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Constriction-limited detection efficiency of superconducting nanowire single-photon detectors Andrew J. Kerman Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, MA, 02420 Eric A. Dauler, Joel K.W. Yang, Kristine M. Rosfjord, Vikas Anant, and Karl K. Berggren Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, 02139 7 0 G. Gol’tsman and B. Voronov 0 Moscow State Pedagogical University, Moscow 119345, Russia 2 (Dated: February 2, 2008) n We investigate the source of large variations in the observed detection efficiencies of supercon- a ductingnanowiresingle-photondetectorsbetweenmanynominallyidenticaldevices. Throughboth J electricalandopticalmeasurements,weinferthatthesevariationsarisefrom“constrictions:”highly 2 localized regions of the nanowires where the effective cross-sectional area for superconducting cur- 2 rent is reduced. These constrictions limit the bias current density to well below its critical value over the remainder of the wire, and thus prevent the detection efficiency from reaching the high ] t valuesthat occur in these devices only when they are biased near thecritical current density. e d PACSnumbers: 74.76.Db,85.25.-j - s n i Superconducting nanowire single-photon detectors is to understand (and reduce) the large observed varia- . s (SNSPDs) [1, 2, 3, 4] provide a unique combination of tion of detection efficiencies for nominally identical de- c high infrared detection efficiency (up to 57% at 1550nm vices [2], which would set a crippling limit on the yield i s [2] has been demonstrated) and high speed (<30 ps tim- of efficient arrays of any technologically interesting size. y ingresolution[3,5],andfew-nsresettimes afteradetec- In this Letter, we demonstrate that these detection effi- h tionevent[4]). Applicationsforthesedevicesalreadybe- ciency variationscan be understood in terms of what we p ingpursuedincludehighdata-rateinterplanetaryoptical call“constrictions:”highlylocalized,essentiallypointlike [ communications [6], spectroscopy of ultrafast quantum regionswherethenanowirecross-sectioniseffectivelyre- 2 phenomena in biological and solid-state physics [7, 8], duced, and which arenot due to lithographic patterning v quantum key distribution (QKD) [9], and noninvasive, errors (line-edge roughness). 0 high-speed digital circuit testing [10]. The electrical operation of these detectors has been 6 2 In manyof these applications,largearraysofSNSPDs discussed previously by several authors [1, 2, 3, 4, 13], 1 would be extremely important [5]. For example, exist- so we only summarize it here. The NbN nanowires are 1 ingSNSPDshaveverysmallactiveareas,makingoptical biased with a DC current Ibias slightly below the crit- 6 coupling relatively difficult and inefficient [7, 11]. Their ical value IC. An incident photon of sufficient energy 0 small size also limits the number of optical modes they can produce a resistive “hotspot” which in turn disrupts / s can collect, which is critical in free-space applications the superconductivityacrossthe wire,producingaseries c where photons are distributed over many modes, such resistance which then expands in size due to Joule heat- i s as laser communication through the atmosphere (where ing [4, 13]. The series resistancequickly becomes ≫50Ω, y turbulence distorts the opticalwavefront)andin fluores- andthecurrentisdivertedoutofthedeviceandintothe h cence detection. Furthermore, it was shown in Ref. [4] 50Ω transmission line connected across it, resulting in a p : that the maximum count rate for an individual SNSPD propagating voltage pulse on the line. The device can v decreasesasitsactiveareaisincreased,duetoitskinetic thencoolbackdownintothe superconductingstate,and i X inductance, forcing a tradeoff between active area and the current through it recovers with the time constant r high count rates. Count rate limitations are particularly Lk/50Ω, where Lk is the kinetic inductance [4]. a important in optical communications and QKD, affect- The