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[Shunichiro Kittaka](https://orcid.org/0000-0002-5440-4831), [Yohei Kono](https://orcid.org/0000-0001-7613-3721), Toshiro Sakakibara, [Naoki Kikugawa](https://orcid.org/0000-0003-3975-4478), [Shinya Uji](https://orcid.org/0000-0001-9351-6388), Dmitry A. Sokolov, Kazushige Machida

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[High-Resolution Magnetostriction Measurements of the Pauli-Limited Superconductor Sr                    <sub>2</sub>                    RuO                    <sub>4</sub>](https://mdr.nims.go.jp/datasets/dc29afc7-96c5-4cfc-bb83-0cb71a12a633)

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High-resolution magnetostriction measurements of the Pauli-limited superconductor Sr2RuO4Journal of the Physical Society of JapanHigh-Resolution Magnetostriction Measurementsof the Pauli-Limited Superconductor Sr2RuO4Shunichiro Kittaka1,2 *, Yohei Kono2, Toshiro Sakakibara3, Naoki Kikugawa4, Shinya Uji4,Dmitry A. Sokolov5, and Kazushige Machida61Department of Basic Science, The University of Tokyo, Meguro, Tokyo 153-8902, Japan2Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551,Japan3Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan4National Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan5Max Planck Institute for Chemical Physics of Solids, Nothnitzer Str. 40, 01187 Dresden, Germany6Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, JapanWe performed high-resolution magnetostriction measurements on the Pauli-limited superconductor Sr2RuO4 usinghigh-quality single crystals. A first-order superconducting transition, accompanied by pronounced hysteresis, was ob-served under in-plane magnetic fields, where the relative length change of the sample, ∆L/L, was on the order of 10−8.To ensure the reliability of the measurements, particular attention was paid to minimizing the influence of magnetictorque, which can significantly affect data under in-plane field configurations, via field-angle-resolved magnetostriction.Within the hysteresis regime, slightly below the Pauli-limited upper critical field, a hump-like anomaly in the magne-tostriction coefficient was identified. Furthermore, a characteristic double-peak structure in the field-angle derivative ofthe magnetostriction provides additional support for this anomaly. Although these findings may reflect a lattice responseassociated with the emergence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in Sr2RuO4, the possibility of abroadened first-order transition cannot be excluded. Notably, this magnetostriction anomaly qualitatively deviates fromthe FFLO phase boundary suggested by previous NMR measurements, highlighting the necessity for further experimen-tal and theoretical investigations to elucidate the nature of the FFLO state in this material.1. IntroductionTheoretical proposals for a spatially modulated supercon-ducting state induced by the Zeeman effect, known as theFulde-Ferrell-Larkin-Ovchinnikov (FFLO) state,1, 2) have in-spired extensive experimental efforts to observe this exoticphase. Despite these efforts, experimental evidence for theFFLO state has been reported only in a limited number ofcandidate materials, such as the organic superconductor κ-(BEDT-TTF)2Cu(NCS)2,3, 4) the heavy-fermion superconduc-tor CeCoIn5,5) and iron-based superconductors.6, 7) This limi-tation is likely due to the stringent conditions required for theemergence of the FFLO state:8) (i) a highly clean system, (ii)low dimensionality that enhances Fermi-surface nesting, and(iii) a strong Pauli-paramagnetic effect that surpasses the or-bital depairing. In superconductors with strong Pauli limiting,the nature of the superconducting transition under magneticfield B changes from second order to first order upon cooling,typically below 0.56Tc.8)These criteria are fulfilled in the layered perovskite super-conductor Sr2RuO4.9–12) First, the material exhibits excep-tionally high purity, with a mean free path of approximately1 µm,13) which far exceeds the in-plane coherence lengthξa of 660 Å. Second, its Fermi surface topology,14, 15) alongwith the intrinsic anisotropy in the coherence length (ξa/ξc ∼60),16–18) highlights its pronounced two-dimensionality, an es-sential feature for Fermi-surface nesting. Third, when a mag-netic field is applied nearly parallel to the ab plane, the up-per critical field Bc2 is significantly limited at low tempera-tures, and a first-order superconducting