Description of intrinsic Josephson junctions by the inductive coupling theory

Physica C 362 (2001) 1±9

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Description of intrinsic Josephson junctions by the inductive coupling theory Shigeki Sakai *, Hirotake Yamamori Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 305-8568, Japan Received 23 August 2000

Abstract The inductive coupling theory for intrinsic Josephson junctions is presented. The bias current forms in both one- and two-dimensional models are discussed. As an example of two-dimensional cases, behavior of an intrinsic Josephson junction cylinder is numerically shown. Under magnetic ®eld application, the ®eld penetration length, and the critical ®eld for maintaining the Meissner states in long stacked junctions are described analytically and numerically. The in…N † phase-mode characteristic length kN of N-fold stacks as well as kab is an important scaling length to explain such phenomena. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 74.80.Dm; 74.72.Hs; 74.72. h; 85.25. j Keywords: Intrinsic Josephson junction; Stacked Josephson junction

1. Introduction There has been considerable interest in physics and application of the intrinsic Josephson e€ect in highly anisotropic supercondcutors such as the Bi system and Tl system. In the case of a Bi2 Sr2 CaCu2 Ox single crystal, the minimum molecular unit along the c-axis, 1.5 nm, is the period of stacked intrinsic Josephson junctions [1]. In this periodic structure there are superconducting (S) layer and insulating (I) tunnel-barrier layer. It has been modeled that the Cu±Ca±Cu layers are superconducting and the Sr±Bi±Bi±Sr layers are insulating and thus assumed that the S layer thickness, t, is 0.3 nm and the I layer one, d, is 1.2

*

Corresponding author. Fax: +81-298-54-5476. E-mail address: [email protected] (S. Sakai).

nm [1]. The period, d ‡ t ˆ 1:5 nm, can be exactly determined because this is based on the crystalline structure. In the assignment of t and d values, there may be a point to be investigated carefully, but the S layer thickness is at maximum of the order of 1 nm. The penetration length of magnetic ®eld is anisotropic; kab is around 100±200 nm and kc is of the order of 100 lm [2]. These are huge compared to this molecular layer scale. In an S/I/S/ I/. . ./S/I/S stacked Josephson junction, the magnetic ®eld of an I layer cannot be screened well in the ultrathin S layer contacting to this I layer because of t  kab , and thus the e€ect of the magnetic ®eld in this I layer may appear in the adjacent I layer. The inductive coupling theory [3] for stacked Josephson junctions takes into account these cases and can thus describe the properties of intrinsic Josephson junctions in many cases. Flux ¯ow with and without cavity resonance, and

0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 6 3 9 - 6

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S. Sakai, H. Yamamori / Physica C 362 (2001) 1±9

plasma oscillation and resonance are important subjects for physics and THz device application, and they have been discussed by this inductive coupling model [4±6]. In this paper, we describe several fundamental properties of intrinsic Josephson junctions within the framework of the inductive coupling theory. Section 2 presents the theory. First the outline of the theory in Ref. [3] is reviewed. This theory is written on the form of one-dimensionally long stacked Josephson junction and is useful in most case, in particular for electronic device application. Next a two-dimensional model is presented and the relationship to the one-dimensional model is described. A selected example of the two-dimensional cases is also shown. Section 3 describes mostly static response of the magnetic ®eld and supercurrent under application of the magnetic ®eld. The screening length, and the critical ®eld over which quantized ¯ux (¯uxon) penetration starts entering are described analytically and numerically. It is shown that the scaling length along the ab plane is not …N † simply kc but kN as a function of the stack number N. The size of a ¯uxon is also discussed. Section 4 summarizes the content.

2. Theory 2.1. Outline of the inductive coupling model Let us consider an N-fold stacked Josephson junction that is along in the x-direction. This consists of N insulating and N ‡ 1 superconducting layers. Using the ith S layer thickness, ti , the ith I layer thickness, dii 1 , and the magnetic penetration length ki in the ith S layer, the ith Josephson junction phase /ii 1 , i.e., the phase di€erence between the ith and …i 1†th S layers, i.e., is represented by [3]  o/ii 1 h ˆ si 1 Bi 1i 2e ox for i ˆ 1; 2; . . . ; N ;

