations in Supplementary Fig. 8b, and 1 per every 1/0.047 ≈ 21 vertical streaks for orange dislocations in Supplementary Fig. 8d.
In marked contrast to this behavior, zero dislocations are observed across a 226-unit-cell-wide field of view in Supplementary Fig. 8b. Furthermore, although a significantly higher density of dislocations is present across the 204-unit-cell-wide field of view in Supplementary Fig. 8d, there are nearly equal numbers of edge dislocations having Burgers vectors of −c (orange) and +c (green), respectively, and the dislocations are rather uniformly distributed throughout the entire thickness of the film. These observations imply that throughout a sizable volume fraction of the superconducting film, the crystal structure is, on average, much closer to the commensurately-strained limit than to the fully-relaxed limit. We note that this agrees well with the distribution of x-ray scattering intensities plotted in the RSMs for this sample and others in Supplementary Figs. 9-10. Based on these data, we suggest that it is appropriate to consider the local strain gradients that inevitably accompany the nucleation of dislocations in RuO 2 (110) to be sampledependent perturbations to significantly larger average components of the substrate-imposed strain fields that are present throughout all films shown in the manuscript. Because superconductivity is an essentially mean-field phenomenon, we believe that the latter average components of the strain fields in RuO 2 (110) are the key ingredients for stabilizing superconductivity with transition temperatures at least an order of magnitude larger than in bulk RuO 2 ; finer details of the local strain gradients probably determine finer details of the superconductivity, such as the exact sample-dependent T c s mea-sured by non-bulk-sensitive probes of superconductivity, such as resistivity. Since our platform for applying strain enables scanning-probe measurements of the superconducting condensate, future experiments may be able to provide direct experimental evidence to support these general expectations of mesoscale or nanoscale strain inhomogeneity resulting in spatially inhomogeneous superconductivity [24]. In Supplementary Fig. 9a-b, we plot x-ray reflectivity (XRR) data taken at low incident angles, and XRD data taken near the 110 Bragg peaks of the film and substrate, both acquired along the specular CTRs using Cu-Kα radiation. Finite-thickness fringes are present over a wide range of angles in both data sets, evidencing (from reflectivity) atomically abrupt interfaces of the films with the substrates and with vacuum, and (from diffraction) comparable levels of crystallinity along the out-of-plane direction across samples. Furthermore, the spacings between secondary maxima on either side of the primary film Bragg peaks in XRD match the spacings between the low-angle XRR fringes, suggesting that the crystal structures of all films are essentially homogeneous along the out-of-plane direction. The film thicknesses t listed in Fig. 2b-c of the main text and in Supplementary Figs. 9-10 are obtained by directly fitting the XRR data in Supplementary Fig. 9a using a genetic algorithm, which yields sub-nanometer roughnesses in all cases in the refined models.
Given that there are no obvious differences in film morphology or out-of-plane crystallinity between RuO 2 (110) samples with different t, an alternative explanation that may account for the thickness-dependent superconducting T c s is the proliferation of misfit dislocations in thicker films that progressively relax the epitaxial-i.e., in-plane-strains; in this scenario, it may be that partially strain-relaxed RuO 2 (110) films have higher (average) superconducting T c s compared with fully commensurately strained RuO 2 (110) films. To investigate this possibility, in Supplementary Fig. 9c- Figure 9. Evolution of crystal structure and electrical transport behavior as a function of film thickness for RuO2/TiO2(110). a, X-ray reflectivity and b, x-ray diffraction data along the specular CTR show comparable levels of flatness and crystalline order along the out-of-plane direction for all samples. c -e, Average line cuts versus q || through the 220, 310, and 332 RSMs (fully q-resolved data are shown in Supplementary Fig. 10) indicate that all samples with t > 5.8 nm exhibit partial strain relaxation. f -g, Zero-field ρ(T ) data show that thinner films generally have higher residual resistivities ρ0 and lower superconducting Tcs. Tcs in g are taken as the temperatures at which ρ(T ) crosses 50% of the normal-state ρ0, and error bars indicate the temperatures at which ρ(T ) crosses 90% and 10% of ρ0, respectively. The horizontal dashed line in g represents the base temperature attainable in our cryostat (0.4 K), and the gray-shaded region indicates the range of superconducting coherence lengths (ξ = 12 − 22 nm) extracted from magnetoresistance measurements of the upper critical fields for ten different RuO2(110) thin films. Comparisons of these ξ with the mean free paths (top axis of g) that correspond to the measured ρ0 (bottom axis of g) indicates that superconductivity persists in the dirty limit < ξ.
