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Structure of vortex liquid phase in irradiated Bi Sr CaCu O crystals 2 2 2 8−δ N. Morozova, M.P. Maleya, L.N. Bulaevskiia, V. Thorsmøllea A.E. Koshelevb, A. Petreanb and W.K. Kwokb aSuperconductivity Technology Center, Los Alamos National Laboratory, Los Alamos, NM 87545 b Materials Science Division, Argonne National Laboratory, Argonne, Illinois60439 9 (February 1, 2008) 9 9 The c-axis resistivity in irradiated and in pristine Bi2Sr2CaCu2O8−δ crystals is measured as a 1 functionofthein-planemagneticfieldcomponentatfixedout-of-planecomponentB inthevortex ⊥ n liquidphaseatT =67K.Fromthisdataweextractthedependenceofthephasedifferencecorrela- a tionlengthinsidelayersonB andestimatetheaveragelengthofthevortexlinesegmentsconfined ⊥ J insidecolumnardefectsasafunctionofthefillingfactor f =B⊥/BΦ. Themaximumlength,about 2 15 interlayerdistances, is reached near f ≈0.35. 2 74.60.Ge, 74.25.Fy, 74.62.Dh ] n Thestructureofthevortexliquidinhighlyanisotropic realized,to extractthe pancakedensity correlationfunc- o layered superconductors with columnar defects (CDs) tion using data for JPR frequency as a function of B c k - produced by heavy ion irradiation is one of the most at fixed B⊥ [9]. In this Letter we determine the aver- r intriguing questions in the current study of the vortex age length of vortex line segments as a function of B , p ⊥ u stateinhightemperaturesuperconductors. Forthemost i.e. of the filling factor f, from transport measurements s anisotropic Bi2Sr2CaCu2O8−δ (Bi-2212) superconductor on single crystals of Bi-2212 with and without CDs. For . t without strong disorder, neutron scattering and Joseph- thiswehavedevelopedamethodforextractingthephase a m son plasma resonance (JPR) data provide evidence in difference correlationfunction along the layer from mea- favor of a pancake liquid with very weak correlations of surements of the c-axis conductivity, σ , as a function - c d pancakes in different layers [1–3]. When CDs are intro- of the magnetic field component parallel to the layers, n ducedintothesecrystalsalargedecreaseinthereversible B ,atfixedB . ThecomponentB establishesthevor- k ⊥ ⊥ o magnetization is observed [4], indicating that pancakes tex state to be studied, while the component B serves c k arepredominantlysituatedontheCDs,evenintheliquid as a probe of this state, as described below. Knowing [ state. Recent studies of JPR [5,6] and c-axis transport the phase difference correlation length, we estimate the 2 [7,8]inirradiatedBi-2212crystalsrevealenhancementof pancake density correlation length along the c axis. v c-axis correlations in some interval of out-of-plane mag- In Josephsoncoupled superconductors in the presence 5 6 netic fields, B⊥, below the matching field, BΦ, at tem- ofac-axiscurrent,voltageisinducedbyslipsofthephase 1 peratures T ∼ 70 K. In other words, pancakes at these difference between layers, ϕn,n+1(r,t), as described by 9 fieldsandtemperaturesappeartoformalignedsegments the Josephson relation Vn,n+1 = (h¯/2e)ϕ˙n,n+1. Here n 0 of vortex lines inside CDs, while outside of this region labels layers, r = x,y are coordinates in the ab plane, 8 they are better described as a liquid of weakly c-axis- and t denotes the time. The c-axis conductivity in the 9 / correlated pancakes, as in pristine crystals. Both c-axis vortexliquidstateisdeterminedbytheKuboformula[3] t a transport and JPR involve the flow of Josephson cur- ∞ m rents and are thus sensitive to the misalignment of pan- σ (B ,B )=(sJ2/2T) dt drG(r,t), (1) c ⊥ k 0 Z Z - cakevorticesbetweenadjacentplanes. CDs alignedwith 0 d thec-axiswillpromoteinterplanepancakealignmentina G(r,t)=2hsinϕn,n+1(0,0)sinϕn,n+1(r,t)i n o region of temperature where