Dynamic scaling for 2D superconductors, Josephson junction arrays and superfluids Stephen W. Pierson,∗ Department of Physics, Worcester Polytechnic Institute (WPI), Worcester, MA 01609-2280 Mark Friesen,† 9 2Physics Department, Purdue University, West Lafayette, IN 47907-1396 9 9 1 S. M. Ammirata,‡ 3Department of Physics, Ohio State University, Columbus, Ohio 43210 n a J Jeffrey C. Hunnicutt and LeRoy A. Gorham Department of Physics, WPI, Worcester, MA 01609-2280 6 2 (February 1, 2008) The value of the dynamic critical exponent z is studied for two-dimensional superconducting, ] n superfluid,andJosephson Junction array systemsin zero magnetic fieldviatheFisher-Fisher-Huse o dynamicscaling. Wefindz≃5.6±0.3,arelativelylargevalueindicativeofnon-diffusivedynamics. c Universalityofthescalingfunctionistestedandconfirmedforthethinnestsamples. Wediscussthe - validity of the dynamic scaling analysis as well as the previous studies of the Kosterlitz-Thouless- r p Berezinskii transition in these systems, the results of which seem to be consistent with simple u diffusion (z=2). Furtherstudies are discussed and encouraged. s . t a m I. INTRODUCTION The KTBtransitionis drivenby the unbinding ofvor- - tex pairs. Below the transition temperature T vor- KTB d n The dynamics of two-dimensional (2D) zero-field sys- tices are thermally induced and can only be excited in o tems have been studied continually over the last few pairsthathaveafiniteenergyandnotas“free”vortices, c decades,1–6 usually in the context of the Kosterlitz- which have an infinite energy in an infinite system. As [ Thouless-Berezinskii7–9 (KTB) transition. The dynamic the temperature is increased, the number andsize of the 1 critical exponent z characterizes the critical behavior of vortexpairsincreaseandthesepairsscreenoneanother’s v the dynamics of a transition. Despite the ongoing stud- interactions. At the transition temperature, the vortex 5 ies of KTB dynamics, the value of z is usually not ques- pairs start to unbind and free vortices are formed. It is 9 tioned to be anything but the value that describes sim- thelargestpairsthatunbindfirst,leavingafinitedensity 2 ple diffusion: z = 2. This paper, through an analysis of smaller pairs above the transition temperature. The 1 of various transport data sets from systems including sizeofthelargestpairsdecreaseasonegoesfurtherabove 0 9 superconductors (SC’s), superfluids (SF’s), and Joseph- TKTB. As we review below, for T >TKTB, the free vor- 9 sonJunctionArrays(JJA’s)usingthedynamicscalingof tices result in an ohmic I-V curve at low currents while / Fisher, Fisher and Huse,10 presents ample evidence that the smaller pairs result in a non-linear I-V relationship at the value of z in these systems is much higher: z 5.6. at larger currents. m The purpose of this paper is to convince the reade≃r that Scaling techniques applied to zero-field I-V data from - despite the many previousreportsconsistentwith z =2, SC’s or JJA’s have been reported in the past but with- d the question of the value of z is still an open one. Per- out finding the results reported here. In Ref. 11, Wolf n haps the single element that distinguishes this analysis et al. scaled their I-V data from granular 30˚A thick o frompast analysesis that z is not takena priori to have films. There, however, the dynamic universality class c : a value 2. In the dynamic scaling analysis, varying z to was not explicitly studied and no value of z was deter- v optimize the description of the data is standard. mined. D.C.Harriset al.12 reportedz =2 inadynamic i X This paper is organized as follows. We begin in this scalinganalysisofJosephsonjunctionarrayI-V data,al- r sectionwithageneraldescriptionoftheKTBtransition, though it appears that they did not allow z to vary. We a previousscalingattemptsofI-V data,andpreviousfind- address that data in Section IIIB. L. Miu et al.13 men- ings of anomalousvortexdiffusion. In Section II, we will tion peripherally a dynamic scaling analysis on BSSCO discuss the various length scales whose competition re- (2223) in which it is found that z 4.5 (assuming a 2D ≃ sultsintheinterestingcriticalbehaviorofthistransition. system). However, those results were not pursued fur- In Section III, we present scaling results on SC’s, JJA’s, ther by those authors. In the case of superfluids, Brada and SF’s and also check for universality of the scaled et al.14 have performed a finite size scaling analysis (in data. We discuss the validity of the dynamic scaling ap- contrasttoadynamicscalinganalysis)oftheirfrequency proach as well as that of the conventional approach in dependent superfluid density and dissipation data