Universal inhomogeneous magnetic-field response in the normal state of cuprate high-Tc superconductors
Z. Lotfi Mahyari, A. Cannell, E.V.L. de Mello, M. Ishikado, H. Eisaki, Ruixing Liang, D.A. Bonn, J.E. Sonier
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Universal inhomogeneous magnetic-field response in the normal state of cupratehigh- T c superconductors Z. Lotfi Mahyari and A. Cannell
Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
E.V.L. de Mello ∗ Instituto de F´ısica, Universidade Federal Fluminense, Niter´oi, RJ 24210-340, Brazil
M. Ishikado
Research Center for Neutron Science and Technology, Tokai, Naka, Ibaraki, Japan 319-1106
H. Eisaki
National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki, Japan 305-8568
Ruixing Liang and D.A. Bonn
Department of Physics and Astronomy, University of British Columbia,Vancouver, British Columbia, Canada V6T 1Z1 andCanadian Institute for Advanced Research, Toronto, Canada M5G 1Z8
J.E. Sonier
Department of Physics, Simon Fraser University,Burnaby, British Columbia, Canada V5A 1S6 andCanadian Institute for Advanced Research, Toronto, Canada M5G 1Z8 (Dated: July 21, 2018)We report the results of a muon spin rotation ( µ SR) study of the bulk of Bi x Sr − x CaCu O δ , aswell as pure and Ca-doped YBa Cu O y , which together with prior measurements reveal a universalinhomogeneous magnetic-field response of hole-doped cuprates extending to temperatures far abovethe critical temperature ( T c ). The primary features of our data are incompatible with the spatiallyinhomogeneous response being dominated by known charge density wave (CDW) and spin densitywave (SDW) orders. Instead the normal-state inhomogeneous line broadening is found to scale withthe maximum value T max c for each cuprate family, indicating it is controlled by the same energy scaleas T c . Since the degree of chemical disorder varies widely among the cuprates we have measured,the observed scaling constitutes evidence for an intrinsic electronic tendency toward inhomogeneityabove T c . PACS numbers: 74.72.-h, 74.25.Ha, 76.75.+i
I. INTRODUCTION
Experiments probing the normal state of high- T c cuprate superconductors have provided evidence for thepresence of electronic nematicity, fluctuating stripes, CDW fluctuations, weak magnetic order and su-perconducting fluctuations (SCFs) or phase-incoherentCooper pairs.
At present there is a quest for com-monality amongst these findings, and debate over thetemperature range of SCFs above T c .The initial discovery of a vortex-motion contributionto the Nernst signal in the normal state of cupratesuperconductors, advocated the occurrence of phase-fluctuating superconductivity up to temperatures sev-eral times T c . This finding was subsequently supportedby high-resolution torque magnetometry experiments,which detected field-enhanced diamagnetism at equallyhigh temperatures. More recently, precursor super-conductivity persisting up to 180 K has been inferredfrom the infrared c -axis response of R Ba Cu O y ( R = Y, Gd, Eu). Yet other studies have argued that a no-ticeable contribution of SCFs to the Nernst response ofcuprates is present only at temperatures 10 to 25 K above T c . A narrow region of SCFs above T c has also beenconcluded from AC conductivity studies. Distinct from these bulk measurements is a scanningtunneling microscopy (STM) study of the surface ofBi x Sr − x CaCu O δ (BSCCO), which shows the nu-cleation of pairing gaps in nanoscale regions persistingto temperatures above the detection of diamagnetism. Although the lack of consensus about the temperatureextent of SCFs above T c may be attributed to varyingdegrees of sensitivity of different techniques to inhomo-geneous superconducting correlations, BSCCO, and moreso its surface, are highly disordered. Whether the samenanoscale electronic inhomogeneity observed at the sur-face of BSCCO above T c is also present in the bulk, andwhether it has any relevance to other cuprate supercon-ductors are questions of fundamental importance.Expanding on earlier measurements, we have usedtransverse-field (TF) µ SR to investigate the situation inthe bulk of BSCCO and optimally-doped and Ca-dopedYBa Cu O y (YBCO) single crystals. The implanted pos-itive muon is a pure local magnetic probe, and relaxationof the time-dependent TF- µ SR signal results from a dis-tribution of internal magnetic field n ( B ). The line widthof n ( B ) and the corresponding relaxation rate are re-duced as the internal magnetic field becomes more uni-form, but are also diminished by fluctuations of the localfield. II. EXPERIMENTAL DETAILS
