# Fileset [2312.05130v2.pdf](https://mdr.nims.go.jp/filesets/29a227b0-d89c-49d5-9b33-24ed94a0a594/download) ## Creator [Eli Mueller](https://orcid.org/0009-0002-7351-3926), Yusuke Iguchi, [Fabian Jerzembeck](https://orcid.org/0000-0003-0003-8140), Jorge O. Rodriguez, [Marisa Romanelli](https://orcid.org/0000-0002-1745-2332), Edgar Abarca-Morales, Anastasios Markou, [Naoki Kikugawa](https://orcid.org/0000-0003-3975-4478), Dmitry A. Sokolov, Gwansuk Oh, Clifford W. Hicks, Andrew P. Mackenzie, Yoshiteru Maeno, Vidya Madhavan, [Kathryn A. Moler](https://orcid.org/0000-0002-0233-8123) ## Rights ## Other metadata [Superconducting penetration depth through a Van Hove singularity:

{Sr}_{2} {RuO}_{4}

under uniaxial stress](https://mdr.nims.go.jp/datasets/57d5f29c-75c3-4946-ad69-f10bcc116f7b) ## Fulltext Superconducting Penetration Depth Through a Van Hove Singularity: Sr2RuO4 Under UniaxialStressEli Mueller,1, 2 Yusuke Iguchi,1, 3 Fabian Jerzembeck,4 Jorge O. Rodriguez,5 Marisa Romanelli,5 EdgarAbarca-Morales,4 Anastasios Markou,4, 6 Naoki Kikugawa,7 Dmitry A. Sokolov,4 Gwansuk Oh,8, 9 CliffordW. Hicks,4, 10 Andrew P. Mackenzie,4, 11 Yoshiteru Maeno,8, 12 Vidya Madhavan,5 and Kathryn A. Moler1, 2, 31Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory,2575 Sand Hill Road, Menlo Park, California 94025, USA2Department of Physics, Stanford University, Stanford, California 94305, USA3Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA4Max Planck Institute for the Chemical Physics of Solids, Nöthnitzer Straße 40, Dresden 01187, Germany5Department of Physics, University of Illinois, Urbana, Illinois 61801, USA6Department of Physics, University of Ioannina, 45110 Ioannina, Greece7National Institute for Materials Science, Tsukuba, Ibaraki 305-0003, Japan8Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan9Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Republic of Korea10School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom11Scottish Universities Physics Alliance, School of Physics and Astronomy,University of St. Andrews,St. Andrews KY16 9SS, United Kingdom12Toyota Riken - Kyoto University Research Center (TRiKUC), Kyoto University, Kyoto 606-8501, JapanIn the unconventional superconductor Sr2RuO4, uniaxial stress along the [100] direction tunes the Fermilevel through a Van Hove singularity (VHS) in the density of states, causing a strong enhancement of thesuperconducting critical temperature Tc. Here, we report measurements of the London penetration depth λ asthis tuning is performed. We find that the zero-temperature superfluid density, here defined as λ (0)−2, increasesby ∼15%, with a peak that coincides with the peak in Tc. We also find that the low temperature form of λ (T )is quadratic over the entire strain range. Using scanning tunneling microscopy, we find that the gap increasesfrom ∆0 ≈ 350 µeV in unstressed Sr2RuO4 to ∆0 ≈ 600 µeV in a sample strained to near the peak in Tc. Witha nodal order parameter, an increase in the superconducting gap could bring about an increase in the superfluiddensity through reduced sensitivity to defects and through reduced non-local effects in the Meissner screening.Our data indicate that tuning to the VHS increases the gap throughout the Brillouin zone, and that non-localeffects are likely more important than reduced scattering.Strontium ruthenate (Sr2RuO4) is a highly studied sys-tem in the field of unconventional superconductivity [1–8].The metallic normal state from which superconductivity con-denses is one of the simplest and best understood amongunconventional superconductors [9], but after nearly threedecades of strenuous effort, a complete understanding of thesuperconducting order parameter remains elusive.The low-temperature behavior of the London penetrationdepth λ (T ) provides information on the existence of nodes inthe superconducting gap ∆(k). The majority of experimentalevidence suggests the presence of line nodes [10–15]. In thelimit of large κ = λ/ξ , where λ and ξ are the penetrationdepth