Current-driven effects in disordered arrays of Josephson junctions

ELSEVIER

Physica B 222 (1996) 320-325

Current-driven effects in disordered arrays of Josephson junctions P.L. Leath Department of Physics and Astronomy, Rutgers University, Piseataway, NJ 08855-0849, USA

In this talk, I shall review the work of our group on disordered (diluted) arrays of Josephson junctions [1-3] and the related works of other researchers. Most of our recent studies have been on the statistics of the fracture and breakdown of heterogeneous materials and thus we approached the study of Josephson junction arrays (JJA) as a unique model for the effects of nonlinearity at breakdown in such materials. We consider n x m square arrays of JJAs which are diluted by the random removal of a fraction 1 - p of the junctions in the array. A typical 25 x 25 array, with p = 0.90 is shown in Fig. 1, where the external current is put in across the top row and removed across the bottom row. Each junction is considered to be a resistively shunted Josephson junction, such that the current through the junction connecting sites 0 and 1 is given by h lol = ~

(~1 - - ~ 0 ) + J sin(C1 - q~o),

(1)

where 4~, is the superconducting phase at site n, ¢, is the time derivative of this phase, J is its Josephson current (its maximum supercurrent), and R is its normal resistance. The first term corresponds to the normal current and is R 1 times the voltage across the junction. Imposing current conservation at each node (site) of the array we obtain the array equations i, = Z (~, - q~,,) + • (~b, - q~,,), m

(2)

m

where the applied external current i at each site is measured in units of J and where time is measured

Fig. 1. A typical configuration of random missing junctions in a 25 x 25 array at p = 0.90.

in units of h/(2eJR). We then follow the classic numerical techniques by Shenoy [4], Mon and Yeitel [5], and Chung et al. [6] to solve this set of equations for the time-dependent ¢,(t) at each site of the array. For a perfect array (no defects) 4,(t) is a constant up to the critical current at i = 1, at which point the entire array begins to pulse in unison as a giant single junction with the Josephson oscillations first given long ago for a single junction by Aslamazov and Larkin [7], namely the

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P.L. Leath / Physica B 222 (1996) 320-325 I11 FII FII

,Q •

••,Q

Fig. 2. A perfect 35 x 34 array with a single defect cluster of 11 adjacent missing junctions. The x ' s mark those junctions that carry a voltage and are transversed by the moving vortices for /just above ic.

voltage (Ad) across the sample is given by i cos 2 6

v(t)=l + s i n 6 s i n ( t c o s 6 ) '

fori>l,

(3)

where cos 6 = x/1 - i-2, so that the average voltage ( v ) is zero for i ~< 1 and x/i 2 - 1 for i > 1, in these dimensionless units.

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Next we consider a perfect array with a single cluster of n missing adjacent vertical bonds in a row (see Fig. 2). This cluster represents one of the worst kinds of defects for lowering the critical current. For this kind of sample, we find the ( v ) versus i curve typified by Fig. 3(a), with a series of transitions. Just above the first transition at io, where a voltage first appears, the supercurrent distribution versus time is illustrated, for n -- 10, by the series of time snapshots shown in Fig. 4. The edges of the defect, around which the supercurrent is deflected, become the source of vortices (and antivortices) much as the tips of an airplane wing generate wind vortices. The vortices (and antivortices) progressively depin and move across the central row of the sample, perpendicular to the applied current, toward the edges, where they annihilate with the antivortices (vortices) on the other side (due to the periodic boundary condition chosen here). It is the flow of these vortices which produces the q~ and hence the voltage drop across the central row. At the higher critical current it3, vortices are created not only in the central row but also in the two adjacent rows, so that the three central rows now have a voltage. The vortices in the three rows can be seen in Fig. 5, where they interact to produce a complicated time dependence in the voltage The dependence of the critical current ic on n, the size of the defect cluster, is expected to be

\ •\

o

35X16

O

.3

(a) 0.4 0.5

0.6

0.7

o.8 0.'9

(b) 10°

10 t

10 =

(A/a) Fig. 3. (a) Voltage current curve for a perfect array with a single defect cluster of 9 missing junctions. (b) Critical current ic/io (data points) versus A/a, the number of adjacent missing junctions. The solid line has a slope of - 12.

