Local observation of linear-T superfluid density and anomalous vortex dynamics in URu_2Si_2
Yusuke Iguchi, Irene P. Zhang, Eric D. Bauer, Filip Ronning, John R. Kirtley, Kathryn A. Moler
LLocal observation of linear- T superfluid densityand anomalous vortex dynamics in URu Si Yusuke Iguchi , , , Irene P. Zhang , , Eric D. Bauer , FilipRonning , John R. Kirtley , and Kathryn A. Moler , , Department of Applied Physics, Stanford University, Stanford, California 94305, USA Stanford Institute for Materials and Energy Sciences,SLAC National Accelerator Laboratory,2575 Sand Hill Road, Menlo Park, California 94025, USA Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Geballe Laboratory for Advanced Materials,Stanford University, Stanford, California 94305, USA
Abstract
The heavy fermion superconductor URu Si is a candidate for chiral, time-reversalsymmetry-breaking superconductivity with a nodal gap structure. Here, we micro-scopically visualized superconductivity and spatially inhomogeneous ferromagnetismin URu Si . We observed linear- T superfluid density, consistent with d -wave pairingsymmetries including chiral d -wave, but did not observe the spontaneous magnetiza-tion expected for chiral d -wave. Local vortex pinning potentials had either four- ortwo-fold rotational symmetries with various orientations at different locations. Takentogether, these data support a nodal gap structure in URu Si and suggest that chi-rality either is not present or does not lead to detectable spontaneous magnetization. a r X i v : . [ c ond - m a t . s up r- c on ] F e b he heavy fermion superconductor URu Si has been extensively studied to reveal the or-der parameters of the enigmatic hidden order (HO) phase (with critical temperature T HO =17.5 K) and the coexisting unconventional superconducting (SC) phase (with critical temper-ature T c = 1.5 K)[1, 2]. In the HO phase of URu Si , the small size of the (possibly extrinsic)magnetic moment previously detected by neutron scattering measurements is inconsistentwith the magnitude of the large heat capacity anomaly at the transition[3]. Recent, thoughcontroversial, measurements of the magnetic torque[4], the cyclotron resonance[5], and theelastoresistivity[6] imply that HO phase has an electronic nematic character, reducing thefour-fold rotational symmetry of the tetragonal lattice structure to two-fold rotational sym-metry. Although the crystal lattice is also weakly forced to transform into an orthorhombicsymmetry in ultra-pure samples[7], the structural phase transition temperature differs from T HO at hydrostatic pressure[8]. In response to these experiments, many theoretical modelsfor the order parameter in the HO phase have been proposed, such as multipole orders[9–13], but this order parameter is still not well understood. Further, although the HO phasecoexists with the SC phase, it is unclear whether and how these phases are correlated.Recent studies suggest that the SC order parameter of URu Si most likely possesses achiral d -wave symmetry[2]. Knight shift measurements[14, 15] and upper critical field H c measurements[16] both suggest a spin singlet state. Further, nodal gap structures were indi-cated by point contact spectroscopy measurements[17], electron specific heat[18–20], NMRrelaxation rate [21], and thermal transport measurements[22, 23]. Thermal conductivitymeasurements suggested the presence of a horizontal line node L H in the light hole bandand point nodes 2 P at the north and south poles in the heavy electron band[22, 23]. Simi-larly one electronic specific heat measurement also suggested the presence of point nodes inthe heavy electron band[19], but a recent experiment detected the line node L H in the heavyelectron band[20]. In addition, a largely enhanced Nernst effect has been observed above T c ,which was explained as an effect of chiral phase fluctuations[24]. Spontaneous time-reversalsymmetry breaking in the SC phase was revealed by a polar Kerr effect measurement[25].In addition, ferromagnetic (FM) impurity phases also have been implicated by nonlocalmagnetization measurements[26, 27] and one polar Kerr effect measurement[25].Here we sought to clarify the local time-reversal symmetry, the correlation between theHO and the SC phases, and the SC pairing symmetry in URu Si by examining local mag-netic fluxes and local superfluid responses. We used a local magnetic probe microscope called2 