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On Bose–Einstein condensation in Josephson junctions star graph arrays
Physics Letters A 378 (2014) 655–658
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
On Bose–Einstein condensation in Josephson junctions star graph arrays M. Lorenzo a , M. Lucci a , V. Merlo a , I. Ottaviani a , M. Salvato a,1 , M. Cirillo a,∗,1 , F. Müller b , T. Weimann b , M.G. Castellano c , F. Chiarello c , G. Torrioli c a b c
Dipartimento di Fisica and MINAS Lab, Università di Roma “Tor Vergata”, 00133 Roma, Italy Physikalisch Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany IFN-CNR, Via Cineto Romano, I-00156 Roma, Italy
a r t i c l e
i n f o
Article history: Received 9 October 2013 Received in revised form 19 November 2013 Accepted 17 December 2013 Available online 27 December 2013 Communicated by A.R. Bishop
a b s t r a c t We report on the evidence of anomalous currents in graph-shaped arrays of Josephson junctions along peculiar branches of the networks. The specific case of a star-shaped array is considered and the evidence of the anomalies is achieved by comparing the current-voltage characteristics of the arrays embedded in the star structure with those of “reference” arrays which are fabricated by-side the network structure and are dc-isolated from these. The experimental data are consistent with the results of a theoretical model predicting gradients of the populations of Cooper pairs on the islands situated in proximity of the central superconductive island as a result of a Bose–Einstein condensation process. © 2013 Elsevier B.V. All rights reserved.
Arrays of Josephson junctions have puzzled the fundamental and applied superconducting community in the past two decades. From the fundamental point of view the interest has been triggered by the intriguing properties of the Josephson junctions as nonlinear oscillators [1,2]. The existence of a solid state device whose Hamiltonian is analogous to that of the classical pendulum [3] is a feature which has stimulated the interest of several theoretical and experimental groups. The applications instead have been motivated by the need of radiation sources for the millimeter and submillimeter region of the electromagnetic spectrum [4], but linear arrays of junctions [5] are also considered at present as components for digital electronics [6]. Moreover, it is established today that the maintenance and setting of international voltage calibrations rely on the Josephson junctions series arrays [7]. The development of quantum statistical properties of discrete systems [8] led a group of theorists to formulate a model for graph-shaped networks composed by superconducting sites connected by Josephson junctions. Such a network could exhibit peculiar properties concerning the distribution of Cooper Pairs (bosons) on the sites generating the network itself [9]. Although the model was simplified as far as the boson interactions are concerned (only boson-hopping between the islands was considered), the results were clear in the interpretation and on the possible effects to be observed: the population of Cooper pairs on the superconductive islands of the array would result not to be uniform because bo-
* 1
Corresponding author. Tel.: +39 0672594518; fax: +39 062023507. E-mail address: [email protected] (M. Cirillo). Also CNR-SPIN, Italy.
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.12.032
son hopping would generate density gradients due to the peculiar adjacency matrix of the network. As a consequence of this phenomenon the Josephson current existing between islands located in different areas of the networks could also have gradients, due to the different numbers of charge carriers on the islands connected by the junctions. Recently a mathematical theory by Fidaleo et al. [11] investigating the Bose–Einstein Condensation phenomenon in graph arrays has shown that in these networks bosons condensate in both momentum and configuration space. In previous papers experimental results showing that measurements on a “comb” graph do evidence anomalies in the distribution of the Josephson currents in the graph arrays have been reported [12,13]. On a comb array the theory predicts a uniform distribution of bosons on the superconductive islands located along the “backbone” line of the comb while an exponentially decaying population of bosons is predicted along the finger lines [9,13]. In the present paper we describe measurements on star-shaped graphs which were performed on arrays having very limited scatter of the junction parameters. As we said above a local increase (resp. decrease) of Cooper pairs population on the superconductive islands could generate a local increase (resp. decrease) of the Josephson currents between these islands. Our experimental technique consists in probing the maximum Josephson currents of all the junction of an array constituting a specific branch of the graph. Technically, we are just measuring the current-voltage characteristics (IV curves) of the series array; from these IV we realize whether the Josephson currents of the junctions of the array are uniform or not. A consequence of the series current biasing is that the junction having the lowest critical currents will switch first to the voltage state (the gap voltage when
656
M. Lorenzo et al. / Physics Letters A 378 (2014) 655–658
Fig. 1. (a) A scheme of a star-shaped array with 8 rays; (b) optical image of one of the samples that we have fabricated where is indicated the location of the junctions and of the different islands. The arrows at the end of the arrays indicate the points were we feed the current and read the voltage: in the star we measure the series connection of two “rays” of the star.