and coherence length respectively, it is well-known thatan order parameter with line nodes yields a T -linear depen-dence in the low-temperature form of λ (T ) for clean samples,and a T 2 dependence in the presence of strong impurity scat-tering [16]. Measurements of λ (T ) in Sr2RuO4 have repeat-edly shown a T 2 form [17–19]. However, it would be oddfor the effects of impurity scattering to be strong in Sr2RuO4because it is one of the cleanest correlated electron materialsknown, with the best available samples having a mean freepath l exceeding 1 µm [20]. Alternatively, it was suggestedin Ref. [17] that a λ (T ) ∼ T 2 dependence may be a result ofnon-local electrodynamic effects [21] because the supercon-ductivity in Sr2RuO4 is only marginally in the local limit withκ ≈ 2.9 at low temperature [17].FIG. 1. Quasi-two-dimensional Fermi surface calculations ofSr2RuO4 under representative εxx strains [22]. (a) Fermi surface un-der zero strain, showing the two electron-like bands γ and β and thehole-like band α . Points labeled X and Y indicate high-symmetrypoints. (b) Fermi surface with the system strain-tuned to the Lifshitztransition. The VHS occurs on the γ band at the Y points. (c) Fermisurface for the system strained beyond the Lifshitz transition.The advent of strain-tuning presents a new frontier forstudying λ (T ). Uniaxial stress along the [100] direction drivesthe γ Fermi surface sheet through a Lifshitz transition (seeFig. 1 and Ref. [23]) and an associated Van Hove singularity(VHS) in the Fermi level density of states. In addition, thesuperconducting critical temperature Tc undergoes a dramaticenhancement from an unstressed value of Tc0 = 1.5 K to apeak of 3.5 K [22, 24] at the VHS which is accompanied by a20-fold increase in the out-of-plane critical field Hc2 [22, 25].arXiv:2312.05130v2 [cond-mat.supr-con] 11 Dec 20232FIG. 2. Local Tc maps of (a) sample 1 and (b) sample 2. Markersindicate the positions at which the temperature and strain dependenceof the penetration depth was measured.In a weak-coupling picture, the enhanced superconductivityunder strain is most intuitively explained by the increased den-sity of states around the VHS [22, 26].There are a few possible hypotheses for how the T → 0superfluid density λ (0)−2 might change as the Lifshitz tran-sition is traversed. Theoretical work [27] suggests that thequasiparticle mass renormalization on the γ sheet is a conse-quence of the proximity of the Fermi level to the VHS. Tun-ing to the VHS could cause a further increase in the effectivemass m∗, thereby decreasing the superfluid density throughthe relation λ−2 = 4πnse2/m∗c2, where ns is the number den-sity of superconducting electrons. Alternatively, an increasein the superconducting gap ∆0 could reduce the effects of im-purities and of non-locality in the Meissner screening, lead-ing to an increase in the superfluid density. If the coherencelength is strongly reduced by tuning to the Lifshitz transition,then it is also possible that the T 2 form of λ (T )−λ (0) in un-stressed Sr2RuO4 would become T -linear. We note that a sub-set of the present authors previously reported scanning super-conducting quantum interference device (SQUID) measure-ments of the penetration depth on uniaxially stressed Sr2RuO4[28]. In those measurements, the focus was to test for ananomaly in the superfluid density associated with a transitionbelow Tc [29] and the maximum applied strain ε was wellbelow the Lifshitz transition. In this letter, we report scan-ning SQUID measurements of the strain- and temperature-dependent changes of the penetration depth λ (ε,T )−λ (0,0)in Sr2RuO4 across the Lifshitz transition. To aid interpretationof these data, we also report measurements by scanning tun-neling microscopy (STM) of ∆0 on Sr2RuO4 strained to nearthe Lifshitz transition.The main components of our SQUID susceptometers [30]include a flux-sensitive pickup loop with an inner radius rPLof 1 µm and a concentric field coil with an inner radius rFC of2.5 µm. A low-frequency (893 Hz) alternating current is ap-plied to the field coil, which generates a local field that couplesto the SQUID pickup loop. Near a superconducting sample,the Meissner response screens the field from the field coil andreduces the mutual inductance M of the pickup-loop/field-coilpair, which we measure in units of Φ0/A, where Φ0 = h/2e isthe flux quantum. Recording M as a function of temperatureallows us to measure the local Tc by measuring the onset ofdiamagnetism (M < 0). In the regime where the penetrationFIG. 