RSMs are plotted in Supplementary Fig. 10 using logarithmic false color scales; the line cuts in Supplementary  Fig. 9c-e are averaged over the ranges of q ⊥ between the dashed white lines in Supplementary Fig. 10. All of the samples except the thinnest film exhibit diffuse scattering surrounding the CTRs, indicating that partial strain relaxation onsets between film thicknesses of 5.8 nm and 11.5 nm for the growth conditions used in this work to synthesize RuO 2 /TiO 2 (110) samples. Since the in-plane lattice mismatches between RuO 2 and TiO 2 are highly anisotropic for the (110) orientation, it might also be expected that the substrate-imposed compressive strain along [001] (−4.7%) starts to relax at smaller film thicknesses than the tensile strain along [110] (+2.3%) [17,19]. The off-specular RSMs in Supplementary Fig. 10b-c Figure 10. RSMs for RuO2/TiO2(110) samples with different film thicknesses, t. a, RSMs along the specular CTR near the 220 Bragg reflections for films with increasing t, moving from left to right. q || is aligned with [001] in all panels, although the phenomenology is similar with q || along [110], cf. Supplementary Fig. 4. b, Thickness-dependent RSMs near the off-specular 310 Bragg reflections where q || is purely along [110]. c, Same as b, but near the 332 Bragg reflections where q || is purely along [001]. White, red, and orange squares represent the central peak positions expected for bulk TiO2, bulk RuO2, and commensurately strained RuO2(110), as in Supplementary Fig. 3. The line cuts plotted in Supplementary  Fig. 9c-e are averaged over the ranges of q ⊥ of the RSMs between the horizontal white dashed lines. thickness fringes can still be observed along the CTRs in the RSMs near 310 for films up to at least t = 17.2 nm, whereas only the t = 5.8 nm film shows a contribution to the coherent CTR scattering in the RSMs near 332 that clearly rises above the contributions of the substrate.
Although signatures of scattering from partially strainrelaxed RuO 2 (110) are manifestly present in the data for all of the superconducting samples in Supplementary  Fig. 9-namely, broader distributions of intensity versus q || that asymmetrically gain weight towards the positions expected for bulk RuO 2 as the film thickness increasesit remains somewhat ambiguous whether this data can be interpreted in a straightforward manner to gain insight into what levels of strain optimize the superconducting T c s in RuO 2 . Strain relaxation in oxide thin films often occurs inhomogeneously, with a mixture of commensurately strained and partially relaxed material [25]. Indeed, examining the transport data for these same samples in Fig. 2b of the main text and in Supplementary  Fig. 9f, it is tempting to ascribe the multi-stage behavior of the superconducting transitions to temperaturedependent Josephson coupling of regions of the films under different amounts of strain with correspondingly different "local" T c s; similar behavior has been described theoretically [26] and observed experimentally in patterned niobium islands on gold substrates [27]. However, because of the close proximity of the substrate Bragg peaks along q ⊥ (d 110 = 3.248Å) with the positions expected for commensurately strained RuO 2 (110) (d 110 = 3.241Å), it is difficult to disentangle their respective contributions to the total scattering observed in XRD.