from Section IV. We summarize the paper in Section V. helium films. They however did not include a study of 1 the value of z in their work. the KTB transition and critical behavior as well as the There has been experimental work that points to- approachesusedtostudythemisneeded. Inthissection, ward anomalous vortex diffusion. Théron et al.15 used we will review this background, directing our discussion impedance measurements on weakly frustrated JJA’s to primarilyatsuperconductingsystems. (Fordiscussionon provide evidence of non-conventional vortex dynamics. JJA’s or superfluids, see the reviews listed below.) We Unfortunately, this work did not probe the value of z. will discuss the various relevant length scales, the crite- There has also been theoretical16 and simulational6,17 ria for a phase transition, and the approaches one can workthatindicatesanomalousvortexdiffusionhavebeen take to study the KTB dynamic critical behavior. We made, but none conclude that z > 2 at T = T . In don’t intend our review to be comprehensive and refer KTB this work,in orderto limit the scope ofthis alreadylong the reader to any of the many excellent reviews.22–24 paper, we will not discuss the theoretical and simulational work, but concentrate only on experimental data. In comparing the “conventional” approach with the A. Length Scales dynamic scaling approach, we will make use of many of the formulas derived from the former. However,because The competition of the length scales25 in the system the dynamic scaling approachindicates that the dynam- determines the critical behavior of the KTB transition. ics are non-diffusive, the assumptions used to derive the For this reason it is important to review each of them. formulas for the dynamics in the conventional approach One can subdivide the length scales of the system into may be incorrect. For that reason, we will use those for- two categories: intrinsic and extrinsic. By extrinsic, we mulas only to address the validity of the conventional mean those length scales that are determined by an ap- approach and not the validity of the dynamic scaling. plied current or magnetic field. It is useful forus to clarify ouruse of the phrase “con- Intrinsic ventional approach.” Because the dynamic scaling has The intrinsic length scales include the vortex correla- only been used sparingly in analyses of the type of data tionlengthξ(T),the 2Dpenetrationdepth λ =2λ2/d 2D that we look at here, we view this approach as non- (where λ is the London penetration depth and d is the conventional. We will therefore refer to any other ap- sample thickness), and sample size. proachto analyzingthis data as“conventional”ortradi- There are three important aspects of the correlation tional. length that we discuss here. The first is its distinc- Thevalueofz thatisfoundhererunscontrarytosome tivetemperaturedependence fortemperaturesabovethe oftheconclusionsofpreviousstudies. However,westress transition temperature T :8 KTB thatourresultsarenotinconsistentwithalloftheresults in the literature. In particular, the dynamic scaling re- ξ (T) exp[ b/(T/T 1)] (1) + KTB sults presented here do not contradict (or even pertain ∝ − p to) the conventional findings for static behavior of the wherebisanon-universalconstant. Thisuniquetemper- KTB transition. Further, our results differ from only aturedependenceisincontrasttothecommonpower-law some of the experimental studies of the dynamics. This dependence one finds, for example, in Ginzburg-Landau will be discussed in Section IVB but we mention some theory. of those now. If the measurement does not involve z The second aspect is the behavior of ξ (T) below − or if the measurement does involve z but can measure T . Because the susceptibility below the transi- KTB only the product bz (where b is a material-dependent tion temperature is infinite, Kosterlitz originally defined constant that enters through the correlation length [see ξ (T) to be infinite. Based on the critical behavior of − Eq.