3. (a) and (b) Temperature and strain dependence of the changein penetration depth λ (ε,T )−λ (0,0) for strains (a) less than εc and(b) greater than εc for sample 2. Solid lines are T 2 fits to the data.We take λ (0,0) to be the T = 0 K intercept of a λ (ε,T ) ∼ T 2 fit tothe -0.06% strain dataset. (c) Strain dependence of local Tc measuredon sample 1 (red) and 2 (black). (d) Strain dependence of the zero-temperature superfluid density λ (ε,0)−2 for sample 1 (red) and sam-ple 2 (black). For each dataset, the zero-temperature superfluid den-sity at the minimum strain is set to λ−2 = 27.7 µm−2 (λ = 190 nm[17]) and we estimate changes in λ (ε,0)−2 by extrapolating the T 2fits from (a) and (b) to T = 0 K.depth is smaller than rFC and the height of the field coil fromthe sample surface z0, the measured M can be used to estimatethe quantity λ (ε,T )+ z0 [31]. In our measurement, we fix z0by bringing the SQUID sensor into light mechanical contactwith the sample surface to prevent drift of the SQUID so thatchanges in M are due to changes in λ (ε,T ).For the penetration depth measurements, in situ tunablestrain was applied using a piezoelectric-driven device similar3to that described in previous reports [32, 33]. The device con-tains a capacitive sensor of applied displacement; to convertto strain, we set the strain to be zero at the minimum in Tc andto εc =−0.44% at the peak in Tc [34]. To check reproducibil-ity, we measured two samples cut from separate batches intobars ≈2 mm in length. Maps of local Tc were obtained nearzero strain by taking spatial scans of M at a series of tempera-tures near the bulk Tc [35]. Sample 1 [Fig. 2(a)] shows a loweroverall Tc of 1.36–1.4 K with a few small pockets of local Tcvalues as high as 1.5 K. Sample 2 [Fig. 2(b)] is more homoge-neous and shows higher overall Tc values of 1.51–1.54 K. Theapparent stripe pattern in the local Tc map of sample 2 is mostlikely due to periodic changes in the solidification conditionduring sample growth [36][37].Figures 3(a) and (b), show the temperature and strain de-pendence of the change in penetration depth λ (ε,T )−λ (0,0)plotted against (T/Tc)2. For fixed T/Tc, the curves show aclear reduction in penetration depth with increasing strain be-low εc [Fig. 3(a)] and a subsequent increase in penetrationdepth for strains beyond εc [Fig. 3(b)]. The T 2 fits (solidlines) agree well with the data for T < 0.5Tc over the entirestrain range.In Fig. 3(c), the Tc at two selected points is plotted againstapplied strain. The local Tc for each value of strain was de-termined from M(T ) by the temperature at which the maxi-mum in dM/dT occurs [35]. The Tc versus strain curves forsample 1 and sample 2 approximately track each other; how-ever, the difference between Tc of the two samples shrinksas Tc increases, consistent with the general expectation thathigher Tc corresponds to reduced sensitivity to defect scat-tering. Figure 3(d) shows the strain dependence of the zero-temperature superfluid density λ (ε,0)−2. Under increasingcompression, λ (ε,0)−2 increases monotonically up to εc,where it reaches an approximately 15% enhancement, thendecreases for strains beyond εc.For the STM measurements, due to the restricted samplespace, strain was applied using a differential thermal expan-sion cell as in Ref. [23]. We estimate that the applied strainin the sample was 0.4±0.1% using digital image correlationof the sample platform at low temperature; additional detailsare given in Ref. [35]. Data from the STM