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Characterization of anomalous pair currents in Josephson junction networks

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 215701 (6pp)

doi:10.1088/0953-8984/26/21/215701

Characterization of anomalous pair currents in Josephson junction networks I Ottaviani1, M Lucci1, R Menditto1, V Merlo1, M Salvato1,5, M Cirillo1,5, F Müller2, T Weimann2, M G Castellano3, F Chiarello3, G Torrioli3 and R Russo4 1

  MINAS-Dipartimento di Fisica, Università di Roma Tor Vergata, I-00173 Roma, Italy   Physikalisch Technische Bundesanstalt, Nanostructuring Group, Bundesallee 100, D38116 ­Braunschweig, Germany 3   IFN-CNR, Via Cineto Romano, I-00156 Roma, Italy 4   CNR, Istituto di Cibernetica ‘E. Caianiello’, Via Campi Flegrei 34, I-80078, Pozzuoli, Italy 5   CNR-SPIN Institute, UOS di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano, Italy 2

E-mail: [email protected] Received 10 January 2014 Accepted for publication 27 March 2014 Published 2 May 2014 Abstract

Measurements performed on superconductive networks shaped in the form of planar graphs display anomalously large currents when specific branches are biased. The temperature dependences of these currents evidence that their origin is due to Cooper pair hopping through the Josephson junctions connecting the superconductive islands of the array. The experimental data are discussed in terms of theoretical models which predict, for the system under consideration, an inhomogeneous Cooper pair distribution on the superconductive islands of the network as a consequence of a Bose–Einstein condensation phenomenon. Keywords: Josephson effect, superconductivity, Bose–Einstein condensation (Some figures may appear in colour only in the online journal)

Patterns of superconductive islands connected by Josephson junctions (often referred to as Josephson junctions networks or arrays) have attracted noticeable interest in the past decades both from the fundamental [1] and applied physics [2] point of view. Series arrays of thousands of Josephson junctions, due to the striking current uniformity of the tunnel barrier that can be achieved with the available technology, represent today the basis for the settlement of the international voltage standard [3]. Recently, however, theoretical investigations have pointed out that in Josephson networks shaped in the form of graphs an inhomogeneous distribution of bosons could be generated as a consequence of a Bose–Einstein Condensation (BEC) process [4–6]; the non uniform distribution can affect in turn the uniformity of the Josephson currents between the superconductive islands of the network. In a Josephson network the nodes of the graph are the superconductive islands on which bosons (physically represented by Cooper pairs) are located. In the theoretical models the topology of the network [7]6,

namely the specific connectivity map between the nodes, plays a paramount role for generating the BEC through the adjacency matrix regulating the hopping of the bosons between neighbour superconductive islands [4–6]. These theoretical predictions attracted our attention because in a Josephson network of ‘point junctions’ (junctions whose size is smaller than the Josephson penetration depth [8]) which does not contain superconducting loops, or other coupling structures, the amplitude of the supercurrents should not depend on the connectivity of the network, namely on how many junctions depart from a specific island. The simplest exemplification, on a plane, of a ‘symmetric comb’ is shown in the inset of figure 1: the topology of this kind of graph is defined by the existence of nodes connected to four neighbour units (these nodes form the ‘spine’ or ‘backbone’) and nodes connected only to two neighbour (located on structures departing by side the backbone, called ‘fingers’). The theory predicts a well defined non homogeneous distribution of bosons on the symmetric comb graph [5] and measurements have pointed toward qualitative agreement

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  See in particular section 4 of chapter 2 for the definition graph and ­topological properties of network in the sense herein addressed. 0953-8984/14/215701+6$33.00

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© 2014 IOP Publishing Ltd  Printed in the UK