pancakes remain largely lo- ≈hcos[ϕn,n+1(r,t)−ϕn,n+1(0,0)]i, (2) calizedonCDs. Evenintheliquidstate,wherepancakes c : are mobile, filling of the available CD sites should lead where J0 is the Josephson critical current and s is the v to enhanced c-axis correlation from statistical consider- interlayer distance. Time variations of the phase differ- i X ations alone. In previous c-axis transport measurements ence are caused by mobile pancakes [10] induced by B⊥ r [7], a dip in the magnetoresistance, ρ(B ), of a Bi-2212 andbymobileJosephsonvorticesinducedbytheparallel a ⊥ crystalwasobservedatamagneticfieldcorrespondingto componentB . InthelowestorderinJosephsoncoupling k a filling factor of CDs, f =B⊥/BΦ ≈1/3. wesplit[ϕn,n+1(0,0)−ϕn,n+1(r,t)]intothecontribution So far there has been no technique available for pro- induced by pancakes and that caused by the unscreened viding quantitative information on the degree of c-axis parallelcomponentB . Assumingthat B is alongthe x k k correlationor onthe lengthofpancakeline segments. In axis, we obtain a simple expression for the contribution addition, it is unclear whether vortex interactions play of the parallel component to the phase difference: a significant role in enhancing correlation in the liquid state. Previously, a method was proposed, but not yet ϕn,n+1(0,0)−ϕn,n+1(r,t)≈ [ϕn,n+1(0,0)−ϕn,n+1(r,t)]Bk=0−2πsBky/Φ0. (3) 1 Inserting this expression into Eqs. (1) and (2) we obtain of lines inside CDs. Then only ends of segments contribute significantly to the phase difference ϕn,n+1(r,t) σc(B⊥,Bk)=(πsJ02/T)Z drrG˜(r,B⊥)J0(αBkr), (4) wanhdileletahdeteoffescutpopfrepsasinocnakoefsJinosnepeihgshobnorcionugpllainygerasnidnsiσdce, the segments is much smaller. where J0(x) is the Bessel function, α=2πs/Φ0 and For our experiments, high quality Bi-2212 crystals ∞ (Tc ≃ 85K) of about 1 × 1.5 × 0.02 mm3 were used. G˜(r,B⊥)=Z dtG(r,t,B⊥). (5) The irradiation by 1.2 GeV U238-ions was performed on 0 the ATLAS accelerator (ANL). According to TRIM cal- The functionG(r,t) describesthe dynamicsofthe phase culationstheseionsproduceinBi-2212continuousamor- difference caused by mobile pancakes. If g(B ,B ) = phoustrackswithdiameter4-8nmandlength25-30µm. ⊥ k σ (B ,B )/σ (B ,0) is known in the vortex liquid, the Below, we present the results for the samples irradiated c ⊥ k c ⊥ correlation function G(r) = G˜(r)J2Φ2/4πsTσ (B ,0) withadensityofCDscorrespondingtothematchingfield 0 0 c ⊥ maybefoundusingtheinverseFourier-Besseltransform: BΦ =2 T and for a reference pristine sample. Our measurements of c-axis conductivity were carried outinacryostatwithtwosuperconductingmagnetspro- G(r,B⊥)=Z dBkBkg(Bk,B⊥)J0(αBkr). (6) vidingmagneticfieldsinorthogonaldirections. Themag- netsarecontrolledindependentlyandprovidefieldsupto The correlation lengths R and R1 of this function are 8Tinonedirectionandupto1.5Tintheotherdirection. defined by the relations: Samples canbe orientedalongthe axisofeither magnet. Misalignmentofthecrystalcaxiswithrespecttotheper- ∞ rG(r) ∞ r3G(r) R2 = dr , R2 = dr . (7) pendicular component B was detected by the asymme- Z G(0) 1 Z R2G(0) ⊥ 0 0 tryofρ (B )withrespecttothesignofB . Weadjusted c k k Note that R1 is related to the coefficient of the Bk2 term tthryebdeirloecwti5o%noafttBhefi>el2dTcoamnpdobneelnotws,1p0r%oviadtinlogwaesryfimemldes-. in the expansion of g(B ,B ) in B2: k ⊥ k k The normalized conductivity g(B ) was calculated using k the average resistivity ρ (B ) = [ρ (B )+ρ (−B )]/2. g(B⊥,Bk)≈1−[πsR1(B⊥)/Φ0]2Bk2. (8) Two silver contact padscwerke deposictedkon bocth sidkes of the sample using a mechanical shadow mask. The mask Thus R1(B⊥) can be obtained independently from data provided a clean surface rim ∼0.1 mm from the sample for σ (B ,B ) at