Magnetic field-dependence of the basal-plane superconducting anisotropy in YBa2Cu4O8 from small-angle neutron scattering measurements of the vortex lattice
Jonathan S. White, Charlotte J. Bowell, Alistair S. Cameron, Richard W. Heslop, Joël Mesot, Jorge L. Gavilano, Simon Strässle, Lars Mächler, Rustem Khasanov, Charles D. Dewhurst, Janusz Karpinski, Edward M. Forgan
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec Magnetic field-dependence of the basal-plane superconducting anisotropy inYBa Cu O from small-angle neutron scattering measurements of the vortex lattice Jonathan S. White, − Charlotte J. Bowell, Alistair S. Cameron, Richard W. Heslop, Jo¨el Mesot, , , Jorge L. Gavilano, Simon Str¨assle, Lars M¨achler, Rustem Khasanov, Charles D. Dewhurst, Janusz Karpinski, and Edward M. Forgan Laboratory for Neutron Scattering, Paul Scherrer Institut, CH-5232 Villigen, Switzerland School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK Institute of Condensed Matter Physics, Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland Laboratory for Solid State Physics, ETH Z¨urich, CH-8093 Z¨urich, Switzerland Physik-Institut der Universit¨at Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen, Switzerland Institut Laue-Langevin, 6 rue Jules Horowitz, 38042 Grenoble, France (Dated: January 29, 2018)We report a study of the basal-plane anisotropy of the superfluid density in underdopedYBa Cu O (Y124), showing the effects of both the CuO planes and the fully occupied CuO chains.From small-angle neutron scattering measurements of the vortex lattice, we can infer the supercon-ducting (SC) properties for a temperature ( T ) range T = 1.5 K to T c and magnetic induction B from 0.1 to 6 T. We find that the superfluid density along a has a simple d -wave T -dependence.However, along b (the chain direction) the superfluid density falls much more rapidly with T andalso with increasing field. This strongly suggests the suppression of proximity-effect induced su-perconductivity in the CuO chains. In addition, our results do not support a common frameworkfor the low field in-plane SC response in Y124 and related YBa Cu O , and also indicate that anymagnetic field-induced charge-density-wave order in Y124 exists only for fields above 6 T. PACS numbers: 74.25.Uv, 74.72.-h, 61.05.fg 74.25.Ha
I. INTRODUCTION
The unifying structural constituent of all cuprate su-perconductors is the 2D CuO planes. The YBaCuOfamily, YBa Cu O − δ (Y123) and YBa Cu O (Y124),are special since they also host 1D CuO chains along thecrystal b -axis. For structurally well-ordered chains theassociated electronic states contribute to the Fermi sur-face. This makes YBaCuO a model system for studyinglow dimensional conductors in close proximity. Indeed,it has long been proposed that by proximity to the CuO planes, the CuO chain states become superconducting(SC) below T c . This is supported by the experimentallyobserved ab -plane SC anisotropy where the London pen-etration depth is shorter for currents flowing along the b -axis than the a -axis. Moreover, the observed anisotropyis larger in Y124, which displays two CuO chains perunit cell, than single-chained Y123.While proximity-effect (PE) models provide an expla-nation for the in-plane SC anisotropy, a single frameworkfor both Y123 and Y124 is not supported by experimen-tal evidence. In Y123, a clear electronic anisotropy inthe CuO plane reported from transport and ARPES studies implies SC chain states are not required to ex-plain the SC anisotropy. Measurements of the Londonpenetration depth λ give further information, since for acrystal axis i , n i ( T ) ∝ λ − i ( T ). In Y123, these show thatthe superfluid density n along both a and b axes has a d -wave temperature- ( T -) dependence, which disagreeswith the expectations of PE models. For Y124, the picture is somewhat unclear. It has been suggested that a positive low- T curvature of both n a ( T )and n b ( T ) observed by low field µ SR is evidence for atwo gap SC state in the CuO plane. However, fromother penetration depth measurements only n b ( T ) wasobserved to display a positive curvature, while n a ( T ) dis-played d -wave behaviour. These results were argued asevidence for PE-induced SC chain states in Y124.Here we present small-angle neutron scattering(SANS) measurements of the vortex lattice (VL) in Y124.Our measurements are conducted over a wide range ofmagnetic fields and temperatures, and the results castimportant light on the plane-chain interplay in YBaCuOcompounds. SANS experiments are a bulk probe of boththe VL structure and the microscopic field distribution,each of which depend on the SC length-scales. Our singlecrystal samples of Y124 are naturally twin-free, and thestoichoimetric oxygen content makes the CuO chains ef-fectively infinite in length. These properties suppress thevortex pinning effects seen in some Y123 samples, suchas those due to twin-planes and oxygen vacancies.