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junctions were defined by electron beam lithography. In figure 1(a) we show an optical microscope image zooming a region of a sample and the inset shows schematically a symmetric comb graph where the dots represent the superconductive islands and the lines are the connections between these for which Josephson junctions are responsible. The central pattern of the structure on which every island is connected to four neighbours through Josephson junctions generates what we call backbone array (BA), while the lines departing perpendicularly from it give rise to what we call fingers arrays (FA). The particular of figure 1(a) shows four cross-shaped superconducting islands of the BA and eight FA islands departing from these; some islands are generated by the base electrode (those like A and D) and others (like B and C) by the wiring electrode [8]; the dashed rectangles indicate where the Josephson junctions are located. The samples that we fabricated contain 400 backbone superconducting islands and 400 finger arrays; the high number of fingers was designed in the attempt to approach the thermodynamic limit of the number of bosons considered by the theory [5, 6]. It is worth noting that the present design contains a relevant correction with respect to previously fabricated samples [9, 10]: the BA was now designed in a way to meet more closely the requirements of a symmetric comb topology. In fact we see in figure 1(a) that each superconducting island of the backbone has four first nearest neighbour islands, as it should be (see inset), while in the previous experiments half of the islands of the BA had only two first nearest neighbour islands. Each island of the fingers, instead, is connected only to two first nearest neighbour islands. This difference in the number of neighbors revealed to be one of the main causes of the topology-induced condensation phenomenon on graphs [4–6]. A main concern when dealing with the theory reported in [3] and [4] is the fact that the calculated effects (nonuniform distribution of Cooper pairs on the islands) should be observable when the Josephson coupling energy Ej [8] between the islands is of the order of the thermal energy, namely when Ej = ϕ0Ic/2π ≅ kBT (see equation (18.3) of [9]), where Ic is the maximum Josephson pair current, ϕ0 = 2.07 × 10–15 Wb is the flux-quantum and kB = 1.38 × 10−23 J K−1 is the Boltzmann constant. We have to bear in mind that our bosons exist only below 9.2  K, the superconducting transition temperature of niobium, and therefore in order to preserve the physical sense of the theory, the Josephson energy must be adequately tuned. For a maximum pair current of 1 μA, the Josephson energy would be 3 × 10−22  J and comparable with the thermal energy which, in the (1–9) K temperature interval ranges in the (10−22–10−23) J interval. We conclude that a critical current of the order of few microamperes could be appropriate for the experimental observations; for this reason indeed in previous investigations [9, 10] the interval (1–10) μA for the maximum critical currents of the junctions was targeted8. We must point out, however, that when measurements are performed biasing Josephson junctions with an external current

Figure 1. (a) Optical microscope image of a portion of our combshaped array: in the inset the dots represent the superconductive islands. In the main panel (A and B)-like cross-shaped features are backbone islands; (C and D)-like islands belong to fingers. The dashed areas indicate junctions location; (b) the height of the washboard potential of the Josephson junctions plotted as a function of the external bias current for different values of the maximum critical current Ic. When the potential curves fall below the straight horizontal like, corresponding to a temperature of 9 K, thermal excitations compete with Josephson potential energy.

with the experimental reality [9, 10]7. We have undertaken a systematic analysis of inhomogeneous distribution of Cooper pairs on a comb graph by measuring the current-voltage characteristics (I–V curves) of specific branches arrays and determining temperature dependences of the curves. We check the real incidence of the topology on the observed phenomena by reporting also results obtained on arrays of junctions geometrically identical to those embedded in the graph structures, but having a different connectivity map. The collected data enable us to perform the first quantitative comparisons with the theoretical expectations. Our experiments are made on arrays of high quality Josephson tunnel junctions based on the niobium trilayers (Nb-AlOx-Nb) technology [11]; for these specific experiments, in order to minimize the scatter in the geometrical parameters of the junctions, a hybrid electron beam-optical technology was used and, in particular, the areas of the tunnel

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  Naturally the conditions for the experimental observations would be even more favourable lowering the value of the critical current. However, making uniform series arrays with maximum Josephson currents below 1 µA is a quite challenging technological task.

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  For the derivation of equation 18.3 in [9] see also [14]. 2

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Figure 2. (a) Current-voltage characteristics of three arrays of the sample CB428 at 4.2 K. The expansion of the areas indicated by the

zooming squares 1 and 2 are reported respectively in (b) and (c). In (b) we have added the currents of RFA array to compare it with the RBA. In (d) we show the feature observed in (c) for different values of the temperature.