The samples are all plate-like single crystals. The pureand Ca-doped YBCO single crystals were grown by a self-flux method in fabricated BaZrO crucibles, as describedelsewhere, and assembled into mosaics of 6 to 10 sin-gle crystals from the same growth batch. Typical samplesizes were 5 × × . − . . Zero-field (ZF) µ SR mea-surements on some of the YBCO samples are presentedin Ref. 22. The BSCCO samples studied are of similardimensions, and consist of 1 or 2 single crystals grown bythe traveling-solvent-floating-zone method. Sample com-positions of Bi x Sr − x Ca Cu O δ with x = 0 . O , SrCO , CaCO ,and CuO as starting materials. After pre-melting thepolycrystalline rod, crystal growth was carried out in airand at a feed speed of 0.15 to 0.20 mm/h for about 3weeks. The doping level was adjusted by tuning the ex-cess oxygen content. The underdoped BSCCO samplewas annealed at 570 ◦ C under flowing N gas with lessthan 10 ppm oxygen concentration for 72 hrs. Overdop-ing was achieved by annealing at 400 ◦ C under an oxy-gen partial pressure of 2.3 atm for 72 to 250 hrs. Theoptimally-doped BSCCO sample was annnealed in airat 720 ◦ C for 24 hrs. A superconducting quantum in-terference device (SQUID) magnetometer was used formeasurements of T c .The TF- µ SR experiments were carried out on the M15surface muon channel at TRIUMF (located in Vancouver,Canada) using the so-called HiTime spectrometer, whichfeatures ultra-low background and high magnetic field ca-pabilities. The magnetic field H was applied parallel tothe c axis of the sample by a 7.0 T superconducting split-coil solenoid. Nearly 100 % spin-polarized positive muonswere implanted into the sample with the initial muon-spin polarization P (0) transverse to the direction of theapplied field (see Fig. 1). The muon magnetic momentundergoes Larmor precession at a frequency proportionalto the local internal magnetic field ( i.e. ω = γ µ B , where γ µ = 851 . P ( t ) reflectsthe distribution of internal magnetic fields experienced bythe muon ensemble, and is monitored through the detec- Electronic clockMuondetectorCuprateSingle Crystal(s) zxy s P (0) H m ++ e Right edetector + Left edetector + Up RightDownLeft VetoSample
FIG. 1: (Color online) Schematic of the TF- µ SR experimentalarrangement. The µ + beam passes through a thin plastic scin-tillator muon detector with its initial muon-spin polarization P (0) transverse to the direction of the applied magnetic field H , which creates a start pulse for an electronic clock. Themuons are subsequently implanted one-by-one in the sample,where they come to rest and Larmor precess in the local mag-netic field B . The time evolution of the muon spin polarization P ( t ), which is affected by both static and fluctuating internalmagnetic fields, is monitored via the detection of the decaypositrons (e + ). The detection of a decay positron stops theelectronic clock, and the corresponding elapsed-time bin of thepositron-detector histogram is incremented. The positron de-tectors are arranged in pairs on opposite sides of the sample.In our experiments, two pairs of positron detectors surround-ing the sample (Left and Right, and Up and Down) were used.In addition, the sample was mounted directly on a Veto de-tector, used to eliminate muons that missed the sample fromcontributing to the TF- µ SR signal. The sample covered ap-proximately 40 to 60 % of the Veto detector (The lower figureis a depiction of the counter arrangement facing the µ + beam).With the exception of the incoming muon detector, all detec-tors were contained with the sample inside a helium-gas flowcryostat. The TF- µ SR asymmetry spectrum A ( t ) = a P ( t ) isformed by combining the accumulated histograms of opposingpositron detectors.