P.L. Leath / Physica B 222 (1996) 320 325

322

(a )

I. =880

(b)

1" " 7 ~ 0

(C)

"r =7~o

/

t/

/

J

(d)

r ==~o

J

(e)

1- =870

(f)

T =ago

J

Fig. 4. (a) (f) Snapshots of vortices in a 35 × 34 array, with a single defect cluster of 11 missing junctions at i > %, at different times indicated by z. Pictures (f) and (a) are the same and the process repeats periodically. The vertical axis is the magnitude of the supercurrent in the horizontal junctions versus position in the array. The external current i is applied vertically as indicated by the arrows.

4b T m 2970

Fig. 5. Snapshots of vortex movement (in the same array and plots as in Fig. 4) for i > %, when vortices appear in the central row and in the two adjacent rows.

P.L. Leath / Physica B 222 (1996) 320-325

Fig. 6. A perfect 34 x 51 array with a funnel defect.

proportional to n-1/2 as is discussed in Ref. [1], and this is seen numerically [Fig. 3(b)]. Of course, for the square lattice Rzchowski et al. [8] have shown that vortices are pinned by an infinite perfect lattice array at a critical current of/pin 0.1. From Fig. 3(b), one can see that ic will approach O.1 for n ~ 200, so that this n- 1/2 decrease crosses over to the constant pinning current of the underlying lattice (which is zero for a continuous medium, and will be very small for real samples). Datta et al. [9] have recently managed to see this limit numerically by going to thin rectangular 16 × 256 arrays where the supercurrent is squeezed by the top and bottom boundaries of the sample such that the supercurrent enhancement at the ends of the defect are proportional to n, hence i~ oc n- 1, so that i~ reaches 0.1 at about n = 74. Next, we consider the effect of current focussing by considering the funnel (or bow-tie) defect shown in Fig. 6. Here we naively expected that, due to the funnel, the vortices would depin in the center of the sample before they do from the edges of the sample (outside the funnel). What we found is that the current at the center points do indeed reach the local critical current first, but then are fixed, while the external current is increased further, such that all the excess current is shunted to the other edges until supercurrent on the outer corners of the bowtie defect is also at the local critical current and the vortices on all the corners depin at once, creating a voltage and a normal path across the sample. Clearly, in disordered samples, there is also this =

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kind of self-organized criticality, where one region after another becomes critical and excludes further current increase until there is a critical path (or crack) across the sample (analogous to the critical crack in a mechanical fracture) at which point vortices begin to flow along this breakdown line. This self-organized criticality along with the presence of vortices themselves are signal properties of such nonlinear systems. Next, we have considered randomly diluted JJAs, like that shown in Fig. 1. Here, we generate a randomly diluted sample and measure its critical current. When the vortices first depin from the defects, we see them traveling along the critical breakdown path like that shown in Fig. 7. The plots of voltage q~(t) versus t are very noisy and irregular. We have, in particular, been interested in the statistics of the critical current versus concentration of missing junctions p, and sample size L × L. So numerical results were averaged over an ensemble of randomly diluted samples for each size L x L. The behavior of the average critical current versus L can be understood qualitatively by a Lifshitz argument as follows. Assume the critical

Fig. 7. Vortex or breakdown path for i > ic in the 25 x 25 array of Fig. 1. The square in the third column from the right marks the initial point of vortex motion which moves in an irregular m a n n e r along this path.

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P.L. Leath /Physica B 222 (1996) 320-325

breakdown is nucleated by the worst defect in the sample, which is assumed to be the longest horizontal linear cluster of adjacent missing bonds in the sample. (This is a very good approximation near p ~ 1.) The probability of a linear defect of size n in a L × L sample is of the order of L2(1 - p)". To find the order of magnitude of the longest defect r/max in the sample, we set L 2 ( 1 - p ) .... ~ 1 to obtain r/ma x ~ --2 In L/ln(1 -- p). But i~ ~ n -1/2 in two dimensions, so that the critical current i~ ,-~ [ln(1 - p)/ln L ] 1/2,

(4)

goes to zero logarithmically in the thermodynamic limit. Such logarithmic behavior is seen numerically in mechanical fracture and is consistent with the data we found in this JJA case. An easier result to check is the distribution of critical currents F(i¢), where we expected and found a modified-Gumbel (double-exponential) distribution of critical currents [1]. A complete discussion of these results and the critical current distribution is found in Ref. [-1]. We also studied the behavior of the average sample voltage ( v ) versus applied current in an ensemble of 25 × 25 samples and found ( v ) ~ (i - i~)", where = 3.10 _+ 0.10 [1].