at their ends, as shown in the inset of figure 1(a). The I–V curves refer to three arrays and we notice that switching current distributions and subgap resistance indicate very good quality and uniformity of the samples. We have expanded the regions of the characteristics indicated by the zoom-squares 1 and 2 in figure 2(a) respectively in figures 2(b) and (c). In the zoom of figure 2(b) we see the Josephson current distributions of BA and FA and we observe that the currents of BA are roughly 0.5 μA above those of the FA. In this figure we also show the currents of two ‘reference’ arrays which are two arrays having the same geometry of the backbone and finger arrays but are not embedded in the graph structure and are isolated from it. We call these Reference Backbone Array (RBA) and Reference Finger Array (RFA), respectively; the superconductive islands of the RBA (and the junctions) are identical to those of the BA, however, the RBA is missing the fingers connections meaning that each cross-shaped superconductive island of this array is connected only to two neighbour islands. Thus, the geometry of the measured junction series connection over BA and RBA is the same but their topology is different. It is worth noting that we also bias the RBA from the beginning to the end and therefore the number of biased junctions is the same as for the BA. The RFA is just the connection of two fingers having in common one island

I (as in our case) a relevant energy to be compared with thermal excitations is not the bare Ej, but rather the height of the ⎡ I I ⎤ Josephson potential ΔU = 2Ej⎢ 1 − ( I / Ic ) − cos−1 ( ) ⎥ [8]. ⎣ Ic Ic ⎦ In figure 1(b) we plot ΔU as a function of I for different values of the maximum Josephson current Ic; in the figure the horizontal straight line represents kBT for T = 9 K. We see that for high values of the bias currents, for all the curves, ΔU decreases below the thermal threshold; since when measuring current-­voltage characteristics of specific branches-arrays we trace switches from the highest values of the Josephson current, this experimental technique is such that along current-biased branches a lowering of the Josephson potential is generated. It is likely that this technique eases the observation of the theoretically predicted nonuniform distribution of bosons along the biased branches which can be there independently upon our biasing conditions. What we pointed out above is just that our measurement technique employs a ‘probe’ with very low energy which can enable us to detect current differences induced by boson-hopping. In figure 2(a) we show the current-voltage characteristics of the arrays of the sample CB428. The characteristics are traced biasing the arrays through four contacts pads placed 3

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Figure 4. (a) The anomalous currents plotted versus Josephson currents: each point corresponds to a different temperature; (b) decay of the anomalous excess current as a function of the voltage distance from the gap-sum voltage for two samples (CB427 and CB422). The straight line behavior in this semi-log plot indicates and exponential functional dependence.

Figure 3. (a) The anomalous excess current at the gap-sum voltage

of the sample CB422. We see that the FA (finger array) currents become much greater than the BA (backbone array currents). (b) dependence of the excess current upon temperature (squares) and of Icmin (the maximum Josephson current). The curves fitting the data are obtained from the Ambegaokar–Baratoff (AB) functional relationship.

arrays (BA and FA) and their counterpart reference arrays and these cannot be ascribed to measurements and design uncertainties. As illustrated in figure 2(b), the currents of the FA are slightly lower than the currents of the BA, a condition which is observed for most of the junctions of the arrays; in ­figure  2(c), however, we see that close to the gap sum voltage the currents of the finger array become substantially higher than the currents of the backbone. We also show in figure 2(d) that this anomalous current increases substantially lowering the temperature; we remark that for sample CB428 apparently only two junctions of the FA exhibit an anomalous large current which, at 500 mK is of the order of 4 μA corresponding to 25% of the Josephson critical current. We have observed the anomalous increase of finger currents close to the gap sum voltage on several samples and we notice that this effect was also visible on the experiments reported in ref. 10 (see f­igure  3 of the paper), but in that work the phenomenon was not mentioned; a similar effect has also been measured on star-shaped graph arrays [12]. In ­figure  3(a) we show a striking example of anomalous currents in the finger array for the sample CB422 having a current density of 480 A cm−2 and a maximum Josephson

of the backbone which in turn is not connected to the rest of the backbone. All the islands of the RFA have two neighbours (except for the ending island which only have one neighbour) while only one island of the FA, the one shared with the backbone, has four. We note in figure 2 that the two reference arrays, although their geometry is substantially different, have current amplitudes equal within the experimental error, while BA and FA have currents higher respectively of 1 μA and 0.5 μA than the reference arrays. The differences between ‘comb-­embedded’ and ‘reference’ arrays were pointed out for the first time in [9] and were well characterized in [10], however now, due to fact that in our new design the BA is ‘topologically’ consistent with the theoretical model we record very uniform currents for all the backbone junctions at all temperatures, which was not the case in [10] (see figure 2(b) of the paper). We remark that in the present case the areas of the junctions have a very limited spread (about 1%), therefore geometrical factors can be safely ruled out as cause of the observed differences in critical currents of the arrays: indeed we see that the reference arrays have currents identical within experimental uncertainty. Differences exist between comb-branches 4

