dynamics and chaos of a current driven two-dimensional josephson junctions array under 13Φ0...

Physica A 261 (1998) 409–416

Dynamics and chaos of a current driventwo-dimensional Josephson junctions array

under 13�0 magnetic �eld

Rafael Rangela;b;c;∗, A. Gim�enez b; c, M. OctaviocaUniversidad Sim�on Bol��var, Depto. de F��sica, Apdo. 89000, Caracas 1080 A, VenezuelabUNEXPO, Direcci�on de Investigaci�on y Postgrado. Torre Domus, Caracas, Venezuela

cIVIC, Centro de F��sica, Apartado 21827, Caracas 1020 A, Venezuela

Received 1 April 1998

Abstract

We derive the equations of motion for a two-dimensional capacitive Josephson junctions arrayin the presence of both a DC current and a magnetic �eld f= 1

3 of the quantum ux �0 . Theground state symmetry of an N × N array is assumed to hold for all currents, then by usingthe resistively and capacitively shunted junction equations, a model system of four-coupled non-linear second-order di�erential equations is derived. The system has the form �cx′′+x′+∇U =0,where U is a four-dimensional potential and �c is the Stewart–McCumber parameter. The dynam-ics can be viewed as the motion of a massive particle sliding under the action of the potentialin a four-dimensional con�guration space with a friction proportional to its speed. There arethree distinct branches: one below the critical current Ic where the static zero voltage solutionis stable; the second branch which originates from the static solution through a Hopf bifurcationand where a total voltage develops along the direction of the applied current and across the array(instantaneous Hall voltage), (the latter means vortices moving perpendicular to the current andconstitutes a ux- ow like regime); and a third branch above the synchronization current Is,where the motion of the junctions synchronizes and the motion of the vortices ceases with zeroHall voltage. For a wide range of �c, the second branch shows chaotic dynamics of extremelyrich complexity. A pervasive feature is the presence of antimonotonicity, i.e., reversals of perioddoubling cascades. c© 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

Arrays of Josephson junctions is a matter of considerable interest. The very importantcase of two-dimensional Josephson junctions arrays (TDJJA) have been intensively

∗ Corresponding address. IVIC, Centro de F��sica, Apartado 21827, Caracas 1020 A, Venezuela. Tel.: 58-2-9063603; fax: 58-2-9063601; e-mail: [email protected].

0378-4371/98/$ – see front matter c© 1998 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(98)00240 -4

410 R. Rangel et al. / Physica A 261 (1998) 409–416

studied in recent years [1,2]. Some work has been done in underdamped arrays (�nite�c) as the experimental work of van der Zant et al. [3] and the theoretical workof Octavio et al. [4]. There are many reasons to study TDJJA for any value of thefrustration index f=m=n. We study the case f= 1

3 , i.e., one-third of the ux quantumper plaque [5].

2. Equations of motions for the f = 13 ground state

We assume that the ground state symmetry of the systems for f= 13 as shown in [6].

There is numerical evidence that this type of solution remains stable for all currents.In Fig. 1, we show the superlattice unit cell and the de�nitions of the phase variables�, �, , �0, � and �. The external DC current ows in the x direction. Each junctionis assumed to obey the Resistively–Capacitively–Shunted Junction (RCSJ) equation:

�cd2�d�2

+d�d�+ sin�=

i�ic; (1)

where � is the gauge-invariant phase di�erence, i� is the total current owing throughthat junction, ic is the critical current, �=(2eicR=˜)t is the dimensionless time and�c=(2eicR2=˜) is the Stewart–McCumber parameter (R and C are the shunt resistanceand capacitance, respectively), ˜ is the Planck’s constant and e is the electron charge.We assume all the junctions to have identical ic. Between the ith and jth nodes in thearray of Fig. 1, the gauge-invariant phase di�erence obeys the equation: �=�i−�j−

Fig. 1. The superlattice unit cell for f= 13 with the de�nitions of the phase variables. The external DC

current ow in the x direction. The magnetic �eld is in the z direction. We show the nodes A and B, andthe loops II and III.

