Transient dynamics of Josephson-coupled multilayers

Applied Superconductivity Vol. 6, Nos 7±9, pp. 285±290, 1998 0964-1807/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved

PII: S0964-1807(98)00050-7

TRANSIENT DYNAMICS OF JOSEPHSON-COUPLED MULTILAYERS SUSANNE LOMATCH and EDWARD D. RIPPERT Department of Electrical and Computer Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3118, U.S.A. AbstractÐA novel model is proposed to allow the exploration of the short-time (transient) dynamics of a superconductor±insulator multilayer with Josephson coupling between the layers. This model treats the charge on the layer interface planes as a dynamic variable, whose evolution is determined via the interlayer charge-current continuity equations. The high frequency current responses of both the superconducting and the insulating layers are included in the proper time domain form, derived for the general case of nonuniform layers from standard BCS theory. We present the model equations for the response of the system to an initial charge distribution, with the focus of determining the Josephson switching properties of an overdamped, nonuniform multilayer. Such structures may have potential applications in superconducting ¯ux quantum electronics. # 1999 Elsevier Science Ltd. All rights reserved

INTRODUCTION

In an earlier paper [1] we explored some of the basic issues concerning the transient response of a Josephson-coupled multilayer to an ultra-short voltage pulse, taking into account disorder among the layers. The motivation for this study followed from the potential use of multilayer structures as active switching devices in superconducting ¯ux quantum digital circuitry, where they o€er such potential advantages as increased integration density, operating voltage and frequency, and multi-level logic design [2, 3]. Recall that for overdamped junctions (bc I1) used in SFQ digital applications, the important measure of junction response is the time it takes the junction to switch into a voltage state and return to the same nonvoltage state such that the phase di€erence across the junction undergoes a transition from 0 to 2p. This stable state switching depends not only on the intrinsic response time to an applied driving pulse, but also on how the junction reacts dynamically after the pulse via restorative processes. Therefore, we are most interested in stable state switching of an entire multilayer of overdamped junctions, and a proper model to probe the dynamics of this ultra-fast switching. A goal will be to look for coherent switching in a multilayer (i.e. the phase di€erences across each junction in the multilayer switch in near synchronicity from 0 to 2p). In our ®rst paper [1] we con®ned our study of the response to a very simpli®ed model for both shunted and unshunted multilayer structures, where the layer spatial dependence is excluded, along with the high frequency and nonequilibrium behavior of the superconducting layers. Also, junction coupling was only provided via the external load of a simple driving circuit. In this paper we introduce a novel model that includes the high frequency behavior of the superconducting layers through a nonlocal treatment involving the charge-current conservation equations, and attempts to provide internal coupling between the junctions in the stack. In short, with this new model we no longer view the multilayer as a stack of independent junctions, but as a single dynamic unit. For now we restrict our treatment to multilayers for which conventional superconductivity (isotropic s-wave symmetry of the order parameter) is applicable. MODEL EQUATIONS

A Josephson-coupled multilayer consists of alternating layers of superconducting (S) and insulating (I) materials. Each layer has an interface with adjacent layers. Our model begins with identifying the charge on each layer interface as a dynamical variable, evolvable in time. Referring to Fig. 1, we de®ne QLj (t) and QU j (t) as the charge on the lower and upper interface plane of the jth layer. If IIj (Q, t) is de®ned as the tunneling supercurrent across the jth barrier, 285

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Fig. 1. Graphic of a Josephson-coupled multilayer structure.