small B . c ⊥ k k edges and 25µm separation between current and poten- Such a procedure to obtain the function G(r,B ) is ⊥ tial pads. The area of the contact was ≈ 0.75mm2 for validifthevortexstatedependsweaklyontheJosephson current and ≈ 0.05 mm2 for potential terminals. The coupling and hence on the probe field B (which affects k resistance of the current pads at room temperature was Josephson coupling). Then this method is nondestruc- ≈ 2 Ω. A current of 1 mA, driven through the sam- tive. Let us check first under what conditions we can ple, provides an ohmic I-V regime. For very anisotropic neglect the effect of Josephson coupling on the equilib- Bi-2212,in our range of the magnetic field and tempera- rium vortex state. The energy of Josephson coupling in tures, in-plane conductivity, σ ∼103σ [11], providing the correlated area πR2 is πE0R2/λ2J, which should be nearlyequipotentialcurrentdiasbtributioncin theab-plane, much smaller than the temperature T to be treated as a at least in the central area of the sample. Thus the con- perturbation[9]. HereE0(T)=Φ20s/16π3λ2ab(T). ForBi- tribution of σ to the anisotropic conductivity is weak 2212, with γ ≈ 300 and λ (0)≈2000 ˚A, the Josephson ab ab and can be neglected, in the limit of small pad separa- coupling in the correlated area is ≈ 0.2T at the maxi- tion. Thus a standard 4-probe method can be used for mum value of R found below andat T >60K. Thus the ρ measurement instead of the complicated multitermi- effect of Josephson coupling, and hence B , on the equi- c k nal Montgomery analysis. The temperature was stabi- librium vortex state may be neglected. For dynamical lized with an accuracy ±50 mK. The resistivity ρ (B ) parameters, such as σ , higher order terms in Josephson c k c was measured in the B interval where g at maximum coupling describing dynamic screening of B omitted in ⊥ k B drops with B at least to the value 0.2. Eq.(3),maybeimportant. Thiswillbediscussedbelow. k k In Fig.1 we presentρ as a function ofB2 atdifferent We anticipate that in the pancake vortex liquid state c k B for a) irradiated and b) pristine crystals. For irradi- in pristine crystals the characteristic lengths R and R1 ⊥ atedcrystalsatlowB theresistanceincreasesquadrati- of the correlationfunction G(r) are of orderof the inter- k vortex distance, a = (Φ0/B⊥)1/2, because each pancake callywithBkasprescribedinEq.(8). Incontrast,forthe pristine crystal we observed that ρ increases quadrati- hereismobileandinducesphaseslippageasdescribedin c cally at high B , but exhibits a minimum at low fields Ref. [10]. In irradiated crystals with CDs we anticipate k that is not described by Eq. (8). This low field behav- much bigger R and R1 if pancakes form long segments 2 ior is caused by dynamic screening of Bk that results in via R ≈ Φ0/B˜ks to the magnetic field B˜k which charac- additional dissipation at low fields, due to the combined terizes the scale of the drop of σ with B . Here B˜ c k k effectsofmotionofJosephsonvorticesinducedbyBk and is the magnetic field at which flux in the area Rs is of pancake vortices. These effects are not important at ≈ Φ0. If we define the characteristic field B˜k, as given high fields. They do not appear in the irradiated crystal by: g(B˜ ) = 0.1, then from Fig.2 at B = 0.2 T, we k ⊥ due to pinning of the Josephson vortices. estimate for the pristine crystal R/a ≈ 2.5, while for irradiated one R/a ≈ 8. As one can see from Fig. 2, at the high fields B used in our measurements, the depen- k aa b dence g(B ) is close to exponential, and we use this to 1100 3 k extrapolate g(B ) to higher fields. Then we determine k 88 11..22 TT G(r) in the interval r < r ≈ 0.5 µm by use of Eq. (6). m mm 2 Accuracy of our experimental data and of the inverse rrW(cid:215)W(cid:215), c, ccc4646 B^ = 0.3 T 1c, cmrW(cid:215) FGo(urr)ieart-Bre>ssreml t.ransformation is not sufficient to obtain 22 0.1 T 00 BB^^ == 00..33 TT 0 00 55 1100 1155 2200 0 10 20 30 40 ((BB(cid:231)(cid:231)(cid:231)(cid:231) ))22,, TT22 (B(cid:231)(cid:231) )2, T2 1.0 a b FIG. 1. Dependence of resistivity ρc versus Bk2 for per- B^ = 0.6 T pendicular components B , increasing sequentially with the ⊥ step 0.1 T in irradiated (a), and pristine (b) crystals. 