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[Jacob P. Ruf](https://orcid.org/0000-0002-4426-2884), [Hilary M. L. Noad](https://orcid.org/0000-0002-3380-668X), [Romain Grasset](https://orcid.org/0000-0003-4769-8947), [Ludi Miao](https://orcid.org/0000-0002-4083-8735), Elina Zhakina, [Philippa H. McGuinness](https://orcid.org/0000-0001-5938-6452), Hari P. Nair, Nathaniel J. Schreiber, [Naoki Kikugawa](https://orcid.org/0000-0003-3975-4478), [Dmitry Sokolov](https://orcid.org/0000-0003-3282-7144), Marcin Konczykowski, [Darrell G. Schlom](https://orcid.org/0000-0003-2493-6113), Kyle M. Shen, [Andrew P. Mackenzie](https://orcid.org/0000-0001-8000-4949)

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[Controllable suppression of the unconventional superconductivity in bulk and thin-film <math>  <mrow>    <msub>      <mi>Sr</mi>      <mn>2</mn>    </msub>    <msub>      <mi>RuO</mi>      <mn>4</mn>    </msub>  </mrow></math> via high-energy electron irradiation](https://mdr.nims.go.jp/datasets/74db4125-8f3b-4708-8db9-a58bb76390a3)

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Controllable suppression of the unconventional superconductivity in bulk and thin-film ${\rm Sr}_{2}{\rm RuO}_{4}$ via high-energy electron irradiationPHYSICAL REVIEW RESEARCH 6, 033178 (2024)Editors’ SuggestionControllable suppression of the unconventional superconductivity in bulk and thin-film Sr2RuO4via high-energy electron irradiationJacob P. Ruf ,1,2 Hilary M. L. Noad ,1,* Romain Grasset ,3 Ludi Miao ,2,† Elina Zhakina,1 Philippa H. McGuinness ,1,‡Hari P. Nair,4 Nathaniel J. Schreiber,4 Naoki Kikugawa ,5 Dmitry Sokolov ,1 Marcin Konczykowski,3Darrell G. Schlom ,4,6,7 Kyle M. Shen,2,6,§ and Andrew P. Mackenzie 1,8,‖1Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany2Department of Physics, Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA3Laboratoire des Solides Irradiés, CEA/DRF/IRAMIS, École Polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau, France4Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, USA5National Institute for Materials Science, Tsukuba 305-0003, Japan6Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA7Leibniz-Institut für Kristallzüchtung, 12489 Berlin, Germany8SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16 9SS, United Kingdom(Received 1 March 2024; accepted 11 July 2024; published 16 August 2024)In bulk Sr2RuO4, the strong sensitivity of the superconducting transition temperature Tc to nonmagneticimpurities provides robust evidence for a superconducting order parameter that changes sign around the Fermisurface. In superconducting epitaxial thin-film Sr2RuO4, the relationship between Tc and the residual resistivityρ0, which in bulk samples is taken to be a proxy for the low-temperature elastic scattering rate, is far lessclear. Using high-energy electron irradiation to controllably introduce point disorder into bulk single-crystal andthin-film Sr2RuO4, we show that Tc is suppressed in both systems at nearly identical rates. This suggests thatpart of ρ0 in films comes from defects that do not contribute to superconducting pairbreaking and establishes aquantitative link between the superconductivity of bulk and thin-film samples.DOI: 10.1103/PhysRevResearch.6.033178I. INTRODUCTIONConclusive identifications of the order parameter and of theinteractions that induce superconductivity in Sr2RuO4 haveremained elusive despite receiving intense research interestfor more than 25 years [1,2]. Among the first evidence forthe unconventional nature of superconductivity in Sr2RuO4was the observation that the transition temperature Tc couldbe completely suppressed with a minute concentration ofnonmagnetic impurities [3], indicating that the phase of thesuperconducting order parameter changes sign in momentumspace and, with sufficient scattering, averages to zero [1]. Sub-sequent studies confirmed this extreme sensitivity to disorder,*Contact author: hilary.noad@cpfs.mpg.de†Currently at Department of Physics, New Mexico State University,Las Cruces, New Mexico 88003, USA.‡Currently at Institut für QuantenMaterialien und Technolo-gien, Karlsruher Institut für Technologie, 76344 Eggenstein-Leopoldshafen, Germany.§Contact author: kmshen@cornell.edu‖Contact author: andy.mackenzie@cpfs.mpg.dePublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI. Openaccess publication funded by Max Planck Society.either by introducing variable defect densities via intentionalchemical substitutions during growth [4–6] or by taking ad-vantage of native impurities and structural defects that exist inall nominally stoichiometric crystals [7,8].While the basic fact of the unconventionality is well es-tablished, determining the specific symmetry of the orderparameter in Sr2RuO4 continues to be challenging, not leastbecause of the intrinsically low energy scales of the prob-lem, with Tc � 1.5 K. In principle, using epitaxial thin filmsof Sr2RuO4 as a materials platform would offer a new an-gle of attack with numerous potential advantages: films canbe patterned using standard optical lithography into devicesand junctions for phase-sensitive magnetization and electricaltransport measurements [9,10], and are naturally compatiblewith sophisticated scanned-probe microscopies [11,12] andspectroscopic techniques [13,14]. Indeed, in the cuprate high-temperature superconductors, scanning SQUID microscopyof thin-film samples provided conclusive evidence of thedx2−y2 pairing symmetry [15].The strong sensitivity of Sr2RuO4 to disorder meant that itrequired many years of effort following the initial discoveryof superconductivity in Sr2RuO4 single crystals [16] for thematerials science community to realize epitaxial thin filmsof Sr2RuO4 that were electronically clean enough to exhibitsuperconductivity [17]. Now that superconducting thin-filmsamples can be reproducibly synthesized [18–21], there is anurgent need to establish a quantitative link between the phys-ical properties of single crystals and the best thin films. The2643-1564/2024/6(3)/033178(17) 033178-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-4426-2884https://orcid.org/0000-0002-3380-668Xhttps://orcid.org/0000-0003-4769-8947https://orcid.org/0000-0002-4083-8735https://orcid.org/0000-0001-5938-6452https://orcid.org/0000-0003-3975-4478https://orcid.org/0000-0003-3282-7144https://orcid.org/0000-0003-2493-6113https://orcid.org/0000-0001-8000-4949https://ror.org/01c997669https://ror.org/05bnh6r87https://ror.org/015x8rz21https://ror.org/05bnh6r87https://ror.org/026v1ze26https://ror.org/037p86664https://ror.org/02wn5qz54https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.6.033178&domain=pdf&date_stamp=2024-08-16https://doi.org/10.1103/PhysRevResearch.6.033178https://creativecommons.org/licenses/by/4.0/JACOB P. RUF et al. PHYSICAL REVIEW RESEARCH 6, 033178 (2024)essential challenge in doing so is that the correlation betweenTc and the residual resistivity ρ0—taken in bulk samples tobe a proxy for the low-temperature elastic scattering rate—ofas-grown films is not nearly as robust as that observed insingle crystals.One hypothesis to explain the difference between the twoforms of Sr2RuO4 is that epitaxial thin films contain extendedin-plane defects, such as small-angle grain boundaries, thatincrease the electrical resistivity but have a much smallereffect on the intrinsic intragrain quasiparticle scattering rateand hence on Tc. To test this hypothesis would require a meansof introducing point disorder to such films while, hopefully,generating as small a change as possible to the resistivity ofthe extended defects.Here, we use high-energy electron irradiation to con-trollably introduce point disorder into bulk single-crystaland thin-film Sr2RuO4. We establish that the disorder fromirradiation suppresses Tc in bulk Sr2RuO4 in the sameway as chemical substitution or native point defects, butwith far greater control over the disorder axis. We thendemonstrate that thin films respond to irradiation-induceddisorder in essentially the same way as bulk samples,with Tc being suppressed in crystals and films at nearlyidentical rates.II. METHODSA. FIB-sculpted single crystals of Sr2RuO4To prepare samples of bulk, single-crystal Sr2RuO4 thatwere compatible with high-energy electron irradiations andthat enabled reliable measurements of the low-temperatureresistivity, we used a modified version of the epoxy-freemethod of mounting and preparing microstructures describedin Ref. [22]. Procedures for the growth of the Sr2RuO4 crys-tals are described in Ref. [23].From an oriented slice of the parent crystal, we cleaveda piece of Sr2RuO4 having 〈100〉-oriented edges to expose aflat, (001)-oriented surface which later became the top surfaceof the finished microstructure. To make electrical contact,we sputtered 200 nm of gold onto the cleaved surface andannealed the crystal for 5 to 6 min at 500◦C in air. We thenused a focused ion beam (FIB) to extract lamellae (typicaldimensions about 200 µm × 160 µm × 6 µm), working fromthe uncleaved side inwards, and transferred these lamellae exsitu onto mica substrates with the Au-coated surface facingupwards. After depositing 10 nm of Ti and another 150 nmof Au onto each lamella, we deposited Pt bridges in a xenonplasma-FIB to join the top of each lamella to the mica sub-strate. We then deposited a final 10 nm Ti/150 nm Au toensure good electrical contact between the lamella and thegold film on the mica substrate.After mounting and contacting the lamellas, we used agallium FIB to pattern each into a meandering microstructuresuitable for four-point resistance measurements, as shownin Fig. 1(a). From the dimensions of each microstructuredSr2RuO4 bar [24], we calculated the appropriate geometricalfactors to convert the measured resistances into resistivities.Such FIB-sculpted bars of Sr2RuO4 have a well-defined shapeand the positions of the voltage-sensing contacts on the crystalFIG. 