R. Rangel et al. / Physica A 261 (1998) 409–416 411

(2�=�0)∫ ji Adl , where �0 = hc=2e is the ux quantum, �i(�j) is the phase on node

i (j) and A is the vector potential. In the Landau gauge A=Hxy; H is the magnitudeof the magnetic �eld and y is the unitary vector in the y direction. From the uxoidquantization condition, we obtain for loops II and III, respectively (we refer to Fig. 1for the explanations of the symbols):

�+ + �0 + �=2�3(mod 2�); �+ � − �− �= 2�

3(mod 2�) : (2)

From charge conservation, we obtain for nodes A and B, respectively,

�c( ′′ + �′′ − �′′ − �′′0 ) + ( ′ + �′ − �′ − �′0)+(sin + sin �− sin � − sin �0)= 0 ; (3)

�c(�′′ − �′′ + �′′ − �′′) + (�′ − �′ + �′ − �′)+(sin � − sin �+ sin �− sin �)= 0 : (4)

We have to impose the condition of zero current in the y direction:

�c(�′′ − �′′0 + �′′) + (�′ − �′0 + �′) + (sin � − sin �0 + sin �)= 0 : (5)

Because the total current entering the cell on the left boundary is Itot , we obtain:

�c( ′′ − �′′ − �′′) + ( ′ − �′ − �′) + (sin − sin �− sin �)= Itot=ic ; (6)

2�c( ′′ + �′′) + 2( ′ + �′) + (sin + sin �− sin � − sin �0)= 0 ; (7)

2�c(�′′ − �′′) + (�′ − �′) + (sin � − sin �− sin �− sin �)= 0 : (8)

Eqs. (5)–(8) suggest to the choosing of variables: y=(y1; y2; y3; y4)= (x; y; z; u)

x= + �− 2�=3; y= � − �− �=3 ;z=1=2(� − �0 + �); u=1=2( − �− �) : (9)

Using Eqs. (2) and (9), the gauge invariant phase di�erences �=(�; �; �0; ; �; �) canbe easily obtained.

�=(x − y − 2u+ �); �= 13(2z − x + y + �) ;

�0 = 13 (−2z − 2x − y + �) ;

= 13(2x + y + 2u+ �); �=1=3(2z − x − 2y); �=(x + 2y − 2u) ; (10)

Observe that u′=Vx=2 and z′=Vy=2, where Vx is the normalized instantaneous voltageacross the array in the direction of the external current and Vy is the correspondingvoltage in the perpendicular direction (Hall voltage).The system of di�erential equations one obtains using the previous de�nitions can

be written in the following manner:

�cy′′j = − gj(y1; : : : ; y4)− y′j (11)


with j=1; 2; 3; 4; with y=(y1 = x; y2 =y; y3 = z; y4 = u); �=(�1 = �; �2 = �;�3 = �0; �4 = ; �5 = � and �6 = �), one has

gj(y1; : : : ; y4)=12

6∑

k = 1

!jk sin�k(y1; : : : ; y4)− Itot2ic�j4 : (12)

Here �j4 is the Kronecker delta function. The values of !jk are 1, 0 or −1 and canbe obtained from Eqs. (5)–(8) with the de�nitions above. One proves that: @gj=@yi 6=@gi=@yj; i; j=1; 2; 3; 4. We are seeking a potential U , therefore we make a variabletransformation:

xi=4∑

j= 1

Dijyj : (13)

Multiplying Eq. (11) by the coe�cient Dij and summing over j, we obtain:

�cx′′i =−fi(x1; : : : ; x4)− x′i ; (14)

fi=6∑

k=1

aik sin�k − Itot2icDi4 ; (15)

aik =12

4∑

j= 1

Dij!jk : (16)

Our main idea now is to apply the Poincar�e’s theorem to the one form de�ned byF =f1dx1 + f2dx2 + f3dx3 + f4dx4, by imposing the condition that dF =0, i.e.,the one form is close. Then, we invoke the Poincar�e theorem and seek a 0-formsuch that dU =F , which implies that ddU =0. From this last condition we obtainEq. (18) which de�nes the matrix D. This procedure is general and can be used forany frustration index f=m=n. We obtain

fi=6∑

k = 1

@�k@xi

sin�k − Itot2icDii4; aik =

@�k@xi

: (17)

The last relation is the neccesary condition for the equality of mixed second partials,or ddU =0, and can be wirtten in the matrix form

12DD!=�@yk : (18)

We choose D in the form given by Eq. (19), the calculation turns out to be mucheasier:

x1 =A(y1 + y2); x2 =B(y1 − y2); x3 =Cy3; x4 =Ey4 : (19)

