. £45....‘wrh‘ré 4.4%.»? 25, . .. . . ,... a: i.......Mv..r.uL...¢.m.fis.n..c‘x.r.~.. Wfikfim .....figuY...“ .7 _ ..u . . .... .. 1 ~«...1......u..hn...m_........w a. ram... . . .... . . . ., ., ,. Wm...) i.» . fl .. H . 3;... ....fimw. . to. . v t. . ...h..1...w 3k". 3 '3 ‘ i3. I .n .ivrrhmio; ... . . . . ... .vofhvzé v ... . . .. . . v . v .. .. . ... ,. . V p . . . . .. . . . \ h. . . . . «1...... . -1...» .. . . V. U . .. . .. . . . ,. ...!vtnrushmmwurvzz ...m... , . . . . .. 5.... J ._ in: . .. .. . _ .. --.”.inhautsh. . 1.. ... ..Afi . . . . . . . . .2.) I... = . an. . . .mw , . . ... .... . . ., . 1.....5... .. . ....uvmv .. . avwflnv.lvhu..uou . . . . . . . . . . . . Jinn}! ..uu..qkh.ldv..t..K .. . . . . .. . . . .V . . . . , . .. .. . . . .. 1 l £413”... 1 . . u .. n. t . , ... mummy Law. .. .. mwauéwx . .In . n. :5. 1.1... .. ..NK. ..v ..nmummhf 5?... .4... c . . . . ...... -... .....- ... .... . ..., .. . .. ..v J ...”.fiwmdgéws 3w . . .. .. ,. _ u was "......3... . -.....“ ....wlfi..§.h.ua.. ,. . . H . . . . .mwuu‘mumrmmvflmh. , ....fiifi: . .. . r . . .. . .. Hit... a .Jllb .. . . .. . . . V. . . . . . ...»Qwfipg. ..ul¢ .4 . .r . l ......4‘uv: f 1": 31:01? ... ... i .. a: n . ... . 1%...." ....2: ....fl U.» . ... . . "uh- . variedhs . 04.11. Human” q». .oh. .. ,.....mvflvmmmwi‘flhuuuofiu c... , .t .u 0. b .. 4 fl. .. .. . .I...r ' 1‘, '3 .... ...I ...7 .uflv........§t....:..lr, .. .. . u ....l... 9:2... v .-. . . . . 53.6.me ... . .. , ... .... .. .. . . -\..!... Q ..... A? ..-..t. . . unn.%l.fibnuhfl.otu..lk . ... . . . ... .t. ...!J ...”... vhf”! ..li .. . . a. n. 1.... .3“. a.-. .n Sun... I 3 y 8 n . . . . 00-6.6. vcu‘vi .... .2 . -... fiNWftWhnhki. an... .9. I . .u. . . . V ... . .. .. . I..- .... . - 2 . o; 1.195.... [lucréwovuunhulh ..nivfil .. .. l ..-.I . . . . . . 1 Liana-0». .. I... ...... .H «A. I up! .... ...I ..-..I .10 5 . . .. . lief-z}: . . . . . . . .3. . X H L- . . | Ilatflclc‘lu.o 0' ‘ ... t . ....Iyw. . . . . J 3: . .. .. v. .I4.ptv . linélxrv‘- . {hub-1.. ...v. 1:». «...... , . l . . v - . .. ,. . .. . . ., ....L... 2.7.7.1, .-lu . ... . w ... .... . . ‘Iitmnuvl¢. .. ... I. u .5 to“, O. . cl! . . 1 . . .u .srLIA. \Cl L! c. .. vatvILvVVrto§AVHD - fl. . ‘1 . .n . “I a» I .l .10.... . 4))... r! :13 :<‘u.pvvr|...[....n .I. 15.01:.‘IP civuozol. s . I .313... .... . .. o.}&x's.wla . .. Lu v!" . . v .11. ‘1"?! LI , . 1% ......,...U.cl.vr..|.a.1v. . ‘orlwf . . ... ......“ iv... ... - ..nfiob: Voila... .1thqu .'f;xE 1 ,-,. ";’="!i ~' 3;: , ‘ “11?; . unv’lll-Q‘OOOW un ., .Iu.H>9; .. . . v..01,.... ..I...”..‘.9V.n‘l\.0 . V . . U-.. «M‘st .- lulu: .! a . . . 1 , u . - ......nvl‘J: -.'.v.os.nh.lu:.tholi|.r ...:1..-.l_. .. v a . .‘vuyh. . . . In... .mn...\. ~v ,v n . ..dI attilntarl luv... . . ... .. . .. . - - v .. l . . . . . .- . , .. . .I. .... 7. v 1-... . . .. . .. r .Latifuflukxillt .. . . . . ......mw............. - i- .... I...~...l.c mvwnn—gg .u-;‘.; ‘_ . . o. pout-I‘Hno'.)blub b .... 6.0!! , ..I . .. . .vflfi tut. ...., mun“. J. ....v .. H .44.. ’n Illl”INIIUIIIHIIHHHllilllllHIIHHIIUIIUHHIIHIW 31293 01570 6520 This is to certify that the dissertation entitled Models of Superconductivity: A Josephson Junction with Thin Norma] Layer presented by Joan Cecile Remski has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics //-L/ Major professor C; 1.4 Iv e, b L( Date April 21. 1997 MS U is an Affirmatiw Action/Equal Opportunity Institution 0-12771 LIBRARY Mlchlgan State . Universlty PLACE ll RETURN BOX to remove We checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE ”l MSU Is An Afflnnetlve Action/Equal Opportunity Inetltwon mm: MODELS OF SUPERCONDUCTIVITY: A JOSEPHSON JUNCTION WITH THIN NORMAL LAYER By Joan Cecile Remski A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1997 ABSTRACT MODELS OF SUPERCONDUCTIVITY: A JOSEPHSON JUNCTION WITH THIN NORMAL LAYER By Joan Cecile Remski When a thin layer of normal (non-superconducting) material is placed between layers of superconducting material, a Josephson Junction is formed. This paper con- siders a model for a Josephson Junction based on the Ginzburg—Landau equations as the thickness of the normal layer tends to zero. The model is first derived formally by averaging the unknown variables in the normal layer. Uniform bounds and conver- gence are established, as well as an estimate for the order of convergence. Numerical results are shown for a one-dimensional junction. ACKNOWLEDGMENTS I would like to thank my advisor, Professor Qiang Du, for his guidance, support, and patience. I am also very grateful to Professor Patricia Lamm for her advice and encouragement. iii TABLE OF CONTENTS LIST OF FIGURES INTRODUCTION 1 Superconductivity 2 3 1.1 1.2 1.3 Ginzburg—Landau Free Energy ...................... Ginzburg—Landau Equations ....................... Properties of Solutions .......................... Josephson Junctions 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Notation .................................. The Free Energy of a Josephson Junction ................ Equations for Josephson Junctions ................... Gauge Transformation .......................... Formal Derivation of the Model ..................... Next Order Correction .......................... An Equivalent Formulation ........................ Uniform Bounds .............................. Passing to the Limit ........................... 2.10 Time-Dependent Equations ....................... 2.11 Order of Convergence ........................... Numerical Results 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Finite Difference Equations ....................... A Special Case of the Difference Equations ............... Convergence of the Difference Equations ................ Convergence of the Leading Order Equations .............. Comparison of Solutions for Various n ................. Comparison of Solutions for Various (1 ................. Numerical Solutions of a Josephson Junction .............. iv vi “03.51% CO 9 10 11 12 13 17 19 20 23 26 30 38 38 41 43 45 48 CONCLUSION BIBLIOGRAPHY 63 64 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 LIST OF FIGURES Comparison of |\II€| for two different grids. Left: At = 10"3 and A2: = 0.05, Right: At = 5 x 10‘4 and Ax = 0.025. .............. Error in approximating by the leading order term at time=.05. Left: 6 = .02, Right: 6 = .01 .......................... Error in approximating by the leading order term at time=2.0. Left: 6 = .02, Right: 6 = .01 .......................... Error in approximating by the leading order term at steady-state. Left: 6 = .02, Right: 6 = .01 .......................... Error in the steady-state solution after the correction term has been added. Left: 6 = .02; Right: 6 = .01. .................. Comparision of the order parameter for e = .1 in type I materials. T0p Left: Leading order solution, |\II|, Top Right: Corrected solution, |\II + e‘Il(1)|, Bottom: SNS solution i\II£| ................. Comparision of the order parameter for e = .05 in type I materials. Top Left: Leading order solution, |\II|, Top Left: Corrected solution, |\II + 611ml, Bottom: SNS solution, |\I'€| ................ Comparison of the magnetic field for e = .05 in type I materials. Left: SNS solution, 6062/6113, Right: Leading order solution, Bag/6x . . . . Comparision of the order parameter for e = .1 in type II materials. Top Left: Leading order solution, |\II|, Top Right: Corrected solution, |\II + 611ml, Bottom: SNS solution, |\Ile| ................ Comparision of the order parameter for c = .05 in type II materials. T0p Left: Leading order solution, |\I'|, Top Right: Corrected solution, |\II + diml, Bottom: SNS solution, [\Ilel ................ Comparison of the magnetic field for 6 = .05 in type II materials. Left: SNS solution, 6a,,2/6x, Right: Leading order solution, Bag/6:1: . . . . Comparision of the order parameter for e = .05 and a = 5. Top Left: Leading order solution, |\II|, Top Right: Corrected solution, |\Il+e\II(1) I, Bottom: SNS solution, |\II€| ....................... vi 44 46 49 3.13 Comparision of the order parameter for c = .05 and a = .5. Top Left: Leading order solution, |\11|, Top Right: Corrected solution, |\II+£\II(1)|, Bottom: SNS solution, |\Ilel ....................... 56 3.14 |\IJ£| at steady-state for different applied fields. TOp Row: Hext=.005; Middle Row: Hezt=.05; Last Row: He$t=.1 ............... 58 3.15 |\II¢| with Hm = .2 ............................ 59 3.16 |\IJ6| at steady-state for varying sample lengths ............. 60 3.17 Comparison of |\II,| for different values of Is. Top: n = .5, Middle: n = 5.0, Bottom: K. = 10.0 ........................ 62 vii Introduction Superconductivity occurs when certain metals, such as mercury, tin and lead, are cooled to below a critical temperature, usually on the order of -450 degrees Fahren- heit. It was first observed in 1908 by H. Kammerlingh—Onnes and is characterized by perfect conductivity and perfect diamagnetism. Perfect conductivity means that there is no electrical resistance in a sample; at M.I.T., Collins observed current flow- ing in a closed loop of material for over two and a half years without decay [13]. Diamagnetism implies that superconducting materials will repel external magnetic fields with magnitude below a critical external field. Before the physics of superconductivity was understood, a number of phenomeno- logical models were pr0posed. Macrosc0pic models of superconducting phenomena have been proposed by London and London in 1935, Pippard in 1950 and by Ginzburg and Landau also in 1950. In 1957 Bardeen, Cooper, and Schrieffer published a widely accepted microscopic model that was later shown to be equivalent, under an appro- priate limit, to the Ginzburg-Landau model. For a more complete discussion of these models we refer the reader to [12], [13] and [14]. Because of the rigidity of superconducting materials at such low temperatures, samples consisting of both non-superconducting (normal) and superconducting ma- terials are of interest in both the study of superconducting phenomena and the design of superconducting devices. A sample in which a layer of normal material is sand- wiched between two layers of superconducting material is called a Josephson junction or SNS junction. In these devices the proximity effect is observed near the interface of the two materials: the presence of the normal material hampers the pairing of the electrons in the superconducting layer and the presence of the superconducting layer promotes the formation of electron pairs in the normal material. In other words, near the interface, we observe a non-zero density of superconducting electrons in the normal material and a reduced density of superconducting electrons in the supercon- ducting material. The strength of the proximity effect is determined by the physical properties of the normal layer. Interesting quantum mechanical effects also occur in an SNS junction due to the fact that the Cooper pairs can tunnel through the interface. These effects are named after B. D. Josephson who was given the Nobel prize in physics for this discovery. The quantum effects became a basis for many applications of superconducting devices such as SQUIDS [12]. Thus it is of great interest to be able to numerically simulate the physical pr0perties of such junction devices. In this paper, we consider a model for a Josephson junction based on the Ginzburg- Landau theory and take advantage of the fact that the normal layer is very thin in order to derive a simplified model. The asymptotic behavior of the solutions of the Ginzburg—Landau equations as the layer thickness tending to zero has been studied much in the literature. In [3], a thin superconducting film is studied. Even for SNS junctions, discussions of the super-current across the junction have already been made in [2] for one-dimensional junctions. In [9], more detailed studies were made for junctions that have very thin normal layer. Nevertheless, the model equations derived in [9] remain to coupled nonlinear equations in the leading order. It is the purpose of this paper to show that, to leading order, the thin normal layer induces negligible influence on the junction device and thus the resulting equations could be even simpler. Higher order corrections, rigorous convergence theory for the steady state models, analysis on the order of convergence for the time-dependent models as well as numerical simulations are also presented here. Chapter 1 contains a brief overview of the Ginzburg-Landau theory and is orga- nized as follows. Section 1 contains a brief description of the free energy for a super- conducting sample proposed by Ginzburg and Landau. In section 2, the dimension- less free energy is given along with the corresponding Ginzburg-Landau equations. The last section discusses various properties of solutions of the Ginzburg-Landau equations that will be used in Chapter 2. In Chapter 2, new results are presented for the Josephson junction. Section 1 describes the notation that will be used throughout. In section 3, we describe the Ginzburg-Landau type models for the superconducting—normal-superconducting junc- tions. In section 4, a formal derivation of the simplified model is given in the steady state case and a rigorous convergence proof is given in the section 5. The time- dependent equations are considered in the section 6, together with an estimate on the order of convergence to the leading order equations. Results of various numerical experiments are presented in Chapter 3. The numer- ical scheme, along with solutions of the Josephson junction problem are included. CHAPTER 1 Superconductivity 1.1 Ginzburg-Landau Free Energy Ginzburg and Landau postulated the the Gibbs free energy density, in the presence of an applied magnetic field was given by [H]2 _ H - Hm 87r 47r g = f~+a|‘1’|2+§|‘I/|4+ 2 +1 2 ms (—z‘hV —- as?) \I: where \II is the complex-valued order parameter such that IIIII2 represents the density of superconducting charge carriers, A is the vector-valued