measurements areshown in Fig. 4. As in previous works [38–42], we do not ob-serve a superconducting gap on the top strontium oxide (SrO)surface, possibly because of the surface reconstruction[43–45]. However, we do observe a superconducting gap in-side several nanometer-scale trenches, most probably createdwhen the sample was cleaved. In these trenches both the sur-face SrO layer and the RuO2 layer directly below it are re-moved, thereby exposing the lower SrO layer which providesa window into the bulk properties [see Fig. 4(a)].Figures 4(b) and (c) show typical topographies as wellas spectroscopic linecutes [Fig. 4(d) and Fig. 4(e)] in thesetrenches in an unstrained and strained sample, respectively.In the unstrained sample [Fig. 4(d)], the spectra inside thetrench show a well-defined gap with 2∆0 equal to approxi-mately 700 µV, as denoted by the dashed lines marking theposition of the coherence peaks. We note that this value of2∆0 is consistent with previous measurements on unstrainedFIG. 4. Superconducting gap under the surface SrO layer inSr2RuO4. (a) Schematic diagram of the crystal structure of Sr2RuO4showing subsurface layers (b) Topographic image of a defect (darkblue area) on an unstrained sample exposing the RuO2 layer at adepth of ≈ 2.1 Å (c) Topographic image of a defect (green area) on astrained sample exposing the lower SrO layer at a depth of ≈ 3.4 Å.See Ref. [35] for details for determining the depth of the cleavage de-fects. The topographic depth indicated by the colorbar is measuredrelative to the surface SrO layer (light blue area) (d) Waterfall plotof dI/dV spectroscopy data measured at the points indicated in (b).(e) Waterfall plot of dI/dV spectroscopy data measured at the pointsindicated in (c). The superconducting gap is determined from thedistance between the pair of dashed lines in panels (d) and (e). a.u.:arbitrary units.Sr2RuO4 [46, 47]. In contrast, as illustrated by the dashedlines in Fig. 4(e), on the strained sample, we find a muchlarger value for 2∆0 of approximately 1.2 mV. We note thatthe gap measured by STM is likely to be dominated by the αand β sheets. As discussed in Ref. [46], the α and β sheetsare dominated by xz and yz orbital weight and the tunnelingconductance is likely to be much higher for the xz and yz or-bitals than the xy orbital due to their greater extent along the z4FIG. 5. Superfluid density enhancement at εc plotted against the scat-tering rate Γ determined from the unstrained Tc as described in thetext and normalized by kBTc0. The labels at each marker correspondto the location on the samples indicated in Fig. 2(a) and (b). Point7 was measured outside the scan range of Fig. 2(b). The error barsindicate the range of values observed over several measurement cy-cles and the data points represent the average value. The dotted lineshows the prediction from reduced sensitivity to impurities, and thedashed line from reduced non-locality, both under an assumption that∆0 increases in proportion to Tc.axis.Importantly for discussion of the superfluid density, thegap on the α and β sheets is well away from the Van Hovepoint in k-space (points labeled Y in Fig. 1); due to the lowFermi velocity, carriers near the Van Hove point will not con-tribute strongly to the superfluid density. We therefore restrictour analysis of the superfluid density results to regions of theFermi surface away from the Van Hove point. An enhancedm∗ should increase λ (0), thereby reducing the superfluid den-sity. In contrast, our results show an increase in superfluiddensity, suggesting that any enhancement in m∗ near the Lif-shitz transition is too small to have a dominant effect. Angle-resolved photoemission spectroscopy (ARPES) data show thatthere is no substantial change in band renormalization as theγ sheet is tuned to the Lifshitz transition by substitution [48],and ARPES data under uniaxial stress show, similarly, no dra-matic change [23]. Therefore, we suppose that m∗ on sectionsof the Fermi surface away from the Van Hove point