0 ) G ( 0.2 T b =InBFig/.(B2 wBe s)h1o/2wftohreirdreapdeinadteedncaengd=prσisct(ibn)e/σsca(m0)ploens G ( r ) / 0.1 1.3 T r10C0 , W cm k s ⊥ at different B⊥. Here Bs = Φ0/2πs2. For the pristine 10-3 crystal the values σc(B⊥) at Bk = 0 were determined 0.2 T 10-6 by extrapolation of ρc(Bk) from the high field quadratic 0 1 B^ , T dependence to zero as shown in Fig. 1b by the dashed 0 1 2 3 4 5 0 1 2 3 r / a r / R lines. Note that for the pristine crystal all three curves 1.00 pristine T=67 K FIG. 3. The functions G(r/a) (left), and G(r/R) (right) B^ = 0.3 T as extracted from g(Bk) for irradiated (solid) and for pris- 0.2 T tine (dashed) samples. Inset: ρ (B ) at B = 0 from Ref. 7. ( 0 )c 0.1 T Arrows indicate B⊥ values for tche pcresentekd G(r) curves. s ) / B(cid:231)(cid:231) 0.10 irradiated theInsaFmige.3daatwaevsehrosuwstxhe=furn/cRti,ownsheGr(ert/hae)ascnadliinngFleign.g3thb s ( c B^ = 00..32 TT R(B⊥,BΦ) is defined by Eq. 7. For the pristine crystal at B = 0.1 T and 0.2 T we obtain R(B ) ≈ a. The 0.4 T ⊥ ⊥ curve for B = 0.2 T is plotted by dashed line and it ⊥ 0.01 practically coincides with that at B = 0.1 T. For the ⊥ 0 0.2 0.4 0.6 0.8 1 1.2 irradiated sample we show the curves corresponding to B / (B(cid:215)B )1/2 (cid:231)(cid:231) s ^ regions below, above, and at the position of the dip in FigF.2,I NG. M.oro2zov. Dependence g(B ) = σ [B /(B B )1/2]/σ (0) in the magnetoresistance curve, ρc(B⊥,0) from Ref. [7] as k c k s ⊥ c shownintheinset. ItisevidentinFig.3athattherateof irradiated and pristine crystals. decay of correlations with r is not a monotonic function of B and is a minimum ata value correspondingto the coincide at high fields, demonstrating scaling of the cor- ⊥ dip in ρ (B ). Notably all these curves quite accurately relation length R with a. Such a scaling is not observed c ⊥ merge in Fig 3b, providing a universal function G(x) for for the irradiated crystal at B ≤ 1.2 T, because the ⊥ bothpristineandirradiatedsampleswithasinglescaling average distance between CDs, aΦ = (Φ0/BΦ)1/2, gives lengthR,althoughtheirradiatedsampleischaracterized another length scale in addition to a. More importantly, generally by two lengths a and aΦ. This means that we see that g drops with B much faster for the irra- k G(x)is determinedmainlybythe staticeffectofvortices diated sample. The correlation length R(B ) is related ⊥ on Josephson coupling [3] though it is extracted from 3 the dynamic quantity σ . In comparison with pristine traction of pancakes in different layers is smaller by the c crystals CDs simply diminish the effective concentration factor s/λ . The random distribution of CDs is impor- ab of pancakes acting on Josephson coupling by the factor tant. Repulsion between vortices in the same layer leads a2/R2. toasignificantincreaseofpancakeenergyinsideCDssit- The correlationlength R as a function of B is shown uated near those already occupied by pancakes [12]. As ⊥ in Fig. 4. It exhibits a distinct maximum, R/a ≈ 4, at B increases,someCDsbecomemorefavorableforfilling ⊥ f ≈0.35,againcoincidingwiththe positionofthedipin by pancakes, while others remain unoccupied. Another ρ (B ). At f = 0.35 the ratio of R(B ) for irradiated importantpoint is that favorableconfigurationsaresim- c ⊥ ⊥ and pristine crystals is about 4 times. ilarinalllayersduetothegeometryofCDs. Thusrepul- sion of pancakes inside randomly positioned CDs in the R/a same layer leads to enhancement of c-axis correlations. T = 67 K, BF = 2 T Another mechanismfor pancakealignmentinside CDs is magnetic attraction of pancakes in adjacent layers. 