2

‡ dii0 1 Bii

1

‡ si Bi‡1i …1†

where Bii 1 …i ˆ 1; . . . ; N † is the y component of the magnetic ®eld in the ith I layer, and B0 1 and BN ‡1N are those just outside the 0th and N th S

layers, respectively. dii0 1 ˆ dii 1 ‡ ki 1 coth…ti 1 = ki 1 † ‡ ki coth…ti =ki † and si ˆ ki = sinh…ti =ki † are the e€ective barrier thickness of the ith junction and the inductive coupling strength of the ith S layer, respectively. By using Eq. (1) and the continuity condition among the currents in the S layer and the Josephson junctions, and the external bias currents, a set of equations of the Josephson junction phase is obtained [3]: h o2 /10 0 Z ˆ d10 …J10 2el0 ox2 h o2 /ii 2el0 ox2

1

Z IB † ‡ s1 …J21

ˆ si 1 …JiZ 1i

for i ˆ 1; 2; . . . ; N h o2 /NN 2el0 ox2

1

IB † ‡ dii0 1 …JiiZ

2

Z ‡ si …Ji‡1i

IB †;

1

IB †

IB †;

1;

ˆ sN 1 …JNZ

…2a†

…2b† IB †

1N 2

0 Z ‡ dNN 1 …JNN

1

IB †:

…2c†

IB is the bias current. JiiZ 1 …i ˆ 1; 2; . . . ; N † is the resistively and capacitively shunted Josephson current, i.e., JiiZ 1 ˆ …hCii 1 =2e†…o2 /ii 1 =ot2 † ‡ …h  Gii 1 =2e†…o/ii 1 =ot† ‡ Jii 1 sin /ii 1 , where Cii 1 , Gii 1 and Jii 1 is the capacitance, conductance, and Josephson critical current of the ith junction. The imposed boundary condition [3] under the application of magnetic ®eld Ba is …h=2e†…o/ii 1 = ox†xˆ0;L ˆ …si 1 ‡ dii0 1 ‡ si †Ba . An N -fold stack has N kind cavity resonance modes of the ¯ux ¯ow [4], N characteristic velocities of electromagnetic waves [7] and N plasma dispersion curves [6]. In the case of an N -fold stack of identical layers and junctions, the velocities and dispersion relation are analytically expressed such 1=2 as cm…N † ˆ kJ xp0 =‰1 2s cos…mp=…N ‡ 1†=d 0 Š and 2 …N † 2 …x=xp0 † ˆ 1 ‡ ‰k…km †Š for m ˆ 1; 2; . . . ; N . Here unnecessary subscripts can be eliminated, e.g., dii0 1  d 0 and ki  k because of the identical layer 1=2 and junction assumption. xp0 ˆ …2eJ =hC† is the 1=2 plasma angular frequency, and kJ ˆ …h=2el0 d 0 J † is the called Josephson penetration length. km…N † ˆ kJ =‰1

2s cos…mp=…N ‡ 1††=d 0 Š1=2

…3†

S. Sakai, H. Yamamori / Physica C 362 (2001) 1±9

h 2el0



o2 /10 o2 /10 ‡ ox2 oy 2

3

 0 Z Z J10 ‡ s1 J21 ; ˆ s0 IB0 …x; y† ‡ d10

…5a† h 2el0



o2 /ii ox2

o2 /ii ‡ oy 2

1

ˆ si 1 JiZ 1i

2

‡ dii0 1 JiiZ

for i ˆ 2; 3; . . . ; N h 2el0



o2 /NN ox2

ˆ sN 1 JNZ …N †

Fig. 1. Characteristic length of the in-phase mode, kN and of …N† the anti-phase mode, k1 as a function of N .

is the scaling length of mth plasma mode that connects the plasma frequency and characteristic velocities (cm…N † ˆ xp0 km…N † ) [8]. The mode of m ˆ N is the in-phase mode with the highest velocity and m ˆ 1 is the anti-phase mode with the lowest ve…N † …N † locity. In Fig. 1, kN and k1 are plotted as a function of N . In the in®nite N limit, k…1† 1 ˆ kc ˆ …N † 1=2 … h=2el0 …d ‡ t†J † . kN plays important roles as a scaling length under static magnetic ®elds that is shown in Section 3. Note that, in the limit of t  k, d 0  2k2 =t, s  k2 =t, and d 0 ‡ 2s  d ‡ t. 2.2. Two-dimensional case By the similar procedure for obtaining Eq. (1), o/ii 1 =oy has the following relationship with the magnetic ®eld in the I layers:  o/ii 1 h ˆ si 1 Bi 1i 2;x ‡ dii0 1 Bii 2e oy for i ˆ 1; 2; . . . ; N ;

