Insulating, metallic and superconducting behavior in a single nanowire
Jan Nicolas Voss, Yannick Schön, Micha Wildermuth, Dominik Dorer, Jared H. Cole, Hannes Rotzinger, Alexey V. Ustinov
IInsulating, metallic and superconducting behavior in a single nanowire
Jan Nicolas Voss, Yannick Sch¨on, Micha Wildermuth, Dominik Dorer, Jared H. Cole, Hannes Rotzinger,
1, 3, ∗ and Alexey V. Ustinov
1, 4, 5 Physikalisches Institut, Karlsruher Institut fr Technologie, 76131 Karlsruhe, Germany Chemical and Quantum Physics, School of Science,RMIT University, Melbourne, Victoria 3000, Australia Institute for Quantum Materials and Technologies,Karlsruher Institut fr Technologie, 76021 Karlsruhe, Germany National University of Science and Technology MISIS, Moscow 119049, Russia Russian Quantum Center, Skolkovo, Moscow 143025, Russia (Dated: June 12, 2020)In systems with reduced dimensions quantum fluctuations have a strong influence on the electronicconduction, even at very low temperature. In superconductors this is especially interesting, since thecoherent state of the superconducting electrons is strongly interacting with these fluctuations andtherefore is a sensitive tool to study them.In this paper, we report on comprehensive measurements of superconducting nanowires in thequantum phase slip regime. Using an intrinsic electromigration process, we have developed amethod to lower the resistance of lithographically fabricated highly resistive nanowires in situ andin small consecutive steps. At low temperature we observe critical (Coulomb) blockade voltagesand superconducting critical currents, depending on the nanowire’s normal-state resistance, in goodagreement with theoretical models. Between these two regimes, we find a continuous transitiondisplaying a nonlinear metallic-like behavior.The reported intrinsic electromigration technique is not limited to low temperatures as we find asimilar change in resistance that spans over three orders of magnitude also at room temperature.Aside from superconducting quantum circuits, such a technique to reduce the resistance may alsohave applications in modern electronic circuits.
When the dimensions of electronic circuits are reducedto nanometer scales their transport characteristics canchange fundamentally. A superconducting wire, whichis approximately as narrow as the variation length scaleof the superconducting order parameter, responds to anapplied electrical field in dramatically different ways: Thebehavior of a short wire with a low normal-state resis-tance resembles, at temperatures well below the criticaltemperature, the well-known behavior of a bulk super-conductor. The order parameter is well defined and theresistance vanishes. A long wire with a sufficiently highnormal-state resistance, instead, does not even reveal su-perconductivity at a first glance, since the wire does notconduct electrical current at low applied voltages.This intriguing phenomenon has its origin in the strongconfinement of the superconducting condensate whichleads to a highly fluctuating order parameter, and there-fore its phase ’slips’ [1–4] (for an overview of the effectsee e.g. [5]). At temperatures close to the transition tem-perature, the electrical response is governed by thermallyactivated phase slips [6, 7]. At very low temperatures,however, the origin of these phase slips has a quantumnature. Between the two extreme states, superconductingand insulating, the response shows a nonlinear metallic-like, behavior at small applied voltages (in the followingfor simplicity denoted as ’metallic’).Until now, it was not possible to access these threedifferent regimes at low magnetic field with a single wire[8–10], since the intrinsic properties, like coherence length or nanowire resistance, were fixed by the preparation ofthe wire.