(1)]), then our findings do notcontradictthose mea- the dielectric constant, Ambegaokar et al. (AHNS)1 de- surements. An example of the first type of measurement fined a finite diverging correlationlength for T W. Thisbegsthequestionofwhyoneshouldsee 2D where the vortex interaction strength has been ex- criticalbehavioratallifthereisnotruephasetransition pressedintermsofthesuperconductingparametersq2 = infinitesizesystems. Theanswerofcourseisthatcritical πn2sD¯h2/2m (n2sD = nsd is the areal superfluid density, behavior can be seen if there is a diverging length. This ns is the superfluid density, and m is the mass of a occurs in the KTB system, provided that ξ±(T)rc, the linear, current- superconductors induced term dominates and so the interaction is repul- sive. As a result, the interaction energy peaks at AvarietyofapproachescanbeusedtostudyKTBdy- namicbehavior. Herewereviewthetwomainapproaches r =4k T We/π¯hI (7) c B KTB used to study the dynamics of superconducting and JJA films and that therefore determine a value for the dy- where we have written the interaction strength in terms of the transition temperature,29 πn2D¯h2/2m = namic critical exponent z. Brief mention of the methods s used for superfluid helium systems will be made in Sec- 4k T . The non-linear I-V relationship originates B KTB tions IIIC and IVB2. There are other approaches for inthermally-activatedhoppingoverthisbarrieratarate investigating KTB behavior that don’t uniquely deter- Γ which depends on the value of the vortex pair energy mine a value for z; we will not review here but we will 3 discuss them in Section IVB in the context of previous R n . (14) f ∝ evidence that z =2. In this section, we will first review the conventional Substituting Eq. (12) and Eq. (13) into Eq. (14) and results and its derivation. We will then generalize these using V = IR(T,I) one arrives at Eqs. (9) and (10) re- formula to a general value of z. The dynamic scaling spectively. Kadin et al.28 have made extensions of this analysis will then be introduced and finally, the connec- worktofinitecurrent. Manyworkersinthefieldhowever tions between the two approaches will be discussed. take Eq. (9) to be valid over wide ranges of I. Eq. (14) is based on the Bardeen-Stephen flux-flow formula,38 (as stated by subsequent authors.18,24) For 1. Conventional approach subsequent discussion, it will be useful to outline the derivation of this equation,29 whose starting point is the electrodynamic Josephson relation, In the first, more “conventional” approach, the current-voltage (I-V) isotherms are measured and an- V =[h¯/2e]d∆φ/dt, (15) alyzed in terms of their I 0 limit:24,29 → V/I Iα(T)−1. (9) where ∆φ is the change in the phase of the supercon- ∝ ducting order parameter across the width of the sample. d∆φ/dt is proportional to the number of vortices that The signature of a KTB transition is a jump from cross the width of the sample per unit time, non-linear behavior below the transition temperature to ohmic above the T : KTB d∆φ/dt =2πLn v , (16) F D | | | | V/I R(T) exp[ 2 b/(T/T 1)] (10) ≡ ∝ − KTB− wherev isthevortexdriftvelocity. Finally,oneassumes D where b is a non-universal pconstant. [Minnhagen24 has v =µπ¯hI/eA, (17) generalized this to take into account the underlying su- D perfluid. See Eq.(22).] In particular,the exponentα(T) where µ is the vortex mobility and e is the electron will decrease linearly with increasing temperature until charge. Because the vortex mobility is taken to be local thetransitiontemperatureisreachedatwhichpointα(T) andthereforeindependentofn ,Eqs.(15)-(17)resultin will, in the I 0 limit, jump from 3 to 129 because of F the “universal→jump” in the superfluid density.35 Indeed, Eq. (14): R≡V/I ∝nF. The linear relationship between R(T) and n in the condition α(T = T ) = 3 is commonly used to f KTB Eq. (14) presumes single vortex diffusion. If one were determine the transition temperature. to allow more complicated dynamic or critical behavior, Eq. (9) is derived by determining R(T,I) above and then Eq. (14) must be modified. The general expression below T and using V = IR(T,I). This derivation KTB forR(T)inthecriticalregiondependsuponthedynamic is well documented28,29,36 but we will highlight the key exponent z: points. To find R(T,I), the density of free vortices n f must be determined. R ξ−z nz/2. (18) Below the transition temperature, free vortices in the ∝ ∝ f limit of a weak current are produced by thermal activa- Of course the two expressions are identical for z = 2. tionoverthe barrierin E(r ) asmentioned above[below c Using Eq. (18) in place of R n in the conventional f Eq. (6)]. This is done using the kinetic equation for the derivation,18,28,29 one would fi∝nd that α(T) will jump rate of change in the number of free vortices: from z+1 to 1 in the I 0 limit and that → dn /dt=Γ(T,I) n2. (11) f − f R(T) exp[ z b/(T/TKTB 1)]. (19) ∝ − − As mentioned above, Γ(T,I) is the rate at which vortex p We will discuss the ramifications of using z = 2 in pairs are unbound [Γ(T,I) exp E(r )/k T .] The 6 ∝ {− c B } Eq. (18) in Section IIC3. second term in Eq. (11) takes into account free vortices combining to form pairs. In steady state, 2. Dynamic scaling approach n =Γ(T,I)1/2 Iq2/2kBT, (12) F ∝ Dynamical scaling is motivated by the observation of Above the transition temperature, critical