Despite these complications in quantitatively analyzing the XRD results, we can use the values of t obtained from XRR to plot the normalized resistance versus temperature curves from Fig. 2b of the main text in terms of absolute resistivities, as shown in Supplementary Fig. 9f. From these data, a robust correlation between the superconducting T c s and the residual resistivities ρ 0 immediately becomes apparent, as displayed in Supplementary Fig. 9g. As noted in the main text, this correlation may suggest that the primary effect of reducing t is to enhance the relative importance of elastic scattering off disorder near the substrate-film interfaces, which is known to decrease T c in numerous families of thin-film superconductors, both conventional [28] and unconventional [29]. It is largely outside the scope of this paper to contribute meaningful data to ongoing research efforts investigating the mechanism underlying thickness-induced suppressions of T c that are ubiquitously observed for superconducting films in the two-dimensional limit [30]; however, in passing we note that the residual resistivity (ρ 0 = 60 µΩ-cm) of the thinnest (t = 5.8 nm) nonsuperconducting (T c < 0.4 K) RuO 2 (110) film shown in Supplementary Fig. 9 corresponds to a sheet resistance of R s = ρ 0 /t = 0.10 kΩ/ . This value of R s is about 60 times less than the Cooper pair quantum of resistance h/(2e) 2 = 6.45 kΩ/ that was empirically noted to separate insulating from superconducting ground states in ultrathin films of numerous elemental metals [31,32]; this indicates that quantitatively different physics is likely operative here in suppressing T c , which may place ultrathin films of RuO 2 (110) in closer proximity to the anomalous metal regime that was shown to occur at weaker levels of disorder in Ref. [33].
We believe that identifying the exact mechanism underlying the strain-stabilized superconductivity in RuO 2 (110) is also well beyond the scope of the current paper. Phase-sensitive measurements of the superconducting order parameter-and/or momentumresolved measurements of the superconducting gap magnitude-are notoriously challenging in multi-band materials with small (sub-meV) gaps, and a definitive answer to whether the pairing is (un)conventional will have to wait until such data become available. With that qualification in mind, it is natural to consider whether the T c versus ρ 0 behavior displayed in Supplementary Fig. 9g can shed any light on the answer to this question. To address this possibility, we need to convert measured properties of the normal-metal and superconducting states into comparable characteristic length scales. In unconventional low-temperature superconductors with sign-changing order parameters, such as Sr 2 RuO 4 [34], superconductivity is completely suppressed by non-magnetic impurity scattering whenever the normal-state mean free path is comparable with the clean-limit superconducting coherence length ξ 0 -i.e., T c → 0 K if ≈ ξ 0 . In conventional superconductors, by contrast, superconductivity persists even in the dirty limit << ξ 0 .
On the top horizontal axis of Supplementary Fig. 9g, we indicate approximate values of corresponding to the measured values of ρ 0 on the bottom horizontal axis; these numbers are computed following the analysis of Glassford et al. [3], who used the DFT-computed plasma frequencies and Fermi velocities for bulk RuO 2 to obtain [nm] = 3.6·35/ρ [µΩ-cm]. Because of various uncertainties implicit in these estimations, we neglect any straindependent changes in Fermiology that will, of course, quantitatively renormalize the precise relationship between and ρ for RuO 2 (110). To compare with , we also include a gray-shaded region on Supplementary Fig. 9g corresponding to the range of average in-plane superconducting coherence lengths ξ we extracted experimentally from magnetoresistance measurements of the perpendicular upper critical magnetic fields H c⊥ for ten different superconducting RuO 2 (110) samples, following the procedures detailed in Supplementary Note 2. As noted previously, these values of ξ likely represent a lower bound for what the clean-limit ξ 0 would be in the absence of extrinsic defects in the films that impede vortex motion. In any case, we find that superconductivity robustly persists in RuO 2 (110) even when < ξ < ξ 0 ; for example, the samples shown here with T c = 0.9 − 1.8 K have measured residual resistivities corresponding to mean free paths = 4.0 − 9.6 nm, which are all less than the range of measured superconducting coherence lengths, ξ = 12 − 22 nm. Therefore, whatever the superconducting pairing mechanism is in RuO 2 (110), these empirical considerations demonstrate that it is rather insensitive to defect scattering.
Conceptually, perhaps the most straightforward test of our proposal that substrate-imposed strains are an essential ingredient in stabilizing superconductivity in RuO 2 (110) would be to perform electrical transport measurements for a thick RuO 2 (110) film where the epitaxial strains are almost completely relaxed. In reality, however, such efforts are complicated by the observation that the +2.3% tensile strain along [110] is not released in sufficiently thick films of RuO 2 (110) by the nucleation of misfit dislocations; instead, cracks form in such samples (in our studies, for t 30 nm) that propagate through the entire thickness of the film and even tens of nanometers into the substrate.