Sample Ainsula-ting metallic superconducting (a) I p (c)(d)(b) 20nm L w FIG. 1. (a)
Normal state resistance of an oxidized (granular)aluminum nanowire as a function of the applied pulse current I p measured at 25 mK. The resistance of the 20 nm wideand 1000 nm long nanowire is stepwise (inset) lowered from900 kΩ to 2.5 kΩ. At low bias values an insulating, metallicand superconducting behavior is observed. (b) Scanningelectron micrograph of a lithographically fabricated nanowire(coloured in red) on a sapphire substrate. (c)
Illustrationof the wire structure. Isolated aluminum grains (red, size (cid:39) (d)
Proposed microscopic process: The current pulses lead to anintrinsic electromigration process (IEM) merging grains and /or clusters of grains. a r X i v : . [ c ond - m a t . s up r- c on ] J un In this paper, we present a technique which allows forthe permanent reduction of the normal-state resistanceof single wires by three orders of magnitude compared toits initial resistance at low temperatures. We use currentpulses of increasing amplitude to alter the internal struc-ture of oxidized (granular) aluminum (AlO x ) nanowires byan intrinsic electromigration (IEM) process. By doing so,we observe a transition of the electrical response from theinsulating through the metallic to the superconductingstate.The studied thin films consist of a network of smallaluminum grains with a diameter of about 4 nm coveredby a thin aluminum-oxide insulator (see TEM picturein supplementary material of Ref. [11]). This inter-grainmatrix plays a dominant role in the transport propertiesof the material and, at very low temperatures where thealuminum is in the superconducting state, is usually de-scribed as a network of Josephson weak links. The chargetransport between the grains and thus also the normal-state resistance are therefore strongly dependent on thethickness of the insulating barriers. In the fabricationprocess of the films, the thickness of the insulating layerbetween the grains can be influenced by the amount ofoxygen added to the deposited pure aluminum. Thesefilms can have a normal sheet resistance up to severalkΩ and exhibit low intrinsic microwave losses in the su-perconducting state, making them a versatile materialfor high impedance superconducting quantum circuits(see e.g. [11–14]). The kinetic inductance L k of a wiremade of such film may exceed the geometric inductanceby orders of magnitude. It is convenient to describe itby L k = (cid:126) R n /π ∆ = 0 . (cid:126) R n /k B T c , where R n , ∆, T c arethe normal-state resistance, the (BCS) superconductingenergy gap and the transition temperature (1 . − K forgranular aluminum)[11].Due to the Josephson-coupling between the grains,the transport properties of AlO x films are inherentlynon-linear. However, if the number of parallel grains in awire is large, the Josephson non-linearity is washed outand becomes only visible at large electrical currents. Theoretical background
Neglecting the microscopicdisorder, the transport dynamics of superconductingnanowires in the insulating regime has been proposedto be dual to the transport dynamics of a Josephson junc-tion in the superconducting regime [15]. Following [16],a measure of the strength of the phase fluctuations in awire is given by the phase slip energy E s = α (cid:18) L w ξ (cid:19) k B T c R q R n exp (cid:18) − β R q L w ξR n (cid:19) (1)where L w , ξ , R q = h/ e , are the length of the wire,the effective superconducting coherence length and thesuperconducting resistance quantum, respectively. Theempirical constants α and β are of the order of 1 [5]. The TABLE I.
Parameters of the samples measured at mKtemperatures . All AlO x nanowires have a width and athickness of 20 nm. R , R En , denote the initial and final normal-state resistance (before and after altering the nanowire), E and E Es are the corresponding phase slip energies. R mn , E ms and R sn , E ss are the largest metallic and superconducting normal-state resistances / phase slip energies. w R R E n R mn R sn R R En E h E ms h E ss h E Es h nm kΩ kΩ kΩ kΩ GHz GHz MHz HzA 1000 900 2.5 37 16 360 200 2.5 1 10e-21B 750 500 3.7 28 17 135 164 3.0 34 10e-7C 250 12.4 1.5 12.4 4.7 8.3 0.5 0.5 2.5e-3 10e-16 phase slip energy exponentially depends on the normal-state resistance R n . Thus, E s can vary by orders ofmagnitude within the R n range studied in this paper.The inductive energy E L = Φ / L k also plays an im-portant role in the electrical response of a wire. If E L ismuch larger than E s , the superconducting phase differencealong the wire is well defined showing a superconductingbehavior. Here coherent transport of Cooper pairs leadsto a vanishing voltage drop up to a critical current I c ; fora review see e.g. [17]. In the insulating regime, where E s is much larger than E L , no conductance is observed upto an applied critical voltage of V c = 2 π/ e E s . Whenaltering R n , both E s and E L are changed, E L however,only linearly [16] E s E L = α (cid:18) L w ξ (cid:19) . π exp (cid:18) − β R q L w ξR n (cid:19) . (2)As a result, already small changes in R n have a highimpact on the ratio between E s and E L and thus on thetransport properties of the wire. Following Mooij et al.[16], the transition from an insulating to a superconduct-ing behavior should happen at E s ≈ E L . RESULTS AND DISCUSSION