transition occurs be-*kittaka@g.ecc.u-tokyo.ac.jplow approximately 0.8 K (∼ 0.53Tc). This transition has beenconfirmed through various thermodynamic measurements, in-cluding specific heat,19) magnetocaloric effect,20) magnetiza-tion,17) and magnetic torque.17, 21) Indeed, thermodynamic es-timation of the Pauli-limiting field, BP = Bc/√χn − χsc ∼1.4 T, is in good agreement with the observed limitation ofBc2 for B ‖ ab. This estimation adopts a thermodynamic crit-ical field of Bc = 0.02 T, a spin susceptibility in the normal(superconducting) state of χn = 0.9×10−3 (χsc = 0) emu/mol,and a molar volume of Vm = 57.5 cm3/mol. Such consistencystrongly suggests that the Pauli-paramagnetic effect is crucialfor Sr2RuO4.Sr2RuO4 has long been regarded as a promising candi-date for a spin-triplet superconductor with a chiral p-wave or-der parameter9, 10) primarily based on NMR Knight shift andspin-polarized neutron scattering experiments that indicatedan invariant spin susceptibility across Tc for all magnetic-field directions.22–24) However, this interpretation has becomeincreasingly controversial following the discovery of a first-order transition at Bc2,20) accompanied by a magnetizationjump.17) More critically, in 2019, Pustogow et al. reported asignificant reduction in the NMR Knight shift in the supercon-ducting state under low radio-frequency pulse powers.25) Thisresult was subsequently confirmed by Ishida et al.26) and fur-ther supported by polarized neutron scattering experiments.27)A similar B–T phase diagram, featuring both a first-orderphase transition and Knight-shift reduction, has also been ob-served in uniaxially strained samples with an enhanced Tcof approximately 3.4 K.25, 28) These findings strongly suggestthat the first-order phase transition for B ‖ ab originates fromthe Pauli-paramagnetic effect, in stark contrast to the long-1arXiv:2511.06938v2  [cond-mat.supr-con]  14 Nov 2025https://arxiv.org/abs/2511.06938v2J. Phys. Soc. Jpn.standing chiral p-wave scenario. Consequently, a paradigmshift is underway in the understanding of the superconductingorder parameter in Sr2RuO4.11, 12)Thus, an intriguing question arises as to whether the FFLOstate is realized in Sr2RuO4. A fourfold oscillation in the spe-cific heat under an in-plane rotating magnetic field at low tem-peratures was found to be abruptly suppressed around 1.3 T,followed by its drastic development above 1.4 T with an op-posite sign,29) which may reflect the field-angle anisotropyassociated with the FFLO state. Recently, NMR measure-ments have provided compelling evidence for the emergenceof the FFLO phase in Sr2RuO4.30) Nevertheless, clear ther-modynamic signatures of such a double transition remain elu-sive, despite long-standing discussions linking its possible ex-istence to chiral triplet pairing,31–33) particularly the splittingof degenerate order parameters in the chiral px±ipy state.34–36)2. MethodsIn this study, we performed high-resolution field-angle-resolved magnetostriction and thermal-expansion measure-ments on Sr2RuO4 as a powerful experimental approach. Thistechnique offers a significant advantage for probing first-orderphase transitions, as recently demonstrated in CeCoIn5.37)We utilized a home-built capacitively-detected dilatometer,which achieves a resolution significantly better than 0.01 Å.The relative change in sample length along the i direction,∆Li(T, B) = Li(T, B) − Li(T0, B0), was determined from thechange in capacitance, ∆C = C(T, B) − C(T0, B0), betweenthe movable and fixed electrodes using the relation ∆Li =ε0A∆C/[C(T, B)C(T0, B0)], where A = 25π mm2 is the elec-trode area and ε0 is the vacuum permittivity. A typical capaci-tance value is approximately 13 pF, corresponding to a capaci-tor gap of about 0.05 mm. The sample was gently sandwichedbetween the movable component and an adjustment screw, thelatter being firmly fixed to the outer frame with a locking nut.To prevent sample rotation due to magnetic torque, the sam-ple was bonded to the adjustment screw using GE varnish.For comparison, the specific heat cp was measured using thequasi-adiabatic heat-pulse method in a dilution refrigerator.In all measurements, the magnetic-field orientation was pre-cisely controlled in three dimensions using a vector-magnetsystem.38)High-quality single crystals used in this study were grownby the floating-zone method.39, 40) We selected a large sin-gle crystal weighing 50 mg, with approximate dimensionsof 3 mm × 3 mm in the ab plane