In Supplementary Fig. 11a-b, we show HAADF-STEM images of such oriented micro-cracks for the same 48 nm thick RuO 2 (110) film characterized by ARPES and LEED in Fig. 4 of the main text and in Supplementary Figs. 14-15. Although these images evidence strong interfacial bonding between film and substrate-which is certainly crucial to maintain high levels of strain throughout the thinner RuO 2 (110) films we characterize elsewhere in the manuscript-the strongly anisotropic nature of the cracks makes the distribution of current flow through such samples extremely non-uniform. Accordingly, putative measurements of electrical "resistance" R(T, H) displayed in Supplementary Fig. 11c-e should be more pedantically interpreted as the voltage difference measured between two voltage contacts, divided by the total current sourced through two other contacts placed elsewhere on the film. Unsurprisingly, the results thus obtained for R depend in an essential way on the orientation of the voltage contacts relative to the cracks in the film: in Supplementary Fig. 11c-d we observe either a downturn (green trace) or upturn (purple trace) in the apparent R as the temperature is decreased below T c = 1.3 − 1.5 K, followed by plateaus at lower temperatures. In Supplementary Fig. 11e, we show that these temperature-induced anomalies in R can be suppressed in both cases by applying small magnetic fields H c⊥ < 3 kOe at fixed T = 0.45 K, confirming that they result from (inhomogeneous) patches of superconductivity. Irrespective of whether the measured R decreases or increases at (T c , H c ), we emphasize that the fractional changes in resistance induced by superconductivity in this 48 nm thick RuO 2 (110) film are less than 1% of the residual normal-state resistances R 4 K , in marked contrast to the full 100% drops to zero resistance exhibited by all thinner, more highly strained, uncracked RuO 2 (110) films shown in other figures.

Structural relaxations
One of the central themes of this work is the exploration of strain-induced changes to the electronic structure in epitaxial thin films of RuO 2 subject to biaxial epitaxial strains imposed by differently oriented rutile TiO 2 substrates. To model this situation computationally within the framework of density functional theory (DFT), we started by using the Vienna Ab Initio Software Package [35,36] to perform full structural relaxations (of lattice parameters and internal coordinates) to minimize the DFT + U -computed total energy of RuO 2 in the ideal tetragonal rutile crystal structure (space group #136, P 4 2 /mnm). Structural relaxations employed the same exchange-correlation functional and calculational parameters as for the DFT + U (U = 2 eV) calculations described in the Methods section of the main text, and forces were converged to < 1 meV/Å. Throughout the main text and supplementary information, we refer to DFT results for this minimum energy structure as "bulk RuO 2 ". The actual lattice parameters for this structure, (a bulk = 4.517Å, c bulk = 3.130Å), overestimate the experimentally measured lattice parameters at 295 K for RuO 2 single crystals of (a = 4.492Å, c = 3.106Å) by < 1%, due to well-established deficiencies of the generalized gradient approximation. With the former as the bulk reference structure, we then simulated biaxial epitaxial strains to (110)-oriented TiO 2 substrates by performing constrained structural relaxations for RuO 2 in which the in-plane lattice parameters c = (1 − 0.047) × c bulk and d 110 = (1 + 0.023) × d 110, bulk were held fixed, while the out-of-plane lattice constant d 110 and all other internal coordinates of the structure were allowed to relax so as to minimize the total energy. The fixed compression and expansion of c and d 110 , respectively, correspond to the experimentally measured lattice mismatches between TiO 2 and RuO 2 single crystals at 295 K [13,14].
Within this scheme, DFT + U predicts that commensurately strained RuO 2 (110) thin films will have an outof-plane lattice constant d 110 = (1 + 0.017) × d 110, bulk , which compares reasonably well with the 2.0% expansion of d 110 measured experimentally on a 5.8 nm thick RuO 2 (110) film. Because the splitting of d 110 and d 110 in strained RuO 2 (110) breaks the non-symmorphic glide plane symmetry of the parent rutile structure, we used a base-centered orthorhombic structure (space group #65, Cmmm) with lattice constants of c × 2d 110 × 2d 110 for DFT simulations of RuO 2 (110). The primitive unit cell of this Cmmm structure contains the same number of atoms as the parent rutile unit cell, so there is no apparent doubling and/or folding of the bands in spaghetti plots that compare the bandstructures of RuO 2 (110) and bulk RuO 2 , such as in Supplementary Fig. 12 or Fig. 4a of the main text.