ISj (Q, t) as the supercurrent across the jth superconducting layer, and Iin(t) as the input current to the top charge plate, then the charge-current continuity equations for the multilayer are given by: dQLj‡1 …t† ÿ ISj‡1 …Q; t† ‡ IIj …Q; t† ˆ 0; dt

…1a†

dQU j …t† ÿ IIj …Q; t† ‡ ISj …Q; t† ˆ 0; dt

…1b†

dQU N‡1 …t† ÿ Iin …t† ‡ ISN‡1 …Q; t† ˆ 0: dt

…1c†

The currents IIj and ISj are functions of the charges in the system Q 0{QLj (t), QU j (t)}, which form a set of 2N + 1 charges to be solved for, where j runs from 1 to N with N as the number of insulating layers. Note that we are taking these charges and currents to be spatially homogeneous over each planar interface surface (we assume we are dealing with small-area multilayers). We have also chosen a particular coordinate system and direction for current ¯ow (cf. Figure 1). For both the tunneling and layer supercurrents, we include frequency dispersion e€ects by using the currents derived from linear response theory, but transformed to the time domain. In the time domain, the tunneling supercurrents are given by the Fourier transform of the frequency space expressions derived by Werthamer [4] (see also Ref. [5] and Appendix A):   …t fj …Q; t† ‡ fj …Q; t 0 † 1 I I 0 0 V …Q; t† ÿ dt IPj …t ÿ t † sin Ij …Q; t† ˆ RNj j 2 ÿ1   …t fj …Q; t† ÿ fj …Q; t 0 † dt 0 IQPj …t ÿ t 0 † sin ; …2† ÿ 2 ÿ1 where VIj (Q, t) is the charge and time dependent voltage across the jth barrier and fj(Q, t) is the

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287

phase di€erence across the jth barrier; they are related to each other through the second Josephson relation, fÇ j= ÿ 2e/h VIj . The kernel functions IPj and IQPj in Equation (2) are the Cooper pair and quasiparticle currents across the insulating barrier. In general, they are dependent on time, the gap functions of the upper and lower superconducting layers, and temperature: IPj(t, Dj, Dj+1, T) and IQPj(t, Dj, Dj+1, T). At T = 0 these functions are given by the following expressions for uneven gaps: IPj …t; Dj ; Dj‡1 † ˆ 3

Dj Dj‡1 p ‰J0 …Dj t=h†Y  0 …Dj‡1 t=h†  ‡ J0 …Dj‡1 t=h†Y  0 …Dj t=h†Š;  2heR  Nj

…3a†

Dj Dj‡1 p ‰J1 …Dj t=h†Y  1 …Dj‡1 t=h†  ‡ J1 …Dj‡1 t=h†Y  1 …Dj t=h†Š;  2heR  Nj

…3b†

and IQPj …t; Dj ; Dj‡1 † ˆ 3

where J0(1) and Y0(1) are ®rst and second kind Bessel functions of 0th(1st) order, and Dj is the gap function of the jth layer. Equation (3) is derived in Appendix A. The time domain expression for the superconducting layer supercurrents is derived in Appendix B, and is given by: …t dt 0 sSj …t ÿ t 0 †VSj …Q; t 0 †: …4† ISj …Q; t† ˆ ÿ1

VSj

and sSj are the voltages across, and the normalized conductivities{ of, the superconductThe ing layers, respectively. The sSj generally depend on the gap function and temperature: sSj(t, Dj, T). For T = 0 one can show (see Appendix B) that these normalized conductivitites are given by the Fourier transform of the frequency space expressions derived by Mattis and Bardeen [6]: … sNj Dj p 1 du‰J0 …u†Y0 …u† ‡ J1 …u†Y1 …u†Š: …5† sSj …t; Dj † ˆ h Dj t= h The sNj are the normal conductances. Collecting Equations (1), (2) and (4) together we obtain the following set of equations for the charges: …t dQLj‡1 …t† 1 I ˆ V …Q; t† ‡ dt 0 sSj‡1 …t ÿ t 0 †VSj‡1 …Q; t 0 † dt RNj j ÿ1   …t fj …Q; t† ‡ fj …Q; t 0 † …6a† dtPj …t ÿ t 0 † sin ÿ 2 ÿ1   …t fj …Q; t† ÿ fj …Q; t 0 † dtQPj …t ÿ t 0 † sin ; ÿ 2 ÿ1 …t dQU 1 I j …t† ˆÿ V …Q; t† ÿ dt 0 sSj …t ÿ t 0 †VSj …Q; t 0 † RNj j dt ÿ1   …t fj …Q; t† ‡ fj …Q; t 0 † dtPj …t ÿ t 0 † sin ‡ 2 ÿ1   …t fj …Q; t† ÿ fj …Q; t† dtQPj …t ÿ t 0 † sin ; ‡ 2 ÿ1 dQU N‡1 …t† ˆ Iin …t† ÿ dt

…t ÿ1

dt 0 sSN‡1 …t ÿ t 0 †VSN‡1 …Q; t 0 †:

{This normalized conductivity has the units of conductance per unit time; see Appendix B.