no external magnetic fields are applied) when current-biasing the array; after the switching, only the voltage of this junction will appear on the IV of the series array. Increasing further the bias current through the array the junction having a slightly higher Josephson current will switch to the voltage state and at this point the sum of two gap voltages shall be visible in the IV curve (and so forth for increasing bias current). Thus, the junction having the highest critical currents will switch to voltage state for the highest values of the bias currents and its voltage, resulting as the sum of all the others, shall be located at the gap-sum voltage of all the junctions of the array. In principle, we have no information on where in the array the junctions with higher (or lower) critical currents are located; however, we can clearly distinguish a situation of non uniform Josephson current distribution. We did not put extra probing contacts (which would allow us to test specific junctions) on the arrays because we preferred not to perturb the system and to keep the design as close as possible to the theoretical model. Our samples were arrays of Josephson junctions fabricated in the niobium trilayer technology [14]; a peculiarity of our fabrication process was the fact that, in order to minimize the scatter of the geometrical parameters of the junctions, their area was defined by electron beam lithography. A typical scheme of a star graph chip is shown in Fig. 1(a) where the empty boxes indicate the superconductive islands and the lines represent the connection between those operated by the Josephson junctions (crosses on the lines). Below the star array we have on the chip a “Reference Array” (which we will call RA) and one single junction which is used for checking the properties of the tunnel barrier. The arrows indicate the points in which the external bias current can be fed and the voltage is read through a four contact probe scheme. The result of our fabrication process is reported in Fig. 1(b) where we can see the central island and 8 rays originating from it; the location of junctions and superconductive islands is also indicated. We measure only one array of the star graph, namely the one parallel to the reference array and formed by two aligned rays: this array
Fig. 2. (a) Current-voltage characteristic of SA (red in colour and indicated by the arrow) and RA (black in colour). The two current-voltage characteristics are superimposed. (b) Zoom of the gap-sum voltage region where we see the increase of the currents of the SA. In (c) we plot the difference between the SA currents and those of the RA. The variable n, multiplied by 5 mV gives the idea of the distance from the gap and therefore n = 0 is the difference at the gap. We see in particular that for n below 5 the difference is positive meaning that the SA currents become higher than the RA currents (this is clearly visible even in (b)). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
is indicated by the arrow on the star graph of Fig. 1(a). This array, which is indeed generated by the series connection of two rays of the star, shall be called “Star Array” (SA). Since every ray of the star had 300 junctions the SA that we measure has 600 junctions in series. All the measurements herein presented were performed at 4.2 K. The RA also had 600 junctions in series. In Fig. 2(a) we show the current-voltage characteristics of the RA and of the SA of the chip SIS428. We see in the figure that the characteristics of the two arrays coincide at the gap voltage. The arrays were both formed by 600 junctions and we see that the Josephson current of the reference array (black in colour) are roughly 2 μA higher than the currents of the star graph (red in colour). However, making a zoom of the gap region we see that, close to the gap-sum voltage, the height of the SA currents
M. Lorenzo et al. / Physics Letters A 378 (2014) 655–658
657
Fig. 4. Dependence of the amplitude of the Josephson current as a function of the distance from the gap-sum compared with the theoretical expected behaviour which predicts the increase of bosons as we get close to the central island. The parameter p = 2.09 was obtained for the sample SIS428 while p = 2.24 is relative to sample SIS427 The curve for p = 8 indicates the theoretically expected behaviour.