Therefore, the results of the present study on Y124 pro-vide valuable comparison to the intrinsic VL propertiesonly recently observed in twin-free and fully-oxygenatedYBa Cu O (Y1237). As will be seen in what follows, the VL in Y124 dis-plays remarkably different properties to those observed inY1237. This is apparent from measurements of both theVL structure presented in Sec. III A, and the magneticfield- and T -dependence of the VL form factor presentedin Sec. III B. In Sec. IV we discuss the new insights pro-vided by our results in connection with important topicsrelevant for both Y123 and Y124, such as proximity-effectinduced CuO chain superconductivity, the basal planesuperconducting anisotropy, and charge-density-wave or-der. Finally, a summary is presented in Sec. V. In Sec. IIwe begin by detailing the experimental method. II. EXPERIMENTAL METHOD
Single crystals of YBa Cu O were prepared as de-scribed in Ref. 17. Each had approximate size 0.8 x 0.3x 0.05 mm , and the longest side parallel to the b -axis.To obtain a sample mass suitable for neutron scatteringexperiments, 130 single crystals of total volume of 3.95 × − m were mounted onto a thin aluminum plate,each with their c -axis perpendicular to the plate, and a -axis vertical. The resulting mosaic had T c ≃ . T c ≃ T ) of 1.5 K. The crystal c -axis was parallel to the applied magnetic field ( µ H ),with both approximately parallel to the incident neutronbeam. Cold neutrons ( λ n = 0.6 to 1.66 nm, with FWHMspread of 10 %) were collimated over 8-14 m, and afterdiffracting from the sample, were counted on a position-sensitive multidetector placed 8-14 m away. Measure-ments carried out at T > T c were subtracted from thosedone at low T , in order to leave just the VL signal.Due to the intrinsically twin-free and stoichiometricproperties of the crystals, VL pinning is expected to besuppressed in our sample. Nonetheless, weak pinningdue to residual crystal defects may still be expected. Inthese circumstances, cooling through T c in a magneticfield that is weakly oscillating around the target field al-lows the vortices to overcome the pinning potential andattain a coordination closer to equilibrium. Therefore, allmeasurements reported in the paper were conducted onVLs prepared using a weakly oscillatory magnetic fieldcomponent of ± T . When at the intended T , thefield was held stationary at the target value when con-ducting the SANS measurements. III. RESULTSA. Vortex Lattice Structure
Fig. 1 shows VL diffraction patterns obtained fromY124 at T =1.5 K, and over the observable field range upto µ H k c = 6.0 T. At all fields, the VL forms a singledistorted hexagonal domain aligned with the crystal axes.We quantify the distortion in terms of the axial ratio of FIG. 1. (Color online) VL diffraction patterns obtained inY124 at 1.5 K, and in µ H k c of (a) 0.2 T, (b) 0.8 T, (c)4.0 T and (d) 6.0 T. Each image is the sum of scattering fromthe VL as the sample is both tilted and rotated so that theBragg condition is satisfied at the detector for the differentdiffraction spots. In each image, solid lines show the recipro-cal VL basis vectors, and dashed line ellipses emphasise theVL anisotropy. The VL opening angle ρ is defined in (a). the ellipse that overlays the Bragg spots, η which is re-lated to the VL opening angle ρ by η = ( √ ρ/ − .The field-dependencies of both η and ρ at T =1.5 K areshown in Fig. 2. With increasing µ H , η reduces andthe VL structure becomes increasingly isotropic. Forcomparison, in Fig. 2 we include equivalent data forY1237. It is clear that the low field-dependence of η is much stronger in Y124 than in Y1237. At higher fieldshowever, the two compounds display more comparablebehavior.For high- κ materials, and low µ H k c , anisotropiclocal London theory gives the VL distortion param-eter η equal to