magnetic potential, and f N = constant-valued free energy of the non-superconducting state, a, ,6 = constants that depend on temperature, H = curlA, the magnetic field, c = speed of light, 6, 2 charge of superconducting charge carriers, m, = mass of superconducting charge carriers, 27rh = Planck’s constant. According to the microscopic theory, the superconducting electrons occur in pairs, so the value as will be twice the charge of an electron and similarly m, will be twice the mass of an electron. This pairing will imply that |‘II]2 will equal half the density of superconducting electrons. Next we briefly discuss the terms in (1.1). For a more complete description of the Ginzburg-Landau theory refer to [12]. Note that fly + %E is the free energy per unit volume associated with the normal (non-superconducting) state. The term —LI'TWIL“ is the work (per unit volume) done by the electromagnetic force induced by the external field H m. Ginzburg and Landau introduced the complex-valued pseudowavefunction \I/ and considered the free energy density near the transition temperature where one would expect \II to be small and vary slowly in space. They assumed that the free energy could be expanded in powers of I‘ll]2 and [V‘II|2. By choosing an expansion in even powers of [\III, we get 9 to be real valued, independent of the phase, and also analytic at \II = 0. Note that the free energy should be penalized for variations in the order parameter, so one would expect a term proportional to |V\II|2. Ginzburg and Landau considered the last term in (1.1) to be the added energy density associated with spatial variations in \P in gauge-invariant form. If our superconducting sample occupies the region Q, then the physical state is given by the minimizer of gut/1) = Ad9 2 2 . +/( 1 (WM)... + IHI LIL)... (1.2) {2 2m, c 87r 47r Note that the constant 6 must be positive in order for g to have a minimum value. The values of a and fl depend on the temperature which is assumed fixed in the Ginzburg-Landau theory. If the temperature T is above the critical temperature Tc, we have a > 0, and one can see that g is minimized by \11 = 0 and curlA = H cm. This corresponds to the non-superconducting state: no superconducting electrons are present and the external magnetic field penetrates the sample. On the other hand, if the temperature T is below the critical temperature Tc, we have a < 0, and the energy is minimized without |\II| identically zero. For this reason, a generalized Ginzburg— Landau model was proposed in [2] that allows the constant a to have different values in the normal and superconducting layers (see section 2.2). 1.2 Ginzburg-Landau Equations By choosing an appropriate length scale, the non-dimensional free energy for a su- perconducting sample occupying the region Q is given by [5], [7]: 1 2 2 z’ 2 9015A) = [92' (m —1) +];V\II+A\II datdydz + [m lcurlA — Hm|2 dasdydz, (1.3) where \II is the complex order parameter such that [‘11]2 represents the number density of superconducting electrons (|\II| = 1 corresponds to the superconducting state, [\III = 0 corresponds to the normal state). A is the vector magnetic potential and h = curlA. H m is the constant external field and re is a material parameter which determines the type of superconducting material; K, < l/x/2 describes a type-I superconductor, K: > l/x/2 describes a type—II superconductor. Using standard techniques from the calculus of variations, the corresponding di- mensionless Ginzburg-Landau equations are 2 (lV—iA) \II = (|\1:|2—1)\I: in $2, (1.4) K n (curl)2A = 5’;(\pv\r—w*vw)—|w|2.4 inQ, (1.5) (curl)2A = 0 in 1R3\Q, (1.6) with boundary conditions [curlA] = 0 , (1.7) [n - A] = 0, (1.8) n- (%V — iA) ‘11 = 0 on 052, (1.9) curl A —> Hm as Ice] —> oo, (1.10) where n denotes the outward unit normal to an, " denotes complex conjugation, z' = \/—1, and [] denotes the jump in the enclosed quantity across 89. 1.3 Properties of Solutions The existence of solutions of (1.4)-(1.10) has been established in [7]. For the non- stationary model, existence, uniqueness and regularity results may be found in [4]. Other results, which will be used in the next chapter, are given below. Since the order parameter ‘1! behaves like a wavefunction, Ginzburg and Landau constructed their free energy functional to be gauge invariant. In other words, if (\II, A) is a minimizer of 9, then so is G¢(\II, A) where, for any ab 6 H 2(9), we define G¢(\I’) A) = (C: Q) if C=\Ilei"¢ and Q=A+V¢. This property demonstrates the existence of equivalence classes of minimizers to (1.3) so that by choosing a particular gauge, one determines a member of the equivalence class. The next result which may be found in [7], gives a useful bound on the order parameter. Lemma If (\II,A) is a solution of the Ginzburg-Landau equations (1.4)-(1.10), then [‘III S 1 almost everywhere. Note that since [\I'I represents the density of superconducting electron pairs, we have |\II| = 1 corresponds to the ideal superconducting state, while I‘ll] < 1 corresponds to the non-superconducting state. CHAPTER 2 Josephson Junctions 2.1 Notation We begin this section by introducing some notation that will be used below. Through- out, for any non-negative integer k and domain ’D C IR", n = 2 or 3, H "(D) will denote the Sobolev space of real-valued functions having square integrable deriva- tives of order up to k. The corresponding spaces of complex-valued functions will be denoted by ’H"(D). Corresponding spaces of vector-valued functions, each of whose components belong to H ”(D), will be denoted by H"(D), i.e., H"(D) = [H "(D)]". Norms of functions belonging to H k (D), H"(’D), and ”H" (D) will all be denoted, with- out any possible ambiguity, by ” - “km or [I - ”k. The latter notation will be used when there is no chance of confusion. For details concerning these spaces, one may consult [1]. We will also use the H(IR3) = the closure of the set C3° (1R3) 1/2 under the norm (/ 3 |VA|2d93 dy dz) IR and H(div, 1R3) = {A e H(IR3) : divA = 0)} 10 2.2 The Free Energy of a Josephson Junction Let D be a bounded region in 1R2. We consider a three dimensional Josephson junction occupying the domain (2 which is symmetric with respect to the (x,y)-plane given by n = { (mm) e R“ = (23,31) e 0. —d(a:,y) < z < d 0 for all (x, y) E D. The Josephson (or SNS) junction 9 is formed by sandwiching a normal layer, SIN, between layers of superconducting material occupying (25," for m=1, 2. More specifically, m = {(x,y,z)eIR3 : (x,y)eD, —edo(x.y) 0 is a constant determined by the material in the normal layer, QN. For detailed discussion on the value of a for various junctions, we refer to [2]. 11 2.3 Equations for Josephson Junctions The physical state is described by (\IIuAe) which is the minimizer of 96. Such a minimizer satisfies the following dimensionless Ginzburg-Landau equations which hold in Q [2]. (iv—info. = (m2— 1) x11. in 95",, m=1, 2 , (2.1) (fir/4.492% = my. in $2”, (2.2) (curl)2A, = Egg-(rye; — \rgvr.) —|\II.