944 946 948 950 9520.000.020.040.060.08 (c) R ea l A m p li t ude Frequency (MHz) T = 2.7 K
944 946 948 950 9520.00.10.20.30.4 (d) T = 120 K R ea l A m p li t ude Frequency (MHz) (b)
T = 120 K A sy mm e t r y Time ( s) -0.10-0.050.000.050.10 (a)
T = 2.7 K A sy mm e t r y Time ( s)
FIG. 2: (Color online) Representative TF- µ SR signals. (a)TF- µ SR asymmetry spectrum for optimally-doped ( p = 0 . H = 7 T and T = 2 . ∼ ◦ , come from the two pairs of opposingpositron detectors shown in Fig. 1 ( i.e. Up and Down, andLeft and Right). The solid black curves through the datapoints are fits described in the main text. (b) Same as (a),but for T = 120 K. (c), (d) Fourier transforms (with Gaus-sian apodization) of the asymmetry spectra, which providevisual depictions of the internal magnetic field distribution n ( B ) sensed by the muon ensemble. The frequency (horizon-tal axis) is related to the local internal magnetic field via therelation f = ( γ µ / π ) B . tion of the decay positrons of the implanted muons.Calibration measurements on pure Ag for H = 7 Tand H = 0 . µ SRsignal. Measurements on the cuprate samples at tem-peratures below T c were performed under field-cooledconditions, to generate the most uniform vortex lat-tice. At temperatures above T c , the TF- µ SR signal forall samples was found to be independent of whetherthe measurements were recorded under field-cooled orzero-field cooled conditions. Typical TF- µ SR signals forYBa Cu O . are presented in Fig. 2. For T < T c the re-laxation of the TF- µ SR signal is dominated by the spatialfield-inhomogeneity created by the vortex lattice, whichresults in a broad asymmetric internal magnetic field dis-tribution n ( B ). Conversely, the reduced relaxation ob-served for T > T c is associated with a narrow and sym-metric n ( B ). III. DATA ANALYSIS AND RESULTS
The TF- µ SR spectra were fit to A ( t ) = a P ( t ) = a G ( t ) cos( ω µ t + φ ) , (1) where a is the amplitude, φ is the phase angle betweenthe axis of the positron detector and the initial muon-spin polarization P (0), ω µ is the Larmor frequency and G ( t ) is a function that describes the relaxation of theTF- µ SR signal. In particular, G ( t ) = G nuc ( t ) G other ( t ),where G nuc ( t ) is a temperature-independent function dueto the distribution of random nuclear dipole fields, and G other ( t ) is a phenomenological function describing thesignal relaxation by other internal sources. For YBCOand BSCCO, G nuc ( t ) = exp( − ∆ t ), as is usually thecase for a dense system of randomly oriented magneticmoments. However, recent ZF- µ SR measurements showthat the nuclear contribution for LSCO deviates some-what from a pure Gaussian function, and the relax-ation of the TF- µ SR signal at T = 200 K has the func-tional form G ( t ) = exp[ − (Λ t ) β ], with 1 . ≤ β ≤ . Consequently, we assume G nuc ( t ) = exp( − λt ) n for LSCO,where λ and n are the values of Λ and β determined at T = 200 K in Ref. 20.For temperatures T < . T c , the internal mag-netic field distribution n ( B ) for all samples is asym-metric due to an arrangement of vortices, and satis-factory fits of the TF- µ SR signal were achieved with G other ( t ) = exp[ − (Λ t ) β ], where 1 . ≤ β ≤ .
8. How-ever, at higher temperatures n ( B ) is symmetric, and G other ( t ) is a pure exponential relaxation function, i.e. G other ( t ) = exp( − Λ t ). Hence we fit the TF- µ SR signalfor all samples above T c to A ( t ) = a G nuc ( t ) exp( − Λ t ) cos( ω µ t + φ ) . (2)Figure 3 shows representative data for the temperaturedependence of Λ. Below T c the dominant contributionto Λ is the spatially inhomogeneous field created by vor-tices, which depends on the inverse square of the in-planemagnetic penetration depth λ ab , the anisotropy, and thespatial arrangement of vortices. As shown in Fig. 4(a),the hole-doping dependence of Λ for YBCO and BSCCOat low temperatures resembles the doping dependenceof λ − ab . The value of Λ is significantly smaller forBSCCO, partly because of extreme anisotropy that per-mits significant wandering of the vortex lines along theirlength. For LSCO the observed decrease of Λ with increasedhole-doping in the range 0 . ≤ p ≤ .
176 [Fig. 4(a)] isopposite to the behavior of λ − ab ( p ). This is explainedby a recent small-angle neutron scattering study of LSCOshowing enhanced vortex-lattice disorder below p ∼ . Such random frozen disorder of the rigid vor-tex lines in LSCO broadens n ( B ) and enhances Λ. Theupturn of Λ at higher doping also opposes the behav-ior of λ − ab ( p ), which decreases beyond p ∼ . Thisis due to Curie-like paramagnetism that dominates Λin heavily-overdoped LSCO, but is also present in theunderdoped regime.