the in-plane penetration depth anisotropy γ ab = λ a /λ b . The sign of the observed VL distor-tion in Y124 shows that λ a > λ b , so the supercur-rent density is larger along the CuO chain direction.Moreover, η is larger in Y124 than Y1237 for the samefields, thus confirming the more anisotropic SC state inthe double-chained compound. For both materials, the µ H -induced suppression of η implies a reduction in γ ab within local theory. However, care must taken if assum-ing η = γ ab across the entire T - and µ H -range. In par-ticular, at high µ H the equality becomes increasinglyinvalid due to nonlocal effects. Nonetheless, at low µ H where local theory is most valid, the two compounds dis-play clearly different behavior. In Y1237 η is constantup to a kink at a VL structure transition at ∼ In contrast, η in Y124 varies smoothly over the entire field range,and falls quickly at low fields. These observations showthat even close to the local regime, the nonlocal inter- H (T) µ η ab η = ab H (T) µ O pen i ng A ng l e , ( D eg s . ) ρ ρ FIG. 2. (Color online) The µ H -dependence of the VL dis-tortion parameter η , defined by the inset sketch. Results forY124 are shown with filled symbols. Empty symbols denotesimilar Y1237 data. Inset: the µ H -dependence of the VLopening angle ρ in Y124. All lines are guides for the eye. actions present in each compound lead to markedly dif-ferent VL properties. In turn, this evidences the verydifferent basal-plane SC responses for Y124 and Y1237.Since local theory provides no constraint on the VLorientation for µ H k c , even a weak additional interac-tion can give a preferred VL alignment. In anisotropicmaterials like YBaCuO, this can be due to nonlocal inter-actions between the VL and the system anisotropies suchas those of the Fermi surface and the SC gap. De-termining which is most influential requires first-principlenumerical calculations that include the details of bothanisotropies. These calculations will also shed light onwhy the single VL orientation observed in Y124 is seenonly in an intermediate field range 2 T < µ H < Furthermore, in Y1237 a high-field transition,proposed to be driven by the d -wave gap, separatesthe intermediate field structure from a rhombic one thatevolves smoothly to become almost square by 10.8 T. Nosign of a similar transition is observed for µ H ≤ . B. Vortex Lattice Form Factor
Next we discuss measurements of the VL form factor, F ( q ) which is the Fourier transform of the magnetic fieldmodulation in the mixed state. Experimentally, F ( q ) atthe wavevector q is obtained from the integrated inten-sity I q of a VL diffraction spot as it is rotated throughthe Bragg condition at the detector. Fig. 3 shows thetypical angular variation of the diffracted intensity (the −1.5 −1 −0.5 0 0.5 1 1.500.20.40.60.811.21.41.61.82 x 10 Tilt Angle ( o ) I n t en s i t y ( C t s ./ S t d . M on . ) Top Right Spot at 0.3 TBottom Left Spot at 0.3 T
FIG. 3. Typical examples of the angular dependence of thediffracted intensity (rocking curves) obtained from the VL inY124 at µ H = 0 . T =1.5 K. Dashed lines correspondto a fit of a Lorentzian lineshape to each curve. The tilt anglecorresponds to the rotation angle of the sample and cryomag-net around the horizontal axis. The inset sketch shows theVL structure, with two of the VL spots denoted by symbolsthat match those of the associated rocking curves. rocking curve) measured from the low field VL in Y124.The quantity I q is obtained by integrating the area un-derneath the Lorentzian line shape used to fit the curve,and is related to | F ( q ) | by I q = 2 πφ ( γ/ V λ n Φ − q − | F ( q ) | . (1)Here φ is the intensity of the incident neutron beam, γ isthe neutron magnetic moment in nuclear magnetons, Φ is the flux quantum, V is the sample volume, and λ n theneutron wavelength.