To simulate the electronic structure of commensurately strained (101)-oriented RuO 2 thin films, we adopted a slightly different approach, since it is not straightforward to perform constrained structural relaxations with DFT in such a low-symmetry situation. Specifically, we took the rutile b axis to be under +2.3% tension, i.e. b = (1 + 0.023) × b bulk , as dictated by the lattice mismatch of RuO 2 with TiO 2 along this direction (cf. Supplementary Fig. 3d). On the other hand, the lengths of the rutile a and c axes are free to adjust their lengths, but are subject to the simultaneous constraints: Supplementary Eq. (3) ensures that the film is lattice matched to the TiO 2 substrate along the [101] in-plane direction (cf. Supplementary Fig. 3c), and Supplementary Eqs. (4)-(6) ensure that the d-spacings for the HKL = 202, 103, and 402 Bragg reflections reproduce the values we measured experimentally for a commensurately strained RuO 2 (101) film in Fig. 2a of the main text, in Supplementary Fig. 3c, and in analogous RSM data taken around 402 (a proper, i.e. highly overconstrained, lattice constant refinement would of course include data for many more reflections). Note that in deriving these equations, we assumed for simplicity that the angle between the rutile a and c axes remains 90 • in epitaxially strained films; small deviations away from this limit should be expected in reality, since this orthogonality is not guaranteed by any symmetry or constraint of the system. Nonetheless, finding the best-fit solution to Supplementary Eqs. (3)-(6) gives lattice constants of (a = 4.501Å, c = 3.077Å) in absolute units; dividing through by the experimentally measured lattice constants of bulk RuO 2 yields a = (1 + 0.002) × a bulk and c = (1 − 0.009) × c bulk as appropriately scaled inputs for DFT simulations. With a = b = c and all angles between the primitive unit cell translations equal to 90 • , the crystal structure for RuO 2 (101) DFT simulations was taken as the primitive orthorhombic space group #58, P nnm. Supplementary Table 1 summarizes all parameters of the crystal structures used in DFT simulations for bulk RuO 2 , RuO 2 (110), and RuO 2 (101).

Effects of adding +U
In Supplementary Fig. 12, we show the effects of including an ad hoc static mean-field +U term on the Ru sites in DFT calculations. Adding such a phenomenological term to the Kohn-Sham Hamiltonian shifts the bands relative to each other (up/down in energy) so as to force the orbital occupancies towards integer fillings, rather than also shrinking the bandwidths of the quasiparticle excitations, as would occur in a more realistic  Figure 12. Strain dependence of the electronic structure of RuO2, according to DFT (+ U ).
theory that includes dynamical electron-electron interactions. The blue and red traces are reproduced from the non-magnetic DFT + U (U = 2 eV) results presented in Fig. 4a of the main text; the purple and orange traces are the results of repeating GGA + SOC calculations for the same RuO 2 (110) and bulk RuO 2 crystal structures, respectively, but now setting U = 0. Irrespective of U , both sets of calculations show a shift of the d || -derived flat bands towards E F and concomitant enhancement of the density of states (DOS) near E F when the amount of caxis compression is increased upon going from bulk RuO 2 to RuO 2 (110), as indicated by the gray arrows. While these strain-dependent trends in the electronic structure are robust against fine-tuning of parameters employed in the calculations, Supplementary Fig. 12 also suggests that the calculated positions of the peaks in the DOS and the exact values of the DOS near E F should not be taken too seriously, as there are considerable theoretical uncertainties in these quantities, depending on the choice of U . We leave a complete treatment of the effects on the electronic structure of commensurate Q AFM = (1 0 0) antiferromagnetic spin-density wave order [13,37] to future studies [38], because it is not possible in standard DFTbased approaches to stabilize self-consistent solutions for the spin densities that have small values of the ordered magnetic moment comparable to those measured experimentally (≈ 0.05 µ B /Ru) [13].