This contribution is evident inthe temperature dependence of Λ for optimally-dopedLSCO presented in Fig. 3(g). There is a similar Curieterm discernible in the bulk magnetic susceptibility of
50 100 150 200 2500.00.20.40.0000.0350.0700.1050.0000.0350.0700.105 50 100 150 200 250 c ( - e m u / g ) c ( - e m u / g ) T c LSCOp = 0.15
T (K) (g)(f) T c YBCOp = 0.165 (e)
BSCCOp = 0.16 T c
40 80 120 160 200 2400.000.020.040.060.080.10
YBCOp = 0.103p = 0.165Ca-YBCOp = 0.205p = 0.214
T (K) (d)
YBCOp = 0.103p = 0.165Ca-YBCOp = 0.205p = 0.214 ( s - T (K) (c)
40 80 120 160 200 240 2800.000.020.040.060.080.10
BSCCOp = 0.094p = 0.16p = 0.186p = 0.197 (b)
BSCCOp = 0.094p = 0.16p = 0.186p = 0.197 ( s - (a) FIG. 3: (Color online) (a) Temperature dependence of Λ at H = 7 T (solid circles) for BSCCO. (b) Blow-up of the high-temperature data from (a). Also shown is data for the p = 0 .
094 sample at H = 0 . T c determined by bulk magnetic susceptibility. (c), (d) Representative results for pure andCa-doped YBCO for H = 7 T. The data at p = 0 .
103 is from Ref. 18. (e), (f), (g) High-temperature behavior of Λ (solid circles)for optimally-doped BSCCO, YBCO, and LSCO. Also shown is the bulk magnetic susceptibility (light-blue curves) for theBSCCO and LSCO samples for H = 7 T applied parallel to the c axis. the p = 0 .
197 BSCCO sample (see Fig. 5), which maycontribute somewhat to Λ( T ) at this doping. The bulkmagnetic susceptibility of our BSCCO samples at H = 7Tresembles previous high-field measurements. This in-cludes the Curie-like contribution to the p = 0 .
197 sample,which is a common feature of heavily-overdoped cuprates.Shifting attention to the normal state, Fig. 4(b) showsa comparison of the hole-doping dependence of Λ at T = 1 . T max c . Despite BSCCO possessing a higher degreeof chemical disorder than YBCO, there is good agree-ment between the data sets for these two compounds.As shown previously, the Curie-like contribution to Λfor LSCO exhibits a dominant p -linear dependence above T c , with a slope ( d Λ /dp ) that weakens with increasing T .Measurements at higher temperature show that the p -linear contribution extends to lower doping. Figure 6shows the hole-doping dependence of Λ for LSCO above T c (solid red circles) obtained from fits of the TF- µ SRasymmetry spectra of Ref. 20 to Eq. (2). Here we stressthat the values of Λ differ from those in Ref. 20 due tothe removal of the nuclear dipole contribution to G ( t ),which must be done to compare the residual relaxation rate to the values for YBCO and BSCCO. The lineargrowth of Λ above p = 0 .
19 at T = 40 K is visible below p = 0 .
19 at high temperatures, and arises from the Curie-like paramagnetism.
Subtracting this linear contribu-tion from the hole-doping dependence of Λ at each tem-perature (yielding the open red circles in Fig. 6) results ina collapse of the Λ data for LSCO onto a universal curveshared with BSCCO and YBCO (Fig. 7). Note that forthe range of comparison 0 . ≤ p ≤ . H c for all three compounds. Theuniversal behavior for Λ suggests that T c and the sourceof the inhomogeneous magnetic-field response above T c are controlled by the same energy scale. IV. DISCUSSION
It may be surprising to some that a significant inho-mogeneous magnetic response occurs in YBCO above T c ,given that the mean free path of optimally-doped YBCOsingle crystals has been reported to be as large as 4 µ m. (b) Ca-dopedYBCOBSCCO ( s - ) Hole doping, p
T = 1.3T cmax
LSCO (a)
Ca-dopedYBCO BSCCO ( s - ) T = 0.11T cmax
LSCO
FIG. 4: (Color online) (a) Doping dependence of Λ for H =7 T and T = 0 . T max c , where T max c = 90, 94.1 and 38 K,for BSCCO, YBCO, and LSCO, respectively. The YBCOdata for p ≤ .
141 are from Ref. 18. For visual purposes therelaxation rate for BSCCO has been multiplied by a factor of2. (b) Results for T = 1 . T max c , where Λ is a pure exponentialrelaxation rate. The BSCCO (blue diamonds) and YBCO(green circles) data are for T = 120 K, and the LSCO data(solid red triangles) for T = 50 K. Unlike in (a), the BSCCOdata is not rescaled. The dashed line is a linear fit of the LSCOdata for p ≥ .