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current Ic = 43 μA at 4.2 K: the specific record of figure 3(a) was taken at a temperature of 1 K, and we see in this case that the anomalous extra current almost reaches the level where the normal state resistance takes over in the conduction process above the gap. At this temperature the span of the anomalous current above the average Josephson critical currents is 27 μA and this corresponds to being 56% of the critical currents of the arrays, more than twice the percentage observed for the sample CB428. We have systematically investigated the response of the anomalous excess currents to temperature variations. The amplitude of these currents is obtained as the difference between a maximum and a minimum defined as follows: the maximal current is determined by the value where the last 2.5 mV switch before the gap-sum occurs, and the minimum (Icmin) is the extrapolated value (from the main array switching line) of the Josephson current at the corresponding voltage. The results that we obtain for the specific case of the sample CB422 are shown in figure 3(b); in this plot we also show the dependence upon temperature of Icmin (which is essentially the Josephson current). Both the measured temperature dependencies have been fitted, see the lines through the data in the figure, with the Ambegaokar–Baratoff functional relationship [8] in which we approximated the gap by the equation ⎛ T −T ⎞ Δ ( T ) = Δ (0)tanh ⎜ A c ⎟ [8, 13] where A is a constant. T ⎠ ⎝ For the fits of figure 3(b) we have Tc = 7.28 K and A = 1.45 for the anomalous current (squares) and Tc = 8.47 K and A = 1.3 for the Josephson current (triangles). Since the only significant difference in the fitting of figure 3(b) is the value of the condensation temperature Tc we speculate that the two phenomena are related to different condensation processes in which Cooper pairs are involved. It is worth noting that the ‘anomalous’ currents appear at a temperature of 7.28 K which is 1.2 K below the superconducting condensation temperature of the samples. In other words, below the BCS condensation temperature an additional condensation occurs at a lower characteristic temperatures but this phenomenon has nothing to do with the birth of the Cooper pairs: according to the theoretical model it is generated by a nonuniform distribution of pairs on the islands. The data of figure 3(b) show that the extra tunnelling currents observed on the finger arrays have a pair-current nature whose temperature dependence is regulated by a BCS functional relation. Moreover, since the Ambegaokar–Baratoff relationship gives, close to the critical temperature, a dependence of the anomalous current upon temperature like (1−T/Tc) we recover, in this limit, a prediction of Burioni et al [5]. In figure 4(a) we plotted for different temperatures, the amplitude of the anomalous current versus the maximum Josephson current for CB422 and another sample, CB427 having a current density of 200 A cm−2. In the figure we report, for different temperatures, the maximum Josephson current on the horizontal axis and the amplitude of the anomalous currents on the vertical axis: the two straight line dependencies indicate that the ratio of the physical quantities is a constant not depending upon temperature. If we

assume that the observed anomalous currents are generated by boson hopping between the islands and associate it with the ‘filling factor’ of the theory [5, 6] we recover here a theoretical prediction, namely that the ratio between filling factor and Josephson energy is a constant not dependent upon temperature [5, 6]. The theory [5, 6] also indicates that the population of bosons on the backbone islands is maximal and therefore one would expect that the currents of the finger array could reach the value of the backbone currents: the FA shares one superconductive island (see figure 1(a)) with the backbone and two junctions of the fingers could have value of the current close to that of the backbone. Instead, in figures 2 and 3 we see FA Josephson currents having currents substantially higher than all the backbone currents. This phenomenon can be interpreted as follows: the total current flowing between islands of the arrays along one direction can be written as dqi ∂ qi ∂ qi dx i = + , where qi represents the total charge on ∂t ∂ xi dt dt ∂q each superconductive island, i the charge gradient between dx ∂ x i neighbouring islands and i the speed of the charge flow. As dt far as the BA is concerned, according to the theoretical model, the second term on the right hand side of this equation is zero since the charge distribution on all the BA islands is uniform [4, 5]. On the FA instead, the islands close to the backbone experience a variation of charge carriers density and therefore ∂q the spatial derivative of the charge i , performed where the ∂ xi FA ‘crosses’ the backbone, can give a relevant contribution to the total current. Since [4] and [5] report the specific spatial— dependency of the density of bosons on the islands of the FA, the spatial derivative of the current(s) due to Cooper pairs hopping between the islands can be calculated, resulting in a exponential decay. It is worth noting that on reference arrays there are no effects that one could ascribe, as above, to charge non uniformities: reference arrays present uniform Josephson currents and related standard properties. We remark now that the decrease of current in figure 3 takes place in correspondence of voltage steps which are, on the average, of the order of 5 mV meaning that two junctions of the FA share roughly the same current (as it was the case in figure 2 for sample CB428): this result is consistent with the fact that the FA that we are measuring is symmetric with respect to the backbone line, therefore each current step corresponds to a jump generated by two junctions situated at the same distance from the BA. The dependence that we extracted from the samples CB422 and CB427 is shown in the semi-log plot of figure 4(b) where we see that the functional form is a decreasing exponential. Since we do not have access to all the junctions of the arrays, in principle, we cannot distinguish whether the currents of the pairs of junctions originating the results of figure 3 correspond to pairs located at the two sides of the BA and at the same distance from it. However, as ‘singular’ (and centre of symmetry) points of the FA we can identify only the one crossing the BA: the ends in fact generate no effects on the Josephson current distribution in other arrays (BA and reference arrays) and 5