nanowires used in this work were patterned at ingtheachievablereceiversensitivityordatarate[6,12]. thesharedscanning-electron-beam-lithographyfacilityin Detector arrays could provide a solution to these prob- theMITResearchLaboratoryofElectronicsusingapro- lems, giving larger active areas while simultaneously in- cess described in Refs. [2, 4, 14], on ultrathin (∼ 5 nm) creasing themaximumcountratebydistributingtheflux NbN films grown at Moscow State Pedagogical Univer- overmanysmaller(andthereforefaster)pixels. Largear- sity [15]. The majority of the wires were on average 90 rays could also provide spatial and photon-number reso- nm in width, and were fabricated in a meander pattern lution. Although few-pixel detectors have been demon- with 200 µm line pitch (45% fill factor), subtending an strated [5, 9, 11, 12], fabrication and readout methods active area of 3 × 3.3 µm or 10 × 10 µm. Some devices scalable to large arrays have not yet been discussed. with54nmaveragewidthand150nmpitch(36%fillfac- AfirststeptowardsproducinglargearraysofSNSPDs tor)werealsofabricated-seeFig. 2(a). Thedeviceshad 2 20 100 andpreventsthecurrentdensityeverywherebutnearthe (a) (b) constrictionfromeverapproachingthecriticalvalue(and 10-1 Number of devices 1105 111000---432 Detection efficiency hlfooecrnIafcalaelylplpontriheeneavtrleninkatthenseocwtochnoiernsetswsrtiwricriteceirtoeifnorion,dm)ew.nethiwacavoliunilngdaaellxhdpiiegmchtenDthsEiaotnesxifcsaethpveet 5 data of Fig. 1(b) were plotted vs. absolute current Ibias 10-5 (rather than Ibias/ICobs) it would all lie on a single, uni- 0 10-6 versal curve, with the data for more constricted devices 0.00 0.05 0.10 0.15 0.20 0.6 0.7 0.8 0.9 1.0 Detection efficiency Bias current [I /Iobs] simply notextending to as highcurrents. This turns out biasC to be approximately true, but variations from device to FIG. 1: Figure 1: (color online) Variations in SNSPD detec- device across the chip either in film thickness or in the tion efficiency. (a) Histogram of the DEs measured for 132 nanowirewidth obscurethis essentialfeature of the data devices from a single fabrication run, on a single chip. The whenitisplottedinthissimpleway. Instead,wepresent devices were 3×3.3 µm in size, and composed of a 50 µm ourresultsasshowninFig. 2. Inpanel(a), theDE data longnanowireinameanderpatternwith45%fillfactor. The foreachof170deviceswith90nmwidewires(acrosstwo measurementsweremadeatT =1.8K,andIbias=0.975ICobs. chips fabricated in separate runs) is shown superposed Note that the peak DE of 22% increases to 57% with the (filled circles indicate T = 1.8K, and crosses T = 4.2K). addition of an optical cavity, as described in Ref. [2] (b) MeasurementsoftheDEvs. Ibias/ICobs foraselectionofthese AasllinogfltehuendiavtearsfaolrceuarcvhetbeymspcearliantgurtehecacnribtiecamlacduerrteontlieIoobns devices. C foreachdevicebyanadjustablefactor(whichisjust1/C: IC = ICobs/C). The very fact that data from this many criticaltemperatures TC ∼ 9−10K,and criticalcurrent devicescanbesowellsuperposedisalreadysuggestiveof densities JC ∼5×1010 A/m2 at T =1.8K. auniversalshape. However,wecannowtaketheC value TheexperimentswereperformedatMITLincolnLab- foreachdeviceextractedusingthisscalingprocedureand oratory,usingtheproceduresandapparatusdiscussedin cross-check it. Based on our previous discussion, if all detail in Refs. [2, 4]. Briefly, the devices were cooled to wires were identical save for constrictions, we would ex- as low as 1.8 K inside a cryogenic probing station. Elec- pect C = ICobs/IC, i.e. C to be exactly proportional to trical contact was established using a cooled 50 Ω mi- ICobs. Duetothevariationsacrossachipdiscussedabove, crowaveprobeattachedtoamicromanipulator,andcon- thisisonlypartiallytrue. However,wecannormalizeout nected via coaxial cable to the room-temperature elec- these variations using a very simple method: instead of tronics. We counted electrical pulses from the detectors comparing C directly to ICobs, we instead compare it to using low-noise