1. Effects of high-energy electron irradiation on the elec-trical resistivity and superconducting Tc of Sr2RuO4. (a) Scanningelectron microscope image of a single crystal of bulk Sr2RuO4 thatwas microstructured with a focused ion beam. (b) Optical micro-scope image of many resistivity bridges lithographically patternedon an epitaxial thin-film sample of Sr2RuO4. (c) Resistivity versustemperature at T � 2 K for the crystal shown in panel (a), before andafter the sample received a 0.45 C/cm2 dose of 2.5 MeV electronirradiation. The irradiation causes an increase in ρ0 from 0.13 to0.45 µ� cm and a decrease in Tc from 1.35 to 0.97 K. (d) Resistiv-ity versus temperature data measured for a representative thin-filmbridge, before and after a 0.45 C/cm2 dose of 2.5 MeV electronirradiation. Inset shows a zoomed-in view of ρ(T ) for T < 2 K:irradiation induces an increase in ρ0 from 1.14 µ� cm to 1.56 µ� cmand a decrease in Tc from 1.40 K to 0.79 K. Although the as-grownTcs of the single-crystal and thin-film samples are within 0.05 K ofeach other, the 2 K resistivity of the film is nearly a factor of 10higher. The larger resistivity in the film is likely due to extended in-plane defects that are characteristic of even the best superconductingthin-film Sr2RuO4, such as antiphase boundaries at step edges inthe substrate or higher-order Ruddelson-Popper intergrowths [10,18–21,33–36].are independent of the wires and epoxy used for connectingthe sample to external electronics equipment in subsequent re-sistance measurements. The latter point is important becausethe wires attached to the mica substrate must be removed andremade every time the sample is transferred between the cryo-stat used for low-temperature electrical transport experimentsand that used for electron irradiation experiments. Having astable, reproducible geometry for the resistance measurementis essential for detecting small, irradiation-induced changes inthe absolute resistivity.B. Epitaxial thin films of Sr2RuO4For our experiments on thin-film Sr2RuO4, we synthesizeda (001)-oriented film of Sr2RuO4 on a (110)-orientedNdGaO3 substrate (Crystec, GmbH) by molecular033178-2CONTROLLABLE SUPPRESSION OF THE … PHYSICAL REVIEW RESEARCH 6, 033178 (2024)beam epitaxy following the substrate preparation andadsorption-controlled film growth procedures detailed inRefs. [19,25]. After preliminary structural characterization ofthe Sr2RuO4/NdGaO3(110) wafer via x-ray diffraction [24],we used standard photolithography, sputter deposition, andion milling techniques to pattern the Sr2RuO4 film into manystandalone devices for electrical resistivity measurements, asdescribed in Ref. [10]. As shown in Fig. 1(b), the region of thewafer patterned in this way is about 6 mm × 4 mm in lateralextent, and each device occupies a total areal footprint of1 mm × 1 mm, thus allowing many bridges to be fabricatedfrom the same film.Each resistivity bridge consists of two Pt (25 nm)/Ti(5 nm) contact pads at its ends for current injection andremoval, and two Pt/Ti pads in the middle for attach-ing voltage-sensing wires, separated by a center-to-centerdistance of L = 200 µm. Mismatch between the lattices ofSr2RuO4 and the (110) surface of NdGaO3 imposes in-plane,uniaxial and biaxial strains on the Sr2RuO4 film; at 295 K,the two Ru-O-Ru bonding directions along x and y are com-pressed by −0.39% and −0.16% relative to unstrained, bulksingle crystals of Sr2RuO4 (a = 3.8694 Å) [26,27], corre-sponding to in-plane A1g and B1g strains of −0.28% and0.23%. The longest dimension of every resistivity bridge isaligned with the slightly longer of the two in-plane Ru-O-Rubond directions. Since the NdGaO3 substrate is a wide-band-gap electrical insulator, the cross-sectional area through whichthe current is constricted to flow through is the Sr2RuO4film thickness, t = 28.1 ± 1.3 nm, multiplied by the widthof the resistivity bridges as determined by the lithography,w = 80 µm. All of the physical dimensions of the thin-filmresistivity bridges are significantly larger than the character-istic mean free paths of the charge carriers in Sr2RuO4 alongthe corresponding crystallographic directions, so we expectOhmic conduction to dominate the charge transport responseat all temperatures.We diced the patterned wafer into individual pieces con-taining one or two devices and mechanically polished eachof these smaller pieces down from the backside until theNdGaO3 substrate was reduced to a thickness of 100 to200 µm. Each piece could then be handled and irradiatedseparately from the others; the substrate-thinning step allowedthe irradiation to penetrate the full thickness of the film andsubstrate without causing excessive attenuation of the beam.Throughout the main text and Supplemental Material [24], werefer to individual Sr2RuO4 thin-film resistivity bridges sepa-rated out in this way from the same original wafer as distinctas-grown samples. Comprehensive electrical characterizationof these as-grown samples is described in the SupplementalMaterial [24].C. Electrical transport measurementsWe measured four-point resistances of the Sr2RuO4 epitax-ial thin films and FIB-sculpted single crystals using standardac techniques at excitation frequencies less than 200 Hz,employing a multichannel lock-in amplifier (Synktek) anddual-ended current sources with active common mode rejec-tion [28]. Unless otherwise noted, we used bias currents ofIrms � 10 µA for resistance measurements.We controlled the sample temperature and external mag-netic field using a Physical Properties Measurement System(PPMS; Quantum Design) equipped with a helium-3 insertand a 14 Tesla superconducting magnet. To ensure uniformthermalization of the samples upon traversing the supercon-ducting transitions, we incremented the temperature graduallyat low temperatures: typical cooling/warming rates for ther-mal cycles between 0.4 and 2 K were about 1 K/h. Detailedexplanations of the definitions and analysis methods that weused to extract ρ0 and Tc from resistivity versus tempera-ture data can be found in Ref. [24]. To remove variation inρ(300 K) of the thin-film bridges caused by systematic un-certainties in sample preparation and measurement, we havescaled the thin-film ρ(T ) presented in the main text to themean value of ρ(300 K) measured across all thin-film resis-tivity bridges. Further details can be found in Sec. III A of theSupplemental Material [24].D. High-energy electron irradiationWe performed high-energy electron irradiations (Eincident =2.5 MeV) at the SIRIUS Pelletron linear accelerator inPalaiseau, France, following the procedures described inRef. [22]. The FIB-sculpted single-crystal and epitaxialthin-film samples all had the c axis of the Sr2RuO4 crys-tal structure aligned with the out-of-plane direction, alongwhich the electron beam propagated. The samples wereimmersed in a bath of liquid hydrogen at a temperatureof 22 K during irradiation to produce a random distribu-tion of vacancy-interstitial Frenkel pairs [29]. Maintaininglow temperatures during electron irradiation promotes theformation of immobile, pointlike defects, without apprecia-ble defect clustering, cascades, columnar defects, or otherextended spatial correlations [30]. Measurements of theupper critical field of films before and after irradiation(Appendix B) confirm that the irradiation-induced defects arepointlike.Using NIST tables [31] to calculate the electron stoppingpower of Sr2RuO4 in the continuous-slowing-down approx-imation, we estimate the stopping length of the 2.5 MeVelectrons to be 2.9 mm in Sr2RuO4, meaning that theirradiation-induced defects should be homogeneously dis-tributed throughout the entire thickness of all samples studiedhere. According to the standard electron-scattering formalism,we expect that 2.5 MeV electrons transfer sufficient energy tothe atoms in the crystal structure to create vacancy-interstitialFrenkel pairs on all four unique sublattices in Sr2RuO4 (i.e.,strontium, ruthenium, equatorial oxygen, and apical oxygen),albeit with higher production rates for the cations than foreither oxygen site based on likely values for the interactioncross-sections (Appendix A).The electron irradiation facility in Palaiseau allows for insitu electrical resistance measurements to be conducted dur-ing irradiation, with the sample held at constant T = 22 K.Data of this kind are shown in Appendix A for representativeepitaxial thin-film and FIB-sculpted single-crystal samplesof Sr2RuO4, and allow for perhaps the most straightforwardcomparison with theories that attempt to model cross sec-tions for Frenkel pair production, based on tabulated crosssections for electron scattering from different nuclei and the033178-3JACOB P. RUF et al. PHYSICAL REVIEW RESEARCH 6, 033178 (2024)displacement energies of the nuclei away from the uniquechemical bonding sites in question.All low-temperature (T < 22 K) electrical transport dataon irradiated samples shown in this work, however, were ac-quired after ex situ transfers of the samples from the cryostat atthe irradiation facility to different cryostats at the Max PlanckInstitute for Chemical Physics of Solids in Dresden, Germany,which required warming up the samples to room temperature.This initial “room-temperature annealing” step causes recom-bination of some of the irradiation-induced Frenkel pairs, aswell as migration of some of the interstitials created duringirradiation to various other sinks in the materials, such assurfaces, grain boundaries, and/or dislocations. Nevertheless,quasiequilibrium populations of irradiation-induced vacanciesand interstitials remain after the initial warmup to room tem-perature that are robust against further thermal cycles between0 and 300 K. It is the cumulative effect of the latter pointlikescattering centers on the charge transport and superconductiv-ity in Sr2RuO4 that we attempt to quantify when investigatingthe dose dependencies of ρ0 and Tc.III. RESULTSIn Figs. 1(c) and 1(d), we show that a moderate e−dose of 2.5 