1. Field-dependence of the form factor
Fig. 4 (a) shows the µ H -dependence at T =1.5 K ofthe VL form factor in Y124. In many strongly type-IIsuperconductors, the observed fall off with µ H can berepresented by the anisotropic London model extendedby a gaussian cutoff which represents the finite size ofthe vortex cores. The cutoff leads to an expectedexponential reduction in F ( q ) with µ H . The model isvalid for both κ ≫ H ≪ H c2 , and is in general is T -dependent: F ( q , T ) = h B i exp (cid:0) − . (cid:0) q x ξ b ( T ) + q y ξ a ( T ) (cid:1)(cid:1) x λ a ( T ) + q y λ b ( T ) ) , (2)where h B i is the internal induction. ξ i ( T ) and λ i ( T ) re-spectively denote the GL coherence lengths and London H (T) µ | F ( q ) | ( m T ) ( a ) q horiz q vert H (T) µ ( n m ) λ b ( b ) H (T) µ γ a b ( c ) FIG. 4. (Color online) (a) A semilog plot of the µ H -dependence of the VL form factors, | F ( q vert) | and | F ( q horiz) | at T = 1.5 K. The inset sketch of the VL struc-ture defines the two types of Bragg spot, the form factors ofwhich are treated separately. Black lines are fits of Eq. 2 to thecurves using a µ H -independent parameter sets (see text fordetails). (b) and (c) respectively show the field-dependenceof λ b and γ ab , after varying λ b in Eq. 2 to obtain a good de-scription of the low field | F ( q vert) | data in (a) (see text fordetails). Dashed lines are guides for the eye. penetration depths along directions i . q x and q y denotecomponents of q parallel to b ∗ and a ∗ , respectively. Forall fits we used the experimentally observed q -values.We note that the horizontal | F ( q horiz) | spots, whichhave q k b ∗ , lie close to a straight line in the semilogplot in Fig. 4 (a) and so can be fitted with constant val-ues of just two parameters ξ b and λ a in Eq. 2. However,the top/bottom | F ( q vert) | spots have an anomalous be-haviour for µ H < | F ( q horiz) | spots, we obtain ξ b = 3.6(2) nmand λ a = 293(3) nm at 1.5 K. The value of ξ b implies H c ∼
25 T, which is below the recently reported value of44 T, and suggests a contribution of weak VL disor-der to the variation of the form factor. Extending the analysis to the | F ( q vert) | spots, it isclear that above 0.8 T the model can be applied to fitthem too, and we obtain in addition λ b = 170(4) nm and ξ a = 3.7(2) nm. To capture the behaviour of | F ( q vert) | for µ H < ξ i has little influence at low fields. Therefore, to describethe data λ b must become smaller at low field, which cor-responds to an increase in the superfluid density for cur-rents along b . Fig. 4 (b) shows the low µ H -dependenceof λ b which, for all other parameters µ H -independent,gives calculated | F ( q vert) | values consistent with the ex-perimental data. Using the µ H -dependent values of λ b ,and λ a = 293(3) nm, the low µ H -dependence of γ ab at1.5 K is shown in Fig. 4 (c). | F ( q ) | ( m T ) ( a ) H = 0.25 T µ | F ( q vert)| − Peak| F ( q vert)| − Rock| F ( q horiz)| − Peak| F ( q horiz)| − Rock | F ( q ) | ( m T ) ( b ) H = 0.25 T µ | F ( q horiz)| − Peak d −wave fit N o r m a li z ed S upe r l f u i d D en s i t y ( c ) H = 0.25 T µ n a ( T ) / n a (0) n b ( T ) / n b (0) A n i s o t r op y P a r a m e t e r ( d ) H = 0.25 T µ γ ab from form factor η from VL structure FIG. 5. (color online) (a) A warming T -dependence at µ H =0 .