and 1.1 mm along the caxis, exhibiting an onset Tc of 1.525 K. Both surfaces alongthe ab plane were polished, while the bc plane was left as-cut. Magnetostriction and thermal-expansion measurementswere performed along the c and a axes in a 3He refrigera-tor or a dilution refrigerator, where Lc ∼ 1.1 mm and La ∼2.7 mm. To obtain a large Sr2RuO4 sample with a high Tc,a small amount of bilayer Sr3Ru2O7 inclusions was unavoid-able. These inclusions were identified via specific-heat mea-surements and polarized light optical microscopy of the pol-ished surface (see Supplemental Material41)). Since cp/T ofSr3Ru2O7 in the Sr2RuO4-Sr3Ru2O7 eutectic system is knownto be temperature-independent below 2 K,42) we subtract itscontribution from the data presented below.41) The correctedspecific heat is referred to as c214. Here, the phonon contribu-tion, assuming a Debye temperature of 410 K, and the nuclear20406080100 1  1.2  1.4  1.6c214 / T (J / mol K2 )B (T)φ = 27°θ = 90°0.2 K0.3 K0.5 K0.8 K(b)0204060 0  0.5  1  1.5c214 / T (J / mol K2 )T (K)0 T(a)1.41.51.6−45  0  45  90Bc2 (T)φ (deg)[100][110]0.3 K(c)uc214∆LcFig. 1. (Color online) (a) Temperature dependence of the zero-field spe-cific heat of the Sr2RuO4 component, c214, divided by temperature. (b) Mag-netic field dependence of c214/T at several temperatures, measured at φ = 27◦and θ = 90◦. Each dataset at the same temperature is vertically shifted by10 mJ/(mol K2) for clarity. The blue open and red closed circles representdata obtained during decreasing and increasing field sweeps, respectively. (c)Field-angle φ dependence of the upper critical field Bc2 at θ = 90◦ (i.e., withinthe ab plane) at 0.3 K, determined from specific-heat and magnetostriction∆Lc measurements during increasing field sweeps. The inset in (a) depictsthe definition of the field angles, φ and θ.specific heat, given by CN(T, B) = (0.08 + 0.14B2)/T 2 µJ /(mol K),29) are subtracted along with the addenda heat ca-pacity, which is always less than 5% of the sample heat ca-pacity. Moreover, Sr3Ru2O7 inclusions are expected to haveonly a minor effect on the magnetostriction, as no anomalyhas been reported for Sr3Ru2O7 in the low-field region below2 T.43) To confirm the reproducibility of our results, we alsomeasured ∆Lc for a relatively small, pure Sr2RuO4 sample(12 mg), which was free of Sr3Ru2O7 inclusions.41)3. Results and Discussion3.1 Specific heatFigure 1(a) shows the temperature dependence of the zero-field electronic specific-heat data, c214/T , for the Sr2RuO4component. A clear specific-heat jump is observed at Tc =1.483 K (midpoint), with an onset temperature of 1.525 K.The width of the jump is less than 0.09 K, and no additionalanomalies or kinks are present. These results demonstrate thehigh quality and narrow Tc distribution of the present sample,despite the relatively large sample size.Figure 1(b) presents the magnetic-field dependence ofc214/T at several temperatures under an in-plane magneticfield (θ = 90◦) with φ = 27◦. Here, as illustrated in the in-set of Fig. 1(a), φ and θ denote the azimuthal and polar an-gles of the magnetic field, measured from the [100] and [001]axes, respectively. These angles were defined based on theanisotropy of Bc2 [Fig. 1(c)] and verified using Laue diffrac-tion images. Despite the relatively large sample size, a clearhysteresis is observed in c214(B) near Bc2 at low temperatures,indicating the presence of a first-order superconducting tran-2J. Phys. Soc. Jpn.sition. This observation confirms that the Sr2RuO4 componentin the present sample possesses sufficiently high quality.As shown in Fig. 1(b), a specific-heat peak appears near Bc2in c214(B) at moderate temperatures. This peak complicatesthe identification of possible internal phase transitions withinthe superconducting state, particularly at intermediate temper-atures. In addition, the specific-heat data exhibit some scatter,and the measurement sensitivity is not sufficiently high to re-solve subtle anomalies, such as those expected from the FFLOphase transition.3.2 Thermal expansionTo investigate possible internal superconducting phasetransitions, we focus on the lattice response of the sam-ple using a home-built, capacitively detected subpicometerdilatometer. Figure 2 presents the thermal-expansion data. Aclear kink is observed at Tc = 1.515 K in both Lc(T ) andLa(T ), consistent with the specific-heat results. Here, the lin-ear thermal-expansion coefficient along each axis is definedas αi = (∂Li/∂T )/Li. Across the superconducting transition,the coefficient exhibits a discontinuity ∆αi. As indicated bythe dashed and dotted lines in Fig. 2, we estimated the dis-continuities in thermal expansion coefficients at Tc as ∆αc =−(7.5 ± 0.5) × 10−8 K−1 and ∆αa = −(5.0 ± 0.7) × 10−8 K−1.For a second-order phase transition, the Ehrenfest rela-tion ∂Tc/∂σii = −TcVm∆αi/∆c should be satisfied. Usingthe obtained values of ∆αi and the specific heat jump at Tc,∆c ∼ 30 mJ mol−1 K−2, in zero field, we estimate ∂Tc/∂σxx ∼0.1 K/GPa and ∂Tc/∂σzz ∼ 0.15 K/GPa in the zero-pressurelimit for Sr2RuO4. Assuming hydrostatic pressure as a com-bination of uniaxial components, the pressure dependenceof Tc is given by dTc/dP ∼ −(2∂Tc/∂σxx + ∂Tc/∂σzz) ∼−0.3 K/GPa, which agrees well with the previous experi-mental observations (≈ −0.2 K/GPa).44, 45) However, the es-timated value of ∂Tc/∂σxx exhibits a sign opposite to that ob-served experimentally,46) whereas the estimate for ∂Tc/∂σzz isin good agreement with recent uniaxial strain measurementsalong the c axis.47) These contrasting results suggest that thestrain dependence of Tc in Sr2RuO4 is more complex than ex-pected from simple thermodynamic relations.48)3.3 Effect of magnetic torque on our magnetostriction mea-surementsFigure 3(a) shows the relative change in the dilatometercapacitance, ∆C, for L ‖ c, as a function of the polar an-gle θ of the magnetic field (B = 1 T) for various azimuthalangles φ. Although magnetostriction is expected to be sym-metric with respect to θ = 90◦ due to the tetragonal crystalsymmetry, ∆C exhibits clear asymmetry except near φ = 27◦.This asymmetric behavior is attributed to magnetic torque inthe superconducting state, which reverses sign at θ = 90◦,49)and is likely due to the anisotropic mechanical response ofour home-built dilatometer, where the movable electrode issuspended by crossed phosphor-bronze wires. Indeed, a qual-itatively similar θ dependence of magnetic torque has been re-ported for Sr2RuO4,17) originating from the large anisotropyin the coherence length (ξa/ξc ∼ 60).50) It is speculated thatthe torque induces a slight rotation of the sample, depend-ing on the adhesion strength of the mounting paste. This ro-tation may shift the position of the movable electrode by ap-proximately 0.1 Å, which is comparable to the magnetostric--30-20-100 0  0.5  1  1.5  2∆Lc / Lc (10-8 )T (K)0 Tnormal state(a)-20-100 0  0.5  1  1.5  2∆La / La (10-8 )T (K)0 Tnormal state(b)Fig. 2. (Color online) Temperature dependence of the relative lengthchange, ∆Li/Li, along the (a) c and (b) a axes. The red circles represent zero-field data. The blue squares in (a) indicate normal-state data measured underan in-plane magnetic field of 2 T at φ = 27◦. The open and closed squaresin (b) correspond to normal-state data measured under in-plane fields of 1.45and 1.8 T, respectively, at φ = 90◦. The dashed (dotted) lines represent linearfits to the zero-field data just below (above) Tc.−4−20 1  1.2  1.4  1.6∆C (10−5 pF)B (T)φ = 27°0.3 K θ = 90°(b)L || cLc Bc−50 1  1.2  1.4  1.6∆C (10−5 pF)B (T)φ = 90°0.12 K θ = 90°(c)L || aLaBc−1012 85  90  95∆C (10−5 pF)θ (deg)0.3 K, 1 T0.05 Å(a)L || cφ = 122°77°32°27°−13°012 1.2  1.3  1.4  1.5∆Lcsc / Lc (10−8 )B (T)0.3 K0.02 Å(d) B1stFig. 3. (Color online) (a) Polar-angle θ dependence of the capacitancechange relative to the normal-state value (taken at θ ∼ 95◦) for L ‖ c, mea-sured at 0.3 K and 1 T for various azimuthal angles φ. (b), (c) Magnetic-fielddependence of the capacitance change relative to the value at 1.7 T, measuredat low temperatures for two different sample orientations: (b) L ‖ c at 0.3 Kwith φ = 27◦ and θ = 90◦, (c) L ‖ a at 0.12 K with φ = 90◦ and θ = 90◦ .The blue open and red closed circles denote data obtained during decreasingand increasing field sweep. The green triangles show normal-state responsemeasured at θ = 85◦ for each configuration, where Bc2 is below 0.8 T. Thedashed lines are fits to the normal-state data using a cubic polynomial func-tion. (d) Superconducting contribution to the normalized magnetostriction∆Lscc /Lc for L ‖ c, obtained by subtracting the background [dashed line in(b)]. The characteristic field B1st, at which the hysteresis in ∆Lscc vanishes, isindicated by an arrow.tion signal ∆Lc observed in the present sample. Therefore, inthe following analysis, we focus on magnetostriction data ob-tained at φ = 27◦, where the magnetic-torque effect is fortu-itously minimized for L ‖ c in this