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JETP Letters, Vol. 72, No. 1, 2000, pp. 26­29. Translated from Pis'ma v Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, Vol. 72, No. 1, 2000, pp. 39­44. Original Russian Text Copyright © 2000 by Ishikaev, Matizen, Ryazanov, Oboznov, Veretennikov.

Magnetic Properties of Two-Dimensional Josephson Networks: Self-Organized Criticality in Magnetic Flux Dynamics
S. M. Ishikaev*, è. V. Matizen*, V. V. Ryazanov**, V. A. Oboznov**, and A. V. Veretennikov**
* Institute of Inorganic Chemistry, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent'eva 3, Novosibirsk, 630090 Russia ** Institute of Solid-State Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432 Russia e-mail: ryazanov@issp.ac.ru
Received June 7, 2000

The field dependence of the magnetic moment of square (100 â 100) Josephson networks was examined with the use of a SQUID magnetometer. The field dependence of the magnetic moment was found to be regular with features corresponding to integer and half-integer numbers of flux quanta per cell. At temperatures below 5.8 K, jumps in the magnetization curves associated with the entry and exit of avalanches of tens and hundreds of fluxons were observed. It was shown that the probability distribution of these processes corresponded to the theory of self-organized criticality. An avalanche character of flux motion was observed at temperatures at which the size of the fluxons did not exceed the size of the cell, that is, when a discrete vortex structure occurred. © 2000 MAIK "Nauka / Interperiodica". PACS numbers: 85.25.Cp; 74.25.Ha; 74.50.+r; 74.60.Ge

Two-dimensional regular Josephson junction networks arouse intense interest because of specific features of vortex dynamics in these discrete superconducting systems [1, 2] and because of the possibilities of their practical use as sources of coherent millimetric radiation [3] and elements of logic units [4]. Even though a number of theoretical articles devoted to the magnetic properties of such networks have been published [5, 6], experimental studies of the magnetic properties of Josephson junction networks are actually lacking. We studied square 100 â 100 Josephson junction networks with a cell of size a2 = 20 â 20 µm2 (a network fragment is schematically depicted in Fig. 1). Underdamped Nb­NbOx­Pb Josephson tunnel junctions [7] had an area of 7 µm2 and the following characteristics at T = 4.2 K: critical current Ic 150 µA and normal resistance Rn 10 (see inset in Fig. 1). The measurement of magnetic dynamics even in such relatively large, two-dimensional arrays of Josephson junctions with 104 cells required that the SQUID magnetometer at our disposal had to be essentially updated. The pickup coils of the flux transformer were made in the form of a symmetric second-order gradiometer [8]. As distinct from the classical circuit, the central coil was divided into two identical separated coils [9]. This offered some preferences, providing, in particular, a significantly weaker dependence of the signal on the position of the sample. The astaticism of the pickup coils was about 3 â 10­4. An additional coil of several copper wire turns was used for fine compensation. A

current proportional to the solenoid current was passed through this coil during operation. The slope of the magnetization curves can be varied by varying the proportionality factor. This allowed the weak signal ~10­10 A m2, which is directly related to the dynamics of the magnetic field in the network, to be distinguished. Thus, the intrinsic contribution of the superconducting Nb and Pb films in the structure under study was compensated to a maximum extent in the magnetization curves presented below. To diminish drifts and interferences, the liquid

Fig. 1. Geometry of a Josephson Nb­NbOx­Pb junction network. The inset shows the voltage­current characteristic of an individual junction at T = 4.2 K.