scanning Superconducting QUantum Interference Device (SQUID) microscope (Fig. 1a).Scanning SQUID microscopy (SSM) has been used to scan the local magnetization of candi-date chiral superconductors, which provided limits on the size of chiral domains by comparingexperimental noise with theoretically expected magnetization[28], and to image the mag-netism in the superconducting ferromagnet UCoGe[29]. SSM also revealed stripe anomaliesin the susceptibility along twin boundaries near T c in iron-based superconductors[30, 31] anda copper oxide superconductor[32]. Recently, local anisotropic vortex dynamics along twinboundaries were observed via SSM in a nematic superconductor FeSe[30]. In addition, thelocal London penetration depth λ can be estimated by the scanning SQUID height depen-dence of the local susceptibility[33]. Therefore, SSM provides information of spontaneousmagnetism, rotational symmetry of lattice structures, and superconducting gap structuresin situ.We used SSM to locally obtain the dc magnetic flux and ac susceptibility on the cleavedc-plane of single crystals of URu Si (Figs. 1b,c) at temperatures varying from 0.3 K to 18 Kusing a Bluefors LD dilution refrigerator[34]. Bulk single crystals of URu Si were grown viathe Czochralski technique and electro-refined to improve purity[25]. Our scanning SQUIDsusceptometer had two pickup loop (PL) and field coil (FC) pairs (Fig. 1a) configured witha gradiometric structure[35]. The PL provides the local dc magnetic flux Φ in units of theflux quantum Φ = h/ e , where h is the Planck constant and e is the elementary charge.The PL also detects the ac magnetic flux Φ ac in response to the ac magnetic field He iωt ,which was produced by an ac current of | I ac | = 3 mA at 150 Hz through the FC, using anSR830 Lock-in-Amplifier. Here we report the local ac susceptibility as χ = Φ ac / | I ac | in unitsof Φ /A and the local flux Φ as φ = Φ / Φ .We cooled samples from T = 5 K to T = 0.5 K with a dc magnetic field to producethe vortices. Then we observed inhomogeneity in the local susceptibility (Fig. 1d, sample2) and the local magnetic flux (Fig. 1e, sample 2). Strong diamagnetic susceptibility dueto the Meissner effect was only detected inside the sample(Fig. 1d); the inhomogeneity of χ mainly results from surface roughness(Fig. 1d). In contrast to the almost homogeneousMeissner effect observed on the whole sample, we detected FM domains on the right side ofthe sample, and many vortices on the left side(Fig. 1e, sample 2).We also observed local vortex dynamics of sample 1(Figs. 1h,i) and of sample 2(Fig. 1l).Figures 1f and 1g schematically show the values of φ and χ expected for an isolated vor-3ex if the vortex pinning potentials U are anisotropic or isotropic, respectively. Local vortexpinning potentials can be inferred from scanning SQUID measurements of isolated vortex dy-namics by modeling a simple quadratic pinning potential U (∆ x, ∆ y ) = ( k x ∆ x + k y ∆ y ),where k x and k y are the vortex pinning force constants and ∆ x and ∆ y are the displacementof the vortex center from the equilibrium point[30]. Note that screening from the SC shields FIG. 1: SSM imaged inhomogeneous magnetic fluxes, superfluid response, and anomalous isolatedvortex dynamics. (a) PL and FC of our SQUID susceptometer are covered with superconductingshields except for the loop area to detect local magnetic flux. (b),(c) Optical images of (b) sample 1and (c) sample 2. We examined scanning SQUID measurements at flat regions A, B and C. (d),(e)In sample 2, (d) χ and (e) φ values acquired at T = 0.5 K. (f),(g) SSM directly images isolatedvortex dynamics. Schematics of SSM measuring φ and χ over an isolated vortex, where U is (f)anisotropic or (g) isotropic. (h),(i) χ values over an isolated vortex acquired at (h) the star mark ofregion A at T = 1.0 K, and (i) the plus mark of region A at T = 1.2 K. (j),(k) Simulated χ valuesobtained by using (j) k x = 107.7 nN/m, k y = 19.9 nN/m, and (k) k x = k y = 17.6 nN/m to capture(h) and (i), respectively. (l) Local rotational symmetry of U varies randomly on a microscopicscale. χ values in region C at T = 1.2 K. Black open circles and red double ended arrows indicatethe isotropic and anisotropic vortex dynamics, respectively, which were observed at 1 K in differentcooling cycles. The full scale variation in χ in images of (h)-(l) is 0.7 Φ /A. Black single endedarrows indicate a axis.