slowing down near a continuous transition. This n ξ−2, (13) phenomenon is well established for the KTB transition, f ∝ + and is marked by the divergence of the relaxation time since ξ is the average distance between free vortices. scale, τ. The dynamic scaling hypothesis asserts that + The final step in determining Eqs. (9) and (10) is to criticalslowingdownisrelatedcruciallytothedivergence relate R(T,I) to n :28,29,37 of the static correlationlength:39 τ ξz. f ∝ 4 In general, many types of dynamics may be associ- theothershouldbeconsideredforanarbitraryvalueofz. atedwithaparticularstaticuniversalityclass,andthese [For the generalization of the conventional theory to an should fall into distinct dynamicaluniversality classes.40 arbitrary value of z, see the discussion around Eqs. (18) For two-dimensional systems, including SC’s, SF’s, and and (19)]. JJA’s,theconventionalKTBdynamicaltheory1,2 iscon- Taking these considerations into account and looking sistent with model A universal dynamics,40 and z = 2. at the asymptotic limit I 0 of the dynamic scaling → For bulk superconductors however, the correct dynami- function, one finds that the two approaches are indeed cal universality class is presently unclear, and seems not compatible: (i) both theories predict that the critical to belong to model A.41 isotherm (T = T ) should be a power-law V Iz+1 KTB ∝ In light of this situation, it is important to test the [i.e., α(T )=z+1], (ii) for T T , both theories give α(T) = 1, with KTB tions are a priori unspecified. z is then determined by R(T) defined as in Eq. (19). collapsing different data sets onto a single curve. Al- though conventional KTB dynamical theories do not support such a general approach, a scaling ansatz has III. SCALING RESULTS recently been proposed by Fisher, Fisher, and Huse10 (FFH) for superconducting critical phenomena. This In this section we will apply the scaling theory successful ansatz has been applied to a wide variety of [Eq.(21)]ofFisher,Fisher,andHuse10 totransportdata systems and transitions, including simulations of KTB fromsuperconductors,JosephsonJunctionarraysandsu- dynamics.42 We expect that dynamical scaling will be perfluids. Theuniversalityofthescaleddatawillthenbe appropriate in the same regimes where the static KTB checked. Preliminary results on both high-temperature theory is applicable. superconductors (HTSC’s) and low-temperature (con- In the FFH theory, for 2D superconductors, the I-V ventional) superconductors (LTSC’s) were presented in curves should scale as Ref. 43 where it was concluded that z 5.7 0.3. ≃ ± A preliminary estimate for the value of z indicates V =Iξ−zχ (Iξ/T), (20) ± that z 2 for these systems. The critical I-V isotherm ≫ (T = T ) is easily identified on a log-log plot by the where χ (x) is the scaling function for temperatures KTB +(−) fact that it is straight (i.e., α(T ) is independent of above(below)T . Thetwoimportantasymptoticbe- KTB KTB current). The slope of this isotherm [α(T ) = z+1] haviorsofχ(x) arelim χ (x)=const. (ohmic limit), KTB x→0 + gives anestimate for z. A visual check for this condition and lim χ (x) xz (critical isotherm). The uni- x→∞ ± ∝ ontheI-V dataofRepacietal.,34 Vadlamannatietal.,44 versal jump appears as the difference in (log-log) slopes Matsudaetal.,45,oranyoftheothersclearlyshowsthat, between the two asymptotic limits of χ (x), as we will + indeed, z 2. With this initial evidence,we moveonto discuss below. ≫ a more rigorous scaling analysis of the data. It is convenient to rewrite the Eq. (20) as Toperformthescalinganalysis,ξ (T)inEq.(21)must ± be specified. This can be done for superconductors and 1/z I I Josephson junction array by exploiting the ohmic limit =ε (Iξ/T) (21) T V ± of Eq. (21): R(T) ξ−z. (For superfluids, the thermal ! ∝ + conductance K is used in place of the resistance: K where ε (x) x/χ1/z(x). The advantage of Eq. (21) ξ+z.) For ξ−, we will assume that the vortex correlatio∝n ± ≡ ± length is symmetric about the transition (modulo some over Eq. (20) is that one can better judge the scaling, pre-factor) in this section. We will explore the validity because only the x-scale is stretched in Eq. (21). In of this assumption in Sec. IVA1. Eq.