Dynamics of a fluxon in a stack of coupled long Josephson junctions is studied numericallv. Based on the numerical simulations, we show that the dependence of the propagation velocity c on the external bias current Î³ is determined by the ratio of the critical currents of thc two junctions J

Susanto, H.

van Gils, S. A.

Visser, T. P. P.

English

Journal of Mathematical and Fundamental Sciences

PROC. lTB. VOL. 35, NO. 2&3,2002 325ON KINK.DYNAMICS OF STACKED.JOSEPHSONJUNCTIONSH. Susanto, T.P.P Visser & S.A. van GilsDepartment of Applied Mathematics, University of Twente,P.O. Box 217,7500 AE Enschede, The NetherlandsA bstractDynamics of a fluxon in a stack of coupled long Joscphson junctions is studiednumericallv. Based on tbc rumerical simulations, wc show rhat t,he dcpcndcnce of thcpropagation velocity c on thc exterDal bias current 1 Ls deterurined by thc ratio of thccritical currents of thc two iunctions J.1 IntroductionThe stacked-Josephsorr junctions we consider here consist of three slabs of superconductingmaterial which have an insulating barrier between two superconductors. An importantapplication of these long Josephson junctions (LJJ) is the magletic flux quanta (fluxon^s).The fluxon. *'hich is a circulating current, can move a)ong the junctions if tlrere is a biasedcurrent appl ied.The one dinrensional stacked-LJJ cal be modelled b1' a perturbed sine-Cordon erluaticrnwhich, in nornral ized form, may be nri t ten as [9]:o' , ,  -  ol,  -  sirr dl -  56': ,  :  odl - 1.6), -  r?, * sirr62 f J - SO:, = a6| - ' tl lere d' is t l re superconducting phase dif ference across t lrc jurrct ion r [ l ] .  Tht '  t ,er:at o61models the damping due to quasi part icle tuulel l i rrg. o > 0. The driving force, correspoudingto the normalized bias current '1, typical lv l ies irr the interval 0 S f S 1. S e l-1.0] is thecouplirrg parameter of the two junctions and J is the ratio of the critical current of of thetwo junctions.One of the solut ions of the unperturbed-sine-Gordon equation (6,, -  d,,  = sin d) is th€solitolrd ,  =  4  a r *ar r  lexp  ( i# ) ] .  (? )which is moving with velocit j 'c,  lcl  < 1. In the u;;; ;"0 equation this parameter is notdetermined. The wave speed will become a function of 1 when o and S do not vanish arrdthe damping is balanced by the driving force. Therefore, c is a function of 'y for given a& 5. The free parameter (0 determines the initial position of the iluxon. In physics, thissoliton solution is related with units of the flux quantum or fluxon. The system (1) has,wben ,S  :  a :1=  0 ,  the  so lu t ion  (d0 ,0)  in  the  pr imary  s ta te  { l l0 l .In this paper, we investigate the relation of c and 1 for spesi6c values of a & S in theprimary state [ll0]. To represent the configuration, the notation [nlm] is used to denotethe state where n solitoDs are located in the fust junction and m solitons in the other.Negative numbers are usd for antisolitons. In Sec. 2 we consider the case of identicaljunctions. Unequal junctions are considered in Secs. 3 and 4. The junctions considered intbese sections differ only on the critical currents. In Sec. 5 we draw conclusions.( l )326and2 Identical JunctionsWe look at travelling wave solutions to the equation (1). So we assume that the solutiononly dcpends on f: s - ct. The partial differential equation is thus reduced to a sysrem ofordinary diferential equations( t  - , ' , )dle -  s in/r  -  sdt(  :  -ca6l -  " ' r ,(1 - ,1)d?; -s in42lr  -3 'Oi ,  - - ,Lo?-r  (3)C<-rnsidcring (3) in (/ '  ,O'e,6,,df ),  ph"cspacc, thc problcm of c.rnputi 'g sol irary wavcprofi les and specds for (1) with I l l0] state is equivalent to f inding heterocl inic orbits con-nccting two 6xed points:H. Susonto, T.P.R Visser & S. A. von Gils@t , Or) = (arcsin 7, 0)( 0 ,  , 0 , ) :  ( a r c s i n T  *  2 r , 0 ) .(4 )( 5 )We can calculate the parameter combinations for which there exist such a heteroclinic solu-tion numerically using the program AUTO [8]. First, we consider thc casc whcn J: L Inthis casc, the system of cquations (3) can be rewritten t<rf c ' ll r r  ^ 2 \  v  |  / r : 2  . . : -  r l . 2  S  :  t 2 t  . 9  ,  , 2 . 1l ( t - c " )  -  |  _ i l O ; ;  -  s i n @ ' '  -  , - s i n @ 2 r : 7 1 7 O r A i ' - t ) + r , c 6 l "  - ?  ( 6 )r.o- *tri.h *" r"" u singularity at | - c2= *.9 or l.+l : y't +T. Thcsc valucs of c arccallcd the Sutharl uelocity. Note that lcll < lc_l because S is negativc.In Fig. l, the numerically obtained (c,7) picture (in the futurc we call thcse kind ofpictures lV-characteristics) is shown. Thc numerical values uscd in all calculations arcS:  -0 .25  and o  =  0 .17 .