We obtain straightforwardly the matrix D de�ning the following transformation to thenew variables x=(x; y; z; u):

x→ x + y; y→ 13(x − y); z→ 2z√

3; u→ 2√

3u : (20)


The phase di�erences are now given in terms of the new variables:

�=1√3y − 1√

3u+

�3; �= − 1√

3y +

1√3z +

�3;

�0 = − 12x − 1

2√3y − 1√

3z +

�3;

=12x +

1

2√3y +

1√3u+

�M3; �= − 1

2x +

1

2√3y +

1√3z ;

�=12x − 1

2√3y − 1√

3u : (21)

We obtain for i=1; 2; 3; 4, where x1; = x, x2; =y, x3 = z and x4 = u are given byEq. (20), the functions fi which are obtained from the potential given by

U (x1; : : : ; x4)= −6∑

k = 1

cos�k(x1; : : : ; x4)− Itot√3icx4 ; (22)

where �1 = �; �2 = �; �3 = �0; �4 = ; �5 = � and �6 = �

�cx′′i = − @U@xi

− x′i : (23)

These equations have the form we searched for, i.e., �cx′′+x′+∇U =0, and describethe motion of a particle of mass �c in a four-dimensional con�guration space slidingdown the gradient of the potential U , its speed being limited by a velocity-proportionalforce. Notice also that global coupling appears now in contrast to the nearest-neighborscoupling of the equations as initially formulated for the TDJJA in the � variables.Therefore, synchronization behavior can be expected [7].

3. Simulation of the system and I–V characteristics

The system of equations (23) is solved by using a fourth-order Runge–Kutta al-gorithm with uniform time steps. We recorded the values of the variables and their�rst derivates every time the voltage along the array Vx reached a maximum. Thisprocedure yields values of the voltage maxima and the time intervals which re ectsthe nature of the attractor. We also performed the time average of u′ and z′, whichis twice the averaged voltage in the x and y directions. Eq. (23) exhibits the follow-ing symmetry: xi⇒−xi; �=3⇒−�=3 and Itot⇒−Itot , i.e., this transformation leavesEq. (23) invariant, which means that an averaged voltage can exist only in the x di-rection. The reason for this is the term originating in the total current Itot . If we varythe current, we obtain the bifurcation diagrams Vx max vs. I . For values of �c¡16as well as for very high values, there are only periodic solutions. In between, thereexist very complicated chaotic dynamics. Hysteresis is another feature of the bifurca-tions diagrams. It is small for small values of �c and increases monotonically with


Fig. 2. The bifurcation diagram for the maxima of the Vx-voltage vs. the drive current. �c =16. The threedistinct branches are clearly observed. At Ic =0:2667, the instantaneous Hall voltage appears. Beyond Is,around 0.5467, the synchronization sets in and the Hall voltage is essentially zero.

it. The critical current calculated numerically agrees with the one calculated by Benzet al. [8], Itot=3ic·=0:2667. It can be calculated by solving the relation @U=@xi=0 andestablishing the maximum value of the current up to which values of xi exists. Thesecalculated values of xi represent a stationary solution which is linearly stable up tothe critical current. There is a Hopf bifurcation at Ic to a ux-fow regime, where theinstantaneous Hall voltage appears. The complexity of the dynamics for �c=16 andpresented in Fig. 2 is easily appreciated. Some of the period-doubling bifurcations arenon-universal, which is a very interesting �nding. (Another bifurcation to identify intheir mechanism are the reversal of period-doubling cascades.) There is an enormousamount of complexity in the bifurcation features as the parameter �c is increased be-yond 16. One can understand this complexity at least qualitatively by exploiting theexistence of the potential in the equation of motion (23) and deducing generic fea-tures of the particle motion. One can see that the four-dimensional potential makesup a very complicated landscape and for su�ciently small currents the particle maystay trapped in this landscape, i.e., the �rst term in Eq. (22) dominates the behavior,whereas for high enough currents it receives mostly energy from the second or cur-rent term in Eq. (22). In this case, the particle moves essentially in a line with small uctuations in the remaining three dimensions of the con�guration space. In otherwords, the motion is quasi-one-dimensional along x4 = u. It now becomes clear that asthe particle becomes heavier (greater �c), the last feature appears for smaller currents,