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References

there is no reason why the FA should behave differently. We believe at this point that the exponential decay shown in figure 4(b) is generated by the predicted decrease of population of bosons [5, 6] when moving from backbone islands toward the end of the fingers. In conclusion, we have reported evidence that topologyinduced effects in arrays of Josephson junctions can generate spatial gradients in the distribution of Cooper pairs on the superconductive islands of the arrays. Our results confirm that non uniform distributions of currents embedded in the graph structures exist and that these are not due to geometrical factors but just to connectivity of the superconductive islands. We have found that the experimental results follow the expectations of the theory as far as the distribution of Josephson currents on the ‘finger’ arrays of the graph is concerned. The dependence of the extra anomalous current upon temperature evidences that the anomalous extra currents are indeed pair currents which only appear below a temperature which is well distinguishable from the superconducting niobium condensation temperature.

[1] Jain A K et al 1984 Phys. Rep. 109 309 Ustinov A V et al 1995 Phys. Rev. B 51 3081 Barbara P et al 1999 Phys. Rev. Lett. 82 1963 Fazio R and van der Zant H 2001 Phys. Rep. 335 235 [2] Pagano S et al 1989 IEEE Trans. Magn. 25 1080 [3] Kautz R L 1996 Rep. Prog. Phys. 59 935 Mueller F et al 2009 IEEE Trans. Appl. Supercond. 19 981 [4] Burioni R et al 2000 Europhys. Lett. 52 251 [5] Burioni R et al 2001 J. Phys. B 34 4697 [6] Fidaleo F et al 2011 Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 149 [7] Van Valkenburg M E 1974 Network Analysis (Englewood Cliffs, NJ: Prentice Hall) [8] Barone A and Paternò G 1982 Physics and Applications of the Josephson Effect (New York: Wiley) Van Duzer T and Turner C W 1999 Principles of ­Superconductive Devices and Circuits (Englewood Cliffs, NJ: Prentice Hall) [9] Cirillo M et al 2006 Spatial Bose–Einstein condensation in Josephson junctions arrays Quantum Computation in Solid State Systems ed B Ruggiero et al (New York: Springer) p 147–53 [10] Silvestrini P et al 2007 Phys. Lett. A 370 499 [11] Gurvitch M, Washington M A and Huggins H A 1984 Appl. Phys. Lett. 55 1419 [12] Lorenzo M 2010 Effetti topologici in arrays di giunzioni Josephson MSc Graduation Thesis Università di Roma Tor Vergata Lorenzo M et al 2014 Phys. Lett. A 378 655 [13] Senapati K et al 2011 Nature Mater. 10 849 [14] Giusiano G et al 2004 Int. J. Mod. Phys. B 18 691

Acknowledgements We acknowledge fruitful discussions with R Burioni, D Cassi and F Fidaleo. Our interest in the topic of the Josephson graph-arrays was stimulated by Mario Rasetti and Pasquale Sodano and we wish to mention their role in originating the work herein presented.

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Characterization of anomalous pair currents in Josephson junction networks.

Measurements performed on superconductive networks shaped in the form of planar graphs display anomalously large currents when specific branches are b...
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