Fig. 3. The features reported in Fig. 2 for another sample of a different fabrication batch.
increases and therefore the difference between RA and SA currents decreases. Indeed, a more enlarged zoom would show that the currents of the SA become even higher than those of the RA. This effect is shown in Fig. 2(c) where we have plotted the difference between the current of the SA junctions and those of the RA. On the horizontal axis the integer number n indicates essentially the distance of the measured critical current of the junctions of the array from the gap measured in units of 5 mV. In particular, n = 0 corresponds to the difference measured at the gap and we see that when we get closer to the gap sum voltage the current difference between the currents of SA and RA decreases. Very close to the gap the SA currents become even higher than the RA currents as we can see from the fact that their difference turns positive. Similar results, shown in Fig. 3, were obtained on the sample SIS427. Here we just report the zoom of the characteristics at the gap (in Fig. 3(a)) and a dependence of the difference between the SA currents and the RA currents. Even in this case we see that when voltages close to the gap are attained the difference between the two currents tends to zero. It is worth noting that the voltage “unit” for the integer variable n is 5 mV because it corresponds to the sum of two voltages: due to the symmetry we observe, as expected, that couple of junctions, placed on the two rays of the SA at the same distance from the central island have the same Josephson current. We notice that, both in Fig. 2 and in Fig. 3, the switching currents of the RA are roughly 2 μA above those of the SA currents. Since the areas of the junctions are all defined by electron beam lithography, the differences between the junctions of SA and RA (which are nominally identical) are defined by the writing error of the electron beam, which is of the order of 1%. Moreover, the measurements of the areas of the junctions after the fabrication process confirm that this estimate of the error was correct. It is not unreasonable then to postulate a phenomenon of migration of the Cooper pairs toward the central superconducting island [10]
which would generate what we record: a decrease of the Josephson currents in all the junctions of the SA but an increase of the currents of the junctions between the islands in proximity of the central one. We also wish to point out that for the comb array the situation is opposite as far as the current of reference array and backbone array is concerned [12,13]. In particular, in this case the reference array currents are always lower than the backbone array, as one would expect from the theory predicting a uniform increase of bosons along the backbone islands. Our conclusion is that geometrical effects due to uncertainty in junction area definition can be ruled out as origin of the phenomena that we observe. The theoretical model proposed by Burioni et al. [10] indicates that, for a star graph shaped an increased number of bosons should accumulate on the central island. When we bias in series one array of the graph as indicated in Fig. 1(a) by the arrows we cannot identify the current of each particular junction because we read the voltage at the ends of the array. However, we can realize if there are junctions with a higher Josephson current because these will switch to the voltage state (from the top of the Josephson current) for higher bias current values. In Fig. 2 one can see that few junctions of the SA have currents larger than all the others and larger than the reference array currents. Since the only “singular point” of the SA is the point where the “rays” join the central island, we believe that the junctions with larger currents are those connected, or very close, to the central island. Figs. 2 and 3 provide an indication of the anomalous behaviour of arrays embedded in the star structure. Burioni et al. [10] calculated a specific functional form for the dependence of the number of carriers upon the position of the islands on the branches and therefore we have attempted to compare their theoretical predictions with our data. In Fig. 4 we show a comparison of our data with the theory. The number of bosons on the islands of a ray of the star is characterized by the function Ψ (n) =
p −2 2p −2
exp(−n/ξ )
where ξ = 2/ log( p − 1) and the discrete variables p and n indicate respectively the number of branches (8 in our case) and the position of the island (n = 0 represents the central island of the star). What we measure are currents and if we estimate the current as proportional to the spatial gradient of the pairs number we should also get an exponential decay. As we see in Fig. 4 the experimental data follow the exponential decay predicted by the theory, the lowest curve in the plot, obtained for p = 8, but the coefficient in the argument of the exponential function is slightly different, as if the star has a smaller number of rays. This result could be attributed to the theoretical assumption which neglects
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M. Lorenzo et al. / Physics Letters A 378 (2014) 655–658
Fig. 5. The exponential growth of the Josephson currents along the finger array of a comb-lattice graph. The data are extracted from Ref. [12]. In this case the growth is due to an increase of boson population along the backbone.