Fig. 3. The time dependence of the time derivatives of the gauge-invariant phases whose sum equals thevoltage Vx . �c =16. See Eqs. (9) and (20). The lower curves correspond to I =0:5465, the solid line is Vx .Upper curves correspond to I =0:5470. The almost constant line by 0.8 is Vx .

i.e., the synchronization current Is becomes smaller and the local stability of this solu-tion goes even to currents much less than the Is. This explains without trivially tediouscalculations the increasing of the hysteresis as mentioned above. Fig. 3 shows the timederivatives of the gauge-invariant phases whose sum equals the voltage Vx=2= u′, (seeEq. (9)), for Itot =0:5470 (upper curves). One can see the perfect synchronization ofthe rapid oscillations and the negligible time uctuations of Vx=2 (top slashed curve). Ithas a value which is just the balance of the last two terms in Eq. (23). The bifurcationat Is is such that the phase space shrinks practically from dimension 8 to 2. The Hallvoltage Vy =2z′ is such that 〈Vy〉 =0 (numerically 10−10 ) with very small uctuations〈�Vy〉 =10−6. All variables execute periodic rapid oscillations at the scale �c such thatin Eq. (23) 〈�cx′′i + �iU + x′i〉 = 〈�cx′′i 〉 + 〈�iU 〉 + 〈x′i〉 =0, and also such that onecan interchange the time average of the sine functions with the sine of the time av-erage of the phase variable. In particular, for i=3, each term of Eq. (23), averageszero, i.e., 〈�〉 = 〈�0〉 = 〈�〉 =0. One also has that 〈x1〉 =2�=3, 〈x2〉 = −�=3, 〈x3〉 =0and 〈x4〉 = �. In order to understand the chaotic dynamics one has to remember thatEq. (23) represents a system of four coupled damped driven pendula and that onlyone pendulum (x4) is subject to a driving torque. Besides this, the potential de-�nes a gradient vector �eld. Both aspects can help to analyze the chaotic dynamics.Fig. 3 also shows the time derivatives of the gauge-invariant phases for a currentI = Itot=3ic=0:5465 in the chaotic branch not far from Is (lower curves). The solid


line is Vx=2. One notices the di�erent time scales of the oscillations as well as thestrong non-harmonicity.In conclusion, we have derived the equations of motion for a Josephson array with

frustration index f= 13 . We �nd three distinct branches in the I–V characteristics. One

of these right above Ic, which is related to a Hopf bifurcation exhibits an extremely richdynamic behavior including non-universal period-doubling cascades and chaos. Moredetails will be published elsewhere [9].

References

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[2] S. Chung, K.H. Lee, D. Stroud, Phys Rev. B 40 (1989) 6570; K.H. Lee, D. Stroud, J.S. Chung, Phys.Rev. Lett. 64 (1990) 962; J.U. Free, S.P. Benz, M.S. Rzchowski, M. Tinkham, C.J. Lobb, M. Octavio,Phys. Rev. B 41 (1990) 7267; D. Dominguez, J.V. Jos�e, A. Karma, C. Wiecko, Phys. Rev. Lett. 67(1991) 2367; N. Grfnbech-Jensen, A.R. Bishop, F. Falo, P.S. Lomdahl, Phys. Rev. B 45 (1992) 10 139;S. Das, S. Dattta, D. Sahdev, R. Mehrotra, Phys. Rev. E 55 (1997) 2228.

[3] H.S.J. van der Zant, F.C. Fritschy, T.P. Orlando, J.E. Mooij, Phys. Rev. Lett. 66 (1991) 2531.[4] M. Octavio, C.B. Whan, U. Geigenmuller, C.J. Lobb, Phys. Rev. B 47 (1993) 1141.[5] R. Rangel, A. Gim�enez, M. Octavio, Bull. Am. Phys. Soc. 38 (1993) 803.[6] M.S. Rzchowski, L.L. Sohn, M. Tinkham, Phys. Rev. 43 (1991) 8682.[7] Hisa-Aki Tanaka, Allan J. Lichtenberg, Phys. Rev. Lett. 78 (1997) 2104.[8] S.P. Benz, M.S. Rzchowski, M. Tinkham, C.J. Lobb, Phys. Rev. B 42 (1990) 6165.[9] R. Rangel, A. Gim�enez, to appear.

dynamics and chaos of a current driven two-dimensional josephson junctions array under 13Φ0...

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