4n the probe provide an additional asymmetry, which we reproduce in our numeric simu-lations. Thus, local ac susceptibility scans reveal the local rotational symmetry of pinningpotentials. We observed two types of χ images around an isolated vortex in different loca-tions of region A (Fig. 1h,i). The anisotropic data (Fig. 1h) look similar to the anisotropicvortex dynamics ( k x (cid:54) = k y ) observed by our similar measurement of SSM in FeSe[30], buton the other hand, the isotropic data (Fig. 1i) look similar to the isotropic vortex dynamics( k x = k y ) numerically simulated in [30]. Our simulations reproduced the experimental data(Fig. 1j, anisotropic; Fig. 1k, isotropic)[34]. Our measurements and simulations revealedthat vortex pinning potentials had four-fold or two-fold rotational symmetries at differentlocations in the same sample on a microscopic scale(Figs. 1h-l).Two types of vortex dynamics, anisotropic (Fig. 1h) and isotropic (Fig. 1i), were observedwith various orientations at different locations of sample 1. The observed vortex pinningpositions were not ordered. The observed vortex responses to an applied force are modeled bysimulations with isotropic pinning potentials (Fig. 1j) and two-fold rotationally symmetricpinning potentials (Fig. 1k). One scenario, which causes locally isotropic and anisotropicvortex dynamics, is that local strain caused by local defects in the tetragonal crystal structuredrives the anisotropic vortex pinning forces. This scenario is consistent with our data:the susceptibility images acquired near T c did not show the stripes along potential twinboundaries (Figs. 2b,c and Supplemental Figs. 1a,c[34]) that were previously reportedin copper oxide[32] and iron-based superconductors[30, 31]. The sample may have had aslightly orthorhombic crystal structures, but if so, its effect on the local vortex dynamicswas so small that we could not detect it. Thus, we suspect that our observed anisotropicvortex pinning force may have been caused by local strain from point defects in our URu Si samples.Next, in order to examine correlations of the superconductivity, the ferromagnetism andthe HO in URu Si , we determined the temperature dependence of χ and φ at region A ofsample 1 (Figs. 2a-l). In the HO phase, χ and φ were homogeneous at T = 16.4 K, butFM domains appeared in the upper right area below the HO transition. In this region anincrease in the susceptibility χ was observed at 16.1 K (Figs. 2e,k), followed by a nearlyconstant magnetization φ below 15.0 K (Fig. 2d). In the coexisting SC + HO phase anegative χ appeared uniformly at 1.44 K (Fig. 2c). It is surprising that the FM domaincontinued to exist across T c and that it persisted even at 0.36 K, where the whole area5howed strong diamagnetic χ (Fig. 2a). When we plotted the temperature dependence of χ at two specific points, the FM domain showed a sharp peak at 16.1 K (Fig. 2m). Thedirection of magnetic flux at the FM domain could be reversed by cooling the sample in asmall applied dc magnetic field (Fig. 2n). There was no anomaly at T HO (Figs. 2m,n).Although our investigations uncovered FM domains, we detected no spontaneous currentpropagating along sample edges or chiral domains. The expected spontaneous magnetiza-tion, which is carried by the chiral edge current, may be estimated by considering the orbitalangular momentum of ¯ hl per Cooper pair, where ¯ h = h/ π and l =1 ( p -wave), 2 ( d -wave),or 3 ( f -wave)[36–38]. This estimate neglects Meissner screening and surface effects, whichwill reduce the size of the effect. If the superconducting gap of URu Si has chiral d -wavesymmetry, the spontaneous magnetization M c is given by e ¯ hln/ m ∗ (cid:39)
200 A/m, where n is the carrier density and m ∗ is the effective mass[22, 34, 39–42]. More careful calculationsof the chiral edge current based on Bogoliubov-de Gennes analysis[37, 38, 43] showed that FIG. 2: Superconductivity and ferromagnetism coexist locally. (a)-(l) Appearance of ferromagneticdomains and superconducting state visualized in (a)-(f) χ images and (g)-(l) φ images at T = 0.36-16.4 K in region A of sample 1. (m) Ferromagnetism did not suppress superconductivity. χ above 15 K were plotted as 5 times experimental values to make these data easily viewable. (n)Ferromagnetic domain fields were oriented along the c -axis. χ and φ measured after field cooling,where µ H ∼ µ is the permeability of free space. l ≥ c axis including Meissner screening[44]. For large domains ( > ∼ µ m), the scanning SQUID could resolve individual domainboundaries. The expected magnetic flux along the domain boundary for our experimentalsetup is estimated as ∼