25 T of the VL form factors. Filled and empty symbolsrespectively denote full rocking curve, and rocking curve peakmeasurements. (b) The T -dependence of | F ( q horiz) | . Theline is a d -wave model fit to the data with ∆ (0)= 24(2) meV.(c) Normalized superfluid densities n a ( T ) and n b ( T ). Thevalues for n b ( T ) are extracted as described in the text. Thesolid line is a guide to the eye. The dashed line is the idealcurve for n a ( T ) obtained from the fit in panel (b). (d) The T -dependences at µ H = 0 .
25 T of the anisotropy parameters γ ab and η . From our analysis of the µ H -dependent form factorat 1.5 K, there are apparently two disagreements withanisotropic London theory: a) γ ab is always > η at thesame field (compare Fig. 4 (c) with Fig. 2), and b) theform factors for the two spot types do not become equalat low fields. Both these discrepancies may arise if theVL structure is T -dependent, and becomes pinned so thatthe value of η at 1.5 K does not represent the true SCanisotropy, which is γ ab . To demonstrate that this is thecase, we consider low field results at higher T .
2. Temperature-dependence of the form factor T -dependent measurements of the VL form factorat low field provide direct insight concerning both theanisotropy of the superfluid density and the underlyinggap structure. The intensive nature of these measure-ments means that there was insufficient neutron beam-time to record full rocking curves, and hence I q , at each T . Therefore, measurements were done just at the Braggangle (at the peak of the rocking curve), with full rock-ing curve measurements done at selected T s to confirmthe T -independence of the rocking curve width. All T -dependent measurements were done by warming scansconducted after an initial oscillation field-cool to 1.5 K.Fig. 5 (a) shows the T -dependence of the form factorsat µ H = 0 .
25 T. We see that on warming the base T form factor anisotropy is suppressed, so that the form fac-tors eventually become equal as expected within Londontheory. To show how both γ ab and η compare at higher T ,we firstly need to calculate λ a ( T ) from the | F ( q horiz) | data, and then subsequently we can extract λ b ( T ) fromthe | F ( q vert) | data. For the first step, we see from Eq. 2that q y = 0 for the | F ( q horiz) | spots, and so they aresensitive only to the T variation of ξ b ( T ) and λ a ( T ) fromtheir previously established base T values. The T vari-ations of both ξ b ( T ) and λ a ( T ) are calculated followingthe same approach reported in Ref. 9. For calculating λ a ( T ), we compute the T -dependent quasiparticle spec-trum expected over a standard d -wave SC gap on a sin-gle quasi-cylindrical sheet, and for which the zero T gapmagnitude, ∆ ( T = 0) is a free parameter. In principle therefore, the fit of the | F ( q horiz) | datais dependent on three parameters, ξ b (0), λ a (0) and∆ (0). We found that the fit was insensitive to ξ b (0),and so this parameter was fixed at 3.6 nm as deter-mined in Sec. III B 1. As shown in Fig. 5 (b), the T -dependence of | F ( q horiz) | is well described by the d -wave model for the superfluid density, and the two re-maining free parameters are fitted to be λ a = 290(3) nmand ∆ (0)= 24(2) meV. The good fit and agreement ofthe parameter values with those reported elsewhere confirms n a ( T ) to be controlled by a single d -wave SCgap.Next we extract the T -dependence of n b ( T ). By us-ing the | F ( q vert) | data shown in Fig. 5 (a), the d -wavemodel fit for n a ( T ), and the zero- T values for ξ i obtainedin Sec. III B 1, we can solve for the only remaining un-known in Eq. 2 which is λ b ( T )( ∝ / p n b ( T )). The ex-tracted T -dependence of n b ( T ) is shown in Fig. 5 (c). Thepositive curvature observed below ∼ T c / d -wave behaviour, and instead evidences amulti-gap SC response along b . We also note that theextrapolated value λ b (0)= 145(2) nm at this field agreeswith that shown in Fig. 4 (b).Fig. 5 (d) shows the T -dependence of the SC anisotropyparameters at µ H = 0 .