|2AE , in as, UQN U952, (2.3) (curl)2A¢ = 0 in IR3\Q, (2.4) with boundary conditions [curlA.]an = O , (2.5) [Aslan = 0 v (2-6) 1 n- (EV — iAe) \II,E = 0 on 00, (2.7) curl A. ——> Hm as [an] —> 00, (2.8) and interface conditions [A6]” = 0, l‘I'clrm = 0, for m=1, 2 , (2.9) [(i—V — iAe) \IIe - n] = 0, for m=1, 2 , (2.10) I‘m 0, for m=1, 2 . (2.11) [curlA. x "l1“... 12 Here, the interface is given by Pm = OS," 0 ON for m = 1 ,2, n is the unit outward normal to the indicated boundary or interface, []1‘ denotes the jump in the enclosed quantity across the boundary, I‘, and ‘ denotes complex conjugation. The existence and uniqueness of solutions to the time dependent version of equa- tions (2.1)-(2.11) are established in [9] as well as regularity results and a leading order approximation. Some of these results are restated in Chapter 2, section 10. 2.4 Gauge Transformation Using the gauge transformation from [7], Q = A6 — if, ‘116 = feix with f, X real (X = 0 in R3\Q) equations (2.1)-(2.11) can be rewritten 1 . EVZ’f = f3—f+f]Q[2 anSm,m=I,2, 1 —,W = af+lel2 mm, K? div (fi’Q) = 0 (mm = 4262 (curIVQ = o H = curlQ, with boundary conditions [curl Q] on [Q] on 5f (in in 08; UQN U952 , III 951 UQNUQSZ, in IR3\Q , in $25, UQN U052 , 0, 0, 0, on 80, Hm as lml —+oo, (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) 13 and interface conditions [62],,“ = 0, mm = 0, for m=1, 2 , (2.22) [curlQ x n]1.m = 0, for m=1, 2 , (2.23) [2:] = 0, for m=1, 2 . (2.24) 811. Pm 2.5 Formal Derivation of the Model We consider the formal asymptotic limit of the solutions to the Josephson junction equations as e -) 0 with I»: fixed. The existence of various limits and the validity of the limiting processes which arise in this derivation will be justified later in the chapter. In order to determine the leading order equations, we will use the scaling Z =- d(z — edo)/(d — edo) on 95,, and define {2291) ={(:1:,y,Z) :0 < E < d(a:,y)}. By the change of variables 2 —> E in (2.12), (2.14), and (2.15) and letting e ——> 0, we get the following equations 1 . givzfm) : (f(0))3 _ f(0) + fm) |Q(0) [2 m (2)501), (2.25) div ((f<°>)2Q(°>) = 0 in 9‘59), (2.26) (curl)2Q‘°’ = -(f‘°’)2Q‘°) in 9203, (227) where ( f (0), (2(0)) denote the leading order behavior of (f, Q). Note that by a similar 14 scaling argument we find that (f (0), Q(0)) satisfy (2.25) - (2.27) on the scaled domain: 52‘5".) = {(22.22) : -d(z,y> < 2 < 0}. Now we must consider what happens on the interface of the two new regions (2201) and (la). If our leading order solutions are smooth across the interface, then the Ginzburg-Landau equations for a superconducting sample will be satisfied. In other words, up to leading order, the solution behaves as though there is no normal layer present. In this section, we define (fsm, Qsm) = (f, Q) for (x, y, z) 6 Q5," and m:1, 2, and (fly, QN) = (f, Q) for (:r, y, z) E (IN. Denote the average value of a function g by _ 1 .2. d g—EI—edogz. Integrating (2.13) from —edo to edo, we obtain: 1 6f” (do +V2f -a7_+f IQ I2 2526d0 62 _(d0 N_ N N N ’ where — 3 3 ‘7—(55155) So assuming fN, Q N, and I72fN are bounded we have an ‘d" —£do Using the interface condition (2.24) we have afsm = _a_fli i CVfN . V510, on Fm,m:1, 2 an 32 which implies fla 6n edo Soase—>0, _ 612. an _6 do 012?) 62 ___ afé‘l’ 15 62 «do +€Vdo ° (Vlecdo + Vle—edo) 0(6) oz (2.28) 0- N ow we consider the interface condition for Q. Using (2.17) and writing out (2.15) in component form 8h“, Bth ( By - 8h N,1 Ohms ( (92 _. mag 3hN,1 ( 0x — ) : —fl2VqN,l a (2'29) ) = “'f12VCIN,2, (2-30) ) : _f12VqN,3 :1 (231) where hNJ and qNJ- represent the jth component of H N and Q N respectively. Also, since H N is divergence free: aim,1 ahN,2 ahN,3 8x+8y+az = 0. (2.32) Integrating equations (2.29)-(2.30) and (2.32) from —ed0 to edo we have BhN fI2VqN,1 — ay,3 1 —— 8h~ 3 2 1 6:1: 03/ 16 Letting e -—> 0 we have edo H53) = 0(6). Using the interface conditions (2.23), this implies curngOl) x 13]de — curngl) = 0(6). X11 —6 Or,ase—>O: curng? x k[0+ = curngl) x k]0_ , where Ic denotes the unit vector in the z-direction. So the leading order equations are (dropping the superscript (0)): 231721 = f3—f+f|Q|2 magi. m=1,2, div (sz) = 0 in 9‘50], m=1,2, (curl)2Q = —f2Q in 93:, m=1,2, (curl)2Q = 0 in IR3\Q, H = curlQ, inf)???“ m=1,2, with boundary conditions [curl Qlan = 0, lQlan = 01 8_f = 0 onBQ, m=1,2, an H —> Hm as]:c|—>oo, (2.33) (2.34) (2.35) (2.36) (2.37) (2.38) (2.39) (2.40) (2.41) (2.42) 17 and interface conditions llezo :- 01 [f]z:0 = 01 (243) Of _ [52].-. ‘ 0’ (“4) [curlQ X k]z=0 = 0. (2.45) Remark 1. The above equations imply that, to leading order, there is no coupling between the solutions of the superconducting layers and the normal layer as the normal layers thickness tends to zero (6 —> 0). So, at least to the leading order, the Josephson junction is not affected by the properties of the middle normal layer in this limit. 2.6 Next Order Correction Having determined f (0) and Qm) from the leading order equations, we may determine the next order correction by first scaling equations (2.1)-(2.11) and then expanding f = f‘°’+ef“’, Q : Qm) + €Q(1). It can be shown that f (1) and Q“) satisfy the following equations in 9?; for m=1, 1 9W“) = f“) (30”)? —— 1+ IQ‘0’I2) + 2Q(1) °f(0)Q0, Vf(0) . (2f(1)Q(0) + f(0)Q(1)) = 0, 18 curmm = 20>(282524.828), H“) = curlQm, and curlell) = 0 in IR3\f2, with boundary conditions [curl QM]an — 0, llean : 0’ Of“) 6n = 0 on 89 , H(1)—> 0 as]:c|—>oo, and interface conditions 081.- [c231 z=0 ) lafl1 i = 218261001 +1lf(0)(93ay,0) 2:0 —2I€2d0(f(0))3(xa ya 0): [curlQm x k] = 0. Remark 2. The observation of the above equations illustrates how the SNS junction might be affected by the properties of the middle layer in the next order correction when the normal layer’s thickness goes to zero. The material parameter oz also appears in the interface conditions. 19 2.7 An Equivalent Formulation Similar to [3], define B = B (:0, y, 2) such that V-B 0 in Q, (2.46) curlB = Hm in R3, (2.47) (e.g. for Hm = (H1,H2,H3) with H,- = constant for i=1,2,3, then B = (zHg,xH3,yH1) satisfies (2.46)-(2.47)) We can then define an equivalent formulation of the free energy: 2 :Z/s... %(—|\p|’~’ + )lng + A1: + 3111 dxdydz Z 2 / —|\II]2 + — dxdydz nN +/1R3 IcurlA|2le3. (2-48) For convenience, we also use 2 2 221/05 m;((|\II|2— ) dxdydz 1: 2 +/ (% (—|111|2 1)l::[;V\II+A\II+B\II )dzdydz + / IcurlAlzdxdydz, IRS We consider the following variational problem 1 ” 11: A . () weH1(nr)n,iileH(m3)g( ’ ) 20 Using a gauge transformation, the above problem is equivalent to (H) 55011.14) - min \IIEHlm) ,AeH(div,IR3) Let (\IIC,A,) denote a minimizer of §£(\II,A) in 711(9) x H(div,IR3). Note that (‘11,,AE + B) is then a minimizer of g,(\II, A). 2.8 Uniform Bounds Here, we derive some uniform bounds, independent of e, for the minimizer (\II,, A,). First, since (\II, A) = (0,0) is feasible for (11), we get: Lemma 1 For any 6 > 0, ~ 9411104.) 