Effect of gap suppression by superfluid current on nonlinear microwave response of d-wave superconductors E.J. Nicol∗ Department of Physics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada 6 J.P. Carbotte† 0 Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S 4M1, Canada 0 (Dated: February 6, 2008) 2 Recentlyseveralworkshavefocusedontheintrinsicnonlinearcurrentinpassivemicrowavefilters n a as a tool for identifying d-wave order parameter symmetry in the high Tc cuprates. Evidence has J beenfoundford-wavepairinginYBCOandfurtherworkhasensued. Most ofthetheoreticalwork hasbeenlimited tolow temperaturebecauseithasnotincludedtheeffect ofthesuperfluidcurrent 4 on the energy gap. We find that this effect leads to important corrections above T ∼ 0.2Tc, while leaving the1/T low temperaturebehavior intact. A twofold increase in the nonlinear coefficient at ] n temperatures of order ∼ 0.75Tc is found, and as T → Tc the nonlinearity comes entirely from the o effectsofthesuperfluidcurrentonthegap. Impurityscatteringhasbeenincludedand,inaddition, c signatures for thecase of d+s-waveare presented. - r p PACSnumbers: 74.20.Rp,74.25.Nf,74.72.Bk u s . I. INTRODUCTION cess has led to further calculations to examine the issue t a of nonlocal effects for this quantity[11] (thought to be m the possible reason for not seeing the NLME proposed Interest in generating novel methods for probing the - by Yip and Sauls[12]) and the d-wave signature remains d orderparametersymmetryinsuperconductorshasdriven robust. More recently, measurements of the nonlinear n thedevelopmentofnewtechniquesandexperimentalcon- o figurations, such as angle-resolved electron tunneling[1] current have been made near Tc on a range of different c films and the data has been analyzed in terms of the DS or angle-resolved magnetic field dependence of specific [ approachwith impurity scattering being used to explain heat[2],whichcanprovideunambiguoussignaturesofor- variations between the films[13]. However, in this work 1 der parameter nodes and their location. One such pro- v posalis associatedwith the measurementofthe intrinsic and others[14, 15] which extrapolate the DS theory to 0 high temperature, the effect of the superfluid current on nonlinear current in passive microwave filters. Such de- 8 the gap has been neglected as an approximation. We vices give rise to to third order intermodulation effects 0 show here that inclusion of this effect has significant im- which, while detrimental for practical applications in 1 0 superconducting communication filter technology, make pact on the nonlinear coefficient for T & 0.2Tc, requir- ing a reanalysis of previous results. Indeed, when the 6 these devices idealfor examining issues oforderparame- current-dependence of the gap is considered, the nonlin- 0 tersymmetry. Excellentprogressinthefieldofhightem- t/ perature superconductivity has been made in this area. ear coefficient shows a twofold increase at T ∼ 0.75Tc a over the value calculated using the approximation of a Initially,YipandSauls[3,4]proposedtheexamination m current-independent gap. of the nonlinear Meissner effect (NLME) for evidence of - Finally, we note that other theoretical works have ex- d-wave gapsymmetry in the cuprates. Their predictions d amined the intrinsic nonlinear current for two-band su- n were not confirmed by experiments at the time[5], how- o ever, Dahm and Scalapino (DS)[6, 7] proposed to exam- perconductorsandMgB2[15,16]andforone-bands-wave superconductors[4, 16]. Effects of the superfluid on the c ine a related quantity: the intermodulation current or : gap have been included in Refs. [4, 16]. v distortion (IMD) which arises from the nonlinear induc- Ourpaperisstructuredasfollows: inthenextsection, i tance resulting from a quadratic dependence of the pen- X we summarize our theoretical