Cooperpairinsulatorinamorphousfilmsinducedbynanometer-scalethicknessvariations S.M.Hollen,1 H.Q.Nguyen,1 E.Rudisaile,1 M.D.Stewart,Jr.,1 J.Shainline,1 J.M.Xu,1,2 andJ.M.Valles,Jr.1 1Department of Physics, Brown University, Providence, RI 02912 2Division of Engineering, Brown University, Providence, RI 02912 (Dated:February1,2011) Unusualtransportpropertiesofsuperconducting(SC) (seeFig. 1(a)and(b)insets). Theholeysubstrateshavenon- materials, such as the under doped cuprates[1], low di- uniform topography that causes thickness and, consequently, 1 mensional superconductors in strong magnetic fields[2], superconducting coupling constant variations throughout the 1 and insulating films near the Insulator Superconductor film. TheinsulatingNHCa-BifilmsneartheISTexhibitthe 0 Transition (IST)[3], have been attributed to the forma- characteristic transport signatures of the CPI phase[18, 19] 2 tion of inhomogeneous phases. Difficulty correlating the . Moreover, the insulators are confirmed to contain Cooper n behaviors with observations of the inhomogeneities make pairsthroughLittle-Parkslikemagnetoresistanceoscillations a these connections uncertain. Of primary interest here induced by the hole array[18]. These films provide a unique J are proposals that insulating films near the IST, which exampleofasystemdevelopingaCPIphaseasaresultofthe 8 show an activated resistance and giant positive magne- introduction of inhomogeneities. Establishing that well de- 2 toresistance, contain islands of Cooper Pairs (CPs)[4–7]. finedislandsformattheISTwillputimportantconstraintson ] Here we present evidence that these types of inhomo- modelsoftheCooperPairInsulatorphase. n geneitiesareessentialtosuchaninsulatingphaseinamor- Here we focus on the systematics of thickness tuned ISTs o phous Bi (a-Bi) films deposited on substrates patterned inthesefilms. Wepayparticularattentiontohowthecritical c - with nanometer-sized holes. The patterning induces film thicknesses for the onset of conduction (d ) and supercon- Gc r p thicknessvariations,andcorrespondingcouplingconstant duction(dIST)varywithsubstratequalitiesandalsopointout u variations, that transform the composition of the insula- therelativeinvarianceofthecriticalnormalstatesheetresis- s tor from localized electrons to CPs. Analyses near the tance (R ). The ISTs, shown in Fig. 1(a) and (b), differ . IST t thickness-tunedISTsoffilmsonninedifferentsubstrates significantly from those of their unpatterned, or “reference,” a m show that weak links between SC islands dominate the counterparts(seeRef. [8]foranexample). TheR(T)evolve transport. Inparticular, theISTsalloccurwhenthelink withdepositedthickness,ddep,fromamonotonicriseasT - → d resistance approaches the resistance quantum for pairs. 0thatisexponentialratherthanlogarithmic,todevelopareen- n TheseobservationsleadtoadetailedpictureofCPslocal- trant dip not seen in reference films, to finally drop steadily o izedbyspatialvariationsofthesuperconductingcoupling intotheSCstate. Theexponentialriseindicatesthermallyac- c [ constant. tivated transport, R(T) eT0/T[18]. For all the ISTs, the A central issue in the field of insulator to superconduc- normal state sheet condu∝ctance, G(8K) = 1/R(8K) grows 1 tor transitions concerns the nature of the insulating phase. linearlywithdepositedfilmthickness,ddep,startingatacrit- v In most homogeneously disordered thin films, the insulator ical thickness, d (Fig. 1(c)). This linear behavior reflects 2 Gc 4 is composed of unpaired electrons that are weakly localized homogeneousratherthangranularfilmgrowth, whichwould 6 bydisorder[8–11]. Alessfamiliarstatewhichsupportsrem- appear as an exponential dependence[20]. Consistently, the 5 nants of superconductivity appears in other homogeneously slope, dG(8K)/dddep, is smaller and the intercept, d , is Gc . 