the interactions between bosons of the islands which limits the condensation of bosons on the central island: more experiments performed on stars having different numbers of rays will help us in the future to clarify this aspect. It is worth noting that a definite consistency can be deduced when comparing the present data with those of the comb arrays. In particular, it has been observed that for the comb array the distribution of Josephson currents along the junctions of the backbone array is rather uniform [13]. However, when the “finger” arrays are concerned, the theory predicts a non uniform distribution of bosons (Cooper pairs) on their superconductive islands [9]. Thus, even on the fingers of the comb arrays we should observe a phenomenon similar to our star array branches, namely an increase of the currents close to the gap-sum voltage. Even for the comb arrays, when biasing one finger, few junctions close to the backbone islands have larger current than all the others because of the increased number of bosons on the backbone islands and even in this case the dependence of the population upon the position is exponential. Indeed, this phenomenon has been observed on experiments performed on comb arrays [12,13]. From the data of Fig. 3 of Ref. [12] we have extrapolated the plot reported in Fig. 5 showing the dependence of the difference in current height between the “finger” array Josephson currents and reference array as a function of the distance from the gap voltage. The line fitting the data corresponds to what one can expect from the theory which predicts an exponential decay of the boson population as we move, along the fingers, away from the backbone [9]. We must point out that in order for the phenomena investigated by Burioni et al. [9,10] to take place in our junctions the Josephson energy should be comparable with the thermal energy. In our measurements the zero bias Josephson energy (of the order of 10−20 J) is two orders of magnitude above k B T at 9 K (the transition temperature of niobium below which we can have bosons on
our superconductive islands). However, as it was recently pointed out [13], when we bias the junctions close to the maximum of the supercurrent, the height of the Josephson washboard potential can be substantially lower than the Boltzmann energy. Since our probing technique (measurement of current voltage characteristics obtained sweeping a current bias through the junctions) implies that the junctions switch to the voltage state from the top of the Josephson where the washboard potential [1,13] height is very low, it is likely that this probing allows us to distinguish the variation of Cooper pairs populations on the islands. These gradients can be there independently upon our specific probing technique. In conclusion, we have reported on the observation of dishomogeneities of the Josephson currents in specific arrays of star-shaped networks of Josephson junctions. These effects have been predicted on the basis of theoretical calculations which indicate that a “topological” Bose–Einstein Condensation might take place in graph-shaped arrays of Josephson junctions. Although the theoretical model does not take into account the interactions between Cooper pairs excitations on the superconductive islands, we remark that none of the theoretical predictions results in conflict with the experiments. Acknowledgements Helpful discussions with Raffaella Burioni, Davide Cassi and Francesco Fidaleo are gratefully acknowledged. Our interest toward this topic was stimulated by our colleagues Mario Rasetti and Pasquale Sodano. References [1] A. Barone, G. Paternò, Physics and Applications of the Josephson Effect, J. Wiley, NY, 1982. [2] R.L. Kautz, Rep. Prog. Phys. 59 (1996) 935; M. Cirillo, J. Appl. Phys. 60 (1986) 338; A.R. Bishop, J.A. Krumhansl, S.E. Trullinger, Physica 1D (1980) 1. [3] P.W. Anderson, in: E.R. Caianiello (Ed.), Lectures on the Many Body Problem, vol. 2, Academic, New York, 1964, p. 132. [4] V.P. Koshelets, A.V. Shchukin, S.V. Shitov, L.V. Filippenko, IEEE Trans. Appl. Supercond. 3 (1993) 2524; I. Ottaviani, M. Cirillo, M. Lucci, V. Merlo, M. Salvato, M.G. Castellano, G. Torrioli, F. Mueller, T. Weimann, Phys. Rev. B 80 (2009) 174518. [5] M. Cirillo, B.H. Larsen, A.V. Ustinov, V. Merlo, V.A. Oboznov, R. Leoni, Phys. Lett. 183A (1993) 383; M. Cirillo, V. Merlo, J. Appl. Phys. 78 (1995) 5053. [6] V.K. Semenov, G.V. Danilov, D.V. Averin, IEEE Trans. Appl. Supercond. 13 (2003) 938. [7] F. Mueller, et al., IEEE Trans. Appl. Supercond. 19 (2009) 981. [8] R. Burioni, D. Cassi, Phys. Rev. Lett. 76 (1996) 1091. [9] R. Burioni, et al., Europhys. Lett. 52 (2000) 251. [10] R. Burioni, et al., J. Phys. B 34 (2001) 4697. [11] F. Fidaleo, et al., Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (2011) 149. [12] P. Silvestrini, et al., Phys. Lett. A 370 (2007) 499. [13] I. Ottaviani, et al., Characterization of anomalous pair currents in Josephson junction networks, arXiv:1309.2836 [cond-mat]. [14] M. Gurvitch, M.A. Washington, H.A. Huggins, Appl. Phys. Lett. 55 (1984) 1419.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
On Bose–Einstein condensation in Josephson junctions star graph arrays M. Lorenzo a , M. Lucci a , V. Merlo a , I. Ottaviani a , M. Salvato a,1 , M. Cirillo a,∗,1 , F. Müller b , T. Weimann b , M.G. Castellano c , F. Chiarello c , G. Torrioli c a b c