Reducing the normal-state resistance
Our exper-iments focus on a scheme of applying current pulses andmeasuring the changes in the normal-state resistance R n of a wire together with the current-voltage ( I − V ) charac-teristics in the superconducting state. Figure 1 (a) showsa typical resistance R n vs. pulse amplitude I p measure-ment at 25 mK here for sample A (length 1000 nm). Weapplied current pulses with increasing amplitudes rangingfrom I p (cid:39) µ A to I p = 380 µ A in 240 steps. Once areduction of R n at a certain threshold current I p is ob-served, applying pulses with an amplitude below the nextthreshold does not change R n . This behavior is illustratedin the inset of Fig. 1 (a). After the resistance is changedto a certain R n it remains stable, also after thermal cy-cling of the cryostat to room temperature. In addition,to ensure that the altered resistance values, as well as thetransport characteristics, are stable in time, test measure-ments were performed over days. No recovery of eitherthe normal-state resistance or the transport response wasobserved. We have applied the described measurementscheme at various temperatures to about 25 nanowiresamples of different length and from different fabricationbatches, with similar results. Due to the character of thechange in resistance, we name the method as intrinsicelectromigration (IEM).We suggest the following microscopical origin for thealteration of the resistances, illustrated in Fig.1 (d). Witha current applied to the nanowire, a local voltage developswhich mainly drops over the insulating grain-to-grain in-terlayers. At a threshold current I p the tunnel junction ispinched out, merging two or more grains with the weakestinsulating barrier [18, 19]. We observe that the currentpulses can be either applied at room temperature or atlow temperatures with very similar results, leading toa permanent change in resistance. It is advantageous,especially at low temperatures, to apply relatively shortcurrent or voltage pulses to avoid unnecessary heating.Our findings indicate that this procedure creates a newnetwork of more strongly connected grains, which effec-tively leads to an increase in the wire conductance, asseen in Fig. 1 (a).We note that both, the magnitude of the R n changesand the adjusting accuracy, are strongly dependent on thewire length. A qualitative explanation for this behaviormay lie in the random distribution of barrier thicknesses.To first order, the number of junctions in the networkscales linearly with the length of the nanowire. Theprobability of having a few very weak internal junctionsdominating R n therefore also increases quickly with thelength (see Table I). As a consequence, the first changesin resistance are very steep (Fig. 1 (a)). However, with afurther reduction of resistance, we observe smaller steps.Figure 2 (a) displays the distribution of resistance stepsd R n for sample A, hosting a few thousand separatedaluminium grains. Insulating regime
By using the IEM method, theresistance values initially change rapidly (Fig. 1 (a)) andwe observe larger gaps between the measured resistancevalues (order of 10 kΩ), visible as larger steps in thedistribution tail in Fig. 2 (a). Consequently, also thecritical voltage values reflect these gaps in R n .Prior to applying any current pulses to lower R n , sam-ples A and B, see Tab. I, showed strong Coulomb block-aded behavior with maximal critical voltage V c = 3.6 mV(Fig. 2 (d)). For high blockade voltages, we observed acontinuous rounding of the I − V characteristics at V c which lead to a significant error contribution in the de-termination of V c values > s / h ≈
150 GHz in (c)(d) (a) Vc
900 k37 k (b)
25 k15 k (e)
FIG. 2. (a)
Distribution of resistance steps d R n for sampleA. Steps of d R n >
20 kΩ occur only at very small I p . (b) I − V characteristics for a 1000 nm long nanowire. Darkercurves correspond to higher resistance values, brighter to lower.The Coulomb blockade range is from 3.6 mV for a normal-state resistance of ≈
800 kΩ down to about 0.1 mV for thelowest resistance value ( ≈
40 kΩ). (c)
Ratio between phaseslip energy E s and inductive energy E L as a function of thenormal-state resistance of wires A and B. For both wires, theratio converges for smaller resistances towards the same value.Here, the transition from insulating to metallic behavior occursat E s /E L ≈ . (d) Critical voltages V c as a function ofthe normal-state resistance R n . The comparison of measuredvalues and the predictions from Eq. 1 (b) and Eq. 2 (c) withthe fitting parameters α = 0 . ± .