3 [D 48.244424. Thus, we get: Corollary 1 For any 6 > 0, 1/2 11741.1le43): ([0 d(:v,y)dxdy) . Proof: Since for any A E 08°(IR3) with divA = 0 we have the norm equivalence ||curlA||i2(m3) = ||VA||i2(m3) VA e H(div, 1R3). Cl 21 Using a standard interpolation inequality [10] llAllL6(1R3) S CllVAllL2(IR3) we get, Corollary 2 There exists a constant c > 0, such that for any 6 > 0, llAellL6(m3) S C- Now, we consider the restriction on 52. From Corollaries 1 and 2, and the imbedding L6 (Q) <—) L2(Q), we have the following result: Proposition 1 There exists a constant c > 0, independent of e, such that: ”AallHlm) _<_ C - Next we consider the order parameter. Similar to [4], we can show that: Lemma 2 ll‘I’ellcmm) S 1' By Lemmas 1 and 2, and Proposition 1 we get: Lemma 3 There exists a constant c > 0, independent of e such that: llV‘I’ellc2m) S 6 So by Lemmas 2 and 3: 22 Proposition 2 There exists a constant c > 0, independent of e,such that: ll‘pcllulm) S C- Using the above estimates, we get Proposition 3 There exists a constant c > 0, independent of e,such that: ][‘I’€A£[]w1,3/2(Q) S C. Proof: By Proposition 1, and Lemmas 2 and 3 we have: 2 3 2 3 ll‘I’eAcllyCm/zm) 5 C + ”V‘I’e ° Aellcé/zm) S C + llV‘I’ellcz(n)llAell56(n) |/\ c.[:] Using the standard elliptic regularity results and bootstrapping argument, we can show that Proposition 4 There exists a constant c > 0, independent of e, such that H‘I’dlwm) S C - Proof: By taking the £3/2(Q) norm of equations (2.1)—(2.2) and using Lemma 2, we get: [[Vzlpellcs/2(Q) _<__ C ‘l‘ “V‘pe ’ A6[[£3/2(Q) . So from Proposition 3: V2111. e 133/262) 23 and W. e w1’3/2(o) «—+ Me). Now by taking the [12(9) norm of equations (2.1)-(2.2), using Lemma 2 and Propo- sition 1: ]]V2\I’€[]£2(n) S C + ”V‘IIE ' Aell£2(fl) < c + ”V‘IIEHL3(Q) llAell£6(0) |/\ c. D (2.49) For magnetic vector potential, by considering equations (2.3)-(2.4) over the whole space and using the previous lemma, we get Proposition 5 There exists a constant c > 0, independent of e, such that “ACHH2(Q) S C . Now by taking the [1"(52) norm of equations (2.1)-(2.2) for some 6 2 r 2 2 and using Proposition 3 we get: Proposition 6 For any 6 Z r _>_ 2, there exists a constant c > 0, independent of e, such that ll‘I’cllwwm) S C - 2.9 Passing to the Limit By Propositions 1-4 we have Theorem 1 There exists a subsequence {ck} with e), —> 0 as k —> 00 such that 24 111., _. W in 742(0) , 21., .1 71‘ in H2(o) 0 HOW) . By Sobolev compact imbeddings theorem, after further extracting a subsequence, we can assume that Corollary 3 There exists a subsequence {6);} with 6]; ——> 0 as k —) 00 such that \Ilék —> W in 711(9), and A“ ——> A in H1(Q). Using the estimates in the last section we get from Hdlder’s inequality that .9. 2 _l 2 _ 2) V0” (2 Iw‘kl 2(|\I'.,.| 1) 61-734de S 66k) where c is a constant independent of ck. Therefore, we get 9141/)... A..) — 9(4).“ A.,,) = 0. lim (jg-)0 It is not difficult to check that the functional Q is weakly lower semi-continuous in 711(0) x H(IR3). Thus we obtain 613310 91,, (won A61) 2 9(21) We now show that Theorem 2 (EA) is a minimizer of g. 25 Proof: Suppose that there exists a minimizer (C, Q) of 9 such that 9&3) > 9+6: 9(C,Q)+5 for some 6 > 0. Similar to lemma 2, we have that llCllcoom) S 1- Then for k large, we have " _ 9f 2 _ l 2 _ 2 2.3.0) — gee) — f9” (2141 200 1) )dxdydz < 6/2. Meanwhile, for 6,, small, we may also have g~€k ($61“ Aék) — g($,Z) > __6/2 So, 91" (work, Ark) > 9(E1Z) _ 6/2 > 9(C,Q)+5/2 > G..(C,Q) 2 min 91k . This is a contradiction. The above theorem illustrate that the leading order approximation of We“ A6,) is given by the minimizer of g which satisfies the corresponding Euler-Lagrange equa- tions given in (1.4)-(1.10). 26 2.10 Time-Dependent Equations The time-dependent Ginzburg-Landau equations are often used to study the motion of vortices in type-II superconductors [6]. The equations for Josephson-junctions de- veloped in [2] were generalized to the time-dependent case in [9]. For simplicity, we ignore the effect of the exterior of (2. From [9] under an appropriate gauge transfor- mation, the time dependent equations are given by: 6AE 8t — Ver + ll‘Pel2Ac with boundary conditions, 6‘11E 6n interface conditions, -0, AE-n curlAE x n (V — 88.43211. 11!. (1 — 02.)?) , in QSmT,m=1, 2 (V — 18.432111. 01‘1’6, in QNT V(V . A.) + |\II.|2A. i — * we _ c * 25 (‘1’.V 11/ V111,) 0, in QSmT,m=1, 2, 1' — 111* 11:. — 11:. * 2f€( 6V V‘IIC) 0, in 0NT, 0 on 69 x (0,T), Hext x n on 89 x (0,T), (2.50) (2.51) (2.52) (2.53) (2.54) (2.55) 27 [AflI‘mT = O, [‘IIE]PmT = 0, for m=1, 2, (2.56) [V - A£]rmT = 0, for m=1, 2, (2.57) V — MAE \IIE = 0, for m=1, 2, 2.58) I‘mT [curlA.E x "ll“mr = O, for m=1, 2 , (2.59) and initial conditions: @5090) 2 W200, A€(X,0) : ASK"), where OT 2 Q x (O, T), QsmT = 93m x (0, T), for m=1, 2, QNT = 9N x (O, T), and the interface is given by I‘mT = I‘m x (0, T) for m=1, 2. Using the same method as section 3, one can show the leading order equations are: (9:11 — in‘IlV - A) — (V - MA)2 ‘11 = 11(1— |\II|2) , in 991T, m=1, 2 (2.60) 797+curlA — V(V-A)+|\I'|A i * $ + 576-01! V\P — \IIV\II ) = 0, in 993T, m=1,2 (2.61) with interface conditions, [A]{z=0}X(0,T) = O ’ [\P]{z=0}x(0,T) = 0, (2-62) 28 [V ' A]{z=0}x(0,T) = 0, (2-63) [(58- — inag) \Il] = 0, (2-64) 2 {z=0}x(0,T) [curlA x k] {2:0}X(O,T) = 0, (2.65) boundary conditions, g—il-=O,Aon = 0 onBQX(O,T), (2.66) curlA x n = H x n on 69 x (0, T) , (2.67) and initial conditions \I'(x, 0) = \II°(x), A(x,0) = A°(x). Let (\116, A.) denote the solution of (2.50)-(2.53). To show convergence, we quote the result from [9]: Lemma 4 Assuming smooth conditions for the data: 8DEC2, ‘1’26 H1(Qsm)fl£°0(flsm),A2 E H1(Qsm), m: 1, 2 V? e 711(QN) n c°°(nN) ,AS 6 H1(QN), 29 there exists a unique solution to (2.50)-(2.53) such that A. E H1(0, T; L2(9)), we 6 H1(0,T; 32(9)) ‘7 £°°(9T), A. E L2(0,T;H2(9)), ‘1’. E L2(0,T; H261» , and II‘I’ellcme) s ma${1,ll‘1’°l|c°°(m} - Proposition 7 There exists a subsequence {6);} with 61c —-> 0 as k —> 00 such that A6, _. K in mm, T; L2(Q)), ‘11., _. E in H1(0,T; 3(9)), and A“ —-* X in L2(0,T; H2(Q)) , in, —~ V in L2(0,T; 212(9)) . Using compactness of the imbedding H1(0, T; L2(Q)) ‘—> L2 (O, T; L2(Q)) we get: Corollary 4 There exists a subsequence {6);} with 61c —> O as k —> 00 such that A6,: -—> K in L2(O, T; L2(Q)) , \Ilek ——) if in £2(O,T; £2(Q)). 30 Similar to the steady state case, we may show that the limit is the solution of the original time-dependent Ginzburg-Landau equations (with no middle normal layer present). Based on the uniqueness of the solution for the time-dependent equations, we see that the whole sequence actually converges to the same limit. 2.11 Order of Convergence In this section, we show that the convergence is of order 6. Using the identity curlzu :- V(V - u) -— Au, the weak form of the equations (2.50)-(2.53) for a Josephson junction are: ‘1’ as — 55V ~ 21.6.6 + (V — mum. - (V + 55A.)¢}dxdt 2 T .2213/0 98m x1141 — I‘I'fl )¢dxdt— a/ / \Ilodxdt, (2.68) n+VA..Vn + |V.’