100 mΦ from the expected spontaneous magnetization of M c =200 A/m and could be as low as 20 A/m [ ∼
10 mΦ ] after accounting for surface effects andmultiple current modes[34]. For random domains of size of L = 1 µ m, the expected magneticflux would have a random varying sign (depending on the local domain orientations) witha magnitude of about 4.1 mΦ [0.4 mΦ ] for M c = 200 A/m [20 A/m][34]. The observedmagnetic flux far from the FM domains was ∼ in the PL, and its magnetic fluxdensity was 3.5 × − T. For the expected spontaneous magnetization of M c = 200 A/m [20A/m], we obtain a domain size limit of L ≤
250 nm [1.1 µ m], which is comparable to the sizeof our PL. It would be surprising to find domains that are so similar in size to the naturallength scales of the superconductivity. Therefore, our measurements set an upper limit onspontaneous magnetization that suggests that chiral superconductivity, if present, does notresult in the estimated magnetic flux. However, some effects, such as surface effects or smalldomain structures, may have suppressed the spontaneous magnetization to levels below oursensor’s detection limit.A FM signal was previously studied as an impurity effect[26, 27]. Amitsuka et al . useda commercial SQUID magnetometer to detect three FM phases in URu Si , T ∗ = 120K, T ∗ = 35 K, and T ∗ = 16 . T ∗ phase was caused by thestacking faults of a Q = (1 , ,
0) antiferromagnetic phase with a small moment. High-pressure scattering measurements revealed that the small-moment antiferromagnetic phasewas spatially separated from the HO phase, and that the small moments originated fromthe small volume of the antiferromagnetic phase depending on the lattice ratio c/a [45, 46].Here, we clearly visualized that the T ∗ phase makes FM domains but find no evidence ofeither T ∗ or T ∗ phases (Figs. 2j-l). The FM domains are spatially inhomogeneous, becausepositive peaks in susceptibility were only observed locally (Fig. 2e). In the SC phase, the FM7omains coexist with superconductivity (Figs. 2a-c,g-i). It is difficult to obtain a zero-fieldcondition due to the long-range magnetic fields ( ∼ Si is robust againstFM domains and disorders such as impurities and local strain, which are believed to beresponsible for the FM T ∗ phase. It remains possible, however, that the SC and the FMphases are spatially separated on a nanoscopic scale. FIG. 3: Isotropic or anisotropic vortex dynamics were enhanced near T c , which are well explained byour simulation. (a)-(c) Temperature dependence of isotropic vortex dynamics in (a) experimental χ in region A of sample 1 (location denoted by a cross in Fig. 2f), and in (b) simulated χ with penetration depth obtained from the fitting of (c) the observed vortex field and pinningforce constants ( k x = k y ). (d)-(f) Temperature dependence of anisotropic vortex dynamics in (d)experimental χ at region C of sample 2, and in (e) Simulated χ with the penetration depths from(f) the observed vortex field and various constants ( k x (cid:54) = k y ). We experimentally obtained isotropic and anisotropic vortex dynamics (Figs. 3a,d) andvortex fields (Figs. 3c,f) and numerically simulated vortex dynamics (Figs. 3b,e). Weobtained the local London penetration depth by fitting the magnetic flux from an isolatedvortex[35] (Supplemental Fig. S3[34]). The simulations of the vortex dynamics have asystematically shorter spatial extent than experiments (Figs. 3 and Figs. S3[34]). Weignored these difference in the simulation, which may be caused by the error in the SQUIDsensor height. By applying a χ -test, we calculated the pinning force constants k x , k y as 5-50nN/m at T = 1 . − . k x = 1-30 nN/m and k y /k x = 5-108t T = 1 . − . k x > k y . All obtained isotropic pinningforce constants in regions A, B, and C had the same temperature dependence (Fig. 4a)[34].The temperature dependence of an isolated vortex pinning force has been discussed onlyin non-local measurements at small fields[47, 48], but here we directly measured it. Weuse the hard core model [47], where an isolated vortex cylinder core is pinned at a normalconducting small void, to fit the temperature dependence of an isolated vortex pinning forcewith constants k ∝ (1 − ( T /T c ) ) m , where m depends on the dimensions of the small void.We obtain m = 2 from the best fit in Fig. 4(a), which indicates that our samples includesmall voids of roughly the same size as the coherence length[47], ∼