…6b†

…6c†

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To solve Equation (6), one needs to specify how the the voltages relate to the charges. Since either voltage relates directly to the electric ®eld by VIj (Q, t) = ÿ EIj (Q, t) dj or VSj (Q, t) = ÿ ESj (Q, t)tj, where dj and tj are the thicknesses of the insulating and superconducting layers, this amounts to a speci®cation of the ®elds EIj (Q, t) and ESj (Q, t) as functions of Q. A very simple model is to take these ®elds to be those due to in®nite planes of charge: EIj …Q; t† ˆ

j j ‡1 X X QU QLk …t† N QLk …t† X QU k …t† k …t† ‡ ÿ ÿ ; 2AE0 kˆj‡1 2AE0 2AE0 kˆ1 2AE0 kˆj‡1 kˆ2

ESj …Q; t† ˆ

N ‡1 X

N ‡1 X

j jÿ1 ‡1 U X Qk …t† X QLk …t† X QU QLk …t† N k …t† ‡ ÿ ÿ : 2AE 2AE 2AE 2AE 0 0 0 0 kˆ2 kˆ1 kˆj‡1 kˆj

…7a†

…7b†

Here A is the planar area of the layers. Note that we have assumed that the permittivity of each layer is the vacuum permittivity, E0. This is an approximation of the frequency dependent permittivity E(o) = E0+Ecorr. We assume that at very high frequencies Ecorr 40. It is important to point out that we have assumed that the electric ®elds in each layer are uniform (see Appendix B for more comments). DISCUSSION

Two issues will be discussed in this section: the features and solution of Equation (6), and the relation of our work to the recent work of others. Our model for the short-time dynamics in a multilayer, as represented by Equation (6), has two important features. First, the nature of the coupling is such that each layer in the stack depends on the dynamics of all the other layers in the systemÐthe time evolution of a given charge distribution depends on the time evolution of all the others. This is precisely what is needed to investigate possible coherent coupling mechanisms for a transient perturbation, as discussed in Section 1. Second, since we have derived the high frequency response kernels for both the tunnel barriers and the superconducting layers as functions of the gaps Dj and Dj+1, there is a parametric relationship of each layer to one another through the gap function. This should be useful for including nonequilibrium e€ects through the modulation of the gap function [7] (at present we take the gaps to be constants). Equation (6) forms a set of 2N + 1 coupled nonlinear ®rst-order ordinary Volterra integrodi€erential equations. Their solution is nontrivial and is best done by numerical methods. Stability of the numerical solution is an issue, since the method chosen must take into account the fast response of the supercurrents ISj . It is easy to show that these currents peak very quickly in time and then die o€ slowly in response to a short-time voltage pulse [8]. Both the numerical treatment and the comprehensive analysis of the dynamics of these equations will appear elsewhere [8]. The focus of our numerical analysis will be on a multilayer with overdamped characteristics, or parameters consistent with unshunted high Jc tunnel junctions. It is this limit where we will investigate stable state switching by tracking the interlayer phase di€erences. We would like to comment on how our model di€ers from models presented for Josephsoncoupled, layered superconductors by others (see for example, Ref. [9]). In Ref. [9], charge-current conservation is used to derive a system of equations for the phase di€erences across all barrier layers, taking into account the charge density of the superconducting layers. In our model, we neglect the charge density of the superconducting layers, and deal only with the charge distributions on the layer interface planes. Also, Ref. [9] does not include dispersion e€ects in their calculations. Here our focus is to include these high frequency e€ects, and to probe the transient, short-time dynamic response of the system. The primary focus in Ref. [9] is the calculation of Josephson plasma modes of the system, which are a steady-state e€ect. Thus our work is uniquely di€erent. Finally, it should also be mentioned that although our focus has been on the treatment of layered superconductors with conventional superconductivity, the model can be adapted for unconventional superconductors, such as the high Tc compounds (e.g. single crystals of Bi2Sr2CaCu2O8+x which show intrinsic Josephson e€ects [10]). Recently, the high frequency