amplifiers and a gated pulse counter. To theproductICobsRn =(C×JCAcs)(ρnl/Acs)=C×JCρnl, optically probe the devices, we used a 1550 nm mode- whereAcs andRn arethe cross-sectionalareaandroom- locked fiber laser (with a 10 MHz pulse repetition rate temperature resistance of each nanowire, JC and ρn are and ≤1 ps pulse duration) that was attenuated and sent the critical current density and room-temperature resis- intothe probingstationviaanopticalfiber. The devices tivityofNbN,respectively,andl isthetotalwirelength. were illuminated from the back (through the sapphire Thisproductdependsonthewiregeometryonlythrough substrate) using a lens attached to the end of the fiber l(whichisfixedlithographicallyanddoesnotvaryappre- which was mounted to an automated micromanipulator. ciablybetweendevices)andnotoneachwire’sindividual The focal spot had a measured e−2 radius of ∼ 25 µm. Acs. Figures2(b)and(c)showacomparisonbetweenthe C values extracted from the data in Fig. 2(a) and the Figure 1 illustrates the detection efficiency (DE) vari- ations observed on a single chip of 132 devices of the ICobsRn product. The data lie on a straight line through the origin, indicating that these two independent mea- same geometry,fabricatedin a single run. These devices sures of C are mutually self-consistent. are the same ones reported in Ref. [2] before the optical cavities were added (maximum DE after the addition of Our data can also be used to infer that the constric- cavities was 57%). In panel (a) we show a histogram of tions are essentially pointlike (i.e. very short in length). the measured detection efficiencies at Ibias = 0.975ICobs TheopencirclesinFig. 2(a)aredatafor15deviceswith (where Iobs is the observed critical current of each de- 54nmwidewires,andclearlyshowadramaticallydiffer- C vice), and (b) shows some representative data of the ob- ent shape (i.e. high DE persists to much lower currents servedDEasafunctionofIbias/ICobs. Notethattheshape thanforthewiderwires). Thebrokenlinesareestimates, of these curves also varies significantly. As we show be- based on the data for 90 nm and 54 nm wide wires, of low, these data can be explained with the hypothesis, what one would expect for a device having 90 nm wide firstsuggestedinRef. [16],that somedevices have“con- wire, except at a single constriction 54 nm wide (cor- strictions:” regions where the (superconducting) cross- responding to C ∼ 0.6) with a length of either 0.5 µm sectional area Acs of the wire is reduced by a factor we (dashed line) or 50 nm (dotted line) long. These curves label C. This effectively reduces the observed critical haveadifferentshapefromthedatafor90nmwidewires, current by that same factor (ICobs = JCAcsC = ICC), because at low currents the DE is dominated by the re- 3 (a) (b) 1.0 Hence, the kinetic inductance increases as J → JC [17]. 10-1 0.9 Since the kinetic inductivity locallyincreaseswith J/JC, Detection efficiency 111000---432 (c) 000001......678890 C obtained from (a) tdipuchelsaaeitlncetegrvo.matalaiunnWleeekteioiwnfvmeoettrhreikcaetsaihcnuneudraerulwrdycehztnteaothrnle,edcbeewiynnoisdrofietbutyhscoeteriarswvnoiicinnnreldegyeoptehfardoetovnpuoiedhrnaeaernssealatonhocwoefawalaciyizrrreiteetdos-- 10-5 0.7 flected microwave signal as a function of frequency. We 10-6 0.6 then fit this data using a suitable electrical model to ex- 0.4 0.5 0.6 0.7 0.8 0.9 0.6 0.7 0.8 0.9 1.0 Bias current [I /Iobs x C] Normalized Iobs R product tract the inductance value. A bias tee was inserted into bias C C n the signal path to superpose the desired Ibias with the FIG.2: Figure2: (coloronline)Constrictionvaluesextracted network analyzer output. The phase contributions from using DE vs. Ibias data. (a) Universal DE curves for devices thecoaxialcable,biastee,andmicrowaveprobewerere- with 90nm wide wires, at T = 1.8K (•) and T = 4.2K (+). movedby probinganin situ microwavecalibrationstan- Data from 170 devices distributed over two separate fabrica- dard(GGBIndustriesCS-5). Themicrowavepowerused tion runs is shown superposed. By rescaling the ICobs of each in this measurement corresponds to a peak current am- device such that all data lies on a single universal curve as plitude of ≤0.5µA, and the critical current measured in shown,theconstrictionC (whichindicatesthefractionofthe the presence of the microwaves was within 10% of that wire’scross-sectionalareathatissuperconductingatthecon- measuredintheirabsence(∼20µAfortypicaldevicesat stricted point) can be obtained. Also shown (◦) are data for T =1.8K). 