01 and β = 0 . ± . (e) I − V characteristics of sample A inthe metallic regime ( ≈
25 kΩ -15 kΩ). In the vicinity of zerobias current, the slope increases when the wire resistance isreduced. the QPS model). We attribute this rounding to an effectof the relatively low impedance of the environment andtherefore elevated temperature due to dissipation in thenanowire [20]. However, at lower V c the I − V character-istics become very steep, but not hysteretic in the currentvalues.As an important consequence of the lowering of R n weexpect the granular structure to be altered, forming anetwork of more and more galvanically connected grains(see Fig. 1 (d)). Due to reduced grain boundary scattering[21] a longer mean free path l also results in an increasedcoherence length ξ eff = √ l ξ . Experimentally, this hasalso been observed in other granular systems which havebeen treated by classical electromigration [22]. The datapresented in Fig. 2 (c,d) are best fitted assuming a linear ξ span from ξ = 8 nm (R n = 900 kΩ) to ξ = 12 nm( R n = 37 kΩ).Figure 2 (d) displays the extracted critical voltagesas a function of the altered normal-state resistances forboth wires. The overlaid curve is a fit to the measureddata using Eq. 1. The two extracted parameters α = 0 . ± .
01 and β = 0 . ± .
03 are common to thedata of sample A and B and in good agreement with thevalues given in Ref. [16]. Figure 2 (c) shows excellentagreement with the QPS theory, Eq. 2, for samples Aand B. Here we also took into account the change in theinductive energy. The ratio E s /E L drops almost linearlyto values close to unity where we find a transition to themetallic regime. Metallic regime
We observe a metallic regime forall samples with R n values between 40 kΩ and 16 kΩ,characterized by a linear response for small bias values anda non-linear response at larger bias values. For sample A,this is shown in Fig. 2 (e), see also supplementary material.The exact nature of the transition from insulating tometallic, and ultimately to the superconducting phase, isnot currently clear.From the perspective of the QPS model, using theparameters evaluated above, in this regime the I − V characteristic is associated with E s in the range betweenE s /h ≈ s /h ≈
30 MHz, while E L changesbetween E L /h ≈
150 GHz and E L /h ≈
250 GHz. Inthis intermediate regime, E s reduces to values where onaverage neither the localization of charges in the wire northe phase coherence across the wire is dominating.Several observations in our experiments are consistentwith previous work on 2D granular films [23], and 1Darrays of Josephson junctions. For a detailed discussionon the I-M-S transition in 2D granular superconductors,see [24–26]. More recently studies of 1D chains and 2Darrays of Josephson junctions [27] have reported similarbehavior with transitions between insulating, metallic(quasiparticle dominated) and Cooper-pair dominatedtransport, as a function of temperature and magnetic field[28–31]. The interplay between fundamental energy scalesin these systems is similar to the granular aluminum films,and so one would expect similar transport properties.If one considers the model of a network of Josephsonjunctions (illustrated in Fig. 1 (c)), there are two energyscales of interest. The finite charging energy of thegrains sets the energy scale associated with localizedcharges in the system, whereas the Josephson energy setsthe energy scale of the delocalization which underpinsthe superconducting state. One can argue that thereshould be a transition (even at zero temperature)associated with the crossover from Coulomb dominatedto Josephson dominated dynamics, as is typicallydiscussed in the context of Josephson junction arrays[27]. As more links are connected by the IEM, thecharging energy per grain is reduced eventually allowing conduction pathways to form, resulting in a metallic state. Superconducting regime
For values of R n /L W (here L W is the wire length) smaller than about 20 Ω / nm, the I − V characteristics display a transition into a supercur-rent state with no voltage drop up to a critical current I c . In Fig. 3 (a) this is shown for sample A. At thelargest R n and at current values close to I c the voltagedrop across the wire develops rather smoothly (inset),very similar to the phase-diffusion behavior of small ca-pacitance Josephson junctions. Larger I c values show avoltage discontinuity which develops in magnitude as R n decreases as it is common for superconducting wires.Under the assumption that the QPS model is still validand that the above determined empirical parameters α and β are unchanged, we can compute E s to be of the orderof 1 MHz, while E L can be a few hundred MHz. Due tothe smallness of E s , however, it is more useful to describethe wire as a narrow superconducting filament. Thereforewe compare the measured I c to a model for short weaklinks in the dirty limit (mean free path l (cid:28) L link ) (Kulikand Omel’Yanchuk, KO1) [32] and to the expectation thatthe wire would behave like a Josephson tunnel junction(Ambegaokar and Baratoff, AB) [33]. For details seesupplementary material.For both models, I c as a function of the normal-stateresistance is given by: (cid:104) I c (cid:105) = g (cid:48) ( π (cid:52) BCS / e ) (cid:104) R n (cid:105) − with g (cid:48) = 1 .