approach which allows for etration depth on the superfluid current. In their work, r a signature of the d-wave gap would be found in an up- both the inclusion of the current dependence in the gap a and impurity scattering (from unitary to Born limit). turn in the temperature dependence of the nonlinear co- Strong electron-boson coupling effects are also available efficient at low temperatures, in contrast to exponential inthisformalism. InSectionIII,wediscusstheresultsil- decayforans-wavegap. Suchevidenceofanupturnhas lustratingthecorrectionstothesimplifiedtheoryatfinite been found in YBCO films by several groups[8, 9, 10] temperature and we revisit the experimental situation, with excellent agreement with the DS theory. This suc- including the issue of impurities. We end by providing predictions for an admixture of d- and s-wave symmetry ashasbeenrecentlysuggestedbyangle-resolvedelectron ∗Electronicaddress: [email protected] tunneling experiments on YBCO[1]. We form our con- †Electronicaddress: [email protected] clusions briefly in Section IV. 2 II. THEORY an approximate manner, a parameter g is introduced to represent that the interaction in the ω-channel could Inthiswork,weevaluatethefullcurrentfromthestan- be different from that in the ∆-channel in the case of a dard expression given for the imaginary axis Matsubara momentum-dependentinteractionthatwouldgiveriseto representationandmodifiedforad-waveorderparameter a d-wave order parameter symmetry. Likewise, there is ∆(θ)=∆cos(2θ) in two-dimensions:[4, 16, 17, 18, 19] no cos(2θ) factor in the numerator of the ω-channel as there is in the ∆-channel reflecting that the interaction 2en +∞ 2π dθ in the renormalization channel is taken to be isotropic js(qs,α) = πT to first order.[21, 22] Thus, the electron-boson spectral mvF n=X−∞Z0 2π function, which we denote by α2F(Ω), enters λ(n−m) × i[ω˜n−iscos(θ−α)]cos(θ−α) (.1) as follows: [ω˜n−iscos(θ−α)]2+∆˜2ncos2(2θ) ∞ Ωα2F(Ω) q λ(m−n)≡2 dΩ. (5) Thisexpressioncontainsboththecondensatecurrentand Z0 Ω2+(ωn−ωm)2 the quasiparticle current due to excitations.[20] The an- Here,totakethelimitoftheseequationstogivethestan- gle α measures the direction of the current with respect dard BCS result for d-wave in our numerical evaluation, to the order parameter antinode, with α = 0 indicat- we take the electron-boson spectrum to be a delta func- ing the current in the antinodal direction and α = π/4 tionathighfrequency. Likewise,weexcluderenormaliza- for the nodal direction. The other notation is standard tion effects which are not based on impurities by taking with e the electric charge, m the electron mass, T the g to be zero. This gives the BCS gap ratio in d-wave to temperature, n the electron density and vF the Fermi be 2∆0/kTc = 4.28. When we wish to consider strong- velocity. The superfluid momentum qs enters through coupling effects corresponding to 2∆0/kTc = 5, for ex- s =vFqs, and, importantly for the results of this paper, ample,thenwetakeg tobeafinite valueanduseadelta it also enters the equations for the Matsubara gaps and function at lower frequency for the boson spectrum[22]. renormalizedfrequenciesand,asaresult,thecurrentwill To extract the nonlinear coefficient that is relevant to decay the gap. The equations for the Matsubara gaps the passive microwave filters and hence can be used as ∆˜n = Zn∆n and renormalized frequencies ω˜n = ωnZn a sensitive probe of order parameter symmetry, we can modified for d-wave symmetry are: assume that js can be expanded to third order for small +∞ 2π dθ qs as: ∆˜n = πT λ(m−n) × mX=−∞ ∆˜mcZo0s2(22θπ) (2) js =j0(cid:20)nsn(T)(cid:18)q∆sv0F(cid:19)−β(T)(cid:18)q∆sv0F(cid:19)3(cid:21), (6) q[ω˜m−iscos(θ−α)]2+∆˜2mcos2(2θ) where j0 = ne∆0/(mvF). Note for strong coupling, we replacewe replacethe qsvF inthis formulabyqsvF/(1+ and λ), where λ is