Supplementary Note 6: DETERMINATION OF OUT-OF-PLANE MOMENTA PROBED BY ANGLE RESOLVED PHOTOEMISSION SPECTROSCOPY
Figure 3d-f of the main text compares the electronic structure of a 7 nm thick RuO 2 /TiO 2 (110) sample experimentally measured by angle-resolved photoemission spectroscopy (ARPES) with the results of DFT + U simulations. To make this comparison, it is necessary to determine what range of out-of-plane momenta k z = k 110 in the initial state are probed by ARPES at a given finalstate kinetic energy and momentum. This was established by plotting the DFT-computed E(k) dispersions on top of the experimentally measured spectra along several one-dimensional cuts through momentum space measured over a small range of kinetic energies corresponding to near-E F states at the given photon energy (21.2 eV), and allowing k z to vary in the calculations so as to best match the experimental data. Supplementary Fig. 13 shows representative examples of this procedure for experimental spectra taken along  Fig. 13a, the k F s and electron-like character of the band crossing E F are best fitted by calculations with k z in the range −0.2 → 0.0 π/d 110 . Likewise, for the panels in Supplementary Fig. 13b, k z values in the range −0.6 → −0.3 π/d 110 best reproduce the measured spectra, although the results here are more ambiguous because of the insensitivity of the flat-band energies to the precise value of k 110 . Therefore, we took the range of reduced initial-state out-of-plane momenta probed at normal emission (k x = k y = 0) to be k z,i = −0.1 ± 0.1 π/d 110 . Assuming a free-electron-like model of final states, the final-state k z,f is given by the expression where m e is the free-electron mass, E k is the kinetic energy of the photoelectrons, θ is the emission angle relative to the surface normal, V 0 is the inner potential, and 2d 110 is the spacing between equivalent lattice points along the out-of-plane direction (N can adopt any integer value). Substituting E k = 16.6 ± 0.3 eV, θ = 0 • , k z,i = −0.1 ± 0.1 π/d 110 , and d 110 = 3.23Å into Supplementary Eq. (7), we find that an inner potential of 13.7 ± 2.3 eV is compatible with our determination of k z,i . Taking this same value of V 0 and setting θ = 30−35 • in Supplementary Eq. (7)-as is appropriate for the experimental data in the panels displayed in Supplementary  Fig. 13b-yields k z,i = −0.35 ± 0.17 π/d 110 ; visual inspection of the DFT bands for this range of k z,i show that the calculations also reproduce the experimental spectrum reasonably well in this region of the Brillouin zone. The curved green planes drawn in the Brillouin zone schematic in Fig. 3c of the main text are constructed by evaluating Supplementary Eq. (7) with V 0 = 13.7 eV and N = 3 for all (k x , k y ), and accounting for an intrinsic uncertainty of ≈ 0.2 π/d 110 in k z owing to the finite elastic escape depth of photoelectrons, which we take to be ≈ 5Å.

Supplementary Note 7: SURFACE LATTICE CONSTANT REFINEMENT
In Supplementary Fig. 14, we present the results of a surface lattice constant refinement for two different RuO 2 /TiO 2 (110) films of different thicknesses, 7 nm and 48 nm. For both samples we acquired many low-energy electron diffraction (LEED) images at normal incidence using incident energies ranging from E = 100 − 300 eV in 2 eV steps; for examples of the raw data, the insets in Supplementary Fig. 14b contain representative images taken at 200 eV. For each image, we located the positions of all visible spots and indexed the spots according to their in-plane momentum transfer values q || = 2π(H/c, K/2d 110 ), where H and K are integers and, by our convention, H defines the magnitude of q || along [001] (nearly horizontal in the images in Supplementary Fig. 14), and K defines the magnitude of q || along [110] (nearly vertical). We then calculated the distance of all spots from the specular q || = (0, 0) reflection and converted these image distances D (in pixel space) to scattering angles sin(θ) (where θ is the angle of each diffracted electron beam relative to the surface normal) based on D → sin(θ) calibrations that were independently determined from reference measurements on SrTiO 3 (001) surfaces having a known lattice constant. Note that these calibrations absorb the overall scaling factor that depends on the working distance between the LEED screen and the sample (and the camera image magnification factor), as well as some higher-order distortions of the spot patterns that result from the sample not being positioned precisely at the center of curvature of the LEED screen (and the screen itself being slightly aspherical).
