ELSEVIER Physica B 222 (1996) 326-330

Chaos and dynamics of vortices in Josephson junction arrays

Ravi Mehrotra National Physical Laboratory, Dr. K.S. Krishnan Rd., New Delhi 110 012, India

Abstract

The approach to spatiotemporal chaos in two-dimensional (2D) Josephson junction arrays (JJAs) subjected to DC current drives is reviewed. The role of vortices in describing the various transitions and types of dynamical behaviour that occur is explored. Some analogies with hydrodynamics are found.

1. Introduction

Vortices play an important role in normal fluid turbulence [1], superfluid and x-y turbulence [2], and spatially varying chaos in certain coupled maps [3]. Vortex-like structures have been ob- served in numerical simulations of models for dis- cretized hydrodynamics [4] and chemical reaction chaos [5]. Vortices also occur naturally in the equilibrium Berezinskii-Kosterlitz-Thouless [6] transition of a 2D Josephson-junction array. The role of vortices in turbulence is still far from being understood well. An important question is if vor- tices (or in general, topological excitations) can be considered to be the relevant collective variables whose dynamical instabilities can describe the transition to turbulence.

Josephson-junction arrays (JJA) driven by ex- ternal currents are natural systems used to investi- gate such questions because of several reasons: (i) vortices are natural thermal excitations and the dynamic response has a natural interplay between length and time scales, (ii) the nonlinearities are simple and physically determined (sinusoidal), (iii) the model is naturally discretized in space, (iv) the arrays can be realized physically in the laboratory, (v) many control parameters apart from current drives like external magnetic field, capacitive iner- tial terms, disorder, etc., add to the richness of the phase diagram.

In this paper, we describe the behaviour of a 2D JJA driven by external currents [7]. The dynamics of vortices close to the quasi-periodic to chaotic transition is emphasized. Analogies with hydrodyn- amics are also made.

2. The model

We consider an Nx × Ny square array (see inset in Fig. 1) of resistively shunted Josephson junctions (RSJ) driven by an external current drive. The choi- ces of RSJ, the form of the current drive, and the size of the array are guided by the following factors.

The junctions are noncapacitive so that single junctions are not chaotic [8, 9]. Thus, chaos can only result from many-body couplings between junctions [10].

There is a no-slip viscous boundary condition in hydrodynamics according to which the tangential velocity component of the fluid at the surface van- ishes. In a pipe this yields a parabolic velocity profile perpendicular to the flow direction. There is no such condition for the tangential superflow at the boundary in JJAs. Hence, we consider an im- posed y-gradient (AI = / m a x - - Irnin) of drive current flowing in the x-direction. The region of the array thus considered is then like the boundary layer in hydrodynamics where changes take place in the flow at the transition to turbulence.

R. Mehrotra / Physica B 222 (1996) 326-330 327

We find that the critical value AIc for which the array shows choatic behaviour is independent of N~ for Nx/> 4. Thus, Nx = 4 is a "long-enough" pipe and we study arrays with Nx = 4. We also find that AI¢ scales with N r ("pipe-diameter") as dis- cussed later and we study arrays with 2 ~< N r ~< 14. Thus, the arrays with these limited size are large enough for our studies.

At zero temperature, the dynamical equations using the RSJ model and total current conservation [11] are given by

1 . ] Ix(y) (1) ~=~I dd~O~J+T~smOij =6x, o I---£--'

where 0i is the superconducting phase on a grain i, 0i j= 0~- 0j are defined over the interval - 7t < 0~ ~< n, and ~ runs over the nearest neigh-

bours of grain i. The current drive Ix(y) is restricted to the x = 0 edge and varies along the y-direction. Currents are measured in units of the single junc- tion critical current Ic and time in units of

= (lm/e) [R/(h/2e2)], where R is the shunt resist- ance of a junction and I m = ½(Imin -+- /max) is the mean drive current. The array is shorted at the right edge by a bus bar. The control parameters are /max and AI. Vortices at the centres of plaquettes can be located as usual by measuring the discrete sum )~0~j around each plaquette.

The coupled set of equations can be written in the form ~-ca-~0 = D, where £?-1 is the discrete Laplacian with free boundary conditions and D is the divergence of the current at all grains. The equations can be solved either by inverting the Laplacian or much more efficiently by Fourier transform [12] and cyclic reduction methods [13]. For the case of bus bars or missing bonds in the array, efficient algorithms have been developed by Datta et al. [14]. The equations can then be numer- ically integrated.

3. Phase diagram

The phase diagram in the (AI,/max) space can be obtained by measuring the voltage versus time be- haviour at any grain and studying its Fourier spec- trum. This is plotted in Fig. 1 for a 4 x 3 array.