the mass renormalization parameter[16]. +∞ From this Dahm and Scalapino define[6] ω˜n =ωn+gπT λ(m−n)Ωm+πΓ+ Ωn , (3) c2+Ω2 mX=−∞ n b(T)≡ β(T) , (7) [ns(T)/n]3 with 2π dθ ω˜n−iscos(θ−α) thesquareofwhichisrelatedtothethirdorderintermod- Ωn = , ulationpowerinmicrowavefilters. Hence,measuringthe Z0 2π [ω˜n−iscos(θ−α)]2+∆˜2ncos2(2θ) intermodulationpowerprovidesameasureofthe nonlin- q (4) ear coefficient. As we calculate the full qs dependence of where the Matsubara frequencies are ωn = πT(2n−1), js, using Eqs. (1)-(5) with no approximation of taking for integer n, and we have included the possibility of the gap to be independent of qs, as was done in previ- impurity scattering from the unitary (c = 0) to Born ousworks,itiseasiesttoextractthesequantitiesdirectly (c→∞)limitviathe lastterminEq.(3)whichrequires from our numerical data. To do this we form the quan- self-consistency through Ωn. Here, Γ+ is proportionalto tity of js/qs versus qs2 which is a straight line at low qs the impurity scattering rate and c is related to the the and from this we obtain the superfluid density from the s-wave scattering phase shift.[13] Note that an impurity intercept and the nonlinear coefficient from the slope. term does not appear in Eq. (2) in d-wave as it does in By this method, we have confirmed previous results at s-wave. This is because in d-wave it averages to zero. lowtemperatureandcanproceedtoexaminetheissueof Finally, in general, the kernel of these equations would higher temperatures where the current reduces the gap normally be based on a momentum-dependent electron- even at low qs. We have also demonstrated this method boson spectrum. To mimic this unknown spectrum in for one-band and two-band s-wave superconductors.[16] 3 III. RESULTS portanttonote thatthe sameprocedurewhichgivesrise to these curves also provides the normalized superfluid density which is shown as the solid curve in the inset In Fig. 1, we show the current as a function of qs in of Fig. 3. This superfluid density curve is independent the two major directions, along the node and antinode, of the direction of the current as expected as it arises atbothlowandhightemperatures. Thiswasdoneusing theequationsaboveandillustratesthatwecanreproduce from the qs →0 limit and this curve is exactly the same as that which is calculated by standard formulas for the correctly the T = 0 results in the literature[23, 24] and that we can also evaluate the current at high tempera- penetration depth. Here it is extracted from our js ver- ture in this formalism. The Matsubara formalism is also sus qs curve as explained in the theory section, verifying the accuracy of our method. ideally suited for including impurity scattering. There are few points to note in this figure. The qs in the order parameterisessentialtoobtainthesecurvesanditisthe decay of the order parameter by the superfluid current that causes the current js to drop dramatically beyond the peak (otherwise, if ∆(qs =0) is used, the curves de- cay slowly to zero as qs → ∞, for example, as 1/qs for T =0 BCS s-wave). Furthermore, the effect of the qs in thegapbecomesevenmoreimportantatlowqs whenthe temperatureisapproachingTc. Finally,forhightemper- atures near Tc, the current is fairly independent of the angle α. FIG.2: Thenonlinearcoefficient β(T)asafunctionofT/Tc, shown for two directions: α = π/4 (solid line) and α = 0 (short-dashed). The dotted curves are for the same two di- rections but with the approximation of neglecting the qs de- pendence in the gap, i.e. ∆(qs = 0). The inset illustrates viaalog-logplotthatthelowtemperaturebehaviorvariesas ∆0/24T forα=0and∆0/12T forα=π/4(theseexpressions are shown as thelong-dashed lines in both cases). AlsoshowninFig.2bydottedlinetypearethecurves for β(T) for α = 0 and π/4 when the approximation of FIG. 1: The normalized current js/j0 as a function of ∆(qs = 0) is taken (i.e., the s is set equal to zero in qsvF/∆0, where j0 =ne∆0/(mvF) and ∆0 is the energy gap Eqs.