MeV electron irradiation (D = 0.45 C/cm2 =2.8 × 1018 e−/cm2) subtly perturbs the electrical resistivityρ(T ) of Sr2RuO4. At low temperatures, where other, presum-ably inelastic, charge-carrier scattering mechanisms becomefrozen out by phase-space constraints, the effects of irradia-tion on ρ become proportionally more relevant and noticeable.The irradiation-induced modification of the elastic transportscattering rate, as measured by the change in residual re-sistivity �ρ0, reduces the superconducting Tc while largelypreserving the sharpness of the superconducting transitions,indicating that the irradiation-induced defects are distributedhomogeneously throughout the samples over the characteris-tic coherence length scales {ξab, ξc} of the Cooper pairs. Theinitial rates of ρ0 increase and of Tc suppression with e− doseare similar for the FIB-sculpted single crystal of Sr2RuO4and for the 28-nm Sr2RuO4/NdGaO3(110) epitaxial thin-filmsample.Magnetoresistance measurements of the upper criticalfields for superconductivity and of Shubnikov-de Haasoscillations for electron-irradiated Sr2RuO4 thin films (Ap-pendix B) provide additional evidence for the spatial unifor-mity and pointlike nature of the irradiation-induced defects,and show that the primary effect of this additional disorderis to boost the rate at which the momentum of the charge-current-carrying excitations is relaxed, while preserving theitinerant charge-carrier densities (i.e., band occupancies) aswell as other salient features of the normal-state electronicstructure near the Fermi level, such as the quasiparticlevelocities.Having established the qualitative effects of high-energyelectron irradiation on some of the basic properties of thesuperconducting and correlated-metal states in Sr2RuO4, wenow turn our attention to a detailed investigation of e−-dose-dependent changes to ρ0 and Tc. We leave direct microscopicvisualization of the irradiation-induced disorder in Sr2RuO4to future studies, because detecting dilute concentrations ofFIG. 2. Irradiation dose dependencies of ρ0 and Tc in Sr2RuO4epitaxial thin films and single crystals. (a,b) Relative change in resid-ual resistivity �ρ0, irr(D) ≡ ρ0, after irr(D) − ρ0, as grown induced per unite− dose D of 2.5 MeV electron irradiation in (a) microstructured bulksingle crystals and (b) thin films after a single round of irradiation.Each data point represents a physically distinct sample. Linear fitsof all available data points are shown as solid gray lines; the shadedregion bounded by dashed lines indicates ±2σ uncertainties in thefitted slope of the best-fit line for each data set. (c), (d) Resistivelymeasured superconducting Tcs versus accumulated e− dose D forthe irradiated samples displayed in panels (a), (b) (teal markers), aswell as the as-grown Tcs for the same samples (dark blue markersclustered at D = 0). Here Tc is defined as the temperature at whichρ(T ) crosses the 50% threshold of ρ0, and vertical error bars oneach thin-film Tc square in panel (d) indicate the temperatures atwhich ρ(T ) crosses the 10% and 80% thresholds of ρ0; error barsdefined analogously for the single-crystal Tc in panel (c) are smallerthan the heights of the corresponding midpoint Tc squares. In panels(c), (d), irradiated samples having Tcs below the base temperaturesof the measurement cryostats (horizontal black dashed lines) aredrawn as open squares, and were not included in the linear fits ofTc(D) drawn as solid gray lines [dashed gray lines have the samemeaning as in panels (a), (b)]. The vertical and horizontal scalesof the single-crystal graphs in panels (a), (c) are identical to thecorresponding thin-film graphs in panels (b), (d) to facilitate visualcomparisons between the data sets.pointlike defects, on the order of one part per thousand, isa notoriously challenging task, well beyond the current ca-pabilities of state-of-the-art microstructural probes such asscanning transmission electron microscopy [32,33].Given that substitutional disorder on the Ru site has beenshown to increase ρ0 at approximately 10× the rate of sub-stitutional disorder on the Sr site [4–6], we infer that Ruvacancies (produced at a rate of 1.2 to 2.6 × 10−3 displace-ments per Ru atom per 1 C/cm2 of irradiation) account formost of the observed modifications to ρ0. We provide a de-tailed discussion of the conversions between D and Frenkeldefect densities, and an analysis of the likely contributions ofthese defects to ρ0, in Appendix A.In Fig. 2(a), we plot the relative irradiation-inducedchanges in ρ0 measured across four distinct FIB-sculptedsingle-crystal samples of Sr2RuO4 as a function of the033178-4CONTROLLABLE SUPPRESSION OF THE … PHYSICAL REVIEW RESEARCH 6, 033178 (2024)TABLE I. Linear fits to the dose dependencies of ρ0 and of Tc plotted in Figs. 2(a), 2(b) and Figs. 2(c), 2(d) respectively. Error bars on thefit coefficients are ±2σ . Units: ρ0 [µ� cm], Tc [K], D [C/cm2].Sample type �ρ0(D) Range of fit in D �Tc(D) Range of fit in DBulk single crystal (0.70 ± 0.02)D [0.3, 1.18] (1.34 ± 0.02) − (0.86 ± 0.07)D [0, 0.6]28 nm Sr2RuO4/NdGaO3(110) films (1.00 ± 0.13)D [0.2, 1.0] (1.38 ± 0.02) − (1.22 ± 0.06)D [0, 0.6]accumulated dose D of 2.5 MeV electron irradiation, and inFig. 2(c) we plot the resistively measured Tcs versus D for allof these samples, before and after irradiation. In Figs. 2(b)and 2(d), we plot the same quantities as in Figs. 2(a) and 2(c),except now for nine distinct 28-nm Sr2RuO4/NdGaO3(110)thin-film samples. Raw ρ(T ) curves for all thin-film samplesthat underlie the data points in Figs. 2(b) and 2(d) arepresented in the Supplemental Material [24]. For both typesof Sr2RuO4 samples, there are clear systematic effects of Don both �ρ0, irr and Tc, that further corroborate the trendssuggested by the data and samples highlighted in Fig. 1:precisely controlled increases in D cause nearly monotonicincreases in ρ0 that are accompanied by precisely controlled,smooth decreases in Tc.For the FIB-sculpted Sr2RuO4 single-crystal sample thatreceived the highest irradiation dose in Figs. 2(a) and 2(c)(D = 1.18 C/cm2), such a dose was sufficient to reduce Tcfrom its as-grown value of 1.52 K to a post-irradiation valuethat was less than the base temperature of the cryostat (inthis case, an adiabatic demagnetization insert for the PPMS),Tc < 0.25 K. Accordingly, we schematically display this sam-ple in Fig. 2(c) as open square markers consistent with a rangeof possible Tcs between 0 and 0.25 K. This sample was notincluded in fits to Tc(D) described below, but it provides anillustrative example that highlights the stringent requirementson sample purity for observing superconductivity in Sr2RuO4:an extremely small (in absolute terms) increase of the quasi-particle scattering rate by �ρ0 = 0.83 µ� cm is sufficient tosuppress Tc from near its maximally achievable value to al-most zero. Likewise, for the Sr2RuO4 thin-film samples thataccumulated the highest irradiation doses in Figs. 2(b) and2(d) (D = 0.80 C/cm2 and 1.00 C/cm2), we did not observesuperconductivity in ρ(T ) data collected down to the basetemperature of the helium-3 cryostat, 0.4 K. We do not in-clude these points in subsequent fits to Tc(D), and we leavea more comprehensive exploration of the regime of very lowTcs to future work.To provide the simplest-possible quantitative parametriza-tions of these dose dependencies, we assume that �ρ0, irr andTc should initially vary linearly with D over the investigatedranges of e− doses in Fig. 2, because all D here correspondto dilute concentrations of irradiation-induced defects, of theorder of 0.1% (Appendix A). Fitting the data points in Fig. 2to lines, and taking systematic error bars on the fit coeffi-cients to be ±2σ surrounding the best-fit values, we obtainthe curves summarized in Table I. Alternatively, the explicitdependencies of �ρ0 and �Tc on D can be eliminated bytaking the ratios of the slopes of the best-fit lines, which yields�Tc/�ρ0 = (−1.24 ± 0.08) K/µ� cm for bulk crystals and(−1.23 ± 0.19) K/µ� cm for the thin films.In other words, although the individual magnitudesof both of the rates �ρ0/�D and |�Tc/�D| for theSr2RuO4/NdGaO3(110) epitaxial thin-film samples aregreater, in a statistically significant sense, than the correspond-ing rates for the FIB-sculpted single crystal Sr2RuO4 samples,these differences essentially compensate each other when tak-ing the ratio of these quantities, resulting in initial rates of Tcsuppression with increasing residual resistivity, �Tc/�ρ0, forthe two types of Sr2RuO4 samples that are indistinguishablewithin experimental precision.IV. DISCUSSIONIn Fig. 3, we plot 22 (ρ0, Tc) data points extracted from theliterature as gray circles for bulk single crystals of Sr2RuO4containing natively occurring (as-grown) defects [3], delib-erate substitutions of Ti or Ir on the Ru site [5], as well asdeliberate substitutions of La on the Sr site [6]. Results forelectron-irradiated FIB-sculpted single crystals of Sr2RuO4from the present work are overlaid as dark purple squareson these reference data points. For the e−-irradiated crystals,several repetitions of the cycle of {measure ρ(T ) → irradiate→ remeasure ρ(T )} allowed us to use the same physicalsamples to acquire multiple (ρ0, Tc) data points—hence thelarger number of data points shown in Fig. 3 (11 in total) thanin Figs. 2(b) and 2(d)—which, taken together, span nearlythe entire range of disorder over which superconductivity isobserved in unstrained bulk single crystals of Sr2RuO4. Theoverall trend of the previously reported Tc(ρ0) dependence isreproduced well, with the more precisely clustered data fromthis work suggesting a slightly steeper slope of the best-fitline than previously, as seen by comparing the purple and graysolid lines in Fig. 3.The Tc data points for all Sr2RuO4 single-crystal samplesin Fig. 3 are plotted versus the absolute residual resistivityρ0, without any shift in the zero point of this scale. Implicitin this