15 devices having 54 nm wide wires (and 36% fill factor) at T = 1.8K, indicating that narrower wires exhibit a very dif- In Fig. 3(a), we show the measured inductance vs. ferentuniversalcurveshape. Thesedataprovideevidencefor current of two devices; one which has nearly the highest thelocalizednatureoftheconstrictions,sinceanyappreciably detection efficiency observed on this chip (22% - filled longsectionofwirehavingasmallercross-section shouldsig- circles), and the other having one of the lowest (0.1% - nificantly alter theshape of the curves, making it impossible filledtriangles). AlsoshownisthepredictionforLk(Ibias) tosuperposethemasshown. Thisisillustratedbythebroken from Ginsburg-Landau theory (see, e.g., [17]), with no lines,whichshowasimpleestimatefora90nmwidewirehav- free parameters (solid line). The data for the high-DE inga54nmwideconstrictionwithalengthof0.5µm(dashed device show good agreement with this prediction, indi- line) or 50 nm (dotted line). These estimates are obtained catingthatthisdeviceisindeedunconstricted. However, simply by adding together the universal curves for the two wire widths, in a ratio given by the length of wire with each for the low-DE device the inductance does not increase width,i.e. (0.5µm/50 µm)and(0.05µm/50 µm). Notethat as much as predicted, which is precisely what we would these broken lines terminate at Ibias/ICobs×C =0.6=54/90 expect within the constriction hypothesis; the current due to the assumed 54 nm constriction of the 90 nm-wide density is only nearcriticalat one localizedplace (which wire. (b) and (c) Constriction values C obtained from the constitutes a negligible fraction of the total wire length) data in (a) (for (b) T = 1.8K and (c) T = 4.2K) vs. those whereaseverywhereelsethecurrentdensityislower,pro- obtained using the ICobsRn product. The C values in both ducing a smaller increase in inductance. The factor by cases are normalized absolutely using Lk(Ibias), as described which Iobs must be rescaled for a given device so that below (see Fig. 3). The solid lines are straight lines through C the Lk(Ibias) data matches the prediction constitutes an theorigin with slope 1; no fittingwas used. absolute measurement of C. One such measurement for a single representative device, from among a large set of nominally identical devices, then allows us to correctly gionofwireneartheconstriction,whileathighercurrent normalize the C values obtained using either of the pre- it becomes dominated by the contribution from the rest vious methods described above (see Fig. 2) for all other of the wire length. This very different shape should be devices in that set. distinguishableifitwerepresent,andshouldpreventthe datafrombeing superposedontoasinglecurve. The ab- We can also verify that the observed Lk(Ibias) and senceofthisinourdataatanylevelabovethenoiseindi- ICobsRn product give mutually consistent results. To catesthattheconstrictedregionsarelikelymuchshorter check this, for each device we measured the inductance than ∼0.5µm. ratio RL ≡L(I0)/L(0), where I0 ≈0.9ICobs. Using the C So far, we have in fact only measured constriction in obtained from the normalized ICobsRn product, we also a relative sense; that is, we have no way to tell if our obtain I0/IC = I0C/ICobs for each device. The inset best devices in fact have C =1. To address this, we can to Fig. 3(a) shows RL vs. I0/IC (filled squares), and exploit the known dependence of the kinetic inductance the dataareinreasonableagreementwiththe Ginsburg- on