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expressions for the interlayer tunneling currents and conductivities were derived from the microscopic theory, taking into account d-wave pairing symmetry of the order parameter [11]. These frequency space expressions should be transformed to the time domain for inclusion in our model.

CONCLUSION

We have introduced a novel model for investigating the dynamics of a superconducting-insulating Josephson-coupled multilayer in response to an ultra-short voltage pulse. This model includes the high frequency behavior of the superconducting layers, and by its very design has internal coupling of each layer to all other layers in the system. Likewise, there is a parametric relationship of each layer with one another through the gap function. A numerical solution of this system should provide some useful insight into the switching dynamics of a single multilayer with overdamped characteristics, a focus being the desired coherent stable state switching necessary for active switching in ¯ux quantum electronics. It should also serve as a guide for future experimental measurements of the switching dynamics (transient response) of Josephson-coupled multilayer structures. AcknowledgementsÐThis work was supported by the National Science Foundation under grant no. ECS-9500279.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

S. Lomatch and E. D. Rippert, IEEE Trans. Appl. Superconduct. 7, 2394 (1997). S. Lomatch and E. D. Rippert, IEEE Trans. Appl. Superconduct. 5, 3147 (1995). S. Lomatch and E. D. Rippert, J. Appl. Phys. 76, 1711 (1994). N. R. Werthamer, Phys. Rev. 147, 255 (1996). R. E. Harris, Phys. Rev. B13, 3818 (1976). D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412 (1958). G. M. Eliashberg, Sov. Phys. JETP 34, 668 (1972). S. Lomatch and E. D. Rippert. unpublished. T. Koyama and M. Tachiki, Phys. Rev. B54, 16183 (1996). R. Kleiner and P. MuÈller, Phys. Rev. B49, 1327 (1994). S. E. Shafranjuk, M. Tachiki and T. Yamashita, Phys. Rev. B53, 15136 (1996). M. Tinkham, Introduction to Superconductivity. 2nd edn, Chap. 3. McGraw-Hill, New York (1996).

APPENDIX A In this Appendix section we derive the expression for the tunneling currents (Equation (2)) and the kernel functions (Equation (3a) and (3b) in the time domain, speci®cally for uneven gaps of the upper and lower layers. From microscopic BCS theory in the linear response limit, Werthamer [4] showed that the tunneling current for a general tunnel junction is given by: …1 0 IIj …t† ˆ Im do do 0 ‰Uj …o†Uj …o 0 †eÿi…o‡o †t IPj …o 0 ; T† ÿ1 …A1† 0

‡ Uj …o†Uj …o 0 †eÿi…oÿo †t IQPj …o 0 ; T†Š; where o is the frequency variable and   …1 … ÿie t do Uj …o†eÿiot ˆ exp…ifj …t†=2† ˆ exp dt 0 VIt …t 0 † : h ÿ1 ÿ1 It is straightforward to show that the time domain expression for this current, Equation (2), results from applying the following Fourier transform convention to eqn (A1): … 1 1 I…t† ˆ dt I…o†eÿiot : …A2† 2p ÿ1 o,T) are given by Equation (13) in Ref. [4] for The complex temperature-dependent frequency space kernels IPj(QPj)(o uneven gaps D1 and D2 at T = 0. The time domain expressions are obtained by taking the Fourier cosine or sine transD2=D D in forms of the real or imaginary parts of the frequency space kernels. This was originally done for even gaps D1=D Ref. [5], but it is worthwhile to outline the calculation for uneven gaps.

