Spontaneous Fluxoid Formation in Superconducting Loops
SSpontaneous Fluxoid Formation in Superconducting Loops ∗ R. Monaco
Istituto di Cibernetica del CNR, 80078, Pozzuoli,Italy and Unit `a INFM Dipartimento di Fisica,Universit `a di Salerno, 84081 Baronissi, Italy † J. Mygind
DTU Physics, B309, Technical University of Denmark, DK-2800 Lyngby, Denmark ‡ R. J. Rivers
Blackett Laboratory, Imperial College London, London SW7 2AZ, U.K. § V. P. Koshelets
Kotel’nikov Institute of Radio Engineering and Electronics Russian Academy of Science,Mokhovaya 11, Bldg 7, 125009, Moscow, Russia. ¶ (Dated: October 29, 2018) Abstract
We report on the first experimental verification of the Zurek-Kibble scenario in an isolated su-perconducting ring over a wide parameter range. The probability of creating a single flux quantumspontaneously during the fast normal-superconducting phase transition of a wide Nb loop clearlyfollows an allometric dependence on the quenching time τ Q , as one would expect if the transitiontook place as fast as causality permits. However, the observed Zurek-Kibble scaling exponent σ = 0 . ± .
15 is two times larger than anticipated for large loops. Assuming Gaussian windingnumber densities we show that this doubling is well-founded for small annuli.
PACS numbers: 11.27.+d, 05.70.Fh, 11.10.Wx, 67.40.Vs a r X i v : . [ c ond - m a t . s up r- c on ] J u l he Zurek-Kibble (ZK) scenario [1, 2, 3] proposes that continuous phase transitions takeeffect as fast as possible i.e. the domain structure after the quenching of the system initiallyreflects the causal horizons. This proposal can be tested directly for transitions whosedomain boundaries carry visible topological charge. In this letter we shall show that, withqualifications, this scenario is strongly corroborated by the behaviour of superconductingloops, for which the topological defect is a fluxoid i.e. a supercurrent vortex carrying onemagnetic flux quantum Φ = h/ (2 e ) (cid:39) . × − Wb.The basic scenario is very simple. Consider a planar low- T c superconductor in whicha hole has been made of circumference C . In the Meissner state the order parameter forthe superconductor is a complex field ψ with phase φ , ψ = ρe iφ , where | ρ | measures thedensity of Cooper pairs. On quenching the system from the normal to superconductingphase, causality prevents the system from adopting a uniform phase. If, on completion ofthe quench, we follow the periodic phase φ ( x ) (mod 2 π ) along the boundary of the hole(co-ordinate x ), we can define a winding number density: n ( x ) = dφ ( x ) /dx/ (2 π ). The totalnormalized magnetic flux through the hole is, in units of Φ , the winding number: n = (cid:90) C n ( x ) dx = ∆ φ π , (1)where ∆ φ is the change in φ . In the absence of an external magnetic field, on average (cid:104) n (cid:105) = 0, but it will have non-zero variance (∆ n ) = (cid:104) n (cid:105) , which is what can be measured interms of the probabilities f ± m to trap ± m flux quanta: (cid:104) n (cid:105) = (cid:80) ∞ m = −∞ m f m . Accordingto Ref.[2], on completion of a thermal quench having a given inverse quench rate τ Q = − T c / ( dT /dt ) T = T c , the phase φ is correlated over distances 2 π ¯ ξ , where ¯ ξ was predicted todepend allometrically[2] on the quench time τ Q :¯ ξ ≈ ξ (cid:18) τ Q τ (cid:19) σ . (2)¯ ξ , also called the ZK causal length, is defined in terms of the cold correlation length ξ andthe Ginzburg-Landau relaxation time τ of the long wavelength modes. The ZK scalingexponent σ is determined by the static critical exponents of the system and, in the mean-field approximation, σ = 1 / π ¯ ξ then, for a hole of radius r , circumference C (cid:29) π ¯ ξ ,2 n (cid:105) ≈ C π ¯ ξ = rξ (cid:18) τ Q τ (cid:19) − σ . (3)For small rings with C < π ¯ ξ the likelihood of seeing two or more units of flux is smalland (cid:104) n (cid:105) ≈ f +1 + f − = f , the probability of single fluxoid trapping. It is plausible toextrapolate Eq.(3) to: f ≈ (cid:104) n (cid:105) ≈ rξ (cid:18) τ Q τ (cid:19) − σ , (4)showing allometric behaviour of f with the same exponent. We note that the ZK argumentmakes no assumptions about the rest of the superconductor, equally valid for the phasechange along the inner circumference of an annulus as it is for the phase change around asingle hole in a superconducting sheet. In 2003 the first experiment with superconductingloops [4] was performed to test Eq.