(a )  (b )Figure l: The relation between 1 (vertical axis) and c (horizontal axis). Picture (b) magnifiesa part of thc plot (a). It contains the center of the spiral.Somehow, near the cdge of the IV-characteristic, we have a spiral shaped curve. Prcvi-ously, this kind ofspiral in the IV-characteristic has been observed in a single long Josephsonjunction by Brown et al. l4l. Later on, it is shown in [10, 11] that such a spiral exists inthe sirrgle Joscphson junction if the dissipation due to surface resistance, i.e. adding a terrnfl$',|i, in Eq. (l), is takcn into account.However, there is a difference between the spiral we have in this case and spiral in asingle long Josephson junction. The spiral in the single long Josephson junction has itsON KINK-DYNAMICS OF STACKED-JOSEPHSON JUNCTIONS 327center located at lcl = 1 which is the maksimum velocity possibly achieved by a fluxon. Inour cas€ here, the center is below the lowest Swihart velocity lcal : y4TT [3]. The firstfour turning points are givel in Table 1. The plot of the solitons related to those turrrinspoints is giverr in Figure 2.Table l: The first four turning points of the curve in the 1- c space.Coldobin et al.  15) give a'proof 'which slrows that the f luxon in the state I l l0l  withJ = I cartnot pass the lowest Swihart velocity lc1l. The result says that if lc*l is reache<i,there is at lea^st a fluxon in the second junction which then gives the {ll1l state.( a )Figure 2: @t (a) and d' (b) * functiorrs of {solut ion corresponds to the f irst turning point.tlte vertical directiolr.at the 6rst four turning points. The lowestThe i-th solut ion is shif ted ( i  -  l )  unit(s) in3 Unequal Junctions: J > 1An ittterestilrg situation occurs if the two junctiorrs forrning tire stack are not identical.We assunte that only the critical currents of the two are unequal, whiclr translates to thecondit ion that J in Eq.(3) is not equal to 1.Goldobin et al.  16) mention for the 6rst t ime that there is a'backbending'phenomenonin the lV-clraracteristic. With AUTO we can continue the backbending such that we getthe full-branch of the IV-characteristic. The result is shown in Fig. 3.Goldobin et oI. l5l also show that the velocity lcal cannot be reached at this case.In Fig. 4, two solutions for the same value of .y are plotted. In thc first junction thereis hardly any change, but in the second junction there is a big difference. on the leftbranch there is only a small 'irnage' in the second junction, which then if we follow theIV-characteristic, this image has grown to a large hurnp.The stability of solutions on the right branch has been tested numerically in I I l]. Bytaking the solution as calculated with AUTO and using this as initial solutiou to the PDE-solver [12], we conclude that all solutions on the right branch are unstable. All solutions at0.5121 004328 H. Susonto, T.P.P Visser & S. A. von GilsFigure 3: The IV-characterist ic for J ) l .the left  branch up to the point wlrcre dlfdc:0 are stable. Solut ions on the r ight branchare irr the domaiu of attraction of the solution for the sarne value of 1 orr the left brauch.The instabi l i ty of the solut ion carr be seen clearcr whcn we fol low the r ight brarrch inthe <l irect ion of decreasing 1. In thc l i rnit  'y tcnds to 0, there is a f luxon-auti f luxon pair injunctiorr 2 or the stack is irr  i l l l ,  -1] [5] Bccause the l luxon and arrt i f luxon arc attract ingeach other, if we apply a bias crrrrcnt, they will collkle and a new dilferent, state is forrned-This is the physical argument to show that this state is urrstable.t  4. ,F'igrrre 4: Comparison of two solut ions to (3) for J = 1.5 arrd "t  :0.77 on the left  and r ightbranch.4 Unequal Junctions: -r < 1A sornewhat different behavior happens when we look at case the "/ < 1. While irr tire case-/ > 1, the velocity is bounded by c.. ,  lc l  < lc1l,  in the case../  < 1 we have that lcl  canexceed the value lc1l. Wben fluxons move with velocities above the lowest Swihart velocity,Cherertkou radiation takes place.Cherenkov radiation is the errrission of waves behind the rnoving fluxon. Behind thefluxon we will find an oscillating tail which is the emitted waves. E. Goldobin et al. 16, 7lobserve Cherenkov radiatiorr happening in the [110] state numerically arrd experimentally.The lV-characteristic of the state is given in Fig. 5. Unfortunately, the IV-characteristicstops at c* because our equatiorr becomes singular at that value and AU'IO cannot con-ON KINK-DYNAMICS OFSTACKED-JOSEPHSON JUNCTIONS 329Figure 5: The lV-characteristic for J < 1.tinue tbe calculation through the singularity. Therefore, all solutions that lie in this IV-characteristic do not have such enritted wave behind the lluxon as we see in Fig. 6.If we keep increasing the applied bias current, the velocity of the fluxon increases up tothe maximum velocity l"lll"oJl. 