sample.3J. Phys. Soc. Jpn.02468 1.1  1.3  1.5∆Lcsc / Lc (10−8)B (T)0.120.30.40.50.60.70.80.9(a)0246 1.1  1.3  1.5∆Lasc / La (10−8)B (T)0.120.30.40.50.6(b)Fig. 4. (Color online) Magnetic-field dependence of the normalized mag-netostriction ∆Lsci/Li at several temperatures for (a) i = c (φ = 27◦ andθ = 90◦) and (b) i = a (φ = 90◦ and θ = 90◦). Each dataset is verticallyshifted by 1× 10−8 for clarity. The blue open and red closed circles representdata obtained during decreasing and increasing field sweeps, respectively.The arrows indicate the characteristic field B1st, at which the hysteresis loopcloses. The numbers labeling each dataset indicate the temperature in K.3.4 Magnetostriction along the c axisFigure 3(b) shows the magnetic-field dependence of ∆C(B)when L ‖ c, measured during increasing and decreasing fieldsweeps at 0.3 K with φ = 27◦ and θ = 90◦. To estimate thenon-superconducting background contribution, including themagnetostriction of the dilatometer itself, we also measured∆C(B) in the normal state at θ = 85◦ (green triangles), whereBc2 < 0.8 T.51) By subtracting this background signal from thedata at θ = 90◦, we isolate the superconducting contribution tothe length change of the sample along the i direction, definedas ∆Lsci≈ ε0A[∆C(θ = 90◦)−∆C(θ = 85◦)]/[C(T0,H0)]2. Theresulting normalized magnetostriction, ∆Lscc /Lc, is plotted inFig. 3(d). Remarkably, ∆Lscc /Lc in Sr2RuO4 is on the orderof 10−8, which is two orders of magnitude smaller than thoseobserved in CeCoIn537, 52, 53) and CeCu2Si2,54) both of whichexhibit values around 10−6. These results highlight the im-portance of high-resolution magnetostriction measurementsand careful consideration of parasitic magnetic-torque effectswhen investigating superconductivity in Sr2RuO4.As shown in Figs. 3(b) and 3(d), the onset of Bc2 clearlydiffers between increasing and decreasing field sweeps. Thishysteresis indicates that the first-order superconducting tran-sition at Bc2 can be sensitively detected via magnetostriction.As indicated by the arrow in Fig. 3(d), we can precisely de-fine the characteristic field B1st, at which the hysteresis loopin ∆Lsci(B) closes within the superconducting phase. Qualita-tively similar behavior is observed in ∆Lscc for another samplewith Lc = 0.7 mm, as shown in the Supplemental Material.41)Figure 4(a) shows the magnetic-field dependence of∆Lscc /Lc at several temperatures. The data at 0.12 K were mea-0481216202428 1.1  1.3  1.5λ c (10−7 /T)B (T)0.120.30.40.50.60.70.80.9(a)0481216 1.3  1.4  1.5λ a (10−7 /T)B (T)0.120.30.40.50.6(b)Fig. 5. (Color online) Magnetic-field dependence of the magnetostrictioncoefficient λi = (∂∆Lsci/∂B)/Li at several temperatures for (a) i = c (φ =27◦ and θ = 90◦) and (b) i = a (φ = 90◦ and θ = 90◦). Each dataset isvertically shifted by 4 × 10−7 T−1 for clarity. The blue open and red closedcircles represent data obtained during decreasing and increasing field sweeps,respectively. The arrows indicate a possible hump-like anomaly at BK. Thenumbers labeling each dataset indicate the temperature in K.sured in a dilution refrigerator, while those for T ≥ 0.3 K wereobtained using a 3He refrigerator. The former dataset exhibitsnoticeable scatter, primarily due to heating of the sorptionpump in the refrigerator. As the temperature increases, B1stshifts toward lower values. Above 0.6 K, it becomes difficultto determine B1st precisely because of the resolution limit ofour dilatometer. The magnetostriction coefficient, defined asλi = (∂∆Lsci/∂B)/Li, along the c axis is plotted in Fig. 5(a).3.5 Magnetostriction along the a axisFor L ‖ a, to avoid torque-related artifacts, we restrict ouranalysis to data taken at φ = 90◦, where the magnetic-torqueeffect is minimal. Figure 3(c) shows ∆C(B) for L ‖ a, mea-sured during increasing and decreasing field sweeps at 0.12 K,with φ = 90◦ and θ = 90◦. The superconducting contributionto the magnetostriction,∆Lsca /La, and the corresponding mag-netostriction coefficient λa at several temperatures are shownin Figs. 4(b) and 5(b), respectively. All data were obtainedusing a dilution refrigerator.Theoretically, an abrupt enhancement of the zero-energyquasiparticle density of states, N(E = 0), has been pre-dicted at the transition between the FFLO and Abrikosovstates.37, 55, 56) Such an anomaly should be observable throughlow-temperature thermodynamic quantities that reflect N(E =0). Indeed, previous specific-heat19) and entropy20) measure-ments at 0.3 K may have captured this