remainsapproximately constant as stress is applied.We suppose further that away from the Van Hove point,the gap increases in proportion to Tc (the gap is likely sub-stantial at the Van Hove point [49], but the superfluid den-sity is not sensitive to this due to the low Fermi velocity).For a nodal superconductor, an enhanced gap can lead to anincreased superfluid density in two ways: (1) reduced sen-sitivity to impurity scattering and (2) reduced non-local ef-fects in the Meissner screening. For the case of impuritiesin a d-wave superconductor [16], it was shown that λ (0) in-creases from its value in the clean, local limit λ0 by an amountthat depends on the magnitude of the superconducting gap:λ (0)−λ0 ∼ λ0√Γ/∆0, where Γ is the nonmagnetic impurityscattering rate. Studies of Tc on samples with varying levelsof impurities [50–53] show a suppression of Tc with increas-ing impurity concentration that follows an Abrikosov-Gorkov(AG) relation, ln(Tc0/Tc) = ψ(1/2 + Γ/2πkBTc)− ψ(1/2)[54, 55], where ψ is the digamma function and Tc0 is theclean-limit Tc. Therefore, taking Tc as a measure of Γ, wewould expect a greater enhancement of the superfluid den-sity at the Lifshitz transition for samples that have lower un-stressed Tc.In Fig. 5, we show the increase in superfluid density at εcfor each point measured on the samples (see Fig. 2) plottedagainst Γ/kBTc0, where Γ is determined from the AG rela-tion applied to Tc(ε = 0). Measurements on sample 1, whichhas lower Tc(ε = 0), indeed show a larger increase in super-fluid density at the Lifshitz transition. However, the trend ofthe data does not match the expectation from impurity effectsalone: the increase in superfluid density in sample 2 remainslarge, even though its Tc(ε = 0) is very close to the clean-limitvalue.At sufficiently low temperature, non-local effects in theMeissner screening are expected to become important in allnodal superconductors because the coherence length, definedas ξ (k) = h̄vF(k)/π∆(k), diverges along the nodal directionssuch that ξ (k) > λ0 is satisfied near the nodes [21]. Simi-lar to the effect of impurities, the low-temperature behaviourof λ (T ) becomes quadratic and λ (0) increases from λ0. Ina d-wave superconductor [21], the correction to λ (0) can beestimated as λ (0)−λ0 = λ0π√2/16κ . Taking λ0 = 190 nmand a zero-temperature coherence length of ξ0 =66 nm [2],the non-local correction to λ (0) is approximately 10% of λ0in unstressed Sr2RuO4. Tc increases by a factor of 2.3 betweenε = 0 and the Lifshitz transition. If we take κ to increase bythis same factor, we find that λ (0) decreases by ≈ 10 nm;λ (0)−2 increases by ≈ 3 µm−2. This increase is shown bythe dashed line in Fig. 5 and is consistent with the superfluidenhancement observed on sample 2. We note that the twenty-fold increase in Hc2 [22, 25] indicates a greater decrease in ξ0by a factor of√20. However, the highly anisotropic electronicstructure of Sr2RuO4 tuned to the Lifshitz transition should berecalled: this increase in Hc2 could be driven by carriers in theimmediate vicinity of the Van Hove point. The value of thepenetration depth measurements is that they provide informa-tion on how portions of the Fermi surface away from the VanHove point, where the Fermi velocity is larger, respond to thetuning to the Lifshitz transition.In summary, the zero-temperature superfluid density inSr2RuO4 undergoes a ∼15% enhancement as the systemis strain-tuned through the VHS, and the penetration depthshows a λ (T )∼ T 2 dependence throughout the Lifshitz tran-sition. In addition, the superconducting gap increases withstrain going from ∆0 ≈ 350 µeV in an unstrained sample to∆0 ≈ 600 µeV in a sample strained to near the Lifshitz transi-tion. Our analysis indicates that the increase in superfluid den-sity with tuning to the Lifshitz transition is primarily driven bya decrease in non-local effects in the Meissner screening. Wenote that Ref. [56] reports measurements of Hc1 in unstressedSr2RuO4 that also highlight the importance of non-local