(20),boththex-scaleandy-scalearestretchedmak- Note that in the dynamic scaling theory, there are no ingithardertojudgeacollapseofthescaleddata. (Com- requirements for the temperature dependence of ξ (T). pare Fig. 4 of Ref. 12 with Figs. 1-4 here.) ± Inthis work,we willassume that the KTB form, ξ(T) ∝ exp[ b/(T/T 1)], provides the most efficient pa- KTB − rameterization of the correlation length. It is through 3. Connections between the two approaches p this assumption that the explicit connection with KTB theory is made. Any other temperature dependence for Whilenotapparentatfirstglance,theconnectionsbe- R(T) could be used to check the scaling collapse, which tweenthedynamicscalingapproachandtheconventional would leave the type of transition more ambiguous. approachbecomeclearwhenoneconsidersthefollowing. We determine R(T) in two ways where possible. The First, one must keep in mind that the conventional ap- first method is to extract it from the ohmic part of the proach,asdescribedinSectionIIC1,isvalidonlyinthe I-V curvesandthesecondmethodistodigitizetheR(T) limit I 0 while the dynamic scaling approach is valid → data. Thetworesultsarethencomparedtooneanother. for finite currents. Secondly, the relationship of one to 5 In the case of discrepancies, the R(T) determined from a 12 ˚A thick YBCO mono-layer and includes isotherms theI-V isusedsincethermalequilibriumismorelikelyin rangingintemperaturesfrom10Kto40K.Theresistance that case. Another advantage of using R(T) determined was fit over the range 25K to 34.5K yielding, along with from the I-V is that one is assured that the R(T) is the other two criteria, the following parameter values: ohmic. The disadvantage of course is that fewer data T = 17.6 K, b = 7.79, and z = 5.9 0.3. The KTB ± pointsareavailableforthisR(T). Itshouldalsobenoted scaled data set denoted by c in Fig. 1 and covering the that we fit R(T) only over the temperature range over temperature range [2.6 K:3.4 K] corresponds to a con- which we have I-V isotherms. ventional, 100 ˚A thick, superconducting sample28 (Hg- The fitting parameters for Eq. (21) are z (universal), Xe alloy). The parameters which led to this collapse are T and b (non-universal). Three requirements must T = 3.04 K, b = 3.44, and z = 5.6 0.3. The resis- KTB KTB ± be fulfilled self-consistently in our scaling procedure: (i) tance was fit over the temperature range [3.3 K:3.4 K]. V Iz+1, along the critical isotherm T = T ; While this collapse is not as complete as those of the KTB (ii)∝R(T) ξ−z, in the high temperature range; and othersinthis figure,weemphasize thatthe bestcollapse ∝ (iii) scaling collapse of the I-V isotherms, according to was obtained for the reported value of z. Eq. (21). Condition (i), which says that the I-V curves are straight on a log-log scale at T , is used first to KTB estimate a value of T and z. That value of T KTB KTB 10-3 is then used in (ii) to fit the ohmic resistance data to obtainanexpressionfor ξ±(T). Finally, condition(iii) is s) (a) checked. Because there may be a couple isotherms that nit arepppeeaatredto, ibnetshteramigahntnoenr othfeShloagw-loetgaslc.a19le,,ftohristhperoaccecsesptis- b. u 10-6 (b) r a abIlenrsaonmgee ooffTthKeTBd’astatosesatstistfhyatalwl tehreexeamcoinndedit,iotnhse. I-V z] ( curves crossedoverto anohmic regionat largeI.32 This 1/V 10-9 (c) T behaviorisnotduetothecriticalbehaviorofthevortices but rather to a breakdown of the underlying superfluid. 1/z/[ + For that reason, we have omitted such data from our 1I analysis. 10-12 10-10 10-5 100 105 Note that in the following, we have displayed all the scaling results (except for a few noted exceptions) with- Iξ/T (arb. units) outfirstmakingajudgmentofthequalityofthedata. As FIG. 1. The I-V curves from thin superconducting films a result, the quality of the scaling also varies. Nonethe- scaled with Eq. (21): (a) 24 ˚A thick YBCO layers in a multi-layer structure from S. Vadlamannati et al.;44 (b) 24 less, we stress thateachscaling resultdisplayedhere has ˚AthickYBCOmono-layerfromRepacietal.;34 and(c)100˚A been optimized for the best collapse and not for a value thick Hg/Xe film from Kadin et al.28 (Data sets (b) and (c) ofzinagreementwiththeothersamples. Thismakesthe havebeenshiftedarbitrarily.) Thelowerbranchoftheseplots resultthatallofthecollapsesoccurforroughlythesame correspondtoT >TKTB. Thelimitsofthisbranchareohmic value of z all the more striking. Furthermore, we believe in the weak current limit to V ∝ Iz+1 in the high current that there is a strong correlation between the quality of limit. It is these limits that represent the jump in the expo- the raw I-V data and the scaled data. But we leave this nent α. for the reader to judge. A few features of the scaled data in Fig. 1 should be pointedout. First,theupperbranchcorrespondstotem- A. Superconducting films peraturesbelowthetransitiontemperatureandthelower branch to temperatures above the transition tempera- Fig. 1 shows the scaling of three separate data sets ture. Secondly, an ohmic I-V relationship here is repre- to Eq. (21), (previously reported in Ref. 43 but plotted sented by a slope 