choice is the assumption that all sources of quasiparti-cle scattering that contribute to ρ0, both intrinsically presentfrom the original crystal growth and induced by irradiation,also contribute proportionally to the Cooper-pair-breakingenergy scale h̄/τpb, and thus to the suppression of Tc. Thisassumption is justified by the limited scatter in the Tc(ρ0)behavior observed across different studies, and by the inferredextrapolation of the experimental data to a common rangeof zero-disorder Tc0 = 1.5 to 1.55 K, which are furthermoreconsistent with the highest values of Tc found in the literaturefor actual Sr2RuO4 crystals under ambient conditions.In contrast, the relatively large scatter in the initial absolutevalues of ρ0 across nine distinct as-grown Sr2RuO4 thin films(1.1 to 1.5 µ� cm [24]) is incommensurate with the narrowdistribution of as-grown Tcs for these same samples (1.35 to1.41 K). An assumption of direct proportionality between ρ0and h̄/τpb is therefore not appropriate for the thin-film sam-ples. Nevertheless, while there is no discernible correlation033178-5JACOB P. RUF et al. PHYSICAL REVIEW RESEARCH 6, 033178 (2024)FIG. 3. Universal impact of elastic, momentum-relaxing scat-tering on superconductivity in Sr2RuO4. Gray circles representpreviously published data for bulk single-crystal samples of Sr2RuO4containing native defects [3], as well as intentionally substitutedchemical impurities on the Ru and Sr sites—viz., Sr2(Ru1−xTix )O4[5], Sr2(Ru1−xIrx )O4 [5], and (Sr2−yLay )RuO4 [6]. Dark purplesquares represent results from this work for e−-irradiated Sr2RuO4single crystals containing native defects plus irradiation-inducedvacancy-interstitial Frenkel pairs on all atomic sublattices. For allsingle-crystal data shown here, the measured residual resistivity inabsolute units, ρ0 [μ� cm], is utilized as a proxy assumed to bedirectly proportional to the Cooper-pair-breaking energy scale, h̄/τpb,with no adjustments of the common zero point of these scales (bot-tom axis). However, results from this work for e−-irradiated 28-nmSr2RuO4/NdGaO3(110) epitaxial thin-film samples are plotted asred squares against the irradiation-induced change in residual resis-tivity, �ρ0, irr [μ� cm] (top axis). As explained in the main text, arange of rigid shifts of the zero point of the top axis relative to thebottom axis are, in principle, compatible with the results of this work;here we chose an offset of +0.5 µ� cm for visual clarity, indicatedby the red arrow in the upper left corner. Color-coded straight linesare linear fits to each family of Sr2RuO4 samples and the systematicuncertainties in the best-fit rates of Tc suppression (±1σ ) are indi-cated by the color-coded shaded regions between dashed lines.between absolute values of ρ0 and Tc, the changes in ρ0 andTc after being subjected to different doses of 2.5 MeV elec-tron irradiation, �ρ0 and �Tc, respond in a highly correlatedfashion, in a way that is similar to the dose-dependent trendsthat we observed on FIB-sculpted single crystals. The mostintuitive way to reconcile these observations is to assume thatmany of the natively occurring extended defects in as-grownSr2RuO4 thin films [10,18–21,33–36] contribute to ρ0 butdo not contribute appreciably to Tc suppression, whereas anypointlike defects, including those created by electron irradia-tion, cause both ρ0 and h̄/τpb to rapidly increase in lockstep,just as in bulk crystals.The natural interpretation of the thin-film data as reflectinga separation of independent sources of scattering that coupledifferently to ρ0 and Tc therefore suggests that our irradiationexperiments should still accurately probe the Cooper-pair-breaking rate of Tc suppression, �Tc/�(h̄/τpb) ∝ �Tc/�ρ0,in thin films, provided that the relative change in ρ0 isTABLE II. Error bars on the fit coefficients are ±1σ .Sample type �Tc/�ρ0 (K/µ� cm)Single crystal (Refs. [3,5,6]) −1.15 ± 0.11Irradiated single crystal (this work) −1.34 ± 0.09Irradiated films (this work) −1.20 ± 0.07measured pairwise for the same thin-film sample before andafter each sample accumulates a given dose D of electronirradiation (Appendix C and Fig. 7). Following this lineof reasoning, we combine the data for the as-grown andelectron-irradiated 28-nm Sr2RuO4/NdGaO3(110) thin-filmsamples previously shown in Figs. 2(b) and 2(d) and replotthese 18 Tc data points as red squares against the top axis ofFig. 3, �ρ0, irr, thereby eliminating the explicit dependenceon the dose D of 2.5 MeV electron irradiation.Applying this procedure reveals a striking similarity be-tween the best-fit slopes �Tc/�ρ0 of the single-crystal andthin-film data sets, as depicted by the color-coded solidlines in Fig. 3 and summarized in Table II. This resultstrongly suggests that momentum-relaxing elastic scatteringfrom pointlike disorder plays the same microscopic role inCooper-pair breaking in thin-film and single-crystal Sr2RuO4,a conclusion which is further supported by a comparisonof the Tc versus ρ0 behavior observed here with previousimpurity-scattering studies of bulk Sr2RuO4 single crystals[3–6,8]. Despite those studies encompassing numerous dis-tinct realizations of “disorder”—i.e., variable densities ofchemically different pointlike scattering centers, some ofwhich couple weakly and some of which couple strongly tothe charged quasiparticle excitations near the Fermi level—there is remarkable consistency across all studies when thesemicroscopic details of the scattering processes are subsumedinto their combined impact on the residual resistivity, and ρ0is treated as the relevant independent variable that ultimatelycontrols the dependent variable, Tc.Quantitatively determining the appropriate rigid shift be-tween the zero points of the top and bottom axes inFig. 3 would require more information than we have ob-tained in these experiments, since we do not know theclean-limit Tc0 of these thin-film samples with certainty be-cause the films are subject to biaxial (A1g) and uniaxial(B1g) in-plane strains imparted by lattice matching to thesubstrate. Indeed, the simplest-possible empirical estimates,described in Appendix C, place a maximum range on Tc0for Sr2RuO4/NdGaO3(110) of 1.4 to 2.7 K; the illustrativechoice of the 0.5 µ� cm horizontal offset employed in Fig. 3corresponds to Tc0 = 2.0 K, and is furthermore consistent witha comparison to expectations from Abrikosov-Gor’kov pair-breaking theory (Appendix D).Regardless of this remaining quantitative uncertainty indetermining the exact Tc0 of Sr2RuO4/NdGaO3(110) on a trueplot of Tc(h̄/τpb), the results displayed in Fig. 3 offer com-pelling evidence that quantitatively very similar mechanismsof superconductivity are operative in epitaxial thin films andsingle crystals of Sr2RuO4. We finish with a brief discussionof what insights are offered by the robust connection betweenTc and ρ0 revealed by this work regarding open questions033178-6CONTROLLABLE SUPPRESSION OF THE … PHYSICAL REVIEW RESEARCH 6, 033178 (2024)surrounding the materials science and the physics of super-conducting Sr2RuO4 thin films.Although the in-plane B1g orthorhombic strain impartedby the NdGaO3(110) substrates on Sr2RuO4 likely boostsTc0 of these thin films slightly above that of unstrained bulktetragonal Sr2RuO4 [19,37], our data indicate that any strain-induced enhancement of the clean-limit pairing scale in ourthin films [and concomitant symmetry-imposed anisotropiesin the superconducting gaps caused by the reduction of thecrystal point-group symmetry from tetragonal (D4h) to or-thorhombic (D2h)] are modest effects. They likely result fromsubtle changes in the superconducting order parameter thatbecome irrelevant in the experimentally accessible range ofquasiparticle scattering rates, because samples of either vari-ety having as-grown Tcs of about 1.4 K exhibit nearly identicalrates of Tc suppression with increasing ρ0 when additionalpointlike defects are introduced via high-energy electronirradiation.Turning this observation around, the relatively “high” Tcsachieved for the as-grown films, which approach the clean-limit value for bulk Sr2RuO4 of 1.50 to 1.55 K, indicateminuscule concentrations of native in-plane defects that areknown to be strongly pair-breaking (for example, of order0.01% Ru vacancies). This attests to the remarkable de-gree of vacancy control that has been achieved in the bestSr2RuO4 thin films, which sets a benchmark for modern oxidemolecular-beam epitaxy. This result is also encouraging froma physics point of view, particularly for ongoing research ef-forts using Sr2RuO4 thin films to gain new perspectives on theoutstanding puzzles regarding the order parameter and mech-anism of the enigmatic unconventional superconductivity inthis material. Finally, we note that the smooth progressionfrom “high” to “low” Tcs in all cases is consistent with thepresence of a sign-changing gap �(k) on all bands (likelywith symmetry-enforced, nonaccidental nodes) if intrabandscattering dominates the pair breaking [3], or with the absenceof large differences between the average magnitudes of |�(k)|on distinct bands if interband scattering dominates the pairbreaking (see also Appendix D, Figs. 8 and 9).In conclusion, the results displayed in Figs. 1–3 showthe insights that can be gained by precisely tuning the pointdefect concentrations in Sr2RuO4 using high-energy electronirradiation. These experiments can be performed equallywell on epitaxial thin films and single crystals of Sr2RuO4,and have the extremely high precision demonstrated here.Other methods commonly used to locate samples along ahorizontal “disorder” axis, such as x-ray spectroscopy inelectron microprobes [3,38], require large sample volumes toaccurately quantify dilute impurity concentrations, and wouldnot have the sensitivity to detect such minute deviations fromideal stoichiometry. The precision available in low-volumestoichiometry techniques based on transmission electronmicroscopy [22,33], Raman scattering [32], or Rutherfordbackscattering spectrometry, is lower still, so the high-energyelectron irradiation enables investigation of a part ofparameter space that cannot currently be reached by anyother technique. Moreover, the ability in irradiation studies todirectly compare measurements performed on the same sam-ples before and after irradiation renders moot concerns aboutinevitable sample-specific peculiarities that could obscurethe effects of the pairbreaking point disorder that we wishto study.With a controlled and reproducible knob now availablefor finely adjusting the quasiparticle scattering rate and Tc inSr2RuO4, it may be possible to definitively resolve some of thelongstanding controversies regarding the superconductivity,such as the interpretation of heat capacity data for T < Tc[39–41] in light of other thermodynamic probes that evidencea multicomponent order parameter [42–44], as well as theinterpretation of signals in muon spin relaxation [45–47] andpolar Kerr effect data [48], which evidence some kind oftime-reversal symmetry breaking onsetting near Tc. Lookingforward, it will also be interesting to see what new experi-ments Sr2RuO4 thin films enable in this field; for example,performing irradiations using a collimated e− beam insidean electron microscope [49,50] may provide a straightfor-ward way to spatially pattern the disorder landscape to createsuperconductor/normal-metal all-Sr2RuO4 tunnel junctionsfor use in phase-sensitive measurements.Data plotted in Figs. 1–9 of the main text and Figs. S1–S5 of the Supplemental Material are available at Ref. [51].Additional data available upon reasonable request.ACKNOWLEDGMENTSWe thank Cyrus Dreyer for useful discussions. J.P.R.,H.M.L.N., E.Z., P.H.M., D.S., and A.P.M. thank the MaxPlanck Society for financial support. H.M.L.N. was supportedby the Alexander von Humboldt Foundation Research Fellow-ship for Postdoctoral Researchers. N.K. is supported by JSPSKAKENHI (Grants No. JP18K04715, No. JP21H01033, andNo. JP22K19093). Research in Dresden benefits from the en-vironment provided by the DFG Cluster of Excellence ct.qmat(EXC 2147, Project ID No. 390858490). Electron irradiationexperiments performed at the SIRIUS beamline were sup-ported by the EMIR&A French network (FR CNRS 3618).This work was supported by the National Science FoundationPlatform for the Accelerated Realization, Analysis, and Dis-covery of Interface Materials (PARADIM) under CooperativeAgreement No. DMR-2039380. This work was also supportedby NSF Grant No. DMR-2104427 and Air Force Office of Sci-entific Research Grant No. FA9550-21-1-0168. This researchwas funded in part by the Gordon and Betty Moore Founda-tion’s EPiQS Initiative through Grants No. GBMF3850 andNo. GBMF9073 to Cornell University. Sample preparationwas facilitated in part by the Cornell NanoScale Facility, amember of the National Nanotechnology Coordinated Infras-tructure (NNCI), which is supported by the National ScienceFoundation (Grant No. NNCI-2025233). J.P.R. and H.M.L.N.contributed equally to this work.APPENDIX A: CONVERSIONS BETWEENACCUMULATED DOSE AND FRENKEL DEFECT DENSITY1. In situ resistivity experimentsIn Figs. 4(a) and 4(b), we show the resistivity ρ(D) mea-sured in situ (during 2.5 MeV electron irradiations) at T =22 K as a function of accumulated e− dose, for representative(a) FIB-sculpted single-crystal samples and (b) epitaxial thin-film samples of Sr2RuO4, respectively. Increasing D causes033178-7JACOB P. RUF et al. PHYSICAL REVIEW RESEARCH 6, 033178 (2024)FIG. 4. Dependence of ρ(22 K) on irradiation dose, cross sec-tions for creating Frenkel defects, and the effects of these defectson the residual resistivity in Sr2RuO4. (a), (b) ρ(D, 22 K) measuredin situ during 2.5 MeV e− irradiations, for a representative (a) FIB-sculpted single-crystal and (b) epitaxial thin-film sample of Sr2RuO4.(c)–(e) Calculated cross sections for creating vacancy-interstitialFrenkel pairs from (c) strontium, (d) ruthenium, and (e) oxygenatoms via electron irradiation. Each panel considers a range of pos-sible displacement energies Edisp for creating each type of Frenkelpair (light to dark green color scale). (f) Data for FIB-sculptedSr2RuO4 single crystals showing the experimentally measured in-crements in residual resistivity after accumulating variable doses ofEkin = 2.5 MeV e− irradiation (black markers). These data are to becompared with the expected changes in ρ0 that are attributable toirradiation-induced Sr vacancies (blue dashed lines), Ru vacancies(red dashed lines), and the sum of Ru + Sr vacancies (purple dashedlines), given the Frenkel pair production rates displayed in panels (c),(d). The region between each pair of dashed lines represents how thesystematic uncertainties in Edisp convert into ranges of possible �ρ0versus Deff behavior; the physically relevant parameter ranges arelikely Edisp ranging from 7.5 to 15 eV for all sublattices in Sr2RuO4.monotonic, nearly linear increases in ρ(D, 22 K). The best-fitlines to data collected over the range D: [0, 0.60] C/cm2 areρ(D, 22 K) [µ� cm] ={4.8 + 1.8D [C/cm2] (films),2.4 + 1.5D [C/cm2] (crystals).After both samples accumulated e− doses of 0.60 C/cm2,we thermally cycled the samples to 300 K and back to 22 Kwithout removing the samples from the irradiation cryostat,and remeasured the resistivities indicated by the blue mark-ers in Figs. 4(a) and 4(b). This initial “room-temperatureannealing” of the irradiated samples caused ρ(22 K) to de-crease from 3.3 to 3.0 µ� cm (5.89 to 5.62 µ� cm) forthe FIB-sculpted single-crystal (thin-film) Sr2RuO4 samples,respectively, or to about 91% (95%) of the starting val-ues. Such reductions of ρ are commonly observed uponinitially warming up electron-irradiated crystals, includingmaterials that have similar solid-state chemistries to Sr2RuO4(e.g., SrRuO3 [52,53] and high-Tc cuprates [30,54]), andare typically attributed to some fractions of the originalconcentrations of irradiation-induced defects becoming mo-bile at elevated temperatures. Finally, we note that due totechnical difficulties with conducting in situ transport mea-surements during some irradiation beamtimes, the thin-filmsample measured in Fig. 4(b) is a resistivity bridge patternedon a 24-nm Sr2RuO4/LSAT(100) sample, rather than the28-nm Sr2RuO4/NdGaO3(110) samples characterized else-where in the manuscript. Although this likely affects certainquantitative details of the measurements, such as the rate ofresistivity increase �ρ(22 K)/�D, we believe that all qual-itative features of the observed ρ(D) behavior in Fig. 4(b),including the annealing of some defects upon the first ther-mal cycle to 300 K, would also be observed for irradiatedSr2RuO4/NdGaO3(110) samples measured in situ.2. Conversions between accumulated electron irradiationdose and Frenkel defect densitiesWe used the methods outlined in Ref. [22] to estimate therates at which Frenkel defects (i.e., vacancy-interstitial pairs)are produced in Sr2RuO4 by high-energy electron irradia-tion. Creating a defect requires a certain amount of energy,known as the displacement energy Edisp, which is specificto a given atomic site and type of defect. The probabilitythat such a defect will be formed during irradiation is givenby the relativistic cross section for the transfer of E � Edispfrom an electron having kinetic energy Ekin to a nucleus ofthat atom. In our irradiation experiments, the probability thata high-energy electron (Ekin ≈ Eincident = 2.5 MeV) interactswith any atom in the crystal is extremely low; we are operatingin the “long mean free path” limit. Therefore, we calculateirradiation-induced defect concentrations for each atomic siteindependently, using the same global, measured irradiationdose in each case, without self-consistently accounting forenergy losses and/or multiple scattering events that reduceEkin below Eincident as the electron beam propagates throughthe samples.In Figs. 4(c)–4(e), we plot the calculated cross sections forcreating Frenkel pairs σFP on the strontium, ruthenium, andoxygen sites, respectively. The dependence of σFP on theelectron kinetic energy Ekin (horizontal axis) is modeled usingvalues for the relativistic Mott scattering cross section tab-ulated in the literature [22,55,56], and the simulations areperformed considering a range of displacement energies Edisp(individual traces, green colorscale) for the formation of eachtype of Frenkel pair. Density-functional-based calculationsof Frenkel defect energetics suggest that in Sr2RuO4,Edisp, Sr � 7.6 eV and Edisp, Ru � 8.2 eV (C. E. Dreyer, privatecommunication). Meanwhile, previous electron irradiation033178-8CONTROLLABLE SUPPRESSION OF THE … PHYSICAL REVIEW RESEARCH 6, 033178 (2024)studies of high-Tc copper oxides (specifically, YBa2Cu3O7−δ)that are structurally similar to Sr2RuO4 have found thatEdisp, eqO = 10 eV [54] or 8.4 eV [49] for creating Frenkeldefects on the planar oxygen site. Accordingly, we proposethat displacement energies in the range of Edisp = 7.5 to 15 eVare the physically relevant parameter regimes for producingFrenkel defects on each of the strontium, ruthenium, andoxygen sublattices; for simplicity, we neglect any distinctionbetween Edisp, O for equatorial and apical oxygen sites.Over these ranges of Edisp, the cross sections for Frenkelpair production at Ekin = 2.5 MeV can be read off fromFigs. 