32 (KO1) or g (cid:48) = 1 (AB). Our measurements donot allow the determination of the current-phase relationof the wire at a given resistance and thus cannot becompared with both models in this respect. Fig. 3 (b)shows the calculated values for both models, togetherwith the I c s from all three samples. (a) (b) FIG. 3. (a) I − V characteristics of sample A as a function ofnormal-state resistance. Here R n is extracted from the resistiveslope above I c at finite voltages. (b) Critical currents, whenthe wires are in the superconducting regime, compared withthe predicted values from the KO1 (black solid line) and AB(black dashed line) theory.
Connecting the picture with Josephson networks, wecan interpret the M-S transition as the point at whichthe average Josephson energy is larger than the Coulombenergy and the usual Bose gas/glass description wouldnow apply. If the superconducting gap is suppressed inthe smallest grains [34], the fusing of grains will also havethe effect of reducing the suppression, increasing thegap and further strengthening the superconducting phase.
Phase diagram
We construct a tentative phase dia-gram by taking the smallest R n values of sample A whichstill show a zero current state and the largest R n valuewhich just shows a zero voltage state and estimate the E s / E L ratio according to the QPS theory. By assumingthat this ratio is the same for other samples, the thickblack lines in Fig. 4 indicate the phase transition fornanowires with differences in length and in R ξ (normal-state resistance per coherence length). The shortest wire,sample C, shows no insulating phase, which is consistentwith the phase diagram. For comparison, we added thedata from Bollinger et al. [35] together with the anal-ysis from Mooij, et al. [16]. The measurement data ofBollinger et al. do not strictly distinguish between insu-lating and metallic, thus the data appear in Fig. 4 in boththe insulating and the metallic phase. For the studiedwires, the transitions from superconducting (red dots) tometallic (gray dots) and further up to insulating state(blue dots) nicely coincides with the solid lines drawn forconstant E s /E L ratios of 10 − and 2 · − . Except forvery short wires, the agreement is remarkable, especiallyif one takes the large parameter space, the very differentmaterial systems and employed techniques into account. CONCLUSION AND OUTLOOK
In this paper, we have demonstrated that the normal-state resistance of narrow nanowires made from oxidized(granular) aluminum can be reduced in-situ at low tem-peratures by using current pulses. This tuning allowedus to observe a transition from insulating to supercon-ducting behavior in a very controlled way, showing goodagreement with theoretical models. We also report onthe observation of a pronounced metallic behavior char-acterized by strongly non-linear conductance which islocated between the insulating and the superconductingphase. The demonstrated intrinsic electromigration pro-cess (IEM) of the granular aluminum also provides apowerful new method of probing the superconductor toinsulator transition. Using the QPS model we proposeda tentative phase diagram, predicting the transition pa-rameters.Apart from basic questions addressing superconductingnanowires, the results of this work may have impactson more applied topics. The demonstrated strong non-linearity of the wires – in conjunction with their low lossand their large kinetic inductance – make them promisingelements for superconducting quantum circuits [3, 14, 20,36, 37], qubits and metamaterials. Furthermore, particledetectors such as microwave kinetic inductance detectors insulating (1000 nm) metallicsuperconducting
Mooji, et al. (750 nm)B(250 nm)C A
FIG. 4. Tentative phase diagram for nanowires made fromoxidized (granular) aluminum. For the black lines, the ratiobetween the phase slip energy and the inductive energy of thewires is constant for different geometries and specific resis-tances. The dots represent the altered normal-state resistancesand the colour the low temperature state (blue = insulating,gray = metallic, red = superconducting). Assuming a con-stant wire geometry and only small changes in the coherencelength, the ratio between E s and E L decreases together with R ξ . From the smallest measurable critical voltages, we finda constant ratio of E s /E L ≈ .