(2)-(4)sothatthegapisnotmodifiedbythesuper- atT =0. ShownarethelowtemperatureBCScurvesfortwo fluid current). This is a central point of our paper, that directions: the current in the antinodal direction with α=0 whilethisapproximationworkswellatlowtemperatures, (dashed line) and the nodal direction with α = π/4 (solid), one sees from the comparison of the dotted curves with given for a reduced temperature t = T/Tc = 0.1. The in- their respective short-dashed and solid ones, that this set shows that the curvesoverlap for T near Tc (in this case, approximation breaks down for T & 0.2Tc and produces t=0.95). significant deviations as T → Tc. Indeed, from physical grounds one does not expect the nonlinear current to go This latter feature is seen moreclearly inFig. 2 where tozeroasT →Tc,butrathertoincrease. Fromthepoint we show the nonlinear coefficient β(T) for d-wave in the of view of device applications which would typically op- clean limit as a function of temperature for the two di- erateatabout0.5Tc orhigher,thiseffectofdecayingthe rections just discussed (solid and short-dashed curves). gap by the superfluid can introduce a factor of 1.5-2 in- Once again, it is seen that while the two curves are dif- creaseinthenonlinearresponseofthedevice. Finally,we ferent at low T, as T → Tc, the anisotropy is reduced confirminFig.2 thatthe originaluseofthis approxima- and disappears at Tc. Indeed, we can obtain an analyt- tionforlowtemperaturesisrobustandthe1/T signature ical value for β(T) at T = Tc which is 0.651 and this is ofthed-wavegapdiscussedbyDahmandScalapino[6,7] confirmedby the numerics showninFig.2. It is alsoim- remainsintact. Wefindbyourprocedure,thesameresult 4 as determined by Dahm and Scalapino analytically, that β(T,α = 0) ≃ ∆0/24T and β(T,α = π/4) ≃ ∆0/12T for T →0, which is illustrated by the log-log plot in the insert of Fig. 2. FIG. 4: Plot of log(Ab2av(T)) versus log(T/Tc) for s-wave (dashed curve), BCS d-wave with 2∆0/kTc = 4.28 (solid), andstrong-coupling(SC)d-wavewith2∆0/kTc =5(dotted). The curves are compared with the experimental data (solid dots) taken from Ref. [9]. The inset shows the same curves for b(T) versusT/Tc. FIG.3: Thenonlinearcoefficientβ(T)versusT/Tc0 forvary- ing impurity scattering in the unitary limit (c=0) and with α = π/4. The inset shows the corresponding curves for the has been shown to be related to b2(T), i.e. superfluid density ns(T)/n. Curves are given for: the pure limit (solid), Γ+/Tc0 = 0.0082 (short-dashed), 0.0404 (long- PIMD ∝b2(T) (8) dashed),0.0697(dot-short-dashed),0.1432(dot-long-dashed). and hence we plot b2(T) in the figure and use an ad- justable factor A to match the data with the theory as Turningtothe caseofimpuritieswhichwerediscussed was done by Oates et al.. Likewise, we have averaged previously by Dahm and Scalapino[6] and Andersen et the two directions for α=0 and α=π/4 after the man- al.[13], we comment on the modifications that occur due ner of Oates et al. as their measurement averages over to the current in the gap. Again, the results of Dahm all directions. The low temperature part of this log-log and Scalapino which are given for T < 0.2Tc remain ro- plot does not change from previous d-wave calculations bust, however,the resultsofAndersenet al. whichfocus butthehightemperaturepartisincreasedandtheagree- on high temperature are necessarily modified when the mentwiththedataisnotasgoodinthisregime(wehave approximationofaqs-independentorderparameterisre- chosentofitthelowtemperaturepartofthecurvetothe moved. ThisisshowninFig.3,whereweshowbothβ(T) data). However,comparedto the s-wavecalculation,the and ns(T)/n for varying impurity content (here we use evidence in support of d-wave remains striking. Oates theunitarylimitforillustration,aswasdoneinRef.