bias current. Kinetic inductance arises from energy Landau prediction (solid line). stored in the effectively ballistic motion of Cooper pairs; In addition to providing evidence for the constriction obs as the current density is increased towards the critical hypothesis,themeasurementofLk(Ibias)andIC Rnpro- value, the density of Cooper pairs is depleted, forcing vide a powerful diagnostic tool, since they constitute a theremainingpairstospeedup(andthereforestoremore purely electrical measurement of C, which can then be kinetic energy per unit volume) to maintain the current. used to predict the detection efficiency, as indicated in 4 Kinetic inductance ratio [L(I)/L(0)] 111...012 0.0(a) 0.2 I0/IC0o.4bs x C0.6 0.8 111...012L(I)/L(0) 0 (((dcb))) 024612024600000000 Number of devices afmfPpdnorrnfaimaooetdnbnmminoeoewt(nltlIsd(ihoa(Cdbo)fbl(b.e)(c)sdCc)RFwae)snovhanif3rrdoion,ce×rwdntes(hsoseco3(rar)dodne.m,c3nae)hwaatµassalhsehmi2ezsfito0coteis,nw0hrdavogtnniuaflrhaemdrsedeeisLane(usdpvdngkeldiit)(otctasIimc1oetbshsif0fiioioa,,nndsn×rwabg)ealvaluell1aeyit0sccosaehawbibµddsmitsipemefatosrp,nhicl.onltauremiaeTlitclbdceoahhFteflmeiCd,aiv3gvdeeea1.viaxenb0aasv1hgruoli(iuaecbvar9daeeee)i0ssts--;. 0 very similar C distributions (though some yield fluctu- 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 Bias current [I /Iobs] Normalized C from Iobs R ation, which we commonly observe, is evident). Panels bias C C n (c) and (d), however, clearly show a lower average C FIG. 3: Figure 3: (color online) Absolute measurement of for the larger-area devices, qualitatively consistent with constriction using bias current-dependence of kinetic induc- some fixed area density of constrictions. This implies tance. (a) The kinetic inductance of our nanowires should that at the present state of film growth and nanowire increase with Ibias, due to the depletion of the Cooper-pair patterning,theyieldforhigh-DEsingledevicesordevice density near the critical current density (solid line). A de- arrayscovering 10 × 10 µm areas or larger will likely be tector with the highest observed detection efficiency on the chip(22%) behavesasexpected(•),with nofreeparameters. quite low. However, a detector with much lower DE (0.1%) does not As a final note, we remark on the obvious question (filled triangles). This is due to a constriction, which pre- of the origin of these constrictions. The most natural ventsthecurrentdensityfrom approachingcritical anywhere explanation would be lithographic patterning errors; for butnear thisone localized point. Inset: theinductanceratio RsqLua≡resL)(,Ip0lo≈tt0e.d9IvCosb.s)/IL0/((0I)Cobms/eCas)ur=edIf0o/rICe,acwhhdeerevicCe (isfilolebd- eexxapmospulree, acosumldalrlepsualrtticinleaorlodceafleicztedinnathrreowressiestctbioenforoef tained from the ICobsRn product. These data agree with the wire. However, we have performed extensive scanning prediction (solid line), indicating that RL and ICobsRn give electron microscopyof devices that were measuredto be mutually consistent results for the constriction. (b), (c), and severely constricted (e.g. C ∼ 0.5) and no such errors (d): Distribution ofC valuesobtained usingpurelyelectrical were observable. This suggests that constrictions in our measurements, for (b) the same set of devices shown in Fig. devicesresulteitherfromthicknessvariationsormaterial 1, and (c),(d) for 310 additional devices on a separate chip, defects, which may have been present in the film before withactiveareasofboth(c)3×3.3µmand(d)10×10µm. patterning, and may even be due to defects present in Thesedatawereobtainedfrom ICobsRn,withtheirnormaliza- the substrate itself before film growth. tion set by single measurements of Lk(Ibias) like that shown in (a), one for each of the three sets of devices shown in (b), Inconclusion,wehaveverifiedbothopticallyandelec- (c),and(d). Thedevicesfrom(b)and(c)arenominallyiden- trically that the large variations in detection efficiency tical in all respects, though they were fabricated on different between