4(c)–4(e) as σFP, Sr = 339 to 159 barn, σFP, Ru = 410 to193 barn, and σFP, O = 76 to 37 barn.Note that 1 barn = 10−24 cm2; thus, to convert these crosssections to the number of displacements per atom (dpa) fora specific dose D of 2.5 MeV electron irradiation, we simplyneed to multiply by D given in units of [number of electronsper unit area]. It follows that Frenkel defect densities (nFP) areproduced on the unique sublattices per unit dose D at rates ofdnFP, SrdD= (2.1 to 1.0) × 10−3 dpaC/cm2(× 2 Sr/unit cell),dnFP, RudD= (2.6 to 1.2) × 10−3 dpaC/cm2(× 1 Ru/unit cell),dnFP, OdD= (0.47 to 0.23) × 10−3 dpaC/cm2(× 4 O/unit cell).As mentioned in the main text, direct visualization of suchdilute pointlike defect concentrations, nFP � O(0.1%) = onepart per thousand, is a formidable challenge, well beyondthe current capabilities of state-of-the-art microstructuralprobes such as scanning transmission electron microscopy[22,33]. Nonetheless, by making a few more assumptions,it is possible to indirectly confirm the general consistencyof these estimated defect densities by relating nFP to oneof the observables measured in our current study—namely,the residual resistivity in the absence of superconductivity,ρ0 ≡ ρns(T → 0 K).In Fig. 2(a) of the main text, we displayed the results ofex situ measurements of the irradiation-induced incrementin residual resistivity �ρ0 for four distinct FIB-sculptedsingle crystals of Sr2RuO4 that accumulated variable dosesD of 2.5 MeV e− irradiation. As discussed above, somefraction of the defects created by irradiation are lost whena sample is first warmed to room temperature followingirradiation. We model this loss as a simple multiplicativereduction of the dose received by each sample, calculatingthe reduction factor using the 22 K in situ data [red points inFig. 4(a)] and the resistivity measured at 22 K after thermallycycling the sample to room temperature and back [blue pointin Fig. 4(a)]. In Fig. 4(f), we re-plot the data points fromFig. 2(a) of the main text versus the effective irradiationdose received, Deff = 0.62 × D, where 0.62 is the calculatedreduction factor. We used the reduction factor determinedfrom Sample C for all four samples because most of theother samples had incomplete in situ resistivity datasets.Although this adjustment procedure implicitly appeals toMatthiessen’s-Rule-type reasoning, which is known to notbe strictly valid in perovskite-based ruthenates [14,52], theerrors associated with this approximation are small comparedwith other systematic uncertainties in these estimates of nFPand in how efficiently the equilibrium nFP for Sr, Ru, and Oare transduced into corresponding increases in ρ0.Previous studies of Sr2RuO4 single crystals that were in-tentionally doped with chemical impurities on the Ru site,namely, Sr2Ru1−x(Ti, Ir)xO4, demonstrated that such defectsact as nearly unitary-limit scatterers that increase the resid-ual resistivity at a rate of dρ0/dx ≈ 425 µ� cm, where x isthe fractional dopant density per unit cell [4,5]. Likewise,previous studies of Sr2RuO4 single crystals that were inten-tionally doped with chemical impurities on the Sr site, namely,Sr2−yLayRuO4, demonstrated that such defects act as nearlyBorn-limit scatterers that increase the residual resistivity ata much smaller rate of dρ0/dy ≈ 40 µ� cm, where y is thefractional dopant density per unit cell [6]. Assuming that Srand Ru vacancies scatter the charge-carrying quasiparticles inan identical fashion to chemically substituted impurities onthe respective atomic sites, we can then multiply the aboveestimates of dnFP, Sr/dD (dnFP, Ru/dD) by the empirical valueof dρ0/dy (dρ0/dx) to obtaindρ0dD= (0.17 to 0.079) µ� cmC/cm2(from Sr vacancies),dρ0dD= (1.1 to 0.51) µ� cmC/cm2(from Ru vacancies),dρ0dD= (1.3 to 0.59) µ� cmC/cm2(summed total).These simulated rates of residual resistivity increase perunit of 2.5 MeV e− irradiation dose are drawn as dashedlines in Fig. 4(f). The experimentally measured �ρ0 versusDeff data points for FIB-sculpted single crystals of Sr2RuO4agree reasonably well with these simulations, which suggeststhat the estimated cross sections for Frenkel defect productionare realistic. We note that in arriving at these simulated ratesof dρ0/dD, we neglected any possible contributions of Ruand Sr interstitials to increases in ρ0, and we ignored entirelyall irradiation-induced defects on the O site. Although thesechoices were made out of convenience—namely, becausethere are not any available data in the literature to indicatehow much such defects might contribute to ρ0 in Sr2RuO4—Fig. 4(f) indicates that considering these additional types ofirradiation-induced disorder is not strictly necessary to ac-count for the experimentally observed rate dρ0/dD.It remains an open question to determine whether theapparent level of agreement between simulation and exper-iment in Fig. 4(f) is somewhat artificial, in the sense thatthe true displacement energies for Sr and Ru are actuallytowards the higher end of our estimated range (15 eV) andthat the contributions of interstitials and oxygen defects to�ρ0 are not negligible, or whether it is suggestive that theeffects on �ρ0 of the explicitly neglected defects are alreadyimplicitly accounted for by the Deff → D “room-temperatureannealing” conversion of the experimental data. Along theselines, we note that the extent to which Frenkel defects onthe oxygen sites affect the resistivity and superconductivityin Sr2RuO4 might be addressable in a more direct fashion infuture irradiation studies that utilize lower incident electronkinetic energies, which would transfer sufficient energy to033178-9JACOB P. RUF et al. PHYSICAL REVIEW RESEARCH 6, 033178 (2024)create defects on the O sublattices, but not to create defectson the Sr or Ru sites [cf. the low Ekin � 0.5 MeV regionsof Figs. 4(c)–4(e)]; a similar approach has been employedpreviously for high-Tc cuprates [49].APPENDIX B: MAGNETORESISTANCE OF THIN FILMS1. Superconducting upper critical fieldIn Fig. 5, we show that a moderate e− dose (D =0.45 C/cm2) of 2.5 MeV electron irradiation substantiallyreduces the upper critical magnetic fields required to suppresssuperconductivity in thin-film Sr2RuO4. For these electricaltransport measurements, the external magnetic field is di-rected along the crystallographic c axis of Sr2RuO4, which isalso the out-of-plane direction of epitaxial thin-film samplesstudied here [see inset to Fig. 5(c)]. Thus, the critical fieldprobed is Hc⊥ in the terminology of Refs. [57,58], which isessentially equivalent to Hc2 (with H || c) for a bulk specimenof a type-II superconductor such as Sr2RuO4. Hereafter werefer to Hc⊥ and Hc2||c interchangeably.The simultaneous decreases of both Tc and Hc2||c uponhigh-energy electron irradiation provides further evidence thatthe irradiation-induced defects are pointlike scattering centersthat uniformly suppress the characteristic intragranular con-densation energy scale of the superconductivity in Sr2RuO4.By contrast, if the electron irradiation created columnarand/or other types of extended defects, then we would gener-ally expect Hc2||c to remain unchanged, or perhaps even to beenhanced, due to increased pinning of vortices at the extendeddefects, while properties that depend on the details of theintergranular Josephson network of superconducting grains[30], such as critical currents, would change.Without a clear understanding of the effective pairing inter-actions that cause superconductivity to emerge in Sr2RuO4, itis difficult to quantitatively interpret the observed magnitudesof Tc and Hc2||c reduction upon irradiation, since both of thesequantities depend on some appropriately weighted averagesof the superconducting order parameter over the quasiparti-cle states near the Fermi level EF . Nevertheless, making thestandard assumption that Hc2||c is orbitally limited, we canconvert the measured upper critical fields into phenomenolog-ical in-plane superconducting coherence lengths (ξab) usingthe Ginzburg-Landau relation:μ0Hc2||c = φ02πξ 2ab�⇒ ξab [Å] =√329106Hc2||c [kOe],where φ0 = h/(2e) is the superconducting flux quantum.To account for the temperature dependence of the quanti-ties in this expression, we follow previous studies of epitaxialthin films of Sr2RuO4, which have shown that the nearlylinear scaling of Hc⊥ versus Tc—observed in Fig. 5 for Tdown to 0.4 K, i.e., down to T/Tc = 0.3 for the as-grownsample—in fact persists over at least another decade in T ,down to the lowest temperatures accessible in a dilution re-frigerator, O[0.06 K] [59]. Accordingly, we fit all availableHc⊥ versus Tc data points for the as-grown and irradiated28-nm Sr2RuO4/NdGaO3(110) samples to separate lines, andextrapolate the fit results to zero temperature, as shown bythe gray lines in Fig. 5(c), to obtain ξab(T → 0 K) = 500 ±FIG. 5. Effect of electron irradiation on the superconducting up-per critical fields of Sr2RuO4 thin films. (a) Normalized resistivityversus temperature data for an as-grown Sr2RuO4 thin-film resistivitybridge, acquired at external magnetic fields ranging from 0 to 1.1kOe in discrete steps of 0.1 kOe. (b) Same measurements as inpanel (a), but after the sample had received a 0.45 C/cm2 dose of2.5 MeV electron irradiation. Note that the external magnetic fieldsteps in panel (b) are half of those displayed in panel (a), and that themaximum applied field is now merely 0.4 kOe, which is sufficientto suppress Tc to substantially below the base temperature of thehelium-3 cryostat, ≈0.4 K. Currents of 1 µA, rather than the usual10 µA, were used in the acquisition of the data shown in panels (a),(b). (c) Upper critical magnetic fields for superconductivity (Hc⊥),before and after irradiation, extracted from the resistivity data in pan-els (a), (b), along with linear fits (grey solid lines) used for estimatingHc⊥(T → 0). The inset schematically depicts the measurement ge-ometry and orientation of H with respect to the Sr2RuO4 crystalstructure. Solid markers indicate the Tcs where ρ(T, H ) crosses the50% threshold of ρ(2 K, 0 kOe) [bold dashed line in panels (a), (b)],and the horizontal error bars on these Tcs represent where ρ(T, H )crosses the 10% and 80% thresholds of ρ(2 K, 0 kOe) [dashed linesin panels (a), (b)].20 Å for as-grown films, and 790 ± 30 Å after a dose of0.45 C/cm2.The error bars quoted here on ξab refer to the spreadof Hc⊥(T → 0 K) values obtained by employing 10% or80% thresholds of ρ(2 K, 0 kOe) to define Tc, rather than a033178-10CONTROLLABLE SUPPRESSION OF THE … PHYSICAL REVIEW RESEARCH 6, 033178 (2024)50% threshold, and