1 on the log-log scale. One can see in on a different scale here.) The first data set44 (marked eachofthethreecurvesinthisfigure(andthescaleddata a in Fig. 1) covers a temperature range [30.06 K:46.09 in the following figures,) the lower (T > T ) branch KTB K] and is from a YBCO/PrBa2Cu3O7−δ multi-layer in is ohmic atlowvalues ofthe scalingvariablex (typically which the YBCO layers have a thickness of 24 ˚A and low currents)and curves over and approaches a horizon- are reported to be nearly electrically isolated from one tallineasxisincreased. Thehorizontallinecorresponds another by PrBa2Cu3O7−δ (PBCO) barrier layers. The to the I-V relation, V Iz+1. scaling procedure leads to the results T = 32.0 K, ∝ KTB b = 14.0, and z = 5.6 0.3 where the resistance was fit ± over the range [43.5 K:47.0 K]. (See Figure 8.) Curve b in Fig. 1 is the scaled I-V data from Ref. 34 taken on 6 scale in the same way as the thinner samples in Section IVA3. A scaling analysis of the Hebard and Fiory I-V data on a In/InO film18,47 was not possible since the temperatures of their isotherms were not published. s) 10-4 (a) t ni u B. Josephson junction arrays . b r (a (b) I-V characteristicsofJosephsonjunctionarraysareex- z] 10-7 pected to be similar to that of superconducting films.33 1/V In this section, we apply Eq. (21) to data from two JJA (c) T systems.48,49 A primary difference for these systems is [ / z that their resistance is not described by Eq. (19) be- 1/ 1+ causethetemperatureisrenormalizedanddependsupon I 10-10 the temperature-dependent criticalcurrent ic(T). In the 10-8 10-3 102 data that we analyze below, we did not have access to Iξ/T (arb. units) ic(T)andsowecouldnotdetermineifR(T)followedthe FIG.2. The scaled I-V curves of (a) Ref. 45 on a YBCO KTB behavior for JJA’s. Nonetheless, following the dis- mono-layer; (b) Ref. 46 on a In/InO 150 ˚A thick composite cussionatthe beginning ofthis Section (III), we willuse film; and (c) Ref. 43 from a 1000 ˚A thick BSCCO crystal. the Eq.(19)tocheckfor acollapseofthe dataaccording (Data set (b) and (c) has been shifted arbitrarily.) The col- to Eq. (21). As a result, the value of b will not have the lapse of curves (a) and (b) in this figure are not as complete significance it had in Section IIIA. as that of curve (c) or curves (a) and (b) of Figure 1. Yet, thecollapse could not beimproved using othervalues of z. For this paper, we have extended this analysis to two 10-2 more2Dsuperconductingfilms. Thecollapseisshownin Fig.2. Thecurvemarkedainthatfigureisfromamono- layerof YBCO sandwichedbetween two PBCOlayersof ) s differentthicknesses45 andcoversisothermsrangingfrom nit (a) 16.32K to 41.31K. For the parameters, we find TKTB = u 18.3 K, b=31.04,and z =5.3 0.5. The value of TKTB b. issimilartothatofthe YBCO±mono-layerofRef.34but ar 10-6 (b) ( thevalueofbisnearly4timeslarger. CurvebinFig.2is ] fromI-V dataon150˚AthickIn/InOcompositefilmfrom 1/z (c) V Ref. 46. The I-V isotherms cover a temperature range T [3.010 K:3.182 K]. Here, the resistance data determined /[ z fromtheI-V curvescoveredonlyalimitedrangeanddid +1/ 1 not match well the R(T) data from Fig. 3 of that paper. I 10-10 For these reasons, the scaling criteria (ii) was not fully 10-9 10-6 10-3 100 met. Nevertheless, the scaling collapse occurred for a Iξ/T (arb. units) valueofznearthatoftheothersamples: T =2.97K, KTB FIG.3. ThescaledI-V curvesfromJosephsonjunctionar- b=10.21, and z =5.2 0.5. We also applied thi±s scaling analysis43 to a thicker rays: (a) YBCO/Ag weak links from Ref. 49; (b) Nb-Ag-Nb proximity-coupledjunctionsfrom Ref.12;and(c)Nb/Nbar- (1500 500 ˚A) Bi Sr CaCu O crystal. For such a 2 2 2 8+δ rays of Ref. 48. (Data sets (b) and (c) has been shifted arbi- ± thick crystal of a layered material, one would expect a trarily.) crossover to 3D behavior near the critical temperature anda failure of the 2D scalingandperhaps a breakdown Curve a in Fig. 3 is scaled I-V data from an high- inthescaling. Wehoweverdidnotseeanybreakdownin temperature superconducting Josephson junction array the scaling for this sample. Indeed, as shown in curve c (YBCO/Ag)49 where the parameters used were T = KTB of Fig. 2, we found a good collapse of the data with the 74.3 K, z = 5.8 0.4 and b = 0.72. The resistance was 2Dscalingformwithz =5.6 0.3,T =78.87K,and ± KTB fit over the temperature range: [78K:90 K]. One can see ± b = 0.20. (See also Fig. 2 of Ref. 43.) A difference be- that the collapse is very good except for the isotherms tweenthisscaleddataandtheotherswasfoundwhenwe furthest below T . This breakdown could be because KTB looked at universality. We examine this issue in Section thoseisothermsareoutofthecriticalregion. Anattempt