A semi-linear elliptic pde model for the static solution of Josephson junctions

Abstract. In this study we derive a semi-linear Elliptic Partial Differential Equation (PDE) problem that models the static (zero voltage) behavior of a Josephson window jUllction. Iterative methods for solving this problem are proposed, analyzed and
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  Purdue University  Purdue e-Pubs Computer Science Technical ReportsDepartment of Computer Science1993  A Semi-Linear Elliptic PDE Model for the StaticSolution of Josephson Junctions  J. G. CaputoN. FlytzanisE. A. Vavalis Report Number: 93-049 is document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact foradditional information. Caputo, J. G.; Flytzanis, N.; and Vavalis, E. A., "A Semi-Linear Elliptic PDE Model for the Static Solution of Josephson Junctions"(1993). Computer Science Technical Reports. Paper 1063.hp://  A Semi-Linear Elliptic PDE Model for the StaticSolution of JosephsonJunctions J.G.Caputo!, N. Flytzanis 2 and E.A.Vavalis 3 CSD-TR-93-049 July 1993  A SEMI-LINEARELLIPTIC PDE MODEL FOR THE STATIC SOLUTION OFJOSEPHSON JUNCTIONS J. G. CAPUTO·, N. FLYTZANISt AND E.A. VAVALJS~ Abstract. Inthis study we deriveasemi-linearEllipticPartialDifferentialEquation (PDE) problem that modelsthe static (zerovoltage)behavior of aJosephsonwindowjUllction. Iterativemethods forsolvingthisproblemareproposed,analyzed and theirconvergenceanalysisispresented. The preliminary computational results that aregiven,show the modelingpower of our approach and exhibititscomputational efficiency. 1. Introduction. Josephsonjunctiondeviceshavebeenextensivelyusedin many applications likeverysensitive magneto-meters, high frequencyoscillators,fastswitches etc. To study and analyzesuch devices,differential equationproblems that model them have been widelyused. Inthispaper we present asemi-linearelliptic PDE problem whicheffectively and accurately models the static behavior of a2-dimensional Josephson window junction. The existence of the solutions and the analysisof their smoothness andstability are not addressedhere.Efficient and stablenumerical methods areproposed to solve this PDE problem and asoftware infrastructure that can be usedasa simulationengine for Josephson junction devicesis presented. The rest of the paper isorganizedasfollows. In Section2we present the derivation of thePDE problem that models the Josephson junction. A brief discussion about the underlying physics and alistofapplicationsbasedon the Josephson junctionare alsogiven. In Section3 numericalalgorithms forsolving the derived PDE problem arcpresented, their convergenceanalysisisdiscussed andcertainimplementation issues are addressed.InSection4, preliminarynumericalexperiments that confirm the convergence properties of the method, aswell its efficiency, arepresented. A list of plotsconcerning the characteristics of the computed solutionsfordifferent junction geometries and boundary conditions are alsogiven. Ourpreliminary conclusions arepresented in Section5 togetherwithour future researchplans.2. Derivation of the Josephson junction PDE model. AJosephson junction isaweak linkbetween two super-conductors allowing[or the couplingof electron pairs. Josephson [6] showed that the electrodynamics of such adevice aredescribed not by currentand voltage but by their integrals,charge and phase whichheshowed to beproportionalto the phase of the macroscopicwavefunctionsof the electronpairs inthe twosuper-conductors. He showed thatthe current of pairs across the link dependedon the sineof that phase difference. These tworelationsallow the device to act asafrequency-voltageconvertorleading to applicationslikehighfrequencyoscillators, fast switches or the definitionofavoltage standard ((14]'[7]' [2]). Another effectis •Laboratoirede Mathematiques, Instilut deSciencesAppliquees,BPS,76131Mont-Saint-AignanCedex,France. ( t Physics Department, University or Crete,Heraklion,Greece. (/ I Purdue University, Computer Science Department,West Lafayette,IN47907,USA. ( Work supported in part byNationalScience Foundationgrant CCR 92-02536. 1 I i I  FIG. 1. AwindowJosephsonjunction. : ,.' .. - ._.- .-. ~ ... _._ ... _ . _._.; .. , the existence ofa current of pairsin the absenceof voltage. This current can be modulated intoan interference pattern by an externalmagnetic field makingthese devicesverysensitive magneto-meters [2]. For anisolatedjunction theseeffects remain small energetically andan idea toincrease theenergy output isto have thelll drive an electro-magnetic cavity.In practice this is done by makingthetop andbottom super-conducting layers much larger than the junction area. This design leads toverywell controlled specifications and thejunctions do notdeterioratewithtime. A view of sucha device is presentedin Figure 2. In this workwe model such type of junctionand find thestatic (zerovoltage)solutions. The properties of these solutions can be verified experimentally and from adifferent point of viewthese static solutionsprovide adapted initial conditionsfor the time-dependentproblem. The governing equations of suchadevice are Maxwell's equationstogether with the Josephson equationsmentioned above.Designissues and the fact that the thickness of the weak link (oxidelayer)isvery small, force the fieldsto be in aplane.One can then model eachsuper-conductinglayer by anarray of inductancesand their couplingbya non-linearelement givenby the Josephson equations ill parallel witharesistor representing thecurrent of normal electrons and a capacity [7]. Writing the equations for the phase difFerence between thetopand the bottom layer, and assuming a perfect symmetry for the parameters we get the modelpresented in Figure 2. The Kirchofflawsfor thisarray can be condensed into the following discrete evolution equation for the phase (fl, (integral of the voltage) at eachllodei of the array: (I) L neMesl neighbors <Pi-<Pi C·· <PO. (<Pi) 1'" L .. + (fli+ L .\2 sm - = i· j l] ,A !PO Thisequation expresses the fact that the algebraic sum of all currents iszero at agivennode. The first term corresponds to thecurrent in the iuductances (surface current), 2  FIG.2. Thcequivalentcircuitmodel. the secondis the ern-rentin the capacitanceand thethird is the current of pairs whichexistsonlyin the junction region. The last term If xl is an external currentappliedsymmetrically to the boundary of the array and it can alsodescribe the effect ofanexternalmagnetic fieldifappliedantisyrnetrically [9]. The physical parameters are, L ij the inductance in branch ij , C the capacitance at node i, A the Josephsonlength, L; theinductancein eachbranch, <Po thequantum offlux. The time independent problem givesrisetoadiscretesemi-linearellipticequationwhichreducesin the continuum limit to: 1 1.. (2)\7(L( )\7u)-«x,y)" ( )sm(u)=O III il, x,y A L x,y where ((x, y) is the indicator functionof n J. i.e. () _{l if(x,y)Eilj t: x,y - o otherwise au an =f(x,y) on ail, and where :n denotesthe outward normalderivative.Whenconsideringadevicesuch as the onein Figure 1 with a totalarea nand ajunetion area n J the function L assumesa constant value in n andanotherconstant value in njn J. Then equation (2) reduces to the boundary value problem whichcan be expressedformally as (3)\7([< + k(l- <)]\7u) - «x,y) ;, sin(u) = 0 in il, au an = f(x,y) on oil, (4) where k = LLin and where Lin and Lout are the inductances inside and outside the "., junction n J respectively. This PDE problem isequivalent tothe compositesystem: .LlUin = sin(Uin) >.' 3 , , I r
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