[13]). et al. obtained a better fit with the d-wave theory be- Theβ(T)curvesareshownonlyforα=π/4. Onceagain, causeDahmandScalapinousedalargergapratiointheir for T &0.2Tc, deviations occur in the manner discussed BCStheorywhichissupportedbyotherexperiments. To before. The significant feature is that the impurity scat- illustrate this within our formalism, we can do a strong- tering reduces the Tc relative to the pure case which has coupling calculation where we take g = 1 in Eq. 3 and Tc0 and hence for fixed temperature of T ∼ 0.75Tc0, for move the electron-boson spectral function to lower fre- example,withvaryingimpuritycontent,thedecayofthe quency. In this manner, we can obtain 2∆0/kTc = 5 in gap by the superfluid current has an even greater effect d-wave corresponding to a mass renormalization param- for increasing impurity content and so it must not be eter λ=5. In this case, the result (shown as the dotted neglected. curve and plotted with a new A to fit the low T region) We nowturnto are-examinationofthe comparisonof does indeed move toward a better fit with the data over theorywithsomeofthedatathatexistsintheliterature, thefulltemperaturerange. Asatechnicalnote,toobtain maintaining the same analysis that was done previously. a gap ratio of 2∆0 = 6kTc in this Eliashberg formalism, First, we consider the clean limit YBCO data presented wewouldhavetotakeλtoextremelylargevaluesandwe by Oates and coworkers[9]. In Fig 4, we reproduce this wouldthenbe in the asymptotic regime whichlimits the datafor the normalizedIMDalongwithour calculations valueofthe gapratiotogonohigherthanabout6.5.[22] for BCS d-wave, where 2∆0 =4.28kTc. The IMD power Amuchbetter fitisobtainedwitha gapratioofabout6 5 but in our model the λ is unphysical. To overcome this, gap should cause significant changes (our Fig. 3 should in orderto produce higher gapratioswith morephysical be contrasted with Fig. 3 of Ref. [13]) and this was not valuesofλ,wouldrequireaspin-fluctuationcalculationof included in the original analysis. In Fig. 5, we show our thetypegiveninRef.[25],whichwouldincludefeedback calculation at T = 0.75Tc0 for the variation of the non- on the electron-boson spectral density itself, which sup- linearcoefficientwithrespecttothevariationofthepen- pressesitatlowωandintroducescouplingtoanewreso- etration depth when unitary scattering is included (like nantmodethatgrowsinamplitudeasT isreduced. This Andersenetal.,wehavecheckedBornscatteringandfind is beyond the scope of this work. Other work which has there is essentially no difference in the result presented discussedthisdatahasbeenthatofAgassiandOates[11] here). We have also taken α = π/4 as at this temper- where they considerednonlocal effects, however,the cal- ature the anisotropy is almost completely gone, which culation is a BCS one with no inclusion of the effect of we have checked. It should be noted that, once again, the superfluid current on the gap and so it is difficult thereis anadjustable parameterjc whichcanbe usedto to say how both this and strong coupling might modify scale the theory to overlap with the data. Andersen et their results. al. kept the theory fixed and used a value of jc to ad- Finally in Fig. 4, in the inset, we give for reference just the data. Here we choose to keep the scaled data as the unscaled b(T) versus T curves. The 1/T divergence it was presented in the original paper and so we adjust is clear in comparison with the exponential decay of the our theory by a scale factor to overlap with the data (if s-wave case and the strong-coupling calculation is above we had adjusted the data instead, this would have in- theBCSoneasisexpectedfrompreviousworkons-wave creasedthe jc value used for scaling the data by a factor