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Submitted on 1 Jan 1978
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SOLITARY PHENOMENA IN DISSIPATIVE DRIVEN
JOSEPHSON JUNCTIONS
E. Ben Jacob, Y. Imry
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplPment au no 8, Tome 39, aoat 1978, page C6-569
E. Ben Jacob and Y. Imry
TeZ-Aviv University, Department of Physics and Astronomy, Ranat Aviv, Israel
Rdsum6.- Les solutions de solitons, pour un modPle discret de jonctions Josephson ponctuelles cou- pldes, incluant les termes de dissipation et les forces extdrieures, ainsi que les caractdristiques I-V rdsultantes, sont obtenues. Les diffgrences avec les solitons dans les modPles id6alis6s et les moyens de leur observation experimentale sont discutds.
Abstract.- Solitary solutions for a discrete model of coupled Josephson point junctions including dissipation and driving terms, and the resulting I-V characteristics are obtained. Differences from the solitons in idealized models are discussed and ways for experimental observation of these effects suggested.
The space-time dependence of the relative pha- se @(x,t) in an extended Josephson-Junction is go- verned by a Sine-Gordon-type equation/]/. In a rea- listic case the situation is complicated by dissi- pative terms, effects of the driving currents and finite boundaries. The knowledge of +(x,t) and its dependence an e.g. static magnetic fields or impo- sed microwave radiation, determines the measurable properties of the system such as current-voltage characteristics, switching and response to external fields. The Sine-Gordon equation for an infinite one dimensional system with no dissipation and no dri- ving forces has solitary solutions /2/ where a lo- calized change of +(x,t) by 28 propagates with no change of form with an arbitrary velocity v
5
c
wherec
is the light velocity in the junction. How- ever, the existence and properties of solitary so- lutions for a realistic system is an important un- solved problem / 3 , 4 / . In addition to the fundamen- tal interest in the problem, it may be of relevan- ce to experimentally realizable systems. The model that we shall discuss below describes configurations which appear in D.C. S.Q.U.I.D.S./5/ and logic/6/curate representation if N is taken to beaL/XJ, where
XJ
is the Josephson penetration depth. An understanding of the small N case will give us use- ful insights on the problem as well as ways for predicting and estimating the I-V characteristics of general junctions and their dependences on the junction's parameters and external fields.Theequations of the model are (see figure 1
i = 1, N is obviously changed)
8. is the 'phase difference of the ith junction
a '-I,
KIE @ /ZfiIJ ; L is the inductance of the ith ring.
0
I. is the driving current. Oexi is the external ma- gnetie'dlux in units of @ /28, G
=A
/FRC and IJ the0 J
Josephson current.
It is seen that the equation for each junc- tion is identical to that of a S.Q.U.I.D. 151. The characteristic solutions for a S.Q.U.I.D. have been obtained and interpreted physically. These enable us to predict the general forms of the solutions of the N-Junctions problem. We have also generated numerical solutions to the latter problem which are devices.
in good qualitative agreement with the expectations. We consider a system of N Josephson point
In the case of two junctions /4/ three gene- junctions /4,7/ where nearest neighbor junctions
ral types of solutions exist, resulting in three are coupled inductively and where each of the junc-
branches of the D.C. I-V characteristics : A static tions is described by a simple lumped circuit model
solution where = O, a running solution where In addition to the interest in the specific case
and 02 increase steadily, which has the highest where N = 2, which is the simplest one where the
voltaee and a "beatine" solution of intermediate
-
spatial dependence of the phase exists, the modelvoltage where the two junctions interchange static may be viewed as a numerical simulation of a con-
and running roles. The "beatinn" solution can be
-
-
tinuous junction of length L. This should be an ac-viewed as being of a solitary type, a phase change
+ Partially supported by the Commission for Basic
Research of the Israeli Academy of Sciences of an integer multiple of 28 (which can also be
C6-570
regarded as n flux units) runs across the system.
Fig. 1 : The voltages of the various junctions, in arbitraty units, a s functions ofdime, measured in units of wo '= XJ/c = (C$o/2nIJ)
.
N=4. la) Abound two soliton pair, for G=0.8, K=1.8/24nir,(cor- responding to e.g. AJ=10 2cm, IJ = 10 'A, C = c/30 C = npF, R=4fi, L = 1.2 x 10 'H) 11=3.21J, I =I =I 2 3 4 =O Ib) Two correlated solitons, for the same para- metersexcept K = 1.816~. The straight lines are
drawn to guide the eye through each soliton.
The solitary nature of the "beating1' type so- lution is more vividly seen in the case where N>2. Here a larger number of "modes" may exist. In fi- gure la a bound two soliton-like solution is pre- sented. It is seen that the phase change of 4n is propegated along the chain. The form of this solu- tion is not exactly invariant with the motion but a strlct periodicity in the motion of the whole system exists. After one soliton crosses the chain, a new one is generated in the first junction and launched along the chain. Other types of solitary solutisns are possible. A double (phase change of 2 x 2n) soliton, a bound pair of solitons,etc. In the latter case two solitons move at each instant along the chain see figure Ib.
Each type of solution corresponds to a branch of the I-V characteristics (figure 2), where hyste- retic jumps among the branches may occur. From our investigation an understanding of the conditions for getting the various solutions and their limits of stability (including cases where only multiple solitons are possible) are obtained. Also, the de- pendence on external fields is understood and can
be estimated analytically in agreement with the numerical data.
Fig. 2 : DC I-V characteristics, traced with in- creasing or decreasing current (shown by the arrows) I=Il,12=T3=14=0. The smooth lines are for $ex=O, the dashed ones for $ex=l/4@o a : same parameters as in lb b: G=8, K=1-8112~.
To summarize : Solitary-type solutions are shown to exist in dissipative driven Josephson sys- tems. However, they differ in important respects from the ideal case and do not seem to be just a small modification of the latter. For example, the propagation velocity is fixed for a given junction. Experimentally, the most direct observations of the- se effects would be through the voltage peaks that occur upon the passing of the solitary solution at
a given junction. A less direct method to observe these effects would be through the low V branches of the I-V characteristic where theoretical infor- mation on the limits of the branches and their de- pendence on external fields exists. It is hoped that a systematic study of the effects discussed here may elucidate these interesting questions, and help in the design of related devices.
References
/1/ Josephson,B.D., Adv. Phys.3 (1965) 419 /2/ Scott,A.C., Proc. IEEE
61
(1973) 1443/3/ Fulton,T.A. and Dynes,R.C., Solid State Commun. 12 (1973) 57
-
141 Imry,Y. and Marcus,P.M., IEEE Trans. Mag.E (1977) 868
/ 5 / Tesche,D.C. and Clarke,J., Low Temp. Phys.
9
(1977) 301161 Zappe,H.H., IEEE Trans. Mag.