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For IP(o o) 4 IP(t), taking the Fourier sine transform of the imaginary part of IP(o o) in Equation (13) of Ref. [4] we obtain: …1 … ÿD1 2D1 D2 1 IP …t† ˆ Im do eÿiot do 0 0 2 ÿ D2 †1=2 ‰…o 0 ‡ o†2 ÿ D2 Š1=2 peRN …o D2 ÿo …D1 ‡D2 † 1 2 …1 …1 ÿiu 0 t ÿiut 2D1 D2 e e ˆ Im du 0 0 du peRN …u 2 ÿ D21 †1=2 D2 …u2 ÿ D22 †1=2 D1 …A3†      2D1 D2 ip ip …2† …2† ÿH0 D2 t ˆ Im ÿH0 D1 t 2 2 peRN ˆ

pD1 D2 ‰J0 …D1 t= h†Y  0 …D2 t= h†  ‡ J0 …D2 = h†Y  0 …D1 t= h†Š;  h2 2eRN

where in the second line we have switched integration limits and variables, and in the last line we have properly normalized the gap functions by factors of h.  The function H(2) 0 is a third kind Bessel function of 0th order. The expression for IQP(t) is obtained in a similar manner.

APPENDIX B In this Appendix section we derive the expression for the layer supercurrents (Equation (4)) and the normalized conductivity kernels (Equation (5)) in the time domain. In general, the supercurrent density for a given superconducting layer can be written in momentum (q) and frequency (o o) space as: JSj …q; o; T† ˆ sSj …q; o; T†ESj …q; o†: sjS(q,

…B1†

ESj (q,

o, T) is the temperature-dependent complex conductivity and o) is the electric ®eld. If we assume that the dirty limit lW x0 applies to the layer material, where l and x0 are the mean free path of the electrons and the coherence length of the pair ®eld, respectively, then the q dependence of the conductivity can be dropped [12] and we get: JSj …q; o; T† ˆ sSj …o; T†ESj …q; o†:

…B2†

The dirty limit is equivalent to estimating the current density at a point in space by a spatial average of ®eld strengths in a region around that point of size x0. The resulting q-independent complex conductivity in this limit is given by s1jSÿis s2jS, where the frequency-dependent expressions for the real and imaginary parts s1jS and s2jS are given by sjS=s Equation (3.9) and (3.10) in Ref. [6] multiplied by the normal-state conductivity sjN, which is taken to be a constant in our model. Transforming eqn (B2) from momentum to position space and taking both the current density and the electric ®eld to be uniform along the layer axis (c-axis) we obtain: … JSj …o; T† ˆ sSj …o; T† dz ESj …z; o† ˆ sSj …o; T†VSj …o†: …B3† A ®nal transform to the time domain using the Fourier transform convention eqn (A2) and a normalization of both sides by the layer thicknesses tj divided by the cross-sectional area A of the multilayer yields Equation (4). The assumption of ®eld uniformity is consistent with the derivation of the time-dependent tunneling current derived by Werthamer (cf. eqn (A1)), who also assumed such spatial uniformity. The conductivity kernels (Equation (5)) are found by taking either the cosine transform of s1jS or the sine transform of s2jS; we show the procedure for the former: …1 2 do s1 …o†eÿiot s…t† ˆ Re p ÿ1 …1 2sN eRN 1 Re ˆ do ‰IP2 …o† ‡ IQP2 …o†Šeÿiot o p ÿ1 …B4† …1 …1 2sN eRN t Re i ˆ dx do‰IP2 …o† ‡ IQP2 …o†Šeÿiotx p 1 ÿ1          2 …1 oN tpD Dtx Dtx Dtx Dtx Y ‡ J Y : ˆ dx J 0 0 1 1 h h h h h2 1 The second line shows that the real component of the conductivity can be written as a combination of the imaginary components of the pair and quasiparticle current kernels in frequency space; a similar relation can be made for the complex component of the conductivity in terms of the real components of the pair and quasiparticle current kernels. The last line is valid for T = 0, and a change of variable ®nally gives Equation (5).