12

8

~ ' ' / / / ' , , m~. ~ A C

• I41sIsl " ~//E x x

max ~ 0 0

% Bo o / ~ o o B ° OP3 QP2

I m ~ x

Fig. 1. Phase diagram for the 4 x 3 array. The injected vorticity x = A I ( N r + 1) is plotted against lmax. Only lm~n > 0 region is considered. The regions with different dynamical behaviour are marked (S) steady state, (QP2, QP3) quasi-periodic, and (C) chaotic. Between the S and QP3 regions, there are narrow periodic (P) and intermittent chaotic (IC) regions. The lines separating the regions are only a guide to the eye. The various symbols denote • (S), A(P), + (IC), O (QP2, QP3), and x (C). Points A, B, D and E are examples of P, IC, QP3 and C behav- iour, respectively. The inset shows the geometry of the array. The lattice sites denote superconducting grains. The x = 0 edge is driven and x = Lx edge is shorted out by a bus bar.

Here, x = AI(Ny + 1) is chosen as a variable be- cause it turns out to be independent of N r as dis- cussed later. Various kinds of behaviour can be observed: (i) steady state where all voltages are zero and only supercurrents flow in the array; (ii) peri- odic state where the voltages vary periodically in time; (iii) intermittenly chaotic where the voltages are periodic in time with random chaotic bursts; (iv) quasi-periodic where the voltage signal contains exactly Nx - 1 (number of columns of plaquettes) frequencies; and (v) chaotic where the voltage spectra show a large noisy background at all fre- quencies with some peaks.

For a 4 x 3 array, three frequencies can be clearly seen in the voltage Fourier spectra in the QP3 region whereas only two frequencies can be dis- cerned in the QP2 region. Two frequencies are so close to each other in the QP2 region, that it is not possible to resolve them even in very long computer r u n s (106 integration steps).

The simplest configuration which can sustain a vortex is a triangular plaquette with two vertices driven by unequal currents [15]. With overdamped junctions, a triangular plaquette cannot show chaos as the trajectories lie on a 2D torus. The other regions of the phase diagram for larger arrays

328 R. Mehrotra / Physica B 222 (1996) 326 330

can be seen in this elementary plaquette including periodic tongues extending from the steady-state boundary all the way to infinity [15(a)]. It has been shown that all regions of the phase diagram have a straightforward interpretation in terms of a vor- tex variable [-15(b)].

4. Vortex dynamics

A measurement of the total number of vortices of either sign present in the array as a function of time t, and their Fourier spectra show the same frequen- cies as the voltage spectra. This leads us to examine the possibility of understanding the dynamical be- haviour of the array in terms of vortex collective variables. We study the QP ~ C transition in detail.

Vortices appear and disappear in the array with time. In Fig. 2 we plot the time averaged vortex occupancy of positive and negative vortices, N+ and N_, versus AI in a 4 × 3 array. A sharp increase in N + and a maximum in N_ are seen at the transition. This shows that vortices are involved in the transition.

A further pointer that velocity is the key to the transition is in the nature of the size dependence of AIc at the QP ~ C boundary. A plot of AIc versus Ny is shown in Fig. 3. While AIc falls with Nr, xc = AIc(Ny + 1) ~ 8 is a constant at the transition, independent o f N r. The drive AI = - OI(x)/~y with distances scaled in system size L r is an injected vorticity density. The total injected vorticity x is then AI multiplied by the number of relevant plaquetes in the y-direction. The current continuity boundary condition implies image charges at two extra image plaquettes on either side of the N r - 1 rows of real plaquettes. Thus, tc = AI / (N r + 1) and the vortex instability occurs at a critical value K c ~ - - - 8 .

In terms of unscaled times and currents Kc = (AI/I~) (N r + 1)(2e2R/h). It is interesting to note that the dimensionless flow parameter for pipe tur- bulence is the Reynolds number, R e ,~ U L / v , where U is a typical velocity, L is say the pipe diameter, and v is the kinematic viscosity. Turbulence sets in for R~ greater than a critical value ~ 10 3. For SNS arrays, R ~ 200mf~, A I ( N r + 1) ~ 8, one has the critical value of x~ ~ 10-3.

0 . 6

0 . 5

0 . 4

Z 10.3

0 . 2

0 . t

0

0

I I I I

( 0

0

0 • 0

o , J

2 4 6 8

K

0 . 1 2

0 . 1 0

O . O B

0 . 0 6 Z +

0 . 0 4

0 . 0 2

t 0

Fig. 2. The time averaged values of the total vortex number of each sign in a 4 × 3 array, (©) N_ and (0) N+ versus • for lm,x = 4.0. Note the different y-scales for N and N+.

2 . 5 ' I ' I ' 1 ' I ' ~ ' I ' I t0

2.0

i . 5 u I--,I

t . 0

0 . 5

0 . 0

0

! , 0

l a l l 0

0 0 0

I 1 , I , I , I , I z I

2 4 6 8 l O 1 2

Ny

g

t'l 7

o 6

I , 5 i4 i6

Fig. 3. A plot of injected vorticity density AI, (©) and total injected vorticity x, (solid bars) versus N r showing the size dependence of the QP ~ C boundary for Nx = 4 and lmax = 3.0. The vertical size of the symbols corresponds to the error bars.