1 disordered films, such as In Oxide and TiN[12, 13]. This largerfortheNHCfilmscomparedtoreferencefilms. T de- 0 0 so-called Cooper Pair Insulator (CPI) state exhibits activated creases continuously on approaching the IST (also shown in 1 transport and a giant positive magnetoresistance, which sug- Refs. [18, 19, 21]). We define the critical thickness for the 1 gestthatlocalizedCooperpairsparticipateintransport. How IST, d , as the thickness at which T linearly extrapolates : IST 0 v this bosonic insulating state develops has not been deter- to zero. Notice that similar to d , d is larger for NHC Gc IST i X mined. Theoretical descriptions of the CPI phase often pre- films than reference films. Finally, RIST shows little varia- sumethatdisorderormagneticfieldcreatesislandsofCooper tionoverninedifferentsubstratesthatexhibitawiderangeof r a Pairs(CPs)embeddedinanon-SCbackground[7,14–16]. In- d (seeFig. 1(e)). TheR sclusteraroundR 3R , IST IST IST Q ≈ deed,scanningtunnelingspectroscopyexperimentsshowevi- whereR = h/(2e)2,thequantumofresistanceforpairs,is Q denceofincipientpuddlinginsomefilmsclosetotheIST[17]. thecriticalsheetresistanceforSCinreferencefilms. SinceCooperpairsarenotevidentinmostotherinsulatingho- TopographicalanalysesofAAOsubstratesusingAFMim- mogeneouslydisorderedthinfilms,thefactorsresponsiblefor ages show that the minimum film thickness in critical NHC thechangeinbehavioraregenerallyunclear. films(atddep = d andd )correlateswithcriticalthick- Gc IST Recently, it was established that amorphous Bi films can nesses in reference films, dref and dref . As pointed out Gc IST be induced to form a CPI state instead of the unpaired insu- previously[22],theNHCsubstratetopographyh(x,y)shows lator state by depositing them on an anodized Al oxide sub- regular variations around holes, which makes the thermally stratepatternedwithananohoneycomb(NHC)arrayofholes evaporatedfilmthicknessafunctionofposition,d(x,y). For 2 a c 20 60 a ddep = 0.76 nm = d c 1.5Ca 103 G ) 40 Gc 1 lcu 102 01000/G eight (nm−22000 0.89 nm d 1100..55lated Film ) 1 H−40 stance (kΩ 11000 d 120T (K) Surface −601200 e 1100..55 Thickness esi 3 b b 1.07 nm = d 0.5 (n R 10 0 0 IST 0 m et ddep (nm) 0 20Trace4 D0istan0ce (nm2)0 40 ) e 2 Sh 10 5 101 e 34ISR m) 0.8 f T 2/R (n 100 1Q linksd 0.6 0 0.1 1 10 0.5 1 1.5 0.4 Temperature (K) d (nm) IST 0.6 0.8 1 1.2 1.4 d (pink), d (blue) (nm) Gc IST Figure 1: Characteristics of thin films on nanohoneycomb sub- strates (a) and (b) Insulator Superconductor Transitions tuned by Figure2:NHCfilmtopography(a)AFMimageshowingthetopog- thicknessofa-Bionmoreandlessorderedholearrays.Insets:post- raphyofthesubstrateinFig. 1(a). Thescalebar(whiteline)spans experiment SEM images. Center-to-center hole spacing: 100 ± 5 100nm. (b)Heightprofilealongthetrace(blackline)in(a). (c)-(e) nm.Radii:27and28±3nm.Thescalebarspans200nm.