0021-3640/00/7201-0026$20.00 © 2000 MAIK "Nauka / Interperiodica"


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helium in the volume containing the flux transformer, the solenoid, and a superconducting magnetic screen was transferred to a superfluid state by pumping the vapor out. A set of magnetization curves for the Josephson junction array measured at various temperatures is shown in Fig. 2. A regular structure is well pronounced in all the curves at temperatures below Tc of Pb (7.2 K) with a temperature-independent magnetic-field period equal to approximately 60 mOe. With allowance made for the screening of the solenoid field by the superconducting film structure, this value satisfactorily corresponds to the flux quantum per cell H = 0/a2 50 mOe, where 0 is the magnetic flux quantum and a is the network period. Small features are also apparent that correspond to a magnetic flux equal to one-half the quantum per cell. Large hysteresis loops at temperatures of 6.0 and 6.8 K were obtained by double passage. To demonstrate the reproducibility, two loops with a smaller field swing are also presented. It is evident that all these curves are perfectly superimposed on each other. The behavior of the magnetic moment of the Josephson junction network at temperatures below 5 K is of special interest. Pronounced jumps in the magnetic moment appear in the hysteresis loops. These jumps increase with decreasing temperature. The jumps with the maximum amplitude form periodic compact groups arranged in the vicinity of the magnetic flux values corresponding to an integer number of quanta per cell ("integer frustrations" f = /0 = 0, 1, 2, ...). Magnetization curves recorded in detail at T = 2.1 K in the region f = 0 upon varying the external field within the range ±25 mOe, which encloses one group of flux jumps, are shown in Fig. 3. It is evident that jumps occur at random field values, and their amplitudes have a significant scatter. Such behavior, even though it resembles thermal noise, is not of this kind, if only because the processes of origination and relaxation of unstable states are enhanced with decreasing temperature. Each of the jumps in the magnetic moment noticeable in the figure corresponds to the simultaneous entry (or exit) of avalanches of tens and hundreds of flux quanta in the network. To illustrate this fact, in addition to the main scale, a magnetic flux quantum scale is presented in the plot. Distinctive features of the flux jumps are a very short time of the transient process (much less than the characteristic response time of our recording system, 0.1 s) and their definite direction. The character of the jumps observed does not depend on the magnetic field sweep rate within three orders of magnitude 0.01-10 mOe/s. (At a rate of 10 mOe/s, the recording system already had no time to follow the jumps and all acute angles in the curves adjacent to the vertical drops became smoothed and fuzzy.) The dependence of the critical current on the magnetic field was directly measured in transport studies of 10 â 10 networks with close parameters [10]. The
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Fig. 2. Set of magnetization curves for a Josephson network measured at various temperatures.

Fig. 3. Hysteresis loop in the region of small magnetic fluxes (0­0/2 per cell) that encloses one group of avalanche breakdowns of the flux in the network.

curves obtained for temperatures above 5 K are qualitatively identical with the curves M(H) presented in Fig. 2. The latter curves are also in good agreement with the calculations [5]. These curves are manifestations of the specific critical state occurring in this system. The periodic magnetic moment peaks correspond to the sharp increase in the critical current of fluxon


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Fig. 4. Histogram of the dependence of the number of jumps on their amplitudes confirming the power law of the probability distribution of processes.

pinning (depinning current) at integer-valued frustrations, when the flux state in the network is most stable. In resistive studies [10], as well as in magnetization curves, small features are also manifested at half-integer values of f, when the flux fills the network cells in a staggered order [11]. The fluxon depinning current at half-integer frustrations can be evaluated from the halfwidths of hysteresis loops with the use of the simplified assumption that currents in the network flow along concentric square circuits: Ip(6.8 K) = 0.9 µA, Ip(6.0 K) = 2.5 µA, Ip(5.8 K) = 2.9 µA, Ip(5.5 K) = 3.3 µA, Ip(4.3 K) = 4.6 µA, and Ip(2.1 K) = 4.7 µA. It is evident that the real current lines cut off the corners of the squares; therefore, the value obtained from the simplest model is somewhat overestimated. The excess depinning current at the peaks arising at integer f over the values for intermediate f ("at the pedestal") decreases with decreasing temperature. This is associated with the enhancement of the self-field effects of currents [12, 13] that arise as the fluxon radius = 0/2µ0Ic approaches the network parameter a. At temperatures when exceeds a, each fluxon extends over several cells and the self-fields of fluxons are small. For the structure under study, (T) becomes markedly smaller than a = 20 µm at /a = 5.5 K. At this temperature, the depinning current at the peaks is nearly twice as large as its value at the pedestal and is about 0.3Ic (Ic is the critical current of an individual junction). This is in agreement with the results [12] for /a = 0.5. At low temperatures (below 4.3 K), the height of the peaks in reference to the pedestal becomes insignificant, which corresponds to the transition to a discrete system of fluxons. The fluxon avalanche breakdowns described above are a specific feature of the low-temperature dynamics of discrete fluxons. Below, we will show that these breakdowns resemble sand pile growth dynamics [14], which was the first model subject of the self-organized