19, which describes the Curie-like contribution. p = 0.094 p = 0.186p = 0.16 c ( e m u / g - ) T (K) p = 0.197 BSCCO H = 7 T FIG. 5: (Color online) Temperature dependence of the bulkDC magnetic susceptibility of the p = 0 . H = 7 T applied parallel to the c -axis. T = 40 K (a) (d)
T = 80 K (b)
T = 50 K ( s - ) (e) T = 100 K (c)
T = 60 K p (f) T = 120 K p FIG. 6: (Color online) Hole-doping dependence of the expo-nential relaxation rate Λ for LSCO from fits to Eq. (2) (solidred circles). The solid black line in each panel is the best fitof a straight line to the data at p ≥ .
19. Subtracting thestraight line fit from Λ yields the open red circles.
However, this estimate is at low temperatures far below T c and in zero applied magnetic field. One of the reasonsthat the mean free path is so long at low temperatures isthat the superconducting gap reduces the available phasespace for impurity scattering. In fact there is a dra-matic increase of the scattering rate of optimally-dopedYBCO above 20 K, indicating a large reduction of themean free path with increasing temperature well beforethe normal state is reached. In addition, experiments ob-serving quantum oscillations in the electrical resistanceof underdoped YBCO indicate a significant reduction ofthe mean free path in the presence of an applied mag-netic field. For example, at 35 T, the mean free path ofYBa Cu O . at low temperatures is roughly estimatedto be only 0.016 µ m. Previous measurements on YBCO revealed a clear re-duction of Λ near p = 0 . where recent neutron scat-tering measurements for H = 0 show enhanced incom-mensurate CDW correlations, and nuclear magnetic res-onance experiments at H = 28 . Enhanced incommensurate SDWorder is also observed in LSCO near p = 0 . How-ever, slowing down of charge or spin fluctuations shouldincrease the value of Λ. Moreover, while Λ above T c in-creases with H [Fig. 3(b)], even fields well in excess of 7 Thave no effect on CDW or SDW fluctuations above T c . Hence CDW or SDW correlations, which are confined to anarrow range of doping and seem to compete with super-conductivity, cannot be the primary source of the resid-
T = 1.7T cmax
Hole doping, p
T = 1.3T cmax ( s - ) YBCOBSCCOLSCO T = 1.1T cmax
Ca-doped
FIG. 7: (Color online) Doping dependence of Λ for T = 1 . T max c . The p -linear dependent contribution to thedata for LSCO has been subtracted, as described in the maintext. The solid curves are guides to the eye. ual inhomogeneous line broadening shown in Fig. 7 thatis observed over the wide doping range 0 . < p < . when the applied magnetic field is strong(as in our experiments), the external field direction be-comes a natural quantization axis for the nuclear spinsystem. In this case the second moment of the nucleardipolar field distribution is independent of the magnitudeof the external field and is not altered by changes in thelocal EFG.Figure 8 shows that Λ persists above the pseudogaptemperature T ∗ at high doping, but vanishes below T ∗ in the underdoped region, indicating an origin not solelyrelated to the pseudogap phase. Instead the doping de-pendence and universal scaling of Λ with T max c suggestthat the inhomogenous field response above T c is asso-ciated with superconducting correlations. Like the vor-tex Nernst signal and diamagnetism observed by torquemagnetometry, Λ for
T > T c is reduced with in-creasing temperature and enhanced by the external field.The diamagnetism and vortex Nernst signal have beenattributed to SCFs, which are observed over a narrowrange of temperature above T c on a frequency scale of0.01 - 10 THz. At somewhat higher temperatures, (b) Bi Sr CaCu O T c T dia T p,max = 0.02 s -1 T ( K ) Hole doping, p
T* ( s -1 ) T* (a) ( s -1 ) T ( K ) T c T SCF = 0.02 s -1 Y Ca x Ba Cu O y FIG. 8: (Color online) (a) Contour bar graph of the variationof Λ with temperature and hole doping in pure and Ca-dopedYBCO for H = 7 T, achieved by interpolation of the Λ vs. T data sets. The black connected circles indicate the arbi-trary value Λ = 0 . µ s − . Also shown is the onset temper-ature T SCF for SCFs at p = 0 .