(3). The experiment consisted of taking an isolated arrayof thin-film wide rings and making it undergo a forced phase transition by heating it aboveits superconducting critical temperature and letting it to cool passively back towards the LHe temperature. Once the thermal cycle is over, the rings are inspected by a scanningSQUID and the number and polarity of any trapped fluxoids determined. These rings, ofamorphous Mo Si thin films, had thickness almost one order of magnitude smaller than thelow temperature London penetration depth. Although this provides favorable conditions forthermally activated phenomena it drastically increases the likelihood that nucleated vorticesescape through the ring walls during the fast quench. In fact, the experimental outcomewas totally at variance with the allometric scaling above. However, the prediction Eq.(3)presupposes that we can ignore the contribution to the flux from the freezing in of thermalfluctuations of the magnetic field [5] and the results of [4] could be explained in terms of thefreezing of thermally activated fluxoids in a similar spirit to Ref.[5].In this paper we shall present results from a new experiment with high-quality Nb filmrings with r = 30 µ m, two times thicker than their low temperature London penetrationlength λ L,Nb ; the film thickness and composition were chosen to reduce the thermal activationof fluxoids and, at the same time, ’the washing out’ of fluxoids generated by the conventionalcausality mechanism. In our case the contribution ∆ f to the probability of finding a unitof flux from thermal magnetic field fluctuations is approximately[5]:3 f (cid:46) ( k B T c ) rµ / Φ ≈ × − , (5)and can be safely ignored. We therefore look for scaling behaviour in τ Q .Here a different way of counting both the number and the polarity of generated defectshas been adopted. It is based on the detection of the persistent currents J s circulatingaround a hole in a superconducting film, when one or more fluxoids are trapped inside thehole. The circulating currents screening the bulk of the superconductor from the trappedflux induce a magnetic field H in the volume around the ring, such that J s = ∇ xH . Byplacing a Josephson tunnel junction (JTJ) along the perimeter of the hole in the area wherethis field passes, any trapped fluxoid will result in a modulation of the JTJ critical current,similar to the effect of an external field applied perpendicular to the ring. Indeed, thismethod is strongly inspired by the results found investigating the effects of a transverse fieldon Josephson junctions of various geometries[6]. The geometry of our experiment is sketchedin Fig.1; the black wide ring is a 200 nm thick Nb film, which also acts as the common baseelectrode for two JTJs whose top electrodes are depicted in grey. The JTJs have the shape ofgapped annuli and the bias current is supplied in their middle point: to our knowledge, thisgeometrical configuration has never been realized before and is characterized by a peculiarmagnetic diffraction pattern: for small magnetic fields the critical current increases both forpositive and negative field values. The original purpose of having two counter electrodeson the base ring was that any screening current circulating on the outer ring circumferencewill preferentially affect the outermost JTJ, and vice versa for the screening current onthe inside of the ring. Since the persistent currents due to trapped flux mainly flow inthe inner ring circumference [7], in this experiment we only used the innermost JTJ. Thelayout shown in Fig.1, with a few key differences, bears remarkable topological similarity tothe one used in a series of experiments by us to demonstrate the ZK scaling behaviour ofEq.(3) in annular JTJs [8, 9, 10, 11, 12]. The most obvious difference in the design is theinclusion of the two junction counter electrodes on top of the ring-shaped base electrode.The second change is the removal of a small section of the full annular junction to leave agapped annular junction with the purpose of avoiding fluxons created inside the JTJ at theJosephson phase transition. Should any be produced, they will simply migrate through thejunction extremities driven by the applied bias current needed to overcome eventual pinningpotentials. This leaves the experiment only sensitive to the fluxoids produced in the ring at4 IG. 1: Sketch of a superconducting loop (black) used as a base electrode for two gappedNb/AlOx/Nb annular Josephson tunnel junctions (whose top electrodes are in gray). The ringinner and outer radii are r = 30 and R = 50 µ m, respectively, while the top electrodes width is5 µ m. the phase transition.As with our previous experiments, the present one relies on a fast heating system, obtainedby integrating a Mo resistive meander line on the 4 . × × . τ Q could be continuously varied overmore that four orders of magnitude (from 20 s down to 1 ms) by varying the width and theamplitude of the voltage pulse across the integrated resistive element. In order to determinethe quench time with high accuracy, the ring temperature was monitored exploiting thewell known temperature dependence of the gap voltage of high-quality Nb/AlOx/Nb JTJsalready described in Ref.