(c)Nor- Film thicknesses along the trace in (a) for ddep at (c) the onset of malizedconductance(G00 = 1/81kΩ)and(d)activationenergies conduction, (d)arepresentativeinsulatingfilm(boldinFig. 1(a)), forfilmsonfourdifferentNHCsubstrates. Similarsymbolsrepre- and(e)theIST.Pinkandbluelinesindicatethecriticalthicknesses sentdatafromexperimentsonside-by-sideNHCsubstrates.Unfilled forconductivityandSC,respectively,forareferencefilm. (f)Aver- andfilledsymbolsrepresentdatafrommoreandlessorderedholear- agelinkthicknessforNHCfilmsattheirtransitionstoconductivity rays,respectively.Datafromtheexperimentin(a)and(b)areshown (dGc,pink)andSC(dIST,blue)forsixdifferentsubstrates. Critical ascircles. Forsquares,rhole =16(unfilled)and15(filled)±3nm. thicknessesforSCareshownbyplottingthelinkthicknessesofthe Reference film data are shown as grey triangles in (c). The dotted lastinsulatingandfirstSCfilmsspanningtheIST. linesareleast-squareslinearfitstothedata,andthearrowsindicate critical thicknesses for conductivity and SC. Open arrows indicate criticalthicknessesonareferencefilm: 0.5nmforconductivityand 0.7nmforSC.(e)Normalstate(T = 8K)criticalsheetresistance critical thicknesses, d and d , the minima coincide with forthelastinsulatingandfirstSCfilmsoftheISTofninedifferent Gc IST critical thicknesses in reference films, dref (pink line at 0.5 NHCsubstrates.Thehorizontaldashedlineat3RQisaguidetothe Gc eye. nm) and drISefT (blue line at 0.7 nm). These observations inti- matethatNHCfilmsconductorSCwhenthethinnestregions, the weak links, reach the appropriate critical thickness. Cal- culations of the weak link thicknesses averaged over each of auniformfluxofatomsimpingingparalleltotheaveragenor- the six substrates analyzed, d¯links, strongly support this sug- maltothesubstratethesearegivenby: gestion. Fig. 2(f)comparesd¯links atNHCfilmcriticalthick- 1 nesses with the critical thicknesses for conductivity and SC d(x,y)=ddep(cid:112) . (1) ofreferencefilms. Itisevidentthatforallbutoneofthesub- 1+( h)2 ∇ stratesd¯linksatcriticalddepcorrespondstocriticalthicknesses Fig. 2(a) shows the topography of the AAO substrate. Six inreferencefilms. peaks, visible as bright spots, surround each of the holes. Full two-dimensional maps of the local film thickness, A line scan between two neighboring peaks shows a single d(x,y) reveal a compelling picture: SC and normal regions central minimum in the height (Fig. 2(b)). Using Eqn. 1, coexist in NHC films through the IST (Fig. 3). These maps we calculate thickness profiles along this link corresponding weregeneratedbyapplyingEqn. 1totheAFMimageinFig. to the onset of conduction (ddep = d ), an insulating film 2(a)fortheseriesofdepositionsinFig. 2(c),(d),and(e). The Gc (d < ddep < d ,boldinFig. 1(a)),andtheonsetofSC color scale indicates regions that are SC, resistive, and insu- Gc IST (ddep = d ) (see Figs. 2(c-e)). All show two minima that latingaccordingtotheirbehavioratthosethicknessesinaref- IST correspondtothesteepestregionsoftheheightprofile. Atthe erencefilm. ItisevidentthatSCislandsformandgrowwhere 3 ) m n1.5 s ( a b c 3 s e n ck 1 2 T m Thi 01 c (K ) Fil0.5 d e at cul 0 ddep = 0.76 nm = dGc 0.89 nm 1.07 nm = dIST Cal e Rlink d 00.51.510202.5 11.5 0.50 Tc 1 00.5 0.5 0 001112.51.5 01000.5.5 1000.520.5110.51.521201..515 11.50.5000.5111.50.152.05 10.5.511 00.50110.51.51.52100.50.5121.5011.50.5100.500.5101.52 00.05.511.0501 Figure3: FilmthicknessvariationsinNHCfilmsCalculatedthicknessvariationsinfilmsdepositedontheNHCsubstrateofFig. 