12 inferred from Nernst-effectmeasurements, and T ∗ (solid red line) extrapolated to higherdoping (dashed red line) from Ref. 2. (b) Similar schematicphase diagram for BSCCO, where T dia is the temperature atwhich torque magnetometry measurements detect the onset ofdiamagnetism and T ∗ is from Ref. 41. The T p,max curve in-dicates the temperature above which STM measurements on BSCCO find less than 10 % of the sample containingnanometer-sized regions with pairing gaps. our experiments show Λ varying on the order of 0.01 -0.1 µ s − (Fig. 7). Considering for a moment Λ to be adynamic relaxation rate due to fast fluctuations of thelocal field, Λ = γ µ h ( δB ) i /ν , where ν is the fluctuationfrequency and h ( δB ) i is the second moment of n ( B ) inthe static limit ( ν → . < Λ < . µ s − corresponds to a static linewidth of 0 . < p h ( δB ) i < .
037 T, which exceeds theline width of n ( B ) associated with the frozen vortex lat-tice below T c . Hence the vortex liquid inferred from theNernst signal above T c is not detectable by µ SR. Thisconclusion is supported by a µ SR study of the sizeablevortex-liquid regime of BSCCO below T c , where theTF- µ SR line width is severely narrowed by thermal vor-tex fluctuations. In stark contrast to the field dependenceof Λ observed above T c [Compare the data of p = 0 . H = 0 . H = 7 T in Fig. 3(b)], the TF- µ SR linewidth previously measured in the vortex-liquid phase ofBSCCO is reduced by stronger applied magnetic field.Although homogeneous SCFs do not produce µ SR linebroadening, spatially varying SCFs will. The local pair-ing observed by STM on BSCCO above T c is char-acterized by a distribution of gap sizes and a partialsuppression of the density of states at the Fermi energy N ( E F ), which vanish inhomogeneously with increasing T . In a conventional metal the Pauli susceptibility χ associated with the conduction electrons is proportionalto N ( E F ). Spatial variations in the magnitude of χ as-sociated with nanoscale regions of pairing cause inhomo-geneous broadening via the hyperfine coupling betweenthe µ + and the spin polarization of the surrounding con-duction electrons. Yet any such contribution to Λ mustbe minor, since above T c the depletion of N ( E F ) in theunderdoped regime is dominated by the spatially inho-mogeneous pseudogap.Regular inhomogeneous regions of SCFs may occur inthe bulk from a competition with some other kind of frag-ile order. One candidate is fluctuating stripes, which arecharacterized by a dynamical unidirectional modulationof charge, or both spin and charge. In this environmentinhomogeneous line broadening will result from muons experiencing distinct time-averaged local fields h B ( t ) i indifferent parts of the sample. Such will be the case formuons stopping inside regions with different strengthsof fluctuation diamagnetism. However, even if the re-gions in which SCFs persist are comparable throughoutthe sample, inhomogeneous line broadening results frommuons stopping in intermediate or surrounding areas.These muons experience the expelled time-averaged field,which diminishes in magnitude with increased stoppingdistance away from the diamagnetic regions. As shownin Fig. 8(b), Λ tracks the pairing gap coverage in BSCCOobserved by STM — where the latter is presumably sim-ilar regardless of whether the regions of SCFs maintaina regular pattern or are broken up into