|2A.n+2 (\11; pr— —\IJV\II:-) n}dxdt = [T /5( 11)....” ndet, (2.69) where <15 6 £2(O,T;H1(9)) , 77 = (771,772,773)T 6 132(0, T; H1(Q))- Similarly, the weak form for the leading order equations are: [OT / {—¢ — mV AIMS + (V _ WAMP ' (V + inA)¢}dxdt = 2 [OT f 20— le2)¢dxdt, (2.70) 35,?) f [a {— “VA V77 + IwIZAn+i(w*Vw—wvw*)-n}dxdt = [OT/an( H)e,,.txn ndet, (2.71) 31 Subtracting (2.70) from (2.68) and (2.71) from (2.69), writing \Ild = ‘116 — it and Ad = A.E — A and adding and subtracting appropriate terms we obtain the following. [T / {%t—‘I"‘¢ + (V—inAJWd-(V+inA.)¢+in(V—inA)\IJ.Ad¢ — iK(V—iI€A£)¢Ad\I’E — molly . Ad + de - A)¢}dxdt 2 T _ _ 3 3 _ 213/0 fnsmlw“ \II€+¢}¢dxdt t + [O/QN{(—~a—1)\IIE+\IIE}¢dxdt. (2.72) 6A / fa, {— d + Vlad-www.12—IwI2)A.n+I«/712Adon + (wngd + 213% — qug — \IIdV7p*)}dxdt = o. (2.73) Let 45 = ”0,011; in (2.72), where X(0,t) is the characteristic function for the interval (0, t). Then (2.72) can be written as: f [6; —q:;dx xdt + /( [(V —- 575A )\Ild(V +mA )xpgdxdt = 45/0 fn( V—mAw-Adxp; +in/Ot/Q(V—in)\IIQ-Adw t +575 / / (1:.V - Adxp; + |\Ild|2VA) + Z [0‘ (a... {17.12 +‘I'2(1/23 — 73)} +£7sz {(—a — 1)\I:. + V3} \II;. (2.74) Labeling the integrals in (2.74) [1,12,...I7, we now estimate each integral. Since \Ild(x, 0) = 0 we have: 12 32 all.) t 1 I=//—"d =—/\P2ddt. 1 onat‘pdxdt andl .IL' 2 By H61der’s inequality: lot [6 (V ‘ We) ‘I’a - (V + 75A.) 11365.77 ft] |V\Ild _ inAe‘pdlzdxdt 0 Q [at [a ||V\Ild| — nlAewdllzdxdt. (2.75) t t t 25/ / |prd||A.\1:d|dxdtg -1- f / |V\Ild|2dxdt+2n2/ / |A€\Ild|2dxdt. o n 2 o n o :2 Substituting this last expression into (2.75): 12 IV IV lReUsll |/\ |/\ t t -1- f / |V\Ild|2dxdt—n2/ / IAEKIId|2dxdt 2 o n o n It t — V\112ddt— 2/ 21.2... [xi/26m. 213/“)de «on llumnldlx t (75/0 fnw - Adv; — mmp . Adm t [0 IIV¢||c4(n)llAdllc4(a)llwdllczm) “:2 t A A 2 [$2 t \Il*2 3(L|-5+3(Lwa t c t p/o ||Ad||2L4m> + 5 [0 ”57664761756276 K2t 762‘ —— A2... [A2 —//\I:2. +2/0n “57.501 51+, 0 nIdl 33 Note that the last inequality holds for any p > 0 and c is an arbitrary constant. Now using the imbeddings Ad 6 H1(Q) 9) L4(Q) and Vw 6 711(0) '—+ [14(0) we have for anyp>0z t 2 C t 2 2 IRe(Ia)I s Pf. “mummy/0 11771122764)de "2 t 2 2 "2 t 2 +—,—/0 “Alums/am.) +—,—/0 fnl‘I’dl . t |Re(14)| = Re{in/ (2(V—m)\II;-Ad¢}l 0 1 t t < __ 2 [f 2 - 16/71/szlv‘pdl+c o nlAdwl t t +c//|A€-Ad|2+c//|\P;wlz 0 Q 0 Q 1 t t t < — 2 fl 2 / / 2 _ 15/0 [5,va.) +c0 and) +c 0 IIAIIL ..., n1A.!) Similar to the estimate for 13 for all positive p: t |Re(15)| = Re{m/[no.V-AdVHIVdVV-A}. 0 t 5//|2.V-Ad\p;| 0 Q t c t 'A2 _ 2 p/O/nlv ..I +p/0/nlwdl t Ct A2 +-//\112 p/O u an...) p 0 0I ..1 Clearly, from the L°°(QT) estimate on 7/), ‘11.: |/\ |/\ |/\ t lRe(Ie)| s c / f In): 0 Q t mew): 3 Cf. 5 I‘m 0 34 Now, we estimate 17 by letting 7(2) = [D I‘I'dlzdwdy e w1’3/2(—1, 1) e £°°(—1, 1) Then by Hiilder’s inequality, 1 1 Ilfllwla/2(_ 1,, s c [_1 [D |\Ild|3dxdydz+c' [_1 [D |\IIdV\IId|3/2dxdydz 3 2 3 2 Cll‘pdllifln) + C'll‘I’dllc/fim)“V‘I’dllc/2m) |/\ |/\ Cll‘I’dllifim) + C’llV‘I’dll322(n) |/\ CII‘I’dllism) (275) where the last inequality follows from the imbedding 70(9) % 116(0). So, t 5 1/2 lRe(17)| s cx/E/ (_ fdz) dt 3 c/ ”fit/i -.., “/Omllfllla/fiwq 11) |/\ So that lRe(I7)| S cell‘I’dllcz(o,t;w(n))- Putting these estimates together, we obtain the following. 1 7t — \112 —//V\112 2/nldl + 160 9' dl t C t s 372/0 llAdllipm)+;/O (|l¢||%2(n)+llAell2Loo(m) [9:252 35 t t +c/0 (llAllimm) + Hana...) [0 1A.)? +C€|I‘I’d”52(o,t;ul(g)) (2.77) Similarly, letting 17 = X(0,t)Ad in (2.73) we have: $412.12 + U. IVAd|2+ [[0 5712111.)? = ttf..("1’2'2"2'2)A2‘Ad ' t +227: [0 [a (xpngd + wgVu — 11:.ng — WM“) 'Ad- Labeling the integrals on the right hand side above J1 and J2, we now estimate them using arguments similar to the previous ones. t |Re(J1)| g M) A(xp.xp;+¢*xpd)A..Ad t S f0[aII‘I’dllcz(n)llAeIIL4(n)llAd||L4(n) t c t s Pf. IIAdlli4m)+;/0 llAelli4(n)ll‘I’dlliz(n) S t c t pf. ”nutmeg/0 ”Asia... [9111612. 1 t mean) 3 fife (Iva-Adwlwdwml} 1 t 2 t 2 16/0 [‘IIV‘I’dI +C/0 fnlAdl t C t 2 2 2 +P/0 llAdllH1(Q) + E/0 “Wham/Dwell - l/\ Putting these estimates together gives: 1 t -... // A2 .1.) .7 + ...)w 36 t 1 t S 210/0 [[Adllh1(n)+1g/0 [alv‘l’dl2 C t 4-5/0 (llAellioom) + [ll/’[lii2(n)) [0 Wall2 +0 [0‘ [a 1A.)? (2.78) Now, adding (2.77) and (2.78) and choosing the parameter p sufficiently small, we obtain: é/‘JI‘I’dIQ + [Adl2} + S/Ot/n {[V‘I’arl2 + IVAdl2} t 2 2 3 Cf. hon/“hm +|Ad|} +Céll‘1’dllc2(o,t;w(n)) - (2-79) where h“) = 3UP(llAelli,oo(n) + [ll/’ll372(n)1llAlli.oo(o)+ “Aellioomo E L1(O,T). Ignoring the gradient terms on the left hand side of inequality (2.79) and applying a generalized version of Gronwall’s inequality gives: |/\ t C€||‘I’d||c2(o,x;ul(n)) (1+ / h-”<2>ds) 0 S Cfll‘I’dllcz(o,x;u1(n)) . (2-80) fn{l‘1’d|2 + [Adl2} where H (t) = f0t h(s)ds. Now, by substituting this last expression into the right hand side of (2.79) and integrating over [0, T], we obtain: ”‘I’dll22(o,r;ul(n)) + [[Adllinoxmlm» S Cfll‘pdllcszmlmn which gives the desired order of convergence. Theorem 3 Assuming the same smooth conditions for the data as in Lemma 4. Then 37 for small enough 6, there exists a constant c > 0, independent of e, such that [I‘Pc - ¢’|lc2(o,r;w(n)) + ”As - A||L2(0,T;H1(n)) S 66 - The order of convergence can be also observed in the numerical experiments pre- sented in the next chapter. CHAPTER 3 Numerical Results In this chapter we present the finite difference equations for a general one dimensional Josephson Junction as well as the difference equations for a simplified setting, and use these equations to run various numerical experiments. The simulations are grouped as follows. The convergence of the numerical scheme is discussed in section 3. In section 4, we verify of the order of convergence that was proved in Chapter 2, section 11 and show the validity of the higher order corrections. In section 5, we compare the solution of the junction problem with small values of e to the solution of the leading order equations (with and without correction) for different values of the Ginzburg-Landau parameter 75. Then in section 6 we again compare the true solution with our approximation for various values of a, the material parameter for the normal layer. The last sections in this chapter demonstrate how the solution varies as we vary the external applied field, Hm, the length of the sample, L, and the Ginzburg—Landau parameter, [5. 3.1 Finite Difference Equations Assuming 7,!) = 1/J(x), A = A(x), we will partition [-L,L] into equal subintervals of length Ax: {—L = x1 < x2 < 233... < xn = L}, letting xM = —e and xN = 6. Using 38 39 centered difference for boundary and interface conditions, the one dimensional finite difference equations describing a Josephson Junction are given below. Equations for \II = \Ilre + i\II,-m: (1 + 2x + Atn2|Aj 12) ‘14. — r1173? -— mic-1 — 215111280 — WP) - nqa’l (‘14.? - @3721) = 1142:”. j e [2,M— 1]U[N+1,n— 1) (1 + 2r + Atn2|Aj|2) 1113,, — r1133 — r1173; — 21151113,, (1 — 111111?) + 1152. (17:2 — 12:2) = W351”. j e [2,M—1]U[N+1,n— 1] (1 + 2x + AtnglAj 12) 1111,, — r1113? - r1171: + AtfiWZe — nqal (11321 — 1232‘) = wig”, j e [M+1,N— 1] (1 + 27" + AtnzlAjlz) 1113'", -— 71113,? — r1113; + Atsxpg'm + nqai (\Ilifil — \Ilie‘l) = 111132“ je [M+1,N—1] 3m ’ Equations for A = (a1, a2, a3): (1+ 21" + At|\I'j|2) ai — ra[+1— raJlV1 + £13414? — 11:21) K. - 531.6132 — 13.7.1) = 6113"“, j e [2,n — 1] (1+ 27‘ + At[\IlJ[Z) at) — rag?"1 — Tag—1 2 01,6151, j E [2,n — 1] 40 (1+ 21‘ + At|\IIj|2) a5 — rag,+1 — rag—1 = 6?”, jE [2,n—1] Boundary Conditions at x = —L: (1+ 27‘ + Atn2|A1|2)\Ilie — 2r\I!