ef-fects. Both the STM and the penetration depth measurements,which are sensitive primarily to regions of Fermi surface awayfrom the Van Hove point, show that tuning to the Lifshitz tran-5sition increases the gap throughout the Brillouin zone, an ob-servation that might be a useful clue on the pairing mechanismin Sr2RuO4.This work was supported by the Department of Energy,Office of Basic Energy Sciences, Division of Materials Sci-ences and Engineering, under contract DE-AC02-76SF00515.N. K. is supported by JSPS KAKENHI (No. JP18K04715,No. JP21H01033, and No. JP22K19093). YM wassupported by JSPS KAKENHI (Nos. JP15H05851 andJP22H01168). GSO was supported by JSPS KAKENHI (Nos.JP15H05851, JP15K21717) during his stay at Kyoto Univer-sity. We thank Gwansuk Oh for his contribution to the crys-tal growth. F.J., A.P.M. and C.W.H. acknowledge the sup-port of the Max Planck Society and the German ResearchFoundation (TRR288-422213477 ELASTO-Q-MAT, ProjectA10). 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Gor’kov, Contribution to the theoryof superconducting alloys with paramagnetic impurities, Zhur.Eksptl’. i Teoret. Fiz. 39 (1960).[55] P. Hirschfeld, P. Wölfle, and D. Einzel, Consequences of reso-nant impurity scattering in anisotropic superconductors: Ther-mal and spin relaxation properties, Phys. Rev. B 37, 83 (1988).[56] J. F. Landaeta, K. Semeniuk, J. Aretz, K. Shirer, D. A. Sokolov,N. Kikugawa, Y. Maeno, I. Bonalde, J. Schmalian, A. P.Mackenzie, and E. Hassinger, Evidence for vertical line nodes7in Sr2RuO4 from nonlocal electrodynamics, arXiv preprint arXiv:2312.05129 (2023).Supplementary Materials for: Superconducting Penetration Depth Through a Van HoveSingularity: Sr2RuO4 Under Uniaxial StressEli Mueller,1, 2 Yusuke Iguchi,1, 3 Fabian Jerzembeck,4 Jorge O. Rodriguez,5 Marisa Romanelli,5 EdgarAbarca-Morales,4 Anastasios Markou,4, 6 Naoki Kikugawa,7 Dmitry A. Sokolov,4 Gwansuk Oh,8, 9 CliffordW. Hicks,4, 10 Andrew P. Mackenzie,4, 11 Yoshiteru Maeno,8, 12 Vidya Madhavan,5 and Kathryn A. Moler1, 2, 31Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory,2575 Sand Hill Road, Menlo Park, California 94025, USA2Department of Physics, Stanford University, Stanford, California 94305, USA3Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA4Max Planck Institute for the Chemical Physics of Solids, Nöthnitzer Straße 40, Dresden 01187, Germany5Department of Physics, University of Illinois, Urbana, Illinois 61801, USA6Department of Physics, University of Ioannina, 45110 Ioannina, Greece7National Institute for Materials Science, Tsukuba, Ibaraki 305-0003, Japan8Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan9Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Republic of Korea10School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom11Scottish Universities Physics Alliance, School of Physics and Astronomy,University of St. Andrews,St. Andrews KY16 9SS, United Kingdom12Toyota Riken - Kyoto University Research Center (TRiKUC), Kyoto University, Kyoto 606-8501, JapanI. ADDITIONAL INFORMATION: STMUniaxial strain was applied in the STM measurement using a thermal expansion cell described in Ref. [1] and adapted fromthat used in Ref. [2]. The STM strain cell was calibrated using digital image correlation (DIC) and elastoresistivity measurementsdescribed in Ref. [1] which estimated the strain of the sample platform to be approximately -0.47% to -0.5%. The surface of thesample is under less strain than the sample platform itself and based on profilometry measurements of the sample thickness aftercleaving, we estimate the strain at the sample surface to be approximately -0.4%.We determined the topographical depths of the cleavage defects by flattening the three dimensional topographic dataset intotwo dimensions, via T (x,y) → T (x), along the vertical dimension. The topographic values on this restructured array are thenbinned by frequency for a particular value of x. The lower panel of Fig. S1(b) shows this projected and binned data, while theupper panel shows