1;x

‡ si Bi‡1i;x …4†

where Bii 1;x is the x component of the magnetic ®eld in ith I layer. Bii 1 in the previous sub-section should be read as Bii 1;y here in the two-dimensional model. By combinig Eqs. (1) and (4), and by using the Maxwell equation r  H ˆ J ‡ oD=ot inside the stacked Josephson junction and for the bias current inputs to the top and bottom S layers, we obtain,

1



1

Z ‡ si Ji‡1i

1;

o2 /NN ‡ oy 2

1N 2

1

1

…5b† 

0 Z ‡ dNN 1 JNN

1

‡ sN IBN …x; y†;

…5c†

where IBN …x; y† and IB0 …x; y† as arbitrary functions of x and y are the bias current density entering into the top (i ˆ N ) S layer and ¯owing out of the bottom (i ˆ 1) S layer, respectively. The total current of the entering and ¯owing out must be balanced, of course. The applied magnetic ®eld and the self-®eld by current applications are taken into account as the boundary condition of …o/ii 1 = ox, o/ii 1 =oy† at peripheries. 2.3. Bias current in one- and two-dimensional theory Eqs. (5a)±(5c) of the two-dimensional theory have intuitively understandable form with respect to the bias current, since the bias currents relate to Eqs. (5a)±(5c) through only the top and bottom S layers. In the case of the one-dimensional theory, on the other hand, the bias current in Eqs. (2a)± (2c) appears together with each junction current JiiZ 1 …i ˆ 1; 2; . . . ; N †. This is also intuitively correct, because the self-magnetic ®eld produced by the bias current may give rise to the screening current to each junction. The origin of this screening current is the bias current itself. This matter is discussed more preciously as follows. As shown in Fig. 2, let us consider an in®nitely long stacked-Josephson-junction with a ®nite small width W . It is assumed that the bias current whose current density is IB is applied uniformly. The magnetic ®eld for y < 0 is Bx ˆ l0 WIB =2 and that for y > W is Bx ˆ l0 WIB =2. Thus the boundary conditions at y ˆ 0 and y ˆ W are …h=2el0 †  …o/ii 1 =oy† yˆ0 ˆ …si 1 ‡ dii0 1 ‡ si †WIB =2. The width yˆW

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S. Sakai, H. Yamamori / Physica C 362 (2001) 1±9

Fig. 2. One-dimensionally long stacked Josephson junction with a uniform bias current. The bias current makes the magnetic ®eld along x.

is so small that a linear approximation to o/ii 1 =oy can be made. This gives … h=2el0 †…o2 /ii 1 =oy 2 †  0 …si 1 ‡ dii 1 ‡ si †IB . The equation obtained by putting this equation into Eqs. (5a)±(5c) of the two-dimensional theory exactly coincides with Eqs. (2a)±(2c) of the one-dimensional theory. 2.4. Selected example for the two-dimensional theory Let us consider a cylinder of stacked Josephson junctions where the bias current Icen is applied to the center of the circles of the top and bottom S layers. (See Fig. 3.) By the cylindrical symmetry, we shall ®nd solutions of the same symmetry. Then in Eqs. (5a)±(5c), o2 /ii 1 =ox2 ‡ o2 /ii 1 =oy 2 for i ˆ 1; 2; . . . ; N is replaced to o2 /ii 1 =or2 ‡ r 1 o/ii 1 =or, and both IB0 …x; y† and IBN …x; y† become Icen d…r†. The boundary conditions are …o/ii 1 =or†rˆ0 ˆ 0 and …o/ii 1 =or†rˆR ˆ …2e=h†…si 1 ‡ dii0 1 ‡ si †…l0 Icen = 2pR†, where R is the radius of the cylinder. Fig. 4 shows numerical simulation results. Typical parameters of Bi2 Sr2 CaCu2 Ox were used. That is, t ˆ 0:3 nm, d ˆ 1:2 nm, k ˆ 90 nm, J ˆ 2

Fig. 3. Cylinder of stacked Josephson junctions. The current Icen is biased to the center of the top and bottom surfaces.