criticality theory [15]. This universal theory describes the behavior of a wide range of complex interactive systems attaining a critical state in their evolutionary process. This critical state becomes subsequently selfsustaining; that is, it requires no precision adjustment of external parameters for its existence. Up to now, experimental data on self-organized criticality have been obtained for a limited set of physical objects [14, 16, 17] in spite of the great interest in this problem and a great number of theoretical works (for example, [18­ 20]), some of which are devoted to self-organized criticality in superconducting systems [21­25]. A conventional critical state in hard superconductors, in principle, can possess the properties of self-organized criticality as was shown in model [21], which takes into account the discreteness of the effects of pinning centers on the Abrikosov vortices. However, the Josephson network is undoubtedly the best candidate for a discrete version of a type II superconductor. It was shown in [22­25] that, in the case when V = a/ 1, the continuum approximation is inapplicable and the Josephson medium transforms to a system in which pinning is accomplished in elementary circuits within one network cell. Taking into account that the inductance L of a Josephson network cell equals µ0a by the order of magnitude, one may rewrite the condition V 1 as 2LIc 0, which corresponds to the ability of the cell to retain several magnetic flux quanta. In this case, provided that the size of the Josephson junction network is sufficiently large, the ensemble of fluxons in the network is a complex system of interacting elements possessing a great number of metastable states, which is equivalent to the classical Abelian sand pile model [20]. In a slowly varying magnetic field, the fluxon system attains an unstable state as the currents approach their critical value, from which it passes to one of the numerous metastable states under the action of random perturbations. The final state in which the system ends up after each of these jumps is determined by the number of metastable states in the system, their configuration, and the dynamics of collective fluxon motion in the network. Here, we note once again the fact that the jumps have the largest amplitudes predominantly in the vicinity of the field values at which the depinning current is a maximum. This allows the system to attain strongly nonequilibrium states, from which it recovers with the formation of large avalanches (Fig. 3). A histogram of the dependence of the number of jumps on their amplitudes constructed for T = 2.1 K demonstrates a power law that serves as the "trademark" of self-organized criticality. The power n = ­1.9 ± 0.1. Close values, i.e., -1.75 and 1.80, were obtained in [22] by a computer simulation of a one-dimensional Josephson network (one-dimensional multiple-contact SQUID) for structures containing 256 and 128 cells, respectively. Note in conclusion that Josephson junction networks are in principle perfect model subjects for studying self-organized criticality. The networks can be prepared with precisely specified parameters like period,
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number of cells, and critical current of junctions, whereas the magnitude of the critical current can be varied within some limits directly during the experiment by varying the temperature. In addition to studying the magnetic and resistive properties of Josephson networks, electromagnetic radiation due to magnetic flux motion can also be measured. Comprehensive quantitative measurements and a detailed statistical analysis are necessary for a reliable corroboration and further investigations of processes of self-organized criticality in these structures. This analysis should include, for example, studying the scaling properties of the fluxon system in Josephson networks upon varying their sizes. The authors are grateful to P.P. Bezverkhioe and V.G. Martynets for help in manufacturing photomasks of the networks and also to M.A. Lebedkin for useful discussions and comments. This work was supported by the State Program High-Temperature Superconductivity and by INTAS, project no. 97-1940. REFERENCES
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Translated by A. Bagatur 'yants

JETP LETTERS

Vol. 72

No. 1

2000