superconductors[16]. of about 1.35). Doing so, we find with the solid curve that there is reasonable agreement between theory and data. However,we note, referring the readerto the inset of the figure, that the range of variation in penetration depth, if explained via impurity scattering, would imply areductioninTc ofabout15-20%fromthe purecase[26] and yet the experimental data indicates a Tc variation of only about 2%. This discrepancy was not noted in Ref. [13] and it does pose a problem for this interpreta- tion. However,continuingwiththisexplanation,wehave compared our result with the case of taking ∆(qs = 0), shown as the dashed curve and using the same jc scale factor,andwefindthatwhilesomedeviationmaybecor- rected by choosing a different jc, which would shift the curve, the slope is also different. While the interpreta- tion of the data via impurity scattering may be open to question due to the large variation in Tc required, more experimental data may help to reduce the scatter and provide further tests of the theory. Recently, a novelprobe of angle-resolvedelectrontun- neling was used to determine the order parameter sym- FIG.5: Plotoflog(j2(T)/jc)versusλ(T)/λ0forT =0.75Tc0. metry in YBCO and this work has suggested that the Andersen et al.[13] have defined a quantity (j2/jc)2 which is order parameter is not pure d-wave but an admixture of equal to our 1/b(T). The data (open circles) is taken from d+s with about15%ofs-wave. This is not the first time Ref.[13]. Thesolid curveisforBCS d=wavewith∆(qs)and thatit has been suggestedthat YBCO mayhave a small thedashedisforthecasewheretheapproximation∆(qs =0) s-wave component and it is important to assess the con- has been used. The inset shows the reduction in Tc from the pure value of Tc0 that is implied by the values of λ(T) at sequencesfortheintrinsicnonlinearcurrentandwhether T =0.75Tc0 normalized to thepurevalue for T =0, λ0. there would be any signatures unique to a d+s-wave or- der parameter. As a result, we now briefly consider such a symmetry, where we assume that the s-wave compo- We revisit the issue of impurities in Fig. 5. This fig- ure is based on a similar one presented by Andersen et nent is small compared with the d-wave part, such that al.[13]. Severalfilms wereexaminedat75Kandthe non- nodes still exist but they are shifted from π/4. Taking ∆(θ)=∆[acos(2θ)+c],weevaluateanalyticallythemain linear coefficient was found to vary from sample to sam- ple, as did also the penetration depth. The analysis in featuresexpectedinthelimits ofT →Tc andT →0. As the paper used the DS theory with impurity scattering T →Tc, we find that we can define a quantity relatedto b(T): to argue that the data could be understood by assuming varying impurity content from one film to another. We 3 re-examine this issue because at this high temperature T ¯b(Tc)≡ lim b(T) 1− , (9) the pairbreaking effect of the superfluid current on the T→Tc (cid:18) Tc(cid:19) 6 ∆0/24T as is shown in the lower frame of Fig. 6, we see that the fourfold d-wave anisotropy shifts to twofold for d+s-wave and also exhibits a shift in the position of the maxima (indicating a shift in the position of the nodes). The magnitude of the anisotropy, which was a factor of 2 for d-wave,is now even greaterin d+s-wave. The ana- lytic formula for this anisotropy, assuming c<a, is 2 2 β(T) = ∆0 1− c cos(2α) + 1− c sin2(2α) 24T(cid:18)(cid:20) a (cid:21) (cid:20) (cid:18)a(cid:19) (cid:21) (cid:19) 1 × . (14) 1−(c/a)2 p Thus,thereshouldbe observablesignaturesofans-wave component should it be possible to do angle-resolved measurements of intrinsic nonlinear current. IV. CONCLUSIONS Insummary,wehavecalculatedtheintrinsicnonlinear currentofad-wavesuperconductors,including the effect of the superfluid current on the order parameter. We find that the low temperature 1/T behavior