kA/cm2 , C ˆ 3 lF/cm2 , and G ˆ 100 kS/cm2 . The stacked junction sizes are N ˆ 8 and R ˆ 10 lm. Fig. 4(a) shows the static response state when a weak bias Icen ˆ 1:2  105 A/cm2 was applied. At the central area the phases were pushed, but no Josephson vortices entered. The phase rotations of the top and bottom junctions are large. With the increase of Icen , Josephson vortices entered in the top and bottom junctions, but they stayed at some places in 0 < r < R. The further increase of Icen brought vortex (¯ux) ¯ow states. Fig. 4(b) is a snapshot of this state (at Icen ˆ 6  106 A/cm2 ), where the ¯ow in the top and bottom junctions can be seen. Vortices generating at the center disappeared at the edge. The core magnetic ®eld line of a vortex makes a self-closed line (ring) in the cylindrical junctions. These behavior may take place because of the very thin top and bottom layers of 0.3 nm. Quite di€erent behavior is expected in the case of very thick S layers. We changed the thickness of only the top and bottom S layers from 0.3 to 900 nm. (This may be the case where traditional superconductors such as Nb and Pb are deposited to the top and bottom.) As shown in

S. Sakai, H. Yamamori / Physica C 362 (2001) 1±9

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Fig. 4. Numerically simulated distribution of the normalized-supercurrent (sin /) for a stack N ˆ 8. Graphs from i ˆ 1 to 4 are omitted, because the graphs of i ˆ 1; 2; 3; 4 are the same as those of i ˆ 8; 7; 6; 5; respectively. Used parameters are k ˆ 90 nm, t ˆ 0:3 nm, d ˆ 1:2 nm, J ˆ 2 kA/cm2 , C ˆ 3 lF/cm2 , and G ˆ 100 kS/cm2 , R ˆ 10 lm for (a) and (b) and R ˆ 100 lm for (c), where R is the radius of the cylinder. (a) Zero voltage states under a weak bias current Icen ˆ 1:2  105 A/cm2 and (b) ¯ux ¯ow states in the top and bottom junctions when Icen ˆ 6  106 A/cm2 . (c) Supercurrent distribution in case that the top and bottom S layer were 900 nm thick. The thicknesses, 0.3 nm, of the inner layers were not changed.

Fig. 4(c), large supercurrent ¯ows at the periphery. The bias current given at the center ¯ows at the surface of the thick S layer from the center to the peripheries. 3. Fundamental properties under magnetic ®eld 3.1. Screening length at the Meissner state In this section we come back to one-dimensional cases. At a sucient inside of the junctions, o/ii 1 =ox ˆ 0 may be expected, and we have si 1 Bi 1i 2 ‡ dii0 1 Bii 1 ‡ si Bi‡1i ˆ 0 for i ˆ 1; 2; . . . ; N from Eq. (1). Under applied magnetic ®eld Ba , B0 1 ˆ Ba and BN ‡1N ˆ Ba . These are regarded as driving terms and the magnetic ®eld Bii 1 may be obtained as the response (a solution to a set of simple linear equations). Fig. 5 is a calculation result for a 150 nm thick (N ˆ 1000) Bi2 Sr2 CaCu2 Ox . The decay length along z (magnetic penetration length kab ) is 230 nm from the curve in the ®gure. The penetration length (k  ki ) in the S layers used in the calculation was 100 nm. Between k in the discrete model theory and the macroscopic quantity kab obtained from experiments, kab =k ˆ 1=2 ‰…d ‡ t†=tŠ should be satis®ed. t ˆ 0:3 nm pWhen  and d ˆ 1:2 nm, this ratio is 5, the calculation result in Fig. 5 certainly ful®lls this relationship.

Fig. 5. Magnetic ®eld at each junction in an intrinsic Josephson junction …N ˆ 1000† at a place where o/ii 1 =ox ˆ 0 is satis®ed. This is the solution to a set of linear equations (Eq. (1)).