propagating these systematic uncertaintiesthrough the linear fits and extrapolations of the measuredHc⊥(Tc) phase boundary [cf. gray-shaded regions in Fig. 5(c)].While the sensitivity of ρ(T, H ) to percolating, but otherwisefilamentary, superconducting pathways implies that electricaltransport measurements inherently tend to overestimate uppercritical fields compared with bulk-sensitive measurements ofsuperconductivity [57,60], such considerations only affect theabsolute values of Hc2 that are sensed by different probes; bycontrast, the relative change observed here in Sr2RuO4 uponelectron irradiation is unambiguous: increasing the disorderscattering rate suppresses Hc2||c (enhances ξab). This result isqualitatively consistent with previous studies of Sr2RuO4 thatdeduced such a correlation by analyzing Hc2||c data across acollection of different as-grown samples, in both bulk [8] andthin-film form [59].2. Effect of electron irradiationon Shubnikov-de Haas oscillationsTo place the irradiation-induced modifications to ρ0, Tc,and Hc2||c in proper context, it is important to note thatcertain features of the normal-state near-EF electronic struc-ture in Sr2RuO4, such as the sizes and shapes of the Fermisurfaces, are not appreciably affected by the doses of high-energy electron irradiation administered in this work. Toinvestigate the Fermiology in detail, in Fig. 6 we plot low-temperature magnetotransport data acquired for an as-grown28 nm Sr2RuO4/NdGaO3(110) sample, and for the same sam-ple after accumulating a moderate e− dose (D = 0.60 C/cm2)of 2.5 MeV electron irradiation. The external magnetic field isagain directed out-of-plane, along the crystallographic c axisof Sr2RuO4, and the temperature is held constant at a nom-inal value of T = 0.45 K. We used rms excitation currentsof I = 100 µA for transport measurements performed duringthese field sweeps, to further enhance the signal-to-noise ratio.The overall magnitude of the normalized magnetoresis-tance (MR) in the normal state (i.e., for B = μ0H 
 μ0Hc⊥)is somewhat reduced upon increasing D [Fig. 6(a)]. This be-havior is generically expected whenever more than one decayrate enters the charge-carrier momentum relaxation dynam-ics (see, e.g., Ref. [22] and references contained therein),and indeed time-domain THz spectroscopy measurements onthese Sr2RuO4 epitaxial thin films have directly shown thatthe low-energy optical conductivity at low temperatures con-tains contributions from more than one Drude-like oscillator[14]. Although detailed accounts of how electron irradiationcouples to the multiple transport lifetimes that determine thedc MR [as well as the ac σ (ω) for small ω �= 0] are beyondthe scope of the present work, there is another feature of thedata that is simpler to interpret quantitatively: namely, rapidlyoscillating Shubnikov-de Haas (SdH) contributions to the MR.To better isolate and visualize the SdH signal, we fit theraw MR data within the 12 to 14 Tesla field range [dashedbox in Fig. 6(a)] to the locally quadratic form ρbg(H ) =A0 + A1H + A2H2, and then compute the background-subtracted and background-normalized MR, ρ̃/ρ0 ≡ [ρ(H ) −ρbg(H )]/ρbg(H ). We plot the results thus obtained for ρ̃/ρ0for the as-grown and electron-irradiated Sr2RuO4 samples inFIG. 6. Effect of electron irradiation on the magnetoresistanceof Sr2RuO4. (a) Normalized magnetoresistance (MR) plotted versusthe externally applied magnetic field for an as-grown Sr2RuO4 thin-film resistivity bridge (blue), and the same sample after receivinga 0.60 C/cm2 dose of 2.5 MeV electron irradiation (teal). Here wedefine MR [%] ≡ 100 × [ρ(H ) − ρ(0)]/ρ(0), where ρ(0) is the ex-trapolated resistivity as H → 0 in the absence of superconductivity.Currents of 100 µA, rather than the usual 10 µA, were used in theacquisition of the data shown in this figure. (b), (c) Zoomed-in viewsof the resistivity measured over the high-field range indicated bythe dashed box in panel (a), after subtracting and normalizing toempirically determined quadratic polynomial backgrounds ρbg(H )for both data sets: ρ̃/ρ0 ≡ [ρ(H ) − ρbg(H )]/ρbg(H ). Superimposedon these slowly and smoothly varying backgrounds are rapidly os-cillating Shubnikov-de Haas contributions to the magnetoresistance,that are periodic in inverse field and have dominant frequencies Fαcharacteristic of charge carriers completing phase-coherent cyclotronmotion around the α-sheet Fermi surface of Sr2RuO4 [cf. upper leftinset to panel (a)].Figs. 6(b) and 6(c), respectively. Both ρ̃/ρ0 traces clearlyexhibit SdH oscillations that are periodic in inverse field.The dominant oscillation frequency in the data collectedat the temperatures and fields shown here is Fα , which resultsfrom charge carriers completing phase-coherent cyclotronorbits around the α-Fermi surface of Sr2RuO4. Countingpeaks and troughs in Figs. 6(b) and 6(c), we measureN = 37 complete oscillation periods over the field rangeμ0H = 11.880 to 13.964 Tesla for both the as-grown and theelectron-irradiated samples, corresponding to oscillation033178-11JACOB P. RUF et al. PHYSICAL REVIEW RESEARCH 6, 033178 (2024)frequencies of Fα = N/(1/11.880 T − 1/13.964 T) =2945 ± 20 Tesla for both samples. In other words, moderatee− doses do not modify Fα , within the experimental resolutionof approximately 1%. We estimated the systematic error barsquoted here on Fα by our ability to detect ±1/4 of anoscillation period of the α-sheet-derived SdH signal over the(inverse) magnetic field range probed in Figs. 6(b) and 6(c).In the inset to Fig. 6(a), we schematically sketch the well-known {α, β, γ } Fermi surfaces of Sr2RuO4, projected ontothe two-dimensional first Brillouin zone, which is bounded bysolid black lines. The area Ak enclosed by each Fermi surfacein k-space follows directly from the corresponding primaryquantum oscillation frequency Fi (i ∈ {α, β, γ }), accordingto the Onsager relation Ak,i = (2πe/h̄)Fi. Therefore, the in-variance of Fα noted above implies that electron irradiationdoes not alter the k-space area enclosed by the α-sheet Fermisurface, i.e., the gray-shaded region drawn in the inset toFig. 6(a). Invoking Luttinger’s theorem, this observation alsoimplies that moderate doses of electron irradiation do not alterthe itinerant carrier density in Sr2RuO4.APPENDIX C: ESTIMATING CLEAN-LIMITTc0 FOR THIN FILMSIn the discussion surrounding Fig. 3 of the main text, wenoted that in Sr2RuO4 thin films, Tc responds sensitively torelative changes in the residual resistivity induced by elec-tron irradiation, �ρ0, irr, yet Tc appears to depend much lessregularly on ρ0 when all residual resistivities for as-grownand irradiated thin-film samples are plotted in absolute units.Figure 7 illustrates both of these observations graphically. All18 (ρ0, Tc) red square markers for Sr2RuO4 thin films plottedin Fig. 7 are extracted in absolute units from the correspond-ing ρ(T ) data traces plotted in Fig. S4 of the SupplementalMaterial [24]. To serve as side-by-side references, the same(ρ0, Tc) gray points and purple squares for Sr2RuO4 bulksingle crystals from Fig. 3 of the main text are also includedin Fig. 7.Viewed merely as a collection of independent points, the Tcversus ρ0 behavior of the thin-film data in Fig. 7 appears ratherscattered, and it is difficult to establish any clear correlationbetween these two variables. By contrast, when one drawslines between each pair of (ρ0, Tc) points that correspond tothe same physical sample, measured before and after irradia-tion, visual inspection of the solid lines in Fig. 7 reveals thatelectron irradiation has very similar, proportional effects on�Tc and �ρ0, irr across all distinct Sr2RuO4 thin-film samples.Furthermore, as emphasized in the main text, collecting theseindependent measurements of �Tc/�ρ0 into a single numberfor the rate of Tc suppression with increasing quasiparticlescattering rate results in a number that is quantitatively verysimilar between these electron-irradiated Sr2RuO4 thin-filmsamples and single-crystal Sr2RuO4 samples containing vari-ous types of defects (Table II).In Fig. 3 of the main text, we proposed that the mostnatural interpretation and framing of the raw Tc(ρ0) data forthin films can be summarized as follows: (i) the quantitativesimilarity of Tc across all as-grown thin-film samples (1.35to 1.41 K) implies that their h̄/τpb values are nearly identical;(ii) based on the detailed comparisons between Sr2RuO4 thinFIG. 7. Tc versus absolute ρ0 for Sr2RuO4 single crystals and epi-taxial thin films. For Sr2RuO4 single crystals, purple square markersfrom the electron-irradiation experiments detailed in this work andgray circular markers extracted from impurity-scattering studies inthe literature [3,5,6] are reproduced from Fig. 3 of the main text. ForSr2RuO4 thin films, the red square markers plotted here are obtaineddirectly from ρ(T ), without the shift in the horizontal axis used inFig. 