[10]. After each ring thermal quench the critical current of theinnermost JTJ is automatically stored and an algorithm has been developed for the countingof the trapped fluxoids. Finally, all the measurements have been carried out in a magneticand electromagnetically shielded environment. During the thermal quenches all electricalconnections to the heater as well as to the JTJs were disconnected. While more details onthe measurement setup and on the fabrication process can be found in Ref.[13] and Ref.[14],respectively, an extensive description of the chip layout, the experimental setup and thesystem calibration will be given elsewhere[15].The experimental results shown in Fig. 2 were obtained using a ring with inner andouter radii, r = 30 µ m and R = 50 µ m, respectively. Similar samples have shown thesame behaviour. We note that wider rings prevent fluxoids from tunneling out of the ring,5lthough their smaller normal self-inductance L n makes fluxoid formation energetically moreunlikely. In our case, the field energy E = Φ / L n associated with a single flux quantumΦ is several orders of magnitude larger than the thermal energy k B T c / T c [16].Magnetostatic numerical simulations implemented in the COMSOL Multiphysics 3D Elec-tromagnetics module showed that, when a single flux quantum is trapped in such rings, theradial magnetic field induced by the circulating currents at the ring inner border is as largeas 1A/m, a value easily detectable by the JTJ. For our rings, the number of trapped flux-oids was small, usually no more than one; indeed we measured the probability of trappinga single up-fluxoid (field up) f +1 and a single down-fluxoid f − .Fig. 2 shows on a log-log plot the measured frequency f = f +1 + f − = ( n +1 + n − ) /N = n /N of single fluxoid trapping, obtained by quenching the sample N times for each value ofa given quenching time τ Q , n being the number of times that one defect or one anti-defectwas spontaneously produced. N ranged between 250 and 300 and n was never smallerthen 10, except for the rightmost point for which n = 5. The vertical error bars gives thestatistical error f / √ n . The relative error bars in τ Q amounting to ±
10% are as large asthe dot’s width. As expected we had n +1 ≈ n − , but slightly larger than n − indicating thepresence of a small residual stray field in our apparatus (see inset of Fig.2).To test Eq.(2), we have fitted the data with an allometric function f = a τ − bQ , with a and b as free fitting parameters. An instrumentally weighted least-mean-square fit of f vs. τ Q ,represented by the continuous line in Fig. 2, yields a = 0 . ± .
02 (taking τ Q in ms) and b = 0 . ± .
15. The large fit correlation coefficient R = 0 .
987 indicates that the allometricbehaviour is reliably confirmed, however the scaling exponent b is about two times largerthan expected for large loops.A doubling of the large-loop ZK exponent for small loops has a possible explanation in theframework of the Gaussian correlation model introduced in Ref.[12] in which it was assumedthat the winding number n ( x ) is a Gaussian variable until the transition is complete, wherebyall correlation functions are determined by the two-point correlation function g ( x − x , C ) = (cid:104) n ( x ) n ( x ) (cid:105) . As a result [17]: (cid:104) n (cid:105) = (cid:90) C (cid:90) C (cid:104) n ( x ) n ( x ) (cid:105) dx dx = 2 C (cid:90) C g ( x, C ) dx. (6)For C (cid:28) π ¯ ξ , we can assume a correlation function of the form g ( x, C ) =¯ g ( x/ π ¯ ξ, C/ π ¯ ξ ) / (2 π ¯ ξ ) , so that: 6 IG. 2: Log-log plot of the measured frequency f of trapping single fluxoid versus the quenchingtime τ Q for a Nb ring having inner radius r = 30 µ m, outer radius R = 50 µ m and thickness d = 200 nm (cid:39) λ L,Nb . Each point corresponds to hundreds of thermal cycles. The vertical errorbars gives the statistical error, while the relative error bars in τ Q amounting to ±
10% are as largeas the dots’ width. The solid line is the best fit to an allometric relationship f = a τ − bQ whichyields a = 0 . ± .
02 (taking τ Q in ms) and b = 0 . ± .