the original topography. From this analysis, we see that the difference in height between the upper and lowersurfaces is approximately 3.4 Å, in good agreement with the established value of 3.8 Å for the separation between two SrOlayers and much larger than the 2.1 Å separation between SrO and RuO2 layers, as illustrated in the schematic of the defect inFig. S1(a).II. ADDITIONAL INFORMATION: SCANNING SQUIDThe local Tc maps shown in Fig. 2 of the main text were made by taking spatial scans of M at various temperatures near theunstressed bulk Tc of the sample and at the base temperature Tbase ≈500 mK with a fixed scan height. At each temperature, themaps of M(T ) are converted to maps of λ (T )+ z0. We assume constant z0 for all scans so that temperature dependent changesin λ (T )+ z0 correspond to changes in λ (T ). For each pixel in the temperature series scans, we set ∆λ (T ) = λ (T )−λ (Tbase)and we calculated the normalized superfluid density ρ(T ) = λ 20 /(λ0 +∆λ (T ))2 using λ0 = 190 nm. We performed a T -linearfit of ρ(T ) near the bulk Tc and set the local Tc to be the temperature of the ρ = 0 intercept. We note that the calculation ofρ(T ) here assumes a constant and spatially uniform penetration depth of λ0 = 190 nm for 0 < T < Tbase; however, varying λ0 by±20 nm does not substantially change the values of ρ(T ) very close to the bulk Tc and therefore does not significantly changethe resulting local Tc map.Data on the strain- and temperature-dependent changes of the penetration depth shown in Fig. 3(a) and (b) of the main text wereobtained by bringing the SQUID susceptometer into mechanical contact with a selected point on the sample and recording Mwhile sweeping the temperature from Tbase ≈500 mK to above Tc for a series of applied strains. Figure S2 shows the temperaturedependence of the mutual inductance M(T ) measured under a series of compressive strains through the Lifshitz transition onsample 1 [Fig. S2(a) and (b)] and sample 2 [Fig. S2(c) and (d)]. We observed rounding of the superconducting transition underincreasing strain due to strain inhomogeneity within the measurement volume. The local Tc at each strain is defined by thetemperature at which dM/dT was maximal and is indicated by the red data points in S2.arXiv:2312.05130v2 [cond-mat.supr-con] 11 Dec 20232The main text shows data on the temperature and strain dependence of the change in penetration depth λ (ε,T )− λ (0,0)measured on sample 2. The data measured on sample 1 are shown in Fig. S3. Consistent with measurements on sample 2, thepenetration depth shows a ∼ T 2 dependence for T < 0.5Tc thoroughout the Lifshitz transition.FIG. S1. (a) Schematic diagram of the large crystallographic defects on the surface of strontium ruthenate. We identify two different types,which expose a different crystal layer, SrO and RuO2. (b) Topographic image (upper panel) of a defect on the surface of strontium ruthenate.The density histogram (lower panel) of the topographic image shown, projected along y and binned, allows us to better visualize the meandepth of 3.4 ÅFIG. S2. (a) and (b) M(T ) measured on sample 1 for strains below εc and above εc respectively. (c) and (d) M(T ) measured on sample 2 forstrains below εc and above εc respectively. Red data points indicate where the slope of M(T ) is maximal and the temperature at which we taketo be the local Tc3FIG. S3. Temperature and strain dependence of the penetration depth measured at point 4 on sample 1. Panels (a) and (b) correspond to strainvalues below and above εc respectively.[1] J. Olivares Rodriguez, Stress and crystal imperfections: Tools for the exploration of unconventional superconductivity via scanning tun-neling microscopy, Ph.D. thesis, University of Illinois Urbana-Champaign (2022).[2] V. Sunko, E. Abarca Morales, I. Marković, M. E. Barber, D. Milosavljević, F. Mazzola, D. A. Sokolov, N. Kikugawa, C. Cacho, P. Dudin,et al., Direct observation of a uniaxial stress-driven Lifshitz transition in Sr2RuO4, npj Quantum Mater. 4, 46 (2019).