1(a)and Fig.2(a).Filmthicknesscolorsrepresentbehavioronareferencefilm:whiteisnon-conductive(d<dref),pinkisconductive(d>dref),and Gc Gc bluesuperconduct(d>dref ).(a)-(c)Filmthicknessvariationplots(a)attheonsetofconduction,(b)forarepresentativeinsulatingfilm,bold IST inFig. 1(a),and(c)attheIST.ThecriticaltemperaturesforSCinreferencefilms(Tc)ofthesethicknessesareshownontheright. Tc = 0 markstheISTandthemaximumTc seenineachofthepanelsis(a)0.4K,(b)1.3K,and(c)2.2K.(d)Tc (inKelvin)contoursshownfora sectionofthefilmatddep = dIST. (e)Illustrationofawirearrayofresistors,Rlink,connectingSCislands. Theblacksegmentsshowthe basicelementofthearray. thesubstrateisflattest,suchthattherearegenerally12islands shuntedJosephsonJunctionsintheextremequantumlimitun- surroundingeachhole,oneforeachpeakandvalley. Through dergoanISTforR R .Applyingthesedissipationdriven N Q ≈ the IST, the SC islands are larger than the coherence length IST models to NHC films seems plausible as the SC islands inthesefilms,ξ 15nm. Therightaxisofthefigurefurther aresmallenoughtobeintheextremequantumlimitandthe ≈ convertsd(x,y)toalocalT map,T (x,y),usingtherelation- resistive regions surrounding the islands can act as the dissi- c c shipbetweenTref anddref observedhereandinmanyamor- pativeelements. c phousfilmsystems[8,23]:Tref =Tbulk(1 dref /ddep).For We have shown that the CPI phase in NHC a-Bi films is c c − IST a-Bi, Tbulk = 6 K and dref = 0.7 nm[24]. Note that the composed of superconducting islands. These islands result c IST T mapsarebetterthoughtofascouplingconstantmapssince from variations in the film thickness on scales larger than ξ c theydonotincludeproximityeffects. Eachoftheislandshas thatleadtoinhomogeneitiesinthesuperconductingcoupling a larger T at its center than on its edges and resistive film constant. Our observations support conjectures that CP is- c regions surround the islands. With increasing thickness, the landing gives rise to the activated resistance and giant mag- intervening resistive regions shrink until the SC islands coa- netoresistance observed in this and other thin film systems, lesce (see Supplemental Information). The T contour plot, such as InOx and TiN. For these structure-free films, how- c Fig. 3(d), emphasizes that T in the regions between islands ever, an alternative mechanism for island formation is re- c attheircoalescenceisfinite,butmuchlessthanthemaximum quired. Whether this mechanism is spontaneous electron- attheislands’centers. Thus,theseregionslikelyactasweak electron interaction driven phase separation[27], the fractal links. nature of the electronic wavefunctions close to the metal to insulator transition[6], a random Gaussian potential[14], un- The color maps suggest a network model for NHC films derlyingstructuralinhomogeneitiesinthematerial[5,28], or that provides a clue to the origin of the apparently universal yet to be determined is an open question. We anticipate that normal state resistance at the IST. In the model, the SC is- the detailed picture of the superconducting inhomogeneities lands become highly conductive regions connected by resis- presentedherewillaidmicroscopiccalculationsoftheseand tive weak links as shown in Fig. 3(e). The resistance of the other characteristics, such as the tunneling density of states, basic building block of this network, which is equivalent to oftheCPIstate. the macroscopic sheet resistance, is 3Rlink. Thus, according toFig. 