anomaly, althoughthe signature was not particularly pronounced. Furthermore,recent NMR studies have revealed a characteristic double-horn spectrum at temperatures below 0.3 K and magneticfields above approximately 1.2 T, indicating a spatial mod-4J. Phys. Soc. Jpn. 0 1 2 88  90  92∆La / La (10-6 ) 0.12 K, 1.4 T(a)-3 0 3 88  90  92λ θ,a  (10-6 / deg)θ (deg)0.12 K, 1.4 T(b)Fig. 6. (Color online) Field-angle θ dependence of (a) the magnetostric-tion ∆La/La and (b) its coefficient λθ,a = (∂La/∂θ)/La , measured at 0.12 Kunder a magnetic field of 1.4 T rotated within the bc plane at φ = 90◦. Theblue open and red closed circles represent data obtained during decreasingand increasing θ sweeps, respectively. The solid line in (b) represents a fitto the data obtained during increasing θ, using a function composed of twoantisymmetric Gaussian components, shown by the dashed and dotted lines.ulation of spin density intrinsic to the FFLO state. However,no corresponding anomaly was detected around 1.2 T in thepresent magnetostriction measurements. Instead, a hump-likeanomaly was detected in λa(B) at BK, as indicated by the ar-rows in Fig. 5(b), above B1st.To investigate the possible anomaly at BK in Sr2RuO4, wemeasured the field-angle dependence of magnetostriction at0.12 K and 1.4 T, where the sample is likely in the Abrikosovstate at θ = 90◦ for B ‖ b. Figures 6(a) and 6(b) present themagnetostriction data, ∆La(θ)/La, and its field-angle deriva-tive, λθ,a = (∂La/∂θ)/La, respectively. Tilting the magneticfield away from the a axis toward the c axis suppresses su-perconductivity around θ = 90◦ ± 1.5◦. The solid line inFig. 6(b) shows a fit using the sum of two antisymmetricGaussian functions, g1(θ) + g2(θ), where each gi(θ) is definedas the difference between two Gaussian functions centeredsymmetrically at θ = 90◦ ± θi. Here, we obtain θ1 = 1.05◦and θ2 = 0.62◦. These components are shown as dotted anddashed lines, respectively, indicating the presence of two dis-tinct anomalies associated with Bc2 and BK. This result sug-gests that the anomaly at BK vanishes when the in-plane mag-netic field is rotated by approximately 1◦ toward the c axis.3.6 Thermodynamic relation associated with the magne-tostriction jumpFor a first-order phase transition, the strain dependenceof Bc2 for B ‖ a, denoted as B‖ac2, is governed by theClausius-Clapeyron relation: ∂B‖ac2/∂σxx ≈ ∆εxx/∆M‖a, where∆εii and ∆M‖a correspond to the discontinuities across B‖ac2in ∆Li/Li and the a-axis component of the magnetization,respectively. Using the experimentally observed values of 1.3 1.4 1.5 0  0.2  0.4  0.6  0.8B (T)T (K)Bc2 BKB1st 1.3 1.4 1.5 0  0.2  0.4  0.6  0.8B (T)T (K)B || b,  L || aFig. 7. (Color online) Field–temperature phase diagram determined frommagnetostriction measurements for L ‖ a. The closed (open) circles repre-sent Bc2 obtained during increasing (decreasing) magnetic field sweeps. Thesquares indicate BK, where a hump-like anomaly is observed during increas-ing field. The triangles mark B1st, the field at which the hysteresis in ∆Lsca (B)disappears.∆La/La ≈ −2 × 10−8 and 4π∆M‖a ≈ 0.7 G,17) we estimate∂B‖ac2/∂σxx ≈ −0.3 T/GPa in the zero-strain limit. This es-timate is of the same order as the value (∂B‖ac2/∂σxx)|σ→0 ≈−0.1 T/GPa, which is inferred under the assumption of a scal-ing relation (∂B‖ac2/∂σxx)|σ→0 ∼ (∂Tc/∂σxx)|σ→0(B‖ac2/Tc)|σ→0,using the experimental observation of (∂Tc/∂σxx)|σ→0 ≈−0.1 K/GPa.46)3.7 Possible origins of BK anomalyThe high-resolution magnetostriction measurements reveala possible anomaly at BK in λa(B), slightly below the Pauli-limited upper critical field Bc2 in Sr2RuO4. This feature, alongwith the double-peak structure observed in λθ,a, suggests alattice response that may be linked to the emergence of theFFLO phase. Notably, the anomaly at BK, situated withinthe hysteresis region above B1st, is reminiscent of FFLO sig-natures reported in CeCoIn5, which have been attributed tochanges in the spatial modulation of the superconducting or-der parameter.37) In CeCoIn5, the FFLO phase boundary isinterpreted to correspond to the lower onset field of the hump-like anomaly.However, a significant discrepancy exists between theFFLO phase boundary inferred from the magnetostrictiondata and that identified in recent NMR studies. While theNMR results indicate a FFLO boundary with a positive slopein the B–T phase diagram,30) the magnetostriction measure-ments reveal negative slopes in both the B1st(T ) and BK(T )lines, as shown in Fig. 