10 nm[49]. The existenceof nano-scaled voids supports our hypothesis that the local strain causes anisotropic andisotropic vortex dynamics at different locations of a URu Si sample. FIG. 4: (a) Vortex pinning force constants at three regions had the temperature dependenceof (1 − ( T /T c ) ) . (b),(c) Superfluid density had a linear- T dependence at low temperature. (b)Temperature dependence of the penetration depth at three points. (c) Temperature dependence ofnormalized superfluid density from the penetration depths in (b), with λ (0) = 1.0 µ m. The dottedblack line is the single s-wave gap BCS model for reference. The solid green line and dashed blueline are the two-band models for k z ( k x + ik y ) (light hole, heavy electron) with the indicated gapenergies ∆ h,e in a unit of k B T c to capture the experimental data. The local London penetration depth λ was determined by fitting the height dependence9f susceptibility[33] (Supplemental Fig. S4[34]). λ at P2, P3, and P4 each saturated toapproximately 1.0 ± µ m at zero temperature (Fig. 4b). These λ values are quantita-tively consistent with previous reports of λ = 0.7-1.0 µ m from measurements of muon spinrelaxation[14] and the estimate λ = (cid:113) m ∗ /µ ne = 1.1 µ m[34]. We calculated the localsuperfluid density n s = λ (0) /λ ( T ) from the experimentally obtained λ at P2, P3, and P4(Fig. 4c). Here we determined T c at P2, P3, and P4 as 1.50, 1.41, and 1.34 K, respectively,by defining these as the temperatures where the superfluid density becomes almost zero.The superfluid density varied spatially near T c , but all superfluid density values linearlyincreased as the normalized temperature decreased with temperature (Fig. 4c).The temperature dependence of the superfluid density in unconventional superconductorsis estimated by the semi-classical approach with an anisotropic gap function[34, 51]. Ourresults deviate from the numerically calculated superfluid density of the single band isotropic s -wave pairing symmetry model (BCS model)(Fig. 4c). The calculated curves for d -wavemodels are roughly consistent with our experimental results (Supplemental Fig. S5)[34, 50].However, they do not completely capture the behavior near T c . In order to explain thisdifference, we used the two-band model n s = xn h + (1 − x ) n e , where x = 0 .
87 is the ratioof the electron and hole mass, n h is the light hole band superfluid density, and n e is theheavy electron band superfluid density[51, 52]. Here we fit the experimental data witha model using chiral d -wave symmetry on the light hole and heavy electron bands withtwo free parameters of superconducting gaps ∆ h (hole band)and ∆ e (electron band)[34],which well explain the experimental results (Fig. 4c, Supplemental Fig. S6a[34]). Thefits to all d -wave symmetry two band models showed nearly identical results with differentparameters (Supplemental Figs. S6b-e,S7a-b,S8[34]), but the two-band isotropic s -wavemodel’s fitting results were markedly different from the experimental results (SupplementalFig. S7c,S8)[34]. In particular, the values of ∆ h and ∆ e , which were used in Fig. 4c, arealmost same as values of ∆ h = 1 . k B T c and ∆ e = 4 k B T c that were obtained from fits to thelower critical field H c along the a axis, which was measured with a Hall bar measurement[52].While we expect n s to exhibit the same temperature dependence as H c along the c axis, theHall bar measurement report an anomalous kink structure at 1.2 K [52], which we did notobserve in Fig. 4c. This difference may be a benefit of local measurements. For example,FM domains may affect H c measurement only along the c axis; here, FM domain fields hadmagnetic anisotropy along the c axis (Figs. 1e, 2g-l, and Supplemental Fig. S1[34]) and the10mplitude of a FM domain field is of the same order as the amplitude of H c along the c axis at 1.3 K[52]. Thus, our model and experimental data clearly suggest the existence ofnodal gap structures in URu Si , but it is difficult to distinguish distinct types of nodal gapstructure by our data because the slope of linear- T superfluid density can be adjusted bythe gap energies, which are free fit parameters in our model.In summary, we have locally observed FM domains coexisting with superconductivity,local pinning potentials, and linear- T superfluid densities in URu Si on a microscopic scale.This superconductivity coexists robustly with inhomogeneous ferromagnetism on a micronscale, although we cannot tell if they coexist in the same physical volume on nanometerscales. Further, we detected no spontaneous magnetization associated with chiral domainsin the SC phase. The obtained linear- T superfluid density is well explained by d -wave models,but not by s -wave models. Taken