02 (see Fig. 2 (b)) for theinsulating to metallic transition (upper black line). The ratioat which the transition, metallic to superconducting, occurs( E s /E L ≈ − , lower black line), is determined from thesmallest R n values in the superconducting regime. The blackand gray squares represent the data from Bollinger et al. [35] or superconducting nanowire single-photon detectors maybe optimised using the adjustability of the nanowire’sresistance, as described in this work. METHODS
The nanowires are fabricated from a 20 nm thick AlO x filmdeposited on a sapphire substrate. See Fig. 1 (b) for a scanningelectron microscope image. The film has a sheet resistanceof 2.7 kΩ and T c = 1 . x leads with a width of 0.5 µ mand 2.5 µ m length, that contribute to an inductance of 20 nH.With a stray capacitance of about 45 fF we estimate an en-vironmental impedance of about 0.6 kΩ, not considering theinductance of the nanowire itself.The measurements were carried out in a dilution refriger-ator at a temperature of 20 mK. The electric bias leads arefiltered at several stages from room temperature to the base temperature with copper powder, π and RCR-low pass filtersleading to an effective measurement bandwidth of about 5 kHz.In the insulating regime, however, the I − V characteristicsare measured for a better signal to noise ratio with a mini-mal sampling time of about 0.02 s. For the superconducting(low impedance) and insulating (high impedance) regimes, weuse different amplifier readout schemes (see supplementarymaterial for additional information).A computer-controlled measurement protocol was carriedout for all samples in the following way: First, the I − V characteristics were measured in either a voltage bias scheme(insulating regime) or a current biased scheme (metallic andsuperconducting regime). Then I p was applied for about 20 msfollowed by at least a few seconds waiting time, to allow thesamples to recover into thermal equilibrium. The nanowireresistance R n was determined with an excitation current below I p . This then was followed by the next I − V measurementcycle, and so on [38].The wire resistance is given by R n = R tot − R L − R th , where R tot is the total value of the resistance measured, R L = 26 . R L = 39 kΩ (sample B) are the resistancesof the on-chip leads connecting the nanowires. R L is in verygood agreement with estimates using the AlO x sheet resistanceand the geometry of the leads.We also recognised a ’thermal’ resistance offset R th at largercurrent bias values which is of the order of 15 kΩ and can beexplained by considering Joule heating [9].In the insulating and metallic regime the resistances weredetermined from I p . In the superconducting regime however, R n was extracted from the resistive slopes of the I − V char-acteristics above I c which conveniently allows to determine R L and R th .The effective coherence length ξ was extracted from temper-ature dependent measurements of the upper critical magneticfield H c2 for several µ m wide AlO x wires with a sheet resis-tance ranging from 2.0 kΩ to 5.1 kΩ. We found a constant H c2 ( T = 0) = 4 . ± . /µ , consistent with Ref. [39]. Fromthis result, we get ξ = 8 ± . ξ n asa starting point for unaltered highly resistive nanowires.We assume furthermore, that the wire cross-section is suffi-ciently homogeneous, that the wire’s E s is not dominated by anarrow constriction in the nanowire. However, possible defectscannot be ruled out completely. In electron microscopy scansthe wires appear smooth and uniform with an edge roughnessof the order of 1-2 nm. ACKNOWLEDGMENT