7. This inconsistency apparently under-scores the probe-dependent nature of the FFLO phase mani-festation. Moreover, the absence of a clear thermodynamicsignature of the FFLO transition in specific-heat and entropymeasurements implies the subtle nature of this phase.An alternative interpretation for the origin of the BKanomaly is the broadening of the first-order transition. In-deed, the presence of eutectic boundaries involving Sr3Ru2O7may influence the nucleation and stability of domains inwhich normal and superconducting states coexist. Althoughthe FFLO state remains an intriguing possibility in Sr2RuO4,its realization has yet to be firmly established and warrantsfurther investigation.5J. Phys. Soc. Jpn.4. SummaryIn this study, we performed high-resolution magnetostric-tion and thermal-expansion measurements on the Pauli-limited superconductor Sr2RuO4 using high-quality singlecrystals. Our results revealed a clear first-order superconduct-ing transition under in-plane magnetic fields, accompanied bypronounced hysteresis and a subtle lattice response on the or-der of 10−8. A hump-like anomaly in the magnetostriction co-efficient and a double-peak structure in its field-angle deriva-tive were identified slightly below Bc2, suggesting a possiblelink to the emergence of the FFLO phase. However, the ob-served features may also be interpreted as a broadening of thefirst-order transition. Notably, no corresponding anomaly wasdetected in the magnetostriction measurements at the mag-netic fields where NMR studies reported signatures of theFFLO phase. This discrepancy underscores the need for fur-ther experimental and theoretical investigations to clarify itsrealization in Sr2RuO4.AcknowledgmentsWe thank A. P. Mackenzie, T. Terashima, K. Ishida, Y.Maeno, and Y. Shimizu for fruitful discussions. In particular,we are grateful to A. P. Mackenzie for kindly supporting in thegrowth of high-quality single crystals in Dresden. 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Jpn. 80, 123706 (2011).56) K. M. Suzuki, K. Machida, Y. Tsutsumi, and M. Ichioka, Phys. Rev. B101, 214516 (2020).7J. Phys. Soc. Jpn.Supplemental Material forHigh-resolution magnetostriction measurements of the Pauli-limitedsuperconductor Sr2RuO4Shunichiro Kittaka,1,2 Yohei Kono2, Toshiro Sakakibara,3 Naoki Kikugawa,4 Shinya Uji,4Dmitry A. Sokolov,5 and Kazushige Machida61Department of Basic Science, The University of Tokyo, Meguro, Tokyo 153-8902, Japan2Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan3Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan4National Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan5Max Planck Institute for Chemical Physics of Solids, Nothnitzer Str. 40, 01187 Dresden, Germany6Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan(Dated: November 17, 2025)I. Specific heat of the samples used in this studySr2RuO4 is the n = 1 member of the so-called Ruddlesden-Popper (RP) series ruthenates Srn+1RunO3n+1. Large singlecrystals of these ruthenates can be grown by using a Ru self-flux floating-zone technique. Due to evaporation of RuO2 fromthe solvent at the melting point, precise control of the growth parameters is essential for obtaining high-quality single crystalsof Sr2RuO4.1, 2) As a result, Ru lamellae and/or Sr3Ru2O7 inclusions are frequently observed in single crystal rods of Sr2RuO4.Figure S1(a) shows the temperature dependence of cp/T at 0 T for three samples, labeled #3-2, #3-5, and #3-B. Sample #3-2was used in a previous specific-heat study,3) while samples #3-B and #3-5 were used for magnetostriction measurements in thepresent study. In the main text, we present the results obtained using sample #3-B.As shown in Fig. S1(a), the Sommerfeld coefficient γe, i.e., the normal-state value of cp/T , is unexpectedly enhanced forsample #3-B. It is important to note that cp in Fig. S1(a) was evaluated using the following equation:cp =craw(m/M214), (1)where craw is the measured heat capacity in J K−2, m is the sample mass, and M214 is the molar mass of Sr2RuO4. If the n = 2member of the RP series, Sr3Ru2O7, is unintentionally included in the sample, the apparent normal-state cp/T is enhanced,since the Sommerfeld coefficient of Sr3Ru2O7 (γ327 ∼ 0.22 J mol−1 K−2) is larger than that of Sr2RuO4 (γ214 ∼ 0.04 J mol−1K−2). Indeed, polarized light optical microscopy images of the polished plane of sample #3-B revealed a small amount ofSr3Ru2O7 inclusions, as shown in the inset of Fig. S1(b). To evaluate the