irregular-shapedpatches by disorder.While our findings strongly favor an interpretation in-volving inhomogeneous SCFs, current instrument detec-tion limits prevent us from determining whether the mag-netic response above T c is diamagnetic. Even so, ourresults show that there is an intrinsic electronic propen-sity toward inhomogeneity in the normal-state of high- T c cuprate superconductors, where even weak disorder ofthe kind found in YBCO is sufficient to spatially pin thenon-uniform electron liquid.We gratefully acknowledge informative and insightfuldiscussions with S.A. Kivelson, W.N. Hardy, A. Yazdaniand L. Taillefer. This work was supported by the NaturalSciences and Engineering Research Council of Canada,the Canadian Foundation for Innovation, the CanadianInstitute for Advanced Research, and CNPq/Brazil. ∗ Current Address: Department of Physics, Simon FraserUniversity, Burnaby, British Columbia, Canada V5A 1S6 V. Hinkov, D. Haug, B. Fauqu´e, P. Bourges, Y. Sidis, A.Ivanov, C. Bernhard, C.T. Lin, and B. Keimer, Science , 597-600 (2008). R. Daou, J. Chang, D. LeBoeuf, O. Cyr-Choini`ere, F. Lal-ibert´e, N. Doiron-Leyraud, B.J. Ramshaw, R. Liang, D.A.Bonn, W.N. Hardy, and L. Taillefer, Nature , 519-522(2010). C.V. Parker, P. Aynajian, E.H. da Silva Neto, A. Pushp,S. Ono, J. Wen, Z. Xu, G. Gu, and A. Yazdani, Nature , 677-680 (2010). G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N.B. Brookes, G.M. De Luca, A.Frano, D.G. Hawthorn, F. He, T. Loew, M. Moretti Sala,D.C. Peets, M. Salluzzo, R. Sutarto, G.A. Sawatzky, E.Weschke, B. Keimer, and L. Braicovich, Science , 821-825 (2012). J. Chang, E. Blackburn, A.T. Holmes, N.B. Christensen,J. Larsen, J. Mesot, R. Liang, D.A. Bonn, W.N. Hardy, A.Watenphul, M.V. Zimmermann, E.M. Forgan, and S.M.Hayden, Nature Physics , 871-876 (2012). B. Fauqu´e, Y. Sidis, V. Hinkov, S. Pailh`es, C.T. Lin,X. Chaud, and P. Bourges, Phys. Rev. Lett. , 197001(2006). Y. Li, V. Bal´edent, N. Bari˘si´c, Y. Cho, B. Fauqu´e, Y. Sidis,G. Yu, X. Zhao, P. Bourges, and M. Greven, Nature , 372-375 (2008). J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, andI. Bozovic, Nature , 221-223 (1999). Z.A. Xu, N.P. Ong, Y. Wang, T. Kakeshita, and S. Uchida,Nature , 486-488 (2000). K.K. Gomes, A.N. Pasupathy, A. Pushp, S. Ono, Y. Ando,and A. Yazdani, Nature , 569-572 (2007). Y. Wang, L. Li, M.J. Naughton, G.D. Gu, S. Uchida, N.P.Ong, Phys. Rev. Lett. , 247002 (2005). L. Li, Y. Wang, S. Komiya, S. Ono, Y. Ando, G.D. Gu,and N.P. Ong, Phys. Rev. B , 054510 (2010). A. Dubroka, M. R¨ossle, K.W. Kim, V.K. Malik, D. Mun-zar, D.N. Basov, A.A. Schafgans, S.J. Moon, C.T. Lin, D.Haug, V. Hinkov, B. Keimer, Th. Wolf, J.G. Storey, J.L.Tallon, and C. Bernhard, Phys. Rev. Lett. , 047006(2011). O. Cyr-Choini`ere, R. Daou, F. Lalibert´e, D. LeBoeuf, N.Doiron-Leyraud, J. Chang, J.-Q. Yan, J.-G. Cheng, J.-S.Zhou, J.B. Goodenough, S. Pyon, T. Takayama, H. Takagi,Y. Tanaka, and L. Taillefer, Nature , 743-745 (2009). F. Rullier-Albenque, R. Tourbot, H. Alloul, P. Lejay, D.Colson, and A. Forget, Phys. Rev. Lett. , 067002 (2006). M. S. Grbi´c, M. Poˇzek, D. Paar, V. Hinkov, M. Raichle,D. Haug, B. Keimer, N. Bariˇsi´c, and A. Dulˇci´c, Phys. Rev.B , 144508 (2011). L.S. Bilbro, R. Vald´es Aguilar, G. Logvenov, O. Pelleg, I.Boˇzovi´c, and N.P. Armitage, Nature Physics , 298-302 (2011). J. E. Sonier, M. Ilton, V. Pacradouni, C. V. Kaiser, S.A. Sabok-Sayr, Y. Ando, S. Komiya, W. N. Hardy, D. A.Bonn, R. Liang, and W. A. Atkinson, Phys. Rev. Lett. , 117001 (2008). G. J. MacDougall, A. T. Savici, A. A. Aczel, R. J. Birge-neau, H. Kim, S.