25 T. Here γ ab is determined ateach T using the absolute n i ( T ) data, while η is obtaineddirectly from the VL structure. Above ∼
30 K, γ ab and η agree well, as expected within anisotropic London the-ory. For T <
30 K however, a clear difference between γ ab and η emerges, as η varies only weakly on cooling,while γ ab increases smoothly. This behavior, and thatof Fig. 5 (a), is explained most simply if the VL struc-ture becomes frozen on cooling below ∼
30 K and is thusunable to evolve further on cooling. Importantly then,the intrinsic low T SC anisotropy is only given by γ ab as evaluated using λ i values obtained from analysing theform factor. From our T -dependent data, we find thatat 0 .
25 T, γ ab =1.97(4) by 1.5 K, and from data at 0 . γ ab =1.80(2). These values agree withthose obtained independently from the field-dependentform factor analysis [Fig 4 (c)]. Also from Fig. 4 (c),our extrapolation to zero field of γ ab =2.57(5) agrees wellwith the value of 2.5 determined by far infra-red spec-troscopy. IV. DISCUSSION
Our analysis of the T -dependent form factor at 0.25 Tshows n a ( T ) to be mainly sensitive to a single d -waveSC gap, while n b ( T ) requires a multi-gap description.Evidence for a contribution to the multi-gap responseof n b ( T ) is provided by ARPES where a ∼ This ties the origin of the gapstrongly to plane-chain hybridization as expected withinthe PE models.
The field-dependence of λ b at 1.5 K[Fig. 4 (c)] likely reflects the quenching of the 5 meV gap,or an as yet unobserved part of the gap structure. Wealso expect that the sharp change in the µ H -dependenceof the form factor ratio observed at ∼ µ H SC responses of Y124and Y1237. In Y124, the T -dependent forms for both n a ( T ) and n b ( T ) at µ H = 0 .
25 T [Fig. 5 (c)] are qual-itatively similar to those reported at very low field inRef. 12, and are consistent with a PE model where theplane-chain coupling is mediated via single electron tun-nelling. However, this model can not explain low fieldSANS data collected on Y1237, where both n a ( T ) and n b ( T ) display d -wave-like T -dependencies. Moreover,the direct observation of a PE-induced SC gap analogousto that seen in Y124 is not reported for Y1237. To ex-plain this suprising difference between the two materials,a possibility is that the chain states in Y1237 are non-SC and the SC anisotropy is intrinsic to the planes. Onthe other hand, a d -wave behavior for each of n a ( T ) and n b ( T ) is consistent with calculations that consider intrin-sically SC chains coupled to the planes by a Josephson-type pair tunnelling. If the latter is true, the d -wave T -dependence of λ b in Y1237 implies that there is a nodein the SC gap on the chain FS. Both new ARPES exper-iments and detailed calculations can shed light on theseproposals.For µ H > µ H -dependence of the low- T VL properties in Y124 and Y1237 is more compara-ble, and indicates the two compounds display high-fieldSC regimes that become more similar. In both materi-als γ ab is always > plane anisotropy, or is even related to the Fermi surfacereconstruction in these materials, remains an impor-tant open question.Finally, we comment on an implication of our studyconcerning the interplay between co-existing SC andcharge-density-wave (CDW) orders observed in a range ofunderdoped Y123 samples. The CDW and SC orderscompete since when applying a magnetic field to suppresssuperconductivity, the CDW order is observed to grow. In Y124, no CDW order has yet been observed at zerofield. Nevertheless, underdoped Y123 and Y124 both dis-play comparable quantum oscillations frequencies, and negative values of the Hall coefficient at low T . This indicates similar µ H -driven reconstructions of theFermi surface to occur in both Y123 and Y124, and whichall likely involve CDW