together, these results provide new evidence for a nodalgap structure and robust superconductivity coexisting on micron scales with inhomogeneousferromagnetism and place limits on the size of possible chiral domains in URu Si .The authors thank Ian R. Fisher and Steven A. Kivelson for fruitful discussion. Thiswork was primarily supported by the Department of Energy, Office of Science, Basic EnergySciences, Materials Sciences and Engineering Division, under Contract No. DE- AC02-76SF00515. Work at Los Alamos was performed under the auspices of the Department ofEnergy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division.Y.I. was supported by a JSPS Oversea Research Fellowship. [1] J. A. Mydosh and P. M. Oppeneer, Rev. Mod. Phys. , 1301 (2011).[2] T. Shibauchi, H. Ikeda and Y. Matsuda, Philos. Mag. , 3747 (2014).[3] C. Broholm, H. Lin, P. T. Matthews, T. E. Mason, W. J. L. Buyers, and M. F. Collins, Phys.Rev. B , 12809 (1991).[4] R. Okazaki, T. Shibauchi, H. J. Shi, Y. Haga, T. D. Matsuda, E. Yamamoto, Y. Onuki, H.Ikeda, and Y. Matsuda, Science , 331 (2011).[5] S. Tonegawa et al. , Phys. Rev. Lett. , 036401 (2012).[6] S. C. Riggs, M. C. Shapiro, A. V. Maharaj, S. Raghu, E. D. Bauer, R. E. Baumbach, P.Giraldo-Gallo, M. Wartenbe, and I. R. Fisher, Nat. Commun. , 6425(2014).
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The authors declare no competing financial interests. Correspondence and requests formaterials should be addressed to Y. I. ([email protected])14 upplemental Material for“Local observation of linear- T superfluid density and anomalousvortex dynamics in URu Si ”by Iguchi et al.
1. Chiral superconductivity and ferromagnetism in URu Si on scanning SQUIDmicroscopy We used scanning SQUID microscopy to locally obtain the dc magnetic flux and acsusceptibility of single crystals of URu Si at temperatures varying from 0.3 K to 18 Kusing a Bluefors LD dilution refrigerator. The dimensions of samples 1 (Fig. 1b) and 2 (Fig.1c) were ∼ . × . × .
05 mm and ∼ . × . × .
05 mm , respectively. The widestsurface on each sample was the cleaved c-plane. The pickup loops and field coils are coveredwith Nb superconducting shield layers (Fig. 1a); thus, only the magnetic flux going throughthe pickup loops in the SQUID loop was detected, and the external magnetic field was onlyapplied by the FCs. The inner radius of the pickup loop was 0.3 µ m, and the distancebetween the PL and the sample surface was ∼ µ m when the SQUID tip was touchingthe sample surface.The expected spontaneous magnetization, which is carried by the chiral edge current, maybe overestimated by the orbital angular momentum of ¯ hl per Cooper pair, where ¯ h = h/ π and l =1 ( p -wave), 2 ( d -wave), or 3 ( f -wave)[36–38]. For chiral d -wave symmetry, thespontaneous magnetization M c is given by M c = 12 V | (cid:90) r × jdV | = e ¯ hln m ∗ , (1)where current density j = nep/ m ∗ , p is the angular momentum, n is the carrier density, V is a sample volume, and m ∗ is the effective mass. By using m ∗ h ∼ m (hole band), theaveraged m ∗ e ∼ m (electron band), and a carrier density of 2 . × m − for both bands,we obtain M c = e ¯ hn (1 /m ∗ h + 1 /m ∗ e ) / m is the free electron mass[22, 39–42]. For large domains magnetized along the c axis where the domains are larger than thepenetration depth in size, the expected magnetic field produced by M c sgn ( x ) oriented in15 IG. S1: Superconductivity coexisted with ferromagnetism on a microscopic scale. (a),(b) Tem-perature dependence of (a) susceptibility and (b) magnetic flux images in region B of sample 1.(c),(d) Temperature dependence of (c) susceptibility and (d) magnetic flux images in sample 2.Dashed lines are guides for the eye to separate the FM domain from the paramagnetic domain. the z -direction at height z is given by[44] B z ( x, z ) = µ M c π (cid:90) dk − ike ikx e −| k | z (cid:113) /λ + k (cid:16) | k | + (cid:113) /λ + k (cid:17) . (2)For random domains magnetized along the c axis where the domains are comparable to orsmaller than the penetration depth in size, the expected magnetic field near the domainboundary at x = 0 produced by M c at height z is given by B c = 15 πV λ ( z + λ ) (cid:18) µ M c π (cid:19) , (3)16 IG. S2: Ferromagnetic signals originated from the ferromagnetic domains. Temperature depen-dence of (a) magnetic flux and (b) susceptibility at the four positions. Inset indicates the positionson the magnetic flux image in region A at 15.0 K from Fig. 2b. where V is the domain volume[44]. For our experimental setup, we used z = 0.5 µ m and λ = 1.0 µ m to roughly estimate the limit of domain size of L .