nominally identical superconducting nanowire NbN films, and at different times. Their C distributions are single photon detectors are the result of localized con- are therefore quite similar. The difference between (c) and strictionswhichlimitthedevicecurrent. Furtherworkis (d) indicates that larger devices have a significantly higher ongoing to pin down the exact source of these constric- probability of a sizeable constriction. tions,withthehopeofeventuallyfabricatinglargearrays of these detectors. Figs. 2(b) and (c). Since optical testing of large num- Thisworkissponsoredbythe United StatesAir Force bers of detectors is significantly more difficult and time- under Air Force Contract #FA8721-05-C-0002. Opin- consuming than electrical testing, this is potentially an ions, interpretations, recommendations and conclusions important screening technique. An example of data ob- arethoseofthe authorsandarenotnecessarilyendorsed tained with this technique is shown in Figs. 3(b), (c), by the United States Government. [1] G. Goltsman, O. Okunev, G. Chulkova, A. Lipatov, A. V. Anant, B.M. Voronov, G.N. Gol’tsman, and K.K. Dzardanov, K. Smirnov, A. Semenov, B. Voronov, C. Berggren, Opt.Express. 14, pp. 527-534 (2006). Williams, and R.Sobolewski, IEEE Trans. Appl.Super- [3] J. Zhang, W. Slysz, A. Verevkin, O. Okunev, G. cond.11,pp.574-577(2001); A.Engel,A.Semenov,H.- Chulkova,A.Korneev,A.Lipatov,G.N.Gol’tsman,and W.Hu¨bers,K.Il’in,andM.Siegel,J.Mod.Opt.51,pp. R. Sobolewski, IEEE Trans. Appl. Supercond. 13, pp. 1459-1466(2004);B.Delaet,J.-C.Vill´egier,W.Escoffier, 180-183 (2003). J.-L.Thomassin,P.Feautrier,I.Wang,P.Renaud-Goud [4] A.J. Kerman, E.A. Dauler, W.E. Keicher, J.K.W. Yang, andJ.-P.Poizat,Nucl.Inst.Meth.Phys.Res.A520,pp. K.K. Berggren, G.N. Gol’tsman, and B.M. Voronov, 541-543 (2004). Appl. Phys.Lett. 88, p. 111116 (2006). [2] K.M.Rosfjord,J.K.W.Yang,E.A.Dauler,A.J.Kerman, [5] E.A.Dauler,B.S.Robinson,A.J.Kerman,J.K.W.Yang, 5 K.M.Rosfjord,V.Anant,B.Voronov,G.Gol’tsman,and Appl. Phys.Lett. 88, p. 261113 (2006). K.K. Berggren, IEEE Trans. Appl. Supercond., to be [12] E.A. Dauler, B.S. Robinson, A.J. Kerman, V. Anant, published. R.J. Barron, K.K. Berggren, D.O. Caplan, J.J. Carney, [6] B.S. Robinson, A.J. Kerman, E.A. Dauler, R.J. Barron, S.A.Hamilton,K.M.Rosfjord,M.L.Stevens,andJ.K.W. D.O.Caplan, M.L. Stevens,J.J. Carney, S.A.Hamilton, Yang, Proc. SPIE, to bepublished. J.K.W.Yang,andK.K.Berggren,OpticsLett.31,p.444 [13] J.K.W.Yang,A.J.Kerman,E.A.Dauler,V.Anant,K.M. (2006). Rosfjord,K.K.Berggren,IEEETrans.Appl.Supercond., [7] R.H. Hadfield, M.J. Stevens, S.S. Gruber, A.J. Miller, to be published. R.E. Schwall, R.P. Mirin, and S.W. Nam, Opt. Express [14] J.K.W. Yang, E. Dauler, A. Ferri, A. Pearlman, A. 13, pp.10846-10853 (2005); Verevkin, G. Gol’tsman, B. Voronov, R. Sobolewski, [8] M.J. Stevens, R.H. Hadfield, R.E. Schwall, S.W. Nam, W.E. Keicher, and K.K. Berggren, IEEE Trans. Appl. R.P.Mirin,andJ.A.Gupta,Appl.Phys.Lett.89,031109 Supercond. 15, pp.626-629 (2005). (2006). [15] S. Cherednichenko, P.Yagoubov, K. Il’in, G. Gol’tsman, [9] M.A. Jaspan, J.L. Habif, R.H. Hadfield,and S.W. Nam, and E. Gershenzon, in Proceedings of the 8th Inter- Appl.Phys.Lett. 89, 031112 (2006). national Symposium On Space Terahertz Technology, [10] A. Korneev, A. Lipatov, O. Okunev, G. Chulkova, K. Boston, MA, 1997, p.245. Smirnov,G.Goltsman,J.Zhang,W.Sl ysz,A.Verevkin, [16] J. Zhang, W. Sl ysz, A. Pearlman, A. Verevkin, R. R.Sobolewski, Microelec. Eng. 69, p. 274 (2003). Sobolewski, O. Okunev, G. Chulkova, and G. N. Golts- [11] W. Sl ysz, M. W¸egrzecki, J. Bar, P. Grabiec, M. Grska, man, Phys. Rev.B 67, p. 132508 (2003). V.Zwiller,C.Latta,P.Bohi,I.Milostnaya,O.Minaeva, [17] T.P. Orlando and K.A. Delin, Foundations of Applied A. Antipov, O. Okunev, A. Korneev, K. Smirnov, B. Superconductivity, Addison-Wesley,New York (1991). Voronov, N. Kaurova, G. Goltsman, A. Pearlman, A. Cross, I. Komissarov, A. Verevkin, and R. Sobolewski,

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