of the non- FIG. 6: The effect of a d+s-wavegap on theanisotropy near linearcoefficient,proposedasatestford-wavesymmetry Tc as measured via ¯b(Tc) (upper frame) and at low tem- by Dahm and Scalapino, remains unchanged. However, perature as measured by β(T) normalized to ∆0/24T (lower at temperatures above about 0.2Tc, the approximation frame). Theanisotropyisplottedasafunctionofthecurrent of taking the gap to be qs-independent fails and large direction relativetotheantinode,α. Curvesareshown fors- corrections are found. Indeed, at temperatures of order wave(dotted),d-wave(short-dashed)andvaryingpercentage ofs-wave: 10%(dot-dashed),15%(solid),20%(long-dashed). 0.75Tc, there is a twofold increase in the nonlinear co- efficient over that obtained with a qs-independent gap. This could have implications for the technological appli- which gives (Tc/∆0)2¯b(Tc) = 0.0599 for d-wave and and cation of the cuprate materials as passive microwavede- vices for the communication industry. We also find that 0.0266 for two-dimensional s-wave. In both cases, this thesecorrectionsremainlargeinthepresenceofimpurity quantity is isotropic with respect to the direction of the scattering. current, however,for d+s-wave, we have Are-examinationofthecomparisonoftheoryanddata ¯b(Tc)(cid:18)∆Tc0(cid:19)2 = 76ζ4(π32)γ13(γγ122−γ3γ3)2, (10) frbreeommroevqtehudeiriefindntedtrsomtoohbdatutaliansttiroaonnfipgtoowcvoeeurrpwltiihntehgetehnffitseircaetpstpewrmooxpuiemldraattsuitoirlnel range. However,thisrequirescalculationsofsuchsophis- where tication that they are beyond the scope of this paper. 2π dθ The signature in support of d-wave symmetry remains γ1 = Z0 2π∆2(θ), (11) clear, regardless. More importantly, a recent analysis of 2π dθ data from several films assumes that the explanation of γ2 = ∆4(θ), (12) the variationinthe nonlinearcoefficientandpenetration Z0 2π depth results from impurity scattering. As the data is 2π dθ taken at high temperature, where the decay of the gap γ3 = Z0 2π cos2(θ−α)∆2(θ), (13) bythecurrentissignificant,wefinduponreevaluationof thetheorythattherelationshipbetweenthesetwoquan- and there arises an anisotropy as a function of α in the titiescanvarymarkedlyandadifferentslopeisobtained nonlinearcoefficientasT →Tc,whichisnotthereinpure alongwithadifferentoverallscalefactor. Unfortunately, d-wave. Indeed, it can be sizable as shownin Fig. 6 (up- we must point out that, while a possible fit to the data perframe)fordifferentpercentagesofthes-wavecompo- stillremains,thebasicassumptionthatimpurityscatter- nent. For 15% s-wave, the anisotropy is 20%. Note also ingisthesourceofthechangesseeninthedataimpliesa that the anisotropy is twofold. At low temperature, a d- largevariationintheTc ofthedifferentsamples,whichis waveorderparameteralreadyshowsafourfoldanisotropy notseenexperimentally. Furtherexperimentsalongthese inβ(T)asafunctionofα. Ifweplotβ(T)normalizedto lines might help to resolve this issue more completely. 7 Finally, due to a recent experimental observation of the directionofthecurrentcanbe variedwithrespectto possible d+s-wavegapsymmetry inYBCO measuredby the antinodes or nodes, then we predict that signatures angled-resolvedelectrontunneling,wehaveexaminedthe of a d+s-wave order parameter would be observable. nonlinear coefficient at both low T and T → Tc to de- termine signatures of the s-wave component. Indeed, at T → Tc, the isotropic behavior found for pure s- and pure d-wave is lost and a large anisotropy develops for Acknowledgments d+s-wave as a function of the direction of the current in the plane. At low temperatures, the 1/T dependence remainsbutthefourfoldanistropyofthecoefficientisal- This work has been supported by NSERC (EJN and tered to twofold and the magnitude is modified. There- JPC) and the CIAR (JPC). 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