From the symmetry we can discuss solutions such that behavior of each junction is fully identical for the stacks of N ˆ 2 and 1. By putting /  /ii 1 in Eqs. (1) and (2a)±(2c), we obtain h o2 / hC o2 / hG o/ ‡ J sin / ˆ ‡ 2el0 C ox2 2e ot2 2e ot

…6†

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S. Sakai, H. Yamamori / Physica C 362 (2001) 1±9

with the boundary conditions of h o/  ˆ …d 0 ‡ 2s†Ba 2e ox xˆ0

…7†

xˆL

under the magnetic ®eld Ba . Here C ˆ d 0 ‡ 2s for N ˆ 1, C ˆ d 0 ‡ s for N ˆ 2, and additionally C ˆ d 0 for N ˆ 1. Let us discuss the screening length when weak ®eld is applied. The static solution on the approximation of sin /  / is given by / ˆ …2e= h†…d 0 ‡ 2s†KBa exp… x=K† with K ˆ 1=2 … h=2el0 CJ † . Thus K is the magnetic penetration length along the x-direction. By comparing with Eq. (3), K for N ˆ 1 is equal to k…1† 1 ˆ kc ˆ 1=2 … h=2el0 …d ‡ t†J † , and K for N ˆ 2 and 1 is equal …2† …1† to k2 and k1 ˆ kJ , respectively. Thus the magnetic penetration length for N ˆ 1; 2 and N ˆ 1 coincides with their in-phase-mode characteristic length. This can be understood easily because a fully identical solution among the junctions is a kind of in-phase modes. In the case of ®nite number of stack N P 3, the assumption that all junctions have identical phases is not valid, because of symmetry consideration. Nonetheless, let us predict that the characteristic length of the in…N † phase mode, kN , may approximate the penetration length of N -fold stacks. The Josephson junction phases were solved numerically under the magnetic ®eld Ba (ky) by applying a ®nite di€erent method with respect to x and t to Eqs. (2a)±(2c). Meantime after application of Ba , static solutions were obtained. Fig. 6 shows

Fig. 7. Magnetic ®eld along z for stacks with p various thickness. The ®eld decay length agrees with kab ˆ 5k.

the contour map of the magnetic ®eld in an intrinsic Josephson junction stack that is L ˆ 0:5 mm long and 750 nm …N ˆ 500† high. Parameters used are t ˆ 0:3 nm, d ˆ 1:2 nm, J ˆ 2 kA/cm2 , and k ˆ 50 nm. The corresponding kab is 112 nm as we have discussed above. Fig. 7 shows the decay of the magnetic ®eld along z at the center (x ˆ L=2) for various number of stacks. As shown in the dashed curve in Fig. 7, the decay length of the simulation agrees well p with the theoretical penetration length kab ˆ 5k. Fig. 8 shows the Josephson junction phase along x for three kinds of stacks, i.e., N ˆ 2, 10, and 100. The phases at the central junction (i ˆ N =2) were plotted. In the ®gure, predicted exponential decay curves using the characteristic length …N † of the in-phase mode, kN , are also drawn. The penetration length k…N † as a function of N is summarized in Fig. 9. We can con®rm, in this ®gure, the validity of the prediction such that the …N † characteristic length of the in-phase mode, kN , approximates well the penetration length of N -fold stacks. 3.2. Critical ®eld of ¯uxon entering

Fig. 6. Contour map of the magnetic ®eld in an intrinsic Josephson junction. N ˆ 500, k ˆ 50 nm and Ba ˆ 5  10 4 T were assumed.

On Meissner states under weak applied magnetic ®eld Ba , Eq. (6) with a semi-in®nite (L ˆ 1) structure has analytic solutions such as [9]

S. Sakai, H. Yamamori / Physica C 362 (2001) 1±9

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Fig. 8. Josephson junction phase at i ˆ N =2 along x for stacks of N ˆ 2, 10, and 100. The decay length are compared to the theo…N † retically predicted value of kN .

Fig. 9. Plot of the phase decay length along x as a function of …N † N . Data points are on the theoretical prediction curve kN .