Dipartimento di Fisica and MINAS Lab, Università di Roma “Tor Vergata”, 00133 Roma, Italy Physikalisch Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany IFN-CNR, Via Cineto Romano, I-00156 Roma, Italy
a r t i c l e
i n f o
Article history: Received 9 October 2013 Received in revised form 19 November 2013 Accepted 17 December 2013 Available online 27 December 2013 Communicated by A.R. Bishop
a b s t r a c t We report on the evidence of anomalous currents in graph-shaped arrays of Josephson junctions along peculiar branches of the networks. The specific case of a star-shaped array is considered and the evidence of the anomalies is achieved by comparing the current-voltage characteristics of the arrays embedded in the star structure with those of “reference” arrays which are fabricated by-side the network structure and are dc-isolated from these. The experimental data are consistent with the results of a theoretical model predicting gradients of the populations of Cooper pairs on the islands situated in proximity of the central superconductive island as a result of a Bose–Einstein condensation process. © 2013 Elsevier B.V. All rights reserved.
Arrays of Josephson junctions have puzzled the fundamental and applied superconducting community in the past two decades. From the fundamental point of view the interest has been triggered by the intriguing properties of the Josephson junctions as nonlinear oscillators [1,2]. The existence of a solid state device whose Hamiltonian is analogous to that of the classical pendulum [3] is a feature which has stimulated the interest of several theoretical and experimental groups. The applications instead have been motivated by the need of radiation sources for the millimeter and submillimeter region of the electromagnetic spectrum [4], but linear arrays of junctions [5] are also considered at present as components for digital electronics [6]. Moreover, it is established today that the maintenance and setting of international voltage calibrations rely on the Josephson junctions series arrays [7]. The development of quantum statistical properties of discrete systems [8] led a group of theorists to formulate a model for graph-shaped networks composed by superconducting sites connected by Josephson junctions. Such a network could exhibit peculiar properties concerning the distribution of Cooper Pairs (bosons) on the sites generating the network itself [9]. Although the model was simplified as far as the boson interactions are concerned (only boson-hopping between the islands was considered), the results were clear in the interpretation and on the possible effects to be observed: the population of Cooper pairs on the superconductive islands of the array would result not to be uniform because bo-
* 1
Corresponding author. Tel.: +39 0672594518; fax: +39 062023507. E-mail address: [email protected] (M. Cirillo). Also CNR-SPIN, Italy.
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.12.032
son hopping would generate density gradients due to the peculiar adjacency matrix of the network. As a consequence of this phenomenon the Josephson current existing between islands located in different areas of the networks could also have gradients, due to the different numbers of charge carriers on the islands connected by the junctions. Recently a mathematical theory by Fidaleo et al. [11] investigating the Bose–Einstein Condensation phenomenon in graph arrays has shown that in these networks bosons condensate in both momentum and configuration space. In previous papers experimental results showing that measurements on a “comb” graph do evidence anomalies in the distribution of the Josephson currents in the graph arrays have been reported [12,13]. On a comb array the theory predicts a uniform distribution of bosons on the superconductive islands located along the “backbone” line of the comb while an exponentially decaying population of bosons is predicted along the finger lines [9,13]. In the present paper we describe measurements on star-shaped graphs which were performed on arrays having very limited scatter of the junction parameters. As we said above a local increase (resp. decrease) of Cooper pairs population on the superconductive islands could generate a local increase (resp. decrease) of the Josephson currents between these islands. Our experimental technique consists in probing the maximum Josephson currents of all the junction of an array constituting a specific branch of the graph. Technically, we are just measuring the current-voltage characteristics (IV curves) of the series array; from these IV we realize whether the Josephson currents of the junctions of the array are uniform or not. A consequence of the series current biasing is that the junction having the lowest critical currents will switch first to the voltage state (the gap voltage when
656
M. Lorenzo et al. / Physics Letters A 378 (2014) 655–658
Fig. 1. (a) A scheme of a star-shaped array with 8 rays; (b) optical image of one of the samples that we have fabricated where is indicated the location of the junctions and of the different islands. The arrows at the end of the arrays indicate the points were we feed the current and read the voltage: in the star we measure the series connection of two “rays” of the star.