-J. Kim, T. Ito, J. A. Rodriguez, P. L.Russo, Y. J. Uemura, S. Wakimoto, C. R. Wiebe, and G.M. Luke, Phys. Rev. B , 014508 (2010). C.V. Kaiser, W. Huang, S. Komiya, N.E. Hussey, T.Adachi, Y. Tanabe, Y. Koike, and J.E. Sonier, Phys. Rev.B , 054522 (2012). R. Liang, D.A. Bonn, and W.N. Hardy, Physica C ,105-111 (1998). J.E. Sonier, V. Pacradouni, S.A. Sabok-Sayr, W.N. Hardy,D.A. Bonn, R. Liang, and H.A. Mook, Phys. Rev. Lett. , 167002 (2009). W. Huang, V. Pacradouni, M.P. Kennett, S. Komiya, andJ.E. Sonier, Phys. Rev. B , 104527 (2012). J.E. Sonier, S.A. Sabok-Sayr, F.D. Callaghan, C.V. Kaiser,V. Pacradouni, J.H. Brewer, S.L. Stubbs, W.N. Hardy,D.A. Bonn, R. Liang, and W.A. Atkinson, Phys. Rev. B , 134518 (2007). W. Anukool, S. Barakat, C. Panagopoulos, and J.R.Cooper, Phys. Rev. B , 024516 (2009). E.H. Brandt, Phys. Rev. Lett. , 3213 (1991). T.R. Lemberger, I. Hetel, A. Tsukada, M. Naito, and M.Randeria, Phys. Rev. B , 140507(R) (2011). J. Chang, J.S. White, M. Laver, C.J. Bowell, S.P. Brown,A.T. Holmes, L. Maechler, S. Str¨assle, R. Gilardi, S. Ger-ber, T. Kurosawa, N. Momono, M. Oda, M. Ido, O.J. Lip-scombe, S.M. Hayden, C.D. Dewhurst, R. Vavrin, J. Gav-ilano, J. Kohlbrecher, E.M. Forgan, and J. Mesot, Phys.Rev. B , 134520 (2012). T. Watanabe, T. Fujii, and A. Matsuda, Phys. Rev. Lett. , 5848-5851 (2000). P.M.C Rourke, I. Mouzopoulou, X. Xu, C. Panagopou-los, Y. Wang, B. Vignolle, C. Proust, E.V. Kurganova, U.Zeitler, Y. Tanabe, T. Adachi, Y. Koike, and N.E. Hussey,Nature Physics , 455-458 (2011). G. Grissonnanche, O. Cyr-Choini`ere, F. Lalibert´e, S. Ren´ede Cotret, A. Juneau-Fecteau, S. Dufour-Beaus´ejour, M.-`E. Delage, D. LeBoeuf, J. Chang, B.J. Ramshaw, D.A.Bonn, W.N. Hardy, R. Liang, S. Adachi, N.E. Hussey,B. Vignolle, C. Proust, M. Sutherland, S. Kr¨amer, J.-H. Park, D. Graf, N. Doiron-Leyraud, and L. Taillefer,arXiv:1303.3856. A. Hosseini, R. Harris, S. Kamal, P. Dosanjh, J. Preston,R. Liang, W.N. Hardy, and D.A. Bonn, Phys. Rev. B ,1349-1359 (1999). T.S. Nunner, and P.J. Hirschfeld, Phys. Rev. B , 014514(2005). N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois,J.-B. Bonnemaisson, R. Liang, D.A. Bonn, W.N. Hardy,and L. Taillefer, Nature , 565-568 (2007). C. Jaudet, J. Levallois, A. Andouard, D. Vignolles, B. Vi-gnolle, R. Liang, D.A. Bonn, W.N. Hardy, N.E. Hussey, L.Taillefer, and C. Proust, Physica B T. Wu, H. Mayaffre, S. Kr¨amer, M. Horvati´c, C. Berthier,W.N. Hardy, R. Liang, D.A. Bonn, and M.H. Julien, Na-ture , 191-194 (2011). S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane,Phys. Rev. B , 172501 (2001). B. Lake, G. Aeppli, K.N. Clausen, D.F. McMorrow, K.Lefmann, N.E. Hussey, N. Mangkorntong, M. Nohara, H.Takagi, T.E. Mason, and A. Schr¨oder, Science , 1759-1762 (2001). O. Hartmann, Phys. Rev. Lett. , 832-835 (1977). S.L. Lee, M. Warden, H. Keller, J.W. Schneider, D. Zech,P. Zimmermann, R. Cubitt, E.M. Forgan, M.T. Wylie,P.H. Kes, T.W. Li, A.A. Menovsky, and Z. Tarnawski,Phys. Rev. Lett. , 922-925 (1995). J.K. Ren, X.B. Zhu, H.F. Yu, Y. Tian, H.F. Yang, C.Z. Gu,N.L. Wang, Y.F. Ren, and S.P. Zhao, Scientific Reports ,248 (2012). S.A. Kivelson, I.P. Bindloss, E. Fradkin, V. Organesyan,J.M. Tranquada, A. Kapitulnik, and C. Howard, Rev.Mod. Phys.75