‘:'e — At‘I’ie (1 — [\Illl2) = (PLOW (1+ 21" + Atn2|A1|2) 173,, — 2mg, — Atwgm (1 — [1121(2) : ‘Illpld a1 = 0, (1 + 21° + At|\P1|2) a; — 2mg + 2qh3 = (11,0171, (1 + 21" + At|\II1[2) a; — 2mg — 2qh2 = (11,0171, Interface Conditions at x = —e: \Px+1—2\II£§+1II£§‘1 = 0, 1119:,+1—211734,+1p§;,-1 = 0, where ‘Ilie = \Ilre(xj,t), @1280” = \Ilre(xJ-,t — At), Hm = (h1,h2,h3), r = At/(Ax)2, and q = At/ Ax. Note that a similar boundary condition holds at x = L and a similar interface condition holds at x = 6. By using initial conditions with [\Ill = 1 in the superconducting region and [M = 0 in the normal region, such as j re 2 .8, 3'6 [1,M]U[N,n] t=0 1123'... .6, 3'6 [1,M]U[N,n] t=0 Wrelt=0 = \I’imltzo : 0: jE [M + 171V _1] 41 aitzo = 0, jE[1,n] solutions of the SN S Junction are obtained via Newton’s Method. 3.2 A Special Case of the Difference Equations In this section, we denote the solution of the Josephson Junction by \II6 = \Ilre + i\II,-m, A6 = (a1, a2, a3), and H ext = (0,0, h3), where \I16 and A. depend only on x and H m is constant. If the initial conditions have the form \Ilc(x, 0) = \Il,e(x) , A.(:v. 0) = (0. 112(90). 0). then by uniqueness, the solution of the SNS junction will have the form \Il£(x,t) = \Ilre(x,t), Add?) t) : (01 a2(x1t)10) : where ‘1' = ‘11", and a = a2 satisfy 2 %‘%+~2a22 = 1110—112), 11196.7 m=1,2, 2 %t‘3_%§.+n2a2\p == 01‘1’, inQNT, 2 %%_%%+ql2a : 0, anSITUQNTUQSzTa with boundary conditions: B‘II_O 6a 5?"a_x” —h3 on 89 X (0,T) , 42 and interface conditions as before. In this section the sample occupies {2 = (—L, L) with superconducting material in both 523, = (e, L) and {252 = (—L, —e) and with normal material in SIN = (—e, 6). By partitioning (2 into equal subintervals of length Ax ({—L = x0 < x1 < < 23,, = L}), with x N = e and xM = —e and using standard difference techniques, we obtain the following difference scheme: —r\IIj+1+(1 + 21' + K2Ata12)‘llj — rig--1 (3.1) — At\II,-(1— 117;) = 111;”, j e [1, M] u [N,n — 1], (3.2) —T‘I’j+1 + (1 + 27‘ + n2Ata§)\Ilj — T‘I’j_1 — aAt‘Ilj = 111;”, j e [M + 1, N + 1] , (3.3) —raj+1 + (1 + 2x + At\II§)aj — raj--1 = a3”, j E [1,n — 1] , (3.4) with boundary condition at x = —L: (1+ 27‘ + nzAta3)\Ilo — 2r1111 — 73211100 — 1113) = 1113” (3.5) (1 + 27' + Atwgfllo — 27'01 — 2TA$h3 = (1.3“ (3.6) and interface condition at x = —e: ‘I’M+1 — 2‘I’M + ‘I’M—l = 0 Here, \Ilj = \I!(xj,t) and \IIS?” 2 \Il(xj,t — At). Note that similar boundary and interface conditions hold at x = L and x = e. 43 3.3 Convergence of the Difference Equations Using different mesh sizes, we can show the finite difference scheme is convergent. Figure 3.1 shows the solution of the SNS Junction for two different grids. Note that for the leading order equations that correspond to equations (3.1)-(3.6) convergence, stability and an error estimate are established in [11]. . =1- ///"’ 100 200 0.95 t=§.0 100 200 300 400 0.95- 0.9» steady-state 100 200 300 400 0.951 0.9» Figure 3.1. Comparison of |\II€| for two different grids. 0.05, Right: At = 5 x 10‘4 and Ax = 0.025. 44 t=1.0 ," 200 400 600 “\600 . t=§.o . 2 200 400 600 800 steady-state 200 400 600 300 Left: At = 10‘3 and Ax = 45 3.4 Convergence of the Leading Order Equations In this section we use the finite difference equations to verify that the convergence is order 6. Figures 3.2, 3.3 and 3.4 show the error between the leading order solution, [117], and the solution of the Josephson Junction, [WC], with e = .02 and e = .01 respectively. By comparing the graphs, we see the convergence of the leading order terms is indeed order 6. c=o.s c=0 5 °~°15‘ 0.007» 0.014» 0.0055» 0.013 0.005» 0'012’ 0.0055» 0‘011’ 0.005» 2 400 500 0 1000 0-00452 0.009» - A . ‘ 200 400 500 800 1000 Figure 3.2. Error in approximating by the leading order term at time=.05. Left: 6 = .02, Right: 6 = .01 46 :=2.o 0.006» 0.012 0.0055» 0.011 0.005» 200 00 60 800 1000 0.0045, 0.009» 0.004 0.008» 0.0035» 0.007 0 005 h—*"' 200 400 600 800 “~1§bo Figure 3.3. Error in approximating by the leading order term at time=2.0. Left: 6 = .02, Right: 6. = .01 steady-state steady-state 0.00652 0.013 0.0062 0.012 0.00552 0.011» 0.0052 200 400 600 800 1000 0.0045 0.009 0.003. 2 400 600 0 1000 0.0035 0.007 Figure 3.4. Error in approximating by the leading order term at steady-state. Left: 6 = .02, Right: 6 = .01 47 The next figure shows the error in the steady-state solution for Re(\II) when we add the correction term to the leading order solution (i.e., Figure 3.5 show Re(\II(°)) + eRe(‘II(°)) — Re(‘Il€)). Note that by comparing the graphs for e = .02 and e = .01, the error is order 62. steady-state steady-state 0.0012» 0.0003» 0.0011» ‘ - . 0.0002» 200 40 00 800 1000 0.0009 0.0001» 0.0003» - - - - 200 400 ’1’ 500 000 1000 Figure 3.5. Error in the steady-state solution after the correction term has been added. Left: 6 = .02; Right: 6 = .01. 48 3.5 Comparison of Solutions for Various [5 In this section we comsider both type I and a type II superconducting materials and compare how the leading order solution approximates the true solution for small values of e. The next set of figures compares the solution of the Josephson junction problem for a type I material (with [C = .5) to both the solution of the leading order equations and to the corrected leading order solution for e = .1 and e = .05. The magnetic field, curlA, may also be approximated by the leading order equa- tions. Figure 3.8 compares the solution of the junction problem to the leading order solution for e = .05. 49 V S 11/ or N 0 95 0.95» 0.9» 0 9 o 85 0 85 0 8 0.8 0 75» 0 75 0 4 -10 0 10 20 0'? —10 0 i0 20 ~10 0 10 20 Figure 3.6. Comparision of the order parameter for e = .1 in type I materials. TOp Left: Leading order solution, |\II|, Top Right: Corrected solution, [\Il+e\Il(1)|, Bottom: SNS solution |\Ile| 50 1f i “r V w 0 95» 0 95» 0.9» 0 9 0 85» 0 85 0 8 0.8 0 75» 0 75 0 -10 0 10 20 0'? ~‘10 0 10 20 -10 0 f0 20 Figure 3.7. Comparision of the order parameter for e = .05 in type I materials. Top Left: Leading order solution, [\II], Top Left: Corrected solution, [‘11 + 611(1)], Bottom: SNS solution, |\II£| 51 0.5» 0 s 0 4 0.4 0.3» 0 3 0.2» 0 2 0 1 0.1 -10 0 1‘0 70 -10 o 10 20 Figure 3.8. Comparison of the magnetic field for e = .05 in type I materials. Left: SNS solution, flaw/6x, Right: Leading order solution, 602/611? The next set of figures shows the same comparison of the true solution to that of the leading order for devices with type II superconducting material. In this experi- ment we have taken K. = 2.0 and note the simultaneous existence of both supercon- ducting and normal states in the sample. Again, we see that the leading order term is sufficient in approximating the behavior of the field. 