no external magnetic fields are applied) when current-biasing the array; after the switching, only the voltage of this junction will appear on the IV of the series array. Increasing further the bias current through the array the junction having a slightly higher Josephson current will switch to the voltage state and at this point the sum of two gap voltages shall be visible in the IV curve (and so forth for increasing bias current). Thus, the junction having the highest critical currents will switch to voltage state for the highest values of the bias currents and its voltage, resulting as the sum of all the others, shall be located at the gap-sum voltage of all the junctions of the array. In principle, we have no information on where in the array the junctions with higher (or lower) critical currents are located; however, we can clearly distinguish a situation of non uniform Josephson current distribution. We did not put extra probing contacts (which would allow us to test specific junctions) on the arrays because we preferred not to perturb the system and to keep the design as close as possible to the theoretical model. Our samples were arrays of Josephson junctions fabricated in the niobium trilayer technology [14]; a peculiarity of our fabrication process was the fact that, in order to minimize the scatter of the geometrical parameters of the junctions, their area was defined by electron beam lithography. A typical scheme of a star graph chip is shown in Fig. 1(a) where the empty boxes indicate the superconductive islands and the lines represent the connection between those operated by the Josephson junctions (crosses on the lines). Below the star array we have on the chip a “Reference Array” (which we will call RA) and one single junction which is used for checking the properties of the tunnel barrier. The arrows indicate the points in which the external bias current can be fed and the voltage is read through a four contact probe scheme. The result of our fabrication process is reported in Fig. 1(b) where we can see the central island and 8 rays originating from it; the location of junctions and superconductive islands is also indicated. We measure only one array of the star graph, namely the one parallel to the reference array and formed by two aligned rays: this array
Fig. 2. (a) Current-voltage characteristic of SA (red in colour and indicated by the arrow) and RA (black in colour). The two current-voltage characteristics are superimposed. (b) Zoom of the gap-sum voltage region where we see the increase of the currents of the SA. In (c) we plot the difference between the SA currents and those of the RA. The variable n, multiplied by 5 mV gives the idea of the distance from the gap and therefore n = 0 is the difference at the gap. We see in particular that for n below 5 the difference is positive meaning that the SA currents become higher than the RA currents (this is clearly visible even in (b)). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
is indicated by the arrow on the star graph of Fig. 1(a). This array, which is indeed generated by the series connection of two rays of the star, shall be called “Star Array” (SA). Since every ray of the star had 300 junctions the SA that we measure has 600 junctions in series. All the measurements herein presented were performed at 4.2 K. The RA also had 600 junctions in series. In Fig. 2(a) we show the current-voltage characteristics of the RA and of the SA of the chip SIS428. We see in the figure that the characteristics of the two arrays coincide at the gap voltage. The arrays were both formed by 600 junctions and we see that the Josephson current of the reference array (black in colour) are roughly 2 μA higher than the currents of the star graph (red in colour). However, making a zoom of the gap region we see that, close to the gap-sum voltage, the height of the SA currents
M. Lorenzo et al. / Physics Letters A 378 (2014) 655–658
657
Fig. 4. Dependence of the amplitude of the Josephson current as a function of the distance from the gap-sum compared with the theoretical expected behaviour which predicts the increase of bosons as we get close to the central island. The parameter p = 2.09 was obtained for the sample SIS428 while p = 2.24 is relative to sample SIS427 The curve for p = 8 indicates the theoretically expected behaviour.