15. For comparison purposes, the dashedline is the prediction of Eq.(9)) with χ (see text) set to 0 . f ≈ C π ¯ ξ (cid:90) C/ π ¯ ξ ¯ g (¯ x, C/ π ¯ ξ ) d ¯ x ≈ (cid:18) C π ¯ ξ (cid:19) ¯ g (0 , C/ π ¯ ξ ) . (7) provided ¯ g (¯ x, C/ π ¯ ξ ) is analytic at ¯ x = 0. This suggests that Eq.(4) should be replaced bythe scaling behaviour: f ≈ κ (cid:18) C π ¯ ξ (cid:19) = κ (cid:18) rξ (cid:19) (cid:18) τ Q τ (cid:19) − σ , (8)with a proportionality constant κ of the order of unity. To buttress this suggestion, it is notdifficult to show that, in the Gaussian approximation, the value of (cid:104) n (cid:105) along a small ringin a 2D superconductor is proportional to the area enclosed by the ring [15].This doubling of the scaling exponent in Eq.(8) has the price of coming with a lower prob-ability, but leaves us with some freedom with the ZK prefactor. Indeed, the value of theprefactor a obtained from the allometric best fit of the experimental data in Fig.2 is about κ = 4-5 times larger than the predicted value ( r/ξ ) √ τ = 0 .
04 obtained using the values r = 30 µ m, ξ ≈
30 nm and τ = π (cid:126) / k B T c ≈ .
16 ps and taking τ Q in ms. As a bound we7nly expect agreement in the overall normalization of the prefactor a to somewhat betterthan an order of magnitude, largely confirmed by experiment. We point out that the de-pendence of the prefactor a on the ring width remains to be investigated both theoreticallyand experimentally. [We note that, if Eq.(3) were true, then (cid:104) n (cid:105) would be ≈ .
6, i.e., 20times larger than the experimental value for τ Q = O (10 ms ), say.]In the opposite case of large circumferences, C (cid:29) π ¯ ξ , g ( x, C ) is controlled by thecorrelation length ¯ ξ of the winding number at the time of unfreezing and does not dependon C , i.e., the effect of periodicity for large rings is small. With g ( x ) = ¯ g ( x/ π ¯ ξ ) / (2 π ¯ ξ ) ondimensional grounds, we justify the random walk assumption of Eq.(3), (cid:104) n (cid:105) = Cπ ¯ ξ (cid:90) C/ π ¯ ξ ¯ g ( z ) dz ≈ Cπ ¯ ξ (cid:90) ∞ ¯ g ( z ) dz = χ C π ¯ ξ , (9)with χ = O (1). The dashed line in Fig.2 is the prediction in Eq.(9) with χ set to 0 . . ± .
15. This is obviously at variancewith the extrapolation (4) of the Zurek prediction Eq.(3) to small rings, for which we expect σ = 0 .
25. We have suggested that it be given by Eq.(8) with twice the exponent, as aconsequence of the Gaussianity of the Cooper pair field phase (before truncation by back-reaction), an assumption supported in a slightly different context by the behaviour of JTJsin an external field [12]. As Gaussianity permits the instabilities from which defects form togrow as fast as possible, in general it gives the same scaling exponents as the ZK scenariofor large systems. With this qualification we see our result as providing strong support forZurek-Kibble scaling over a wide range of quenching time τ Q . We stress that this experimentis the only one to date to have confirmed the Zurek-Kibble causality scenario for singleisolated superconducting rings (as distinct from Josephson junctions). Further experimentsto investigate the transition to the random walk regime and the effect of the ring width areplanned. For example, a test of Gaussianity is that f (cid:46) . C [12].Given that the original ZK scenario was posed to demonstrate the similarity in the role ofcausality at transitions in the early Universe and in condensed matter systems we have seenthat the finiteness of the latter systems requires careful disentangling from the underlyingprinciples before we can draw any quantitative conclusions.In the same vein we conclude with a speculation concerning the small-annulus JTJ exper-8ments [9, 10, 11, 12] that, hitherto, have been the only superconductor experiments to showscaling behaviour. In that case also, the observed exponent was twice that anticipated fromlong annuli [8]. However, in the case of JTJs there is an ambiguity in their fabrication thatis sufficient to double the exponent, according as the ’proximity effect’ enables otherwisesubcritical behaviour of the Josephson current density to dominate near the transition [10].We had assumed that this was the reason for the discrepancy. We shall now reexamine theseearlier experiments with the above analysis in mind.The authors thank P. Dmitriev for the sample fabrication and testing, A. Gordeeva foruseful discussions and M. Aaroe for the help at the initial stage of the experiment. ∗ Submitted to Phys. Rev. Letts. † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected][1] W.H. Zurek,
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