2. Isolated vortex dynamics on scanning SQUID microscopy
We observed local vortex dynamics of sample 1(Figs. 1h,i and Figs. 3a,c) and of sample2(Fig. 1l and Figs. 3d,f). When we simulated vortex dynamics induced by ac magnetic fieldfrom the FC with these anisotropic and isotropic pinning potentials by calculating Φ ac = dφdx ∆ x + dφdy ∆ y , our simulations reproduced the experimental data (Fig. 1j, anisotropic; Fig.1k, isotropic). The gradient of vortex field dφdx in our simulation was numerically calculatedusing the experimentally obtained London penetration depth[35]. The displacement of ∆ x satisfies the equilibrium condition of forces k x ∆ x = F x , where F x is the Lorentz force on anisolated vortex produced by an ac magnetic field[30]. Here the four-fold rotational symmetryin the isotropic data (Fig. 1i) was broken, but this symmetry breaking was explained by thescreening effect of the asymmetric superconducting shields in our simulations (Fig. 1k). Itwas also verified that the shield screening effect could not qualitatively change the apparent17xis of anisotropic vortex dynamics in [30]. The temperature dependence of pinning forceconstants at region A, B, and C were plotted by using the local T c = 1.48, 1.35, and 1.4 Kwere defined as the temperature where χ becomes zero (Fig. 4a). FIG. S3: Isotropic vortex dynamics model captured the observed isotropic χ images. (a) Isotropicsusceptibility images were acquired over (e) the isolated vortex at T = 1.0, 1.1, 1.2, and 1.3 K inregion A of sample 1 (location denoted by a cross in Fig. 2f). (b) Simulated susceptibility valueswere calculated by using (c),(d) χ -test with penetration depths obtained from (f) the fitting ofthe experimental data and various spring constants ( k x = k y ) from Fig. 4a. Two scenarios cause locally isotropic and anisotropic vortex dynamics. In the first sce-nario, local strain caused by local defects in tetragonal structure drives the anisotropic vortexpinning forces. As mentioned in the main text, this scenario is consistent with our data. Inthe second scenario, the twin boundaries drives the anisotropic vortex pinning forces, butthe vortex pinning forces inside a large twin domain are isotropic. This second scenario18s not consistent with our observation that the anisotropic vortex pinning forces were ori-ented along various directions at different locations (Fig. 1l). The anisotropy caused bytwin boundaries and pinning locations in orthorhombic structure is expected to be along thetetragonal [100] or [010] directions[30]. In addition, this second scenario is not consistentwith the lack of strip anomaly in our susceptibility scans as mentioned above. However, ananomaly of susceptibility in URu Si could be smaller than anomalies in iron-based super-conductors, because URu Si has an order of orthorhombicity that is two orders of magni-tude smaller than that of BaFe As -based iron-pnictide superconductors[7]. Therefore, ourobserved anisotropic vortex pinning force could be caused by local strain.