3 irradiated In conclusion, we extract the universal phase difference correlation function using measurements of c-axis resistivity as a function of the parallel component of the 2 magnetic field at fixed perpendicular component. We estimated the correlation length of the pancake density pristine correlationfunction along the c axis as a function of the 1 fillingfactorf ofcolumnardefects. Itfirstincreaseswith 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 f, reaches a maximum at f ≈ 0.35, and then drops as B^ / BF pancakes start to occupy positions outside of CDs. We FIG.4. Dependenceof the correlation length R as a func- arguethatenhancementofthe alignmentofpancakesin- tion of filling factor, f = B⊥/BΦ, of columnar defects for sidecolumnardefectsiscausedbyinteractionofpancakes irradiated and pristine samples. Dashed lines are a guide. confined inside columnar defects. Useful discussions with V.M. Vinokur are greatly ap- The length R1 obtained from the function G(r) using preciated. We thank P. Kes and T.-W. Li for providing Eq.(7)is≈2RatallvaluesB studied. Thesamelength ⊥ Bi-2212 single crystals and R. Olsson for technical assis- determined directly fromthe function g(B ) at smallB k k tance. This work was supported by the U.S. DOE. using Eq.(8) is ≈3R. Thus we estimate the accuracyof extracting G(r) as ≈30 %. In order to determine the c-axis correlations of pancakes and to explain scaling, we note that at small f pancakesarepositionedmainlyinsideCDsandhencethe drop in the phase difference correlations is caused by in- terruptionsinpancakearrangementinsideCDs. Namely, [1] R. Cubbit and E.M. Forgan, Nature 365, 407 (1993). wesupposethatthephasedifferenceϕn,n+1(r)isinduced [2] O.K.C. Tsui et al., Phys. Rev. Lett., 76, 819 (1996); when a columnar defect in the layer n is occupied by a Y. Matsuda et al., Phys.Rev. B, 55, R8685 (1997). pancake, but the site inside the same CD in the layer [3] A.E. Koshelev, Phys.Rev.Lett. 77, 3901 (1996). n + 1 is empty (and vise versa). At larger f the un- [4] R.J. Drost et al.,Phys. Rev.B58, R615 (1998). pinned pancakes also contribute to the decay of corre- [5] M. Kosugi et al., Phys.Rev.Lett. 79, 3763 (1997). [6] M. Sato et al., Phys. Rev.Lett. 79, 3759 (1997). lations. Scaling for both pristine and irradiated crystals [7] N. Morozov et al.,Phys. Rev.B 57, R8146 (1998). meansthatendsofvortexsegmentsinsideCDsactonthe [8] L.N. Bulaevskii et al.,Phys. Rev.B 57, R5626 (1998). phase difference correlations as unpinned pancakes. The [9] A.E. Koshelev et al.,Phys.Rev. Lett.81, 902 (1998). net concentration of interruptions and unpinned pan- [10] A.E. Koshelev, Phys.Rev.Lett. 76, 1340 (1996). cakes is 1/R2, i.e. the averagelength of vortex segments [11] R.A. Doyle et al., Phys. Rev. Lett. 77, 1155 (1996); is L/s ≈ R2/a2. In the most aligned vortex liquid, at J.H. Cho et al.,Phys. Rev.B 50 6493 (1994). f ≈ 0.35, we estimate L/s ≈ 15. This is much larger [12] A. Wahl et al., Physica C, 250, 163 (1995); C. Wengel than L0/s ≈ 1/(1−f) ≈ 1.5 in the model of noninter- and U.C. Täuber, Phys.Rev.Lett. 78, 4845 (1997). acting pancakes positioned randomly on CDs. Thus we conclude that interaction of pancakes is important for enhanced alignment inside CDs. InthehierarchyofinteractionsinthepresenceofCDs, both the pinning energy per pancake and the intralayer repulsion energy of pancakes are characterized by the same energy scale E0. The scale of magnetic pair at- 4