We thank L. Radtke and S. Diewald of the KIT Nanos-tructure Service Laboratory for the support concerningthe sample fabrication. The work was funded by theInitiative and Networking Fund of the Helmholtz As-sociation, the Helmholtz International Research Schoolfor Teratronics (JNV, YS) and the Landesgraduiertenfo-erderung (LGF) of the federal state Baden Wuerttemberg(MW). Further support was provided by the Ministry ofEducation and Science of the Russian Federation in theframework of the Program to Increase Competitiveness ofthe NUST MISIS (contract No. K2-2020-017). JHC ac- knowledges the support of the Australian Research Coun-cil Centre of Excellence funding scheme (CE170100039)and the NCI National Facility through the National Com-putational Merit Allocation Scheme. ∗ [email protected][1] N. Giordano, Phys. Rev. Lett. , 2137 (1988).[2] A. Bezryadin, C. N. Lau, and M. Tinkham, Nature ,971 (2000).[3] A. M. Hriscu and Y. V. Nazarov, Phys. Rev. B , 174511(2011).[4] O. V. Astafiev, L. 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SUPPLEMENTARY MATERIALExperimental details
To ensure a low noise impact on the samples, alsoseveral noise-reducing devices were installed for all leads,going down to the sample at ≈ π -filters at room temperature, then followed byRCR-low pass filters at the 4K stage of the refrigerator.The resulting measurement bandwidth is about 5 kHz.Additionally, meter long copper powder filters are installedto suppress high frequency noise.A typical schematic diagram of the bias schemes weused is shown in Fig. 5: To measure and reduce thenormal-state resistance of the wires and to record the I − V characteristics in the metallic and superconduct-ing regime, a current bias scheme was used (depictedin red). As current source ( I bias , I p ), we use a voltage-controlled, in-house made tunnel electronic which uses aTexas Instruments OPA2111 operational amplifier andcurrent dividers as main elements. The input voltagesignal is generated by a Keithley 2636A source meter andpre-filtered by a Stanford Instruments SR560 preampli-fier. The output signal ( V out ) is amplified by a low-noiseinstrumentation amplifier INA 105KP and measured withthe second channel of the source meter. In between theamplifier and the source meter, the signal is filtered by aStanford Instruments SR560.
20 nm V out V out I bias V bias
300 K / I p Sample Cmetallic superconducting (b)(a)
FIG. 5. (a)
Measurement setup used to measure the I − V characteristics and to reduce the normal-state resistance ofthe wire. Either the current bias scheme (red) or the voltagebias scheme (blue) was used. To automatise the procedure, weimplemented several computer-controlled relays. I p representsthe applied pulse for the reduction of the wire resistance. (b) R n tuning for sample C as a function of the applied pulsecurrent I p . The measurements were performed at 25 mK. Forsmall bias values, the electrical response was metallic (12.4 -4.7 kΩ) and superconducting (4.7 - 1.5 kΩ). For samples A and B and being in the insulating regime,we used a voltage bias scheme (see Fig. 5, blue circuit).Here, one side of the sample is set to ground while on theother side, a FEMTO transimpedance amplifier (DDPCA-300) is applied. As for the current bias, the input voltagesignal ( V input ) is generated by a Keithley 2636A and then filtered by a Stanford SR560. The output signal ( V out ) ismeasured in the same way as described before. Coherence length
To determine the initial coherence length (before re-sistance tuning) for our samples, we have measured H c2 ( T ) for three µ m wide AlO x wires with differentsheet resistances (see Fig. 6). For a type 2 supercon-ductor, the relation between H c2 ( T ) and the GinzburgLandau coherence length ξ GL is given by: H c2 ( T ) =Φ / (2 πξ GL ( T ) ) [42]. Therefore ξ GL ( T = 0) is directlyrelated to H c2 ( T = 0). With the universal relation: H c2 (0) = 0 . T c ( dH c2 /dT ) T = T c for a one-gap supercon-ductor in the dirty limit (Werthammer at al., [43]), H c2 (0)can be extracted from the slope of H c2 ( T ) at T = T c . Thecritical temperatures were approximately 2 K for all sam-ples. From the linear fits to the H c2 ( T ) measurements(see Fig. 6), together with the measured T c values, wereceive a coherence length ξ = 8 ± . c r i t. f i e l d ( T ) [1-(T/Tc) ]X1X3X4 FIG. 6. Dependence of the critical field H c2 on the reducedtemperature 1 − ( T /T c ) . X1, X3 and X4 denote three different µ m wide AlO x wires. Sample X1 has a sheet resistance of R X1sq = 2.0 kΩ, X3 of R X3sq = 5.1 kΩ and X4 of R X4sq = 4.5 kΩ.The critical field H c2 is approximately the same for all samplesat a certain temperature. Critical currents