'ffrir maximum velocity is a function of "I, S, a [6]. By now,the analytic form of the maximum velocity for this state is still unknown.Another branch is found when we use one solution on the left branch of the IV-characteristicfor J > I as the initial solution of AUTO. Using this initial condition, then we decrease theparameter J up to J : 0.5. In fact, we get a backbending curve for the IV-characteristic ofthe case J < l. The curve is given irr Fig. 7. If for J > 1, we see hardly any changes whencomparing solutions irr the first junction, while sornethilg different happen^s in tbe secondjunction (Fig. 4), for the case we have the opposite result. In junction 2 we almost haveseen nothing, while there is a difference in the solutions in the first junction. In Fig. 8 wecompare two solutions orr the left and right branch for the sarne value of 1.A further analysis is needed to see the stability of solutiorrs lying on this branch. Weconjecture that all the solutions are unstable. The same argurnerrt rnight be used as inthe case J > I before. Nunrerical simulatiol cotrfirnrs as well that the solutions go to thecorresponding solutions at the other branch. We have dorre that also using PDE-solverwhere we use a solution resulted bv AUTO as the irritial solution of the PDE-solver.( b )for ./ = 0.5 and ''l = 0.17.Figure 6: Solution to (3)330 H. Susonto, T.P.P Visser & S. A. von GilsFigure 7: Another brarch for the IV-characteristic of J < l.(a )  (b )Figure 8: Comparison of two solut ions to (3) for J = 0.5 and 1 : 0.17 on the left  and r ightbranch.5 ConclusionWe have given several calculatiorrs showing that stacked-Josephson jurtctiorrs have very richbehavior. Just with [110] state and changing ./, we already have qualitatively differentIV-characteristics, wbile there are rnany rnore states that cart exist in the junctions.How the lV-characteristics corresponding to the state [nlrn] change when J passesthrough I needs to be ilvestigated further. Also the stability of the various patterns ttee<lsto be determined.Goldobin et cl. [6] state that for the [110] state not only the conditiorr J < i leads tol.l > l.*1, but also the asynnretry of the applied bias current of the two-fold stack, i.e.when we have 7t > -y2. Therefore, it might be interesting to consider also in the near futurethe possibility of having branches of the system under condition'yt + 12.References[l] A. Barone & G. Paterno, Physics and Applications o! the Josephson Effcct. Wiley,New York, 1970.[2] A. lVal lraff ,  E. Goldobin & A.V. Ustinov, J. Appl. Plss.80(11):6523-6535,1996.ON KINK-DYNAMICS OF STACKED-JOSEPHSON JUNCTIONS 33I[3] A. V. Ustinov, Il. Kolstedt, M. Cirillo, N. F. Pedersen, G. Hallmanns & C. Heiden,Phys. Reu. B 48:10614, 1993.[4] D.L. Brown, M.G. Forest, B.J. Nl i l ler & N.A. Petersson, SIAM J. Appl. Math.54(4) :  1048-1066,  1994.[5f E. Goldobin, B. A. IMalomed & A. V. Ustinov, Phgs. Letters A26(t:67-75,2000.[6] E. Goldobin, A. Wallraff& A. V. Ustirrov, J. Lou Ternp. Phys. 119(516):589-614,2000.[7] E. Goldobin, A. Wallraff ,  N. Thyssen & A. V. Ustinov, Phys. Reu. B 57(l):130-133,r998.[8] E.J. Doedel, R.C- Paffenroth, et. al., AUTO2000: Continuation ond, BifurcotionSoftwarc for Orlinary Difierential Equutions (with HomCont/, Concordia University,Canada, ftp. cs- concordia. ca/pub/ doedel/ outo.[9] M. B. Mineev, G. S. Mkrtchjan & V. V. Schmidt, J. Low Temp. Phys.45:497,1981 .[10] J.B. van den Berg, S.A. van Gils & T.P.P. Visser, Pororneter Dependonrc ol HornoclinicSolutions in a Single Long Josepltson Juuction, submitted to Nonlinearity.[11] T.P.P Yisser, Modelling and Analysis of Long Josephson Junctions, Twente UniversityPress, 2002.[12] T.P.P. Visser, PDE-soluer, Twente University, the Nethertands.

On Kink-Dynamics of Stacked-Josephson Junctions

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On Kink-Dynamics of Stacked-Josephson Junctions

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