IIID and discuss the reason why a thicker crystal may to optimize the T T collapse by letting the corre- KTB ≪ 7 lation length be asymmetric in accordance with Eq. (2) weareabouttopresentcanonlybesaidtobe consistent was unsuccessful. withsuchavalueofz butcannotbetakentobeevidence The scaled data denoted by curve b in Fig. 3 is from that z 5.6 for two reasons. First, there is no reliable a Nb-Ag-Nb proximity-coupled junction array.12,50 (We therma≃l conductance K data in the “ohmic” limit (i.e., used the “100%” data from Ref. 50.) The parameters T >T zeropowerQ˙ limit.) Furthermore,onehasto KTB which produced the best fit are TKTB = 6.84 K, z = account for the conductance of the gas Kg surrounding 5.8 0.3 and b = 0.32 and the resistance fit was over the film which can only be approximated to within a ± the temperature range: [7.3K:7.8 K]. The scaled I-V’s factoroftwo. Thesetwopointsallowusmorefreedomto coveredtherange: [6.9K:7.8K].Onecanseethathe6.9K obtainthe optimalscaling. Nonetheless,the best scaling data set which does not scale well for low currents. We doesyieldaz whichissimilartothatofsuperconductors believe that this is not a real effect since all of that non- and Josephson junction arrays. collapsing data has a voltage V < 10−9V and does not The collapsed data in Fig. 4 marked curve a is from parallelthebehavioroftheotherdatainthatrange. The Ref. 53. Instead of varying the temperature, those au- data do scale well for V > 10−9V. Harris et al.12 have thors varied film thickness d which in turn varies the previously checked the dynamic scaling of this data and transition temperature. The independent variable then concluded that z = 2. We believe that they would have is d and not T and so the correlation length depends reachedthesameresultasfoundherehadtheyallowedz upon d d instead of T T where d is the c KTB c | − | | − | tovarytooptimizethescalingandhadtheyusedEq.(21) thickness whose T corresponds to the temperature KTB instead of Eq. (20). of the experiment. (See Eq. (17) of Ref. 53.) By ad- Curvec inFig.3 isthe scaledI-V datafromaNb/Nb justing d, they were able to obtain Q˙-∆T data for both Josephsonjunctionarray.48Theparametersusedtoopti- above and below the transition temperature. The pa- wmhizeerethτis=wekrBeTτK/[T(h¯B/=2e)0i.c5(1TK)],.zA=c5o.l5la±ps0e.5oafntdhebs=ca5l.e7d, raanmdeKtegrs=th8a.0t we10o−b4taWine/dK.wWeree adscs=um5e.d4,tzha=t K5.4g±wa0s.4a, data could be obtained over a relatively large range of z constantovert×heparameterspacethattheQ˙-∆T curves and T . We attribute this to the fact that the I-V’s KTB coveredandestimateditbasedonthesmalldbehaviorof isotherm only covered 2-3 decades of voltage and that K. This placedsomelimits onthis quantitybutwewere there were no isotherms for temperatures below T . KTB stillabletovaryitbyafactoroftwo. We foundthatthe The resistance was fit over the range τ = [0.8:1.3]. scaling collapse was relatively insensitive to the value of Inaconventionalanalysis,Abrahametal.51 reporteda K because most of the power was flowing through the g jumpintheI-V exponentαof3to1forPbBi/Cuarrays, film. Asonecansee,thescalingstartstobreakdownfor aresultwhichimpliesz =2. Asfaraswecouldtell,that thicknesses (d 9 layers) much larger than d . c data was not published and so was not available for our ∼ analysis. WetriedtoapplyEq.(21)tosample6-18-81of Ref.52butno definitiveconclusionswerereacheddue to the fact thatthe I-V isothermsonly covered2-3decades of voltage and a limited temperature range. 105 ) C. Superfluid 4He films ts ni u (b) For superfluid 4He films, the analog of electrical con- b. r ductance and I-V curves are thermal conductance and a Q˙-∆T curves, where Q˙ is power through the film and ( 102 ] z ∆T is the temperature gradient across the film. These 1/ T measurements are done by injecting heat at one end of ∆ (a) [ a thin superfluid film adsorbed on a surface (e.g. my- z/ 1/ lar). Because of dissipation from vortex pairs, there is + 1. Q a temperature gradient across the film that is measured 10-1 and is the analog of the voltage in the superconducting 10-5 100 105 measurements. In reality, Q˙ is not the heat through the . Qξ (arb. units) film but rather represents 4He mass flow from the cool FIG.4. The scaled Q˙-∆T curves from superfluid 4Helium end of the film to the warmend, which occurs to replen- ish the 4He which evaporates from the warmer end at a films of (a) Ref. 53; and (b) Ref. 54. (Data set (b) has been shifted arbitrarily.) quickerratethanfromthe other end. This is thoroughly discussed in Ref. 53. CurvebinFig.4isfromRef.54. Itcontainsmoredata As we shallshortlysee,these results do showthatz 