For wires that are in the superconducting regime, wehave compared the critical currents with the expectationsof two different models (see Fig. 3 (b)): Firstly, the modelof Kulik and Omel’Yanchuk for a weak link in the dirtylimit(KO1). Secondly, the model for a Josephson tunneljunction by Ambegaokar and Baratoff (AB).Both comparisons require a justification. In case of theAB model, we argue that, grossly simplified, the internalstructure of the wires is very similar to a disordered chainof Josephson junctions where the weakest junctions aredominating I c . In contrast, the KO1 model considersthe distributed character of a homogeneous weak linkand is applicable up to an upper limit of the ratio wire/ coherence length of about 3.49 which our wires exceed.We argue that, due to the internal granular structure,the comparison is still possible: The phase drop alongthe wire is not homogenous but instead it drops mainlyover the oxide barriers between the nanoscale grains. Itis therefore reasonable to introduce an effective length L eff (cid:39) L w /
5, assuming an average grain size of 4 nm andan oxide barrier of 1 nm.
Phase transitions
As described in the main part of this paper, at lowtemperatures and small bias values we distinguish betweenthree different phases, namely insulating, metallic andsuperconducting. The phase diagram (Fig. 4) reflects thetransition through the different phases for all samples,driven by the intrinsic electromigration process (IEM).Even though it is possible to go in small resistance stepsvery smoothly through the different phases, we observea sharp border, separating the phases. Here, the I − V characteristics change dramatically between two IEMs.Fig. 7 shows this abrupt change in the transport behaviorfor sample A at both transitions, insulating-metallic (a),metallic-superconducting (b).In contrast to samples A and B, sample C initiallyshowed a metallic behavior (see Fig. 8 (a)). This is ingood accordance with the reduced length, compared tothe other samples, and the relatively small initial normal-state resistance R n of about 12 kΩ. The deviation of themeasured R n from the expected normal-state resistance R ∗ n >
33 kΩ, assuming a sheet resistance of 2.7 kΩ, canbe addressed to a location-dependent variation of thefilm sheet resistance. However, for this sample, we wereable to reduce R ξ very smoothly through the metallic- superconducting transition illustrated in Fig. 4. Thechange in R n is shown in Fig. 5 (b). In the metallic regime,we have noted a substantial difference in the shape of the I − V characteristics, depending on R n . For the highestresistances, the slope in the vicinity of zero bias increaseswith increasing bias current and therefore conductance. In contrast, for the smallest R n values the slope decreasesmonotonously with the increase of the bias current. At R n ≈ ≈
440 nA for the smallest R n (see Fig. 8 (b)). (a) (b) FIG. 7. Change in the I − V characteristics of sample A atthe phase transition lines (see solid black lines in Fig. 4). (a):For R n >
50 kΩ, the response is insulating (blue line), belowit is metallic (black line). (b) The second transition appearsat R n ≈
18 kΩ. Here the behavior again changes abruptly, butnow from metallic (black line) to a superconducting behavior(red line).
Sample C (a)
Sample C (b)
FIG. 8. I − V characteristics of sample C (250 nm, for a detailedoverview of the parameters see Table I). (a) Metallic regime(12.4 - 4.7 kΩ): Sample C, in distinction from samples A andB, initially showed a metallic behavior. (b)
Superconductingregime: For small bias currents, up to a critical value I cc