3. Superfluid density measurements and analysis on scanning SQUID microscopy
In order to obtain the local London penetration depth, we fitted the SQUID height de-pendence of local susceptibility (Supplemental Fig. S4) by using the theoretical equation (7)from Kirtley et al. [33], which includes the London penetration depth as a fitting parameter.Here the SQUID height z included the thickness of the shield layer as 0.4 ± . µ m, theerror of which equally shifts the penetration depth by ± . µ m. However the error in thesuperfluid density was sufficiently small to be ignored.The electric current density of electrons and holes is given by j = − e (cid:32) n e m ∗ e + n h m ∗ h (cid:33) A (4)where we assume all electrons and holes form Cooper pairs and averaged canonical momen-tum is zero, and n e , n h are the carrier densities of electrons and holes. Substituting j intothe Maxwell equation, we obtain the London penetration depth by λ = (cid:115) m µ e n e m /m ∗ e + n h m /m ∗ h . (5)The temperature dependence of the superfluid density in unconventional superconductorsis estimated by the semi-classical approach with an anisotropic gap function ∆( k , T ) = g ( k )∆ ( T ), where ∆ ( T ) = ∆ (0) tanh ( πk B T c / ∆ (0) (cid:113) a ( T c /T − (0) and a [51]. We calculatedthe superfluid density for all d -wave symmetry gap functions in tetragonal symmetry[50].The chiral d -wave symmetry model k z ( k x + ik y ) was given by g = 2 sin θ cos θ , k x k y model by19 = sin 2 φ , k x − y model by g = sin 2 φ − π/ k x k z model by g = sin θ sin φ , and k z ( k x + k y )by g = sin θ sin φ − π/
4. The calculated superfluid density of single band s -wave and d -wavesymmetries were shown in Supplemental Fig. S5.We also fitted the experimental data with the two band model with two free parameters∆ h and ∆ e , where ∆ ( T ) = ∆ i tanh ( πk B T c / ∆ (0) (cid:113) a ( T c /T − i = e,h. The calculatedcurves for the model of chiral d -wave symmetry at both electron and hole bands well explainthe experimental result at P2 with ∆ h = 1 . k B T c and ∆ e = 5 k B T c , and the results at P3 andP4 with ∆ h = 1 . k B T c and ∆ e = 10 k B T c (Fig. 4c, Supplemental Fig. S6a). We also fittedthe data with other two-band chiral d -wave models (such as L H + 2 P at the light hole bandand 2 P at the heavy electron band; Supplemental Figs. S6b-e), other d -wave symmetrymodels (such as k x k z (or k z ( k x + k y )) and k x − k y (or k x k y ); Supplemental Figs. S7a,b), anda two-band isotropic s -wave model (Supplemental Fig. S7c). FIG. S4: Penetration depth was calculated from the relation between the normalized susceptibilityand the height of the pickup loop. Touchdown measurements of P2 at 1.5, 1.4, 1.0, and 0.35 K.Closed circles are experimentally obtained data. Solid lines are fitting curves. Inset shows theschematic of our experimental setup. The green area denotes the thickness of the shield layer. IG. S5: Simulated superfluid density temperature dependence from single band models did notcapture details of temperature dependence of experimental data. ( The red squares, blue trian-gles, and green circles are measured data at P2, P3, and P4, respectively. All gap energies wererepresented in the unit of k B T c . IG. S6: Two band gap models of chiral d -wave k z ( k x + ik y ) fitted well our experimental resultsof the temperature dependence of superfluid density. Solid lines are total superfluid densities ofhole and electron bands. Dashed lines are superfluid densities of hole band. Dash-dotted lines aresuperfluid densities of electron band. (a) L H + 2 P (hole band) and L H + 2 P (electron band). (b) L H + 2 P (hole band) and 2 P (electron band). (c) L H + 2 P (hole band) and L H (electron band). (d) L H (hole band) and L H + 2 P (electron band). (e) L H (hole band) and 2 P (electron band). ( Thered squares, blue triangles, and green circles are measured data at P2, P3, and P4, respectively.All gap energies were represented in the unit of k B T c . IG. S7: Two band gap models of time-reversal symmetric d -wave symmetries fitted better ourexperimental results of the temperature dependence of superfluid density than two band modelsof isotropic s -wave. (a)-(c) Simulated superfluid density temperature dependence from two-bandgap models. Solid lines are total superfluid densities of hole and electron bands. Dashed lines aresuperfluid densities of hole band. Dash-dotted lines are superfluid densities of electron band. (a)Two band models of k x k z or k z ( k x + k y ). L H + 2 L V (hole band) and L H + 2 L V (electron band).(b) Two band models of k x − k y or k x k y . 4 L V (hole band) and 4 L V (electron band). (c) Twoband models of isotropic s (hole band) and isotropic s (electron band). ( The red squares, bluetriangles, and green circles are measured data at P2, P3, and P4, respectively. All gap energieswere represented in the unit of k B T c . IG. S8: Fittings of d -wave models were better than that of isotropic s -wave models for two bandgap models. (a),(b) Differences between simulated data and measured data at (b) P2 and (c)P4. All gap energies were represented in the unit of k B T c . (c) Error mean square of difference ofsimulated data and measured data at P2 and P4 for two-band gap models.. (c) Error mean square of difference ofsimulated data and measured data at P2 and P4 for two-band gap models.