1(f),attheISTRlink R . Interestingly,theory[25] Methods: IST ≈ Q and experiment[26] maintain that arrays of small, resistively NHCa-BithinfilmsaregrownbythermallyevaporatingSb 4 and then Bi onto an Anodized Aluminum Oxide (AAO)[29] [16] V. F. Gantmakher, M. V. Golubkov, V. T. Dolgopolov, G. E. substrate held at 8K in a dilution refrigerator cryostat and Tsydynzhapov,andA.A.Shashkin,JETPLett.68,363(1998). measured in situ. Reference films are grown simultaneously [17] B.Sacepe, C.Chapelier, T.I.Baturina, V.M.Vinokur, M.R. Baklanov, and M. Sanquer, Phys. Rev. Lett. 101, 157006 on a fire-polished glass substrate. Film thicknesses are mea- (2008). sured with a quartz crystal microbalance. The semiconduct- [18] M.D.StewartJr.,A.Yin,J.M.Xu,andJ.M.VallesJr.,Science ing Sb underlayer (d 1 nm) promotes the surface wetting (cid:39) 318,1273(2007). of the subsequently deposited Bi. Previous STM analyses [19] H. Q. Nguyen, S. M. Hollen, M. D. Stewart, J. Shainline, ofquenchcondensedmetalfilmsonsemiconductorsindicate A.Yin,J.M.Xu,andJ.M.V.Jr,Phys.Rev.Lett.103,157001 thatthemetalatomsstickwheretheylandtoformahomoge- (2009). neousamorphousfilm[30]. Thus,thefilmsassumetheholey [20] R.C.Dynes,J.P.Garno,andJ.M.Rowell,Phys.Rev.Lett.40, 479(1978). geometryoftheunderlyingAAOsubstrate(seeinsetFig.1(a) [21] M.D.StewartJr.,A.Yin,J.M.Xu,andJ.M.VallesJr.,Phys. and (b)). In the nine different AAO substrates used in these Rev.B77,140501(R)(2008). experiments the average hole radii range from 14 to 28 3 [22] M.D.StewartJr., H.Q.Nguyen, S.M.Hollen, A.Yin, J.M. ± nmwithaveragecenter-to-centerholespacingof100 5nm Xu,andJ.M.VallesJr.,PhysicaC469,774(2009). ± ineachsubstrate. Theorderintheholearrangementwasin- [23] S.J.LeeandJ.B.Ketterson,Phys.Rev.Lett.64,3078(1990). tentionally varied using two different anodization processes [24] J. A. Chervenak, Ph.D. thesis, Brown University, Providence, (seeinsetsFig. 1(a)vs. (b)). Sheetresistancesasafunction RI(1998). of temperature, R(T), were measured on 1.5 mm square ar- [25] R.FazioandH.vanderZant,Phys.Rep.355,235(2001). [26] Y.Takahide,R.Yagi,A.Kanda,Y.Ootuka,andS.Kobayashi, eas defined by predeposited Ge/Au contacts using four point Phys.Rev.Lett.85,1974(2000). ACandDCmethods. Thegeometryofeachoftheninehole [27] V.M.Galitski,G.Refael,M.P.A.Fisher,andT.Senthil,Phys. arrayswascharacterizedusingScanningElectronMicroscope Rev.Lett.95,077002(2005). imagesandthesurfacetopographyofsixoftheninewasmea- [28] A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. Lett. 81, suredusingtappingmodeAFMovera1µm2area. 3940(1998). [29] A. J. Yin, J. Li, W. Jian, A. J. Bennett, and J. M. Xu, Appl. Phys.Lett.79,1039(2001). Acknowledgements: [30] K.EkinciandJ.Valles,Phys.Rev.B58,7347(1998). Wearethankfultoacknowledgehelpfulconversationswith D. Feldman, N. Trivedi, T. Baturina, and A. Goldman. This work was supported by the NSF through Grant No. DMR- 0605797 and No. DMR-0907357, by the AFRL, and by the ONR. [1] A.HacklandM.Vojta,NewJ.Phys.12,105011(2010). [2] S.Ujietal.,Phys.Rev.Lett.97,157001(2006). [3] V. F. Gantmakher and V. T. Dolgopolov, Phys.-Usp. 53, 1 (2010). [4] D. Kowal and Z. Ovadyahu, Solid State Commun. 