52 1) 1 0.8» 0 8 0.6» 0 6 0.4» 0 4 0.2» 0 2 -10 0 1‘0 2A0 -10 0 10 20 1» 0.8 0.6 0.4 0.2 -10 0 1‘0 20 Figure 3.9. Comparision of the order parameter for 6 = .1 in type II materials. Top Left: Leading order solution, |\Il|, Top Right: Corrected solution, [\II+6\II(1)|, Bottom: SNS solution, [‘11,] 53 1 1 0 8 0 8 0.6» 0 6 0.4» 0 4 o 2) 0 2 -10 0 1‘0 2‘0 -10 0 10 2‘0 1) 0.8 0.6 0.4 0.2 —10 0 10 20 Figure 3.10. Comparision of the order parameter for 6 = .05 in type II materials. Top Left: Leading order solution, [‘11], Top Right: Corrected solution, [‘11 + 6\I1(1)|, Bottom: SNS solution, [\IJCI 54 Figure 3.11. Comparison of the magnetic field for 6 = .05 in type II materials. Left: SNS solution, 8a.,2/6x, Right: Leading order solution, Bag/0x 55 3.6 Comparison of Solutions for Various 04 The next set of figures shows how varying the normal layer’s material parameter, a, affects the approximation. For a fixed value of 6, we see that the smaller the value of a, the better our approximation. 1 1 o 8 0.8 0.6» 0 6 0.4» 0 4 0 2 0.2 -10 0 1‘0 2‘0 -10 0 1‘0 2‘0 11 0.0 0.6 0.4 0.2 -10 0 1‘0 20 Figure 3.12. Comparision of the order parameter for 6 = .05 and a = 5. Top Left: Leading order solution, |\Il|, Top Right: Corrected solution, [\I’+€\I’(1)[, Bottom: SNS solution, [\Ilel 56 -10 0 1‘0 2‘0 -10 0 10 2‘0 - 10 0 1‘0 20 Figure 3.13. Comparision of the order parameter for 6 = .05 and oz = .5. T0p Left: Leading order solution, |\I!|, Top Right: Corrected solution, [\Il+6\II(1)|, Bottom: SNS solution, |\II£| 3.7 Numerical Solutions of a Josephson Junction By changing the external field, Hm, one may obtain different steady-state solutions to the SNS problem. Figure 3.14 shows that as the external field is increased, we no longer have perfect superconductivity in the normal region. For larger fields, we also 57 lose superconductivity at the boundary. Figure 3.15 shows that if the applied field is too large, all superconductivity is eventually lost and in the steady-state, the mate- rial is non-superconducting. Both figures 3.14 and 3.15 were obtained using constant initial conditions and the following parameters: 0.2 5 10.0 10"3 10“1 Figure 3.14. [\Ilel at steady-state for different applied fields. Top Row: Middle Row: 58 steady-state 50 100 150 200 steady—state / 5‘0 100 150 N30 steady-state 50 100 150 200 0.92 0.7] Hext=.05; Last Row: Hext=.1 Hut-£005; 59 50 100 t=50.0 150 200 0 so 100 150 200 steady—state 50 100 150 200 0 50 100 150 200 Figure 3.15. [\Ilel with Ham 2 .2 60 The solution of the SNS Junction also depends on the length of the sample. F ig- ure 3.16 shows steady-state solutions for different sample lengths, L, with 6 changing prOportionally. Again, constant initial conditions were used with the parameters: lfilhxrt At Ax 0.1 10‘3 .05 L=S. eps=.2 ‘ 200 A 400 , eps=. O \O O {D O x) L=15 6 r 100 200 300 N10 , eps=. L=10 100 0.92 L=20, 4 200 300_“‘\\\:00 eps=.8 100 0.9» 200 300 “\<:0 Figure 3.16. |\II.| at steady-state for varying sample lengths 61 By varying rs, we obtain different steady-state solutions as shown in the Figure 3.17. Note that Is: < 1/\/2_ implies the material behaves like a type I superconductor, while It > 1/\/2 behaves like a type II superconductor. The pictures show the exis- tence of both superconducting and normal states in the type II materials as well as the nucleation of the superconducting electrons for increasing values of 75 [11]. Values of various parameters are given below. E L Hext At ALE .2 20.0 0.15 10‘3 .05 62 100 200 300 400 Figure 3.17. Comparison of |\IJ€| for different values of 74. Top: 76 = .5, Middle: 76 = 5.0, Bottom: It = 10.0 Conclusion Based on the Ginzburg-Landau model of superconductivity, the equations that de- scribe a Josephson junction form a coupled, non-linear system of partial differential equations. This paper has shown, both analytically and numerically, that as the thickness of the middle normal layer tends to zero, the solution of the SNS problem converges in the appropriate space to a solution which satisfies the Ginzburg-Landau equations without a normal layer present. In other words, if the normal layer is thin enough, it has no influence on the superconducting pr0perties of the device. It has also been shown that the convergence is order epsilon, where epsilon is the thickness of the normal layer, and a correction term may be determined by an asymptotic ex- pansion in epsilon. Finite difference techniques have also been applied to verify these results numerically. Since many of these junction devices are manufactured by applying thin layers of both superconducting and normal material, it seems appropriate to consider next what happens as both the superconducting and normal layers tend to zero. This analysis may lead to investigating devices with multiple layers of different materials. Another open problem involves allowing the material coefficient for the normal layer (which has been taken to be constant so far) to vary with temperature in order to simulate the effect of thermal fluctuation and vortex pinning. 63 BIBLIOGRAPHY BIBLIOGRAPHY [1] R. A. ADAMS; Sobolev Spaces, Academic, New York, 1975. [2] S. J.CHAPMAN, Q. DU, AND M. D. GUNZBURGER; A Ginzburg-Landau type model of superconducting/ normal junctions including Josephson J unctions,Euro. J. Appl. Math, 6, 97—114,1995; [3] S. J .CHAPMAN, Q. DU, AND M. D. GUNZBURGER; A variable thickness thin film model for superconductivity, ZAMP, 47,pp410—431, 1995; [4] Z. CHEN, K. H. HOFFMANN, AND J. LIANG: On a non—stationary Ginzburg- Landau superconductivity model, Math Methods in the Appl Sciences, 16, 855- 875, 1993. [5] S. J. CHAPMAN, S. D. How1s0N, AND J. R. OCKENDON; Macroscopic models of superconductivity, SIAM Review 34, 1992, 529-560. [6] Q. DU; Global existence and uniqueness of solutions of the time-dependent Ginzburg—Landau equations in superconductivity, Applicable Anal, 52, pp1-17, 1994; [7] Q. DU, M. D. GUNZBURGER, AND J. S. PETERSON; Analysis and approxi- mation of the Ginzburg-Landau model of superconductivity. SIAM Review 34, 1992, 54-81. [8] Q. DU, M. D. GUNZBURGER, AND J. S. PETERSON; Computational simula- tions of type-II superconductivity including pinning mechanisms, Phys. Rev. B 51, 16194-16203, 1995; [9] K. H. HOFFMANN, L. J IANG, AND W. YU: Models of Superconducting Normal Superconducting Junctions, CNS Preprint Series 023, May, 1994. [10] O. A. LADYZHENSKAYA, The Mathematical Theory of Viscous Incompressble Flow, Gordon and Breach, New York, 1969. 64 65 [11] S. Y. LIN AND Y. YANG; Computation of Superconductivity in Thin Films, J. Comp. Phys. 89, 1990, 257-275.a [12] M. TINKHAM; Introduction to Superconductivity, McGraw-Hill, New York, 1975. [13] C. KUPER; An Introduction to the Theory of Superconductivity, Clarendon Press, Oxford, 1968 [14] P. G. DEGENNES; Superconductivity of Metals and Alloys, W. A. Benjamin, New York, 1966. HI HIGRN STRTE U 11.111111)11111117111 “