5.6. However,wearecarefultopointoutthattheresul≃ts corresponding to T >TKTB but doesn’t cover as wide a range of thicknesses overall as the data of Maps et al.53 8 The collapse was found to be consistent with the results FIG. 5. The JJA data of Fig. 3a49, the YBCO SC data we have presented in this paper: d =5.2, z =5.6 0.5, of Fig. 1b34, and the Hg/Xe SC data of Fig. 1c28 plotted c andK =5.0 10−4W/K.LiketheJJAdataofRe±f.48, together. The scaled data sets have been shifted to show g the error bars×here are large because the data extends universality of thescaling function. only over a couple orders of magnitude. In Figure 6, we plot all of the SC, JJA, and SF data that we have presented here in a single plot to test uni- D. Universality versality. As in the previous figure, we adjusted each data set within the error bars so that z = 5.6. We also removedsomeoftheindividualscaledisothermsthatdid One of the fundamental properties of critical behavior not scale well with the other scaled isotherms from the and scaling is universality: the idea that the same func- same sample. This includes two of the isotherms from tion ε (x) in Eq. (21) and the same value of z describes ± theMatsudaet al.45 data(the secondandthirdfromthe allofthedataindependentofthesystemormaterial. For right in data set (a) of Fig. 2), parts of two of the data the T > T branch, we found universality for nearly KTB sets from data set (b) of Fig. 2 (second and third from all of our scaled data. For the T < T branch, on KTB the right), and part of one data set from Harris et al.12 the other hand, the same function ε (x) described the ± (the rightmost set labeled (b) in Fig. 3). One can see data sets of the thinnest 2D samples but not that of the clearly that the T >T data scales well and strongly layeredmaterials43,44 orthe superfluidHelium.53 We ex- KTB suggests universality for this temperature regime. The plore these issues in this section. Garlandand Lee46 data is the weakestof these since it’s In Figure 5, the HTSC data from the YBCO mono- lowest temperature isotherm does not lie on the other layer,34 the LTSC data from the Hg/Xe thin film,28 and scaled data as we have pointed out in this figure. theYBCOJJAdata49 areplottedtogether. Becausethe scaling functions are dimensionless, we multiply the x and y axes by non-universal constants which enter the scaling theory to account for the sample dependence. In ordertotestuniversality,allthedatasetsmustbescaled with the same value of z. So we haveadjusted eachdata set within the error bars of the parameters so that each 10-3 Vadlamannati ) has z = 5.6. As one can see, the collapse is very good, s Maps t and bolsters the evidence that this data scales and that ni u z 5.6 0.3. The agreement of the Repaci et al.34 and . th≃e Herb±ert et al.49 data is particularly impressive since rb a Ammirata both data sets have extensive T < T branches and ( KTB 10-5 becausetheycomefromtwodifferentsystems: JJA’sand z] 1/ SC’s. V Garland T [ / z 1/ + 1 I 10-2 10-7 10-7 10-2 103 Iξ/T (arb. units) ) FIG.6. Thedatafromallofthedatasetsplottedtogether. s unit 10-5 TityheofTth>eTsKcaTliBngdafutancctoiollnapfoservtherayt wteemllpseurgagtuesrteinrgeguimniev.erTsahle- b. T T data and some of the T < T data, it KTB KTB mono-layer systems examined in Section IIIA varied by did not hold for all of the latter. In this Section, we will a factor of 4,34,45 which is concerning. However, we do examine other aspects of the dynamic scaling analysis not think it is troubling that the b varied from mate- presented here and its validity. rial to material by an order of magnitude. The systems consideredhere arediverse fromone another. For example, the electron density can vary significantly from the 1. Vortex correlation length conventional superconductors to the HTSC’s. Another interpretation is that the value of b is a phenomenologi- Therearevariousaspectsofassumptionsthatwemade cal one and not equivalent to its true asymptotic critical regardingthe vortex correlationlength that may detract value.42,57 fromthedynamicscalinganalysis. Herewemakeconsid- erations on its symmetry, the value of the non-universal constant b, and the expected range of the validity of the 2. Universal jump: 6.6 to 1 temperature dependence. Throughout Section III it was assumed that the vor- Ifoneistobelievethatz 5.6inthesematerials,then tex correlation length was symmetric about the transi- ≃ tiontemperatureeventhoughAmbegaokaretal.2predict oneexpectstoseeajumpintheI-V exponentαof6.6to 10

Dynamic scaling for 2D superconductors, Josephson junction arrays and superfluids

Similar Dynamic scaling for 2D superconductors, Josephson junction arrays and superfluids