Fig. 3. The features reported in Fig. 2 for another sample of a different fabrication batch.
increases and therefore the difference between RA and SA currents decreases. Indeed, a more enlarged zoom would show that the currents of the SA become even higher than those of the RA. This effect is shown in Fig. 2(c) where we have plotted the difference between the current of the SA junctions and those of the RA. On the horizontal axis the integer number n indicates essentially the distance of the measured critical current of the junctions of the array from the gap measured in units of 5 mV. In particular, n = 0 corresponds to the difference measured at the gap and we see that when we get closer to the gap sum voltage the current difference between the currents of SA and RA decreases. Very close to the gap the SA currents become even higher than the RA currents as we can see from the fact that their difference turns positive. Similar results, shown in Fig. 3, were obtained on the sample SIS427. Here we just report the zoom of the characteristics at the gap (in Fig. 3(a)) and a dependence of the difference between the SA currents and the RA currents. Even in this case we see that when voltages close to the gap are attained the difference between the two currents tends to zero. It is worth noting that the voltage “unit” for the integer variable n is 5 mV because it corresponds to the sum of two voltages: due to the symmetry we observe, as expected, that couple of junctions, placed on the two rays of the SA at the same distance from the central island have the same Josephson current. We notice that, both in Fig. 2 and in Fig. 3, the switching currents of the RA are roughly 2 μA above those of the SA currents. Since the areas of the junctions are all defined by electron beam lithography, the differences between the junctions of SA and RA (which are nominally identical) are defined by the writing error of the electron beam, which is of the order of 1%. Moreover, the measurements of the areas of the junctions after the fabrication process confirm that this estimate of the error was correct. It is not unreasonable then to postulate a phenomenon of migration of the Cooper pairs toward the central superconducting island [10]
which would generate what we record: a decrease of the Josephson currents in all the junctions of the SA but an increase of the currents of the junctions between the islands in proximity of the central one. We also wish to point out that for the comb array the situation is opposite as far as the current of reference array and backbone array is concerned [12,13]. In particular, in this case the reference array currents are always lower than the backbone array, as one would expect from the theory predicting a uniform increase of bosons along the backbone islands. Our conclusion is that geometrical effects due to uncertainty in junction area definition can be ruled out as origin of the phenomena that we observe. The theoretical model proposed by Burioni et al. [10] indicates that, for a star graph shaped an increased number of bosons should accumulate on the central island. When we bias in series one array of the graph as indicated in Fig. 1(a) by the arrows we cannot identify the current of each particular junction because we read the voltage at the ends of the array. However, we can realize if there are junctions with a higher Josephson current because these will switch to the voltage state (from the top of the Josephson current) for higher bias current values. In Fig. 2 one can see that few junctions of the SA have currents larger than all the others and larger than the reference array currents. Since the only “singular point” of the SA is the point where the “rays” join the central island, we believe that the junctions with larger currents are those connected, or very close, to the central island. Figs. 2 and 3 provide an indication of the anomalous behaviour of arrays embedded in the star structure. Burioni et al. [10] calculated a specific functional form for the dependence of the number of carriers upon the position of the islands on the branches and therefore we have attempted to compare their theoretical predictions with our data. In Fig. 4 we show a comparison of our data with the theory. The number of bosons on the islands of a ray of the star is characterized by the function Ψ (n) =
p −2 2p −2
exp(−n/ξ )
where ξ = 2/ log( p − 1) and the discrete variables p and n indicate respectively the number of branches (8 in our case) and the position of the island (n = 0 represents the central island of the star). What we measure are currents and if we estimate the current as proportional to the spatial gradient of the pairs number we should also get an exponential decay. As we see in Fig. 4 the experimental data follow the exponential decay predicted by the theory, the lowest curve in the plot, obtained for p = 8, but the coefficient in the argument of the exponential function is slightly different, as if the star has a smaller number of rays. This result could be attributed to the theoretical assumption which neglects
658
M. Lorenzo et al. / Physics Letters A 378 (2014) 655–658
Fig. 5. The exponential growth of the Josephson currents along the finger array of a comb-lattice graph. The data are extracted from Ref. [12]. In this case the growth is due to an increase of boson population along the backbone.