From these values of sin(θ), the electron energies E at which each LEED pattern was recorded, and the (H, K) indices, we compiled lists of lattice constants corresponding to each fitted spot position. For simplicity in analysis, we restricted our attention to spots having q || purely aligned with [001] or [110]. Elastic scattering and conservation of momentum modulo translations of the surface reciprocal lattice together require that: which for Bragg spots of the type q || = 2π(H/c, 0) and 2π(0, K/2d 110 ), reduces to:  (110) sample in red shows broader LEED spots, indicating lower surface crystallinity than the 7 nm thick sample, which results in wider distributions of the extracted lattice constants. Furthermore, the centers of mass of the red distributions, (c = 3.07 ± 0.06Å, 2d 110 = 6.39 ± 0.11Å), are displaced away from the blue distributions towards the values expected for bulk RuO 2 ; this is why in Fig. 4b-c of the main text we suggest that the surface electronic structure of this 48 nm thick sample measured by ARPESwhich probes the film over a comparable depth to LEED, within < 1 nm from the top film surface-should be more representative of bulk RuO 2 .

Supplementary Note 8: EXTRACTING THE NEAR-EF DENSITY OF STATES FROM ARPES MEASUREMENTS
In Supplementary Fig. 15 we present LEED and ARPES data taken on three different RuO 2 thin-film samples: 19 nm thick RuO 2 (101), 7 nm thick RuO 2 (110), and 48 nm thick RuO 2 (110). In Supplementary Fig. 15a, all of the samples show LEED spot patterns with the periodicities expected for unreconstructed (101)-and (110)oriented rutile surfaces, respectively; furthermore, the sharpness of the patterns suggest high degrees of surface crystallinity, such that in-plane momentum should be a nearly conserved parameter in photoemission. Given that strained RuO 2 (110) samples superconduct at measurable T c s, while RuO 2 (101) samples and bulk RuO 2 do not, the question we wanted to address using ARPES was: how does the density of states near the Fermi level, N (E F ), evolve between these samples? Recall that based on the LEED lattice constant analysis described in Supplementary Fig. 14, most of the substrate-imposed epitaxial strains are relaxed at the top surface of the 48 nm thick RuO 2 (110) sample, such that its electronic structure probed by ARPES is a reasonable proxy for that of bulk RuO 2 . Specifically, the quantity of interest as it relates to the low-energy physics is: where A(k, ω) is the single-particle spectral function, integrated over all momenta k in the Brillouin zone (BZ) and over some limited range of energies ω near E F (δ is some small parameter). Two separate factors make it extremely challenging to quantitatively extract the total N (E F ) directly from data taken with our lab-based ARPES system. First, our inability to continuously vary the photon energy-or equivalently, the kinetic energy of the photoelectrons at E F -implies that only regions of the Brillouin zone with specific k z can be probed, cf. Supplementary Eq. (7). Therefore the full integration over k in Supplementary Eq. (10) cannot be performed using a lab-based ARPES setup, which is especially problematic in a material such as RuO 2 that has a highly three-dimensional electronic structure depending strongly on k z (cf. Supplementary  Fig. 13). Second, even if the entire Brillouin zone could be mapped exhaustively, the intensity measured in ARPES is not the initial-state spectral function A(k, ω), but rather this quantity multiplied by probabilities (i.e., matrix elements) for photoemission, which are difficult to account for theoretically.