order. We speculate that the ef-fect on the VL due to µ H -induced CDW order in Y124may be similar to that due to µ H -induced spin-density-wave (SDW) order in La − x Sr x CuO , x = 0 . Thereit was reported that the slope of the µ H -dependent VLform factor increases sharply at the onset of SDW order,after which the form factor falls with µ H more rapidlythan describable using conventional models. The in-crease in slope was explained as caused by a disorderingof the VL, which evidences the competition between theSC and SDW orders. Since in Fig. 4 (a) we observeno sharp change in slope of the form factor beyond thatmore easily understood in terms of a quenching of thePE, our results appear to limit any µ H -induced CDWorder in Y124 to fields > V. SUMMARY
In summary we have studied the VL in YBa Cu O (Y124) with µ H k c . At all fields, the VL structureis distorted due to the in-plane SC anisotropy. The VLdistortion is suppressed with increasing µ H which mostlikely reflects a quenching of a proximity-effect-inducedSC gap involving chain states. Our results rule out acommon framework for the low field in-plane SC responseof Y124 and YBa Cu O , and also indicate any µ H -induced CDW order in Y124 exists only for µ H >
ACKNOWLEDGEMENTS
We acknowledge discussions with J. Chang, A.T. Holmes,M. Ichioka and K. Machida. SANS experiments wereperformed at the ILL, Grenoble, France, and the Swissspallation neutron source, SINQ, PSI, Switzerland. Weacknowledge financial support from the EPSRC of theUK, the University of Birmingham, the Swiss NCCR andits program MaNEP, and from the European Commis-sion under the 6 th Framework Programme though theKey Action: Strengthening the European Research Area, Research Infrastructures, Contract No. RII3-CT-2003-505925.
Appendix A: Temperature-dependent form factor at µ H = 0 . T Here we describe the analysis of T -dependent VL formfactor data measured at µ H = 0 . µ H = 0 .
25 T where the T -dependence ofboth | F ( q vert , T ) | and | F ( q horiz , T ) | form factors wasmeasured, at µ H = 0 . T -dependent | F ( q vert , T ) | data measured at µ H = 0 . T -dependence of n b ( T ) at 0.4 Tfrom the data shown in Fig. 6(a). We follow the sameapproach used when analyzing the µ H = 0 .
25 T data[Sec. III B 2], where we solve Eq. 2 at each T to find λ b ( T )( ∝ / p n b ( T )). In this instance, the calculation of λ a ( T ) is done using the values λ a (0) = 290 nm, ∆ (0) =24 meV determined from the fit of the | F ( q horiz , T ) | data shown in Fig. 5(b). For the zero T coherencelengths, we again assume that ξ a (0) = 3 . ξ b (0) = 3 . T -dependence of n b ( T ) at µ H = 0 . µ H = 0 .
25 T, n b ( T )at 0.4 T displays a positive curvature in the low T regionwhich is incompatible with a simple d -wave supercon-ducting (SC) gap function.Using the both the extracted T -dependence of n b ( T ),and the assumed form of n a ( T ), in Fig. 6(c) we plot the T -dependence of the anisotropy parameter γ ab . For com-parison, we also show the T -dependence of η determinedfrom direct measurements of the T -dependent VL struc-ture. Again similarly as seen at µ H = 0 .
25 T, a cleardisagreement between γ ab and η emerges for T .
30 Kthus marking the irreversibility T at this field. The val-ues of γ ab at low T are thus larger than would be deducedsolely by equating γ ab = η as expected in the local Lon-don approximation, though comparatively smaller thanat µ H = 0 .
25 T. This reflects the µ H -induced suppres-sion of n b ( T ), which is shown in Fig. 6(d). Here we makea relative comparison between the n b ( T ) curves extractedat both µ H = 0 .
25 T and µ H = 0 . n b (0) at µ H = 0 .
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25 T and µ H = 0 . µ H = 0 . T value of n b (0) at µ H = 0 .
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