3. Red lines connect pairs of (ρ0, Tc ) points measured beforeand after 2.5 MeV electron irradiation for the same physical thin-filmsample: solid lines for irradiated samples having measurable Tc, anddashed lines terminating at Tc = 0.2 K for irradiated samples havingTc below the base temperature of our cryostat, 0.4 K. The two-pointslopes suggested by this pairwise grouping of the data are remark-ably consistent, despite differing “initial conditions” (i.e., differingabsolute ρ0 values); moreover, the �Tc/�ρ0 Cooper-pair-breakingeffect induced by irradiation is at least 10 times larger in magnitudethan whatever subtle trend may be discernible in the Tc(ρ0) behaviorobserved across distinct as-grown thin-film samples, which are allencircled by a red oval.films and single crystals presented in the main text, usingelectron irradiation is a valid way to produce pure (quasielas-tic and quasi-isotropic) Cooper-pair-breaking scattering,and thereby suppress Tc at an initial rate of ≈1.2 K/µ� cm;thus, (iii) the remaining outstanding question is to determinethe relative rigid offset ρ0, offset of these as-grown thin-filmsamples relative to the zero point of an appropriately definedh̄/τpb Cooper-pair-breaking energy scale. Equivalently, sinceour electron irradiation studies have now precisely establishedthe magnitude of Tc suppression for a given change in ρ0that also proportionally increments h̄/τpb, determining ρ0, offsetrequires knowledge of the clean-limit Tc0 ≡ Tc(h̄/τpb → 0) ofSr2RuO4/NdGaO3(110).At present, the simplest-possible empirical estimates con-strain ρ0, offset to be within the following extreme limits: (i)ρ0, offset = 0, in which case none of the experimentally mea-sured ρ0 in as-grown thin-film samples actually contributes toh̄/τpb (i.e., Tc0 of Sr2RuO4/NdGaO3(110) is equal to the val-ues measured in this work on as-grown samples, ≈1.4 K), and(ii) ρ0, offset = 1.1 µ� cm, which is equal to the experimen-tally measured ρ0 for as-grown Tc ≈ 1.4 K thin-film samplesexhibiting the lowest absolute ρ0s in the present work [24].In this case, the hypothetical Tc0 of Sr2RuO4/NdGaO3(110)would equal 1.4 K + (1.1 µ� cm × 1.2 K/µ� cm) ≈ 2.7 K.033178-12CONTROLLABLE SUPPRESSION OF THE … PHYSICAL REVIEW RESEARCH 6, 033178 (2024)FIG. 8. Abrikosov-Gor’kov pair-breaking theory simulations ofTc(ρ0) for bulk Sr2RuO4. Purple square markers are data collected inthis work for as-grown and electron-irradiated FIB-sculpted Sr2RuO4single crystals, reproduced from Fig. 3 of the main text. The orangecurves and shaded region are simulations obtained by solving the im-plicit AG-like equation for Tc as a function of ρ0, varying {Tc0, h̄ωpl}while keeping χ = 1 fixed; for quantitative values of the parametersemployed here, see the text of Appendix D.In Fig. 3 of the main text, we arbitrarily chose a rigid shiftof the top axis by ρ0, offset = 0.5 µ� cm relative to the bottomaxis, about halfway between these two extremes, correspond-ing to a Sr2RuO4/NdGaO3(110) Tc0 ≈ 2.0 K.Based on the method proposed here of interpreting thethin-film data, it would be possible to further narrow downthe range of potential Tc0 values for Sr2RuO4/NdGaO3(110)in the future either by synthesizing such Sr2RuO4 thin-filmsamples with appreciably higher as-grown Tc > 1.4 K, orby maintaining “high” Tc ≈ 1.4 K in as-grown samples withappreciably lower as-grown ρ0 < 1 µ� cm. While some lim-ited evidence for the former exists in the literature, e.g.,Ref. [19] reported a broad (about 0.4 K-wide) resistivelymeasured superconducting transition centered at a midpointTc = 1.8 K in a 55-nm-thick Sr2RuO4/NdGaO3(110) sam-ple, to the best of our knowledge, there are no examples inthe literature of Sr2RuO4 thin-film samples synthesized onany perovskite-based substrate that exhibit the latter property.The latter observation suggests that the levels of ρ0 currentlyachieved are pushing up against a fundamental length scale forcharge-carrier momentum-relaxing scattering from extendedstructural defects in heteroepitaxial thin film/substrate mate-rial platforms [10].APPENDIX D: COMPARISONTO ABRIKOSOV-GOR’KOV THEORYIn Fig. 8, we show that the results of our electron irradiationexperiments on FIB-sculpted bulk single crystals of Sr2RuO4can be described well by theoretical Tc(ρ0) curves calculatedaccording to the standard Abrikosov-Gor’kov (AG) Cooper-pair-breaking theory of Tc suppression [61], applied to thecase of nonmagnetic point-defect scattering in an effectivelysingle-band unconventional superconductor [62–66].In such theories, Tc is found by solving an implicit equa-tion of the general form (see, e.g., Eq. 16 of Ref. [63] or Eq. 35of Ref. [66]):ln(Tc0Tc)= χ[�(12+ h̄/τ4πkBTc)− �(12)], (D1)where � is the digamma function, h̄ is Planck’s constant, 1/τis the quasiparticle scattering rate, kB is Boltzmann’s constant,Tc0 is the so-called “clean-limit” superconducting transitiontemperature in the disorder-free limit h̄/τ → 0, andχ = 1 − 〈�(k)〉2FS〈�2(k)〉FS(D2)is a constant that describes the anisotropy of the supercon-ducting order parameter (SCOP), 〈averaged〉 over the Fermisurface (FS). For specific limiting examples of the latter,a purely sign-changing (e.g., “d-wave”) SCOP that aver-ages to zero over the Brillouin zone (〈�(k)〉FS = 0) hasχ = 1, whereas a purely uniform/isotropic (�(k) = constant,“s-wave”) SCOP has χ = 0.Although more complicated relationships are often ob-served in real materials, in theory the single-particle scatteringrate that parameterizes the Cooper-pair-breaking energy scalein Eq. (D1) is typically assumed to be directly proportionalto the experimentally measured planar residual resistivity ρ0,1/τ = ρ0(ω2pl/4π ), where h̄ωpl is a suitably averaged char-acteristic plasma frequency describing in-plane motion of thecharge carriers [64,66].The quantitative parameters that describe the range oftheoretical Tc(ρ0) curves for bulk Sr2RuO4 drawn in Fig. 8are χ = 1, Tc0 = 1.50 K to 1.60 K, and h̄ωpl = 1.40 eV to1.34 eV. Taken together, these parameter choices determinecritical scattering rates ρcrit.0 = 0.86 µ� cm to 1.00 µ� cm.For ρ0 > ρcrit.0 , only the trivial solution Tc = 0 exists to theabove AG-like equation with χ = 1.We consider the agreement observed here between theoryand experiment to be essentially phenomenological in nature;there are too many outstanding unknowns about the supercon-ductivity of Sr2RuO4 for this type of effective description tobe interpreted too literally. In the regime of “small” ρ0 <<ρcrit.0 where many of the experimental (ρ0, Tc) data pointswe have collected exist, there are redundant dependenciesof the initial rate of Tc suppression [62,63,65,66], both inhow increases in ρ0 transduce into increases in h̄/τ (i.e., onthe effective ωpl ) [64,66], and in how the fully k-resolvedSCOP of a multiorbital system such as Sr2RuO4 reduces toan effectively single-band SCOP anisotropy χ [67,68]. Nev-ertheless, while the solutions presented here are by no meansuniquely determined, the overall phenomenological Tc(ρ0)curves shown in Fig. 8 may still prove useful as initial guessesfor modeling strain-induced changes to Tc0 across differentthin-film variants of Sr2RuO4, as discussed in more detailbelow.In principle, the systematic deviations from purely linearTc(ρ0) behavior expected in AG pairbreaking theory shouldallow one to infer the proximity of a given set of (ρ0, Tc)data points to the critical ρcrit.0 at which superconductivity issuppressed completely. This method of discerning ρcrit.0 in turnallows for an absolute determination of Tc0, because increases033178-13JACOB P. RUF et al. PHYSICAL REVIEW RESEARCH 6, 033178 (2024)FIG. 9. Abrikosov-Gor’kov pair-breaking theory simulations ofTc(ρ0) for Sr2RuO4 thin films. Square markers are data col-lected in this work for as-grown and electron-irradiated 28-nmSr2RuO4/NdGaO3(110) thin-film samples. Differently colored setsof markers are the same underlying Tc versus �ρ0, irr data set re-produced from Fig. 3 of the main text, plotted with different adhoc rigid shifts of the horizontal axis, ρ0, offset, as indicated in theinset legend. Solid color-coded lines are simulations obtained bysolving the implicit AG-like equation for Tc as a function of theCooper-pair-breaking energy scale h̄/τpb ∝ ρ0 ≡ ρ0, offset + �ρ0, irr,varying Tc0 while keeping both h̄ωpl = 1.29 eV and χ = 1 fixed.in Tc0 cause linear increases in ρcrit.0 in simple Cooper-pair-breaking theories that are effectively single-band, with onedominant superconducting pairing scale, |�max.|.To apply this type of reasoning to our Sr2RuO4 thin-filmelectron-irradiation data, we started with a phenomenologi-cal AG-like Tc(ρ0) curve similar to those that describe wellthe (ρ0, Tc) data points we collected for electron-irradiatedSr2RuO4 FIB-sculpted single crystals in this work, Fig. 8.To optimize the general agreement of these calculated Tc(ρ0)curves with the experimental thin-film (ρ0, Tc) data points,we adjusted the effective plasma frequency h̄ωpl = 1.37 eVused for the bulk Sr2RuO4 AG simulations in Fig. 8 toh̄ωpl = 1.29 eV for the thin-film AG simulations in Fig. 9.The effective SCOP anisotropy χ = 1 was held at the samefixed value as in the case of bulk samples, although as westressed earlier, within AG-like pair-breaking theories thereare redundant dependencies of the calculated Tc(ρ0) behav-ior on {χ, ωpl}, so other solutions would be possible thatwould fit the existing data equally well. Finally, we assumedthat any epitaxial-strain-induced increase (or decrease) inTc0 for Sr2RuO4/NdGaO3(110) relative to the Tc0 of bulkSr2RuO4 results from a k-independent multiplicative en-hancement (or suppression) of |�(k)|, along with negligiblestrain-dependent changes in the Fermi surface (cf. Fig. 6).Under this assumption, we uniformly scaled up (or down)the phenomenological AG-like Tc(ρ0) curve based on varioushypothetical Tc0 values, keeping h̄ωpl = 1.29 eV constant, toproduce simulated Tc(ρ0) curves against which to compare theSr2RuO4 thin-film (ρ0, Tc) data points.Figure 9 illustrates the results of this type of analysis. Forhypothetical Tc0 curves towards the high end of our allowedrange of Tc0, e.g., for the blue simulated curve with Tc0 =2.6 K (ρcrit.0 = 1.77 µ� cm), the two irradiated thin-film sam-ples with Tc = 0.63 K and 0.68 K (which both received a2.5 MeV e− dose of 0.6 C/cm2) would be small enough frac-tions of Tc0 (i.e., Tc/Tc0 < 0.3) that they would be well withinthe range of a given AG-like pair-breaking-theory curve wherenonlinearities in the Tc(ρ0) behavior would become obvious.The absence of any such nonlinear features in the experimen-tal data suggests that Tc0 for Sr2RuO4/NdGaO3(110) is likelyless than 2.2 K, provided that our assumption of χ ≈ 1 is, infact, realized.However, the red simulated Tc(ρ0) curve having Tc0 =2.0 K (which combined with {χ = 1, h̄ωpl = 1.29 eV} re-sults in ρcrit.0 = 1.36 µ� cm) matches the red experimentaldata squares (plotted with an ad hoc ρ0, offset = 0.57 µ� cm)quite well. 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