With these qualifications in mind, there is a route to answering the simpler question of whether N (E F ) increases in strained RuO 2 (110) compared with strained RuO 2 (101) or bulk RuO 2 : we simply need to determine where the flat bands with d || orbital character are located in energy relative to E F . DFT calculations suggest that if these bands move closer to (further away from) E F , the total N (E F ) will increase (decrease), respectively. To approximately determine the positions of these bands experimentally, we integrated the photoemission intensity over the color-coded slabs in the Brillouin zone schematic in Supplementary Fig. 15a, plotted the resulting energy distribution curves (EDCs) in Supplementary Fig. 15b, and found the maxima in the EDCs as indicated by the dashed lines. The regions colored yellow in the Brillouin zone denote where the near-E F wavefunctions have greater than 90% d || orbital character, according to our DFT + Wannier90 calculations; since all slabs lie in this region, we expect that the dominant contributions to the measured EDCs are from d || initial states. Note that the region of k z = k 110 probed by ARPES with He-Iα (21.2 eV) photons on the (110)-oriented samples is well-constrained by analysis of the E(k) dispersions as outlined in Supplementary  Fig. 13; however, for the (101)-oriented sample the region of k z = k 101 probed by ARPES with He-IIα (40.8 eV) photons is merely calculated from the free-electron final state model in Supplementary Eq. (7), using the same inner potential as for RuO 2 (110), and thus is subject to greater experimental uncertainties. Nonetheless, the results of this analysis qualitatively agree with the straindependent trends anticipated by DFT ( Supplementary  Fig. 15c): in highly strained RuO 2 (110) films (blue), the flat bands move closer to E F compared with either more strain-relaxed RuO 2 (110) films (red) or commensurately strained RuO 2 (101) films (purple). This modification of the effective d orbital degeneracies boosts N (E F ), which-as proposed in the main text-likely contributes to the enhanced superconducting T c s observed in highly strained RuO 2 (110) samples. As described in the Methods section of the main text, we performed first-principles DFT-based electronphonon coupling calculations of the isotropic Eliashberg spectral function α 2 F (ω) and total electron-phonon coupling constant λ el-ph (integrated over all phonon modes and wavevectors) for bulk RuO 2 and commensurately strained RuO 2 (110). From these quantities, we estimated the superconducting transition temperatures using the semi-empirical McMillan-Allen-Dynes formula: McMillan obtained a formula resembling Supplementary Eq. (11) by numerically solving the equations of finite-temperature Migdal-Eliashberg theory using the experimentally measured spectral function of niobium [40]; Allen and Dynes improved the agreement of McMillan's formula with experimentally measured α 2 F (ω) and T c s for a variety of conventional superconductors by introducing an appropriately weighted average over α 2 F (ω) in the prefactor of the exponential, rather than using the Debye temperature [41]. In essence, Supplementary Eq. (11) can be considered an extension of Eq. (1) from the main text that identifies phonons as the bosonic modes that mediate Cooper pairing (i.e., λ = λ el-ph ), and which remains valid even in the limit of stronger couplings (i.e., λ > 1) by virtue of Migdal's theorem for the electron-phonon interaction.
For bulk RuO 2 (strained RuO 2 (110), respectively), we obtained λ el-ph = 0.685 and ω log = 34.1 meV (λ el-ph = 1.97 and ω log = 7.59 meV). The large strain-induced enhancement in λ el-ph and shift of ω log to lower frequencies is caused by substantial phonon softening that occurs under c-axis compression in the rutile structure. In fact, we found that some of the calculated zone-boundary phonon frequencies even become imaginary under strain in RuO 2 (110), possibly indicating an incipient structural instability. A more detailed account of this phenomenon will be described in a future publication; for the purposes of this work, we omitted such phonon modes in subsequent electronphonon coupling calculations, and neglected any errors this may cause in λ el-ph and ω log .
To convert the calculated values of λ el-ph and ω log to T c s via Supplementary Eq. (11) requires knowledge of the appropriate value(s) of the screened Coulomb interaction µ * between quasiparticles. Typically µ * is chosen in an ad hoc fashion to match the experimentally measured T c of a given material. Because bulk RuO 2 is not known to be superconducting at experimentally accessible temperatures, we cannot employ such a prescription here; nevertheless, we can use the experimentally measured least upper bound on T c for bulk RuO 2 (T c < 0.3 K [39]) to place a lower bound on µ * (µ * > 0.30), as illustrated by the dashed lines in Supplementary Fig. 16. For this range of µ * , inserting the values of λ el-ph and ω log calculated for RuO 2 (110) into Supplementary Eq. (11) predicts T c < 7 K, which agrees reasonably well with the experimentally measured values. Because several uncontrolled approximations enter into these estimates of T c , we consider this level of agreement as suggestive, although not conclusive, evidence for a phonon-mediated mechanism of superconductivity; in any case, it is clear that reducing the axial ratio c/a in appropriately strained variants of RuO 2 robustly boosts λ el-ph , in good agreement with expectations based on steric trends for other rutile compounds [42,43]. Any effects of strain on µ * are ignored for the purposes of this work.