specific heat of the Sr2RuO4 component in sample#3-B, we define c214 as:c214 =craw − γ327T (m327/M327)(m − m327)/M214, (2)where m327 and M327 denote the mass and molar mass of the Sr3Ru2O7 inclusions, respectively. We found that assumingm327 = 5.5 mg (approximately 10% of m) yields c214/T ∼ 0.04 J mol−1 K−2 in the normal state for sample #3-B, as shownin Fig. S1(b). For comparison, m327 is assumed to be zero for samples #3-2 and #3-5. Based on these analyses, we concludethat the enhancement of cp/T in sample #3-B originates from Sr3Ru2O7 inclusions. Nevertheless, the quality of the Sr2RuO4component in sample #3-B is comparable to that of samples #3-2 and #3-5, as evidenced by the good agreement among thec214 data for all three samples in Fig. S1(b). To focus on the superconducting properties of the Sr2RuO4 component, we adoptc214 in the main text.II. Effect of Tc distribution on the magnetostrictionTo verify the reproducibility of the results, we performed magnetostriction measurements for L ‖ c on sample #3-5 in adilution refrigerator, in addition to those on sample #3-B shown in main text. Sample #3-5 (m = 12 mg and Lc = 0.7 mm)is smaller than sample #3-B (m = 50 mg and Lc = 1.1 mm). Although the onset Tc of sample #3-5 is comparable to that ofsample #3-B, the specific heat jump at Tc is significantly sharper in sample #3-5, as shown in Fig. S1(b); the jump widths forsamples #3-5 and #3-B are approximately 0.05 and 0.09 K, respectively. These observations suggest that the Bc2 distribution isconsiderably narrower in sample #3-5.Figures S2(a) and S2(b) show the relative capacitance change, ∆C = C(T, B)−C(T, 1.7 T), for sample #3-5, measured at 0.3and 0.09 K, respectively, under an in-plane magnetic field applied at φ = 45◦, where the magnetic-torque effect is negligible.The superconducting component of the relative magnetostriction,∆Lscc /Lc, is plotted in Figs. S2(c) and S2(d) at 0.3 and 0.09 K,respectively. The corresponding magnetostriction coefficient is shown in Figs. S2(e) and S2(f). Qualitatively similar behaviorto that observed in sample #3-B (see main text) was obtained. A hump-like anomaly was not prominently detected in sample#3-5 for L ‖ c, either. After this measurement, sample #3-5 was found to be cleaved, suggesting the presence of internal cracksthat may have contributed to signal scattering. Nevertheless, these results demonstrate that the magnetostriction behavior forL ‖ c is reproducible across different samples, supporting the intrinsic nature of the observed features.8J. Phys. Soc. Jpn.References1) Z. Q. Mao, Y. Maeno, and H. Fukazawa, Mat. Res. Bull. 35, 1813 (2000).2) J. S. Bobowski, N. Kikugawa, T. Miyoshi, H. Suwa, H. Xu, S. Yonezawa, D. A. Sokolov, A. P.Mackenzie, and Y. Maeno, Condens. Matter 4, 6 (2019).3) S. Kittaka, S. Nakamura, T. Sakakibara, N. Kikugawa, T. Terashima, S. Uji, D. A. Sokolov, A. P.Mackenzie, K. Irie, Y. Tsutsumi, K. Suzuki, and K. Machida, J. Phys. Soc. Jpn. 87, 093703 (2018).9J. Phys. Soc. Jpn. 0 20 40 60 80 0  0.5  1  1.5cp / T (mJ / mol K2)(a)#3-2#3-5#3-B 0 20 40 60 80 0  0.5  1  1.5c214 / T (mJ / mol K2)T (K)(b)1 mm#3-2#3-5#3-BFig. S1. Temperature dependence of the specific heat data, (a) cp/T and (b) c214/T , in zero magnetic field for samples #3-2,#3-5, and #3-B. The inset in (b) shows a polarized light optical microscopy image of a polished surface of sample #3-B. Thedarker (brighter) area corresponds to Sr2RuO4 (Sr3Ru2O7).10J. Phys. Soc. Jpn.012 1  1.2  1.4  1.6#3−5φ = 45°0.09 Kθ = 90°(b)B1st0123 1.3  1.4  1.5  1.6#3−5φ = 45°0.09 Kθ = 90°(d) B1st012 1  1.2  1.4  1.6∆C (10−5 pF)#3−5φ = 45°0.3 Kθ = 90°(a)B1st0.05 Å0123 1.3  1.4  1.5∆Lcsc / Lc (10−8 ) #3−5φ = 45°0.3 Kθ = 90°(c)B1st−50 1.3  1.4  1.5λ c (10−7 / T)B (T)#3−5φ = 45°0.3 Kθ = 90°(e)−50 1.3  1.4  1.5  1.6B (T)#3−5φ = 45°0.09 Kθ = 90°(f)Fig. S2. Magnetic-field dependence of the capacitance change relative to the value at 1.7 T for sample #3-5, measured at (a)0.3 and (b) 0.09 K with φ = 45◦ and θ = 90◦, during increasing (closed circles) and decreasing (open circles) field sweeps. Thedashed lines present linear fits to the data in the normal state, which are assumed to reflect the non-superconducting backgroundcontribution. (c), (d) Superconducting component of the relative magnetostriction, ∆Lscc /Lc, obtained by subtracting the back-ground contributions corresponding to the dashed lines in (a) and (b). (e), (f) Magnetostriction coefficient λc = (∂∆Lscc /∂B)/Lcestimated from the data in (c) and (d).11