/ ˆ 4 arctanfexp‰ …x

X0 †=KŠg

where X0 ful®lls the equation including Ba ,   2 X0 2e sech ˆ …d 0 ‡ 2s†Ba : K h  K

…8†

…9†

Eq. (9) exhibits a threshold condition of the Meissner state, because the left-hand side cannot exceed 2=L. Thus this critical ®eld is BK;cr ˆ

h : eK…d 0 ‡ 2s†

…10†

Eq. (6) can be used in the case that behavior of each junction is fully identical. From the symmetry, this is allowed for N ˆ 1; 2 and 1. In such

Fig. 10. The critical magnetic ®eld over which a Josephson vortex start to enter in junctions. Data points by numerical simulation are well on the theoretical prediction curve. …N †

cases, K in Eq. (10) is replaced by kN . In particular, in the N ˆ 1 case, the critical ®eld B…1† is cr …2† equal to h=‰ekc …d ‡ t†Š or 2l0 kc J . Bcr is much lar…2† ger than B…1† is small. Again we may cr , since k2 predict the critical ®eld of arbitrary N stacks as …N † BK;cr ˆ h=‰ekN …d 0 ‡ 2s†Š. Numerical simulation results as a function of N is shown in Fig. 10. It is found that the points by the simulation stay on the analytic prediction curve. The usefulness of the prediction was con®rmed. 3.3. Single ¯uxon size An isolated single ¯uxon size was evaluated by numerical simulation. We assumed annular

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S. Sakai, H. Yamamori / Physica C 362 (2001) 1±9

geometry along x and one ¯uxon was put, as the initial condition, in the middle junction of the stack. Since no bias condition, the ¯uxon stops meantime after computation. In order to observe suciently isolated status, we had to bring very large size. As shown in Fig. 11, the length was 1 mm and the stack number N was 300. Smaller k…ˆ 10 nm† than the real cases of Bi2 Sr2 CaCu2 Ox was used in order to save the size and computation

times. Fig. 11(a) shows three-dimensional picture. We ®nd the magnetic ®eld of the ¯uxon decayed suciently for both x and z directions. Fig. 11(b) and (c) are the cross-sections with respect to x and z, respectively. The decay length of the exponentially decaying part was in good agreement with …N † the characteristic length of kN at N ˆ 300 along x (Fig. 11(b)), and the penetration length of kab ˆ p  5k along z (Fig. 11(c)), respectively.

Fig. 11. Magnetic ®eld of a stationary ¯uxon. Numerically simulated conditions are N ˆ 300, L ˆ 1 mm, and k ˆ 10 nm. (a) Threedimensional view, and cross-sections along (b) x and (c) z. The decay length in the x direction and z direction is in good agreement with …N† kN (N ˆ 300) and kab , respectively.

S. Sakai, H. Yamamori / Physica C 362 (2001) 1±9

4. Summary The inductive coupling theory has been presented. The relationship between one-dimensional and two-dimensional cases has been clari®ed, in particular, with paying attention to the bias current. As an example of the two-dimension case, Josephson vortex motion with a self-closed ¯ux line was demonstrated in a cylinder of intrinsic Josephson junctions. The characteristic length under magnetic ®eld applied parallel to the ab plane was discussed analytically and numerically. On the Meissner state under weak magnetic ®elds, the ®eld decay length along c-axis is explained well by kab . However, the phase decay length in the ab …N † plane is not simply kc , but is expressed by kN ˆ 1=2 kJ =‰1 2s cos…N p=…N ‡ 1††=d 0 Š . That is, the decay length depends on N . The in-phase-mode …N † characteristic length (kN ) of the N stack also plays an important role in the critical magnetic ®eld for

9

maintaining the Meissner states and the size of an isolated ¯uxon. For device application, the size and shape of intrinsic Josephson junctions, the way how to put bias current leads and so on will become important. References [1] R. Kleiner, F. Steimeyer, G. Kunkel, P. M uller, Phys. Rev. Lett. 68 (1992) 2394. [2] J.R. Cooper, L. Forro, B. Keszei, Nature 343 (1990) 444. [3] S. Sakai, P. Bodin, N.F. Pedersen, J. Appl. Phys. 73 (1993) 2411. [4] R. Kleiner, Phys. Rev. B 50 (1994) 6919. [5] M. Machida, T. Koyama, A. Tanaka, M. Tachiki, Physica C 330 (2000) 85. [6] N.F. Pedersen, S. Sakai, Phys. Rev. B 58 (1998) 2820. [7] S. Sakai, A.V. Ustinov, H. Kohlstedt, A. Petraglia, N.F. Pedersen, Phys. Rev. B 50 (1994) 12905. [8] S. Sakai, N.F. Pedersen, Phys. Rev. B 60 (1999) 9810. [9] S. Sakai, M.R. Samuelsen, Appl. Phys. Lett. 50 (1987) 1107.