t=

28090

8250

More recently, we have numerically simulated [3] the below-gap photoresponse [10] of disordered thin films of high-To superconductors by an ensemble of disordered JJAs at finite temperatures. The temperature was simulated by the addition of a random, white-noise current in each junction 1-11, 12]. As a result we were able to see the creation of vortex-antivortex pairs near the Kosterlitz-Thouless transition Tk. As is shown in the snapshots of Fig. 8, the random temperature fluctuations (or hot spots) act like temporary defects, around which the supercurrent must flow so that vortices and antivortices are created on each end of the fluctuations. In addition, the below-gap photoresponse was simulated by the addition of an applied AC-current and the average voltage across the sample was measured. The results of the photoresponse versus temperature T, frequency f, and bias current i are in general qualitative agree with the behavior seen in the real thin films of high-To superconductors 1-10], Detailed results are given in Ref. [2]. Finally, very recently, we have studied the fluctuations that occur on the risers of the Shapiro steps in perfect JJAs at finite temperature [3]. It is well

4p' t=

28130

8260

t =

t =

28170

28270

Fig. 8. Snapshots of the creation and annihilation of a vortex-antivortex pair in a perfect 16 x 16 array at T = 0.25, id¢ = 0.70, iac = 0.25, and f = 0.2. The time t (in Monte Carlo steps) is indicated on each picture. Plots of the magnitude of the supercurrent in the horizontal junctions.

P.L. Leath / Physica B 222 (1996) 320-325

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where x = 0.46 + 0.05 and nv is the average number of vortices per plaquette seen in snapshots taken at different times. Clearly this is consistent with a square-root singularity. We also have considered the noise in the voltage near the steps by studying the power spectrum. We found a hint of 1 / f noise over about a decade of frequencies. A more complete discussion can be found in Ref. [3]. T=O.O01

References i

I

06

0.8

I

I

!

1.2

iDC

Fig. 9. The number of vortex antivortex (xl00) per lattice plaquette (averaged over 40000 snapshots taken at different times) versus idc for iac = 0.5, f = 0.2, and T = 0.001 in a 16 x 16 array.

known that as the temperature rises the Shapiro steps corners become rounded due to thermal fluctuations. We have used our simulations to observe a dramatic increase (apparent singularity) in the number of vortices (and antivortices) on the risers of these steps as the sample is making the transition from one harmonic to the next in going up to the next step, as is shown in Fig. 9. The apparent singularities at the first nv step at il, is seen to be of the form nv ~ a + b(i - i t ) - x ,

(5)

[1] W. Xia and P.L. Leath, Phys. Rev. Lett. 63 (1989) 1428; P.L. Leath and W. Xia, Phys. Rev. B 44 (1991) 9619. [2] Y. Cai, P.L. Leath and Z. Yu, Phys. Rev. B 49 (1994) 4015. [3] P.L. Leath and Y. Cai, Phys. Rev. B 51 (1995) 15638. [4] S.R. Shenoy, J. Phys. C 18 (1985) 5163. [5] K.K. Mon and S. Teitel, Phys. Rev. Lett. 62 (1988) 673. [-6] J.S. Chung, K.H. Lee and D. Stroud, Phys. Rev. B 40 (1989) 6570. [7] LG. Aslamazov and A.I. Larkin, Pis'ma Zh. Eksp. Teor. Fiz. 9 (1969) 150 [JETP Lett. 9 (1969) 87]. [8] M.S. Rzchowski, S.P. Benz, M. Tinkham and C.J. Lobb, Phys. Rev. B 42 (1990) 2041. I'9] S. Datta, S. Das, D. Sahdev and R. Mehrota, unpublished. [,10] M. Leung, U. Strom, J.C. Culbertson, J.H. Claassen, S.A. Wolf and R.W. Simon, Appl. Phys. Lett. 50 (1987) 1691; U. Strom, J.C. Culbertson, S.A. Wolf, S. Perkowitz and G.L. Carr, Phys. Rev. B42 (1990) 4059. 1'11] V. Ambegaokar and B.I. Halperin, Phys. Rev. Lett. 22 (1969) 1364. [,12] A. Barone and G. Paterno, Physics and Applications of the Josephson Effect (Wiley, New York, 1982).