the interactions between bosons of the islands which limits the condensation of bosons on the central island: more experiments performed on stars having different numbers of rays will help us in the future to clarify this aspect. It is worth noting that a definite consistency can be deduced when comparing the present data with those of the comb arrays. In particular, it has been observed that for the comb array the distribution of Josephson currents along the junctions of the backbone array is rather uniform [13]. However, when the “finger” arrays are concerned, the theory predicts a non uniform distribution of bosons (Cooper pairs) on their superconductive islands [9]. Thus, even on the fingers of the comb arrays we should observe a phenomenon similar to our star array branches, namely an increase of the currents close to the gap-sum voltage. Even for the comb arrays, when biasing one finger, few junctions close to the backbone islands have larger current than all the others because of the increased number of bosons on the backbone islands and even in this case the dependence of the population upon the position is exponential. Indeed, this phenomenon has been observed on experiments performed on comb arrays [12,13]. From the data of Fig. 3 of Ref. [12] we have extrapolated the plot reported in Fig. 5 showing the dependence of the difference in current height between the “finger” array Josephson currents and reference array as a function of the distance from the gap voltage. The line fitting the data corresponds to what one can expect from the theory which predicts an exponential decay of the boson population as we move, along the fingers, away from the backbone [9]. We must point out that in order for the phenomena investigated by Burioni et al. [9,10] to take place in our junctions the Josephson energy should be comparable with the thermal energy. In our measurements the zero bias Josephson energy (of the order of 10−20 J) is two orders of magnitude above k B T at 9 K (the transition temperature of niobium below which we can have bosons on
our superconductive islands). However, as it was recently pointed out [13], when we bias the junctions close to the maximum of the supercurrent, the height of the Josephson washboard potential can be substantially lower than the Boltzmann energy. Since our probing technique (measurement of current voltage characteristics obtained sweeping a current bias through the junctions) implies that the junctions switch to the voltage state from the top of the Josephson where the washboard potential [1,13] height is very low, it is likely that this probing allows us to distinguish the variation of Cooper pairs populations on the islands. These gradients can be there independently upon our specific probing technique. In conclusion, we have reported on the observation of dishomogeneities of the Josephson currents in specific arrays of star-shaped networks of Josephson junctions. These effects have been predicted on the basis of theoretical calculations which indicate that a “topological” Bose–Einstein Condensation might take place in graph-shaped arrays of Josephson junctions. Although the theoretical model does not take into account the interactions between Cooper pairs excitations on the superconductive islands, we remark that none of the theoretical predictions results in conflict with the experiments. Acknowledgements Helpful discussions with Raffaella Burioni, Davide Cassi and Francesco Fidaleo are gratefully acknowledged. Our interest toward this topic was stimulated by our colleagues Mario Rasetti and Pasquale Sodano. References [1] A. Barone, G. Paternò, Physics and Applications of the Josephson Effect, J. Wiley, NY, 1982. [2] R.L. Kautz, Rep. Prog. Phys. 59 (1996) 935; M. Cirillo, J. Appl. Phys. 60 (1986) 338; A.R. Bishop, J.A. Krumhansl, S.E. Trullinger, Physica 1D (1980) 1. [3] P.W. Anderson, in: E.R. Caianiello (Ed.), Lectures on the Many Body Problem, vol. 2, Academic, New York, 1964, p. 132. [4] V.P. Koshelets, A.V. Shchukin, S.V. Shitov, L.V. Filippenko, IEEE Trans. Appl. Supercond. 3 (1993) 2524; I. Ottaviani, M. Cirillo, M. Lucci, V. Merlo, M. Salvato, M.G. Castellano, G. Torrioli, F. Mueller, T. Weimann, Phys. Rev. B 80 (2009) 174518. [5] M. Cirillo, B.H. Larsen, A.V. Ustinov, V. Merlo, V.A. Oboznov, R. Leoni, Phys. Lett. 183A (1993) 383; M. Cirillo, V. Merlo, J. Appl. Phys. 78 (1995) 5053. [6] V.K. Semenov, G.V. Danilov, D.V. Averin, IEEE Trans. Appl. Supercond. 13 (2003) 938. [7] F. Mueller, et al., IEEE Trans. Appl. Supercond. 19 (2009) 981. [8] R. Burioni, D. Cassi, Phys. Rev. Lett. 76 (1996) 1091. [9] R. Burioni, et al., Europhys. Lett. 52 (2000) 251. [10] R. Burioni, et al., J. Phys. B 34 (2001) 4697. [11] F. Fidaleo, et al., Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (2011) 149. [12] P. Silvestrini, et al., Phys. Lett. A 370 (2007) 499. [13] I. Ottaviani, et al., Characterization of anomalous pair currents in Josephson junction networks, arXiv:1309.2836 [cond-mat]. [14] M. Gurvitch, M.A. Washington, H.A. Huggins, Appl. Phys. Lett. 55 (1984) 1419.