90, 783 (1994). [5] Y.Dubi,Y.Meir,andY.Avishai,Nature449,876(2007). [6] M. V. Feigel’man, L. B. Ioffe, V. E. Kravtsov, and E. A. Yuzbashyan,Phys.Rev.Lett.98,027001(2007). [7] V.M.Vinokur,T.I.Baturina,M.V.Fistul,A.Y.Mironov,M.R. Baklanov,andC.Strunk,Nature452,613(2008). [8] D.B.Haviland, Y.Liu, andA.M.Goldman, Phys.Rev.Lett. 62,2180(1989). [9] J.M.VallesJr.,S.Y.Hsu,R.C.Dynes,andJ.P.Garno,Physica B197,522(1994). [10] A.M.Finkelshtein,JETPLett.45,46(1987). [11] D.Belitz,Phys.Rev.B.40,111(1989). [12] G.Sambandamurthy,L.W.Engel,A.Johansson,andD.Sha- har,Phys.Rev.Lett.92,107005(2004). [13] T.I.Baturina,A.Y.Mironov,V.M.Vinokur,M.R.Baklanov, andC.Strunk,Phys.Rev.Lett.99,257003(2007). [14] V.L.Pokrovsky, G.M.Falco, andT.Nattermann, Phys.Rev. Lett.(2010). [15] Y. Dubi, Y. Meir, and Y. Avishai, Phys. Rev. B 73, 054509 (2006). 5 ddep (nm) π(ξ/2)2 A cell 0.8 1 1.2 1.4 110100 ddep=0.76 nm a 75 b 100film arDomin ea an 100 0.89 nm 50 frat S ) 10 cC action (% 110100 1 nm 50Number IST 0 tion (%) island ea fr 1 of Is m ar 11000 1.07 nm = dIST 25land cc 1.5 Fil 1 s d 5 0 % 100 1 /d 10 1.2 nm IST 1 0 0.5 101 102 103 104 105 106 0.8 1 1.2 1.4 Superconducting island size (nm2) d (nm) SIT SupplementaryFigure1: Coalescenceofsuperconductingislands(a)SCislanddistributionsshowingindividualislandsize,thenumber ofislandsofthatsize,andthefractionofthefilmareaoccupiedbyislandsofthatsizeforthesubstrateofFigs. 1(a),2(a-e),and3. SCfilm regionsareidentifiedford(x,y)>dref =0.7nmona1x1µmimage.Fivedifferentddepareshown,threeofwhichmatchthoseinFigs.2 IST and3. ddep =1.07nmistheIST.Verticaldashedlinesareplottedtoindicateacoherencearea,π(ξ/2)2withξ=15nm,andthehole-array unitcellarea,Acell =9900nm2. (b)FilmareacoverageofthelargestSCislandforeachdepositionin(a)versusddep. TheISTisshownas averticaldashedlineatddep=1.07nm.Theshadedregionindicatesa10%uncertaintyinthethicknessmeasurement.(c)Theanalysisof(a) and(b)isrepeatedforfilmsonallsixdifferentNHCsubstratesandhereweplotthedepositionthicknessatwhich50%ofthefilmareais coveredbyasingleSCisland,d50%,comparedtotheexperimentalvaluedISTforeachsubstrate.Theopencirclerepresentsthesubstratein (a)and(b). SupplementaryInformation TheevolutionofthedistributionofSCislandsizeswiththicknessprovidesfurthersupportfortheislandingpicture. Supple- mentaryFig. 1(a)presentshistogramsoftheSCislandsizesfortheexperimentofFig. 1(a)forfivethicknesses. Forinsulating films with ddep < 0.89 nm, the island size is relatively uniform and on the order of the estimated coherence area. There as manyas71islandsinonebinat75nm2thattogethermakeup0.7%ofthefilmareaatddep =0.76nm. Assumingtheislands areround(seeFig. 3(a)and(b)),theirdiametersare 12nmand25nmforddep =0.76and0.89nm,respectively. Asddep ∼ nearsd ,individualislandscoalescetoformlargerislands. AttheIST,ddep =1.07nm,asingleSCislandoccupies 50% IST ’ ofthefilmarea. Thisislandgrowsrapidlywhenddep d asshowninSupp. Fig. 1(b). Denotingd asthethickness ’ IST 50% atwhichthedominantislandcovers50%ofthefilmarea,wefindthatd d forfilmsonallsixsubstrates(seeSupp. 50% ’ IST Fig.1(c)).Thus,thecoalescenceofregionsthatarethickenoughtosupportSC,asdefinedbyareferencefilm,isnecessaryfor theonsetofSCintheNHCfilms. Finally,becausecolorationofSCregionsinthethicknessvariationimagesofFig. 3andthe determinationofSCislandsinSupp. Fig. 1arebasedonlyond > dref anddonotconsiderrelevantcoherencelengths,phase c coherencebetweenislandsisnotcompulsoryandisnotuniversallyrealizeduntild=d . IST 1