From the above it is clear that vorticity is con- nected with the transition. One can go on to show that a change in the vortex behaviour is the QP --* C transition and the vortex variables are the "collective variables" sufficient to describe the es- sential features of the transition.

The array contains overdamped junctions so that no inertial oscillations or single junction chaos can be sustained. Moreover, the array is DC driven so that any frequencies or chaos have to come from the collective behaviour of the array as a whole. Since vortices produce phase slips and voltages, it is natural to examine vortex movements and ap- pearances as a source for the AC voltages.

R. Mehrotra/Physica B 222 (1996) 326 330 329

It is found that in the QP region, vortices in the ith column appear and disappear with a frequency ~. The frequencies decrease as one moves away from the drive edge into the bulk of the array. Moreover, the separation between frequencies in adjacent columns decreases rapidly as one moves away from the drive edge and also as 1max increases, so much so that frequencies f~ for i > 3 can be resolved only in extremely long runs ( > 10 6 integ- ration steps). Thus, AC voltages frequencies are directly generated by vortex appearances and are associated with specific locations in the array.

At the QP ~ C transition, the distribution of appearance times of the vortices broadens. Vortices which appeared and disappeared regularly in each column, now do so at random intervals. Since vor- tices move both in space and time, this leads to a broad range of voltage frequencies, i.e., chaos. Thus, vortices a r e the objects that generate the frequencies and chaos in this DC driven system.

We next examine the spatial dependence of n+(r), the time-averaged vortex occupancy. Fig. 4 shows a plot of n_ and n+ in plaquettes 1 and 2 (labels as in inset of Fig. 1) as a function of • for Imax = 4.0 in a 4 x 3 array. In plaquette 1 there are no positive vortices for ~c < ~¢. These appear as soon as K exceeds K¢. Negative vortex occupancy increases in the (QP) region with K, shows a max- imum at K¢ and then drops with ~ in the (C) region. In plaquette 2, it is n+ which shows a peak at x = ~ and n_ shows a sudden increase at the transition.

The following physical picture of the transition emerges from these observations. For ~ < K¢, the negative vortices injected into the system by the external drive remain at the x = 0 edge bound by their image charges. These move in the y-direction (due to the Lorentz force from x-direction current). There are no positive vortices in the edge layer. There is a partially compensating positive charge in the second column. As K increases, more negative vortices are injected until at K = K~, no more nega- tive vortices can be squeezed in. A rearrangement of charges then takes place. The negative vortices deadsorb or depin from the drive edge and move into the bulk of the array whereas positive vortices move in from the bulk towards the drive edge. There is thus mixing of the charges.

|

r-

0 . 3 -

0 . 2

0 . !

0

0 . 0 4

d 0 . 0 2

I I

PLAQUETTE t

o

o

PLAGUETTE 2

o o o I I I

0 2 4

K

I I ,I

o•O o • ( o 0 o

o • o $ .--->

• o

• o

o

• .--- o

0 008 • e •

I I

6 8 10

- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

"0 - 0 . 0 0 6

- 0 . 0 0 4

- 0 . 0 0 2

- 0

"1 4"

"1 4-

Fig. 4. The t ime-averaged vor tex occupancy for negat ive vor-

tices n_ and posi t ive vort ices n+ in p laque t tes 1 and 2 versus

x for Imax = 4.0 in a 4 x 3 array. Symbols used are (©) negat ive

vort ices and (Q) posi t ive vortices.

It is interesting to note that in hydrodynamic turbulence in pipes, there is a sudden change [1] in the eddy behaviour in the boundary layer. Surface fluid flies out ("eruption") whereas fluid from the pipe centre rapidly moves in ("inrush"). The turbu- lence in the both cases is thus associated with mixing of flows between high and low vorticity regions.

5. Conclusions and futnre work

Josephson-junction arrays driven by nonuniform DC current drives show a rich dynamic behaviour. Vortices are the collective variables in terms of which the transition to chaos can be described. There exist similarities with hydrodynamic turbu- lence despite the fact that the JJA has different nonlinearities, quantized vortex charge and is dis- cretized in space.

Future work may include (i) experiments on SNS arrays to detect such chaos, (ii) work on larger

330 R. Mehrotra /Physica B 222 (1996) 326-330

arrays to study bulk behaviour in addition to the boundary layer behaviour, (iii) studies on flow around defects consisting of missing bonds [16] forming a superconducting "wind-tunnel", (iv) studies on JJA including capacitance, magnetic fields, bond-disorder, etc. which would lead to a much richer phase diagram. Here, turbulent be- haviour could be enhanced by nucleation of vor- tices at disorder sites, but could be hindered at the same time by pinning of vortices. (v) Studies in three dimensons is another area.

Acknowledgements

It is a pleasure to acknowledge discussions with Subodh R. Shenoy, Deshdeep Sahdev, Sujay Datta and Shantilal Das.

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